Elements of Chenzicn1 Reaction Engineering
PRENTICE HALL PTR INTERNATIONAL SERIES IN THE PHYSICAL AND CHEMICAL ENGINEERING SCIENCES NEALR. AMUNDSON, SERIES EDITOR, University of Houston
ANDREAS AcRI\~OS, Stanford University JOHN DAALER, University of Minnesota H.SCOTTF%LER, University of Michigan THOMAS J. H A N R A University ~, of Illinois JOHN M. PRAUSNITZ. University of California L.E.S C R ~ I ~University EN, ofMinnewta
Chemical Engineering Thermodynamics BALZHISER, SAMUEI-S, AND EWASSEN BEQUETTEProcess Controi: Modeling, Design, and Simulation BEQUETTE Process Dynamics BIEGLER, GROSSIWA?~~. AND WESTERBERGSystematic Methods of Chemical Process Design RRosILow A N 5 JOSEPH Techniques of Model-based Control CQNSTAN~NXDES AND MOSTOUFI Numerical Methods for Chemical Engineers with MATLAB Applications CROWLAND LOUVAR Chemical Process Safety: Fundamentals with Applications, 2nd edition Problem Solving in Chemical Engineering with Numerical CUTLIP AND SHACHAM Methods DENY Process Fluid Mechanics ELLIOT AND LIRA Introductory Chemical Engineering Thermody narnics F ~ G L E RElements of Chemical Reaction Engineering, 4th edition HEMMELBLGUAND RIGGS Basic Principles and CalcuIations in Chemical Engineering, 7th edition H J N EAND ~ MADDOX Mass Transfer: Fundamentals and Applications PRAUSNITZ, LICHTENTHALER, AND DE AZEVEDO Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd edition PRENTICEEIectrochernical Engineering Principles SHULER ASD KARGI Bioprocess Engineering, 2nd edition STEPHANOPOUU~S Chemical Process Control TESTERAND MODELL Thermodynamics and Its Applications, 3rd edition TURTON, BAILIE,WHITING.AND SHAEIWITZAnalysis, Synthesis, and Design of Chemical Processes, 2nd edition WII.KES Fluid Mechanics for Chemical Engineers, 2nd edition
Elements of Chemical Reaction Engineering Fourth Edition
H. SCOTT FOGLER Arne and Catherine Vennema Professor of Chemical Engineering The University of Michigan, Ann Arbor
Prentice Hall Professional Technical Reference PRENTICE
.-HALL
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FogIer. H. Scott. Elements of chemical reaction engineering I A. Scott Fogler4th ed.
p. cm. Includes bibliographical references and index. ISBN 23-13-047394-4 (alk. paper) 1. Chemical reactors. I. Title.
Copyright O 2006 Pearson Education, Inc. All rights reserved. Printed in the United States of Amwics This publication is pmtected by copyright. and permission must k obtained from the publisher prior to any prohibited reproductiod, storage in a retrieval system. or transmission in any form or by any means, electronic, mechanical. photocopying, recording, or likewise. For information r~gard~ng permissions. write to:
Pearson Education, Inc. Rights and Contracts Department
One Lake Street Upper Saddle River. NJ 07458 ,
ISBN 0-13-047394-4 Text printed in the United States on recycled paper at Courier in Westford. Massachusetts. First printing. August 1005
Dedicated fo rke tIrerno9 of Professors
Gi useppe Parravano Joseph J. Martin Donald L, Katz
of the University of Michigan whose standards and lifelong achievements serve to inspire us
Contents
PREFACE
1.I 1.2 1.3 J .4
1.5
The Rate of Reaction, 4 The Genera1 Mole Balance Equation 8 Batch Reactors 10 Continuous-Row Reactors 12 1.4.J Continuous-Stirwd Tank Reactor, 1.4.2 Tubular R~aclor 14 1.4.3 Packed-Bed Reacror 17 Industrial Reactors 21 Summary 25 CD-ROM Material 26 Questions and Problems 29 Supplementary Reading 35
12
2 CONVERSION AND REACTOR SIZING 2.1
2.2 2.3
2.4
Definition of Conversion 38 Batch Reactor Design Equations 38 Design Equations for Flow Reactors 48 2.3.1 CSTR (alsr~known as a Backmix Reactor or Vat) 43 2.2.2 Tubular Flow R ~ Q C I O {PFR) P 44 2.3.3 Packed-Bed Rearm 45 Applications of the Design Equations for Continuous-Flow Reactors 45
Contents
2.5
2.6
in Series 54 CSTRs in Series 55 PFRs i n Series 58 Cr~tnbinarionsof CSTRs and PFRs in Series 60 Comparing rhe CSTR alid PFR Reuctor filrrmes ~ m l i Reactor Seqitencitlg 64 Some Further Definitions 66 2.6.1 Spacelime 66 2.6.2 Space Vekocic 68
Reactors 2.5.1 2.5.2 2.5.3 2.5.4
Summary 69 CD-ROM Materials 71 Questions and Problems 72 Supplementary Reading 77
PART 1 Rate Laws
80 Basic Definitions 80 3.1. I Relative Rates of Reaction 81 3.2 The Reaction Order and the Rate Law 82 3.2.1 Power Lnw Models nnd Elementnry Rate Ln~w 82 3.2.2 Nonelemenrav Rate Lnws 85 3-2.3 Reversible Reacrions 88 3.3 The Reaction Rate Constant 91 3.4 Present Status of Our Approach to Reactor Sizing and Design 98 PART2 Stoichiometry 99 3.5 Batch Systems 100 3.5.i Equations fur Batch Concentrations 102 3.5.2 Constant- Volcfme Bnfch Reaction Systems 103 3.6 Flaw Systems 106 Eqttations for Concenfrarions in Flow 3.6.1 Systems 107 3.6.2 Liquid-Phase Concmtmtions 108 3.6.3 Change in the Total Number of Moles with R~enctionin rhe Gas Phase 108 Summary 124 CD-ROM Material 126 Questions and Problems 131 Supplementary Reading 141 3.1
4 ISOTHERMAL REACTOR DESIGN PARTI
Mole Balances in Terms of Conversion 144 Design Structure for Isorherma! Reactors 144 3.1 Scale-Up of Liquid-Phase Batch Reactor Data to the Design o f a CSTR 148 4.2.1 Batch Opemfion 148 4.3 Design of Contincous Stirred Tank Reactors (CSTRs) 156 4.3,J A Single CSTR 157 4.3.2 CSTRs in Series 158 4.3.3 CSTRs in PrrroIIeI 160 4.3.4 4 Second-Order Reoctiott irt n CSTR 162 4.4 Tubular Reactors 168 4.5 Pressure Drop in Reactors 175 4.5.1 Presslire Drop a~rdthe Rate Law 175 4.5.2 Flow Throirgh a Packed Bed 177 4.5.3 Pressure Drop in Pipes 182 4.5.4 Analvfica! Solution for Reaction with Presstire Drop 185 4.5.5 Spherical Packed-Bed Reactors 196 4.6 Synthesizing the Design of a Chemical Plant 196 PART2 Mole Balances Written i n Terms of Concentration and Molar Flow Rate 198 4.7 Mole Balances on CSTRs, PFRs, PBRs. and Batch Reactors 200 4.7.1 Liquid Phase 300 4.7.2 Gas Pi~nse 200 4.8 Microreactors 201 4.9 Membrane Reactors 207 4.10 Unsteady-State Operation of Stirred Reactors 215 4.10.1 Startup of a CSTR 216 4.10.2 Sernibcrrct?Reactors 217 4.10.3 Writing the Semibatch Reactnr Equa [ionsin Terms qf Cancentrntions 219 4.10.4 Wriring the Semibnrch Reacror Equations in Terns of Conversion 223 4.1 1 The Practical Side 226 Summary 227 ODE Solver Algorithm 230 CD-ROM Material 231 Questions and Problems 234 Some Thoughts on Critiquing What You read 249 Journal Critique Problems 249 Supplementary Reading 253
4.1
Contents
5
COLLECTION AND ANALYSIS OF RATE DATA 5.1
5.2
5.3 5.4 5.5 5.6 5.7
The Algorithm for Data Analysis 254 Batch Reactor Data 256 5.2.1 Differential Method ofAna!ysis 257 5.2.2 bztegral Mefhod 267 5.2.3 Nonlinear Regression 271 Method of Initial Rates 277 Method of Half-Lives 280 Differential Reactors 281 Experimental Planning 289 Evaluation of Laboratory Reactors 289 5.7. i Criteria 289 5.7.2 Types of Reacrors 290 57.3 Su171mayqf Reactor Ratings 290 Summary 291 CD-ROM Material 293 Questions and Proble~ns 294 Journal Critique Problems 302 Supplementary Reading 303
Definitions 305 6.I. I Types ?f Renctio~u 305 Parallel Reactions 310 6.2.1 Moxilni: b g rhe Desired Product,for Oize Renciant 311 6.2.2 Reartor Selection n11d Opemfing Corrdiflirfons 31 7 Maximizing the Desired Product in Series Reactions 320 Algorithm for Solution of Complex Reactions 327 6.4.1 Mole Boln~lces 327 6.4.2 Npt Rures ?f Reaction 329 6.4.3 Stnrclrinmerp: Co~~r~oerr~rurio~~s 334 Multiple Reactions in a PFWPBR 335 Multiple Reactions in a CSTR 343 Membrane Reactors to Improve Selectivity in Multiple Reactions 347 Complex Reactions of Ammonia Oxidation 351 Sorting Jt All Out 356 The Fun Part 356 Summary 357 CD-ROM Material 359 Questions and Problems 361 Journal Critique Problems 372 Supplementary Reading 375
253
Contents
7 R EA CTION MECHANISMS, PA THWAYS, BIOREACTZONS, AND BIOREACTORS 7.1
7.2
7.3
7.4
7.5
Active Intermediates and Nonelementary Rate Laws 377 7.1.1 Pseudo-Sready-State Hypothesis (PSSH) 379 7.1.2 Searching for a Mechanism 383 7.1.3 Chain Reactions 386 7.1.4 Reaction Pathways 391 Enzymatic Reaction Fundamentals 394 7.2.1 Et~z~rneSuhsrrate Complex 395 7.2.2 Mechanisms 397 7.2.3 Micheelis-MentenEquation 399 7.2.4 Batch Reactor Calcularionsfor Enzyme Reactions 404 lnhibi tion of Enzyme Reactions 404 7.3.1 Comperirive Inhibirion 410 7.3.2 Uncomperitive Inhibition 412 7 3.3 Noncclmpetitive Inhibition (Mixed Inlzibi~ion) 41 4 7.3.4 Substrate Inhibition 416 7.3.5 Multiple Enzyme and Substrare Systen~s 417 Bioreactors 418 7.4.1 CelI Growth 422 7.4.2 Rare Laws 423 7.4.3 Stoichiometiy 426 7.4.4 Mass Balances 431. 7.4.5 Chemosrafs 434 7.4.6 Design Equations 435 7.4.7 Wash-out 436 7.4.8 0.rqgen-Limited Growth 438 7.4.9 Scale-up 439 Physiologically Based Pharmacokinetic (PBPK) Models 439 Summary 447 CD-ROM Material 449 Questions and Problems 454 Journal Critique Problems 468 Supplementary Reading 469
8 STEA DY-STATE NONISOTHERMA L REACTOR DESIGN 8.1 8.2
Rationale 472 The Energy Balance 473 8.2.1 First Law of Tl~erpnodynamics 473 8.2.2 E\3aluarit~gthe Work Tern 474 8.2.3 O\?en,ien4 of Ellel-gy BaIa~~ces 476 8.2.4 Dissecti?t,q the Stead!-Srate Molar Flow Riir~s to Okrain !he Hear of Reaction 479 8.2.5 Dissecring h e Enrhalpies 48 1 8.2.6 Relating AHR,IT1, AHOR, (TR1- and AC, 483
Adiabatic Operation 486 8.3.1 Adiabafic Energy B~lance 486 8.3.2 Adiabatic Tirbular Reactor 487 Steady-State Tubular Reactor with Heat Exchange 495 8.4.1 Deriving the Dtergv Balance for a PFR 495 8.4.2 Balance on tire CovInnt Heat Transfer Fl~rirl 499 Equilibrium Conversion 511 8.5.1 Adiabatic Temperature and Equilibrilam Conversion 5t2 8.5.2 Optimum Feed Temperature 520 CSTR with Heat Effects 522 8.6.1 Hear Added to the Reactot; 6 522 Multiple Steady States 533 8.7.1 Heat-Removed Term, R(TI 534 8.7.2 Heat of Generation, G ( T ) 534 8.7.3 Ignition- Extinction Curve 536 8.7.4 Runaway Reactions in a CSTR 540 Nonisotherrnal Multiple Chemical Reactions 543 8.8.1 Energy Balance for Mulriple Reactiorls in Plrcg-Flow Rencdors 544 8.8.2 Energy Balance for Multiple Reactions in CSTR 548 Radial and Axial Variations in a Tubular Reactor 551 The Practical Side 561 Summary 563 CD-ROMMaterial 566 Questions and Problems 568 Journal Critique Problems 589 Supplementary Reading 589
9
UiVSTEADY-STRTE NOAXSOTHERMAL REA CTOR DESIGN 4.1
9.2
9.3 9.4
9.5 9.6
The Unsteady-State Energy Balance 591 Energy Balance on Batch Reactors 594 9.2.1 Adinbatic Operation of a Batch Reactor 594 9.2.2 Batch Reactor with Intermpred Isothermal Operation 599 9.2.3 Reactor Safety: The Use ofthe ARSST ro Find AH,,, E 605 and to Size Pressure Relief Valves Semibatch Reactors with a Heat Exchanger 614 Unsteady Operation of a CSTR 619 9.4.1 Startup 619 9.4.2 Falling Offthe Steady State 623 Nonisothermal Multiple Reactions 625 Unsteady Operation of Plug-Flow Reactors 628
xiii
Summary 629 CD-ROM Material 630 Questions and Problems 633 Supplementary Reading 614
645 D.cjniri017s 646 10.1.2 Cnrctl.;st Properties 648 10.I . 3 CdassiJicationof Crrml?r.sts 652 Steps in n Catalytic Reaction 455 10.2.1 Step I Overview: Difllsion fram tile Btdk to the External Transport 658 10.2.2 Step 2 Overview: Int~rnalDiffiision 660 10.2.3 Adsorption Isotherms 661 10.2.4 Surfnce Reection 646 10.2.5 Desorption 668 10.2.6 The Rate-Litniring Step 669 Synthesizing a Rate Law, Mechanism, and Me-Limiting Step 671 10.3.I Is the ddsorprion of Curn.me Rate-Limi~ing? 674 10.3.2 Is the Scttface Reaction Rate-Limiting? 677 10.3.3 IS the De.wrprion of Benzene Rate- Limiting? 678 10.3.4 Summary of the C~rmeneDecomposition 680 10.3.5 Reforming C ~ ~ t a l y , ~ r s681 10.3.6 Rate Lnws Derived from the Pseudo-Steady S r ~ f Hypothesis e 684 10.3.7 Terrtperuture Dependence of the Rare Caw 687 Heterogeneous Data Analysis for Reactor Design 688 10.4.1 Dediicing a Rare h w f m r n the E~perirnentulDara 689 10.4.2 Finding n Mechanism Consistent with Experimental Observations 691 10.4.3 Evnluation of the Rare Law Pammeters 692 10.4.4 Reactor Design 694
10.1 CataIysts 10.1.1
0
10.3
10.4
10.5
10.6
10.7
Reaction Engineering in Microelectronic Fabrication 698 IQ5.I Overview 698 10.5.2 Etching 700 10.5.3 Chemical Vapor Deposition 701 Model Discrimination 704 Catalyst Deactivation 707 10.7.1 Types of Catalyst Deactivation 709 10.7.2 Tenaperatlire-Erne Trajectories 721 10.7.3 Moving-Bed Reactors 722 10.7.4 sf might-TElmugh Tmnsporr Reactors (STTR) 728
Contents
Summary 733 ODE Solver Algorithm 736 CD-ROM Material 736 Questions and Problems 738 Journal Critique Problems 753 Supplementq Reading 755
11 EXTERNAL DIFFUSION EFFECTS ON HETEROGENEOUS REACTIOM 11.1 Diffusion Fundamentals 758 1I . 1. I Defilitions 758 11.1.2 Molar Flux 759 11.1.3 Fick'sFirsrLaw 760 1 1.2 Binary Diffusion 761 J 1.2.1 Eiralrtatirag The Molar F l u 761 11.2.2 Boundary Corlditions 765 11.2.3 Modeling Difusion Withorcr Reaction 766 1 I . 2.4 Temperature art$ Pres.ture Depend~nce of DAB 770 I J.2.S Modeling Difision with Chemical Reaction 771 11.3 External Resistance to Mass Transfer 771 1 1.3.1 TIne Mass Transfer Coeficient 771 11.3.2 Mass Transfer Coeficient 773 11.3.3 Correlario~~s for fhe Mass Transfer Co~firienr 774 11.3.4 Mass Transfer to a Single Particle 776 11.3.5 Mnss Transfer-Limited Reactions in Packed Beds 780 11.3.6 Robert the Worrier 783 1 1.4 What If. . . ? (Parameter Sensitivity) 788 1 1.5 The Shrinking Core Model 792 11 -5.1 Cara!\.sr Regenerarion 793 11.5.2 Phanl~acokinetics-DissoIufinn qf Monodispers~d SoIid Particles 798 Summary 800 CD-ROM Material 801 Questions and ProbIems 802 Supplementary Reading 810
12 DIFFUSION AND REACTION 12.1 Diffusion and Reaction in Spherical Catalyst Pellets 814 12.1.I Efccti~,eD~fif~rrsil'iry 814 12.1.2 Deri~nfionqf rhe D$fer@nrialEquatinn D~scribing Diffusinrr artd R~ucrion 816 12.1.3 Wririrrg the Equarion in Dimensionless f i m n 819
757
Contents
12.1.4
12.2 12.3 1 2,4 12.5
12.6 12.7
12.8
12.9 12.10
Solution to the Dlferential Equation for a First-Order Reaction 822 Internal Effectiveness Factor 827 Falsified Kinetics 833 Overall Effectiveness Factor 835 Estimation of Diffusion- and Reaction-Limited Regimes 838 12.5.1 wei.Ti-Prsate Crilerion for I~tterna!Diffusion 839 12.5.2 Mearx' Crirerion for External Difusion 841 Mass Transfer and Reaction in a Packed Bed 842 Determination of Limiting Siruations from Reaction Data 848 Multiphase Reactors 849 12.8.1 SlurnRenctors 850 12.8.2 Trickk Bed Reactors 850 Fluidized Bed Reactors 851 Chemical Vapor Depwi tion (CVD) 851 Summary 853 CD-ROM Material 852 Questions and Problems 855 Journal Article Problems 863 Journal Cririque Problems 863 Supplementary Reading 865
13 DTSTRIBUTZOM OF RESIDENCE TIMES FOR CHEMICAL REACTORS 1 3.1
868 PARTI Characterislics and Diagnostics 868 13.I. J Reside~rce-TirneDidribulion (RTD)Functior? 870 13.2 Measurement o f the RTD 871 13.2.1 Puhe Irrput E~prrit~ient 871 13.2.2 Step Tracer E.rl?erinzenr 876 13.3 Characteristics of the RTD 878 13.3.S Jniegr-a1R~lnrir~tlshil~s 838 13.3.2 Mearr Residenre Tinw 879 13.3.3 Orher Mor~lerrtsof the RTD 881 13,3.4 Not.rlla/ted RTD F~o~crion. E(O) 884 13.3.5 I ~ ~ r e ~ n o l - A Di.~~rihuriorr, ge I(a) 885 13.3 RTD in Ideal Reactors 885 13.4.1 RTDs i ~ Batch t and Plug-Flow RPCICIOKT 885 13.4.2 Single-CSTR RTD 887 13.4.3 Lcrrlri~lc~r FICJM* Reocror ( L F R ) 888 13.5 Diagnoqtics and Troubleshnoting 891 13.5.1 Gniewl Cnn~rlrenrs 891 12.5.2 Si171plcDiog\~os~ic.~ o ~ Tt~~~lhlesho(~fit7g d U S ~ I the FR KTD for kIenl Rericrors 892 1.q.5.3 PFR/CSTRSeriesRTD 897 General CIlaracterislics
867
PART2 Predicting Conversion and Exit Concentration 902 1 3.6 Reactor Modeling Using the RTD 902 3 3.7 Zero-Parameter Models 904 13.7.I Segr~gnrionM o d ~ l 904 13.7.2 bfLL~humMi.redne.7~Mode/ 915 ~~l 13.7.3 Comparirzg Segregarion and M o x i r n ~ / iWi.xedness Predictions 922 I 3.8 Using Software Packages 923 13.8 1 Heot Eflect.? 927 1 3.9 RTD and Multiple Reactions 927 13.9.1 Segregafion Model 927 13.9.2 Ma~itnurnbIixedizess 928 Summary 933 CD-ROM Material 934 Questions and Probkms 936 Supplementary Reading 944
14.1
14.2 14.3 14.4
14.5 14.6 14.7
14.8
14.9
14.10
Some Guidelines 946 14.1.I One-Pornmeter Models
947 14.1.2 Two-Parnmer~rModels 948 Tanks-in-Series (T-1-51 Model 948 Dispersion Model 955 Flow. Reaction, and Dispersion 957 14.4.1 Balance Eqlrnrinns 957 14.4.2 Bouadcd~Conditions 958 14.4.3 Finding D, and the Pecler Number 962 14.4.4 Dispersion in a Ethular Reactor with Laminar Florv 962 14.4.5 Correlationsfor D, 964 14.4.6 Experimental Determination of D, 966 14.4.7 Slopp?: Tracer Inputs 970 Tanks-in-Series Model Versus Dispersion Model 974 Numerical Solutions to Flows with Dispersion and Reaction 975 TWO-ParameterModels-Modeling Real Reactors with Combinations of Ideal Reactors 979 14-7.1 Real CSTR Modeled Using Bypassing and Deadspace 979 14.7.2 Real CSTR Modeled as Two CSTRJ with Interchange 985 Use of Software Packages to Determine the Model Parameters 988 Other Models of Nonideal Reactors Using CSTRs and PFRs 990 AppIications to Pharmacokinetic Modeling 991
Contents
Summary 993 CD-ROM ~Vnterial 994 Questions and Problems 996 Supplementary Reading 1005
Appendix A
NUMERICAL ECHIVIQ UES
Appendix B
IDEAL GAS CONSTANT AND COWERSION FA CTORS
Appendix C
THERMODYNAMIC R ELA TIOIVSHIPSZ W L VING THE EQUILIBRIUM CONSTANT
Appendix D
MEASUREMENT OF SLOPES ON SEMILOG PAPER
Appendix E
SOFTWARE PACKAGES
Appendix
F
NOMENCLATURE
Appendix G
RATE LAW DATA
Appendix H
OPEN-ENDED PROBLEMS
Appendix I
HOW TO USE THE CD-ROM
Appendix J
USE OF COMPUTATIONAL CHEMISTRY SOFTWARE PA CKAGES
INDEX ABOUT THE CD-ROM
xvii
Preface
The man who has ceased to learn ought not to be allowed to wander around Ioose in these dangerous days.
M. M.Coady
A. The Audience This book and interactive CD-ROM is intended for use as both an undergraduate-level and a graduate-level text in chemical reaction engineering. The level will depend on the choice of chapters and CD-ROM Prufessionaf R@ference Shelf (PRS) material to be covered and the type and degree of difficulty of problems assigned.
B. The Goals B.4.
To Develop a Fundamental Understanding of Reaction Engineering
The first goal of this book is to enable the reader to develop a clear understanding of the fundamentals of chemical reaction engineering (CRE). This goal will be achieved by presenting a structure that allows the reader to solve reaction engineering problems through reasoning rather than through memorization and recall of numerous equations and the restrictions and conditions under which each equation applies. The algorithms presented in the text for reactor design provide this framework, and the homework problems will git~e practice at using the algorithms. The conventional home problems at the end 05 each chapter are designed to reinforce the principles in the chapter. These problems are about equally divided between those that can be solved with a
XX
Preface.
calculator and those that require a personal computer and a numerical snftware package such as Polymath, MATLAB, or FEMLAB. To give a reference point as to the level o f nnderst:inding o f CRE required in the profession. a number of reaction engineering problems from the California Board of Registration for Civil and Professional Engineers-Chernical Engineering Examinations (PECEE) are included in the text.] Typically. these problems should each require nppmximately 30 minutes to solve. Finally, the CD-ROM should greatly facilitate learning the fundamentals of CRE because it includes summary notes o f the chapters, added examples, expanded derivations, and self test>. A complete description of these knrnirrg resorrrces is given in the 'The Integration of the Text and the CD-ROM" section in this Preface. 8.2.
To Develop Critical Thinking Skills
A second goal is to enhance critical thinking skills. A number of home problems have been included that are designed for this purpose. Smratic questioning is at the h e m of critical thinking, and a number of homework problems draw from R. W.Paul's six types of Sacsatic questions2 shown in Table P-I. 1 4 1) Q~resticmsjbfirr ck~r$uatiun: Why do you fay that7 Hoic does
1 (2)
th~qrelake
to our d i ~ u s s ~ o n ?
*'Are you going to include diffusion In )our mole balance equations?" Quasrionr rhnr pmhc nssrrmpnons: What could we assume instead? How can you verify or disprove that assumption?
"Why are you neglecting rddial diffusion and including only ~ ~ l diffu~ionT' a l
( ( 3 )Q~trsrionsrhar p m k
(
reasons and evirleucu: What would be an example?
"Do you think that diffusion is respnnsibIe For the Iower cnnvers~onr'
((4) Quesrions about viewpoinrs and perspctrl-e~:Whar would be an alternative?
"With all the bends in the pipe. from an industriallpracticaI
querrion:
wo neglected diffusion?"
was the
point of thrc
question9 Why do you think I
asked th~squestion?
"Why do you thlnk diffusion i s important'?'
permission for use of these problems, which. incidentally, may be obtained from the Documents Section, California R o d of Regiaration For Civil and Professional Engineers-Chemical Engineering, 1004 6th Street, Sacramento, CA 95814, is gratefully acknowtedged. (Note: These problems have been copynghred by the California Board of Registration and may not be reproduced without its permission). 2 R. W. Paul, Crirical Thinking (Santa Rosa, Calif.: Fbundation for Critical Thinking, 1992). I The
Sec. B
The Goais
XX~
Scheffes and RubenfeId1'l expand nn the practice of cntical thinking skill, discussed by R. W. Paul by using the activities. statements, and question5 shown
in Table P-2. T ~ LP-? E
CR~ICA THIYKINC L SKLI 5'
'
Analyzing: separating or brcaking a whole into pnrts to discover their nature. function, and atio ion ship^ "I~tudiedir piece hy pirce." "'1 sorted thing%out."
.
Applying Standards: judging nccatding to e\tablishcd pr.wnal. profc~rional.or rocid rules or criteria "Ijudged it according to. ..."
Discriminating: recognizing differences and similariries among things or situarirln\ and &tinguishing sarcfully as to catepr): or n n k "Irank ordered rhe various. ..." "Igrouped things together."
Enformation Seeking: searching for evidence. facts, or knowledge by idel~tifyinprelevant sources and gathering objectibe, subjcctire, historical, rind current data from those wurces " 1 knew I needed ro look uplstudy ...:' "I kept .searching for dntn."
Loplical Reasoning: drawing inferenits or conclusions that are supported in or ]ustitied by evidence "I deduced from the information that ...Y "My rdtionale fur the conclusion was.. .."
Predicting: envisioning a plan and its convqurnces "I envisioned the outcome would he.. ..+' "1 wns prepared for...." Transforming Knowledge: cb:~ngingor convening the condition. nature. form. concepts among calltexts "Iimproved on the bilbics by.. .:' "I wondered if that would fit the siuuation or ...."
or fi~nctionof
I have found the best way so develop and practice critical thinking akills is to use Tdbles P-I and P-2 to help students write a question on any assigtled homework problem and then to explain why the question involves critical thinking. More information on critical thinking can be found on the CD-ROM in the section on ProbI~rnSolving. 8.3.
To Develop Creative Thinking Skills
The third goal of this book is to help deveIop creative thinking skills. This goal will be achieved by using a number of probiems that are open-ended to various degrees. Here the students can practice their creative skills by exploring the example problems as outlined at the beginning of the home problems of each Courtesy of B. K. Scheffer and M. G.Rubenfeld. "A Consensus Statement on Critical Thinking in Nursing." Jorrntnl 01Niirsing Ehcntion. 39, 352-9 (20001. Courtesy of B. K. Scbeffer and M. 6 . Rubenfeld, "Critical Thinking: What b It and How Do We Teach It?" Ctirrenr Issltes in A11rK~it~g (200 1).
xxii
Prefs-ce
chapter and by making up and solving an original problem. Problem P4-I gives some guidelines for developing original problems. A number of techniques that can aid the sntdents in practicing and enhancing their creativity can be found in FogIer and CeBlanc5 and in the Thoughts on Problem Solving section on the CD-ROM and on the web site wwu:engifiumich.edu/-ere. We will use these techniques, such as Osborn's checklist and de Bono's lateral thinking (which involves considering other people's views and responding to random stimulation) to answer add-on questions such as those in Table P-3. (1 ! Brainstorm ideas to a~kanother question or suggest another calculation that can be made for this homework problem
(2) Bralnstorin ways you could work this homework problem incorrectly. ( 3 )Brainslorn ways to maie this problem easier or more difficult or more exciting. (4) Brainstorm a lisl of thlngs you learned from working this homework problem and what ynu thtnk the p i n t of the problem is.
(5) Brainstorm the reafons why your calculations overpredicted the conversion that was mearured when the reactor was put on slrearn. Assume you made no numerical errors on your calculat~ons. ( 6 )"What if...'' questions: The "What i f .. " questions are particularly effective when used with the Linng Emntple Problems where one varies the parameters to explore the problem and to carry out a sensitivtty analysis. For example, w11rrt i f s o r n ~ o nsuggcrrtd ~ rltor vou shmrld douhle tht caraly.~!pnrriclc d~umtrerel;wltat w odd you s o j '
One of the major goals at the undergraduate level is to bring students to the point where they can solve complex reaction problems, such as multiple r a c tions with heat effects, and then ask "What if ...*'questions and look for optimum operating conditions. One problem whose solution exemplifies this goal is the Manufacture of Styrene, Problem P8-26. This problem is panicuIarly interesting because twa reactions are endothermic and one is exothermic. Endofhennic {I ) Ethylbenzene 4 Styrene + Hydrogen: Endothem~ic (2) Ethylbenzene 4 Benzene + Ethylene: (3) Ethylbenzene + Hydrngen + Toluene + Methane: Exo~hermic To summarize Section B. it is the author's experience that both critical and cseative thinking skills can be enhanced by using Tables P-I, P-2. and P-3 to extend any of the homework problems at the end of every chapter.
C. The Structure The strategy behind the presentation of materia1 is to build continually on a few basic ideas in chemical reaction engineering to solve a wide variety of problems. These ideas. referred to as rhe PiIlacr ~ ; l f Clze~nicalR~actionEngineeri~lg. . -
W. S. Fogler and S. E. LeBlanc. S~rnregiesfi)rCreative Pmblem Sol\.ing (Upper Saddre River. N.J.: Prentice HaH, 1995).
Sec. C
xxiii
The St:ucfvre
are the foundation on which different applications rest. The pillars holding up the application of chemical reaction engineering are shown in Figure P- I .
Figure P-I Pillars of Chemical Reaction Engineering.
From there Pillars we construct our CRE algorithm: Mole balance + Rate laws
+ Stoichiornetry + Energy balance + Combine
With a few restrictions, the contents of this book can be studied in virtually any order after students have mastered the first four chapters. A flow diagram showing the possible paths can be seen in Figure P-2.
]
CH ~ - C $ V E ~ S I W T & N O
FIEllCTOR SUING
I
CH 3 - RATE LAWS AND
-1
H H HRe+1
yrlTH-, CW 7 810RELCTQNS A tEcs :;M5
STAT?
+iE&T
CATALYTIC fllhtf OR5
VMSTEACY
EYEQNuL
NONIDEAL
STATE HEAT
DIFFUSbOV EFFFC'S
REKTORS
BIOREACTQPS
EFFECTS
MULTLPLE REK~lOYS
cEki~15
EEFECTS
IN POROUS
--
WITH H E P F " F C C r - Cu
Figure P-2
Sequences for \tidying the text
DtSTRI0~ION
xxiv
Preface
Table P-4 shows examples of topics that ran be covered i n n graduate course and an undergradua~ecourse, In a four-hour undergraduate course at [he University of Michigan. approximately eight chapters are covered in the following order: Chapter? 1, 2, 3. 4, and 6; Sections 5.1-5.3; and Chapters 7. 8, and parts of Chapter 10. TAHLE P-4.
U\IDERGRADCATHGRADL'ATE COVERAGE OF CRE
Gmdul~re,Wo!eri~~i/Co~t rw
Uttdp.ryrndrmr~.Marerinl/C~llrce Mole Balances (Ch. I ) Smog in LOFAngeles Basin IPRS Ch. 1) Reactor Staging (Ch. 2) Hippopotamus Stomach (PRS Ch. 2) Ralr L w s (Ch. 3) Stoichiometry ICh.31
Rc~ctors(Cb. 4): Batch. PFR. CSTR. PER. Sernibntch. Membrane Dara Analysis: Regresalon (Ch 5 1 Multiple Reactions (Ch. 6) Blood Congulatron (SN Ch. 6)
Bioreaction Engineering (Ch. 7) Steady-State Heat Effects (Ch.8): PFR and CSTR with and without a Hear Exchanger Multiple Steady Stetea Unsteady-State Heat Effects (Ch. 9) Reactor Safety
Catnlysit (Ch. 10)
Shon Re\-iew ICh. 1-4. 6. 8 ) Collision Theory (PRS Ch. 3) Tran~itionState Theory (PRS Ch. 3) Molecular Dynarn~csIPRS Ch 3) Aero~olRwctors CPRS Ch 4) Multiple Reactrons (Ch 6): Fed Membrane Rracton Binreaction%and reactors (Ch.7. PRS 7.3.7.4. 7.51 Pnlymeri7~tion(PRS Ch 7) Co- and Counter Current Heat Exchange (Ch. 8) Radial and Axial Gndientx In a PFR FEMLAB ICh. 8 ) Reactor Stabil~tyand Safety tCh. 8. 9. PRS 9.3) Runaway Reactions IPRS Ch. 8 ) Catalyst Deactivation [Ch. 10) Revdmce Time Distributron [Ch 13 1 Models of Real Reactorc [Ch. 14) Applications (FRS): Mult~phaseReactors. CVD Reactors. Bioesctors
The reader will observe that although metric units are used primarily in this text (e.g., kmoVm3, Jlmol), a variety of other units are also employed (e.g.. Ib/ft3). This is intentional! We believe that whereas most papers published today use the metric system, a significant amount of reaction engineering data exists in the older literature in English units. Because engineers will be faced with extracting information and reaction rate data from older literature as well as the current literature, they should be equally at ease with both English and metric units. The notes in the margins are meant to serve two purposes. First, they act as guides or as commentary as one reads through the material. Second, they identify key equations and relationships that are used to solve chemical reaction engineering problems.
D. The Components of the CD-ROM The interactive CD-ROM is a novel and unique part of this book. The main purposes of the CD-ROM are to serve as an enrichment resource and as a professional reference shelf. The home page for the CD-ROM and the CRE web site (www.engin.umich.edu/-cre/fogler&gl4nen) is shown in Figure P-3.
Sec. D
The Components of the CD-ROM
XXV
Figure P-3 Screen shot of the home page of the CD-ROM.
The objectives of the CD-ROM are threefold: ( I) to facilitate the learning of CRE by interactive1y addressing the Feld~r/SolornonInventory nf Learning Sglesh in the Summary Notes. the additional examples, the Interactive Computing Modules (ICMs), and the Web ,Modules; (2) to provide additional technical material for the professional reference sheIf; (3) to provide other tutorial information, examples, derivations, and self tests, such as additionai thoughts or1 probtem solving, the use of computational software in chemical reaction engineering, and representative course structures. The following components are listed at the end of masr chapters and can be accessed from each chapter in the CD-ROM. Learning Resources The Learning Resources gwe an overview of the material in each chapter and provide: extm explanations. examples, and applications to reinforce the basic concepts of chemical reaction engineering. The learning resources on the CD-Rob1 include the following:
Summary Notes
I . Srrrnmap Notes The Summary Notes give an overview of each chapter and provide on-demand additional examples, derivations, and audio comments as well as self tests to assess each reader's understanding of the material. 2. Web Mod;rles The Web Modules, which apply key concepts to both standard and nonstaodard reaction engineering problems (e.g., the use of wetlands to degrade toxic chemicals, cobra bites), can be loaded directly from the CD-ROM.
xxvi
0 Computer Modules
Preface
Additional Web Modules are expected to be added to the web site (wwn~.engin.un~ich,edu/-rre) over the next several years. 3. Interactiiu Conlpufer Modules (ICMs) Students have found the Interactive Computer M d u l e s to be hoth f u n and extremely useful to review the important chaprer concepts and then apply them to real problems in a unique and entertain~ngfashion. In addition to updating all the ICMs from the last edition, two new modules. The Gwaf Rauc (Ch. 6) and Enzyme Mon (Ch. 7)- have been added. The complete set of 1 1 modules foll~ws:
-
Quiz Show I (Ch.1 ) Reactor Staging (Ch.7) Quiz Show H (Ch.3) Murder Mystep (Ch.4) TICTac (Ch. 4 ) * Ecoloey (Ch 5 )
solved Problems
Q
CL :-2 ~5
W
A
L ~ v ~ nExample g Problem
Refzrence Shelf
The Great Race (Ch.61 Enzyme Man {Ch. 7) Heat Effect\ 1 (Ch. 8) Heat Effects 11 tCh. 8 ) CataIyslr (Ch. JO)
4 . Solved Probler~rs A number of solved problems are presented along with problem-solving heuristics. Problem-solving strategies and additional worked example problems are available in the Pt~lbletnSollirlg section of the CD-ROM. Living Example Problems A copy of Polymath is provided on the CD-ROM for the students zo use to salve the homework problems. The example problems that use en ODE salver (e g.. Polymath) are referred to a< "living example problems" because students can load the Polymath program directly onto their own computers In order to study the problem. Students are encouraged to change parameter values and to "play with" rhe key variables and assumptions. Using rhk Living Example Problems to explore the problem and asking "What if.. ." questions pro\ ide students with the opportunity to practice crttical and crea~ivethlnking skills. Professional Reference Shelf This section of the CD-ROM contains I. Matertal ahat was in previous editions .(e.g., polymer~zation,rlurry reactnn. and chern~cal vapor disporition reactors) that har been omitted from the pr~nledversion of the fourth edition 2. New topics such as coll~sioriartd rrarrsition smte rltenr?: aerosol rroc!orq, DFT. and r-urlmvo,v mcriorts, which are commonly found In graduate courses 3. Material that is important ro the practicing engineer, such as derails of the industrial reactor design for the oxidation of SO? and design of spllericnl reactors and other ~na~esial that is typically not included in rhe majority of chemical reaction engineering courses = Software Toolbox on the CD-ROM P f ~ / ~ f n n fThe ! l . Polymath software rncludes an ordinary drfferen~ial equation (ODE) ~olver, a nonlinear equation \nlver, and nonl~near regression. Ac with prejious ed~tions.Polymath is inchded with t h ~ sedition to explore the example problems and to sol\.e the home problems. A special Polymath web sile ( r r ~ ~ ~ r ~ : ~ ~ ~ I ~ n t ~ t I ~ - ~ nhar f i hcen ~ ~ o ~sel ' ~up . cfor o ~ ~this ~ fbook n ~ I ~byr Polymath authar\ Ct~tlipand Shacham Thi5 weh site provides inlrrnarion an how to ohlain an updated \t.rsion of Polymath at a di
Sec. E
The Integration of the Ted and the CD-ROM
XXV~~
FEMLAB. The FEMLAB software inciudes a partial differential equation solver. This edition includes a specially prepared version of FEMLAB on itc; own CD-ROM.With m L A B the students can view both axial and radial temperature and concentration profiles. Five of rhe FEMLAB modules are:
*
Isothermal operation Adiabatic operation Heat effects with constant heat exchange fluid temperature Heat effects with variabre heat exchanger temperature Dispersion with Reaction using the Danckwerts Boundary Conditions (two cases)
As with the Polymath programs, the input parameters can be varied to learn how they change the temperature and concentration profiles.
Instructions are included on how to use not only the software packages ei Polymath. MATLAR, and FEMLAB. but also on how to apply ASPEN PLUS to solve CRE problems. Tutorials with detailed screen shots are provided for Polymath and E M L A B . =
Other CD-ROM Resources FAQs. The Frequently Asked Questions (FAQs) are a compilation of questions collected over the years from undergraduate students taking reaction engineering.
Pmblem Sol~+ing.In this section, both critical thinking and creative thinking are discussed along with what TO do if you get "stuck" on a problem. Viswul E~rc~clopedia r!f Equip~r~ctnr. This section was developed by Dr. S u w ~ Montgomery at the Universify of Michigan. Here a wealth of photographs and descriptions of real and idea1 reactors are given. The students with visual. actibe, sensing, and intuitive leamlng styles of the FelderlSolomon Index will pafticuEarly benefit from this section.
Rearlor Lob. Developed by Professor Richard Hem at the University of California at San Dlego, this interactive tml will allow students not only to test their comprehension of CRE material but also to explore different situations and combinations of reaction orders and types of reactions.
p.c 1i
G r c ~ nL~~llgiileering Home Pml?lfnrs. Green engineering problems for virtually every chapter have k e n developed by Professor Roben Hesketh at Rowan Unlxersity and Professor Martin Abraham at the University of Toledo and can be found at ~'~'~:mwa~l.~du/gwcnrngrr~cer!ng. These problems also accompany the hook by David Allen and David Shonnard. Green Chernicul Enginwring: En1.imnt~terrmllyCart.sciatrs Design of CI~emicnlProcess~s(Prentice HaII. 2002).
Green engineering
E. The Integration of the Text and the CD-ROM E.I.
The University Student
There are a number of ways one can use the CD-ROM in conjunction with the text. The CD-ROM provides enrichment resources for the reader in the form of interactive tutorials. Pathways on how to use the materials to learn chemical
xxviii
Preface
reaction engineering are shown in Figure P-4. The keys to [the CRE learning Aow sheets include primary resources and enrichment resources:
= Primary resources
0 CD
= Enrichment resources
In developing a fundamental understanding of the material, students may wish to use only the primary resources without using the CD-ROM line.,using only the boxes shown in Figure P-4), or they may use a few or all of the interactive tutorials in the CD-ROM (i.e.. the circles 4hown in Figure P-4). However, to practice the skills that enhance critical and creative thinking, students are strongly encourr~gedto use the Lii-irlg E . ~ t t n p iPivhletns ~ and vary the model patarnetem to ask and answer "What if. .." questions.
lnteracflve Computer
Summary Notes
t =
Start
Text
Homework ~ectures
Problems
4
Modules
Problems
Figure P-4 A Student P a t h ~ a y20 Integrate the Class. the Text. and'rhe CD.
Note that even though the author recommends studying the Living Example Problems before working home problems. they may be bypassed, as is the case with all the enrichment resources. if time is short. However. class testing of the enrichment resources reveals that they not only greatly aid in learning the material but also serve to motivate students through the noveI use of CRE principles.
E.2.
For the Practicing Engineer
A figure similar to Figure P-4 for the practicing engineer Is given in the CD-ROM Appendix.
Sec G
Vhat's New
XX~X
E The Web The web site (wu,\r:er?gh.~~tnicI~.e~Ett/-cre) will be used to updare the text and the CD-ROM. I t will ~dentifytypographical and other errors In the hrst and later printings of the fourth edition of the text. In the near future. additional matenal will be added to include more solved probIem5 as well as additiana1 Web Modules.
G. What's New Pedagogy. The fourth edition of this book maintains all the strengths of the previous additions by using algorithms that allow students to Iearn chemical reaction engineering throueh logic rather than memorization. At the same time it provides new resources that allow students to go beyond solving equations in order to get an intuitive feel and understanding of how reactors behave under different situations. This understanding is achieved through more than sixty interactive simulations provided an the CD-ROM that is shrink wrapped with the text. The CD-ROM has been greatly expanded to address the Fttlder/SoIornon Inventory of Different Learning Styles7 throush interactive Summary Notes and new and updated Interactive Computer Modules (ICMs). For example, the Global Learner can get an overview of the chanter material from the Summary Notes: the Sequential Learner can use all the hot buttons: and the Active h earner can interact with the ICM's arid use the hot buttons in the Summazy Notes. ~i new pedagogical concept is introduced in this edition through expanded emphasis on the example problems. Here, the students simply load the Living Example Problems (LEPs) onto their computers and then explore the problems to obtain a deeper understanding of the implications and generalizations before working the home problems for that chapter. This exploration helps the srudents get an innate feel of reactor behavior and operation, as well as develop and practice their creative thinking skills. To develop critical thinking skills, instructors can assign one of the new home problems on troubleshooting, as weIl as ask the students to expand home problems by ashrig a reIated question that involves critical thinking using Tables P-1 and P-2. Creative thinking skills can be enhanced by exproring the example: problems and asking "what if. . ." questions, by using one or more of the brainstorming exercises in Table P-3 to extend any of the home problems, and by working the open-ended problems. For example, .in the case study on safety, studtats can use the CD-ROM to c m y out a post-mortem on the nitroanaline explosiod in Example 9-2 to learn what would have happened if the cooling had failed for five minutes instead of ten minutes. Significant effort has been devoted to developing example and home probIems that foster critical and creative thinking.
h r t p : / h t v w . n ~ . ~eddfeldcr-pu 1~, hlic/llSrlir/scies. hon
Sec. H
Acknowledgments
xxxi
contributions to the first, second. and third editions (see Introduction, CD-ROM). For the fourth edition, 1 give special recognition as follows. First of all. I thank my colleague Dr. Nihat Giimen who coauthored the CD-ROM and web site. His creativity and energy had a great impact on this project and reaIly makes the fourth edition of this text and associated CD-ROM special. He has h e n a wonderful colleague to work with. Professor Flavio F. de Moraes not only translated the third edition into Portuguese, in col~aborationwith Professor Luismar M. Porto, but also gave suggestions, as well as assistance proofreading the fourth edition. Dr. Susan Montgomery provided the Msuul Et~cyclopedia of Equiprnenr for the CD-ROM. as well as support and encouragement. Professor Richard Herz provided she Reaoior Lab portion of the CD-ROM. Dr. Ed Fonres. Anna Gordon, and the folks at Comsol provided a special version of FEMLAB to be included with this book. Duc Kguyen. Yongzhong Liu, and Nihat Giirmen also helped develop the FEMLAB material and web modules. These contributions are greatly appreciated. EIena Mansilla Diaz contributed to the blood coagulation model and, along with Kihat Gurmen. to the pharmacokinetics model of the envenomation of the Fer-de-Lance. Michael Breson and Nihat Giirmen contributed to the Russell's Viper envenomation model, and David Umulis and Nihat Giirmen contributed to the aicohol metabolism. Veerapat (Five) Tanla yakom contributed a number of the drawings. along with many other details. Senior web designers Nathan Cornstock, Andrea Sterling. and Brian Vicente worked tirelessly wltb Dr. Giirmen on the CD-ROM. as wtlI as with web designers Jiewei Cao and Lei He. Professor Michael Cutlip. along with Professor Mordechai Schacham, provided Polymath and a special Poiymarh web site for the text. Brian Vicente took major responsibili~yfor the solution manual, while Massimiliano No15 provided solutions to Chapters 13 and 14. Sornbuddha Ghosh also helped with the manual's preparation and some web material. 1 would also like to thank colleagues at the University of Colorado. Professor Will Medlin coauthored the Molecular Reaction Engineering Web Modules (Dm), Professor Kristi Anseth contributed to rhe Tissue Engineering Example. and Professor Dliirlakar Kornpnla contributed to the Profe~sionaI Reference Shelf R7.4 Mul~ipEeEn7yme~lMvlul~iplc Substrates. I also thank MI> Ph.D. graduate students-Rama Venkate5an. Duc Kugyen. Ann Piyarat Wattana. Kris Paso. Veerapat (Five) Tantayako~m,Ryan Hartman, Hyin Lee. Michael Senra. Lizzie Wany, Pr;~chan~Singh. and Kriangkrai Kraiwa~tannwong-for their patience and understanding during the period while I was wriring t h i ~hook. In i~ddition. he supporl prclr~rdedby the staff and colleagues at the departments. of chemical ensineerinp at University College London and the University uf Colr~radowhile 1 finiched the final details of the Text ic greatly appreciated. Boll1 are very stimu1;iling and art _ereat places to work and €0spend a rahbaticr~l. The stimulating discussions with Pmfessors Roben Hesketh. Phil Savage. John Falconer. D. B. Rattacharia. Rich %lacel,Eric McFarland, Will Medlin. ;ind Krisii hnceth are greatly appreciated. t also appreciate the fr~endchipand insight? provided by Dr. Lec Brown on Chapters 13 and 11.Mike Cutiip not
xxxii
Preface
only gave suggestions and a critical rending of many section'; but. most importantly, provided continuous support and encouragement throughout the course of this project. Don MacLaren (cornpositon) and Yvette Raven (CD-ROM user interface design) made large contributions to this new edition. Bernard Goodwin (Publisher) of Prentice Hall was extremely helpful and supportive throughout. There are three people who need special mention as they helped pull everything together as we rushed to meet the printing deadline. Julie Nahil, Full-Service Production Manager at Prentice Hall. provided encouragement. attention to detail, and a great sense of humor that was greatly apprecinred. Janet Peters was nor only a meticulous proofreader of the page proofs, but also added many valuable editorial and other comments and suggestions. Brian Vicente put out extra effort so help finish so many details with the CD-ROM and also provided a number of drawings in the text. Thanks Julie, Janet, and Brian for your added effort. Laura Bracken is so much a part af this manuscript. I appreciate her excellent deciphering of equations and scribbles, her organization. and her attention to detail in working with the galley and copy edited proofs. Through all this was her ever-present wonderful disposition. Thanks, Radar!! Finally, to my wife Janet, love and thanks. Without her enormous help and support the project would never have been possible. HSF Ann Arbor
For updates on the CD, new and exciting appIicatiws, and typographical errors for this printing. see the web site:
www.engin,umich.edd-cre or
ww.engin.umich.edrJ-cre/fogEerdGgumen
Mole Balances
The first step to knowiedge is to know that we are ignorant. Socrates (470-399
HOW ir
n chcrnrcal
engilleer different fmm other
0.c.)
The Wide Wide Wild World of Chemical Reaction Engineering Chemical kinetics is the study of chemical reaction rates and reaction mechanisms. The study of chemical reaction engineering (CRE) combines the study of chemical kinetics with the reactors in which the reactions occur. Chemical kinetics and reactor design are at the heart of producing almost all industrial chemicals such as the manufacture of phtharic anhydride shown in Figure 1-1. It is primarily a knowledge of chemical kinetics and reactor design that distinguishes the chemical engineer from other engineers. The selection of a reaction system that operates in the safest and most efficient manner can be the key to the economic success or failure of n chemical plant. For example, if a reaction system produces n large amount of undesirable product, subsequent purification and separation af the desired product could make the entire prwess economically unfeasible.
Mole Balances
Chap. 1
Fwd W e r Sham
Ftgurt 1-1 Manufacture of phthalic anhydride.
The Chemical Reaction Engineering {CRE) principles learned here can also he applied in areas such as waste trearment. microelectronics. nanoparticles and living syrtems in addition to the more traditional areas of the manufacture o f chemicals and phamaceutica1s. Some of the exainples that illustrate the wide application of CRE principles are shown in Figure 1-2. These examples include modeling smog in the L.A. basin (Chapter I), the digestive system of a hippopotamus (Chapter 2), and molecular CRE (Chapter 3). Also shown is the ~nanufactureof ethylene glycol (antifreeze), where three of the most common types of industrial reactors are used (Chapter 4). The CD-ROM describes the use of wetlands to degrade toxic chemicals (Chapter 4). Other examples shown are the solid-liquid kinetics of acid-rock interactions to improve oiI recovery (Chapter 5 1: pharrnacokinetics of cobra bites and of drug delivery (Chanter 6): free radical scavengers used in the design of motor oils (Chapter 7), enzyme kinetics. and pharmacokinetics (Chapter 7): heat effects, runaway reactions, and plant safety (Chapters 8 and 9); increasing the octane number o f gasoline (Chapter 10): and the manufacture of computer chips (Chapter 12).
Sec f .l
The Rate of Reachon. -r,
Hippo Digestion (Ch. 2) Q
1
Vinyl Ally1 Ether (arrows ~ndicate
4-Pentenal
Smog (Ch. 1, Ch. 7)
Transitron Stale (dashed llnes show transillon State eIeclron belocallzatlon) Molecular CRE (Ch. 3)
-
Chemical Plant for Ethylene GlycoI (Ch. 4)
a
-2
Wetlands Remediation of Pollutants (Ch. 4)
Effecl~veLubrtcani
Design Scavsnglng Free Rad~mls
ACID
Oil Recovery (Ch. 5) Pharmsmkmetics 01 Gobm B~tes Mult~pleReact~ons m a ~alch (Bodvl Reactor Cobra kltes (Ch. 6)
N~troanaltnePlant Explosron Exoiherm~cReact~onsThat Run Away Ptant Safety (Ch.8, Ch.9)
Pharmacokhetics (Ch. 7)
Mic~mlectronicFabrication Steps (Ch. 10, Ch. 12)
Figure 1-2 The wide uorlcI uf applications of CRE
Mole Balances
Chap. 1
Overview-Chapter 1. This chapter develops the first building block of chemical reaction engineering, mole balances. that will be used continually throughout the text. After completing this chapter the reader will be able to describe and define the mte of reaction, derive the general mole balance equation, and apply the general mole balance equation to the four most common types of industrial reacrors. Before entering into discussions of the conditions that affect chemical reaction rate mechanisms and reactor design, it is necessary to account for the various chemical species entering and leaving a reaction system. This accounting process is achieved through overall mole balances on individual species in the reacting system. In this chapter, we develop a general mole balance that can be applied to any species (usually a chemical compound) entering, leaving, and/or remaining within the reaction system volume. After defining the rate of reaction, -rA, and discussing the earlier difficulties of properly defining the chemical reaction rate. we show how the general balance equation may be used to develop a preliminary form of the design equations of the most common industrial reactors: batch, continuous-stined tank (CSTR), tubular (PFR), and packed bed (PBR). In developing these equations, the assumptions pertaining to the modeling of each type of reactor are delineated. Finally, a brief summary and series of shon review questions are given at the end of the chapter.
1.I The Rate of Reaction, -b The rate of reaction tells us how fast a number of moles of one chemical species are being consumed to form another chemical species. The term chemrcnl species refers to any chemicnI component or element with a given identic. The identity of a chemical species is determined by the kincl, nirrnbet; and conjigitrrrrinn of that species' atoms. For example. the species nicotine Ea bad tobacco alkaloid) is made ar up configuntioa. of a fixed number specitic shown atoms iniIlustmtes a definitethe rnolecular arrangement The ofstnicture kind,
8 GH,
Nicotine
number. and configuration of atoms in the species nicotine (responsible for "nicotine fits") on a molecuiar level. Even though two chemical compounds have exactly the same number of atoms of each element, they could still be different species because of different configurations. For example, 2-butene has four carbon atoms and eisht hydrogen atoms; however, the atoms in this compound can form two different
arrangements.
H
\ C , =C
CH,
H
H
/ \
' 3 3
cis-2-butenc
and
\ C , =C
CH,
CH3
/
\
H
trii~ts-2-butene
Sec 1 1
When hac a chemical reaction tnken pliice'?
The Rate of R99ctl99
5
-5
As a consequence of the different configurations, these two isomers display different chemical and physrcal propertres. Therefore. we consider them as two different xpecie~even thnuph each has the same number af atoms of ezch element. We say that a chemical reactiort has tnken place when a detectable number of mulecufes of one or more species have lost their identity and assumed a new form by a change in the kind or number of atoms in the compound and/or
by a change in structure or configuration of these atoms. In this classical approach to chemical change, it is assumed that the total mass is neither created nor destroyed when n chemicill reaction occurs. The mass refemd to is the totar collective mass of a11 the different species in the system. However, when considering the individual species involved in a particular reaction, we do speak of the rate of disappearance of mass of a particular species. The ,artJ nf disoppeorolzce of n species, say species A, i s the trtrntber ($A mnlrc~ries[hot lose their chemical i d t w t i ~per ~rtrirt i r l l ~per ~rnirvc~iumefhrn~rghfhe breaking and slrbreqiienr re-forming of ckentical bonds drrring the course of the reartion. In order for a particular species to '"appear" in the system, some prescribed fraction of another species must lose its chemical identity, There are three basic ways a species may lose its chemical identity: decomposition. combination. and isomerization. In deconrposiriot~.the molecule lose4 its identity by being broken down into smaller molecuIes. atoms. or atom fragments. For example. if benzene and propylene are formed from n cumene molecule.
A species r a n low its identill, by decomposition. combination, ur isomcrizaIion.
curnene
benzene
propylene
the cumene molecule has lost its identity [i.e., disappeared) by breaking its bonds to form these molecuies. A second way that a molecule may lose its species identity is through conlbinarion with another molecule or atom. In the example above. the propylene molecule would Lose its species identity if he reaction were carried out in the reverse direction so that i t combined with ben-
zene to form curnene. The third way a species may lose its identity is through isorneri-ization, such as the reaction
Here, although the molecule neither adds other molecules to itself nor breaks into smaller molecules, it still loses its identity through a change in configuration.
6
Mole Balanoes
Chap. 1
To summarize this point, we say that a given number of molecules (e.g., mole) of a particular chemical species have reacted or disappeared when the molecules have lost their chemical identity. The rate at which a given chemical reaction proceeds can be expressed in several ways. To illustrate, consider the reaction of chlorobenzene and chIoraI to produce the insecticide DDT (dichlorodiphenyl-trichlomthane) in the presence of fuming sulfuric acid.
Letting the symbol A represent chloral, be H 2 0 we obtain
B be chlorobenzcne, C be DDT, and D
The numerical value of the rate of disappearance of reactant A, -r,. is a positive number (e.g., -r, = 4 mol Aldm3+s). What is -r,?
The rate of reaction, -r,, i s the number of moles of A (e.g.. chloral) reacting (disappearing) per unit time per unit volume (mol/dm3.s).
A+ZB-+C+D The convention r~
-rs r,
= 4 rnol Aldm' F = 8 mol Bldrn' r = 4 mol B/dm3 a
What ir r ; ?
The symbol r j is the rate of formation (generation] of species j. If species j is a reactant, the numerical value of r, will be a negative number (e.g.. r , = -4moles Atdm3.s). Ifspecies j is a product. then r, will be a positive number (e.g.. rc = 4 moles C/dm"s), In Chapter 3. we will delineate the prescribed relationship between the rate of formation of one species. r, (e-g., DDT[C]), and the rate of disappearance of another species. -r, ( e . ~ . chlorobenzene[B]), , in a chemical reaction. Heterogeneous reactions involve more than one phase. In heterogeneous reaction systems, the rate of reaction is usually expressed in measures other than volume, such as reaction surface area or catalyst weight. For a gas-solid catalytic reaction, the gas molecules must interact with the qolid catalyst surface for the reaction to take place, The dimensions o f this heterogeneous reaction rate, r; (prime). are the n a n ~ h e rof rnoic.7 of A rraciirig per eni, rims per enit rrlar.7 of carnlysi (tnol/s+gcatalyst). Most of the introductory di5cussions on chemical reaction engineering in this hook focus on homogeneous'sy stems. The mathematical definition of a chemical reaction rate has been a source o f confusion in chernlcal and chemical engineering literature for many years. The origin of this confusion stems from laboratory bench-scale experiments t h a ~were carried out to obtain chemical reaction raw data. These early txperiments were batch-type. in which the reaction vessel was closed and rigid: consequently, the ensuing reaction took place at constant volume. The reactants were mixed together at time t = O and the concentration of one of the reactants. CA, was measured at iarious rimes I. The r;tte of reactinn \+as deter-
7
The Rats of Reaction, -r,
See. 1.1
mined from the slope of a plot of C, as a function of time. Letting r-, be the rate of formation of A per unit volume (e.g., mol /s.dm", the investigators then defined and reported the chemical reaction rate as
dl
r* = dC*
Surnma:y Notes
The ~ i t cInw doe5 nor depcnd on the I>p of rescrur u\cd?!
However this "definition" is wrong! Ft is simply a mole balance that i s only valid for a constant volume batch system. Equation (1-1) will not apply to any continuous-flow reactor operated at steady srate. such as the lank (CSTR)reactor where the concentration does not vary from day to day (i.e.. the concentration is not a function of time). For amplification on this point, see the section "Ir Sodium Hydroxide Reacting?'iin the Summary Note!, for Chapter 1 on the CD-ROM or on the weh. In conclusion. Equation ( 1 - 1 I is not the definition of the cheinical reaction rate. we shall simply say that I . is the rate #ffirrnulir)~?of .sfwc.ifbj j w r . aaii i?o!unir.i r is the number of rno/es of species j generated per unit volume per unit rime. The rate equation (i.e., rate law) for rj i s an algebraic eqzlation that is solely a function of the properties of the reacting materials and reaction conditions (e.g., species concentration, temperature. pressure, ar type uf catalyst, if any) at a pnint in the system. The rate equation is independent of the type of reactor le.g., batch or continuous flow) in which the reaction is carried out. However, because the propertiec and reaction conditions of the reacting materials may vary with position in a chemicaI reactor. r, can in [urn be a function of position and can vary from point to point in the system. The chemical reaction rate law is esrentially an algebraic eq~~iitiun involving concentration, not a differential equation.[ For example. the alpehraic form of the rate law for -r, fur the reaction
may be a linear function of concentratinn,
or. as shown in Chapter 3. it may he some other algebraic fi~nctionof concen-
rration, such as
- -- - -
-
.
' For furllicr elaboration on
his poinl. w e CIJPJPI G i c . .Tr.i., 2.T. 337 (19711): R. L. 1. I (New Yt11.k: AIChE. 19811. :tnd R. 1. Kahcf. "R;~te\." Cher~r.[ti<. Cr>tr~r~rroiY. I 5 (19x1 ).
C ~ ) n c \ and
H.S. Foplcr. crl5 . AIChE .Llrrdrrlrtr I r f \ ~ r l t r f i o r i 5l~r.ic.rE: XKuzc..r.
.
Mole Balances
The convention
k,C, =-
-r4
The rate law is an a l ~ e b r a ~equation. c
Chap. 1
I +klCA
For a given reaction. the particufar co-xentrarion dependence that the rate law follows (i.e.. -r, = kc, or - r A = kc; or ... ) must be determined from experirnenral observation. Equation (1-2) states that the rate of disappearance o f A i h equal to a rate constant k (which is a function o f temperature) times the square of the concentration of A. By convention. r, is the rare of formation of A; consequently. -,-, is the rate uf disappearance of A. Throughout this book. the phrase mte qf gerlerution means exact1y the same as the phrase rntr qf'jiwmntion, and these phrases are used interchangeably.
1.2 The General Mole Balance Equation To perform a mole balance on any system, the system boundaries must first be specified. The volume enclosed by these boundaries is referred to as the sTstern v o l ~ m e We . shall perform a mole balance on species j in a system volume. where species j represents the particular chemical species of inrerest, such ax water or NaOH (Figure 1-3). J
Figure 1-3
System Vorurne
Balance on system volume.
A mole balance on species j at any instant i n time. t. yields the following
1 1-1
equation:
Rate of flow ofjinto the system (moles1time
Mole balance
In
50
-
Rate o f flow ofjoutof]+ the system (molesftime)
+
a
Out
Fi
Rate of generation
ofreaction j by chemical within the system (molesJtime) -
=
I
o f j wirhin
the system
1
+ Generation
= Accumulation
+
-
G,
3 dt
(1-3)
Sec. 1.2
The General Mole Balance Equation
9
where N, represents the number of moles of species j in the system at time t. If all the system variable5 l e . ~ . ,temperature. catalytic activity, concentration of she chemical species) are spatially uniform throughout the system volume, the rate of generation of specie? j, G,. is j u r t the product of the reaction volume. V. and the rate of formation of species j, r,.
Suppose now that the rate of formation of species j for the reaction varies with the position in the system volume. Thai is. it has a value r,, at location I. which is surrounded by a small volume, A V l , within which the rate is uniform: similarly, the reaction rate has a value x,? at location 2 and an aswciated volume, AV, (Figure 1-4).
Figure 1-4 Dividing up the system volume, K
The rate of generation, AG,,, i n terms of r j l and subvolume A V t , is AG,, = r j , h V 1 Similar expressions can be written for AG,? and the other system subvolumes, A V , . The total rate of generation wirhin the system volume is the sum of all the rates of generation in each of the subvolurnes. If the total system volume is divided into rM subvoIumes, the total rate of generation is
10
Mole Balances
Chap. 1
By taking the appropriate limits (i.e., let M + w and A V + 0 ) and making use of the definition of an integral, we can rewrite the foregoing equation in the form
From this equation we see that r, will be an indirect function of position, since the properties of the reacting materials and reaction conditions (e.g., concentration. temperature) can have different values at different locations in the reactor. We now replace G,in Equation (1-3)
hy its integral form to y~elda form of the general inole balance equation for any chemical species j that is entering, leaving, reacting. andlor accumulating
within any system voluine 1! This i s a baxic equation for chemical reac~ion engineeri~ig.
From this general mole balance equation we can develop the design equations for the various types of industrial reactors: batch, semibatch. and continuous-flow. Upon evaluation of these equarions we can determine the time (batch) or reaclor volume (cominuous-flow) necessary to convert a specified amount of the reactants into products.
f .3 Batch Reactors is a hatch reactor u\rriq
@\ \
/
Reference Shelf
A hatch reactor is used for small-scale operation. for testing new procesces that have not been fully de\elnped. for the manufacture of expensive products. and for processes that are difficult 10 conImento continuous operations. The reactor can he charged (i.e., filled) through the holes at the top (Figure 1-S[al). The batch reactor has the advantngc of high convercions that can be obtained by leaving the reactant in the reactor for long periods of time. but it also has the disadvantages of high labor costs-per batch, the vxiabiliry of products from batch to batch. and the difficulty of large-scale production (see Profecsional
Reference Shelf [PRS]).
Sec. 1.3
Figure I-5(a) Simple batch homogeneous reactor. [Excerpted by special permission from Chem. Eng, (5.?(10), 2 1 I (Oct. 19561 Copyright 1856 by McGraw-Hill. Inc., New York, h'Y 10020.1
-
II
Batch Reactors
Figure 1-5(6) Batch reactor m~xingpatrems. Further descr~ptionrand pho~osof the batch reactors can be found in both the Vr~~rul Ewc~clopcdraof Equrpnretrl and in the Pmfe~sinnrtlRcf~rtr>rtShelf on the CD-ROM
A batch reactor has neither inflow nor outflow of reactants or products while the reac~ionis being carried out: F,o = F, = 0. The resulting general mole bal-
ance on species j is
If the reaction mixture is perfectly mixed (Figure 1 -5[b]) so that there is no variation in the rate af reaction throughout the reactor volume. we can take r, out of the integral, integrate. and write the mole balance in the form
Perfect mixing
Let's consider the isomerization of species A in a batch reactor
As the reaction proceeds. the number of moles of A decreases and the number of moles of B increases, as shown in Figure 1-6.
Mole Balances
0
ti
Figure 1-6
t
0
Chae. 1
1 '
Mole-time rrajectories.
We might ask what time. t , , is necessary to reduce the initial number of moles from NAo to a final desired number N A I .Applying Equation (1-5) to the isomerization
rearranging.
and integrating with limits that at r = 0. then N, = N,, NA = we obtain
and at t = I , , then
This equation is the integral form of the mole balance on a batch reactor. It gives the time, r,. necessary to reduce the number o f moles from /VAoto N,, and also to form rYB1moles of B.
1.4 Continuous-Flow Reactors Continuous flow reactors are almost always operated at steady state. We wiil consider three types: the continuous stirred tank reactor (CSTR), the plug flow reactor (PFR), and the packed bed reactor (PBR). Detailed descriptions of these reactors can be found in both the Professional Reference Shelf IPRS) for Chapter 1 and in the Visrral Encyciopeclin of Equiprnenr on the CD-ROM, 1.4-1 Continuous-Stirred Tank Reactor
What is a CSTR
uxd
A type of reactor used commonly in industrial processing is the stirred tank operated continuously (Figure 1-7). It is referred to as the continuo~u-stirred tnrlk renrtor (CSTR)or vat, or backmix reactor; and i s used primarily for liquid
phase reactions. I t i s normafly operared a t steady state and I \ a\surned to be perfectly mixed: conrequenily, t hers is no ti me dependence or pokition dependence o f the temperature. [he concentration, or the reaction rnte inside rile CSTR. That is, every variable i> the wme at every point inside the reactor. Becat~sethe temperature md concenrration are identical everywhere u itliin the reaction vessel, they are the same at the exit point as they are elsewhere in the tank. Thus the temperature and concentralion in the exit stream are modeled a5 being the same as those jnside the reactor. In systems where mixing is highly nonideill, the weII-mixed model i* inadequate and we must resort to other modeling techniques, such ns rebidence-time distributions, to obtain meaningful results. This topic o f nonideal mixing is discussed in Chapters 13 and 14.
What reac~ion systems use a CSTR?
The idtal CSTR i v assumed trr ht. perrec~lymixed.
Figure 1-7(a) CSTRharch reactor. [Courtesy of Pfnudlcr. Inc.1
Figure I-71bk CSTR mixing patterns. Alw see the Vir~rnlEnr~cluped~n oJ Eqwp~nenton the CD-ROM.
When the general mole balance equation
is applied to a CSTR operated at steady state (i.e., conditions do not change with time),
14
Mole Balances
Chap. 1
in which there are no spatial variations in the rate of reaction (i.e., perfect
mixing),
it takes rhe familiar form known as the design eqrrarion for a CSTR: F~~
Al
The CSTR design equation gives the reactor volume I{necessary to reduce the entering flow rate of species j, from Fj, to the exit flow rate F,,when species j i s disappearing at a rate of -rj. We note that the CSTR is modeled such that the conditions in the exit stream (e.g.. concentration, temperature) are identical to those in the tank. The molar flow rate 6is just the product of the concentralion of species j and the volumetric flow rate u :
I Fl = C j +v
& p
time
-
moles
volume
volume time
Consequently. we could combine Equations (1-7) and ( 1-8) to write a balance on species A as
1.4.2 Tubular Reactor
When is reactor mnsi u,ed?
In addition 10 the CSTR and batch reactors, another type of reactor commonly used in industry is the ruhlliar rmr.ror: It consists of a cylindrical pipe and is normally operated at steady state, as i~ the CSTR. Tubular reactors are used most often for gas-pha~ereactions. A schematic and a photograph of industrial tubular reactors are shown in Figure 1-8. In the tubular reactor, the reactants are continually consumed as they flow down the length of the reactor. In modeliny the tubular reactor. we assume that the concentration varies continuoucly in the axial direction through the reactor. Con~equcntly.the reaction rate, which is a function of concentration for all but zero-order reactions, will also vary axially. For the purposes of the material presented here. we consider systems in which the flow field may be modeled by that of a plug flow profile l e g . . uniform veIocity as in turbulent floa). as shewn rn Figure 1-9. Thar is. there i h no radial variation it1 reaction rate and the reactor is referred to as a plug-fiow rcactor [PFR). (The laminar flow renctor is diwurced in Chapter 13.)
Sec. l.d
15
Contintlous-Flow Reactors
ee PRS and Enr~cloprEqltip~~~~nt.
Figure 1-8(a) Tubular reoclrw wherna~ic Longitudinal tubular reactor. [Excerpted b! spctnl permission from CIINII.Ella, h3( 10). 2 1 1 (Oct. lq56). Copyright 1956 by McCrau -Hill. Inc , Kcu York, Y 1' 10010.1
Figurc 1-8(bl Tuhulw i-ellclur photo Tubular reac~orfor p~vductlonof Dlmerwl G. [Photo Colrnecy of Editions Technvq Institute Frrrncois du Pc~rolj
Plug tlr>ur-l~o radral iarinlionr in vclncity.
Reactants
Products Fiaure 1-9 Pluy-flow
The general
t u l l ~ ~ l are;lctor. r
moIe balance equation i s
given by Equation ( 3 4 ) :
The equation
we will use to design PFRs at steady
variations in reaction rate within this volume. Thus the generation term. AG,,i s
Figure 1-14) Molc balance cm \pc.r~t=cj tn trrlutnc A \ ' .
16
Mole Balances
Molar flow
Molar flow ,
In
-
Chap. 7
Generotinn of species j
of species j within A V +-g en era ti on = Accumulation within AV
Out
Dividing by A V and rearranging
the term in brackets resembles the definition of the derivative
Taking the limit as AV approaches zero, we obtain the differential form of steady state moIe balance on a PFR.
Tubular mador
We could have made the cylindrical reactor on which we carried out our mole balance an itregular shape reactor, such as the one shown in Figure 1-1 1 for reactant species A.
Picasso's reactor
Figure 1-11 Pablo Picasso's reactor.
However, we see that by applying Equation (1-10) the result would yield the same equation (i.e., Equation [I-1 I]). For species A, the mole balance is
Sec. 1.4
27
Continuovs-Flow Reactors
Consequently. we see that Equation ( I - I I ) applies equally well to our model o f tubular reactors of' variable and constant cross-sectional area, although ir is duubtful that one would hnd a reactor of the shape shown in Figure I - ll unIess it were designed by Pablo Picasso. The conclusion drawn from the
applicarion of the design equation to Picasso's reactor is an important one: the degree of completion of a reaction achieved in an ideal plug-flow reactor (PFR) does not depend on its shape, only on its total volume. Again consider the jsornerizntion A + B, this time in a PFR.As the reactants proceed down the reactor. A is consumed by chemical reaction and B is produced. Consequent!y, the molar Raw rate of A decreases and that of B increases, as shown i n Figure 1 - 1 2.
Figure 1-12 Profiles of molar flow rates in a PFR.
We now ask what is the reactor volume V , necessary to reduce the entering molar flow rare o f A from FA,to FA!.Rearranging Equation ( I 12) in the f orrn
-
and integrating with limits at V = 0, then FA= FA,,, and at V = V,. then FA= FA,.
V, is the volume necessary to reduce the entering molar flow rate FA, to some specified value FA1 and also the volume necessary to produce a molar flow rate o f B of FBI. 1.4.3 Packed-Bed Reactor
The principal difference between reactor design calculations involving homogeneous reactions and those involving fluid-solid heterogeneous reactions is that for the: batter, the reaction takes place on the surface of the catalyst. Consequently, the reaction rate is based on mass of solid catalyst. W, rather than on
d8
Mole Balances
Chap. 1
reactor volume, K For a fluid-solid heterogeneous system, the rate of reaction of a substance A is defined as
-r; = mol A reacted/s.g catalyst The mass of solid catalyst is used because the amount of the catalyst is what is important to the rate of product formation. The reactor volume that contains rhe catalyst is of secondary significance. Figure 1-1 3 shows a schematic of an industrial catalytic reactor with vertical tubes packed with catalyst. Product gas
t
Feed gas
Figure 1-13 Longitudinal cataly~icpacked-bed reactor [From Cmpley, American Institute of Chemical Engineers. 8612).?4 t I99U). Reproduced with perrnik~ionof the American In~tituteof Chemical Enpinccrs. Copynpht O 1 9 0 AIChE. All right\ reserved ]
PRR MoleB~~~~~~
In the three idealized types of reactors just discussed (rhe perfectly mixed batch reactor. the plug-flow tubular reacror 1PFR). and the perfectly mixed continuous-stirred rank reactor (CSTR), the design equations (i.e.. mole balances) were developed based nn reactor volume. The derivation of the design equation For a packed-bed catalytic reacror (PRR)will be carried out in a manner analogous to the development of the tubular design equation. To accomplish this derivation, we simply replace the voFume coordinate in Equation (1-10) with the catalyst weight coordinate W (Figure 1-14),
Figure 1-14
Pncherl-lxcf reactor srhem;~tic.
Sec. 1.4
19
Continuous-Flow Reactors
As with the PFR,the PBR is assumed to have no radial gradients in concentration, temperature, or reaction rate. The generalized mole balance on species A over catalyst weight AW results in the equation
In -
+
Out
Generation = Accumulation
The dimensions of the generation term in Equation (1-14) are (rL)AWz
!notes A
A .(mass of catabsr) = moles -
(time)(rnass of caraly.~r)
~irf~e
which are. as expected, the same dimensions of the molar flow rate FA. After dividing by AW and taking the limit as A W -+ 0. we arrive at the differential form of the mole balance for a packed-bed reactor:
Use differential form of d m g n equation for cataIy
When pressure drop through the reactor (see Section 4.5) and catalyst decay (see Section 10.7) are neglected, the integral form af the packed-cataIyst-kd design equation can be used to calculate the catalyst weight. Use integral fnrm only for no AP and no catalyst decay.
W i s the catalyst weight necessary to reduce the entering molar flow rate of species A. F,,. to a flow rate FA. For some insight into things to come, consider the following example of how one can use the tubular reactor design Equation ( 1 - 1 I). Exumpk 1-1 How Large Is
it?
Consider the liquid pha.re ris - rrrrrls isomerizafion of 3-burene
runs-2-bu tene
cis-2-butene
which we will write symbolically as
1
A - R
The first order (-r, = kc,)reaction is carried nut in a rubular reactor in which the volulnezric flow mte, c, I F constanl. 1.e.. I? = E ) , ~ .
20
Mole Balances
Chap.
'
1. Sketch the concentration prof le. 2. Derive an equation relating the reactor volume to the entering and exitrng concentrations of A , the rate constant R, and the volumetric Row rate v . 3. Determine the reactor volume necessary to reduce the exiting concentration t~ 10% of the entering concentration when the volumetric ffow rate is I ( dm3/rnin (i.e., litenlrnin) and the specific reaction rate, k. is 0.23 mrrr-'
.
1. Speciec A is consumed as we move down the reactor, and as a result. both the molar flow rate of A and the concentration of A will decrease as we move. Because the volumetric flow rate is constant, v = v , , one can use Equation (1-8) to obtain the concentration of A, C, = F , ~ U and ~ , then by compariwn with Figure 1-12 plot the concenrration of A as a function of rertctor volume as shown in Figure El-1.1.
Figure EI-1.1
Concentration prufile.
2. Derive an equation relating Y
v,, k, CAo,and CA.
For a tubular reactor, the mole balance on species A Cj = A) was shown to be given by Equation (1-1 1). Then for species A (j= A) results
For a fist-order reaction, the rate law (discussed i n Chapter 3) is
Reactor sizing
Because the volumetric flow rate. u , is constant ( u = uo). as it is for most liquidphase reactions,
Multiplying both sides of Equation (EI-1.2) by minus one Equation (E 1-1. I ) yields
and then substituting
Sec
1
I
A+B
r
5
Industriat Rsac!ors
U ~ i n gthe conditions a ihc entrance ef the reactor that when V = 0, Ben C , = C,,,,
- 5f:d5 , J:( ,V k r
V
C,= C,,,exp
C,
Carrying oul the integration of Equation (E I- 1.4) gives
-(kV/~*l
I 3. We want to find the vo:olume. V , , at which C , = --C,, I0 c = 10 dmJlmin.
,,
I
for k = 0.23 min-I and
Substituting C*,,, C,. y,. and k in Equation (El-1.5). we have
We see that a reactor volume of 0. E rn' is neceshary to cnnven 9 0 4 of species A entering into product B for the parameters given.
In the remainder of this chapter we look at slightly more detailed drawings of some typical industrial reactors and point out a few of the advantages and diradvantages of each.'
1.5 Industrial Reactors When is a batch reactor u r ~ d ' '
d Links
Be sure to view actual photographs of industrial reactors on the CD-ROM and on the Web site. These are also links to view reactors on different web sites. The CD-ROM also includes a portion of the K.slml Ei~cyclopen'ia of Equipment-Chemical Reactors developed by Dr. Susan Montgomery and her students at University of Michigan. [I] Liquid-Phase Reactions. Semibatch reactors and CSTRs are used primarily for liquid-phase: reactions. A semibatch reactor (Figure 1- 151 has essentially the same disadvantages as the batch reactor. However. it has the advanrages of temperature control by regulation of the feed rate and the capabifity of minimizing unwanted side reactions through the maintenance of a Iow concentration of one of the reactants. The semibatch reactor is also used for two-phase reactions in which a gas usually is bubbled continuousty through the liquid. Chern. Eng., 63(10),2 l 1 ( 1956). See also AlChE MoritlEar hstn~crionSeries
(1984).
E, 5
Mole Balances
Chap. 7
Flus gas
t
Heater or m l e r
and product
, Naphtha and recycle gas
Reactant B
Compressed alr
Figure 1-15(s) Semibatch reactor. [Excerpted by spec~alpermis\ion rrom Chc~n.Errg., fi.?(lO~. 21 1 rocl. 19561. Copyright 1956 by McGrau-H111. Inc., Yew York. NY 10020.1
M'hat are the advanrages and disadvantaFer of a CSTR?
Furnace
Figure I-15fi) Fluidized-bed catalytic reactor. [Excerpted by ~pecialpermiss~onfrom Chem. Grx., h3( 10). 21 1 (Oct. 1956). Copyright 1956 by McCraw-Hirl. Inc., New York, NY 1 Ml2O.l
A CSTR is used when intense agitation is required. Figure I-71a) showed a cutaway view of a Pfaudler CSTRhatch reactor. Table 1-1 gwes the typicaI sizes (along with that of the comparable size of a familiar object) and costs for batch and CSTR reactors. All reactors are glass lined and the prices include heatinglcooling jacket, motor, mixer. and bafffeq. The reactors can be operated at temperalures between 20 and 450°F and at pressures up to 100 psi.
Iblurrn'
Pljce
!~)/IIIIIP
5 Gallnnr (was~eba~ket I
S19.01KI
I CIUO Gallons 12 Jacuzzis)
585.IKX1
50 Gallonr
S38.M)
4000 GaIlom (8J ~ c u z z i r l
$150.000
S70,W
ROW G a l l o n ~ [pafnline lanker)
5280.000
(yarhuge
can 1
5()0 Gallon4
(Jacuzzi)
Price
The CSTR can either be used by itceIf or. in the manner shown in Figure 1-16, as part of a series or battery of CSTRs. It is relatively easy to mainrain good temperature conrrol with a CSTR because it is well mixed. There is. however. the disadvantage that the cornersion of reactant per volume of reactor is the smallest of the flow reactors. Consequen~ly.very large reactors are neces-
Sec. I5
23
Industrial Reactors
sary to obtain high conversions. An industrial flow sheet for the manufacture of nitrobenzene from benzene using a cascade of CSTRs is shown and described in the Professional Reference Shelf for Chapter 1 on the CD-ROM. F
Feed
cooling jackets
Product
Figure 1-16 Baltery of mrred tank< IExcerplcd by special permrssion from C l r ~ ~ Org.. n. 631 10). 21 I (Oct. 1956) Copyright 1956 by McGraw-HIII, Inc.. New York. NY 1010 1
If you are not able to afford to purchase a new reactor. it may be possible find a used reactor thul may fir your needs. Previously owned reactors are much less expensive and can he purchased from equipment clearinghouses such ns Aaron Equipment Company ( ~ ~ w ~ : a u r n ~ ~ e q u i p n ~ cor n f .Loeb co~n) Equipment Supply ~ w ~ ~ ~ : l o e b e g ~ i p ~ ~ ~ e n f . c o ~ ~ ~ f l . to
Lrrkr
What are the a d l ~ n f a ~ and es d~qadbantage
CSTR: liquids
PFR, gnce\
/2J Gas-Phase Reactions. The tubular reactor (i.e.. plug-flow reactor [PFR]) is relatively easy to maintain (no moving partc), and it usually produces thc highest convesqion per reactor volume of any of the flow reactors. The disadvantage of the tubular reactor is that it i s difficult to control temperature within the reactor. and hat spots can occur when the reaction i s exothermic. The tubular reactor is commonly found either in the form of one long rube or as one of a number of shorter reactors arranged in a tube hank as shown in Figures I -R(a) and (h). Most homoyeneaus liquid-phase flow reacrors are CSTRr. whcreoq moxt homogeneous pis-phase flow reactors are tubular. The cosrq of PFRs and PBRs (without catalyst) are similar to the costs of heat exchangers and can be found in Planf Desipl and Erot~or~~ics~for Cl~rr?lrcnl ErrgErreer.5. 5th cd.. by M. S. Peters and K. D. Timmerhaus (New York: McGraw-Hill, 2002). Frnln Figure 15-12 of the Peters and Tirnmerhaus book. one can get an estimate of the purchase cost per foot of $ 1 for a I -in.pipe and 52 per foot for a ?-in. pipe for single tube? and approximately 520 to $50 per square font of \usface area for fixed-tube sheet exchangers. A packed-bed (also called a hxed-bed) reactor is essentially a tubular renctor that is packed w~rhsolid satatyst panicles (Figure 1-1 3). Th~sIieterogcneous reaction rystenl i\ moG1 often used to c;1(:11y~e gas 1.eactionx. This reactor has the same difiicultiec with temperature conlrol a\ nther tubular reactors: in addition. he c:~lab\! is 11rui1lly troubIc\i~ine10 repl;~ce.On occasion. channeling of thc
24
Refe:enceChe1f
sclu& Roblemr
Mole Balances
Chap. 1
gas flow occurs. resulting in ineffective use of piins of the reactor bed. The ad%antageof the packed-bed reactor is that for most reactionr it give^ the highe5t conversion per weight o f catalyst of any catalytic reactor. Another type of catalytic reactor in common use is the Ruidized-bed (Figure I - 15[b]) reactor, which is analogous to the CSTR in that its contents. though heterogeneous, are well mixed. resulting in an even temperature distribution throughout the bed. The fluidized-bed reactor can only be approximately modeled as a CSTR (Example 10.3): for higher precision it requires a model of its own (Section PRS12.3). The temperature is relatively uniform throughout, thus avoiding hot spots. This type of reactor can handle large amounts of feed and solids and has good temperature control; consequently. i t is used in a large number of applications. The advantages of the ease of catalyst replacement or regeneration are sometimes offset by the high cost of the reactor and catalyst regneration equipment. A thorough discussion of a gas-phase industrial reactor and process can be f w n d on the Pmfessional Reference Shelf of the CD-ROM for Chapter I. The process is the manufacture of paraffins from synthesis gas (CO and H1)in a straight-through transport reactor (see Chapter 10). In this chapter. and on the CD-ROM, we've introduced each of the major types of industrial reactors: batch. semibatch, stirred tank, tubuIar, fixed bed (packed bed), and fluidized bed. Many variations and modifications of these commercial reactors are in current use; for further elaboration, refer to the detaiIed discussion of industrial reactors given by Walas.-' The CD-ROM describes industrial reactors. along with typical feed and operating conditions. In addition, two solved example problems for Chapter 1 can be found on the CD.
Closure. The goal of this text is to weave the fundamentals of chemical reaction engineering into a structure or algorithm that is easy to use and apply ro a variety of problems. We have just finished the first building block of this algorithm: mole balances. This algorithm and its conesponding building blocks will be developed and discussed in the following chapters: Mole BaIance, Chapter 1
Rate Law, Chapter 3
-
Stoichiometry, Chapter 3 Combine, Chapter 4 Evaluate, Chapter 4
Energy Balance, Chapter 8
With this algorithm, one can approach and solve chemical reaction engineering probIems through logic rather than memorization.
3
S. M. Walas, Reaction Kinetics for CIzemical Engineers (New 1959). Chapter I 1.
York: McGraw-Hill.
25
Sun?mary
Chap. 1
SUMMARY Each chapter summary giles the Lcy points of the chapter that need to be remembered and carried into succeeding chnpters. I. A
a
more balance on species j. u h ~ c henters. leaves, reacts. and accumuIntea in system volume F: is c
F,"-F,+/
dN sfl==i dr
(51-I)
Ik nndon1.v iJ: the contents of the reactor are well mixed, then a mole balance (Equation S 1- 1 1 on specie?A grves
2. The kinetic rate law for r, is:
Solely a function of properties of reacting materials and mactirm cond~tions (a.g., concentration [activities], temperature, pressure, catalyst or solvent [if any11 The rate of formation of species j per unit volume (e.p., molls~dm'l An intensive quantity Ii.e.. it does not depend an the total amount) An algebraic equation. no? a differential equation (e.g., -rA = kc,. -rq = kc:) For homogeneous catalytic systems. typical units of -5 may be gram moles per second per liter: for heterogeneous systems, typical units of $ may be gram moles per second per gram of catalyst, By convention, -rF, is the n t e of disappearance of species A and r, is the rate of formation OF species A.
3. Mole balances on species A in four common reactors are as FoIlow~: TABLE
Reacror
CSTR
SUM~~IARV OF REACTORMOLEBALAYCES
Cornnwnr
Mole hrlunce Diflerenrrd firm
No spatid variations.
-
steady state
Steady state
PBR
S-I
Steady state
A!srhrrrir f i r m
Intqroi Fnrrlr
26
Mole
Batrnces
Cclap. T
CD-ROM MATERIAL Learning Resources 1
2 Summa. y Motes
.
S~irntlrar~ tdorrs Web Miireriol A. Problem-Solving Algnr~rhin B. G e t ~ ~ nUnstuck p on a Problen~ Thic \ire on the web and CD-ROM gives t i p on how to overcome mental krrriers in pmbleni snl'r In$. C. Stnng in L.A. bacin
R. G ~ t t i n rUnstuck
3. Ir~rerncrr~,c Co)~iprrrer. Morltilet A. Quiz Show I
C. Smog in L.A.
Chap 1 4.
L ~ v i n gExample Problem
CD-ROt! Material
27
Solved Pmhlerlw A. CDPI-AB Batch Reactor Calculations: A Hint of Things to Come B. PI-14, Modeling Smog in the L.A. Basin FAQ [Frequently Asked Questions]-In VpdateslFAQ icon sectlon Professional Refemnee Shelf I . Photos of Real Reactors
Smog in L A .
Reference 9 e l f
2. Reactor Section of the Usual Er~cj*clopediaof Eqrripme~ri This sectlon of the CD-ROhI shows industria1 equipment and disct~ssesits operation. The reactor portlon of thir encyclopedia i s ~ncluded on the CD-ROM accompanj ing this h o k
28 E~~implrs of ~nrlu\tr~rtlreacr~ons
Mole Balances
Chap.
'
3. The productitm of nitrobenzene exnmple problem. Here thc procesh Ilo~r rheet
ir
prven. don: with operating cr>ntl~tinris.
and reactors N~trobanzens
Crude
I
I
Vacuum
Vapors -,
I
I
I
C
I
nrlrobenzerre Recorrcenlraled 0c1d
Sulhr~cacid pump lank
Ntrrrc acid Makeup sulfune acrd
Figure PRS.A-I
Condensare
Steam
I Flou~sbce~ For the rnanuhcture of nitrobenzene.
4. Fischer-Tropsch Reaction and Reactor Example. .4 Fischcr-Tropsch reaction carried out in o typ~caEstraight-through trancppr reactor {Riser).
Figore PRS.B-I Thc reactor Ir 3.5 m in d~amcterirnd 38 m tnl[. [Schematic and phoro courtesy o f SasollSastech F T Limirrd.]
Chaa. 1
29
Questions and Probl~rrrs
Here photograph\ and ichernatics of the equiprnenr along with the feed rate\. reactor sizes. and principal renotlons
are also discussed in the PRS.
CIUEST1ONS A N D P R O B L E M S
I wish I had an answer for that, because I'm getting tired of answering that question. Yogi Berra. New York Yankees Spnrrs Illi~str(lterl,June 1 I, 1 984 The subscript to each af the problem numben ind~catesthe level of difficulty: A, leaxt difficult: D. most dlfficulr.
In each of the questions and problems below, rather than just drawing a box around your enwer. write a sentence or two describing how you solved the problem, the assurnptions you made, the reasonableness of )our answer, what you learned, and any other facts that you want to include. You may wish no refer to W./.trunk and E. 8. White. The El~ntenrsof SMe. 4th Ed. (NewYork: Macrnillan, 2000) and Joseph M. Williams, Stvke: Ten Lessons in Clarity & Gmcc, 6th Ed. (Glcnview, Ill.: Scott, Foresman, 1999) to enhance the quality of your sentences.
P_'Refore solving the problems, smtr or
re~ultsur trendq.
PI-lA
= Hint on the web
(a) Read through the Pmhcc. Write a paragraph describing buth the content goals and the intellectual goals OF the course and text. Also describe what's on the CD and how the CD can be used with the text and course. (h) List the areas in Figure 1-1 you are most looking forward to studying. (c) Take n quick look at the weh modules and list the ones that you feel are the most novel applicntions of CRE. (dl Visit the problem-solving web site, www.engin.umich.edid-cw/pmbsolv/ciaserlkep.hrrn, to find way? to "Get Unstuck" on a problem and to review the "Pmblem-Soivtng Alpotithm." List four ways that might help yo11 in your solutions to the home problems. (a) After reading each page or two ask yourself a question. Make a I~st~f the four best questions for this chapter. (b) Make a list of the five most imponant things you learned from this chapter. Visit the web site on Critical and Creative Thinking, ~urvw.enpin.rimich.edu/ -ce/probsoIv/stmtegv/crit-11-cwat, hrm. (a) Write 3 paragraph describing what "critical thinking" i s and how you can develop your critical thinkrng skills. (b) Write a paragraph describing what "'creative thinking" is and then list four things you will do during he next month that will, increase your creative thtnking skills.
Mole Balances
Chap. 1
(c) Write a question based on the material in this chapter that involves critical thinking and explain why it involves cntical thinking.
( d ) Repeat (c) for creative thinking. Brainstorm a Iist of ways you could work problems P-XX (to be specified by your instructor--e.g., Example El, or PI-15,) incorrectly. Surf the CD-ROMand the web (www.engin.umrch.edd-cw). Go on a scavenger hunt using the summary notes for Chapter E on the CD-ROM. (a) What Frequently Asked Question (FAQ)is not really frequently asked? (e)
hot button leads
(c) What
(d) Whai
IC%f Quiz Shnw
PI-l*
to a picture of a cobra?
ISellTe;;/l hot button leads to r picture 4101 buffnn
o f r rabbit?
leads to a picture of a hippo?
(e) Review the oblecrives for Chapter 1 in the Summary Notes on the CD-ROM. Write n paragraph in which you descr~hehow well you feel you me1 these objec~ives.Diqcuss any difficulties you encountered and three ways (e.p.. meet with professor. claqsmarec) you plan to address removing these difficulties. (f) Loub at the Chemical Reactor section of the Vi.rual E~lc?.rinl~rdin of Eyuiprnenr on the CD-ROM. Write a paragraph describing what you learned. (g) View the photos and schematics on the CD-ROM under Elements of Chemical Reaction Englneerrng--Chapter 1. Look at the qulcktime videos. Write a paragraph describing two or more of the reactors. What similarities and differences do you observe be~weenthe reacton on the weh (e.g.. 1r~tr~n.Ir~ebeqrtipi71er1t.con~). on the CD-ROM. and in the text? How do the used reactor prices compare aith those In Table 1-I? Load the Interactive Computer Module (ICM) from the CD-ROM. Run the module and then record your performance number for the module which indicates your mastery of the material. ICM Kinetics Challenge 1 Performance # Example 1-1 Calculate the volume of a CSTR for the conditions used to figure the plug-lion reactor volume in Example E -1. Which volume i s larger. the PFR or the CSTR'? Explain why. Suggest two ways to work t h ~ sproblem ~ncorrec~ly. Calculate the rime ra reduce the number o f moles of A to 19 of its initial value in a constant-volume batch reactor for the reaction and data in Example 1 - 1 . \%'hat assumptions were made in the dcrivat~onof the design equation for: la) the batch reactor? (b) the CSTR? {c) the plug-Row reacior (PIT)? Id) thc packed-bed reactor IPBR)? (e) State in words the meanings of -r,. -r; . and X r . Is the reaction nte -r, an extensive quantity'! Explain. Use the mole balance to derive an equation analogous to Equi~rion( 1-7 1 for n fluidifrd CSTR contalnrnp catalyst particles in terms of the catalys~we~ght.IV and other appropriate term\. Hirlr: Scc rnaq~nfigure.
Chap. 1
PI-10,
1
PI-12,
31
Ouestions and Problems
How can you conven the general mole balance equation for a given species, Equation { 1-4)- to a general mass balance equation for that species? We are going to consider the cell as a reactor. The nutrient corn steep liquor enters the cell of the mrcroorganlsm Penirillfunl chnsngenum and is decomposed to form such products as amino acids. RNA. and DNA. Write an unsteady mass balance an (a) the corn steep liquor. Ib) RNA. and (c) pencillin. Assume the cell is well mixed and that RNA remains inside the cell.
The United States produced 32.59 af the world's chemical products in 2002 according to "Global Top 50." Cherniraf rtnd Engrrleerin~Nen:~. July 28* 2003. Table PI-12.1 lists the 10 most produced chemicals in 2002. TABLE PI-I? I .
CHEUICALP R O I ) L ~ ~ I C J %
Reference: ClnentirrrI urrrl E r ~ g i ~ r t ~ r iNPII,S. t l p July 7 . ?Oil?,
htl/~://p11/7.~.(1~.~.0~fi/c't~~d
(a) What
were fhc I 0 rnoxt pmduced chemicals for the year that just ended7 Were there an\: sipnifica~lt chanfes from the 1995 qtatist~cs? (See Chapter I o f 3rd edi~ionof El~rnentso j CRE.) The Fame issue of C&E Nm:r ranks chenircal companies as riven in Table PI 12.2. (b) What I0 companies were tops in
-
oxide. and benzene? (e\ Wh?. do you su\pect
there are \r) few organic chcniicals rn the top 10?
Mole Balances
Chap.
Dew Chemical Dupont ExxonMobil
General Electnc Hunt~manCorp. PPG lndustrie~ Equistar Chemicals Chevron Phil lips Enstrnsn Cheti~lcal Prarair Refemnres: Rank 1002. Chrmirul ortrl E~tprneerirrxN ~ I I ' May s, Rank 2001 : CE~erniuul~ t r dE t ~ , q [ r ~ c c rNeam3, i n ~ May Rank 2000 Chctnrtnl r i r ~ dErrgrne~rirzgAVr~:ci's. May Rank 1994: Cl~emicfllot~rlE n # ~ n r c r i r ~Ncu.~, g May Itttp //psbs.ncs-or.y/retd
P1-13', Referring to the
text
I?. 2003. 13, 2002. 7 , 200 1 . I . ?W.
material and the additional references on comrnerciz chapter. fill in Table P I - t 3.
reactors given at the end of this TABLT PI.1.1
??pe qf Reortor
B
Halt of Fame
kind.^ of Phuscs
Chrrmct~risfics
Prrvrnr
Use
Batch
- -
CSTR
--
PFR
- - -
PBR
Ucmbe-
COMPARISON OF REACTORTYPES
PI-14,
Ar1rfl1~1rrgt.s
Disodrctntugrs
Schematic diagrams of the: Lor; Angeles basin are shown In Figure P I - 14. Thr basin R o o t covers approximately 300 square miles ( 2 x 10'"ff2) and is almos completely surrounded by mountain ranges. 1F one assumes an inversior height in fhe basin of 2000 ft. the corresponding volume of air in the basin i! 4 x 1013ff3.We shall use this system volume to mode1 the accumulation anc depletion of air pollutants. As a very rough first approximation. we shall trea the Lus Angeles basin as a well-mixed container (analogous to a CSTR) ir which there are no spatial variations in pollutant concentrations.
Chap. 1
Ouestions and Problems
Q ~ ~ c a n t manklim s or hills
C.A.
vo
Wlndfm
SFde view Figure PI-14 Schematic diagmrns or the Loc AngeIes basin.
living Efample Prob!em
We shall perform an unsteady-stale mole balance on CO as it is depreted from the basin area by a Santa Ano wind. Snnta Ana winds nre hrgh-velocity winds that originate in the Mojave Desert just to the northeast of Lw Angeles. Load the Smog in Los Angeles Raslin Web Module. Use the data in the module to work part 1-14 (a) through (h) given in the module. Load the living example polymath code and explore the problem. Fur part (I), vary the parameters D,,,n, and b. and write a paragraph describing what you find. There i s heavier traffic in the L.A. basin in the mornings and in the evenings as workers go to and from work in downtown L.A. ConscquentIy, the flow of CO into the L.A. basin might be better represented by the sine function over a 24-hour period.
PI-ISB The reaction is to be carried out isothermnl!y in a continuous-flow reactor. Calculate both the CSTR and PFR reacmr volumes necessary to consume 99% of A (i.e.. C4 = O.OICA,) when the entering molar flow rate is 5 molfh, assuming the reaction rate -rA is: mol (a) -r, = k with k = 0.05 (Am.: V = 99 dm3) h . dm'
(b) -rA = LC, (c) -rA
=
kc:
= 0.0001 s-I with k = 3 dm' with k
moE .h
(Ans.:
r/csre = 66,000 dm")
The entering volumetric flow rate is TO ddlh. (Note: FA = C,u. For a constant volumetric flow rate v = v , , then FA= C,u,,. Also. C,,, = F,dv, = [5 rnol/hl/llQ drnJlhJ= 0.5 molldrnJ .) (d) Repeat (a). (hl, and Ic) tocalculate the time necessary to consume 99.9% of species A in a 1000 dm3 constant volume batch reactor with CAo= 0.5 molldm3. P1-16* Write a one-pangraph Pummary of a journal articIe on chemical kinetics or reaction engineering. The articIe must have heen published within the lasr five years. What did you I e m from this article? Why 2s the article important?
34 P1-17,
Mole Balances
Chap. 1
F m e r Oat's property. Use Polymath or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 500 days. The predator-prey relationships are given by the following set of coupled ordinar). differential equatims:
(a) There are initially 5 0 0 rabbits fx) and 200 foxes (y) on
Constant for growth of rabbits k, = 0.02 d a y ' Constant for death of rabbits k2 = 0 aKX34/(day x no. of foxes) Constani for p m h of foxes after eating labbits k, = O.N04/(dayx no. of rabbitq) Constant for death of foxes Ir, = 0.04 duj-' What do your results look like for the case of k, = 0.00004/(day x no. of rabbits) and r,,,, = 8I)O daysq A l ~ oplot the number of foxes versus the number of rabb~ts.Explain why the curves look the way they do. Vary the parameters k,, kz. k7, and k,. Discuss which parameters can or cannot be larger than others. Write a paragraph describing what you find. (b) Use Polymath or MATLAB to solve the following set of nonlinear algebraic equations: Polymath Tutorial
Summary Notes
PI-18,
PI-19,
with initial guesses of x = 2. = 2. Try to become familiar with the edit keys in Polymath MATLAB. See the CD-ROM for instructions. Screen shots on how to run Polymath are shown at the end of Summary Notes for Chapter 1on the CD-ROM and on the web. What if: (a) the benzene feed stream in Example R1.3-1 in the PRS were not preheated by the product stream'?What would be the consequences? (b) you needed the cast of a 6000-gallon and a 15.000-gallon Pfaudler reactor? What ~eouldthey be7 {c) the exit concenrration of A in Example 1-1 were specified at 0.1% of the entering concentration? (d) ot~lyone operator showed up to run the nitrobenzene plant. What would be some of >our firs1 concemc? Enrim Ferrni (1901-19541 Problems (EEP). Enrico Ferrni was an Ftalran physicist who received the Nohel Prize for his work on nuclear processes. Fermi was famous for h ~ "Back s of the B~velopeOrder of Mqnltude CaicuIntion" to obtain an estimate' of the answer through logic and making reasonable assumptions. He used a process to set bounds on the answer hy saying i t is probably larger than one number and smaller than another and amved at an answer lhat was aithin a factor of 10. hrtp://tnarhfon~t~~. 0r~/wurk~hop1/~~in~96/i~~t~rdi.rc~/s~~1~~2. h1m1 Enrico Fcrmi Problem IEFP) # l How many piano tuners are there In the city o f Chicago? Show the steps in yonr reasoning. I. Populat~un01' Chicago 1. Ntlrnber of people per hourchold
Supplementary Reading
Chap. 1
Numkr of households Households wlth pianos 5. Average number of tunes per year 6. Etc. An answer is given on the web under Summary Notes for Chapter 1. PI-20, EFP #2. How many square meters of pizza were eaten by an unde~raduare student body population of 20,000 during the Fail term 2W? PI-21, This problem will be used in each of the following chapters to help develop critical-thinking skill$. (a) Write a question about this problem that involve critical thinking. (b) What generalizations can you make about the results of this problem? (c) Write a question that will expand this problem. PI-22 New material for the 2nd printing the following changesladditions have been made to h e 2nd printing. 3.
4.
NOTE TO INSTRUCTORS: Add~tionalproblems (cf. those from the preceding ed~tions)can be found in the solurin~ismanual and on the CD-ROM. These problems could he photocopied and used ro lielp reinforce the fundamental princrples discusred in [hi\ chapler.
CDPT-A, CDP143,
Calculate the time to consume 80% af species A in a constant-volume batch reactor for a first- and a second-order reaction. (Includes SoIution) Derive the differential mole balance equation for a foam reactor. [Znd Ed. PI-loB]
Lolved Problems
SUPPLEMENTARY READING 1 . For further elaboration of the development of the general balance equation. see not only the web site wnn~.e~;gin.utnich.~dul-cre hut also
FELDER, R. M., and R. W. ROUSSEALJ,El~nterltot? Prfnciplex of Clzcnr~colProcesses, 3rd ed. New York: Wile>/,2200, Chapter 4. HIMMELBLAU.D. M,, and J. D. Riggs. Basic Prinripl~sand Calcirlariorts in Chen~icnlE~tgi?~eeriltg, 7th ed. Upper Saddle Rwer, N.J.: Prentice Hall. 2004. Chapters 2 and h. SANDERS. R . 1.. The Alrafninx ofSkzing, Denver. CO:Golden Bell Press, 1974. 2. A detailed explanation of a number a f topics in ?his chapter can be found in
H. S. FOCLEH,eds.. AlCItE Mndulular I~t.~rrucriorlSerz~rE. K ~ I I E Y ~VOIS. T S . 1 and 2. New York: AlChE, 1981.
C R Y ~ E R. S , L.. and
3. An excellent description o f the various types of commercial reactors used in industry is found in Chapter I I of WAIAS.S. M., Rencfiotl Kinerfc.sfor CCheicuI Brgil~eers.New Y
d McGnw-HiU, 1959.
1. A discussion nf some of the most important industr~alprocesses i s presented by
MEYERS,R.A., Handbook of Chcnlicols Pmd~rcriott Processes. New York: McGraw-H111. 1986 See also
MCKETTA. J. 1.. ~ d . .Etrryrloprriiu oj" Cl~e~rricnl Pmrrsse.~urld Design. New York: Marcel Dckker. 197h.
36
Mola Balances
Chap.
.4 simiIar book. which describes a larger nuniber o f processes, i s
C. T.. S l ~ r e ~ ~Clirrnicnl e'~ PTOCF.TF Indrrstrirs. 5th ed. New Yorb McGraw-Hill, 1984.
AL'STIY.
d LlnRr
5. The following journals may be useful in obtaining infomation on chemical reac tiun engineering: Internotional Jo~~rnul of CS~ernEcrlEKinetics, Joltrnnl of Car~ilyx Journal of Applied Catul~r.~is. AlChE Jolurr ma/. CEwmical Engineering Scie~c Canadian Journal of Chemical Etigineering, Chemical Drgzr~eering Comrnunica tro~ls,Joi~rnnlof Physical C.hettiisrr?: and Jndusrrrol nnd Engit~cerirrg Chemisrc Reseami1.
6. The price of chemicals ern be found in such journals as the Ch~ttiicoiMarketin! Reporter, Cherniml Weekly3and Chemical Engrr~eeringNews and on the ACS wet site hrrp://pubs. acs.or~/c~n.
Conversion and Reactor Sizing
2
Be more concerned with your character than with your
reputation, because character is what you redly are while reputation is merely what others think you are. John Wooden, coach, UCLA Bruins
Overview. In the first chapter, the general mole balance equation was detived and then applied to the four most common types of industrial reactors. A balance equation was developed for each reactor type and these equations are summarized in TabIe S-I. In Chapter 2, we will evaluate these equations to size CSTRs and PFRs. To size these reactors we first define conversion, which is a measure of the reaction's progress toward completion, and then rewrite all the balance equations in terms of conversion. These equations are ofren referred to as the design equations. Next, we show how one may size a reactor line., determine the reactor volume necessary to achieve a specified conversion) once the relationship between the reaction rate, - r ~ and , conversion, X,is known. In addition to being abIe to size C S R s and PFRs once given -r, =Am, another goal of this chapter is to compare CSTRs and PFRs and the overall conversions far reactors arranged in series. It is also important to arrive at the best arrangement of reactors in series. After completing this chapter you will be able to size CSTRs and PFRs given the rate of reaction as a function of conversion and to calculate the overall conversion and reactor volumes for reactors arranged in series.
38
Conversion and Reactor Sizing
Chap. 2
2.1 Definition of Conversion In defining conversion, we choose one of the reactants as the basis of calculation and then late the other species involved in the reaction to this basis. In virtually all instances it is best to choose the limiting reactant as the basis of calculation. We develop the stoichiometric relationships and design equations by considering the general reaction
The uppercase letters represent chemical species and the lowercase letters represent stoichiometric coefficients. Taking species A as our basis of c~lcularion, we divide the reaction expression through by the stoichiometric coefficient of species A. in order to arrange the reaction expression in the form
to put every quantity on a "per mole of A basis. our limiting reactant. Now we ask such questions as "How can we quantify how far a reaction [e.g., Equation (2-2)] proceeds to the right?" or "Mow many moles of C are formed for every mole A consumedr' A convenient way to answer these questions is to define a parameter called conversion. The conversion XA is the number of moles of A that have reacted per mole of A fed to the system:
of A reacted x* = Moles Moles af A fed
Definition of X
Because we arre defining conversion with respect to our basis of calculation [A in Equation (2-211, we eliminate the subscript A for the sake of brevity and let X = X, . For irreversible reactions, the maximum conversion is 1.O, i.e., complete conversion. For reversible reactions, the maximum conversion is the equilibrium conversion & (i.e., X,, = X,).
2.2 Batch Reactor Design Equations In most batch reactars. the longer a reactant stays in the reactor, the more the reactant is converted 10 product until either equilibrium is reached or the reactant is exhausted, Consequently. in batch systems the conversion X is a function of the time the reactants spend in the reactor. If WAO i s the number of moles of A initiaIly in the react* then the total number of moles of A that have reacted after a time r is [h'A,lXj
I
Moles of A reacted r
7
Moles of A I reacted j
=
I.Wqul
[XI
(2-3)
Sac. 2.2
39
Batch Reactor Design Equations
Now, the number of moles of A that remain in the reactor after a time r, N,, can be expressed in terms of NAOand X:
Moles of A that
The number of moles of A in the reactor after a conversion X has been achieved 1s
When no spatial variations in reaction rate exist, the mole balance on species A for a batch system is given by the following equation [cf. Equation (1-5)J:
This equation is valid whether or not the reactor volunle is constant. In the general reaction, Equation (2-2). reactant A is disappearing: therefore, we rnultiply both sides of Equation (2-5) by -1 to obtain the mole balance for the hatch reactor in the form
The rate of disappearance of A. -r,, in this reaction might be given by a rate law similar to Equation (1-2), such as - r , = kCACB. For batch reactors. we are interested in determining how long to leave the reactants in the reactor tn achieve a certain conversion X . To determine this length of time, we write the mole balance. Equation (2-5). in terms of conversion by differentiating Equation (1-4)with respect to time, remembering that NAo is the number of moles of A initially present and is thereforc a conqtant with respect to time.
Combining the above with Equation ( 2 - 5 ) yields
For n batch reactor. the dedgn equation in differentia! form is
40
Conversion and Reactor Sit~ng
Chap.
We call Equation (2-6) the differential form of the design equation for batch reactor because we have written the mole balance in terms of conversior The differential forms of the batch reactor mole balances. Equations (2-5) an1 (2-6). are often used in the interpretation of reaction rate data (Chapter 5 ) ant for reactors with heat effects (Chapter 9), respectiveIy. Batch reactors are fre quently used in industry for both gas-phase and liquid-phase reactions. Thl laboratory bomb calorimeter reactor is widely used for obtaining reaction rat1 data (see Section 9.3). Liquid-phase reactions are frequently carried out ii batch reactors when small-scale productton is desired or operating difficuftie,
mle out the use of continuous flow systems. For a constant-volume batch reactor. V = V,,, Equation (2-5) can bc arranged into the form
Constant-volume batch reactor
As previously mentioned. the differential form of the mole balance, e.g.. Equa tion (2-7). is used for analyzing rate data jn a batch reactor as we will see ir Chapters 5 and 9. To determine the time to achieve a specified conversion X, we first separate the variables in Equation (2-6) as follows
Batch time t to achieve e conversion X
u Batch
Design Equation
This equation is now integrated with the limits that the reaction begins at time equal zero where there is no conversion initially (i.e., t = 0,X = 0). Carrying out the inteption, we obtain the time t necessary to achieve a conversion X in a batch reactor
The longer the reactants are left in the reactor, the greater will be h e conversion. Equation (2-6) is the differential Form of the design equation. and Equation (2-9) is the integral form of the design equation for a batch reactor.
2.3 Design Equations for Flow Reactors For a hatch reactor. we saw that conversion increases with time spent in the reactor. For continuous-flow systems, this time usually increases with increasing
Sec. 2.3
41
Design Equatio~sbr F!ow Reactors
reactor volume. e.,a.. the biggert'lonper the reactor, the more time it will take the reactant5 to Row conipleteIy through the reactor and thus, the more time to react. Consequently, the conversion X is a Function of reactor volume V. If FA,, i s the molar flnw rate of specres A fed to a system operated at steady state. the molar rate at which species A is reacting rr3ithirrthe entire system will be F,d.
Moles of A fed, Moles of A reacted lime Mole o f A fed Moles of A reacted '4 = time
LF*ol.L.\1=
The molar feed rate of A ro the system minus the rate of reaction of A within the system eqltnls the moIar flow rate of A leaving the system FA. The preceding sentence can be written in the form of the following mathematical statement:
Molar flow rate fed to the system
Molar rate at
Molar flow rate
consumed within
the system
Rearranging gives
The entering molar flow rate of species A. FA, (mol/s), is just the product of the entering concentration, CAo(mol/dmf ), and the entering volumetric flow rate, u, (drn31s):
?%IJ = c~n uo
Liquid phase
For liquid systems, C ,,
is commonly given in terms of molarity, for example, CAO= 2 rnol/drn3
For gas sysrerns, CAocan be calculated from the entering temperature and pressure using the Ideal gas law or some other gas law. For an ideal gas (see Appendix B):
Gas phase
42
Conversion and Reactor Sizing
Chap. 2
The entering molar flow rate is
where C,, = entering concentration, mol /dm3 = entering mole fraction of A
y,
P, = entering total pressure, e.g., kPa PA, = .v,,Po
= entering partial pressure of A,
e.g., kPa
To = entering temperature, K R
= ideal gas constant
kPa ' mol * K
see Appendix
B
1
The size of the reactor will depend on the flow rate, reaction kinetics, reactor conditions, and desired conversion. Let's first calculate the entering molar flow rate. Exumpk 2-1
U~ittgthe Ideal Gas Law to Caicuhfe CAl and FA*
A gas of pure A at 830 kPa (8.2 atm) enters a reactor with a volumetric flow rate, v* of 2 dm% at 500 K. Calculate the enterlng concentration of A, C,,, and the entertng molar Bow rate.
rho.
Solurioil
U'e again recall that for an ideal pas:
where Po = 8.70 k% (8.3 atm) YM) = 1 . 0 ( P u ~ A ) To = in~traltemperature = 500K R = 8.3 14 dm3 1;Palniol , K (Appendix B ) 4
Substituting the given paraineter values inlo Equation (E2-1. I ) yields .c.40 =
mol I = 0.20dm3 (8.334 dm? kPdrnol . KJ(500K) (1)(830 kPa)
We could also solve for the partial pressure in terms of the concentration:
Sec. 2.3
Design Equations for Flow Readors
43
pure A enters, the total pressure and partial pressure entering are the same. The entering molar flowrate, FA,, is just the p d u c t of the entering concentration, C,,, and the entering volumetric flow rate, vo: However, since
I
FA, = CA,vo = (0.2 mal/dm3)(2 dm3Is) = (0.4 rnol/s)
This feed rate (FA, = 0.4 moYs) is in the range o f that which is necessary to form several million pounds of product per year. We will use this value of FA, together with either Table 2-2 or Figure 2-1 to size and evaluate a number of reactor schemes in Examples 2-2 through 2-5.
Now that we have a relationship [Equation (?-lo)] between the molar flow rate and cenversion, it is possible to express the design equations (i.e., mole balances) in terms of conversion for the flow reactors examined in Chapter I .
2.3.1 CSTR (also known as a Backmix Reactor or Vat) Recall that the CSTR is modeled as being we11 mixed such that there are no spatial variations in the reactor. The CSTR mole balaoce, Equation ( I -7), when applied to species A in the reaction
can he arranged to
We now substitute for FA in terns of FAOand X
and then substitute Equation (2- 12) into (2-1 1)
- F,d v= 5 0 - (50 - r* Simplifying, we see the CSTR volulne necessary to achieve a specified conversjon X is FA ruU)Im
Perfec~mixing
(2- 13)
44 Evalrlate -r,
at
CSTR exit.
Conversion and Reactor Sizing
Chap. .
Because the reactor is perfprrk mixer/, the exit composition from the reactor i identical to the composition inside the reactor, and the rate of reaction is eval uated at the exir conditions.
2.3.2 Tubular Flow Reactor (FFR) We model the tubular reactor as having the fluid flowing in plug flow. i.e.. nr radial gradients in concentration, temperature, or reaction rate.' As the reac [ant$ enter and flow axially down the reactor, [hey are consumed and the con version increases along the length of the reactor. To develop the PFR desigr equation we first multiply both sides of the tubular reactor design equatior (1- 123 by - I . We then express the mole balance equation for species A in tht reaction as
For a flow system, FA has previously been given in terms of the entering molat Row rare FM and the conversion X
differentiating
dFA= - F A d X
and substituting into (7-141, gives the differential form of the design equation for a plug-flow reactor (PFR):
4
blgn
PFR cquntlon
b,
We now separate the variables and integrate with the limits V = 0 when X = 0 to obtain the plug-flow reactor voluine necessary to achieve a specified conversion X:
To carry out the integrations in the batch and plug-Row reactor design equations (2-9) and (2-16). as welI as to evaluate the CSTR design equation (2-23), we need to know how the reaction rate - r ~varies with the concentration (hence conversion) of the reacting species. This relationship between reaction rate and concentration is developed in Chapter 3.
This constraint will be removed when we extend our analysis to nonideal (industrial) reactors in
Chapters 13 rind
Id.
Sm. 2.4
Apolicatiors cl ths Design Equaticns for Continuous-Flow Fleactors
45
2.3.3 Packed-Bed Reactor
Packed-bed reactors are tubular reactors filled with catalyst particles. The drrivation of the differential and integral forms of the design equations For packed-bed reactors are analogous 10 those for a PFR {cf. Equations ( 2 - 15) and (2- 1611. That is, substituting Equation (2- 12) for FA in Equation ( 1 - 15) gives PBR design equation
The differential Form o f the design equation [i.e., Equation (2-17)J must be used when analyzing reactors that have a pressure drop along the length of the reactor. We discuss pressure drop in packed-bed reactors in Chapter 4. I n the abserlce of pressure drop, i.e., AP = 0. we can integrate (2- 17) with Iimits X = 0 at W = 0 to obtain
Equation (2-181 can be used to determine the catalyst weight W necessary to X when the total pressure remains constant.
achieve a conversion
2.4 Applications of the Design Equations for Continuous-Flow Reactors In this section. we are going ro show how we can size CSTRs and PFRs (i.e., determine their reactor volumes) from knowledge of rbe rate of reaction. -r,. as n function of conversion, X. The rate o f disappearance of A. -r,. is almost aIways a function of the concentrations of the various species present. When only one reaction is occurring. each of the concentrations can be expressed as a function of the conversion X (see Chapter 3); consequently, -r, can be expressed as a function of X. A particularly simple functional dependence, yet one that occurs often, is the first-order dependence
Here. k is the specific reaction rate and is a function only of temperature, and CA0is the entering concentration. We note in Equations (2-13) and (2-16) the reactor volume in a function of the reciprocal of -r,. For this first-order dependence, a plot of the reciprocal rate of reaction (I/-r,) as a function of conversion yields a curve similar to the one shown in Figure 2-1, where
46
Conversion and Reactor Sizing
Chap. 2
To illustrate the design of a series of reactors, we consider the isotherrnaI gas-phase isomerization A-B We are going: to the laboratory to determine the rate of chemical reaction as a function of the conversion of reactant A. The laboratory measurements given in Table 2-1 show the chemical reaction raze as a function of conversion. The temperature was 500 K.(440"FI. the total pressure was 830 kPa (8.2 atm), and the initial charge to the reactor was pure A.
If we know -r, as a function of X,we can size any isothermal mcrion system.
Recalling the CSTR and PFR design equations, (2-13) and (2-I&}, we see that the reactor volume varies with the reciprocal of -r,, (I/-r,4f. e.g., V=
(%)(F*&').
Consequently, to size reactors, we conven the rate data in
Table 2- I to reciprocal rates, ( 1 I-rA). in Table 2-2.
These data are used to arrive at a plot of (I/-r,) as a function of X. shown in Figure 2- 1 . We can use chis figure to size Row reactors for different entering molar flow rates. Before sizing flow reactors let's first consider some insights. If a
Sec. 2.4
Appl~cat~ons of the Desrgn Equations for Contfnuous-Flow Reactors
Figure 2-1 Processed data -I.
reaction is carried out isothermally, the rate is usually greatest at the start of the reaction when the concentration of reactant is greatest (i.e., when there is negligible conversion SX E 03). Hence I 1/-rAf will be small. Near the end of the reaction, when the reactant has been mostly used up and thus the concentration of A is small (i.e., the conversion is large), the reaction rate will be small. Consequently. (I/-rA)is large. For all irreversible reactions of greater than zero order (see Chapter 3 for zero-order reactions), as we approach complete conversion where all the limiting reactant is used up. i.e., X = 1. the reciprocal rate approaches infinity as does the reactor volume. i.e.
As X
A+BtC "To infinity and beyond" --Buzz Lightyear
A#B+C
I +x + I . - I : , + 0 , thus, -
-rA
therefore If
-+
Consequently, we see that an infinite reactor volume is necessary to reach complete conversion, X = 1.0 For reversible reactions (e.~., B), the maximum conversion is the - A equilibrium conversion X,. At equilibrium. the reaction rate is zero ( r , s 0). Therefore. As
I + X + X,.- 7, -+ 0 . thus. -r,
and therefore 'I
+
~3
and we see that an infinite reactor volume would also be necessary to obtain the exact equilibrium conversion, X = X,. To size a number of reactors for the reaction we have been considering, uRewill use FA, = 0.4 moI/s (calculated in Example 2- 1) to add another row to the processed data shown in Table 2-2 to obtain Table 1-3.
Sec. 2.4
Apal~caltronsof the Design Equat~onsfor Cont~nuous-FfowReactors
49
I
(8)
Quation (2-13) gives the volume o f a CSTR as a function of FA,,.X,and -r,,:
I n a CSTR, the composition, temperature, and conversion of the effluent stream an: rdentical ta that of the Ruid within the reactor, because perfect mixing is assumed. Therefore, we need to hiid the value of -rA (or reciprocal thereof) at X = 0.8. From either Table 2-2 or Figure 1-I , we see that when X = 0.8, then
Substitution into Equation (2-13) for an entering molar flow rate. FA* of 0.4 mol A/s and X = 0.8 gives
(b) Shade the area in Figure 2-2 that yields the CSTR vocllume. Rearnnging Equa-
tion (2- 13) gives
In Figwe E2-2.1, the volume is equal to the area of a rectangle with a height (X= 0.81. This rectangle is shaded in the figure.
(FAIj-rA = 8 rn3) and a base
(E2-2.2) V = Levenspiel rectangle area = height x width
The CSTR volume necessary to achieve 80% conversion is 6.4 m7when operated at 500 K. 830 kPa (8.2 am), and with an entering molar flow rate ~f A of 0.4 rnolh. This volume corresponds to a reactor about 1.5 rn in diametervand 3.6 rn high. It's a large CSTR, but this is a pas-phase reaction, and CSTRs ate normally not used for gas-phase reactions. CSTRs are used primarily for liquid-phase reactions.
Conversion and Reactor Siting
Chap. 2
Plots of Il-r* vs. X arc sometimes referred to a<
Lcvensprel plots (after Octave
kvensp~el) --
I4
ILf
M
06
Od
Id
Conversion. X
figure EZ-2.1 Levenspiel CSTR plot.
Example 2-3 Sizing a PFR The reaction described by the data in Tables 2-1 and 2-2 is to k carried out In a PI%. The entering molar flow rate of A is 0.4 rnofls. (a) First. use one of the integration formulas given in Appendix A.4 to determine the PER reactor volume necessary to achieve 80% conversion. (b) Next. shade the area in Figure 2-2 that would give the PFR the volume necessary to achieve 80% conversion. (c) Finally, make a qualitative sketch of the conversion. X, and the rate of reaction. -rA, down the length (volume) of the reactor.
Solution We start by repeating rows ( I ) and (4) of Table 2-3.
(a) For the
PFR,the differential form of the mole balance is
Rearranging and integrating gives
Sec. 2.4
I [
Applications of the Design Equations lor Continuous-Flow Reactors
51
We shall use the jive poin~quadrature formula (A-23) given in Appendix A.4 to numerically evaluate Equation 12-16), For the five-point formula with a final conversiw of 0.8, gives for four equal segments between X = 0 and X = 0.8 with a 0.8 = 0.2. The function inside the integral is evaluated at segment length of AX = 4
Using values of FAJ-rA) in Table 2-3 yields
The PFR reactor volume necessary to achieve 80% conversion is 2165 dm3. T h i s volume could result from a bank of 100 PERs that are each 0.1 m in diameter with a length of 2.8 m (e-g.. see Figures 1-8(a) and {b)). (b) The integral in Equation (2-16) can also be evaluated from the area under the curve of a plot of (FAd-vA)versus X.
* 100 P f Rs in parallel
=
J
2 d ~
= Area under the curve between X = 0 and X = 0.8 (see appropriate shaded area in Figure E2-3.1)
PFR
Conversion, X
I
Fl'lgure E2-3.1 Lebenspiel PFR p101
52
Conversion and Reactor Sizing
Chao.
The area under the curve ail1 gibe the tubular reactor volume necessary to achiex the specified conversion of A . For 80% conversion. the shaded area is roughly equ,
to 2165 dm'(2.165 m"). (c) Sketch the pmf l a of -r, and X down the length of the reactor.
We know that as we proceed down the reactor and more and more of the reactant i consumed. the concentration of reactant decreases, as does the rate of disappearanc of A. However, the conversron increases as more and more reactant 1 s converted t product. For X = 0.2. we calculate the corresponding reactor volume using S i m ~ ?on's rule (given in Appendix A.4 as Equation [A-211) with AX = 0. I and the dat in rows 1 and 4 in Table 2-3,
For X = 0.4. we can again use Simpson's rule wirh A X = 0.2 to find the react01 volume necessary for a conversion of 40%.
We can continue
in this manner to arrive at Table E2-3.1.
The data in Table E1-3.1are plotted in Figures E2-3.2 la) and (b). One observes that the reaction rate, -XA, decrease^ as we move down the reactor while the conversion increases. These plots are typical for reactors operated
isothomally.
Sec. 2.4
1
For isnthcr~nnl reacllons. lhs C ~ ~ ~ ~ eIncrease< r~ion and the nte derrta<e< v,e miwe down the
53
Apal~catjonsof The D~srgnEauatlons b r Cont~nuous-Flav~ Reactors
118
PFR. X
--
01
-
02
.-.
-
.
- -
.
---
-1
.
00
a
HXI
IWO
1%
ZODD
o
~m
Figure E2-3.2ia) Conver.cion protile.
5m
IWO
1 4 ~ 2wo
2 5 ~
V (dm3)
V (dm?
Figure EZ-3.7(b) Reaction rate protile
Example 2-4 Comparing CSTR and PFR Sizes It E S interesting to compare the volumer of a CSTR and a PFR required for the same job. To make thls comparison. we shall use the data in Figure 2-2 to learn which reactor would require the smaller volume to achieve a conversion of 80%: a CSTR or a PFR.The entering molar Row rate FAo = 0.4 molls, and the feed conditions are
the same in both cases.
The CSTR volume was 6+4m" and the PI% voluine was 2. I65 m3. When we combine Figures El-2.1 and E2-3. I on the same graph. we see that the crosshatched area above the curve is the difference in the CSTR and PFR reactor volumes. For isothermal reactions greater than zero order (see Chapter 3 for zem order), the CSTR volume will urually be greater than the PFR volume for the same conversion and reaction conditions (temperature, Bow rate, etc.). We see that the reason the isotherma[ CSTR volume is usually greater than the PFR volume is that the CSTR is always operating at the Iowest reaction rate (e-g.. -rA = 0.05 in Figure E2-4.IIb)).The PFR on the other hand stans at a high rate at the entrance and gradually decreases to the exit rate, thereby requiring less volume because the volume is inversely proportional to the rate. However, for autocatalytic reactions. product-inhibited reactions. and nonisothermal exothermic reactions. these trends will not always be the case. as we will see in Chapten 7 and 8.
54
Conversion and Reactor Sizing
:
Chap. 2
between CSTR & PFR
1
Figre El-4.l(a) Comparison of CSTR and
(b) -r, as a function of X.
PFR reactor sizes.
2.5 Reactors in Series Many times, reactors are connected in series so that the exit stream of one reactor is the feed stream for another reactor. When this arrangement is used, it is often possible to speed calculations by defining conversion in t e n s of location at a point downstream rather than with respect to any single reactor. That is, the conversion X is the total nunrber of moles of A that have reacted up to that point per mole of A fed to thejirst reactor. Only valid for For reactors in series NO side streams
X- = Total moles of A reacted up to point 1
i
Moles of A fed to the first reactor
However, this definition can only be used when the feed stream only enters the first reactor in the series and there are no side streams either fed or withdrawn. The molar flow rate of A at point i is equal to moles of A fed to the first reactor minus all the moles of A reacted up to point i:
For the reactors shown in Figure 2-3, X, at paint i = 1 is the conversion achieved in the PFR, Xz ar point i = 2 is the total conversion achieved at this point in the PFR and the CSTR, and X, is the totat conversion achieved iby all three reactors.
Sec. 2.5
55
Reactors in Series
To demonstsate these ideas, let us consider three different schemes of reactors in series: two CSTRs,two PFRs, and then a combination of PFRs and CSTRs in series. To size these reactors, we shall use laboratory data that gives the reaction rate at different conversions. 2.5.1 CSTRs in Series The first scheme to be considered is the two CSTRs in series shown in Figure 2-4.
Figure 2-4
Two CSTRs in series.
For the first reactor, the rate of disappearance of A is -r,, at conversion XI. A mole balance on reactor 1 gives In - Out Reactor 1:
at
+
=O
(2- 19)
FA,=EAo-FAOXI
(2-20)
FAo-FAl
The molar flow rate of A
+ Generation = 0 r,,V,
point 1 is
Combining Equations (2- 19) and (2-20) or rearranging Reactor I
lo the second reacror, the rate of disappearance of A. - r ~ ?is. evaluated at the conversion of the exit stream of reactor 2, X2. A mole baEance on the second reactor
En - Out Reactor 2:
+ Generation = 0
FA,-FA2 +
rA2v
=o
The molar flow rate of A at point 2 is = F ~ F~J2 ~ -
(2-22)
56
Conversion and Reactor Sizing
Chap.
Combining and rearranging
Reactor 2
For the second CSTR recall that -r,: is evaluated at Xz and then use (X2-X, to calculate V2 at X2. In the examples that follow, we shall use the molar Row rate of A we cal culated in Example 2-1 (0.4 ma1 Ms) and the reaction conditions given it Table 2-3. Example 2-5 Comparing Vohntes for CSTRs in Series
For the two CSTRs in series, 40% convenion is achieved in the first reactor. Wha is the volume of each of the two reactors necessary to achieve RO% overall conver sion of the entering species A?
Snlrtlion
Fur reactor 1, we observe from either Table 2-3 or Figure E2-5.1 then
then
For reactor 2, when Xl = 0.8. then =
I
(?) r~
= S.O X = 0.8
V2 = 3200 dm3 (liters)
mi
that when X = 0.4.
Sec. 2.5
Readon in Series
To achreve the same over~ll convenlon, the
tr~tal.olume fur two CSTRs In series is l e ~ than s {hat required
for one CSTR. 0.2
0.0
0.4
0.8
0.i
1.0
Conversion X
Figure EZ-5.1 Two CSTRs in .series.
Note again that for CSTRs in series the rate - r ~ ,i s evaIuared at a conversion of 0.4 and rate -r+ is evaluated at a conversion of 0.8. The rota1 volume for these two reactors i n senes i s
We need only -r.4 = Jk) and FA" 10 size reactors.
By comparison. the volume necessary to achieve 80% conversion in one CSTR is
Notice in Example 2-5 that the sum of the two CSTR reactor volumes (4.02 m3) in series is less than the volume of one CSTR (6.4 m3) to achieve the same conversion.
Approximating a PFR by a large number of CSTRs in series Consider approximating a PFR with a number of small. equal-volume CSTRs of V, i n series (Figure 2-5). We wanr to compare the total volume of all the CSTRs with the volume of one plug-flow reactor for the same conversion, say 808.
uuuuu Figure 2.5
Modeling a PFR with CSTRs in series.
58
Conversion and Reactor Sizing
Chap. 2
The fact that we can m d e l a PFR with a large number of CSTRs is an imporrant result.
Conversion, X Figure 2-6 Levenspiel pla showing cornpanson of CSTRs in series with one PFR.
From Figure 2-6. we note a very important observation! The total volume to achieve 80% conversion for five CSTRs of equal volume in series is roughly the same as the volume of a PFR. As we make the volume of each CSTR smaller and increase the number of CSTRs,the total vofume of the CSTRs in series and the vol. ume of the PFR will become identical. That is, we can m d r i a PFR urirln o large number of CSTRs in series. This concept of using many CSTRs in series to model a PFR will be used later in a number of situations, such as modeling catalyst decay in packed-bed reactors or transient heat effects in PFRs.
2.5.2 PFRs in Series We saw that two CSTRs in series gave a smaller total volume than a single CSTR to achieve the same conversion. This case does not hold true for the two plug-flow reactors connected in series shown in Figure 2-7.
Figuw 2-7 Two PFRf in series,
Sec. 2.5
Rsactors In Series
We can see from Figure 2-8 and from the following equation
PRF in series
that it is immaterial whether you place two plug-flow reactors in series or have one continuous plug-flow reactor; the total reactor volume required to achieve the same conversion is identical!
The overall conversion o f twa PRFs in series rs the same as one PRF with the same total volume.
Converslon. X
Figure 2.8
I
Levenspiel plot for two PFRs in series.
Exarnple 2-6 Sizing Plug-Flow Reacfors in Series Using either the data in Table 2-3 or Figure 2-2, calculate the reactor voIumes V, and V2 for the plug-flow sequence shown in Figure 2-7 when the intermediate con\?ersion is 40% and the final conversion is 80%. The entering molar flow rate is the same as in the previous examples, 0.4 molts.
In addition to graphical integration, we could have used numerical methods to size the plug-flow reactors. I n [his example, we shall uqe Stmpson's rule (see Appendix A.4) to evalnate the integral%.
60
Conversion and Reactor Sizing
Chaa.
Simpson's
three-pvint rule
For thefi,:rt
recrctur: X,,=
0. X, = 0.2, X2 = 0.4, and A X = 0.2.
Selecting the appropriate vatue5 from Table 2-3. we get
For the second renctnr;
picq The rotill volume i s then
Note: This is the same volume we calculated for a single Pl% to achieve 8809 ror version in Example 2-4.
2.5.3 Combinations of CSTRs and PFRs in Series The final sequences we shall consider are combinations of CSTRs and PFRs i series. An industrial example of reactors in series is shown in the photo in Fig ure 2-9. This sequence is used to dimerize propylene into isohexanes. e.g..
Sec. 2 5
61
Reactors in Serie?
Figure 2-9 Dimtrsol C; (an orgdnometall~ccatnly~t)unlt ~ t * oCSTRs and one tubular reactor in serler) to d~merizepmpylene ~ n t oisohesanes. In%tltutFmnqaic du Pitrole process. [Photo courtesy of Ed~t~nns Technip (Insntut Fran~aisdu Pitrole) ]
A schematic of the industrial reactor system in the Figure 2-9 is shown in Figure 2-10,
Figure 2-10
Schematic of a real system.
For the sake of illustration, let's assume the reaction carried out in the vs. X curve given by Table 2-3.
reactors in Figure 2- 10 fo~lowsthe same
The voIumes of the first two CSTRs i n series (see Example 2-51 are: In this series arranpement -r,, is eva1;ated at xZ'?or the second CSTR.
Reactor 1 -
Reactor 2
r,\ l
v, = f'~o(X2- XI1
Staning with the differential form of PFR design equation
(2-24)
62
Conversion end Reactor Sizing
Chap. 2
Rearranging and integrating between limits, when V = 0, then X = X2,and when V = V3. then X = X,.
Reactor 3 The corresponding reactor volumes for each of the three reactors can he found from the shaded areas in Figure 2-1 1,
CSTR l
Csm 2
PFR 0
X,
XI
xu
Conversion. X
Figure 2-11 Leven~pielplot to determine the reactor volumes V,. V:, and V,.
The FAd-rA curves we have been using in the previous examples are typical of those found in isothermal reaction systems. We wilt now consider a real reaction system that is camed out adiabatically. Isofherma1 reaction systems are discussed in Chapter 4 and adiabatic systems in Chapter 8. Exomple 2-7 An Adiabatic tiquid-Phase lsomeniation The isomerization of butane
was carried out adiabatically in the liquid phase and the data in Table E2-7.1 were obtained. (Example 8.4 shows how the data i n Table E2-7.1 were generated.)
See. 2.5
63
Reactors in Series
Don's worry how we got this data or why the (11-rA) looks the way it does, we will see how to construct this table in Chapter 8. It is real data for a real reaction carried out adiabaticaliy, and the reactor scheme shown in Figure E2-7.1IS used.
Figure E27.I Reac~orsin series.
Calculate the ~olurneof each of the reactors for an entering molar flow rrtre of 50 kmollhr.
n-butane of
Soluriori
Taking the reciprocal of -r, and multiplying by FAo we obtain Table E2-7.2.
(a) For the first CSTR, when X = 0.2, then
F ~-o 0.94 - I-*
I
(b) For the PFR,
I
64
Conversion and Reactor Sizing Using Simp.;onVsthree-pclint formula with
IX = 10.6 - 0.7)/2 = 0.2.
Chap.
and X , = 0.
X: = 0.4, and X3= 0.6.
I (c) For the
V2 = 0.38 m'= 380drn31
(El-7..
Iast reactor and the second CSTR, moIe balance
on A for the CSTF
In - Out + Generation = 0
F*z-F*3+
~*3V3
=Q
Rearranging
1
Simplifying
We find from Tnble E?-7.2 that at X3 = 0.65, then
V3= 2 rn3 (0.65
-
=
2.0 tn'
- r ~ ~ 0.6) = 0.1 m'
A Levenspiel plot of (FAd-rA)vs. X i s shown in Figure E2-7.2
2.5.4 Comparing the CSTR and PFR Reactor VoIurnes and Reactor Sequencing
If we look at Figure E7-2.2, the area under the curve (PFR volume) betwee
X = 0 and X = 0.2, we see that the PFR area is greater than the recranguIar are
corresponding to the CSTR volume, i.e., V,, > Vcm. However, if we corn pare the: areas under the curve between X = 0.6 and X = 0.65, we see that th
Sec. 2.5
65
Reactors in Series
0.5
0
o
0.1
o2
0.3
0.4
05
o6
0.7
1100 dm3)
Cwe'slon X
Figure Et-7.2 Levenspiel plot for ndiabatic reactors in series. llrhicharrangement
area under the curve (PFR volume) is smaller than the rectangular area corre-
is bestq
sponding fo the CSTR volume, i.e., Vcsm 4 VPFR.This result often occurs when the reaction is carried out adiabatically, which is discussed when we Iook at heat effects in Chapter 8. In the sequencing of renctors one is often asked. "Which reactor should go first to give the highest overalI conversion? Should it be a PFR folIowed by a CSTR, w two CSTRs, then a PFR, or ...?*' The answer is "It depends." It depends not only on the shape of the Levenspiel plots (FA',d-rA)versus X , but also on the relative reactor sizes. As an exercise, examine Figure E2-7.2 to learn if there is a better way to mange the two CSTRs and one PRF. Suppose you were given a Levenspiel plot of (FA&-r,) vs. X for three reactors in series dong with their reactor volumes VCSTRl= 3 m3,VCSTRI= 2 m3*and VPFR= I .2 r d and asked to find the highesr possible conversion X. What would you do? The methods we used to calculate reactor volumes all apply, except the procedure is reversed and a trial-and-ermr solution is needed ro find the exit overall conversion from each reactor. See Problem P2-5,. The previous examples show that if we know the molar flow rate to the reactor and the reaction rate as a function of conversion. then we can calculate the reactor volume necessary to achieve a specified conversion. The reaction rate does not depend on conversion alone, however. It is also affected by the initial concentrations of the reactants, [he temperamre, and rhe pressure. Consequently, the experimental data obtained in the laboratory and presented in Table 2-1 as -r, as a function of X are useful only in the design of full-scale reactors that are ro be operated at the identical conditions as the laboratory experiments (temperature, pressure, initial reactant concentrations), However,
3 L
or
66
Conversion and Reactor Sizfng
Chap. 2
such circumstances are seldom encountered and we must revert to the methods we describe in Chapter 3 to obtain -rA as a function of X. It is important to understand that if the rate of reaction is available or can be obtained solely as a function of conversion, -r, = flX),or if it reactors can be generated by some intermediate calculations, one can design a variety of reactors Or 8 combination of reactors. Ordinarily, laboratory data are used to formulate a rate law, and then the Chapter 3 shows reaction rate~onversionfunctional dependence is determined using the rate how to find -r, =RW. law. The preceding seaions show that with the reaction rate-conversion relationship, different reactor schemes can readily he sized. In Chapter 3, we show how we obtain this relationship between reaction rate and conversion from rate law and reaction stoichiometry.
2.6 Some Further Definitions Before proceeding to Chapter 3, some terms and equations commonly used in reaction engineering need to be defined. We also consider the special case of the plug-flow design equation when the volumetric flow rate is constant. 2.6.1 Space Time
The space time, z. is obtained by dividing reactor volume by the volumetric flow rate entering the reactor: 7 is an
imponant quantity!
The space time is the time necessary to process one reactor volume of fluid based on entrance conditions. For example, consider the tubular reactor shown in Figure 2-1 2, which is 20 m long and 0.2 m V n volume. The dashed line in Figure 2-12 represents 0.2 m b f fluid directly upstream of the reactor. The time it takes for this fluid to enter the reactor completely- is the space time. 1t i s also called the holding rinw or llzeiil~tvsidence rime.
b
3
1-20rn+20rn-j
Time or mean residence time, t = VFu,,
- - - - - - - - - - - - - - - - J
Spa-
II <
V-0 2
m3
-
Reactor
I - - - - - - - - - - - - - - -
Figure 2-12 Tubular reactor chnwing den tical volumc upstream.
For example. if rhe iolumetric flow rate were 0.03 m3/s. it would take the upstream volume rhoun by the dashed lines a time r
Sec. 2.6
67
S m e Further Definitions
to enter the reactor. In other words, it would take 20 s for the fluid at point a to move to point b, which corresponds to a space time of 20 s. In the absence of dispersion, which is discussed in Chapter 14, the space time is equal to the mean residence time in the reactor, t,. This time is the average time the molecules spend in the reactor. A range of typical processing times in terms of the space time (residence time) for industrial reactors is shown in Table 2-4.
Mean Re.ridcnce
Rtacrur Tvpc
Time Range
P d u c r i o n Copacitr.
Batch
15 min to 20 h
Few kglday to 100,000 tondyes
CSTR
10 min to 4 h
10 to 3,000.000 tondyear
Tubular
0.5 s to I h
$0 to 5,000,000 tonslyear
Table 2-5 shows space times for six industrial reactions and reacrors.
Reaction 11)
'hpical indus1rial reactron space
Reacfor Temperoruw
Pres~um arm
Space Time
CTH~+C~%+H~
PFRt
860°C
2
I s
(2) CH3CH10H + HCHICOOH -+ CH,ICH~COOCH~ + H1O
CSTR
100DC
1
2h
(3)
Catalytic cracking
PBR
490°C
20
(51
CO+HzO+CQ2+HZ
PBR
300°C
26
4.5 s
CSTR
50°C
I
20 min
time$
1sc~<400s
reactor i s tubular but the Row may or may not be ideal plug Row.
'Trambouze, Landeghem, and Wauquier, ChcrnicoI Reac~nrs,p. 154, (Pans: Editions '
Technip, 1988: Houston: Gulf Publishing Company, 1988). %'alas, S . M.Chemical Reactor Data, Chemicnl E~lgirleerin~. 79 (October 14, 19851.
Conversion and Reactar Sizing
Chap.
2.8.2 Space Velocity The space velocity (SV). which is defined as
might be regarded at first sight as the reciprocal of the space time. Howeve there can be a difference in the two quantirles' definitions, For the space tim the entering volumetric flow rate is measured at the entrance conditions, bl for the space velocity, other conditions are often used. The two space velocitit commonly used in industry are the liquid-hourly and gas-hourly space veloc ties, LHSV and GHSV, respectively. The entering volumetric flow rate, u,, i the LHSV is frequently measured as that of a liquid feed rate at 60°F or 75". even though the feed to the reactor may be a vapor at some higher temperatun Strange but true. The gas volumetric flow rate. un, in the GHSV is normnll measured at standard temperature and pressure (STP).
[
Erample 2-8 Reactor Space Times arrdSpace Velocities
Calculate the space time, T. and space velccities for each of the reactors in Exan ples 2-2 and 2-3
From Example 2-1, rve recar1 the entering votumetric Row rate was given r 2 dm% /s10.002 m'ls), and we calculnted the concentration and rnolar flow rates fc the conditions given to k CAn= 0.2 mol/dm3 and F,, = 0.4 moVs. From Example 2-2, the CSTR volume was 6.4 m%nd the correspondin space time and space velocity are
From Example 2-3, the PFR volume was 2.165 m3,and the correspondin space time and space velocity are
Chap. 2
69
Summary
These space times are the times For each uf the reactors to take one reactor volumc of fluid and put it into the reactor.
The CRE Algonthrn -Mole Balance. Ch 1 .Rate Law. Ch 3 Stoichiomeuy. Ch 3 *Combine, Ch 4 *Evaluate, Ch 4 -Energy Balance. Ch 8
To summarize these last examples, wc have seen that in the design of reactors that are to be operated at conditions (e.g., temperature and initial concentration) identical to those at which the reaction rate data were obtained, we can size (determine the reactor volume) both CSTRs and PFRs alone or in various combinations. In principle, it may he possible to scale up a laboratory-bench or piIot-plant reaction system solely from knowledge of -r, as a function of X or C,. However. for most reactor systems in industry, a scale-up process cannot be achieved in this manner because knowledge of -r, solely as a function of X is seldom. if ever. available under identical conditions. In Chapter 3, we shall see how we can obtain -r, = AX) from information obtained either in the laboratory or from the literature. This relationship will be developed in a two-step process, In Step 1 , we will find the rate law that gives the mte rts a function of concentration and in Step 2. we will find the concentrations as a function of conversion. Combining Steps I and 2 in Chapter 3, we obtain -r, =AX).We can then use the methods developed in this chapter along with integral and numerical methods to size reactors.
C~OSUR In this chapter, we have shown that rr you are given the rate of reaction as a function of conversion. i.e., -r, =AX>,you will be able to size CSTRs and PFRs and arrange the order of a given set of reactors to determine the 3esr overa11 conversion. After completilng this cb,apter, the reader shl ible to a. define the parameter comers1ion and rewrite the mole bala nces in terms of conversion show that by expressing -TA iis a function of con.version X a number of reactors and reaction systems cam be size{3 or a conversion calculated fmm a given reactor size arrange re!actorsin series to a m conversion for a given t levenspiel plot
SUMMARY 1. The conversion X is the moles of A reacted per mole of A fed. For batch systems:
For flow syterns:
-F A ,Y= Fi.0 FA,
70
Conversion and Reactor Sizing
Chap. 2
For reactors in series with no side streams, the conversion at point i is
X,= Total moles of A reacted up to point i
'
(S2-3)
Moles A fed to the first reactor
2. In terms of the conversion, the differential and integral. forms of the reactor design equations become: -
Diffewnrial Form
Batch
dX NAodl
Algcbmic Form
-
hllgral Form
- -rAY -
CSTR
PBR
FAOdX = -rr dW
A
3. If the rate of disappearance is given as a function of conversion, the following graphical techniques can be used to size a CSTR and a plug-flow reactor. A. Graphical Integration Using Levenspiel Plots
CSTR
Conversion, X
Chap. 2
CD-ROM Materials
71
The PFR integral could also be evaluated by
B.
Numerical Integration See Appendix A.4 for quamtuE formulas such as the five-point quadrature formula with AX = 0.8/4 of five equally spaced points, X , = 0, X2 = 0.2, X, = 0.4,X4 = 0.6,and X5 = 0.8. 4. Space time. T, and space velocity, SV, are given by
sv = 3v (at STP) CD-ROM MATERIALS Learning Resources I . Summary Notes for Chapter 2 Surnma~yMotes
2. Web Module A. Hippopotamu~Dige5tiue System
Lavenrporal PIM b r eiulo~alaycD~gesr~m m a CSTR
3. Interactive Computer Modules A. Reactor staging
Conversion and Reactor Sizing
Chap.
4. Solved Problem?
A. CDP2-As More CSTR and PFR Calculations-No
Memorization
= FAQ [Frequently Asked Questions]
Professional Reference Shelf
R2.1 ~WodifiedLevenspiel Plots
For hquids and constant volume batch reactors, the mole balance equatior can be modified to
Solved Prohlfirns
A plot of (l/-r,) versus CAgives Figure CD2-1
Reference Chelf
LA!
Flgure CD2-1
Determining the space time, 7.
One can use this plot m study CSTRs. PFRs. and batch reactors. This materia using space time as a variable is given on the CD-ROM.
QUESTIONS AND PROBLEMS
The subscript to each of the problem numbers indicates the Ievel of difficulty: A, leas1 difficult; D. most difficult.
P2-IA
-
P2-2,
ClomeworR ~r3hlems Before solving the problems, state
or
sketch qualitativefy the expected results or trends.
Without referring back, make a list of the most important items you leamec in this chapter. Nlat do you believe was the overall purpose of the chapter? Go to the web site www.engcncsu.edu/Iearnin~s~Iedil~~eb. html (a) Take the Inventory of Learning Style test, and record your learning style accordine to the SolomonFelder inventow.
Chap. 2
Links
73
Questions and Problems
(h) Atter checking the web 51te 1 c ~ ~ ~ ~ ! ~ n g i n . u n ~ i c h e ~ I r J - c r erces. / 3 ~ htm>. TRe.r0~1 suggest two ways to facilitate your learning style in each of the four categories. (c) Visit the problem+solving web site ww~t:en~in.s~nich.eriu/-rre/probroit'/ closed/cep.htm to find way? to "Get Unstuck" when you get stuck on a problem and to review the "Problem-Solving Algorithm." List four ways that might help you in your solution to the home problems.
0,~: /
hkb Mint
Hal o f Fame
*fiL
(d) What audio, . from the first two chapters sounds like Arnold Schwarzenegger (e) What Frequently Asked Question (EAQ) would you have asked7 P2-3, ICM Staging. Load she Interactive Computer Module (ICM) from the CD-ROM. Run the module and then record the performance number, which indicates your mastery of the material. Your professor has the key to decode vour performance number. Nore: TOmn this module you must have Windows i0a0 or a later version. ICM Reactor Staging Performance # P2-4, (a) Revisit Examples 2-1 through 2-3. How would your answers change if the flow rate, FAo, were cut in half? If it were doubled? (b) Example 2-5, How would your answers change if the two CSTRs (one 0.82 m3 and the other 3.2 rn) were placed in parallel with the flow, FA*, divided equally to each reactor. (c) Example 2-6. How would your answer change if the PFRs were placed in parallel with the flow, FA*,divided equally 10 each reactor? {d) Example 2-7. (11 What wouid be the reactor volumes if the two intermedlate conversions were changed to 208 and 50%. r'espe~fively,(2) What would be the conversions, XI,X?, and X3. if a11 the reactors had the same volume of 100 dm3 and were placed in the same order? (3) What i s the worst possible way to m n g e the two CSTRs and one PFR? (e) Example 2-8. The space time required to achieve 806 conversion in a CSTR i s 5 h. The entering volumetric flow rate and concentration of reactant A are 1 ddlmin and 2.5 molar, respectiveiy. IF possible. determine (1) the n t e of reaction, -r, = , (2) the reactor volume, V = , (3) the exit concentratron of A. C, = , and (4) the PFR space time for 80% conversion. P2-iTD You have two CSTRs and two PFRs each with a volume of 1.6 m3. Use Figure 2-2 to cnlcuhte the conversion b r each of the reactors in the following arrangements. (a) Two CSTRs in series.
'
(bj Two PFRs in series. (c) Two CSTRs in parallel with the feed,
FAIl, divided equally between the reactors. (d) Two PFRs in parallel with h e feed divided equally between the two reactors. (e) A CSTR and a PFR in p d I e I with the Row equally divided. Also calculate the overall conversion, Xm
two
FA,=
(4 A PFR followed by a CSTR.
FRO - (1 -Xm) 2
Conversion and Reactor Sizing
(g) A CSTR followed by
P2-6,
P2-7R
Chap. 2
a PFR.
Ih) A PFR followed by two CSTRs. Is this arrangement a good one or is there a better one? Read the chemical reaction engineer of hippopotamus on the CD-ROM or on the web. [a) Write five sentences summarizing what you learned from the web module. (b) Work problems ( I ) and (2) on the hippo module. (c) T h e hippo has picked up a river fungus and now the effective volume of the CSTR ctornach compartment is onl? 0.2 &. The hippo needs 30% conversion to sunlive? Will the hippo survi\~e. (d) The hippo had to have surgery to remove a blockage. Unfonunately, the surgeon. Dr. No, accidentally reversed the CSTR and the PER during the operation. Oops!! What will be the conversron with the new digestive arrangement? Can the hippo survive? T h e exothermic reaction
was carried out adiabatically and the following data recorded:
P2-8,
The entering molar Row rate of A was 300 mollmin. (a) What are the PFR and CSTR volumes necessary to achieve 40% convession? (VPFR = 72 dm.'. V,,,, = 24 dm') (b) Over what range of conversions would the CSTR and PFR reactor volumes be identical? (c) What i\ the maximum conversion that can be achieved in a 10.5-dm.' CSTR7 Ed) What conr*ersioncan be achieved if a 7 2 - d m V P F is followed in series by a 24-dm3 CSTR? (e) What conversion can be achieved if a 24-dm" CSTR i q followed in a series by a 77-dm' PFR? (fl PIor the cnnversion and rate of reaction as a function of PFR reactor volume up lo a volume of 100 dm.'. In bioreactors. rhe growth is autocatalytic in that the more cells you have, the greater the gro\vth rate Cells + n u t r i ~ n ~ rcells, more cells + product
The cell prnuth rate, s,. and the rate of nutrient consumption, r:,, are direcrly pmponional to the concenrration o f cells for a given 5et of condirions. A
Chap. 2
Questions and Problems
75
Levenspiel plot of (11-r,,) a function of nutrient convenion X, = (Cm C3)/Cmis given below in figure P2-8.
Figure P2-8
-
Levenspiel plot for bacteria growth.
For a nutrient Feed rate of lkghr with Cm = 0.25 g/drn3. what chernostat (CSTR) size is necessary to achieve. (a) 40% conversion of the substrate. (h) 80% corlversion of the substrate. (c) What conversion could you achieve with an SO-dm"STR? An 80-dm3 PFR? i d ) How could you arrange a CSTR and PFR in series to achieve 80% conversion with the minimum total volume? Repeat for two CSTRs in series. ( e ) Show that Monod Equation for celI growth
along with 'the stoichiometric relationship between the cell concentrarion. C, and the substrate cancentntio~,C,,
P2-9,
is consistent with Figure P2-8,. The adiabatic exothermic irreversible gas-phase reaction 2 A + B +2C i s to he carried out in a flow reactor for an equimuiar feed of A and B. A Levenspiel plot for this reaction i s shown in Figure P2-9. (a) What PFR v ~ l u m eis necessary ro achieve 508 conversion? (b) What CSTR volume is necessary to achieve 50%canversion? (c) What is the volume of a second CSTR added in series to the first CSTR (hrt B) necessary to achieve an overall conversion of 80%? Id) What PFR volume must be added to the first CSTR (Part B) to raise the conversion to 80%? (el What conversion can be ach~evedin a 6 x lo4 rn3 CSTR and also in a h x 10" m3 PFR? (fl Critique the shape of Figure P2-9 and the answers [numbers) to this problem.
Convmrs~onand Reactor Sizing
Chap :
Figure P2-9 Levenspiel plot.
P.2-10, Estimate the reactor volumes of the two CSTRs and the PFR shown in t h photo in Figure 2-9. P2-lID Don'[ calculate anything. Just go home and relax. P2-12, The curve shown in Figure 2- 1 is typical of a reaction carried out isothermallq and the curve shown in Figure P2-12 is typical of gas-solid catalytic exother mic reaction carried out adiabatically.
.2
.4
.5
.8
1.0
Conversion, X
Figure PZ-12 Levenspiel plot for an exothermic reaction.
Rate Laws and Stoichiometry
Success is measured not so much by the position one has reached En life. as by the obstacles one has overcome while q i n g to succeed. Booker T. Washington
Oveniew. In Chapter 2, we showed that i f we had the rate of reaction as a function of conversion, - r ~= fo, we could calculate reactor volumes necessary to achieve a specified conversion for flow systems and the time to achieve a given conversion in a batch system. Unfortunately, one is seldom, if ever, given - r ~= XX) directly fram raw data. Not to fear, in this chapter we will show how to obtain the rate of reaction as a function of conversion. This relationship between reaction rate and conversion will be obtained in two steps. In Step 1, Part 1 of this chapter, we define the rate law, which relates the rate of reaction to the concentrations of the reacting species and to temperature. In Step 2, Par? 2 of this chapter, we define concentrations for flow and batch systems and develop a stoichiometric table so that one can write concentrations as a function of conversion. Combining Steps 1 and 2, we see that one can then write the rate as a function conversion and use the techniques in Chapter 2 to design r e a c h systems. After completing this chapter. you will be abIe to write the rate of reaction as a function of conversion for both liquid-phase and gas-phase reacting systems,
Rate Laws and Storchiometry
PART1
Chap.
RATELAWS
3.1 Basic Definitions
3pes
A homogeneous reacriotl is one that involves only one phase. A heterogeneou reaction involves more than one phase, and the reaction usually occurs at zh interface between the phases. An irreversible reaction is one that proceeds i only one direction and continues in that direction until the reactants ar exhausted. A ~ver.sihleseacrion, on the other hand, can proceed in eithe direction, depending on the concentrations of reactants and products relative st the corresponding equilibrium concentrations. An irreversible reaction behave as if no equilibrium condition exists. Strictly speaking, no chemical reaction i cornpleteEy irreversible. However, for many reactions, the equilibrium poin lies so far to the product side that these reactions are treated as i~eversibl reactions. The molecularity of a reaction is the number of atoms, ions. or molecule involved (colliding) in a reaction step. The terms unirnolecular, bimoteculat and termoleculnr refer to reactions involving, respectively. one. two, or thre~ atoms (or molecules) interacting or colliding in any one reaction step. Thl most common example of a unimoleculnr reaction is radioactive decay, such a, the spontaneous emission of an alpha parlicIe from uranium-238 to give tho rium and helium:
The rate of disappearance of uranium (U) is given by the rate law
The true bimoleculnr reactions that exist are reactions involving free radical! such as Br
+ C,H,-+ HBr +C,H,
with the rate of disappearance of bromine given by the rate law
The probability of a remolecular reaction occurring is almost nonexistent, and in most instances the reaction pathway foIlows a series of bimolecular reactions as in the case of the reaction
The reaction pathway for this "Hall of Fame" reaction is quite interesting and i s discussed in Chapter 7 along with similar reactions that form active intermediate complexes in their reaction pathways.
Sec. 3.1
3.1.1
Basic Definitions
Relative Rates of Reaction
The relative rates of reaction of the various species involved in a reaction can be obtained from the ratio of stoichiometric coefficients. For Reaction (2-2),
we see that for every m d e of A thaf is consumed, c/a moles of C appear. In other words,
Rate of formation of C = C (Rate of disappearance of A ) CI
Similarly, the relationship between the sates of formation of C and D is
The relationship can be expressed directly from the stoichiometry of the reaction,
for which
Reaction stoicbiometry
or
For example, in the reaction
we have
If NO? is being formed at a rate of 4 mol/m3/s. i.e., rNo2 = 4 mol/m3is
Rate Laws and Stoichiornetry
Chap. 3
then the rate of formation of NO is ZNO + O2+ 2N02 r N =~4 moVm3/s -r,,
= 4 moVm31s
-ro, = 2 rnol1~15
-2 rNo = - rNO2= - 4 r n o ~ r n ~ i s
2
the rate of disappearance of NO is
and the rate of disappearance of oxygen, Q2,is
3.2 The Reaction Order and the Rate Law In the chemical reactions considered in the following paragraphs, we take as the basis of calculation a species A, which is one of the reactants that is disappearing as a result of the reaction. The limiting reactant is usually chosen as our basis for calculation. The rate of disappearance of A, -r,, depends on temperature and composition. For many reactions. it can be written as the product of a reaction rcrrP co~rstantk, and a function of the concentrations (activities) of the various species involved in the reaction:
The rate law gives the r e i a t ~ ~ n -
chip tlon rate and concentratio*.
The algebraic equation that relates -r, to the species concentrations is called the kinetic expression or rate law. The specific rate of reaction (also called the sale constant). kA, like the reaction rate -r,, always refers to a particular species in the reaction and normally should be subscripted with respect to that species. However. for reactions in which the stoichiornetric coefficient is I for all species involved in the reaction. for example.
INaOH + IHCl
+ J
INaCI
+ 1 N,O
we shall delete the subscript on the specific reaction rate, (e.g., A in k,), to let
3.2.1 Power t a w Models And Elementary Rate Laws The dependence of €he reaction rate. -r,. on the concentrations of the species present. fn(C,), is almost without exception determined by experimental observation. Although the functional dependence on concentration may be postuIated from ~heory,experiments are necerrary to confirm the proposed fornl. One of the most common general forms of this dependence is the power law model. Here the rate law 1s the product of concerlrrations of the individual reacting species. each of which is raised to a power. for example.
Sec. 3.2
83
The Reaciion Order and the Rate t a w
0-31 The exponents of the concentrations in Equation (3-3) lead to the concept of reaction order. The order of a reaction refers to the powers to which the concentrations are raised in the kinetic rate law.' In Equation (3-31, the reaction is a order with respect to reactant A. and order with respect to relzcranr B. The overall order of the reaction, n, is Overall reaction nrder
n=a+/3 always in terms of concentration per unit time while
The units af -r, tire the units of the specific reaction rate, k,, will vary with the order of the reaction. Consider a reaction involving only one reactant, such as A +Products
with a reaction order n. The units of the specific reaction rate constant are
k=
on cent ration)^ - " Time
Consequently, the rate laws corresponding to a zero-, first-, second-, and third-order reaction, together with typical units for the corresponding rate constants, are:
( k ) = mol/dm3. s
First-order (n = 1 ):
-rA =
kACA:
(k) =s
-I
( k ) = dm?mol s
(3-5)
(3-6)
Strictly speaking, the reaction rates should be written in terms o f the activities, u,, (a, = y,C,. 1s the activity cwffic~ent).Kline and FogIer, ICIS, 82. 93 (19811: ihid., p. 103: and Ind. Eug. CChern hndompnrols 20, 1 55 ( 198 1 ). I
where y,
r a p
- r A = kAaAaR
However, for many rcacting sysrerns, the actlvlry cocficients, y,. do not change appreciably during Ihe course nf the reaclion, and they are adsorbed into the specific reaction rate:
84
Rate Laws and Stoichiometry
{k)
=
(drn?lmol)'.s-'
Chao
(3-7
An elernaltny rtaotioa is one that evolves a single step such as t h ~ bimoIecuIar reaction between oxygen and methanol
Om+ CH,OH+CH,O
+ OH*
The stoichiometric coefficients in this reaction are identic.nl to the powers ir the rate law. Consequently, the rate law for the disappearance of rnolecula oxygen is
Reference Shelf
Collision theory
The reaction is first order in molecular oxygen and first order in methanol therefore. we say both the reaction and the rate law are elementary. This fom of the rate law can be derived from Colli.~iiot~ T h e o as ~ shown in the Profes. sion Reference Shelf 3A on the CD-ROM. There are many reactions where the stoichiornetric coefficients in the reaction are identical to the reaction orders but the reactions are not elementary owing to such things as parhways involv. ing active intermediates and series reactions. For these reactions that are no1 elementary but whose stoichiometric coefficients are identical to the reaction orders in the rate law, we say the reactionfollows nn elernenran. rote Eaw. For example, the oxidation reaction of nitric oxide discussed earlier.
is not elementary but follows the elementary rate taw Note: the rate constant, k, is defined with respecr to NO.
Another nonelementary reaction that follows an elementary rate law is the gas-phase reaction between hydrogen and iodine with
In summary,far many reactions involving multiple steps and pathways. the powers in the rate laws surprisingly agree with the stoichiometric coefficients. Consequently, to facilitate describing this class of reactions, we say a reaction f01lm.s an elementary rate law when the reaction orders are identical with the stoichiometric coeficients of the reacting species for the reaction ns written. It is i m p o m t to remember that the tate laws are determined by experimenhf observation! They are a function of the reaction chemistry and not the type of reactor in which the reactions occur. Table 3-1 gives examples of rate laws for a number of reactions.
SRC 3 2
{Vhere do you tind rate lawr
'
The Reactloo Order and !he Rate Law
85
The values of specific reaction rates For these and a number of other reactions can be found in the Dcltc~Bn.w found on the CD-ROM and on the web. The activatron energy, frequency factor, and reaction orders for a large number o f gas- and liquid-phase reactions can be: found in the National Bureau of Standards' circulars and supplements.? Also consult the journals listed at rhe end of Chapter I .
A. First-Order Rate Laws
B. Second-Order Rate Laws
' See Problem P3-13, important refyou
should also look rn
the other literature hcfom going to the
lab
and Section 9.2.
Kinetic data for larger number of reactions can be obtained on floppy disks and CD-ROMs provided by Nntionnl lnstitlrre of Standanis and Technology (NIST). Standard Reference Data 22t/A320 Gaithersburg, M D 20899; phone: (301) 975-2208. Additional sources are Tables of Chemical Kinetics: I?'omogeneous Reacrions, National Bureau of Standards CircuIar 510 (Sept. 28, 1951); SuppI. 1 (Nov. 14. 1956); Suppl. 2 (Aug. 5 , I960): Suppi. 3 ISept. 15, 1961) (Washington, D.C.:U.S. Government Printing Office). Cl~emicalKinetics and Photochemicnl Dnta for Use m Stratospheric Modeling, Evaluate No. 10. JPL Pubtication 92-20 (Pasadena, Calif.: Jet Propulsion Laboratories, Aug. €5. 1992).
86
Rate laws and Stoichiometry
Chap. 3
C. Nonefernenbry Rate Laws
Cumene (C)jBenzene (8)+ Pmpylene (P)
13. Enzymatic Reactions (Urea (U) + Urease (E)) +H7O NH,CONH2 + Urease 4 2NHJ + C02 + Urease
E. Biomass Reactions Substrare (S) + CelIs 1C) + More Cells + Product
NO!?: The rate constant, k. and activation energies for a number of the reactions in these exampIes are given in the Dara Base on the CD-ROM and Summary Notes.
3.2.2 Nonelementary Rate Laws A large number of both homogeneous and heterogeneous reactions do not follow simple rate laws. Examples of reactions that don't follow simple elementary rate Iaws are discussed below.
Homogeneous Reactions The overall order of a reaction does not have to be an integer. nor does the order have to be an integer with respect to any individual component. As an example. consider the gas-phase synthesis of phosgene,
in which the kinetic rare Inw is
This reaction is first order with respect to carbon monoxide, three-halves order with respect to chlorine. and five-halves order overall.
Sec. 3.2
The Reachon Order and the Rate Law
87
Sometimes reactions have complex rate expressions that cannot be separated into solely temperature-dependent and concentration-dependent portions. In the decomposition of nitrous oxide,
the kinetic rare law is
Apparent reaction orden
lmponant resources for m e laws
Both kNlo and k' are strongly temperature-dependent. When a rare expression such as ihe one given above occurs, we cannot state an overaIl reaction order. Here we can only speak of reaction orders under cestain limiting conditions. For example, at very low concentrations of oxygen, the second term in the denominator would be negligible WIT 1 (1 >> k'Co, ), and the reaction would be "apparent" first order with respect to nitrous oxide and first order overall. However. if the concentration of oxygen were large enough so that the number 1 in the denominator were insignificant in comparison with the second term. k'Co (ktCO7>> I), the apparent reaction order would be -1 with respect to oxvE& and first order with respect to nitrous oxide an o\,eralI apparenr zero order. Rate expressions of this type are very common for liquid and gaseous reactions promoted by solid catalysts (see Chapter 10). They also wcor in homogeneous reaction systems with reactive intermediates (see Chapter 7). 11 is interesting to note that although the reaction orders often correspond to the stoichiometric coefficients as evidenced for the reaction between hydrogen and iodine, the rate expression for the reaction between hydrogen and another halogen, bromine. is quite complex. This nonefernentary reaction
proceeds by a free-radical mechanism, and its reaction rate law i s
In Chapter 7. we will discuss reaction mechanisms and pathways that lead to nonelementary rate laws such as rate of formation of HBr shown in Equation (3-8). Heterogeneous Reactions In many pas-solid catalyzed reactions. it historically has been the practice to write the rate law in terms of partial pressures rather than concentrations. An example of a heterogeneous reaction and corresponding rate law is the hydrodcmethylation of toluene (T) to form benzene ( B j and methane (MIcarried out over a solid catalyst.
88
Rate Laws and Stoichiometry
Chap.
The rate of disappearance of toluene per mass of catalyst, - r ' , , follou Langmuir-Hinshelwood kinetics (Chapter 10). and the rate law was foun experimentally to be
where Kg and KT are the adsorption constants with units of kPa-I (or atm-I and the specific reaction rate has units of [kl
=
mol toluene kg cat - s .k ~ a '
To express the rate of reaction in terms of concentration rather than partia pressure, we simply substitute for P, using the ideal gas law
The rate of reaction per unit weight catalyst, -rA, (e,g., -r;), and [hi rate of reaction per unit volume, - s, , are related through the bulk density p, (mass of solidlvolume) of the cafalyst particles in the fluid media:
(-) (
moles = mass time. volume volume
)
moles trme mass
.
In fluidized catalytic beds, the bulk density is normally a function of the volumetric flow rate through the bed. In summary on reaction orders, they cannot be deduced from reactior stoichiometry. Even though a number of reactions foIlow elementary rate laws, at least as many reactions do not. One must determine the reaction order from the literature or from expedmenrs.
3.2.3 Reversible Reactions A11 rate laws for reversible reactions must reduce to the thermodynamic refationship relating the reacting species concentrations at equilibrium. At equilibrium, the rate of reaction is identically zero for all species (i-e., - r , = O ). That is, for the general reaction
the concentrations ax equilibrium are related by the thermodynamic relationship for the equilibrium constant Kc (see Appendix CI. Themcdyriarnic Equilibrium Relationship
Sec. 3.2
89
The Reaction Order and the Ra:e Law
The units of the thermcdynamic equilibrium constant. Kc. are Kc, are ( r n ~ l / d r n ? ) ~- 'h(- " , To illustrate how to write rate laws for reversible reactions. we will use the combination of two benzene molecules to form one molecule of hydrogen and one of diphenyl. In this discussion, we shall consider this gas-phase reaction to be efementary and reversible:
or, symbolically,
The specific reaction must be defined wn a particurar species.
The forward and reverse specific reaction rate constants, k, and k - , respectively, will be defined with respect to benzene. Benzene (B) is being depleted by the forward reaction
.
2C6H6---%C I I H I o + H , in which the rate of disappearance of benzene is
If we multiply both sides of this equation by - 1, we obtain the expression for the sate of formation of benzene for the forward reaction: r~,forward =
RcB'-
(3-1 1)
For the reverse reaction between diphenyl (D)and hydrogen (Hz ).
the rate of formation of benzene is given as
Again, both the rale constants k, and k-, are defined with respect to bencene!!! The net rate of formation of benzene is the sum of the rates of formation from the forward reactiorl [i.e.. Equation (3- 1I)] and the reverse reaction [inen, Equation [3-12)]:
" TB, ncr = 'B.
fo-d
+ 'B,
reverse
90
Rate Laws and Stoichiometry
Chap. 3
Multiplying both sides of Equation (3-13) by -1. we obtain the rate law for the rate of disappearance of benzene, - r , : EIementary reversihle A * B
Replacing the ratio of the reverse to forward rare law constants by the equilibrium constant, we obtain
where
k~ = Kc = Concentration equilibrium constant k-B The equilibrium constant decreases with increasing temperature for exothermic reactions and increases with increasing temperature for endothermic reactions. Let's write the rate of formation of diphenyl. r ~ in, terms of the concentrations of hydrogen, H2, diphenyl. D, and benzene, B. The rate of formation of diphenyl, r,, must have the same functional dependence on the reacting species concentratjons as does the rate of disappearance of benzene. -r,. The rate of formation of diphenyl is
Using the relationship given by Equation (3-1) for the general reaction This is just stoichiornetn:
we can obtain the relationship between the various specific reaction rates.
k,, k , :
Comparing Equations (3- 15 ) and (3-1 6). we see the relationship between the specrfic reaction sate with respect to diphenyl and the specific reaction rate with respect to benzene is
Sec. 3.3
91
The Reaction Rate Constant
Consequently, we see the need to define the rate constant, k, wrt a particular species. Finally, we need to check to see if the rate law given by Equation (3-14) is thermodynamicaIly consistent a! equilibrium. Applying Equation (3-10) {and Appendix C ) to the diphenyl reaction and substituting the appropriate species concentration and exponents, thermodynamics tells us that
At equilibrium- the rate law must reduce
Now let's look at the rate law. At equilibrium, - r ~r 0, and the rate law given by Equation (3-14) becomes to an equation
consistent wth thermodynamic equilibrium.
Rearranging, we obtain, as expected, the equilibrium expression
Cndolhrnk
I
which is identical to Equation 13-17) obtained from thermodynamics. From Appendix C, Equation (C-9), we know that when there is no change in the total number of moles and the beat capacity term, ACp = 0 the temperature dependence of the concentration equilibrium constant is
[A?(;, --I)-:
K c ( T ) = K,(T,) exp
T
Therefore, if we know the equilibrium constant at one temperature, T, [i.e.,Kc (T,)], and the heat of reaction, AHRn,we: can calculate the equilibrium constant at any other temperature T For endothermic reactions, the equilibrium constant, Kc, increases with increasing temperacure: for exothermic reactions, Kc decreases with increasing temperature. A further discussion of the equilibrium constant and its themlodynamic relationship is given i n Appendix C.
3.3 The Reaction Rate Constant The reaction rate constant k is not truly a constant: i t is merely independent of the concentrations of the species involved En the reaction. The quantity k is referred to as either the specific reaction rate or the rate constant. It is almost always strongly dependent on temperature. It depends on wherher or not a catalyst is present, and in gas-phase reactions, it may be a function of total pressure. ln liquid systems i t can also be a function of other parameters, . filch as ionic strengrh and choice of solvent. These other variables normally exhibit much Iess effect on the specific reaction rate than temperature does with the exception of supercritical solvents, such as super critical water.
92
Rate Laws and Stoichiometry
Chap.
Consequently. for the pilrposes of the material presented here. it will L. assumed that A-, depends only on temperature. This assumption is valid in rno laboratory and industrial reactions and seems to work quite well. Ir was the great Swedish chemist Arrhenius who first suggested that tl. temperature dependence of the specific reaction rate, kA, could be correlated b an equation of the type Arrhenius equation
where
A = preexponential factor or frequency factor E = activation energy. J/mol or callmol R = gas constant = 8.3 14 Jlmol K = 1.987 cailmol K
-
-
T = absolute temperature, K T(K)
Equation (3-18), known as the Arrhetlius eylrution, has been verified empir caily to give the rernperature behavior of most reaction rate constants withi experimental accuracy over fairly large temperature ranges. The Arrheniu equation is derived in the Professional Reference Shelf 3.A: Colli.rion Theor on the CD-ROM. Why is there nn activation energy? If the reactants are free radicals th essentially react imtnediately on collision. there usually isn't an activatio energy. However. for rnort atoms and moleculer undergoing reaction, there i an activation energy. A couple of the reasons are that in order to react, 1 . The molecules need energy to distort or stretch their bonds so that the break them and thus form new bonds. 2. The steric and electron repulsion forces must be overcome as th reacting motecuEes come close together. The activation energy can be thought of as a barrier to energy transfe (from the kinetic energy to the potential energy) between reacting molecule that must be overcome. One way to view the barrier to a reaction is throug the use of the relaction coordinates. These coordinates denote the potenti: energy of the system as a function of the progress along the reaction path a we go from reactants to an intermediate to products. For the reaction A+EC
A-B-C
+AB+C
the reaction coordinate is shown in Figure 3-1. Figure 3- I(a) shows the potential energy of the three atom (or molecule system. A, 8 . and C, as well as the reaction progress as we go from reactar specie5 A and BC to products AB and C. Initially A and BC are far apart an1 the system energy is just the bond energy BC. At the end of the reaction, th products AB and C are far apart, and the system energy is the bond energy AE As we move along the reaction coordinate (x-axis) to the right in Figure 3-l(a' the reactants A and BC approach each other, the BC bond begins to break. an1 the energy of the reaction pair increases vntil the top of the barrier is reachec At the top, the transition srure is reached where the intermolecular distance between AB and between BC are equaI (i.e., A-B-C). As a result. the potenria energy of the initial three atoms (molecules) is high. As the reaction proceed
Sec. 3.3
93
The Reaction Rate Constant
I
reaaants pradtists Reaction wordinate
1.3
2.1
23
2.5
27
CH3 -I Bond Distance in Angstroms
(b)
Figure 3-1 Progress along reaction path, la) Syrribolic reaction: Ib) Calculnted from computationaI ~oftwnreon the CD-RDSI Chapter 3 14'eb Modutc
further. the distance between A and B decreases, and the AB bond begins to form. As we proceed further, the distance between AB and C increases and the energy of the reacting pair decreases to that of the AB bond energy. The calculations to arrive at Figure 3-l(b) are discussed in the CD-ROM web module, and transition state theory is discussed in the CD-ROM ProfessionaI Reference Shelf R3.2 Transition State Theory. We see that for the reaction to occur, the reactants must overcame an energy barrier, Es, shown in Figure 3-1. The energy barrier, EB, is related to the activation energy, E. The energy barrier height, EB, can be catcutated from differences in the energies of formation of the transition state molecule and the energy -. of formation oT the reactants. that is,
(3- I 9 ) The energy of formation of the reactants can be found in the literarure while the energy of formation of transition state can be calculated using a number of software programs such as CACHE, Spartan. or Cerius2. The activation energy. EA, is often approximated by the barrier hei%ht,which is a good approximation in the absence as quantwm mechanical tunneling. Kow that we have the general idea for a reaction coordinate ler's consider another real reaction system:
H + C2Hb HI + C2Hs
-.
RzfeferenceChef
The energy-reaction coordinate diagram for the reaction between a hydrogen atom and an ethane molecule is shown in Figure - 3.2 where the bond distorlions, breaking, and forming are identified. One can also view the activation energy in terms of collision theory (Professional Reference Shelf R3.1). By increasing the temperature, we increase the kinetic energy of the reactant molecules. This kinetic energy can in turn be transfemd through molecular collisions to internal energy to increase the stretching and bending of rhe bonds. causing them ro reach an activated state, vulnerable to bond breaking and reaction (cf. Figures 3-1 and 3-2).
Rate Laws and Stoichiometrl,
Chap. 3
Figure 3-2 A diagram of the orbital dtstortions during the reaction H + CH3CH3+ Hz+ CHICH3 The d i a p m shows only the interaction w ~ t hthe energy state o f ethane (the C-H bond). Other molecular orbitals o f the ethane also dirton. [Courtesy of R.Masel. Clzemical Kinetics (McGraw Hill, 2002). p. 594.1
Re,erence chc!f
The energy of the individual molecules falls within a distribu~ionof energies where some molecules have mare energy than others. One such distribution is shown in Figure 3-3 whereJE,T) is the energy distribution function for the kinetic energies of the reacting molecules. It is interpreted most easily by recognizing ($. d Q as the fraction of molecules that have an energy 'between E and (E + dm. The activation energy has been equated with a minimum energy that must k possessed by reacting molecules before the reaction will occur. The fraction of the reacting molecules that have an energy EAor greater is shown by the shaded areas in Figure 3-3. The molecules in the shaded area have sufficient kinetic energy to cause the bond to break and reaction to occur. One observes that, as the temperature increases, more molecules have sufficient
Fracl~onof colltstons
f(E,T)
at T2 that have energy
l EA
\
Fraction of col~is~ons at T, that have energy EA or greater
Figure 3-3 Enerr? dirtnbuf~onof rcnctlnp ~noleculec
Sec. 3.3
The Aeaction Rate Constant
95
energy to react as noted by an increase in the shaded area, and the rate of reac-
Calculation of the
tian, -r,, increases. Postulation of the Arrhenius equation, Equation (3-Is}, remains the greatest single step in chemical kinetics, and retains its usefulness today, nearly a century later. The activation energy, E, is determined experimentally by carrying out the reaction at several different temperatures. After taking the natural logarithm of Equation (3-18) we obtain
Ssrndw PIM
(3-20) CI 0 01
slow=-$ OW250033
and see that the activation energy can be found from a plot of In k , as a functionof (ltn. Example 3-1 Determination of the Activalion Energy Calculate the activation energy for the decomposition of benzene diazoninm chloride to give chlorobenzene and nitrogen:
using the information in Tahle E?-I . l for this first-order reaction.
We start by recalling Equa~ion(3-20)
Summary Notes
Tutorials
We can use the data in TabIe E3-I . I to determine the activa~ionenergy, E. and fiequency factor, A. in two d~fferentways. One way is to make a semilog plot of k vs. (llr) and determ~neE from the slope. Another way is to use Excel or Polymath to regress the data. The data In Table E3-1 .I was entered in Excel and i s shown in Figure E3-1.1 which was then used to obtain Figure E3-1.2. G step-by-step tutorial to construct both an Excel and a Polymath spread sheet is given in the Chapter 3 Summary Nates on the CD-R0.V.
96
Rate Laws and Stoichiometry
I I
0.00355 0.00717
l
0.00305
-5.64 -4.94
Chap.
l
0.00300 v
Figum E3-1.1 Excel spreadsheet.
k a
df
-
.#A
-
74
-
$t-?!!L*nlt
--.
-
I*
3m~~ooo3mam~ororm1~0rm~s00a3;0000111
WZ
1~6')
r a)
Figure E3-1.2
1
'*"' 1T 1 ~ i ' ~ 1 '*11
(b)
( a ) Excel semilog plot: (b) ExceI normal plot.
( a ) Graphical Sol~ition
Figure E3-1.2(a) shows the semilog plots from which we can calculate the nctivntio energy. From CD-ROM Appendix D,we show how to rearrange Equation (3-20) i the form
log
k,--E 1 - 1 --- 1.3R (T2 TI)
k,
Rearranging
use the decade method, choose l l T , and 11T2 so that k, = O.lk, . Ther log(k,/k,)= 1.
To
When
k, = 0.005:
1= 0.003025 TI
and when kz = 0.0005:
1= 0.00319 T,
Sec. 3.3
The Reaction Rate Constant
Therefore,
E= =
I
2.303R I T - I T
I I6
- (2.303) (8.31 4 Nmol . K ) (O.OU319-0.0N3025)JK
kJ w 28.7 kcal/rnol mol
The equation for the best-fit of the data
is also shown in Figure E3-1.2(b). From the slope of the line given in Figure 3- I .2(b)
1 1
From Figure El-1.2(b)and Equation (U-1.3).r e see
taking the antilog
The rate dws not always double For s There temperature increase of 10"c. e v e q IO°C
Reference Shelf
is a rule of thumb that states that the rate of reaction doubles for increase in temperature. However, this is tme only for a specific combination of activation energy and temperature. For example, if the activntion energy is 53.6 kJlmol. the rate will double only if the temperature is raised from 300 K to 310 K. If the activation energy is 147 kJ/mol. the rule will be valid only if the temperature is raised from 500 K to 510 K. (See Problem P3-7 for the derivation of this relationship.) The larger the activation energy, the more temperature-sensitiveis the rate of reaction. M i l e there are no typical values of the frequency factor and activation energy for a first-order gas-phase reaction, if one were forced ro make a guess, values of A and E might be IOl3 s-I and 200 kJlmoP. However, for families of reactions (e.g., halogenation), a number of corre1ation.s can be used to estimate the activation energy. One such corntation is the Polanyi-Semenov equation, which relates activation energy to the heat of reaction (see Professional Reference Shelf 3.1). Another correIarion relates activation energy to
98
Rate Laws and Stolchiomet~
Chap. 3
differences in bond strengths between products and reactantsn3While activation energy cannot be currently predicted a priori. significant research efforts are under way to calculate activation energies from first principle^.^ (Also see
Appendix J.) One final comment on the Arrhenius equation, Equation (3-18). It can be put in a most useful form by finding the specific reaction rate at a temperature To, that is,
and at a ternperaturc T
A most useful form of k l T )
and taking the ratio to obtain
This equation says that if we know the specific reaction rate ko(To)at a temperature, "r,,, and we know the activation energy, E. we can find the specific reac~ionrate k ( T ) at any other temperature, T. for that reaction.
3.4 Present Status of Our Approach to Reactor Sizing and Design In Chapter 2, we showed how it was possible to size CSTRs. PFRs, and PBRs using the design equations in Table 3-2 (page 99) if the rate of disappearance o f A is known as a function of conversion. X: Where are we?
In general, information in rhe form -rA = g(X)is not available. However, we have seen in Section 3.2 that the rate of disappearance of A , - r , , is normally expressed in terms of the concentration of the reacting species. This functionality, - r ~= [k,(T)l[fnIC,.C,.
...I
(3-2)
- r & = fIC,I
is called a rurp low. In Part 2, Sections 3.5 and 3.6. we show how the concenc, = l & , { X ) [ration of the reacting species may he written in terms of the conversion X. 1 C, = h, (X) (3-22) +
-? ',
= p 1x1
and then we can design ~rorhermal rerrctors
M. Boudart, Kinrfics of Cl~en~irnl Ptnrrssrx (Upper Saddle River. N.J.: Pi-cntice Hall. 1968). p. 168. J. N'. Moore and R. G. Pearlon, Xirretic.7 and Mr,cl~nlrisnts,3rd ed. (New York: Wiley. 1981 ), p. 199. S. W . Renwn. TherntorEt~micnlKit~~rrrs. 2nd ed. (New York: Wiley. 1976). S. M.Senkan, Defnilrd Cl~rr?~ir.r~l Kirlerir. Murkclirrg: Clicinirnl Hrnrfi(11~ Enginepring of the Fufrrre. Ad\'ances in Chemical Engineering. Vol. 18 (San Diego: Academic Precq. 19921, pp. 95-96.
Sec. 3.4
TABLE 3-2.
Diflerential Form
Batch
The design equations
N40Fr-rAV
( PFR )
Packed bed
WBR)
DESIGNEQUATIONS lntegml Form
Algebmic
Fonn
(2-6)
Backmix tCSTR)
Tubular
99
Present Slatus of Our Approach to Reactor Siz~ngand Design
v=- F*"
- FA
(2-15)
(2-17)
((2.1
3)
dX
V=F,,( .n -r, dX
W = Fhu
(?-!6j (2-18)
A
With these additional relationships, one observes that if the rate law is given and the concentrations can be expressed as a function of conversion. r k ~ nit7 fact we have - r A as a ft~ncrion of X and rhis is nll ,ha1 is needed io eraluore ?he design .equarions. One can use either the numericai techniques described in Chapter 2, or. as we shall see in Chapter 4, a table of integrals, andlor software programs le.g.. Polymath).
Now that we have shown how the rate law can be expressed as a function of concentrations. we need only express concentration as a function of conversion in order to carry out calculations similar to those presented in Chapter 2 to size reactors. If the rate law depends on more than one species. we must relate the concentrations of the different species to each other. This relationship is most easily established with the aid of a stoichiometric table. This table presents the stoichiometric relationships between reacting molecules for a single reaction. That is. it tells us how many rnolecuIes of one species will be formed during a chemical reaction when a given number of molecules of snorher species disappears. These ~Iationshipswill be developed for the general reac~ion
T h ~ s~oich~ometnc c ~lallonshiprelating reaction rates WIII he used in Pan 1 of Chapter 4.
Recall that we have already used stoichiametr-jl to relate the reIatrve rates of reaction for Equation (2-1 ):
100
Rate Laws and Stoichiometrj
Chap.
:
In formulating our stoichiornetsic table, we shall take species A as o u ~ basis of calculation (i.e.. limiting reactant) and then divide through by the stoichiometric coefficient o f A.
in order to put everything on a basis of "pet mole of A." Next, we develop the stoichiometric relationships for reacting species thal give the change in the number of moles of each species li.e.. A. B, C. and D).
3.5 Batch Systems Batch reactors are primarily used for the praduction of specialty chemicals and to obtain reaction rate data in order to determine reaction mte laws and rate law parameters such as k, the specific reaction rate. Figure 3-4 shows an artist's rendition of a batch system in which we will carry ot~tthe reaction given by Equation (2-2). At time t = O, we will open the reactor and place a number of moles of species A, B. C, D,and I (NAO,Ng0, N,,, N,, and N,, respectively) into the reactor. Species A is our basis of calculation, and NAois the number of moles of A initially present in the reactor. Of these. NA& moles of A are consumed in the system as a result of the chemical reaction, leaving (NAo- NA& moles of A in the system. That is, the number of moles of A remaining in the reactor after a conversion X has been achieved is We now will use conversion in this fashion to expresc the number of moles of B, C, and D in terms of conversion. To determine the number of moles of each species remaining after N,,X moles of A have reacted, we form the stoichiometric table (Table 3-3). This stoichiornetric table presents the foIlowing information: Components of the stoichiornettic table
Column I: Column 2: Column 3: CoIumn 4:
the particular species the number of moles of each species initially present the change in the number of mojes brought about by reaction the number of moles remaining in the system at time t
To calculate the number of moles of species B remaining at time t, we recall that at time t the number of moles of A that have reacted is N A o X . For every mole of A that reacts, bla moles of B must react; therefore, the total: number of moles of B that have reacted is
moles B reacted
=
-
reacted rnoIes A reacted moles A reacted
Sec. 3.5
Batch Systems
Figure 3.4
Batch reactor. (Schematic with permission by Renwahr.1
TABLE 3-3.
STOICH~OMETRICTMLEFOR A
It~irtuil~
Specirs A
(mol)
N%o
BATCHSYFTEM
CItcrrrge
Rennining
(moll
(moll
-X
W , = NAG- ,V+,,XI
Because B is disappearing from the system, the sign of the "change" is negative. NBO is the number of moles initially in the system. Therefore, the number of moles of B remaining i n the system. N, , at a time f, is given in the Iast column of Table 3-3 as
102
Rate Laws and Stoichiomefry
Chap. 3
The complete stoichiometric tabIe delineated in Table 3-3 is for a11 species in the general reaction
Let's take a took at the totals in the last column of Table 3-3. The stoichiometric coefficients in parentheses (dla -k c / a - bla - 1) represent the increase in the total number of moles per mole of A reacted. Because this term occurs so often in our calculations, it is given the symbol 8:
The parameter 6 tells us the change in the total number of moles per mole of A reacted. The total number of moles can now be calculated from the equation
we C,= " , [ X I
We recall from Chapter 1 and Part 1 of this chapter that the Enetic rate law (e.g., - r , = kc:) is a function solely of the intensive properties of the reacting system (e-g.. temperature, pressure, concentration, and catalysts, if any). The reaction rate, - r , . usually depends on the concenrration of the reacting species raised to some power. Consequently, to determine the reaction rate as a function of conversion X, we need to know the concentrations of the reacting species as a function of conversion.
3.5.1 Equations for Batch Concentrations
The concentration of A is the number of moles of A per unit volume: Batch concentration
C, = N 1 V After writing similar equations for B. C, and D. we use ;the stoichiometric table to express the concentration of each component in terns of the conversion X:
Sec. 3.5
103
Batch Systems
We further simplify these equations by defining the parameter O, , which allows us to factor N,, in each of the expressions for concentration:
C, =
we
to
ohlain C, = I I , ( X ) .
[OD+ ( d / a ) X ] NDO , with OD=
v
-
NAQ
1% now need only to find volume as a function of conversion to obtain the species concentration as a function of conversion.
3.5.2 Constant-Volume Batch Reaction Systems Some significant simplifications in the reactor design equations are possible when the reacting syrtem undergoes no change in volume as the reaction progresses. These systems are called constant-volume. or constant-density, because of the invariance of either voIume or density during rhe reaction process. This situation may arise from several causes. In gas-phase batch systems, the reactor is usually a sealed constant-volume vessel with appropriate instsuments to measure pressure and temperature within the reactor. The volunle within this vessel is fixed and will not change. and is therefore a constant-\.olume system ( V = V,,). The laboratory bomb calorimeter reactor i s a typical example of this type of reactor. Another example of a constant-volume gas-phase isothermal reaction occurs when the number of moles of products equals the number of moles of reactants. The water-gas shtft reaction. important in coal gasification and many other processes, is one of these:
In this reaction, 2 mol of reactant forms 2 mol of product. %'hen the number of reactant molecules forms an equal number of product molecules at the s m ~ i e temperature and pressure, the volume of the reacting mixture will not change if the conditions are such that the ideal gas law is applicable. Qr if the compressihiliry factors of the products and reactants are appmximateIy equal. For liquid-phase reactions taking place in solution. the solvent usually dominates the situarron. As a rewlt. changes in the denrity of the rolute do not
104
Rate Laws and Stolchiometry
Chaw
affect the overall density of the solution significantly and therefore it is essen tially a constant-volume reaction process. Most liquid-phase organic reaction do not change density during the reaction and represent still another case t which the constant-volume simplifications apply. An important exception t this general rule exists for polymerization processes. For the constant-volume systems described earlier, Equation (3-25) ca be simplified to give the following expressions relating concentration and con version:
Concentration as a function of conversion when no volume chanee occurs with reactron
l-----l
f ( N ~ ~ /-N(b/a)Xl~ ~ ) - 40 [@,-(blalXl = & ( O B - : X ) CB= NAo
[ ( N , , / ~ , , 3 + (c/a)Xl CC= NAO yo
To summarize for liquid-phase reactions (or as we will soon see for isotherma and isobaric gas-phase reactions with no change in the total number of moles) we can use a rate law for reaction (2-2) such as -r, = kACACBto obtair - r , =AX)% that is,
-r , x
vn
v~
(
= kCACB=
kc:*(1 - X)
Substituting for the given parameters k. CAO,and OB,we can now use the tech niques in Chapter 2 to size the CSTRs and PFRs for liquid-phase reactions. Example 3-2
Expressing
= hj(X) for a Liquid-Phase Reaction
Soap consists of the sodium and potassium salts of various fatty acids such as oleic stearic, patmitic, lauric, and my~isttcacids. The saponification for the formation o soap from aqueous caustic soda and glyceryl stearate is
Letting X represent the conversion of sodium hydroxide (the moles of sodiurr hydroxide reacted per mole of sodium hydroxide initially present), set up a stoichio metric table expressing the concentration of each species in terms of its initial con centration and the conversion X.
Sec 3.5
105
Batch Systems
Because we are taking sodlurn hydroxide as our basis, we divide through by the stoichiometric coefficient of sodium hydmx~deto put the reaction expression in the form Choosing a bacis of calculation
We may then perform the calculations shown in Table E3-2.1. Because this ir a liquid-phase reaction, the density p is considered to be constant; therefore, V = I.',.
TABLE E3-2.1. S~Y)ICHIOMETRICTABLEFOR Specie1
Symbol Initially
NaOH
A
Water (inen)
I
NAO
LIQUID-PHASE SOAPREALTION
Chan~e
Remaining
Concentration
-maax
N~o(l-x)
C~o(1-x)
Stoichiomerric table (batch)
/
%
-
AfT,,
0
Nto NT= NTO
&lo
Example 3-3 What i s Be Limiting Reactant? Waving set up the stoichiometric table i n Example 3-2, one can now readily use it to calculate the concentrations at a given conversion. If the initial mixrute consists solely of sodium hydroxide at a concentmtion of 10 rnol/dmJ (i.e., 10 rnol/L or 10 kmollrn3 5 and of plyceryl stearate at a concentration of 2 molldm3, what is the concentration of glycerine when the co'nversion of sodium hydroxide i s (a) 20% and (b) 90%?
[
Solution
Only the reactants NaOH and (Cl,H35COO)3C3FE5are initidly present: therefore. 0,= 0,= 0.
Rate Laws and Stoichiometry
1
Chap. 3
(a) For 20% conversion of NaOH:
(h) For 909 conversion of NaOH:
]
The bass or calculal'on chould he Ihe limirinf
reaclant.
Let us find C,:
Oops!! Negative concentration-impossible! What went wrong? Ninety percent conversion of NaOH is not possible. because glyceryl stearate is the limiting reactant. Consequently, all the glyceryl stearate i s used up before 90% o f the NaOH could be reacted. It i s irnprtant to choose the Ilrnlting reactant as the basis of calcuIation.
3,6 Flow Systems The form of the stoichiometric table for a continuous-flaw system (see Figure 3-5)is vir~uallyidentical to that for a batch system (Table 3-3) except (hat we replace N!o by q;:,and N, by F, (Table 3-4). Taking A as the basis. divide Equation i2-1) through by the stoichiornetric coefficient of A to obtain
Entering
F~~
1
Figure 3-5 Flou- reactor.
Sec. 3.6
F ~ e dRate lo Reacfox
Species
(molltime)
Change wirhin Reactor (molltirm)
A
FA0
- FAOX
B
F~~ = @ B ~ A O
--b FAOX
C
Fco " @cFm
:
Sro~chiometric table (Row)
107
flow Systems
Eflueni Rote fmm Reocro r (mu1{time) FA = FA0( I
- X)
a
FAOX
where
and Bc, OD. and 8,are defined similarIy. 3.6.1 Equations for Concentrations in Flow Systems
For a flow system, the concentration C, at a given point can be determined from the molar flow rate F A and the volumetric flow rate v at that point: Definition of concentration for a flow system
Units of u are typically given in terms of liters per second, cubic decimeters per second, or cubic feet per minute. We now can write the concenrrations of A, B, C, and D for t h e general reaction given by Equation (2-21 in terms of their respective entering molar flow rates (F,,, F R O .F,,. F,,), the conversion X. and the volumetric flow rate, v .
108
Rate Laws and Stoichiometry
Chap.
:
3.6.2 Liquid-Phase Concentrations
For liquids, volume change with reaction is negligible when no phase changer are taking place. Consequently, we can take
For IfquidS Then Cq=C*o(I- X I
Therefore, for a given rate law we have -rA = (X)
C,=C,,
:I
(@,--X
etc.
Consequentiy, btsing arty one oj' the rare lows in Port I of this chapr~l;cve con now Jnd -r, = A X ) for liquid-phase reactions, However, for pas-phase reactions the volumetric flow rate most often changes during the course of the reaction because of a change in the total number of moles or in temperature or pressure. Hence, one cannot always use Equation (3-29) to express concentration as a function of conversion for gas-phase reactions.
3.6.3 Change in the Total Number of Moles with Reaction in the Gas Phase
In our previous discussions, we considered primarily systems in which the reaction volume or volumetric flow rate did not vary as the reaction progressed. Most batch and liquid-phase and some gas-phase systems fall into this category. There are other systems, though, in which either V or u do vary. and these will now be considered. A situation where one encounters a varying flow rate occurs quite frequently in gas-phase reactions that do not have an equal number of product and reactant moles. For example, in the synthesis of ammonia,
4 rnol of reactants gives 2 mol of product. In ffow systems where this type of reaction occurs, the molar flow rate will be changing ar the reaction progresses. Because equal numbers of moles occupy equaI volumes in the gas phase at the same temperature and pressure. the volumetric flow rate wilI also change.
Another variable-volume situation, which occurs much less frequently, is in batch reactors where volume changes with time. Everyday examples of this situation are the combustion chamber of the internal-combustion engine and the expanding gases within the breech and barrel of a fiream as it is fired. In the stoichiometric tables presented on the preceding pages, it was not necessary to make assumptions concerning a volume change in the first four coIumns of the table (i.e.. the species, initial number of moles or molar feed
See. 3.6
109
Flow Systems
rate, change within the reactor, and the remaining number of moles or the molar effluent rate). All of these columns of the stoichiometric table are independent of the volume or density. and they are irfenticnl for constant-volume (constant-density) and varying-volume (varying-density) situations. Only when concentration is expressed as a function of conversion doer variable dens~ty enter the picture.
Batch Reactors with Variable Volume Although variable volume batch reactors are seldom encountered because they are usually solid steel containers. we wiIl develop the concentrations as a function of conversion because (1) they have been used to collect reaction data for gas-phase reactions, and (2) the development of the equations that express volume as a function of conversion w ~ i lfacilitate analyzing flow systems with variable volumetric flow rates. Individual concentrations can be determined by expressing the volume V for a batch system, or volumetric flow rate v for a flow system, as a function of conversion using the following equation of state: PV = ZN,RT
Equation of state
(3-30)
i n which V = volume and N , = total number o f moles as before and
T = temperature. K P = total pressure, atm &Pa; t atm = 101.3 kPa) Z = compressibility factor R = gas constant = 0.08206 dm" aatmtmol K
-
This equation is valid at any point in the system at any time t. At time r = 0 (i.e., when the reaction is initiated). Equation 13-30) becomes
Dividing Equation (3-30) by Equation (3-31) and rearranging yields
We now want to express the volume V as a function of the conversion X. Recalling the equation for the total number of moles In Table 3-3,
where
S = Change in total number of moles Mole of A reacted
110
Rate Laws and Stoichiometry
Chap. 3
We divide Equation (3-33) through by N,:
Then
'T =I NTQ R c [ a l i ~ n s h between i~ 8 and e
+EX
where yAo is the mole fraction of A initially present, and where E = ~ + n - ~ - ~ N, ) ~ = J b A o ~
Equation (3-35) holds for both batch and flow systems. To interpret rearrange Equation (3-34)
Interwiati*n of
(3-34)
(3-351
E,
let's
at complete conversion, (i .e.. X = I and N , = NTf)
- Change in total number of moles for complete conversion Total moles fed
If all species in the generalized equation are in the gas phase. we can substitute Equation (3-34) with Equation (3-32) to arrive at
In the gas-phase systems that we shall be studying, the temperatures and pressures are such that the compressibility factor will not change significantly during the course of the reaction: hence Z,=Z. For a hatch system, the volume of gas at any time I is Volume of gas for a variable volume batch reaction
Equation (3-38) applies only to a 1~arinble-1~r)fu?nr hatch reactor, where one can now substitute Equation (3-38) into Equation (3-25) to express r, =PX). HOWever, if the reactor is a rigid steel container of constant volume, then of course
Sac. 3.6
Ill
FIOW Systems
V = V,. For a constant-volume container, V = I],, and Equation 13-38] can be used to calculate the gas pressure inside the reactor as a function of temperature and conversion.
Flow Reactors with Variable Volumetric Flow Rate. An expression sirnifar to Equation (3-38) for a variable-volume batch reactor exists for a variable-volume fiow system. To derive the concentrations of each species in terns of conversion for a variable-volume flew system, we shall use the relationships for the total concentration. The total concentration, CT. at any point in the reactor is the total molar flow rate, 6, divided by volumetric flow rate v [cf. Equation (3-27)J. In the gas phase, the total concentration is also found from the gas law. Cr = P E R 1 Equating these two ~Iationshipsgives P ZRT
c,=F'=v
At the entrance to the reactor,
Taking the ratio of Equation (3-40) to Equation (3-39) and assuming negligible changes in the compressibility factor, we have upon rearrangement
We can now express the concentration of species j for a flow system in terms of its flow rate, 5, the temperature, T, and total pressure. P.
Use t h ~ sform for membrane reactop (Chapter 4) and for rnuEiipfe reaction.:
(Chapter 61
The total moiar flow rate is just the sum of the molar flaw rates o f each of the species in the system and is
112
Rate Laws and Stoichiometry
Chap
One of the major objectives of this chapter is to learn how to express any giv rate law - r , as a function o f conversron. The schematic diagram in Figu 3-6 helps to summarize our discussion on this point. The concentration of tl key reactant. A (the basis of our calculations), is expressed as a function conversion in both flow and batch systems, for various conditions of temper ture. pressure, and voIume.
Flow
v
NO Phase Change
J
NO Phase Change OR
NO Sern~penneableMembranes
4
Isothermal
Ifr Neglect Pressure Drop c, =
c*,(e, - $ x ) 1+EX
Flgure 3-6 Expressing concentration as a function of conversion.
Sec. 3.6
113
Flow Svsterns
not used in this sum. The molar flow rates, F,, are found by solving the mole balance equations. Equation 11-42) wiIl be used for measures ohher than conversion when we discuss membrane reactors (Chapter 4 Pan 2) and multiple reactions (Chapter 6). We wiil use this form of the concenfration equation for multiple gas-phase reactions and for membrane
We see that conversion is
reactors.
Now let's express the
concentration in terms of conversion for gas flow systems. From Table 3 4 the total niolx Row rate can be written in terms of conversion and is
FT = F f f l+ F,408 X
13-43}
Substituting for F , in Equation (3-41) gives U
= Uo
FTO+ f,, 5X P, T
Fm
(F)E
Gns-phase volumetric flow
rate
The concentration of species j is
The molar flow rate of species j is
where v, is the stoichiometric coefficient, which is negative for reactants and positive for products. For example, for the reaction
v, = -1,
v, = -bla, v, = c / a . v D = d / a , and O j = FplFAw
Substituting for v using Equation (3-42) and for F,, we have
114
Rate Laws and Stoichiometry
C h a ~ 3.
Rearranging Gasphase concmtration as a
function of conversion
C ,,
Recall that yAo= F,,/F,,, E=
= y,,Cm,
and
E
from Equation (3-35)(i.e.,
?'A06).
The stoichiornettic table for the gas-phase reacrion (2-2) is given jn Table 3-5.
v
Wc now have C, = h , ( X , and -r,= R(X)
- FC - F,, I(-), + ( r l n ) X l u
c - y
for variable-volume gas-phase reaction<
-FD -F4~[8~C(d!u)Xl D--
-
( TO
FA(, + ( I . / N J X I V ~ ( I + ~ X )
)
=
@ , + ( r / a ) X T, p I+EX
Q, f ( d l a ) X
)F[g) 7, p
L'
Exumple 3 4 Maniprlatiotr of the Equation for
1
P
C, = hj (XI
Show under H hat condirions and manipulation the expression for CB for a gas Row system reduces to that given in Tahle 3-5. Soll~tinrr
For a flow system the concentration is dcji~~cd as
From Tahle 3-3, the molar Row raIe and conversion are related by
Cornhininy Eqi~ations(E.1-4.1) and (E3-4.2) yields
Sec. 5.6
This equalion for u is only for a gasphase reaction
II
Flow Systems
Using Equation (3-45)gives us
to substitute for the volumetric flow rate gives
(
which is identical to the concentration expression for a variable-volume batch reactor.
I
Example 3-5 Determining Cj = hi(XIfor a Gas-Phase Reaction A mixture of 28% SO, and 72% air is charged to a flow reactor in which SO, is oxidized.
2so2+ 0, ----4 2 S 0 , First. set up a stoichiometric table using only the symbols (i.e.. O , , F , ) and then pzpare a second stoichiometric table evaluating numericalry as many symbols as possible for the case when the total pressure is 1485 kPa (14.7 atm) and the temperature is constant at 227'C.
-
Taking SO: as the basis of calculation. we divide the reaction through by the stoichiometric cmfficient of Our chosen basis of calculation:
I
+
SOz f02
SO?
The initial stoichiometric table is given as Table E3-5.1. Initially, 72% of the total number of moles i s air containing (21% O2 and 79% N 2 ) along with 2 8 8 SO?.
From the definition of conversion, we ~ubstitutenot only for the molar flow rate of SO, ( A ) in tenns of conver~ionhut a150 for the volumetric flow rate as a function of conversion.
116
Rate Laws and Stoichiometry
Species
Smhf
so2
A
FA,
-FAUX
FAF,=F,,,(l-X)
SO,
C
0
+FAfiX
Fc = FAOX
Initially
Chmge
Chap. ;
Remnining
Recalling Equation (3-451,we have
NegIecting pressure drop.
Neglecting pressure drop in the reaction. P = P,, yields
P = Po
If the reaction is also carried out isothermally. T = T o . we obtain
isotherma[ operation, T = To
The concentration of A initially is equal ta the mole fraction of A initially multiplied by the total concentration. The total concentration can be calculated from an eauation of state such as the ideal gas taw. Recall that y~~ = 0.28, To = 500 K. and Po= 1485 Wa.
Sec, 3.6
Flow Systems
I
The total concentration i s
I
We now ewlunte e .
.
The concentntions of different species at various conversions are calculated in Table E3-5.2 and plotted in Figure E3-5.1. Note that the concentntion of N2 is changing even though it is an inert species in this reaction!! TABLE E3-5.1.
COXCESTRATIOY 45 A F U Y ~ I O OFN CONVERSIOS
C, (molldm') Species
SO, The concentration of the inert is nor constant!
X=O.O
a?
C, = C, =
SO,
C, =
0.100 0.054 0.000
X=0.25 0078
0.043 0.026
X=0.5
0.054 0.031 0.054
X=8.75
X=
1.0
0.028
0.000
0.018
0.005 0.1 16
0.084
We are now in a position to express - r , as a function of X.For example, ifthe nte law for this reaction were first order in SO, (i.e., A) and in 0, (i+e., S), with k = 280 dm3/mol s , then the rate law becomes
-
118
Rate Laws and Stoichiometry
Chap. 3
Nore: Because the voiurnetic Row rate varies with conversion, the
concentration of inert5 (N2) i s not
constant.
Now use techniques presented i n Chapter 2 to size reaclors.
Figure E3-5.1 Concentration as a function of conversion.
-1
Taking the reciprocal of - r , yields
1 Need to first cllrculate xr
We see that we could ~itea variety of combinations of i s o r h e m i reactors using the techniques discussed in Chapter 2.
Thus far in this chapter, we have focused mostly on irreversible reactions. The procedure one uses for the isothermal reactor design of reversible reactions is virtually the same as that for irreversible reactions. with one notable exception. First calculate the maximum conversion that can be achieved at the isothermal reaction rernperature. This value is the equilibrium conversion. In the following example it will be shown how our algorithm for reactor design is easily extended to reversible reactions.
I
Example 3-6 Calcuhting the Equilibrium Conversion The reversible gas-phase decomposition of nitrogen tetroxide, N,O,. to nitrogen dioxide, NO2,
Sec. 3.6
f19
flm Systems
is to be carried out at constant temperature. The feed consists of pure NzO, at 340 K and 202.6 kFa (2 a m ) . The concentration equilibrium constant. Kc. at 340 K is 0.1 molldm". (a) Calculate the equilibrium conversion of N,O, in a constant-volume batch reactor. (b) CalcuIate the equilibrium conversion of N20, in a flow reactor. Assuming the reaction is elementary, express the rate of reaction soleIy as a (c) function of conversion for a Row system and for a batch system. Id) Determine the CSTR volume necessary to achieve 80% of the equilibrium conversion.
At equilibrium the concentrations of the reacting species are relaled by the relationship dictated by thermodynamics [see Equation (3-10) and Appendix C]
(
(a, Batch system-constant
volume, V = Y o .See %Me E3-6.1.
For batch systems C, = N, / V , Living Example Problea
C - !'~oPo
*'
RT,,
( I ) ( ? atm) (0.082 atm.dm3/rnol +K)(340K)
,
At equilibrium. X = J,,.and we substitute Equations (E3-6.2) and (E3-6.31 illto Equation (E3-6. I ) .
120
Rate Laws and Stoichiometry
Chap.
We will use Polymath to solve for the equilibnum conversion and let xeb repfese the equilibrium conversion in a constant-volume batch reactor. Equation (E3-6.m written in Polymath format becomes
f (xeb) = xeb - [kc*(l
- xeb)/(?*cao) J
"0.5
The Polymath program and solution are given in Table E3-6.1.
When looking at Equation (E3-6.4). you probably asked yourself. "Why not use tl quadratic formula to solve for the equilibnum convesston in both batch and flo syrterns?' That is,
Batch:
There is a PoIymath tutorial in the summary Notes of Chapter 1
I X, = -[(-I + JF+ l6CAOIKc)/(CA,JK c ) ] 8
Flow: X, =
[(E- I ) +J(G-
I ) ~ * ~ I F + ~ C ~ ~ / K ~
2 ( +~4 C A o l K c )
The answer is that future problems will be nonlinear and require Polymath solution: and I wanted to increase the reader's ease in using Polymath. TABLE E3-6.2.
POLYMATH
PROGRAM AND
miable Xeb Xe f Kc
SOLLTEOY FOR BOTHBATCH4x0 FLOWSYSTEMS
f (XI
4.078E-08 2.622E-10
Ini
G u w
0.5 0.5
0.1
NLES Report Wenewt) Nonlinear equations .I: f(Xeb) = Xeb-(Kcm(l-Xeb)l(4"Cao))"O.S= 0 i i : f(Xef) = Xef-(Kc'(1-Xef)'(t+eps*Xef)I(CCao)~.S = O
Explicit equations i - ] Kc=O.l if.;Cao=0.07174 : 3 ; eps= 1
The equilibrium conversion in a constant-volume batch reactor is
Sec. 3.6
R,lymath ~
121
Flow Systems
Nore: A ~tutorial~of Polymath can ~ i ~ be lfound in thc summary notes uf Chapter I .
~
Chapter I
(b) Flvw system. The stoich~ometrictable is the same as that for a batch s y ~ t e m except that the number of moles of each species, .V,, is replaced by the tnofar flow rate of that species. F,. For constant temperature and pressure, the volumetric f f o ~ rate is u = v o / l + e x ) , and the resuIting concentrations of species A and B are
At equilibrium, X = X,.and we can substitute Equations (E3-6.5) and (E3-6.6) into Equation (E3-6.1) to obtain the expression
I
Simplifying giver
1
Rearnnging to use Polymath yields
For a flow system with pure N,O, feed, e = y , ~6, = 1 (2 - I ) = I . We shall let Xef represent the egullibriurn conversion in a flow system. Equation (E3-6.8) written In the Polymath format becomes
I
f(Xef) = Xef
- [kc*(l - Xef)*[ l + eps*Xe~/4/cao]"0.5
This solution is alro shown in Table E3-6.2 (X,,= 0.51). Note that the equilibrium conversion in a flow reactor tine..X, = 0.5 1 ). with negligible pressure drop, is greater than the equilibrium convenion in a constant-volume batch reactor (X, = 0.44 ). Recalling Le Chitelier's principle, can you suggest nn explanation for this difference in X, ? (c) Rate laws. Assuming that the reaction fallows an eIementar)r rate law, then
]
1. For a constant volume ( V = Vo)batch system.
122
Rate Laws and Stokhiomatry
Chap. 3
Here C, = N A I V, and C, = & 1 Vo. Substituting Equations (E3-6.2) and (E3-6.3) into the rate law, we obtain the rate of disappearance of A as a function of conversion:
2. For a flow system. Here C, = FAlv and Ca = Fs/u with v = vU ( I t. m. Consequently. we can substitute Equations (E3-6.5) and (E3-6.6) ~ntoEquation (B-6.9) 10 oblain - r , = J - ( X ) for a flow reactor
As expected, the dependence of reaction rate on conversion for a constantvolume batch system [i.e., Equation (E3-6.lo)] is different than that fur a Row system [Equation (E3-6.11)] for gas-phase reactions. I f we substitute the values for CAO,KC, E, and k = 0.5 min-' in Equation (E3-6.11). we obtain -rA solely as a function o f X for the flow system.
We can now form our Levenspiel plot. W e see (I/-r,) goes to infinity as X approaches X,.
I
Figure E3-6.1 Levencpiel plot Tor a flow %!\tern.
5ec. 3.6
123
Flow Systems
(d) CSTR volume. Just for fun let's calcuIate the CSTR reactor voIume necessary to achieve 80% of the equilibrium conversion of 50% {i.e., X = 0.8X,) X = 0.4 for a feed rate of 3 rnollmin.
1
The CSTR volume necessary to achieve 40% conversion is 1.7 1 m3.
Closure. Having completed this chapter you should l x able to write the rate law in terns of concentration and the Arrhenius temperature dependence. The next step is to use the stoichiometric table to write the concentrations in terms of conversion to finally amve at a relationship between the rate of reaction and conversion. We have now completed the fiat three basic building blocks in our algorithm to study isothermal chemical reactions and reactors. The CRE Algorithm mole Balance. Ch 1 .Rate Law. Ch 3
S toichiornetry
mSto~chiorneiry. Ch 3 *Combine. Ch 4 'Evaluate, Ch 4 ' E n e ~ yBalance. Ch 8
Rate Law
I
Mole Balance
1)
In Chapter 4. we will focus on the curnbine and evaluation building blocks which will then complete our algorithm for isothermal chemical reactor design.
Rate Laws and Stoichiometry
Chao.
SUMMARY
PART 1 1. Relative rates of reaction for the generic reaction:
The relative rates of reaction can be written either as
1. Re(rc~criono d r is determined from experimental observation:
The reaction in Equation (S3-3)is a order with respect to species A and order with respect to species B , whereas the overall order, n. is a + p. Rea tion order is determined From experimental observation. XF a = I and P = we would say that the reaction is first order with respect to A, second ord with respect to B, and overall third order. We say a reaction follows an el m e n t q sate law if the reaction orders a g m with the stoichiornetric coeA cients for the reaction as written. 3. The temperature dependence of a specific reaction rate is given by the Arrh nius equntion,
where A is the frequency factor and E the activation energy. If we know the specific reaction rate, k, at a temperature, To, and the acl vation energy, we can find k at any temperature. T,
Similarly from Appendix C, Equation (C-9),if we know the equilibrium co stant at a temperature, TI. and the heat of reaction, WF can find the equilil riurn constant at any other temperature
Summary
Chap. 3
PART 2 4. The sroichiomerric table for the reaction given by Equation 6 3 - 1 ) being carried out in a flow system is Species
Entering
F~
A
having
Chunge
- F~&f
where
FAo(1
6=
- A')
d
e
b
5. In the case of ideal gases, Equations (S3-6) and (S3-7) relate volume and voIumetric flow rate to conversion. Batch constant volume: V= (S3-6,
vo
Flow systems:
T
Gas:
Liquid:
v = u,
For the general reaction given by (S3-11, we have
6 = Change in total number of moles Mole of A reacted Definitions of 6 and E
and
e=
Change in totaI number of moles for complete conversion Total number of moles fed to the reactor
t26
Rate Laws and Stoichiometry
Chap. 3
6. Far pas-phase reactions, we use the definition of concentration (C, = FJu) along with the stoichiometric table and Equation {S3-7) to write the concentration of A and C in terms of conversion.
Q,irh
@,
=
& - 'c" - 3
F A , CAO ! . ~ n 7. For incompressible liquids. the concentrattons of species A and C i n the reaction gtven by Equatlon rS3-1) can be writ~enas
c, = C,
i
..I
Oc + - X
FZuations (S?- 17 ) and (S3-13) also hold for gas-phase reactions carried out at constant volume in batch systems 8. I n terms ul' pas-phace molar Row rates, the concentration of species r is
-
F To Ci = C," Fi FT" Po T CD-ROM MATERIAL Learning Resuurce 1 . Sun119ran' N o ~ fc~r s Cliupicr 3
2 We11Mvdtt/e.~ summa^:, Nmes
A. Cooking a P o ~ u The chemical reaclion eng~ncrringis 3ppIred Fa cookin: a polato
k Starch (c~stalline)-+Starch amorphous
Chap. 3
CD-ROM Material
127
3. MolecuIar Reaction Engineering Molecular simulators (Spiman. Ceriuq21 are used to make predictions of the activation energy. The fundamentals of density fi~nctionalare presented along with specific examples.
3. Inreracfi1,eCrnrprrrrr Modrrler A. QUIZ Show I1
Solved 7rob!ems
Snh~rdPmhler~ir A. CDP.1-A, Actirnt~nnEnerg! h r a Beetle Pw
Rate Laws and Stoichiometry
Char
Professional Reference ShcIf R3.1. Coiii~iotiThmy In h i s section, the fundamentals of collision theory
Schematic of collision cross wclion
are applied to the reaction A+B+C+D to arrive at the following rate law
The activation energy, E,, can be estimated from the Pofanyi equation R3.2. Transition Srclre Theoy
In this section, the nte law and rate Iaw pmmeters are derived for the reacti
using transition state theory. The Figure P3B-1 shows the energy of the ma ecules along the reaction cmrdinate which measures the progress of the reactic
Figure P3B-I Reaction coordinate for (a) SH2reaction,and (b) generalized reaction. (c) 3-1 energy surhce for generalized reaction.
We will now use statistical and quantum mechanics to evaluate KACto arri at the equation
m
Chap. 3
CD-ROM Material
129
where q' r:, overnIl the partilion function per unit volume and i< the product or tranklational, vibrat~on.rotational. and electnc partition functions: that is. 4' = q',q'vq',q',
The indivrdual panitron Functions to bc evaluated areS Translatiou
Reference Shelf
Rotation
The Evring Fquatioq
R3.3. MokcuIur Dynamics The reaction trajectories are calculated to determine the reaction cross section of the reacting molecules. The reaction probability is found by counting up the number of reactive: trajectories after Karplus."
Nonreactive Traiectom
R. Masel. Chemical Kinetics: (New York McGmw Hill, 2002), p. 594. M. KarpIus. R.N. Porter. and R.D. S h m a , 1 Chem. Phys., 43 (9).3259 (3965),
130
Rate Laws and Stoichiometry
Chap. 3
.-.-. .-' Reference Shelf '1
t~me
From these trajectories. one can cnlcufate the following reaction crosq section. ST. shown for the case where both the tibrational and rotarronal quantum nuinhers are zero:
The specific reaction m e can then he calculated fmm first principle for simple mnlecules.
R3.4. ,I.IEn~icresOrher TI~otlCnrn.eniorr Gut P l r r r ~ p OVorr: T h i s toptc will be covered in Chapter 4 but for r h o ~ ewho want to use 11now. look un the CD-ROM.) For membrane reactors and gas-phase rntlltiple reactions. it is much more cc~nvenientto work In term5 of the number of molec bV4. hrT,)or molar Row rates (FA.F,, etc.) rather than con\er\inn.
R3.5. Rcurtiotrs
~ 7 1 1 1Co~t(irt~ \(rtio~t We now concider a pas-pira~e reaction in which uundensntron occurs. An example of thic clav of reaclions is
Here we uill deielnp our \toichiometric table for reaction5 nith phase chanre When one nf (he product5 conrlene:. durrnp the c o i t r ~of a reaction. ciilculaticln of the change In inlumc n t ~olumctricAON raw nutht be undcrtnhen in a \ l ~ g h ~ lryl ~ t l ' c r cmanner. ~ ~ ~ Plois of the lnular Ron mlc< of cnndcn\.!te D I h t tol;il, togelher nlih tht' recrprt~alrate. are huwn hcre a\ :I fu~lutinnof cnnir.r\ioi~
Chap. 3
@ :---,/: ,'
Ouestions and Problems
QUESTIONS AND PROBLEMS The subscript to each of the problem numbers indicates the level of difficulty: A, least difficult; D. most difficult.
Ciomcwork 7-oblems
A=.
B=.
C = + D=++
List the impmiant concepts that you learned from this chapter. What con-
cepts are you not clear about? Explain the strategy to evaluate reactor design equations and how this chapter expands on Chapter 2. Choose a FAQ from Chapers 1 through 3 and say why it was the most helpful. Listen to the audios on the (23. Sclect a topic and explain it.
a
Creatlvs Thinking P3-2,
Read through the ~ e l f ~ c sand t s Self Assessments for LRctures 1 through 4 on the CD-ROM.Select one and critique it. (See Preface.) (0 Which example on the CD-ROM Lecture nnres for Chapters I throuyh 3 was most helpful? Is) Which of the JCMs for the first three chapters was the most fun? (a) Example 3-1. Make a plot of k versus T. On this plot also sketch k versus f l l 7 l for E = 240 kllmol, for E = 60 kllmol. Write a couple of sentences describing what you find. Next write a paragraph describtng the actlvation, how it affects chemical reaction rates. and what its origins are? Example 3-2. Would the example be correct if water was considered an ~nert?
Example 3-3. How would the answer change if the initial concentration or glyceryl sterate were 3 molldm3? Example 3-4. What is the srnallesr value of QB = (MBl,dNA,)for which the concentration of B will no, become the Iimiting reactant? Example 3-5. Under what conditions will the concentration of the inen nitrogen be constant'?Plor Equation (E3-5.2) of (I/-r,) as a function of X up to value of X = 0.94. What did you find? Example 3-6.Why is the equilibrium conversion lower for the batch sysrem than the flow system? Wlll this always be the case for constant volume batch systems? For the case in which the total concentrarion Cm is to remain constant as the inerts are varied, plot the equilibrium conversion as a function nf mole fraction of inerts for both PFR and a constant-volume batch reactor. The pressure and temperature are constant at 2 atm and 340 K. Only N:O, and inen 1 are to he fed. Collision Theory-Profeqsional Reference Shelf. Make an outline of the steps !hat were u ~ e dto derive
a
(h) The rote law for the reaction (7.4 + B -+ C) i s - r ~= k , ~ :C, with k, = 1S(drnYmol)~ls. What arc k , and A,?
(i) At low temperatures the rate law for rhc reartian (;A
+ ;B +C ) ir
- r , = kACACR. If the reaction is reversible at htgh temperatures, what is Klncur r L'hallrnpc I I
P3-3, ?#
XRI
?IXI
the rate law? h a d the interactive Computer Mudthe (ICMI Kinetrc Challenge from the CD-ROM. R u n the moduIc. and then record yot~rperformance rlumber for \he lnodule uhich indrcate\ your mastering of the mater~nl.Your professor has
f
32
P3-4,
Rate Laws and Stoichlometry
Chap.
the key to decode your performance number. ICM Kinetin Chnlleng Performance The frequency of flashing of fireflies and the frequency o f chirping of cricket as a funchon of temperature follow [J. Clrc~l~. ELILI~ .. 5. 333 ( 1972) Reprinte by permission.].
For fireflies:
T ("C)
21.0
25.0
300
Ear crickets:
The running speed of ants and the flight speed of honeybees aa a function c temperature are given below [Source: B. Heinrich, Thr Hot-Bloorlrd Insecl (Cambridge, Mass.: Harvasd University Press. 1993)j.
For ants:
For honeybees:
T ("C) V (cds)
30
35
30
0.7
1.8
3
1
in common? What are their cliffel ences? (b) What i c the velocity of the honeybee at 40°C?At -YC? (c) Do the bees, ants, crickets. and fireflies have anything in common? If st what is it? You may also do n pairwise comparison. (d) Would more data help clarify the relationships among frequency, speec and temperature? If so. in what temperature rhould the data be obtained Pick an insect, and expIain how you would carry out the experiment t obtain more data. [For an alternative to thi? problem. see CDP3-A,.] Troubleshooting. Corrosion of high-nickel stamless steel plates was found t occur in 3 distillation column used at DuPont to separate HCN and water. Sur furic acid i s always added at the top of the column to prevent polyrnerizatio of HCN. Water collects at the bottom of the coiumn and HCN at the top. Th amount of corrosion on each tray is shown in Figure P3-5 as a function r plate location in the column. The bottom-most temperature of the column is approximately 125"l and the topmast is IWC. Tne cormsion rate is n function of temperature an the concenrration of an HCN-H2S04 complex. Suggest an explanation for th observed corrosion plate profile in the column. What effect wouid the colurn operating conditions have on the corrosion profile? (a) What do the firefly and cricket have
P3-5,
2.5
Chap. 3
Questions and Problems
Too
I
HCN
b$o,
mills
Figure P3-5
P3-6,
P3-7,
Inspector Sgt A m k c m b y of htland Yard. It is believed, although never proven. that Bonnie murdered her first husband, Lefty, by poisoning the tepid brandy they drank together on their first anniversary. Lefty was unware she had coated her glass ~ i t an h antidote before she fiIled both grasses wrth the poisoned brandy. Bonnie married her second husband, Clyde, and some years later when she hod tired of him. she called him one day to tell him of her new promoziofi at work and to suggest that they celebrate with a glass of brandy that evening. She had the fatal end in mind for Clyde. However. Clyde sug. gested that instead of brandy. they celebrate with ice cold Russian vodka and they down i t Cossack style, in one gulp. She agreed and decided to follow her prev~ouslysuccessful plan and to put the poison in the vodka and the antidote in her glass. The next d ~ y both , were found dead. Sgr. Amhercromby arrives. What were the first three questions he asks? What are two possible explanations? Bnsed on what you learned from this chapter. what do you feel Sgt, Arnbercmmby suggested as the most logical explanation? [Professor Flavio Marin FIores, ITESM. Monterrey, Mexico] (a) The rule of thumb that the rate of reactran doubles for a 10°C increase in temperature occurs only at a specific temperature for a given activation energy. Develop a relationship between the temperature and activation energy for ~ h i c hthe rule of thumb holds. Nglecr any variation of concentration with tempenlure. (b) Determine the activation e n e w and frequency factor from the followtng data:
(c) Write a paragraph explaining activation energy, E. and how it affects the
P3-8,
chemical reaction rate. Refer to Section 3.3 and especially the Professional Reference Shelf sections R3.1, R3.2. and R3.3 if n e c e s s q Air bags contain a mixture of NaN,. KNO,, and SiO?. When ignited. the folTowing reactions take place: Z N a , 4 2Na + 3?1? (1) (2) lONa + 2 m O 3 + K20-+ 5Na20 + N, ( 3 ) K,O + Na,O + SiO, + a h l i n e silicate glass
134
Rate Laws and Stolchiometry
Chap. 3
Reactions (2) and (3) are necessary to handle the toxic scdjum reaction product from detonation. Set up a stoachiome?ric table solely in terms of NaN, (A), KNOj (B), etc., and the number of moles initially. IF 150 g of sodium azide are present in each air bag, how many grams of KN03 and SiOz must be added to make the reaction products safe In the form of alkaline silicate glass after the bag has inflated. The sodium azide is in itself toxic. How would you propose to handle all the undetonatd air bags in cars piling up in the nation's junkyards. P3-9, Hot Potato. Review the "Cooking a Potato" web module on the CD-ROM or on the web. (a) It took the potato described on the web 1 hour to cmk at 350°F. Builder Bob suggests that the potato can be cooked in half that time if the oven temperature is raised to 60Q°F.What do you think? (b) Buzz Lightyear says, "No Bob," and suggests that it would he quicker to boil the potato in water at 100°C because the heat transfer coefficient is 20 times greater. What are the rradeoffs of oven versus boiling? (cj Ore Ida Tater Tots is a favorite of one of the ProctorslGraden in the class, Adam Cole. Tater Tots are 1112 the size of SI whole potato but approximately the same shape. Estimate how long it would take to cook a Tater Tot at 400°F?At what time would it be cooked half way through? (rlR = 0 3 7 P3-10, (a) Write the rate law for the following reactions assuming each reaction follows an elementary rate law.
(b). Write the rate law for the reaction
P3-11,
if the reaction [ I ) is second order in B and overall third order. 12) is zero order i n A and first order in R, (3) is zero order in both A and B. and (4) is first order in A and overall zero order. (c) Find and upritethe rate laws for the followrng reactions Hz+ BT?-4 ?HBr (1 ) (2) H, + I2 + ?HI Sel up a stuichiometric table for each of the following reactions and express the concenlratton of each species in the reacrion as a function of conversion
Chap. 3
auestions and Probrems
135
evaluating all constants {e.g., E. Q). Then, assume the reaction follows an elementary rate law. and write the reaction rate solely as a function of conversion. i.e., -rA = fl). (a) For the liquid-phase reaction
the initial concentra~ionsof ethylene oxide and water are I Ib-rnol/fk3 and 3.47 Ib-mollf~l162.4I Iblft? + 18), res~xctive3y. If k = 0 1 drn'lmol . s a1 300 K with E = 12,500 cal/mol. calculate the space-lime volume for 90G conversion at 3OQ K and at 3.50 K. Ib) For the isothermal, isobaric gas-phase pyrolysis
pure ethane enters the flow reactor at 6 at111 and 1 I(H) K, How nould your equation for the concentration and reaction rale change if the reaction were to be carried out in a constant-volume batch reactor? (c) For the isothermal, isobaric, catalyttc gas-phase oxidation
the feed enters a PER at h atm and I?fiO°C and is a ~toichiometricnithture of only oxygen and ethylene. (d) For the isothermal, isobaric. catalytic €a<-phase reaction is carried out In a PBR
the feed enters a PBR a1 6 nrtn and 170°C and i s n s~o~cliinmetl.ic ~nirture. Whdl catalyst weight is required to reach 80% conrervon In a Ruidized CSTR at 1 70rC and 27floC? The rate conctarlt I \ detined wrt benzene and vn = 50 dm'hnin kB=
53 mol at 200 K u ith E = 80 !d/niol kgcat . min atm
P3-12, There were 5430 million pounds of ethylene oxrde produced i n the United Stares in 1995. The flowcheet for the camniercial production of ethylene oxide (EO) hy oxidation of ethylene is shown In Eigilre Pl-I?. We note that the process essentrally conqists of two systems. a reaction system and a sepnrillion kyqtem. Discuss the flowcheet and how your anhwer' to P3-E 1 (c) would change if alr IS wed In a \to~cliiometricreed. Th15 reactlon I \ studled Further in Chapter 4.
Rate Laws and Stoichiometry
EO
EO
€0
reactor
absorber
stripper
Light-ends rejection
Chap.
€0 refiner
Sta~t
EOtwaler
fssdwabr
Orbased EO reaction
gfycol plant EO fernvery and refining
Figure P3-12 ED plant Rowsheel. IAdapred from R. A. Meyers, ell.. Hor~dbool, r$ C/~erniculPtrnii~crir~n PPrnte.f~cr.C/~eler~ricrrl Pmc.e\s Trrhnolrrx! H
P3-13, The formation o f nitroanalyine (an important intermediate in dyes+alle fast orange) is formed From the reaction of nrthon~trochlorobenzene(ONCE and aqueous ammonia. (See Table 3-1 and Example 9-2.)
(a)
(b)
(c) (d) (e)
If) (g)
The liquid-phase reaction is first order in both ONCB and ammonia wit! k = 0.00I7 d k m o l . min at 188'C with E = 1 1.173 caltinol. The iniria enterlng concentrations of ONCB and ammonia are 1.8 krnollrnJ and 6.1 kmollrn'. respectively (more on this reaction in Chapter 9). Wrire the rate Taw for the rate of disappearance of ONCB in terms o concentration. Set up a stoichiometric table for this reaction for a flow system. Explain how part (a) would be different for a batch system. -r, = Write -rA solely as a function of conver~ion. What is the initiaE rate of reaction (X = 0) at 188"C? -r,* = at 25°C:' -rA = at 288"C? -r, = What is the rate of reaction when X = 0.90 at 18S°C? -7, = at 2S°C? -rA = at 28X°CT -r, = What would be the corresponding CSTR reactor volume at 2S°C tr achieve 90% conversion at 25°C and at 288°C for a molar feed rate of: moUmin at25"C7 at
28R°C?
V= Y=
Chap. 3
137
Ouestions and Probtsms
P3-1dR Atlnpted from M. L. Shuler and E Kargi, Bioyrncr.~~ En~irrecrin~. Prenticr Hall (20021. Cell grouth tithes place in b~ore:~ctors called chemostnt?.
A substrate such as glucose is used to grow cells and produce a product.
Substrare
Cell% ---4
More Cells Ibiomass) + Prcduct
A gcneric molecule formuIn for the biomass is C41H73NUM01 Consider the growth of n generic organism on glucose Experimentally, it was shown that for this organism, the cells convert 213 of carbon st~bstrateto btomass. (a) Calculate the staichiornetric coefficients a, b, c, d, and e (Hint: c a w 0111 atom balanceh [Atlr: c = O 911). (b) Calculate the yjeId coefficients Yo (g cells/g substrate) and Y,,, (g cellslg 0:).The grom of cells are dry weight (no water-gdwj IAns: Y, = 1.77 gdw cellslg 021(gdw = g n m s dry weight). P3-ISa The gas-phase reaction %, ,
!N1+iH,
4
NH,
is to be carried out isothermally. The molar feed is 50% H, and SO% N?, at a pressure of 16.4 atm and 227". (a) Construct a complete stoichiornetric table. (b) What are . ,C 8. and E ? Calculate the concentrations of ammonia and hydrogen when the conversion of H21s 60%. (Ans: C,*= 0.1 rnol/dm3) (c) Suppose by chance the reaction is elementary with k,; = 40 dmVmolls. Wrtre the rate of reaction F O I Pas~ a function of conveision for ( I ) a flow system and (2) a constant volume batch system. P3-1fin Calculate the equilibrium codversion and concentrations for each of the following reactions. (a) The liquid-phase reaction
with C,40= Cgo = 2 rnoIldrnSand Kc = 10 dm3/mol.
(b) The gas-phase reaction
carried out in a flow reactor with no pressurn drop. PureA enters at a temperature of JOa K and 10 atm. At this temperature, Kc = 0.Z5(dm3/mol)~.
138
Rate Laws and Stoichiometry
Chap. 3
(c) The gas-pha~ereaction in part (b) carried out in a constant-volume batch
reactor,
(dl The gas-phase reaction in part (b) carried out in a constant-prescure batch reaction.
P3-17, Consider a cylir~dricalbarch reacror thar has one end fitted with a frictionless piston attached to a spring (Figure P3- 17). The reaction with the rate expression
is laking place in this type of reactor.
I
I Reoclion occurs in hare
(a) Write the rate law solely as a function of convenioli. numerically evaluating all posslbIe symbols. (Atls.: -r, = 5.03 X 10-" I(i - X )3/( 1 + 3X)'"] Ib niollft7.s ) (h) What is the conversion and rate of reaction when I' = 0.2 ft3'?(Arts.: X = 0.259. -rA = 8.63 X lo-"' lb rnollft?-s.) Addifionnl irtfnr~t~nriiir~:
Equal moles of A and B are present at I = 0 Initial \?olume: 0.15 ft3 Value of k,: 1.0 ~ft'llbrno1):-s-I The relationship between the volume of the reactor and pressure within the reactor 1s
V = (0.1)(P)
P3-18,
(Vin ft3. P i n atml
Temperature of ayrtem (considered constant): 140°F G3\ constant: 0.73 ftl.atm/lh rn~l.'R Read the section\ o n Colliqion Theory. Transition Stale Theory. and Molecular nynamics in the Professional Reference Shelf on the CD-ROM. la) Use colIlsicln theory to outline the derivation of the Polanyi-Sementlv equation. ~ { h i c hcan he uxed to estimate activation energles from the heats of reac~ion,AH,, according 10 the equation
thE Why i q thiq 3 reasonable correla~inn?(Hitlt: See Pnlfewinnal Reference Shelf 3R. 1 : Cnlliriort T l t e n ~ ) (c) Concider the ToHon ~ n gfanilly of reactions:
Chap. 3
Questions and Problems
Estimate the activation energy for the reaction
CH, #
Web Mtnt
+ RBr
CH,Br
+R
#
which has an exothermic heat of reaction of 6 kcallrnoi (i.e., AH, = -6 kcallmol). (d) What parameters v a y over the widest range in calcularing the reaction rate constant using transition state theory ( i t . , which ones do you need to focus your anention on to get a good approximation of the specific reaction rate)? (See Professional Reference Shelf R3.2.) (e) List the assumptions made in the molecular dynamics simulation R3.3 used to calculate the activation energy for the hydrogen exchange reaction. (T) The volume of the box used to calculate the translational partition function for the activated complex was taken as I dmq.True or False? (g) Suppose the disrance between two atoms of a linear rnolecuIe in the transition state was set at half the true value. WouId the rate constant increase or decrease over that of the true value and by how much (i.e., what factar)? (h) List the parameters you can obtain from Cerius2 to calculate the molecular partition functions. P3-19D Use Spartan, CACHE, Cerius2. Gaussian. or some other chemical computational software package to calculate the heats of formation of the reactants, products, and transition stare and the activation barrier, EB.for the foIlowing reactions: ( 8 ) CH30H + 0 4 CH30 + OH (b) CH,Br + OH --+ CH,OH + Bt
P3-20B It is proposed 20 produce ethanol by one of two ~eactions:
C2HsCI + OH- w ClH50H + C1-
(1)
Use Spartan (see Append~xJ) or some other software package to ansuw i k foIlwhg: (a) What is he ratio of the rates of reaction at 25'C? IOO°C?SOO°C? (b) Wh~chreactran scheme would you choose to make ethanol? (Hint: Consult Cltemicol Markfir~gReporrer or w~.~:clnemn~cek.com for chemical prices). fProkssor R. Baldwin. Colorado School of M~nesl
Additinna! Homework Problems on CD-ROM Temperature Effects
CDP3-An CDM-Bn Solved F:oblcms
CDP3-CR
Estimate how fast a Tenebrionid Beetle can push a ball of dung at 41.5"C. (Solution Included.) Use the Polanyi equatron ro calculate activation energies. (3rd Ed. P3-?OR] G1ve11the irreversible rate law at low temperature, write the reverhihle rate law at high temperature. [3rd Ed. P3-]OR]
Rate Laws and Stoichiornetry
Chap.
Stoichiornetry
CDP3-D,
CDP3-E,
CDP3-Fa Hall of Fame
Set up a stoichiometric table for
in terns of molar Row rates. (Solution included.) Set up n stoichiometric table for the reaction
[Znd Ed. P3-10,) The elemenfary reaction A(gj + B(1) PC(g) takes place in square duct containing liquid B, which evaporates into the gas react ing with A. [2nd Ed. P3-20,]
Reactions with Phase Change
CDP3-GB
Silicon is used in the manufacture of micro~lectronicsdevices. Set ul a stochiometric table for the reaction (Solution included.)
CDP3-H,
Reactions with condensation between chlorine and methane. [3rd Ec P3-2 1 ,I. Reactions with condensation
[2nd Ed. P3-16,] CDP3-I,
C,H,(g) -+ 2Br(g)
CDP3-J,
CZH4Br:(1, g) + 2HBtfg)
[3rd Ed. P3-22,] Chemical vapor deposition 3SiH,(g) + 4NH3(g)
--t
SiN,(s)
+ l ZH,lgl
[3d Ed. P3-23B]
0P3-KB
Condensation occurs in the gas-phase reaction:
CzM&)
+ 2CI(g) d CH2C12(g,0 + 2HCl(g)
[2nd Ed. P3-17,]
New Problems on the Web CDP3-New From time to time new problems relating Chapter 3 material to weq day interests or emerging technologies will be placed On the wet Solutions to these problems can be obtained by emailing the author.
Chap 3
Suoplementary Reading
SUPPLEMENTARY
READING
1 . Two references relating to the discuwion of activation energy have already been ctted in this chapter. Acrikation energy i s usually discursed in terms nf either collision theory or trans~tion-state theory. A concise and readable account of there two theorle~can be found in Masel, R.. Chemical Kinerics, New York: McGraw Will, 2002. p. 594. LAIDLER. K. J. Chrreicnl Kitlctics. New York: Harper & Row. 1985. Chap. 3
An expanded but stjlI elenlentary presentation can be found i n
MOORE,J. W., and R. G.PEARSOY, Kinetics nttd Mecl~nrrism,3rd ed. New York: Wiley. 198 1. Chaps. 4 and 5. A more advanced treatise of activation energies and collision and transition-state theories is
S. W., Tire Fowldutions of ChemicnI Kinetics. New York: McGrawHill. 1960. STEIXFELD. J. 1.. J. S. FRANCISCO, W. L. HASE, Chemrcal Kinetics and Dynamics, 2nd ed. New Jersey: Prentice Hail, 1999. BEWOK,
2. The books listed nbove also give the rate laws and activation energiec for or number of reactions: in addition, as mentioned earher in this chapter, an extensive listing of rate Iaws and activation energies can be found in NBS circulars:
Kinetic data for larger number of reactions can be obtained on Floppy Disks and CD-ROMs provided by National Institute of Standards and Technology (NIST). Standard Reference Data 221fA320 Gaithersburg, MD 20899: ph: (301) 975-2208 Additional sources are Tables of Chemical Kinetics: Homogeneous Reactions. Nntional Bureau of Standards Circular 510 (Sept. 28, 1951): Suppl. 1 (Kov. 14. I956): Suppl. 2 (Aug. 5. 1960): Suppl. 3 (Sept. 15. 1961) (Washington. D.C.: U.S. Government Printing Office). Chem~calKinetics and Photochemical Data for Use in Srratosphenc Modeling. Evaluate No, LO, JPL Publication 92-20. Aug. 15. 1992. Jet Propulsion Labontories. Pasadena Calrf. 3. Also consult the current chemistry literature for the appropriate algebraic form of the rate law for a given reaction. For example, check the JournoE of Physical Chemist. in addition to the journals listed in Section 4 of the Supplementary Reading section in Chapter 4.
Isothermal Reactor Design
Why, a four-yeas-old child could understand this. Someone gel me a four-year-old child. Groucho Marx --
&sf! -
,
L,
~~i~~ eve,er).tl,ing together
Overview. Chapters I and 2 discussed mote balances on reacror,~and the manipulation of these balances to predict reactor sizes. Chapter 3 discussed reactions. In Chapter 4, we combine reactions and reactors as we bring a11 the material in the preceding three chapters together to arrive at a logical structure for the design of various types of reactors. By using this structure, one should be able to solve reactor engineering problems by reasoning rather than by memorizing numerous equations together with the various restrictions and conditions under which each equation applies (i.e., whether there is a change in the total number of moles, etc.). In perhaps no other area of engineering is mere formula plugging more hazardous; the number of physical situations that can arise appears infinite, and the chances of a simple formula being sufficient for the adequate design of a real reactor are vanishingly small. We divide the chapter into two parts: Part 1 "Mole Balances in Terms of Conversion," and Part 2 "Mole Balances in Terms of Concentration, C,.and Molar Flow Rates, F+" In Part 1, we wiIl concentrate on batch reactors, CSTRs,and PFRs where conversion is the preferred measure of a reaction's progress for single reactions. In Part 2, we will analyze membrane reactors. the startup of a CSTR. and semibatch reactors, which are most easily analyzed using concentration and molar flow rates as the variables rather than conversion. We will again use mole balances in terms of these variables (C,. F,)for multiple reactors in Chapter 6.
144
Isothermal Reactor Design
Chap.
This chapzer focuses attention on reactions that are operated isothermally. We begin the chapter by studying a liquid-phase reaction to form ethylene gIycol in a batch reactor. Here, we determine the specific reaction rate constant that will be used to design an industrial CSTR to produce ethylene glycol. After illustrating the design of a CSTR from batch date, we carry out the design of a PFR for the gas-phase pyrolsis reaction to form ethylene. This section is followed by the design of a packed bed reactor with pressure drop to form ethylene oxide from the partial oxidation of ethylene. When we put ail these reactions and reactors together, we will see we have designed a chemical plant to produce 200 million pounds per year of ethylene glycol. We close the chapter by analyzing some of the newer reactors such as microreactors. membrane reactors, and, on the CD-ROM, reactive disdllation semibatch reactors.
PARTI
Mole Balances in Terms of Conversion
4.1 Design Structure for Isothermal Reactors ~ o g ~ One c FS.
Memorization
Use the algorithm mher than nzing equations.
of the primary goals of this chapter is to solve chemical reaction engineering (CRE) problems by using logic rather than memorizing which equation applies where. It is the author's experience that following this structure. shown in Figure 4-1, will lead 10 a greater understanding of isothermal reactor design. We begin by applying our genera! mole balance equation (level O) to a specific reactor to arrive at the design equation for that reactor (Ievel Q). If the feed conditions are specified (e.g., NAoor FA,), alf that is required to evaluate the design equation is the rate of reaction as a function of conversion at the same conditions as those at which the reactor is to be operated (e.g.. temperature and pressure). When -r, =flm is known or given, one can go directly from Ievel O to level @ ta determine either the batch time or reactor volume necessary to achieve the specified conversion. If the rate of reaction is not given explicitly as a function of conversion, we must proceed to level @ where the rate law must be determined by either finding ir in books or journals or by determining it experimentalty in the laboratory. Techniques for obtaining and analyzing sate data to determine the reaction order and rnte constant are presented id Chapter 5. After she rate law has been established. one has only to use stoichiumetry (level @) together with the conditions of the system (e.g., constant volume, temperature) to express concentration as a function of conversion. For liquid-phase reactions and for gas-phase reactions with no pressure drop (P= Po), one can combine the information in IeveIs @ and B,to-express the rate of reaction as a function of conversion and arrive at level @. It is now ~ossibleto determine either the time or reactor volume necessnrv to achieve the desired conversion by substituting the retationship linking conversion and sate of reaction into the appropriate design equation (level @).
Sec. 4.1
Design Structure for Isothermal Reactors
i
Q The general mule balance equation:
Chapter I
FAn- F,* + j v r A d~
d1
I
@ Apply mole balance to specific reactor dewgn qualions
Chapter 2
Batch:
dX NAO-= -rhV
CSTR:
v=- F~17
dt
-R
dX F;lo-=-r4 dV dX FA, -=-rA
Plug flow.
PacW bed:
,
dW
\
Yes
Chapter 2
'*
'
'
Evaluate the algebnic (CSTR) or rntegral (lubulrtrl equations elrhrr nurnericaily or analytically to determine the reactor volume or the processing time, or con~ersion, A
1
I
Detcrm~nethe rate law in terms of the concenrntion of the
Chapter 3
1
reacting sprrles
Chapter 4 Chapter 4 LTre stoichiometry to express concentration m a Function of conversion.
Chapter 3
Liquid-phase or constant volume batch: CA = CAO(l- X) t L
Gas phase: T = To
-
CA =Chn i
(I-X)
P -
( ~ + E X I6,
Liqutd Combine steps@ and Q to obtain -ri
=f(Xj
~to~ch~umetry.
@ Gar-phase reactions with pressure drop.
Chapter 4 Semibatch reacton:
v=v*+v,t
tnnspon law, and
I
Figure 4-1 Isothermal reaction design algorithm For conversion
pressure drop term ~nan ordinm differentid e4uation solver (ODEsolver Polvmathr.
146
Isothermal Reactor Design
Chap. 4
For gas-phase reactions in packed beds where there is a pressure drop,
we need to proceed to level 8 to evaluate the pressure ratio (P 1 Po) in the concentration term using the Ergun equation (Section 4.5). In level @, we combine the equations for pressure drop in level 6 with the information in levels 6 and a,to proceed to level 19where the equations are then evaluated in the appropriate manner (i.e., analytically using a table of integrals, or numerically using an ODE solver). AIthough this structure emphasizes the determination of a reaction time or reactor volume for a specified conversion, it can also readily be used for other types of reactor calculations. such as determining the conver-
The Algorithm
'. lrfrrle
?. Rite law 3. Stolchiomc{y 4. Combtne
5 Eialuate
sion for a specified volume. Different manipulations can be performed in level @ to answer the differenl types of questions mentioned her.i The structure shown in Figure 4-1 allows one to develop a few basic concepts and then to arrange the parameters (equations) associated with each concept in a vafery of ways. Without such a structure, one is faced with the possibility of choosing or perhaps memorizing the correct equation from a 11~1tS!itude of ~qcmrio~ls that can arise for a variety of different combinations of reactions, reactors. and sets of conditions. The challenge is to put everything together in an orderly and logical fanhion so that we can proceed ta arrive at the correct equation for a given situation. Fortunately. by using an algorithm to formulate CRE problems, which happens to be analogous to the algorithm for ordering dinner from a fixedprice menu in a fine French restaurant. we can elirninale virtually ail memorization. In both of these algorithms. we must make choices in each category. E(lr example, in ordering from a French menu, we begin by choosing one dish from the & p ~ ~ i z e rlisted. s Step 1 in the anaIog in CRE is to begin by choosing the appropriate male balance for one of the three types of reactors shown. In Step 2 we choose the rate law { e ~ ~ t r P and e ) . in Step 3 we specify whether the reaction is gas or liquid phase ( c l ~ e e or ~ ~drsserr). e Finally. in Step 4 we combine Steps 1. 2, and 3 and either obtain an analytical solution or solve the equations using an ODE solver. (See the complete French menu on the CD-ROM). We now will apply thi? algorithm to a specific situation. Suppose that we have. as shown in Figure 4-2. mole balances for three reactors. three rate laws. and the equation\ for concentrarionr; f i ~ rboth liquid and gas ph:ises. In Figure 4-2 we see how the algm.ithm is wed to forrnula~ethe equation to calcblate the PFK r.rJtrctor~ ~ o l i i t r i r~,fi~st-orrlr~re~~~r ~ n s - j ~ l ~ (rcoc~ioi?. rre Thc pathway to arrive at rhis equation is s h o ~ nby the ovals connected to the dark lines through the algorithm. The dashed lines and the bnxes represent other pathways for solutions to other situation$. The algorithm for the pathway shown ic 1. Mole balance, chooce species A reacting in a PFR
2. Ratc law, choose the irretersihle tirsr-order reaction 3. Staichiometry choose the gas-phaxe concentration 4. Cornhine Steps I . ?, and 3 to amve at Equation A 5. Evaluate. The col~lhinestep can be eraluatrd either a. Analyr~cnll! Appendix AI) b. Gr;iphicallj (Chapter 2 )
Sec. 4.1
Design Structure lor Isothermal Reactors
147
1 . MOLE BALANCES
,..-s,
LIQUID Constant flow rate L'
= U*
IDEAL GAS Vanable flow rate
,..o.... * LIQUID OR GAS
IDEAL GAS "' Variable volume
P T r > = c o [ i + ~ ~ v) . ~v -~
Constant volume
v =vo
( I + E x ) ~ I
P To
P To
u CA= CAO(1 - X)
4. COMBINE ( F ~ nOrder i Gas-Phase Reaction in a PFR)
From mole balance
/
From stolch~ornet~y
Integratmgfor lhe case of constant temperature and pressure gives
lorithm for
isothermal reactors,
I
Isothermal Reactor Destgn
Substitute pararnc. ter valuec in Fteps 1 3 only if they are zero.
Chap.
4
c. Nurnerically(Appendix Ad), or d. Using software (Polymath)
In Figure 4-2 we chore to integrate Equation A for constant temperature and pressnre to find the volume necessary to achieve a specified conversion (or calculate the conversion that can be achieved in a specified reactor volume) Unless the parameter values are zero. we typically don't substitute numerical values for parameters in the combine step until the very end. We can solve the For the case of isothermal operation with no pressure drop, we were able equalloris in the to obtain an analytical solution, given by equation B, which gives the reactot combine ftep e~rher volume necessary to achieve a conversion X for a first-order gas-phase reaction I . Analyticnlly (Append~xA I ) carried out isothermally in a PFR. However, in the majority of situations, ana2. Graphlcdl) lytical solutions to the ordinary differential equations appearing in the combine (Chapter 2) step are not possible. ConseguentIy, we include Polymath. or some other ODE 3. Nun~erically (Appendix A4) solver such as MATLAB, in our menu in that it makes obtaining solutions tc 4. Using Software the differential equations much more paIalable. !Poljmathl.
4.2 Scale-Up of Liquid-Phase Batch Reactor Data to the Design of a CSTR
One of the jobs in which chemical engineers are Involved is the scale-up of labratory experiments to pilot-plant operation or to fulP-scale production. In the past, a pilot plant would be designed based on laboratory data. However. owing to the high cost of a pilot-plant study, this step is beginning to be surpassed in many instances by designing a futl-scale plant from the operation of a laboratory-bench-scale unit called a microplant. To make this jump successfully requires a thorough understanding of the chemicaI kinetics and transpotl limitations. In this section we show how to analyze a laboratory-scale batch reactor in which a liquid-phase reaction of known order is being carried out After determining the specific reaction rate, k, from a batch experiment, we use it in the design of a full-scale flow reactor. 4.2.1 Batch Operation
In modeling a batch reactor, we have assumed that there is no inflow or outflow of material and that the reactor is wet1 mixed. For most liquid-phase reactions, the density change with reaction is usually small and can be neglected (i.e., Y = V,). In addition. for gas-phases reactions in which the batch reactor
volume remains constant. we also have V = Vo. Consequently. for constantvolume (V = Vo) (e.g.. closed metal vessels) batch reactoss the mole balance
can be written in terms of concentmtion.
Sec. 4 2
Scale-UP of L~qu~d-Phase Batch Reactor Data to the Design of a CSTR
149
Generally. when analyzing laboratory experiments. it is best to process the data in terms of the measured variable. Because concentration is the measured variable for most liquid-phase reactions, the general mole balance equation applied to reactions in which there is no volume change becomes
dc,- - r ~ dt
Used to analyze batch reaction dara
This is the form we will use in analyzing reaction rate data in Chapter 5. Let's calculslte the time necessary to achieve a given conversion X for the irreversible second-order reaction
The mole baIance on a constant-volume, V = Vo, batch reactor is Mole balance
The rate law is Rate law
-rA =
kCi
(4-3)
From stoichiometry for a constant-volume batch reactor. we obtain Stoichiometw
-m
CA= CAO{~
13-29)
The mole balance. rate law, and stoichimetry are combined to obtain Combine
g =kcAn(1 - x)" dr
14-41
To evaluate, we rearrange and integrate
Initially, if t = 0, then X = 0. If the reaction is carried aus isothermally, k will be constant; we can integrate this equation (see Appendix A.1 for a table of Integrals used in CRE applications) to obtain
Evaluate
Isothermal Reactor Design
Chap. 4
Second-order,
(4-5)
isothermal. constant-volume batch reaction
This time is the reaction time t (i.e., tR) needed to achieve a conversion X for a second-order reaction in a batch reactor. It is important to have a grasp of the order of magnitudes of batch reaction limes, t,, to achieve a given conversion, say 90%, for different values of the product of specific reaction rate, k, and initial concentration, CAo.Table 4-1 shows the aIgorithrn to find the batch reaction times, I,, for both first- and a second-order reactions carried out isothermally, We can ohtaio these estimates of t, by considering the first- and second-order irreversible reactions of the form
dx=XV
Mole balance
d r ~ hTAn
Rate law
Fint-Order
Evaluate (Integrate)
t, =
Second-Order
1 -1 ln -
k
I-X
For first-order reactions the reaction time to reach 90% conversion (i.e., X = 0.9) in a constant-volume batch reactor scales as t - -1 l n - I - - 1 l n 1 - - 2.3 '-k I-X k 1-0.9 k
The time necessary to achieve 90% conversion in a batch reactor for an irreversible first-order reaction in which the specific reaction rate is 110-+'ss-') is 6.4 h. For second-order reactions, we have
Sec. 4.2
Scale-Up ot Liquid-Phase Batch Reactor Data to the Design of a CSTR
151
We note that if 99% conversion bad besn required for this value of kc,,, the reaction time, 1,. would jump to 27.5 h. Table 4-2 gives the order of magnitude of time to achieve 90% conversion for first- and second-order irreversible batch reactions. TAELP4-2. Rcacrion 7i3ne f~
BATCHREACTIOST I M ~ S Ftrsr-Order k (5-I)
Hours Minutes Estimaling Reaction Times
Secortd-Onles kCAo ( 5 - ' 1 10-J
10-2
Seconds
1
Millisecond%
IMK)
10-I 10 I0 . 0 ~
Flow reactors would be used for reactions with ulauraererirtic reacrinri rimes, r., of minutes or less. The times in Table 4-2 are the reaction time to achieve 90% conversion (i.e., to reduce the conccntsalion from C,, to 0.1 C,,,). The total cycle time In any batch operation is considerably longer than the reaction time. I ~ as . one mu51 account for he time neces7ary to fill 0,) and heat (r,) the rcactor together with the time necessary to clean the reactor between batches, t,. In some cases. the reaction lime calculated from Equation (4-5) may be only a small fraction of the total cycle time, r,.
Typical cycle times for a batch polymerization process are shown in Table 4-3. Batch poiymerization reaction times may vary between 5 and $0 hours. Clearly. decreasing the reaction time with a 60-hour reaction i s a critical prob[em.As the reaction time i s reduced (e.g.. 2.5 h for ri liecond-order reaction with kC,, = 10?; SKI),it becomes important to use large lines and pumps to achieve rapid transfers and to utilize efficient sequencing to minimize the cycle time.
Batch operation lfrnes
I. Charge feed ro the reactor and agitate, r, 2. Heal to reacrron temperalure. r, 3. Carry out reacl~on.tR 4. Empty and clean ~c:~ctr>r.r, Total time excluding reaction
1.5-3.0
0.2-2 0 (varieq)
0.5-i .(I
3.C-6 0
Usually one has In optimize the reactinn time ~ i t the h procelising times listed in Table 4-3 rcl produce the maximum number of hatches (i.e., pound\ of produ c t ~in a day. See Problems P4-h(f) and P4-T(c).
Isothermal Reactor Design
Cha
In the next four examples, we will describe the various reactors nee( to produce 200 million pounds per year of ethylene glycol from a feedstock ethane. We begin by finding the rate constant. k. for the hydmlysis of ethyl( oxide to form ethylene glycol. Example &I
Dsteminiltg k from BatcIi Data
It is desired to design a CSTR to produce 200 million pounds of ethyEene gLycoE year by hydrolyzing ethylene oxide. However, before the design can be carried c it is necessar?, to perform and analyze a batch reactor experiment to determine specific reaction rate CQnStant, k. Because the reaction wiIl be carried out koth mnlly, the specific reitct!on rate will need to be determined only at the renctlon te perature of rhe CSTR. At high temperatures there is a sipn~ficant by-prd formation. while at temperatures below 40°C the reaction does nor proceed at a s nificant rate; consequently. a temperature of 55°C has been chosen. Because water i s u~ualfypresent in excess, its concentration mny he considered constant d ing the course of the reaction. The reaction is first-order in ethylene oxide.
I n the laboratory experiment, 500 mL of a 2 M solution ( 2 kmollrn'f of e8 ylene oxide in water was mixed with 5 0 0 mL of water containing 0.9 wt Ic sulfu acid, w h x h i s a cataIyst. The tempemure was maintained at 55'C. The concent] tton of ethylene glycol was recorded as a function of time (Table E4-1. I). Using the data In Table E4- 1. I,determine the specific reaction rate at 55°C
Time (rnin)
Concenrrarion of'E ~ l t ~ I e j ~ e Glxcol ( k r n ~ l l r n ~ ) ~
Check I0 types of
homework problems on the CD-ROM for more solved examples using t h ~ salgorrthm.
I
In this example we use the problem-solving algorithm (A through G )that is givt in the CD-ROM and on the web www.engin.umich.edd-pmb1emsoIi1ing.You m; wish to Follow this algorithm in solving the other examples in this chapter and tl pmblems glven at the end of the chapter. However, to conserve space it wiIl not 1 repeated for other example problems. A. Problem statement. Determine the specific reaction rate. k,.
Sec 4 2
Scale-Up of tiqtlld-Phase Batch Reactor Data to the Des~gnot a
CSTR
153
C. ifenrib: C I . Relevant theories Check 10 types of I
1
i/
homework problems on the
CD-ROM for more S O I V ~ examples .~~
u~ingthis algunthrn.
Solved Problcmr
Mole balance:
3 = r,4v dr
Rate law: - r , = k,C, C2. Variables Dependent: concentrations, CA,C,, and Cc Independent: time, t C3. Knowns and unknowns Knowns: concentration of ethylene glycol as a function of time Unknowns: 1. Concentration of ethylene oxide ns a function of time, C, = ? 2. Specific reaction rate. kA = ? 3. Reactor volume, V = ? C4. Inputs and outputs: reactant fed all at once to a batch reactor C5. Missing infomation: None: does not appear that other sources need to be sought.
D.Assumprions nnd ~lpproxirnafions: Assurnprions 1. Well mixed
2. A11 reactants enter at the same time 3. No side reactions
4. Negtigible filling time 5 . Isothermal opention
Approximations I . Water in excess so that its concentration is essentially constant, (i.e,, CB cBO)E. Sp~rifirotmn.The
nrnhlern i c n r i r h ~ rnvprcwrifird
nnr r!nrIrrrnpr;G..rl
154
Isothermal Reactor Design
Chap. 4
'
Relared mererial. This problem uses the mole balances developed in Chapter 1 for a batch reactor and the stoichiometry and rate laws developed in Chapter 3. G. Use on algorirhln. For an isothermal reaction, use the chemical reaction engineering algorithm shown in Figures 4- 1 and 4-2.
1. A mole balm& on a batch reactor that is well-mixed is
*
Mole aalance
2. The rate law is
Because water is present in such excess. the concentration of water at any r~mef i s \irtually the same ns the initial concentrat~onand [he rate law 1s independent of the concentntion of H 2 0 . (CB= CHI,.) 3. Stoichiometry. Liquid phase, no volume change, V = V, (Table E4-1.2):
Table for ~ 0 l s f . Volume h
Recall O, is the initial number of moles of A to B (i.e..
BO). =N A'nrr
- N* A - - - V
-
N*
V"
1. Combining the rate law and the mole balance. we have
-dC,4 = kc, dr
5. Evaluate. f ( ~ ~\othernial r operidtion, A. is constant, so we can intcgra~ethik
eqii;~tiun( E l - l .3)
Sec. 4.2
Scale-Up of Liquid-Phase Batch Reactor Data to the Design of a CSTR
155
using the initial condition hat when t = 0,then C, = CAo.The initial concentration of A after mixing the two volumes together is 1.0 hol!m3 (1 rnoyL). Integrating yields m
The concenrration of ethylene oxide at any time r is
The concentration of ethyytene glycol at any time I can be obtained from the reaction staichiometry:
For liquid-phase reactions V = Vo,
Rearranging and taking the logarithm of both sides yields
We see that a plot of In[(C,, - Cc)/CA(l]as a function oft will be a straight line with a slope -k. Using Table E4-I. I. we can construct Table E4-I .3 and use Excel to plot Jn(CA0- C,)/C,, as a function of r.
156 Evaluating the spec16c reaction rate from bfltcl~reactor concentmtiontime data
1
Isothermal Reactor Design
From the slope of a plot of In[(C,,, the Excel Figure EJ-I. I .
-
C,)/C,,,] vrnus t , we
Chap
can Rod k as shown
-3.500 0.0
2.0 4.0
6.0
8.0
10.0 12.0
t (rnin)
Figure E4-1.1 Excel plot of data.
Slope = -k = -0.31 1 rnin-I
The rate law becomes -rA = 0.31 1 min-IC,
Summary Notes
The rate taw can now be used in the design of an industrial CSTR.For those wl prefer to find k using semilog graph paper. this type of analysis is given in Chapter Summary Notes on the CD-ROM. An ExceI tutorial is also given in the Summa Notes for Chapter 3.
4.3 Design of Continuous Stirred Tank Reactors (CSf Rs) Continuous stirred tank reactors (CSTRs). such as the one shown here sch matically, are typically used for Iiquid-phase reactions.
Sec. 4.3
Qes~gnof Contrnuous Strrred Tank Reactors (CSTRs)
f 57
In Chapter 2. we derived the following design equation for a CSTR: M o l e halance
which gives the volume V necessary to achieve a conversion X. As we saw in Chapter 2, the space time, T, i s a characteristic time of a reactor. To obtain the space time. T. as a function of conversion we first substitute for FA0= uDCAO in Equation (2- 1 3)
and then divide by uoto obtain the space time, T, to achieve a conversion X in
a CSTR
This equation applies to a single CSTR or to the first reactor of CSTRs connected in series. 4.3.1
A Single CSTR
Let's consider a first-order irreversible reaction for which the rate law is -rh = kCA
Rate law
For liquid-phase reactions, there is no volume change during the course of the reaction, so we can use Equation (3-29) to relate concentration and conversion, We can combine mole balance Equation (4-7). the sate law and concentration, Equation (3-29) to obtain Combine
CSTR Relationship between swce time and convekian for a first-order liquidphase reaction
Rearranging
3 58
Isothermal Reactor Design
Chap. 4
We could also combine Equations (3-29) and (4-8) to find the exit reactor concentration of A, C,,
Da =
-x*ov FAO
For a first-order reaction. the product ~k is often referred to as the reaction DamkGhler number, Da, which is a dimensionless number that can give us a quick estimate of the degree of conversion that can be achieved in continuousflow reactors. The Damkohler number i s the ratio of the rate of reaction of A to the rate of convective transport of A at the entrance to the reactor.
Da=
-r,oV - Rate -
F AO
of reaction at entrance = Entering flow rate of A
Ractjon mte " f
r
rate"
~
The Damkohler number for a first-order irreversible reaction is
For a second-order irreversible reaction, the DarnkGhler number is
'Da %
lo
It is imponant to know what values of the Damktjhler number. Da, give high and low conversion in continuous-flow reactors. A value of Da = 0.1 or less will usually give less than 10% conversion and a value of Da = 10.0 or greater will usually give greater rhan 90% conversion: that is. the rule of thumb is if D a < 0.1. then X ~ 0 . 1 Equation (4-8) for a first-order liquid-phase reaction in a CSTR can also be written in terms of the DamkGhler
4.3.2 CSTRs Fn Series
A first-order reaction with no change in the volumetric flow rate be carried out in two CSTRs placed in series (Figure 4-3).
(0
= v0) is to
Sec.4.3
Design of Continuous Stirred Tank Reactors (CSTRs)
159
Figure 4-3 Twu CSTRs in series.
The emuent concentration of reactant A from the first CSTR can be found using Equation (4-9)
with z, = Vl/%. From a mole balance on reactor 2.
- F A I - F A Z= vo(C.ql-C42) - VA? k:C,*
2 -
Solving for CA2.the concentration exiting the second reactor. we obtain First-order reaction
If both reactors are of equal pile ir, =
= r ) and operate at the saine ternper-
ature ( k l = k, = k), lhen
If instead of two CSTRs i n series we had n equal-sized CSTRs connected in e r e ( t = r = - = r,, = r,= ( V , l c , ) ) operating at !he same ternperarure ( A , = k2 = ..-= k,, = k 1, the concenrra~ionleaving the last reactor would be
Subsrituting for C,,,, in terms of conversion
isothermal Reactor Design
C,,( I
-
Chap.
'AO
X) = (1
+ Da)"
and rearranging. the conversion for these n tank reactors in series will be Conversion as a function of the number of tanks in series
A plot of the conversion as a function of the number of reactors in series for first-order reaction is shown in Figure 4-4 for various values of the Darnkohl1
CSTRs in series
Conversion as a function of the number of anks in series for different Damktihler numbers for a Rrst-order reaction.
Figure 4-4
Economics
number rk. Observe from Figure 4-4 that when the product of the space tim and the specific reaction rate is relatively large, say, Da 2 1. approximatel 90% conversion is achieved in two or three reactors; thus the cast of d d i n subsequent reactors might not be justified. When the product rk is smal Da 0.1, the conversion continues to increase significantly with each react< added. The rate of disappearance of A in the nth reactor is
-
4.3.3 CSTRs in Parallel
We now consider the case in which equal-sized reactors are placed in paraltf rather than in series, and the feed is distributed equally among each of th reactors (Figure 4-5). The balance on any reactor. say i, gives
Sec. 4.3
Design of Continuous Stirred Tank Reactors (CSTRs) x1
Figure 4-5 CSTRs In parallel.
the individual reactor volume
Since the reacictors are of equal size, operate at the same temperature, and have identical feed rates. the conversion wilI be the same for each reactor:
x,= X, = ... = X" = x as wiIl be the rate of reaction in each reactor
The volume of each individual reactor. V,, is related to the total volume. the reactors by the equation
1! of all
A sirnilv relationship exists for the total molar Row rate, which is equally divided:
Substituting these values into Equation (4-12) yields
Isothermal Reactor Design
Chap. 4
Conversion
for tanks in parallel. 1s this reiult surprising?
This result shows that the conversion achieved in any one of the reactors in parallel is identical to what would be achieved if the reactant were fed in one stream to one large reactor of volurne Y ! 4.3.4 A Second-Order Reaction in a CSTR
For a second-order liquid-phase reaction being carried out in a CSTR, the combination of the rate law and the design equation yields
Using our stoichiometric table for constant density u = u, , CA= CAo(1 - X), and F,,X = v , , C , , X . then
Dividing by un,
We solve Equation (4-15) for the conversion X:
Conversion for a second-order liquid-phase
reaction in a CSTK
The minus sign must be chosen in the quadratic equation because X cannot be greater than 1. Conversion i s plotted as a function of the Damkbhler parameter, tkC,,, in Figure 4-6. Observe from this figure that at high conversions (say 67%) a 10-fold increase in the reactor volume (or increase in the specific reactioil rate by raising the temperature) will increase the conversion only to 88%. This observation i s a consequence of the fact that the CSTR operates under the condition of the lowest value of the renc1enl concentration (i.e.. the exit concentration). and a m ~ e q u e n t l ythe smallest value of the rate of reaction.
Sec. 4.3
163
Design of Continuous Stirred Tank Reaclors (CSTAs)
Ryre 4-6 Convemiod as
a function of the Damkohler number
(tkCAo) for a
second-order reaction in a CSTR.
Example 4-2 Producing 200 Million Pounds per Ytar in a CSTR Uses and economics
Scale-Up of
Batch Reaaor Data
Close to 12.2 billion metric tons of ethylene glycol (EG) were p d u c e d in 2000, which ranked it the twenty-sixth most produced chemical in the nation that year on a total pound basis. About one-half of the ethylene glycol is used for antifreey while the other half is used in the manufacture of polyesters. In the polyester category. 88% was used for fibers and 12% for the manufacture of bottles and films. The 2004 selling price for ethylene glycol was $0.28 per pound. It is desired to produce 200 million pounds per year of EG. The reactor is to be operated isothermally. A 1 lb moVft%olution of ethylene oxide (EO) in water is fed to the reactor (shown in Figure E4-2.1) together with an equal volumetric solution of water contarning 0.9 wt 8 of the catalyst H,SO,. The specific react3011rate constant is 0.31 I mio-', as determined in Example 4-1, Practical guidelines for reactor scale-up were recently given by Mukeshband by Warstee12. If 80% conversion is to be achieved, determine the necessary CSTR (a) volume. ( ) If two 800-gal reactors were arranged in parallel, what is the corresponding conversion? If two 800-gal reactors were arranged in series, what is ihe comspond(c) ing conversion?
Asrumpfion: Ethylene glycol (EG) i s the only reaction product formed.
' D. Mukesh. Chemical Engirreering. 46 (January 5002); ruww.CHE.com.
' J. Warsteel. C/lenlictl/ Engrn~eringPmgre.~s,(June 1 0 ) .
64
Isothermal Reactor Design
Chap
Figure Ed-2.1 Single CSTR.
The specified Ethylene Glycol (EG) production rate in lb mollmin is
F c = 2 x 108Ibx I yr 365 days
xx
24 h
x LX 1 h x 6 . 1 3 7 - Ib mol 60 min
62 lb
mln
From the reaction stoichiometry
Fc = FADX we find the required molar flow rate of ethylene oxide to be
(a) We now calculate the single CSTR volume to achieve 80% conversio using the CRE algorithm. 1. Design equation:
v="
F X - r ~
2. Rate law:
-r* = kc* Following the ~lgorithnr
3. Stoichfometry. Liquid phase (v = u,) :
Sec. 4.3
1
Design of Continuous St~rredTank Reactors (CSTRs)
4. Combining:
5. Evaluate: The entering volumetric flow rate of stream A. with C,,
= Ib mol/ft3 before
mixing, is
I
From the problem statement uno = vAo
I
The total entering volumetric flow raze of liquid is
Substituting in Equation (W-2.4),recal3ing that k = 0.31 1 min-I,yields
= 1480 gal (5.6 m3) A tank 5 ft in diameter and approximately 10 ft tall is necessary to achieve 80% conversion. (b) CSTRs in paraIlel. For two 800-gal CSTRs arranged in parallel (as shown in Figure E4-2.2) with 7.67 ft3/mio ( ~ ~ 1 fed 2 ) to each reactor, the conversion achieved can be calculated by rearranging Equation (E4-2.4)
to obtain
where
The DamkohIer number is
Da = rk = 13.94 rnin x
1
min
0.31 1 = 4.34
Substituting into Equation (E4-2.5) gives us
166
!sothermal Reactor Design
Chap. 4
Figure E4-2.2 CSTRs in parallel
The conversion exiting each of the CSTRs in parallel is 8 1 8 . (c) CSTRs in series. If rhe 800-gal reactors are arranged in series, the conversion i n the first reactor [cf. Equation (E4-2.5)] is
where 7.48 gal First CSTR
15.34 ft'lmin
= 6.97 min
The Damkohter number is Da, = .c,k = 6.97 min X
0.311 = 2.167 min
To calculate the conversion exiting the second reactor. we recall that V, = V2 = V and u,, = uo2 = v , ; then 5,
= '1 = T
A mole balance on the second reactor i s
In
-
Out
+
Generation
=
0
Basing the conversion on the total number of moles reacted up to a point per mole of A fed to the first reactor,
Sec. 4.3
Design of Continuous Stirred Tank Reactors (CSTRs)
FAI=FAD(l-XI)
and
FA2=FA,(t-X,)
Rearranging
Combining the mole balance on the second reactor [cf. Equation (2-24)] with the rate law, we obtain
V=
FAOI X1 - X I ) - C.40~'~(X2-X1)
tE4-1.7)
- r ~ 2
Solving for the conversion exiting the second reactor yields
Second CSTR
x 2 = -X-I-+Da I+Da
-XI
-
I+tk
0.684+2.167
0,90 -
1 +2.167
The same result could have been ob~ainedfrom Equation (4-1 1 ):
Two hundred million pounds nf EG per year can be produced using two ROO-gal (3.0-m3) reactors in series. Conversion in the series arrangemen1 is grea\er than i n parallel for CSTRs. From our discussion of reaclor staging In Ch~pler2 , we could have predicted thal [he ~ e r i e ~ arrangement wouId have gn'en the higher converclon.
u Figure E4.2.3
CSTRE in serieq.
The two equal-sired CSTRs in serles (shown in Figure E4-2.3) will give a higher conversion than two CSTRs in parallel of the same size when the reaction order i s greater than zero. Saie~q consideratlnnF
We can find information about the safety of ethylene glycol and other chemicals from the World Wide Web (H'WW) (Table 4-4). One source i s the Vermont Safety Information on the Internet (Vermont SIRI). For example. we can learn from the Calrrrol Meosur-rs that we should use neoprene gloves when handling the material. and that we should avoid breathins the vapors. tF wc click on "Dow Chemical USA" and scrolf the Rearri\?ih D m ,we would find that ethylene glycol will ignite in ;1Ir at 413°C.
Isothermal Reactor Design
Chap
-
Safety Information MSDS
I . Type in htlp.llwww.siri.or@ 2. When the first screen appears, click on "Material Safety Data Sheet<,"("MSDS") 3. When the next page appear%,type in the chemical you want to find.
(ethylene]
Example: F~ n d Yhen click on Enter. 4. The next page will show a list of a number oTcompanies that provide the data on ethylen
g1ycol. MALLINCKRODT BAKER FISHER DOW CHEMFCAL USA etc.
Let's clrck on "Mnllinckrodt Baker." The materials safety data sheet pmrided will nppear 5. Scroll "ethylene glycol" For ~nformationyou dmire. I. P n ~ l ~ tIdenri$cn:nrio~r ct 2. Composirio~L/Infomtarioaon Ingredients 3. Hu;nrd~fdenrlj5crzrior1 4. Firsr Aid Mcrrums 5. fire Fighting Measures 6. Accidental Release Measures 7. Handling and Storage 8. E.rposum Con!rols/Personal Pmtection
9. Physical nnd ChernicnI Pmperti@s 10-16. Other fnfonnr~rion
4.4 Tubular Reactors Gas-phase reactions are carried out primarily in tubular reactors where the flo i s generally turbulent. By assuming that there is no dispersion and there are n radial gradients in either temperature, velocity, or concentration, we can mod1 the ffow in the reactor as plug-flow. Plug flow-no radial variations in velocity. perature, or reaction rate
Reactants
Products Figtare l - P iRcvtslted) Tubular reactor.
h n w reactors are discussed in Chapter 13 a d dispersion effects in Chapter 1~ The dierential form of the PFR design equation Use this differenctial form of the PFWPBR mote balances when there is AR
(2-15
Sec. 4.4
169
Tubular Reactors
must be used when there i s a pressure drop in the reactor ar heat exchange between the PFR and the surroundings. I n the absence o f pressure drop or heat exchange, the integral form of the plrrg-flow design equation is used.
As an example, considerthe reaction A
--+ Products
for which the rate law is 2
Rate law
-'A
= kc,
We shall first consider the reaction to take place as a liquid-phase reaction and then to take place as a gas-phase reaction. Liquid Phase v = uo
The combined PFR mole balance and rate law is
If the reaction is carried out in the liquid phase, the concentration is Stoichiometry [I~quidphase)
c,4 =c~n(1 -XI and for the isothermal operation. we can bring K outside the integral
Combine
kcAoI-X
(1-x)'
This equation gives the reactor volume to achieve a conversion Q (t = M%) and solving for conversion. we find
X = =kC*o 1 + zkC,,
-
X. Dividing by
Da2 I -tDa,
where Da2 is the DarnkShler number for a second-order reaction. Gas Phase
(T = To)and constant-pressure (P = Po) gas-phnse reactions, the concentration is expressed as a function of conversion:
For constant-temperature
Stoichiometry (gas phase)
170
isothermal Reactor Design
Chap. d
and then combining the PFR mole balance. rate law, and sroichiometry
v = FA,
Combine
The entering concentration CADcan be taken outside the integral sign since it is not a function of conversion. Because the reaction is carried out isothermally, the specific reaction rate constant, k . can also be taken outside the integral sign. For an isothermal reaclion, A. is
constan I.
From the intesral equations i n Appendix A. 1. we find rhat Reactor volume for s second-order gas-phase reacrion
Using Equation (4-17). a plat of conversion along the length (i.e., volume) of the reactor is shown for four different reactions. and values of E are given in Figure 4-7 for the same value of [ ~ ( J k cto~illunrate ,] the effect of volume change with reaction.
The term 2
l,,n:[
is The Eame for each renction.
FLg~rre4-7
Con\er\~onaI; a functlon o f distance down the rcaclor.
We now look at the effect of the change in the number of moles in the gasphace on the relationship between converrion and voIurne. For constant temperature and pressure. Equation (3-451 becomes
u = c l l ( l+EX)
(3-4.51
Let's now consider three types of reactions. one in which F = 0 (6 = O ) , one En ~ h i c hE < O (6 < 0). and one in which E > 0 (6 > 0 ) .When there i s no changc in the number of moIec with reaction, le.g., A -+ B E6 = fl and E = 0: then the
Sec. 4.4
Tubular Reactors
171
fluid moves through the reactor at a constant voIumetric flow rate {u = vo) as the conversion increases. When there is a decrease in the number of moles (6 < 0,E < 0)in the gas phase (e.g., 2A t B), the volumetrjc gas flow rate decreases as the conversion increases; for example,
Consequently, the gas molecuIes will spend longer in the reactor than they would if the flow rate were constant, v = vo. As a result, this longer residence time wouId result in a higher conversion than if the flow were constant at &. On the other hand, if there is an increase in the total number of moles (6 > 0.E > 0) in the gas phase (e.g.. A -+ 231, then the volumetric flow rate wiII increase as !he conversion increases; for example, and the molecules wiIl spend less time in the reactor than they would if the volumetric flow rate were constant. As a result of this smaller residence time in the reactor the conversion will be less than what would resuIz if the volumetric flow rate were constant at vo.
The importance of
changes in volumetric flow rate ri.e., & + 0) with reaction
Figure 4-8 Change In ga~-phasevolumetric flow rate down the length of the reactor.
Figure 4-8 shows the volumetric flow rate profiles for the three cases just discussed. We note that, at the end of the reactor, virtually complete conversion has been achieved. Example 4-3 Producing 300 MiIlirrion Pounds per Year of Ethylene in a PlugFlow Reactoc Design of a Full-Scale Tubular Reactor T h e economics
Ethylene ranks fourth in the United States in total pounds of chcmicaIs produced each year, and it i s rhe nurnher one organic chemical produced each year. Over 50 billion pounds were produced in 2000. and it sold for 50.27 per pound Sixty-five percent of the ethyEene produced is used in the manufacture of fabricated plastics,
172
Isothermal Reactor Design
Chap.
208 for ethylene oxide. 169 for ethylene dichloride and ethylene glycoI. 5% fc fibers. and 5% for solvents. Determine the plug-Row reactor volume necessary to produce 300 rnillio pounds of ethylene a year from cracking a feed stream of pure ethane. The rear tion is irreversible and fallows an elementary rate law. We want to achieve 8OC conversion of ethane. operating the reactor isothermally at 1100 K at a pressure r 6 atm.
The uses
Solution
Let A = C2H6, B = C2H,.and C = Hz. In symbols.
Because we want the reader to be farniIiar with both metric units an Engtish units, we will work some of the examples using English units. The molar flow rate of ethylene exiting the reactor is j=,
-300 x
1061bx lyear x ld_ay x Ihx lbmol year
365 days
24 h
3600 s
28 Ib
Next caIculate the molar feed rare o f ethane, F,,, to produce 0.34 Ib molls c ethylene when 80% conversion is achieved,
I
1. Plug-Row design equation:
Mole balance
Rearranging and integrating for the case of no pressure drop and isotherm; operation yieids
2. Rate law:'
Ind. Eng. Chem. Process Des. Dev., 14, 218 (1975): Ind. Eng. Chem., 59(5), 7 (1963).
Sec. 4.4
Tubular Reactom
Rate law
r
I73 -
kc,
with k = 0.072 s-' at 1000 K
(E4-3.2)
The activation energy is 82 kcaWg rnol. 3. Stoichiometry. For isothermal operation and negligible pressure drop, the concentration of ethane is calculated as follows: Gas phase, constant T and P: Stolchiometry
c,
=
c*, x (1 + EX)
(E4-3.4)
4. We now combine Equations (E4-3.1) through (E4-3.3) to obtain Combining the drsipn equation, rate law, and
stoichiometry
5. Evaluate. Since the reaction is carried our isothermally, we can take k outside the integral sign and use Appendix A.1 to carry out our integration. Anal?.ttrraF solution
6. Parameter evaluation:
, C = Y A O ~ T O= YAQPO -=
RTo
= 0.00415
6 atm (0.73 ft3.atm/lb mol - O R )
X
(1980°R)
-
lb rnol (0.066,mo~dm') ft
'
Oops! The rate consmnt k is given at 1OOO K, and we need to calculate k at reaction conditions, which Is 1100 K.
isothermal Reactor Design
k(T,) = k ( T , ) exp
Chap. d
I),;
- -- [f; (,:
Substituting into Equation (E4-3.6) yields
For X = 0.8,
It was decided to use a bank of 2-in. schedule 80 pipes in paraIlel that are 40 ft in length. For pipe schedule 80, the cross-sectional area, 4, is 0.0205 ft2. The number of pipes necessary is The number of PFRs in parallel
To determine the concenuations and conversion profiles down the length of the reactor, x, we dtvide the volume Equation (E4-3.8) by the cross-sectional area, Ac,
Equation (Ed-3.9) was used along with A, = 0.0205 ft2, and Equations (E4-3.8)and (E4-3.3)were ured to obtain Figure E4-3.1. Using a bank of 100 pipes will give us the reactor volume necessary to make 300 rniflion pounds per year of ethylene from ethane. The concentration and conversion profiles down any one of the pipes are shown in Figure E4-3.1.
Ssc. 4.5
Pressure Drop in Reactors
ta: 0 5
10
15 20
25
30 35 40 45
50
D~stancedown the reactor z (It)
Figure E4-3.1
Pressurc drop 15 for liquid-
phaw kinetics calculations
Converxion and concentration profiles.
4.5 Pressure Drop in Reactors In liquid-phase reactions, the concentration of reactants is insignificantly affected by even relatively large changes in the total pressure. Conseqvenlly. we can totally ignore the effect of pressure drop on the rate of reaction when sizing liquid-phase chernicaI reactors. However. jn gas-phase reactions, the concentration of the reacting species is proportional to the total pressure; consequently, proper accounting for the effects of pressure drop on the reaction system can, in many instances, be a key factor in the success or failure of the reactor operation. This fact is especially true in microreactors packed with solid catalyst. Here the channels are so small (see Section 4.8) that pressure drop can limit the throughput and conversion for gas-phase reactions. 4.5.1 Pressure Drop and the Rate Law
For gas-phase reactions, precsure drop may be very lmponant
We now focus our attention on accounting for the pressure drop in the rate law. For an deal gas. we recall Equation (3-46) to write the concentrarion of reacting species i as
176
Isothermal Reactor Des~gn
Chap.
.
F,, where 0,= -
E = yA06and v is the stoichiornetric coefficient (e.g., v, =-I F*n va = -Ma). We now must determine the ratio PIPo as a function of the vol umc. or the catalyst weight, W, to account for pressure drop. We then ca combine the concentration. rate law, and design equation. However, wheneve accounting for the effects of pressure drop, the diflerential form of the moi balance (design equation) must be used. If. for example, the second-order isomerization reaction
When P + P, one mst use the differential forms
is being carried out in a packed-bed reactor, the differential form of the mol, balance equation in terms of catalyst weight is
of the PFRIPBR design equations.
dX
FAOd W
=
-rL
gram moles gram catalyst. min
1
(2-17
The rate law is
From stoichiometry for gas-phase reactions (Table 3-5),
and the rate law can be written as
Note from Equation (4-20) that the larger the pressure drop (i.e.. the smaller P from frictional losses, the smaller the reaction rate! Combining Equation (4-20) with the mole balance (2-17) and assumin; isothermal operation (T = To) gives
Sec. 4.5
Pressure Drop in Reactors
Dividing by FA" line., u,C,,) yields
For isothermal operation (T = To). the right-hand side is a function of only conversion and pressure: Another equation is needed I P .P~=.f(U1)). .
dx- - F,(X,P)
dW
We now need to relate the pressure drop to the catalyst weight in order to determine the conversion as a function of catatyst weight. 4.52 Flow Through a Packed Bed
The majority d gas-phase reactions are catalyzed by passing the reactant through a packed bed of catalyst particles.
The equation used most to calculate pressure drop in a packed porous bed is the Ergun equation:" I
\ r
Tcrm 1
Term 2
7
Ergun equation
Term I is dominant for laminar Row. and Term 2 is dominant for turbulent flow.
a
R. 8. Bird. W. E. Stewart, and E.N. Lightfoot, Transpun Phenomena, 2nd ed. (New York: Wiley. 2001), p. 200.
178
where
Isothermal Reactor Design
Chap. 4
P = pressure, lb,/ft2 (kPa)
& = porosity = I-+=
of = Void fraction total bed volume:
volume of solid total bed volume
g, = 32.174 Ib,
- Als2~lb,(convession factor)
~4.1X 7 lQX Ib;ft42.Ebf
(recall that for the metric system g, = 1.0)
D, = diameter of particle in the bed, fi (m) CI. = viscosity of gas passing through the bed. lb,/ft
- h(kg/m s)
,- = length down the packed bed ofpipe, ft (m) u = superficial velocity = volumetric flow + cross-sectional
area of pipe, Wh ( d s ) p = gas density, lb!ft"(kg/m')
G = pu = superficial mass velocity. 1b,/ft2- h ( k g h ~s)~ . In calculating the pressure drop using the Ergun equation, the only parameter that varies with pressure on the right-hand side of Equation (4-22) i s the gas density, p. We are now going to calculate the pressure drop through the bed. Because the reactor is operated at steady state, the mass flow rate at any point down the reactor. ~ j l(ksls), i s equal to the entering mass flow rate, ii,, (r.e., equation of continuity),
Recalling Equation 13-41), we have
Combining Equations (3-12)and 14-23) gives
Sec.4.5
179
Pressure Drop in Reactors
Simplifying yields
where fro is a constant down the reactor that depends only on the properties of the packed bed and the entrance conditions.
For tubular packed-bed reactors, we are more interested in catalyst weight rather than the distance :down the reactor. The: catalyst weight up to a distance of :down the reactor is
-[
Weight of catdyst
] [ =
1'
Volume o solids
Density of [ d i d cataIyst]
(4-26)
where A, is the cross-sectional area. The bulk density of the catalyst, p, (mass of catalyst per volume of reactor bed). is just the pmduct of the density of the solid catalyst particles, p,. and the fraction of solids, (1 - 4) : Bulk densiry
Pb
P C ( ~
-4)
Using the relationship between z and W [Equation (4-26)] we can change our variables to express the Ergun equation in t e n s of catalyst weight: Use this form for multiple reactions and membrane reactors.
Further simplification yields
\;
= P l Po, then
Isothermal Reactor Design
Chap.
We will use Equation (4-28) when multiple reactions are occurring ( when there is pressure: drop in n membrane reactor. However. for single real tions in packed-bed reactors, it is more convenient to express the Egun equ: tion in terns of the conversion X. Recalling Equation (3-43) for FT,
and dividing by FTO
where, as before.
Differential form of Ergun equation for the pressure dmp In packed beds.
Substituting for the ratio (FdFm), Equation (4-28) can now be written as
(4-3(
We note that when F is negative, the pressure drop AP will be less (i.e higher pressure) than that for E = 0.When E is positive, the pressure drop A will be greater than when E = 0. For isothermal operation, Equation (4-30)is only a function of convel sion and pressure:
dP -F, (X,Pj dW Two coupled equations to be solved numerically
Recalling Equation (4-2I ), for the combined mole balance, rate law, an stoichiometry,
dx-- F,(X,P )
dW
we see that we have two coupled first-order differentia1 equations, (4-31) an (4-21), that must be solved simultaneously, A variety of software packages an numerical integration schemes are available for this purpose.
I
Sec. 4.5
181
Pressure D r o ~ in Reactors
Analytical Solution. If E = 0. or i f we can neglect EX^ with respect to 1.0 (i.e.. 1 * E X 1. we can obtain an analytical solution to Equation (4-30) for isothermal operation (it..T = To). For isothermal operation with E = 0, Equation (4-30) becomes Isothermal with
I
Rearranging gives us
I
Taking y inside the derivative, we have
Integrating with y = 1 {P = Po) at W = 0 yields ( y ) 2 = 1 - aW
Taking the square root of both sides gives !
Pressure ratio only for e
=0
Be sure not to use this equation if thermaIly, where again
E#
O or the reaction is not carried out iso-
Equation (4-33) can be used to substitute for the pressure in the rate law. in which case the mole balance can be written solely as a function of conversion and catalyst weight. The resulting equation can readily be solved either analytically or numericalty. If we wish to express the pressure in terms of reactor length z, we can use Equation (4-26) to substitute for W i n Equation (4-33). Then
Isothermal Reactor Design
Chap. 4
4.5.3 Pressure Drop in Pipes Normally, the pressure drop for gases flowing through pipes without packing can be neglected. For flow in pipes, the pressure drop along the length of the pipe is given by
where D = pipe diameter, cm u = average velocity of gas, cm/s f = Fanning fnction factor G = pu, g'crn2.s
The friction factor is a function of the Reynolds number and pipe roug3ness. The mass velocity, G, is constant along the length of the pipe. Replacing u with Glp, and combining with Equation (4-23) for the case of constant temperature, T, and total molar flow rate, FT,Equation (4-35) becomes PdP
G2dP PdL
P"zIntegrating with limits P
=
Po when L
=
+
2fcI
-0
D 0, and assuming that f does not vary,
we have
Neglecting the second term on the right-hand side gives upon rearrangement
where CY
=
4fG2 .%POPOD
For the ROW conditions given in Example 4-4 in a 1000-ft length of I ;-in. schedule 40 pipe (a, = 0.01 18). the pressure drop is less than 10%. However, for high volumetric flow rates through microreactors, the pressure drop may be significant.
SM. 4.5
Pressure Drop in Reactors
Example &f
183
Cakulating Pressure Drop in o Packed Bed
Plot the pressure drop in a 60 ft length of I f -in. schedule 40 pipe packed with catalyst pellets ;-in. i n diameter. There i s 104.4 lblh of gas passing through the bed. The temperature is constant along the length of pipe at 260°C.The void fraction is 4.59 and the pmpenies of the gas are similar to thase of air at this temperature. The entering pressure is t O atm.
At the end of the reactor, r = L and Equation (4-34) becomes
Evaluating
the pressure drop paramews
For I -in. schedule 40 pipe. A, = 0.01414 f t 2 :
G=
1104.4 Ib,/h
-
= 7383.3 Ibm
0.01414 ft2
haft2
For air at 260°C and I0 atm.
From the problem statement.
D, = j
1
in. = 0.0208
fi, & = 0.45
Suhstiti~ti~~g there values into Equatlan 14-23) giver
184
Isothermal Meactor Design
[m -Term t
Ib,. h Po = 0.01244 ft . lb,
Chap.
Term 2
+
12.920.8) ]
Lbm Ib = 164.1 ft2 . h ft3
-'
(E4-4.4
We note that the turbulent flow term. Term 2, is dominant.
Po=
164.1
1 ft2 9X X 144 in.2 f13
1 atrn
14.7 Ibdin.!
Unit Conversion
for P o
I
I atm = 333kPa
ft
m
(
, = P =1 Po
- S L
)1"
0 155
=
- 1 x 6.0775' a t d R x 60 ft)' 10 atm
P = 0.26SP0 = 2.65 atm (268 kPa) AP = Po - P = I D - 2.65 = 7.35 atm (744 kPa)
,,,,,, (E4-4.7
Now let's use the data to plot the pressure and the volumetric flow rate profiles. Recalling Equation (4-34) for the case E = 0 and T = To
Equations (4-34) and (E4-4.8) were used in the construction of Table E4-4.1. TABLE E4-4.1.
For p, = 120 lb/ft3
P AND V PROFILU
Sec. 4.5
I
Pressure Drop in Reactors
The values in Table E4-4.t were used to obtain Figure Ed-4.1.
Figure E4-4.1 P and v profiles.
4.5.4
Analytical Solution for Reaction with Pressure Drop
We will first describe how pressure drop affects our CRE algorithm. Figure 4-9 shows qualitatively the effects of pressure drop or reactor design.
Figure 4-9 Effect of pressure drop on P (a), C, (b), -rA (c), X ( d l and v (el.
These graphs cornpare the concentrations. reaction rates, and conversion profiles for the cases of pressure drop and no pressure drop. We see that when there is pressure drop in the reactor, the reactant concentrations and thus reaction rate for reaction (for reaction orders greater than 0 order) will always be
186
Isothermal Reactor Design
Chap. 4
smaller than the case with no pressure drop. As a result of this smaller reaction rate, the conversion will be less with pressure drop than without pressure drop. Now that we have expressed pressure as a function of catalyst weight [Equation (4-33)]. we can return to the second-order isothermal seacdon,
to relate conversion and catalyst weight. Recall our mole balance, rare law, and stoichiometry.
FAo dx - - r; dW
I. Mole balance: 2. Rate law:
-r;
=
kc:
(2- 1 7) 14- 19)
3. Stoichiometry. Gas-phase isothermal reaction ( T = T,)with E = 0. From Equation 13-45], u = udy FA - CA,,(I - X ) ? . C, = u
Only for c=O
(4-33)
Using Equation (4-33) to substitute for y in terms of the catalyst weight, we obtain
4. Combining:
--FA0
dW
-(I-mW)dH? 5. Separating variables: F'" dx kCio (1 - X I * Integrating with limits X = 0 when W yields
Solving for conversion gives
=
0 and substituting for FAo = C,,,v,
Sm. 4.5
Pressure Drop in Reactors
Solving for the catalyst weight, we have Catalyst weight
for second-order reaction in PFR with
dP
I
Example 4-5
Eflect of Pressure Drop on the Conversion Profile
Reconsider the packed bed in Example 4-4 for the case where a second-order reaction
is taking place in 20 meters of a 1f schedule 40 pipe packed with catalyst. The flow and packed-bed conditions in the example remain the same except that they are canverted to S I units; that is, Po = 10 atm = 1013 kPa. and Entering volumetric flow rate: vo = 7.1 5 m3/h (252 ft31h) Catalyst pellet size: D, = 0.006 m (ca. i -inch)
We wed ro be able
work e~ther mehic. 5.1.. or English units. to
I
Solid catalyst denshy: p, = 1923 kg/& (120 lblfts) Cross-sectional area of I f -in,schedule 40 pipe: Ac = 0.0013 m2 Pressure drop parameter: Po = 25.8 kPdm Reactor length: L = 20 m We will change the particle size to learn it5 effect on the conversion profile. However, we will assume that the specific reaction race. k, is unaffected by particle s n e . an assumption we know from Chapter 12 1s valid only for small panicles. ta) First. calculate the conversion in the absence of pressure drop. (b) Next, calculate the conversion accounting for pressure drop. (c) Finally, determine how your answer to (b) would change if the catalyst particle diameter were doubled. The entering concentration of A is 0.1 krnoVm3 and the specific reaction rate is
k=
12mb
kmol . kg cat h
Isothermal Reactor Design
Char
Sohrrion
Using Equation (4-38)
Fur the bulk catalyst density,
The weight of catalyst in the 20 rn of I; -in. schedule 40 pipe is
(a)
First calculate the conversion for AP = 0 (i.e., a = 0)
IX (b)
= 0.82
I
Next, we calculate the conversion with pressure drop. RecaIling Equati (4-29) and substituting the bulk density pb = ( 1 - $11p, = 1058 kdm3
then
Set. 4.5
(c)
22
189
Pressure Drop in Fieactors
We see the prediccd conversion dropped from 82.2% to 69.39 because of pressure drop. It would be not only embarrassing but also an economic disaster if we had neglected presqure drop add the actual conversion had turned out to be significnntEy smaller. Robert the Worrier wonders: What if we increase the catalyst size by a factor of 2? We see fmrn Equation (E4-4.51that the second term in the Ergun equation is dominant: that is,
[.75~>>-
(84-5.5) DP
Therefore from Equation (4-25)
we have
We will learn more about Roberr !he Worrier i n Chapter t I .
We see for the conditions given by Equation (E4-4.4) that the pressure drop parameter varies inversely with the particle diameter
and thus
Fw Case 2. D4 = 2D,,
= 0.0 185 kg'
190
1
Isothermal Reactor Design
Substituting this new value of
a in Equation
Chap. 4
(GI-5.4)
By increasing the panicle diameter we decrease the pressure drop parameter and thus increase the reaction rate and the conversion. Howwer, Chapters 10 and 12 explain that when interpa~~icle diffusion effec~sare imponant in the catalyst pellet, this increase in conversion with increasing particle size wit1 not always be the case. For Imger panicles, it takes a longer time for a given number of reactant and product molecules to diffuse in and out of the catalyst particle where they undergo reaction (see Figure 10-6). Consequently. the specific reaction rate decreases with increasing particle size k ]ID, [see Equation (12-35)], which in turn decreares the conversion. At smaII panicle diameters. the rate constant, C, is large, and at its maximum varue. but the pressure drop is also large. resulting In a low rate of reaction. At large particle diameters, the pressure dmp is small, but so is the rate constant, k, and the rate of reaction, resuIting in low conversion. Thus, we see how a low conversion at both large and small particle diameters with an optimum in between. This optimum is shown in Figure E4-5.1.See Problem P4-23.
-
The variation
k-- I
D, is discussd in detail in Chapter 12. Internal d i i l m
Pressure drop dwninatss
X
D~oplimurn
DF
Fi~ureEd-5.1 Finding the optimum panicle diameter. Problems wtth large diameter t u b 3 (1) Bypassing of
catalust (2) ~ r n a i e heat r Iranqferarea
Ef pressure drop is to be minimized. w h y nor pack the ca~alysrinto n larger dinmerer rube to decrease- rhe superficial velocity. G. thereby reducing #? There are two reasons for rtof increasing the tube diameter: ( 1 ) There is an increased chance the Eas could channel and bypass most of the catalyst. resulting in little conversion (see Figures 13-2 and 13-10); {') the ratio of the heartransfer surface area to reactor volume (calalyst weight) wiIl be decreased. thereby making heat transfer more difficult for highJy exothermic and endothermic reactions. We now proceed {Example 4-6) to combine pressure drop with reaction i n a packed bed when we have volume change with reaction and therefore cannot obtain an anaIy tical solution.
Sec. 4.5
1 I The economics
The uses
191
Pressuw Drop in Reactors
Example -4-6 Calculating X in o Reactor with Pressure Drop Approximately 7 billion pounds of ethylene oxide were produced in he United States in 1997. The 1997 selling price was $0.58 a pound, amounting to a commcrcis1 value of $4.0 billion. Over 60% of the ethylene oxide produced is used ro make ethylene glycol. The major end uses of ethylene oxide are antifreeze (30%). polyester (308),surfactants (1091, and solvents (59). We want to calculate the catalyst weight necessary to achieve 60% conversion when ethylene oxide is to be made by the vapor-phase catalytic oxidation of ethylene with air.
Ethylene and oxygen are fed in stoichiometric proponions to a packed-bed reactor oprated isothermally ar 160'C. Ethxlene is fed a1 a rate o f 0.30 Ib motis at n pressure of 10 atm. It is proposed to use 10 banks of 1 !-in.-diameter schedule 40 tubes packed with catalyst with 100 tubes per bank. ConsequenrIy. the molar flow rJte to each tube IS 10 be 3 X Ib niol/s. The propenies of rhe reacting fluid am to be considered identical to hose of air at this temperature and pressure. The density of the -in.-catalyst particles i s 120 Ib/fi7and the bed void fraction is 0.45.The rate law is
- ri = kPA!Pi3
Ib mol!lb cat - h
withs
1. Differential moIe balance:
Fol;o!v~rg:he Algcrithm
(
The algnnrhm
2. Rate law:
3. Staichiornetry. Gas-phase. isothermal v = {I,,( l
F,--C,,,(1 - m f p CA -U
' 11;rl. En,?. Chetn.. 4.7. 234 (I953
_C,,,(l -XI!.
& X ) ( P , ,J P) :
P where r. = Po
1.
(E-1-6.4)
Sec. 4.5
Pressure Drop in Reactors
-
Ib X 28 -= 56.84 Ibth Ib mol
rir,, = 1.OX Ib mol X 25 Ib - 30.24 Ibth h lb rnol ?h, = 0.54 Ib mol X 32 Ib = (7.28 I b h h ib mol
h,,, = 2.03
h
The rotul mass flow rate is
Ah ha! The superficial mass velocity. temperature. and pressure are the same as in Example 4-4.Consequently. we can use the value of Po calculated rn Example 4-4. to calculate a
-- '+' 66 (a= 3.656 X I Ib cat
cat)
6. Summary. Combining Equation (E4-6.1) and (E4-6.8) and summarizing
k' = 0.0266
Ib rnol h Ib cat
-
F,40= 1.08 Ib Ib mol h a=- 0.0 f 66 I b cat k = -0.15
We have the boundary conditions W = 0,X = 0,and y = 1.0, and W,= 60 Ib. Here we are guessing an upper limit of the integration to be 60 Ib with the expectation that 60%conversion wiII be achieved with~nfhis catalyst weight. If 60% conversion is not achieved, we will guess a higher weight and redo the caiculation. A large number of ordinary differential equation soIver software packages (i.e., ODE solvers). which are extremely user friendly, have become available. We
194
isothermal Reactor Design
Chap. 4
shall use Polymathb to solve the examples In the printed text. With Polymath, one simply enters Equations lE4-6.9) and (E4-6.101 and the corresponding parameter values [Equations (4-6.1 1) through (4-6.14)J into the computer with the boundary conditions and they are solved and displayed as shown in Figures E4-6.l and Ed-6.2. Equations (E4-6-9) and (E4-b.IO) are entered as differential equations and the parameter values are set using explicit equations. The rate law may be netered as an explicit equation in order to generate a plot of reaction rate as it changes down the length of the reactor. using Polymath's graphing function. The CD-ROM contains all of the MatLab and Polymath solution prograins used to solve the example problems, as well as an example using ASPEN. Consequently. one can load the Pol)math program directly from the CD-ROM, ahich has programmed Equations (E46.9) through (E4-6.14), and run the propram for different parameter valueh. It is also interesting to learn what happens to the volumetric flow rate along the length of the reactor. Recalling Equation 13-45),
Program examples
Polymath. MATLAB can be loaded from the CD-ROM (see the Inlroduction).
v = v , ( I + E X ) - P-o- T - voC1 + E X ) I T I T ~ ) PIP, p 7-0
1 Volumetric Row rate increases with increasing
We let $be the ratio of the volumetric flow rate, v. to the entering volumetric fiow rate. v , . st any point down the reaclor. For isothermal operation Equat~on(3-45) kcome5
preqsvre drop.
TABLE E4-6.1 POLYMATH PROGRAM
ODE REPORT (STIFF)
Di~erantialequations as entered by the user [ I ;; d(X)ld(W) = -raprimelFao [ 2 1 d(y)/d(W) = -alpha"(l+eps*X)#y Explicit equations as entered by the user [I] eps=-0.15 12j kprime = 0.0266 [ I ! Fao= 1.08 t 3 1 alpha = 0.0166 [ 5 j raprime = -kprime8(l-X)/( t +eps*X)*y [ 6 j f = (1ceps'X)ly !7 1 rate = -raprime
Living Example Problem There is a Pa!ymath ODE tvlor~illin Chapier I Sr~rnmnryXcltes.
-
-
" Developed by Proferhor M. Cutlip of the University of Connec~icut,and Profersor M Shacharn of Ben Guriun Unizzrsi~y.Available from the CACHE Corporation. P.0. Box 7939. Aust~n.TX 7R713.
Sec. 4 6
Using C, (liquid) and F, (gas) in the Mole Balances and Rate Laws
195
Figure E4-6.2 shows X,y (it., y = PJP,), and f down the length of the reactor. We see that both the conversion and the volumetric flow increase along the Iength of the reactor while the pressure decreases. For pas-phase reactions wrth orders greater than zera, this decrease in pressure will cause the reaction rate to be less than in the case of no pressure drop. Program examples Polymath, MATLAB can be loaded from the CD-ROM (see the Inrroduciionf. $cak. l-BM) Y: to2 Key: 1.m
-rate 0.600
Figure M 4 . l
Reaction rate prnfile down the PBR.
Figure Ed-6.2
Output in graphical form from Polymath.
From either the conversion profile (shown in Figure E4-6.2) or the Pol?niath table of results (nor shown in text. but available on the CD), we find 60% conversion is achieved with 44.5-Ib catalyst in each tube. We note from Figure E4-6.2 that the catalyst weight necessary to n i s e the conversion the last 1% from 65% to 66% (3.5 Ib) 1s 8.5 times more than that (0.41 Ib) required to raise the conversion 19 at the reactor's entrance. Also, during the last 5 % increase in converston, the pressure decreases from 3.8 atm to 2.3 atm. This catalyst weight of 44.5 lbltube corresponds to a pressure drop of approximately 5 atm. If we had erroneously neglected pressure drop, the catalyst weight would have been found by integrating equalion (E4-6.9) with y = 1 to give
Effecl of added cataryst on conversion
pressure drop results in poor des~gn(here 5 3 9
I Embarrassing !
= 35.3 Ib. of cataryst per tube {neglecting pressure drop) ( I 6 k p h b e )
IF we had used this catalyst weight tn our reactor we would have had insufficrent catalyst to achieve the desired conversion. For this catalyst weight li.e., 35.300 Ib total. 35.3 lbJtubel Figure Ed-6.2 grves a conversion of only 53%.
196
Isothermal Reactor Design
Ck
4.5.5 Spherical Packed-Bed Reactors
I
Let's consider carrying our this reaction i n a spherical reactor similar tc one shown in the margin and discussed in detail in the CD-ROM. In a sp! caI reactor, the cross section varies as we move through the reactor ar greater than in a normal packed-bed reactor. Consequently, the superficial r velocity G = tn/Ac will be smaller. From Equation (4-22), we see th smaller value of G will give a smaller presrure drop and thus a greater cor sion. If 40,000 Ib of catalyst in the PBR fn Example 4-6 had been used - spherical reactor, 67% conversion tvouId have been achieved instead of I conversion. The equations for calcuIating conversion in spherical reac along with an example problem are given in the Professio~~al Rqferencc 5 R4.1 for Chapter I on the CD-ROM.
4.6 Synthesizing the Design of a Chemical Plant Reference Shelf
Synthesizing a
chernrcal plant
Always
boundaries of the
S$$S
CarefuI study of the various reactions, reactors, and molar Rows of the n tants and products used in the example problems in this chapter reveals they can be arranged to form a chemlcal plant to produce 200 million pou of ethylene glycol from a feedstock of 402 million pounds per year of ethi The flowsheet for the arrangement of the reactors together with the molar f rates is shown in Figure 4-10. Here 0.425 lb molls at' ethane is fed t o 100 t t 1ar plug-flow reactors connected in pardlel; the total volume is 81 ft3 t o I duce 0.34 Ib moYs of ethylene (see Example 4-3). The reaction mixture is t fed to a separation unit where 0.04 Ib mays of ethylene is lost in the separal process in the eLhane and hydrogen streams that exit the separator. This cess provides a molar flow rate of ethylene of 0.3 lb molls, which enters packed-bed catalytic reactor together with 0.15 lb rnol/s of Or and 0.564 molJs of N2. There are 0.18 Ib molls of ethylene oxide (see Example 4-6) duced in the 1 0 0 pipes arranged in parallel and packed with silver-coated r alyst pellets. There is 60% conversion achieved in each pipe and rhe ct catalyst weight in all the pipes is 44,500 Ib. The effluent stream is passed t separator where 0.03 lb molls of ethylene oxide is lost. The erhylene ox stream is then contacted with water in a gas absorber to produce a I-Ib mol solution of ethylene oxide in water. In the absorption process, 0.022 !b mr of ethylene oxide is lost. The ethylene oxide solution is fed to a 197-ft3 CS together with a stream of 0.9 wt 41c H2S0, solution to produce ethylene gly at a rate of 0.102 lb molls (see Example 4-21, This rate is equivalent to apprl irnately 200 miIlion pounds of ethykne glycol per year. The profit from a chemical plant will be the difference between income fr sales and the cost to produce the chemicals. An approximate formula might be Profit =Value of products - Cost of reactants
- Operating cost - Separation costs
Seo. 4 6
1
1
Stream
Using GA (Ifqujd)aEd
~ornponenta
Air
6 (gas) in the Mole Balances and
Rate Laws
197
F ~ W rate {ib mob's)
(
Stream
CaqonenP
FIW rats im mows)
1
11.714
]
P
EG
0.102
1
I
1
=EG, elhylene glycol: EO, ethylene oxlde.
Figure 4-10 Production of ethylene glycol.
The operating costs incIude such costs as energy, labor, overhead. and depeciation of equipment. You will learn more about these costs in your senior design course. While most, if not all. of the streams from the separarors could be recycled, lets consider what the profit might be if the streams were to go unrecovered. Also, let's conservatively estimate the operating and other expenses to be $8 million per year and calculate the profit, Your design insmcror might give you a better number. The prices of ethane, sulfusic acid, and ethylene glycol are $0.04, 30.042, and $0.38 per pound, respectively. See www.chemweek. c o d for current prices.
198
Isothermal Reactor Design
Chap. 4
For an ethane feed of 400 million pounds per year and a production rate of 200 million pounds of ethylene glycol per year:
-
Ethylene glycol cost
Profit =
Ethane cost
year
year
-
-
~ulfbricacid cost fJ
Operating cost
,\
#-
= $76.000.000 - 5 16,000,000 - $54,000 - $8,000,000
$52 million Using $52 million a year as a rough estimate of the profit, you can now make different approximations about the conversion. separations. recycle streams, and operating costs to learn how they affect the profit.
PART2 Mole Balances Written in Terms of Concentration and Molar Flow Rates h4ultipIe rxns Membranes
There are many instances when it i s much more convenient to work in terms of the number of moles (N,, NB) or molar flow rates (FA, F,, etc.) rather than conversion. Membrane reactors and multiple reactions taking place in the gas phase are two such cases where molar flow rates are preferred rather than conversion. We now modify our algorithm by using concentrations for liquids and molar flow rates for gases as our dependent variables. The main difference between the conversion algorithm and the molar flow ratelconcentration algorithm is that. in the conversion algorithm, we needed to write a mole halance on only one specie.^, whereas in the molar flow rate and cancentrarion algorithm, we muzt write a mole balance on each and eve? .~pecies.This algorithm is shown in Figure 4-1 1. First we write the mole balances on all species present as shown in Step Q. Next we write the rate law. Step @. and then we relate the mole halances to one another through the relative rates of reaction as shown in Step @. Steps O and G are used to relate the concentrations in the rate law to the molar flow rates. In Step 8. all the steps are combined by rhe ODE soIver (e.g., Polymath).
Sec. 4.6
Using CA(liquid) and F, (gas) in the Mole Balances and Rate Laws
Mole Balance
(a Write mole balance on each species7
@ ( Writt rate law in terms of concenrration
Rale Law
199
)
f @ Relate the rates of reaction of each species to one nno~her
Stoichiometry
Stoichiometry
f
@
Ia) W r ~ t ethe concentrations in term7 of molar ffna, rates for i\orhermal gas-plmre reactions
1 Pressure Drop
with FT=FA+Fa +F;.
[ dwtyF~,,1 @
Wntc the gas-phase pressure drop term ~n term+ of molar flow rates
-dy
=-a F,.with
=-
l:
Conibine
I
i
@ Use i an ODE solver or a nunlinear equation solver tc.p..
Polymath) to combine Steps @ through @ to sr,lve for. for example, the profiles of molar flow ram,concentration and pressure.
Figure 4-11
I~otherrnalreac~iondesign algorith~nfor mole balances.
200
Isothermal Reactor Design
Chap
4.7 Mole Balances on CSTRs, PFRs, PBRs, and Batch Reactors 4.7.1
Liauid Phase
For liquid-phase reactions in which there is no volume change. concentratic is the preferred variable. The mole balances for the generic reaction
are shown in Table 4-5 in terms of concentration for the four reactor types rf, have been discussing. We see from Table 4-5 that we have only to specify tf parameter values for the system (CAO,v0, etc.) and for the rate law paramete (e.g., k,, a, p) to solve the coupled ordinary differential equations for eithc PFR, PBR, or batch reactors or to solve the coupled algebraic equations for CSTR.
TABLE4-5.
MOLEBALANCES FOR LIQUID-PHASE REA~OSS
PFR
v,- d C < - r*
PBR
dV
and
dCa PO-
dV
- b-or ,
dC, br' I
and
' c i ~ a "
4.7-2 Gas Phase
The mole balances for gas-phase reactions are given in Table 4-6 in terms o number rnoIes (batch) or molar Row rates for the generic rate law for t h
Sec 4.6
Usmg CA(hqu~d)and F, (gas) In the Mole Balances and Rate Laws
201
generic reaction Equalion 12-1). The molar flow rates for each species F, are obtained from n mole balance on each species, as given in Table 4-6. For example, for a plug-flow reactor d FI -
Murr write a maIc balance on each species
dL'
The generic power law rate law is B
- r A = kAc:cB
Rate law
To relate concentrations to molar flow rates, recall Equation (3-421, with = P/P"
The pressure drop equation, Equation (4-28). for isothermal operation (T= Tu)
is -dv = - A-a
dW
2y
FT
F,,
The total molar flaw rate is given as the sum of the flow rates of the individual species:
when species A, B, C,D, and I are the only ones present. Then FT=FA+FB+Fc+FD+Fl We now combine a11 the preceding information as shown in Table 4-6.
I
4.8 Microreactors Microreactors are emerging as a new technology in CRE. Microreactors are characterized by their high surface area-to-volume ratios in their microstructured regions that contain tubes or channels. A typical channel width might be 100 prn with a length of 20,000 p n (2 cm). The resulting high surface area-tovolume ratio (ca. 10,W mZJm3reduces or even eliminates heat and mass
Isothermal Reactor Design TABLE 4-6.
ALGORITHMFOR GAS-PHASE REAC~IOSS
oA+bB
+c C + d D
1. Mole balances:
Bulurch
CSTR
PFR
2, Rate L a w : -rA =
kAqcI
3. Stoichiometry:
Relative raes of reaction:
then
Concentrationr:
Gas phaw
Total
molar Rev rate F , = F , + S ,
-
Fc- F ,
4. Combine: For an icothemal operntion nf a
+ F,
PBR urihno AP
1. Sprcifv paramcrrr ralues: I;,,. C , , , . u ,0. T,.n.b, r,.rl
1. Specify cniering numbers: F,.,.f I . Use an ODE solver.
,,,,.F , , , .F,,
and final vnluea: t',ir,da
Chap. 4
Sec. 4.6
Using C, (tiqu~d)and F, (gas) In the Mole Balances and Rate Laws
203
transfer resistances often found in larger reactors. Consequent1y, surfacecatalyzed reactions can be greatly facilitated, hot spots in highly exothermic reactions can be eliminated, and in many cases highly exothermic reactions can be carried out isothermally. These features provide the opportunity for rnlcmreactors to be used to study the intrinsic kinetics of reactions. Another Advantages of microreactors advantage of microreactors is their use in the production of toxic or explosive intermediates where a leak or microexplosion for a single unit will do minimal damage because of the srnali quantities of material involved. Other advantages incIude shorter residence times and narrower reridence time distributions. Figure 4-12 shows (a) a microreactor with heal exchanger and (b) a microplant with reactor, valves, and mixers. Heat, Q,is added or taken away by the fluid flowing perpendicular to the reaction channels as shown in Figure 4-12(a). Production in microreactor systems can be increased simply by adding more units in parallel. For example, the catalyzed reaction
required only 32 microreaction systems in parallel to produce 2000 tonslyr of acetate!
Microreactor (a) and Micraplnnt (b, Couneq nf Ehrfeld. IIes~el,and Cdne. Micn)rcacrors: IVPw ;rfrltnnkofl fnr h f d rrn CI?ern~ rrc ( Wiley-VCH. 2000).
Figure 4-12
Microreacrors are also used for the production of specialty chemicals, combinatorial chemical screening. lab-an-a-chip. and chemical sensors. In modeling micmreactors. we will assume rhey are either i n plug flow for which the mole balance i s
204
isothermal Reactor Design
Char
or in laminar flow, in which case we wiil use the segregation model discuss in Chapter 13. For the plug-flow case. the algorithm is described in Fig1 4-1 I. Example 4-7 Gas-Phase Reaction in a Mfcroreactor--Molar Flow Rates
The gas-phase reaction
is camed out at 425°C and 1641 kPa (16.2 atm). Pure NOCl is to be fed, and t reaction follows an elementary rate law? It i s desired to produce 20 tons of NO I year in a microreactor system using a bank of ten microreactors in parallel. Ea microreactor has 100 channels with each channel 0.2 mrn square and 250 rnm length.
Plot the molar flow rates as a function of volume down the length af the reactor, T i voEume of each channel is le5dm'. Additional Infomarion
To produce 20 tons per year of NO at 85% conversion would require a feed rate 0,0226 moVs of NOCl, or 2.26 x ID-* molls per channel. The rnte constant is
Solurion
For one channel.
Find K
fi
J. B. Butt, Reaction Kinefics and Reactor Design, 2nd ed. (NewYork: Marcel Dekke 2001). p. 153.
Alrhnugh this particular problem could be solved using conversion, we shatl illustrate how it can also be solved wing molar flow rates as the variable in the mole balance. We f i r ~ twrite the reaction i n symbolic form and then divide by the srotchinrnetric coefficient of the limiting reactant, NOCI.
SNOC I -2NO
+ Ct,
2A+2B + C A+B+;C
1. Mole balances on species A, B, and C:
2. Rate law:
FT 3. Stoichiometry: Gas phase with T = To and P = Po,then v = v, a. Relative ntes F~~
b.
Concentration Applying Equation (3-42) to species A, 3,and C,the concentrations ;Ire
with FT = F A + F, + Fc
206
Isothermal Reacfor Design
Chap. 4
4. Combine: the rate law in terns of molar R o w rates is
1
combining all
One can skip the combine step when using Polymath because Polymath or simirar ODE solvers combine everything for you: mole balance, rate law, and stoichiometry.
5. Evaluate:
cT O -- - P- o RT"(8314
( 1 64 1
kPa)
emol) - ,K O R
mol
= 0.28b7
=
0.286 mmol
dm-
cm
'
When using Polymath or another ODE solver, one does not have to actually combine the mole balances, rate laws, and stoichiornetry as was done in the combine step in previous examples in this chapter. The ODE soIver will do that for you. Thanks, ODE solver! The Polymath Program and output are shown in Table E4-7.1and Figure E4-7.1. TABLEEd-7.I .
POLYMATHPROGRAM
ODE REPORT (BKF45) Differential equations as entered by the user : 11 d(Fa)ld(V)= ra 12 d(Fb)/d(V)= rb t 3 ] d(Fc)/d(V) = rc
Explicrt equations as entered by the user t : ] T=698 i 2 l Cto= 164118.314rT L '1 I E = 24000 i ;] f t = FatFb+Fc I5 I Ca = Cfo'Fa/ft I5 I k = 0.29'exp(E/l.987*(1/500-1 n)) , :I Fao = 0.0000226 [4:
I
YO
= FaoICto
=5
\'(dm')
Figure FA-7.1 Profile? of microreactor rnotar flow rates
Sec. 4.9
Membrane Reactors
207
TABLE m-7.1.POLYMATH PROGRAM (CO~TINUED) ODE REPORT (RKF45) (Continued)
4,9 Membrane Reactors
By having ane o f Ihe Pmducts
PaFs
throughour the membrane. we driGC the reactlon towad completion
Membrane reactors can be used to increase conversion when the reaction is therrnodynamicaIly limi led as well as to increase the selectivity when multiple reactions are occurring. Therrnodynamically limited reactions are reactions where the equilibrium lies far to the left (i.e.. reactant side) and there is little conversion. If the reaction is exothermic, increasing the temperature will only drive the reaction further to the left, and decreasing the temperature will result in a reaction rate so slow that there is very little conversion. If the reaction is endothermic. increasing the temperature will move the reaction to the right to favor a higher conversion: however, for many reactions these higher remperatures cause the catalyst to become deaclivated. The term metnbmne reacfnr describes a number of different types of reactor configurations that contain a membrane. The membrane can tither provide a barrier to certain components while being permeable to others, prelent certain ComponenZs such as particulates from contacling the catalyst, or contain reactive sites and be a catalyst in itself. Like reactive distillation. the membrane reactor is another technique for driving reversible reactions to the right toward completion in order to achieve very high conversions. These high conversions can be achieved by having one of the reaction products diffuse out of a semipermeable membrane surrounding the reacting mixture. As a result, the reverse reaction wilI not be able tn take place. and the reaction will continue to pmceed to the right toward completion. Two of the main types of catalytic membrane reactors are shown in Figure 4-13. The reactor in Figure 4-13(b) is called an inert nro?tbratte rencror ~ i r cafal.wt l ~ p ~ l l e r son rhe feed side (TMRCF). Were the membrane is inert and senes as a harrier to the reactants and some of the products. The reactor in Figure 3-1 3(c) is a catnlj?ic ~nellrbranerencfmr (CMR j. The catalyst is deposited directly on the membrane, and only specific reaction products are able to exit the permeate side. For example, in the reversible reaction
Permeate EH21
Inert Membrane Feed
IC,H,,) Catalyst Particles --
IMRCF
Permeate u-2)
Catalyst Membrane
Feed
IC,H,z)
CRM
V R,
Figure 4-13 Membrane reactor%.(Photo courtesy of Coon ceramic^, Golden, Colorrtdo.l (a) Photo of ceramic reactors. (b) cross section of 1MRCE ( c ) cross section of CRM. (d) schemat~cof IMRCF for mole baIance.
Sec. 4.9 H: diffurec through the nrernbrane u h ~ l eC,H, doc\
,,
209
Membrane Reactors
the hydrogen molecule is small enough to diffuse through the small pores of the membrane while C,H,, and C,H, cannot. Consequently, the reaction continues to proceed to the right even for n small value of the equilibrium constant. Sweep
Gas
-
--,
Hydrogen. species 8, Rows out through the sides of the reactor as it flows down the reactor with the other products, which cannot leave until they exit the reactor, In analyzing membrane reactors. we only need to make a small change to the algorithm shown in Figure 4- 1 1. We shall choose the reactor voIume rather than catalyst weight as our independent variable for this example. The catalyst weight, W. and reactor volume, I! are easily related through the bulk catalyst density, p, (i-e.. W = p,W. The mole balances on the chemical species that stay within the reactor, nameIy A and C.are shown in Figure 4-1 1 and also in Table 4-6.
The mole balance on C is carried out in an identical manner to A. and the resulting equation is
However. the mole bafance on B (Hz) must be modified because hydrogen leaves through both the sides of the reactor and at the end of the reactor. First we shall perform mole balances on the volume element AV shown in Figure 4-121~).The mole balance on hydrogen (B)is over a differential volume AW shown in Figure 4-12(d) and it yields Bnhncc on B in the caral~ricbed: Out Now there arc two " O W terms For species B.
[by
!OW]
F~l"
- [by:;bw] - F + -
'
-
I
+[Generation] =[Accumulation]
[by diffusion]
R,~v'
+ A
=o
210
Isothermal Reactor Design
Chap. 4
where RB is the molar rate of B leaving through the sides the reactor per unit volume of reactor (molldm3 s). Dividing by AV and taking the limit its AV -+ 0 gives +
The rate of transport B out through the membrane R, is the product of the molar flux of B, U7,, and a, the surface area, per unit volume of reactor. The molar flux of B. WE in (mollm' s) out of the reactor is a mass transfer coefficient times the concentration driving force across the membrane. 4
7
8
4
LB
Where kl, is the overall mass transfer coefficient in m l s and CBSis the concentration of B in the sweep gas channel (mol/drn3). The overall mass transfer coefficient accounts for all resistances to transpon: the tube side resistance of rhe membrane, the n~emhraneitself, and on the shell (sweep gas} side resistance. Further elaboration of the mass transfer coefficient and its correlations can be fuitnd in the literature and in Chapter 1 1. In general. this coefficient can be a function of the membrane and fluid properties. the fluid velocity, and the tube diameters. To obtain the rate of removal of B. we need to multiply the flux through the membrane by the surface area of membrane in the reactor. The rate at which 3 is removed per unit volume of reactor. RB.is just the flux \I/, times thc surface area membrane per volume of reactor, a (m2/m"): that is,
The membrane silrface area per unit volume of reactor is
Area a=----vO'ume
-
7iDL
-
4
D
T R-~ 4
Le~tingk , = k;. n and assuming the concentration in the sweep gas is essentially zero (i.e.. CBS= 0). we obtain Rate of R out
thmuph rhe s ~ d e ~ .
k-%cl
where the units of k , are s-I. More detailed modding of the transport and reaction steps in membrane reactors is beyond the scope of this text hut can be found in Menihrane Rmcrnr Techr?oEr~~y' The salient features. however, can be illustrated by the follo~)in& example. When analyzing membrane reactors. i t is much inore convenient to use molar flow ratec than conversion. +
'R. Govind. and N.Itoh. ed\ . Mr~rnhrrrrrc.R~rrrl o x fi~rl~irolng~: AIChE S>mpo\iuin Scrim Nn. 268. Vol XS i 19891.T.Sun and S. Khanp. Irtrl. file. Thetrl Re\.. 177. 1 13h ( 19881.
Sec. 4.9
According to the DOE, 10 trillion BTUlyr could bc saved by using membrane reactors.
I
Membrane Reactors
fiatttpb M I
Membmne Reactor
According to The Department of Energy (DOE),an enerWy saving of 10 trillion BTU per year could result from the use of catalytic membrane reactors as replacements for conventional reactors for dehydrogenation reactions such as the dehydrogenation of ethylbenzene to styrene:
and of butane to butene:
The dehydrogenation of propane is another reaction that has proven successful with a membrane react~t.'~
All the preceding dehydrogenation reactions above can be represented symbo~ically as
I
and will take place on the catalyst side of an IMRCF. The equilibrium constant for this reaction is quite small at 227°C (e.g., Kc = 0.05 mol/dm3).The memhrane i s permeabfe to B (e.g., HI) hut not to A and C. Pure gaseous A enters the reactor at 8.2 atm and 227% at a rate of 10 moWmln. We will take the rate of diffusion of B out of the reactor per unit volume of reactor, R,, to be proportional to the concentration of B (i.e.. RE = kCCB). (a) P e r f o n differential mole balances on A, B,and C to amve at a set of coupled differential equations lo sohe. (b) Plot the molar flow rates of each species as a function of space time. (c) Calculate the conversion. Additional information: Even though this reaction is a gas-solid camlytrc reactton. we win use the hulk catalyst density in order to write our balances in terms of reactor volume rather than catalyst weight (recall -r, = --rAp,). For the balk catalyst density of p, = 1.5 dcm3 and a 7-crn inside diameter of the tube containing the cafaIyst pellets, the specific reaction rate. k. and the transport coefficient, kc, are k = 0.7 mln-I and kc = 0.2 min-I respectively.
.
1
We shall choose reactor volume rather than catalyst weight a5 our independent vanable for thir example. The catalysl weight, lY and reactor volume. I< are easily
212
Isothermal Reaclor Design
Chap.
related through rhe bulk catalyst density, ph. (~.e.,W = p,V). First. we shall perfor mole balances on the volume element A Y shown in Figure 4- l3(d). 1 . Mole balances: Bulnrlce
on A in rhc ccltalytic bed:
[ by Inflow] [
1
Out
by flow]
[ ene era ti on] = [ ~csumulation]
Dividing by & Y and taking the limit as AV+Q gives
Balance on B in the catalytic bed: The balance on B is gwen by Equation (4-41).
where RB is the muIar Row of 3 out through the membrane per unit volun of reactor. The mole balance on C is carried out in an identical manner to A ar
the resulting equation is
2. Rate law:
3. Transport out of the reactor. We apply Equation (4-42) for the case which the concentration of B of the sweep side is zero. C,, = 0, to obtain
where kc is a tnnspon coemcient. In this example, we shall assume that 11. resistance to species B out of the membrane is a constant and. consequent1 kc is a constant. 4. Stoichiometry. Recalling Equation ( 3 4 2 ) for the case of constant temper: ture and pressure, we have for isothermal operation and no pressure drc ( T = T , , P=PJ.
Sec. 4.9
I
Membrane Reactors
Concentrations:
C, = C ,,
FB -
c,=c,-
Fr FT
Fr
F, = F , + F , t F ,
Relative rates:
I
5. Combining and summarizing:
Summary of equations describing Row and reaction in a
membrane reactor
6. Parameter evaluation:
7. Numerical solution. Equations (E4-8.1)through (E4-8.10)were solved using Polymath and MATLAB. another ODE solver. The profiles of the molar flow rates are shown here. Table E4-8.1 shows the Polymath programs.
214
Isothermal Reactor Design
Chap. 4
md Figure E4-8.1 shows the results of the numerical solution of the initial (entering) conditions.
TABLE E4-8,l
POLYMATH PROGRAM
ODE REPORT (RKF45) Differential equations as entered by the user [ 11 d(Fa)/d(V) = ra r z d(Fb)/d(V) = -ra-kc'Cto'(FhlFt) [ 3 1 d(Fc)/d(V) =-ra
r
Explicit equations as entered by the user [ 1 1 kc= 0.2 I21 Cto=0.2 [ 3 j Ft = Fa+Fb+Fc [dl k=0.7 [ S J Kc=0.05 r 6 1 ra = -k'Cto9((FdFt)-CtolKc'(Fb/Ft)'(FclFt))
Fc
-
F, (moVmin)
-..--.._ FB
---**-.-.._ -.-.. --=.-.__
0.000
I
I
0.000 100.000
I
I
I
I
I
200.000 300.000 400.000
500.00
V (dm3)
Figute W3.l Polymath solution. (c)
From Figure Ed-8.1 we see the exit molar flow rate of A is 4 mollmin. for w h ~ c hthe correspondtng conversion is
Sec. 4.10
Unsteady-State Operatton d Stirred Reactors
215
Use of Membrane Reactors to Enhance Setedivity. In addition to species leaving the membrane reactor, species can also be fed to the reactor through the membrane. For example, for the reaction A+B-+C+D A could be fed only to the entrance, and B could be fed only through the membrane as shown hex.
As we wiIl see in Chapter 6, this arrangement is often used lo improve selectivity when multiple reactions take place. Here B is usually fed uniformly through the membrane along the length of the reactor. The balance on B is
where R R = Fs& with FRO the molar feed rate of B thmugh sides and V, the total reactor volume. The feed rate of B can be controlled by controfling the pressure drop across the reactor membrane.If
4. I0 Unsteady-State Operation of Stirred Reactors In this chapter, we have already discussed the unsteady operation of one type of reactor. the batch reactor. In this section. we discuss two other aspects of unsteady operation: startup of a CSTR and semibatch reactors. First, the startup of a CSTR is examined to determine the time necessary to reach steady-state operation [see Figure 4-141a)l. and then semibatch reactors are discussed. In each of these cases, we are interested in predicting the concentration and conversion as a function of time. Clo%ed-formanalyl~calrolutions to the differential equations arising fmm the mole balance of these reaction type< can be obtained only for zero- and first-order reactions. ODE solvers muss be used for other reaction orders.
T'he velocity of B through the membrane, U,, i s given by Darcy's law
-
U s = K f P , P,) where K i s the rnemhrane permeability and P, i~ the shell-~idepieqaurr and P, the reactor side pressure. , RB , F D , = C,,,,rrU" I?, = R B V , where, as before, rr IS the membrane surface area per unit ~olume.CH,,i \ the enlertng concentratir~nof 3. and V, 1s the ~oialreaclor \~olunie.
Isothermal Reactor Design
Ch,
Figure 4-14
Semibatch reactom. (a) Reactor startup, (h) semihatch with cwling. and ( c ) reactive dist~llntion.[Excerpted by specla1 permission frum Chrm. Efl~.. 631 10) 2 1 1 (Ocl. 1956). Copyright 1956 by McGraw-Hill. Inc.. New York. NY !oo2n.1
There are two basic types of semibatch operations. In one type. ont the reactants in the reaction
An expanded version of thrs sectton can be found on the CD-ROM.
summary Notes
(e.g., B) is slowly fed to a reactor containing the other reactant (e.g.. A). wf has already been charged to a reactor such as that shown in Figure 4-14 This type of reactor is generally used when unwanted side reactions occu high concentrations of B (Chapter 6) or when the reaction is highly exothen (Chapter 8). In some reactions, the reactant B is a gas and is bubbled conti ously through Iiquid reactant A. Examples of reactions used in this type semibatch reactor operation include arnrnonolysis. c/~Iurinatinn,and hydmly The other type of semibatch reactor is reactive distillation and is shown sc maticaIly in Figure 4-14rc). Here reactants A and B are charged simu neously and one d the products is vaporized and withdrawn continuou Removal of one of the products in this manner (e.g., C) shifts the equilibri toward the right, increasing the final conversion above that which would achieved had C not been removed. En addition, removal of one of the p m d ~ further concentrates the seacmnr. thereby producing an increased rate of re tion and decreased processing time. This type of reaction operation is cal reactive distillation. Examples of reactions carried out in this type of reac include ace@lation reaciions and e,rter$cc1tion reactions in which water
removed. 4.1 0.1 Startup of a CSTR
The startup of a fixed voIurne CSTR under isothermal conditions is rare, bu does occur occasionally. We can, however, carry out an analysis to estimate time necessary to reach steady-state operation. For the case when the react01 well mixed and as a result there are no spatial variations in r,, we begin m the general mole balance equation applied to Figure 4-I4(a):
Sec. 4.10
21'7
Unsteady-State Operat~onof St~rredReactors
Conversion does not have any meaning in startup because one cannot separate the moles reacted from the moles accumulated in the CSTR. Consequently, we
must use concentration rather than conversion as our variable in the balance equation. For liquid-phase ( u = u,) reactions with constant overflow ( V = V o ) , using r = Vo/u,, we can-transform Equation (4-45) to
F1r5t-order
For a first-order reaction (- r, =
Equation (4-46) then becomes
which, for the initial conditions CA = CADat t = 0 solves
Letting t, be the time necessary tion, CAs:
to
TO
reach 99% of the steady-state concentra"
Rearranging Equation (4-47) for CA = 0.99Cks yields
For slow reactions with small k ( I >> tk):
For rapid reactions with large k ( ~ r> k 1): Time to reach rtendy Qtate in an t=,othemal CSTR
Fur most first-order systems. steady smte is achieved in three times.
to
four spnce
4. t 0.2 Semibatch Reactors
Motivation One of the best reasons to use semibatch reactors is to enhance selectivity in liquid-phase reactions. For example, consider the following two simuitaneous reactions. One reaction produces the desired product D
lsofhermal Reactor Design
Chap. 4
with the rate law I
r,
=
kc, CB
.
and the other produces an undesired product U
with the rate law
ru
= ~,C,C;
The instantaneous selectivity SDR! i s the ratio of the relative rates We want SDW as large as possible.
and guides us how to produce the most of our desired product and least of our undesired product {see Section &I). We see from the instantaneous selectivity that we can increase the formarion of D and decrease the formation of U by keeping the concentration of A high and the concentration of B low. This result can be achieved through the use of the semibatch reactor. which is charged with Pure A and to which B is fed slowly to A in the vat. Of the two types of semibatch reactors, we focus attention primarily on the one with constant molar feed. A schematic diagram of this semibatch reactor is shown in Figure 4-15. We shall consider the elementary liquid-phase reaction
A + B-C
-u Figure 4-15
in which reactant
Semihatch reactor.
B is slowly added to a well-mixed
vat containing reactanl.4.
Sec. 4.10
Unsteady-Skate Operation of Stirred Reactors
A mole balance on specie A yields
Three variables can be used to formulate and solve semibatch reactor problems: the concentrations, Cj,the number of moles, 4, and the conversion, X. 4.10.3Writing the Semibatch Reactor Equations in Terms of Concentrations Recalling that the number of moles of A, N,, is just the product of concentration of A, C,, and the volume, I!we can rewrite Equation (4-51) as
We note that since the reactor is being filled, the volume, X varies with time. The reactor volume at any time r can be found from an overall mass balance of all species: OvewlJ mas< balance
[+];;[-]:R[
"akof]=[
generation
hk0f
]
accumulation
(4-53)
For a constant-density system, p, = p , and
with the initial condition V = Vo at I = 0,integrating for the cast of constant volumetric flow rate uo yields Sem~batch reactor votume as a Function of tlme
-
(4-555
Substituting Equation (4-54) into the right-hand side of Equation (4-52) and rearranging gives us
220
Isothermal Reactor Design
Chaf
The balance on A &e.. Equation 14-52)] can be rewritten as (4-5
Mole balallce on A
A mole balance on B that is fed to the reactor at a rate FBOis
In
+
Out
+
Generation
=
Accumulation
Rearranging
Substituting Equation (4-55) in terms of V and differentiating, the mole bi a w e on B becomes Mole balance on B I
1
At time r = 0, the initial concentration of B in the vat is zero, CB,= 0. The conce (ration of B in the feed is CBwIf the reaction order is other than zero- or first-od or if the reaction is nonisothermal. we must use numerical techniques to detemi the conversion as a function of time. Equations 14-56]and (4-58) are easily solvq with an ODE soiver. h m p l e 4-9 Isothermal Semibatch Reactor with Second-Order Reaction
The production OF methyl bromide is an imversibtt liquid-phaqe reaction that fc lows an elementary rate law. The reaction
is carried out isothermnlly in a semibatch reactor. An aqueous solution of meth amine (B)at a concentration of 0.025 rnoVdm3 1s to be fed at a rate of 0.05 d m to an aqueous solution of bromine cyanide (A) contained in a glass-Iined react( The initial volume of fluid in a vat is to be 5 dm) with a bromine cyanide conce tntion of 0.05 rnolldrn3.The specific reaction rate constant is
-
k = 2.2 dm31s mof
Solve for the concentrations of bromine cyanide and methyl bromide and the ratem reaction as a function of time.
Sec. 4.10
I
Unsteady-Slate Operailon of Stirred Reacton
Solution
Symbolically. we write the reaction as
/
The reaction ir elementary: therefore, the rate law is
Rote Law
Substituting the rate Iaw in Equations (4-56)and (4-58) gives Combined mole
balances and rate laws on A, B,
C, and D Polymath will cornbine for you. Thank you. Polymath!
Similarly for C and D we have
]
Then
and
I
We could also calculate the conversion of A.
I
The initial conditions are r = O: C,, = 0.05 moUdm3, C, = C,. = C, = 0, and V, = 5 dm3.
222
Isothermal Reactor Design
Chap. 4
Equations (Ed-9.2) through (E4-9.10) are easily solved with the aid of an ODE solver such as Polymath (Table E4-9.1). TABLEFA-9.1POLYMATH PROGRAM
ODE REPORT (RKF45)
Differentialequations as entered by the user [ 1 1 d(Ca)/d(t) = -k9Ca'Cb-vo'Ca [ 2 1 d(Cb)/d(t) = -k'Ca'Cb+vo'(Cbo-Cb)N 1 3 I ,d(Cc)/d(t) = k*CaeCb-vo*CcJ\C Explicit equations as entered by the user I l j k=2.2 121 vo=0.05 13 1 Cbo = 0.025 143 V o = S Is I Cao = 0.05 1 6 I rate = k'Ca"Cb t 7 1 V = Vwvo't [ R I X = (Cao'V*Ca9V)/ICao'Vo)
Living E~amptcProblem
The concentrations of bromine cyanide (A) and methyl amine are shown as a function of time in Figure M-9.1, and the rate is shown in Figure E4-9.2.
Scale:
5.000
Y: x102 4.000
3.000 mol
Why does thc concentration of
CH3Rr (C)go through a maximum wn time?
Figare FA-9.1 Polymath
output:
Concentration-time ~rajectories
Set. 4.10
223
Unsteady-State Operation of Stirred Reactors
-
0.00020
-
Y
m
E 0.00015 -
2
-
-E $ 0.ooOlO-Lf C
0
2
0.00005
0
u
0.0000 0
t
1
50
Figure E4-9.2
I
I
100
,
]
150
,
I
200
250
Time Is) Reaciion rate-time tqieciory.
4.Y0.4 Writing the Semibatch Reactor Equations in Terms of Conversion
Consider the reaction
in which B is fed to a vat containing only A initially. The reaction is first-order in A and first-order in B. The number of males o f A remaining at any time, r, is The limiting reactan1 i s the one In the bat.
- - N.4
-
N,,,
-
N*[J
where X i s the ~nolesof A reacted per mole of A initially in the vat. Similarly, for species B.
[
Number of moles o f B in
the vat at time
i=[Numbgrd]+[2tr; ~i-~mberdm&] moles of B in
<
the vat initially
added to the va?
of B reacted up to time r
(4-60)
224
Isothermal Reactor Design
Cha
For a constant molar feed rare and no B initially in the vat. Ns
=
F,,t
-
NAoX
(4-
A mole balance on species A gives
The number of moles of C and D can be taken directly from the stoichiome table; for example,
For a reversible second-order reaction A + B rate law is
C + D for which
Recalling Equation (4-55), the concentrations of A, B, C, and D are Concent~tion of reactants as a function
of conversion and time
Combining equations (4-62), (4-64), and (4-65), substituting for the concent tions, and dividing by hO, we obtain
Equation (4-66) needs to be solved numerically to determine !he conversion a function of time. The third variable, in addition to concentration and conversion, we c use to analyze semibatch reactors is number of mole NA,NB, etc. This meth Cummary Meter; is discussed in the Summary Notes on the CD-ROM,
Equilibrium Conversion. For reversible reactions carried out in a semibat reactor, the maximum attainable conversion (i.e., the equilibrium conversic will change as the reaction prcceeds because more reactant is continuou: added to the reactor. This addition shifts the equilibrium continually to t right toward more product.
Sec. 4.10
Unsteady-StateOperatton of St~rredReactors
225
a
If the reaction A + 3 C -t D were aIlowed to reach equilibrium after feeding species B for a time r, the equilibrium conversion could be calculated as foIlows at equilibrium [see Appendix Cj:
The relationship between conversion and number of moles of each species is she same as shown in Table 3-1 except for species B, for which the number of moles is given by Equation (4-61). Thus
-
NAox:
El - X,>(Feof - NAoX,)
Rearranging yields
1 -xe
t
Equilibrium conversion
in a semibatch reactor
X, =
(4-70) I Reactive distillation is used with thermodynamically limited reversible liquidphase reactions and is particularly attractive when one of the products has a lower boiling point than the reactants. For reversibIe reactions of this type. ? ( K c - 1)
L
the equilibrium lies far to the left. and Ijttle product is formed. However, if one or more of the product (e.g., Dl is removed by vaporization, as shown in Figure 4- 16,
Isothermal Reactor Design
-
Chap. 4
--.
Figure 4-16 Reactive disrillation with B fed lo a val containing A and D vaporizing.
the reaction will continue toward completion. The equilibrium constraint is removed, and inore product will be formed. The fundamentals of reactive distillation are given on the CD-ROM web module.
4.1 1 The Praciical Side The material presented in this chapter has been for isothermal ideal reactors. We will build on the concepts developed in this chapter when we discuss nonideal reactors in chapters- 13 and 14. A number of practical guidelines for the operation of chemical reactors have been presented over the years, and tables and some of these descriptions are summarized and presented on the CD-ROM and web. The articles are listed in Table 4-7. TAKE4-7
LITERATERE THAT G ~ V PRACTICAL E GUIDELINES FOR REACTOR OPERATION
D. Muhech. Chenr. Enr.. 16 (January 2002). S. Dutta and R. Gualy. CEP. 17 (October 20001: C&EdV.8 (January t0. 2W03 S. Jayakurnar. R. G Squires. G. V RekEal~i~. P. K. Andersen. and L.R. Rnin, Clrerrr Enx. Edur.. 136 [Spr~ng199.11. R. W Cusack. C ~ P I IEI .I I R ,88 {Fehn~aq2000). A. Balker. A. H. Hoidari. and E \4. Manhall. CEP. 10 (December 3101 ) t! Tramhouzc. CER 23 #February 1990). G Scholwsky and B. Loftus-Kwh. Cltnrt. Otg.. 96 (Febmap 21KKI).J. H. wowell. CEP, 55 (June 2000). 1 H WorsteII. CEP. 68 (March 2001) 5 Dutta and R. Gualy. Cl!rt~z.Etiy 72 (June 2000), A. Ahu-Khalaf. Chern EIIR Ed~rr..-18 IWlnlcr 1943).
.
For example. Mukesh gives relationships between the CSTR tank diameter. T. impeller bite diameter. D. tank height. H. and rhe liquid level. I. To scale up a pilot plant ( 1 to a full scale plant (2). the following guidelines are given
Chap. 4
227
Summary
And the rotational speed, N,. is
where values of n for different pumping capacities and Froude numbers are given in Mukesh's article.
Closure. This chapter presents the heart of chemical reaction engineering for isothermal reactors. After completing this chapter, the reader should be abIe to apply the algorithm building blocks
Evaluate
Combine Stoichiornetry
Rate Law
to any of the reactors discussed in this chapter: batch reactor, CSTR, PFR, PBF, membrane reactor, and semibatch reactor. The reader should be able to account for pressure drop and describe the effects of the system variables such as panicle size on the convmion and explain why there is an optimum in the conversion when the catalyst particle size is varied. The reader should be able to use: either conversions (Part I ) or concentration and molar flow rates (Art 2) to solve chemical reacrion engineering problems. Finally, after completing this chapter, the reader should be able to work the California Professional Engineering Exam Problems in approximately 30 minutes [cf. P4-I1, through P4-15,] and to diagnose and troubleshoot malfunctioning reactors [cf. P4-8B].
SUMMARY I . SoTution algorithm-Conversion a. Design equations (Batch. CSTR. PFR, PBR):
228
Isothermal Reactor Design
Cha
b. Rate law: For example,
d A + -bB + ~ c + - D
c. Stoichiometry: (1)
Gasphase.
a
a
a
(33
v = vo(f f E X ) - -
F*o(l-X) C.4 = -F -A - ~ * o ( l - X l - U W 0 . r 1 EX,
P To- co .).-(
(K).
1-x
To
7
.';
For a packed bed u=
A,( 1 - ~ I P C P ,
and 8, = U = Uo
(2) Liquidphuse:
c,
= C*,(f -X)
d. Combining for isothermal operation Gas:
2
(l-x12
-rpf = kcAO (1
Liquid: -r; = k&(
2
+ 1-x
)~
e. Solution techniques:
(1) Numerical integratiorr-Simpson's rule (s4 (2) Table of integrals (3) Software packages (a) Polymath (b) MATLAB 2. Solution algorithm-Measures other than cornersion When using measures other than conversion for reactor design. the mole b ances are written for each species in the reacting mixture: Mole balances
Chap 4
229
Summary
The mole baIances are then coupled through their relative rates of reaction. If
-r,% = k C " , ~ z
Rate Inw
for aA
(S4-9)
+ bB + cC + dD,then rB
Sroichiometry
b
c
a
a
= - r A , rC = - - r A $
Concentration can atso be expressed in terms and in molar flow rates (flow).
Liquid:
d
r ~ --rA =
(S4- 10)
II
of the number of moles {batch)
C, = FA -
(S4- 15)
"0
Combine
3. An ODE solver (e-g.. Polymath) will combine all the eqrttions for you. Variable density with
E
= 0 or EX G 1 and isothermal:
4. For membrane reactors the mole baIances for the reaction
A
0+C
when reactant A and pnoduct C do not diffuse out the membrane
-dV
dV
d F -~ r B- R,, and dFc =
rC
(54-17)
with
and k, is the overall mass transfer coefficient. 5. For semibatch reactors, reactant B is fed continuously to a vat initially containing only A:
Tsofhcrmal Reador D e s i ~ n
Chair. 4
The combined mole balance, rate law, and stoichiomerry in terms of conwrsion ir
When usinp an ordinary differential equalion (ODE)soher such as %!path or MATLAB, it is usually eas~erto leave the mole balances, rate laws, and concen-
uations as separate equations rather than combining them into a single equation as we did lo obtain an analytical solution. Writing the equations separately leaves it to the computer to combine them and produce a solutron. The formulations for a packed-bed reactor ~ i t preqsure h drop and a semibatch reactor are given below for two efernentary reactions carned out isothermally.
,4 + B + 2 C
Packed-Bed R~ocror
(where
=
PIP,,)
Seniibarclr Rencrnr
231
CD-ROM Malerial
Chap. 4
CD-ROM MATERIAL Learning Resources
I . S I ~ ? I I I Inote.^ TU~ 2. U'eb Mnd~tles A. Wetlands
Summarv hlotes
3. I ~ ~ r ~ ~ c r cComprdter rire Morlrc1e.r A Murder Mytery
$
B. Tic-Tac
I
Con?puterModule.
C. Reac~orLab Modules The I'ollols ing reactor Lab Modules have heclr developed by Profe\sor Richard Herz in the Chcmical Enginceriny Depanmc 111 at lllc U I Iersi ~ ~t! of California. San Diego. They are copyriglited hy LTCSD ntld Prnfe~~or Henr and are wed here with their permis\~or~.
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232
Solved Probftmr
Q
Isothermal Reae!or Desigr!
Cha
4 . Solt,cd P 1 ~ ) h k 1 ? t ~ A. CDP4-A, A S i n i w r Gentleman Messing with a Batch Reactor E. Solution to Cueifornia Registration Ewm Problem C. Ten Type\ of Home Problems: 20 Sol(cd Problemc 5. Attcrlogy of CRE Algitritlirns ro o Mer~irin o Fiw Frrr~rh Resrrf~tmi~t 6. Algnrttl11n for Gus Phrrs Renrriotl Living Example Problems Example 4-6 Calculnting X in a Reactor with Pressure Drop Example 4-7 Gas-Phase Reaction in Microreactor-MoIsr Flow Rate Example 4-8 Membrane Reactor
ExampIe CDR4.I Spherical Reactor Example 4.3.1 Aerosol Reactor Example 4-9 Isothermal Semibatch Reactor T;~"LS LlvrngExamp!< Prob!-m * frofe5essional Reference Shelf RJ. 1. Sph~rirrrlPackrd-Brd Reactors When small catalyst pellets are required, the pressure drop can be significi One type of reactor that minimizes pressure drop and i s also inexpenswe build is the spherical reactor, shown here. In thiq reactor. called ultraformer, dehydrogenation reactions such as
6
k-?.
Paraffin
-
Aromatic
+ 3H:
are carried out. Feed
141
Spherical ultr~formerrenctor. (Councsy of Amuco Petroleum Productk.1 T h ~ sreactor is one in a senes of ~ i used x by A ~ n ~ cfor o reforming prtroleum naphtha. Photo by K. R. Renicker. St
Analysis of a spherical reactor equation along with an example problem carried out on the CD-ROM.
Green engineering
;
R4.2 RecycIe Reactors Recycle reactors are used (1) when conversioil of unwanted (toxic) prdul is required and they are recycled to extinction, r2) the reaction is autocil lytic or (3) it is necessaq to malntain isothermal operation. To design recyc reactors, one simply follows the prwedure deveioprd in this chapter and th adds a little additional bookkeeping.
233
CD-ROM Materral
Chap 4
x, vo
Fw
3
Fa? etc.
"01'
X, '~3
lr
etc.
R vr, FAR,F B ~FGR, , elc.
Fm
-
F03
F ,
Recycle rcacmr. A.c shown in the CD-ROM,two conversions are usually associated with recycle reactors: the owrall tnnversion, Xo. and the conversion per pass, Xs.
R4.3. A E ~ O F Renrtor~ O/ Aerosol reactors are used to synthesize nanu-size panicles. Owing to their size, shape, and high rpecific surface area, nanopartlcles can he used in a number of appIrcntions such as In prgments in cosmetics, membranes. photocatalytic reactus, cntoIysts and ceramics. and catalytic reactors. We use the production of aluminum particles as an example of an aerosol plug-flow reactor (APFR) operation. PI sfream of argon pas saturated with A! vapor is cooled.
Carrier
Monomers Nucle~ Part~cles
Nuclsat~on Part~cle Flocculat~on
Aerosol reactor and temperature profile.
As the gas is cooled, it k o m e s supersaturated. leading to the nucleation of particles. This nuclearion is n T ~ S of U molecules ~ colliding and agglomerating until a critical nucleus size is reached and a particle is formed. As these particles move down, the supersaturated %as molecules condense on the particles causing them to prow in size and then to flocculate. (IIthe development on the CD-ROM.we will model the formation and growth of aluminum
nanopanicles in an AFPR.
R4.4 Cririqrrinp Jolrrnal ArttcIes After graduation, your textbooks will be, in part, the professional journals that you read. As you read the journals, it is important that you study them with a critical eye. You need to learn if the author's conclusion is supported by the data. if the article IS new or novel, it' it advances our understanding. and 1f the analysis is current. To develop this technique, one of the major assignment? used in the graduate course in chemical reaction engineering at
234
tsothermal Reactor Desrgn
Chap 4
[he University of Michigan for the past 25 years has been an ~n-depthanalp i s and critique of a journal article related to the course material. Significan~ effort is made to ensure that a cursory or superficial review is not carned our. The CD-ROM gives an example and some guidelines about critiquing journal articles.
Q U E S T I O N S AND P R O B L E M S
@
The subscript to each of the problem numbers indicates the level of difficulry: A. least dificult: D. most difficult. .4=.
%, b
Momewcr!. Problems
B=.
C=4
n=++
In each o f the following questions and problems. rather than just drawing a box around your answer, write a aentence or two describing how you salted the problem, the assurnprions you made. the reasonableness of your answer, what you learned. and any orher facts that you want to include. You may wish to refer 10 W. Strunl, and E. B. White. The Elen~enrsof Style. 4th ed. (New York: Macmillan. 2000) and Joseph M. Williams, Syie: Tert L ~ S S O i~r I I TClarin. & Gmw, 6th cd. (Glenview, 111.: Scott, Foresman. 1999) to enhance the quality of your sentences. See the Preface for additiona1 generic parts (x), .b I Z ) to the home problems. P4-1,
Read through all the problems at the end of this chapter. Make up and solve an origirlul probleni based on the material In this chap~er.(a) Lse reaI data and reaction<. (b) Make up a reaction and data. (c) Uce an example from everyday Iife (e.g.. making toast or cooking spaphettt). In preparing your orig~nalprohEem, first list the principles you want to get across and why the prohlem IS important. Ask yourself how your example wll be different from those in the
or lecture. Other things for you to consider when choosing a problem are relevance, interest, impact of the solution. time required to obtain a soIution. and degree of difficulty. Look through some of the journals for data or to get some ideas for indurtrially important reaction< or for novel applrcations of reaction engineering principles (the environment, food processing, etc.). At the end of the problem and colution describe rhe creative process used to generate the idea for the problem. {dl Write a question based on rhe material in this chapter that requires cnticaI thrnking. Explarn why your que~rinnrequrreq critical thinh~ng.[Nr~tr:See Preface. Section B.21 (el Listen to the audiuq on the CD Lecture Noter. pick one. and descr~behow you m~ghtexplain text
fi
differently. What if..you . were a\ked to explore the exampie prohlcms in t h i ~chapter tfl learn the erect5 of varl,ing.the different parameters? Thir sensitivity a n a l y ~ s can be carrled out by either downloading the examples from rhc web or h!) loading the progrim\ from the CD-ROM supplied x ith the text. For each of the example problems you ~nve\tipnte,write a parqraph decrihing your findings. (a) What if you were asked to give examples of the material in this hook that are found in everyday life9 Whai would !nu \a!? Ib) Example 4-1. U'hat would be the error in 1 ~f the hatch reactor uere o n 1 HOT filled w ~ t hthe same cornpsition of reaclants instead o f hcrrlg conlplctel!: lilled a\ in the example:' What generalizar~onhcan you draw Ctoln t h i ~euample'? it
P4-2#,
Hefore solv~ngthe
y~ohlelnr.ctate or rhetch qu~litatrr,cly
the expected r e w l ~ ~ or
trend5
Questions and Problems
Chap. 4
235
reactor volume change if you only needed 50% conversion to produce the 200 million pounds per year required? What generalizations can you draw from this example? (d) Example 4-3, What would be the reactor volume for X = 0.8 if the pressure were increased by a factor of 10 assuming everything else remains the same? What generalizations can you draw from this example? (e) Example 4-4. How would the pressure drop change if the particle diameter were reduced by 25%? What generalizations can you draw from this (c) Example 4-2. How would your
example?
I f ) Example 4-5. What would be the conversion
(g)
(h) (ik
ti)
(k)
(1)
(rnl
with and without pressure drop ~f the entering pressure were increased by a factor of 1 O? Would the optimum diameter change? If so, how'? What would the conversion be if the reactor diameter were decreased by a factor of 2 for the same mass flow rare? Example 4-6. Load the Living Example Pmblern 4-6 from the CD-ROM. How much would the catalyst weight change if the pressure was increased by a factor of 5 and the partide slze decreased by a factor of 5? (Recall a is also a function of Po)? Use plots and figures to describe what you find. Example 4-7. Load the Living Example Pmblem 4-7 from the CD-ROM. Hen, would thc results change if the pressure were doubled and the temperature was decreased IO°C? Example 4-8. h a d the Living Exatnple Pmhlem 4-8 from the CD-ROM. Vary parameters (e.g., kc), and ratios of parameters (klkcl. (hC,dK,). etc., 3nd write a paragraph describing what you find. What ratio of parameters has the greatest effect on the conversion X = ( F A , - F,)/F,, ? Example 4-9. Load the Living Example Prnhlan 4-9 from the CD-ROM. The temperature i s to be lowered by 35°C so that the reaction rate constant is now (1110) its original value. (i) If the concentration of B is to be mainrained at 0.01 mol/drn3 or below. what is the maximum feed rate of B? (ii) How would your answer change if the concentralion of k were tripled? Web Module on Wetland# from the CD-ROM.Load the Polytnath proRrnrn and vary a number of parameters such a< rainfall, evaporation rate, atnzine concentration, and liquid flow rate, and write a paragraph describing what you find. This topic is a hot Ch.E, research area. N i b Modrrle on Reactive DissilEntion Smm the CD-ROM. Load the Polv111nthprY>Rrmn and vary the parameters such as feed rate, and evaporarinn rate, and writc a paragraph descrrbing whal yot~find. U'eb Module on AernsnI Reactors from the CD-ROM. Load the Pohsninth prngrarn and ( I ) vary the parameters such as cooling rate and flow rate. and describe their effect on each of the regimes nucleation. growth and flocculation. Write a paragraph deccribing what you find. (2) It i s proposed ro replace the carrier gas by helium (i) Compare your plots (He versus Ar) of the number of Al particles as a funclion of lime. Explain the shape of the plols. lii) How doe< the final value of d, compare with that when the carrier gas was argonVxp1ain. ( r ~ i lCompare h e time at which rhe rate of nucleation reacher a peak in the two cases [carrier gas = Ar and He1 Discuss the comparison.
236
Isothermal Reactor Design
Chaf
Data for a He moIecule: Mass = 6.64 x 101" kgg,Volume = 1.33 x IO-?' : Surface area = 2.72 x EO-"'mm!, Bulk density = 0.164 kglrn3, at non temperature (25°C) and pressure (F atm).
+ Intemc!fve
>:
6-
P4-3,
2;
Compater Modules
P4-4,
Appllcetin Pending for Problem Hall of
P4-5,
(n) Vary some of the operating costs, conversions. and separations in Fig 4-10 to Ieam how the profit changes. Ethylene oxide, used to make e ylene glycol, sells for $0,56/1b while ethylene glycol sells for 10.381 Is this a money-losing proposition? Explain, ( 0 ) What should you do if some of the ethylene glycol splashed out of reactor onto your face and clothing? (Hint: Recall www:siri.org/.) (p) What safety precautions should you take with the ethylene oxide fom tion discussed in Example 4-h? With the brom~necyanide discussed Example 4-9? (9) Load reactor lab on to your computer and call up Dl Isotherfrml Rpr ~ors.Detailed instnictions with screen shots are given in Chapter 3 of i Summary Notes. (1) For LI Nth Order Reactions. Vary the parameters E. T for a batch, CSTR, and PFR. Write a paragraph discussing I trends (e.g., first order versus second order) and describe what you f i (2) Next choose the "Quiz" on membrane at the top of the screen, a find the reaction order (3) and turn in your performance number. Performance number: (s) The Work Self Tests on the Web. Write a question for this problem tl involves criticai thinking and explaining why it involves critical thinkir See examples on the Web Summary Note for Chapter 4. Load the Interactive Computer Modules (ICM) from the CD-ROM.Run t modules and then record your performance number, which indicates yc mastery of the marerial. Your instructor has the key to decode your perf1 mance number. (a) ICM-Mystery Theater-A real "Who done it?". see Pulp ond Paper, (January 1993) and also Pulp at10 Paper, 9 (July 1993). The outcome the murder trial is summarized in the December 1995 issue of Papt maker, page 12. You will use Fundamental chemical engineering FK Sections 4.1 to 4.3 to identify the victim and the murderer. Performance number: (h) ICM-Tic Tac-Knowledge of all sections is necessary to pit your r against the computer adversary in playing a game of Tic-Tnc-Toe. Performance number: If it takes !I minutes to cook spaghetti In Ann Arbor, Michigan. and 14 mi utes in Boulder, Colorado, how long would it take in Cuzco. Peru? Discu ways to make the spaghetti more tasty. If you prefer to make a creatlve SF ghetti dinner for family or friends rather than answering this question, tha OK, too; you'll get full credit-but onIy if you turn in your receipt and bri~ your instructor a taste. (Ans. t = 21 min) The liquid-phase reaction
A+B
-
C
follows an elementary rate law and is carried out isothermally in a flow sy tern. The concentrations of the A and B feed streams are 2 M before mixin The volumetric flow rate of each stream is 5 dm31rnin, and the enteri temperature is 300 K. The streams are mixed immediately before enterin
Chap. 4
Hall of Fame
237
Questrons and Problems
Two reactors are available. One is a gray 200.0-dd CSTR that can be heated to 77°C or cooled to O'C. and rhe other is a white 800.0-dm' PFR operated at 300 K that cannot be heated or cooled but can he painied red or biack. Note k = 0.07 dm3/mol-min at 300 K and E = 20 kcaI/mol. la) Which reactor nnd what conditions do you recommend? Explain the reason for your choice ( e . , ~ .color, . cost, space available, weather conditions). Back up your reasoning with the appropriate calcuInttons. (b) How long would it take to achieve 90% conversion in a 200-dm3 batch reactor with CAD = CBO= I M after mixing at a temperature of 7I0C? (c) What would your answer to part (b) be if the reactor were cooled to O°C? (Ans. 2.5 days) Id) What conversion would be obtained if the CSTR and PFR were operated at 300 K and connected in series? In paraIlel with 5 mournin to each9 (e) Keeping Table 4-1 in mind, what batch reactor volume wourd be necessary to process the same amount of species A per day as the flow reactors while achieving 90% conversio~?Refemng ro Table 1-1, estimate the cost of the batch reactor. (0 Write a couple of sentences describing what you learned from the probIcm and what you believe to be the point of the problem. P4-6R Dibutyl phthalate (DBP),a plasticizer, has a potential market of 17, million Ib/yr (AIChE Studenr Conresr Ploblern) and is to be produced by reaction of n-butanol with monobutyl phthaiate (MBP). The reaction follows an erementaw rate law and is catalyzed by H,SO, (Figure P4-6).A stream contalninp MBP and buenol is to be mixed with the H,SO, catalyst immediateIy before the stream enters the reactor. The concentration of MBP in the srream entering the reactor is 0.2 lb rnollft-'. and the molar feed rate of butanol i s five times that of MBP. The specific reaction rate at l O 0 T is 1.2 ft3/lb mol- h . There is a IWgaIIon CSTR and associated peripheral equipment available far use on this project for 30 days a year (operating 24 Wday).
National AlCHE
Contest hoblem
(MBP)
(n-butanol)
(DBPE
(a) Determine the exit conversion in the available 1000-gallon reactor if you were to produce 33% of the share (i.e.. 4 million Ib/yr) of the predicted market. (Ans.: X = 0.33) (b) How might you increase the conversion for the same Fa? For example, what conversion would be achieved if a second ICW-gal CSTR were placed either in series or in parallet with the CSTR? [X?= 0.55 (series)]
238
lsolhermal Reador Design
Chap. 4
Ic) For the same temperature as pan (a). what CSTR volume would be nec(d)
(el (f)
(g)
(h)
P4-7,
The
essary to achieve a conversion of 85% for a molar feed rate of MBP of 1 Ib mollmin? If possible. calculate the tubuIar reactor volume neceshaq ro achieve 85% conversion, when the reactor is oblong rather than cylindrical, with a major-to-minor axis ratio nf 1.3 : 1 .O. There are no radial gradients in either concentration or velocity. If it i s not po~sibleto calculate V,, explain. How would ynur results for parts (a) and (b) change if the temperature were raised to 150aF where k is now 5.0 ft3/lb mol . b but the reaction is severs~bleu lth Kr = 0.3? Kecping in mind the times given in Table 4-1 for filling, and other operations. how many 1000-gnIlon reactors o ~ r a t e din the batch mode would be necessary to meet the required production of 4 mhlion pounds in a 30day period? Es~imatethe co\t of the reactors in the system. Nure: Present in the feed stream may be some trace impurities, mhich you may lump as heranol. The activation energy 1s believed to be sflrnewhere around 25 kcallmo2. Hint: Plot number of reactors a< a function o f conversion. (At? Ans.: 5 reactors) What generalizations can you make about what you learned in this problem that would apply to other problems? Write a question that requires critical thinking and then explain why ynur quesr~onrequires critical thinking. [Hinr: See Preface. Section B.11 elementary gas-pha~ercacrion
ApplicaYin For Problem Hall of
i s carried out isothermally in a flow reactor with no pressure drop. The specific reaction rate at 50°C is LO-4 min-I (from perico5ity data) and the activation energy is 85 Wlmnl. Pure di-[err-butyl peroxide enters the reactor at 10 otm and 127'C and a molar flow rate of 2.5 mollrnin. Calculate the reactor volume and space time to achieve 9 0 9 conversion in: (a) a PFR (Arts.: 967 dm?) (b) a CSTR (Aiir.: 1700 dm') (c) Pmsure drop. Plot X. ?; as a function of the PFR volume when a = O 001 dm-". What are X,and y at 1' = 500 din'? (d) Write a quec~ionthat requires cri~icalrhinking. and explain wh! it involve.es cntical thinking te) If this reacrlm is to he carried out soth hem ally at 137°C and an initial pressure of 10 atm in a cnnqtant-volume batch mode with 90% conversion, w h a t rcnctor size and cost would hc required to process (2.5 ~nollmin X 60 mlnh X 2.1 hldny) 3600 mol of di-ten-but>\ perox~deper day? (Hirlr. Recall Table 4- I .1 (fl Assume that the reaction is reverrrblc with Kr = 0.025 rnol'ldmh. and calculate the equilibr~umconversion; lhen redo l a ! through (c) to achieve a con\erainn thal i\ 90% of the equilibrium conver4nn. (g) Membranc reactor. Repeat Part ( f ) for the care when C:H, flows out through the .side\ ol' lhe reactor and !he tr:in%porr coefficient kr = 0.08 \-I.
Chap. 4
I
239
Questions and Problems
P4-88 ~ u b l e s h o o t i n g (a) A
J'Q
liquid-phase isomerization A
B is carried out in a 1000-gal
CSTR that has a single impeller located halfway down the reactor. The liquid enters at the top of the reactor and exits at the bottom. The reaction is second order. Experimental data taken in a hatch reactor predicted the CSTR conversjon should be 5056.However, the conversion measured in the actual CSTR was 579. Suggest reasons for the discrepancy and suggest something that wouid give closer agreement between the predicted and measured conversions. Back your suggestions with cnkulatioos. F.S. It was raining that day. (b) The first-order g-as-phase isornerization reaction
Creative Thlnking
B with k = 5 min-1
A
is to be carried out in a tubular reactor. For a feed of pure A of 5 drn31min. the expected conversion in a PFR is 63.2%.However, when the reactor was put in operation. the con\~ersionwas only 5R.h%. U'e should note that the straight tubular reactor would not fit in the available space. One engineer suggested that the reactor be cut in half and the two reactors he put side by side with equal feed to each. However. the chief engineer overrode this suggestion saying the tubular reactor had to be one piece so he bent the reactor in a U shape. The bend was not a good one. Brainstorm and make a list of things that could cause this off-design specification. Chaose the most logical explanationlmodel, and carry out a calculation to show quantitatively that with your model the conversion I< 5X.6&. (An Ans: 57% of the total)
Mall c f Fame
(c) The liquid-phase reaction
was carried out in a CSTR.For an entering concentration of 2 rnol/dm3. the conversion was 4 0 9 . For the qame reactor volume and entering conditions as the CSTR, the expected PFR conversion is 48.6%. However. the PFR conversion was amazingly 50% exactly. Bninqtonn reasons for the disparity. Quantitatively show how these conversions came about (i.e., the expected conversion and the actual conversion). (d) The pas-phase reaction
A-tB
P4-9,
C+D
is carried out in a packed bed reactor. When the panicle size was decreased hy 15%. the conversion remained unchanged. When the particle size was decreased by 20%. the conveaion drcreaced. When the original particle size was increaced by 15'3, the conver
240
lsc!herrnal Reactor Design
Ck
AHR, = -25,IXX) callnol. Assurntng that the batch data taken at 31Jo F accurate and that E = 15,000 ctll/mol, what CSTR temperature do you re mend to obtain maximum conversion? Hinr: Read Appendix C assume ACp = 0 in the appendix Equation (C-8): Kc(.)
AH,
= ~~X K (T~)P!(
1
-
j)]
Use Polymath to make a plot of X versus T. Does it go through a rnaxin If so, explain why. The growth of bacteria to form n product. P,is carried out i n a 25 dm' C (chernortat). The bacteria (e.g., Zjwononos) consunres the nutrient sub, (e.g., to generate more cells and the desired product--ethanol)
P4-10
I
'
Substrate
Calls
More Cslls + Pmdwt
1
The CSTR was initially inoculated with bacteria and now has reached s state. Only substrate (nutrient) is fed to the reactor at a volumetric ra 5 dm3/%and a concentration of 30 @dm3. The growth law r, (glhr dm")
and the rate of substrate consumption is related to growth rate by
with the stoichiometric refationship
(a) Write a mass balance on the cells and the substrate concentration i CSTR operated at steady state, (b) Solve the cell mass balance for the substrate concentration and cslc ,-,
0
Cs.
(c) Calculate the cell concentration, Cc.
(d) How would your answers to {b) and (c) change if the volumetric rate were cut in haif? (e) How would your answers to (b) and (c) change if the CSTR volume reduced by a factor of three? ( f ) The reaction is now carried out in a 10 dm3 batch reactor with initial , = 30 @dm3and cells of CEO= 0.1 g/dn centrations of substrate C Plot C,,C,, r,, and -rs as a functron of time. (g) Repeat (f) for a 100 dm3 reactor. Additional Enformat~on: hax = 0.5hr-', K, = 5 gldrn3 Y, = 0.8 g cell Formedlg substrate consumed = IIY,,c
Chap. 4
P4-11,
Questions and Problems
24 1
The gaseous reaction A d B has a unimolecular reaction rate constant uf 0.00 IS min-I at 80°F. This reaction i s to be camled out In pnrcrilel tithes I O ft long and I in. inside diameter under a pressure of 132 psi: at 2Cf1°F. A production rate of \OW lblh of £3 is required. Assum~ngan activation energy of 25.000 callmol. how many tubes are needed if the conversion of A i s to be 9 0 9 ? Assume perfect gas laws. A and B each have molecular weights of 58.
{From California Profe~sionalEngineers Exam.) (a) The irreversible elementary reaction 2A 4 B takes place in che gas phage in an i~othermaittthrlur (plrtg-Jow) rencror. Reactant A and a diluenr C are fed in equimolat ratio, and conversion of A is 80%. If the molar feed rate of A is cur in half, what is the conversion of ,4 assuming that the feed rate of C is left unchanged? Assume ideal behavior and that the reactor temperature remains unchanged. What was the point of this problem? (From Califomla Professional Engineers Exam.) (b). Write a que5tion that requires critical thinking, and explain why it involves critical thinking. P4-13, Compound A undergoes a reversible isornerizrttion reaction. A 8 B , over a supported metal catalyst. Under pertinent conditions, A and I3 are liquid, miscible, and of nearIy idenrical density; the equilibrium constant for rhe reaction (in concentration units) is 5.8. In afu;ed-bed isothemaljlow rencror in w h ~ hbackmixing is negligible (i.e.. plug Row), a feed of pure A undzrgoes a net conversion to 0 of 55%. The reaction is elementiiry. If a second, identical flew reactor at the same temperature i s placed downstream from the first, what overall conversion of A would you expect if: (a) The reactors are directly connected in series? (Ans.: X = 0.74.) (b) The products from the first reactor arc separated by appropriate processing and only the unconverted A is fed to the second reactor? (Fmm California Professions! Engineers Exam.) FJ-Idc A total of 2500 galh of metaxylene is being isomerized to a mixture of orthoxylene, rnetaxylene. and paraxylene in a reactor containing 1000 ft3 of catalyst. The reaction is being carried out at 75VF and 300 psig. Under these conditions, 3 7 6 of the metaxylene fed to the reactor is isornerized. At a flow rate of 1667 gaVh. 50% of the rnetaxylene is isomerized at the same temperature and pressure. Energy changes are negligible. It is now proposed that a second plant ke built to process 5 5 0 0 gal171 of rnetaxylene at the s m e ternpenture and pressure ;is described earlier. What size reactor (I.c.,what volume af catalyst) is required if conversion in the new plant 2s to be 46% instead of 37QT Justify any assumptions made for the ale-up calculation. (Ans.: 2931 ft3 of catalyst.) (From California Professional Enpineen Exam.) Make a list of the things you learned From this problem. P4-15* 1t is desired to carry out the gaseous reaction A +B in an existing ruhular rencror consisting of 50 parailel rubes 40 ft loo$ with a 0.75-in. inside diameter. Bench-scale experiments have given the reaction nte constant for this first-order reaction as 0.00152 s-I at 200°F and 0.0740 s-I at 3Q0°F. At what temperature should the reactor be operated to give a conversron of A of 80% with a feed rate of 500 Ibh of pure A and an operating pressure of l a ) psig? A has a molecular weight of 73. Departures fromperfect gas behavior may be neglected, and the reverse reaction is insignificant at these conditions. 1Ans.: T = 275T.) (Fmm CaIifornia Professional Engineers Exam.)
P4-12,
Chap. 4
Ouest~onsand Problems
243
(6) What is the conversion exiting the last reactor? (c) What is the pressure at the exit o f the packed bed?
P4-19
( d ) How would your answers challge if the catalyst diameter were decreased by a factor of 2 and the PBR diameter were increased by 5 0 5 assuming turbolent flow? A microreactor similnr to the one shown i n Figure P4-19 from the MIT group is used to produce phosgene in the gas phase.
The rnlcroreactor ic 20 rnm Iong, 500 prn in diameter. and pached with catalyst particles 35 pm in diameter. The entering prcsqure is 830 kPa 18.2 am). and rhe entering flommto each mtcroreactor i s equin~olar.The molar flow rate of C O is 2 x 111-5 nioVs and the ~olu~nctrrc flori is. 2 83 x I()-* rn7/s. The weight o f catallsf In one iii~croi~actt~r: 1V = 3.5 x 10" Ip. The reactor ts kept isother~nalat 120°C. Because the catalypt ts also sliphtly differeni than rhe one in Figure P4-19.11ic rate lrrw is difierent as u~ell.
(a) Plnt the molar flow ratcc FA. FB,and Fc. the uonrerrion X. and pre\sure rntrcl alony the length of the reaclor. I h ) Calculate the nurnkr nf microreactnrf In parallel In pmduce 1IE.MIR hgtyenr phoyene. ( c ) Repeat pan f a ) for the case when the catalyst ueight remaim ~ h ccame hut the panicle diameter i< ctit rn half. I f pokcible cnlnpare your ansiscr u i t h part (a) nntl dc~cribeu h a ~y c x ~find. nclting anyth~ngunusual ( d l I i n a would your ancirrrc to pan (;I) chanpe if t l ~ cre,lchon licre revers~ b l emith K,. = O 4 dni'/rnt~l?Dcsc~.ihru hat !nu iind
244
Isothermal Reac:or Design
CR:
(e) What are lhc advantages and disadvantage5 of using an array of mi reactt>r~over ucing one conventional packed bed reactor that provider same yield and convers~on'? ( 0 Write a que5tion that involves critical thinking. and explain wh involves criticaI th~nking. (g) Discuss what you learned from t h ~ sproblem and mhnt you believe t! the p i n t of the problem. AddEtionrrI inforn~lntion: a = 3.55 x 10Slkg catalyst (based on properties of air and b = 0.4) k = 0.0M rnhlrnol . L . ks catalyst at 120°C u, = 2 83 . IV7 &Is, p = 7 kglrn3, p = 1.94 l o 5 kg/m . s, A, = 1.96 - IP7m2,G = 10.1 kglm2 . s
P4-20c The elementary gas-phase reaction
is carried out in a packed-bed reactor. Currently, catalyst panicIes 1 mn diameter are packed inro 4-in. schedule 40 pipe (A, = 0.82126 dm". value of p, in the pressure drop equation is 0.001 stmldm. A stoichiomt mixture of A and B enters the reactor at a total rnolnr flow rate of 10 mollr a temperature of 590 K, and a pressure of 20 atm. Flow i s turbulent througt the bed. Currently, only 12% conversion is achieved w ~ t h100 kg of catalys It is suggested that conversion could be increased by changing the alyst panicle d~arneter.Use the following data to correlate the specific rr tion rate as a function of particle diameter Then use this correlatior determine the catalyst size that glves the highest conversion. As you will in Chapter IZ,k' for first-order reaction is expected to vary according to following relationship
Mall of Fame
.
where rl, varies directly with particle diameter, @ = uD,. Although the reac is not first order, one note3 from Figure 12-5 the functionality for a s e a order reaction is similar to Equation (P4-20.1). (a) Show that when the flow 1s turbulent
and that g, = 0.8 x IIT atrnlkg and also show that r = 75 min-I. (b) Plot the specific reaction rate k' a? a function of D,,and compare
I.
Figure 12-5. (c) Make a plot of ronversion as a function of catalyst size. Id) Discuss how your answer would change if you had used the effectiver factor for a second-order reaction rather than a hrst-order reaction. (e) How would your answer to (b) change if both the particle diameter pipe diameter were increased by 509 when ( I ) the flow is laminar. (2) the flow is turbulent. (f') Write a few sentences describing and explaining what would happel the pressure drop parameter a is varied.
Chao. 4
245
0ues:ions and Problems
(g) What genenl17ations can you make about what you lenrned in thik pmbl e ~ nthat would apply to other problem.;? (h) Drscus$ what you learned from this problc~nand what you beliekc Icr be the point of the problem. Additiotlul ~rrforitrrrr~o~t:
Void Fraction = 0.35 Solid catalyst density = 2.35 kgldrnJ Bulk density: pB = (I - 0 ) pc = (0.35)(2.35) = 0.823,
k'(dmVmol.min. kg cat) 0.06 0.12 0.30 I.?,
'
2.M 3.W
[Hill!: You c o ~ ~ luse d Equation (P4 20-I), which would include D, and an unknown proport~onalityconstant that you could evaluate from the data. Fnr very small values of the Thiele modulus we know 7 = I , and for very large values of the Thiele rnodulu~we know that 7 = 3/9= 3/cD,.] P4-2I,, Nutri~ionis an important part of ready-to-eat cereal. To make cereal healthier, many nutrients are added. Unfortunately, nutrients degrade over time, making ~t necessary to add more than the declared amount to assure ennugh for thc life of the cereal. Vitamin V , is declared at a level of 20% o f the Recommended Daily AiIowance per serving size (serving size = 30 g). The Recommended Daily Allowance i s 6500 ILr ( I .7 X loh IU = 1 g). It has been found that the degradation of thiq nutrient is fint order in the amount of nutrients. Accelerated storage tests have been conducted on this cereal, with the following results:
Temperature { O C )
]
4S
55
65
fact that the cereal needs to have a vitamin level above the declared d u e of 6500 IU for I year at 25°C. what IU should be present In the cereal at the time tt is manufactured? Your answer may also be reported in percent ovemse: (Am. 13%)
(a) Given this information and the
%OU =
C ( r = O l - C ( r = 1 yr)
C ( t = I yr)
(b) At what percent of declared value of 6500 I U must you apply the vitamin? If 10.000.000 lhlyr of the cereal t s made and the nutrient c o ~ is t $100 per pound, how much wilt this overuse cost'? Ic) If this were your factory. what percent overuse would you actually apply and why?
4-22*
Id) How would your answers change if you stored the material in a Bangkok warehouse for 6 months, where the daily temperature is 4VC, before moving it to the supermarket? (Table of results of accelerated storage tests on cereal; and Problem of vitamin level of cereal after storage courtesy of General MiIIs. Minneapolis. MN.) A very proprietary industrial waste reaction. which we'll code as A + B + S is to be carried out in a 10-dm? CSTR followed by 1 0-dm3PFR. The reaction is eIernentary, but A. which enters at a concentration of 0.001 rnol/drn3 and a molar flow rate of 20 mollmin. has trouble decomposing. The specific reaction rate at 42°C (i.e., morn temperature in the Mojave desert) is 0.0001 S K I .
246
Isothermal Reactor Design
Chap. 4
However, we don't know the activation energy; rherefore. we cannot carry out this reaction in the winter in M i c h i p . Consequently this reaction, while
important, is not worth your time to study. Therefore, perhrtps you want to go watch a m o v ~ esuch as Dances with Wnlres ( a favorite of the author), Bride and Prejudice, or finding Neverland. The productionof ethylene glycol from ethylene chlmhydrin and sodium b i a h a t e take a break and
P4-23,
CH,OHCH,CI
+ NaHCO,+
(CH20H)2+ NaCI -+ CO,
is carried out in.a sernibatch reactor. A 1.5 molar solution of ethylene chlorohydrin is fed at a rare 0.1 molelminute to 1500 d r n b f a 0.75 molar solution of sodium bicarbonate. The reaction is elementary and carried out isochermally at 30cC where the spec~ficreaction rate is 5.1 dm3/rnolJfi. H~ghertemperatures produce unwanted side reactions. The reactor can hoId a maximum of 2500 dm3 of liquid. Assume constant density. (a) Plot the canversion, reaction rate. concentration of reactants and products. and number of moles of glycol formed as a function of time. (b) Suppose you could vary the flow rate between 0.01 and 2 mollmin, what flow a rate would and holding time you choose to make the greatest number of moles of ethylene glycol in 24 hours keeping in mind the downtimes for cleaning, filling, etc., shown in Table 4- 1. ( c ) Suppose the ethylene chlorohydrin i s fed at a rate of 0.15 rnolhin until the reactor i s full and then shut in. Plot the conversion as a function of time. ( d ) Discuss what you learned from this problem and what you be3ieve to be the point of this problem. P4-24c The following reaction is to be carried out in the liquid phase
NaOH + CH3COOC,H,
---+
C H , C O O - N ~+~ C,H,OH
The initial concentrations are 0.2 M In NaOH and 0.25 M in CH,COOC2Hs with k = 5.2 x rn3/mol-s at 20°C with E = 42,810 JJmol. Design a set of operating condition$ to produce 200 mollday of ehanol in a semibatch reactor and not operate above 35'C and below a concentration of NaOH of 0.02 molar.'? The semrbatch reactor you have available is 1.3 m in diameter and 2.5 m tall. P4-2Sc (Membmne rrrrr*tor)The first-order, reversible reaction
is taking place in a membrane reactor. Pure A enters the reactor, and I3 diffuses rhrough the membrane. Unfortunately, some o f the reactant A also diffuses thmugh the membrane. (a) Plot the Row rates of A. B. and C down the reactor. as well as the flow rates of A and R through the membrane. (b) Compare the convenion profiles of a conventtonal PFR with those of an IMRCF.What generalizat~onscan you make7
'' Manual of Chemical Engineering Laboratory.
University of Nanc!.. Nancy. France. 1994. rrrc@rs?.~~r~r-srur!gort.de. 2 i . r l . ~ :~~rhio.drVAlCHE
248
Isothermal Reactor Des~gn
Cha
Good Alternatives on the CD and on the Web
P4-28 P4-29 P4-30
P4-31
The following problems are either similar to the ones already pre5ented use different reactions or have a number of figures that would requlre a 10 text space. Consequently. the fulI problem statements are on the CD-ROM Pressure drop i n a PBR with a first-order reaction using real data: Wha questions asked. [3nl Ed. P4-181 Good Troiibleshooting Problc~n.Inspector Sergeant Ambercrornby inrre gates possible fraud at Worthless ChemicaI. [3rd Ed. P4-91 The first-order reaction
is to be carried out in a packed k d reactor with pressure drop where the r constant varies inversely with k-( IIDp). One can also choose frotn van( pipe sizes to get the maximum conversion. Similar to Problems PJ-22 a P4-23.[3rd Ed. P4-201 Pressure drop in a packed bed reactor to make alkylated cyclohexannls. [: Ed. P4-221
Figure P4-31
P4-32
A semibatch reactor is used to carry out the reaction
P4-33
Similar to problems 4-26 and 4-27. 13rd Ed. P4-261 A CSTR with hvo impellers is modeled as rhree CSTRs in series. [3rd Ed, p4-2
Chap. 4
Journal Critique Problems
SOME THOUGHTS ON CRITIQUING W H A T YOU R E A D
Ref~renccShelf
Your texthnoks after your gndunt~onwill be, ln pan, the professional journals that yotl reild. As you read the journals, it is important that you study them with a critical eye. You need to learn if the author's conclusion is supponed by the data, if the article IS nea or novel, if it advances our understanding. and to learn if the analysil; is current. To deveIop this technique. g e of the major asstgnments used in the gmduate course in chemical reaction engineering at the University of Michigan for the past 25 years has been an in-depth analysis and critique of a journal anicle related to the course material. Si~niticanteffort i s made to ensure that a cursocy or supeficiol review is not carried uut. Students arc asked to analyze and critique ideas rather than ask questions such as: Was the p ~ s s u r emeasured accurately? They have been told that they are not required to find an error or inconsistency in the artide to receive a good grade, but if they do find such things, it just makes the assignment that much more enjoyable. Beginning with Chapter 4, a number of the problems at the end of each chapter in this hook are based on students' analyses and critiques of journal anicles and are designated with a C le.g.. P4C-I). These problems involve the analysis of journal articles that may have minor or major inconsistencies. A discussion on critiquing journaI articles can be found in Professional Reference Shelf R4.4 on the CD-ROM.
JOURNAL CRITIQUE PROBLEMS P4C-1
PIC-2
In Wuter Resenvch, 33 (9),2130 (1 999). is there a disparity id the rate law obtained by batch experiments and continuous flow experiments? In the article describing the liquid reaction of isoprene and rnaleic anhydride under p ~ s s u r e[RIChE J., J6(5),766 (1970)l.the authors show the reaction rate to be greatly accelerated by dhe application of pressure. For an equimolw feed they wsite the sewid-oder reaction mte expression in terms of the mole fraction y:
and then show the effect of pressure on k, {s-I). Derive this expression from hrst priociplec and suggest a possible logical expIanntion for the increase in the true specific reaction rate constant k (dm3/mol s) with pressure that is different from the author's. Make a quick check to verify your chaIlenge. P4C-3 The reduction of NO by char was carried out in a fixed bed between 5 0 and 845°C Ilnt. Chem. Eng., 2a2).239 (19801J. It was concluded that the reaction is first order with respect to the concentration of NO feed (300 to 1 0 0 ppm) over the temperature range studied. Pr was also found that activation energy begins to increase at about 680°C. Is frst order the true reaction order? If rhere were dimepancia in this aTticIe, wk3t might be k reasons for h m ' ? P4C-4 In the article describing vapor phase esterification of acetic acid with ethanol to form ethyl acetate and water [Ind. Eng. Chem. Res., ?6/2),198 1 I987)], the pressure drop in the reactor was accounted for in a most unusual manner [i.e.. P = Po(l fX),where f is a constant]. 4
-
250
Isothermal Reactor Design
Chap, 4
(a) Using the Ergun
equation along with estimating some of the parameter values (e.g., C$ = 0.4). calculate the value of a in the packed-bed reactor (2 cm i.d. by 67 cm long). (b) U s ~ n gthe value of a, redo part (a) accounting for pressure drop along the l~nesdescribed in this chapter. (c) Finally, if possible, estimate the value off used in these equations.
*
&? Solved Problems
Additional Homework Problems
CDP4-AB
A sinisier looking gentlemen i s interested in producing methyl perchtorate in a batch reactor. The reactor has a strange and unsettling m e law. [2nd Ed. P4-281 (Solution Included)
Bioreactars and Reactions
CDP4-B,
(Ecological Engineering) A much more complicated version of Problem 4-17 uses actual pond (CSTR) sizes and flow rates in modeling the site with CSTRs for the Des Plaines River experimental wetlands site (EW3) i n order to degrade atrazine. [ See Web Module on CD or
CDP4-C,
The rate of binding ligands to receptors is studied in this application of reaction kinetics to bioerrginecri~rg.The time to bind 50% of the l~gandsto the receptors is required. [2nd Ed. P4-341 J. Llndemann, Univers~tyof Michigan
WWW]
Batch Reactors CDP.1-D,
n
CDP4-E,
A batch reuctor. is used for the bromina~ionof p-chlorophenyl isopropyl ether. Calculate the batch reaction time. [Znd Ed. P1-291 California Professional Engineers Exam Problem. in which the
reaction
CDP4-FA
1s carried out in a batch reactor. [2nd Ed. P4-IS] Verify that the Ilquid-phase reaction o f 5-6-knzquinoline with hydrogen IS psuedo first order. [2nd Ed. P4-71
Flow Reactors
CDP4-Gn
CDP4-H,
CDP4-I,
Radial flow reactors can be used to good advantage for exorhermic reaction5 with large heats of reaction. The radical velocity is
Vary the parameters and plot X as a function of r. [2nd Ed. P4-311 Dehigned to re~nforcethe hasic CRE principles thwuph very strai~htforward calculat~onsof CSTK and PFR volumes and batch reaclor time Thic problem was one of the most frequently assigncd problems Vrmm the 2nd Edition. 12nd Ed. P4-41 Forma~ionof diphen!] In a hatch. CSTR. and PFR. 13rd Ed. P4-It)]
Chap. 4
251
Journal Crrtique Problems
Packed Bed Reactors CW-J* CDP4-Kc CDP4-LB
-
n-Pentane i-pentane in a packed bed reactor. [3rd Ed. P4-211 Packed bed spherical reactor. [3rd Ed. P4-201 The reaction of A I3 is carried out in a membrane reactor where B diffuses out.
Recycle Reactors CDP4-MB CDP4-Nc
The overall conversion is required in a packed-bed reacror wirh r e p cle. [Znd Ed. P4-221 Excellent reversible reaction with recycle. Good problem by Professor H.S.Shaokar. IIT-Bombay. [3rd Ed. P4-281
Really Dimcult Problems
CDP-Oc
DQE April 1999 A + 3 in a PFR and CSTR with unknown order. (30 minutes to solve)
CDP-Pn
A Dr. Prohjot Singh Problem A I3 C Species C starts and ends at the same concentration.
G m n Engineering
New Problems on the Web
CDP4-New From time to time new problems relating Chapter 4 material to everyday interest< or emerging technologies will be placed on the web. Solutions to these problems can be obtained by e-mafling the author. Also, one can go on the web site, n~ww.rnwan.eddgreeneng~neering, and work the home problem specific to this chapter. These Problems Were on CD-ROAMfor 3rd Edition but Not in Book for 3rd Edition
CDP4-Q,
CDP4-RB CDPP-SB
CDP4-T,
The gas-phase reaction A + 2 B 2D has the rate law -r, = 2 5CXSCB.Reactor volumes of PFRs and CSTRs are required in t h i ~multipart problem. (2nd Ed. P4-81 What type and arrangement of flow reactors should you use for a decornpositlon reaction with the rare law -r, = k,CO,'I(I 4kJ,) ? [!st Ed. P4-141 The liquid-phase reaction 2A + B t~C f D is camed out in a sernihatch reactor. Plot the conversion. volume, and specie? concentratrans as a function of time. Reactive distillation is also considered in part (e). I?nd Ed. P3-271 The growth of a bacterium is 20 be carried out in excess nutrrenr. Nutrient + Cells -t More cells + Product
L)[?nd Ed. P4-151
The growth rate law i s rB = p , , , ~ ~-( l
C~max
n
252
Isothermal Reactor Design
CDP4-U, CDP4-V,
Chac
Califumia Rrg~strationExamination Problem. Second-order reacti in different CSTR and PER arrangements. [Znd Ed, P1-I I1 An unremarkable semibatch reactor problem, but it does requ assessing whlch equation to use.
SUPPLEMENTARY READING
DAVIS. M. E.. and R. J. D A V I S ,Fu?~&nlenritlsof Chemicirl Reaction Engine, utg. New York: McGtaw Hill, 2003. HILL.C. G., Art lnrmduction ro CI~emicalEnginrering Kinetics arrcl Renc, Desrgrr. New York: Wiley, 1977, Chap. 8. LEVEMPIEL. 0.. Chnnircrl Rerrrtiun Engineering. 3rd ed. New York: Will l99S, Chaps. 4 and 5. SMITH,j, M.. Chemical Engineering Kincti~.~, 3rd ed. New York: McGra Hill, 1981. ULRICH. G. D.,A Gltide to Chemical Engineering Reactor Design and Kineti Pnnted and bound by Braun-Brumfield. Inc., Ann Arbor. Mich., 1993. WALAS. S. M., Reaction Kinerics for ChernicclE Engineers. New York: McGm HIIZ, 1970.
Recent information on reactor design can usually be found in the followi journals: Chenricnrl Engirreering Science, Chemical Engineering Cutnmunic tions, Industrial rtnd Engit~eeringChemisty Research, Canadian Journal Chemical Engineering, AIChE Joumot, ChemicaI Engineering Progress.
Collection and Analysis + of Rate Data
5
You can observe a lot just by watching. Yogi Berrn, New York Yankees
Overview. In Chapter 4 we have shown that once the rate law is known. it can be substituted into the appropriate design equation, and though the use of the appropriate stoichiornetric relationships, we can apply the CRE algorithm to size any isothermal reaction system. In this chapter we focus on ways of obtaining and analyzing reaction rate data to obtain the rate law for a specific reaction. In particular, we discuss two common types of reactors for obtaining rate data: the batch reactor, which is used primarily for homogeneous reactions, and the differential reactor, which is used for solid-fluid heterogeneous reactions. In batch reactor experiments. concentration, pressure, andor volume are usually measured and recorded at different times during the course of the reaction. Data are collected from the batch reactor during transient operation. whereas measurements on the differential reactor are made during steady-state operation. In expedments with a differential reactor, the product concentration is usualEy monitored for different sets of feed conditions. Two techniques of data acquisition are presented: concentration-time measurements in a batch reactor and concentration measurements in a differential reactor. Six different methods of analyzing the data collected are used: the differential method, the integral method, the method of half-lives, method of initial rates, and linear and nonlinear repession (Ieast-squares analysis). The differential and integral methods are used primarily in anatyzing batch reactor data. Because a number of software packages (e.g., Polymath, MATLAB) are now avaiIable to analyze data, a rather extensive discussion of nonlinear regression is included. We close the chapter with a discussion of experimental planning and of laboratoy reactors (CD-ROM).
254
Collection and Analysls of Rate Data
Chap. 5
5.1 The Algorithm for Data Analysis For batch systems. the usual procedure is to collect concentration time data, which we then use to determine the: rate law. Table 5-1 gives the procedure we will emphasize in analyzing reaction engineering data. Data for homogeneous reactions is most often obtained in a batch reactor. AAer postulating a rate law and combining it with a mole balance, we next use any or ail of the methods in Step 5 to process the data and arrive at the reaction orders and specific reaction rate constants. Analysis of heterogeneous reactions is shown in Step 6. For gas-solid heterogeneous reactions, we need to have an understanding of the reaction and possible mechanisms in order to postulate the rate law in Step 6B. After studying Chapter 10 on heterogeneous reactions, one will he able to postulate different rate laws and then use Polymath nonlinear regression to choose the "btsl" rate law and reaction rare parameters. The procedure we should use to delineate the rate law and rate law parameter is given in Table 5-1.
I. Postulate a rate law. A.
Power law lilodels for homogeneous reactions
- % = LC; ,
-,.,
= kC2 C:
8. Langmuir-Hin~helwoodmodels for hetempeneou~reactions X-PA X-PAP, -r; = -r' I + K ~ P ~ ' . 4 - ( 1 + K~P,+P,)' 2. Select reactor type and corresponding mole balance. A.
If batch reactor [Seclion . 2 ) , use mole balance on Reactant A
- A dC
*-
B.
If differenrial
PRR (Section 5 51,uqe mole =
-,I
A
(TE5-1 .I)
dl
balance on Product P ( A -4P)
P =c AW
I
Aw
(TE5-1 .?I
3, Process your data in terms of rneasurcd variable le.g., NA, CA. or PA>.If nece\s a y rewrlte your mole balance 11: terns of the rnea~uredvar~able(e.g.. P,) 4. L w k For simplifications. For example, ~f one of the reactants in excess. assume its concentration is conqtant. IT the gas phare mole fraction of reaclanl 1s cmall. ser E=O. 5. For a batch reactor. calculate 4, a5 a function of concentration C, to determine reaction order. A.
D ~ f f e r e n ~ i anal! a l
- r , = kC;
I
(TE5-1.3)
Sec. 5.1
The Algorithm for Data Anaiysis
1
and then take the natural log.
(I )Find
255
1
-5 from C, versus r data bj dt
(a) Graphical method (h) Finile differeniial method ( c ) ~uC)JynorninaI
I
(2)P h h
&.
- dC* reiru. dt
InCAa d find rcxiion order
a, which is the slope of
the line fil to the data. (3) Find 6. Integral method For -rA = LC:, the combined mole balance and rate law
IS
(1)Guesq a and integrate Equarion ITES-1.6).Rearrange your equation to obtain the appropriate funcrion of C, which when plotted as a funct~onof lime should he linear. If ~t I S linear. hen the guesced value of a 1s COKeCt and the slope is qpecific reaction rale. k. If il IS not Irnear. pes5 again for a. If you puecs a = I), I, nnd 2 and none of rheses orders fit the data. proceed to nonlinear reression.
(2)Nonlinear regression (Polymofhl Integrate Equation (TES-I ,4) ID oblain
for a + I
(TE5-1.6)
Use Prllynlalh regrewon 10 f nd cc and b. A Prll>math tutorial on regression with screen rhuts ih ~ h u w nin [he Ch;~p~et5 Srrmrllrrr? Nofcs on the CD-ROM and web 6. For differential PER calculate -r', as a function of C, or PA Summary gofer
-ri
C IW
=~
Pa
function of re.lctant concentration.
C,.
A.
Calculate
C.
Use nonlinear regressloo 10 Hnd the be\r model and model parsnjerers. See example on [he CD-ROM Slrmlnun WCI!(V using dafa fbm hctrmpe~~enus catal-
:iq
y w . Ch;~p~cr10 7. Analyze your rate law mudcl Fur "gwdneu uf tit:'Calcula~e n correla~iolrcocftic~ent.
1
256
Collection and Analysis of Rate Data
Chap
5.2 Batch Reactor Data
Process data in terms o f the
measured variable
Batch reactors are used primarily to determine n t e law parameters for horr geneous reactions. This determination is u s u ~ l l yachieved by measuring cc centration as a function of time and then using either the differential, integr or nonlinear regression method of data analysis to determine the reacti order, a , and specific reaction rate constant, k. If some reaction parame other than concentration is monitored, such as pressure, the mote balance mt be rewritten in terms of the measured variable (e.g., pressure as shown in t example in Solved Pmblenls on the CD). When a reaction is irreversible, it is possible in many cases to determi the reaction order a and she specific rate constant by either nonlinear regrt sion or by numericaIly differentiating concenrrntion versus time data. This 1, ter method is most applicable when reaction conditions are such that the rr is essentialty a function of the concentration of only one reactant; for examp if, for the decomposition reaction.
Assume that the rate law is of the
form - r A = kACl
then the differential method may be used. However, by utilizing the method of excess, it is also possible to dett mine the relationship between -r, and the concentration of other reactan That is, for the irreversible reaction
A + B -+ Products with the rate law
where rw and f3 are both unknown, the reaction could first be run in an exce of B so that C , remains essentially unchanged during the course of the rea
tion and
where Method of excms
After determining a , the reaction is carried out in an excess of A, f which the rate law is approximated as
where k" = kACi = kAC;*
Sec. 5.2
257
Batch Reactor Data
Once u and p are determined. A, can be calculated from the measurement of - r , at known concentrations of A and B:
Both a and P can be determined by using the method of excess, coupled with a differential analysis of data for batch systems. 5.2.1
Differentia! Method of Analysis
To outline the procedure used in the differential method of analysis. we consider a reaction carried out isothermally in a constant-volume batch reactor and the concentration recorded as a function of time. By combining [he mole b d once with the rate Inw given hy Eqtrarion (5-1). we obtain Conutartt-volume batch rcuctor
After taking the natural logarithm of both sides of Equation (5-61,
observe that the slope of a plot of In (-dC,ldr) reaction order, a (Figure 5-2 ).
as a function of (la C,} is the
vesus ln C, ro find
a and k,
(a)
~ b )
Figure 5-1 Differential method to determine reaction order.
Figure 5-l[a) shows a plot of [- (dCAJdt)]versus [CAIon log-log paper (or use Excel to make the plot} where the slope is equal to the reaction order a . The specific reaction rate, k,, can be found by first choosing a concentration
258
Cellectlon and Analysis of Rate Data
Chap. 5
in the plot, say CAP,and then finding the corresponding value of I- (dC,ldt)] as shown in Figure 5-l(b). After raising CAPto the a power, we divide it into [- (rdC,/dr),]
Methods for finding
-
2
to determine X., :
To obtain the derivative -dCA/dt used in this plot, we must differentiate the concentration-time data either numerically or graphically. We describe three methods to determine the derivative from data giving the concentration as a function of time. These methods are: Graphical differentiation Numerical differentiation formulas Differentiation of a polynomial fit to the data
from
concentntmn-time data
5.2.1A Graphical Method
krI
gig
t
TKW
See Appendix A.2.
With this method, disparities in the data are easily seen. Consequently, it is advantageous to use this technique to analyze the data before planning the next set of experiments. As explained in Appendix A.2, the graph~cal method involves plotting -ACA/Ar as a function of t and then using equal-area differentiation to obtain - d C , / d l . An illustrative example is also given in Appendix A.2. I n addition to the graphical technique used to differentiate the data, rwo other methods are commonIy used: differentiation formulas and polynomial fitting.
5.2.LB Numerical Method Numerical differentiation formulas can be used when the data points in the independent variable are eguolly spuced. such as t , - t,, = r, - 1 , = : Time lmin)
I r u
r~
1
C o ~ ~ c ~ ~ ~ ~ r ~ z i i o r i l m o UC,,, d m ' ~ CA,
I?
I>
CA,
CAI
C,,
C,,
The three-point differentiation formulas -3C,(,
Initial point:
+ 4CA1- C,? 2Ar
I
B. Carnahan, H. A. Luther, and J 0. Wilke.;, Applied Nrrl~~rricrrl ~Mrrhorir(New Yorh: W~ley,1969). p. 179.
Sec. 5.2
Batch Reactor Data
Interior points:
Last point:
can be used to calculate d C , / d l . Equations (5-8) and (5-10) are used for the first and last data points, respectively, white Equation (5-9) is used for all inrermediate data points. 5.2.1C Polynomial Fit Another technique to differentiate the data is to first fit the concentration-time data to an nth-order polynomial:
Many persona1 computer software packages contain programs that will calculate the best values for the constants a , . One bas only to enter the concentration-time data and choose the order of the polynomial. After determining the constants, a , , one has only to differentiate Equation (5-11 ) with respect to time:
Thus concentration and the time rate o f change af concentration are both known at any time t. Care must be taken in choosing the order of the polynomial. I f the order is too low, the polynomial fit will not capture the trends in the data and nor go through many of the points. If too large an order is chosen. the fitted curve can have peaks and valleys as it goes through most all of the data points, thereby producing significant errors when the derivatives. dCAldf.are generated a t the various points. An example of this higher order fir is shown in Figure 5-2, where the same concentration-time data fit to a third-order polynomial (a) and to a fifth-order polynomial (b). Observe how the derivative for the fifth order changes from a positive value at 15 minutes to a negative value a t 20 minutes.
260
Collection and Analysis of Rate Data
Figure 5-2
Chal
Polynomial fit of concentration-time data.
5.2,lD Finding the Rate Law. Parameters
Now, using either the graphicaI method, differentiation formulas, or the pol nomiaI derivative, the following table can be set up:
Derivative
The reaction order can now be found from a plot of in ( -dC,/dt) as function of In C,, as shown in Figure 5-l(a), since
Before solving an example problem review the steps to determine the reactk rate Iaw from a set of data points (Table 5-1). Example 5-1 Dntetmini~gthe Rate Lnw The reaction of triphenyl methyl chloride (trityl) (A) and methanol (B)
261
Batch Reactor Data
Sac.5.2
was carried out in a soiution of benzene and pyridine at 25°C. Pyridine reacts with HCI that then precrpitates as pyridine hydrochloride thereby making the reaction irreversible. The concentration-time data In Table E5-1.1 was obtained in a batch reactor
Time (min)
0
Concentrarion of A (moVdrn3)x loJ
50
50
38
100
150
200
250
300
30.6
25.6
22.2
19.5
17.4
The initial concentration of methanol was 0.5 mol/dm3. Part (I) Determine the reaction order with respect to triphenyl methyl chloride. Part (2) In a separate set of experiments, the reaction order wrt methanol waq found to be first order. Determine the specific reaction rate constant.
Part (1) Find reaction order wrf trityl. Postulate a rate law. Step 1
Step 2
Process your case is C,.
Step 3
Lwk for simplifications. Because concenmtion of methanot is 10 times
data
in terms of the measured variable, which in this
the initial concentration of lriphenyl methyl chloride. its concentration is essentially constant
I
I I @ &
@&@@ Follo~vingth5 Algorithm
Substituting for Cs in Equation (U-1.I)
Step 4
Apply the CRE algorithm Mole Balance
Rate Law
Stoichiometry: Liquid
v = v, c* , - NA -t6
262
Colleclion and Analysis of Rate Data
Chap. 5
Combine: Mole balance, rate law, and stoichiometry
I
Taking the natural log of both sides of Equation (E5-I.5)
(E5-1.6) The slope of a plot of In
versus In CAwill yield the reaction [-5 drl
order a with respect to triphenyl methyl chloride (A).
as a function of C, from concentration-time data.
We will find
(-%)
by each of rhc hrcc mehods just dircurrd. the
graphical, finite difference. and polynomial methods. Step 5A.Za Graphical Method. We now construct Table E5-1.2. TABLE E5-1.2
PROCESSED DATA
r (min) 0 50 100 150 200
250 300
The derivative -dCAMt is determined by calculating and plotting (-dCAlbt) as 3 function of time, r, and then using [he equal-area differentiation technique (Appendix A.2) to determine (-dCAldt) aq a functton of CA. F ~ n l we , calculate the ratio I-ACA/A!) from the first two columns of Table E5-1.2: the result is written in the third column, Next u e use Table E5-1.2 to plot the third column as n function of the
Sec. 5.2
263
Batch Reactor Data
first column in Figure E5-I. I [i.e., (-AC,/At) versus I]. Using equal-area differentiation, the value of (-dC,ldrf is read off the figure (represented by the arrows); then it is used to complete the founh column of Table E5-1.2.
mol dm3 min
0
50
100
150
300
250
300
350
t (min) Figure ES-1.1 Graphical differentiation.
We now calculate (dC,/dt) using the finite difference formulas 1i.e.. Equations (5-8) through 15- lo)].
Step 5A.lb Finite Difference Method.
- 3C,,+4CA, - C*, 2A 1
= 1- 3(50)4 4(38) - 30.61 X loL.' 100
= -2.86 x !04 moI/drd . rnin
-5 X 10' dr 1
= 2.86mol/drn3 I rnin
= - 1.94 X 1 IE-' molldrn? . min
264
Collactron and Analys~sof Ra!s Data
Chap
= -0.84 x 1 O4 mol/dm3 , rnin
= -0.6 1 x 10-' molldm3 . rnin
= -0.48 x 1W mol/dm3 min 4
.
= -0.36 x 1 p molldm3 rnin
Step SA.Ic Polynomial Method. Another method to determine (dC,/(ir) is to fit tt concentration of A to a polynomial in time and then to differentiate the resultin Summary Notes .4 PoIymnth turonnl Tor fitting data can be found iln the Summary Notes on the
polynomial.
We will use the Polymath software package to express concentrarion as a funcric of time. Here we first choose the polynomial degree (in this case, fourth degree) an then type in the values of C, at various times r to obtain
CD.
C, is in (rnoVdm") and t is in minutes. A plot of C, versus t and the correspondin fourih-degree potynomial tit are shown in Figure E5-1.2.
MU-% C A = a@+ a t 1 s ? ~ ? a p. l : 1
a0 a1
Value 0.0499901 -2.9fBZ-04
+
a
r
~
9 S a lcmZidm~e
3.1P-01 1.762E-05
::"
lon of K~me
-
D,057 0O M
Figure ES-1.2 Polynomial fit.
Sec. 5.2
Batch Reactor Data
/
Differentiating Equation tE5- 1.7) yields
I
Elntp: You can
also obtain Equation tE3-1.9) directly from Polymath.
To find the dcrivatiw at various timer, we substitute the appropriafe time into Equation (ES-1.8)to arrive at the fourth column 10 Table E5-1.4and multiply by (-1). We can see that thee is quite a close agreement between the graphical technique, finite difference, and the polynomial methods.
Finite Differpncs
-
-dC4 x IO.BM
dr
(moUdn' . min)
(molldm! . min)
3.0 1 .86
2.86 1.91
1.20
1 .?-I
0.80 0.68 0.54
0.81 0.61 0.4R
0.42
0.36
We wilt now plot columns 2. 3. and 4 column 5 (CAx 1.000) on log-log paper as shown in Figure E5-1.3.We could also substitute the parameter values in Table W-1.4 into ExceI to find a and k'. Note that most all of the points for all methods fall virtually on top of one another. From Figure ES-1.3. we found the slope to be 2.05 so that the reaction is said to be second order wrt triphenyl methyl chloride. To evaluate k'. we can evaluate the derivative and CAP= 20 x I&? moIEdm3.which is
(
then
As will be shown in Section 5.1.3, we couId also use nonlinear regression on Equation (E5-1.7)to find k': k' = 0.122 dm3/moI . min
(E5-1.11)
266
Collection and Analysis of Rate Data
Chap. 5
mln mol dn?
0.1
1m
10
WI 0' Figure ES-13
{rnotldrnAa)
Excel plot to determine a and k.
-
Summary Motes
The Excel graph shown in Figure E5-1.3 gives a = 1.99 and k' = 0.13 dm3/mol min. We could set ci = 2 and regress again to find 'k = 0.122 drn3/mol . min. ODE Regression. There are techniques and software becoming available whereby an ODE solver can be cornblned with a regression program to solve differential equations, such as
to find k, and
a from concentration-time date.
Part (2) The reaction was said to be first order wrt methanol,
Assuming Cs,,is constant at 0.5 molldrn? and solving for k yields
-
k' CBO
0.122-
dm3 rnol . min
0.5-mol dm3
k = 0 . W (dm31mol)2J min The rate law is
fi = 1,
Sec. 5.2
Batch Reactor Data
5.2.2 Integral Method
The integral uses a tnaland-ermr gmcedure to
find reaction order.
To determine the reaction order by the integral method, we guess the reaction order and integrate the differential equation used to model the batch system. If the order we assume is correct, the appropriate pIot (determined from this integration) of the concentration-time data should be linear. The integral method is used most often when the reaction order is known and it is desired to evaluate the specific reaction rate constants at different temperatures to determine the activation energy. In the integral method of analysis of rate data, we are looking for the appropriate function of concentration corresponding to a particular rate law that is linear with time. You should be thoroughly familiar with the methods of obtaining these linear plots for reactions of zero. first, and second order. For the reaction A -+ Products
~t is important to
know how lo ncnerate linear plots Gf functions of CA versus r for zero-, Rrsl-, and secondarder reactions.
carried out in a constant-volume batch reactor, the mole balance is
For a zero-order reaction, r,
=
-k, and the combined rate
law and mole bal-
ance is
~ - -C -A- k dr
Integrating with CA = C,,
at
t = 0, we have
Zero order
A plot of the concentration of A as a function of time will be linear (Figure
5-3) with slope ( - k ) far a zero-order reaction carried out in a constant-volume batch reactor. If the reaction is first order (Figure 54), integration of the combined mole halance and the rare low
with the limit C, = C,, at 1 = 0 gives First order
Consequently, we see that the slope of a plot of [In (C,,/C,f time is linear with slope k. If the reaction is second order (Figure 5-51, then
3 as a function of
1 Tne idea
1s to
arrange the data so that a
linear
relalronshlp is obtained.
269
Batch Reactor Data
Sec. 5.2
out not to be linear, such as shown in Figure 5-15,we would say that the proposed reaction order did not fit the data. In the case of Figure 5-6, we would conclude the reaction is not second order. It is imponant to restate that, given a reaction rate law, you should be able to choose quickly the appropriate function of concentration w conversion that yields a straight line when plotted against time or space time.
I
I
/
Example 5-2
Integral Method of CRE Dafa Analysis
Use the integral method to confirm that the reaction is second order wrt triphenyl methyt chloride as described in Example 5-1 and to calculate the specific action rate k'
I
Trityl (A) + Merhanol (B)+ Products
Substituting for a = 2 in Equation (E5-t - 5 )
I
we obtain
1
Integrating with CA= C,
1
Rearranging
at
r =0
We see if the reactton is indeed second order then a plor of (l/CA) versus t should be linear, The data in Table ES-1.1 in Example 5-1 will be used to construct TaMe
E5-2.1.
In a graphical solution, the data in Table E5-2.1 can be used to construct a plot of lECA as a function of r, which will yield the specific reaction rate k'. This plot is shown in Figure E5-2.1. Again. one could use Excel or Polymath to find k" from the data in Table E5-2.1. The slope of the line is the specific reaction rate K
270
Collection and Analysis of Rate Data
m
0
1W
1M
200
m
Em
Chap. 5
36(1
(mlnl Plot reciprocal of C, vesut r for a second-order reaction.
Figurn E5-2.1
We see from the Excel anaysis and plot that the slop of the line is 0.12 dm'/mollmin.
k' = 0.12
dm3 ma1 - min
Calculating k.
k=---k'
_ 0.12 dm3 lmollrnin = 0+2q(&)
3 2
CBO 0.5 mol/dm3
mol
The rate law is
We note the integral method tends to smooth the data.
An alternate computer solutron would be to regress
I -
C,
versus t with a software
package such as PoI~math.
1
Let CA inverse = -,
c4
a.
= -,1
and
c*,+
n , = k'
and then enter the data in Table E5-1.1.
Linear Regression Report
Variable
Xlue
95% confidence
From the Polymath output. we obtain k' = 0.125 dm'trnol/min. which yields k = 0.25 dm3Ernollrnin.H'e shall dlscuks regression in Example 5-3.
Sec. 5.2 Integral method
aorrnally used to find k when order is known
271
Batch Reactor Data
By comparing the methods of analysis of zhe rate data presented in Example 5-2, we note that the differential method tends to accentuate the uncertainties in the data, while the integral method tends to smooth the data, thereby disguising the uncertainties in it. In most analyses, it is imperative that the engineer know the limits and uncertainties in the data. This prior knowledge is necessary to provide for a safety factor when scaling up a process from laboratory experiments to design either a pifot plant or full-scale industrial plant. 5.2.3 Nonlinear Regression
In nonlinear regression analysis, we search for those parameter values that minimize the sum of the squares of the differences between the measured values and the calculated values for all the data points.? Not only can nonlinear regression find the best estimates of parameter values. it can also be used to discriminate between different rate law models. such as the Langmuir-Hinshelwocd models discussed in Chapter 10. Many software programs are available to find these parameter values so that all one has zo do is enter the data. The Polymath software will be used 20 illustrate this technique. In order la carry out the search efficiently, in some cases one has to enter initial estimates of the paramerer values close to the actual values. These estimates can be obtained using the linear-least-squares technique discussed on the CD-ROM Professional Reference Shelf. We will now apply nonIineas least-squarer analysis to reaction rate data to determine the rate law parameters. Here we make estimates of the parameter values (e.g., reaction order, specific rate constants) in order to calculare the rate of reaction. r;. . We then search for those values that will minimize the sum of the squared differences of the measured reaction rates. r,,, , and rhe calculated reaction rates, r, . That is, we want the sum of ( r , - r,)' for all data points to be minimum. If we carried out N experiments, we would want to find the parameter values (e.g., E, activation energy. reaction orders) that would minimize the quantity 2 C T I = S
N-K
=
,v C L
(rjrn-ric)~
N-K
where =
C IT?,,,- l . J 2
N = number of runs K = number of parameters to be determined r,, = measured reaction rate for run i (i.e., - r,,,) r,, = calculated reaction rate for run i (i.e., ?
- I-,,,)
See also R. Mezakiki and J. R. Kirtrell. AICIiE J.. I4, 513 (I96R1, and J. R. Kttrrell. 111d.Etrg. Chrm.. 61 151, 76-78 1 1 9691.
272
Collection and Analysis af Rate Data
Chap
To illustrate this technique, let's consider the first-order reaction A
+Product
for which we want to learn the reaction order. a, and the specific reaction rate,
The reaction rate will be measured at a number of different concentrations. P now choose values of k and a and calculate the rate of reaction Ir,,) at eat concentration at which an experimental point was taken. We then subtract tf calculated value (r,,)from the measured vaIue ( 5 , ), square the result. and su the squares for all the runs for the values of k and a we have chosen. This procedure is continued by further varying ol and k until we find the best values, that is, those values that minimize the sum of the squares. Mar well-known searching techniques are available to obtain the minimum val i uii, .-' Figure 5-7 shows a hypothetical plot of the sum of the squares as function of the parameters or and k
Figurr 5-7 Minimum sum aT squares,
In searching to find the parameter values that give the minimum of th sum of squares u2,one can use a number of optimization techniques or so€ ware packages. The searching procedure begins by guessing parameter valuc and then calculating r, and then &for these values. Next a few sets of pararr eters are chosen around the initial guess, and v2 is calculated for these sets 2 well. The search technique Iooks for the smallest value of a ' in the vicinity c
(a) 33. Carnahsn and J. 0. Wilkes, Di~italComputing aad Numerical Methods (Ne Yark: Wiley, 1973), p. 405. (b) D. I. Wlde and C. S. Beightler, Fotindations of Opt mizafion 2nd ed. (Upper Saddle River, N.J.: Prentice Hall. 1979). (c) D.Miller an M. Frenklach, Int. J. Chem. Kincf., 15, 677 (1983).
Sec. 5.2
Batch Reactor Data
k
Figure 5-8 Trajectory
vap the in~tial guessesof parameters to make the sure you true rnlnirnum.
ReferenceShelf
to find the best values of
k and a
the initiat guess and then proceeds along a trajectory in the direction of decreasing cr2 to choose different vaiues and defermine the corresponding rr2. The trajectory is continuaIly adjusted so as always to p m e e d in the direction of decreasing u2 until the minimum value of rr2 is reached. A schematic of this procedure is shown in Figure 5-8, where the parameter vaiues at the minimum are a = 2 and k = 5 s-I. If the equations are highly nonlinear, the initial guess is extremely important. In some cases it is useful to tql different initial guesses of the parameter to make sure that the software program converges on the same minimum for the different initial guesses. The dark Iines and heavy arrows represent a computer trajectory, and the light lines and arrows represent the hand calculations. In extreme cases, one can use linear least squares (see CD-ROM) to obtain initial estimates of the parameter values. A number of software packages are available to carry out the procedure to determine the best estimates of the parameter values and the corresponding confidence limits. AlI one has to do is to type the experimental values in the computer, specify the model, enter the initial guesses of the parameters, and then push the computer button. and the best estimates o f the parameter values aIong with 95% confidence limits appear. If the confidence limits for a given parameter are larger than the parameter itself, the parameter is probably not significant and should be dropped from the model. After the appropriate model parameters are eliminated, the software is run again to d e t e n i n e the best fit with the new model equation.
Concentration-Time Data. We wiIl now use nonlinear regression to determine the rate law parameters from concentration-time data obtained i n batch experiments. We recall that the combined rate law-stoichiometry-mole balance for a constant-volume batch reactor is
274
Collection and Anaiysis of Rate Data
Chap. 5
We now integrate Equation (5-6) to give
Rearranging ro obtain the concentration as a function of time, we obtain C, = C [,
I-u-
( I - c~)kt]"('-~'
(5-1 8)
Now we could use Polymath or MATLAR to find the values of rx and k that would minimize the sum of squares of the differences between the measured and calculated concentrations. That is, for N dara points.
we want the values of ol and k that will make s2 a minimum. Tf Polymath is used. one should use the absolute value for the term in
brackets in Equation 15-19), that is, 2
n
s2 =
1[c,.,
- ((abs[~:;'-
(I
]
- a)kt,]} I 4 l -a)
(5-20,
1=1
Another way to solve for the parameter values is to use time rather than concentratians:
That is. we find the values of k and a that minimize
Finally, a discussion of ~t.eighredIeusr squaws as applied to a first-order react i ~ nis provided in the Pmfessio~znlReference She%R5.2 on the CD-ROM. Reference Shelf
I
Example 5-3 Usc of R~gressionto Find the Rate Law Parornetem We shall use the reactron and data in Example 5-1 to illustrate how to use regression to find a and k'.
See. 5.2
275
Batch Reactor Data
The Polymath regression p r o p is included on the CD-ROM.Recalling Equation (ES- 1.5 j
and integrating with the initial condition when t = 0 and CA= CAO for a r 1.0
Substituting for the initial concentration CAO = 0.05molidm3
Let's do a few calculations by hand to illustrate regression. We will first assume a value of a and k and then calculate t for the concentrations of A given in Table E5-1.1. We will then calculate the sum of the squares of the difference between the measured times I, and the calculated times (i.e.. s2). For At measurements,
Our first guess is going to be a = 3 end k'= becomes
5. with CAo= 0.05.Equation (E5-3.2)
We now make the calculations ibr each measurement of concentration and fill in columns 3 and 4 of Table E5-3.1. For example, when CA= 0.038 molldm3 then =-I
--
'
1°[(0.03S)'
1
400 = 29.2 mtn
which is shown in Table E5-3.1 on line 2 for guess 1. We next calculate the squares of difference (t,, - rCl)' = (50 - 29.2)2 = 433. We continue in this manner for points 2, 3, and 4 to calcutate the sum s2 = 2916. After calcuiating s2 for a = 3 and k = 5, we make a second guess for a and k'. For our second guess we choose a = 2 and k = 5: Equation (E5-3.2) becomes
We now proceed with our second guess to find the sum of (I, - tJ2 to be s? = 49.895, which is far worse than our first guesq. So we continue to make more guesses of a and k and find s?. Let's stop and take a look at t, for gueFses 3 and 4.
Collection an< Analysis OF Rate Data
276
Chap
We shall only use four points for this illustration. REGRESSION OF DATA
TABLE E5-3. [.
We see that (k' = 0.2 drn3Jrnol . min) underpredicts the rime (e.g., 3 1.6 rnin vers 50 minutes). while (k' = 0.1 dni3/mol min) overpredicts the time (e.g., 63 min vr sus 50 minutes). We could continue in thls manner by choosing k' between 0.1 < < 0.2, but why bother to go to all the trouble? Nobody has that much time on th hands, Why don't we just let the Polymath regression program find the values of and a that will minimize s2? The Polymath tutorial on the CD-ROM shows screen shots of how to en the raw data in Table E5-I. 1 and to carry out a nonlinear regression on Equati (E5-3. I). For C, = 0.05 rnolldm3. that is, Equation (ES-3.1)hecomes 4
Summary Note:
A Polymath tutorial on regression I F given on the
CD-ROhl.
We want to minimize the sum to give o: and k'
115'4
-.o.-
Ka.0 vaci-
= 03MYtl
P-I R'I
O.POP97t7
.
OM1IW
278
Collection and Analysis of Rate Data
Chap. 5
Example 5 4 Method of hirial Rdes in Sola-Liquid D f solution Kineks
The dissolution of dolomite. calcium magnesium carbonate, in hydrochloric acid is a reaction of particular imponance in the acid stimulation of dolomite oil reservoirs? The oil is contained in pore space of the carbonate material and must flow through the small pores to reach the well bore. In matrix stirnulatian, HCI is injected into a well bore to dissolve the porous carbonate matrix. By dissolving the solid carbonate, the pores will increase in size, and the oil and gas will be able to flow out at faster rates. thereby increasing the productivity of the well? The dissolution reaction is
The concentration of HCI at various times was determined from atomic absorption spectrophotometer measurements of the calcium and magnesium ions. Determine the reaction order with respect to HCI from the data presented in Figure E5-4.1 for this batch reaction. Assume that the rare law is In the form given by Equation (5- I ) and that the combined rate law and mole balance for HCI can be gwen by Equation (5-61.
An irnpwtant reaction for enhancement of 011 flow in carbonare
reseworrs
Run 1
Run 2
Wormholes etched by acid
tbl Figure ES-4.1 Concentration-time data.
Solution
Evaluating the mole balance on a constant-volume batch reactor at time f = 0 give5
1
Taking the log o f both rider of Eq~ation(E5-4.1).we have
K. Lund, H.S . Fogler, and C. C. McCune, Chern. Eng. Sri.. 28, 691 11973).
' M. Hcefner and H. S. Fogler, AICltE Journal, 3 4 I), 45 (198R).
280
Collection and Analysis of Rate Data
Cha
5.4 Method of Half-lives The method of
T h e half-life of a reaction,
rip, is defined as the time it takes for the concr of the reactant to fall to half of its initial value. By determining half-life of a reaction as a function of the initial concentration. the react1 order and specific reaction rate can be determined. if two reactants ,
Vuires[ration
many experiment\
involved in the chemical reaction, the experimenter will use the method excess in conjunction with the method of half-lives to m n g e the rate Iaw the form
For the imversible reaction Products
A The concept of a half life. f l js very important in b
a mole balance on species A in a constant-volume batch reaction system co bined with the rate law results in the following expression:
drug medication.
Integrating with the initial condition C, = C,o when t = 0. we find that
The half-life is defined as the time required for the concentration to drop half of its initial value; that is, I=
when
C, = iC,,
Substituting for C ,, in Equation (5-23) gives us
There is nothing special a b u t using the rime required For the concent tion to drop to one-half of its initial vaIue. We could just as well use the ti required for the concentration to fall to Lln of the initial value, in which c z
Sec. 5.5
281
Differential Reactors
For the method of half-lives. taking the natural log of both sides of Equation (5-241, plor
a< 4 Function ol C," OF use regression software.
we see that the slope of the plot of lntln as a function of InCA,, is equal to 1 minus the reaction order [i.e.. slope = ( I - u}]: Rearranging: Sfope = 1 - a c
For the plot shown In Figure 5-Q, the slope i s I:
-
a = 1-(-1)=2 In CAO Figure 5-9 Method of half-liws.
The corresponding rate law is -rA = kc:
Note: We also could have used nonlinear regression an the half-life data. However. by plotting the data, one often gets a "better feel" for the accuracy and precision of the data.
5.5 Differential Reactors
M O S ~ commonly
used catalytlr reactor to obtain evperlmental data
Limitations of the differentid reactor
Data acquisition using the method of initial rates and a differential reactor are similar in that the rate of reaction is determined for a specified number of predetermined initial or entering reactant concentrations. A differential reactor is normally used to determine the rate of reaction as a function of either concentration or partial pressure. It consists of a tube containing a very small amount of catalyst usually arranged in the form of 3 thin wafer or disk. A typical mangement is shown schematically in Figure 5-10. The criterion for a reactor being differential is that the conversion of the reactants in the bed is extrerneiy small, as is the change in temperature and reactant concentration through the bed. As a result, the reactant concentration through the reactor is essentially constant and approximately qua1 to the inlet conccntratian. That is, the reactor is considered to be gradientle~s.~ and the reaction rate is considered spatiatly uniform within the bed, The differential reactor is refatively easy to constnlct at a low cost. Owing to the low conversion achieved in this reactor. the heat refease per unit volume wilt be small (or can be made small by diluting the bed with inert solids) so that the reactor operates essentially in an isothermal manner. When operating this reactor, precautions must be taken so that the reactant gas or liquid does not bypass or channel through the packed catalyst, but instead ffows uniformly across the catalyst. If the catalyst under investigation decays rapidly, the differential reactor is not a good choice because the reaction rate parameters at the
' B. Anderson, ed.. Experimental Methods in Catalytic Research (San Diego, Calif.: Academic Press. 1976).
282
Collection and Analysis of Rate Dafa
Chap. 5
start of a run will be different from h s e at the end of the run. In some cases sampling and analysis of the product stseam may be difficult for small conversions in multicomponent systems. lnen 6111ng
Figure 5-11 Differential catalyst bed.
Figure 5-10 Differ~ntialreactor
The volumetric dow rate through the catalyst bed is monitored, as are the entering and exiting concentrations (Figure 5-11). Therefore, if the weight of catalyst, AW, is known, the rate of reaction per unit mass of catalyst, -rA, can be calculated. Since the differential reactor is assumed to be gradientlesx, the design equation will be similar to the CSTR design equation. A steady-state mole balance on reactant A gives Rate of
-
Rate
of
I-.:--
1
Rate of reaction Mass of cat
The subscript e refen to the exit of the reactor. Solving for
-ri,
we have
The mole balance equation can also be written in terms of the concentration -
Differential reactor design equation
or in terms of the conversion or product flow rate F p:
Sec. 5.5
Differential Reactors
283
The t e n FA& gives the rate of formaiion of the product, F p , when the stoichiometric coefficients of A and of P are identical. For constant volumetric Aow, Equation (5-27) reduces to
Consequently, we see that the reaction rate. -rA, can be determined by measuring the product concentration, C, . By using very little catalyst and large voIumetric flow rates, the concentration difference. (CRO - C A r ) can , be made quite small. The rate of reaction determined from Equation (5-29) can be obtained as a function of the reactant concentration in the catalyst bed, C :,,
by varying the inlet concentration. One approximation of the concentration of A within the bed. CAh,would be the arithmetic mean of the inlet and outlet concentrations:
However, since very little reaction takes place within the bed. the bed concentration is essentially equal to the inlet concentration,
so -uA is a function of C,,
As with the method of initial rates, various numerical and graphical techniques
Y
can be used to determine the appropriate algebraic equation for the rate law. When collecting data for fluid-solid reacting systems. care must be taken that we use high Row rates through the differential reactor and srnaIl catalyst particle sizes in order to avoid mass transfer limitations. Xf data show the reacriori to be first order with a low activation energy, say 8 kcal/moles. one should suspect the data is being collected in the mass transfer limited regime. We wilI expand on mass transfer limitations and how to avoid them in Chapters 10, I I , and 12.
284 Example 5-5
Collection and Analysis of Flala Data
Cha
DifSerential Reactor
The formation of methane from carbon monoxide and hydrogen u s ~ n ga nickel alyst was studied by P ~ r s I e y The . ~ reaction
was carried wt at 50O0F in n differential reactor where the effluent concentratb methane was measured. (a) Relate the rate of reaction to the exit methane concentration. (b) The reaction rate law is assumed to be the product of a function of the par pressure of CO.f(CO), and a function of the partial pressure of Hz,g ( H , )
Determine the reaction order with r e s p t to carbon n~onoxide.using the dat: Table E5-5.1. Assume that the functional dependence of rkH, on PC, is of the fc
TAHLE E5-5.1.
P,, is constant in - Rum l , Z , 3 Pco is constant in Runs 4.5. h
Run 1 2
3 4 5
h
P,
(atm) I
1.8 4 08 1.0 1.0 1.O
RAWDATA
P,: (atrn) I .o 1 .O 1 .O 0.1 0.5
4.0
C,,,(mol!drnJj
I 73 x lo-' 4.40 X IOA4 10.0 x lom4 1.65 x 10-a 2.47 X 1.75 x 10-4
The exit volumetric Row rate from a differential packed bed containing 10 g of8 alyst was maintained at 300 drnslrnin for each run. The partial pressures of H, CO were determined at the entrance 10 the reactor, and the methane concentral was measured at the reactor exit. Solution
(a) In this example the product composition, rather than the reactant concentwt is being monitored. -rho can k written in terms of the flow rate of methane fi the reaction,
J. A. Putsley, An Investigation of ltbe Reaction between Carbon Monoxide and Hyr gen on a Nickel Catalyst above One Atmosphere, Ph.D. thesis, University of Michit
Sec. 5.5
I
285
Differential Reactors
Substituting for F,-,jJ in terms o f the volumetric flow rate and the concentration af methane gives
Since u, .Cod, and A W are known for each run. we can calculnte the rate of reaction. For run 1: -r;o =
(?)
lo-' mol,dm? = 5.1 x 10 gcat
1 0 3
mol CHI g cat X min
The rate for runs 2 through 6 can be calculated in a similar manner (T~bleE5-5.2). TABLE €5-5.1.
Run
Po (arm)
RAW AND CAI.CULATED DATA
PHI(atrn)
CCH4Imoltdm~)
Determining the Rate Law Dependence in CO For constant hydrogen concentration, the rate law
r ; ~= ,
.g(P,,>1
can be written as
Taking the log of Equation (ES-5.4) gives us
InIr;,
= Ink'+a InPco
We now plot In(rmJ) versus In Pco for runs 1. 2, and 3 . (b) Runs 1, 2, and 3. for which the H2 concentration is constant, are plotted in Figure E5-5.1.We see from the Excel plot that a = 1.22. Had we included more p i n t s we would have found that the reaction is essentially first order with a = 1, that is,
-rho = k'Pc0 F~nmrhe first three data points where the partial pressure of the rate is linear in partial pressure. of CO.
(E.5-5.5)
Hzis constant. we see
CoflMion and Analysis of Rate Data
Chap. 5
0.001 1
0.1
10
PCO latm) Figure E5-5.1 Reaction rate
as a function of concentration.
Now let's look at the hydrogen dependence
Determining rhe Rote Lnw Dependence on HI
Fmm Table E5-5.2it appears that the dependence of riH4on PHl,cannot be represented by a power law. Comparing run 4 with run 5 and run I w ~ t hrun 6. we see that the reaction rate first increases with increasing partial pressure of hydrogen. and subsequenrly decreases with increasing P,, . That is, there appears to be a concentration of hydrogen at which the rate is mhimum. One set of rate laws that is conslstent with these observations is: 1. At low HZ concentrations whcre rhH4increases as PHI increases, the rate law may be of the form
2. At high H2 concentrations w)here rbHddecreases as PHI increases,
We would like to find one rate law that is consistent with reaction rate data at both high and low hydrogen concentrations. Application of Chapter 10 material 5uggesrs Equations (E5-5.6) and tE5-5.7) can be combined into the form
We wiIT see in Chapter 10 that this combination and similar rare laws which have reactant concentrations for partial pressures) in the numerator and denominator are common in heremgeneous caralysis. Let's see if the resulting ratc law cE5-5.8) is qualitatively consistent with the rate observed. 1 . For condition 1: At low P H 2 .( b ( ~ ~ ~1) ) and ' ~Equation e (E5-5.8)reduces to
Equation (E5-5.9)is consistent with the trend in comparing mns 4 and 5. 2. For candirion 2: At high P H 2 ,b ( ( ~ ~ , ) ' "11) and Equation (E5-5.8) reduces to
where p, > P, . Equation (E5-5.10) 1s consistent with the lrends in comparing runs 5 and 6. Combining Equations (E5-5.8)and (65.5) Typical form of the rate law For heterogeneous
catalysis
We now uqe the Polymath regression program to find the parameter values n, b, Dl. and P2. The results are shown in Table E5-5.3.
I
Model: Rale = a'Pco'PM%lall(l*b'PWeZ)
I
Msx
Polymath rc-shion tutanal 1s in the Chapter 5 Summuy .Vnres.
Nonlinear regrassion M n g s #
iteralions = 134
The corresponding rale law is
h'e could use the rate Ian, as given by Equation (E5-5.17)as is. hut there are only six dara points, and we should be concerned about extrapolating the ratc law over n wider range of partial pre3sures. We could lake more data, and/or w e could carry out a theomtical analysis of the type diqcus9ed in Chapter IO for heterogeneous reactions. If we nwume hydrogen underptxs disiociative ;~dsorption on the catalyst
288
Collection and Analvsis of Pate Data
Cht
surface one would expect a dependence of hydrogen to the % power. Because I 1s close to 0.5, we are going to regress the data again setting = I and 0: = The results are shown in Table E5-5.4.
D,
Modd: Rate = a'Pco'PhZ"O.SI{l+b'PhZ)
The rate law is now
mol/gcat . s and the partial pressure is in atm, also have set Dl = !+ and f12 = 1.0 and rearranged Equa (ES-5.11)in the form where r'&, is in
We
>auld
Linearizing the rate law to determine the rate Iaw parameters
A plot of F ~ ~ Pas a~ function ~ ~ of IPH2 ~should , ~be a straight line with an ir cept of I l a and a sbpe of bla. From the plot in Figure E5-5.2. we see that the law i s indeed consistent with the rate law data.
I 0
I
t
2
3
PH*(arm)
Figure E5-52 Linearized plot of data.
4
Sec. 5.6
Expertmental Planning
5.6 Experimental Planning Four to six weeks in the lab can save you an hour in the library. G. C.Quarderer. Dow Chemical Co.
Reference Shelf
So f a , this chapter has presented various methods of anaiyting rate data. It is just as imponant to knaw in which circumsrances to use each method as it is to knaw the mechanics of these merhods. On the CD-ROM. we give a thumbnail sketch of a heuristic to plan experiments to generate the data necessary for reactor design. However. for a more thorough discussion, the reader is referred to the books and articles by Box and Huntern9
5.7 Evaluation of Laboratory Reactors
1 I
!i
I I
I
Reference Shelf
The successful design of industrial reactors ties primarily with the reliabi!ity of p c experimentally determined pnmmercrr w.ed in the scale-up. Consequently, rt i s imperative to design equipment and experiments that will generate accurate and meaningful data. Unfortunately, there is usually no single cornprehensive laboratory reactor that could be used for all types of reactions and catalysts. In this section, we discuss the various types of reactors that can be chosen to obtain the kinetic parameters for a specific reaction system. We closely follow the excellent strategy presented in the articIe by V. W. Weekman of Mobil Oil. now E x x ~ n M o b i I . ' ~
5.7.1 Criteria The criteria used to evaluate various types of laboratory reactors are listed in Table 5-2. I. Ewe of sampling and pmduct analysis 2. Degree of isothemality 3. Effectiveness of contact between catalyst and reactant 4. Handling of catalyst decay 5. Reactor cost and e s e of construction
Each type of reactor is examined with respect to these cdteria and given (F). or poor (P). What follows is a brief description of each of the laboratory reactors. The reasons for rating each reactor for each of the criteria are given in Pmfessiunal Reference Shelf R5.4 on the CD-ROM. a rating of good (G), fair
G. E. P.Box, W.G.Hunter, and J. S. Hunter. Srnrisficsfor Erperimenrers: A n Inrmdiiction to Design, Dara Analysis, and Model Building (New 'fork: Wiley, 1978). j0V. W. Weekman. AlChE J., 20,833 (1974).
290
Collection and Anatysis of Rate Data
Chap. 5
5.7.2 Types of Reactors
The criteria in Table 5-2 is applied to each of the reactors shown in Figure 5-12 and are also discussed on the CD-ROM in Pmfessional Rderence ShelfR5.4. Reference Shelf
(a) In~egraIreactor
(h~ Stirred batch reactor.
( c ) Stirred contained sol~ds
reacror.
(dl Sold\ in a CSTR.
(el Stra~ght-through transpn reactor
(0 Recirculating transport reactor.
Figure 5-12 [From V. Weekman, AICiw J , 20. 833 ( 1974) with perml~sionof the AIChE. Copyright 0 1974 AIChE. All rights reserved.]
5.7.3 Summary of Reactor Ratings The ratings of the various reactors are summarized in Table 5-3. From this table one notes that the CSTR and recirculating transport reactor appear to be the best choices because they are satisfactory in every category except for construction. However, if the catalyst under study does not decay, the stirred batch and contained solids reactors appear to be the best choices. If the system is not limited by internal diffusion in the catalyst pellet, larger pellets could be used. and the stirred-contained solids is rhe best choice. Ef the catalyst is nondecaying and heat effects are negligible. the fixed-bed (integral) reactor would be the top choice. owing to it< ease of construction and operation. #o\vever. in practice, usually more rhnn nrle reactor type is used in determining the reaction rate law parameters.
Chap. 5
291
Summary
Reac~orfipe Differenrial Fixed bed S t ~ batch d Surred-contained solids Continuous-mmed tank Straight-through transpan
SOI~IP~IR~ and Fltiid-Solid Analyszs Ixothennali~ Contacr
Drcnx~ng Caratur
P-F
F 4
F
G
P-F
F
F
G G G
G
P P P
F-G
P
F-G
F-G
P -F
F-G
G
G F-F
G P
F-G
G E
Reciruular~ngIranspon
F-G F-G
Pulre
G
FG
Enrr of Consrnrction
C G
G F 4 P-F F-G P- F G
%, g d : F. fmr: P, poor.
Closure. After reading this chapter, the reader should be able to analyze data to determine the rate law and rate law parameters using the graphical and numerical techniques as well as software packages. Nonlinear regression is the easiest method to analyze rate-concentration data to determine the parameters, but the other techniques such as graphicd differentiation help one get a feel for the disparities in the data, The reader should be able ta describe the care that needs to be taken in using nonlinear regression to ensure you do not arrive on a false minimum for d . Consequently, it is advisable to use more than one method to analyze the data. Finally, the reader should be abIe to carry mit a meaningful discussion on reactor selection to determine the reaction kine~icsalong with how lo efficiently plan experiments.
SUMMARY
a. Plat -bCA /ill as a function nf r. h. Determine -dC,/dt from t h ~ sphr. c. T~kerhe In of both hides of (55-1) to get
292
Ccl!ection and Analysis of Rate Qa!a
Cha
Plot In[-dCAldt) versus In C., The slope will be the renclion order a . could use finite-difference formutns or software packages to eval (-dC,/df) as a functlon of time and concentration.
2. Ftltegml method
a. Guess the reaction order and inrgrate the mole baEance equation. b. CalcuEnte the resulting function of concentntion for the data and plot i a function o f time. Tf the resulting plot is linear, you have prob: guessed the correct reaction order. c. If the plot i s not linear, guess another order and repeat the procedure. 3. Nortlinear regression: Search for the parameters of the rate law that will n irnize the sum of the squares of the difference between the measured rat1 reaction and the rate of rtaction calculnted from the pnrameter values cho For N experimental suns and K parameters to be determined, use PoIyma
"',. 1"
[r,(measured) - ri(calculatedEl?
=
N-K
(S:
4. Method of initial sates
In this method of analysis of rate data, the slope of a plot of In(-rAo) ve, In CA0will be the reaction order. 5. Modeling fhe diflerentinl reacror: The rate of reaction i s calculated from the equation
In calculating the reaction order, a ,
the concentration of A is evaluated either at the entrance conditions or mean value between C,, and C,,
.
Chap. 5
CD-ROM Maler~al
CD-ROM MATERIAL Learning Resources f ummsrv No:cr
I. s l l l ? l ~ ~ lWoOt ~~ L , . ~ 3. Interactive Comprrrer Modrrles A . Ecology
interactive
f$
-\
Computer Modulcs
Solved Prcblems
Q F L~ 5%
L;v~ngExamu!e Problerv
B. Reactor L a b ( u ~ ~ v ~ ~ : r r a ~rwt) ~ u See r i n hReactor Lab Chapter 4 and P5-3,. 4. Solvrd Ptvblem.~ A. Example Differential Method of Analysis of Pressure-Time Data B. Example Integral Method of Analysis OF Preswre-Erne Data C. Example Oxygenating Blood Living Example Problems I. Ernmplr 5-3 U1.r (fReL~rrssintito Find !he Rore Lcrrc Prrrurnerers FAQ [Frequently Asked Questions]-Tn UpdateslFPIQ icon section Professional Reference Shelf RS. 1 Lmsr Sqlrai-es Annlni.7 r,f the Lrr~euri:edRrrte Lntv The CD-ROM describes how the rate law
i s linearized In(-r,)
= In k + a In C,$+ p In CH
and put in the form Y=llo+aYt+pX1
I
and used to solve for a,p, and k . The etching of a semiconductor. MnO,. is used as an example to illwtrate this technique.
R5.2 A Discussiun of Weigl~tedLeast Sqltnres For the case when the error in measurement is not constant. we must use a weighted least squa~es. R5.3 Erperrrnent~ilP l i l n n i ~ ~ ~ A. Why perform the experiment? B. Are you choosing the correct parameters? C. What is the n n g e of your experimental variables'?
294
Collection and Analysis of Rate Data
Chap. 5
D. Can you repeat the measurement? (Precision) E. Milk your data for all it's worth. F. We don't believe an experiment untiI it's proven by theory. G. Tell someone about your result. R5.4 Evaluation or Lubornrov Reacfors (see Table 5-3)
QUESTIONS AND PROBLEMS The subscript to each of the problem numbers indicates the lwel of difficulty: A, least difficult; D. most difficult. A=.
uome'hto''~
P5-1,
B = l l C = f D=++
(a) Compare Table 5-3 on laboratory reaaors with a similar table on page 269 of Bisio and &el (see Supplementary Reading, listing 1 ). What are the similarities and differences? (b) Which of the ICM5s for Chapters 4 and 5 was the most fun? (c) Choose a FAQ from Chapters 4 and 5 and say why it was the most helpful. (d) Listen to the audios on the CD and pick one and say why it could be eliminated. (el Create an onginaI problem based on Chapter 5 material. (f) Design an experiment for the undergraduate laboratory that demonstrates the principles of chemlcal reaction engineering and wifl cost less than $500 in purchased parrs to build. {Fmm I998 AIChE National Student Chapter Competition) Rule? are provided on the CD-ROM. (g) Plant a number of seeds in different pots (corn works well). The plant and soil of each pot will be subjected to different condit~ons.Measure the height o f the plant as a function of time and fertilizer concentration. Other variables rn~ghtincrude lighting, pH, and room temperature. (Great Grade School or H ~ g hSchool Science Project)
a
Creative Thinking
(h) Example 5-1. Discurs the differences for finding
[-$-I
dC*
shown in
Table E5-3. I by the three techniques. Example - 1 .Construct a table and plot similar to Table E5-7.1 and Figure ES-2.1. aqsuming a zero-order and a first-order reactinn. Looking at the plotr. can e~therof there ordcrs possibly explain the data? Example 5-3. Explain why the reFr&sron had to be carried out twice to find k' and k. (k) Example 5-4. Use regreszion to analyze the data in Table E5-4.1. What do you iind for the renct~onorder? (I) Example 5-5. R e p e r 5 the data to lil the rate lab i
[n:erac?rve
-$
4%@ Comou'er Modules
P5-2,
Whel is the d~fferencern the correlation and sums-of-squares coinpared with those given In Example 5-5? Why was it necessary tn regre55 the data twice. once 10 ohtaln Table E5-5.3 and once to obtain Table E5-5.41 h d h e Interactive Computer Mtdule IICM) Fmrn the CDROM. Run the module and then mord >our pcrfommce number f i r the d u l e which ~ndlcatesyour mastenng of h e malerial. Your pmfesm h x the key 10 decode your p d o r m m numkr. ~
ICM E c o l o ~ yPerformance #
Chap. 5
Visit Reactor Lab
PS-3*
d Links
P5-4,
295
Questions and ProbCems
GO to Professor Herz's Reactor Lab on the CD-ROM or on [he web at wuwreactorlab.nc~.Do (a) one quiz, or (bl two quizzes from D~vision 1. When you first enter a lab. you see all input values and can vary them. In a lab, click on the Quiz button in the navrgation bar to enter the quiz for that lab. In a quiz, you cannot see some of the input values: you need to find those with *'???" hiding the values. In the quiz, perform experiments and analyze your data in order to determine the unknown values. See the bottom of the Example Quiz page at nw~lc:rear!orlnb.nerfor equations that relate E and k. Click on the "???" next to an input and supply your value. Your answer will be accepted if is within +?O% of the correct \glue. Scoring is done with imaginary dollars to emphasize that you should design your experimental study rather than do random experiments. Each time: you enter a quiz. new unknown values are assigned. To reenter an unfinished quiz at the same stage you left, click the [i] info button in the Directory for instruct!ons. Turn in copies of your data. your analysis work. and the Budget Repon. When arterial blood enters a tissue capillary. lr exchanger oxygen and carbon dioxide with its environment. as shown in this diagram.
The kinetics of this dtoxygenation of hemoglobin in blood was studied with the aid of a: tubular reactor by Xakamura and Siaub [J. Phl~iol.,173. 1611.
Although this i s a reversible reaction, measurements west made In the inilia1 phases of the decamposit~onso that the reverse reaction could be neglected. Consider a system similar to the one used by Naknmura and S~aub:the solution enters a tubular reactor (0.158 r m In diameter) that has oxygen electrodes placed at 5-cm interralr dou)n the tube. The solution floq rate into he reactor is 19.6 cdk. Electmde Povtion Percent Decompotition ot HhO,
P5-5,
I
2
3
4
5
h
7
O.OO
1 .Y3
3.82
5.68
7.38
9.25
t 1.00
(a) Using the method of differential analysis af rate data. determine the reaction order and the forward specific rcactjon rate constant d for [he deaxypenation of hemoglobin. (b) Repeat using regression. The liquid-phase irreverqible reaction
296
Collection and Analysis of Rate Data
Char
is carried out in a CSTR. To learn the nte lam the volunietric flow rate, I [hence t = V l u , , ) 1s varied and the elfluent concentratloris of qpecies recorded as a function of the space time t . Pure A enters the reactor at a cc centratton of 2 rnolldrn'. Steady-state conditlonh exist when the rnensu
ments are recorded.
PS-6,,
(a) Determine the reaction order and \pecific reaction n t e . (b) If you were to repeat this experiment to determine the kinetics, wt would you do differently? Would you run at a higher, lower, or the sar temperature? If you were to rake more data. where would you place t measurements (r.g., r )'? (c) 11 is believed that the technician may have mnde a dilution factor-oferror in one of the concentmtiun measurements. What do you think? Hc do your answers compare using regression (Polymath or other softwnr with those obtained by graphical methods? Nore: All measurements were taken at steady-stnte conditions. The redchon
was carried out In a constant-volume batch renetor where the following co centration measurements were recorded as a function of time.
P5-7,
(a) Use nonlinear lrnst squares (i.e.. regression) and one other method determine the reaction order a and the specific reaction rate. (b) If you were to take more data. where would you place the points? Wh: (c) If you were to repeat this experiment to determine the kinetics, wh would you do differentty? Would you run at a higher, tower. or the san temperature? Take different data points? Explain. (d) Lt is believed that the technician mnde a dilution emr in the concentr tion measured at 60 mio. What do you think? How do your answe compare using regression tPoIyrnath or other software) with tho: obtained by graphical methods? The Iiquid-phase reactlon of methanol and triphenyl takes place in n batc reactor at 25°C
For an equal molar feed the following concentration-time data was obtaine for methanol:
Chap. 5
297
Questions and Problems
The following concentration time data was carried out for an initial methanol conuentmtion 0.01 and an initial tnpheoyl of 0.1 :
(a1 Detcrniine the rate law and rate law parameters. (b) if you were to take more data points, what would be the reawnable settlngfi (e.g.. C,<,,. C,$? Why?
P5-Sh
The following data were reported [C. N. Hinshelwocd and P. J. Ackey, Pmc. R. SOL-. (Lond)., ,4115. 1151 for a gas-phase constant-volume decomposit~on of dimethyl ether at 504'C in a bort.h rcnctor. Initially. only (CH,)?O wa\ present.
(a) Why do you think the total pressure measurement at Can you estimate it? (b) Assuming that the reaction
P5-9,
I =
O i s missing?
is irreversible and goes to complet~on,determine the reaction order and specific reaction rate R. (c) What experimental conditions would you suggest if you were to obtain more data? (d) How would the data and your answers change if the reaction were run at a higher or lower temperature'? In order to study the photochemical decay of aqueous bromine in bright sunlight, a small quantity of liquid bromine was dissolved in water conwined in a glass battery jar and placed in direct sunlight. The following data were obtained at 2SeC:
Determine whether the reaction rate is zero. first. or second order in bromine, and calculate the reactron rate constant in units of your choice. (b) Assuming identical exposure conditions. calculate the required hourIy rate of injection of bromine (in pounds) into a sunlit body of water, 25.000 gal in volume, in order to maintain a sterilizing level of bromine of 1.0 ppm. (Ans.: 0.43 Iblh) (c) What experimental conditions would you suggest ifyou were to obtain (a)
more data'?
298
Collection and Analysis of Rale Data
Chap. 5
(Note: ppm = parts of bromine per million parts of brominated water by weight. In dilute aqueous solutions, 1 ppm = 1 milligram per liter.) (From California Professional Engineers Exam.) P5-10c The gas-phase decomposition
is carried out in a consrat~r-volumebarch reacrur. Runs 1 through 5 were carried out at 10O0C, while run 6 was carried out at 1 10°C. la) From the data in Table P5-10, determine the reaction order and specific reaction rate. (b) What is the activation energy for this reaction?
P5-11,
The reactions of ozone were studied in the presence of alkenes [R. Atkinson et al., Inf. J. Cl~em.Kiner., 15(8),721 (1883j1, The data in Table P5-I E are for one of the alkenes studied. cis-2-butene. The reaction was carr~edout isothermally at 297 K. Determine the rate law and the values of the rate law paramelers.
O:UIIP O:or~eRate
Run
Imol/s.drn3x 10')
Gncerirrat~sn (mol/dm7)
Blrrcrrr Cn~tce~~trario~~ [mol:dm3)
(Hi~rr:Ozone also decsmposes by collision with the wall.)
P5-12, Tests were run on a small experimental reactor used for decomposing nitrogen oxides in an aurotnobile exhaust stream. In one series of rests. a nitrogen stream containing various concentratrons of NO: was fed to a reactor, and the kinet~cdata ohtained are shown in Figure P5-I?. Each point represents nne
Chap. 5
299
Questions and Problems
complete run. The reactor operates essentially as an isothermal backrnix ROCtor (CSTR). What can you deduce about the apparent order of the reaction over the temperature range studied? The plot gives the fractional decomposition of NO?fed versus the ratio of reactor volume Y (in cm3) to the NO, feed rate. Fx,, ,(! moVh), at different feed concentrations of NO, (in parts per million by welghtl.
0 700 ppm NO2 in feed
v
3050 ppm N02 in fd
Figure P5-12 Auto exhaust data,
P5-13B Microelccrmnic delicer are formed by first forming SiOl on a silicon wafer hy chemical vapor deposition (Figure PS-131. T h i ~procedure is followed by coating the SiO, with a polymer called a photoresist. The pattern of the electronic circuit Is then placed on the polymer and rhe sample is irradiated with ultraviolet light. I f the polymer is a positive photoresist, the sections that were irradiated will dissolve in the appropriate solvent, and those sections not irradiated will protect the SiO, from further treatment. The wafer is then exposed to strong acids. such as H E which etch (1.e.. dissolve) the exposed SIO,. It is extremely imponant to know the kinetics of the reaction so that the proper depth of the channel can he achieved. The dissolution reaction is
From the foliowing initial rare data. determine the rare law.
A total of 1000 thin wafer chips are ro !x placed in 0.5 dm3 of 209 HF. Ef a spiral channel 10 p m wide and 10 m in length were to be etched lo a depth of 50 p rn on both sider of each wafer, how long shou'ld rhe chips be left in the solution? Aswme that the wlution is well mixed. ( A I ? ~ :330 rnin)
Collection and Analysis of Rate Data
VWX~OSW? P W ~ j s l
Chap
After HF etching
figure P5-13 Semiconductor etching.
PS-14,
The following reaction
is cnrried out in a differenrial reactor, and the flow rate of ethylene is recort as a function of temperature and entering concenttations.
The space time for the differential reactor is 1 minutes. (a) Determine the rate law and rate law parameters. (b) If you were to take more data points, what would be the reason: settings (e.g., CAo)? Why? P5-1SB The following data as obtained in a batch reactor for the yeast Sitcchammnj cerevisicne Yeast Budding
Chap. 5
Questions and Problems
(a) Determine the rate law parameters &, and PC,, assuming the data can be
described by Monod Equation
5 = rs = dt
&+C,
[Hint: It might be best to regress your data taking the reciprocal of the Manod equation in the form (CsC&,) vs, C,.] What is the residual sums of squares? (b) Determine rhe rate parameters p,, and k, assuming the data can be fit by the Tessier Equatton
What is the residual sums of squares? (c) Determine tRe rate law parameters &,,.,. k, and h. assuming the data can be fit by the Mosw Equation
What is the residual sums of squares? P5-Z6c The thermal decomposition of isopropyl isocyanate was studied in a drfferenlial packed-bed rmctor. Fmm the data in Table P5-16, determine the reaction rate law parameters.
Rate
Run
(moVs - dmJ3
Concentmfion (rnoVdm' )
Tempe~ture
IK)
302
Cotlecfion and Analysis 01 Rate Data
Chap. 5
JOURNAL CRITIQUE PROBLEMS PSC-I
A packed-bed reacror was used to study the reduction of nitrlc oxide with erhylene on a copper-silica catalyst [fnd. Eng Chenr. Pmcess DPS.Dn!, 9, 455 (1970)l. Develop the integral design equation in terms of the conversion at various initial pressures and temperatures. Is there a significant discrepancy between the experimental results shown in Figures 2 and 3 In [he article and the calculated results hased on the pmposed rate law? If so, what is the possible source of this deviation? l e Chem, Technol. BEofechnol.. 31. 273 (1981)j is P5C-2 Equation ( 3 ) in' the a ~ ~ i c [J. the rate of reaction and i s incorporated into derign equatlon (2). Rederive the design equation in terms of conversion. Determ~nethe rate dependence on H, hased on zhis new equation. How dms the order obtained compare with that found by the authors? P5C-3 See "Kinetics of catalytic esferification of terephthalic acid with methanol vapoui' IChern. E ~ RSci.. . 28. 337 11973)). When one observes the data points in Figure 7 of this paper for large tinics. it is noled that the l a ~ tdata point always falls significantly off the straight-hne interprelation. Is it posqible to reanalyze these data to determine if the chosen reaction order is indeed correct? Substituting your new rate law inro equation 13). derive a new form of equation (10) in the paper relating time and particle radius. P5C-4 The selective oxidation of toluene and methanol over vanadiu~npentoxide-supported alkali metal sulfate catalyas was studied [AICIiE A, 27( 11, 41 ( 198I!], Examine the experimenlal technique wed (equipment, variables. etc.) in light of the mechankm proposed. Comment on the shortcomings of the analysis and compare with another study of t h ~ ssystem presented In AICJrE 1.. ZX(5 ), 855
(19823.
Additional Homework Problems
CDP5-A, CDP5-B, CDPS-C,
CDP5-D, CDPS-E, CDPS-FR
The reaction of penicillin G with NH,OH is carrled out in a batch reactor. A colorirneter was used to measure the absorbency as a functlon of time. [lst Ed. P5-101 The isomeritation A + B is carried out in a batch reactor. Find a and k . 13rd Ed. P5-38] The ethane hydropenolysis over a commercial nickel catalyst was studied in a stirred con~ainedsolids reaclor. (An.\: X = 0.48 mot . aFm ( k g . h] [3rd ed. PS-Ill The half-life of one of the pollutants. NO. in autoexhaurt i~requ~red. [lst Ed. P5-I 11 The kinetics of a fils-phace reaction A, -+ 2A were studied in a constant-pressure hatch reactor in wh~chthe \olurnc was measured as a function o f time. [ I st Ed. PS-61 Reac~ionkinetlcs in a tubular reactor.
GalnAs in fiber oprics [3rd Ed. P5-h]
Chap. 5
Supplementary Reading
CDPS-G,
Differential reactor data for the reaction CHJH = CH2 + O2 4 CH7= CHCHO + H,O
CDPS-HB CDPS-1, CDPS-,IB
[3rd Ed. P5-131 Lumping of species for hydrocarbon mixtures. [3rd Ed. P5-161 Prepare an experimental plan to find the rale law. [3rd Ed, P5-171 Batch data on the liquid phase reaction
A+B-tC
CDP5-KB
[3rd Ed. PS-181 Analyze data LO see if it fits an elementarj reaction ?A+B-+ZC
[3rd Ed. P5-2 1R ] G~~ ~
~
New~Problemsi on the Web ~
~
~
i
~
~
CDPS-New From drne to time new problems relating Chapter 5 material to everyday interests or emerging technologies will be placed on the web. Solutions to these problems can be obtained by emaifing the author. Also, one can go 10 the web site, ~~u'~:rnwnn.edu/Rre~~~engincesinp, and work the home problem specific to this chapter.
SUPPLEMENTARY READING L~nk
I . A wide va.ety of kxhiqucs for rneawring the c ~ n r e n t r a i n rof rhe reacting apecies may be found in Baro. A,. and R. L. &PEL. Scaleup qf Chemical Puocessr.~. New York: Wiley-Inrerscience. 1985. RORINSON.J. W., Ljndeqradr~nre 1nsfrut)retrtal Attal~sis,5th ed. New York: Marcel Dekker, 1995. SKCXX;, DOUGLAS A., F. JAMES HOLLER,and TIMOTHY A. NIEMAN, Principles of Insrrumental Analysis, 5th ed. Philadelphia: Saurlders College Publishers. Harcourt Brace College Publishers, 1998. 2. A discussion on the methds of interpraation of hatch reaction data can be found in CRYYES, B. L.. and H. S. FOOLER.eds., AlChE Modular lrisrrucriorr Series E: Kinerirs. Vol. 2. New York: .4rnerican Inptlture of Chemical Engineers. 1981. pp. 51-74. 3. The interpretation of data obtained from flow reactors is also discussed in
CHURCHILL. S . !A.' Thc Itlterprerorior~ and Use McGraw-Hill. 1974.
nf
Rare Dora. New York:
4. The design of laboratnry catalytic reactor5 For obtaining sate d a ~ ai~ presented In
RASE,H . F., Cha?nicrrl Reartor. D P F I R .for ~ ~Prncers Plnnrs. VoI. 1 . Keu' 'iork: Wiley. 1983. Chap. 5.
304
Collection and Analysis of Rate Data
Chi
5. Model building and current statistical methods applied to interpretation of rate are presented
In
FROMEKT, G. F., and K. 8 . BISCHOFF.Chenlical Rr(ictor A n r r l ~ r iand ~ Des 2nd ed. New York: Wile?, 1990. MARKERT, R. A,, Instruments! Elemmt and M~ilfi-ElernentAnalysis of P Samples: Mefkods and Appiicatrot~s.N e w York: Wiley. 1996.
6. The sequential design of experiments and parameter estimation is covered in
BOX,G.E.P.,W. G.HUNTER,and J. S. HUKW-R. S ~ ~ f i ~for l i r E.rperimmt .~ An Introducrion to DesiLqn, Dnr~iAnn/ysis, and Model R~liEding.New k; Wiley. 1978.
Multiple Reactions
The breakfast of champions is not cereal, it's your opposition. Nick Seitz
Overview. Seldom is the reaction of interest the only one that occurs in a chemical reactor. TypicalIy, multiple reactions will occur, some desired and some undesired. One of the key factors in the economic success of a chemical plant is the minimization of undesired side reactions that occur dong with he desired reaction. In this chapter, we discuss reactor selection and general mole balances for multiple reactions. First, we descrik the four basic types of multiple reactions: series, parallel, independent, and complex. Next, we define the selectivity parameter and discuss how it can be used to minimize unwanted side reactions by proper choice of operating conditions and reactor selection. We then develop the algorithm that can be used to solve reaction enpneering probIems when multiple reactions are involved. Finally, a number of examples are given that show how the algorithm is applied to a number of .seal reactions.
6.1 Definitions 6.1.1 Types of Reactions
There are four basic types of multiple reactions: series, parallel, complex, and independent. These types of multiple reactions can occur by themselves, in pairs, or all together. When there is a combination of parallel and series reactions, they are often referred to as complex reactions.
306
Multiple Reactions
Chap. 6
ParalEel reactions (also called competing reactions) are reactions where the reactant is consumed by two different reaction pathways to form different products:
yB A
Pxallci reactions
-+A,,
C An example of an indusnially si,&ficant pamllel mction is the oxidation of ethylene to ethylene oxide while a~nidingcomplete cornbudon to a h n dioxide and water.
/2C02
+ 2H20
Serious chemistr)
Series reactions (also called consecwive reac?inrzs) are reactions where the reactant forms an intemiediate product, which reacts funher to form
another product:
A
Series reactions
1.
A
B
L
c
An example of a series reaction is the reaction OF ethylene oxide (EO) with ammonia to form mono-, di-, and triethanolamine:
In recent years the shift has been toward !he production of diethanolamine as the desired product rather than rriethanolamine. Complex reactions are multiple d o n s that invoIve a combination of both s e r k and parallel reactions, such as A + B +C + D A+C
+E
An example of a combination of parallel and series reactions is rhe formation
of butadiene from ethanol: Complex reactions: coupled simultaneous ceries and parallcI reactinns
Independent reactions are reactions that occur at the same time but neither [he products nos reactants reacl with themselves or one another. Independent reactions
Sec. 6.7
307
Definitions
An example is the cracking of crude oil to form gasoline where two of the many reactions occurring are
Desired and Undesired Reactions, Of particular interest are reactants that are consumed in the formation of a desired pmducr. D. and the formation of an undesired pmduct, U. in a competing or side reaction. In the parallel reaction sequence A
L
D
A-%u
nr in he series sequence A & D & u
We want to minimize the formation of U and maximize the formation of D because the greater the amount of undesired product formed. the greater the cost of separating the undesired product U from the desired product D (Figure 6-11.
system VI
Figure 6 1 Reaction-separa~ionsystem producing both desired and undesired products
The economic lncenllve
In a highly eficient and costly reactor scheme in which very little of undesired product U is frrrmed in the reactor, the cost of the separation process could be quite low. On the usher hand, even if a reactor scheme is inexpensive and inefficient resulting in the formation of substantiaI amounts of U,the cost of the separation system could be quite high. Normally, as the cast of a reactor system increases in an attempt to minimize U. the cost of separating species U from D decreases (Figure 6-21. Selectivity tells us how one product is favored over another when we have multiple reactions. We can quantify the formation o f D with respect to U by defining the selectivity and yield of the system. The instantaneous selectivity of D with respect to U i s the ratio of the rate of formation of D to the rate of formation of U.
=
-
rate of formation of
u
Multiple Reactions
Chap.
- -
High
Low Figure 6-2
Efficiency of a reactor system.
In the next section, we will see how evaluating SDRiwill guide us i n the desig and selection of our reaction system to maximize the selectivity. Another definition of selectivity used in the current literature. Sow, given in terms of the flow rates leaving the reactor. Sow is the overall selectivit:
-
Overall selectivity
-
FD= Exit molar flow rate of desired product SDIU= F, Exit molar flow rate of undesired product
(6-2
For a batch reactor, the overall selectivity is given in terms of the numbc of moles of D and U at the end of the reaction time:
Example 6 1 Comparing the Qvemll and Instantaneous Selective$, S,, for a CSTR
Develop a relationship between ,S
and
30,~rcad
for a CSTR.
Sululinn
Consider the instantaneous selectivity for the two parallel reactions just discussed:
The overall selectivity is
A mvlc balance on D for a CSTR yields
and a mole balance on LI yields
Substituting for Fa and Fr i n the overall selectivity Equation 1E6- 1-4) and canceling rhe volume, V
Consequently for a CSTR the overall and instantaneous selectivities are equal; that is,
QED One can carry out a similar analysis of the series reaction
A+D+U TWOdefinitions for selecti>ity and yteId are fot~ndIn the
literature, Inrtnnmneous
Reaction yield, like the selectivity. has two def nirions: one hilsed on the ratio of reaction rates and one based on the ratio of molar ffow rates. In the first case, the yield at a point can be defined as the ratio o f the reacrion rate of a given product to the reaction sate of the key reactant A. This is sometimes referred to as the instantaneous yield.] -
yreid bilsed on reactinn r~tes
16-31
In the case of reaction yield based on molar flow rates. the overall yield, at the end of the reaction to the number of moles of the key reactant. A, that have been consumed.
YD. is defined as the ratio of moles of product formed
For a batch system: Overall yield based on moles
For a flow system: Overall yield based on molar flow rates
I
f . 3. Carberry. in Applied Kinerics and Chernical Reacrion Engineering. R. L. Goning and V, W. Weekman, eds. (Washington, D.C.:American Chemical Society, 19671, p. 89.
31 0
Multiple Reactions
Chap. 6
As with selectivity, the instantaneous yield and the overalI yield are identical
PD
for a CSTR (i.e., = Y, ). Yield and yield coefficients are used extensively in biochemical and biomass reactors. [See P3-14 and Chapter 7.1 Yet other definitions of yield even include the stoichiometric coefficients, As a consequence of the different definitions for selectivity and yield, when reading literature dealing with multiple reactions. check carefully to ascertain the definition intended by the author. From an economic standpoint, the overall selecrivities,
k.
S , and yieIds. are important in determining profits. However, the rate-based selectivities give insights in choosing reactors and reaction schemes that wiIl help maximize the profit, There often is a conflict between selectivity and conversion (yield) because you want to make a lot of your desired product (D) and at the same time minimize the undesired product 1U). However, in many instances, the greater conversion you achieve, not only do you make more D. but you also form more U. 6.2 Parallel Reactions In this section, we discuss various means of minimizing the undesired product, U, through the selection of reactor type and conditions. We also discuss the development of efficient reactor schemes. For the competing reactions A
t"D
(desired)
A
&U
(undesired)
the rate laws are Rate laws for
format~onof desired and undesired products
rD= k , ~ :
(6-6)
ru = kEc?
(6-7)
The rate of disappearance of A for this reaction sequence is the sum of the rates of formation of U and D: -rA = rn+rLT (6-8)
InstanraneouF
selectivity
where a, and a, are positive reacrion orders, We want the rate of formation of D, r , , to be high with respect to the rate of formation of U. r , . Taking the ratio of rhese rates [i.e., Equation (6-6) to Equation (6-7)J. we obtain a rate selectivih parameter, SDm. which is to be maximized:
Sec. 6.2
31 7
Parallel Reactions
6-2.7 Maximizing the Desired Product for One Reactant Maximize the rare selectivity
parameter. a,is the order of the desiredreacr!on, a,, of the undesrred
,
In this section. we examine ways to maximize the instantaneous selectivity, SDm,for different reaction orders of the desired and undesired products. Case 1: a , >a, For the case where the reaction order of the desired product is greater than the reaction order of the undesired pmdvct. let a be a positive number that is the difference between these reaction orders (a > 0):
a,-a, = a Then, upon substitution into Equation (6-101, we obtain
For al>a,.make To make this ratio as large as possible. we want to cany out the reaction C, ac large as in a manner that will keep the concentration of reactant A as high as possithe posT1hle. during the reaction. If the reaction is carried out in the gas phase. we should run it without inerts and at high pressures to keep C, high. If the reaction is in
the liquid phase, the use of diluents should be kept to a minimum.' A batch or plug-flow reactor should be used in this case because, in these two reactors, the concentration nf A stans ar a high value and drops progressively during the course of the reaction. In a perfect/? mixed CSTR. the concentranon of reactant within the reactor is always at its lowest value live.,that of the outlet concentration)and therefore the CSTR should no! be chosen under thew circumstancw.
a2>a, When the reaction order of the undesired product Is greater than that of the desired product,
Case 2:
A & u
Let h = a,-a, , where h is a positive number: then
For the ratio I-dr,, to be high, the concentration of A should be as low ar possible. This low concentration inay be accomplished by diluting the feed with For rx,>al uqe inerts and running the reactor at low concentration^ of A. A CSTR should be a CSTR and dilute used because the concentrations of reactants are maintained at a low level. A the feed stream. recycle reactor in which the product stream acts as a diluent could be used to maintain the entering concentralions of A at a low value. Because the activation energies of the two reactions in cases 1 and 2 are nor given. i t cannot he determined whether the reaction should be run at high For a nurnkr of liquid-phase reac~iclnr,the prcqxr choice of a soI~enIcan enhance selectiv~ty.See. for example. Ir~il.Et~g.Cl~rn~.. rST(9). I h E 1970). In pa<-phase heterogeneous calalyl~creactions. xelect~vityis an ~mportantparametcr nf any parliculnr catalyst.
312
Mul!ipls Reactions
Chap
or low temperatures. The sensitivity of the rate selectivity parameter to temp1 ature can be determined from the ratio of the specific reaction rates. Effect of temperature on
selectivity
I
ED'%
T
I
Eu=-Eo
where A is the frequency factor, E the activation energy, and the subscripts and U refer to desired and undesired product, respectively.
Case 3: E,>E, I n this case, the specific reaction rate of the desired re: tion k, (and therefore the overall rate r , ) increases more rapidly with increi ing temperature than does the specific rate of the undesired reaction h Consequently, the reaction system should be operated at the highest possil
temperature to maximize S,,v.
Case 4: Eu>ED In this case, the reaction should be carried out at a low te perature to maximize Sn,,, , but not so low that the desired reaction does r - proceed to any significant extent.
GI
Example 6-2 Maimiring the S e k c t i v i ~for the Tradauze Reactions
Reactant A decomposes by three simultaneous reactions to f o ~ mthree produt one that is desired, B , and two that are undesired, X and Y. These gas-ph: reactions. along with the appropriate sate laws, are called the Trambouze re, tions [AICIiE J. 5, 384 (1959)l.
3)
k,
A --r Y
3
-r,,= r,= k,C',= 0.008-dll_e mol . s
The specific reaction rates are given at 3W K and the activation energies for renctil (1). (2), and (3) are E, = 10,W kcallmole, Ez= 15,000 kcallmole, and Ej= 20,( kcal.mole. How and under what conditions (e.g., reactor typfs), temperatum, conc trations) should the reaction be carried out to maximize the selectivity of B for entering concentration of A of 0.41W and a volumetic flow rate of 2.0 dm%. Solution The selectivity with respect to 8 is
Sec. 6.2
I
313
Parallel React~ons
Plotting Ss,xu vs. CA*we see there i s a maximum as shown in Figure E6-2.1.
CAlmMd
Figure E6-2.1 Selectivity as a function of the concentration of A.
'
c4.
can see, the selectivity reaches n maximum at a concentration Because rhe concentration changes down the length of a PFR,we cannot operate at this maximum. Consequently. we wiIl use a CSTR and design it to operate at this maximum. To find the maximum, we differentiate SBIXYwrt C,. set the derivative to zero, and solve for C*,.That is, As we
c,
Solving for C ,'
Operate at this
CSTR reacrant concentntion:
We see h m Figure E6-2. t the selectiviy is indeed a marimurn at $L
= 0.112 mol/dm3
C: = 0. l 12 moUdmT.
We therefore want to cany out our reaction in such a manner that C; is always at this value. Consequently, we will use a CSTR operated at d,. The corresponding selectivity at CL is
31 4
Multiple Reactions
Chap. 6
We now calculate this CSTR volume and conversion. The net rate of formation of A from reacr~ons( I ) , (Z),and (3) is
Using Equation (E6-2.5)in the mole balance on a CSTR fm this liquid-phase reaction (v = uo),
CSTR volume to maim!= selectivity SBIXY= S
~
Y
Maximize the selectivity wrt Ieinperature
A1 what
iemperature should we oprate the CSTR?
E +E, Case 1: If ' < E : 2
+L; Case 2: IfE L'E2 2
Run at as high a temperature as possible wlth existing equipment and watch out for other s~dereactions that might occur at higher tempemtures.
{
Run at low temperatures but not so low that a significant con\.ersion ~s not achieved.
For the activation energies given above
So the ~electivit~ for
this combination of activation enegies is independent of tempemare! What is the conters~onof A in the CSTR?
Sec. 6.2
Parallel Reactions
315
If greater than 72% conversion of A is required, then €he CSTR operated with a reactor concenmtion of 0.312 moVdm3 should be followed by a PFR because the concentration and hence sel~tivjtywill decrease continuously from
as we move
down the PFR to an exit concerteation C,,. Hence the system How can we increase the conversion and still have a high selectivity
would give the highest selectivity while forming more of the desired pruduct B,
Ssntv?
beyond what was formed at
1
chin a CSTR.
Optimum CSTR followd by a P m . The exit concentrations of X, Y, and B balances Species X
can now be found from the CSTR mole
Rearranging yields
I I
Fx = uoCx= 0. !56 molls Species B
Rearranging yields
FB = 0.264 molls
316
hrultiple Reactions
Char
Let's check to make wre the sutn of all the species in solutian equals the initial cd centration Chll= 0.4.
The reason we want to use a PFR after we reach the maximum selectiv SBXPis that the PFR will continue to gradually reduce C,. Thus. more B will formed than ~f another CSTR were to follow. If 90% conversion were required tl the exit concentration would be CAI = (1 - 0.9)(0.4 molldmf) = 0.04 moVdm3. The PFR mole balances For this liquid-phase reactlon ( u = u,) are
Dividing uo into V to form T and then combining the previous mole balances H their respective rate laws, we obtain
dr
=- -
~ C -X -rlz
,
'
-
(at
r = 0, then CA= 0.1 12 rnolidrn')
(at z = 0, then C,
= 0,0783 molldrn')
S=&C, dr
(at 7 = 0.then C, = 0.132 mo~/drn')
dc'.sCi d~
(at T = 0,then C, = 0.0786 mol/dm3)
The conversion can be calcuiated from Equation (E6.2-16)
We note that at r = 0. the values of the entering concentrations to the PFR are exit concentrations From the CSTR. We will use Polymath to plot the exit conr trations as a function of r and then determine the volume (V = vnz) for 90% con
An economic
decision
sion (C, = 0.04 moVdm3) and then find C,. CB,and Cy a1 this volume. This v o l ~ turns out to be approximately 600 dm3.The Polymath program along with the concentrations and selectivity are shown in Table E6-2.1. Ar the exit of the PFR, C , = 0.037,Cx = 0.1 1, CB = 0.16. and Cy = 0.05 in rnoLldrn3. The corresponding molar flow rates are Fx = 0.22 molls, FB = ( molls, nnd F y = 0.18 molls. One now has to make a decision as to whether ad( the PFR to increase the conversion of A from 0.72 to 0.9 and the molar flow rat 8 from 0.26 to 0.32 moVs is worth not only the added cost of the PFR, but atso decrease in selectivity. Tnis reaction was cam'ed out isothermafly; nonisothe~ multiple reactions will be discussed in Chapter 8.
Parallel Reactions
Sec. 6.2
- TAR[-F Eh-2. I.
POI.YYAT11 PROGRAM FOR PFR FOLLC)\CING CSTR
V)I,Y MATH Repulb
E u m p k C l Marlml~lnu the Sclralvlty tor th. Tmmbour. -R
Wrr nxr. h s t !I:
I
I
o m aa a l a ~m *so aro
~.a am
m
LBUtSI
(3)
Figure E6-2.2 (a) PFR concentration ~ r o f i t e s :(b) PFR Selectivity protile.
6.2.2 Reactor Selection and Operating Conditions Next consider two simultaneous reactions in which two reactants, A and B, are being consumed to pmduce a desired product. D,and an unwanted product. U, resulting from a side reaction. The rate laws for the reactions
are
378
Multiple Reactions
Chap. 6
The rate selecrivity parameter Instantaneous selectivity
is to be maximized. Shown in Figure 6-3 are various reactor schemes and conditions that might be used to maximize SDm.
Reactor Selection Criteria: Selectivity Yield Temperature conlrol * Safety Cost
Figure 6.3 D~fFerenlreactor< and rcheme\ frlr minirni~inp he zinwdnled produd.
Sec. 6.2
Parallel Reactions
31 9
The two reactors with recycle shown in (i) and Pj) can be used for highly exothermic reactions. Here the recycle stream is cooled and returned to the reactor to dilute and cool the inlet stream thereby avoiding hot spots and tunaway reactions. The PFR with recycle is used for gas-phase reactions, and the CSTR is used for liquid-phase reactions. The last two reactors, (k} and (I), are used for thermodynamically limited reactions where the equilibrium lies far to the left (reactant side)
and one of the products must be removed (e.g., C ) for the reaction to continue to carnpIetion. The membrane reactor (k) is used for thermodynamically limited gas-phase reactions. while reactive distillation (1) is used for liquid-phase reactions when one of the products has a higher volatility (e.g.. C) than the other species in the reactor. Example &3 Choice of Reactor and Conditions to Minimize Unwanted Products For the parallel reactions
consider all possible combinations of reaction orders and select the reaction scheme that will maxim~zeS,, .
,
CaseI: a l > a 2 .p l > p z . Lct a=a,-az and b=p,-p2.whefitnandbarepositive constants. Using these definitions. we can write Equation (616) in h e form
To maximize the rario r d r u . maintain the concentrations of both A and B as high as possibIe. To do this. use A tubular reactor [Figure 6.?(b)] A batch reactor [Figure 6.3(c1] * High pressures (if gas phase). and reduce inerts Case Ik cc,ru:. p,
320
Mult~pleReactions
Chap
To make S,, as large as possible, we want to make the concentration of A hi and the concentration of B low. To achieve this result. use
A semibatch reactor in which B is fed slowly into a large amrlunt of A [F ure 6.31d)l A membrane reactor or a tubular reactor with side streams of B continua fed to the reactor [Figure 6.3(fl] A series of small CSTRs with A fed only to the first reactor and sm amounts of 8 fed to each reactor. In this way B i s mostly consumed befr the CSTR exit stream Rows into the next reactor [Figure 6.3(h)]
Case Xfl: a , < a z ,p,
To make , ,S as large as possible, the reaction should be canied out at low cc centrations of A and of B. Use A CSTR [Figure 6.3(a)E
A tubular reactor in which there is a large recycle ratio [Figure 6.3(i)J A feed diluted with inerts
Low pressure (if gas phase)
CaseIV: a,P,. Let a = % - a , and b = P , - p z , wherea and bare pc hive. constants. U s ~ n gthese definitions we can write Equation (6-16) in the for
.
To maximize S,,. run the reaction at high concentrations of B and low conct rrations of A. Use
A semibatch reactor with A slowly fed to a large amount of B [Figure 6-3(e A membrane reactor or a tuhlar reactor with side streams of A [Figure 6-3( A series of small CSTRs with fresh A fed to each reactor [Figure 6-3{h)]
6.3 Maximizing the Desired Product in Series Reactions In Section 6.1, we saw that the undesired product cmld be minimized
adjusting the reaction conditions (e.g., concentration) and by choosing t proper reactor. For series of consecutive reactions, the most important variat is time: space-time: for a flow reactor and real-time for a batch reactor. To ill1 trate the importance of the time factor, we consider the sequence
A t
in which species
k
~
B
B is the desired product.
~
!
c
S s . 6.3
Maximizing the Desired Produc! in Series Reactions
32 1
If the first reaction is slow and the second rertcrion is Fast, ir w ~ I lbe extremely dimcult to produce species B. IF the first reaction (formation of B ) is fast and the reaction to form C is slow, a large yield of B can be achieved. However, if the reaction is allowed to proceed for a long time in a batch reactor, or if the tubular ff ow reactor is roo long, the desired product B will be converted to the undesired product C . I n no other type of reaction is exactness in the calcuiation of the time needed to carry out the reaction more imponant than in series reactions.
I
Example 6-4 Muximizing the Yield of the Intermediate Product
The oxidation of ethanol to form acetaldehyde is carried out on a catalyst of 4 wt % Cn-2 wt % Cr on AI,O, .>Unfortunately, acetaldehyde is also oxidized on this catalyst to form carbon dioxide. The reaction is camed out in a threefold excess of oxygen and in dilute concentrations Ica. 0 . E% ethanol. 1% Or, and 98.9% N2 ). Consequently, the volume change with the reaction can be neglected. Determine the concentration of acetaldehyde as a function of space-time,
The reactions are irreversible and first order in ethanol and acetnldehyde. respectively.
Solution Because 0, is in excess. we can write the preceding series reaction as A&B&c
The preceding series reaction can be written as
I
Reaction ( I ) A
&
I3
Reaction (2) B
R'
'C
1. iMole balance on A:
a. Rate law:
-rA = k,CA
1
I
6. Stoichiometry (Dilute concentrations: y,, = 0.001) .".u = v,,
FA= C,u,
R.W. McCabe and P. J. Mitchell, Ind. Enp. Chern. Pmce.ss Res. Dm.,22, 212 (2983).
322
1
Multiple Reactions
Chap. 5
c. Combining, we have V,
dCA ----klCA
(E6-4.2)
dW
Let r' = W/v0 = pbV/uO= p b ~where , p, is the bulk density of the catalyst. d. Integrating with C, = CAoat W = 0 gives us
2. Mole balance on &:
a. Rate law (net):
dntl = rim, +rind rLnc1 = kl c* - k2
b. Stoichiometry:
FB= uOCB c. Combining yields
Substituting for C ,, dividing uo into W and rearranging, we have t'
= Pb?
pb = bulk density
d. Using the integrating factor gives us +k,r'
=
ds' There is a iurorial on the inteprating factor in Appendix A,3 ar~d on the web.
I
cA0 elkrkllz'
At the entrance to the reactor, W = 0, r'= W h o = 0, and CB = 0.Integrating. we get
The concentrations of A B, and C are shown in Figure E6-4.1. 3. Optimum yield. The concentration of I3 goes through a maximum at a point along the reactor. Consequently, to find rhe optimum reactor length. we n e d to differentiate Equation tE6-4.7): d&=Ods'
k I CA U ( G-k,
1
+k2e4zr')
Sec. 6.3
Maximizing the Desired Product in Series Reactions
There is a space time at which B is a maximum.
7'1
7'3
714 7'
Flpre E6-4.1 Concentration profile<down a PBR in terms of space time r' = Wtv* INof@:7' = p,(Wlp,)luo = ph(V/uo)= p,zl
Solving for
gives
T&
-
The corresponding conversion of A at the maximum CB i s
Using Equation (E6-4.9) to substitute for,,'T
(E6-4.13 ) 4. Mde Balance on
C:
At the entrance to the reactor, no t'=O
I
Note as r +
-
C is present, so the boundary condition is C,=O
I
then C, = C,4,as expected
We note that the concentration of C, C,. couId have also been obtained from the averall mole balance
324
Multiple Reactions
Chz
We know that the preceding solutions are not valid for k , = k:. What wuuic the equivalent solution for T,~,, FVo,,, and X,, when k l = k:? See Pb-l(c). The yield has been defined as
-
YA = Moles of acetaldehyde in exit .Moles of ethanol fed
and is shown as a function of conversion in Figure E6-4.2.
.,k
k
n L e 2 c
0.1i
Plug flow reoclw
0.6
0
I
-
0.2
0.4
06
08
1.0
Fractional conversion of eihonol (dolo1
Figure E6-4.2 Yield of acetaldehyde as a function of ethanol conversion. Data wcre obtained at 518 K. Data points ( ~ order n of ~ncreasingethanol conversion) were obtained at space velocities of 26,000, 52,OW. 104.000. and 108.000 h-I. The curves were calculated for a first-order senes reaction in a plug-flow reactor and show yield of the intermediate species B as a function of the conversion of reactant for various ratios of n t e constants k, and k, . [Reprinted with permission from Ind. E q . Chem. P d Res. Do..2 2 . 2 12 (1 983). Copyright 8 1983 American Chemical Society.]
Another technique is often used to falIow the progress for two reactil in series. The concentrations of A. B. and C are plotted as a singular poin different space times (e.g., r; , 7; ) on a triangular diagram (see Figure 6The vertices correspond to pure A. B, and C. For (kl/k2) n 1, a large quantity of B can be obtained.
For (k,/k2) * I a very linle B can be obtained.
Figure 6-4 Reaction paths for different values of the specific ntes.
Sec. 6.3
325
Maximizin~:be Desired Prcduc: in Series Reacttons
1
Side note: Btood Clotting
Many metabolic reactions involve a large number of sequential reactions, such as those that occur in the coagulation of blood.
F Clot
Cut + Blood + Clotting
Blood coagulation is part of an important host defense mechanism called hernostasis. which causes the cessation of blood loss from a damaged vessel. The clotting process is ini~atedwhen a non-enzymatic lipoprotein (called the tissue factor) contacts blood plasma because of cell damage. The tissue factor (TF)norma1Iy remains out of contact with the plasma (see Figure B) because of an intact endothelium. The rupture (e.g., cut) of the endothebum exposes the plasma to TF and a cslscade of series reactions proceeds (Figure C). These series reactions ultimately result in the conversian of fibrinogen (soluble) to fibrin (insoluble), which produces the clot. Later, as wound healing occurs, mechanisms that =strict formation of fibrin clots, necessary to maintain the fluidity of the blood, start: working. Red Blold Cell
Red w CdI
FIgYre B. Schematic of separation of TF (A) and
FIgarr A. Normat clot coagulation of
piasma (B)before cut occurs.
blood. Picture
Ffgure C. Cut allows contact of plwisma to initiate coagulation. (A + -+ Cascade)
courtesi of: Mebs, Venomous and Poisonous Animab Medpharm, Stuttgart
*Platelets provide pmcoagulant phospholipids-equivalentsurfaces upon which the complex-dependent reactions of the blood coagulation cascade m localized.
2002. p. 305.)
An abbreviated form ( I ) of the initiation and foIIowing reactions that can capture the clotting process is
I
k,
F + v I J ~ ~ T T - 7 k-1 +WII,
-Fit
In ,order to maintain the fluidity of the bIood, the clotting sequence (2) must be moderated. The reactions that attenuate the clotting process are
326
Multiple Reactions
Chap. 6
wnere I r = nssue ractor, vua = ractor novoseven, x = smart m e r factor, EIa = thrombin, ATIIl = mtithrombin. and XIlIa = factor m a . One can model the clotting process in a manner identical to the series reactions by writing a mole balance and rate law for each species such as
Xa = Stuart h w e r factor activated, JI = prot
etc.
and then use Polymath to solve the coupled equations to predict the thrombin (shown in Figure D) and other s p i e s concentration as a function of time as well as to determjne the clotting time. Laboratory data are also shown below for a TF concentrazjon of 5 pfk4. One notes that when the complete set of equations are used, the Polymath output i s identical to Figure E, The complete set of equations along with the Polymath Living Example Problem code is given on the Solved Pmblems on the CD-ROM. You can Ioad the program directly and vary some of the parameters.
Lfvlng Example Probterr
Figure 33. Total thrombin as a fimctim-of time with an initiating TF conccnrration
of 25 ph4 (after running Polymath) for the abbreviated blwd clotting cascade.
Figure E. Total thrombin as a function of rime with an initiating TF cancentration of 25 pM (figure courresy of M.F.Hockin et a!., " A Model for the Stoichiomeuic Regulation of BIood Coagulation:' The Journal of Biolo~rculCfiemisrr?!277[2 11, pp. 1832218333 [2002]).Full blood cloning cascade.
Sec. 6.4
327
Algorithm for Solution of Complex Reactions
6.4 Algorithm for Solution of Complex Reactions
In compIex reaction systems consisting of combinations of parallel and series reactions, the availability of software packages (ODE solvers) makes it much easier to solve problems using moles or molar flow rates rather than conversion. For liquid systems, concentration is usually the preferred variable used in the mole balance equations. The resulting coupled differential mole balance equations can be easily solved using an ODE solver. In fact, this section has been developed to take advantage of the vast number of computational techniques now available on personal computers (Polymath). For pas systems, the molar flow rates are usually the preferred variable in the mole balance equation.
4
6
6.4.1 Mole Balances
We begin by writing mole balance equations in variables other than conversion, N,, C,, Fi. Table 6-1 gives the forms of the mole balance equation we shall use for complex reactions where r, and r~ are the net rates of formation o f A and B. MOLE BAI.A%CES FOR MULTIPLE REACTIOWS
T~FILE 6-1.
MuCe Balance
1-
d.lfA I
These are the forms of the mule balances we will use for multiple reactions.
MoIar Quantities I Gas ar Liquid)
Retch
1
Szr rl,
-- - r,
dFh
dl
dfu --
dl.
CSTR
- Ik
-
I'= F4u -Fa -rA
y - F,,-Fn
-ro
Semibatch B added to A
d,$lA
-=r41T dr
d.dr\la= F,,,
.t r ,
I I
dNA=r*V
dl
PFWPRR
]
1'
I I I I I I I 1 I I I F I
Concrntration (Lrqujd)
JC" = dr
2 "A
dV
rA
= rR
- ~4 P,,
% dl'= Qc,, j
= L ' C I ~ ' . ~ ~ ~ - C A1 -'-4
I,= U''K-,,-C,I -"B
~C =A r A -uoC, 1'
dl
(IC,= I.R dr
+
l'U[cur,-C~I J'
1
The algorithm for solving complex reactions is apptied to a gas-phase reaclion in Figure 6-5. This algorithm is very similar to the one given i n Chapter 4 for writing the mole balance? in terms uf molar ffow rales and concentralions (i.e..
328
Chap.
Multiple Reactions
Figure 4- 1 1 1. After numbering each reaction. we write a mole balance on eac species sirniIar to those in Figure 4- 1 1. The major difference between the tw algorithms is in rate law step. Here we have four steps (@, 8. @. and a)t find the net rate of reaction for each species in terms of the concentration ( the reacting species in order to combine them with their respective mole ba ances. The remaining steps are andogous to those in Figure 4-8. Multiple Reachons
Reactions (I) A+2B-C
3A+C-ID
(2)
Mole Balance
(Q Wnte mole balance on each and evely spies>
Net Rate Laws
-
Write the net rate of machon for each species
rA -I,'* + r i A j r i - r i B +r;s3 r;
t;= +r&, and
rt, -g+r;,
@ Write rate law for one spcies in every readion r;, --~,,C,C;
and r:,
-
-k,,C,C,
@ In each reaction relate the rates of react1011of each species to one another ria 2r;,,. riC -rrA+ rib O+ r;D 0. r;& riA1 3 and riD - 2J r'2.4
-
-
-
-
-
-
@ Combinenet rates, rate laws, and relative rates to write the net rates in terms of concentrahons
r;
5toichlorneh-y
'
- -~,,c,c; - k,,C,C,,
r;
r; -k,,~,~:-k,,~,~,13).
-
-2k, ,C,C;
rt, -{k,,C,C,
4
For tsotherrnal (T=TO) ps-phase reactions, wnte the concentrahons in termr of molar flow rate3 e g , C A - C T n G Y 3C B - C T , b y ~ t FT-FA+FB*Fc+Fn h
FT
(
Ft
For liquid phase reaction, iust useconcentrations as they are, e.g., C,.
c,)
Wnte the ~ ( a s - p h s epressure drop tern In terms of molar flow rates
&_-a--L-. F T wth dW
2~ F T ~ To
-
y F
Po
I
Combine
@ (
Use an ODE solver Ie.g,, Polymath) to combine stPps@ &ugh to solve for the profiIes of molar flow rates, concentration, and pressure, for example.
Flgure 6-5 Isothermal reaction design algorithm for multiple reactions.
@
3
Sec 6 4
Algorrthv for Solut~cnof Complex React~ons
6.4.2 Net Rates of Reaction h.4.2A Writing the Net Rate of Formation for Each Species Having written the mole balances. the key point for muftiple reactions i s to write the net rate of formation of each species (e.g., A, B). That is, we have to sum up the rates of formation for tach reaction in order to obtain the net rate of formation (e.g.. r, 1. If q reactions are taking place
-
Reactionl:
AtB
Reaction 2:
A+2C
Reaction 3:
2B+3E
4
-%
3C+D 3E
k'B-- 4F
We note the reaction sate constants, k. in Reactions 1 and 2 are defined with respect to A, while k in Reaction 3 is defined with respect to B. The net rates of reaction of A and B are found by summing up the rates of formation of A add B for every reaction that species A and B occur.
Net rates of reaction
When we sum the rates of the individual reaction for a species, we note that for those reactions in which a species (e.g., A. B) does not appear, he rate is zero. For the first three reactions above, r,, = 0 , r,, = 0. and r,, = 0. In general the net nte of reaction foc species j 1s the sum of all rates of the reactions in which species j appears. For q reactions taking place. the net rare of formation of species j.
6.4.2B Rate Laws The rate laws for each of the individual reactions are expressed in terms of concentrations, Cj, of the reacting species. A rate law is needed for one species in each reaction (i-e.,for species j in reaction number i) r,j =
- . . C,. . - CII)
kgJ(CA,CB
330
Multiple Reactions
Chap. 5
Here the reaction rate is shown to be dependent on the concentrations of n species. For example, if Reaction 1 Reaction (1):
kl h
+3 C + D
A*B
followed an elementary rate Iaw, then the rate of disappearance of A in Reaction 1 would be -rl~
=k l ~ C ~ C ~
and in Reaction 2 Reaction (2):
k7~
A + 2 C +3E
it would be
or in terms of the rates of formation of A
In Reaction 1: In Reaction 2 :
=-~IACACB T~~
2
=-kTACACC
Applying Equation (6-IT), the net rate of formation of A for these two reactions is Net rate of reaction of species A
For a general reaction (Figure 6-51.the rate law for the rate of formation of reactant species Aj in reaction i might depend on the concentration of species Ak and species A,. for example, 2
rd. = -k,CkC, reaction number
3
We need to defermine t h rare ~ 1alir"forat least one species in each reurriolt 6.4.2C Stoichiometry: Relative Rates of Reaction
The next step is to relate the rate law for a particuIar reaction and species to other species participating in that reaction. To achieve this relationship. we simply recall the generic reaction from Chapters 2 and 3,
Sec. 5.4
33 1
Algorithm for Sdutlon ol Complex Reactions
and use Equation (3-1) to relate the sates of disappearance of A and El to the rates of formation of C and D:
-- r- ~, B'- - -'C = 3 a b e d In working with multiple reactions. it is usually more advantageous to relate the rates of formation of each species to one another. This relationship can be achieved by rewriting Equation (3-1) for reaction i in the form Relative mtes of reaction
We now wiIl apply Equation (6-18) to reactions 1 and 2
Reaction I I):
A +B
k'*
> 3C + D Given: rIA= -kbACACB
We need to relate the rates of formation of the other species in Reaction 1to the given rate law.
Similarly for Reaction 2. Reaction (2):
A
+ 2C
1'"- 3E
the rate of formation of species E in reaction 2, r ? ~is,
and the rate of formation of C in reaction 2 is T~~
species
reaction n~~rnber
2
Given: rz4=-kzACACc
Multiple Reactions
332
Char
6.4.21) Combine Individual Rate Laws to Find the Net Rate
We now substitute the rate laws for each species in each reaction to obtain t net rate of reaction for that species. Again, considering only Reactions 1 anc Summary Rates
Relative Rntea Rate Laws Net Rates
!I)
(2)
A + B >-
'IA
3C+D
k~~ A+2C + 3E
the net rates of reaction for species A, B. C,
D.and E are
Now that we have expanded step two of our aIgorirhm, let's consider an exal ple with real reactions. Example 6 5 Stoichiomchy and Rate Laws for Multiple Reactions Consider the following set of reactions:
Rate Law,?
5N; + 6 H z 0 -rIbO= k l N O ~ N H , ~ h A (E6-5
Reaction I: 4NH,+6NO
Reaction 2: 2 N 0
NZ+ O?
Reaction 3: N,+20, ---+ 2 N 0 2
2 r ~ " i : =k 2 ~ 2 C ~ ~
-'ro,
(E6-5
~ ~ ! c N(E6-5 ~G
'
Write the rate law for each species in each reaction and then write the net rates formation of NO. 0:. and N,
.
A. Net Rates of Reaction In writing the net rates of reaction, we set the rates to zero For those spec that are not in a given reaction. For example, H,O is not involved in Reactic 2 and 3; therefore. = 0 and r,, = 0. The net rates are
From tortusirnetry data (1 1121200h).
Sec. 6.4
333
Algorithm br Solution of Complex Reactions
8.
Relative ~ a t e sof Reaction The rate laws for Reactions 1. 2, and 3 are given in terms of species NO, Nz,and Os, respectively. Therefore, we need to relate the rates of reactions of other species i n a chosen reacrlon to the given rate laws.
Recalling Equation 16-18), the corresponding rate laws are relafed by
Reaction 1 : The rate Saw wrt NO is
The relative rates are Multiple reaction stoichiometty
Then the rate af disappearance of NH3 IS Net rate NH,
THIO
= r ~ H L=o
vo=k1n:oc,vHJccLB
2 N 0 + N,
Reaction 2:
+ O2
2
rzNl is given (i.e., r,, = kZN2CNO); therefore, 2
+:NO
= Z~ZN:= ~ ~ I H > ~ ? I O 'to,
= r2u2
(E6-5.13)
334
Multiple Reactions
Chap. 6
We now combine the individual rates and the rate taws for each reaction to find the net rate of reaction. Next. let us examine the nP! rate of fonnations. The net rate of formation of NO is
Net rate NO
Next consider N: 3
Net rate
N?
(E6-5.21 ) Finally 0:
= r201+r.ro: = r 2 ~ : + 1 ' 3 0 L
{E6-5.32)
6.4.3 Stoichiomefry: Concentrations
In this step. if the reactions are liquid-phase reactions. we can go directly the combine step. RecaIl for liquid-phase reactions, u = uo and Liquid phase
cI , =F-i ufl
to
Sec. 6.5
Multiple Reactions In a PFWPBR
335
If the reactions are gas-phase reactions, we proceed as follows.
For ideal gases recall Equation (3-42): Gas phaw
where
and
For isothermal systems (T = To}with no pressure drop (P = Po) Gas phase
and we can express the net rates of disappearance of each species (e.8.. species 1 and species 2) as a function of the molar flow rates ( F , ,... ,F; ):
where,fn represents the functional dependence on concentration of the net rate of formation such as that given in Equation (E6-5.21) for N2.
6.5 Multiple Reactions in a PFRlPBR We now insert rate laws written in terms of molar flow rates [e.g., Equation (3-4211 i a o the mole balances (Table 6-1 1. After performing this operation for each species, we arrive at a coupled set of first-order ordinav differential
equations to be solved for the molar Row rates as a function of reactor volume (i.e., distance along the length of the reactor). In liquid-phase reactions, incorporating and soIving for total molar flow rate is not necessary at each step along the solution pathway because there i s no volume change with reaction.
336
Multiple Reactions
Chap
Combining mole balance, rate laws, and stoichiometsy for species through species j in the gas phase and for isothermal operation with no prr sure drop gives us Coupled ODES
For constant-pressure batch systems we would simply substitute (P for F; the preceding equations. For constant-volume batch systems we would use co centrations:
We see that we have j coupled ordinary differential equations that mt be solved simultaneously with either a numerical package or by writing ODE solver. In fact. this procedure has been developed to take advantage the vast number of computation techniques now available on personal compt ers (Polymath, MATLAB). Example 6 6 Combining Male Balances, Rate Caws, and Stoichiomern Jor itiulfiple Reactions
Coosider again the reaction in Example 6-5. Write the mole bafances on a PFR terms of molar Row rates for each species. Reaction 1: 4NH, + bNO
+5N2+6H20 -r,,,
Reaction 2 : 2N0
N,+02
Reaction 3:
N,+20, ---+ 2NOz
IN: = kLN2 ' 3 0 r
SoIurion For gas-phase reactions. the concentmtion of species j is
For no pressure drop and isothermal operation.
= k l w ~ y H (j ~ ~6
(EB-5.
6: (E6-5,
=k302~N2
Sec. 6.5
Multiple Reactions in a PFRIPBR
337
I n combining the mole balance, rate laws, and stoichiometry, we will use our results fmrn Example 6-5.The total molar flow rate of all the gases is
We now rewrite mole balances on each species in the total molar flow rate. Using the results of Example h-5
( 1) Mole balance on N
O
(2) Mole balance on NH,: 1.5 ~ dY A F N H= r w , = r ~ m 323 = - r l v o = - ~ k ~ w o c ~ ~ 3 c ~ o '
Combined Mole balance
Rate law
Stoichiornetry
(3) Mole balance on H,O : 4 4 0 = 1
dy
1.5
r~,o=r~~20=-r~no=k~uoC~~,C~o
(4) Mole balance on
N, :
Multiple Reactions
Do we need the Combine step when we use Polymath or another ODE solver? (Answer:
Chap. 6
( 5 ) Mole balance on 0,:
No) (See
Table EB-6.1)
(6) Mole balance on NO, :
The entering molar flow rates. F,", along with the entertng rernperatuE, T o , and 3 pressure. P,. (C, = P/RTn = 0.2 moUdm ) , are spcified as are the specific reaction rates k,, [e.g.. k,,, = 0.43 (dm~/mol)I5/s, k,, = 2.7 dm3fmol- s, etc.]. Consequently. Equations (E6-6.1) fhrough (E6-6.8) can be solved simulraneously with an ODE solver (e.g.. Polymath, MATLAB). I n fact, with almost all ODE sdvers. the combine step can be eliminated as the ODE solver will do the work. I n th~scase, the ODE solver algorithm is shown in Table E6-6.1. TARLE E6-61. ODE SOLVER ALGORITMMFOR MULTIPLE REACTIONS 1
Note: Polymath will do all the sub~tlmt i m g for you Thank you. Polymath!
339
Multiple Reactions in a PFRlPBR
Sec. 6.5
TABLEE6-6.1. ODE SOLVERA
m FOR M U L ~REACTIONS E (CO~~ED)
m
=
(2 1)
T H ~ O TI H ~ O
(27) F T = FNO+FWH~+FN~+FO,+EH~~+FN~~
fZ2)
NO^ = r 3 ~ 0 2
(28)
(23) C,, = c ,%
(29) k,, = 0.43
PT
(24)
FN: CN,= Cm-
(25)
CNN, =C m d
-
cm= 0.2
(30) kZN2= 2.7
Fr
FNH Fr
(31) kTOl=S.R
For
(26) Coz= Cm-
FT
Summarizing to this point, we show in TabIe 6-2 the equations for species j and reaction i that are to be combined when we have q reactions and n species.
Mole haIances:
ELc dY
(6-26)
Rates: Relative wtm:
2= 3 = k 3 -h,
Q,
c,
d,
Rate laws:
r,,=$,f?(cl.c,.C.l
Net laws:
r, =
(6-16)
4
The basic equations
z 1=
Y,,
(6- 1 7)
I
Stoichiometry: (gag phase)
C , = C ,F~ P - - To FTPOi-
F,=
r.
Fl
(3-42)
(6-20)
,I= I
(liquid phase)
F
C, = A L'o
(6-19)
340
I I I
Multiple Reaeions
Chap
Example 6 7 HydrodeaIkylafion of Mesitylena in a PFR
The production of m-xylene by the hydrodealkylation of rnesitylene over a Houc Detral catalystt involves the folfowing reactions:
m-Xylene can also undergo hydrodealkylation to form toluene:
The second reaction is undesinbie, because m-xyIene selIs for a higher price th toluene: (S 1.32/lb vs. $0.3011b)? Thus we see that there is a significant incentive maximize the prduction of m-xylene. The hydrodealkylation of meqitylene is to be carried out isothermally 15U0°R and 35 atm in a packed-bed reactor in which the feed is 66.7 mol% hyd gen and 33.3 mol% mmesitylene. The volumetric feed rate is 476 ft-'/h and the reac volume (i.e., V = W/p,) is 238 ft3. The rate laws for reactions 1 and 2 are, respectively.
A significant economic incentive
where the subscripts are: M = mesitylene, X = rn-xylene. T = toluene, Me methane, and N = hydrogen (Hz). At 15W0R.the specific reaction rates are Reaction I: k, = 55.20 (ft3Ab r n ~ l ) ~ . ~ / h Reaction 2: k, = 30.20 (ft3Ab r n 0 1 ) ~ ~ ~ ~ l r
The bulk density of' the catalyst has been included in the specific reaction rate (i k , = k; 1. Plot the concentrations of hydrogen. mesitylene, and xylene as a function space time. Calculate the space time where the production of xylene is a maximi
5 6
Fnd Eng. Chem. Process Des. Dm.,4, 92 (1965); 5, 146 ( 1466). November 2004 prices, from Chemical Marker Reporter (Schnett Publishing Cc 265. 23 (May 17, 2004). Also see www.chemweek.com/ and www.icisloccom
Sec. 6.5
M o l e balance on each and even, species
Mul!iple React~onsin a PFRiPBR
Reaction 1 :
M+H
Reaction 2:
X+A
-
X+Me
w T+Me
1. Mole balances:
dF" = rH dl.'
Hydrogen:
Xy lene: Toluene:
-_
~Fu, dp' - r,,
Methane:
2. Rate laws and net rates: Given Reaction 1:
-rl, = klC ~ C ,
Reaction 2:
r:, = ~ , C ; ~ C ,
Relative rates: (1)
-rlH =-riM= rlMe = rlx
(2)
r~ = r2nt = - r 7 ~= - r 2 ~
Net rates: r ~ = r , ~ = - k1:~zCL1 C, 11: rn = rl,+r2, = rI,-rzT =-RICH C,-k2CH1 1C,
rx = r,,+r,, = -rIH-r2,= '%
~,c;c,-~,C;'C,
= rlMe+r~Mc = - r l ~ + T z T = k l ~ l $ +k2cVcX ~M IR
rs = rzr = k2CHCX
3. Stoichiometry The volumetric flaw rate is
342
Mullipla Reactions
Chap. 6
Because there is no pressure drop P = Po fie., y = 1)- the reaction is carried out isothermally. T = To,and there is no change in the total number of moles: consequent1y, V = Uo
Flow rates:
4. Combining and substituting in terms of the space-time yields
If we know C., C,, and C,. then C,, and C, can be calculated from the reaction stoichionetry. Consequently, we need only to sohe the forlowing three equations:
The emergence of user-friend1y ODE solwrs favors this approach over frsctional conversion.
5. Parameter evaluation: At To= 1.500" R and Po = 35 atm, the total concentration is
Sec. 6.6
MuRiple Reactions in a CSTA
We now solve these three equations, (E6-7.22) to (86-7.24).simultaneously using Polymath. The program and output in graphical form are shown in Table E6-7.1and Figure E6-7.1. respectively. However, I hasten to p i n t our that these equations can be solved analytically and the solution was given in the first edition of this text.
s(h) Figure EX-7.1 Concentration profiles: in a PFR.
Dlffermtial quatione as entered by the mer [ 11 d(Ch)ld(tau)= r!h+flh 12 1 d(CmVd(tau)= r l m I3 I d(Cx)ld(tauE = rlxtRx Living Ewamnle Problem
Exp!lci! aquatiom as entered by the user [ 11 kl = 55.2 [ 2 ] k2 = 30.2 1-1 1 r l m = .kl' Crn'(Chn.5) 1 4 1 r2t = k2*Cx'(ChA.5) ~ 5 1r l h c r l m [ h i E!mr:-Rt 171
rlxs-rlm
[ 5 1 RW=.PI 1'31 Qhr-I% -
6.6 Multiple Reactions in a CSTR For a CSTR, a coupled set of algebraic equations analogous to PFR differential equations must be solved.
344
Multiple Reactions
Ck
Rearranging, yields
Recall that r, in Equation (6-17) is a function ($ ) of the species con
trations
After writing a mole balance on each species in the reaction set. we substi for concentrations in the respective rate laws. If there is no volume chi with reaction, we use concentra~ions.C,, as variables. 1f the reactions are phase and there is volume change, we use molar flow rates, as variat The total molar flow rate for n species is
5,
For q reactions occurring in the gas phase, where N different species present, we have the following set of algebraic equations:
We can use an equation solver in Polymath or a similar program to sc Equations (6-31) through (6-33).
I
Example &s Hydrodealkylntion of Mesitylene in a CSTR For the multiple reactions and conditions described in Example 6-7, calculate conversion of hydrogen and mesitylene along with the exiting concentrations mesitylene, hydrogen, and xylene in a CSTR.
Solution As in Example 6-7, we assume u = 0,: for example,
FA= vCA= uOCA,etc.
Sec. 5.8
1.
Multiple Reactions in a CSTR
345
CSTR Mole Bala~ces: Hydrogen:
uoCH,- uoCH= r,V
Mtsitylene:
~OCM - VOCM O = r~ v
Xylene:
v,Cx = rxY
Toluene:
vDCT= rTY ~ O C=M he ~V
Methane:
(E6-8.5)
2. Net Rates The rate laws and net rates of reaction for these reactions were given by Equations (E6-7.12) through (E6-7-16)in Example 6-7. 3. Stoichiornetry: Same as in Example 6-7. Combining Equations (E6-7.12) through (E6-7.16)with Equations (E6-8.1) through (E6-8.3) and after dividing by uo, (T = Vlvo), yields
Next, we put these equations in a form such that they can be readily solved using Polymath.
fl c, )
= 0 = (k,C , ! ~ C , - ~ ~ C ~)~t-CC, ,
(E6-8. I 1)
The Polymath program and solution lo equations (E6-8.9), (E6-8.101, and (E6-8.11) are shown in Table E6-8.1. The ptobfem was sotved For different values of r and the results are plotted in Figure E6-8.1. For a space time of T = 0.5, the exiting concentrations are C, = 0.0089, CM = 0.0029. and Cx = 0.0033. The overall conversion is
OveralI conversion
346
1
Multiple Reactions
Chap. 6
NLES Solutian
I
Variable
Ch Cm
CX tau
Value 0.0089436 0.0029085 0.0031266 0.5
f (x)
1.99%-10 7.834E-12 -1.839E-10
I n i Guess 0.006 0.0033 0.005
NLES Report (safenewt)
Living "ample Problem
Nonlinear equations , f(Ch) = Ch-,021+[55.2'Cm*ChA.5+30.2*Cx*ChA.5)'tau= 0 , f(Crn) = Cm-.0105+(55.2*Cm*ChA.5)'tau= O ; '$ : f/Cx) = {55.2*Crn'ChA.5-30.2CCx"Chh.5)*tau-Cx =0 '
Explicit equations : tau = 0.5 '
To vary p , , , one can v q either v, for a fixed V or very V for a fixed vo.
Figure E6-8.1 Concentrations as a function of space time We resolve Equations (E6-8.6) through (E6-8,Il) for different values of r la
arrive at Figure E6-8.1. The moles of hydrogen consumed in reaction 1 are equal to the moles of mesitylene consumed. Therefore, the conversion of hydrogen in reaction 1 IS
The conversion of hydrogen in reaction 2 is just the overall conversion minus the conversion in reactlon I : that is.
The vleld of xylene from meutklene ha3ed on molar flow a r e s exiting the CSTR lor t =
0.5 i q
Sec. 6.7
DvamIl seleclivity, i,and yield. ?.
Membrans Reactors to Improve Sefecffviin Multiple Reactions
1-1
347
mole mesitylene reacted
The ovemlI selectivity of xylene relative to toluene is
RecalI that for a CSTR the ovenll selectivity and yield are identical with the instantaneous selecrivity and yield.
6.7 Membrane Reactors to Improve Selectivity in Multiple Reactions In addition to using memhrane reactors to remove a reaction product in order to shift the equilibrium toward completion, we can use membrane reactors to increase selectivity in multiple reactions. This increase can he achieved by injecting one of the reactants along the length of the reactor. It is particularly effective in partial oxidation of hydrocarbons, chlorination, ethoxylation, hydrogenation, nitration, and suIfunation reactions to name a few.'
---t
+ CH,
-
+ CH,
W.S . bsher. D.C. Bornberger. and D. L. Huestis. Eljaluatfon o f S R / k No~,elReactor Process Per'pmlixTM(New York:AIChE).
348
Multiple Reactions
Ck
In the top two reactions, the desired product is the intermediate (e.g., C2H,1 However, because there is oxygen present, the reactants and intermediates c be completely oxidized to form undesired products C 0 2 and water. The des product in the bottom reaction is xylene. By keeping one of the reactants at low concentration, we can enhance selectivity. By feeding a reactant thmug the sides of a membrane reactor, we can keep its concentration low.
In the solved example problem on the CD-ROM, we have used a n brane reactor (MR) to continue the hydrodealkylation of mesitylene reactic Examples 6-7 and 6-8. In some ways, this CD example parallels the u: MRs for partial oxidation reactions. We will now do an example for a diffc reaction to ilIustrate the advantages of an MR for certain types of reactiot Solved Problems
I
Example 6-9 Membrane Reactor to Improve Selretivi~in ~WultipbReoctio The reactions
take place in the gas phase. The overall selectivities, S m . are to be compare a membrane reactor (MR)and n conventional PFR. First, we use the instantar selectivity to determine which species should be fed through the membrane
We see that to maximize dm we need to keep the concentration of A high an concentration of B low: therefore, we feed B through the membrane. The moIar rate of A entering the reactor is 4 moYs and that of B entering through the I bnne is 4 molls as shown in figure E6-4.1.For the PFR,B enters along with
The reactor volume is 50 dm3 and the entering total c_oncentmtion is 0.8 moVc Plot the molar flow rates and the overall selectivity, SWTJ, as a function of rr volume for both the MR and PFR.
Sec. 6.7
Membrane Reactors to Improvs Selectivity In Multiple Reactions
Solirfiun
Mole Balances for both the PFR and the .MR
MR
PFR
Species B: (2)
5 =r dV
2
(E6-9.214)
= rB+RB
(E6-9.L[b])
Species C: (3)
Species D:(4)
1
Net Rates md Rate Laws
Transpnrt Law The volumetric flow rate through the membrane i s given by Darcy's Law (see Chap-
ter 4):
where K is the membrane pemeab~lity( d s kPa) and P, (kPa) and P, (kPa) are the she[[ side and tube side pressures, and A, i s the membrane surface area m?.The flow rate through the membrane can he controlled by pressure drop across the membrane (P,- P,). Recall from Equation 14-43) that "a" i s the membrane surface area per unit volume of reactor, +
A, = nVI
The total molar flow rate of B through the sides of the reactor is
I
The molar flow rate of B per unit volume of reactor is
(E6-9.10)
350
Multiple Reactions
Chap. 6
Stoichiometry: Isothermal (T = To)and neglect pressure drop down the length of the reactor (P= Po, y = 1.0) For borh the PFR and MR for no pressure drop down the length of the rector and isothermal operation. the concentrations are Here T = Toand 3P = 0
Combine The Polymath Program will combine the mole balance, net rates, and stoichiornetric equations to solve for the molar flow rate and selectivity profiles for both the conventional PFR and the MR and also the selectivity profile. A note of caution on calculating the overall selectivity
Fool Polymath!
We have to fool Polymath because at the entrance of the reactor F,:= 0. Polymath will Eook at Equation (E6-9.17) and will not run be~avseit will say you are d~vidine by zero. Therefore. we need to add a very small number to the denominator. say 0.0001; that is,
(E6-9. IS) Skach the trends or results you expect before working out the details of the problem.
Table E6-9.1shows the Polymath Program and report sheet. WLYCIATH Results Err~nylr6.9 Membmn* Reactor ID trnprout hIKltvlq lo Mvlriplr Rwctlonr o s - l s . ? ~~, c v 5 . 13:
W.r#nUa apvauoneaa snlerafcymm usmr Varrab-o !L I d ( F n Y ~ V J . 11 v 1: I ~ ; F O Y ~ Y ) .f b + ~ b .F 11 I diFdWrnV1
.
lu
[ I I I:FYII(I,Y] - m
Fb
P?
Inizih: valve
r n ~ n i z a lvalue
.mxima: u e l u c
3
3
:c
1
i.3513875 D 3
4
!:r.al sc
valu.
L J5I1815
1 :5:~n~s 1 15138'5
: 9299793
:9099'89
Ewtlcd s ~ a l m n aM e n w a d Oy h a umr I :, Fl = PbcFb+FO+Fy :2' C 1 0 1 3 8 i l l Xll.2 t i l *21=3 : 5 : Cb.Cm-~w! l a r Ca k ttP'F& 181
-
111;
w-50
: 7 1 I b rn .ktm'cd*Z'C&k2m'Cd.CM
n-s I C I Cd CWFW! N I P ! C u r CWFWFt l t l l rd- kla'Cs*I'Cb I .!: N rn *2aUa*Cb*2 [11' F b 0 . A
1::'
R l r f m
[:&I Sau . F W F u + W l )
We can e a ~ i l ymodif!. the program, Table Eh-9 1. for the PFR simply by setting RH ~(l113~ to zeru (Rg= 0 ) and the initla1 cundi~ionfor B EO be 4.0.
Sec. 6.8
I
Complex Reactions of Ammonia Oxidation
351
Figures E6-9.2(a) and E6-9.2(b) show the molar flow rate profiles for the conventional PFR and MR, respectively.
la) PFR
Ibj
MR
Figure E6-92 Mnlar Row rates.
:Iectivities = 5 dm"
at
mm = 14 pFMDK = 0.65
(a)
tb) MR
PFR Figure E6-9.3 Selecrivity.
Figures W-9.3(a) and Eh-9.Xb) show the selectj\iry for the PFR and MR. One notlces the enormous enhancement in selectivity the MR has over the PFR. Be sure to load this liling example problem and play with the reactions and reactors. With minor modifications. you can explore reactions analogous to paaial oxidations.
]
where oxygen (3)is fed lhmugh the membrane. See Problems P6-9 and P6-19.
6.8 Complex Reactions of AmmonEa Oxidation In the two preceding examples. there was no volume change with reaction: consequently, we could use concentration as our dependenr banable. We now consider a gas-phase reaction with volume change taking place in a PFR. Under these conditions. we must use the molar Row rates as our dependenr variables.
352
Multiple Reactions
Ch:
Example 4-10 Calculating Concentrations as Functions of Position for NH, Oxidation in a PFR The following gas-phase reactions take place simuttaneously on a metal oxide-: ported cataiyst: 1.
4NH3+5OZ----+
2. 2NH3+E.502
3.
2N0+Q2
4.
4NH,+6NO
4N0+6H20
-
WL+3H20
+2N02 5N2+6H,O
Writing these equations in terms of symbls yieids Reaction 1:
Reaction 2: Reaction 3: Reaction 4: withs
414
+ 5B --+
+ 1.5B
2A
2C + B 4.4
+ 6C
4C
----4
-
+ 6D
-r,, = ~,,c,c; 026-1
E + 3D
-r2, = k,,C,C,
2F
-rJB=
5E -I-6D
-r4,
tE6-1
k 3 B ~ l ~ (E6-1 B
= R,,c,c?
k,, = 5.0 (m3/kmol)2/min
k,, = 2.0 m 3 h o l . m i n
k,, = 10.0 (m3/km0l)~/min
k,c = 5.0 (rn3/kmol)m/min
(E6-I
Note: We have convened the specific reaction rates to a per unit volume bask multiplying h e K on a per mass of catalyst basis by the bulk density of packed bed (i.e.. k = k'p,).
Determine the concentrations as s function of position (i.e., volume) in a PI Additional information: Feed rate = 10 drn3/min: volume of reactor = 10 dm3;
C,, = , C = 1 .O moVdm3, C, = 2.0 moVdm3 Solution
Mole balance:
Species B:
3 = rs dV
Reaction orders and rate constants were estimated from perjscosity measurements a bulk cataiyst density of 1.2 kgldm3.
Complex Reactions of Ammonia Oxidation
Sec. 6.8 Sr~lutiunsto ihc3e ellu:ltion5 are ~ i i r l > i
msily obtained with an ODE solwr
-
Species E:
[~FE d - r~
Species P:
5 = ri dV
Total:
FT = FA*FB+FC+FO+FE+FF
Rate laws: See above for r,,, r,,, r,,, and r,,. Stoichiametry: A.
Relative rutes
Reaction I : Reaction 2:
-
- X~~ - r~~
~ I A r~~
4 - 5
4
6
r 2 ------~ r ? ~ r~~ - r
-1.5
-2
1
2 ~
3
(E6-10.13)
Reaction 3: Reaction 4:
B.
- r~~ - r 4 ~
r 4 ~ -r 4 ~
4
-
6
5
6
Concenrmtiorls: For isothermal operation and no pressure drop, the of the molar flow rates by
concentrations are given in terms
Next substitute for the concentration of each species in the rate laws. Writing the rate law for species A in reaction I in terms of the rate of formation, r , , , and molar flow rates, FA and F, , we obtain
Thus
Similarly for the other reactions,
354
Multiple Reactions
Chap. 6
Next, we determine the ne! rate of reaction for each species by using the appropriate stoichiometric coefficients and then summing the rates of the individual reactions. Net rates of formation:
Species A:
L r, = rlA+rza+~r,,
(E6-10.20)
Species B:
r, = 1.2Sr,,+0.75r2,+r,,
(E6-10.2 1)
SpeciesC:
rc=-r,Af2r3s+rnC
(E6-10.22)
Species D:
rD = -1.5rIA- 1.5r2A-r4c
(E6- 10.231
Species
V?A
E:
S p c i e s F:
sE
=
5
-T - :r4c
r F = -2rJR
(E6- 1 0.24) (E6-10.25)
Combining: Rather than combining the concentrations, rate laws, and mole balances to write everything in terms of the molar flow rate as we did in the past. it IS mare convenient here to write our computer solution (either Polymath or our own program) using equations far r , , , F A , and so on. Consequently, we shall write Equations (E6-10. I61 through (E6- 10.19) and (E6-10.5) through (E6-10.11) as individual lines and let the computer combine them to obta~na solution. The corresponding Polymath program written for this problem is shown in Table E6-10.1 and a plot of the output is shown in Figure E6- 10.1. One notes that there is a maximum rn the concentration of NO (i.e.. C) at approximately 1.5 dd.
*b. TJ,'
+*
rJ/P
.
However, there is one fly in the ointment here: It may not be possible to determine the rate lau s for each of the reactions. In this case, it may be necess a y to work with the minimum number of reactions and hope that a rate law can he found for each reaction. That is. you need to find he number of Iinearly independent reactions in your reaction set. In Example 6-10. there are four reactions given [(E6- 10.5) through (€6- 10.811. However, only three of these reactions are independent. as the fourth can be formed from a linear cornbination of the other three. Techniques for determining rhe number of independent reactions are given by AsisnY -.
-
-
' R.Aris,
-
E l r r n ~ ~ ~ rCllr~r~icr~l np Rrmr?ur At~aly.ri.r(Upper Saddle R~ver.N.J.: Prentlce Hall, 1969).
Suing Example Problcm
Sec. 6.8
Complox Reactions of Ammonia Oxidation
355
POLYMATH Rcgultr Fslmplt 6.10 Cllculmllw Carw4~0tloarna hdhdklllw br W Wdrtk.in a PPR
I
Variable
i n i t i a l value
minlmbl ualrre
V
c
0
PA
10
ao
1.501099 ~.&OPO?~P
PC
0
0
FD
0
0
PE
I
?I
C
0
Pt
10
I0
vn
0
XlA
-5
-5
r2A
-2 0
-I -0 5619376 -0.1148551
r4C r3U CA
0
1
rA
rB
rC
-I -7.15 5
0.118876-7 -7.73
-3.2008313 1 2182361
rD
10 1
CL
1
O.Og3074q
rP
0
E
amwmlar s a u d m us e I .I
final valua 10
Q W M Y I = rA
m w ~ h LW s
12 1 d(FBW(Y) = IB J 1 d(FCVfl(VJ= rC !d f d,FDVO[V) m
:
I 5 I dlFEYdlW= rE 1 6 1 d{FFpd(V) e rF
V
Iddl
Figure E6-10.1 Molar Row rates proliles.
356
Multiple Reactions
Ct
6.9 Sorting It All Out In Example 6-9 we were given the rate laws and asked to caIculilte the produc
Gonllnear leart-squxes
tribution. The inverse of the problem descrihed in Example 6-9 must frequent solved. Specifically, the rate Iaws often must be determined from the variati the product distribution generated by changing the feed concentrations. In ! instances this determination may not be possible without carrying out indr dent experiments on some of the reactions in the sequence. The best strate, use to sort oirt all of the rate law parameters will vary from reaction sequen reaction sequence. Consequently. the strategy developed for one system ma be the best approach for other mu1tiple-reaction systems. One general n~l start an analysis by lookins for species produced in only one reaction: next. ! the species involved in only two reactions, then three, and so on. When some of the intermediate products are free radicals, i t may nl possible to perform independent experiments to determine the rate law pa eters. Consequently, we must deduce the rate law parameters from chang the distribution of reaction products with feed conditions. Under these c i a stances. the analysis turns into an optimization problem to estimate the bes ues of the parameters that will minimize the sums of the squares betwee calculated variables and measured variables. This process i s basically the as that described in Section 5.2.3, but more complex. owing to the larger nu of parameters to be determined. We begin by estimating the parameter v using some of the methods just discussed. Next, we use our estimates tc nonlinear regression technrques to determine the best estimates of our parat values from the data for a11 of the experiments.In Software packages are be ing available for an analysis such as this one.
6.10 The Fun Part
I'm not talking about fun you can have at an amusement park. but CRE Now that we have an understanding on how to solve for the exit concentra of multiple reactions in a CSTR and how to plot the species concentra down the length of a PFR or PBR, we can address one of the most i m p and fun areas of chemical reaction engineering. This area. discussed in Se 6.2, is learning how to maximize the desired product and minimize the t sired product. It is this area that can make or break a chemical process f cially. It is also an area that requires creativity in designing the re schemes and feed conditions that will maximize profits. Here you can mi: match reactors, feed streams, and side streams as well as vary the ratios of concentration in order to maximize or minimize rhe selectivity of a parti species. Problems of this type are what I call digitnl-age problemsH bet we normally need to use ODE solvers along with critical and creative thir skills to find the best answer. A number of problems at the end of this ch loSee. for example. Y. Bard, Nonlinaor Pornmeter Estrrnation
(San Diego, Calif.: demic Press. 19741. "H. Scott Fogltr. Teaching Cririca! Thinking. Creative Thinking, and Problem Sc in the Digitnl Age, Phillips Lecture (Stillwater. Okla.: OSU Press, 1997).
Chap. 6
357
Sdmmary
will allow you to practice thew critical and crci~tivethinking skills. These problems offer opportunity to explore many different solution alternatives to enhance selectisity and have fun doing it. However. lo carry CRE to the next level and t t r have a lot more fun solving multiple reaction problems, we will have to be patient a little longer. The reason is that in this chapter we consider only isothermal multiple reactions, and i t is nonisothermal multiple reactions where things really get interesting. Consequently, we will have to wait to carry out schemes to maximize the desired product in nonisothermal multiple reaclctions until we study heat effects in Chapters 8 and 9. After studying these chapters, we will add a new dimenMultiple Rsactlonc sion to muItiple reactions, as we now have another variable, temperature. that ~ ~ heat t hutTecrs is we may or may not be able to use to affect selectivity and yield. In one particunique tu thls h ~ k ularly interesting problem (P8-26). h e will study is the production of styrene from ethylbenzene in which two side reactions, one endothermic, and one exothermic, must be taken into account. Here we may vary a whole slew of variables. such as entering temperature. diluent rate, and observe optima, in the production of styrene. However, we will have to delay gratification of the styrene study until we have mastered Chapter 8.
Closure. After completing this chapter the reader should be able to describe the different types of multiple reactions (series, parallel, compIex, and independent) and to select a reaction system that maximizes the selectivity. The reader should be able to write down and use the algorithm for solving CRE probIems with rnultipIe reactions. The reader should also be able to point out the major differences in the CRB algorithm for the multiple reactions from that for the single reactions, and then discuss why care must be taken when writing the rate law and stoichiornetric steps to account for the sate laws for each reaction, the relative rates, and the net rates of reaction. Finally, the readers should feel a sense of accomplishment by knowing they have now reached a IeveI they can solve realistic CRE problems with complex kinetics.
SUMMARY 1. For the cornpetin5 reactions
Reaction 1:
A+B
--% D
a [ $ (S6-1) rD= A e - ~ d ~A ~ C B
Reaction 2:
A+B
-A O
s, = A
e-~~~ a22? o ~C (56-2)
the instantaneous selectivity parameter is defined as
A
Multiple Reactions
Chap. 6
a. If ED>EU,the selectivity parameter SDnrwill increase with increasing tempwature. b. If a,>a, and PZ> the reaction should be carried out at high concentrations of A and low concentrations of B to maintain the selectivity parameter S,., at a high value. Use a semibatch reactor with pure A initially or a tubular reactor in which B is fed at d i f f e ~ n tlocat~onsdown the reactor. Other cases discussed in the text are (a,> a,,P, > P?). (a2> a,, B2 P,x and (a, > P I > B2)-
Pi,
.
The overall selectivity, based on molar flow rates leaving the reactor, for the reactions gwen by Equations (S6-1) and (S6-2) is
2. The overall yield is the ratio of the number of moles of a product at the end of a reaction to the number of moles of the key reactant that have been consumed:
I . The algorithm:
Mole balances: Following the Algorithm
PFR
CSTR Batch
Membrane ("i" diffuses in)
Liquid-semi batch
Rate laws and net rates: Laws
Chap. 6
CD-ROM Material
Net rates
Relative rates Stoichiornetry:
F. P To- r 55,: c.= C A_-n ~ , p T, r n ~ , T ,
Cur p k n ~e
CD-ROM MATERIAL Learning Resources 1. Szt~lrrnunNorrs
2. Web module.^ A. Cobra Bites
B. Oscillating Reactions
--1 2 .
,. ....... ,
l*
(S6- 14)
360
4L-
.H
Multiole Reactions
Ck
3, {trrpr.rrctir.c2 Cot)+pfr1riA b l o t l e l r ~1ICil.l, The Grzi~tRace
&[+
@ C o ~ g u t e Modules r
4. Reir~torLr~h.her) L L , L I ) . I #R~~~, TI ~O U I TLII Y I~lrrC rg"tlre~rit~rrrui?ri~,e rmrrlprit~r ulmersr:es.
I I (tf ~
C / ~ q l t ~4 'fur r dercri~
5. Solveil Pmhlernr A. Blood Coagulation B. Hydrodealkylation of Mes~tylrnein s Membrane Reactor C. "411Yotl Wanted to Knnw About Making Malic Anhydrtde and More 6. ClnrIficnrinn: PFR wirh frril srrealnr nk~rr,qrkr lrrtpth ofthe rmcror: Living Example Problems Solved Problemr ' I . Exotnple 6-2 Tr~rrrnbotcrRrrrctint~r 1 Ertrrnplr 6-7Hydn)deolk!Enrion r$bfcsih/etl~ in n PFR 3. Exat~~ple 6-8 H!8(/r'drrlr/~n/k~i~r!on qf !fMrsr!vlr~~e in n CSTR 4. Erntnple 15-9 Me~~hrrlrfr Rmctor lo Itliprove Selecririn it1 Mirlrrple Renct 5. E.mnrplr 6- 10 Crrlclrlirri~lgCorrcr~~rnrrio~ls as rr F~rtlr.!imnof Porltior~for, 0-ridatron in a PFR 6. E.wmple weh - Cohm Bite Problern Living Example Problem 7. E . ~ m ! p l eweb - 0sc.illrrtmg Reacricvts Pmblern 8. , ? h : n / ~ l eCD Solved Pmhlemh - H ~ d r ( ~ c I ~ a l X ~ufiW~>.rit?.lenc brio~~ In n M bmne Reclrrer 9. Erfintple CD Sohrrl Plnblrrns - Blood Congrr/ano?~ FAQ [Frequently Asked Questions] In Updates/FAQ icon section Professional Reference Shelf R6,1 Attuir~nhls.Rginn Annlysis IARA) The ARA allowr one to find the optimum reaction system for certain type rate laws. The example used in one of modified van de Vusse kinetics ~eferenceShelf
-
to find the optimum wrt
Lints
B using a combrnation of PFRs and CSTRs
Chap F
Ques:ions 2nd Prob!ens
Q U E S T I O N S AND
PROBLEMS
The rubscr~ptto each of the problem n u m k r s indicates the level of diffictilty: A. least difficult: D, most difficult.
I n each of the following questions and problems. rather than just dnwing a box around yuur answer. write 3 sentence or two describ~nghow you solved the problem. the nssumptrons you made. the reasonableness of your answer. what you learned. and any other Facts that you want to include.
Clornewc~r"~sb!ems
P6-1
2nd solve an ongins1 problem tn illustrate the principles of this chapter. Ser Probleni PJ-l for guidelines. (b) Write a question based on the material rn this chapter that require5 critical thinking. Explain why your question rtquirer critical thinkinc. [Hint: See Preface section 33 2.1 Ic) Choose a FAQ from Chapter 6 to be eliminated and say why it should be elirn~nnted. (d) Listen to the audios , on the CD and then pick one and say why it
(a) Make up
-,,
P6-2,
was the most helpfu1-~~t (e) Which exmple on the CD-ROM kclure Note$for Chapter 6 w a s Iea5t helpful? (a) Example 6-2. ! I ) What wot~ldhave k e n nhe selectivity SBqY nnd conversion, X, if the reaction had been cartied out in a singlc PFR with the same volume as the CSTR'?(2) How would your answers change if the pressure were ~ncreasedby a factor of 100? (hl Exsmpla 6-3. Make a tableAist for each reactor shown i n Figure 6-3 identifying all the types of reactions that would he best carried out in this reactor. For enamplz, Figure 6-3(d) Semibatch: (I)high?) exothermic reactions and ( 2 ) selectivity. for example, to maintain concentntion A high and B Iow. (3) ta control convcr
Multiple Reactions
Chap. 6
(c) Example 6-4. (1) How would zip, change if k, = k2 = 0.25 m31slkg at 300 K? (2) How would r,&, change for a CSTR? (31 What CSTR (with T' = 0.5 kglm3/s) operating temperature would you recommend 10 maximize B for CAo= 5 mol/dm3, kl = 0.4 m3kg . s, and k? = 0.01 m'kg s with E, = 10 kcallmol and E2 = 20 kcal/mol (plot CB versus 7). (d) Example 6-5. How would your answers change if Reaction 2 were reversible and followed an elementary rate Iaw?
.
2NO=N2+O2
with Kc = Kc = (&2c){~I,)/(C',,2) = 0.25
(e) Example 6-6. How would Equations (E6-6.3) and (E6-6.6) change if the reactions were carried out in the liquid phase under high pressure? (0 Example 6-7. (1) How would your answers change if the feed were equaI moiar in hydrogen and mesitylene? ( 2 ) What is the effect of OH on T*~,?
sks?
(g) Example 6-8. Same question as P6-2(f)? (h) Example 6-9, Load the Lrl-ing Exur?lple Pmblcrn from the CD-ROM. ( 1 ) How would your answers change if FRO= 2FAO? ( 2 ) If reaction ( 1 were A+2B -+ D with the rate la* remaining the same* from the CD-ROM. (i) Example 6-10. Load the h b i n p Exo~nplc P1nb1~~1'lrt
Living Ex~mple.Problem
Interac:!ve
&s$k% @ Corr~puterModuler
P6-3,
( 1 ) How would your anrwers change if the reactor volume was cur in half and k,, and k , were decreased by a factor of 4? ( 2 ) !hat reactor schemes and conditrons would you choose ro maximize SC,F,r? Hitlr: Plot SC/F bersus O, as a start. Describe h o pressure ~ drop would affect the selecttvity. 0) Read Solved Problem A, Blood Coagulation. Load the lit ing example. ( 1 ) Plot out some of the other concentrations, such as TF-VIIa and TF-VIIaX (2) Why do the curves look the way they do? What reaction in the cascade is most likely to be inhibited causing one to bleed lo death? (3) What reactions if eliminated could cause one to die of a blood clot? (Hhtr: Look at ATIIII and/or TFPI.) (k) Read Solved Problem B, Membrane Reactor. Load the M ~ i l ~ h r n ~ i e Reorrrir from the CD-ROM.Hnw would your answerc change i f the feed were the same as in Examples 6-7 and 6-8 (i.e.. F,, = Fhq0J7 Vary M'I. * and the reactor volume and describe what you find. (1) Liulng Example I17eh Module: Oscirlating Reactions. Load the Lirrr~e El-~jnrpfePo/~!?lii!I~ P r o ~ r nfor~ ~o
Chap. 6
Questions and ProWerns
363
Read the cobra Web Module on the CD-ROM. (a) Determine how many cobra bites are necessary in order that no amount of anti-venom will save the victim. (b) Suppose the victim was biaen by a harmless snake and not bitten by a cobra and anti-venom was injected. How much anti-venom would need to be injected to cause death? (c) What is the latest possible time and amount that anti-venom can be injected after being bitten such that the victim would not die? (d) Ask another auestion about this problem. Hint: The Living Example Polymath program is on the CD-ROM. P6-5R The foflowing reactions
P6-4c
---P
k,
take place in a batch reactor.
P6-6,
(a) Plot conversion and the concentrations of A, D, and U as a function of time. When wouId you stop the reaction to maximize the concentration of D? (b) When is the maximum concentration of U? (c) What are the equilibrium concentrations of A, D, and U? (d) What would be the exlr concentrations from a CSTR with a space time of 1.0 min? of 10.0 min? of 100 min? Additional infunnation: k! z 1.0 m i d , K,, = 10 k2 = 100 min-I, X?, = 1.5 CAo= l molldm3 (Adapted from a problem by John Falkner, University of Colorado.) Consider the following system of gas-phase reactions:
3 is the desired product, and X and Y are foul pollutants that are expensive to get rid of. The specific reaction rates are at 27°C. The reacrion system IS to be operated at 27°C and 4 atm. Pure A enters the system at a volumetric flow rate of 10 dm31min. (a) Skelch the instantaneom selectivities (S,,.S,, and SB = I k / ( r x + r y ) ) as a function of the concentration of CA. (b) Consider o senes of reactors. What should be the volume of the first reactor? (c) What arc the embent concentrations of A. B. X, and Y from the first reactor.
364
Multiple Reactions
Cb
(d) What is the conversion of A in the first reactor?
P6-7,
Mall of Fame
(e) If 99'3 conversion of A I \ h i r e d . what reaction ~chcrneand re stzes should you use to maximize S,,,? (0 Suppo~ethat El = 2 0 . 0 0 cal/mol. E: = 10.000 caVmol. and E, = 3( callmol. What temperature would you recommend for a single CSTR a space time of I0 min and an entering concentration of A o f 0.1 moll^ (g) If you could vary the pressure between 1 and 100 atni. what pre would you choose? Phitrmncokinetics concerns the ingestion, distribution. reaction, and etimin reaction of drugs in the M y . Consider the application of phamacokineti one of the major problems we have in the Vnited States, drinking and drim Were we shall model how long one must wait to drive after having a tall tini. In most states. the legal ~ntoxicationlimit is 0.8 g of ethanol per I1t8 body fluid. (In Sweden it is 0.5 g/L. and in Eastern Europe and Russia any value above 0.0 g/L.) The ingestion of ethanol into the bloodstream subsequent elimination can be modeled as a series reaction. The rat nb~orptionfrom the gastrointestinal tract into the bloodstream and body first-order reaction with a specific reaction rate constant of 10 h-' . The ra which ethanol is broken down in the bloodstream is limited by regeneratic a coenzyme. Consequently, the process may be modeled as a zero-order I tion with a specific reaction rate of 0.192 g l h - L of body fluid. How would a penon have to wait (al in the United States; (h) in Sweden: and ( Russia if they drank two tall martinis immediately after arriving at a ps How would your answer change if Id) the drinks were taken h apart: (e two drinks were consumed at a uniform rate during the first hour? (0 Sup1 that one went to a part!: had one and a half tall mart~nisright away. and received a phone call saying an emergency had come up and the pz needed to drive home immediately. How many minutes would the indivi have to reach home before helshe k c a m e legally intoxicated, assuming the person had nothing further to drink? (g) How would your answers be ferent for a thin person? A heavy person'' fHirtt: Rase all ethanol conce! tions on the voIume of body fluid. Plot the concentration of ethanol in blood as a function of time.) What generalizations can you make? What is point of this problem*? 24601 = Jean (i.e., who?)
J?
Additional inJormntiot7:
Ethanol in a tnIE martini: 40 g Volume of body fluid: 40 C (SADD-MADD problem) [See Chapter 7 for a more in-depth Iook at alcohol metnbolism.1
P6-8,
Ual a f Fame
a liquid antibiotic that ir taken orally to t infections of the spleen. 1t is effective only if it can maintain a concentra in the blood~trearn(bnsed on volume of body fluid) above 0.4 mg per dm body Ruid. Ideally. a concentration of 1.0 rng/dm3 in the blmd would lik be realized. However. if the concentration in the b l d exceeds 1.5 mgld (Phormacokinerics) Tnrzlon i s
Chap. 6
365
Questions and Problems
harmful s ~ d eeffect? can occur. Once the Tarrlon reaches the stomach, it can proceed i n two pathway. both of which are fint order: (I)I1 can be absorbed into the bloodstream through rhe stomach walls: ( 2 ) it can pa\< nut rhmugh the gastrointesttnal trnot and not be absnrhed into the blond Borh thehe processe3 are fint order In TarzFon concentration in the stomach. Once in the bloodstream. Tanlon attacks bacterial cells and i s subsequently degraded by a zero-order process. Tarzlon can also be removed from the b l d and excreted In urine through a first-order prmess within the kidneys. In the stomach: Absorption into brood
k , =0.15 h-I
Elimination through gastrointestine
k2 = 0.6 h-I
In the bloodstream: Degradation of Tanloti
k, = 0. I mg/drn'. h
Elimination through urine
k, = 0.2 h-1
(a) Plot the concentration of T ~ n l o nin the blood as a function of time when I dose h e . , one liquid capsule] o f Tarzlon is taken. (b) How should the Tartlon be administered (dosage and frequency) over a 48-h weriod to be most effective:' (c) Comment on the dose concentrations and potential hazards. (d) How would your answers change if the drug were taken on a full or empty stomach?
P6-9c
One dose of Tanlon is 250 rng in liquid form: Volume of body fluid = 10 dm3 (Reuctor sekction and operating conditions) For each of the following sets o f reactions describe your reactor system and conditions to maximize the desired product D. Make sketches where necessasy to support your choices. The rates are in (rnoYdrn3 . s). and concentrations are in (molldm").
(2) (cl ( I )
A+B-+U
-rz, = 10' exp(-8,000 W W A C B
+ B -+
-rlA= 10 EXP(- 1.000 K/QCqCB
A
D
Chap. 6
P6-IS,
367
Questions and Problems
is canied out in a 500-dm3batch reactor. The initial concentration of A is 1.6 mol/dm3. The desired product is B. and separation of the undesired product C is very difficult and costly. Because the reaction i s carried out at a relatively high temperature, the reaction is easily quenched. k, = 0.4 h-1 at 100°C k, = 0.01 h-I (a) Assuming that each reaction is irreversible, plot the concentrations of A. B. and C as a function of time. (b) For a CSTR space time of 0.5 h. what temperature would you recommend to maximize B? (El = 10,000 callmol, E2 = 20.000 crtYmol) (c) Assume that the first reaction is reversible with k - , = 0.3 h-I. Plot the concentrations of A, B. and C as s function of time. (d) Plot the concentrations of A. B. and C as a function of time for the case where both reactions are reversible wzth k-2 = 0.005 h-I. (e) Vary k l , k2, k - , , and R-:. Explain the consequence of k , > 100 and k 2 < 0 . 1 withk-, = k - > = O a n d w ~ t h k - , = P , k - , = 0 , a 1 1 d k - ~ = 0 . 2 5 . Note: This problem is extended to include the econnmics [profit) in CDP6-B. TerephthaIic acid (TPA) finds extenswe use In the manufacture of synthetic fibers (e.g., Dacron) and as an intermediate for polyest~rfilms (e.g., Mylar). The formation of potassium terephthalate from potassium benzoate was studied using a tubular reactor [htd. Eng. Chon. Res.. 26, 169 1 ( 1 98711. I t was found that the intermediates (primarily K-phthalatesj formed from the dissociation of K-benzoate over a CdCI, catalyst reacted with K-terephthalate In an autocatalytic reaction step.
R+S
A
2S
Autocatalytic
where A = K-benzoate, R = lumped intermediates (K-phthalatis, K-isophthalates. and K-benzenecarboxylates). and S = K-terephthalate. Pure A is chaqed to the reacror at a pressure of 110 kPa. The specific reaction rates at 410°C are k, = 1.08 x llr' s-I wilh El = 42.6 kcallmol. k, = 1. I9 x lo-' 5-I with E, = 48.6 kcallrnol. k, = 1.59 x I R3dm3/rnol . s with &, = 32 kcallmnl. (a) Plot the concentrations of A, R. and S a< a function of time in a batch reactor at dl 0°C noting when the maximum in R mcurs. (h) Repeat ( a ) for temperature5 of 430'C and 390'C. (c) What would be the exit concentration^ from a CSTR operated at 310'C and a space time of 1200 s. P6-12, The following liquid-phase reactions were carried out in a CSTR at 325 K.
4D+.1C +3E
1-7,:
= kjJf,C<-
k;f =2.0- dm3
mol -min
368; Sketch the trends or results you expect before work1ng out the
details of the problem.
Multiple Reactions
Ch;
The concentratiom measured in.~idcthe reactor were CA = 0.10. CB = C: Cc = 0.51. and Cn = 0.049 all in molldm'. (a) What are r , , . r?,, and sl,? ( r , , = -0.7 mol/drn'.min) {b) What are r I Br, ? ~and . r,,? (c) What are r,,, rZc,and rqC? ( r I C= U 2-1 rnol/dm3-min j (d) What are r,,. rzn. and r3Dq (e) What are r,,, r ? ~ and . rjE? (f) What are the net rates of formation of A, B. C, D. and E? (g) The entering volumetric flow rate ih 100 drnilmin and, the entering cont bation of A Is 3 M. What is the CSTR reactor volume? rAns.: 4(XW dm' (h) Write a Polymath pro-mam to calculate the exit concentrations when the ume is given as 600 dm3. (i) PFR. Now assume the reactions take place in the gas phase. Use the I ceding data to plot the molar flow rates as a function of PFR volume. ' pressure drop panmeter is 0.001 dm-'. the total concenption ent_er the reactor is 0.2 moVdm', and u, = 100 drn7/min. What are SD/Eand Sc Ij) Membrane Reactor Repeat (I)when species C d~fiuseqout of rnr brane reactor and the transport coefficient. kc,is 10 tnin-I. Compare y results with part (I). Ph-13B Calculating the space time for pamllel reactions. m-Xylene is reacted ove ZSM-5 zeolite catalyst. The follow~ngpomllel elementary reactions w ' found to occur [kid. E ~ r g Chem. Res., 27. 947, 1198X)J:
m-Xylene
kt
Benzene
+ Methane
(a) Calculate the PER volume to achieve 85% conversion of m-xylenc
it
packed-bed reactor. Plot the overall selectivity and yields as a Function t.The specific reaction rates are k, = 0.22 s-I and k2 = 0.71 s-I at 673' A mixture of 75% rn-xylene and 25% inerts is fed to a tubuIar reactor volumetric flow rate of 200 dm3/s and a total concentntion of 0. rnolldm'. As a first approximation, neglect any other reactions such the reverse reactions and isornerization to o-xylene. (b) Suppose that E l = 20.000 cnl/mol and E2 = 10.000 caWmol. what tel perature would you recommend to maximize the formation o f p-xyle in a 2000-dm3 CSTR? P6-1dB The following reactions are carried out isoth~rmallyin a 50 dm? PFR:
Addirional in@mation: Liquid phase
k,, = 0.25 dm6!mo12.min
v 0 = 10 dml/rnim
C,, = 1.5 rnol/dm4 k,, = 5.0 dm6/mol2+rnin
CBo = 2.0 mol!dm3
Chap. 6
389
Ouestrons and Problems
la) P l o ~the specbe\ concentration5 and the conversion of A as a function of the disiance (1.e.. volume) dam a 50-dn~'PFR. Note any maxima. Ih) Determine the effluent concentrations and conversion from a 50-dm7 CSTR. (Ans.: C, = 0.61. CB = 0.74. CF = 0.25, and C, = 0.45 rnoVdmJ.) (cl Plot the species concentrations and the conversion af A as a function of time when the reaction i? carried out in a semibatch reactor initially containing 40 dm3 of liquid. Consider two cases: ( 1 ) A is fed to B. and (2) B rs fed to A. What differences do you observe for these two cases7 {dj Vary the ratio of B toA ( I < BB< 10) in the feed to the PFR and descrik what you find. What generalizations can you make from this problem? (e) Rework this problem for the case when the reaction is a gas-phase reac-
Eumma ry Notes
tion. We will keep the constants the same so you won't hake to make too many changes in your Polymath program, but we wfll make vo = LO0 dm3/rnin. Cf, = 0.4 moUdrn3. V = 500 dm3 and equal molar feed of A and B. Plot the molar Row rate? and Scm and SFfi down a PFR. if) Repeat l e ) when D diffuses out through the sides of a membrane reactor where the mass transfer cmfficient, kcD, can be varied between 0.1 min-I and 10 min-I. What trends do you find? (g) Repeat (e) when B is fed through the sides of a membrane reactor. P6-1SB Review the oxidation of formaldehyde to formic acid reactions over a vanadium titanium oxide catalyst [ind. E ~ RChem. . Res., 28, 387 (1989)l shown in the ODE soher nlgorirhm i n the Summary Notes on the CD-ROM. (a) Plot the species concentrations as a function of distance down the PFR for an entering flow rate of iIX) dm"/min at 5 atm and 140°C. The feed Is 66.7% MCHD and 33.3% 02.Note any maximum in species concentrations. (b) Plot the yietd of overall HCOOH yield and overall seiectrvi~of HCOH to CO, of HCOOCH3 to CH,OH and of HCOOH to HCOOCH? a? a function of the Qo1. Suggest some conditions to best produce formic acid Write a paragraph describing what YOU Fmd. (c) Compare your plot in par! la) with a similar plot when pressure drop i s taken into account with u = 0.002 dm-'. (d) Suppose that E, = 10.000 callmol. E, = 30,000 callmol, E3 = 20,M)O cal/mol, and & = 10,000 callrnol. what temperature would you recornmend for a 1000-dd PFR? Ph-16, The liquefaction of Kentucky Conl No. 9 was carried out in a slurry reactor [D. D. Gertenbach, R. M . Baldwin, and R. L. Bain, Ind. Eng. Chetn. Process Des. Dm., 21. 490 (1982)]. The coal panicles, which were less than 200 mesh, were dissolved in a -250°C vacuum cut of recycle oiI saturated with hydrogen at 400°C. Consider the reaction sequence
Caal (C)
kl 7 Preasphaltines (P)
~ s ~ h a l t i n (A) es
-%
Oils (0)
which is a modified version of the one given by Gertenbach et al. All reactions are first order. Calculate the molar Row rate of each species as a function of space time in (a) A plug-flow reactor. (h) A 3-m3 CSTR. (c) What is the point of this problem?
370
MultFple Reactions
Chap. 6
Entering concentration of cosl: 2 h o l l r n J Entering flow rate: 10 dm31rnin At 40Q0C, k, = 0.12 min-I. k2 = 0.046 min-1, k, = 0.020 min-I, k, = 0.034 min-I, Ii, = 0.04 min-I, P6-17, The product~onof acetylene is described by R. W. Wansbaugh [Chem. Eng, 92t16). 95 (1985)J.Using the reaction and data in this article, develop a problem and solution, P6-18t,Read the blood coagulation solved problem on the CD-ROMIWeb. Load the blood coaguIation Livi~tgfionrple Problem from the CD-ROM.Use part (h) for the complete set of coagulation equations. First verify that you can obtain the cunre shown in the shaded Side note on page 325. (a) Plot some of the other species as a function of time. specifically TFVlIa,
TFVIIaX. TFVIIaXa. and TFVIIalX. Do you notice anything unusual about these curves, such as twa maxima? (b) How does the full solution compare with the solution for the abbreviated reactions? What reactions can you eliminate and still get a reasnnnl?k approximation to the thrombin in rime curve? P6-1gC The ethylene epoxydation i s to be carried out using a cesium-doped silver catalyst ill a packed bed reactor.
Along wiih the desired reaction. the complete combustion of ethylene aIso Occllrs
IM.Al-Juaied. D. Lafarga, and A. Varma. Cliem. Ettg. Scf. 56,395 (200I)J. It is proposed to replace the conventional PBR with a membrane reactor in order to improve the select~vity.As a rule of thumb, a 19 Increase in the qelectivity to ethytene oxide translates to an increase in profit of about $2 mdlionlyr. The feed conr;ists of 12% (mole) oxygen. 6% ethylene, and the remainder nitrogen at a temperature of 250" and a pressure of 2 arm. The total molar flew rate is 0.0093 mol/s and to: reactor co~itaining? kg of catalyst. la) What conversion and selectivity, S, are expected in a conventional PFR" (h) What would he the conversion and selectivity if the total molar flow rate were divided and the 12% oxygen stream (no ethylene) were uniforml!, fed througl~the sides of the membrane reactor and 6 9 ethylene (no oyygen) were fed at the ent!ance? Ec) Repeat I b l for a case when ethylene i s fed uniformly through the sides and oxygen is fed at the entrance. Compare with parts (a) and (bj. (dl Repeat (b'r and (c) for the methanol reaction given in problem 5-9 (hj with 1'; in Bar and --I.;CH,OH and -r-;c~,o are in Barls.
Chap. 6
Questions and ProMems
Additional i ~ f o m r i o n :
k,, = 0.15
at 523 K with
ITLO'
El = 60.7k J h o l
kg-s atm'.'"
k2B = 0.0888
moi kg.s
P6-20,
at 523
K with E: = 73.2 k J h o l
For the van de Vusse elementary reactions
.--
Web Mint determine the reactor or combination of reactors that maximize the amount of
B formed. See PRS R6.1. Additional infirnation:
Links
,C ,
Ph-21,
= 2 kmol/m3 and v , = 0.2 m3h
Repeat for k, = 0.002 s-I, The gas-phase reactions take place isothermally in a membrane reactor packed with catalyst. Pure A enters the reactor at 24.6 atm and 500 K and a flow rate of A of 10 moYmin
A-D
rit, = ~ ~ D C A
Only specie? B difftises out of the reactor through the membrane. (a) PIot the concentrations down the Iength o f the reactor. (b) Explain why your curves look the way they do. (c) Vary some of the parameters (e.~.,k,, klc. K,,) and write a paragraph describing what you find. Rddir ional Iq formation: Overall mass transfer coefficient kB = 1.0 dm7I kg cat . min k l r = 2 dm" kg cat . rnin K,,= 0.2 mol I dm' k,,= 0.4 dm3 / kg cat . min k,, = 100 dm3 1 mo12 kg cat . min W f = 100 kg a = 0.008 kg-' +
372
Multiple Reac!ions
Ch;
P6-22c Read over the o.~cillcrrit~~ reooinn Web Mnrl~ile.For the four reactions inv ing I-- and 10-1: (a) What factors influence the arnpIitude and frequency of the oscilIa reaction? What causes these oscillations? (In other words: What m, thrs reaction different than others we have studied so far in Chapter (b) Why do you think that the oscillations eventually cease (in the orie experiment by Belousov, they lasted about 50 minutes)? (c) A 10°C increase in temperature p d u c e d the following observati The dirnensionles$ times at which the oscfllations began and er decreased. The dimensionless period of the oscilln~iooat the stan ot oscillation increased while the dimensionless period near the end of oscillation decreased. What conclusions can you draw about the r tions? Explain your reasoning. Feel free to use plotslsketches or et tions if you wish. Id) What if ... play wound with the Living Erample Polymath Prograrr the CD-ROM-what are the effects of changing the values of k,, k, and k2? Can you make the oscillations damped or unstable? Ph-23, Go to Professor Herz's Reactor Lab on the CD-ROM or on the we1 LVCVM!reactnrlnb.net. (a) t o a d Division 5, Lnb 2 of The Reactor Lab from the CD for the selcc oxidation of ethylene to ethylene oxide. Click the [i] infa button to information about the system. Perform experiments and develop equations for the reactions, Write a technical memo thar reports ! results and includes plots and statistical measurements of how well ) k~neticmodel fits experimental data. (b) Load Division 5, Labs 3 and 4 uf The Reactor Lab for batch reactor whlch paraIlel and series reactions, regpectively, can be carried Investigate how dilution with solvent affects the selectivity for diffe reaction orders. and write a memo describing your findings.
JOURNAL CRITIQUE PROBLEMS P6C-1
P6C-2 P6C-3
P6C-4
In J. of Hazwdous Marerinls. B89. 197 (2002). is there an atgebnic erro the equation to calculate the specific reaction rate kl? If so. what are the r ificatlons later on in the analysis? I n Inr. J. Clicn. Kirier., 35,555 (20031, does the model overpredict the 01 diethyl benzene concenmtion? 1s it possible to extrapolate the curves on Figure 2 [AIChE J., 17, 856 (19' to obtain the initial. rate of reaction? Use the Wesz-Prdter criterion to de mine if there are any diffusion limitations in this reaction. Determine the I tial pressure of the pmducts on the surface based on a select~vityfor ethyl oxide ranging between 51 and 65% with conversions between 2.3 and 3.5 Equation 5 [Chem. Eng. Sci.. 35. 619 (1980)Jis written to express the for tion rate of C (ofefins). As described in equation 2. there is no change in concentration of C in the third reaction of the series:
A+B
B+C
k,
C
--% C
equation 2
Chap. 6
Journal Critique Problems (a) Determine if the rate law given in equation 5 is correct. Ih) Carl equations 8. 4. and I2 be derived from equatron S? (c) I s equation 14 correct? ( d ) Are the adsorption cosfficients 6,and b, calculated correctly?
Good alternative problems on the CD and web. CDP6-24,
The production of maleic anhydride by oxidation with air can be carried out owr a vanadium catalyst in (a) a "fluid~zed" CSTR arid (b) a PBR at different temperatures.
In excess air these reactions can be represented by
CDP6-25,
[3d Ed. P6-I?,] Rework Problems (a) P6-6, (b) P6-7, (c) P6-8 when the toluene formed in reaction CE6-6.2) also undergoes hydrodealkylation to form benzene
[3d Ed. P6-1.51 CDP6-2hB
A series of five hydrodeatkylationreactions beginning with
and ending with
are carried out in a PFR, [3d Ed. P6-161. CDP6-2TB The hydrogenation of benzene
is carried out in a CSTR slurry reactor where the desired product is cyclohexene. [3d Ed. P6-231
374
Multipte Reactions
CD&2Bc
Hall bf fame
CDP6-29,
1
Chap. 6
Industrial data for methanol synthesis reactions is proI2ided for the complex reactions
The reaction is carried out in a PFR. Find the best operating conditions to.produce methanol. [3rd Ed. P6-241 Load the L h i n g Example Pmblem for the hydrodealkylation of mesitylene carried out in a membrane reactor. Optimize the parameters to obta~nthe maximum profit.
PFR
(b)
ta)
Flgui-e P6-30 Ial Cornpanson (b) cost funcrlon.
Additional Hnmework Problems Oldies, but goodies-problems from previous editions.
Series Reactions c D p 6 - A ~ Chlorination of benzene to rnonochl~robenzeneand dichloraben~ne in a CSTR. [lst Ed, P9-141 B, B + C. 8 4 D is CDP6-RE The reaction sequence A carried out in a batch reactor and in a CSTR. [2nd Ed. P9-123 CDPC-C, Isobutylene is oxidized to rnethacroIefn, CO. and CO?. [I st Ed. P9-I 61 Parallel Reactions
CDP6-I), CDP6-E,
CDP6-FB
Catculale conversion in two independent reactions. [3rd Ed. P6-4(d)l c c-~-D Calculate equil~briumconcentrations in reaction 14+ B C+B X + Y [3rd Ed. P6-51 Find the profit for A -+ D and A + U [3rd Ed. P6-101
Chap. 6
375
Supplementary Reading
Complex Reactions Ibuprofen
CDM-Gc CDP6-HB CDP6-I,
Production of ibuprofen intermediate. [3rd Ed. P6-261 Hydrogenation of o-cresol. 13rd Ed. P6-31 Design a reaction system to maximtze the selectivity of p-xylene from methanol and toluene over a HZSM-8 zeolife catalyst. [2nd Ed.
P9-171
n'd Green Engineering
CDP6-Jc CDP6-K, CDP6-& CI)P6-hfB CDP6-N, CDP6-0,
Oxidation of propylene to acmlein [ C h ~ mEng. . Sci. 51, 2189 (l996)]. Complex reactions. [Old Exam] California problem. [3rd Ed. &19B] Fluidized bed. [3rd Ed. Ph-201 Flame retardants. [3rd Ed. P6-2 I ] Oleic acid production. [3rd Ed. P6-251
New Problems an the Web CDPI-New From rime to time new prohlerns relating Chapter 6 material to everyday interests or emerging technologies will be placed on the web. Sc~lutionsto these problems can he obtained by e-mailing the author. Also, one can go to the weh site. ~ihn~n:mwan.edrdgwe~le~~gineeri~l~. and work the home problem on green engrneering specific to t h ~ chapter.
SUPPLEMENTARY
READING
1. Selectivity, reactor schemer. and staging for multiple reactions, together with etaluatinn of the correspond~ngdesign equations, are presented in
DEYBIGH. K. G..and 1. C.R. TLRNER. Chemical Rcurmr- Thron: 2nd ed. Cambridge: Cambridge University Press. 1971 . Chap. 6. LEVENSPIEL. 0.. CIw~nicalEFEarrior~ Engine~ring. 2nd ed. New Ynrk: Wiley. F 972. Chap. 7 . Some example problems on reactor d c ~ i g nfor multiple reactions are pre~entedin Harlr;~~ 0. . A,. and K. M. WATSOV.Chrn~ErnlPrnresr P r i t ~ ~ i p l Part e ~ . 3: Kineric5 and Cum(wis. New York: Wiley. 1947. Chap. XVIII. SMITH.J. M.. Cla~nricnlEngiiieerrrlg Kir~ctirs,3rd ed. Kew York: McGmwHill, lQ80, Chap. 4.
2. Bookr that have
many analytical solutions for parallel, serieq, and cumh~nation
reactions are
CAPELLOS, C.. and B. H. J. BIELSKI,Kiuetir S ~ s f e ~ nNew s . York: Wile!. 1972. WAI.AS. S . M.. Chr~riirulRr,rrrrinrr E~t-qillerr~n~ H r ~ ~ ~ d ~ofnSnlr'ed o k Ptnhl~~ln. Newark. N.J:
Gordon
dnrl
Rreach. 1995.
376
Multiple Reactions
Ch:
3. A brief discussion of a number of peninent references on parallel and series r tions is given in ARIS, R., Elernenrary Chemrcal Reactor Analysi.~.Upper Saddle River. Prent~ceWall, 1969, Chap. 5.
r
4. An excellent example of the determination of the specific reaction rates, k , . in r tiple reactions is given in
BOUDAUT,M., and G.DIEGA-MARIADASSOC, Kinetics of Heremgetleeus C i y r k Reactions. Princeton, N.J.: Princeton University Press, 1983.
Reaction Mechanisms, Pathways; Bioreactions, and Bioreactors The next best thing to knowing something is knowing where
to
find it.
Samuel Johnson (1 709-1784)
Overview. One of the main w a d s that tics this chapter together i s b e pseudestendy-state-hypothesis (PSSH) and the concept of active intermediates, We shall use it to develop rate laws for both chemical and biological reactions. We kgin by discussing reactions which do not follow elementary rate laws and are not zero, first, or second order. We then show how reactions of this type involve a lumber of reaction steps, each of which is elementary. After finding the net rates of reaction for each species, we invoke the PSSH to arrive at a rate Iaw that is consistent with experimental observation. After discussing gas-phase reactians, we apply the PSSH to biological reactions, with a focus on enzymatic reactions. Next, the concepts of enzymatic reactions are extend& to organisms. Here organism growth kinetics are used in modeling both batch reactors and CSTRs (chernostats). Finally. a physiological-based-phmacokinetic approach to modeling of the human body is coupled with the enzymatic reactions to develop concentration-time trajectories for the injection of both toxic and nontoxic substances.
7.1 Active~lntermediatesand Nonelementary Rate Laws In Chapter 3 a number of simple power law models. that is,
378
Reaction Mechan~srns,Pathways, Bioreactions, and Bkoreactors
Chap. 7
were presented where n was an integer of 0. 1, or 2 corresponding to a zero-,
first-. and second-order reaction. However, a large number of reactions. the orders are either noninleger such as the decomposition of acetaldehyde at 500'C
where the rate Iaw is
or of a form where there are concentration terms in both the numerator and denominator such as the formation of HBr from hydrogen and bromine
Rate laws of this form usualIy involve a number of elementary reactions and at least one active intermediate. An rxc.rhheinrrrn~ediareis a high-energy molecule that reacts virtually as fast as il is formed. As a result. it is present in very small concentrations. Active intermediates (e.g., A * ) can he formed by collision or interaction with other molecules.
Properties of
mactive intermediate A*
Here the activation occurs when translational kinetic energy is, transferred into energy stored in internal degrees of freedom. particularly vibrational degrees of freedom.' An unstable molecule (i.e., active intermediate) ir not formed solely as a consequence of the ~noleculemoving at a high velocity (high translational kinetic energy). The energy must be absorbed into the chemical bonds where high-amplitude oscillations will lead to bond ruptures. molecular rearrangement. and decomposition. lo the absence of photochemical effects or similar phenomena. the transfer of translational enerFy to vibrational energy to produce an active ~ntermediarecan occur only as a consequence of moleciiIar collision or interaction. Colliqion theory is discussed in the Profes.rinnal Rqfer4ence Shelf in Chapfer 3. Other types of active intermediates that can be formed are free radicals (one or more unpaired electrons. e.g., CH,.). ionic intermediates hg.. carbon~umionL and enzyme-substrare complexes, to mention a few. The idea of an active intermediate was first postuIared in 1922 by F. A. Lindermann2 wlio used it to explain changes i n reaction order with changes In reactant concentratic~n\.Becauae the active intermediates were zo short Lived
.
--
-.
\V. J. Muore. IJI~y.riccilClr~rairir?:(Readrng. Mass,: licmgman Pi~hlishingGmup, 199s). ! F. A. 1,indermann. Trotlr. fi~mtlu\: Soc:. 17, 39X i 1922 j. I
Sec. 7.1
379
Active Intermedates and Norelementary Rate Laws
and present in such low concentrations, their existence was not really defmitiveIy seen untiI the work of Ahmed Zewail who received the Nobel Prize in 1999 for femtosecond spectroscopyn3His work on cyclobutane showed !he reaction to form two ethylene molecules did not proceed directly, as shown in Figure 7-I(a), but formed h e active intermediate shown in the small trough at the top of the energy reaction coordinate diagram in Figure 7-l(b). As discussed in Chapter 3, an estimation of [he barrier height, E, can be obtained using computational software packages such as Spartan, Cedus" or Gaussim as discussed in the Moleculur Modeling Web Module in Chapter 3.
la)
Figure 7.1
(b)
Reaction coordinate. Counesy Scierrce Nen's. 156, 247 (1999).
7.1.I Pseudo-Steadystate Hypothesis (PSSH)
In the theory of active intermediates, decomposition of the intermediate does not occur instantaneously after internaI activation of the molecvle: rather, there is a time lag, although infinitesimally small, during which the species remains activated. Zewail's work was the first definitive proof of a gas-phase active intermediate that exists for an infinitesimally sban time. Because a reactive intermediate reacts virtualIy as fast as it is formed, the net rate of formation of an active intermediate (e.g.. A*) is zero, i.e., PSSH
r** = 0
(7-1 1
This condition is also referred to as the Pseudo-Steady-State Hypothesis (PSSH). If the active intermediate appears in n reactions, then
To illustrate how rate laws of this type are formed, we shall first consider the gas-phase decomposition of azomethane, AZO, to give ethane and nitrogen:
-
J . Pererson. S C ~ P I INPW. L * ~ 156. 247 (1999).
380
Reaction Mechanisms, Pathways, Bioreadions, and Bioreactors
CI
Experimental observations4 show that the rate of formation of etha first order with respect to A 2 0 at pressures greater than 1 atm (relatively concentrations)
and second order at pressures beIow 50 mmHg (low concentrations):
To explain this first and second order depending on the concentratic A 2 0 we shall propose the following mechanism consisting of three ele tar), reactions.
Mechanism {Reaction 2: [(cH~),N,]" + (CH,),N,
2(CH,),N,
+ (CH,),N
In rencrion I, two AZO molecules collide and the kinetic energy of one . molecule is transferred to internal rotational and vibrational energies o other AZO molecule, and it becomes activated and highly reactive AZO*}). In reaction 2, the activated molecule (AZO*) is deactivated thrl collision with another AZO by transferring its internal energy to increasl kinetic energy of the mdecutes with which AZO* collides. In reacrbn 3, high activated AZO* molecule, which is wildly vibrating, spontanec decomposes into ethane and nitrogen. Because each of the reaction ste elementary. the corresponding rate laws for the active intermediate AZC reactions ( l ) , (21, md (3) are Note: The specific reaction rates, k, are all defined w n the active intermediate AZO*.
(1)
(2)
(3)
These rate laws [Equations (7-3) through (7-5)] are pretty much us' in the design of any reaction system because the concentration of the a intermediate AZO* is not readily measurable. Consequently, we will us< Pseudo-Steady-State-Hypothesis (PSSH)to obtain a rate law in terms of I surable concentrations. We first write the rate of formation of product (with k, = k,,,,)
H.C. Ramsperget, J. Am. Chcrn. Soc., 49, 912 (1927).
Sec. 7.1
Active Intermediates and Nonelernentary Rate Laws
381
To find the concentration of the active intermediate AZO*, we set the net rate of AZOX equal to zcro,' r,,,, = 0.
=k
l ~ i z o- k~CAzo~CAzo-A-,CAzo~ =0
Solving for CAZQ*
Substituting Equarion (7-8) into Equation (7-6)
At low AZO concentrations,
for which case we obtain the following second-order rate law:
Ar high concentrations
in which case the rate expression fotlows first-order kinetics,
In describing reaction orders for this equation, one would say the reaction is apparent firsf order at high azornethane concentrations and apparent second order at low azomethsne concentrations. The PSSH can also explain why one observes so many first-order reactions such as
"or
further elaboration on this secrion. see R.Ark. Am. Sci., 38, 419 ( 19701,
382
Reaction Mechanisms, Pathways, Bioreaclions, and Bioreactors
Chap. 7
SymbolicaIly this reaction will be represented as A going to product P, that is,
A+P with
The reaction is first order but the reaction is not elementary. The reaction proceeds by first forming an active intermediate, A*, from the collision of the reactant molecule and 'an inert molecule of M. Either this wildly oscillating active intermediate is deactivated by collision with inen. M, or it decomposes to form product.
Reaction pathways
The mechanism consists of the three elementary reactions:
Activation
(1)
Deactivation
(2) A' + M
~ecorn~os'ition(3)
A+M
*I
A*+M
'' > A +M A* i 'P
Writing the rate of formation of product
and using the PSSH to find the concentrations of A* in a manner similar to the azomethane decomposition described earlier, the rate law can be shown to be
Because the concentration of the inert M is constant. we let
to obtain the first-order rate law -rA = CCA
Sec. 7.1
First-order rate law for a nonelementary reaction
Aniva lntermediafes and Nonelementarj Rate Laws
Consequently, we see the reaction
A+P follows an elementary rate law but is not an elementary reaction. 7.1.2 Searching for a Mechanism
In many instances the rate data are correlated before a mechanism is found. It is a normal procedure to reduce the additive constant in the denominator to 1. We therefore divide the numerator and denominator of Equation (7-9) by k3 to obtain
General Considerations. The rules of thumb listed in Table 7-1 may be of some help in the devciopment of a mechanism that is consistent with the experin~entalrate law. Upon application of Table 7-1 to the azomethane example just discussed. we see the following from rate equation (7-12): 1. The active intemediate. AZO*. collides with azomethane. AZO [Reaction 21. resulting in the concentration of AZO in the denorninator. 2. AZO* decomposes spontaneously [Reaction 31, resulting in a constant in the denominator of the rate expression. 3. The appearance of AZO in the numerator suggests that the active intermediate AZO* is formed from AZO. Referring zo Reartion 1, we see that this case is indeed true. - - -
.!A
--
I . Species having the concentraricm[s) appearing in the detinminaror of the rate law probably collide with the active intermediate, for example, .4 -t A * + [Collision products]
2. If a constant appears i n the ~lcnornitrator,one of the reaction neps is probably the spontaneous decomposition of the active intermediate. for example. A*
-
[Decompasition products]
3. Species having the concentration(s) appearing in the numerator of the sate law probably produce the active intermediate in one of the reaction steps. for
example, [reactant]
---+
A * t- (Other products]
Finding the Reaction Mechanism. Now that a rate law has been synthesized from the experimentai data, we shall try to propose a mechanism that is consistent with this rate law. The method of artack will be as given in Table 7-2.
384 - --
Once the rate law is found, the search for the mechanism begins.
Reaction Mechanisms, Pathways. Bioreactions, and Bioreactors
Cr
-
I . Assume an active intermediate(5). 2. Postulate a mechanism. utilizing the
rate law obtained from experimei data, if possible. 3. Model each reaction in the mechanism sequence as an elementary reacti 4. After writing rate Iaws for the rate of formation of desired product, w the rate laws for each of the active intermediates. 5 . Use the PSSH. 6. Eliminate the concentration of the intermediate species in the rate laws solving the simultaneous equations developed in Steps 4 and 5 . 7.If the derived rate law does not agree with experimental observat assume a new mechanism andor intermediates and gc~to Step 3 . A szn background in organic and inorganic chemistry is helpful in predicting activated intermediates for the reaction under consideration.
Example 7-1
The Stern-Volmer Equatiu ion
Collapsing cavitation microbubble
-- ---Liquid
Light is given off when a high-intensity ultrasonic wave is applied to water' light results from microsize gas bubbles (0.1 mm) being formed by the ulta wave and then being compressed by it. During the compression stage of the the contents o f the bubble (e.g., water and whatever else is dissolved in the t.g.. CS,, Ot, N,) are compressed adiabatically. This rornpression gives rise to high temperatures and k~neticenergies r gas molecules, which through molecular collisions generate active intermediate cause chemical reactions to occur in the bubble.
The intensity of the light given off. I, is proportional to the rate of deactivati an activated water moleculr that has been formed in the rnicmbubble.
Light intensity (I) x ( - r ti20,) = kc,,1o. An order-of-magnitude increase in the intensity of sonofuminescer observed when either carbon disulfide or carbon tetrachloride is added to the The intensity of Iuminescence, I, for the reaction
A similar result exists for CC1,.
P. K. Chendke and H. S. Fogler, J. Phys.
Chem., 87, 1362 (1983).
Sec 7.t
385
Rctrve Int~rmediatesand Nonelementary Rate Laws
However, when nn alrphatic alcohol, X. is added to the solution, the intensity decreases with increa\ing concentration of alcohol. The data are usually reported in tern? nf a Stern-Volrner plot in which relat~vcilltensity is given as a function of alcohol concentration, C., (See Figure E7-1.1, where 1,) is the sanoluminescence ~ntensity in the absence of alcohol and I is the sonoluminescence intensity in the presence of alcohol.) Suggest a mechanism consistent with erperimental observation.
Stem-Volmer plot
cx(krnol/m3) Figure E7-1.1 Ratio of luminescence intensities as a function of Scavenger concentration.
From the linear plot we know that
where C, = (X).Inverting yieIds
From rule 1 of Tabte 7-1, the denominator suggests that alcohol the active intermediate:
X Reaction Pathways
(X)collides with
+ Intermediate + Deactivation products
(E7-I .3)
The alcohol acts as what is called a scavenger to deactivate the active intermediate. The fact that the addition of CCI, or CS? increases the intensity of the luminescence.
lead? us to postulate (rule 3 of Table 7 - I ) that the active intermediate was probably formed from CS,: M + CS2
where
M is a third body {CS?, H 2 0 , N:,
CS; etc.).
+M
386
Reabton Mechanisms, Pathways, Bioreactiom, and Biomactors
Chap. 7
We also know that deactivation can occur by the reverse of Reaction (E7-I - 5 ) . Combining this information, we have as our mechanism:
The mechanism
Activation:
M
+ CS2
CS; + M
(E7-1.5)
Deactivation:
M
+ CS;
CS2
+M
(E7-1.6)
Deactivation:
X
+ CS;
CSZ + X
(E7-1.3)
CS, + hv
(E7-1.7)
CS;
Luminescence:
I = lE,(CS;)
(E7-2.8)
Using the PSSH on CS; yields r,,;
=0=
I C S : ) ( M f - L:(CS;)(M) - k,(X\(CS:) - k4(CS;)
Solving for CS; and substituting into Equation (E7-1.8) gives us I=
k4 k , (CS,)(W k, (M)+ k, (X) + k,
In the absence of alcohol,
For constant concentrations of CS, and the third M y , M, we take a ratio of Equation (E7-l . 10) to (E7- 1.9):
5I = 1 + k2(")+3 '
k4
(X)= I + X1(X)
which i s of the same form as that suggested by Figure E7-1 . l . Equation (E7-1.11) and similar equations involving scavengers are called Sfern-Volmcr equations. A discussion of luminescence is continued on the CD-ROM Web Module. Glow Sticks. Here, the PSSH ts applied to glow sticks. First, a mechani~mfor the reacttons and luminescence i s developed. Next, mole balance equations are written on each species and coupled with rate law obtained using the PSSH and the resulring equations are solved and compared with experimental data. Glow sticks
3.1.3 Chain Reactions Now, let us proceed to some slightly more complex examples involving chain reactions. A chain reaction consists of the following sequence:
Step? i n a chain reaction
1 . Iniriarion: formation of an active intermediate. 2. PE-opugurionor chai~ltrnng~r:interaction of an active intermediate with the reactant or product to produce another active intermediate. 3. Tcrrinnfinn: deactivation of the active intermediate to form products.
Sec. 7.1
387
Active Intermediates and Nonelementary Rate Laws
/
Example 7-2 PSSH Applied to Thermal Cracking of Ethane
I
The thermal decomposition of ethane to ethylene, methane, butane, and hydrogen is believed to proceed in the following sequence: Initiation:
CzHd % 2CH,*
(1)
(4)
H*+C2H6- 4
C2Hs*+H2
=
r ~ ~ 2 % - k ~ ~ iCzH63 2 ~ 6
r4c,1~,=
-k4 [H I[C?
H61
Termination:
(a) Use [he PSSH to derive a rate law for the rate of formation of ethylene. (b) Compare the PSSH solution in Part fa) to that obtained by solving the complete ser of ODE mole balances.
Solution
Part (a) Developing thc Rate Inw The rate of formation of ethylene (Reaction 3) is
Given the f~llowingreaction sequence: For the active intermediates: CH,* , Cz1-1,. ,H the net rates of reaction are
From reaction stoichiornetry we have
388
Reaction Mechanisms. Pathways, Bioreactions, and Bioreactors
Ch,
Substituting the concentrations into rhe elementary Equation (E7-2.4) gives 2 k I [ C I H h-] kZICH1 ][C2Hh]= 0
(El
Solving for the concentration uf the Free radical [CH,- ] ,
Adding Equations (E7-2.2) and tE7-2.3 t yields -
l
.
1 A I~ - 5 ~~ : " 5 *~=
~0
Substituting for concentrationr in the rate laws kI[CH3a J[C2H6]-k5[C2Hr*12 = 0 PSSH solution
Solving for [ C, H,
, ,r
] gives us
H~ ] =
{x.
[CH,
][C, H6I}[I2 =
I;,
{%[C2H61}
CIZ
' 8 5
(E7
Substituting for C2H,min Equation (E7-2.1) yields the rate of formation of ethj
Next we write the net rate of
H formation in Equation (E7-2.3) in terms of
centration k 3 [ C : H 5 * ] - k 4 [H*lICtHbl = Q
Using Equat~on(E7-2.8) to substitute for (C2H, m ) gives the concentration o hydrogen radical
The rate of disappearance OF ethane is
=-k,[C,H,]-k2[CH,m][ClH6]-k4[H*1[C2N61 (E7-
r=2~6
Substituting for the concentration of free mdicats. the nte law of disappearan1 ethane is
Sec. 7.1
Ac!ive Intermediates and Nonelementary Rate Laws
389
For a constant-volume b ~ t c hreactor, the combined mtlle balilnces and rate laws for disappearance of ethane ( P I ) and the tormatlo~rnf ethylene (PSI are
Combined mole balance and rate law using rhe PSSH
The P in PI (i.e., Cpl1 and P5 (i.e.. C,,) is to remind us rhat we have used the PSSH in arriving ar these balances. At 1000 K the specific reaction rates are C I = 1.5 X 10-I s-', k2 = 2.3 X 10h drn'lmol +s, k, = 5.71 X 1I)' s-[. k, = 4.53 X I V dm31rnol.s. and k, = 3.98 x IOV dtnVmol .s. For an entering ethane concentration of 0.1 molldm' and a temperature of 1 0 0 K, Equations (E7-2.13) and (E7-1.14) were solved and the concentrations of ethane. Cp,, and ethylene, C,,. are
/
Part (bl Testing the PSSH for Ukame Cracking The thermal cracking of ethane i s believed to ofcur by the reaction sequence given in Part (a). The speo~fcreaction rates are given as a function of temperature:
Part (b): Carry out mole baiance on every species, solve. and then plot the concentrations o f ethane and ethylene as a function o f time and compare with the PSSH concentration-time measurements. The initial concentration of ethane i s 0.1 molldm-' and the temperature i s 1 0 K.
Let I = C2H6, 2 = CHJm,3 = CH4. 4 = C2H59, 5 = C2H4,6 = H-, 7 = H2, and 8 = C4Hlo. The combined mole balances and rate laws become
(C2H6):5 =-klC~-k2ClCI-k4ClCn dr
(E7-2.15)
390
Reaction Mechanisms, Pathways, Bloreactions, and Biowacrors
%
(CH.):
(CzH,m):
(C2HA:
(H*):
= kzClC2
Chap. 7
(E7-2.17)
5 = k 2 ~ , ~ 2 - k 3 ~ , i & ~ l ~ 6 - k 5(E7-2.18) ~: dr 5 = k3C4 dr
(E7-2.19)
dt
(E7-2.20)
dC6 =k3C4-k4C,C,
(c4H , ~ ) : 5 dr
=!
(E7-2.22)
2
The Polymath program is given in Table E7-2.1. TABLEE7-2.1.
POLYMATHPROGRAM
Figure E7-2.1 shows the cnncentration time trajectory for CH, (i.e.. Cz3. One notes a flat plateau where the PSSH is valid. Figure E7-2.2 shows a comparison of the concentration-time trajectory for ethane calculated from the PSSH (Cp,)with the ethane trajectoy (C,) calculated from solving the mole balance Equations (E7-2.131 through (E7-2.22). Figure E7-2.3 shows a similar comparison for ethylene C (), and (C5).One notes that the curves are identical, indicating the va3idity of the PSSH under these conditions. Flgure E7-2.4 shows a comparison of rhe concentration-time trajector~esfor methane (C,) and butane (C,). Problem P7-2(a) explores the temperature for which the PSSH is valid for the cracking of ethane.
See. 7.1
Active Intermediates and Nonelememary Rate Laws
Note: Curves for C,and Cp,are
vinually identical.
0.020
Key:
C,
,
CP D
1(d
12
Flgure E7-2.1 Concentntion o f active intermediate CH,*as n function of lime.
m
0.000
3.m
6 CQU
9.m 12.W 15.0K1
FEgure E7-2.2 Cornpanson of concentration-time trajectorieq for ethane.
Note: Curves for
C5 and Cps
are
virtually idenlical.
(3
Key:
0000
30M1
9.000 1 Z W 1 5 . W
6[XX)
Figure E7-2.3 Comparison o f concentration-time trajectop for ethylene
0.000 3 . m 6 . m 8.000 12.000 15.000 Figure E712.4 Comparison of concentration-time frajecioriesfor methane tC3) and butane (C,),
7.1.4 Reaction Pathways
Reaction pathways help see the connection of all interacting species for multiple reactions. We hare already seen two relatively simple reaction pathways, one to explain the first-order rate law, -rA = kcA.(M + A + A * + M) and one for the ~onolurninescenceof CS, in Example 7-1. We now will develop reaction pathways for ethane cracking and for smog generation.
cd>qI,
Figure 7-2
Pa1hu.a~of erhane cracking.
392
Reactron Mechanisms, Pathways, Dioreaclions, and Bioreaclors
Chi
Ethnne Cracking. With the increase in cotnputing power. more and rr analyses involving free-radical reactions as intermediates are carried out u: the coupled set? of differential equations (cf. Example 7-2). The key in such analyses is to identify which intermediate reactions are important in overall sequence in predicting the end products. Once the key reactions identified. one can sketch the parhways in a manner similar to that shown the ethane cracking in Example 7-2 where Reactions 1 through 5 are show1 Figure 7-2.
-
Smog Formation. In Chapter 1, Problem PI 14. in the CD-ROM Sr Web hIoduIe, we discussed a very simple model for smog removal in the L hasun by a Santa Ana wind. We wiH now look a little deeper into the chemi of smog formation. Nitrogen and oxygen react to form nitric oxide in the inder of au~omobileengines. The NO from automobile exhaust is oxidize1 NO2 in the presence of peroxide radicals.
Nitrogen dioxide is then decomposed photochemicaliy to give nascent 0x9
which reacts to form ozone
The ozone then becomes involved in a whole series of reactions with hy carbons in the atmosphere to form aldehydes, various free radicals, and o intermediates. which react further to produce undesirable products in pollution:
Ozone -t Olefin
+Aldehydes + Free radicals
0, + RCHeCHR
'h RCHO + 0
+ HCO
(
One specific example i s the reaction of ozone with I3-butadiene to f acroIein and formaldehyde, which ace severe eye irritants. Eye tmtants
20, + CH2=CHCH=CH2 3
,," k,
> CH,=CHCHO
+ HCHO
I
By regenerating NOz, more ozone can be formed and the cycle continued. ' regeneration may be accomplished through the reaction of NO with the
Sec. 7.1
Active lntsrmediares and Nooelementary
Rate Laws
393
radicals i n the atrnorphere Reaction ( R 1). For example. thc free radical formed in Reaction (R4)can react with O2 to give the peroxy free radical.
The coupling o f the preceding reactions is shown schematically in Figure 7-3. We see that the cycle has h e n completed and that wf th a relatively small amount of nitrogen oxide, a large amount of pollutants can be produced. OF course. many other reactions are taking place. so do not be misled by the bcevity of the preceding discussion: i t does, however, serve to present, in rough outline, the role of nitrogen oxides in air pollution.
Fi~tlre7-3 Reaction pathways in smog formation.
Metabolic Pathways. Reaction pathways find their greatest use in metabolic pathways where the various steps are ca.talyzed by enzymes. The metaboiism o f alcohol is catalyzed by a different enzyme in each step.
Alcohol Dehydrogenase
NADH
2
Acetaldehyde Dehydrogenase
394
Reaction Mechanisms, Pathways, Bioreactions, and Bioreactws
Chap. 7
This pathway is discussed in Section 7.5 Phmacokinetics. However, we first need to discuss enzymes and enzyme kinetics, which we will do in Section 7.2. Check the CD-ROM and the web for future updates on metabolic reaction.
7.2 Enzymatic Reaction Fundamentals An enzyme is a high-molecular-weight protein or protein-like substance that acts on a substrate (reactant molecuIe) to transform it chemically at a greatly accelerated rate, usually 103 to 10" times faster than the uncatalyzed rate. Without enzymes, essential biojogical reactions would not take place at a rate necessary to sustain life. Enzymes are usually present in small quantities and are not consumed during the course of the reaction nor do they affect the chemical reaction equilibrium. Enzymes provide an alternate pathway for the reaction to occur thereby requiring a lower activation energy, Figure 7-4 shows the reaction coordinate for the uncatalyzed reaction of a reactant molecule called a substrate IS) to form a product (PI S+P
The figure also shows the catalyzed reaction pathway that proceeds through an S). called the enzyme-substrate complex, that is.
ncfh?e intermediare (E
Because enzymatic pathways have lower activation energies, enhancements in reaction rates can be enormous. as in degradation of urea by urease where the degradation rate is on the order of 10" hrgher than without urease.
Energy
Figure 7-4 Reaction cnordinate for enzyme c a ~ a l y s ~ s
An important property of enzymes i? that they are specific; that is. n?!e enzyme can usually catalyze only ntFe type af reaction. For example, a protease hydrolyzes nril~hands between specific amino acids in proteins, an am)'lase works on bond< between glucose molecules in starch, and lipase attacks fats. degrading them to fatty acids and glycerol. Consequently, unwanted products are easily controlled in enzyme-catalyzed reactions. Enzymes are produced only by living organisms. and coinn~ercial enzymes are generall?' produced by bacteria. Enzymes usually work (i.e.. catalyze reactions) under
Sec. 7.2
395
Enzymatic Reaction Fundamentals
mild conditions: pH 4 to 9 and temperatures 75 to 160°F.Most enzymes are named in terms of the reactions they catalyze. It is a customary practice to add the suffix -use to a major part of the name of the substrate on which the enzyme acts. For example, the enzyme that cataIyzes the decomposition of urea is urease and the enzyme that attacks tyrosine is tyrosinase. However, there are exceptions to the naming convention, such as a-amylase. The enzyme a-amylase catalyzes starch in the first step in the production of the soft drink (e.g., Red Pop) sweetener high-fructose corn s y t p (HFCS) from corn starch, which is a $4 billion per year business
Corn starch
+ Thinned starch
u-amylase
a~~e",t~ucose%e
HFCS
7.2.f Enzyme-Substrate Complex
Falded enzyme Wm gcffve site
The key factor that sets enzymatic reactions apart from other catalyzed reactions is the formation of an enzyme-substrate complex, E*S. Here substrate binds with a specific acrive sire of the enzyme la form this complexh7Fjgure 7-5 shows the schematic of the enzyme c h y m o ~ p s i n(MW = 25.000 Daltons), which catalyzes the hydrolytic cleavage of polypeptide bonds. In many cases the enzyme's active catalytic sites are found where the various folds or Imps interact. For chyrnotrypsin the catalytic sites are noted by the amino acid numbers 57. 102, and 195 in Figure 7-5. Much of the catalytic power is attributed
Flgure 7-5 Enzyme chyrno~ryp
' M. L. Shuler and F. Kargi, Riopmcess E n ~ i n e r r i nBasic ~ Cnrlceprs. 2nd td. (Upper Saddle River. N.S.: Prentice HaH, 20021.
396
vmax
Reaction Mecha+sms, Pathways. Rtoreartrons, and Ror~rlc!c.s
Ch
to the binding energy of the auhstr~relo the enzyme through mulliple hc with the specific fr~nctionaIgroups on the enzyme (amino aide chains. rr i o n ~ jThe . interactions that stabilize the enzyme-substrate colnplex are hy gen bonding and hydrophobic, ionic. and London van der Wallis forces. 3 f enzyme i s exposed to uxlrcrnc temperature5 or pH environments (i.e., I high and low pH values). i t may i~ntbldlo5ing its active sites. When occurs. the enzyme is said to be d~nar~rt?lf. See Problem P7-15R. There are two models for suhstrate-enzyme interactions: the lock and n~ndeland the itlrf~rcedjhn~mrlt.l, both of which arc shown in Figure 7-6. many years the lock and key model was preferred because of the sterosper effects o f one enzyme acting on one substrate. However. the induced tit ma is the more useful mdel. In the induced tit model both the enzyme mole1 and the substrate molecules i11.t d~atorted.These change.; i n conformation tort one or more of the substrate bonds, thereby stressing and weakening bond to make the n~olzculemore susceptible to rearrangement or artachme
(a) Lock-and-key model
(b) Induced BI W e l
Sec. 7.2
Enzyrnat~cReact~onFundamentals
There are six cIasses of enzymes and only six: 1 . Oxidoreductases 2. Transferases 3. Hydrolase? 4. [somerases 5. Lyases 6. Ligases
More information about enzymes ran be found on the following rwa web sites: Llnks
http://~s.e x p r m org/etr:yt~te/and nvwlc.clretn.qmw.nc. uk/ilrhrnb/en:vrlle These sites also give information abour enzymaric reactions in general. 7.2.2 Mechanisms
In developing some of the elementary principles of the kinetics of enzyme reactions, we shal! discuss an enzymatic reaction that has been suggested by Levine and Lacourse as part of a system that wouId reduce the size of an artificial kidney.x The desired result is the production of an anificial kidney that could be warn by the patient and would incorporate a replaceable unit for the elimination of the nitrogenous waste products such as uric acid and creatinine. In the microencapsulation scheme proposed by Levine and LaCourse, the enzyme urease would be used in the removal of urea from the bloodstream. Here, the cataIytic action of urease would cause urea to decompose into ammonia and carbon dioxide. The mechanism of the reaction is believed to proceed by the following sequence of elementary reactions: 1. The enzyme urease (E) reacts with the substrate urea (S) to form an enzyme-substrate complex (E S). The reaction meuhan ism
NH,CONH,
+ Urease
">
/NH2CONH2 UreaseJ*
(7- 13)
2. This complex {E S ) can decompose back to urea (S) and urease (El: [NHICONH2 Ureasel*
'k
> Urease
+ NH2CONH2
(7- 14)
3. Or it can react with water 1W) to give the products (P)ammonia and carbon dioxide, and recover the enzyme urease (€1. [NH,CONH,
Ureasel*
+
H20
"
)
2NK3 + COI + Urease (7- 15)
We see that some of the enzyme added to the solution binds to the urea, and some remains unbound. Although we can easily measure the total concentration of enzyme, (E,), it is difficult to measure the concentration of free enzyme, (E).
N. Levine and W. C.LaCourse. J. Biomed. Muter: Res.. !, 275 (1967).
398
Reaction Mechan~sms,Pathways, Btoreacttons, end Bloreactors
Chap. 7
Letting E, S, W, E+S,and P represent the enzyme, substrate, water, the enzym~substratecomplex, and the reaction products, respectively, we can write Reactions (7-13), (7-14), and (7-15) symbolically in the forms
E-S+W
Here P = 2NI-I,
k3
)
P-kE
(7-18)
+ CO?.
The corresponding rate laws for Reaction 17-16), (7-17). and (7-18) are
The net rate of disappearance of the substrate, -rs, is This rate law is of not much use to us in making reaction engineering calculations because we cannot measure the concentration of enzyme submare complex (E S). We will use the PSSH to express (E = S ) in terms of measured variables. The net rate of formation of the enzyme-substrate complex is
Using the. PSSH, r,., = 0,we solve equation (7-20) for (E S)
and substitute for (E S } into {Equation (7-I9)J
We need to replace
unbound concentration (El In theratelaw.
We still cannot use thls rate law because we cannot measure the unbound enzyme concentration {E); however, we can measure the total enzyme concentra~ion,E,.
Sec. 7.2
Enqmatic Reaction Fundamentals
399
In the absenm of enzyme denaturization, the total concentration of the enzyme in the system, (E,), is constant and equal to the sum of the concentrations of the free or unbonded enzyme, (E), and the enzyme-substrate complex, (E * S):
($1 oral enzyme concentration = Bound + Free enzyme concentration.
= (E) + (EmS)
17-23]
Substituting for (E * S)
solving for (E)
substituting for (E) in Equation (7-22), the rate law for substrate consumption is m e final form of the rale law
Note: Throughout, Et z IEt) = total concentration of enzyme with typical units (kmol/rnbr g/drn3).
7.2.3 Michaelis-Menten Equation Because the reaction of urea and urease is carried out in aqueous solution. water is, of course, in excess, and the concentration of water i s therefore considered constant. Let + k: kc,, = k3(W) and KM = kcat k,
Dividing the numerator and denominator of Equation (7-24) hy k , , we obtain a form of the Michaelis-Menfern equation:
The parameter kc, is also referred to as the tuntol~rnumber, It is the number of substrate molecules converted to product in a given time on a
single-enzyme rnoEecuIe when the enzyme is saturated with substrate (i.e.. all the active sites on the enzyme are occupied, S>>KM).For example. turnover Turnover number for the decomposition H 2 0 2 by the enzyme catalase is 40 x 1 Oh s-'. That IS. 40 million molecules of H,02 are decomposed every second on a ,,,k,r.,,, single-enzyme molecule saturated with Hz02. The constant K M (mol/dm') is called the Michaelis constant and for simple systems is a measure of the
400 M~chaelis conrtant Kq
Reaction Mechan~sms,Pathways. Bioreactions, and Biorsactors
Cha
attraction of the enzyme for its substrate. so it's also called the nfinio c stant. The Michaelir constant. K,, for the decomposition of H20z discus earlier is I . I M while that for chymotrypsin is 0.1 M.9 If, in addition, we let V,,, represent the maximum rate of reaction f( given total enzyme concentration, Ymex
= kcat(Et)
the Michaelis-Menten equazian takes the familiar form Michaelis-Menten equation
For a given enzyme concentration, a sketch of the rate of disappearance of
substrate i s shown as a function of the substrate concentration in Figure 7-
Figure 7-6 Michaelis-Menten plot identifying the parameters V,,,
and
K,.
A piot of this type is sometimes called a Michaelis-Menten plot. At low s strate concentration, KM (S),
and the reaction is apparent firs.[ order in the substrate concentration. At h substrate concentrations, ( S ) 9 K,,
and she reaction is apparent zero order
-rs = Y,,
13. L. Nelson and M. M. Cox, Lehninger Principles of Biochemistry, 3d ed. (E York: Worth Publishers, 2000).
40t
Enzymahc Reaclron Fundamentals
Sec. 7.2
Consider the case when the substrate concentration is such that the reaction rate is equal to one-half the maximum rate,
then
Solving Equation (7-27) for the Michaelis constant yields Interpretation of
Michael~sconftant
The Michaelis constant is equal to the substrate concentration at wwhich the rate of reaction is equal to one-half the maximum rate. The parameters V,, and KM characterize the enzymatic reactions that are described by Michaelis-Menten kinetics. V,,, is dependent on total enzyme concentration, whereas KM is not. Two enzymes may have the same values for kc, but have different reaction rates because of different values of K,. One way to compare the catalytic efficiencies of different enzymes is to compare the ratio kcJKM. When this ratio approaches 108to lo9 (dm3/mol/s) the reaction rate approaches becoming diffusion-limited. That is. it takes a long time for the enzyme and substrate to find each other, but once tbey do tbey react immediately. We will discuss diffusion-limited reactions in Chapters 11 and 12. ExarnpIs 7-3 Evaluation oJMichaelis-Menten Parameters V,,, and K,, Determine the Michaelis-Menten parameters V,, Urea + Urease
k,
[Urea. Urease]'
tl
and KM for the reaction
& -H*O
2NH,
+ CO, + Urease
The rate of reaction is given as a function of urea concentration in this table. Cu,,(kmollrns)
10.2
Sohtion
Inverting Equation (7-26) gives us
0.02
0.01
0005
0.002
402
Reactton Mechanisms, Pathways, Bmreactlons, and Bioreacfors
Chap 7
A plot of the reciprocal reaction rate versus the reciprocal urea concentration should be a straight line with an intercept llV,,, and slope K M / V - . This type of plot is called a Lineweaver-Burk plor. The data in Table E7-3.1 are presented in Figure E7-3.1 in the form of a Lineweaver-Burk plot. The intercept is 0.75, so
TABLEE?-3.1.
b~AND PROCESSEDDATA
Lineweaver-Burk plot
l'uw
e
Figure E7-3.1 (a)MicheeIis-Wenten plot; (b) Lineweaver-Burk plot.
Therefore, the maximum rate of reaction is
Vma,= 133 krnolirn"~ = 1.33 molldm3-s From the slope. which i s 0.02 s, we can calculate the Michaelis constant, Khl: For enzymatic reactions, the two
key rate-law paramelers are V,,, and K,.
Sec. 7.2 1
Enzymatic Rsactbn Fundamentals
403
Substituting K M and V,,,',,,into Equation (7-26)gives us
-
where C,,, has units of kmol/m3 and -r, has units of kmol/mLs. Levine and Lacourse suggest that the total concentration of urease, (E,),corresponding to the value of V,, above i s approximately 5 g/dm3, In addition ro the Llneweaver-Burk plot, one can also use a Hanes-Wookf pIot or an Eadie-I-fofstee plot. Here S r Cum,and -r, s -r,,,. Equation (7-26)
can be rearranged i n the following forms. For the Eadie-Mofstee f m , Zadie-Hofstee plot
For the Hanes-Woolf form, we have
pz&J I
-
- r ~
'max
'man
I
For the Eadie-Hofstee model we plot -rs as a function of (-r,/S) and for the Hanes-Woolf model. we plot [(S)/-r,l as a function of (dl. The Eadie-Hofqtee plot does no! bias the points at low substrated concentrfltions, while the Hanes-Woolf plot gives a more accurate evatuation of V,,,. In Table E7-3.2, we add two columns to Table E7-3.1 to generate these plots (C,,, r S).
PIotting the data in Table E7-3.2, we arrive at Figures E7-3.2 and E7-3.3.
S
Figure E7-3.2 Hane+Waolf plc~t.
Figure E7-3.3 Eadie-Hofstee plot.
404
I
Reaction Mechanisms, Pathways, Bioteactions, and B~oreactors
Cha
Regression Equation (7-26) was used in the regression program of Polymath with the follow results for V,, and Ku.
I
Madel: rate = Vrnax*Cursal(KmCurea)
Vmax Km
u.busEs 1
rmess 1.2057502
0.02
0.0233322
Nonlinear regression settings Max # iterations = 64 Precision R"2 RA2adj Rrrisd
V, K,t
-
95% 0.0598303 0.003295
= 1.2 mol/dm7 s = 0.0233 moVdm3
= 0.9990611 = 0.9987481
=
0.0047604
variance = 1.888E-04
The Product-Enzyme Complex In many reactions the enzyme and product complex (E P) is formed direc from the enzyme substrate complex (E S) according te the sequence
Applying the PSSH to both (E S) and
(E P), we obtain
Briggs-Haldane Rate Law
-rs =
vm, (C, - C,/Kc3
(7-2
Cs+K,, + K P ~ P
which is often referred to as the Bdggs-Haldane Equation (see Problem P7-1 and the application of the PSSH to enzyme kinetics often called t Bsiggs-Haldane approximatian. 7.2.4 Batch Reactor Cakulations for Enzyme Reactions
A male balance on urea in the batch reactor gives Mole balance
Because this reaction is liquid phase, the mole baIance can be put in the fc lowing form:
Sec. 7.2
405
Enzymatic Reaction Fundamentals
The rate law for urea decomposition is
-r u m -
Rate law
'mil3Cure3
KU +
cures
(7-3 1 )
Substituting Equation (7-3 1 ) into Equation (7-30) and then rearranging and integrating, we get
Integrate
/ = - I nKY Vmax
- Cure2
Cureno + Curco~
Cure,
ym,,
We can write Equation (7-3 1) in terms of conversion as Time to achieve a conversion X i n a batch enzymatic reaction
The parameters K, and V,, can readily be determined from batch reactor data by using the integral method of analysis. Dividing both sides of Equation (7-32) by fKM/Vmaxand rearranzing yields
We see that K, and V,,, can be determined from the slope and intercept of a pIot of l l t In[l/(l - X)] versus Xlt. We could also express the Michaelis-Menten equation in terms o f the substrate concentration S:
where So is the initial concentration of substrate. In cases similar to Equation (7-33) where there is no possibility of confusion, we shall not bother to encEose the substrate or other species in parentheses to represent concentration [i.e., Cs = (S) E S]. The carresponding plot in terms of substrate concentration is shown in Figure 7-8.
Reaction Mechanisms, Pathways, Biomactions, and B~oreactors
Figure 7-7 Evaluating V,
Chap. 7
and K,.
&le 7-4 Batch Enwmtic Reactors
Calculate the time ~Bededto convert 99% of the urea to ammonia and carbon dioxide in a 0.5-dm' batch rcactor. The initial concentration of urea is 0.1 mol/dm3, and the urease concentration is 0.001 g/drn3. The reaction is to be carried wt isotheimally at the same temperamre at which the data in Table E7-3.2 were obtained. Solution
We can use Equation (7-32),
KM = 0.0266 rnolldm~,X = 0.99, and CuWm = 0.1 molldm3,V,,, was 1.33 molldrn3.s. However. for the conditions in the batch reactor, the enzyme concentration is only 0.001 g/dm7compared with 5 g in Example 7-3. Because V, = E;b, V,, for the second enzyme concentration is where
I
K M = 0.0266 m o l l d m ~ a n d X = 0.99 Substituting into Equation (7-32)
! 1
=460s+380s
= 840 s (14 minutes)
Sec. 7.2
Enzymatic React~onFundamentals
407
Effect of Temperature
vmx
T
The effect of temperature on enzymatic reactions is very camplex. ~f the enzyme structure would remain unchanged as the temperature: i s increased, the rate would probably follow the Arrhenius temperature dependence. However. as the temperature increases, the enzyme can unfoId and/or become denatured and lose its catalytic activiry. Consequently, as the temperature increases, the reaction rate, -rs, increases up to a maximum with increasing temperature and then decreases as the temperature is increased further. The descending part of this curve is called temperature inactivation or thermal denaturi~ing.~~ Figure 7-9 shows an example of this optimum in enzyme activity."
Figure 7-8 Catalytic breakdown rate of HID2 depending on temperature. Counecy of 5. Aha. A. E.Humphrey. and N. F. Mills. B t n r h ~ m i c E ~ ln ~ i n e t r r t r Academic ~. Press 11473).
'OM.L.Shuler and F. Kargi, B i o p r o c ~ Engittecring .~~
Basic Conc~prs,2nd ed. (Upper Saddle River. N.J : Prentrce Hall. 20011). p, 77. " S . Aiba, A. E. Humphrey, and N. F. Mills, Biochemical engineer in^ (Nm York: Academic Press, 1473). p. 47.
408
React~onMechanisms, Pathways, Bloreact~ons,and 81oreactors
CI
Side note: Lab-on-a-chip. Enzyme-catalyzed polymerization of nucleoti is a key step in DNA identification. The microfluidic device shown in I ure SN7.I is used to identify DNA strands. It was developed by Frofe! Mark Bums's group at the University of Michigan.
SAMRE LO*L*IVO
Figure SW7.1
GEL
z:g?&
M & g w MlXlM
GEL
L ~ OELECTROP~RESIS M
I
Micmfluidi device to identify DNA. Courtesy of Science. 282. 484 (1998).
In order to identify the DNA, its concentration must be raised to a level t can be easily quantified. T h i s increase is typically accompIished by replic ing the DNA in the following manner. After a biological sample (e.g., pl; fied saliva, blood) is injected into the micro device, it is heated and 1 hydrogen bonds connecting the DNA strands are broken. After breaking
primer attaches to the DNA to form a DNA primer complex. DNA*. , enzyme @ then attaches.to this pair forming the DNA* enzyme complc DNA* E. Once this complex is formed a polymerization reacrion oca as nucleotides (dNTPs-dATP, dGTP, d m , and d m - N ) attach to t primer one molecule at a time as shown in Figure SN7.3,. The enzyme int~ acts with the DNA strand to add the proper nucleotide in the proper ord The addition continues as the enzyme moves down the strand attaching x nucleotides until the other end of the DNA strand is reached. At this poi the enzyme drops off the strand and a dupIicate, double-stranded DNA mt ecule is formed. The reaction sequence is DNA
1
k
-
I
-
.
&-
_
*
*-.-A
.-
- .- -
+
--
DNA'
-*aHeaf
*
-
[-I flpf;
+ Pdmm.r
DPlAStrend
.@
-
DNA' Enzyme
Complex
DNA'
DNA Strand Prlmer Complex & r
Sm. 7.3
Inhibition of Enzyme Reactions
409
The schematic in Figure SN7.2 can be written in terms of single-step reactions where N is one of the four nucleotides. Complex Formation:
DNA + Primer +DNA* Nucleotide additiodporyrnerization
The process then continues much like a zipper as the enzyme moves Bong the sttand to add more nucleatides to extend the primer. The addition of the last nucleotide is
where i is the number of nucleotide moIecuIes on the originaI DNA minus the nucleotides in the primer. Once a complete double-stranded DNA is formed, the poZyrnerizatian stops, the enzyme drops off, and separation occurs.
Here ?DNA strands really r'epresents one double-stranded DNA helix. Once replicated in the device, the length of the DNA molecules can be analyzed by electrophoresis to indicate relevant genetic information.
7.3 Inhibition of Enzyme Reactions In addition to temperature and solution pH, another factor that greatly influences the rates of enzyme-catalyzed reactions is the presence of an inhibitor. inhibitors are species that interact with enzymes and render the enzyme inef-
fective to cataiyte its specific reaction. The most dramatic consequences of enzyme inhibition are found in living organisms where the inhibition of any particular enzyme involved in a priman, rnerabolic pathway will render the entire pathway inoperative, resulting in either serious damage or death of the organism. For example, the inhibition of a single enzyme. cyrochrorne oxidnse, by cyanide will cause the aerobic oxidation process to stop; death occurs in a very few minutes. There are aIso beneficial inhibitors such as the ones used in the treatment of leukemia and other neoplastic diseases. Aspirin inhibits the enzyme that catalyzes the synthesis of prostaglandin involved in the pain-producing process. The three most common types of reversible inhibition occurring in enzymatic reactions are competitive, uncompefitive, and noncomperfrive. The enzyme moIecule is analopous to a heterogeneous cataIyzic surface in that it contains active sites. When competitive inhibition eccurs. the substrate and
410
Reacfion Mechanisms, Pathways, BIoreaCtlons, end Bioreactors
Chap. 7
inhibitor are usually similar molecules that compete for the same sire a n the enzyme. Urzcomperirke inhibition occurs when the inhibitor deactivates the enzyme-substrate complex, sometimes by attaching itself to both the substrate and enzyme molecules of the complex. Noncompe~itiveinhibition occurs with enzymes containing at least two different types of sites. The substrate attaches only to one type of die, and !he inhibitor attaches only to the other to render the enzyme inactive.
7.3.t Competitive Inhibition Competitive inhibition is of particular importance in pharmacokinetics (drug therapy). If a patient were adminisrered two or more drugs that react simuitaneously within the body with a common enzyme, cofactor, or active species, this interaction could lead to competitive inhibition in the formation of the respective metabolites and produce serious consequences. In competitive inhibition another substance, I, competes with the substrate for the enzyme molecules to form an inhibitor-enzyme complex, as shown here. Reaction Steps
Competitive inhibition pathway E+S+E4S
+ I
u
KI
t.1
4E + P
Competitive Inhibition Pathway Active
(1) (2)
E+S E.S
(3) (4)
EaS I +E
(5)
Em1
"
"
>E*S > EE+S
,P + E t'
"
EE.I (inactive)
r E+I
5 G-O-.
0:30 Inactive
(a) Competitive inhibition. Courtesy of D.L. Xelson and M. M. Cox, Lehn~nger Prinriples of Bioch~rnistty~ 3rd ed. (New York: Wonh Publishers, 20(M),p. 265.
In addition to the three Michaelis-Menten reaction steps, there are two additional steps as the inhibitor reversely ties up the enzyme as shown in reaction steps 4 and 5. T h e rate law for the fodiation of product is the same [cf. Equation (7-18A)I as it was before in the absence of inhibitor
S6c.7.3
Inhibition of Enzyme Reactions
41 1
Applying the PSSH, the net rate of reaction of the enzymesubstrare complex is
The net rate of reaction of inhibitor-substrate complex is also zero The total enzyme concentration is the sum of the bound and unbound enzyme concentrations
Combining Equations (7-33, (7-36), and (7-37) and soIvjng for (E and substituting in Equation (7-34) and simplifying
S)
Rate law for competitive inhitrit~on
V,,, and K,, are the same as before when no inhibitor is present, that is, Vmax= k3E, and
-
KM = kz i,+ k3 k,
and the inhibition constant K, (rnol/dm3) js
By letting K; = 4 ( 1 + I/K,II we crm we that the effect of a competitive inhibition is to increase the "apparent" Michaelis consmt. KM.A consequence of the larger "apparent" Mjchaelis constant K M is that a Iarger substrate concentration is needed for the rate of substrate decomposition, -rs, to reach half its maximum rate. Rearranging in order to generate a Lineweaver-Burk plot, I
(7-39)
From the Lineweaver-Burk plot (Figure 7-10'), we see that as the inhibitor ( I ) concentration is increased the slope increases (i.e,, the rate decreases) while the intercept remains fixed.
412
React~onMechanrsms, Pathways. Bioreacttons, and Bioreactors
CP
Increasing inhibitor Concentration ( I ) I
Figure 7-Ill
Line~,rilver-Burk plot for compztitive inhibition.
Side note: Methanol Poisoning. An interesting and important example competitive substrate inhibifion is the enzyme alcohol dehydrogenase (AC in the presence of ethanol and methanol. If a person ingests methanol, A1 will convert it to formaldehyde and then formaze, which causes bIindnc Consequently, the treament involves intravenously injecting ethanol (wh is metabolized at a slower rate than methanol) at a controlled rate to tie ADH to slow the metabolism of melhmol-to-famaldehyde-to-formatt sot the kidneys have time ro filter out the methanol which is then excreted in urine. With this treatment, blindness is avoided. For more on the met. nollethmol competitive inhibition, see Problem W-25,. 7.3.2 Wncompetitive In hibition
Here the inhibitor has no affinity for the enzyme itself and thus does not ( pete with the substrate for the enzyme; instead it ties up the enzyme-subs complex by forming an inhibitor-enzyme-substrare complex. (I E S) w is inactive. In uocompetitive inhibition, the inhibitor reversibly ties enzyme-substrate complex oftfter it has been formed. As with competitive inhibition. two additional reaction steps are addt the MichaeIis-Menten kinetics for uncomvtitive inhibition as shown in r tion steps 4 and 5 .
Sec. 7.3
413
Inhibition ol Enzyme Reactions
Reaction Steps
Uncornpetilive Prithway
Uncompetitive lnh~bitionpathway E+S-'E.S-E+P
E.S.1
(1)
E + S L$ Ems
12)
E*S
(3)
E-S
(4)
I+E*S I-E*S
(5)
Active
"'aE+S ''>P+E "
ila
+ 1eE.S (inactive)
k,
1 0 ~ ~ s Inactive
Rate law for Starting with equation for rate of formation of product, Equation (7-34). uncomperitive and then applying the pseudo-steadystate hypothesis to the intermediate ~nhibirion
(I E S), we arrive at the rate law for uncompetitive inhibition
Rearranging
I
I
The Lineweaver-Buck plot is shown in Figure 7-1 1 for different inhibitor concentrations. The slope (KMIV,,,) remains the same as the inhibition ( I ) concentration is increased, while the intercept (1 + (flIKI) increases.
Increasing Inhibitor Concentration (I)
L
-1 S
Figure 7-11 Lineweaver-Burk plot for uncornpetitive inhibition.
474
Reaction Mechanisms, Pathways, Bioreactions, and Biomctors
Chap. 7
7.3.3 Noncompetitive Inhibition (Mixed Inhibition)? In noncompetitive inhibition, also called mixed inhibition, the substrate and inhibitor molecules react with different types of sites on the enzyme molecule. Whenever the inhibitor is attached to the enzyme it is inactive and cannot form products. Consequently, the deactivating complex (I E S) can be formed by two reversible reaction paths. 1. After a substrate molecule anaches to the enzyme molecule at the substrate site. the inhibitor molecu~eattaches to the enzyme at the inhibitor site. 2. After an inhibitor molecule attaches ta the enzyme molecule at the inhibitor site, the substrate molecule attaches to rhe enzyme at the substrate site. These paths, along with the formation of the product, P, are shown here. In noncompetitive inhibition, the enzyme can be tied up in its inactive form either before or nfer forminy the enzyme substrate complex as shown in steps 2, 3, and 4. Reaction Steps
Mixed inhibition
Noncompetitive Pathway Active
E.l+s-E.svI
( 2 ) E + I 2 1 E (inactive) (3) 1 -+ E * s I * E * s (inactive) (4) S -t I E Z 1 E S (inactive)
z
(5)
E-S
-
os(p(-~ A
G
P
I!
Inactive
Summery Notes
Again starting with the rate law for the rate of formation of product and then applying rhe PSSH to the complexes (I E) and (I E S) we arrive at: the rate law for the noncompetitive inhibition
Rale l a w for noncompetitive inhibition
The derivation of the rate law is given in the S u r n m a ~Notea on the web and CD-ROM. Equation (7-42) is in the form of the rate law that is given for nn enzymatic reacflon exhiQiting noncompetitive in hibition. Heavy metal ions such a$ pb2'. ~ g ' . and Hg-+. as well as inhibitors that react with the enzyme to form chemical derivatives. are typical examples of noncompetitive inhihitors. ' In some text<, mixed inhihition inhibi~ion.
1%
a combination nf competltive and uncompe~itiYe
Sec. 7.3
415
Inhibition of Enzyme Readions
Rearranging
Figure 7-12 Lineweavw-Surk plot for noncompetitive enzyme inhibition.
For noncompetitive inhibition, we see in Figure 7-12 that both the slope
(2[1 + 21)
and intercept
(&[I + $1)
increase with increasing
inhibitor concentration. In practice, uncamperirive inhibition and mixed inhibition are observed only for enzymes with two or more substrates, S , and Sz. The 'tbree types of inhibition are compared with a reaction in which no inhibitors are present on the Lineweaver-Burk plot shown in Figure 7-13. Noncompetitive (both slope and intercept change) Uncompetitive (intercept changes) Competitive (slopechanges) Sumrnaiy plot of YPes of ~nhib~tion
No inhibition
Figure 7-13 L~neweawr-Burk plors for three types of enzyme lnhibition
416
Reaction Mechan~sms,Pathways, Bioreactrons, and Biareactors
Ct
In surnnlary we observe the following trends and relationships: 1. In competitive inhibifion the slope increases with increasing inhi concentration, while the intercept remains fixed. 2. In uncorrrpetifiveinhibition, the y-intercept increases with incre: inhibitor concentration while the slope remains fixed. 3. In noncompetirive inhibition (mired inhihition), both the y-inte and slope will increase with increasing inhibitor concentration. Problem W-14 asks you to find the type of inhibition for the enzyme catal reaction of starch. 7.3.4 Substrate Inhibition I n a number of cases, the substrate itself can act as a inhibitor. In the ca uncompetitive inhibition, the inactive molecules (S E S) is formed b
reaction S
+E
S
+S E S
(inactive)
Consequently we see that by replacing (I) by (S) i n Equation (7-40) thl law for -r, is
We see that at low subsaate concentrations
then
and the rate increases linearly with increasing substrate concentration. At high substrate concentrations (5' / KI) >>(KMc S), then
and we see that the rate decreases as the substrate concentration inc Consequently, the rate of reaction gives through a maximum in the su concentration as shown in Figure 7-14. We also see there is an optimul strate concentration at which to operate. This maximum is found by takj derivative of Equation (7-44) wrt S, to obtain
Inhibition of Enzyme Reactions
Sec. 7.3
D
-5-A. Sub.irrate ~nhlbitron
4s
S
S ma
Figure 7-14 Substrate reaction rate as a function of substrate concentration for subqtrdte inhlb~tinn,
When substrate inhibition is possible. a semibatch reactor called a f e d batch is often used as a CSTR to maximize the reaction rate and conversion.
7.3.5 Multiple Enzyme and Substrate Systems In the preceding section, we discussed how the addition af a second substrate, I, to enzyme-catalyzed reactions could deactivate the enzyme and greatly inhibit the reaction. In the present section, we look not only at systems in which the addition of a second substrate is necessary to activate the enzyme, but also at other multiple-enzyme and mulriple-substrate systems in which cyclic regeneration of the activated enzyme occurs. Cell growth on multiple substrates is given in the Sirrn~nopNotes.
Enzyme Regeneration. The first example considered is the oxidation of glucose (S,) with the aid of the enzyme glucose oxidase (represented as either G.O. o r [E,lj) to give 8-gluconolactone (PI: Glucose
+ G.O.
(Glucose . G.O.)
; ((6glucona'lactone - G.O.H,)
In this reaction. the reduced form of glucose oxidase (G.0.H2), which will be represented by E,, cannot catalyze further reactions until it is oxidized back to E,. This oxidation is usually carried out by adding molecular oxygen to the system so that glucose oxidase, E,, is regenerated. Hydrogen peroxide is also produced in this oxidation regeneration step:
G.Q.H2 + O2
G.O.+ HIOz
Overall, the reaction is written Glucose + OZ
plucosc ox,acc
+
> HzOl 8-Gluconolactone
418
Reaction Mechanisms, Patftways, Bioreactions, and Bioreactors
Chap. 7
In biochemistry texts, reactions of this type involving regeneration are
usuaIly written in the form
Derivation of the rate laws for this reaction sequence is given on the CD-ROM.
Enzyme Cofactors. In many enzymatic reactions, and in particular biologicaI reactions, a second substrate line.,species) must he introduced to activate the enzyme. This substrate. which is referred ra as a cofaaor or coenzyme even though it is not an enzyme as such, attaches to the enzyme and is most often either reduced or oxidized during the course of the reaction. The enzyme-cofactor complex is referred to as a holoengfme.The inactive form of the enzyme-cofactor cnmpIex far a specific reaction and reaction direction is called an apoenzyme. An example of the type of system in which a cofactor is used is the formation of ethanol from acetaldehyde in the presence of the enzyme alcohol dehydrogenase (ADH) and the cofactor nicotinamide adenine dinucleotide (NAD): alcohol dehydrogenase acetaldehyde ( S , ) NADH ( S , ) ethanol (P ,) Derivation of the rate Iaws for this reaction sequence is given in PRS 7.4. ~eferenceShelf
The growth o f biotechnolopy S 16 billion
7.4
Biareactors
A bioreacror is a reactor that sustains and supports life for cells and tissue cultures. VirtuaEly all cellular reactions necessary 10maintain life are mediated by enzymes as they catalyze various aspects of cell metabolism such as the transformation nf chemical energy and the construction. breakdown. and digestion of cellular components. Because enzymatic reactions are involved in the growth of microorganisms, we now proceed to study microbial growth and bioreactors. Not surprisingly. the Monod equation. which describes the growth law for a number of bacteria, is similar to the Michaelis-Menten equation. Consequently, even though bioreactors are nol truly homogeneous because of the presence of living cells. we include them in this chapter as a logical progression from enzymatic reactions. The use of living cells to produce marketable chemical products is becoming increasingly important. The number of chemicals, agricultural products and food products produced by biosynthesis has risen dramatically. In 2003. companier in this seclor raised over S16 biIlion of new financing.12 Both
" CC & E hltlrI'r.January
II . 2MU. p. 7 .
microorganisms and mammalian cells are being used to produce a variety of products, such as insulin, most antibiotics, and polymers. It is expected that jn the future a number of organic chemicals currently derived from petroleum will be produced by living cells. The advantages of bioconversions are mild reaction conditions; high yields k g . , 100% conversion of glucose ta pluconic acid with Aspergillus niger); the fact that organisms contain several enzymes that can catalyze successive steps in a reaction and, most important,act as stereospecific catalysts. A common example of specificity in bioconversion production of a single desired isomer that when produced chemically yields a mixture of isomers is the conversion of cis-proenylphosphonic acid to the. antibiotic (-) cis-1,2-epoxypmpyl-phosphonicacid. Bacteria can also be modified and turned into living chemical factories. For example, using recombinant DNA, Biotechnic International engineered a bacteria ro produce fertilizer by turning nitrogen into niuates.I3 In biosynthesi~,the cells, also referred to as the biorna.~,consume nutrients to grow and produce more cells and important products. Internally, a cell uses its nutrients to produce energy and more cells. This transformation of nutrients to energy and bioproducts is accomplished through a cell's use of a number of different enzymes in a series of reactions to produce metabolic products. These products can either remain in the cell (jntracellular) or be secreted from the cells (extracellular). In the fonner case the cells must be lysed (ruptured) and the product purified from the whole broth Ireaction mixture). A schematic of a cell is shown in Figure 7-15.
Cell Membrane Cylo~h9rn Carl Nlrctasr reglon
_.. .. ..
. -
..
Figure 7-15 (a) Schematic of cell (h) Photo of cell ditiding L rolr. Counesy of D. L. Nelson a ~ i dM M. Cox. Lehninger Pn'nclples of B i ~ c h e r n i s i3rd ~ ~ d. (New York: Worth Publishers. 2000)
The cell consists of a ce!l wall and an outer membrane that encloses cyroplasm containing a nuclear region and ribosomes, The cell wall protects the cell from external influences. The cell membrane provides for selective transport of materials into and our of the cell. Other substances can attach to the cell membrane to carry out important cell functions. The cytoplasm contains the ribosomes that contain ribonucleic acid (RNA). which are irnportanr in the synthesis af proteins. The nuclear region contains deoxyri bonucleic acid
420
Reaction Mechanisms, Pathways. Bioreactions, and BioreacEors
CI
(DNA) which provides the genetic information for the production of prc and other cellular substances and structures. I' The reactions in the cell all take place simultaneously and are clas! as either class (1) nutrient degradation (fueling reactions), class (11) synt of small molecules (amino acids), or class (111) synthesis of large molel (polymerization. e.g., RNA, DNA). A rough overview with only a fractk the reactions and metabolic pathways i s shown in Figure 7-16. A more det model is given in Figures 5.1 and 6.14 of Shuler and Kargi.I5 In the C1 reactions. Adenosine triphosphate (ATP) participates in the degradation o nutrients to form products to be used in the biosynthesis reactions (Class 1 small molecules (e-g.. amino acids), which ate then polymerized to form 1 and DNA (Class 111). ATP also transfers the energy it releases when it lor phosphonate group to form adenosine diphosphate (ADPI
ATP + HzO+ ADT + P + H 2 0+ Energy Nutrient (e.g., Glucose)
Waste (CO:,water, etc.)
Cell
Figure 7-16
Examples of reactions occumng i n the cell
Cell Growth and Division
The cell growth and division typical of mammalian cetls is shown schen cally in Figure 7-17. The four phases of cell division are called GI. S,GZ. M, and are also described in Figure 7-17.
New
-
-
+
01 Phase
S Phase:
G2 Phase:
MP k e :
Cells ~ncrease~n slre RNA ana
DNA dwbles. RNA and pmte~n 5ynlhests
RNA and protein
Mlrosrs. Nuclear reg& dmdss.
occurs.
No ONA
prweln synthesis MYXlrB No DNA synthmus
synlhss& murs
M Pha-:
New
~ywklnssrs CeHs c h v t ~ o n mcum to g w ma OEW cd14
synmsas
Flgute 7-17 Phases OF cell division.
IJM.L. Shufer and F. Kargi, Bioprocess Engineering Basic Concepts, 2nd ed. (U] Saddle River, N.I.: Prentlce Hall. 2002). ISM.L. Shuler and F. Kargi. Biopmcess Engineering Basic Concepts, 2nd ed. (U1 Saddle River, N.J.: Prentice Hall. 2002). pp. 135, 185.
In general, the growth of an aerahic organism follows the equation
[ ~ ~ l+ l[Carbon ~ l source
Cell mult~pl~cation
] [ Nitrogm ] [ Oxygen ] [ Phospl~an ] source sutlrce source +
+
[CO?] + [H?O]+ [Products]
+
+
Culture media
[
MOR cells
1(pH:
conditions temperature,etc.1
+ ,,
.
1 (7-49)
A more abbreviated form oSEquntion (7-49) generally used is
Substrate
C'11S
More cells + Product
(7-50)
The products in Equation (7-50)include CO?, water. proteins. and other species specific to the particular reaction. An excellent discussion of rhe stoichiornetry (atom and mole balances) of Equation (7-49) can be found in Shuler and KargiIh and in Bailey and Ollis." The substrate culture medium contains all the nutrients (carbon, nitrogen, etc.) along with other chemicals necessary for growth. Because, as we will soon see, the rate of this reaction is proportiom1 to the cell concentration, the reaction is autocatalytic. A rough schematic of a simple batch biochemical reactor and the growth of two types of micmrganisrns, cocci (i.e., spherical) bacteria and yeast, is shown in Figure 7-18,
Qum Bacteria
Paddle Dlade
Sparger Oxygen
Batch B10reac:or
Figure 7-18 Batch bioreactor.
I6M. L. S h u h and F,K q i , Binprocess Engineering Basic Concepts, 2nd ed. (Upper Saddle River. N.J.: Prentice Hall. 2001,). ''1. E. Bailey and D. R Ollis, Biochemicd Engineering, 2nd ed. (New York: McGraw-Hill. L987).
Reaction Mechanisms, Pathways, Bioreactions, end Bloreactofs
Chap. 7
7.4.1 Cell Growth
Stages of cell growth in a batch reactor ate shown schematically in Figures 7-19 and 7-20. Initially, a small number of cells is inoculated into (i.e., added to) the batch reactor containing the nutrients and the growth process begins as shown in Figure 7-19. In Figure 7-20, the number of living cells is shown as a function of time.
Time
Figure 7-19
a
-
G W h (11) Phase
Lag (1) Pham
t =O
Increase in cell concentration.
Time
Figure 7-20 P h a e s of bacteria cell
Lap phase
Statlonary (ill) Phase
prowth
I n phase I. called the lag phase, there is little increase in ceIl concentration. During the lag phase the cells are adjus~ingto their new environment, synthesizing enzymes, and getling ready to begln reproducing. During this time the cells c a q out such functions as synthesizing transporl proteins for moving the substrate into the cell, synthesizing enzymes for utilizing rhe new substrate, and beginning the work for replicating the cells' genetic material. The duration of the lag phase depends upon the growth medium from which
the inoculum was taken relative to the reaction medium in which it is placed. If the inoculum is similar to the medium of the batch reactor, the lag phase will be almost nonexistent. If. however, the inmulum were placed in a medium with a different nutrient or other contents, or if the inoculum culture were in the stationary or death phase, the cells would have to readjust their metabolic path to aIlow them to consume the nutrients in their new environment.IY Exponential growth Phase I1 is called the exponential growth phase owing to the fact that the Phax cell's growth rate is proportional to the cell concentration. In this phase the cells are dividing at the maximum rate because all of the enzyme's parhways for metabolizing the substrate are in place (as a resuIt of the lag phase) and the cells are able to use the nutrients most efficiently. Phase 111 is the stationary phase, during which the cells reach a minimum biological space where the lack of one or more nutrients limits cell growth. During the stationary phase, the net grrnvth rate i s zero as a result of the depletion of nutrients and essential metabolites. Many important fermentation prodAnlibiotics ucts, including most antibiotics, are produced in the stationary phase. For produced during example, penicillin produced commercially using the fungus Penjcilliurn the starionary phase c*hq-l.ogenurnis formed only after cell growth has ceased. Cell growth is also slowed by the buildup of organic acids and toxic materials generated during the growth phase. Death phase The final phase, Phase IV, i s the death phase where a decrease in live cell concentration occurs. This decline is a result of the toxic by-products, harsh environments. andlor depletion of nutrient supply. 7.4.2 Rate Laws
-
W i l e many laws exist for the cell growth rate of new cells. that is,
Cells
+ Substrate
More cells -t Product
the most commonly used expression is the Mrv~odequation for exponential growth:
r, =
where
PC,
17-5 I )
P.,= cell growth rate. g/dm3.s C, = cell concentration. g/dmi p = specific growth rate. s - '
The specific cell growth rate can be expressed as
--
IXB. Wolf and H. S. Fogler, "'Alteration of the Growth Rate and Lag Time of hl,conostor. i~~rsenreroiCIrs NRRL-B523." Ein~rchnologj and Binerlgir~cerirrg. 72 (6). 603 (2001). B. Wolf and H. S. Fogler. -'Growth of LPucorrosrnr r~lrsrr~rcrniJ~r NRRL-B523. in Alkaline Med~u~n." Biozechnolo~rnttd Bin~ngir~eerirr~. 89 I 1. 96 (7-OM).
424
Reaction Mechanisms, Pathways. Bioreacfions, and Bioreectorc:
Ch,
wherc F , , = i~ maximum specific growth reaction rate, s-I K, = the Monod constant. _e/drnt C, = subsrrate lie.. nutrient) concentration. g/dm3 Representative valves of p, and K , are 1.3 h-' and 2.2 x 1P5 rnollc respectively, which are the parameter values for the E. culj growth on g h s Combinins Equations (7-51) and (7-52). we arrive at the Monod equation bacterial cell growth rate Monod equation
For a number of different bacteria. the constant K , is small. in which case rate law reduces to
I
rg
l i
~rn;,Xc'
(7The growth rate, r e , often depends on more than one nutrient concentrati however, the nutrient that is limiting is usually the one used in Equar
C.
rR =
(7-53). In many systems the product inhibits the rate of growth. A classic ext ple of this inhibition is in wine-making, where the fermentation of glucosc produce ethanol is inhibited by the product ethanol. These are a number of I ferent equations to account for inhibition: one such rate law takes the empiri form
where Ernplrical form of Monud equation for product inhibition
with
C;
=
product concentration at which all metabolism ceases. gldm3
n = empirical constant
For the glucose-to-ethanol fermentation. typical inhibition parameters are n = 0.5
'
and
Cp* = 93 gldm3
In addition to the Monod equation, two other equations are also commor used to describe the cell growth rate; they are the Tessier equation.
r, = k,,,
[
I - exp
(31 -
C,
Sec. 7.4
Bioteactors
and the Moser equation,
where X and k are empirical constanIs determined by a best fit of the data. The Moser and Tessier growth laws are often used because they have been found to better fit qxperirnental data at the beginning or end of Fermentation. Other growth equations can be found in Dean." The cell death rate is a result of harsh environments, mixing shear forces, local depletion of nutrients and the presence of toxic substances. The rate law is
Doubling tlrncs
where C, is the concentration o f a wbstance toxic to the cell. The specific death rate constants kd and kt refer to the natural death and death due to a toxic substance, respectively. Representative vatues of k, range from 0.1 h-' to less than 0.0005 h-I. The value of k, depends on the nature of the toxin. Microbial growth rates are measured in terms of doubling times. Doubting time is the time required for a mass of an organism to double. Typical doubling times for bacteria range from 45 minutes to I hour bur can be as h s t as 15 minutes. Doubling times for simple eukaryotes. such as yeast, range from 1.5 to 2 hours but may I x as fast as 45 minutes.
Effect of Temperature. As with enzymes (cf. Figure 7-91, there i s an optimum in growth rate with temperature owing to the competition of increased rates with increasing temperature and dennturizing the enzyme at high temperatures. An empirical Iaw that describes this functionality is given in Aiba et al." and is of the form
u T
r
where is the fraction of rhe maximum growth nte, T, is the temperature at which the maximum growth occurs. and p(T,j the growth at this temperature. For the rate of oxygen uptake of Rhicnbium rrifollic, the equation takes the form
The maximum growth occurs az 310K. I9A. R. C.Dean, Growth, Frmction, and Regtifation in Barrerial Cells (London: Oxford University Press, 1964). 20s.Aibn, A. E. Humphrey, and N. F. Millis. Biochemical Engineering (New York:Academic Press, 1973), p. 407.
426
R&on
Mechanisms, Pathways, Bioreactions, and Bioreactors
Chap. 7
The stoichiometry for celI growth is very complex and varies with microorganism/nuhent system and environmental conditions such as pH, temperature, and &ox potential. This complexity is especially true when more than one nutrient contributes to cell growth, as is usually the case. We shall focus our discussion on a simplified version for cell growth, one that is limited by only one nutrient in the medium. In general. we have
Cells + Substrate
+More cells + Product
In order to relate the substrate consumed, new celIs formed, and product generated, we introduce the yield coefficients. The yield coefficient for cells and substrate is YClB
Mass of new cells formed Mass of substrate consumed
=
--ACc Acs
with
A representative value of Y , , might he 0.4 (gig). See Chapter 3, Problem
P3-14Bwhere the value of Y,, was calculated. Product fomatian can take place during different phases of the cell growth cycle. When product formation only occurs during the exponential growth phase, the rate of product formation is r, = y,.';
Growth arwciated
product formation
= yp,',,,pcc=
I.;,,, ~mxccc~ 4 4 C,
(7-63)
where
'' c
Mass of product formed =
- Mass o f new cells formed
-5 AC,
v, that is, (qp= Y,, p) i s often called the specific rate of product formation. y,. (mass productJvoIume/tirne). When the product i s formed during the stationary phase where no cell growth occurs. we can relate the rate of product formation to ssbstrate conw-nption by The product of Ypk and
Nongrowth
rf, = y,,,, (-rr)
(7-65)
a r s m a t e d product
formation
The substrate in this case i s usually a secondary nutrient, which we discuss in more detail later.
--
The stoichiomemc yidd coefficient that relates the amount of product formed per mass of substrate consumed is
- Mass of product formed = S'f - Mass of substrate consumed
AC AC,
(7-66)
In addition to consuming substrate to produce new cells, part of the substrate must be used just to maintain a cell's daily activities. The corresponding maintenance utilization term is
m =
Cell maintenance
Mass of substrate consumed for maintenance Mass of cells Time
u
A typical value i s
m = 0.05 g != 0.05 h-I g dry weight h
The rate of substrate consumption for maintenance whether ar not the cells are growing is
Neglecting cell maintenance
When maintenance can be neglected. we can relate the concentration of cells formed to the amount of substrate consumed by the equation
(7-68)
This equation can be used for both batch and continuous flow reactors. If it is possible to sort our the substrate (S)that is consumed in the presence of cells to form new cells (0from the substrate rhat is consumed to form product (P),that is,
the yield coefficients can be, written as y:j5 =
yp
Mass of substrate consumed to form new cells Mass of new cells formed
(7-698)
Mass of substrate consumed to form product Mass of product formed
(7-69Bl
=
428
React~onMechanisms. Pathways. Bioreactions, and B~oreactors
Ch;
Substrate Utilization. We now come to the task of relating the rate of nl en1 consumption, -r,. to the rates of cell growth, product generation, and maintenance. In general. we can write Substrate accounting
substrate
In a number of cases extra attention must be paid to the substrate balance product is produced during the growth phase, i t may not be possible to sepa out the amount of substrate consumed for cell growth from that consume( produce the product. Under these circumstances all the substrate consume( lumped into the stoichiornctric coefficient. Y,,, and the rare of substrate dis pearance is
The corresponding rate of product formation is Growth-as~miated
(7-
product h m a t ~ o n in the growth phafe
Because there is no growth during the stationaq phase. it is clear that Eq tion (7-70) cannot be used to account for substrate consumption, nor can the I of product fonnation be related to the growth rate [e.g.. Equation (7-63)l. M antibiotics, such ar penicillin, are produced in the stationan, phase. In this phi the nutrient required for growth becomes virtually exhausted. and a different nl ent, called the secondary nutrient. is used for cell maintenance and to produce desired product. Usually, the rate law for product formation during the station phase is similar in fom to the Monod equation, that is. Nongrowthassociated product
formation in the srationary
phaqe
where
k,, = specific rate constant with respect to product. (dm3/g s) C,, = concentration o f the secondary nutrient, gldrn3 C, = cell concentration, g/dm3 (g G gdw = gram dry weieht) K, = Nonod constant, g/dm3 rp = Y p t s n ( - ~ r n ) (gldm3 s)
The net rate of secondary nutrient consumption during the stationary phase In the stationary phase, the: concentration of live cells
is constant.
Sec. 7.4
429
Bioreactors
Because the desired product can be produced when them is no cell growth, it i\ always best to relare the product concentration to the change in secondary nutrient concentration. For a batch system the concentration of product, C,,. formed after a time r in the stahnary phase can be related to the substrate concentration, C,. at that time. Wrgtcct~cell maintenance
We have considered two limiting situations for relating substrate consunlp tion to cell growth and product formation; product formation only during the growth phase and product formation only during the stationary phase. An example where neither of these situations applies is fermentation using Iactobacillus, where lactic acid is produced during both the logarithmic growth and stationary phase. The specific rate of product formation is often given in terms of the Luedeking-Piret equation, which has two parameters cc (growth) and 0 (nonLuedeking-Piret equation for the mte of product forrnatiun
growth) ql, = WB+P
(7-74)
with
The assumption here in using the P-parameter is that the secondary nutrient is In excess.
I
Example 7-5 Esrimate rke Yield Coeficicnts The following data was determined in a batch reactor for the yeast Sncchamm~ces cerevrsine Tmr E E7-5 1.
I
Glucose
Time.
0 t
(hrl 1
2 3
I
RAW DATA
C'11'. More cells + Ethanol
Cells.
Glucose.
Cc @!dm1) 1 i.33 1 87 2 55
Cs (gldm31 2Sfl
245 238.7 229.8
Ethanol, Cp (g/dm3) 0 2.14
5.03 8.96
.,
Determine YPIr. Y Y,,. Y,,, Y,,,,,b and Ks.Assume no lag and neglect rnaintenance at the stan of the growth where there are just a few cells,
(a)
Calculate the substrate and cell ~ i e Mcoeficients. Y,,,and Y,,, Between t = 0 and r = I h
430
Reaction Mechanlsms, Pathways, Bloreactlons, and Bioreactors
Chap. 7
Between t = 2 and t = 3 h
Taking an average
v,,
= 13.3 glg
We could alsa have used Polymath regression to obtain
@)
Similarly for the substrate and product yield rmfici~nfs
1 Up,,= -
Y#,
(c)
1 = 0.459 glg 2.12g!g
The producu'ceI1 ~'ieldcoqflcicnt is
y C'P
1 =-=--
YP,,
5.78 glg
- 0.173 g/gl
We now need to deternine the rare law parameters h,and KT in the M o n d equation
For a batch system
How to regress the Monad equation for p,,, and K,
To find the rate law pararntkxs ha, and K,. we first apply the differenrial formulas in Chapter 5 to columns 1 and 2 o f Table E7-5.1 to find r,. Because C, >> K, initially. i t is best to repress the data using the Hanes-Woolf form of the Monod equation
Using Ptoymarh's nonIinear regression and more data points, we find p, h-I and K, = 1.71dm1.
= 0.33
7.4.4 Mass Balances
There are two ways that we could account for the growth of microorganisms. One i s to account for the number of living cells, and the other is to account for the mass of ~e living ceIls. We shall use the latter. A mass balance on the microorganism in a CSTR (cbemostat) (shown in Figure 7-2 1) of constant volume is Cell Balance
Rate of
Rate of
c,
(7-75)
v-dCc dt
Substrare Balance
=
u0C&
-
vC,
+
(rg-rd)v
The corresponding substrate balance is Rate of
Rate of
Rate of
Rate of
(7-76)
In most systems the entering microorganism concentration C,, is zero.
Batch Operation For a batch system v = v, = 0 and the mass balances are as follows: Cell
v-ddtc ~= r,v-
The mass halances
rdv
Dividing by the reactor volume V gives
Substrate The rate of disappearance of substrate, -r,, results from substrate used for cell growth and substrate used for cell maintenance,
432
Reaction Mechanisms, Pathways. Bioreaciions, and Bioreacton
Ck
Dividing by Y yields the substrate balance for the growth phase
For cells in the stationary phase, where there i s no growth, cell rnainten and product formation are the only reactions to cansume the substrate. U these conditions the substrate balance, Equation (7-76), reduces to Stationay phase
Typically, r, will have the same form of the rate law as r, [e.g.. Quz (7-7 I)]. Of course, Equation (7-79) only applies for substrate concentrat greater than zero.
Product The rate of product formation. r,. can be related to the rate of substrate E sumption through the following balance: Batch stationary growth phase
During the growth phase we couId also relate the rate of formation of prod r,,, to the cell growth rate, r,. The coupled first-order ordinary differer equations above can be solved by a variety of numerical techniques. Example 7 4
Bacteria Growth in a Batch Reactor
Glucose-to-ethanol fermentation is to be carried out in a batch reactor using organism such as Saccknrulrn~cescerertisrae. Plot the concentr~lionsof cells, : strate. and product and growth rates as functions of time. The initial cell concen tion is 1.0 g/dm3, and the substmte (glucose) concentration is 250 g/dm3. Additional doto [partial source: R. Miller and M. Melick, Chem. Eng., Feb. p. 1 13 (t983)l:
C,' = 93 gldrn3 n = 0.52
Y,,,= 0.08g l g Y,, = 0.45 g l g lest.)
I
Sulution
1. Mass balances:
Cells: The algorithm
Substrate: Pmduct:
Y
dC 2
= Y,,,(r,V
2. Rate laws:
'p
= Yp/c'g
4. Combining gives
Cells Substrate hoduc t
These equations were soIved on an ODE equation solver (see Table E7-6.1). The results an shown in Rgun E7-6.1 for the parameter valuer given in the pmb-
lem statement.
POLYMATH Results Example76 Rsetrrln Emwb la s Bowb W
Living Ewample Problem
r #-I&ZLW. Rn3,luz
434
I
Reaction Mechanisms, Pathways, Bioreactions, and Bioreactors
Chap. 7
Figure E7-6.1 Concentrations and rates as a func~ionof time.
The substrate concentration C, can never be less than zero. However, we note that when the subsrrate is completely consumed, the first term on the right-hand side of Equation (E7-6.8) (and line 3 of the Polymath program) will be zero bklt the second term for maintenance, mC,. will not. Consequently, if the integration is carried further in time, the integration program will predict a negative value of C,! This inconsistency can be addressed in a number of ways such as including an ifstatement in the Polymath program (e.p.. if C, i s less than or equal to zero, then m = 0).
7.4.5 Chemostats
Chemostats are essentially CSTRs that contain microorganisms. A typical chemostat is shown in Figure 7-21. along with the associated monitoring equipment and pH controller. One of the most important features of the chemostat is !hat ir allows the operator to control the cell growth rate. This control of the growth rate iq achieved by adjusting the volumetric feed rate (dilution raw).
SZerlle Hedlurn Reservoir
.-.
Fermentor
Figvre f -21 Chemostat system.
7.4.6 Design Equations
CSTR
In this section we return to mass equations on the cells [Equation 17-75)] and substrate [Equation 17-76)] and consider the case where the volumetric Row rates in and out are the same and that no live (i-e., viable) cells enter the chemostat. We next define a parameter common to bioreactors called the dilution rate, D. The dilution rate is
and i s simply the reciprocal of the space time T. Dividing Equations (7-75) and (7-765by V and using the definition of the dilution rate, we have Accurnu1a;tion = In - Out t Generation CSTR mass balances
Cell:
Substrate:
-dr
5 = DC, - DC, + r. dl
Using the Monod equation, the growth rate is determined to be
For steady-state operation we have
DC, = r,
-
r,
and
D(Cm - C,} = r,T
(7-82)
436
Reaction Mechanisms, Pathways, Bioreactions, and Bioreactors
Ch
We now neglect the death rate, Q , and combine Equations (7-51) (7-835for steady-state operation to obtain the mass flow rate of cells out 01 system.
F,. F, = C,u, = r,V = @C,V
Afrer we divide by C,V, Dilution rate
An inspection of Equation (7-86) reveals that the specific growth rate of can be controlled by the operator by controlting the dilution rate Using Equation 17-52] to substitute for in terms of the substrate concen tion and then solving for the steady-state substrate concentration yields
How tocontrol cells cell growth
Assuming that a single nutrient is limiting, cell growth is the only process c tributing to substrate utilization, and that cell maintenance can be negtec the stoichiometry is
-r x = r8 Yslc
(7-
Cr = Y c , s ( C d - Crl
(7 -
Substituting for C, using Equation (7-87) and rearranging, we obrain
To learn the effect of increasing the dilution rate, we combine Equations (7-1 and (7-54) and set r,, = 0 to get
We see that if D > p, then dC,ldt will be negative, and the celE concentrati wil! continue to decrease until we reach a point where all cells will be washed o
The dilution rate at which wash-out wilI occur is obtained from Equati (7-89) by setting C, = 0. Row rate at which
wash-out mcun
Sec. 7.4
437
Bioreactors
We next want to determine the other extsenie for the dilution rate, which is the rate of maximum cell production. The cell production rate per unit volume of reactor is the mass flow rate of cells out of the reactor (i.e., h,. = C,v,) divided by the volume I/, or
Using Equation (7-89) 16 substitute for C, yields
Figure 7-22 shows production rate, cell concentration, and substrate concentration as functions of dilution rate. We observe a maximum in the production raze, and this maximum can be found by differentiating the production rate. Equation (7-92). with respect ta the dilution mre D:
Maximum rate of cell production CDC,)
Dmoxprod
0
Figure 7-22 Cell concentntion and production rate as a function of drlution rate.
Then Maximum rate of cell production
The organism Streptornyces aureofaciena was studied in a 10 dm3chemostat using sucrose as a substrate. The cell concentration, C, (rnglmlj, the substrate concentration, C, (mglml), and the production rate. L)C, (mglrnllh), were
438
Reaction Mechanisms, Pathways, Bioreactions, and Bioreacton
Chap. 7
measured at steady state for different diIutjon rates. The data are shown in Figure 7-23.21
1Symbol [
~ N D (D W va
Figure 7-23 Continuous culture of Strepromvces aureoficiens i n chemostats. (Note: X = C,) Coune~yof S. diba. A. E. Humphrey, and N. F. Millis. Biochemical Engineering. 2nd Ed. (New 'iork: Academic Press, 1973).
Note that the data follow the same trends as those discussed in Figure 7-22. 7.4.8 Oxygen-Limited Growth
Reference Shelf
Oxygen is necessary for all aerobic growth (by definition). Maintaining the appropriate concentration of dissolved oxygen in the bioseactor is impartant far efficient operation of a bioreactor. For oxygen-limited systems. it is neressary to design a bioreactor to maximize the o x y p n transfer between the injected air bubbles and the cells. ~ y p i c a l l ~a ,bioreactor contains a gas sparger. heat transfer surfaces, and an impeller. A chemmtat has a similar configuration with the addition of inlet and outlet streams similar to that shown in Figure 7-18. 7 h e
oxygen transfer rate (Om) is related to the cell concentration by OTR
=
Q,,C,
?'B. Sikyta, J. Slezak, and M. HernId, dppl. Microbial., 9, 233 (1961).
(7-95)
Sec. 7.5
Physiologically Based Pharmacohinetic {PBPK) Models
439
where Q , . is the microbial respiration rate or specific oxygen uptake rate and usually follow~s Michaelis-Menten kinetics (Monod growth, e.g., QO2= YO?,, rg). (See Problem P7-13s.)The CD-ROM discusses the transport steps from the bulk liquid to and within the microorganism. A series of mass transfer correlations are also given.
Scale-up for the growth of microorganisms is usually based on maintaining a constant dissolved oxygen concentration in the liquid (broth), independent of reactor size. Guidelines for scaiing from a pilot-plant biareactor to a commercial plant reactor are given on the CD-ROM. One key to a scale-up is to have the speed of the end (tip) of the impeller q u a 1 to the velocity in both the laboratory pilot reactor and rhe full-scak plant reactor. If the impeller speed is too rapid, it can lyse the bacteria; if the speed is too slow, the reactor contems will not be well mixed. Typical tip speeds range from 5 to 7 mts.
7.5 Physiologically Based Pharmacokinetic (PBPK)Models We now apply the material we have k e n discussing on enzyme kinetics to modeling reactions in living systems. Physiologically based pharmacokinetic models are used to predict the distribution and concentration-time trajectories of medications, toxins, poisons, alcohol. and drugs in the body. The approach is to model the body components (e.g.. liver, muscle) as compartments consisting of PFRs and CSTRs connected to one another with in-flow and out-flow to each organ compartment as shown in Figure 7-24. Blood
F m Lung Richly Pertused
Slow4y Perfused
Liver
~jgui-e 7-2d
(a) camp an men^ model ul human hody. ( h ~ Generic srructurc of of Chew. E ~ y q Pnl,cr.i,.\r. . IM)(15)38 (June 2W1.
PRPK mode\\. Courtcqq
440
React~onMechanisms, Pathways, Bcoreactions, and Bioreac?ors
Ch
Associated with each organ is a certain tissue water volume, TWV, w we will designate as the organ compartment. The TWV for the different or4 along with the blood flow into and out of the different organ cornpartm (called the perfusion rate) can be found in the literature. In the models
cussed here, the organ compartments will be modeled as unsteady well-m C S T h with the exception of the liver, which will be modeled as an unstt PFR. We will apply the chemical reaction engineering algorithm (mole ance, rate law, stoichiomerry) to the unsteady operation of each compann Some compartments with similar fluid residence times are modeled to col of several body parts (skin, lungs, etc.) lumped into one compartment, suc the central compartment. The interchange of material between cornpartmen primarily through blood flow to the various components. The druglrnedica concentrations are based on the tissue water volume of a given compartrr Consequently, an important parameter in the systems approach i s the perfu rate for each organ. If we know the perfusion rate, we can determine exchange of material between the bloodscream and that organ. For exampl organs are connected in series on one in parallel by blood flow as show Figure 7-25,
Blood Ftaw
b0
vi
vl
Organ f
CAI Vo "l
Organ 3
t
v2
Organ 2
~2
z7o
4
Figure 7-25 Physiologically based model.
then the balance equations on species A in the TWVs OF the organs V,.V2, V, are
where r,,, and rA3are the metabolism rates of species A in organs I and 3, respectively, and CAj is the concentration of species A being men
lized in each of the organ compartments j = 1, 2, and 3. Pharmacokinetic models for drug delivery are given in Professional Referr Shetf R7.5.
Sec. 7.5
Physiologically Based Pharmacokinetic (PBPK) Models
I
Example 7-7 Alcohol Mef~bolismin the Bodsz2
/
E7-74.G ~ n a r a l We are going to model the metabol~smof ethanol in the human body using fundamental reaction kinetics along with five compartments to represent the human body. Alcohol (Ac) and acetaidchyde IDe) will flow between these compartments, but the aicohol and aldehyde wrll only be rneraboItzed in the liver compnrtmenf. Alcohol and acetaldehyde are meiabolized in the h e r by the following series reactions.
Example Problerr
The first reaction is catalyzed by the enzyme alcohol dehydrogenease (ADH) and the second reaction is catalyzed by aldehyde dehydrogenease (ADLH). The reversible enzyme ADH reaction is catalyzed reaction in the presence of a cofnctor, nicotinarnide adcnimc dinucleotide ( N A P )
C,H,OH t NAD'
(==?CH,CHO + H++ NADH ADA
The rate law for the disappearance of ethanol foIlows Michaelis-Menton kinetics and i s
where V,, and KM are the Michaelis-Menten parameters discussed in Section 7.2, and CAcand CD,are the concentrations of ethanol and acetadehyde, respectively. For the metabolism of acetaldehyde in the presence of acetaldehyde dehydrogcnast. and NAD*
NAD+ + CH,CHO + H,O lLDH t CHICOOH + NADH e H*
I
the enzymatic rate law is
The parameter values for the rate laws are Vm,,,D, = 2.2 mPYl/(min kg liver). = 32.6 mmoV(min * kg liver), KmADH= 1 rnM. V-, = 2.7 rnmoY{min * kg liver), and KMALDH = 1.2 pM (see Summary Notes). The concentration time trajectories for alcohol concentration in the central compartment are shown in Figure E7-7.I . K a ~ H= 0.4 rnM.,,V
We are going to use as an example a five-organ compartment model for the metabolism of ethanol in humans. We will apply the CRE algorithm to the tissue warer volume in each organ. The TWVs are lumped according la their perfusion rates and
,* Summary Notes
M Urnulis, M. N. Curmen, P,Singh. and A. S. Fogler, Alcohol 35 (I), 2005, The complete paper is presented in the Summary Notes on the CD-ROM.
--Q.
442
Reaction Mechanisms. Pathwsys. Bionactions, and Bioreactors
Chap. 7
residence times. That is, those compartments receiving only small amounts of blood Row will be lumped together (e.g., fat and muscle) as will those receiving large blood flows (e.g., lungs, kidneys, etc.). The following organs will be modeled as single unsteady CSTRs: stomach. gastrointestinal tract, central system. and muscle and fat. The metabolism of ethanol occurs primarily in the liver, which i s modeled as a PFR. A number of unsteady CSTRs in series approximate the PER. Figure E7-7.1 glves a d i a p m showing the connection blood flow (pfusion), and mean residence, T.
The physiologically based mudel
I
Stomach
I
&-----=
m' Muscle & Far 25.76 dm3
=
Central = 11.56 dm'
Liver = I . l dm3 G.I. = 2.4 dm?
LF=d
K~sidenueT t m ~
Muscle & Fat = 27 rnin
Muscle B Fat
Central = 0.9 min h e r = 2.4 rnin
Figure E7-7.1 Companmenr model of human body. The residence times for each organ were obtained from the individual perfusion rate? and are shown in the margin note next to Figure E7.7-1. We will now discuss the balance equation on the tissue water volume of each
of the organslcompartments. Stomach
As a fin1 approximation, we shall neglect the 10% of the total alcohoI ingested that is absorbed tn the sromach because the majority of the alcnhol (90R ) is ahsorbed at the entrance to the fasrmrntestinai (G I.) tract. The contents of the stomach are emptied into the G.1. tract at a rate proponional to the volume of fhe contents in the
stomach.
where i',,is the voluine of the contents of the stomach and Ir, is the rate constant
The flow of ethanol from the \lomach inlo the G.I. tract, where it is absorbed ally instantancov~ly.i\
Sm, 7.5
where CSAc is the ethanol concentration in the stomach, ,k
rAi ,
organ species
'
S = Stomach
G = G.1. Tract
443
Physiologically Based Pharmacokinetic (PBPK) Models
is the maximum enp
tying rate, D is the dose of ethanol in the stomach in (mmoI), and a is the emptying
parameter in (mm01)-~.
Gastrointestinal (G.1,) Trsct Component Ethanol is absorbed virtually instantaneously in the duodenum ttt the entrance of the G.I. tract. In addition, the blood flow to the G.1, compartment from the centraI compartment to h e G.Z. tract is two-thirds of the total blood flow with the other thjrd by-passing the G.I. h c t to the liver, as shown in Figure E7-7.1. A mole mass bslance on ethanol in the G.I. tract tissue water volume (W) VG,gives
C = Central
M = Muscle L = Liver
where CCAc is the concenmtion of alcohol in the G.I. mrnpamenent. B~causethe l%V remains conslant, the mass balance becomes
I
A similar balance on acetaldehyde gives
The centrat volume has the largest W V . Material enters the cenbal companment from the liver and the musclelfat compartments. A balance on ethanol in this corn-
partment is
In
I
[Accumulation1
=
Ethanol: Vc-
= uLCLAc + U CMAE -
d,
"
Out to
Out
uLCCA, - v , ~ OC ' ~(E7-7'8) ~ ~
Similarly the acetaldehyde balance i s Acetaldehyde:
Yc-
dt
(E7-7.9)
444
Reaction Mschan~sms.Pathways. Broreactions, and Bioreactors
C
MuscleCFat Compartment
Very little rnztterial profuses in and out of the muscle and fat compartments pared to the other cornpartlnents. The muscle compartment mahs balances on el nnd acetaldehyde are
Ethanol:
d C, V , w d =uv {Cqc
Acetaldehyde:
Yt,
- C,tt,{)
dt
(E7.
-- -Uw (CcDr- C,tfje)
(E7-
Liver Compartment
The liver will be modeled as a number of CSTRs in series to approximate a aith a voturnr of I . t dm". Approxitnuting a PFR with a number of CSTRs in ! was discusqed in Chapter 2 . The total volume of the liver is divided into Death by alcohol poisoning can occur when the central cornpart-
CSTRs.
ment concentration reaches 2 g/dmJ.
Figure E7-7.2Liver modeled as a number of CSTRs in series
Because the first CSTR receives in-flow from the central compartment (113 u) from the G.I. compartment, it is treated separately. The balance on the first CS? Fiat reactor
Ethanol: AV,-c
2
L =L(
dr
+-)
C
) A L
(E7-7
Aceraldehyde:
*vL5fi
7
= u L ( ~ ~ c b ~ ~ cLm)-rL,c(cL,,p7 ~ G D c -
+rLh(c~ne
(E7-7 where Ct4= is the concentntion of alcohol leaving the Arst CSTR. A balancc
reactor i gives
Later reactors
Ethanol:
dC
b v LdtL c= u~[C,,,,~,- CIA=] + rEAc(CGC, C,D,)AVL
(E7-7
Sec. 7.5
Physiolog~callyBased Pharmacokinet~c(PBPK) Models
Acetaldehyde:
The concentrations exiting the tast CSTR are CLnAk and C,, . Equations (E7-7.1) through (E7-7.15)along with the parameter values are given on %e CD-ROM summary notes and the Polymath living example problem. The Polymath program can be loaded directly Fmm fhe CD-ROIM so that the reader can vary the model parametemt You can print or view the complete Polymath program and read the complete paper [AIcuhol35 ( 1 ), p. 10, 20051 in the Sumttlruy ,Votes on the CD-ROM. Summary dotes
In the near future, E. R. physicians will go to and interact with the computer to run
siniulations to help rreal pat~entsw ~ t h drug overdoses and dntg internction.
I
.!win& Example Problc
+
The Polymath program on the CD-ROMwas written for Polymath Version 5.1. If you use Version 6.0or higher, reduce the number of liver compartments from 10 TO 9 to avoid exceeding the maximum number of equations aIfowed in Version 6.0.
446
Reaclton Mechanisms, Pathways, Bioreactions, and Bioreactors
Chap. 7
Results figure E7-7.3 gives the predicted blood ethanal concentration trajectories and experimentally measured trajectories.The different curves are for different initial doses of ethanol. Mote that the highest initial dose of ethanol reaches a mwimum concentration of 16.5 m M of alcohol and that it takes between 5 and 6 hours to reach a level where it is safe to drive. A comparison of the mode1 and experimental data of Jones et al. for the acetaldehyde concentration is shown in Figure ET-7.4. Because the acetaldehyde concentrations m three orders of magnitude smaller and more difficult to measure, there is a wide range of error bars. The model can predict h t h the alcohol and acetaldehyde concentration trajectories without adjusting any parameters. In s u r n m q , physiologically based pharamacoktnetic models can he used to predict concentration-time trajectories tn the TWV of various organs in the body. These models find wer-increasing application of drug delivery to targeted organs and regions. A thorough discuss~onof the following data and other trends is given in the paper (Ulrnuli~,Gurmen. Singh, and Fogler).
Comparison of model with experimental data
Tme {min) Fjrure EJ-7.3. Blood alcohol-time trajectories from dam of W~lkinsonel al?"
0
20
PO
60
50
100 120 140 tBO 1BO
Time (min)
Figure E7-7.4.Blood nlcahol-lime rrajector~eqfrom data of Jones et a]."
'jP. K. WiIkinson, et a].. '.Pharmacokinetics of Ethanol After Oral Administration in the Fasting State." J'. P h n n ~ i ~ ~ o o kHiophonrr.. ei. 5(3):207-24 ( 1977). "A. W.Jones, J. Neirnan. and 51. Hillhi>rn. "Concentration-time Profiles of Ethanol and Acetaldehyde in Human Volua~eenTreated with the Alcohol-~ensirizingDrug, Cslclurn Carhimide." BK J. CiEtt Phurrnarnl., 25. 2\3-21 (198%).
447
Summaty
Chap. 7
--. - e. The theme running through most of this chapter is the -steady-state hypothesis (PSSH) as it applies to gas-phase reactions and enzymatic reactions. The reader should be able to apply the PSSH to reactions in such problems as W-7 and P7-12 in ordw to develop rate laws. Reaction pathways were discussed in order to visualize the various interactions of the reacting species. After completing this chapter the reader should be able to describe and analyze enzymatic reactions and the different types af inhibition as displayed on a Lineweaver-Burk plot. The reader should be able to explain the use of microorganisms to produce chemical products along with the stages of cell growth and how the Monod equation for celI growth is coupled with mass balances on the substrate, cells, and product to obtain the concentration-time trajectories in a batch reactor. The reader should be able to apply the growth laws and balance equations to a chemostar (CSTR)to predict the maximum product rate md the wash-out rate. Finally, the reader should be able to discuss the application the enzyme kinetics to a physiologicalIy based pharrnacokinetic (PBPK) model of the human body to describe ethanol metabolism.
SUMMARY 1. In the PSSH, we set the wte of formation of the active intermediates equal to zero. If the activc intermediate A * i q involved in rn different reactions, we set
ir
to
This approximation is justified when the active intermediate is highly reactive and present in low conceatrarions. 2. The azomethane (AZO) decomposition mechanism is
2AZO <-AZO L?
AZO*
+ AZO'
(S7-2)
k,
---+ N,-k ethane
By applying the PSSH to AZO*.
we show the rate law. which exhihits first-order dependence nith respect to AZO at high AZO concentrations and second-order dependence with respect to AZO at low AZO cuncentrattons. 3. Enzyme Kinetics: Enzymatic reactions Follov+the sequence
448
Reaction Mechanisms, Pathways, Bioreactions, and Rioreactors
Ck
Using the PSSH for (E S) and a balance on the total enzyme. E,. includes both the bound (E * S) and unbound enzyme (E) conce!ltratian!
we arrive at the Michaelis-Menten equation
is the maximum reaction rate at large substrate concentra (S >> KM)and K, is the Michaelis constant. KM Is the substratc concent1 at which the rate is half the maximum rate (S,,, = KM). 4. The three different types of inhibition-competitive, uncompetitive, and competitive (mixed) inhibition-are shown on the Lineweaver-Rurk plo where V,,,
)
,Noncompetitive(Both slope and intercept change) Uncompetttive (intercept changes) Competitive (Slope changes) No lnhrbrtion
Cells + Substrate
+ More cells + Product
(a) Phases of bacteria growth:
I. t a g
11. Exponential
111. Stationary
(b) M o n d growth rate law:
Ycls
=
Mass of new cells formed Substrate consumed
(d) Unsteady-state mass balance on a chemostat:
IV. Death
Chap. 7
CD-ROM MATERIAL
-r, =
449
Y:,,rcp+YlprPf mC<.= Y,Lrg+mCc
6. Physiologically based phnrmacokinetic mdels
I
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cn
Organ 3
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Ozone tDotson CTnits)
Photo counesy of Goddard Space Flight Center (NASA). See CD-ROM for coIor picture< o f the ozone layer and the glow sticks.
B. Glow Sticks
(ST-8)
450
Reaction Mechanisms, Pathways. Bioreactions, and Bioreactors
D. Fer-de-Lance
C. RusseII's Viper
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,-
Livlp,$Evamr'e Problew
Living Example Pmhlems I . Emt~njdr,7-2 PSSH ;\pp/irii lo Thcnncri Cr~iolii~lg ttf EIIIOIIP 2. E.1-rt17plc7-6 Bnrrrrir; Gmwrh irt a Burr11 Renrror3. Ermliplc 7-7 Alr,ohol MPI(I~IO/~.YIIT 3. E.TIIIII/I~P ~ . e hr~~ndirlr. O:OIIP 5. E I - ~ I I I~I\~. pIh1~10~1ltir: F Glo~~.~ri~.k,~6. E , Y ( J ) ~ Iw~t/h( Jt t : ( ~ / ~ l Ro I: I . + . . '\~ -Viptt~/ 7. E~rrniplri r , ~ hritc~tl~rl~: h,r'rl~-l~~rl~,t, S. Ea~rltpleK7.4 Rr.r-r,/>r(~r E~~c/o(r,r:l,ro.ri.~-
Chaa. 7
Chap. 7
CD-ROM MATERIAL
Profdona1 Reference Shelf P7.1. Polymerization A. Srep Polymerization Mechanism ARB + ARB tAR,B + AB ARB + AR,B +AR3B + A% ARB + AR,B
+AR4B + AB
AR2B + ARIB+ AR,B + AB
Reference Shelf
Rate Law r, =
PIP,-,
- 2kqM
Concentration
Example R7-1 Determining the Concentration of Polymers for Slep Polymerization B. Chain Potymeriza~ions Free-Radical Polymerization
Iz 4 21
Initiation
Ri+M+R,,,
Propagalion
Termination Addition
R, + Rk -+ P,+,
Disproportionation
Ji, + R, -+P,+ P,
Mole Fraction of Pnlymer of Chain Length
h a ? n p l ~R7-2
Parameters of M W Distribution
452
Reaction Mechanisms, Pathways, Bioreactmns, and Bioreactors
Cb
C. Anionic Polymerization Initiation by An Ion AB -A-+B-
Initiation
A-+M+R,
R,+ M + R ,
Propagation
R, +M+Rp1 Transfer to Monomer
R, + M t P, + R ,
Example 7PRS-3 Calculating the Distribution Parameters from Ana
Expressions for Anionic Polymerization Example 7PRS-4 Qetermination of Dead Polymer Disrribution \I Transfer to Monomer Is the Primary Termination R7.2. Oxygen-Limited Fetrnenmrion Scale Up
B Ye&
Reference Shelf
W ~ fM i a h 9 p ~ r ZIt, MiWQ0-S
R7.3. Receptor Kinetics A. Kinetics of signaling
Chap. 7
GO-ROM MATERtAL
B. Endocytosis f
Adapted Fmm D.A. Lauffenburger and J . J. Lineman. Receptors (New York:Oxford University Press. 19931.
R7.4. Multiple Enzyme and Substrate System A. Enzyme Regeneration Example R7.4-I Construct a Lineweaver-Burk Plot for Different Oxygen Concentration B. Enzyme Cofacton (1) Example 7.4-2 Derive a Rate Law for Alcohol Dehydrogenase (2) Example 7.4-3 Derive a Rate Law a Multiple Substrate System (31 Example 7.4-4 CaIculate the Initial Rate of Formation of Ethanol in the Presence of Propanediol
R7.5. Phanacokinetics in Drug Delively Pharmacokinetic models of drug delivery for medication administered either orally or intravenous1y are developed and analyzed. in.
Figurn A. %o-companment mode?.
Figure B. Drug response c u m .
454
Reaction Mechanisms, Pathways, Bioreactions, and Bloreactors
Chap. 7
Q U E S T I O N S AND P R O B L E M S
In each of the following questions and problems, rather than just drawing a box around your answer, write a sentence or two describing how you solved the problem, ClomGworkPsobftmr the assumptions you made, the reasonableness of your answer, what you learned, and any other facts that you want to include, You may wish to refer to W. Strunk and E. B. White, The Elements qf Style, 4th ed. ((New York Macrnillan. 2000) and Joseph M. Williams, Style: Ten Lessons in Clariv & Grace* 6th ed. (Glenview, Ill.: Scott, Foresman, 1999) to enhance the quality of your sentences. See the Preface for additional generic p a t s (x), (y),
W-1
Livlng Example Problem
(2)
to the home problems.
fa) Example 7-1. How would the results change if the concentration of CS, and M were increased? (b) Example 7-2. Over what range of time is the PSSH not valid? Load the Lrving . h ~ m p kProblem. Vary the temperature (800 < T c 1600).What temperature gives the greatest disparity with the PSSH results? Specifically compare the PSSH solution with the full numerical solution. (c) Example 7-3. ( 1 ) The following additional runs were carried out when an inhibitor was present.
(dl
(e)
(fl Liv~ngExample Problem
(g)
What type of inhibition is taking place? (2) Sketch the cutves for no inhibition, competitive, uncornpetitive, noncompetitive (mixed) inhibition, and substrate inhibition on a Woolf-Hanes plot and on an Eadie-Hofstee plot. Exampie 7-4. (1) What wouId the conversion be after 10 minutes if the initial concenmtion of urea were decreased by a factor of 100? (2) What would be the conversion in a CSTR with the same residence trme, T, as the batch reactor? (3) A PFR? Example 7-5. What is the lotal mass of substrate consumed in grams per mass of cells plus what is consumed to form product? Is there disparity here? Example 7-6. Load the Living Example Problem. ( 1 ) Plot the concentration up to a time of 24 hours. Did you observe anything unusual? If so. what? 12) Modify the code ta carry out the fermentation in n fed-batch (semibatch reactor) in whrch the substrate is fed at a rare of 0.5 dm% and at concentration of 5 gfdmhto an initial liquid volume of 1.0 dm3 contain~nga cell mass with an initial concentra~ionof C, = 0.2 rng/dm3 and an initial substrate concentration of C,, = 0.5 rngldm3. Plot the concentration of cells, substrate, and product as a functron of time along with the mass of product up to 24 hours. Compare your result< with (1) above. (3) Repeat (2) when the growth uncornpetitively inhibited by the substrate with K, = 0.7 g/dm3. 14) Set C i = lU.000 gfdm3, and compare your results with the bare case. Example 7-3. This problem IS a gold mine for t h ~ n p slo he learned about the effect of alcohol on the human body Load the Polyn~arl~ LirrtJl: Exa~npleProgranr from the CD-ROM. ( 1 ) Start by varylng the initial doqes of nlcohol. 12) Next comider individuals who are ALDH enzyme
Chap. 7
Explorc the bloodalcohol simulation on the CD-ROM L i ~ i n gExample problem
Llulng E~ampfeProblem
Computer Modules Enzyme Man
455
Questions and Problems
deficient. which includes about 40% to 50% of Asians and Native Americans. Set V,,,,, for acetaldehydes between 10% and 50% of its normal value and compare the concentration-rime trajectories with the base cases. Hint: Read the journal anicle in the Summary Notes [Alcohol 35. p.1 (2005)l (h) Load the Ozone Poipiath Living Example Program from the CD-ROM. Vary the halogen concentrations and describe what you find. Where does PSSH break down? Vary the race constmts and other species concenuatians. (i) Load the GIowsh'cks Livinl: Example Problem from the CD-ROM. Vary the rate constants to learn how you can malie the luminescence last longer. Last shoner. CjJ Load the Russell's ICpm Polymath Living Examglc Program from the CD-ROM. Describe what wou!d happen if the victim rece~vedmore than one bite. In the cobra problem in Chapter 6 we saw that after 10 bites, no amount of antivenom would save the victim. What would happen if a victim received 10 bite5 from a Russell's riper? Replot the concentration-time trajectories for venom, FOP, and other appropriate species. Next, inject different amounts of antivenom to learn if it 1s possible to negate 10 bites by the viper. What is the number of bites by which no amount of antivenom w ~ l lsave the victim? (k) Load the Fer-de-Dance Polj~rnafhLivitrg Example Progmrn from the CD-ROM. Repeal 7- 1 Cj) for the Fer-de-Lance. (1) Load the Receptor Endocytosis Living Example Problem from the CD-ROM. Vary k,,, fR. and ,fL over the ranges in Table R7.31. Describe what you find. When will acute renal failure wcur? (m)List ways you can work this problem incorrectly. (n) How could yow make this problem more difftcult? ICM E;nzyme Man. Load the ICM on your computer and carry out the exercise. Performance number = (a) List ways you can work this probrern incorrectly. (hj How could you make this problem more difficult? (Fiml7e ~rmrrlnrrrs)Hydrogen radicals are important to sustaining combustion reactions. Consequently. if chemical compounds that can scnvange the hydrogen radicals are introduced, the flames can he extinpished. While many reactions uccirr during the cornbusttnn process. we shall choose CO flame< as a rnodcl system to illu~trate the procesq IS. Senkan er al.. Cinhrrrtir~rr n)lrl Flri~ne,69. I I3 (1987)J.In the ahrence of inhibitors
When HCI i< inttvThe last two reactions are rapid compared to the first TWO. duced to the flame, the fnllou rng additional reactions occur: H1 + C1.
+ HCI
H-
C1.
--t
HCI
456
Reaction Mechanisms, Pathways, B~areact~ons, and Bioceactars
Ch,
Assume that all reaction< are elzrncntary and that the PSSH holds for the OH .. and CI . radic;~l\. ( a ) Derive a rate law for the con$umptlon of CO when no retardant i s pre: (b) Derive an equation fur the concentratiun of !-I as - a function or assuming constanr concentration of 0,. CO. and H,O for both unin ited cornbustion and cumbustion with HCI present. Sketch H. vc time for both cases. (c) Sketch a reaction pathway d~agramfor this reaction. (dl List ways you can work (hi5 problem incorrectly. ( e ) How could you make this problem more difficult? More elabornte forms uf t h i ~problem can be found in Chapter 6. where
PSSN is not invoked.
W4, The pyrolysis of acetaldehyde i x believed to take place according to the lowing sequence:
CHO
+ CH,CHQ
"
> CC,.
+ 2C0 +
(a) Derive the rate expression for the rate of disappearance of acetaldeh -rAc.
Ih) Under what conditions does it reduce to Equation (7-3)? (c) Sketch a reaction pathway diagram for this reaction. (d) List ways you can work this problem incorrectly. (e) How could you make this problem mote difficult? W-sR (a) The gas-phase homogeneous oxidation of nitrogen monoxide (NO dioxide (NO:
Hall of Fame
),
is known to have a form of third-order kinetics. which suggests that reaction is elementary as written, at least for low partial pressures of nitrogen oxides. However, the rate constant k actually d f c r e f l ~ e7 ~ increasing absolute temperature. indicating an apparently negative acl tion energy. Because the activation energy of any elementztry reac must be positive, some explanation is in order.
Provide an explanation, starting from the fact that an active interned species. NO3,is a participant in some other known reactions that invr oxides of nrtrogen. Draw the reaction pathway. tb) The rate law for formation of phosgene. COCI2. from chlorine. Cl?, carbon monoxtde, CO, has the rate law
Suggest a mechanism for this reaction that is consistent with this rate and draw the reaction pathway. [Hint:CI formed from the dissociatioi C1, is one of the two active intmediate3.l (E) List ways you can work this problem incorrectly. (d) How could you make this problem more difficult?
Chap. 7
457
Questions and Problems
W-tiR &(me is a reactive gaa that has been ac~ociatedwith resptralory illness an$ decreased lung tunct~on.The follow~ngreactions are involved in ozone formation [D AIfcn and D. Shunnard, Green En~irreesirrg (Upper Saddie River, N.J.: Prentice Hall, 2002)j. Green enfinrering
i q primarily generated by combustion in he automobilc engine. Show that the steady-state concentration of Ozone 1s directly proportional to NOI and inversely proportinnal to NO. (b) Drive an equation for the concentration of ozone in soleIy in rerms of the initial cor~centrationsChO,O, CVO,,, and 0,"and the rate law pamrneteE. (c) In the absence of NO and NOZ,'the rate law for ozone genemtion i s
NO1 (A)
.
Suggest a mechanism. (dE List ways you can work this problem incorrectly. (e) How could you make this problem more difficult? W-7c ( T r i b n l o ~One ) ~f the major reasons for engine oiI degradation is the oxidation of the motor oil. To retard the degradation process, most oils contain an antioxidant [see blrl. Eng. Chrrn. 26, 902 (1987)l. Without an inhibitor to oxidation present. the suggested mechanism ar low temperatures is
Why you need to change the motor oil in your car?
ZRQ, --% inactive whew I? is an initiator and RH is the hydrocarbon in the oil.
fzj Motor Oil
When an antioxidant is added to retard degradation at low temperatures, the following additional termination steps occur:
A-+
R01
inactive
Reaction Mechanisms. Pathways. Bioreactions, and Bioreactors
W-8,
Chap. 7
(a) Derive a rate law for the degradation of the motor oil in the absence of an antioxidant at low temperatures. (bl Derive a rate law for the rate of degradalion of the motor oil in the presence of an antioxidant for low temperatures. ( c ) Hou. would your answer to part (a) change if the radicals I - were produced at a constant rate in the engine and then found the~rway rnto the oil? (d) Sketch a reaction pathway diagram for both high and low ternvratures. with and \\ ithout anrioxida~it. (el See the open-ended probleins on the CD-ROM for rnclre on this problem. (fJ List ways lo11 can work h i s problem incorrectly. (g) Hnw could you make this problem more difficull? Consider the applicarion of the PSSH to epidemiology. We shall Ireat each nf the following steps as elementary in that the rate will he proportional to the number of people In a particular state of health. A healthy person. H, can become ill, I, <pontaneously, such as 'by contracting xniallprrx q ~ ~ s :
H-LI
(P7-8.1 j
or he mag become ill through contact with another ill person: The ill perqon may become healthy: or he may expire: The reaction p e n in Equation (P7-8.4) is normally considered completely irreversrble. although the reverse reaction has h c n reported to occur. (a) Derrve an equatloll for the cleath rare. (b) A1 what concenlration of healthy people dws {he death rate hecome crit~ c a l ?[ A I I . ~When .: [HI = ( k , + k4)/k,.] (c) Comment on the validity of the PSSN under the conditions of part ~ h l id) If kt = In-' h-', k! = IO-Ih (people-!I)-', k c = 5 X 10- I0h, k, = 1 0 ~ ' ' h. and H,,= IOq peopl:. Uce Polymath to plot H, I, and D versus lime. Vay k, and d e r c r i k what you find. Check with your local djsease cn)~lml rtvirer or cearch the WWW ro modify the mndel and/or substitute appropriate ia l u e ~of k,. Extend the model laking into accounz what ytlu learn from other wurcec ( e . ~ .WWW). . ( e ) L I \ ~wa! s !nu can uork this prohlern incorrectly. (f) Hnw could !ou make rhis problem more difticult?
Chap. 7
Chemical Reaction Engineering in rhe Fwd Industry
459
Questions and Problems
(Posracid~frc~on in yagurl) Y0p.m is produced by adding t w strains ~ of bacteria (Lacrobecillusbulgaricnrs Sir-epiococcus thermopkilus) to pasteurized milk. At temperatures of 11O"E the bacteria grow and produce lactic acid. The acid contributes flavor and causes the proteins to coagulate, giving the characteristic propenies of yogurt. When sufficient acid has been produced (about 0.90%). the yogurt is cmled and stored until eaten by consumers. A lactic acid level of 1.10% is the limit of acceptability. One Iimit on the shelf life of yogun is "postacidification," or continued production of acid by the yogurt cultures during storage. The fable that follows shows acid production (% lactic acid) in yogun versus time at four different temperatures. Time IdayrJ
3YF
4PF
4PF
5PF
I
1 02 1.03 1.05 1.09 1.09
1.02 1 .OS 1.a6 1.10 1.12 1.12 1.13 1.14 1.16
1.02 1.14 115
1.02 1.19 1.24 1.26 3.31 1.32 1.32 1.32 I .M
I4
28 35 42 49 56 63 70
1.10
1.09 1.10
1.70
1.22 1.22 1.22 1.24 I.?5
1.26
Acid production by yogurt cultures is a complex biochemical process. For the purpose of this problem, assume that acid production follows first-order kinetics with respect to the consumption of lactose in the yogurt to produce lactic acid. At the start of acid production the lactose concentration is about 1.5%- the bacteria concentsdtion is 1O1keItsldm3,and the acid concentration at which all metabolic activity ceases is 1.4% lactic acid. {a) Determine the activation energy for the reaction. (b) How long would it take to reach 1.10% acid at 3B°F? (c) If you left yogurt out at room remprature, 77T- how long would it take to reach 1.10% lactic acid? ( 6 ) Assuming that the lactic acid I s produced in the stationary state. do the data fit any of the modules develoged in this chapter? [Problem developed by General Mills. Minneapolis, Minnesota] (e) List ways you can work this problem incorr~ctIy. ( f ) How could you make this problem more difficult?
460
React~onMechanisms, Pathways, Biomactions, and Bioreactors
Cl
P7-10, Derive the rate law? for the following enzymatic reactiorn and sketct compare with the piot shown in Figure E7-3.1(a). (a) E + S
(b) E + S
E o S P P+S EE.S EE.P+P+E EE.S 2 EE. P P + E
r
(c) E + S
(d) E+S,
8 .as,
E-S,-t-S, E*S,S,
a E*SlS2
a P+E
(Ems),
(e) E + S (
(0 E + S
a E*S+
a
E+P E ~ P E.S+P EE.S E*P+S E*S (g)E+S p E * S 4 P+E E*P
r
a
((EmS),+P,
( E E . S ) ~ + P ~ E+ (h) Two products E, + S (Glucose)
Eo* S+E, + P, ($-Lactone) 0,+ E,+EoPl +En + P2(H202) (i) Cofactor (C) activation E+S E*S-tP E + C r EE.C E*C+ S+P+E*C 0) Using the PSSH develop rate taws for each of the six types of en? reactions. (k) Which of the reactions (a) through (j), if any. lend themselves to ana by a Lineweaver-Burk plot? W-EIB Beef cataIase has been used to accelemte the decamposiaion of hydrogen oxide to yield water and oxygen [Chem. Eng. Educ., 5, 141 (197 I)]. The centration of hydrogen peroxide is given as a function of time for a rear mixture with a pH of 6.76 maintained at 3VC.
P7-12,
(a) Determine the Michaetis-Menten parameters V,,, and KM. (b) If the total enzyme cancentmtion is tripled. what will the substrate centration be after 20 minutes? (c) How could you make this probIem more difficult? (d) List ways you can work this problem incorrectly. It has been observed that substrate inhibition occurs in the following e
rnatic reaction:
substrate inhibition i s consistent with the in Figure W-12 of -r, (mmol/L.min) versus the substrare concenm
(a) Show that the rate law for
5 (rnmollt).
Chap. 7
461
Questions and Problems
Figure P7-12 Michealis-Menten plot for substrate inhibition.
(b) If this reaction i s c a m 4 out in a CSTR that has a volume of 1OOO dm3,to which the volumewic flow rate is 3.2 dm3/min,determine the three possible steady states, noting, if possibIe, which are stable. The entrance concentration of the substmte is 50 rnrnoYdm?. What is the highest conversion? (c) What would be the effluent substrate concentration if the total enzyme concentration is reduced by 33%? (d) List ways you can work this problem incorrectly. (e) How could you m&e this problem more difficult? W-13R The following data on bakers' yeast in a particdar medium at 23.4'C and various oxygen partial pressures were obtained: QO2
Po:
Q,, (no sulfanilamide)
o,o
0.0
0.5
23.5 33.0
I .O 1.5 2.5 3.5 5.0
(20mg suIfanilamidelmL added to medium)
42.0
0.0 17.4 25.6 30 8 36.4
43.0 43.0
39.6 40.0
37.5
Poz = Oxygen partial pressure. mmHg; Qo! = oxygen uptake rate, pL of O2per hour per mg of cells.
la) Calculate the Q, maximum (V,,), and the Michaelis-Menten
constant
KM. (Anx: V,, = 52.63 pL OaJh-mg cells.) (b) Using the Lineweaver-Burk plot, determine the type of inhibition suifanilarnide that causes O2 to uptake. (c) List ways you can work this problem incomecrly. (d) How couId you make this problem more difficult?
191-14, The enzymatic hydrolysis of starch was carried out with and without maltose and a-dextrin added. [Adapted fmm S.Aiba. A, E. Humphrey, and N.F.Mills. Bdochemicai Engineering (New York: Academic Press, 1973).
462
Reaction Mechanisms. Pathways. Biowactions, and Boreactors
Chap. 7
Starch + a-daxtrin +JLimit dextrin 4 Mdtose
No Inhibition
cs (@dm3)
12.5
-r, (relative) Maltose (I = 12.7 rng/dm3)
lm
Cs(ddm').
10
-r, (relative)
77
a-dexain (I
9.0 92
5.25
62
4.25
70 2 .O 38
1 .O
29 1.67 34
= 3.34 mgldmi)
CJ (@dmJ)
33
10
-r, IreIative)
116
85
3.6 55
1.6
32
Deternine the types of inhibition for maltose and for a-dextrin. W-15, The hydrogen ion, . ' H binds with the enzyme (E-)to activate it in the form EH. H+ also binds with ER to deactivate it by forming EH;
EH+S
xu
EHS -EH+P,
K,=
JEHS) (EH)(S)
Figure W-15 Enzyme pH dependence.
where E- and EH; are inactive. Determine if the preceding sequence can explain the optimum in enzyme activity with pH shown in Figure Pf-15. (a) List ways you can work this problem incorrectly. (b) How could you make this problem more difficult? W-I& The production of a product P from a particular gram negative bacteria fob lows the Monod .growth law
(a)
with p,,,,, = 1 h-' Kv = 0.25 @dm3, and Y,, = 0.5 gig. The reaction is to be carried out in a batch reactor with the ~nitiafcell concentralion of C , = 0.1 g/dm7and substrate concentration of Cd = 20 gldrn3.
Plot -r,, -r,, C,, and C, as a function of time. (b) Redo part (a) and use a logistic growth law.
and plot C,and r, as a function of time. The term C, i s the maximum cell mass and is called the carrying capacity and is equal to C, = 3.0 gJdm3. Can you find an analytical solution for the batch reactor? Cnmpare with part la) for C, = Y,, C, + C.,
C h ~ p7.
Oueslions and Problems
483
The reaction is now to be carried out in a CSTR with Cd = 20 gldrnhand C* = 0. What is the dilution rate at which washout occurs? (d) For the conditions i n pan Ic), what is the dilution rate that wil! give the maximum product rate ( g h ) if Y,, = 0.15 glg? What are the concentrations C, C,, C, and -r, at this value of D? (e) How would your answers to Ic) and (d) change if cell death could not be neglected with k,, = 0.02 h-'? (0 How wouid your answers to (c) and (d) change if maintenance could not be neglected with m = 0.2 @{dm3? (g) List ways you can work this problem incorrectly. (h) How could you make this problem more difficult? P7-17,, Redo Problem P7-16 (31, (c). and (dl ming the Tessier equation (c)
with pmux= I and k = 8 gldm'. (a) List ways you can work this problem incorrectIy. (b) How couId you make this problem more dificwlt? P7-18,$The bacteria X-I1 can be described by a simple Monod equation with,,p = 0.8 h-I and Kkl = 4. Y,, = 0.2 gig, and Yd = 2 g l g . The process i~ carried out in a CSTR in whlch the feed rare is 1000 dm7/h at a substrate concentration of III €/dm'. (a) W h a ~size fernenlor is needed to achieve 90% converqinn of the subctrate? What i s the exitlnp cell concentration? tb) How would your answer to (a) change if i\ll the cells were filtered out and returned to the feed $Iream? (c) Consider now rwo 5000 d n i T S T R s connecl in series. What are the exiting concentrations C,. C,, and C, from each o f the reactors7 (d) Determine. if pos~ible,the volumetrrc RON rate at ahlch wa~h-outoccurs and also the flow rate at which the cell prnduction rate tC, u,) in grams per day is a maximum. (e) Suppse you could use the two 5000-dm' reactors as batch reacrors tha~ take two hours to empty, clean, and fill. What would your producilon rate be in (grams pzr day} rf your initial cell concentration is 0.5 @dm7? How many 500-dm' reactors would you need ro match the CSTR pductlon rate? (0 List waqs you can work t h ~ sproblem incorrectly. [gl Wow could you mahe this problem Inore dir'ficul~? P7-19B Lactic acid is produced by a L.ucrohncill~rsspecier; cultured in a CSTR.TO increase the cell concentration and production rate. most of the cells in the reactor outkt are recycled to the CSTR. such that the cell concenrratlon in the product uream is 10% of cell concontrat~nnin the reactor. Find the oplimum dilut~nnrate that will maximize the rate of lactic acid production in the reactor. Hot%.dms rhis optimu~rrdilu~ionrate change if the exlt cell concentration fraction cs changed? (r, = ( m ~ +i +)C,]
(Contribi~tedby Profe\wr D. S. Komp;~ld.U~iiversikyof Colorado) (a) List way.; yov can work r h l h problem inuor~,ectly 4h) Hnw could ynrl make this prrlhlem more difficult?
464
Reactioe Mechanisms, Pathways, Bioreactions, and Bioreactors
Ct
W-20 (Adapted from Figure 3-20 of Aiba
et al.) M i x e d cultwrer of bacteri~ develop into predictor prey relationhhips. Such a s y w m might be cultur AlcnligenrsJ(~er~o/i.s 4 prey ) and Colpi(1ilirn con~pvltmiC predator) Considel different c e h X I and X? in a chernostat. Cell X , only feeds on the sub! and cell XZ only feed5 on cell X,. X, X I+ Substrate Cells M More XI Cells + Product 1 Cells X,
X2 + X I
+ Product 2
- > More X,CelIs
The growth laws are rgXl = p l C X Iwith p1=
F I rnaxC1 -
k;,,+C,
rate of D = 0.04 h-I, an entering substrate concentwtic 250 rnddrn3, initial concentrations of cells of X,,= 25 mgldn13. XZi rngldrn' and of substrate C,, = 10 mg/drn7, plot C X t .Cx:, r,,, , r and C,35 a function of time. Vary D between 0.0 L h-I and 0. I h-I. and describe what you find. Vary Co and C,, from the base case and delicribe what yott find. List ways you can work this problem incorrectly. How could you make this problem more dificult?
(a) For a dilution
(b) (c)
(d) (e)
Additional Informtint1
vma\,~ 0 . h-l. 5 pmm2 =O.L and Y? =
W-21,
I hL1,KMI= KM2= 10 mgldm3. Y , = yxI,,=It.l~
Y,:, , = 0.5.
The following data were obtained for Pyrorlictiuna ncculrum at 9S°C.RI was carried out 1r1 the absence of yeast extract and run 2 with yeast extl Both mns initially contained Na2S. The vol 9 of the growth product H,S lected above the broth was reported as a function of time. [Ann. N. F.I: A, Sci., 506,52 (1987).] Run 1:
Run 2: i'ime(hj
Cell Densie
Icells/mL) $F
HZS
x
'.' 0.1
0.7
10
I5
20
30
40
50
6
f1
80
250
350
350
250
-
0.7
08
13
4.3
7.5
1 1.0
1
Chap. 7
Quest~onsand Problems
465
(a) What i s the lag time with and without the yeast extract? th) What is the drt'ference in the specific growth rates, pm,,, of the bacteria with and w~thoutthe yeast extract'! Ec) How long is the stationary phase? (d) During which phase does the majarity production of H2S occur? (e) The liquid reactor volume in which these batch experiments were carried out was 0.2 dm3.IF this reactor were converted to a continuous-ffow reactor, what would be the corresponding wash-out mre? (0 List ways yw can work this problem incorrectly. (g) How could you make this problem more difficult? W-25,Cell growth with uncornpet~tivesubstrate inhibition is taking place in a CSTR. The cell growth rate law for this system is
with ,p , = 1.5 h-I, K, = 2 gldm-', K, = SO g/dm3. Cd = 30 gldm3, Y,,,=0.08, CCq= 0.5gldm3. V = 5 0 0 d m 3 , a n d D = 0.75 h-I. (a) Make a plat of the steady-state cell concentration C, as a function of D. What i s the volumetric flow rate (drn31h) for which the cell production rate is a maximum? (b) What would be the wash-out rate if Cm = 07 What is the maximum cetl product~onrate and how does it compare with that in part {a)? (c) Plot C, as a function of D on the same graph as C, us. D? What do you observe? Id) It is proposed to use the 500-dm3 batch reactor with Cfl = 30 g/drn3 and Cd= 0.5 @dm3.Plot C,, C,, r,, and -r, as a function of time? Describe what you find. (el It is proposed to operate the reactor in the fed-batch mode. A 10-dm3 solution i s placed in the 500-dmsreactor with C* = Z.0/dm3 and Cd = 0.5@dm3. Substrate is fed at a rate of 50 dm% and a concentration of 30 g/dm3. Plot C, C, r,. r,, end (VC,)as a function of time. Can you suggest a better valume~ricfeed rate? How do your results compare with pan (d)? (f) L ~ s tways you can work this problem incorrectly. (g) How could you make this problem more difficult? W - a A A CSTR IS k i n g operated at steady state. The cell growth follows the Monod growth Law without inhibition. The exiting substrate and cell concentrations are measured as a function of the volumetric flow rate (represented as the dilution rate), and the resuits are shown below. Of course, meawrernents are not taken until steady state is achieved after each change in the flow rate. Neglect substrate consumption for maintenance and the death rate, and assume that Y,,, is zero. For run 4, the entering substrate concentration was 50 gJdm3 and the volumetric ffow rate of the subsrrale was 2 dm3/s.
466
Reaction Mechanisms, Pathways, Bioreactions, and Bioreactors
Chap 7
(a) Determine the Monod growth parameters ha, and K,. ( b ) Estimate the stoichiometric coeficients, Y,,,and I;,,. (c) List ways you can work this problem incorrectly. (d) How could you make this problem more difficult? W-24c In bio!echnology industry, E. coli is gown aerobically to highest possible concentrations in batch or fed-batch reactors lo maximize production of an intracellular protein product. To avoid substrate inhibition, glucose concentratlon in rhe initial culture medium is restricted to 100 gldrnqin the initial charge of 80-dm3 culture medium in a 100 dm3 capacity bioreactor. After much of thls glucose is consumed, a concentrated glucoqe feed (500 @dm!) will be fed into the reactor at a constant volumetric feed rate of 1.0 dm3h. When the d~ssolvedoxygen concentration in the culture medium falls below a critical value of 0.5 mg/drnq, acetic acid is produced in a powth-associated mode with a yield coefficient of 0.1 g acetic acidg cell mass. The product. acetic acid intibits cell growth linearly. with the toxic concentration (no cell growth) a1 C, 10 p/drnq. Find the optimum volumetric Raw rate that wilf maximix the o~erallrate cell mass production when the hioreactor is filled up and if the feed is turned on after glucose falls below 10 gJdm3. J n ~ u l u r n concentration is I g ce11s/dm3.
Y I , = 0.5gig. Y,, = 0.3 glg, qOl = l.O g oxygen/g cell
Oxygen mass transfer rate, li,a = 500 h-1: saturation oxygen concentration =
rA1 = 7.5 mg/dm3 and
Web Clint
r, =
Illcrease the value of mass transfer rate (up to 1000) or the saturation oxygen concentration ( u p to 40 mgldm7)to see if higher cell densities can he obtained in the fed-hatch reactor, (a) List ways >ou can work rhis problem ~ncorrectly. (b) How could you make this prohlem more difficult? (Conrrihuted by Profes~orD. S. Kompala, University of Color:jdo+) P7-25c (Open-ended probIem) You may have to look up/guess/vary some of the constants. If methanol is ingesred, it ran be metaboltzed ta formaldehyde, which can cause blindness rf the -formaldehyde reaches a concentration of 0.16 gldm" of fluid in rhe body. A concentration of 0.75 gldrn? will be lethal. After all the methanol has been removed from the stomach, the primary treatment is to inject ethanol intravenously to tic up Icornpetitive inhibition) the enzyme alcohol dehydrogenase (ADHE so that methanol is not converted to formaldehyde and is el~rninatedfrum h e body through the kidney and bladder (k,). uill assume as a Sirst approximation that the body i s n wctl-mixed CSTR of 40 dm' (total bud! fluid ). In Sectton 7.5, we applied a more rigarnuc model.
Chap. 7
467
Questions and Problems
The following reaction scheme can be applied to the M y ,
Ethanol (E)
Alcohol Dchydrogenase AIcohoI
Dchydmgensse
Ethanol Methanol FormaIdehyde
Reference Shelf
Acetaldehyde (P,)+ Water Formaldehyde (P1)+ Water
(2)
Excreuon
The complete data set for this reaction is given on the CD-ROM, P7-25 and the open-ended problem H. I U. After mnning the base case. vary the parameters and describe what you find. W-2% Pharmacokinetics. T h e following sets o f data give the concentration of different dmgs that have been administered either intr;lvenously or orally. ( a ) For Giseofulvin and Ampicilin determine the phmacokinetic parameters for a two-compartment model. Plot the drug concentration in each compartment as a function of time. You may assume that VF = QC. (b) Evaluate the mdel parameters for AminophyHine using a two-compartment model for the intravenous injection and a three-compartment model for oral administration. (c) If the distribution phase of a drug has a half-life of 1.2 hours and an elirmnation half-life of 5 hours. plot the concentration in both the peripheral compartment and the central companment as a function of time (I) GiseofirR,in 122-mg intravenous dose (used for fungal infec~ions)
(2) 2-pyridnium adoximine methochloride 10-mgkg intravenous dose
(3) Ampicilin 500-mgintravenous dose (used for bacterial infections)
(4) Diazepnm 1M-mg oral dose (sleeping pills)
( 5 ) R~ninophylline[used for bronchial allergies) r (R) Oral
P (pmoI/dm')
Tntlavenous C {pmolldm')
0.11 2 3 4 6 8 0 5 11 13 IS 12 6.5 26 22 17.5 14 12.5 7 5
468
Reaction Mechanisms. Pathways. Bioreactions, and Bioreactors
Ch.
JOURNAL CRITIQUE PROBLEMS WC-I
Could the mechanism
NO,
I + SO3+-N,Q2 2
+ So2
3
N202+ SO24 N2Q+ SO, NO, +NO + 2 N 0 2
also explain the results in the article AlChE J., 49f 1 ), 277 (2003)' In the anicle "Behavior Modeling: The Use of Chemical Reaction Kinetic Investigate Lordosis in FcmaIe Rats." J. Tkcro. Bioi., 174, 355 (1995). \ would be the consequences of product inhibition of the enzyme? W C - 3 Does the article Ind. Eng. Cltcrn. Res., 39. 1837 (2000). provide conclu evidence that no potassium 0-phthaiate decomposition follows a first-o rate law? WC-4 Compare the theoretical curve with actual data points in Figure 5b [Bier, nol. Bioeng., 24, 329 (1982)], a normalized residenc+time curve. Note the two curves do not coincide at higher conversions. First. rederive the equation and the normalized residence-time equations used by the autk and then, using the values for kinetic constants and lactase concentration c by the authors. see if the theoretical curve can be duplicated. Linearize normalized residence-time equation and replot the data. the theoretical ci in Figure 5b. and a theoretical curve that is obtained by using the const given in the paper. What is the simplest explanation for the results observ WC-5 In Figure 3 [Biotechnol. Bioeng.. 23, 361 /Z981)], 11V was plotted aga (lIS)(IIPGM) at three constant 7-ADCA concentrations. with nn attemp extract V,,, for the reaction. Does the V,,, obtained in this way confom the true value? How is the experimental V, affected by the level of PGh the medium?
P7C-2
Additonal Homework Problems
* Complete Data Set Hospital E. R.
W-25
Methanol poisoning data and questions for the hospital emergency
K
(E.R.). New ProbIems CDP7-A
Anaerobic fermentation of glucose to produce acetic acid [From P D. S. Kim. Chemical Engineering Department, University of Tolei
Oldies, But Goodies-Problems from Previous Editions
CDP7-B
Suggest a mechanism for the reaction
[2nd Ed. W-8,]
Chap. 7
469
Supplernen!av Reading
Enzymes
CDW-C
Determine the diffusion rate in an oxygen fermentor. [2nd Ed. P12-12,l.
Bioreactors CDP7-Dc CDW-E,
CDI7-FA
CDW-6,
CDW-Hc
Plan the scale-up of an oxygen fernentor. [2nd Ed. P12-I&] Assess the effectiveness of bacteria used for denitrification in a batch reactor. [2pd Ed. P11- I g8] An understanding of bacteria transpn is vifal to the efficient operation of water Rooding of petroleum reservoirs. [R.Lappan and If. S. Fogler, 5PE Pmducriun Eng., 7(2), 167 (1992)l. Analyze the cell concentration time data. [3rd Ed. W-28,] Design a reactor using bacilIusJ7nvan to process I0 &/day of 2 M furnaric acid. [3rd Ed. W-31,] Find the inconsistencies jn the design of the hydrolization of fish oil reactor using lipase. [3rd Ed. P7-10,J
Polymerization Anionic polymerization. CalcuIate radial concentration as a function of time. [3rd Ed. P7-183 PoIymerization. Plot distribution of molecular weight using Flory statistics. [3rd Ed. P7- E5] Determine the number average degree of polymerization for free radical polymerization. [3rd Ed. P7-161 Free radical polymerization in a PFR and a CSTR. (3rd Ed. P7-171 Anionic polymerization. Calculate radial concentration as a function of time. [3rd Ed. P7-181 Anionic poIymerization carried out in a CSTR. [3rd Ed. W-191 Anion~cpolymerization. Comprehensive problem. [3rd Ed. P7-201 Use Elory statistics for molecular weight distribution. [3rd Ed. P7-2 t ] Anionic polymerization when initiator is slow to dissociate. [3rd Ed. F7-221 Rework CDW-P, for a CSTR. [3rd Ed P7-23BJ
New Problems on the Web Green Engineering hnblem
CDPZ-New From time to time new problems relating Chapter 6 material to everyday interests or emerging technologies will be placed on the web. Solutions to these problems can be obtained by e-mailing the author. AIso one can go to the web site, nw.mwan.edidgrmengineerin~,and work the home problem on green engineering specific ta this chapter.
11
SUPPLEMENTARY READING Web
Links
Review the following web sites: www.cclls.c~m www.enpmes.corn www.pharmacnkinerics.corn
Steady-State Nonisothermal Reactor Design If you can't stand the heat. get out of the kitchen. Harry S Truman
Oveniew. Because most reactions we not carried out isothermally, we now focus our attention on heat effecrs in chemical reactors. The basic design equations, rate laws, and stoichiomctrjc relationships derived and used in Chapter 4 for isothermal reactor design are still valid for the design of nonisothermal reactors. The major difference lies in the method of evaluating the design equation when temperature varies along the length of a PFR or when heat is removed from a CSm.In Section 8.1, we show why we need the energy balance and how it will be used to solve: reactor design problems. h Sectior18.2,we develop the energy balance to a point where it can be applied to different types of reactors and then give the end result relating temperature and ~on\~ersion or reaction rate for the main types of reactors we have been studying. Section 8.3 shows how the energy balance is easily applied to design adiabatic reactors, while Section 8.4 deveIops the eneey balance on PFRsPBRs with heat exchange. In Section 8.5, the chemical equilibrium limitation on conversion is treated along with a strategy for staging reactors to overcome this limitadon. Sections 8.6 and 8.7 describe the: algorithm for a CSTR with heat effects and CSTRs with multiple steady states, respectively. Section 8.8 describes one of the most important tapics of the entire text. multiple reactions with heat effects, which is unique to this textbook. We close the chapter in Section 8.9 by considering hoth axial and radiat concenrrations and temperature gradients. The Prqf~ssiorrolAefEirnce Shdf R8.4 on the CD-ROM describes a typical nonisothermal industrial reactor and reaction, the SO2 oxidation, and gives many practical details.
472
Steady-State Nonisothermal Reactor Design
CI
8.1 Rationale To identify the additional information necessary to design nonisothermal tors, we consider the following example, in which a highly exothermic seas is carried out adiabatically in a plugflow reactor.
I(
Example 8-1
What Additional hiformation Is Required?
Calculate the reactor volume necessary for 70% conversion.
The reaction is exothermic and the reactor is operated adiabnticarly. As a resul temperature will increase with conversion down the length of the reactor.
-1
1. Mole Balance (design equation):
2. Rate Law:
--I
V
we
know that k is a funct~onoh temperature, T.
3. Stoichiometry (liquid phase):
u = uo
C, = C,(1
v
-X)
(E8-
4. Combining:
Combining Equations (Eg-1. If, (ES-I.2). and (EX-1.4) and canceling the er ing concentration. C,,, yields
Because T varies along the length of the reactor, k will also vary. which not the case for isothermal plug-flow reactors. Combining Equations (ESand (E8-f -6)gives us
Sec. 8.2
I
473
The Energy Balance
Why we need the energy balance
We see that we need another relationship relating X and Tor T and V to solve this equation. The mcrgy bnlance wiIl provide u.7 with this relationship. So we add another step to our algorithm. this step is the energy baIance.
5. Energy Balance: In this step. we will find the appropriate energy balance to relate temperature and convecsian or reaction rate. For example, if the reaction is adiabatic. we will show the temperature-conversion relationship csn be wtitten in a form such as ,
I i I
T,,
= Entering Temperature = Heat of
We now have all the equations we need perature profiles.
React~on
C ,
= Heat Capacity
to solve
for the conversion and tem-
8.2 The Energy Balance
I
8.2.1 First Law of Thermodynamics We begin with the application of the first law of thermodynamics first to a closed system and then to an open system. A system is any bounded portion of the universe, moving or stationary, which is chosen for the application of the various thermodynamic equations. For a closed system, in which no ma2s crosses the system boundaries, the change in total energy of the system, d E , is equal to the heat flow to the system, 6Q,minus the work done by the system on the surroundings, 6W For a closed system, the energy balance is
dk=
EQ-~W
(8-1)
The 6's signify that SQ and FW are not exact differentials of a state function.
Energy balance on an open system
The continuous-flow reactors we have been discussing are open systems in that mass crosses the system bundary. We shalI carry out an energy halance on the open system shown in Figure 8-1. For an open system in which some of the energy exchange is brought about by the flow of mass across the system boundaries, the energy balance for the case of on(v one species entering and leaving becomes 7
-
-
<
Rate of
Rate of flow accumulation o f heat to of energy ' the system within the from the system j surroundings
-
Rate of work done by the system on the surroundings
-
+
-
Rate o f energy added to the system by mass flow into the system
-
-
Rate o f energy leaving system by mass flow out of
the system
A
(8-2) Typical units for each term in Equation (8-2) are (Joulds).
Steady-State Nonisothermal Reactor Dasian
Chap. 8
Figure 8-1 Energy balance on a well-mixed open system: schematic.
We will assume the contents of the system volume are well mixed, an assumption that we could reIax but that would require a couple pages o f text to develop. and the end result would be the same! The unsteady-state energy balance for an open well-mixed system that has n species, each entering and leaving the system at their respective molar flow rates F, (moles of i per time) and with their respective energy Ei (joules per male of i ) , is
The s~aningpoint
We will now discuss each of the terms in Quation (8-3). 8.2.2 Evaluating the Work Term It is customary to separate the work rerm, W , into flow work and orher work, & . The term W,, often referred to as the shaft ~ ~ o rcould k, be produced from such things as a stirrer in a CSTR or a turbine in a PFR. Flo~buwrk is work that is necessary to get the mass into and our of the system. For example. when shear stresses are absent, we write
[Rate of flow work] Flow work and shaft wurk
-
where P i s the pressure (Pa) [ I Pa = 1 Newton/m2 = 1 kg d s 2 / m 2 ] and V, is the specific molar volume of species i (rn3/lmol of i). Let's look at the units of the flow work term. which is
where Fi i s in molJs. P is Pa 1( 1 Pa = 1 Newton/m2), and
-
F;P.Y,
ciis rn3/rnol.
~ o lNenton . , rn' [=I n- - (Newtan*m).-1 = JouEesls = Watts s
?,
mol
S
Set, 8.2
The Energy Balance
475
We see that the units for flow work are consistent with the other terms in Equation (8-2), i.e., Its. Jn most instances, the flow work term is combined with those terns in the energy balance that represent the energy exchange by mass flow across the system boundaries. Substjturing Equation (8-4) into (8-3) and grouping terms, we have
The energy Ei is the sum of the internal energy (U,), the kinetic enerm [u,2/2), the potential energy (gzi), and any other energies, such as electric or magnetic energy or light:
In almost all chemical reactor situations, the kinetic, potential, and "other" energy terms are negligible jn comparison with the entbalpy, heat transfer, and work terms, and hence will be omitted: that is.
Wc recall that the enfhalpy, H,(Jlrnol), is defined in terms of the internal energy U,(Jlmol). and the product PV, (1 %.m3/mol = 1 Jlmol):
Hi = ui+ pQi
Enthalpy
(8-8)
Typical units of Hjare
Ib mol i
J or Btu or cal (Hi) =mol i
mol i
Edthalpy carried into lor out of) the system can be expressed as the sum of the net internal energy carried into (or out of) the system by mass Aow plus
the flow work:
F,H,
=
F,(u,+PV,)
Combining Equations (8-5), (8-T), and (8-8). we can now write the energy balance in the form
The energy of the system at any instant in time, .k,!,, is the sum of the products of the number of moles of each species in the system multiplied by their respective energies. This term will be discussed in more detail when unsteady-state reactor operation i s considered in Chapter 9.
476
Steady-State Nonisothermal Reactor Design
Ch
We shall let the subscript "0" represent the inlet conditions. Un scripted variables represent the conditions at the outlet of the chosen sy:
volume.
F
Ti ' i-
4
,
I
FH.,&~
out
In Section 8.1, we discussed that in order to solve reaction enginec problems with heat effects, we needed to relate temperature, conversion, rate of reaction. The energy balance as given in Equation (8-9) is the most venient starting point as we proceed to develop this relationship. 8.2.3 Overview of Energy Balances What is the Plan? In the following pages we manipulate Equation (8-1 order to apply i t lo each of the reactor types we have been discussing, b PFR, PBR, and CSTR. The end result of the application of the energy bal to each type of reactor is shown in Table 8-1. The equations are used in St af the algorithm discussed in Example E8-1.The equations in Table 8-1 I temperature to conversion and molar Row rates and to the system pararnr such as the overall heat-transfer coefficient and area. Ua, and correspor ambient temperature, T,. and the heat of reaction, AHRnF
(L) = 0) CSTR, PFR, Batch, or PBR. The relationship between con\ sion. XEB,and temperature for W* = 0,constant CpI ' and ACp = 0,is
1. Adiabatic
ITS- I .A
End results of manipulating the energy balance Sections 8.1.1 to 8.4, R 6, and 8.8.
(T8-1 .E For an exothermic reaction (-AHR,) > 0
T
70
-
2. CSTR with h a t exchanger, UA (T, T ) , and large coolant flow rate.
Sec. 8.2
477
The Energy Balance
T.~BLE 8-1.
E ~ E R GBALAXCES Y OF COMMON RE~ZCTORS (~Y)h'Tlhf El))
3A. PER in terms of conversion
38. PFR in terms of conversion
3C. PBR in terms of molar BOW rates
3D. PFR in terms of molar flaw rats
5 . For Semibatch or unsteady CSTR
6.
For multiple teactlons in a PFR
dT
- Ua( T, - T)+ Xr,AH,,, 5 S,CrI
(TS-l.J>
478
Steadystate Nonisothermal Reactor Design TABLE8-1.
Chap. El
ENERGY BALANCES OF COMMONREACTORS (CO-D)
7. For a variable codant temperature, T,
(7-8-1.K) These are the equations that we will use to sokve reaction engineering problems with heat effects. - - - - - - - + - - - - - - - - + - - - - -
[Nomenclaturr: I: = overall heat-transfercoefficient, (Jh? * s K):A = CSTR heat-exchange area, Im2). a = PFR heat-exchange area per volume of reactor. (m21m'l; CS = mean heat capacity of species i,(JlmolK): C p = the heat capacity of the coolant.
(JkJlkglK),ni, = coolant flour rate, tkgls); AH,, = hear of reactLon. Illmol):
AH;, =
kD
a
a
j in reaction i.(JEmoll;
e
b
+ f ~ -;- H i
Q
1
Jimol.4: AH..,, = heat of reaction w n species
= heat added to the reactor, (Jls); and
C -cPD+;cpc
Up=
-
illrno1.A * K ) AII other syrnhols are as dehned in
Chapter 3.1
Examples on How to Use Table 8-1. We now couple the energy balance equations in Table 8-1 with the appropriate reactor mole balance, rate law, smichiometry algorithm to solve reacdon engineering problems with heat effects. For example, recall rate law for a first-order reaction, Equation (Eg-1.5) in Example 8- 1 .
If the reaction is carried out adiabatically. then we use Equation IT&I .B) for the reaction A d B in Example 8-1 to obtain Adiabatic
Consequently. we can now obtai~.-r, as a function of X done by first choosing X. then calculating T from Equation (TE1 .B). then calculating k from Equation (E8-1.3). and then finally calculating (-r,) from Equation (Eg-1.5). Choose X 4 calculate T + calculate k
x Lel'enspiel plot
F~~ + caIcuIate -rA + calculate -
- ?'*
We can use this sequence lo prepare a table of (FA,+-r,) as a function of X . We can then proceed to size P F R ~and CSTRs. In the absolute worst case scenario, we could use (he techniques in Chapter 2 (e.g.. Levenspiel plots or the quadrature formulas in Appendix A ) .
Sec. 8.2
The Energy Balance
479
However, instead of using a Levenspiel plot, we will most likely use Polymath to solve our coupled differential energy and mole balance equations. If there is cooling along the Iength of a PFR. we could then apply Quation (T8-I .€) to this reaction to arrive at two coupled differential equations. Non-adiabatic
PFR
which are easily solved using an ODE solver such as Polymath. Similarly, for the case of the reaction A + B carried out in a CSTR. we could use Polymath or MATLAB to solve two nonlinear equations in X and T. These two equations are combined moIe balance Non-adiabatic CSTR
and the application of Equation (T8-3 .C), which is rearranged in the form
why hother' Here is why'!
From these three cases, (1) adiabatic PFR and CSTR, ( 2 ) PFR and PBR with heat effects. and (3) CSTR with heat effects, ane can see how one couples the energy balances and mole balances. In principle, one could simply use Table 8-1 to apply to different reactors and reaction systems without further discussion. However, understanding the derivation of these equations will greatly facilitate their proper application and evaluation to various reactors and reacfion systems. ConsequentIy, the following Sections 8.2. 8.3. 8.4, 8.6. and 8.8 will derive the equations given in Table 8-1. Why bother to derive the equations in Table 8-1 ? Because I have found that students can a p l ~ these l ~ equations mucll more accurately to solve reaction engineering problems with heat effects if they have gone through the derivation to understand the assumptions and manipulations used in arriving at the equations in Table 8.1. 8.2.4 Dissecting the Steady-State Molar to Obtain the Heat of Reaction
FEow Rates
To begin our journey, we start with the energy balance equation (8-9) and then proceed to finally arrive at the equations given in Table 8-1 by first dissecting two terms.
480
Steady-State Noniscthermal Reactor Design
!. The molar Row sates. F,and FA, 2. The molar enthalpies, Hi. H,,,[H, = lnteractrvc
H,(n,and Ha
Ch
HAT,)]
An animated version of what foIlaws for the derivation of the energy ance can be found in the reaction engineering modules "Heat Effects 1" 'Weat Effects 2" on the CD-ROM.Here equations move around the sc
making substitutions and approximations to arrive at the equations show Table 8-1. Visual learners find these two ICMS a useful resource. We wiIl now consider Row systems that are operated at steady state. Cornpu:~: Modules steady-state energy balance is obtained by setting (dE',,,/dr)equaI to zel Equation (8-9)in order to yield
*~-'v.h
Steady -state energy balance
To carry out the manipulations to write Equation 18-10) in terms of the he reaction, we shall use the generalized reaction
The inlet and outIet summation tems in Equation (8- 10) are expanded. re: tively, to
In:
Z HIoF,o = HAoFAo+ ffBD FBo + Ha F*,, + H m Fm + H10 FIO
IF
and
We first express the molar Row rates in tems of conversion. In general. the molar Row rate of species i for the case of tion and a stoichiometric coefficient v, is
no accun
F, = FA,(Oi -+ v , X ) b SpecificalIy, for Reaction (2-21, A -t - B a
FA= FAD(I -X)
Steady-state operation
+aG C + ad- D , we have
Sec. 8.2
The
fnergy Balance
481
We can substitute thege symbolc for the molar flow rates into Equations (8-1 1 ) and (8-13). then subtract Equation (8-12) From (8-1 1 ) to give
The term in parentheses that is multiplied by FAOXis called the heat of reaction at temperature T and is designated AHR,. Heat of reaction at temperature T
All d the enthalpies (e.g., H A , HB)are evaluated at the temperature at the outtet of the system volume, and, consequently. [AH,,(T)] is the heat of reaction at the specific temperature T.The heat of reaction is always given per mole of the species that is the basis of calculation [i.e., species A coules per mole of A reacted)].
Substituting Equation (8-14) into 18-13] and reverting to summation notation for the species. Equation (8- 13) becomes
Combining Equations (8-10) and (8-15), we can now write the steady-smte live.,(dESy/d! = O)] energy balance in a more usable form: One can use t h i ~ Form of the steadystate energy balance if the enthalpres m ava~lable.
If a phase change takes place during the course of a reaction, this form of the energy balance [i.e., Equation (8-1611 must be used (e.g., Problem 5-4,).
8.2.5 Dissecting the Enthalpies We are neglecting any enthalpy changes resulting from mixing so that the partial rnolal enthatpies are equaf to the mob1 enthalpies of the pure components. The molal enthalpy of species i at a particuIar temperature and pressure, Hi, is usually expressed in terms of an enthulpy offormarion of species i at some reference temperature T,. HI0(TR),plus the change in enthalpy AHQ,, that results when the temperature is raised from the reference temperature, TR.to some temperature T:
482
Steady-State Monisothermal Reactor Design
Chap. B
For example, if the enrhalpy of formation is given at a reference temperature where the species is a solid, then the enthalpy, H(?),of a gas a! tempemre T is
-
-
Enthalpy of
formation
Enthalpy of species Calculating the enthalpy when phase changes are involved
= i
in
-
at
T
1
at
+
I;P -
",:I
Heat of
AHQ in heating
intbtbid S:ptIp,"]
ofspecies
+
Here, in addition to the increase in the enthalpies of the solid, liquid. and gas from the temperature increase, one must include the heat of melting at the melting point, AH,, (T,,), and the heat of vaporizarion at the boiling point. AHvi (Tb). (See Problems P8-4c md P9-4B.) The reference temperature at which HP is given is usuaIly 25OC. For any substance i that is being heated from TI to T2 in the absence of phase change, No phase change
Qpical units of the heat capacity, C, , are ( C p l )=
(moi of i ) (K)
or
cal Btu or (mol of i ) (K) (Ib rnoi of i ) ( O R )
A large number of chemical reactions carried out in industry do no1 involve phase change. Consequently, we shall further refine our energy balance to apply to single-phase chemical reactions. Under these conditions, the enthalpy of species i at temperature T is related to the enthalpy of formation at the reference temperature T, by-
H, = HP(T,)
+J'p, d~
(8-19)
If phase changes do take place in going from the temperature for which the enthalpy of formatron i s given and the reaction temperature T, Equation 18-17] must be used insread of Equation (8-19). The heat capacity at temperature T i s frequently expressed as a quadratic function of temperature, that is.
S ~ C8.2 .
Reference Chef
The Energy Balance
483
However, while the text will consider only constant heat capacities, the PRS R8.3 on the CD-ROM has examples with variable heat capacities. To calculate the change in enthalpy (HI - H,v) when the reacting Ruid is heated without phase change from its entrance temperature, 4,, to a temperature T, we integrare Equation (8-19) for constant C, to write
Substituting for H,and
in Equation (8-16) yieIds
Result of dissecttng the enthalpies r=l
8.2.6 Relating A 4,( T ), A H",A TR),and A c,, The heat of reaction at temperature T is given in terms of the enthalpy of each species at temperature T, that is, Affk,(T) =
b
d
-H~(T)+
where the enthalpy of each species is given by
If we now substitute for the enthalpy of each species, we have
1
+ : H : ( T ~ ) - ~ H ; ( T ,j - x : ( T ~ ) a
d 3 +-B+:c+-D 6
a
a
a
I"
+ ;CP,,+
(8-23)
The first set of terms on the right-hand side of Equation (8-23) is the beat of reaction at the reference temperature T,?
484 One can look up the heats O t formation at TR. then calculate the heat of reaction at
this reference temperature.
Steady-State Nonfsothermal Reactor Desiqn
Cha
The enthalpies of formation of many compounds, HdOITR). are, usually ta lated at 25°C and ran readily be found in the Hnndbook of Chemisfq ( Physics1 and simiSar handbooks. For other substances, the heat of combust (also available in these handbooks) can be used to determine the enthaipy formation. The method of calculation is described in these handbooks. Fr these values of the standard heat of formation, HP (7'') , we can calculate heat of reaction at the reference temperature T, from Equation (8-24). The second term in brackets on the right-hand side of Equation (8-23 the overall change in the heat capacity per mole of A reacted. ACp,
Combining Equations (8-25), (8-241, and 18-23) gives us
AHR,( T ) = AH:, ( T R )+ ACp(T- TR)
Heat of reaction at tempraturc T
Equation (8-26) gives the heat of reaction at any temperature Tin 1e1 of the heat of reaction at a reference temperature (usually 298 K) and the C term. Techniques for determining the heat of reaction at pressures above air spheric can be found in Chem2 For the reaction of hydrogen and nitroger 4WC, it was shown that the heat of reaction increased by only 670 as the pt sure was raised from 1 arm lo 200 atm! Example 8-2
Heat of Reaction
Calculate the heat of reaction for the synthesis of ammonia from hydrogen nitrogen at ISVC in kcallrnol of N, reacted and also i n Wrnol o f Hzreacted. Solution
N2+ 3H2
2NH3
Calculate the heat of reaction at the reference temperature using the heats of for tion of the reacting species obtained from Perry's r-landbook3 or the Handbaol Ckemisrry and Physics.
The heats of formation of the elements
Note:
(HI, N,) are zero at 25°C.
CRC Handbook of Chemistry and Phvsics (Boca Raton. Ra.: CRC Press, 2003). N. H. Chen, Process Reactor Design (Needham Heights, Mass.: Allyn and Bat 19831, p. 26. 3 31. P e q . D.W.Green, and D. Green. eds.. P~rry'sChemicai Engineers' Handbc 7th ed. (New York: McGraw-Hill, 1999). I
Sec. 8.2
The Energy Balance
= 2 ( - 1 1.020)
cal mol N?
= -22,040 callmol N,reacted
AHz..(298 K) = -22.04 kcaltmol N, reacted = -92.22 kJ/mol N, reacted The minus sign indicates the reaction is exothermic. If the heat capacities are constant or ~f the mean heat capacities over the range 25 to 35O"C are readily available, the determination of AH,, at 150°C i c quite simple.
E~othemrcreaction
A ~ =P2ChH3- 3CpH2- CsN2 = 2(8.92) - 3f6.992)- 6.984 = - 10.12callmol N, reacted .K
= -23,3EO calJrnolN, = -23.31 kcatJrnol N? = -23.3 kcal/rnol N, X 4.184 kJ/kcal = -97.5 klfmol N2
(Recall: I kcal = 4.184 kJ)
The heat of reaction based on the moles of H2reacted is
5
AH1, (423 K) = 3 mol Hz(-91.53
= -32.51
&)
kJ at423 K rnol H2
Now that we see that we can calculate the heat of reaction at any temperature, let's substitute Equation (8-22) in terns of AHR(TR)and AC, line., Equation (8-2611. The steady-state energy balance is now Energy balance in terms of mean or constant heat capacities
,
n
Q- WS-FAOC @,Cp,(T-Tid-
.
I= I
[AHlx(Tfi) + ACp(T- T,)]FAJ = 0 (8-27)
486
Steady-Stata Nonisothem1 Reactor Design
Chap. I
Aom here on, for the sake of brevity we will let
unless otherwise specified. In most systems, the work term,
w ~can , be neglected (note the exception in the California Registration Exam Problem P8-5B at the end of this chapter) and the energy balance becomes
In almost all of the systems we will study, the reactants will be entering the system at the same temperature; therefore, T, = Tp We can use Equation (8-28) to relate temperature and conversion and then proceed to evaluate the algorithm described in Example 8-1. However. unless the reaction i s carried out adiabatically, Equation (8-28) is still difficult to evaluare because in nooadiabatic reactors, the heat added ro or removed from the system varies along the length of the reactor. This problem does not occur in adiabatic reactors, which are frequently found in industry. Therefore, the adiabatic tubular reactor will be analyzed first.
8.3 Adiabatic Operation Reactions in industry are frequently carried out adiabaticaIly with heating or cooling provided either upstream or downstream. Consequently, analyzing and sizing adiabatic reactors is an important task.
8.3.1 Adiabatic Energy Balance In the previous section, we derived Equation (8-28). which relates conversion to temperature and the heat added to the reactor. Q. Let's stop a minute and consider a system wlth the special set of conditions of no work, Ws = 0 , adiabatic operation i)= 0 , and then rearrange (8-27) into the form For adiabatic operation. Example 8 I can now be
solved'
(8-29)
In many instances, the hC,(T-T,) term in the denominator of Equation (8-29) is negligible with respect to the AH;, term, so that a plot of X vs. Twill usually be linear, as shown in Figure 8-2. To remind us that the conversion i n
Sec. 8.3
487
Adiabatic Operation
this plot was obtained from the energy balance rather than the mole balance, it is given the subscript EB (i.e., XEsj in Figure 8-2. Equation (8-29) applies to a CSTR, PI%. PBR, and also to a batch (as will be shown in Chapter 9). For = 0 and W, = 0,Equation (8-29) gives us the explicit relationship between X and T needed to be used in conjunction with the mole balance to solve Waction engineering problems as discussed in Section 8.1.
CSTR PFR PBR Batch
Relationship between X and T for udiahuric exothermic reactions
XER
Figum 8-2 Adiabatic temperatuw-conversion relat~nn~hip.
6.3.2 Adiabatic Tubular Reactor We can rearrange Equation (8-29) to solve for temperature as a function of conversion: that is Energy balance for adiabatic operation of
PER
This equation will be coupled with the differential mole balance
to obtain the temperature, conversion. and concentration profifes along the length of the reactor. One way of analyzing this combination is to use Equation (8-30) to construct a table of T as a function of X. Once we have T as a function of X, we can obtain k ( T ) as a function of X and hence -r, as a function of X alone. We could then use the procedures detailed in Chapter 2 ro size the
488
Steady-State Nonisotherrnal Reactor Des~gn
Chal
different types of reactors; however. software packages such as Polymath z MATLAB can be used to solve the coupled energy balance and mole batar differential equations more easily. The algorithm far solving PFRs and PBRs operated adiabatically shown in Table 8-2. TABLE 8-ZA.
ADIABAT~C PFRIPBR ALGORITHM
A
~
B
is carried out in a PFR in which pressure drop i s neglected and pure A enten the reactor.
d X - -r, ' F,,
Mole Balance:
fl8-2. I j
(T8-2.2)
with
Gas, E = 0. P = Po
c,
=
To C*,II -x, -
r
To
CB= C A J7
(T8-2.7) Energy Balance:
To relate temperature and conversion, we apply the energy balance to an adiabatic PFR. If all species enter at the same temperature, To = To. Solving Equation (8-29), with Q = O. W, = 0 , to obtain T as a function of conversion yields
X[-iHi,(TR)]+Zf),CpdToLX.lCpTR
Z qc,, + X AC,
(TR-2.8)
If pure A enters and IR AC, = 0,then
r= T,+ X [ - W , (&)I cpA
(T8-2.9)
Sec. 8.3
TAHLF H-?H The numerrcal technique is presented to provide m i g h t to how the
I
variables (k. K,, etc.) change as we move down the reactor
489
Adiabatic Ooerar~on
SOL^ T f t l l PROCEDCRE
FOR
ADIABATICPER/PBR REACTOR
A. Numerical Technique Inletrating thc PFR mole balance.
I. Set X = 0. 2 . Calculate T unng 'Equalion (T8-3.9) 3. Calculate 8 uslng Equation ITS-? 3). 4. CalcuIate K, uslng Equation (T8-2.4). 5 . Calculate T,,/ 7 (gas phaw). 6 Calculate - r , using Equation (T8-2.7)
from V = 0 and X = 0 to V, and XI.
7. Calculate (F,,I - x , ) . 8. ITXISless then the XI s ~ c i f i e d~ncrementX(i.e.. . X,., = X, t L Y )andgolo Step?,. 9. Prepare table of X VF (FA,/-u, I . 10. Use numerical integration FormuIa~pven in Appendix A. for example.
Lree evaluation techniques discussed in Chapter 2.
X 3
with h = -!
I
B. Ordinary Diffewntiai Equatinn (ODE)Solver
Almost always we will use an ODE solver.
. Kc,, AH,, ( TR).CPAa C,,, To#T I , T? . X = 0, Y = 0 and final value reactor volume, Y = Yr .
5.
Enter parameter values k, E, R,
6.
Enter in inrial vnlufi
490
Steady-State Nontsothermal Reactor Design
Exanqpk 8-3 Liquid-Phase Isomeri&on
Llvlng Example Problt
of N o m l Butane
Normal butane. C,H,, ,is tobe isomerized to isobutme in a plug-flow reactor. Isobutane is a valuable prduct that is used in the manufacture of gasoline additives. For example, isobutane can be further reacted to form jso-octane. The 2004 selling price of n-butane was 72 cents per gallon, while the price of isobutane was 89 cents per gallon. The reaction is to k carried out adiabatically in the Iiquid phase under high p s sure using essentially mace amounts of a Iiquid catalyst which gives a specific reaction rate of 31.1 h- at 360 K. Calculate the PFR and CSTR volumes necessq to process 100.600 gallday (163 kmollh) at 70% conversion of a mixture 90 mol % n-butane and 10 mol 9 i-pcntane, which is considered an inert. The feed enters at 330 K.
Addirional informariorl: The economic incentive $ = 89ctgal cs. 72clgal
AH,, = -6900 Jlmol. butane , Activation energy = 65.5 k.?/mol Kc = 3.03 at 60°C.
Mole Balance: The algorithm
C,, = 9.3 kmol/dm3 = 9.3 kmolIm3
-
FAodX - -rA
dV
Rate Law:
with
Stoichiometry (liquid phase. v = v , ): Cq = C A a ( l - X )
c,
= CAT$'
Combine:
Folto~:'~n& t h e Algorithr
Chap. 8
Integrating Eqi~atinoIER-3. I ) yields
Sec. 8.3
(
1
Adiabatic Operation
Energy Balance: Recalling Equation (8-27), we have
From the problem statement
Adiabatic:
I
491
Q =0
Nowork: fi=0 dcp=cpB-CpA = 141-141 = O Applying the preceding conditions to Equation (8-29) and m a n p i n g gives
Nomanclarute Note
hen= AH;, A
) =A H
Parameter Evaluation
AmH,=
+ A cp(T - TR)
Eat&, = C p A+ @ C -
141
''I-(
I
]
+g 361 0.9
J/mol. K
where T is in degrees Kelvin.
Sub~titu'ingfor the activation energy. T,, and I , i n Equ=iion ( ~ 8 - 3 . 3 )we . obtain
-
k=31.1 exp [6i.:y(3h -- $)](h-~)
"""" .
Substituting for AH,, , T, and K,(T2) in Equation (E8-3.4) yields
Kc = 3.03 exp
Recalling the rate law gives us
-n[:- $1 -- -
(A3
492
Steady-State Nonisothermal Reactor Design
Ch:
Equilibrium Conversion At equilibrium -rA
and therefore we can solve Equation
=0
(E8-3.7) for the equilibrium conversion
Because we know Kc(n,we can find X, as a function of temperature.
PFR Solution It's risky business to ak for 709 conversion in a reversibIe reaction.
Find the PFR volume necessary to achieve 70% conversion znd plot X.X,,-rA.
a1
down the length (volume) of the reactor. This problem statement is risky. H Because the adiabatic equilibrium conversion may be less than 70%! Fortunately not for the conditions here 0.7 < X,. In general, we should ask for the reactor vol to obtain 95% of the equilibrium conversion, Xf= 0.95 X,. We will ~ o l v ethe preceding set of equations to find the PFR reactor vol using both hand calculations and an ODE computer solution. We carry out the ! calculation to help give an intuitive understanding of how the parameters X, and vary with conversation and temperature. The computer solution allows us to rea plot the reaction variables along the length of the reactor and also to study the r tion and reactor by varying the system parameters such as CAoand Tp Solurion by Hand Calcublion ro perhaps give grMFer insight and to build on techniques in Chapter 2.
We will now integrate Equation (E8-3.8)using Simpson's rule after forming a t (E8-3.1) to calculate ( F A , / - r , ) as a function of X. This procedure is similar to descrikd in Chapter 2. We now carry out a sarnpIe calculation tu show how T ES-3.1 was constmcted. For example, at X = 0.2. (a) T = 330 + 43.4(0.2) = 338.6 K
Sample calculation for Table Eg-3.1
Continuing in this manner for other conversions we can complete Table E8-3.1.
Sec. 8.3
493
Adiabatic Operation
Make a Lcvenspiel plot as in Chapter ?.
In order to construct a Levenspiel plot. the data from Table E8-?.I ( F A t r / - rVS. ~ X) was used in Example 2-7 ro size reactors in series. The reactor volume for 70% will be evaluated using the quadrature formulas. Because (FA,,/-r,) increases npidly as we approach the adiabatic equilibrium conversion, 0.7 1. we will break the integral into two pans.
Using Equations (A-24) and (A-22) in Appendix A, we obtain Why are we
doing this hand calculation? If it isn't hstpful. send
3 3 x 0-613.74 + 3 X 2.78 f 3 x 2.50 + 3.88]m3+ -1 x 0.1 -13.88 + 4 X 5.99+ 23.25]rnL
V=
8
3
V = 1.75 m
me an emaiI and you won't see t h ~ sagain.
3
3
+ 0.85 m
2
3
-
You probably will never ever carry out a hand calculation similar to above. So why did we do it? Hopefully, we have given the reader a more intuitive feel OF magnitude of each of the terms and how they change as one moves down the reactor We.. what the computer solution is do~ng),as well as to show how the Levenspiel Plots of (FAd-rA)VS. X in Chapter 2 were constructed. At exit, V = 2.6m3. X = 0.7. Xe=0.715.andT=360K. Computer Solution
PFR We could have also solved this problem using Polymath or some other ODE solver. The Polymath pmgrarn using Equations (ER-3.1). (ES-3.10)- (E8-3.7),(E8-3.11). (E8-3.12), and (Eg-3.13) is shown in Table EX-3.2.
Wldt ~ ~ a t i waaf m t ~t e d by the user 111 caa.9.3 I l l FaO- .9'?83
Living Example Problem
-
T = 330t43.3'X t * 1 Kc = 3 M ' e a p ( + m 3*(r-339y(T339))) I51 k 3 1 . l ' r x u p ~ ~ : ~ ~ 1 y r 3 ~ ) ] t 6 1 XI W(1+KC! ( 7 1 m n -Ir'CsO'{1+\1+1IKc)'X)X) I31
(01 rate a.ca
494
Steady-State Nonisothermal Reactor Design
Chap. 8
Figure EB-3.1 Conversion, temperature, and reaction rate profiles.
h k at me shape of the curves in Figure E8-3.1. Why do they look
The graphical output i s shown in Figure E8-3.1.We see from Figure E8-3.lta) that 1.15 rn3 is required for 40%conversion. The temperature and reaction rate profiles are also shown. One observes that the rate of reaction
the way they do?
goes through a maximum. Near the entrance to the reactor, T increases as does k. causing term A to increase more rapidly than tern B decreases, and thus the rate increases. Near the end of the reactor, term B is decreasing more mpidly than term A is increasing. Consequently,because of these two competing effects, we have a maxi-
mum
in,
the rate of reaction.
CSTR Solution Let's calculate the adiabatic CSTR volume necessary to achieve 40% conversion. Do you think the CSTR will be larger or smaller than the PER? The mole balance is
- Y~ Using Equation (E8-3.7) in the mole balance, we obtain
From the energy balance, we have kquation (E8-3.10):
For 40%conversion
T = 330 + 43.4X T = 330 + 43.4c0.4) = 347.3
Using Equations (E8-3.11)and (E8-3.12) or from Table ER-3.1,
Sec.8.4
I
Steady-State Tubular Reactor with Heat Exchange
Then
The adiabatic CSTR voiumc is less than
the PER vnlurne.
We see that the CSTR volume ( I m") to achieve 40% conversion in this adiabatic reaction is less than the PER volume (1. I5 m3). One can readily see why the reactor volume for 40% conversion is smaller for a CSTR than a PFlZ by recalling the Levenspiel plots Tram Chapter 2. Plotting [FAo/-rA) as a function ofX from the data in TahIe E8-3.1 ~s shown here.
1 The PFR area (volume) is greater than the CSII?I area (volume). 8.4 Steady-State Tubular Reactor with
Heat Exchange
In this section, we consider a tubular reactor in which heat is either added or removed through the cylindrical walls of the reactor (Figure 8-3). In modeling the reactor, we shall assume that there are no radial gradients in the reactor and that the heat flux through the wall per unit volume of reactor is as shown in Figure 8-3.
-9 41 lk \
1
FM To
LFIHr
X#H,
Fu
t
1 1
v
Y+3V
Flguw 8-3 Tuhular reactor with heat gain or loss.
8.4.1 Deriving the Energy Balance for a PFR
We will carry out an energy balance on the volume A V with (8-1 0) becomes
14;= 0. Equation
496
Steady-State Nonisothermal Reactor Design
Cht
The heat Row to the reactor, S Q . i s given in terms of the overall heat tran coefficient, U, the heat exchange area, AA. and the difference between ambient temperature 7;, and the reactor temperature T.
where a is the heat exchange area per unit volume of reactor. For the tubular
reactor
where D is the reactor diameter. Substituting for A Q in Equation (8-. dividing Equation (8-3 1) by Al! and taking the limit as AY -+ O, we get
Expanding
From a mote balance on species i. we have
Differentiating the enthalpy Equation (8- 19) with respect to V
Substituting Equations (8-33) and (8-34) into Equation (8-32). we obtain
Rearranging, we arrive at
This form of the energy balance will 3is0 be applied to multiple reactions.
"Generated" Removed dT ~ A ~ H RU,a-V - To) z F'CP, - 7
which is Equation (T8-1G) in Table 8-t. This equation is caupled with mole balances on each species [Equation (8-33)l. Next we express r, a function of either the concentrations for liquid systems or molar flow rates gas systems as described in Section 4.7.
Sec. 8 4
Steady-State Tubular Reactor with Heat Exchange
497
We will use this form of the energy baIance for membrane reactors and also extend this form to multiple reactions. We could also write Equation (8-35) in terms of conversion by recalling F,= FA,(@, + vjC) and substituting this expression into the denominator of Equation (8-35). PFR energy balance
For a packed-bed reactor dW = ph dV where pb is the bulk density,
PBR energy balance
Equations (8-36) and (8-37) are also given in Table 8-1 as Equations (TS-I€) and (TRI F). As noted earlier, having gone through the derivation to these equations it will be easier to apply them accurately to CRE problems with heat effects. The differential equatron describing the change of temperature with volume (Len,distance) down the reactor. Energy halance Numerical integration of two coupled differential equatrons is wqulred.
Mole balance
= g { x ,T)
must be coupled with the mole balance.
and solved simultaneously. If the coolant temperature varies down the reactor we must add the coolant balance, which is
A variety of numerical schemes can be used to solve the coupled differential equations, (A), (B),and (Cj.
Exarnpb 8-4 Butane Zsome&ztian
Continued---OOPS!
When we checked the vapor pressure at the exit lo the adiabatic reactor in Example 8-3 where the temperature is 360 K, we found the vapor pressure to be about 1.5 MPa far isobutene. which is greater than the rupture pressure of
498
Steady-State Nonisothermat Reactor Design
Chap. 8
the glass vessel being used. Forhmately, here is a bank of ten partially insulated (Ua= 5000 kl/h - m3.K) tubular reactors each 6 m%ver in the storage shed available for use. We are also going to lower the entering temperature to 310 K. The reactors are cooled by natural convection where average ambient temperature in this mpical location is assumed to be 37°C. The temperature in any of the reactors cannot rise a h v e 325 K. Plot X, X,,T, and the reaction rate along the length of the reactor. Does the temperature rise above 325 K?
Solution
For ten reactors in parallel
h
1 = 14.7 kmol A EAo = (0.9)(163 h o r n ) X -
~n
The mole balance, rate law, and stoichiometry are the same as in the adiabatic case previously discussed in Example 8-3; that is, Same as
Mofe Balance:
Example 8-3
Rate Law and Stoichiometry: with
[ (3607 11 '-'
k = 3 l . l exp 7906 T- 360
At equilibrium
Recalling ACp = 0, Equation (8-36) for the partially insulated reactor can be written as "Generated" Removed
dT = I A A H R ~ - U ~ ( To) TF*OGA where and CpI,= Z@,C,, = 159 . kJ/kmol . K, Ua = 5000 kJ/rn7 - h . K. = 310 K, and AH,, = -6900 kllmol. These equations are now solved using Polymath. The Polymath program and profile^ of X . X,, T, and -r, are shown here.
r,
See. 5.4
Steady-State Tubular Reactor with Heat Exchange
Ia)
ODE Rlpw( (RKF45)
llvfng Evample Problem
Figure E8-4.1 (a) Conversion profiles, (bE temperature profile, and (c) reaction rate profile.
We see that the temperature did not rise above 325
8.4.2 Balance
K.
on the Coolant Heat Transfer Fluid
The heat transfer fluid will be a coolant for exothermic reactions and a heating medium for endahermic reactions. If the flow rate of the heat transfer fluid is sufficiently high with reqpect to the heat released for adsorbed) by the reacting mixture, then the heat transfer fluid temperature wiH be constant along the reactor.
-
Transfer Fluid
500
Steady-State Nonisothermal Reactor Design
Ch:
In the materia[ that follows we develop the basic equations for a coo to remove heat from exothermic reactions, however these same equat, apply to endothermic reactions where a heating medium is used to supply h By convention i s the heat added to the system. We now carry balance on the coolant in the annulus between R, and R2 and between V Y + AI! The mass flow rate of coolant is m,. We will consider the case w the outer ndius of the coolant channel R2 is insulated. Case A Co-Current Flow
The reactant and the coolant ff ow in the same direction
The energy balance on the coolant in the volume between Y and (I/ + AV)
Rate of heat added by conduction the inner wall
where To is the coolant temperature, and T is the temperature of the react
mixture in the inner tube. Dividing by AY and taking limit as AY
+0
The change in enthalpy of the coolant can be written as
the variation of coolant temperature T, down the length of reactor is
dT, - Ua(T- T,) m, C p
Sec. 8.4
Steady-State Tubular Reactor with Heat Exchange
50 1
Typical heat transfer fluid temperature profiles are shown here for both exothermic and endothermic reactions
Exothermic
Endothermic
Heat Transfer Fluid Temperature Profiles
Case B Cowter Current Flow Here the reacting mixture and coolant flow in opposite directions for counter current flow of coolant and reactants. At the reactor entrance, V = 0,the reactants enter at temperature To, and the coolant exits at temperature To?.At the end of the reactor, the reactants and producrs exit at temperature T while the coolant enters at T ,.
Again we balance over a differential reactor volume to arrive at reactor volume.
At the entrance V = 0 :. X = 0 and T, = G2. At exit I/ = Vf .: T, = T,. We note that the only difference between Equations (8-40) and (8-41) is
problem to find the exit conversion and temperature requires a trial-and-error procedure.
1. Consider an exothermic reaction where the coolant stream enters at the end of the reactor (V = V,) at a temperature TA say 300 K. We have to carry out a trial-and-error procedure to find the temperature of coolant exiting the reactor. 2. Assume a coolant temperature at the feed entrance (X = 0,V = 0) to the reactor to be Tnz= 340 K as shown in (a).
502
Steady-State Nonisothermal Reaetor Design
Chap. 8
3. Use an ODE soiver to calculate X,T, and T, as a function of V
VI
Y
V
4
We see from Figure (a) that our guess of 340 K far T,? at the feed entrance ( V = 0 and X = 0) gives an entering temperature of the coolant of 310 K (V = Vf), wwhh does not match the actual entering caolant temperature of 300 K. 4. Now guess a coolant temperature at V = 0 and X = O of 330 K. We see from Figure (b) that an exit coolant temperature of Ta2= 330 K will give a coolant temperature at V, of 300 K, which matches the acutal T& We now have all the tools to solve reaction engineering problems involving heat effects in PER for the cases of both constant and variable coolant temperatures. Table 8-3 gives the algorithm for the design of Pms and PBRs with heat exchange for case A: conversion as the reaction variable and case B: molar flow rates as the reaction variable. The procedure in case B must be used when rnuItiple reactions are present. TABLE 8.3. PFRIPBR A E O R ~ MFOR HEAT Emms A. Conversion as the mdion variable A+B I. Mole Balance:
2c
g,--r, Llving E~anplcProblem
t.Rate Law:
dV
-r,=k,
FA,
C C [ A .
3
--
3. Stoichlometry (gas phase, no AP): c * = C A o t l-.qT
Sec. 8.4
Steady-Steie Tubular Reactor wRh Heat Exchange TABLE 8-3.
PFR/PBR ALGORITHMFolr WmT ~
C
503 T
( Sm m ~ m )
4. Energy Balances:
R. Moiar flow tates as the reaction variable 1. Mde Balances:
Follow!ng the Algorithm
2. Rate Law:
fving Euarnpfe Problem P b T-8.3
3. Stoichiometiy (gas phase. no 3P):
4. Energ Balance:
Variable coolanl lernperature
9
s~~~~~~~ hlo?er
If the coolant temperature, To.iq not constant. lhen the energy balance on the cmlant fluid gives = u~cr-T,) Coolant: (TR-3.39) dl' m,C, where m, is the ma<%flow rate of the coolant le g.. kgtc and CS i< the heat capacrly of the
cwlant (e.p ,kJ/kghK!. (See the CD-ROM for the example< in the Chapter 8 Summary Notec and the Pnlymarh Libra? for the cafe when the ambent lemperature is not constani.)
504
Steady-State Nonisothermal Reactor Design
TABLE8-3.
Cb
PFRIPBR ALGORVHMFOR HEATEFFECTS(CONTINUED)
Case A: Co~versiorias the Independent Varinble
with inittaI values
T, and X = 0 at V = 0 and final values: V, = -
Care B: Molar Flow Rates as the Independent Variabb
Same as Case A except the initial values are F,, and FROm specified instead of X at V = Note: The equations in this table have been applied directly to a PER (recall [hat we s~rnply W = phV) using the values for E and AHRxglven in Problem P8-2Im) For rhe L i r b i ~ E.romt; g Problem 8-T8-3 on the CD-ROM. Load this Living Ewmple Pluhlem from the CD-ROM a1 v a q the cooling rate. Bow mte. entering temperalure. and other parameters to get an ~ntuitive cummar! No+cr of what happens in Raw reactors wtth heat effects. After carrying our this exercise. go ro th~ WORKBOOK in the Chapter 8 Summary Notes on the weWCD-ROM and anwer the quest
The following figures show representatave profiles that would result from solving the a h v e t tions. The reader is encouraged to load the Living Example Problem 8-T8-3 and vary a nun o f parameters as discussed In P8 -2 Im). B t sure you can explain why these curve? look the they do. Be sure you can explain why these curves look the way they do.
&warn
T, f ~ n 1 6 1 6 1eliOlhmrY: e raacllen m a PFR wlm haat exchange
I
Example 8-5
%slaol
m
e
T,
m mum
sn a PW mth MIaxch-
Wudsb*. T, 8xolhenc
Va~bln \
mwlsrcurrenl
co.currmt
s m e m
marange
0xcMnge
Production of Atefir Anhydride
Jeffreys: in a treatment of the design of nn acetic anhydride manufacturing faci states that one of the key steps ts the vapor-phase cracking of acetone to ketene methane:
He states fuaher that this reaction is first-order with respect to acetone and that specific reaction rate can be expressed by
where k is in reciprocal seconds and T is in kelvin. In this design it is desired to f 7850 kg of acetone per hour to a tubular reactor. The reactor consists of a bank 1000 I-inch schedule 40 tubes. We will consider three cases:
G. V. J e f i y s , A Problem in Chemical Engineering Design: The Monuflcture Acetic Anhydride, 2nd ed. (London: Institution of Chemical Enginees. 1964).
Steady-State Tubular Fleactor wrth Heat Exchange
Sac. 8.4
505
A. CASE 1 The reactor i s operated ndiaharicalll: B. CASE 2 The reactor lr surrounded by a heat exchanger where the heat-transfer coeffic~entis 11 0 JEm2- s . K,and the temperature of the heating medium, To,is consranr at 1150 K. C. CASE 3 The beat exchanges in Case 2 now has a variable heating medium
Gas-phase endothermic reaction example 1 Adiabat~c 2. Heat exchange T, I S constant
3. Heat exchange T, is variab1e
temperature.
The iniet temperature and pressure are the same for both cases at 1035 K and 162 kPa (1.6 atrn), respectively. Plotthe conversion and temperature along the Iength of the reactor.
I
Solution Let A = CH,COCH,, B = CH,CO. and C = CH,. Rewriting the reaction syrnbolically gives us At8+C 1. Mole Balance:
I1
2. Rate Law:
1
3. Stoiehiometry (gas-phase reaction with no pressure dmp):
1
4. Combining yields
-rA = kch
(E8-5.3)
To solve this differential equation. it is first necessary to use the energy balance to determine T a a function of X. 5. Energy Balance:
/
CASE I. ADIABATIC OPERATION For no work done on the system, *s = 0 , and adiabatic operation, C'= 0 ), Equation (8-36) becomes
Because only A enters,
I: o;c, = CPA and Equation (E8-5.7) becomes
= 0 (i.e..
506
Steady-State Monisotherrnal Reactor Design
Chap. 8
6. Calcdation of Mole Balance Parameters on a Per lhbe Basis:
F -7,850kglh, = 0.135 krnol/h = 0.0376 AO 58 glmol 1,000Tubes
vo =
5 = 2.037 drn3/s cA,
V=
molls
5 m3 = dm3 I000 tubes
7. Calculation of Energy Balance Parameters: a.
AH", ( 5 ):At 29&K. the standard heats of formation are
b. AC, : The mean hear capacities are:
See Table E8-5.1for a summary of the calculations and Table E8-5.2and Figure E8-5.1 for he Polymath program and its graphical output. For adiabatic operation. it doesn't matter whether or not we feed everything to one tube with V = 5 m? or distribute Ihe Row to the 1003 rubes each with V = 5 dm". The temperature and conversion profiles are identical because there is no heat exchange.
Adiabatic PFR
See. 8.4
50'7
Beady-State Tubular Reactor with Heat Exchanm
TAU EB-5.1. SWARY
ADMBA~C O P w n o ~( m m m )
Parameter VaIues ACp = -9 Jfmol K cPA= 163 JImoVAIK Vf= 5 dm3
AHb (T,l = 80.77Ilmol CAo=18.8 moYm3 FAo= 0.376moUs
To= 1035 K TR= 298 K
TABLE E8-5.2. PULYMATH PROGRAM ADMATIC OPERA~ON
OPE Report (R-451 Differential equations as entered by the user r L 1 d(X)/d(V) = -ralFaa [ 2 1 d(T)M(V) = -raq(-deNeH)l(fao'(Cpa+X'delCp)) Living Example Problcm
Expllcit equations as entered by the user 1 1I Feo = .0376 121 Cpaa 163
Adiabatic
endotherm~c reaction in a PFR
(a)
Ib)
Rgure ES-5.1 Conversion and temperature (a) and reaction rate (b) profiles.
As temperature drops. so does
k and hence the rate. -rA, drops to an insignificmt
value. 'a80f a
Note !hat for this adiabatic endothemis reaction. the reachon vinually dies ow after 3.5 m3,w i n g lo the large drop in temperature, and veiy little convcnion is achieved beyond this point. One way to incsease the conversion wwld be to add a
508
Steadystate Nonisothsrrnal Reactor Des~gn
Cht
diluent such as nitrogen, which could supply the sensible heat for this endotbei reaction. However, if too much diluent is added, the concentration and rate wil quite low. On the other hand, if too little diluent i s added, the temperature will 4 and virtually extinguish the reaction. How much diluent to add is left as an exe! [see Problem P8-2(e)]. A bank of 1000 1-in. schedule 40 rubes 1.79 m in length correspond 1.0 m3 and gives 20% conversion. Ketene is unstable and tends to explode, whic a good reason to keep the conversion low. However, the pipe material and scha size? should be checked to lean if they are suitable for these temperatures
pressures.
CASE 2. HEAT EXCHANGE WITH CONSTANT HEATING MEDI TEMPERATURE
Let's now see what happens as we add hem to the reacting mixture. See Figure EX-
Figure EB-5.2 PFR with heat exchange.
1. Mole Balance:
Using the algorithm: (Step 2) the rate law (E8-5.3) and (Step 3) stoichiomt (E8-5.4) for the adiabatic case discussed pteviously in Case I, we (Step 4) CI bine to obtain the reaction rate as
1
/ PFR with heat exchange
5. Energy Balance:
POIthe acetone reaction system,
6. Parameter Evaluation: a. Mole balance. On a per tube basis, v, = 0.002 rn3/s. The concentratio1 acetone is 18.8 mollm3 ,so the entering molar flow rate is
& = ,,u,=
(18.8
$)(2X
lo-'$)=
mol
0.0376 s
Sec. 8.4
Steady-State Tubular Reactor w~thHeat Exchange
The value of k at I035
I
509
K i s 3.58 s-\ ; consequently, we have
b. Energy bdlnncc. From theadiabatic case in Case l, we atready have AC,, Cp4.
The heat-transfer area per unit volume of pipe is
1
Combining the overall heat-transfer coefficient with the area yields Ua = 16,50031rn-'~s.K We now use Equations (EX-5.1)through (E8-5.6), and Equations (E8-5.10) and (E8-5.11)along with Equation ES-5.I2 in Table EX-5.3 in the Polymath program (Table E8-5.4).to determine the conversion and temperarurc profiles shown in Figure
E8-5.3.
I
3
YJ = 0.W1m = 1 dm"
II
We now apply Equation (8-36) to this example to arrive at Equation (ER-5.12). which we then use to replace Equation (E8-5.8)in Summary Table E8-5.1.
with
II
Per tube basis
u,
= 0.002
m3/s
FA, = 0.0376 molls
Vj= 0.001 m 3 = I dm3 Everything etse is the same as shown in Table E8-5.1.
One notes that the reactor temperature goes through a minimum along the length of the reactor. At the front of the reactor, the reaction takes place very rapidIy, drawing energy from the sensible heat of the gas causing the gas temperature to drop because the heat exchanger cannot supply energy at an equai or greater rate. This drop in temperature, coupled with the consumption of reactants, siows the reaction rate as we move down the reactor. Because of this slower reaction rate, the heat exchanger
I
510
Steady-State Nonisotherrnal Reactor Design
Chap. 8
supplies energy at a rate greater than reaction draws energy from the gases and as a result the temperature increases. TABLE E8-5.4. POLYMATHPROGRAMFOR PFR
W m
HMT EXCHANGE
POLYMATH Results h r n p l e 8-3 ProdurUon a€Acetlc hnhydrlde nfL Heu hthnmge (ComumtTn) m - l c m .
~cv5.1.232'
ODE Rewd .(RHFdSl Mtlerent~al e q u a t m aa enbrnd by h a user Ia ; dlXVdW) = .&w l:- 1 d(T)ld(Y) ( U ~ * ~ ~ - T ) + R ~ ~ H ) I ( F & D * ( C P ~ + X * ~ ~ K : ~ ) )
--
E e k l e q u a m as enlorad by the ussr
Living Example Problcn
(11 Fao=.O31f6 [:I Cpa=163 13 1 delCp = -8 I 4 1 Cso=18.R [ 5 1 To-1035 t 6 I & b= ~ aowcp'rr-298) 17 I r8 = -Ca0'3 5BCB%P(34222.(1K*lrr))*(1 .XJ.VOm,(l+K) 181 Ua-113500
[s]
Yam1160
WR with heat exchange
constant coolant temperature T,
vcamgl
~th?
(b)
(a)
Figure ES-5.3 Temperature and conversion profiles in PFR. Temperature and conversion (a) and reaction rate ch! profiles In a PFR with conslant heating medium temperature, T,
CASE 3. HEAT EXCHANGE WITH VARIABLE HEATING MEDIUM
TEMPERATURE Air will a l ~ obe used as a heating stream in a co-current direction entering at a temperature of 1250 K and at molar rate of (0.1 1 rnoUs). The heat capacity of the alr i s 34.5 Jlrnol . K.
For co-current flow dT, - = L1o(T- T,) dl'
ni, C,,.
Sec. 8.5
Equilibrium Conversion
511
The Polymath code i s modified by repking To= 1150K in Tables ES-5.3 and E8-5.4 by Equation (E8-5.7), and adding the numerical values for ni, and Cp r
.
TABLEE8-5.5. SUMMARYHEATE x m O E xml VAWABET,
1
DmmnUal eoualbns an entered InIIm ussr
E@tt aquallma as enkred by Ihe user 1 1) Fan = ,0378
PFR with hear exchange variable ambient
temperature T,
(21 Cpe=163 [ 3 1 delCp c -9 I d ] Cum= 18.8 15) To= 1035 1 6 1 denaH c 80770+delCp'(T-29Of
171 ra = .Ca0'3.68'6~(3d2n'(1TT~1K)~[1 -Xr(TdJYIl+X) If!] U ~ x 1 6 5 0 0 191 me-,111
[ l o 1 Cpc=N.5
The temperature and conversion profiles are shown in Figure E8-5.4.
Figure E8-5.4 (a) Temperature and Ibl conversion profiles in PFR w ~ t ha variable hearing medium temperature, T,
8-5 Equilibrium Conversion For reversible the equiI~brium
x,,
:onveman. is usually ciikulated
first.
The highest conversion that can be achieved in reversible reactions is the equilibrium conversion. For endothermic reactions, the equilibrium conversion increases with increasing temperature up to a maximum of 1.0. For exothermic reactions, the equilibrium conversion decreases with increasing temperature.
512
Steadystate Nonisotherrnal Reactor Desran
Cha
8.5. I Adiabatic Temperature and Equilibrium Conversion
Exothermic Reactions. Figure 8-41a) shows the variation of the concen Lion equilibrium constant as a function of temperature for an exothermic rf tion (see Appendix C), and Figure 8-4(b) shows the corresponding equilibri conversion X, as a function of temperature. In Example 8-3, we saw that fr first-orderreaction the equilibrium conversion could be calculated using Ec tion (E8-3.13) First-order reversible
reaction
Consequently.X, can be calculated directly using Figure 8-4ta).
For exothermic reactions,
equ~libriurn conversion
decreawa wfth
K
~ncreasing
temperature.
Figure 8-4 Variation of equilibrium constant and c o n v ~ s i o nwith temperature for
an
exothermic reaction.
To determine the maximum conversion that can be achieved in an exother reaction carried out adiabatically, we find the intersection of the equilibrium c version as a function of temperature [Figure 8-4(b)] with tempemturexonver relationships from the energy balance (Figure 8-2) as shown in Figure 8-5
If the entering temperature is increased from To to To,,the energy ance line will be shifted to the right and will be parallel to the original lint shown by the dashed line. Note that as the inlet temperature increases, the abatic equilibrium conversion decreases.
.4diabatic
equilihriurn conversion Tor exothermic reactions
Figure 8-5 Gritphlcal solution of equ~hbriurnand energy balance equations to obtain the adiabatic temperature and the adlaktic equilrbriurn conversion X,.
Exampie Bd Calculan'ng the Adiabatic Equilibrium Temperature For the elementary sotid-cntayred liquid-phase reaction make a plot of equilibrium conversion as a function of temperature. Determine the adiabatic equilibrium temperature and conversion when pure A is fed to the reactor at a temperature of 3UO K.
Solutio~t
1. Rate Law:
2. Equilibrium:
Follow!ng tkc Aldorithm
- r,
= 0 ;so
3. Stoichiometry: ( u = uD) yields
514
Steady-State Nonlsatherrnal Reactor Design
Chap. 5
Solving for X, gives
4. Equilibrium Constant: Calculate AC, ,rhen K,(T)
For AC,, = 0. the equilibrium constant varies with temperature according to the relation
AH;, = H," - H i = -20.000 callmol
K,(T)= 100,000 exp
K, = 100,000 exp
1
[-?~q,"? (2i8 -- b)]
I"?-'[
- 33.78 -
Substituting Equation (E8-6.4) into (E8-6.2). we can calculate equilibrium conversion as a function of temperature:
5. Equilibrium Convemion Fmm Thermodynamics Conversion
calculaled from equilibrium
relationship
exp [- 33.78(T- 298)/T] x, = 1 +100,000 100,000 e x p [ - 3 3 . 7 8 ( T - 2 9 8 ) / T j
The calculations are shown in Table E8-6.1.
6. Energy Balance For a reaction carried out adiabatically, the energy balance reduces to
Sec. 8.5
515
Equilibrium GDnversion
Conversion calculated from
energy balance
Data from Table E8-6.1and the following data are plotted in Figure E8-6.1.
F&
a feed temperature of 300
K, the adiabatic equilibrium temperature is
465 K and the corresponding adiabatic equilibrium conversion is only 0.42.
Adiabatic
equilibrium conversion and temperature
Figure W6.1 converjon
finding the adiabatic equilibrium ternpenture (T,I and
(X,).
Reactor Staging with Interstate Cooring of Heating Higher conversions than those shown in Figure E8-6.1 can be achieved for adiabatic operations by connecting reactors in series with interstage cooling:
The conversion-temperature plot for this scheme is shown in Figure 8-6. We see that with three interstage coolers 90% conversion can be achieved compared to an equilibrium conversion of 40% for no interstage cooling.
Steady-State Nonisothermal Reactor Design
C
Interstage cwling used for exothermic reversible reactions
Figure 8-6 Increasing conversion by Interstage cooling. Typical values
Endothermic Reactions. Another example of the need for interstage
'Or gas?ljne transfer in a series of reactors can be found when composltlon
Gasoline CI C6
C, C.
10%10%
20s 25%
upgrading the octane nu of gasoline. The more compact the hydrocarbon molecule for a given nu of carbon atoms is, the higher the octane rating is. Consequently, it is desi to convert straight-chain hydrocarbons to branched isomers, naphthenes. aromatics. The reaction sequence is k1
Cat
The first reaction step (k,) is slow compared to the second step. and step is highly endothermic. The allowable temperature range for which this tion can be c h e d out is quite narrow: Above 530°C undesirable side reac occur. and M o w 430°C the reaction virtually does not take pIace. A typical stcck might consist of 75% straight chains, 155 naphthas, and 10% mmac One arrangement currently used to carry out these reactions is shou Figure 8-7. Note that the reactors are not all the same size. Typical sizes ar the order of 10 to 20 m high and 2 to 5 rn in diameter. A typical feed ra gasoline is approximately 100 m3lh at 2 atm. Hydrogen is usuaIly sepal from the product stream and recycled.
Spring 2005 $2.20/gal Tor octane number (ON) ON = 89 500'
?a Cnlnlyf 1 ~s~ensr~t~on
Figure 8-7 Interstage heating for gasoline production in moving-bed reacton.
Sm. 8.5
51 7
Eqclilibrium Conversion
Because the reaction is endothermic, equilibrium conversion increases with increasing temperature. A typical equilibrium curve and temperature conversion trajectory for the reactor sequence are shown in Figure 8-8.
Interstage hearing
Flgure 8-8 Temperature-conv~rsinntrajectory for interstage heatlng of an
endothermic reaction analopous to F~gure5-6. Example 8-7 Interstage Cooding for Highly Exothemic Reactions What conversion could be achieved in Example 8-6 if two interstage coolers that had the capacity to cool the exit stream to 350 K were available? Also determine the heat duty of each exchanger for a moIar feed rare of A of 40 molls. Assume that 95% of equilibrium conversion is achieved in each reactor. The feed temperature to the first reactor is 300 K.
1. Calculate Exit Temperature We saw in Example 8-6
that for an entering temperature of 300 K the adiabatic equilibrium conversion was 0.42. For 95% of equilibrium conversion (X,= 0,421, the conversion exiting the first reactor is 0.4. The exit temperature is found from a rearrangement of Equation (Eg-6.7): T = 300 + 4OOX = TOO (400)(0.4) (A) T,= 460 K
+
1
We now cool the gas stream exiting the reacror at 460
K down
to
350 K in a heat
exchanger (Figure E8-7.2).
Figure ES-7.1 Determining exit convenion and temptnture in the first stage.
518
Steady-State Nonlsothermal Reactor Design
Chap. 8
2. Calculate the Heat Load There is no work done on the reaction gas mixture in the exchanger, and the reaction dms not lake place in the exchanger. Under these conditions (Fill" = FIl,,,),the energy balance given by Equation (8-10)
F,H, - 0
(8- 10)
= E F, (H,- Hie)
(E8-?.I)
Q- W ~ + ZF,H,,-z for W~ = O becomes Energy balance on the reactron gas mixture in the heat
Q = 1F,Hi - Z Fl,Hm
exchanger
That is, 220 kcaVs must be removed to cool the reacting mixture from 460 K to 350 K for a feed rate of 40 moVs. 3. Calcutate the Coolant Flow Rate We see that 220 kcalls i s removed from the reaction system mixture. The rate at which energy must be absorbed by the coolant stream in the exchanger is
We consider the case where the coolant is available at 270 K but cannot be heated above 400 K and calculate the codant flow rate necessary to remove 220 kcalls fmrn the reaction mixture. Rearranging Equation (E8-7.5) and noting that this coolant has a heat capacity of 18 callmol .K grves Sizing the interstape heat exchanger and coolani flow rate
= 94 moils = 1692 g l s = 1.69 kgls
The necessary coofant flow rate is 1.69 kgls. 4. Calculate the Hmf Exchanger Area k t ' s next determine the counter current heat exchanger area. The exchanger inlet and outlet temperatures are shown in Figure E8-7.2. The rate of heat transfer in a counter current heal exchanger is given by the equation! See page 268 of C. 1. Geankoplis, Transporr Pmcrsxes und Unir Opcrarions (Upper
Saddle River, N.J.: Prentice Hall. 19931.
Set. 8.5
Equilibrlurn Conversion
Bonding with unit opentions
In Thl 35OK Reaction Mixture Exchanger
T,, 270K Coolant
----------+--
Figure ES-7.5 Counter current heal exchanger.
Rearranging Equation (E8-7.71.asquming a value of U of 100 calJs.rn2-K. and then substituting the appropriate values gives Sinng the heat exchanper
A=
-
~ ' ~ ~ T h ? - c ~ ) - ~ ~ t ~ - l To ~o ~&) [l (
4~-~m)-(~5~-27~)]~ s+m2.K
The heat-exchanger surface area required to accomplish this rate of heat transfer is 3.16 m2. 5. Second Reactor Now let's return to determine the conversion in the second reactor. Tne conditions entering the second reactor are T = 350 K and X = 0.4. The energy balance starting from this point is shown in Figure E8-7.3.The correspo~ldingadiabatic equilibrium conversion is 0.63. Ninety-five percent of the equilibrium conversion is 60% and the corresponding exit hmperature is T = 350 + (0.6 - 0.4)400 = 430 K. The heat-exchange duty to cool the reacting mixture fmm 430 K back to 350 K can again be calculated from Equat~on(E8-7.4):
Rearran ping EqualLon for the second rercfor
r2 = 7.
+.Ax(?)
= 350 * 4ODhX
I
Figure FA-73 Three reaclors in series with interstage cooling.
520
Steady-State Nonisothermai Reactor Des~gn
Q
=
F,,,,C,,,(350
= -160
- 430)
=
Ch
)
0 cal (-8Ol - ( 5-
(40 sol)
mol . K
kcal
S
6. Subsequent Reactors
For the final reactor we begin at T, = 350 K and X = 0.6 and follow the line resenting the equation for the energy balance along to the point of intersection the equilibrium convenion. which is X = 0.8. Consequently. the final convel achieved with three reactors and two Interstag coolers is (0.95)(0.8)= 0.76.
8.5.2 Optimum Feed Temperature
We now consider an adiabatic reactor of fixed size or catalyst weigh[ and in tigate what happens as the feed temperature is varied. The reacrion i s reverc and exothermic. At one extreme, using a very high feed temperature. the cific reaction rate will be large and the reaction will proceed rdpldly. but equilibrium conversion will be close to zero. Consequently. very little pro, will be formed. At the other extreme of low feed temperatures, little pro will be formed because the reaction rate is so low. A plot of the equilib conversion and the conversion calculated from the adiabatic energy balanc shown in Figure 8-9, We see that for an entering temperature of 600 K the abatic equilibrium conversion is 0.15. The corresponding conversion pro down the length of the reactor are shown in Figure 8- 10. The equilibrium I version, which can be calculated from an equation similar to Equation (EXalso varies along the length of the reactor as shown by the dashed line in Fi; 8-10. We also see that because of the high entering temperature, the rate is * rapid and equilibrium is achieved very near the reactor entrance.
350
400
450
500
550
600
To
Figure 8-9 Equilibrium conversion for different feed temperatures.
Observe how the temperature proSITe changer a\ the entering temperature i.c
decreased from 603 K. W
Figure 8-10 Adiabatic conversion profiles fnr different feed temperatures.
We notice that the conversion and temperature increase very rapidly over a short distance (i.e., a small amount of catalyst). This sharp increase is sometimes referred to as the "point" or temperature at which the reaction ignites. If the inlet temperature were lowered to 500 K, the corresponding equilibrium conversion is increased to 0.38; however, the reaction rate is slower at this lower ternpentuse so that this conversion is not achieved until closer to the end of the reactor. V the entering temperature were lowered further to 350 K, the comesponding equilibrium conversion is 0.75, but the rate is so slow that a conversion of 0.05 is achieved for the specified catalyst weight in the reactor. At a very low feed temperature, the specific reaction rate will be so small that virtually all of the reactant will pass through the reactor without reacting. It is apparent that with conversions close to zero for both high and low feed temperatures there must be an optimum feed temperature that maximizes conversion. As the feed temperature is increased from a very low value, the specific reaction rate will increase, as wilt the conversion. The conversion will continue to increase with increaing feed temperature until the equilibrium conversion is approached in the reaction. Further increases in feed temperature for this exothermic reaction will only decrease the conversion due to the decreasing equilibrium conversion. This optimum inlet temperature is shown in Figure 8-1 1.
Optimum inlet temperature
Figure 8-11 Finding the optimum Feed temperature.
522
Steady-State Nonisothermal Reactor Design
Chap. B
8.6 CSTR with Heat Effects In this section we apply the general energy balance [Equation (8-22)J to the CSTR at steady state. We then present example problems showing how the mole and energy balances are combined to size reactors operating adiabatically and non-adiabatically. Substituting Equation (8-26) into Equation 18-22), the steady-state energy balance becomes
These are the forms of the balance we wil! use.
[Note: In many calculations the CSTR moIe balance (FAJ = -rAV) will be used to repIace the term following the brackets in Equation (8-27). that is, will be replaced by (-r,.,V).] Rearranging yields the steady-state balance
Although the CSTR is well mixed and the temperature is uniform throughout the reaction vessel, these conditions do not mean that the reaction is canied out isothermally. Isothermal operation occurs when the feed temperature is identical to the temperature of the fluid inside the CSTR. The Q Term in the CSTR 8.6.1 Heat Added to the Reactor, Q
Figure: 8-12 shows schematics of a CSTR with a heat exchanger. The heat transfer fluid enters the exchanger at a mass flow rate m, (e.g., kgls) at a temperature T,, and leaves at a temperature Ta2.The rate of heat transfer fmm the exchanger to the: reactor ish For exothermic reactions
(75ra2>Ta,1 For endothermic reactions
{TIPTQTJ Half-pipe jacket
Figure 8-12 CSTR rank reactor with heal exchanger. [rh) Courtesy of Pfaudler. Inc.1
" Informat~onon lhe overall heat-tran\fer
coefficient may 'be found in C. J. Geankopli.
Transpon Pw)cesscr ur~dUni? Opemrinns, 3rd ed. Englewood Cliffs. N.J., PrentKe
Hall (2003). p. 268.
Sec. 8.6
CSTR wfth Heat Effects
523
The following derivations, based on a c ~ 1 m (exothermic t reaction) apply also to heating mediums (endothermic reaction). As a first approximation, we assume a quasi-steady state fwthe coolant flow and neglect the accumulation term (i.e., dT'ldt = 0).An energy balance on the coolant fluid entering and leaving the exchanger is Rate of heat exchanger
to reactor
by flow
where Cpc is the heat capacity of the coolant fluid and TR is the reference temperature. Simplifying gives us
Solving Equation (8-46)for the exit temperature of the coolant fluid yields
T,, = T - ( T - T,,) txp
-
From Equation (8-46)
Substituting for T,, in Equation (8-48). we obtain Heat transfer to a
CSTR
For large values of the coolant flaw rate. the exponent wilI be small and can be expanded in a Taylor series (e" = 1 - x . . -) where second-order terms are neglected in order to give
+
Then Valid only for large Coolant flow raws!!
where To,2 Taz = TO.
I
524
Steady-State Nonisothetrnal Reactor Design
Chi
With the exception of processes invoIving highly viscous materials s as Problem P8-4c, the California P.E exam problem, the work done by the : rer can usualiy be neglected. Setting W,in (8-27) to zero. neglecting ACp, ! stituting for Q and rearranging, we have the following relationship betw conversion and temperature in a CSTR.
Solving for X -IT-
To)+ SO,C, (T- To)
X=
[-A%AT,?)l Equation (8-52) i s coupled with the mole balance equation
to
size CSTRs. We now will further m a n g e Equation (8-5 1) after letting CO,Cpr= C
Let
KT, + T,
and T, = Then
-XbGK= Cpo(l+ K ) ( T - T,)
(3-
The parameters K and T, are used to simplify the equations for non-adiab; operation. Solving Equation (8-54) for conversion
Forms of the energy
Solving Equation (8-54) for the reactor temperature
balance for a CSTR with heat exchange
Figure 8-13 and Table 8-4 show three ways to specify the sizing o CSTR. This procedure for nonisothermal CSTR design can be illustrated considering a first-order irreversible liquid-phase reaction. The aIgorithm working through either case A (Xspecified), B (T specified), m C (Y specified shown i n Table 8-4. Its application is ilfustrated in the following example.
Sec. 8.6
CSTR with Heat Effects
Algorithm
Example: Etementary ~rreversibleliquid-phase reaction
A+B Given F A OCM, ~ ko, E,CpA,AH~r, ACp=O, q = O CSTR I
Oedgn equation
v = !d
Rate law
+A
Combining
v=
+A
= kcA
FAOX ~ C A(1O- X I
+ I I
X specified: Calculate V and T
Need k(T)
JI
Calculate T
Two quatians and h o unknowns
1 I
I
Calculate k
1
S
E:: Plot X vs. T
I
-----
4 Calculate V
I
X
T Flgure $-13
Algorithm for adiabatic CSTR design
526
Steady-State Nonlsotherrnal Reactor Design TABLE8-4.
WAYSTO SPECIFY THE SIUNG
A
CSTR
A
B
C
Specify X
Find V and T
Spe~ifyT Find X and V
W m d T
Calculate T From Eqn. (8-56)
Calculate X From Eqn. 18-55]
1
1
Use Eqn (8-553 to plat XElgVS. T
Calculate k k = Ap-CtRT
Calculate k k = kC-EIRT
Solve Eqn. (8-53)
L
Spify
L
v
1 J.
for
xus = p n
to find XMnvs.
ce.e., - r ~= kcAO() -XI)
fe.g.. -ra =
kcAo(] - x))
T
le.g., XhlB = TA
expp[-E/(RT)]
1 + 0.exp[-E/(RT)l
J.
S
Calculate -r,(X,TE
Calculate -r,{X,X.T)
1
1
Calculate V
CalcuIate V
y = -Fad: -r~
v=- FAJ -r,
Chap. 8
1
1 Plot XE8 and X,
as a function of T
xp L
T
XMR=conversion calculated from the mole balance XER= conversson calculated from the energy balance
Example
g-8
Prnduction of Proplena Glycol in on Adiabah'c CSTR
Propylene glycol is produced by the hydrolysis o f propylene oxide:
Production, usesa and economics
Over 800 million pounds of propylene glycol were produced in 2004 and the selling price was approximak!y 50.68 per pound. Propylene glycol makes up about 25% of the major derivatives of propylene oxide. The reaction takes place readily at room temperature when catalyzed by sulfuric acid. You are the engineer in charge of ap adiabatic CSTR producing propylene glycot by this method. Unfortunately, the reactor i s beginning to leak, and you must replace it. (You told your boss seveml times that sulfuric acid was corrosive and that mild steel was a poor material for construction.) There is a nice-looking overflow CSTR of 300-gal capacity standing idle: it is glass-lined. and you would l ~ k eto use it. You are feed~ng2500 lblh (43.04 IIb morlh) of propylene oxide (P.O.) to the reactor. The feed stream consists of (1) an equivolumetric mixture af propylene o x ~ d e(46.62 ft'/h) and methanol (46.62 ft'lh). and (2) water contaming 0.1 wt % H2S0,.The volurneuic RON rate of water is 233.1 ftqlh, uhich is 2.5 times the methanol-PO. Row rate. The cornspanding moIar feed rates of methanol and water are 71.87 and 802.8 Ib mol/h, respectively. The water-propylene oxide-methanol mixture undergoes a slight decrease in volume upon mixing
S e .8.6
527
CSTR with Heat Effects
(approximately 38). but you neglect this decrease in your calculations. The temperature of both feed streams is 58°F prior to mixing, but there is an j m d i i r t e 1 7 O F temperature rise upon mixing of the two feed smams caused by the heat of mixing. The entering temperature of all feed streams is thus taken to be 75°F (Figure E8-8.1). pmwlene @xidm
T~ = w
Methanol
;y F~~ Water
To = 75" F
Furusawa et state that under conditions similar to those at which you are operating, the reaction is first-order in prepylene oxide concentration and apparen! zero-order in excess of water with the specific reaction rate
The units of E are Btullb mol. There is an important constraint on your operation. Propylene oxide i s a rather low-boiling substance. With the mixture you are using, you feel that you cannot exceed an operating temperature of 125°F. or you will lose too much oxide by vaporization through the vent system. Can you use the idle CSTR as a replacement for the leaking one if if will be operated adiabatically? If so, what will be the conversion of oxide to glycol?
(All data used in this problem we= obtained from the Handbook uf Clremfsr~ a11d Physics unless otherwise noted.) Let the reaction be represented by
A is propylene oxide (CpA= 35 Btullb mol -OF) 8
B is water (CPB= 18 BtuJlbmol."F) ?
T. Furusawa. H. Nishimura, and T. Miyauchi, 3. Che~n.Eng. Jprr.. 2. 95 119691. .CpA and Cpc are estimated from the observation that the great major~ty of low-molecular-weight oxygen-containing organic liquids have a mas$ hear capacity of 0.6 callg-% 2 1 5 % .
528
Steady-State Nonisothermal Reactor Design
Cha
C i s propylene glycol (Crc= 46 B t u lb ~ mol - "F) M i s methanol ICp = 19.5 Btullb rnol . "F)
In this problem neither the exit conversion nor the temperature of the a1 hatic reactor IS given. By application of the material and energy baIances we solve two equations with two unknowns (X and T). SoIving these coupled equatir we determine the exit conversion and temperature for the glass-lined reactor to if it can be used to replace the present reactor.
I
I. Mole Balance and design equation: FA"-F*+r*V= 0
The design equation in terns of X is
2. Rate Law:
1
3. Stoicbiometry (liquid phase, u = u,): C A = C A , , ( l -q
4. Combining yields Following the Algorithm
1
Solving for X as a Function o f T and recalling that r = V l v , gives
This equation relates temperature and conversion through the mole bnlanl 5. The energy balance for this adiabatic reaction in which there IS neglig energy input provided by the stimr is
Two equations. two unknown
This equation relates X and T through the energy balance. We see that I equations [Equations (EB-8.5) and (ES-8.6)] must be solved with X,, = . for the two unknowns, X and T 6. Calculations: a. Heat of reaction at temperature T : ~
II
AH,, (T)= AH;,(T,)+ AC, (T- TR)
H i (68°F) : -66,600
Stullb moi
-Hland H t are calcurated from heat-of-combustion data.
Sec.8.6
CSTR with Heat Elfects
Calculating the parameter values
Hg (68°F): - 123,000 Btuilb mol H,9(68°F) : -226.000 Btu/lb mol
AH", ((68°F)= -226,000 - ( - 123,000) - (-66,600)
(E8-8.7)
= -36,400 BtuIIb mol propytene oxide ACp = Cpc-CPB- CPA
- =46-I&-35=-7Btullbmol~"F A W l , ( T ) = -36,400- ( 7 ) ( T - 528)
b.
Stoichiometry
T is in "R
(C,,, @, , z ): The total liquid volumetric flow rate entering
the reactor is
For methanol:
(3, = F~~ -- 7E.87 lb m0l/h = FA, 43.0 lb mollh
For water:
lb mollh = 18,65 aB- FFBO -A = 802.8 43.0 Ib moI1h
c. Evaluate mok balance terms: The conversion calculated from the mole balance, XM,,is found from Equation (EX-8.5).
(16.96 X 10rshW1)(0. 1229 h) exp (- 32,4001 1.987T)
Plot XM, as a function of
= 1
+ (11.96X 10il hL1)(0.1229h) exp(-32,40011.987T) (E8-8.10)
temperature. ~ M
(2.084 X 1012) exp ( - 16,3061T) , is = B I + (2.084 X 1012)exp(- 16,3061 T)
d. Evaluate energy balunct terms:
530
Steady-State Nonisoihermal Reactor Design
Chap. B
The conversion cal~uIatedfrom the energy balance, Xm,for an adiabatic reaction is given by Equation (8-29):
Substituting all the known quantities into !he energy balances gives us
Btullb mol - OF)(T - 533°F xm= -[-(403.3 36,400 - 7 ( T - 528)l Btullb mol
I. Solving. There are a number of
different ways to solve these two simultaneous equations. The easiest way is ro use the PoIymath nonlinear equation solver. However. to give insight into the functional relationship between X and T for the mole and energy balances, we shall obtain a graphical solution. Here X is plotted as a function of T for the mole and energy balances, and the inersection of the two curves gives the solntion where both the mole and energy balance solutions are satisfied. In addition, by plotting these two curves we can l e a n if there is more than one intersection (i.e., multipIe steady states) for which both the energy balance and mole balance are satisfied. If numerical mot-finding techniques were used to solve for X and T. ir would be quite possible to obtain only one root when there are actualry mare than one. I f Polymath were used. you couId learn if multiple roots exist by changing your initial guesses in the nonlinear equation solver. We shall discuss multiple steady states further in Section 8.7. We choose T and then calculate X (Table E8-8. I ). The calculations are plotted in Figure EX-8.2. The virtually straight line c m s p o n d s to the energy balance [Equation {E8-8.12)] and the curved line corresponds to the mole balance [Equation (E8-R. lo)]. We: observe from this plot that the only intersection point 1s at 85% conversion and 61 3"R. At this point, both the energy balance and moIe balance are satisfied. Because the temperature must remaln below 125OF (585"R). we cannot use rhe 300-gal reactor as tt 1s now.
Don't give upv Head back ro the storage shed lo check out thc heat exchanFe equipment!
1
Sec. 8.6
CSTR with Heat Effects
'Zhe reactor cannot be used becai~se~t uoiHexceed the specified maximum temperature
of 585%.
T ('RI
Figure E8-8.2
I
Tbe conversions XEa and Xw, as a function of temperature.
EsornpIe 8-9 CSTR wilh o Cooling Coil A cooling coil has been located in equipment storage for use in the hydration of propylene oxide discussed in Example 84. The cooling coil has 40 ft2 of cooling surface and the cooling water flow rate inside the coil is sufficiently l q e that a constant coolant temperature of 85°F can be maintained. A typical overall heat-transfer coefficient for such a coil is 100 Btu/h-ft2."E W111 the seactor satisfy the previous constraint of 125°F maximum temperature if the cooling coil is used?
SnEutidn If u7eassume that the conling coil rakes up negligible reactor volume, the conversion calculated as a function of temperature from the mole balance is the same as that in Example 8-8 [Equation (EB-8.10)]. 1. Combining the mole baiance, stoichiornetry, and rate law, we hav,=. from
Example 8-8.
XWB= ~k - (2 084 X 10") n p (- 16,3061 T) I + t k 1 + (2.084 X 1 OJ2) exp (- 16.306) T )
-
(E8-8,
T i s i n "R. 2. Energy halance. Neglecting the work hy the stirrer, we combine Quations (5-27) and (8-50) to write LTA(To - T ) FA"
I((AHORr(TR)+ ACptT- TH))= SO,Cp,[T- T,)
(Eg-9.1)
I
532
Steady-State Nonisothermal Reactor Design
Cht
SoEcin~Ae enerxy balance for X,, yields Energy Balance
The cooling coil term in Equation (E8-9.2) is
Ud - (IW F~~
-h) Btu ft?."F +
(40 ft2) (43.03 lb moIl'h)
- 92.9 Btu
(E8-
lb mold "F
Recall that the cooling temperahre is
T, = 8S°F
=
545"R
The numerical values of all other terns of Equation (E8-9.2) are identic: those given in Equation (E8-8.12) but with the addition of the heat excha
tern.
We can now use the glass lined reactor
We now have two equations [(ES-8.80) and (E8-9,411and two unknowns. X and The POLYMA~Iprogram and solution to these two Equations (EB-8.10), X and (E8-9.4). XEa.are given in Tables ES-9.1 and E8-9.2. The exiting tempera and conversion are 103.7"F (563.7"R)and 36.46, respectively. i.e..
IT
= 5 6 4 " ~and x = 0.361
Equations: Nonlmear quationr [ II
rrX) = X-(M3.3'(T*5.5uH92.9'fl-MS))I(~7'(T-528)1 =0
I21 IrTI 5 X-~au*W
Uvlng Example Problerr
Ill
1x11
=0.1?29
i2l A = I 6 . W l W l Z 131 E = 3 2 m I41
R = 1. a 7
Sobrion outpur to Polymath program in Table E8-9.1 is shown in Tahk E8-9. TABWE8-9.2. EWLE
8-8 CSTR
WH
HEATEXCHANGE
Variablt
Valuc
flX)
Ini Guesr
X
0 3636087
2.U3E-1 I
0.367
7
563.73893
-5.4118-10
5W
lau
0.1229
A
1.696Ecl3
E
3 . U W
R
1.987
k
4.WF.9843
Sec. 8.7
533
Multiple Steady States
8.7 MutZiple Steady States In this section we consider the steady-state operation of a CSTR in which a first-order reaction is taking place. We begin by recalling the hydrolysis of propylene oxide, Example 8-8. If one were to examine Figure €8-8.2, one would observe that if a parameter were changed slightly, the X,, line could move slightly ta the left and there might be more than one-intersection of the energy and mote balance curves. W e n more than one intersection occurs, there is more than one set of conditions that satisfy both the energy balance and moEe balance: consequently, there will be multiple steady states at which the reactor may operate. We begin by recalling Equation (8-54), which applies when one neglects shaft work and LC, (i.e.. AC, = 0 and therefore AH,, = AH:,).
-XANi,
= C,(1+
K)(T- T,)
(8-54)
where
and
-r Y
Using the CSTR mole balance X= -,
Equation (8-54) may be rewritten as
F.4 0
The left-hand side is referred to u the heat-generated term: I I
C(7)= Heatgenerated term
The right-hand side of Equation (8-58) i s referred to (by flow and heat exchange) R(T>:
R(p =
as the heat-removed
term
(8-60)
Heat-removed term
To study the multiplicity of steady states. we shall pIot both R(T)and G(T) as a function of temperarue on the same graph and analyze the circurnstance.w under which we will obtain multiple intersections of R(T) and G ( 0 .
534
Steady-State Nonisothermal Reactor Design
Chap. 8
8.7.1 Heat-Removed Term, f l T ) Vary Entering Temperature. From Equation (8-60) we see that RIT) increases linearly with temperature, with slope C,(1 + K) As the entering temperature To Is increased. the line retains the same slope but shifts to the right as shown in Figure 8-14.
.
Heat-removed curve R(T)
Figure 8-14 Variation of heat removai line with inlet temperature.
Vary Non-adiabatic Parameter K. If one increases K by either decreasing the molar flow rate FA, or increasing the heat-exchange area. the slope increases and the ordinate intercept moves to the left as shown in Figure 8-15, for conditions of T,< To:
If T, > To, the intercept will move to the right as K increases.
Figure 8-15 Varialion of hear removal line with
K (K
= Url!C,F,,).
0.7.2 Heat of Generation, G( f ) The heat-generated term, Equation 18-59], can be written in terms of conversion. (Recall: X = - r,VI FA,.)
G ( T )= {-AH",,
)X
(8-61 )
Sec. 8.7
535
Multiple Steady States
To obtain a plot of heat generated, CCT), as a function of temperaturr-, we must solve for X as a function of T using the f3TFt mole balance, the sate law, and stoichiometry. For example, for a first-order liquid-phase reaction, the CSTR mole balance becomes
v= FA&--
VOCAOX
kcA,(]- X )
kc*
Solving for X yields I st-order reaction
rk x= I
+ zk
Substituting for X in Quation (8-61), we obtain
Finally, substituting for k in terms of the Amhenius equation, we obtain
Note that equations analogous to Equation (8-63) for G ( T ) c3n be derived for other reaction orders and for reversible reactions simply by solving the CSTR mole balance for X . For example, for the second-order liquid-phase reaction
the corresponding heat generated is
At very low temperatures, the second term in the denominator of Equation (8-63) for the first-order reaction can be neglected so that G(T)varies as LOW
T
G ( T ) = -AHi,zAe-E'Rr (Recall that AH,", means the heat of reaction is evaluated at T,.) At very high temperatures. the second term in the denominator dominates. and G(T)is reduced to
G(T) is shown as a function of Tfor two different activation energies, E, in Figure 8-16. If Ehe flow rate 1s decrea\ed or the reactor volume increased so as to increase 7, the heat of generation term, G(T),c h a n ~ e sas shown in Figure 8-17.
536
Steady-State Nonisothermal Reactor Design
Chal
High E
-
s
S
Figure 8-16
Heat generation curve.
Figure 8-17 Variation of heat generatton curve with space-time
Heat-generated curves. G(Tj
8.7.3 Ignition-Extinction Curve The points of intersection of R(T) and GtT) give us the temperaturn which the reactor can operate at steady state. Suppose that we begin to feed reactor at some relatively low temperature. To,. If we construct our G ( T ) i R ( T ) curves, illustrated by curves y and a, respectiveIy. in Figure 8-18, we that there will be only one point of intersection, point 1. From this point of in section. one can find the steady-state temperature in the reactor, T,,, by follc ing a vertical line down to the T-axis and reding off the temperature as show!
Figure 8- 18. If one were now to increase the entering temperature to T,. the G curve, y, would remain unchanged, but the R(Tf curve would move to the ri) as shown by Iine b in Figure 8- t 8, and will now intersect the G(T)at point 2 : be tangent at point 3. Consequently, we see from Figure 8-18 that there are 1 steady-state temperatures. T,, and T,,, that can be reaIized in the CSTR for
entering temperature TO:.If the entering temperature is increased to ir;,, R(T) curve, line c (Figure 8-19). intersects the G(T) three times and there three steady-state temperatures. As we continue to increase To, we finally re line e, in which there are only two steady-state temperatures. By further incrt ing T, we reach line f, corresponding no T, , in which we have only one tern1 ature that will satisfy both the mole and energy balances. For the six enter temperatures, we can Form Table 8-5, relating the entering temperature to possible reactor operating temperatures. By plotting T, as a function of T,, , obtain the well-known ignition-extinction cune shown in Figure 8-20. FE this figure we see that as the entering temperature is increased. the steady-s temperature increases along the bottom line until To,is reached. Any f n c t i o ~ a degree increase in temperature beyond Tm and the steady-state reactor tern1 ature will jump up to T,,, , as shown in Figure 8-20. The temperature at wk
Sec. 8.7
537
Multrpfe Steady States
+
+-.
5 Both the mole and energy bnlnnccs are mtidicd at the points of intsncction or tangency.
A
Figure I(-I8 F~ndingmultiple steady ~tatekwith T,, varred
Figure 8-19 Finding ~nulriplesteady ~ t a t ew~~ t hT, vaned.
We must exceed
a certain feed temprilture to operate at the upper ~ t e a d ystate where the
temperature and conversion are higher.
this jump occurs is called the ignition temperature. If a reactor were operating at T,,, and we began to cool the entering temperature down from To,, the sredy-state reactor temperature T,, wouId eventually be reached. corresponding to an entering temperature To,.Any slight decrease below To?would drop the steady-state reactor temperature to TF3.Consequently, To? is called the e.~tincfiantemperarure. The middle points 5 and 8 in Figures 8-19 and 8-20 represent unstabIe steady-state temperatures. Consider the heat removal line d in Figure 5- 19 along with the heat-generated curve which is replotted in Figure 8-21.
Steady-State Nonisotherrnal Reactor Desigl
Chap. 8
Figure 8-20 Temperature ignition-extinction curve.
T ~ 7
Figure 8-21
T&
T~
T
Stab~litjon multiple state temperatures
If we were operating at TsB! for example, and a pulse increase in reactor temperature occurred. we would find ourselves at the temperature shown by vertical Iine CZ) be~weenpoints 8 and 9. We see along this vertical line @ the heat-generated curve. G. is greater than the heat-removed line R (G> R).Consequently. the temperature in the reactor would continue to increase until p n t 9 i s reached at the upper cteady state. On the other hand. if we had a pulse decrease helween in temperature from point 8. we would find ourselves a vertical line ~3
Sec. 8.7
539
Multiple Steady States
points 7 and 8. Here we see the heat-removed curve is greater than the heat-generated curve so the temperature will continue to decrease until the lower aeady state is reached. That js a srnajl change in temperature either above or below the middle steady-state temperature, T,, will cause the reactor temperature to move away from this middle steady state. Steady states that behave in the manner are said to be unstable. In contrast to these unstable operating points, there are stable operating points. Consider what would happen if a reactor operating at T, were subjected to a pulse increase in reactor temperature indicated by Iine O in Figure 8-2 1. We see that the heat-removed line (d) is greater than the heat-generated curve (y), so that the reactor temperature will decrease and return to T* On the other hand, if there is a sudden drop in temperature below T* as indicated by line we see the heat-generated curre {y) is greater than the, heat-removed line (d) and the reactor temperatue will increase and return to the upper steady slate at T&. Next let's look at what happens when the lower steady-state temperature at T,, is subjected to pulse increase to the ternperature shown as line 3 in Figure 8-21. Here we again see that the heat removed, R, is greater than the heat generated, G. so that the reactor temperature will drop and return to Ts7. If there is a sudden decrease in temperature below T,, to the temperature indicated by Iine @, we see that the heat genesated is greater than the heat removed, G > R, and that the reactor temperature will increase until i t returns to T,,.A similar analysis could be carried out for ternperature TS1, T,?. T,,, Ts6, T,,,, and T,,, and one would find that reactor temperatures would always return to local sleadv-srare values, when subjected to both positive and negative fluctuations. While these points are locally stable, they are not necessarily globally stable. That is, a perturbation in temperature or concentration. while srnaIl, may be sufficient to cause the reactor to fall from the upper steady state (corresponding to high conversion and temperature such as point 9 in Figure 8-21) to the lower steady state (corresponding to low temperature and conversion, point 7 ) . We will examine this case in detail in Section 9.4 and in Problem P9-16B. An excellent experimental investigation that demonsrrates the muItiplicity of steady states was camed out by Vejtasa and Schmitz (Figure 8-22).They studied the reaction between sodium thiosulfate and hydrogen peroxide:
a,
in a CSTR oprated adiabatically. The multiple steady-state temperatures were examined by varying the Row rate over a range of space times, r, as shown in Figure 8-23. One observes from this figure that at a space-time of 12 s, steady-state reaction temperatures of 4, 33, and 80°C are possible. If one were operating on the higher steady-state temperature line and rhe volwmetric flow rates were steadily increased (i.e., the space-time decreased). one notes that if the space velocity dropped below about 7 s. the reaction temperature would drop from 70°C to Z°C. The flow rate at which this drop occurs is referred to as the ~ I O M ' O velocin'. U~
540
Ch
Steady-State Nonisothermal Reactor Design
100
0" Y,
-
o
f h@oreticul Expenmentalstabk SMtes
P+
Expenmental~nterrwdare states
-
I-
0
X)
40
60
80
0
[I
rl'cl
Figure 8-22 Heat generation and removal functions for feed mixture of 0.8 M NalSIO, and 1.2 M H,Ot at PC.
1
4
I
I B
I I I I 1 2 1
# 6
t I M :
f (5)
Figure 8-23 Multiple steady
states.
By S.A. Vejtasa and R. A. Schmitz. AlChE J., 16 (3). JIS (1970). (Reproduced by permission uf the American Inst~tuteof Chemical Engineers. Copyright Q 1970 AIChE. All right reserved.) See Journal Critique Problem PSC-4.
8.7.4 Runaway Reactions in a CSTR
In many reacting systems, the temperature of the upper steady state ma! sufficiently high that it is undesirable or even dangerous to operate at this I dition. For example, at the higher temperatures, secondary reactions can place, or as in the case of propylene glycol in Examples 8-8 and 8-9, evap tion of the reacting materials can mcur. We saw in Figure 8-20, that we operated at the upper steady state i we exceeded the ignition temperature. For a CSTR,we shall consider tuna (ignition) to wcur when we move from the lower steady sbte to the ul steady state. The ignition temperature occurs at the point of tangency of heal removed curve to the heat-generated curve. If we move slightly off point of tangency as shown in Figure 8-24, then runaway is said to t occurred. At this point of tangency, T,we have not only
Sec. 8.7
Multiple Szeady States
Tc
T*
T
Flgure 8-24 Runaway in a CSTR.
but also the slopes of the R(7)and G(T) curves are also equal. For the heat-removed curve. the slope is
and for the heat-generated curve, the slope is
Assuming that the reaction is irreversibIe and follows a power law model and that the concentrations of the reacting s p i e s are weak functions of temperature. - r ~ = (Aem',
fn(Ci)
then
Substituting for the derivative of (-r,) wrt Tin Equation (8-67)
18-68]
542
Steady-Stele Nonisotherrnal Reactor D ~ i g n Chap. 8
where
Equating Equations (8-66) and (8-69) yields
Next, we divide Equation (8-65) by Equation (8-70) so obtain the fellowing AT value for a CSTR operating at T = T:
I f t h i . ~diflerence herween rhe reacfor temperazure and T,,AT,, is exceeded, transifion m the rrpper sready stare will occur. For many industrial reactions, U F T is typically between 16 and 24, and the reaction temperatures may be between 300 to 500 K. Consequently, this critical temperature difference AT, wiIl be somewhere around 15 to 30°C.
Stability Diagram. We now want to develop a stability diagram that will show regions of stable operation and unstable operation. One such diagram would be a pIot of S* as a function of T,. To construct this plot, we first solve Equation (8-71) for T.the reactor temperature at the point of tangency,
and recalling
T [Equation ( 8 - J Z ) ] , calculate k" at T from rate law. calculate G ( T ) [Equation (8-5411, and then finally calculate S* to make a plot of S* as a function of T, as shown in Figure 8-25. We see that any deviation to the right or below the intersection of Cw ( 1 + K ) and S' will result in runaway. We can now vary 7,. then caIcutate
(k*= Ae-E/RT*),calculate
-r*,
Sec, 8.8
Nonisothermal Mdliple Chrnical Reactions
EYpre 8-25 CSTR stability diagram.
For example, for a first-order reaction, the equation for S is
Shelf
We simply combine Equation (8-72) and the equation for T, and then substitute the resdt into Equation (8-73) and plot S* as a function of Tw From Figure 8-25, we see that for a given value of [C,(I+u)I, if we were to increase the entering temperature To from some Iow-value To',, (T,,) to a higher entering temperature value Tm IT,,), we would reach a point at which runaway would occur. Further discussions are given on the CD-ROM professional reference shelf R8.2. Referring to Equation (8-701, we can infer
we will not move to the upper steady state, and runaway will not occur. How-
ever, if
runaway will occur.
8.8 Nonisothermal Multiple Chemical Reactions Most reacting systems involve more than one reaction and do not operate isothermally. This section is one of the most important sections of the book. It ties together all the previous chapters to analyze multiple reactions rhar do not take place isothermally.
544
Steady+StateNonisolherrnaT Reactor Design
CV
8.8.1 Energy Balance for Multiple Reactions in Plug-Flow Reacts
In this section we give the energy balance for multiple reactions. We begi recalling the energy balance for a single reaction taking place in a PFR H is given by Equation (8-351,
When q multiple reactions are taking place in the PFX and there are m cies, ir is easily shown that Equation (8-35) can be generalized to
i = Reaction number j = Species
The heat of reaction for reaction i must be referenced to the same species i rate, r , , by which AHRxbis multiplied, that is,
[-rqll[-AH~,l
=
s= eI,,,[
Joules "released"'in reactio Moles of j reacted in reactic
Moles o f j reacted in reaction i Volume - time
I
~'releasd" in reaction i Volume time
where the subscript j refers 10 the species, the subscript d refers to the parti reaction, q is the number o f independent reactions, and m is the numb species. Consider the following reaction sequence cartied out in a PFR:
) B
Reaction 1:
A
k'
Reaction 2:
B
k l > C
The PIX energy balance becomes
where AHRxlA = FJlmol of A reacted in reaction 11 and AH,,, = [kllrnol of B reacted in reaction 21.
(:
(:
Sec. 8.8
Nonisothermal Multiple Chern~calReactions
Example 8-18
1
ParalIeL Reoc~bnsin a PFR with Heat
EflecIs
The foilowing gas-phase reaciions occur i n r PFR:
Reaction 1: Reaction 2:
A 2A
&B k,
C
- r , , = k,,C,,
IE8- 10.1)
-rlA = kZ4?*
(E8- 10.2)
Pure A i s fed at a rate of 100 molls. a ternprature of 150PC, and a concentration of
0.I rnolldm3 . Determine4the temperature and Row rate profiles down the reactor.
AHR,,, = -20.000 J/(rnol of A reacted in reaction 1)
AHR,?, = -60,000 J/(mol of A reacted in reaction 2) Living Example Jroblem
Solslrion The PFR energy balance becomes [cf. Equation (8-76)]
Mole balances:
One of the major goais of this text is
I
hat the reader will
be able ta solve multiple reactions with heat effects.
Rate laws, relative rates, and net rates: Rate Eaws
546
Reaction 1:
Steady-State Nonisothermal Fleactor Oestgn
'IA = 5 . -1 I , ~
Chap. 8
=~ ~ A C A
I B
Reaction 2: Ner mtes:
Stoichiometry (gas phase AP = 0):
The algorithm for multiple reactions w t h heat effects
[
I):
k , , = 10 exp 3000 - - - sL1
[
k2* = 0.09 exp 9000
(3;
-- h)]
(3;
(Tin K)
2
Energy balance:
The Polymath program and its graphical outputs are shown in Table E8-10.1 and Figures E8-10.1and E8-10.2.
Sec. 8.8
Nonisotherrnal Multipls Chemical Reactions
547
TABLEE8-10.1. POLYMAWPROGRAM Equarim:
I
Lfving Eram~icProblem
POLYMATH Results Examplt &-111F*tnllcl R n a b
In 4 PFR aILb Hm En=@m-1Mwc.
Ra3.1.u~
Splculrtd values or !C DEO mrlablra initial value
Variablm
v Fd
m FC
-
T kla CTO PC TO Cb.
1
9.7382-06
I00
o
0 D 423 482 .a247 551.05566
35.04326 w.4-1a369 812.19122
2 .?>BE-06 55.04326 22.478369 722.08816
4.4BPE+01 l.4BE*07 0.1
?.4266+04
100 PO3
77.521631
0.1 0.0415941 0.0169B6
2.069E-09 0.04i5941 0.016986
-373 -39077
-5.019E-05
-5.019E-05
-840.11153
-1.591E-11
-1.591E-11
0.1
0.1
100 433
77.521631
123
0.1 0
2.069E-09 0
Cc
0
0
-48.28947 -1.5305566
ria
M ~ value
1
Cb
r21
fL
0
La3 402.8247 553.05566
k21
minim1 valua msximl vaf u t
0 100 0
3.716I+O6 0.1
423
lRKF451
-rt
I)IVWMIMB ~ U * I D M aa meted by me uaer
1:I d(FaYd(W = r l u+Ra i 2 l dffbYdtW = -rla
f xpliil w & t M a as anratedby !he uner I I I k ~ =a ~ O ' ~ ~ ( * O W ' ( I / ~ [ K ~ . I ~ Q ) I 2 l k2e -O.OB'erp(QMXl'(l~9C&1rr)J
-
1 7 1 Cto = 0.1
--
14 I FI Fa+Fb+Fe 151 T o = 4 2 3 1 6 1 Cs Cto'(FaFl)'(Tdr) I?I Cb = Clo'{FWF!).(TdQ
l a ] Cc Cto'{FMt)'(TPrT) 1 9 1 r l a - -hla'Cn I1 0 l Ra P -k2a'Cm
850
Why does the
550
temperalure go through a maximum va rue:'
4M
00
02
04
06
08
V
Figure E8-10.1 Temperature profile.
10 DO
a2
04
v
06
06
10
Figure ES-10.2 Profile of molar flow rates F,.F,,and F,.
548
Steady-State Nonisothermal Reactor Design
Cht
8.8.2 Energy Balance for Multiple Reactions in CSTR RecalI that -F,,,X = r,V fur a CSTR and that AH,,(Ir) = AH;, + AC,(T so that Equation (8-27) for the steady-state energy balance for a single reac may be written as
For g multiple reactions and m species, the CSTR energy balance becomes
1
Energy balance
for multiple reactions in a CITR
1
Substituting Equation (8-50) for Q, negIecting the work term. and assum constant heat capacities, Equation (8-80) becomes
For the two parallel reactions described in Example 8-10. the CSTR ene balance is
Mdor goal of CRE
(8-
One of the major goals of this text is to have the reader solve problems invc ing multiple reactions with heat effects (cf. Problem P&-26c). Example 8-11 Multiple Reactions in a CSTR
The elementary liquid-phase reactions
take place in a 10-dm3 CSTR. What are the effluent concentrations for a volume feed rate of 1000 drn31rnin at a concentration of A of 0.3 molldm3?
The inlet z e r n p t u r e is 283 K.
Additional information:
k, = 3.3 min-I at 300 K. with E, = 9900 cat/rnol k2 = 4.58 min-I
at 500 K,with
El = 27,000 callmol
Sec. 8.8
I
I
hfi',,~, = -55.00fi Jlmol A
The &ons
UA = 40,000 JJrnin-K with
follow elcmentay rate laws
I. Mole Balance on Every Species A: Comb~nedmole balance and rate law for A:
Solving for C, g'~ v e sUS
I
I 1
I
549
Nonisolherrnal Multiple Chemical Reactions
B: Combined mote balance and rate INI f a B:
Solving for C, yields
2. Rate Laws:
3. Energy Balances: Applying Equation (8-82)to this system gives Substituting for r l , and r13 and rearransing, we have
r, = 57'C
550
Steady-State Nonisothermal Reactor Design
Chap. 8
We are now going to write a Polymath program to increment temperature 10
obtain
G(T) and R(r). The Polymath program to plot R ( T ) and G ( T ) vs. T i s shown in Table E8-11.1, and the resulting graph is shown in Figure E8-11.1.
POLYMATH Results Example 8-11 Multiple Reactions in a CS7"R 08-13-zW. Rev5.1.232
Incrementing temperature in this manner is an easy way to generale RU)and G(T) plots
Living Example Problen
Differentialequations as entered by the user [ 1 1 d(T)ld(t) = 2 Expl'iclt equations as entered by the user t l l Cp=200 [ a ] Cao=0.3 [ 3 J Toe283 [ 4 1 tau=.Ol [ 5 ] DHt =-55000 16 1 DH2 = -71500 171 v0=1000 [ 8]
€2 = 27000 =9900
191 El
[10] VA=400M) t l l l Ta=330 [ 12 1 k2 = 4.58'exp((E2/1.987)*(1/500-7 1 13 3 kl = 3.3*exp{(El/l.907)*(11300-1TT)) [ 14 1 Ca = Caol(1+lau'kf ) [ 15 I kappa = UN(vo'Cao)lCp 11 6 1 G = -tau'kl/{l +kl * t a u ~ ~ ~ 1 - k l ' t a u + k 2 " t a u * D ~ 2 ~ ~ ( i + t ~ u ~ k l ) * ( l + ( 17 I TC= (To+kappa'Ta)l( 1+kappa) [ 18 I Cb = tau'kl'Cd(1 + k 2 r ~ u ) [ 19I R = Cp'f 1+kappa)'(T-Tc) 120 I Cc = Cao-Ca-Cb [Ill F=G-R
m)
When F = O
G(T) = R(T) and the steady states can be found.
We see that five steady states ( S S ) exist. The exit concenrrarions and tempemtures listed in Table E8- 1 1.2 were interpre~edfrom the labular output of the ~ o l ~ ~ n a t h program.
Sec. 8.9
Radlal and Axial Variations in a Tubular Reactor
Wow! Five (5) multiple steady states!
71K)
Figure E8-1I.I Heat-removed and heat-generated curvec
We note there are five steady states (SS) whose values are given in Table ES-11.2. What do you think of the value of tau? Is it a realistic number? TABLE ES-I 1.2.
EFFLC'EVI C O ~ ' ~ A T I AND O YTEMPERANRES S
SS
T
cA
CE
cc
1
330 363 449
0.285
2 3
0.189
0015 0.111
4
558
0.033 0.N
0.265 0.163
5
677
0.001
0.005
0 0.0 0.002 0.132 0.294
8.9 Radial and Axial Variations in a Tubular Reactor EMLAB application
In the previous sections we have assumed that there were no radial variations in velocity. concentration, temperature ar reaction rate in the tubular and packed bed reactors. As a result the axial profiles couId be determined using an ordinary differential equation (ODE) solver. In this section we will consider the case where we have horh axial and radial variations in the system variables in which case will require a panial differenrial IPDE) solver. A PDE solver such as FEMLAB. will allow us to solve tubular reactor problems for horh the axla1 and radial profites, as shown on the web module. CVe are going to carry out differential mole and energy balances on the differential cyllndricai annulus shown in Figure 3-26.
Steady-State Nonisothermal Reactor Design
Cha
Front View
w1,
Figure 8-26 Cylindr~calshell of thickness Ar, length h, and volume 2 n r S r k
Molar Flux
In order to derive the governing equations we need to define a couple of ten The first is the moIar flux of species i, Wi {mollm2 s). The molar flux has t cornpsnents. the radial component W,,, and the axial component, W,,.1 molar flow rates are just the product of the molar fluxes and the cross-sectio areas normal to their direction of flow A,. For exampte, for species i flow in the axial (i.e., z) direction
where W,: is the molar flux in the z direction (rnoVm2/s), and A, (m2) is the cross-sectional area of the tubular reactor. In Chapter 11 we discuss the molar fluxes in some detail, but for now us just say they consist of a diffusional component, -D,(aC,/az) , and a ct vective flow component, U:C,
ac.t.upci
wir= - D ~ -
a2
13-r
where D, is the effective diffusivity (or dispersion coefftcienl) (m2/s). and is the axial molar average velocity (mls).Similarly, the flux in the radial din tion i s
ac.+ UrCi
Wir = - D,-
Radial Direction
ar
(8-5
U,( m l s ) is the average velocity in the radiaI direction. For now we n neglect the veiocity in the radial direction, i.e., U, = 0. A mole balance or cylindrical system volume of length Az and thickness Ar as shown in Fig1 8-26 gives where
Rad~aland Rxlal Variations In a Tubular Reactor
Sec 8.9
Mole Balances on Species A
(
Moles of A = i n ) .
(
(
1
Crass-sectionai area = WAr.2~rAr normal t o axial flux
)
*
( M ~ E ~ A ) Moks of* +(M;:~:*)-( out at ( r + At-)
out at ( z + Az)
Dividing by 2xrAr4 and faking the limit as Ar and Az
hz --,O
hfOlesof
)
Similarly, for any species i and steady-state conditions.
Using Equation (8-831 and (8-84) to substitute for W!:and W,,in Equation (8-85)and then setting the radial velocity to zero, U,= 0, we obtain
This =quation For steady-state conditions and assuming 157, does not vary also be d~scussed funher in
Chapter 14.
in the axial direction,
554
Steady-State Nonisothermal Reactor Design
Chap. 8
Energy Flux
Wen we applied the first law of thermodynamics to a reactor to relate either temperature and convtrsian or molar flow rates and concentrati~n.we arrived at Equation (8-9). Negleaing the work term we have for steady-state conditions Conduction
Convection
In terms of the molar fluxes and the cross-sectional area and (q [email protected],)
The q term is the heat added to the system and almost always includes a conduction component of some fom. We now define an energy flux vector. e. (J/rn2 s), to include both the conduction and convection of energy. +
e =e
n e r ~flux J/s.rnZ
e = Conduction
+ Convection
where the conduction term q (kJlm2 . s) is given by Fourier's law. For axial and radial conduction Fourier's laws are 4; =
-k,-aT 8,-
and
q,=-kc-a T
ar
where k, is the thermal conductivity (JJm.s.K). The energy transfer (flour) is the vector flux times the cross-sectional area, A,, normal to the energy flux
Energy flow = e - A, Energy Balance Udng the energy Bux. e, to carry out an energy balance on our annulus [Figure 8-26) with system volume 7nrhr&, we have
(Energy flow in at I.) = e, A,, = e; 2 n r h (Energy flow in at z ) = e,A, = e; 2arAr Accumulat~on o f Energy In VoIumc ( 2 r r ~ r . k )
See. 8.9
Radia! and Axial Vanetlons in a Tubular Reactor
Dividing by 2nrArbz and taking the Ijmit as & and Az
+ 0,
The radial and axial energy fluxes are e, = g,
+ E' W,,HE
e. = g,+CW,, H,
Substituting for the energy fluxes, e, and e;,
and expanding the convective energy fluxes, Z WiHi,
Axial: Substituting Equations (8-93) and (8-94) into Equation (8-92), we obtain upon rearrangement 'I
Recognizing that the term in brackets is related to Equation (8-85) for steady-state conditions and is just the rate of formation of species i. rimwe have 1 a -%- Z H j f j- 3 ?yzmi-o ---(rqr) rar
Recalling
and
ilz
az
(W5)
556
Steady-Stale Nonisothermal Reactor Design
Ch:
we have the energy in the form
Some Initial Approximations
I . Neglect the diffusive term wrt the convective term in the exp sion involving heat capacities
Assumption
With this assumption Equation (8-96) becomes
Energy
with radial and axial grddientq
Assumption 2. Assume that the sum CPm= ZCp,C,= C,,Z8,Cg
is cons1
The energy balance now becomes
Equation (8-98) is the form we will use in our FEMLAB problem. In rr instances, the term C P is~just the product of the soIution density and the capacity of the solution (kJkg K).
Coolant Balance
We also recall that a balance on the coolant gives the variation of coolant t peralure with axial distance where Uhris the overall heat transfer coeffic and R i s the reactor wall radius
For laminasfiow, the veIociv profile is
where Uois the average velocity inside the reactor.
S x . 8.S
Radial and Axial Varia!ions In a Tubular Reacior
Boundary and initial conditions A. Initial conditions ifother than steady state t = O , C,=O. T=T,? f o r z > O a l l s B. Boundary condition 1) Radial
(a) At r = 0, we h a ~ esymmetry a T l d r = 0 and aC,/ar = 0 . (bj At :he tube wall r = R, the temperature flux to the wall on the reaction side equals the convective flux out of the reactor into the shell side of the heat exchanger.
( c ) There is no mass flow through the tube walls aC,/ar = 0 at
Id)
r = R.
At the entrance to the reactor .z = 0,
T = Toand Ci = Cn (e) At the exit of the reactor z = L,
The following examples wiIl solve the preceding equations using FEMLAB. For the exothermic reaction with cooling, the expected profiIes are
Example 8-12 Radial Effects in Tubuhr Reactor This example will highiight the radial effects in a tubular reactor, which up until now have been neglected to simplify the calculations. Now, the effects of panmeters such as inlet Temperature and Row rate will be studied using the software program FEMLAB. Follow the step-by-step procedure in the Web Module an the CD-ROM. We continue Example 8-8, which discussed the reaction of propylene oxide (A) with water ( 8 ) to form propylene glycol (C). The hydrolysis of propySene oxide takes place readify st room temperature when catalyzed by sutfuric acid.
558
Steady-Slate Nonisothermal Reactor Design
Chap. B
This exothermic reaction is approximated as a first-order reaction given that the reaction rakes place in an excess of water. The CSTR from Example .8-8 has been repIaced by a tubular reactor 1.0 m in length and 0.2 m in diameter. The feed to rhe reactor consists of two streams. One stream is an equivolumetric rnlxture of propylene oxide and methanol, and the orher stream is water containing 0.1 wt Yr sulfuric acid. The water is fed at a volumetric rate 2.5 times larger than the propylene oxide-methanol feed. The molar flow rate of propylene oxide fed to the tubular reactor IS 0.1 moUs. There is an immediate temperature rise upon mining the two feed streams caused by the heat of mixing. In these calculations, this temperature rise is already accounted for, and the inlet temperature of both streams is set to 312 K. The reaction rare law i s
with where E = 75362 Jlrnol and A = 16.96 x 10" h-I, which can also be put in the form
With k, = 1.28 h-' ar 300 K. The thermal conductivity kt of the reaction mixture and the diffusivity D, are 0.599W l m K and 1 p rn2/s, respectively. and are assumed to be constant throughout the reactor. In the case where there is a heat exchange between the reactor and its surroundings, the overall heat-transfer coefficient is 1300 Wlm2/K and the temperature of the cooling jacket is assumed to be constant and is set to 273 K. The other propeny data are shown in Table E8- 12.1.
See. 8.9
Radlal and Axial Variations In a Tubular Reactor
Solution
Mole Balances: RecdIing Equation (8-86) and applying it to species A
- !)]c, RT, TJ
-rA = 4 T,) cxp[f(l
(
(E8- 12.2)
Stoichiometry: The conversion along a streamline (r) at a distance z
The overall conversion is
I
The mean concentration at any distance z
I
For plug flow she velocity prohle is
v: = u, The Laminar flow velacily profile is
I
Recalling the Energy Balance
Assumptions .iviog Example Problcm
1. U, is zero. 2. NegIect axial diffusion/dispersionflux wn convective flux when summing the heat capacity times their fluxes.
3. Steady state.
560
1
Steady-State Nonisothermal Aeactor Destgn
Cha
Cooling jacket
- 2aRUh,(T,(z) nl'C, AT0 -1
a=
- T,)
Roundary conditions
At r = R. then ~t
2
= 0,then
aca - 0 and - k , - =ST dr
ar
Lrh,(T,(z)-T,}
C,= C , andT= TO
(E8-12.9
(E8-12. I0
These equations were solved using FEMLAB for a number of cases inctud~ngas batic and non-adiabatic plug flow and laminar flow: they were also solved with without axial and radial dispersion. A detailed accounting on how to change parameter values in the FEMLAB program can be found in the FEMLAB Inst1 t~onssection on the web in screen shots simllar to Figure EX-12. I . Figure E8-I gives the data set in SI units used for the FEMLAB example on the CD-ROM.
Nore There is a ~tep-by-step FEMLAB tutorial uslng screen fhots for th~sexample on
the CD-ROM.
Define expressian
Figure FA-12.1 FEMLAR screen shot of Data Set.
Color surfaces are used to show the concentration and temperature profiles. sirr to the black and white figures shown in Figure ES-12.2. Uqe the FEMLAB pmg on the CD-ROM to develop temperature concentration profiles similar to the r shown here. Read through the FEMLAB web module entitled "Radial and A Temperature Gradient." before running the program. One notes in Figure E&that the conversion is lower near the wall because of the coofer fluid temperat These same profiles can be found in color on the web and CD-ROM in the mdutes. One notes the maximum and minimum in these profiIes. Near the wall. ternperamre of the mixture is lower because of rhe cold wall temperature. Co quently. the rate will be lower, and thus the conversion will be lower. However, r next to the waH, the velocity through the reactor IS almost zero so the react spend a long dme In the reactor; therefore. a greater conversion is achieved as nl by the upturn right next to the walt.
Sec 810
I Result5 of the FEMLAR si~nulation
The Pnctrcat Sree
fa) Tarnwrstura S u k e
Radial Temperature Profllss
Aadrar Cocation rm)
RMaisl Lmdlh (m)
Conversion S m e
Radial Clxrversron Profllm
Radial Locattm (m]
(dl Figure E8-12.2 (a) Temperature surface, (b) ternperamre surface profiles. (c) convenion surface. and Id) radial conversion profile.
8.1 0 The Practical Side Scaling up exothermic chemical reactions can be very tricky. Tables 8.6 and 8.7 give reactions that have resulted in accidents and their causes, respectively.IO
'OCourtesy of J. Singh, Chemical Engineen'ng, 92 (1997) Engineering, 54 (2002).
and B. Venugopal, Chemical
562
Steady-State Nonisothermal Reactor Design
Chap. 8
Conttiburion, Cause
Lack of knowledge of reaction chemistsy
AlkyIation (Friedel-Crafts)
9t 20
Problems with malerial quality
9
Temperature-control problems
19
Agitation problems
10
MIS-chargingof reactants or catalyst
21
Poor ma~ntenance
15
Op~ratorerror
5
More information is given in the Summary Notes and Professional Reference Shelf on the web. The use of the ARSST to detect potential problems will be discussed in Chapter 9. Summary Motes
Closure. Vinually all reactions that are carried out in industry involve heat effects. This chapter provides the basis to design reactors that operate at steady state and involve heat effects. To model these reactors, we simply add another step to our algorithm; this step is the energy balance. Here it is important to understand how the energy bdance was applied to each reaction type so that you will be able to describe what would happen if you changed some of the operating conditions (e.g., To).The living example problems (especially 8T-8-3) and the ICM moduIe will help you achieve a high level of understanding. Another major goal after studying this chapter is to be able to design reactors that have multiple reactions taking place under nonisothemia~conditions. Try working Problem 8-26 to be sure you have achieved his goal. An industrial example that provides a number of practical details is incIuded as an appendix to this chapter. The last example of the chapter considers a tubular reactor that has both axial and radial gradients. As with the other living example problems, one should vary a number of the operating parameters to get a feel of how the reactor behaves and the sensitivity of the parameters for safe operation.
Ghap. 8
Summary
SUMMARY For the reaction
A + -bB + $ C + - Dd a
1.
a
a
T h e heat of reaction at temperature T. per mole oFA, is
2. The mean heat capacity difference. ACp. per mole of A is
where Cp, is the mean heat capacity of species i between temperatures T, and T . 3. When there are no phase changes, the heat of reaction at temperature T is related to the heat of reaction at the standard reference temperature T, by
4.
Neglecting changes in potential energy, kinetic energy, and viscous dissipation, and for rhe case of no work done on or by the system and all species entering at the same temperature, the steady state CSTR eneqy balance is
5.
For adiabatic operation of a PFR,PBR, CSTR, or batch reactor. the temperature conversion relationship is
Solving for the remperature, T.
6.
The energy balance on a PFR/PBR
564
Steady-State Nonisothermal Reactor Design
Ch:
In terms of conversion,
7 . The temperature dependence of the specific reaction rate is given in the fc
8. The temperature dependence of the equilibrium constant is given by v HoFs equation for AC, = 0,
9.
Multiple steady states:
10. The criteria for Runaway Readions accurs when (T,- T,)> R
+
+
s*=G(r)Runaway
E
RT*~
I I. When q multiple reactions are taking ptacc and there are m species.
Chap. 8
565
Summary
12. Axial or radial temperature and concentration gradients. The foIlowing coupled partial differential equations were solved using FEMLAB:
and
ODE SOLVER ALGORITHM Packed-Bed Reactor with Heat Exchange and Pr~ssslreDrop
2.4
aC
Pure gaseous A enters at 5 mollmin at 450 K.
-dX- - -4 dW FA~ dT = UA/p,(T, - T)+ IrA)(AH&) dW
4 = -% ( 1 - 0.5X)(T/TO) dW 2y
-c ~ / K ~ ]
r; = - k [ ~ i
c,
= c+.a[Il - - W ( 1 -o-sx)l(To/T)(~r
C, = ;C,J(T,/ T),v/I I - 0.5m k = 0.5 exp [5032(( 1 J450) - ( 1 / T))]
a = 0.019lkg cat. CN = 0.25 molldm3
U A / h = 0.8 J/kgcat:s.K
c=500K AH;;, = -20.000 J/moE CpA= 40 J/rno!.K FA, = 5.0 molJs To = 450 K
W,= 90 kg cat.
566
Steady-State Nonisothermal Reactor Design
Chap. 8
CD-ROM MATERIAL Learning Rmurces 1. S ~ r ~ n ~ n nNOIPS ? 2. Web Module FEMLA B Rndial and Axiul Grodienrs Summary dotes 0s. 01.
E
a 7.
C P6-
z
P
UP.
-1 C"
2 a> 01Oi.
Rad~a!Locst~on{m)
3. Itlteraclrv~C O I I I ~ I Imodule.^ IPT A. Hcat Effecrs 1
Radial Locafion (mJ
B. Heat Effects I1
Interactiv~,
4. S n l r ~ dPmblems A. Example CDR- I
Functions of Ternperat~~re Second-Order Reactjon Carried Out Ad~abaticalIyin a CSTR 5. P F R P B R Solrtrio~~ Pmced~ii- for n R~verrihlpGas-Plrase Reaction Living Example Psohlems I . E~fi:rrmplcX-3 Rd~rrlx~tir Irrjr~l~t-i;orio~~ of ,\'or~?lul Ruranp 2. Examplp 8-4 Irnrueri,-n!rol~nf Nort~~rtl Hrrtnt~cwillr Herr1 E.~clmn~e .7. Euntrlpfr 8-5 Pmd~tcrionr?f'ArericAt~h\rlrid~ 4. i??~an~l)lt~ 8-9 CSTR ivilh Conlrtt,~Coif 5. Erurnplr 8-10 Prrrrillpl Kerrrlio~rit) rr PFR wirh Hear Efir.!s 6. E ~ n l ~ i p8i r- i f .Ilalii,ilr Rcrafiori ili o C.TTR
3. Example CDS-2
Solved Problems
Q
4,
t.*f?
?I& 'lVng
Prabem
-
AHR,(n for Heat Capacities Expressed as Quadratic
Chap. P
CD-ROM Material
7, Exnmpk 8-1.2 FEMLAB Axial and Radial Gmdienrs 8. Et-ample R8.2-1 Rrrna~~ay Reacrions in a PFR 9. Example R8.4-I Indusmrial Oxidafinn of SO, 10. Example 8-TK-3 PBR with VarrabEe CooIant Temperature. T, Pmfsliiolnal Reference Shelf R8.I . Runaway in Plug Flow Reocror.7 Phase Plane Plots. U'e transform the temperature and concentration profires into a phase plane.
,
Crnsal Tralectory
Temperature Profibs
T
Reference Shelf
Tmz T.,
--
- ------- - - -
Ca! * C m > C m
ZCWI
cm
.- - - - -.-
CM? V
Ternperarure profiles.
TU
fm
Critical trajectory on the
C,, -1;, phase plane plot. The trajecton, going through the maximum of the "maxima curve" is considered to he crrtiral and therefore i s the locus of the crriicrrl inlet condrtions for C, and T corresponding to a given wall temperature.
R8.3. Steady-5taf.e Bifitrcutron Anal~xis. In reactor dynamics, ir i s particularly important to find out if multiple stationasy points exist or if sustained oscillations can arise. We apply bifurcation analysis to learn whether OT not multiple steady states are possible. Both CSTR energy and mole balances are of the form
The conditions for uniqueness are then shown to be those that satisfy the relationship
Specifically. the conditions for which multiple steady Ctates e ~ i smust t satisfy the SoHnwing set o f equations:
R8.3. Vuriobf~ Hen! Ci~pucrr~c.~. Next we want to arrive at a form of the energy halance for the case where heal capaciries are strong functions of temperature over a wide ternpemture range. Undcr thew conditions. the mean value$ of the heat capacrty may no1 he adequate for the rclat~onshipbetween conveerslon and temperature. Comb~ningheat reaction with the quadratic form of the heat capaciry.
568
Steady-State Nonisothermal Reactor Oesipn
CR
we find that
Heat capacity as o function of temperature
Example 8-5 is reworked for the case of variable heat capacities. R8.1. Mnnufac~~rre ojSuwuric Acid. The details of the industrial oxidation of are described. Here the cataIys quantities reactor configuration, opea condttions, are discussed alang with a model to predict the conversion temperature profiles. Reference Shelf
QUESTIONS AND PROBLEMS
Uomewcrt i~roblemr
The subscript to each of the problem numbers indicates the level of difficulty: A, le difficult; D,most difficult. A=.
B=m
c=+
D=++
In each of the questions and problems, rather than just drawing a box arou your answer, write a sentence or two describing how you solved the problem, t assumptions you made. the reasorlabIeness of your answer. what you learned, and a other facts that you want to include. See the Preface for additional generic parts ( (y). (z) to the home problems. PS-1,
Creatlve P-oblems
Read over the problems at the end of this chapter. Make up an original prc lem that uses the concepts presented in this chapter. To obtain a solution: (a) Make up your data and reaction. (b) Use a real reaction and real data. See Problem P4-1 for guidelines. ( c ) Prepare a list of safety considerations for designing and operati1 chemical reactors. (See www.siri.org/graphics.)
Chap. 8
569
Questions and Problems
See R. % Felder. I. Chem. Eng. Edtrc., IY(4). 176 ( 1985). The August 1985 i w c of Cllcntical Engincerrn~P w , y r ~ c smay be uheful for part (c). (d) Choosc a FAQ from Chapter X and say why it was mosf helpful. (e) Liqten to the audios on the CD and pick one and say why it could be eliminated. IQ Read through the Self Tests and Self Assehhrnents for Chapter 8 an the CD-ROM, and pick one thar was most helpful. (g) Which example an the CD-ROMLecture Notes for Chapter 8 was the most helpful4?' (h) What if you were asked to prepare a list of safety comidentions of redesigning and operating a chemical reactor, what would k the tist four Items on your list? (i) What if you were asked to give an everyday example that demonstrates the principles discussed 10 this chapter' (Would sipping a teaspoon uf Tabasco or other hot sauce be one?)
&Fore solving the problems, state or sketch qualitat~vely the e~pectedresults or trends.
-fi
P8-2,
Load the following PolyrnathlMATLABlFEMLAB programs From the CD-ROM whew appropriate: {a) Example 8-1. How would this example change if a CSTR were used instead of a PFR? (b) Example 8-2. What would the heat of reaction be if 50% inert?, (e.g., helium) were added to the system? What would be the Ic error if the b C p term were neglected') (c) Example 8-3. What if the butnne reactinn were carried out in 3. 0.8-rn3 PFR that can be pressurized to very hish pressures? What inlet temperature would you recommend? Is there an optimum ternperature? How would your answer change for a 2-m3 CSTR? ( d ) Example 8-4. ( 1 ) How would the answers change if the reactor were in a counter current exchanger where the coolant temperature was not constant along the length of the reactor? The mass Row rate and heat capacity of the coolant are 50 kg/h and 75 kJlkglK, respectively, and the entering coolant temperature is 3 10 K. Vary the coolant tate, mi . make a plot of X versuq nit . (21 Re eat ( I) but change the parameters Kc, E. l.000 < Ua < 15,0CU(ff hlm IYC) , and AHRx.Write a paragraph describing what you find, notlog any generaiization. {eS Example 8-5. ( I ) How would your answer change if the coolant flow was counter curcent? (2) Make a plot of converrion as a function of FAo for each of the three cases. (3) Make a plot of conversion ss a function of' coolant rate and coolant temperature. (4) Make a plot of the exit conversion and ternperature as a function of reactor diameter but for the same total volume. (0 Example 8-6. How would the result change if the reaction were second order and reversible ?A H 1B with Kc remaining the sane? (g) Example 8-7.How would your answers change if the heat of reaction were three times thar given in the problem mtement? /h) Example 8-8. Describe how your answers would change if the molar flow of methanol were increased by a factor of 4. (i) Example 8-9. Other data show AH,, = -58,700 BTUllbmol and CpA= 2 9 BTU/IbmolleF. How would these values change your results? Make a plot of conversion ns a function of heat exchanger area. [O c A < 200 fr".
P
570
Steady-State Nonisotherrnal Reactor Design
Chap. 8
Q) Example 8-30. How would your results change if there is ( I ) a pressure drop with a = 0.08dmL3,(2) Reaction ( I ) is reversible with Kc = I0 at 450 K. (3) How woutd the selectivity change if Ua is increased? Decreased? /k) Example 8-11. ( I ) How would the results (e.g., ~ B , c )change if the UA term were varied (3500 < UA c 4500 J/m3 - s . k)? ( 2 ) If To were varied between 273 K and 400 K, make a plot of C, versus To. (I) Example P8.4-1. SO? oxidation. Bow would your results change if ( 1 ) the catalyst panicle dlameter were cut in half? (2) the pressure were doubled? Ar what panicle size does pressure drop -become important for the same catalyst weight assuming the porosity duesn't changeq ( 3 ) you vary the initial temperature and the coolant temperature? Write a paragraph describing what you find. (m)Example TS-3. Load the Polymath problem from the CD-ROM for this exothermic reversible reaction with a variable coolant temperature. The elementary reaction
has the following parameter values for the base case.
Clal of Fame
E = 25 kcallrnol
C,
= Cpk= CPC=20 caVmoYK
AHR, = -20 kcallrnol
C,,
= 40 caIlmollK
k = -0.004 dm\
-- 0.5-
mol . kg . s
Summary Notes
pe
cal kg. S. K
Vary the following parameters and write a paragraph describing the trends you find for each parameter variation and why they work the way they do. Use the base case for parameters not varied. Hlur: See Selftests and Workbook in the Summar?. Notes on the CD-ROM. (a) FAn: 15 FA,< 8 moYs (b) 0,:0.51 0, s 4 'Note: The program gives O, = 1.0. Therefore. when you vary @I. you will need to account for the corresponding increase or decrease of C,,, because the zoraI concentration. CTO,is constant. Uu Fa cal (c) -: 0.1 5 - 20.8 Pb Pa kg.s.K d : 3 I0 K S T,, 5 350 K (el T,: 300 K 5 T, 5 340 K
Chap 8
Questtons and Problems
tf) for counter current coolant flow. (h) Determine the conversion in a 5,000 kg fluidized CSTR where UA = 500 caYs.K with T, = 320 K and p, = 2 Irg/m2 (i) Repeat (a), (b), and (d) if the reaction were endothermic with & = 0.01 at 303 and AHR, = +20 kcal/rnoI. (n) Example 8-12. Irzstrucriot~s:If you have not installed FEMLAB 3.1 ECRE, load the FEMLAB 3.1 ECRE CD-ROM and follow the installation instructions. Double-ciick on the FEMLAB 3.1 ECRE icon on your deskrop. In the Model Navigator. select model denoted "4-Non-Isothermal Reactor 11" and press "Documentarion". This will open your wet? browser and display the dofumentation of this specific model. You can also review the detailed documentation for the whole series of models listed on the left-hand s~dein the web browser wrndow. Use chaprer 2 in the online documentation to answer the questions below using the model "4-Non-Isothemial Reactor 11". Select the Model Navigator and press "OK" to open the model. (1) Why is the conucnrntion of A near the wall lower than the concentration near the center? ( 2 ) Where in the reactor do you find the maximum and minimum reaction rates? Why? h~rtrucrions:Click the "Plot Paramelers" butron and select the "Surface" rah. Type "+A" (replace "cA") in the "Expression" edit field to plot the absolute rate of consumptlon of A (moles m--' s-I ). (3) Increase the activation energy of the reaclion by 3%. How do the concentration profiles change? Decrease. Irtsrructions: Select the 'T~onaants" menu item in the "C)ptions" rneou. Multiply the value of " E in the constants list by 1.05 (just type "'1 -05" behind the existing value to increase or multiply by 0.95 to decrease). Press "Apply". Press the "Restart" button in the main toolbar (equal sign with an arrow on top). (4) Change the activat~onenergy back to the oripinal valoe. I~zsfructions:Remove the factor "0.9S3nthe constant list and press "'Apply". (5)Increaqe the thermal conductivity, k f , by a factor of 10 and explaln haw this change affects the temperature profiles. At what radial position do you find the highest conversion? 6~sfr1icriuns: Multiply the value of "ke" in the constants l~stsby 10. Press "Apply". Press "Restart". Ih) Increase the coolant flow rate by a factor of 10 and expEa~n how this change affects the conversion. (7) In two or three sentences. de~cribeyour f ndings when you varied the parameters {for all parts). (8) What would be your recommendarion to maximize the average outlet conversion7 (9) Review Figure E8- 12.2 and explain why the temperature profile goes through a rnaxtmum and why the conversion profile goes through a maximum clmd a minimum. (10) See orher problems in the web module. ( 0 ) ExampIc RX.2-1 Runaway Reactions. t o a d the L i h g Exampk Pn~blenl on runaway trajector~cs.Vary some of the pmneters, such as Po and Tn along with the activation energy and heat of reaction. Write a paragraph describing what you found and what generalizations you can make. (g) Repeat (el and
572 Interacttve
computer ~ ~
PB-3,
PS4C d
Steady-State Nonisothermal Reactor D8sign
Cha
Load the interactive Campurer Module (ICM) from the CD-ROM. Run module. and then record your performance number for the module, wl indicates your mastery of the material. Your professor has the key to dec your performance number. ( a ) ICM Heat Effects Basketball 1 Performance # (b) ICM Heat Effects Simulation 2 Performance # The is~ an excerpt from The Manring News,Wilmington, Dolav ~ following l ~ (August 3, 1977): "Invest~gatorssift through the debris from blast in quesl the cause [that destroyed the new nitrous oxide pIant]. A company spokesl said it appears more likely that the [fatal] blast was caused by another gc ammonium nitrate-used to produce nitrous oxide," An 83% (wt) ammon nitrate and 17% water solution is fed at 200°F to the CSTR operated temperature of about 510°F. Molten ammonium nitrate decomposes d ~ r to produce gaseous nitrous oxide and steam. It IS believed that pressure f tuations were observed in the system and as a result the molten ammon nitrate feed to the reactor may have been shut off approximately 4 min F to the explos~on.(a) Can you explain the cause of the blast? [Hint:See PI lem P9-3 and Equation (8-75).] (b) If the feed rate to the reactor just be shutoff was 3 10 Ib of soIution per hour, what was the enact temperature in reactor just prior to shutdown? (c) How would you stan up or shut down control such a reaction? (d) What do you learn when you apply the runa reactton criteria? Assume that at the time the feed to the CSTR stopped. there was SO0 I ammonium nitrate in the reactor. The conversion in the reactor IS believed at virtually complete at about 99.99%.
AddirianaI information (approximale but close to the real case):
AH;, = -336 Btuflb ammonium nitrate at 500°F (constant) C, = 0.38 Btullb ammonium nitrate - "F C, = 0.47 Btullb of steam."F
where M is the mass of ammonium nitrate in the CSTR (Ib) and k is givet the relationship below.
The enthalpies of water and steam are
H,(2QO0F) = 168 BtuIIb
H,(5WaF) = 1202 Btullb (e) Explore this problem and descrik what you find. B r example, can you a form of R(n versus G(T)?](0Discuss what you believe to be the point ol
problem. The idea for this problem originated from an article by Ben H o r n
Chap. 8
P8-&,
573
Questions and Problems The endorhemic liquid-phase elementary reaction
proceeds, substantially. to completion in a single steam-jacketed, continuous-stirred reactor (Table P8-5). From the following data, calculate the steady-state reactor temperature:
Reactor vaIume: 125 gal Steam jacket'area: 10 ft2 Jacket steam: 150 psig (365.9"Fsaturation temperature) Overall heat-transfer coefficient af jacket. U:IS0 Btul h .ff2 + O F Agitator shaft horsepower: 25 hp Heat of reaction, AH;;, = +20,000 Btullb mol of A (independent o f temperature)
Component
Feed (Ibmollhr) Feed temperncure (T) Specific heat (Btullb mol+"F)' Molecular weight Density (Iblft')
10.0
10.0
80
80 44.0
51.0 128
63.0
0
-
94
47.5 222
67 2
65.0
'Independent of tempemrue. (Ans: T = 1 9 9 O F ~ (Courtesy of the California Board of Registration for sional & Land Surveyors.)
Pmfes-
m-6* The elementary irreversible organic liquid-phase reaction.
is carried out adiabatically in a flow reactor. An equal molar feed in A and 8 enters at 27'C, and the volumetric flow rate is 2 dm3/sand CAo= 0.Ikmolfm3. (a) Calculate the PER and CSTR volumes necessary to achieve 85% conversion. What are the reasons for the differences? (b) What is the maximum tnlet temperature one could have so that the boiling point of the liquid (550 K) would not be exceeded even for complete
conversion? (c) Plot the conversion and temperamre as a function of PER volume (i.e., distance down the reactor). (dl Calculate the conversion that can be achieved in one 500-drnWCST and in two 250-dmJCSTRs in series. (e) Ask another question or suggest another calculation for this reaction.
574
Steady-Stale Nonisothermal Reactor Design
Chap. 8
Additional infommtiun:
P8*TB Use the data and reaction in Problem 8-6 for the following exercices. (8) Plot the mnve~sionand temperature of the PFR profiles up to a reactor volume of I0 dm3 for the case when the reaction is reversible with Kc = 10 rn3/kmol at 450 K. Plot the equilibrium conversion profile. (b) Repeat (a) when a heat exchanger is added, Ua = 2QcaIlrn3/s/K. and thecoolant temperature i s constant at T, = 450 K. (c) Repeat (b) for a co-current heat exchanger for a coolant Aow rate of 50 gls and Cpr= 1 cay& K. and in inlet coolant temperature of T,, = 450 K.V a v the coolant rate ( 1 < m, < 1.000 g/s) .
P8-8,
Id) Repeat (c) for counter current coolant flow. (e) Compare your anywers to (a) through (d) and describe what you find.What generalizations can you make? (f) Repeat (c) and (d) when the reaction is irreversible but endothermic with AH, = 6,000 cal/rnol. (g) Discuss the application of runaway criteria for the irreversible reaction occurring in a CSTR.What value of Towould you recommend to prevent runaway if K = 3 and T, = 450 K? The elementary irreversible gas-phase reaction
is carried out adiabatically in a PFR packed with a catalyst. Pure A enters the reactor at a volumetric flow rate of 20 drn3/s at a pressure of 10 atrn and a temperature o f 450 K. (a) Plot the conversion and temperature down the plug-flow reactor until an 808 conversion ( ~ possible) f IS reached. (The mahimum catalyst weight that can be packed into the PFR is 50 kg.) Assume that AP = 0.0. (b) What catalyst welghf is necessary to achieve 8 0 9 conversion in a CSTR? (c) Write a question that requires critical thinking and then explain why your question requires critical thinking. [Hinr: See Preface Section B.2.1 (d) Now take the pressure drop into account in the PFR.
The reactor can be packed with one of two particle slzes. Choose one. a = O.O19/kg cat, for particle diameter
D,
a = 0.0075,'kgcat. for particle diameter D, Plot the temperature, conversion. and pressurealong the length of the reactor, Vary the parameters rr and P,, to learn the ranger of value5 in which they dramatically affect the conversion.
Chap. $
Questions and Problems Additional infurmotion:
All heats of formation are referenced to 273 K .
P8-gB
Use the data in Problem 8-8 for the case when heat is removed by a heat exchanger jacketing the reactor. The flow rate of coolant through the jacket is suficiently high that the ambient exchanger temperature is contant at T, = 50°C. (a) Plot the temperature and conversion profiles for a PBR with J LIQ = 0.08 Pn s . kg cat. bK where p, = bulk density of the catalyst (kgJm') a = heat-exchange area per unit volume of reactor (rn2/&)
U = overaEl heat-transfer coefficient (Jls. rnz. K) How would the profiles change if Udpb were increased by a factor of 3W? (b) Repeat part (a) for both cc-current and counter current flow with ni,= 0.2 kgk, Cp = 5,000 J k g K and an entering coolant temperature L
of 50°C. (c) Find X and T for a " f l u i d i d CSTR [see margin] with 80 kg of catalyst.
(d) Repear parts (3) and (b) for W = 80.0 kg assuming a reversibIe reaction with a reverse specific reaction rate o f
Vary the entering temperature, To, and describe what you find. (e) Use or modify the data in this problem 10 suggest another questton or calculation. Explain why your question requires either critical thinking or creative thinking. See Preface 8.2 and B.3. PR-10B The irreversible endothermic vapor-phase reaction foIloa's an elementilry rate law
CH3COCH, +JCH7C0+ CH, A-+B+C
and is carried out adiabatically in a 500-dm' PFR. Species A is fed to the reactor at a rate of 10 rnollmin and a pressure of 2 stm. An lnen stream is also fed
Ch
Sleady-State Non+sotRerrnalReactor Design
to the reactor at 2 atm, as shown in Figure P8-10.The entrance temperatu both streams is 1 I 0 0 K.
Figure PS-10 Adiabatic
PFR with inens.
(a) First derive an expression for C,,, as a function of C, and 0,. (b) Sketch the conven~onand tempemre protiles for the case whe inens are present. Using a dashed line. sketch the profiles when a ente amount of inerts are added. Using a dotted line, sketch the prl when a large amount of rnerts are added. Sketch or plot the exit co sion as a functlon of Q,.Qualitative sketches are fine. (c) Is there a ratio of inens to the entering molar flow rate of A ( i . ~ . .I Flo/FAo) at which the conversion is at a maximum? Explain why "is" or "is not" a maximum, Id) Repeat parts (b) and (c) for an exothermic reaction (AH,,= -80 k3/ (e) Repeat pans (b) and (c) for a second-order endothermic reaction. (0 Repeat parts (b) and (c) for an exotherm~c reversible rea (Kc = 2 dm31rnol at 1100 K). (g) Repeat (bj through (fi when the totat volumetric flow rate v,, is held stant and the mole fractions are varied. Ih) Sketch or plot FB for parts (d) through (g) Additional information:
-
k = exp (24.34 - 34,222/ T) drnjlmo! +min C,, = 2M Jlmol K (Tin degrees Kelvin) CpA= 170 Jlmol. K
P8-llc Derive the energy balance for a packed bed membrane reactor. Apply th ance to the reaction in Problem P8-88for the case when it is reversible K, = 0.01 moVdd at 300 K. Species C diffuses out of the membrane. (a) Plot the concentration profiles or diffeferent values of k, when the re: is carried out adiabatically. (b) Repeat part (a) when the heat transfer coefficient is the same as that in P8-9(a). All other conditions are the same as those in Problem F P8-12, The liquid-phase reaction
foliows an elementary rate law and takes place in a 1-rnJ CSTR,to whk volumetric flow rate is 0.5 m3Smin and the entering concentration of A i! The reaction takes place isothermally at 300 K. For an equal molar fr A and B, the conversion is 20%. When the reaction is carrled out adi cally. the exit temperature is 350K and the conversion is 40%.The heat ( ities of A, B, and C are 25, 35, and 60 kJJmol K. respectively. It is prc
Chap. 8
577
Questions and Problems
to add a second reactor of the same size downstream tn series with the first CSTR. There i, a heat exchanger attached to the second CSTR wirh UA = 4.0 kJ/mtn K,and the coolant fluid enten and exits this renctur at virtually the same temperature the coolant feed enters 350 K. (a) What is the rate of heat removal necessary for isothermal operation'! (b) What i s the conversion exiting the second reactor? (c) What would be the conuersian if the second CSTR were repl.tced with n 1-m3 PFR with Ua = 10 kl/m3 min and T, = 300 K? (d) A chemist suggests that at temperatures above 380 K the revenc reaction cannot be neglected. From thermodynamics, we know thnt a1 351> K. K, = 7 dm3/rnch. What conversion c a n be achieved if the entering temperature to the PFR in part (b) is 350 K? (e) Write an in-depth question that extends this problem and involve\ cr~ticul thinking. and explain why it ~nvoluescriticnI thinking. (0 Repeat pan (c) assuming the reaction takes place entirefy in the gas phase (same constants for reaction) with C,, = 0.2 rnoIidm".
-
P8-13* The reaction is carried out adiabatically in a series of staged packed-bed reactors with interstage cooling. The Lowest temperature to which the reactant stream may be cooled i s 27°C. The feed is equal rnoIar in A and I3 and the catalyst weight in each reactor is sufficient to achieve 99.9% of the equil~briurncnnversion. The feed enters at 27'C and the reaction is camed out adiabatically. If four reactors and three coolers are available, what conversion may be achieved? Additional information:
First prepare a plot of equilibrium conversion as a function of temperature. [Partial rms.: T = 360 K, X, = 0.984; T = 520 li. X, = 0.09; T = 540 K, X , = 0.0571 P8-biA Figure 8-8 shows the temperature-conversion trajectory. for a train of reactors with rnterstage heating. Kow consider replacing the intersrage heating with injection of the feed stream in three equal portions as shown here:
P8-15
Sketch the tempenture-conversion trajectories for (a) an endothermic reaction with entering temperatures as shown, and (b) an exothermic reaction with the temperatures ta and from the first reactor reversed, i.e.. To= 450°C. The brornass reaction
Substrate
Cclls
---*More
Celts + Product
is carried out in a 6 dm3 chemostat with a heat exchanger.
578
Steady-State Nonisothermal Reactor Design
Chap. 8
Celt Cell Cyto Cell Ribo
The volumemc Aow rate is 1 dm3/%,the entering substrate concentration and temperature are la> g/dd and 280 K. respectively. The temperature dependence of the growth rate follows that given by Aibe et al.. Equation (7-63'1
and
~ ( 7 1=; ~ ( 3 1 0K31'= PI,,
0.003S*T-exp[21.6-6700/T] Cs I + exp 1 153 - 48000, ] K , + C,
[
(a) Plot G(T) and R(Tj for both adiabatic and non-adiabatic operation assuming a very large coolant rate (i.e., Q = UA (To- 77 with A = 1.1 rn2 and To= 290 K). (b) What IS the hear exchanger area that should be used to maximize the exiting cell concentration for an entering temperature of 188 K' Cooling water is available at 290 K and up to a maximum flow rate of 1 kglminute. (c) Identify any multiple steady states and discuss them in light of what you learned in this chapter. Hint: Plot T, vs. Tofrom Part (a). (d) Vary To, ni,. and T, and describe what you find.
Yo$ = 0.8 g celyg substrate K,7 = 5.0 @dm? plrnax = 0.5h-I (note p = pmax at 310 K) Cps = Heat capacity o f substrate solution including all cells = 74 JIdK m , = Mass
of substrate solution in chemostat = 6.0 kg AHR, = -20,000 Ilg cell5 U = 50.000 Jh/Km2 Cq = Heat capacity of cooling water 74 JJgIK ni, = coofant flow rate (up to 60.000 kglh) ps = solution density = I kg I dm3
Chap. 8
Questions snd Problems
B-16& The first-order irreversible exothermic liquid-phase reaction A 4 3 is to be carried out i n a jacketed CSTR. Species A and an inert I are fed to the reactor in equilrnotar amounts. The molar feed rate of A is 80 mollmin. (a) What is the reactor temperatun: for a feed temperature of 450 K? (b) Plot the reactor temperature as a function of the feed temperature. (E) To what inlet temperature must the fluid be preheated for the reactor to operate at a high conversion? What are the corresponding temperature and conversion of the fluid in the CSTR at this inlet temperature? (d) Suppose that the fluid IS now heated 5°C above the temperature in part (c) and then cooled ZVC, where it remains. What will 'be the conversion? (e) What is the iolet extinction temperature for this reacrion system? (Ans.: To = 87°C.)
Additional i n f o mfion:
Heat capacity of the inert: 30 callg mol."C
T
Heat capacity of A and B: 20 callg mol "C
AHRn= -7500 callmol
UA: 8000 callmin."C
k = 6.6 X 1W3 min-I at 350 K E = 40.000callmol-K
Ambient temperature, To:300 K
= I00 rnin
P8-17c The zero-order exothermic liquid-phase reaction is carried out at 85°C in a jacketed 0.2-rn3CSTR. The coolant ternperature in the reactor is 32°F. The heat-transfer coefficient is 120 W/m2. K . Determine the critical value of the heat-transfer area b l o w which the reactor will run away and explode [Chem. Eng., 91(10),54 (1984)l. Allditionnl information:
k = 3.127 km0l/rn3~min at 40°C
k = 1.421 kmol/m3-min at SO°C
The heat capacity of the solution is 4 JI0CJg.The solution density is 0.90 kpldm'. The heat of reaction is -500 JJg. The feed temperature is 40°C and the feed rate is 90 kglmin. M W of A = 90 glmol. P8-18, The elementary reversible liquid-phase reaction takes place in a CSTR with a heat exchanger. Pure A enters the reactor. (a) Derive an expression (or set of expressions) to calculate C ( T )as a fwnction of heat of reaction, equilibrium constant, temperature. and sa on. Show a sample calculation for C ( T )at T = 400 K. (h) What are rhe steady-state temperatures? (Ans.: 3 10, 377,418 K.) It) Which steady states are locally stable? (d) What is the conversion correspondine to the upper steady state? (e) Vary the ambient temperature 7; and make a plot of the reactor temperature as a function of To. idenlifying the ignition and extinction zem!xratures.
580
Steady-State Nonisothetmal Reactor Design
Ch
(fl I f the heat exchanger in the reactor suddenly fatls (i.e., UA = O), would be the conversion and the reacror temperature when the new u steady state is reached? (An.7.: 43 1 K) (g) What heat exchanger product, UA, will give the maximum conversmn (h) Wnte a question that requires critical thinking and then explain why , question requires critical thinking. [Hint: See Preface Section B.2.1 (i) What i s the adiabatic blowout flow rate, v,, . i Suppose that you want to o p r e at the lower steady state. What paran values would you suggest to prevent runaway?
-
Cp4= Cpg= 40 callrnol K
AH,, =
- 80,000 cart mol A
K,, = I00 at 400 K k = I min-1 at 400 K P8-19,
V = 10 dm3 v , = I dm"min FA,= LO rnollmin
Ambient temperature, T, = 37°C Feed temperature. T,,= 37°C The first-order irreversible liquid-phase reaction is to be carried out in a jacketed CS'R. Pure A i s fed to the reactor at a rat 0.5 g rnollmin. The heat-generation curve for this reaction and reactor systt
is shown in Figure P8-19. (a) To what ~nlettemperature must the fluid be preheated tbr the reacto operate at a high conversion?(Ans.: To 2 214°C .) (b) What i s the corresponding temperature of the fluid in the CSTR at inlet temperature? (Ans.: T, = 16qnC, 184°C.) (c) Siipp~sethat the fluid is now heated 5°C above the temperature in pan and then cooted IO°C, where it remains. What will be the conversi (Ans.: X = 0.9.) (dj What is the exhnction tempemlure for this reaction system? (Am.: To= 2m (e) Write a question that requires critical thinking and then explain why y question requires critical ~hinking.[Hint: See Preface Section B.2.J Additional information:
- 100 cal/g mol A Heat capacity of A and 8:2 callg mol. "C UA: 1 callmin."C Ambient temperature, T, : 100°C
Heat of reaction (constant):
.
Chap. 8
Questions and Problems
m70
-
a m2
50-
i ao-
$30-
t
320
I I I I I I 140 160 lBI,
I
2W T I"C1
I
I I I I 220 2/10
Figure PS-19 G ( n curve.
Pg-20c Troubleshooting. The following reactor system is used to carry out the: reversible catalytic reaction
The feed is equal molar in A and B at a temperature T I of 300 K.
Troubleshoot the reaction system to deduce the problems for an exothemic and an endothermic reaction. Next, suggest measures to comct the problem. You can change ni, , m, , and FA, dong with T, and T,.
TroubEeshoot what temperatures are normal and what are different and what the distinction is. Explain your reasoning in each of the cases below. (a) Exothermic reaction. The expected conversion and the exit temperature are X = 0.75 and T = 4OQ K. Unfortunately, here is what was found in six different cases. Case 1 at the exit X = 0.01, T,= 305 K Case 2 at the exit X = 0.10, T7= 550 K Case 3 at the exit X = 0.20,T7= 350 K Case 4 a! the exit X= 0.5, T7= 450 K Case5atthecxitX=0.01, T7=400K Case 6 at the exit X = 0.3,T, = 500 K
582
Steady-Sfate Nonisothermal Reactor Design
Chap. 8
(b) Endothermic reaction. The expected conversion and the exit temperature are X = 0.75and T, = 350 K. Here is what was found. Case 1 at the exit X = 0.4, T, = 320 K Case 2 at the exit X = 0.02.T, = 349 K Case 3 at the exit X = 0.OQ2,T, = 298 K Case 4 at the exit X = 0.2. T, = 350 K
P8-21 If you have not installed E M L A B 3.1 ECRE, load the FEMLAB 3.1 CD-ROMand follow the installation instructions. (a) Before running the program, sketch the radial temperature profile down a PFR for ( 1 ) an exothermic reaction for a PFR with a cmling jacket and (2) an endothermic reaction for a Pm with a heating jacket. (b) Run the FEMLAB 3.1 ECRE program and conrpare with your results in (a). Double-click on the FEMLAB 3.1 ECRE Icon on your desktop. In the Model Xavigator, select the model denoted "3-Non-Isothermal I" and press OK. You can use this model to compare your results in (1) and (2) above. You can select "Documentar~on" in the "Help" menu in order to review the instructions for this model and other models in E M L A B ECRE. Change the velociry profile from laminar parabolic to plug flow. Select "Scalar Expressions" in the "Expression" menu item in the "Options" menu. Change the expression for uz (the velocity) to " u 0 (replace the expression "2'110*(1-(rlRa)~)*',which describes the parabolic velocity profile). Presr "Apply". You can now continue to vary the input dava and change the exothermic reaction to an endothermic one. (Hint: Select the "Constants" mcnu item in the "Options" menu. Do not forget "Tawthe jacket temperature at the end of the list). Write a paragraph describing your findings. (c) The thermal conductivity in the reactor, denoted "ke" m Figure E8-12.1, is the molecular thermal conductivity for the solution. In a plug flow reactor, the flow is turbulent. In such a reaclor. the apparent thermal conductivity is substantially larger than the molecular thermal conductivity of the fluid. Vary the balue of the thermal conductivity "ke" to Eeam it's influence on the temperature and concentration profile in the reactur. (d) In turbulent flow. the apparent diffusivlty is substantially larger than the molecular diffusivity. Increase he molecular diffu~ivityin the PFR to reflect turbulent conditiuns and study the influence on the temperature and concentration profiles. Here you can go to the extremes. Find something interesting to turn in to your instructor. See other problems in the web module. Ee) See other problems in the web module. P8-2Ic A reaction IS to be carried ON in the packed-bed reactor shown in Figure P8-22
Figure PS-22
Chap. 6
583
Questions a d ProMems
The reactants enter in the annular space between an outer insulated tube and rn inner tube containing the catalyst. NO reaction takes place in the annular region. Heat transfer between the gas in this packed-bed reactor and the gas flowing counter currefltly in the annular space mcurs along the length of the reactor. The overall heat-transfer wfficicnt is 5 W J m 2 K . Plot rhe conversion and temperature as a function of reactor length for the data given in (a) Problem P8-6. (b) Problem P8-9(d). F'$-2&, The irreversible liquid-phase reactions
-
Appliion Pending for Problem Hall of
+ B+ZC
Reaction (1)
A
Reaction (2)
2B+C+D
=
~ I C~ICCACB
r ? =~~ ~ D C B C C
are carried aut in a PFR with heat exchange. The following temperature profile was obtained for the reaction and the coolant stream.
The concentrations of A, B. C,and I) were rntasured at the point down the reactor where the liquid temperature, T, reached a maximum, and they were found to be CA = 0.1, CB = 0.2, CC = 0.5, and C, = 1.5 all in moWdm3.The product of the overall heat-transfer coefficient and the heat-exchanger area per unit volume, Ua, is I0 calls dm3 K. The feed is equal molar in A and B, and the entering molar flow rate of A i s 10 moils. What is the activation energy for Reaction ( l ) ? E = ??calSrnol.
51RK)
k?, = 0.4- dm\
-
mol s
KII-! ksno
rl
584 P8-24
Web hint
Steady-State Nonisothermal Reactor Des~gn
Gh
~Cotnpr~zen,rivr Grtrz Pmblrmj T-Amyl Methyl Ether (TAME) is an exy ated additive tor green gasolines. Bes~desill;uFe as an ocrane enhancer, i t improves the cornb~ist~on o f gasoline and reduces the CO and HC (and, smaller extent, the NO,) nutarnobile exhaust emissions. Due to the env mental concerns related to those emissions, this and other ethers (MI ETBE. TAEE) have been lately studied intensively. TAME i s currently cntc rally produced in the liquid phase by the reaction of methanol (MeOH) anr isoarnylenes 2-methyl- I -butene (2M Z B) and 2-methyl-2-butene (2M2B). T are three s~multaneousequilibrium reactions in the formation and splittin TAME (the two etherification reactions and the isornerization between isoarnylenes):
C
A-B
2M28 + MeOH eTAME
These reactions are to be carried out in a plug-flow reactor and a memb rractor in w h ~ c hMeOH is fed uniformly thmngh the sides. For isothe~ operation:
(a) Plot the concentmtion profiles for a
10-m3 PFR. (b) Vary the entering tempenture, q,, and plot the exit concentntions function of T,. For a reactor wrth heat exchange (U = I0 J m-2 s-' K-'j: (c) Plot the temperature and concentration profiles for an entering temp ture of 353 K. (d) Repeat (a) through ( c ) for a membrane reactor.
.
.
Additional infnrmation:
The data For thls pmbtem is found at the end o f the Additional Ho work Problems for Chapter 8 on the CD-ROM end on the web. [Prob by M. M. Vilxenho Ferreira, I. M. Loueiro. and 0.R. Frias, L'nivet of Porto, Portugal.] M-2Sc (Multiple reactions wirk hrnt effects) Xylene has three major isom m-xylene, o-xylene, and p-xylene. When o-xylene is passed over n Cvc catalyst, the following elementary reactions are observed:
Appricstion Pending for Pmbl~rn
Hall of
The feed to the reactor is equal molar in in both m-xylene and o-xylene (e cies A and BS. For a total feed rate of 2 mollmin and the reaction conditi
Chap. 8
585
Questtons and Problems
below, plot the temperature and the molar Raw rates of each species as a filnction of cataly5t weight up to a weight of I(H) kg. ( a ) Find the lowest concentration of o-xylene achieved in the reactor. (b) Find the highest concentration of m-xylcne achieved in the reactor. (c) Find the maxlmam concenrration of o-xylene in the reactor. Id) Rcpeot parts (a) to (c) for a pure feed of n-xylene. Ie) Vary some of the system parameters and describe what you learn. (fi What do you believe to be the point of thib problem?
Rddirinnnl infirmarton: ' 1 A l l heat capacities are virtually the same at 100 J/rnolnK.
- 1800 Jlmol u-xylenel AHRx3, = - I 100 11rnol o-xylene k , = 0.5 .exp[3( I - 320/T)]dmVkg cat. - min
AHR,,, =
k 2 = k41Kc
k3 = 0.005exp { f4.6(1 - (4601T))lj drn3/kg cat..rnin Kc =
to exp[4.8(4301T - 1.531
To = 330 K T, = 500
K
Unlp, = 16 Jlkg cat:min
a0C
W = 100kg
-
P8-2fic {Compwhensive Pmhlent en rnrrbcple renrtiuns with hens effccrs) Styrene can be produced from ethylbenzene by the following reaction: ethylbenzene
styrene
+ Hz
(1)
However, scvetal irreversible side reactions also occur: ethylbenzene d benzene -t ethylene
ethylbenzene + Hz
toIuene + methane
(2) (3)
[J. Snyder and 0. Subramaniam. Chem. En,q. Sci., 49. 5585 (1994)l. Ethylbenzene is fed at a n t e of 0.OD3U kmolJs to a 10.0-rn7 PFR (PBR)along with inert steam at a total pressure of 2.4 atm. The steamfethylbenzene m a l x ratio i s initially [i.e., parts {a) to (c)] 34.5: 1 but can be varied. Given the following data. end the exiting molar Row ntes of styrene, benzene, and toluene along with Ss,sr for the f~ilowinginlet temperatures when the reactor is operated adiabatically. (sl T ,= 800 K (b) To= 930 K (c) To = I l Q O K
Halt bf Fame
II
Obtained from inviscid pericosi ty measurements.
Steady-Sate Nonisothermal Reactor Design
Chap. 8
( d ) And the ideal inlet temperature for the production of styrene for a stearnlethylbcnzene ratio of 58: E. (Hint: Plot the molar flow rate of styrene versus To.Explain why your curve looks the way it does.) (e) End the ideal steamfethylbenzene ratio for the production of styrene at 900 K. [Hint: See part Id).] ( f ) It is proposed to add a counter current heat exchanger with Ua = 100 kJlminlK where T, i s virtually constant at 1000 K . For an entering stream to ethylbenzene ratio of 2, what would you suggest as an entering rempemture? Plot the molar flow rates and S,,,BT. (g) What do you believe to be the points of this problem? (h) Ask another question or suggest another calculation that can be made for this probIern.
Heat capacities
Methane Ethylene Benzene Toluene
68 Jlmol . K 90 J/mol.K 201 J/rnol.K 249 Jlmol .K
Styrene
273 Jlrnol. K
Ethylbenzene 299 Jlmo1.K Hydrogen 30 J/mol K Steam 40Jlrnol.K
-
p = 21 37 kg/m3 of pellet
d = 0.4 AHRxlE8= I 1B3000kllkmol ethylbenzene
AHRxlEB = 105,200 klEkrno1ethylbenzene mRx3 =E -53.900 B
b2 = -1.302x
kJlkmol ethylbenzene
IP
b3 = 5.051
b3 = 1.302X b, = -4.93 1 X
The kinetic rate laws for the formation of styrene (St), benzene (0). and tnluene (TI.respectively, are as follows. (EB = ethylhenzene)
(P,,)
(kmolJm'.s)
(P,,PH2) (kmolfm7+s) The temperature T is in kelvin.
Chap. 8
Questions and Problems
587
P8-27B compare the profiles in Figure E8-3.1, E8-4.1,E8-5.1,E8-5.3,E8-5.4,8-10. ES-10. I , and E8-10.2. ( a ) If you were to classify the pmfiles into groups, what would they h? What are the common characteristics of each group? (b) What are the similarities and differences in the profiles in the various groups and in the various figures? (c) Describe why Figure E8-5.3and Figure E8-3.1look the way they do. What are the similarities and differences? Describe quditatively how they would change if inerts were added. Id) Repear (c) for Figure EX-10.I and for Figure E8-10.2. TAME
CD-ROMComplete Data Set P8-24c The TAME data set is given on the CD-ROM. This probiem i s a very comprehensive problem: perhaps can be used as a term (semester) problem.
Good Alternative Problems on CD-ROM Similar to Above Problems Halt of F a m ~
PS-28, Industrial data for the reactian 2 vinyl acetylene+ stysene are given. You are asked to make PFR calculatioos similar to those i n Problems to P8-gB.[3rd Ed. P8-9sJ Pa-29, Reactor staging with interstage cooling. Similar ro P8-13H, but shorter because X, versus T i s given. [3rd Ed. P8-1 S,] Pa-30, Use the data in Problems P8-6 and P8-8$0carry our reactions in a radial Row reactor. [3rd Ed. P8- 18,]
P8-31, The reactions are carried our in a CSTR with a heat exchanger. E3rd Ed, P8-28,J
P8-3&, Elementary irreversible reaction
is carried out in a PFR with heat exchange and pressure drop. (3rd Ed., PR-121
PS-33, Liquid-phase reactions
in a CSTR. Maximize D. 13rd Ed. P8-31 )
588
Steady-State Nanisolherrnal Reactor aesrgrr
Ch
Oldies, But Gcmdies--Pmblerns from Previous Editions
The exothermic reaction
is carried out in h t h a plug-flow reactor and a CSTR with exchange. You are requested to plot conversion as a function of I tor length for both adiabatic and nonadiabatic operation as well ; size a CSTR. [2nd Ed. P8-161 Bifnrcation Problems
Use bifurcation theory (R8.2 on the CD-ROM)to determrne the sibIe regions for multiple steady states for the gas reaction with rate law
C2nd Ed., P8-261 In this problem bifurcation theory (CD-ROM R8.2) is used to dt mine if multtple steady states are possible For each of three type catalyst. [2nd Ed., P8-27,J
SO, Oxidation Design Problems for R8.4 This probtem concerns the SO2 reaction with heat losses. [Znd
P8-331 This problem concerns the use of interstage cooling in SO? oxidat [2nd Ed., P8-34aj This problem is a continuation of the SO? oxidation example probl Reactor costs are considered in the analysis cooling. [Znd P8-34(b) and (c)]
Mdtiple Reactions Parallel reactions take place in a CSTR with heat effects. [Ist 1 P9-2LI MultipIe reactions
are carried out adiabatically in a PFR.
Multiple Steady States In the multiple steady state for
The phase plane of C, vs. T shows a sepamtrix. [Znd Ed.. Pa-221 A second-order reaction with multiple steady states is carried o u different solvents.
Chap. 8
a
G PL'
Livtng Example Problem
Journal Critique Problems
Other Problems CDP8-KR
Extemion nf FEMLR8 Example E8-13.
JOURNAL CRITIQUE PROBLEMS P8C-L Equation (8) ln an anicle in J. Chem. Techno[ Bioachnoi.. 31. 273 11981) is the kincttc model of the system proposed by the authors. Starting from Equahon 12), derive thezquation that describes the system. Does i t agree with Equation CS)? By using the data in Figure l , determine the new reaction order. The data in Table 2 show the effen of temperature. Figure 2 itlusrrates this effect. Use Equation (8) and Table 2 to obtain Figure 2. Does it agree with the article's results? Now use Tahlc 2 and your equation. How does the figure obtained compare with Figure ??What i s your new E,, ? P8C-2 The kinetics of the reaction of sodium hypochlorite and sodium sulfate were studied by the Row thermal method in I d . Eng. Chetn. Fundarn., 19, 207 ( 1980). What is the flow thermal method? Can the energy balance developed in this article be applied to a plug-ffow reactor. and if not, what is the proper energy balance? Under what conditions are the author's equations valid? PSC-3 In an article on the kinetics of sucrose inversion by invertase with multiple steady states in a CSTR [Chem. Eng. Commun., 6. 15 1 ( I 98011,consider the following challenges: Are the equations for K, and K, correct? If not, what are the correct equations for these variables? Can an analysis be applied to this system to deduce regions of multiple steady states? P8C-4 Review the article in AIChE J.. 16.4 I5 (1970). How was the G(T)curve generated? Is it miid for a CSTR? ShouId G(T) change when the space-rime changes? Critique the article in light of these questions.
SUPPLEMENTARY READING I . An exceltent development of the energy balance is presented in ARIS, R., Elementar?, Chemical Reactor Andvsis. Upper Saddle River. N.I.: Prentice Hall. 1969, Chaps. 3 and 6. HIMMELBLAU, D.M.,Rnsic Prrnciples and Cnlculutions in ChemicaI Engineering, 7th ed. Upper Saddle River, N.I.: Prentice Halt, 2003. Chaps. 4 and 6. A number of example problems dealing with nonisothennal reactors can be found in FROMEFIT, G.F., AND K. 0. BISCHOFF,Chemical Reacror Analysis and Design, 2nd ed. New York: Wiley, 1990. WALAS.S. M., Chemical Reorriot1 Engineering Handbook of Solved Problems. Amsterdam: Gordon and Breach. 1995. See the foltowing solved problems: PmbEern 4.10.1, page 444: Problem 4.10.08, page 450; Problem 4.10.09, page 451: Problem 4.10.13. page 454: Problem 4.1 1.02. page 456: Problem 4. l1.09, page- 462; Problem 4.1 1.03, page 459; PmbIem 4.10.11. page 463.
590
Steady-State Nonisothermal Reactor Design
Chap. 8
For a thorough discussion on the heat of reaction and equilibrium constant, one might also consult
K. G.,Principles of Chemical Equilibrium, 4th ed. Cambridge: Cambridge University Press, 1981.
DENBIGH.
2. A review of !he multiplicity of the steady state and reactor stability is discussed by
F R O M EG, ~ , F., AND K. E. BISCHOFF,Chemical Reactor Analysis and Design, 2nd ed. New York: Wiley. 1990. P E R L M ~D. R D., . Siabilip of Cherndcnl Rcacrors. Upper Saddle River. N.J.: Prentice Hall, 1972. 3. Partial differential equations describing axial and radial variations in temperature and concentration In chemical reactors are developed in
FROMENT,G. F.,
AKD K. B. BISCWOW, Chemical Reactor Analysis and Design. 2nd ed. New York: Wiley. 1990. WALAS,S. M.. Reacrion Kinerics for Cl~emical Engzncer.r. New Ynrk: McGraw-Hill, 1959, Chap. 8.
4. The heats o f formation, I f , ( T ) GGibbs free energies. G,(T,), and the heat capacities of various compounds can be found in IT)
PERRY,R. H..D, W. GREEN, and J. 0. MALONEY,eds,, Cken~icalEt~gineers' Handbook. 6th ed. New York: McGraw-Hill, 1984. REID. R C..J. M. PRAUSKITZ, and T. K . SHERWOOD, The Properries of Gaaes and Liquids, 3rd ed. New York: McGraw-Hill. 1977. WEAST, R. C., ed,,CRC Handbook of Citemistn and Physics. 66th ed. Boca Ra~on,Fla.: CRC Press. 1985.
Unsteady-State Nonisothermal Reactor Design
9
Chemical Engineers are not gentle people, they Iike high temperatures and high pressures. Steve LeBlanc
Overview. Up to now we have fmused on the steady-state opemtion of nonisothmd reactors. In this section the unsteady-state energy balance will be developed and then appfied to CSTRs, as we11 as well-mixed batch and semibatch reactors. In Section 9.1 we arrange the general energy balance (Equation 8-9) in a more simplified form that can ?xeasily applied to batch and semibatch reactors. In Section 9.2 we discuss the applicaiion of the energy balance to the operation of batch reactors and discuss reactor safety and the reasons for the explosion of an industrial batch reactor. This section is followed by a description of the advanced reactor system screening too1 (ARSST) and how it is used to determine heats of reaction, activation energies, rate constants, and the size of relief valves in order to make reactors safer. In Section 9.3 we apply the energy balance to a semibatch reactor with a variable ambient temperature. Section 9.4discusses s m p of a CSTR and how to avoid exceeding the practical stability limit. W e dose the chapter (Section 9.53 wirh multiple reactions in batch reactors.
9.1 The Unsteady-State Energy Balance We begin by recalling the unsteady-state form of the energy balance developed in Chapter 8.
592
Unsteady-State Non!sotherrnal Reactor Des~gn
Chs
We shall 6rsl concentrate on evaluating the change in the total energy of
system wrt time, dE,,,/dr. The total energy of the system is the sum of products of specific energies, Ei, of the various species in the system volt and the number of moles of that species:
I n evaluating k,,, , we shall neglect changes in the potential and kint energies, and substitute for the internal energy U,in terms of the enthalpy ,
We note the last term on the right-hand side of Equation (9-2) is just the tc pressure times the totat volume, i.e., PL! For brevity we shall write these sums
unless otherwise stated. When no spatial variations are present in the system volume, and ti1 variations in product of the total pressure and volume (PV) are neglected, I energy balance. substitution of Equation (9-2) into (8-91, gives
Recalling Equation (8- 19),
and differentiating with respect to time, we obtain
Then substituting Quation (9-4) into (9-3) gives
Sec. 9.1
The Unsteady-State Energy Balance
The moIe balance on species i is
Using Equation (9-6) to substitute for diVildr, Equation (9-5) becomes
Rearranging, and recalling
1
V.H,
= AHR,T,.we
have
Thh Form of the
energy balance ~houkdbe used when there is a phase change.
Substituting for Hi and Hio for the case of no phase change gives us
- - 0 - K-Z F,~C~,(r-Tlo)+[-AH~xcT)l(-,v) (9-9) &" -
Energy balance on a transient
CSTR or
'
2 Y CP,
di
semibatch reactor.
Equation (9-9) applies to a semibatch reactor as well as unsteady-state operaof a CSTR. For liquid-phase reactions where AC, is small and can be neglected, the following approximation is often made: tion
Z N, Cpls
z
mpBC g cp<
N.0
Cr = NAO
= NAO
where Cps is the heat capacity of the solution. The units of NAoCp3 are (callK) or (BtuPR) and
.'
-
K ) or (Btu/h - O R ) With this approximation and assuming that every species enters the reactor at temperature To,we have
where the units of FA, CpI are (cal/s
1
We see that if heat capacity were given in terms of mass (i.e., C, = callg-K) then 14 both FAoand hr,o would have to be converted to mass:
m ~ n C <= , N ~ ~ c ~ r and 'AoCP,,
=F~fiC~p
but the units of the products would stillbethesame (cal/K) and (cal/s
- K),respectively.
594
Unsteady-State Nonisothermal Reactor Design
Chap. 9
9.2 Energy Balance on Batch Reactors A batch reactor is usually we11 mixed, so that we may neglect spatial variations in the temperature and species concentration. The energy balance on batch reactors is found by setting FA, equal to zero in Equation (9-10) yielding
Equation (9-1 1 } is the preferred form of the energy balance when the number of moles, N,, is used in the mole balance rather than the conversion, X. The number of moles of species i at any X is
Consequently, in terms of conversion, the energy balance becomes
Batch reactor energy Equation (9-12) murr and more bahnce~
be coupled wifh the mole balance
and tlw rate Inw and then s o l i ~ dnumerically,
9.2.1 Adiabatic Operation of a Batch Reactor Batch reactors operated adiabatically are often used to determine the reaction orders, activation energies, and specific reaction rates of exothermic reactions by monitoring the temperature-time trajectories for different initial conditions. Tn the steps that follow, we will derive the ternprature-conversion relationship for adiabatic operation. For adiabatic operation (Q = 0 ) of a batch reactor (F,,= 0) and when the work done by the stirrer can be neglected ( W~= 0), Equation (9-1 1) can be written as
Ssc. 9.2
595
Energy Balance on Batch Reacfors
rearranging and expanding the summation term
+
-AffRx(T)(-rAV) = NAo(CpJ AC&)
dt
(9-13)
where as before
From the mole balance on a batch reactor we have
We combine Equations (9-13) and (2-6) to obtain
I
/
Canceling df, separating variables. integrating, and rearranging gives (see CD-ROM Summap Notes for intermediate steps)
Temperature conuersion relationship for an adiabatic batch reactor (or any
reactor o p t e d adiabaricalry for that rnaaer)
We note that for adiabatic conditions the relationship between temperature and conversion is the same for batch reactors, CSTRs, PBRs, and PFRs. Once we have T as a function of X for a batch reactor, we can construct a table similar to Table E8-3.1 and use techniques analogous to those discussed in Section 8.3.2 to evaluate the folIowing design equation to determine the time necessary to achieve a specified conversion.
I
Example 9-1 Adiabaiic Botch Rtactor
Although you were hoping for a transfer to the Bahamas. you are still the engineer of the CSTR of Example 8-8,in charge of the production of propyfene glycol. YOU are considering the in~tallationnf a new glass-fined 175-gal CSTR, and you decide to make a quick check of the reaction kinetics. You have an insulated inatrumenfed 10-gal stirred batch reactor available. You charge this reactor with 1 gal o f methanol and 5 gal of water containing 0.1 wt % HISt).l. For safely reasons. the reactor is
596
Unsteady-State Nonisothermal Reactor Design
Ch:
located in a storage shed on the banks of Lake Wobegon (you don't want the e plant to be destroyed if the reactor explodes). At this time of year, the initial perature of all materials is 38°F. How many minutes should it take the mixture inside the reactor to rea conversion of 51.5% if the reaction rate law given in Example 8-8 i s correct? F would be the temperature? Use the data presented in Example 8-8. Lfving Example Problem
Solution
1. Design Equation:
Because there is a negligible change in density during the course of this r tion, the volume V is assumed to be constant. 2. Rate Law: - r ~= kcA
(E9-
3. Stoichiornetry:
4. Combining Equations (E9-1. I),(E9-1.2). and (2-61, we have
@= k(l -x) dt
(E9-
From the data i n Example 8-8,
k = (4.71 X iOq) exp
k = (2.73
X
(1.987) (T)
louJ) exp
(E4-
5. Energy Balance. Using Equation (9- 17). the relationship between X and 1 an adiabat~creaction is given by
6. Evaluating the parameters in the energy balance gives us the heat cap: of the solution:
Cps = C @,Cp= OACpA+OBCpB+QcCp,+@ICPI = ( 1 ) ( 3 5 ) + (18.65)(18) = 403 Btullb mol Aa0F
+ 0 + (1.670)(19.5)
Sec. 9.2
597
Energy Balance on Bafch Reactors
From Example 8-8, bC, = - 7 Btullb mol."F and consequently, the second tern on the nght-hand side of the expression for the heat of reaction,
I
is very small compared with the first term [less than (from Example 8-S)].
2%- ar 51.5% conversion
Taking the h a t of reaction at the initial temperature of 515"R. AHR,lTo) = -36.400 - (7)(5 I5 - 528) = -36.309 Btu/lb moI
Because terms containing ACp are very small. it can be assumed that
LC,- 0 In calculating the initial temperature, we must include the temperature rise fmm the hear of mixing the two solutions:
To = (460 + 38) + 17 = 51551
= 515 +90.1 X
(E9-1.6)
A summary of the heat and mole balance equations is given in Table E9-I. I. TAB LEE^-I.!,
SUMMARY
!!LA.(, -,q dr
k = 2.73 X
32 400 1 - 1 [1.987 (535 I )*
10-~ex~
T=515+90.IX where
T is In 'R and t is in seconds.
A table similar to that used in Example 8-3 can now be constructed
A software package (e.g., Polymath) was also used to combine Equations (E9-I.3). (E9-1.4), and (E9-i.6) to determine conversion and temperature as a function of time. Table E9-1.2 shows the program, and Figures E9-1.1 and
E9-1.2 show the solution results.
598
Unsteady-State Noniaothermal Reactor Design
I
POLYMATH Results
[
Example 9-1 Adiabatic Batch Reactor 04-14-2005, Rev5.1.233 Calculated values
a FL~ LlvingExample Problem
Chap 9
ofthe DEQ variables 8lbwlmalvalue f i n e @
t X T
O
3!
515 8.358~-05
515
C'
k
4OOG 0.9994651 605.09685 0.0093229
0
8.3588-0s
4000
0.9999651
605.09685 0.0093229
ODE Report (RKF45) Differential equations as entered by the user
[l] d(X)/d(t) = k*(l - X ) Explicit equations as entered by the user [I] T = 515+90.1'X [2] k = 0.000273*exp(l6306*((f 1535)-(l/r)))
Ffgure E9-1.1
Temperature-time curve.
Figure E9-1.2 Conversion-time cunle.
It is of interest to compare this residence time with the ~esidencetime in the 175-gal CSTR to attain the same conversion ar the same final temperature of 582"R ( k = 0.0032 s-I): T' t=-=--
v,
k(I - X )
OS15 = 332 s = 5.53 minutes (0.0032) (0.485)
This occasion is one when the increase in the reaction rate constant caused by the increase in temperature more than compensates for nhe decrease in rate caused by the decrease in concenrration, so the residence time in the CSTR for this conversion is less than it would be in a hatch or tubular plug-flow reactor.
Sec. 9.2
Energy Balance on Batch Reactors
599
9.2.2 Batch Reactor with Interrupted Isothermal Operation
In Chapter 4 we discussed the design of reactors operating isothermally. This operation can be achieved by efficient control of a heat exchanger, The following example shows what can happen when the heat exchanger suddenly fails.
/
Example 9-2
Safe9 in Chemical Phntx w&h Exathermic Reactions 2
A serious accident mcurred at Monsanta plant in Sauget, Illinois, on August R at 12:18 A.M. (see Fipure E9-2.1). The blast was heard as far as I0 miles away in Belleville, IHinois. where people were awakened from their sleep. The explosion occumd in a batch reactor that was used to produce nitroanaline from ammonia and
o-nitrochloroknzene IONCB):
Llving Example Problem
Fi~ureE9-2.1AFterrlluth of the eiplo\ron (51.L n t ~ ~Gluht. s I)emncral pho~oby Roy Cook. Courresy nf 51. h u i c 3lerconiile Library.)
Wa.: a Par~~rrial This reaction is normaIly carried our ~snthermallyat 175'C and about 500 p ~ i The . Pmhlerfl , 4 l r l t ~ ~ ~ r i ~ r amhlent temperature of the coolrng water in the heat exchanger is 25°C. By adjustrng the coolant rate the reactor temperature could be maintained at 1 75°C. At the rnaxlinum coolant rate the ambient temperature t s 25°C throughout the heat exchanger.
2
Adapted from the problem by Ronald Willey. Sernitlar. nn (1 Ilirrrua,lnl~neRror*tar Rrrprure. Prepared for SACHE, Cenier for Chemical Pmcecr Safe[!. American Institute of Chemical Engineers. New York (1994) A140 see Pn)r.ra S d f i f , ~Prop-es~.VFI. 10,no. 2 (2001 1. pp. 123-1 19. Thc v;~fut!s of AHH, and LiZ were estrmated in the plant data of the tempenrure-time lrajcclory in the ;ir~~cle by G. C V~ncent.h r . Prc~ ibt.rlrion.S. 4&52.
600
UnsteadpState Nonisothermal Reactor Desigq
Ch
Let me tell you something about the operation of this reactor. Over the the heat exchanger would hil from time to time. hut the technicians waul, "Johnny on the Spot" and run out and get it up and running in 10 minutes or so, there was never any problem. One day it i s hypothesized that someone looked a reactor and said. "It looks as i f your reactor 1s only a third full and you still room to add more reactanrs and ta make more product, How ahout filling it u the top so we could triple production?" They did, and you can see what happ in Figure E9-2. I . On the day of the accident. rwo changes in normal operation occurred.
A decision was made to triple product~nn.
1. The reactor was charged with 9 . W kmol of ONCB, 33.0 kmoI of NH,, 103.7 kmol of H,O. Normally, the reactor i s charged with 3.17 kmo ONCB, 103.6 kmol of H,O,and 43 kmol nf NH,. 2. The reaction i s normally carried out isothermnlly at 175'C over a 24-h per Appmximatety 45 min after the- reaction was started. cooling to the rea Failed. but only for I0 rnin. Cooling may have been hafted for 10 rnin o on previous occasions when the normal charge of 3.17 kmol o f ONCB used and no ili effects occurred. The reactor had a nipture disk designed to burrt when the pressure excee approximately 700 psi. If the disk would have ruptured, the pressure in the rea aould have dropped. causing the water to vaporize, and the reaction would k been cooled (quenched) by the latent heat of vaporization. Plot the temperature-time trajectory up to a period of 3 20 rnin after the n tanrs were mixed and brought up to 175°C. Show that the following three conditi had to have been present for the explosion to occur: ( 1 ) increased ONCB cha 12) reactor stopped for 10 min, and (3) relief system failure. Additionai Infomation: The rate law is -roycn= kCoNcBCNH, with
k = 0.00017
m3
kmol . rnin
The reaction volume for the charge of 9.044 kmol of ONCB:
The reaction volume for the charge of 3.17 kmoI of ONCB: V = 3.26 mJ
AH, = -5.9 X 1OS kcallkmoI
Assume that AC, = 0 :
UA = 35'8s kcalwith T, = 298 K
min "C
at 188°C
Sec. 9.2
Energy Balance on Batch Reactors
A+ZB-
C+D
Mole RnIance:
Rate Law: -r..\ = kCACB
Stoichiornetry (liquid phase):
with
Combine:
I
Energy Balance:
For AC, = 0, NCp = 1 N,Cp,= :V.+,CP4+ N,,CpR
+ 1Vw Cpu
Let QR be the heat generated [i.e., Qx = (s, Y)(AH,,) J and fet removed [i.e.. Q,= I I A ( 7 - T,) 1:
Parameter evaluation for day of explosion:
Qr
be the heat
602 j
Unsteady-State Nonisotherrnal Reactor Design
Chap, g
A. Isothermal Operation Up to 45 Minutes We will first carry out the reaction isothermally at 17S°C up to the time the cooling was turned off at 45 rnin. Combining and canceling yields
-
Ar IJ5'C = 448 K, k = 0.0001 167rn3/kmol min. Integrating Equation (E9-2.91 gives us
Substituting the parameter values Thc calculation and resulrs can also he obtamed from the Pulyrnath output on the CD-ROM
tilring Example Problem
45 min =
3.64 - W 5.1 I9 m3 X - In [0 0001 161 rn3:Lmol- rnrn(9.044 h01) (1.~4) 3.64(1 -XI
]
Solving for X, we find that at I = 45 min, then X = 0.033. We will calculate the raw of generation Q, at this temperature and conversion and compare it with the maximum rate of heat removal Q,. The rate o f generation Q, is
At this time (i.e., 1 = 1 5 min. X = 0.033, T = 175°C) we calculate k. then Q, and Q,. A1 175°C. k = 0.0001 167 m31min*kmol.
T h e corresponding maximum cooling rate is
Therefore Everything is OK.
(E9-2.13) The reactinn can he controlled. There would have been no explosion had the cooling not fa~lrd.
Sec. 9.2
603
Energy Balance on ~ a t c Reactor$ t~
B. Adiabatic Operation for 10 Minutes The cooling was off for 45 to 55 min. We will now use the conditions at the end of the period of isothermal operation as our initial conditions for adiabatic operation period between 45 and 55 min: Between r = 45 and 1 = 55 min, Q, = 0.The Polymath program w a s modified to account for the time of adiabatic operation by using an "ifsraremcnt" for Q, i n the
program. i.e., Q, = if (t > 45 and t < 55) then (0)else (UA(T - 298)). A similar "if s~aremm~" is used for isothermal operation, i.e., (dT/dr) = 0. For the 45- to 55-min period without cooling, the temperature rose from 448 K to 468 K, and the conversion jncreased from 0.033 to 0.0424. Using this temperature and cmvmion in Equation (E9-2.1 I ) , we calculate the rate of generation
Q, at 55 min as The maximum rate of cooling at this reactor temperature is found fmm Equation (84-2.12) to be Here we see that
The point o f no return
and the temperature will continue to increase. Therefore. the point of no return has been passed and the temperature will continue to increase, as will the rate of reactlon until the expEosion wcurs.
C. Batch Operation with Heat Exchange
Intemptions in the Mling system have happened before with no ill effects.
Return of the cooling occurs at 55 min. T h e values at the end of the period of adiabatic operation (T = 468 K, X = 0.0423) become che initial conditions for the per~odof operation with heat exchange. The cmling is turned on at its maximum c a p a c i t ~Q = UA(298 - TI. at 55 rnin. Table E9-2.1 gives the Polymath program tn detem~nethe temperature-time trajectory. Note that one can change NAoand NB,, 10 3.17 and 43 kmol in the program and show that if the cooling is shut off for 10 min, at the end of that 10 min Q, wit1 still be greater than Q, and no explosion wiIl occur. The complete temperature-time trajectory is shown in Figure E9-1.2. One notes the tong plateau after the cooling is turned back on. Using the values of Q8 and Qr at 55 rnin and substituting into F~uation(E9-2.g). we find that dT - (659 1 kcal/min) - (6093 kcsl/min) dl
2504 kcal 1°C
= O,ZoC/min
604
Unsteady-State Nonisothermal Reactor Design
Ct
v
-
D'flwsntial s q u m ~ aa 8 entoted by it18user t lI d(T)ld(tJ = fl (t<451thsn (0) else ((Q-WCp) [ 2 1 d(X)ld(t) -ru?lWao
G p M equatlonr m antwed by Lhe w a r
Llvln$ Examplc Proble
111 V-3.285c1.854 I21 N s o = 9 . M I 3 1 UA=35.83 141 k = [ J . W 0 1 7 ' e ~ (t273(1.987).(ll4t-t/TJ) 1 I s ] ilmtaB=33/9.045 t6E T I =T-273 17 1 r& 3 -k~NBdlZ.(l-X)f(kttaB.TX)N"2 [ 8 07 = if(b45 md k55) Ihsn(0) aJ.se (UA'(T-298)) I9F D d t & r * = - s m
1 Qg = ra'V'Del?aHnt I l l 1 NCp-250-4 [lr)
The explosion occurred sbonly after midnight.
Figure E9-2.2 Temperature-time trajectory.
Consequently, even though dT/dt is positive, the tempetgture increases very slc at first, 0.2"Clmin. By 11:45, the temperature has reached 240°C and is begin to increase more rapidly. One observes that 119 min after the batch was startec temperature increases sharply and the reactor expldes at approximately midn If the mass and heat capacity of the stirrer and reaction vessel had k e n inch the NC, term would have increased by about 5% and extended the time until explosion occurred by 15 or so minutes, which would predict the actual time explosion occurred, at 12:18 A.M. When the temperature reached 3WC, a secondary reaction, the decomf tion of nitroaniline to noncondensable gases such as CO,N,, and NOz, occui releasing even more energy. The total energy released was estimated to be 6.8 x 1' which is enough energy to lift the entire 2500-ton building 300 rn (the lengt' hree football fields) straight up.
Sec. 9.2
(
k
Spring .lief
wk.3
Energy Balance on Batch Reactors
D. Disk Rupture We note that the pressure reItef disk should have ruptured when the temperature reached 26j°C (ca. JDO psi) but did not and the temperature continued to rise. If it had ruptured and all the water had vaporized, 106 kcal would have been drawn from the reacting solution, thereby lowering its temperature and quenching it. tf the disk had ruptured at 265°C (700 psi), the maximum mass flow rate, ni,,,. out of the 2-in. orifice ro the atmosphere ( 1 atm) would have been 830 kgJmin at the time o f rupture.
.
= 830
mrn
X
540 &@+ 35.83 *k (538 - 298)K kg K
kcal 4-8604 *k nln min kcal = 4.49 x LO5 min = 4.48 X 105
This value of Q, is much greater than Q,(Q,= 27,460 kcalimin), so that the reaction could easily be quenched. In summary, if any one of the following three things had nor occurred the explosion would not have happened. 1. Tripled production 2. Heat exchanger failure for 10 minutes 3. Failure of the relieving device (rupture disk) In other words, all rhe above had to happen to cause the exptosion. If the reIief had oprated pmpedy, it would not have prevented reaction runaway bur it could have prevented the explosion. In addition to using rupture disks as relieving devices, one can also use pressure relief valves. I ~ many I cases suficiedt care is not taken to obtain data for the reactioo at hand and to use it to properly size the relief device. This data can be obtained using a batch reactor called the ARSST.
9.2.3 Reactor Safety: The Use of the ARSST to Find AHR,, E and to Size Pressure Relief Valves
Use ARSST
lo find:
E,hcr~vationenergy A, frequency factor AH, heat of reaction
The Advanced Reaction System Screening Tool (ARSST) is a caIorimeter that is used routinely in industry experiments to determine activation energies, E; frequency factors, A; hears of reaction, UR,and ; to size vent relief valves For runaway exothermic reactions [Chemical Engineering Progress, 95 (21, 17 (2000)l. The basic idea is that reactants are placed and sealed in the calorimeter which is then electricalty heated as the temperature and pressure in the calorimeter are monitored. As the temperature continues to rise, the rate of reaction aLso increases to a point where the temperature increases more rapidly from the heat generated by the reaction {called the self-heating rate. T S ) than rhe temperature increase by eleclrical heating. The temperature at which this
606
Unsteady-State Monisothermal Reactor Design
Chap. 9
change in relative heating rates occurs is called the omer temperature. A schematic of the calorimeter is shown in Figure 9-1.
Experiments to obtain data to design safer reactors
Figre 9-1 ARSST (a) Schematic of containment vessel and intemals. (b) Tert cell assembly. [Courtesy of Fauske & Associates.]
We shall take as o u r sysrem the reactants, products and inerts inside the spherical container as well as the spherical container itself because the mass of the container may adsorb a little of the energy given off by the reaction. This system is well insulated and does not lose much heat to the surroundings. Neglecting ACp, the energy balance on the ARSST, Equation (9-12), becomes
The total heat added, Q. is the sum of the electrical heat added, Q,,and the convective heat added term, Qc
-
(UA[To- T I )
Because the system is well insula~ed,we shall neglect the heat loss from the calorimeter to the surroundinps. QC-.and further define
Sec. 9.2
Energy Balance on Batch Reactors
607
pr: is called the electrical heating rate. The second term in Equation (9-18) is called the self-heating rale, T S , that is, TS
Reference Shelf
= (-AH,)(-~A~
",cP,
(9-20)
The self-heating rate, which is determined from the experiment. is what is used to calculate the vent size of the relief valve, A, of the reactor in order to prevent runaway reactions from exploding (see PRS R9.4 on the CD-ROM). The electnca1 heating rate is controlled such that the temperature rise, ~g (typically 0.5-2'C/min), is maintained constant up to the temperature when the self-heating rate becomes greater than the electrical heating rate
Again, this temperature is calIed the onsel temperature, T,,. A typical thermal history of data collected by the ARSST is shown in Figure 9-2 in terms of the temperature-t ime trajectory.
Relatively little conversion i s achieved for T < To,,,
Figure 9-2 Typical lernperature history for thermal scan with the ARSST
The self-heating race.
TS. can be easily found by differentiating the tempera-
ture-time trajectov, or T S can be determined directly from the output instrumentation and software associated with the ARSST. We can rewrite Equation 19- 18) in the form
The foElowing example uses data obtained in the senior unit operations labaratory of the University of ~Michigan.
60& Safety Vdlve
Unsteady-State Nonisothermal Reactor D s s ~ g n
Ch
Side note: A relief valve is an instrument on the top of the reactor tl releases the pressure and contents of the reactor before temperature a pressure builds up to runaway and explosive conditions. The safety re1 valve is similar to the rupture disk in h a t once the reactor pressure exec the set pressure Ps, the contents are allowed to flow out through the vent. the pressure falls below the vapor pressure of the reactor contents, the latr heat of vaporization will cool the reactor. The self-heating rate may be us directly to calculate the vent size necessary to release all the contents of 1 reactor. The necessary vent ma, 4 is given by the equation for a vaf system and two-phase flashin- u--~
is a reduction factor for an ideal nouIe, Ps is the relief I pressure (psias, and ms the: mass of the sample in kg and T$ is 1 . . self-heating rate ruscussea in the sample calculation in the Ptofessioml R crence SheIf R9.1 on the CD-ROM. a.
.
Example 9-3 Use of the ARSST We shall use the ARSST to study the reaction between acetic anhydride and v
to form acetic acid
Acetic anhydride is placed in the ARSST to form a 6.7 molar sofution of a1 anhydride and a 20.1 M solution of water. The sample volume is 10 ml. The eIe cal heating is started. and the temperature and its derivative, fi, are recorded function of time by the ARSST system and computer. Analyze the data to finc heat of reacrion AHR,, the activation energy E. and the frequency factor A. and to compare theoretical and experimental temperature-time trajectories. The temperature-time trajectory is obtained directly from the computer l i to the ARSST as is shown in Figure E9-3.1.
Chemical
Den~lr>. (g/mfl
Heat capac~ry (J/gTj
MW
Hent c a p c i y (J/mol'C)
101
189.7
0, I
Acetic anhydride
1. O R H
1.860
Water
1.0000
4.187
I8
75.4
3
0.1474
0.837
-
0 84 Jlg K
-
Glass cell (bomb)
Energy Balance on Batch Reactors
Sec. 9.2
Total volume
10 ml with
Water
3.64 g
Acet~canhydride
6.84 g 10.48 g (10 ml)
Figure E9-3.1Temperature-time trajectory for hydrotysis of acetic anhydride.
Soluiion
Because we are taking our system as the contents inside the bomb as well as the bomb itself, the term HN,C,, needs to be modified to account for the heat absorbed by the bomb calorimeter that holds the reactants. Thus, we include terms for the mass of the calorimeter, mb,and the heat capacity of the calorimeter, C in the sum ZN,Cp,,that is, pb
.
However, for this example: Nl = 0 and we neglect ACp and m b C
to obtain
pa
Z NiCP,= NAOX@ICp, We now apply our algorithm to analyzing the ARSST for the reaction A+B-
2C
(E9-3.1)
610
Unsteady-State Nonisotbml Reacfor Design
BUNCE Epu~no~s
T A ~ LE9-3.1. E
Mole balance: Rate law:
Chap. 9
dN* =v , df rA =
(E9-3.2)
-klCACB
(E9-3.3)
C, = 2CAJ For 8, = 3, as a first approximation we take
(Note: In Problem P9-10,we do not assume CBOis constant.)
Combine:
k = k'CB,
@ d
@@@@ Fullowlng the Algortthm
Enwgy balance:
dr= i;,& dt
Recalling N,= VC,we can substitute Equations (E9-3.5) and (E9-3.6)into Equation (E9-3.1) and then NAozo,Cp, = NAoCpA + NBoCpA +-
Cp,
WPb (E9-3.11)
Neglect
Usually N, = 0, ACp = 0, and rnbCpbarc negligible with respect to the other terms. The self-heating rate IS
611
Energy Balance on Batch Reactors
Sm. 9.2
CalcuIa2ing the Heat of Reaction The, heat of reaction for adiabatic operatim cmi be found from the adiabatic temperature rise for cornpfete conversion X = 1. The onset temperaturr:is taken as the point at which the self-heating rate is greater than the electrical heating rate. From Figure E9-3.1 we see the onset temperature is 55°C and the maximum temperature is 165'C.
Recalling Equation (8-29) with 6 C p = 0
Determining the Activation Energy The self-heating sate Ts is shown as a function of temperature in Figure (E9-3.2). The ARSST software will differentiate the data shown in Figure E9-3.1 directly to give T 5 or (dT,dr, can be found from T versus r data using any of the differentiational techniques discussed in Chapter 5. One notes that the self-heating rare goes to zero at T = 138°C as a result of the reactants having been mnsumed at this point. The electrical heating rate, T E , either is shut off m becomes negligible wrt after the onset temperature To,,, is reached. Applying Equation (9-20) to our reaction, we obtain
We now take the log of the self-heating rate, T s , using Equarion (E9.3-13) to obtain
After the onset temperature is reached, we can obtain the activation energy from the slope o f a plot of [ln(dTEdf)Jas a function of I Iln, neglect~ngchanges in In CA.The slope of the line will be (-EJR). as shown in Figure E9-3.3. From the slope of the plot we f nd the activation to be
E = - K - Slope = 1 . 9 8 7 A k (-7,750 K) mol K +
$72
Unsteady-State Nonisothsrmal Reactor Design
Flgure E9-3.2 Self-heating curve for the hydrolysis of acetic anhydride.
C
Figow E9-33Amhenius pIot of self-hea rate for acetic anhydride.
Calculating the Frequency Factor, A We will now make an approximate calculation to determine the frequency f The intricate details of a more accurate calculation are given in Pmfessional E mce ShcIfR9.1 where the conversion obtained during the electrical heating ph, taken into account. Neglecting the conversion during electricat heating, C, =CAD,and Equation (E9-3.13) can be arranged in the Fom
At the onset, T-, = 5532; at 10 minutes the self-heat rate can ,beestimated fro[ slope of the plot of T versus :shown in figure E9-3.1 to be Ts . ,,, = 5.2 Wrn Evaluating the parameters.
W e now calculate the frequency factor to be
SW. 9.2
Energy Balance on Batch Reactors
613
Now that we have the activation energy E and the Anhenius factor A , we can use Polymath to simulate the equations in Table E9-3.1 and compare with the exprimental results. The Polymath program is shown in Table E9-3.2, and the corresponding output is shown in Figure E9-3.4. TABLE E9-3 2.
POLYMATHPROGRAM
ODE Report IRKF45) Differential e uaticns as entered by the user 111d ( ~ A ) i d ( Y= rA [21 d(T)/d(t) = TedottTsdot
Expl~cite uabons as entered by the user [ I ] CBO =\0.2
[j]A = 3.734e7
I;/
hr :D
8 Tedot = if (T>55+273)then 0 else 2 rA = -k*exp(-E/R/T)TATBo [I 0) Tsdot = I-dHrx)*(-rAYV)/5UMNAOREiCpi
Comments II I d(CA) / d(t) = rA Mole bnhr~ceon Acetic At~hydridc [2] d(T)/d(t) = Tedot+Tsdot
-
Jving Example Problem
Energy Pnlance [3] rA -A*exp(-E/R/TI"CA"CBa Rnk of the reaction-mol/l.rnin [-ll V = 0.01 Volume offhc renctizw nlrifion-1 151 SUMNAO~%E~CD~28
-
-
-
rate mrrslanf-lfrnrn [HIE = 1.5400 cafl~nol R = 1.987 cn1lmof.K (10) Tedot = if fl>55+273) then 0 else 2 (oK/mfn)Aftrr the altset goirri, elertricat fienfrn is only to mlnpnsalcfar Imt loss [I11 T d o t = (-dHr~)*(-rAV)/SUMNAOTHE~&~ Scbf-hentin~rnte (oK/rnin) W h m SUMNAOTHEiCpi =
N ~ , XOic5=28
614
UnsteadyStste Nonisothermal Reactor Design
Chap. g
-
...-~ r p e t i mbi u n -Simulation Resun ustq the kinetics derived tmm experhental rusults I
I
I
I
I
I
I
I
I
T~me(mln)
Figure E9-3.4 Temperature-time trajectory for hydrolysis of acetic anhydride.
-
Reference Shelf
As one can see there is excellent agreement hetween the simulation and the experiment. The decrease in temperature at 13.5 minutes in the experimental data is a result of a small heat loss to the surroundings which was not accounted for in the simulation. The CD-ROM (R9.1)describes how to size the vent size from this data. When using the ARSST in the labontory to actually size the relief vent, follow the procedure in the Professionai Reference Shelf, which accounts for the conversion during the electrical heating and also taking the onset ternFratnre at a point where l"s >> 7 ~Alrio . see the Fauske web site: wltwfauske.com.
9.3 Semibatch Reactors with a Heat Exchanger
'
t i c ci:\tril~ly lutrll wac
In our past discussions of reactors with heat exchanges, we assumed that the ambient temperature T, was spatially uniform throughout the exchanger. This assumption is true if the system is a tubular reactor with the external pipe surface exposed to the atmosphere or if the system is a CSTR or batch where the coolant flow rate through exchanger is so rapid that the coolant temperatures entering and leaving the exchanger are virtually the same. We now consider the case where the cmlant temperature varies along the length of the exchanger while the temperature in the reactor is spadally unifonnThe cooEant enters he exchanger at a mass flow rate m, at a temperature Tel and leaves at a temperature To? (iee Figure 9-3). As a first approximation. we ;issurne a quasi-steady state for the coolant flow and neglect the accumulatian term (i.e.. dT,/dt = O 1. As a result, Equation (8-49) will give the rate of heat transferfrom the exchanger ro the reactor:
Using Equation 18-49) to substitute for
Q in Equation (9-9), we ohtain
Sec. 9.3
Semibatch Reactors with e Heat Emhanger
615
Heat Exchanger Coiled Tubing
F i ~ r 9-3 e Tank reactor with heat exchanger.
At steady state ( l d t = 0 ) Equation (9-21) can be solved for the conversion X as a function of reaction temperature by recalling that FAOX = -rAV and
and neglecting ACp and then rearranging Equation (9-21) to obtain Steady-state energy balance
X=
We are assuming that there is vinually no accumulation of energy in the coolant fluid, that is,
61 6
Unsteady-State Nonisothermal Reactor Desrgn
Cha
The second-order ~aponificationof ethyl acetate is to be carried out in a selnib; reactor shown schematrcally i n Figure E9-4.1.
Living Example Problem
Aqueous sodium hydroxide is to be fed at a coocenaation OF t h l l m B . a tempentun 300 K,and a rate of 0.004 m3/s to an initial volume of 0.2 m" of water and ethyl ace1 The initial concentsat~ons of ethyl acetate and water are 5 kmol/m3 and 3 kmollm" respectively. The reactlon is exothermic, and it is necessary to add a 1. exchanger to keep its temperature below 315 K. A heat exchanger with UA = 331 U s . K is available for use. The coolant enters at a rate of tOO kgls and a temp ture of 285 K. Is the heat exchanger and coolant flow rate adequnte to keep the reactor t r perature below 315 K? Plot temperature, C,4bCB.and C, as a function of time. Additionul informatiun: 3
KC ]03885.44/T -
AH",
=
-79+076kJ/ kmol
CpA= 170.7 J/rnol/K
Cwo=55kmollrn3
Feed: Initially:
C,
= 30.7 kmolJm3
CBn=f.0km01Jm3
CA,= 5 krnol/m3
C,, = O
Figure E94.1 Semibatch reactor with heat exchange.
k from J. M. Smith, Chemical Engineering Kinetics, 3rd ed. (New York: McGw Hill. 1981). p. 205. AHR, and Kc calculated from vatues given in Perry's Chemk Engineem' Hrurdbook, 6th ed. (NewYork: McGraw-Hill. t984), pp. 3-147.
Sec. 9.3
Semrbatch Reactors with a Heat Exchanger
Snl~rtiut~
Mole Balances: (See Section 4.10.2. )
I
Initially, lVwi
= V,Cwr= (0.2)(30.7) = 6.14 kmol
Rate Law:
Stoichiornetry:
(
1 =
However. CpB= CPw: n
1
1
F,Cp, in Equation (9-9).Because only B . P. and water continually Row into the reactor Energy Balance: Next we replace
where
618
Unsteady-State Nonisofherml Reactor Design
Chap. 9
d ~ ~-, C F $ T , ,- T ) I 1 - exp(- UAlk,Cp)l -F,,C,{I + B,)(T- To)+ (r,V) AH, C P / N B + N ~%+NW1f + CPANA (E9-4.10) RecaIIing Equation (8-47) for the outlet temperature of the Auid in the heat exchanger
r m:;pl
T,, = T - ( T - T,,)exp --
The Polymath program is given in Table E9-4.1.The solution results are shown in Figures E9-4.2 and E94.3.
Living Example Problem
Set. 9.4
I
Unsteady Operation of a CSTR
Figure E9-4.2 Temperature-time trajectory in a sernibarch reactor
Figure E943 Concentration-time trajectories in a semihatch reactor.
9.4 Unsteady Operation of a CSTR 9.4.1 Startup Startup of a CSTR
Rcfercnce Sheff
In reactor startup it is often very important faow remperature and concenrrations approach their steady-state values. For example, a significant overshoot in temperature may cause a reactant or product to degrade, or the overshoot may be unacceptable for safe operation. If either case were to occur. we would say that the system exceeded its prtlcrical stabiliv limit. Although we can solve the unsteady temperaturetime and concentration-time equations numerically to see if such a Iimit is exceeded, it is often more insightful to study the approach to steady state by using the temperature-concenrra~io~? phase plane. To illustrate these concepts we shall confine our analysis to a liquid-phase reaction carried out in a CSTR. A qualitative discussion of how a CSTR approaches steady state is given jn PRS R9.4. This analysis, summarized in Figure S-I in the Summary for this chapter, is developed to show the four different regions into which the phase plane is divided and how they allow one to sketch the approach to the steady state.
(
Example 9-5
Startup of o CSTR
Again we consider the production of propylene gIycnl IC) in a CSTR with a heat exchanger in Example 8-8.Initially there i s only water at 75'F and 0.1 wt % H2S04 in the 500-gallon reactor. The feed stream consists o f 80 Ib mallh of prnpylene oxxde (A), 1000 Ib molth of water (3)containing 0.1 wt % H2S04, and 100 rb mollh of methanol (M}. Plot the temperature and concentration of pmpylene oxide as a functron of time, and a concentration vs. temperature graph for different enrering temperatures and initial concentrations of A in the reactor.
620
Unsteady-State Nontsothermal Reactor Design
Chal
The water coolant flows through the heat exchanger at a rate of 5 lbls (1( Ib mollh). The molar densities of pure propyiene oxide (A). water (B), and met no1 (M) are p, = 0.932 Ib mollft3. = 3.45 lb mollft3, and p, = 1.54 rnollft3, respectively.
Solution
Mole Balances: Initial Conditions A:
d%=r,+
Rate Law: Stoichiometry:
dr
(c~, -CA)~, V
- r A = kcA -rA = - r B = rc
Enew Balance:
with
and
Ta2= T - (T- T,,)exp
0
(E9-5
Sec. 9.4
Unsteady Operation of a CSTA
Evaluation of parameters:
Neglecting ACp because it changes the heat of reaction insignificantly over the temperature range o f the reaction, the heat of ofaction i s assumed constant at
-
The Polymath program is shown in Table E9-5.1. TABLE E9-5.1.
POLYMATH~ o G I I A MFOR C S m START~TP
622
Unacceptable sranup
Unsteady-Stale Noniaothsrmal Reactor Design
Chap. 9
Figures (E9-5.1)and (E9-5.2) show the reactor concentration and temperature of propylene oxide as a function of time, respectively, for an initial temperature of 75°F and only water in the tank (LC., C,, = 0). One observes, both the temperatute and concentration oscillate mound their steadystate values (T= 138"E Cn = O.Q.79 Ib rnol/ft'). Figure (E9-5.3) shows the phase plane of tempemture and propylene oxide concentration for three different sets of initial conditions (T, = 75"F, C,, = 0; T, = lSO°F, CAi= 0;and = 160%. CAr= 0.14 Ib mollft3), keeping Toconstant. An upper limit of 180°F should not be exceeded in the tank. This temperature is the practical stahilip limit. The practical stability limit represent a temperature above which it is undesirable to operate because of unwanted side reactions. safety considerations, or damage to equipment. Cansequently, we see if we started at an initial temperature of 160°F and an initial concentration of 0.14 molldrn'. the practical stability limit of 180°F would be exceeded as the reactor approached its steady-state temperature of 1 3 8 T See the concentration-tempemture trajectory in Figure E9-5.4.
Figure E9-5.1 Propylene oxide conceneation as a function of time.
Figure E9-5.2 Ternperamre-time trajectory for CSTR startup.
Figure E9-5.3 Concentration-temperature
Figure E9-5.4 Concentration-temperature phase plane.
Oops! The practical s~abilitylimit was exceeded.
phase-plane ~rajeclon:
.
After about 1 .h h the reactor isoperating at steady state with the foHowing vaIues: -
-
Sac.9.4
Unsteady Opratlan of a CSTR
9.4.2 Falling Off the Steady State
We now consider what can happen to a CSTR operating at an upper steady state when an upset occurs in either the ambient temperature, the entering temperature, the flow rate, the reactor temperature, or some other variable. To ilIusaate, Iet's reconsider the production of propyIene glycol in a CSTR,which we just discussed.
I
Example 9-6 F d h g Oflthe Upper Steady S w e
In Example 9-5 we saw how a 500-gal CSTR used for the production of propylene glycol approached steady state. For the Row rates and conditions (e.g., To = 75"F, T,,= 60°F). the steady-state temperature was 138"E and the corresponding mnversion was 75.5%. Determine the steady-state temperature and conversion that would result if the entering temperature were ro drop from V0F to 7PF,assuming that all other conditions remain the same. First. sketch the steady-state conversions calculated from [he mole and energy balances as a function of temperature before and afrw the drop in entering temperature mcurred. Next, plot the "conversion," concentration of A, and the temperature in the reactor as a function of time after the entering temperature drops from 75°F to 70°F.
)
The steady-state convenions can be calculated from the mote balance.
(
mi h r n tbe energy balance.
before (To = 75°F) and after IT, = 70T) the upset occurred. We shall use the parameter values given in Example 9-5 (e.g., FA, = 80 Ib rnol/h. UA = 16,000 Btulh . "E)lo obtain a sketch of these conversions as a function of temperature, as shown in Figure E9-6. I . We see that for To =7OoF the reactor has dropped below the extinction ternPrature and can no longer operare at the upper steady state. In Problem P4-16. we will see it i s not always necessary for the temperature to drop below the extinction temperature In order to fall to the lower steady state. The equations describing the dynamic drop from the upper steady state to the lower steady smte arc identical ro those given in Example 9-5: only the in~tialcondit~onsand entering temperature are different. Consequently, the qame Polymath and MATLAB programs can be used wtth these modifications. (See Liilirtg ErampIc 9-6 on the CD-ROM.)
624
Unsteady-State Nonisotherrnal Reaclor Design
Figure E9-6.1 Convetxion from mole and energy balances as
Ch,
a Function
of temperature.
Initial conditions are taken from the final steady-state values given in Ex ple 9-5.
C,, = 2.12 Ib mollft~ C, = 0.226 ib molJft' T,= I38.5"F Change T, to 70°F Because the system is not at steady state, we cannot rigornusly define a convers in terms of the number of moles reacted because of the accumulation within the re tor. However, we can approximate the conversion by the equation X = ( 1 - C,IC, This equation is valid after the steady state is reached. Plots of the temperature i the conversion as a function of time are shown in Figures E9-6.2and E9-6.3,r a p tively. The new steady-state temperature and conversion are T = 83.6"Fand X = 0.
Figure E9-6.2 Temperature versus time.
Figure T59-62 Conversion versus time.
We could now see how we can make adjustments for upsets in the reac operating conditions (such as we just saw in the drop in the entering temperatu so that we do not fall to the Iower steady-state values. We can prevent this drop conversion by adding a controller to the reactor. The addition of a controller is d cussed in the Pmfessional Reference Shew R9.2 on the CD-ROM.
Sec. 9.5
625
Nonrsothermal MulttpFe Reactions
9.5 Nonisothermal Multiple Reactions For multiple reactions occurring in either a semibatch or batch reactor, Equation (9-21) can be generalized in the same manner as the steady-state energy balance, to give 4
~ C C P ~ ( T , I - T-exp(-UA/mcCp,)l+ )[~
dT -
1
r,,
I=
c N, CP,
dt
V A H R q ( T ) - Z FjoCp,(T-To)
I
For large coolant rates Equation (9-23) becomes
I
Example 9-7 MuItipk Reaclions in a Semibotch Rebctor The series reactions
Livlng Example Proble
are catalyzed by H2S04.All reactions are first order in the reactant concentration. The reaction ts to be carried our in a semibatch reactor that has a heat exchanger inside with UA = 35.000 callh K and an exchanger temperature, T,, of 298 K. Pure A enters at a concentnition of 4 moI/dm3, a volume&ic flow sate of 240 dm3/h, and a temperature of 305 K. Initially there is a total of I00 dm3in the reactor, which contains 1.0 rnoI/drn3 of A and 1.0 mol/dm3 of the catalyst W2S04.The reaction rate is independent of the catalyst concentration. The initial temperature of the reactor i s 290 K. Plot the species concentrations and temperaturn as a function of time.
626
Unsteady-StnTe NonisoFmrmal Reactor Design
Chap. 9
AaYiriomI information:
-
klA = 1.25 h-' at 320 K with EIA= 9500 caIIrno1
CpA = 30 call mol K
bB = 0.08 h-I
CpB = 60 cal/mol K
at
300 K with
= 7000 cd/mol
AHRxl, = -6500 cal/mol A &IR =, +8000 ,, callmol
Cpc = 20 cat/ mol
B
c%,SO,
SoIuiion
Mole Balances:
Rate Laws: - r , , = RIACA
-rza = k2RCB
Stoichiomern (liquid phase): Use C,, C,, Cr Relative rates: flB=-
=
I -r 2
I*
-3 rZ8
Net rates:
~ ~ , = @ m o ~dm3 x ~ - = gmot mdr n 3
h
h
K
= 35 cal/mol . K
Sac. 9.5
1 ,
Nonisothermal Multl~leReactions
Enecgy Balance:
Equations (E9-7.1through ) (E9-7.3) and (E9-7.8)through (E9-7.12) can be solved simultaneously with Equation (E9-7.14) using an ODE solver. The Polymath propram is shown in Table E9-7.1 and the Matlab program is on the CD-ROM. The time graphs are shown in Figures E9-7.1 and E9-7.2.
kving Example Problem
1
Figure E9-7.1 Cancentrationtime.
Figure E9-7.2 Temperature (K)-time (h).
628
Unsteady-State Nonisothermal Reactor Design
Ch
9.6 Unsteady Operation of Plug-Flow Reactors In the CD-ROM. the unsteady energy balance is derived for a PFR.Neglec changes in tota! pressure and shaft work, the following equation is derive[ Reference Shelf Transient energy balance on a PFR
This equation must be coupled wih the mole balances: Numerical solution required for these three coupled equations
and the rate law,
~ e f e r e n cShelf l
and solved numerically. A variety of numerical techniques for solving eq tions of this type can be found in the book Applied Numerical method.^.^ One can use FEMLAB to solve PFR and laminar flow reactors for time-dependent temperature and concentration profiles. See the FEML. problems and web module in Chapter 8 and on the FEMCAB CD-RE
enclosed with this book. A simpler approach would be to model the PFR a number of CSTRs in series and then apply Equation (9-9) to each CSTR.
Closure. After completing this chapte~ o appl r, the teatIer shoddI be able 1t nd batch Iwctot; the unsteady-state energy balance to CSTRs, selmibatch a~ The reader should be able to discuss R ~ L L U L oalcW usine t w ~*-,.~+..m l r t r l e on s: e ARSST to he1 a case study of an explosion and the other the ihouId be how F prevent explosions, Included in the redader's dL sfart up a reactor so as not to exceed thle practicaJ stability limit. Aft:erread . -- . -. .ing these examples, the reader should be able to descnbe Row to opera1 reactors in a safe manner for both single and multiple reactions. ,-.A
* &. Camahan, H.A. Luther, and I. 0.Wilkes, Applied Nurncrical Methods (New Yo1 Wiley, 1969).
Chap. 9
Summary
SUMMARY I . Unsteady operation of CSTRs and semibatch reactors
For large heat-exchanger coolant rates (To\= To?)
For moderate to low coolant rates
[ ( sf11
Q = m,Cpc(T- T,,) 1 - exp - 2. Batch reactors a.
Nonadiabatic
Where Q is given by either Equation (S9-2) or ( 9 - 3 ) . b. Adiabatic
3. Startup of a CSTR (Figure S-I) and the approach to the steady state (CD-ROM). By mapping out regions of the concentration-temperature phase plane, one can view the approach to steady state and learn if the practical stability limit is exceeded.
Unsteady-State Nonisothermal Reactor Design
Figure S-1
Chap. 9
Startup of a CSTR.
4. Multiple reactions (q reactions and n species)
CD-ROM MATERIAL Learning Resources Summary Motes
Sofved Problems
Living E~ampleProblem
I . Summay Notes 2. Weh links: SACHE Safety web sire www.soche.org. You will need to get the user name and password from your department chair. The kinetics line..CRE) text. examples, and problems are marked K in the product sections: Safety. Health, and the Envimnment (S.H,& E). 3. Solved Problems Example CD9-1 Startup of a CSTR Example CD9-2 Falling Off the Steady State Example CD9-3 Proportional-Integral (PI)Control Living Example Problems I . Example 9-1 Adiuhric Barch Reactor 2. Example 9-2 Safcv ~n CkernicaE PInnrs with Exothermic Reactions 3. h m p S e 9-3 U3e of the A,?SST 4. Example 9-4 Heat Effects in a Semibafch Rcacror 5. Example 9-5 Srartup of a CSTR 6. Exomple 9-15Falling of the Upper Sfeady Stare 7. Example 9-7 Mul~ipleReaclrons in a Semiharch Reactor 8. Example RE9-I Ifiregral Con~ml# f a CSTR 9. Exumplt REP-2 Proportion-it~lexrolConrml of a CSTR 10. Example RE9-3 Lineariz~dSrabiiiy
Chap. 9
CD-ROM Material
Pmfessional Reference Shelf R9.1 The Complete ARSST In this section further details are given to size safety valves
to
prevent run-
away reactions m.
.*.. .
TI.
Refcrcncc Shcff
r*
1M
I#.
E
tm.
Figure W-3.1 Temperature-time trajectory for hydrolysis of acetic anhydride. Controt of a
CSTR
In this section we discuss the use of proponiond (P) and integral (I) control of a CSTR.Examples include I and PI control of an exothemic reaction,
Reactor with control system
Prnprtional integral action
R9.3. Lineoriied Stabili~Theop Zn this section we learn i f a perturbation will decay in an exponential manner ((a\ below) in an oscillatory manner (h). grown exponentialIy (c), grown exponentially with oscillations (d), or just oscillate (e3.
632
Uns!eaby-S!a!e
T ~ B L9C-1 E A.
Tr < 0
Non~so!hermaFReactor Design
Chal
EIGENVALCES OF COUPLED ODES &.&
Det > 0
[Tr2iM)- 1DenM)I > 0
h
(a)
I
~cfercnckShelf
8.
Tr r 0
C.
Tr
Det z 0
(Mj=0
Uns&
Det (MI > 0
R9.4. Approach
10 the Stend!-State Phase-Plane Pbts and Tmjectnrie.~ of Conct rrrlrion versus Temperafur< Were we team if the practical stability is exceeded during startup. Example RE9-4.1 Start Up of a CSTR Example RE9-4.2 Fafling Off the Steady State Example RE9-4.3 Revisit Example RE9-2.
Figure CD9-5 Approach to the steady state.
R9.S. Adiabaric Operution of a Batch Reacfor R9.6. Onrrendy Opemtfon of Plug-Flow Reactors
Chap. !3
Questions and Problems
QUESTIONS AND PROBLEMS W-I,+ Read over the problems at the end of this chapter. Refer to the guidelines given in Problem 4-1. and make up an originat problem that uses the concepts presented in this chapter. To obtain a soIution: (a) Make up your data and reaction. (b) Use a real reaction and real data. Creative Prsblcmr Also, (c) Prepare a list bF safety considerations for designing and operating chemical reactors. See R. M. Felder. Chem. Eng. Educ.. I9 (41, 176 (1985). The August 1985 issue of Chemrcal Engineering Pmgress may be useful for part (c). P9-2, Review the example problems i n this chapter, choose one. and use a software package such as Polymath or MATLAB to carry out a parameter sensitivity analysis, What if.,. (a) Example 9-1. How much time would it take to achieve 90% conversion if the reaction were staned on a very cold day where the initial temperature was 20'F? (Methanol won't freeze at this temperature.) (h) Example 9-2. Explore the ONCB explosion described in Example 9-2. Show that no exploszon would have mcurred if the cooling waq not shut off for the 9.DGkmol charge of ONCB or if the cooling war; shut off for 10 min after 45 rnin of operation for the 3.17-kmol ONCB charge. Show that if the cooling had been shut off for EO min after 12 h of operation, no explosion would have occurred for the 9.04-kmol charge. Develop a set of guidelines as to when the reaction should be quenched should the cooling Bit. Perhaps safe opention could be discussed using a plot of the time after the reaction began at which the cooling failed, t,, versus the length of the cmling failure period, tr, for the different chilrges of ONC8. Parameter d u e s used in this example predict that the reactor will explode at midnight. What parameter values would predict the time the Hall cf Famc reactor wouId explode at the acnral time of 18 rnin after midnight9 Find a set of parameter vaIues that would cause the explosion to occur at exactly 12: 18 A.M. For example. include heat capacities of metal rencror and/or make a new estimate of CIA. Finally, what if a 112-in. rupture disk rated at 800 psi had been installed and did indeed rupture at 800 psi (27OoC)? Would the explosion still have occurred? (Note: The mass flow rate ni varies with the cross-sectional area of the disk. Consequently, for the conditions of the reaction the maximum mass flow rate out of the 112-in. d ~ s kcan be found by comparing it with the mass flow rate of 830 kglrnin of The 2-in. d ~ s k . (c) Example 9-3. What would be the conversion at the o n x t temperature if the hearing rate were reduced by a factor of ID? Increased by a factor of lo?
(d) Example 9-4. What would the X versus t and T versus t trajectories look like if the coolant rate is decreased by a factor of lo? Increased by a factor of SO?
634
Unsteady-State Nonisathermel Reactor Osslgn
(el Example 9-5. Load the Living Exampie Problem for
P9-3,
Chap. 9
Stanup of a CSTR, for an entering temperature of 7VF, an initial reactor temperature of lm,and an initial concentration of propylene oxide oF0.J M. Try other combinations of To, T,,and C,,, and report your results in terms of temperature-time trajectories and temperatureconcentration phase planes. (0 Example 9-6. Load the Living Example Problem for FalIing Offthe Upper Stead-v Ssate. Try varying the entering temperature, To, to between 80 and 68°F and plot the steady-state conversion as a function of T,,, Vary the coolant rate between 10,000 and 400 mollh. Plot conversion and reactor temperature as a function of coolant rate. (g) Example 9-7.What happens d you increase the hear transfer coefficient by a factor of 10 and decrease T, to 280 K? Which trajectories change the most? (h) Example RE9-1. Load the Living Example Problem. Vary the gain, kc, between 0.1 and 500 for the integral controller of the CSTR. Is there a lower value of kc that will cause the reactor to fall to the lower steady state or an upper value to cause it to become unstable? What would happen if Towere to fall to 65°F or 6 0 W (i) Example RE9-2. Load the Living Example Problem. Learn the effects of the parameters kc and 7,. Which combination of parameter values generates the leas1 and greatest oscilIations in temperature? Which values of k,and T, return the reaction to steady state the quickest? Cj) Reactor Safety. Enter the SACHE web site, wfiw.sache.org. [Note you will need to obtain the user name and password for your school from your department chair or SACHE represenrative.] After entering hit the current year (e.g., 2004). Go to product: Safety, Health and the Environment (S,H, & E).The problems are for KINETICS [i.e., CRE). There are some example problems marked K and explanations in each of the above SOH,& E selections. Solutions to the probEems are in a different section of the site. Specifically look at: Loss of Cooling Water ( K - J ) Runowa!. Reocrions (HT-1). Design of Relief Values (D-2). Temperature Cuntrol and Runaway {K-4)and (K-5).and R u n a w u ~and rhe Critical Temperarure Region (K-7). The following is an excerpt from 711e Morning News. Wilmington, Delaware (August 3, 1973): "Investigators sift through the debris from blast in guest for the: cause {that destroyed the new nitrous oxide plant]. A company spokesman said it appears more likely that the [fatal] blast was caused by another gasammonium nltsate-used to produce nitrous oxide." An 83% (wt) ammonium nitrate and 1 7 8 water solution is fed at 200T to the CSTR operated at a temperature of about 5 2 0 T Molten ammon~umnitrate decomposes directly to produce gaseous nitrous oxide and steam. It is beliwed that pressure fluctuations were observed in tht system and as a result the m i t e n ammonium nitrate feed to the reactor may have heen shut off approximately 4 min prior to the explosion. Can you explain the cause of the blast7 If the feed rate to the reactorjust before shutoff war 310 Ib of solution per hour, what was the exact temperature in the reactor ju5t prior to shutdown? UGng the following data. calculate the time it took to explode after the feed was shut off for the reactor. How would you star1 up or shut down and control such a react~on?
Chap. 9
PI-4,
635
Questions and Problems
Assume that at the time the feed to the CSTR stopped, there ms 50Q Ib of ammonivm nitrate in the reactor at a temperature o f 520"E me conversion in the reactor is virtually complete at about 99.998. Additional data for this problem are given in hoblem 8-3. How would your answer change if 100 Ib of solution were in the reactor? 310 Ib? 800 Ib? What if To= 100°F? 5wQp The first-order irreversible reaction is carried out adiabatically in a CSTR into which 100 rnollrnin of pure liquid A i s fed at 400 K. The reacljon goes virtually to completion (i.e., the feed raw into the reactor equals the product of reacrion mtc inside the reactor and &e reactor volume).
CSTR
How
many moles of liquid A are in the CSTR under steadystate condirionsq Plot \he temperature and moles of k in the reactor as n function of time after
the feed to the reactor has been shut off. Additinnall ilefnrmarion:
reaction in Prnhlem P8-5 is to he carried out in a semibatch P9-5, The liq~id~phase reactor. There is 503 mol of A initially in the reactor at 25°C. Species B is fed to the reactor at 50°C and a rate of 10 mollmin. The feed to the reactor is stopped after 500 mol of B has been fed. (a) Plot the temperature and conversion as a function of time when the reactian is carsied out adiaba~ically.Calculare to t = 2 h. (b) Plot the conversion as s function o f time when a heat exchanger (UA = 100 callmin-K) is placed in the reactor and the arnbtent temperature is constant at 50°C.Calculate to I = 3 h. {c) Repeat part (b) for the case where he reverse reaction cannot be neglected.
k = 0.01 (dmVmol -min) at 300 R with E = 10 kcatlmoI Vo = SO dm3, u, = 1 dm3Jrnin, CAn= CB0 = I0 molldm3 Far the reverw reaction: k, = 10 s - I at 300 K with E, = 15 kcallmol
636 P9-6,
Unsteady-State Nonisothermal Reactor Design
Chz
You are operating a batch reactor and the reaction is tirst-order. liquid-ph and exothermic. An inert coolant is added to the reaction mixture to cor the temperature. The temperature is kept constant by varying the flow rat the coolant (see Figure P4-6).
Mhture OF A. 8, and C
(a) Calculate the flow rate of the coolant 2 h after the start of the react] (Anr.: Fc = 3.157 lbls.) (b) It is proposed that rather than feeding a coolant to the reactor, a sol\ be added that can be easily boiled off, even at moderate ternperant The solvent has n heat of vaporization of 1000 Btullb and initially th are 25 Ib mol of A placed in the tank. The initial volume of solvent i reactant is 300 ft3. Determine the solvent evaporation rate as a funct of dme. What is the rate at the end of 2 h? Additional information:
Temperature of reaction: 100°F Value of k at 100°F: 1.2 X s-I Temperature of coolant: 80°F Meat capacity of all components: 0.5 Btu/lb.T Density of at1 components: 50 lblft3 P H L : -25,000 Btutlb mol Initially: Vessel contains only A (no B or C present) CAo:0.5 Ib mollft3 Initial volume: 50 ft3
P9-7, The reaction i s carried out adiabatically in a constant-volume batch reactor, The rate law
Plot the conversion, temperature, and concenmtions of the reacting species a function of time.
Chap. 9
Questions and Pmblems
Additional inJormatinn:
Entering Temperature = 100°C k , (373 K) = 2 x lo-' 5 - 1 kl (373 K) = 3 X lQTF S-I Cka= 0.1 mol/dm3 CBO= 0.125 rnol/drn3 A H L ( 2 9 8 K) = -40,OM) Jlmol A
P9-8,
El = 100 kJlrnol E2 = 150 kJlm01 CpA= 25 Jf m01 .K Cpg = 25 Jfm01.K Cpc = 40 JEmo1.K
The biomass reac%ion Substrate
C"1".
More cells + Product
is carried out in a 25 dm3 batch chernostat with a heat exchanger.
The initial concentration of cells md substrate are 0.1 and 300 g/drn3. respectively. The temperature dependence o f the growth rate follows that given by Aiba et al.. Equation (7-61)5
(7-6 I ) For adiabatic operation and initial temperature of 278 K, ptot 'l; I', r,, &, and Csas a function of time up to 300 hours. Discuss the trends. (b) Repear (a) and increase the initial temperature in 10°C increments up to 330 K and describe what you find. Plot the concentration OF ceIls at 24 hours as a function of inlet temperature. (c) What heat exchange area should be added to maximize the total number of cells at the end of 24 hours? For an initial temperature of 3 10 K and a constant coolan1 temperature of 290 K, what wnuld be the ceI1 concentration after 24 hours? {Ans. Cc = eldrnJ.) (a)
-rs,
AddjIiorral Information:
Yus = 0.8 g celVg substrate
K, = 5.0 ddm3 pIm, = 0.5 h-I (note p = & at 3 10 K and Cs 4 m) Cps = Heat capacity of substrate solution induding all cells
S.Abia. A. E. Humphrey, and N. E Mills. Biochemical Engineering (New York: Academic Press. 1973).
UnsteadyState Nonisotherrnal Reactor Design
Chap. 9
p = density of soIution including ce1Is = 1OOO AHk = -20,000 Jlg e l l s Cpc = Heat capacity of cooling water 74 JIgK
U = 50,000 J k d K h z
P9-%
The fttsr order exothermic liquid-phase reaction is carried out at 85OC in a jacketed 0.2-m3 CSTR.The coolant temperature in the reactor i s 32°F. The heat-transfer coeficient i s 120 WlmZ.K. Determine the critical value of the heat-transfer area below which the reaction will sun away and the reactor wit1 explode [Chem. Eng., 91 (lo), 54 (1984)j. Additional informarion:
Specific reaction rate: k = 1 . 1 min-' at 41PC k = 3.4 min-I at 5VC The heat capacity of the solution is 20 J1g.K. The solution density is 0.90 kg/dm3. The heat of reaction is -2500 Jlg. The feed temperature is 40°C and the feed rate is 90 kglmin. M W of A = 80 glrnol. CAo= 2 M. P9-loc The ARSST adiabatic bomb calorimeter reactor can also be used to determine the reaction orders. The hydrolysis of acetic anhydnde to form acetlc acid was carried out adiabatically
The rate law is postulated to be of the form
The following temperature time data were obtained for
two different critical concentrations of acetic anhydride under adiabatic operation. The hearing rate
was Z0Clmin.
Chap. 9
639
Questions and Problems
Figure P9-10.1 Daa from Undergraduate Laboratory University of Michigan.
la) Assume ACp = 0 and show for complete conversion, X = I, the differenoe between the find temperature, Tf.and the initial temperature, Tb
(b) Show that the concentration of A can be written as
and Cg as
and - r ~as
(c) Show the unsteady energy balance can be written as
(d) Assume first order in A and in B and
that OB= 3
then show
640
Unsteady-State Nonisotharmal Reactor Desrgn
Ct
te) Rearrange Equat~on(P4-10.6) in the form
(f) Ptot the data to obtain the activation energy and the specific reactior
k,. (g) Find the heat of reaction.
Additioml information: Hear cupaciry Chemical
Densit?! ( g h l )
(J/gm0C)
MW
1.0800
1.860
I02
Water
1.0000
4.187
I8
Glass cell (bomb)
0.1474
0.837
Acetic anhydride
Tor4 volume
Water
Hear cupacit (J/molmaC 189.7 75.4 0.84 J/$C
10 ml with
3.638 g
Acetic anhydride
6.871 g (MsCPs = 28.01 2 JPC and $I = 1.CQ4 and msCp,= $ iUs Cps )
P9-llB The elementary irrweraible liquid-phase reaction
is to k carried out in a semibatch reactor in which B i s fed to A. The vo3 of A in the reactor is 10 dm3. the initial concentration of A in the react( 5 rnol/dm3, and the initial temperature in the reactor is 27°C. Species B is at a temperature of 52°C and a concentration of 4 M.It is desired to obtai least 80% conversion of A in as short a time as possible, but at the same I the temperature of the reactor must not rise abDve 130°C. You should tr make approximately 120 mol of C in a 24-hour day allowing for 30 rnin to empty and fill the reactor between each batch. The coolant flow
through the reactor is 2000 motimin. There is a heat exchanger in the reac (a) What volumetric feed rate (drn3/rnin)do you recommend? (b) How would your answer or strategy change if the maximum coolant dropped to 200 moIlmin? To 20 molimin? Additional infontsarion:
AHO, = -55,000 cal/mol A CpA=35cal/mol.K, Cpg =20cal/mol-K,
k = 0.0005
Cpc = 7 S c a l l m o l ~ K
dm6 at 2Y°C with B = 8000 cal/rnol mol2. rnin
caI UA = 2500 with T, = 17°C min.K
Gp(coolant) = 18 callrnoI. K
[Old ex:
Chap. 9
64 1
Questions and Problems
P9-1& Read Section R9.2 on the CD-ROM and then rework Example RE9-I using (al Only a propurtiunal controller.
t ~ntagol~ Contralhr
(b) Only an integral controller. (c) A combined proponional and Integral controller. PP-13, Apply the different types OF controllers to the reactions in Problem P9-I I . P9-14, (a) Rework Example R9-I for the case of a 5°F decrease in rhe outlet temperature when the controlled input variable is the reactant feed rare. (bh Consider a SaF drop in the ambient temperature, T,,, when the controlled variahle is the. inlet temperature, T,. (c) Use each of the controllers (P with kc = 10. I with t, = 1.0 h. D with To = 0.1 h) to keep the reactor temperature at the unstable
to 310 Ib of solution per hour.
la) Plot temperature and
mass of ammonium nitrate in the tank as a function of time when there is no control system on the reactor. Assume that 311 the ammonium nitrate reacts, and show that the mass balance is
There is 500 Ib of A in the CSTR. and the reactor temperature. T, is 516°F at time t = 0. (b) Plot T and /MA as a functia~OF time when a propontonal controller is added to control T, in order to keep the reactor temperature at 5 16°F The controller galn, kc, i s -5 with T,, set at 975 R. (c) Plot T and IW, versus rime when a PI controtler is added with r, = I. (d) Plot T and M A versus time when rwo PI controlrers are added to the reactor: one to control T and a second to conrrol M by manipulating the feed rate
P Member
Hall of Fame
via.
with M,, = 500 lb, kc = 25 h-'. P4-16B The elementary liquid phase reaction A-
TI,
= Ih.
B
is carried out in a CSTR.Pure A is fed at a rate of 200 Ib molJh at 53n R and a concentration of 0.5 Ib molJft3. [M. Shacham. N. Brauner, and M. 8.Cutlip, Ghem. Engr. Edu. 28 (1). 30 (Winter 19941.1 The mass densiry of the solution is constant at 50 lblR3. (a) Plot G(T)and R ( T ) as a function of temperature. (b] What are he steady-state on cent rations and temperatures? [One answer T = 628.645 R, C, = 0.06843 Ib mollft3.J Which ones are stable? What is the extinction tempemure? (c) Apply the unsteady-state mole and energy balances to this system. Consider t b upper steady state. Use the values you obtained in part (b) as your initial values to plot C, and T versus time up to 6 hours and then to plot C, versi~sT. What did you find? Do you want to change any of your answers to part (b)?
642
Unsteady-State Nonisothermal Reactor Desigrl
Chap. 9
(d) Expand your results for part (c} by varying To and To. [Hint: Try To= 5W0R]. Describe what you find. (e) What are the parameters in pan (dl. for the other steady states? Plot T and C, as a function of time using the steady-state values as the initial conditions at the lower steady state by value of To = 550 R and To = 560 R. Start at the lower steady state (T = 547.1, CA= 0.425) and make a CAP pbase-plane plot for the base case in part (a) T,= 530.Now increase To ro 550 and then to 560, and dewrik what you find. Vary To. Explore this problem. Write a paragraph describing your results and wha~ you learned from this problem. (g) Read Section 9.3 on the CD-ROM. Carry out a linearized stability analysis. What were your values of A. B, C, s, J. L, and M? Find the roots for the upper, lower, and middle steady states you found in part (a). (h) Normalize x and 1 by the stesdy-state values, xl = xJCA, and ?*1 = !IT,, and plot sl and !>I as a function of time and also xl as a function of pl Plot xl and j 1 as a function of time for each of the three steady states. [Hinr:First try ~nitialvalues of x and p of 0.02 and 2, sespect~vely.] (f)
Additional information:
v = 400 ft3/hr CAo= 0.50 I b mol/ft3 V = 48 ft3 A = 1.416 X lo'* hh-' E = 30,000 BTU / Ib mol R = 1.987 BTU/lb rnol "R U = 150 BTU/h-ft2-"R
A = 250 ft2(Heat Exchanger)
T, = 530°R T, = 530% AH,, = -30,000 Btullb mol C, = 0.75 BtuJlb "R p = 50 Ibm/ft3
P9-17, The reaction in Example 8-6 are to I x carried out in a 10-dm3 batch reactor. Plot the temperature and the concentrations of A, B, and C as a function of time for the following caqes: (a) Adiabatrc operarlon. (b) Values of C'A of 10.000, 40.000, and 100,000 J/rnin.K. (c) Use UA = 40,000 J1min.K and different initial reactor temperatures. P9-18, The reaclion in Problem P8-34, is to be carried out in a semlhatch reactor. (a) Haw would you carry out this reaction (i.e.. To,v,. T;)? The molar concentrations of pure A and pure B are 5 and 4 moJldm3 respctively. Plot concentrations, temperatures, and the overall selectivity as a function of time for the conditions you chose, (b) Vary the reaction orders for each reaction and describe what you find. (c) Vafy the heats of reaction and describe what you find. P9-19, The following temperature-time data was taken on the ARSST. Determine the heat of reactton and the activation energy. Heat capacity of the solution Cp = 18 caHmollK. P9-20, The forma~ianof high-molecular-weight olefins, for example.
643
Questions and ProMems
Chap. 9
is c a m 4 out in the CSTR shown in Figure P9-20(a) where {=) denotes the molecule is an olefin. The reaction is exothermic, and heat is removed from the reactor by heat exchange with a cooling watet stream as shown in Figure P9-2Wa). hubl~shooting
Feed
Pure
t, + 10
'1
Xme (hours)
Fimre P9-20(b)
Figure P9-M(a)
Initially, there is a tempemtuE mtroIIer that regulates the reactor temperature. At r = I,, the controller 1s set into manual, and a step increase in the water flew rate through the heat exchanger was made. The temperature response is shown id Figure m-2qb).What are the parameters to match this trend? Explain the observed temperature-time trajectory.To make your analysis simpler, you may assume that the only reaction taking place is
C,(=)+ C,t=)
C,(=)
with
Also, you can assume that the exchanger is inside the reactor {to avoid recycle stream calculations). See P9-20 CD-ROM Complete Data S& Additional Homework Problems
CDP9-AB CDP9-Bc
The production of propylene gIycoI discussed in Examples 8-4, 9-4. 9-5,9-6, and 9-7 is camed out in a semihatch reactor. Reconsider Problem P9-14 when a PI controller is added to the coolant stream.
CDPS-CB
CwP9-DB
Calculate time to achieve 90% of equilibrium in a batch reactor. [3rd Ed. P9-8,] Stanup of a CSTR. [3rd Ed, P9-lo,]
Unsteady-State Nonisothermal Reactor Design
C
SUPPLEMENTARY READING I . A number of solved problems for batch and semibatch reactors can be founl WALAS,5. M.. Chemical Reacrion Engineering Handbook. Amsterdam: don and Breach, 1995, pp. 386392, 402, 46W62. and 469.
2. Basic controt textbooks SEBORG, D. E.,T.F. EDGAR.and D.A. MELLICHAMP, Process Dynnmic. Control, 2nd ed. New York: Wiley. 2004. OCGNNAIKE.B. A., and W.H.RAY.Pmcess Dynamics, Modcling rtnd Cn Oxford: Oxford University Press. 1994.
3. A nice historical perspective of prmess control is given in E D G ~ RT. , E, "From the Classical to the Posrmodem Era," C k ~ n ?Eng. . E 31, 12 (1997).
Links
Links
1. The SACHE web site has a great discussion on reactor safety with exan (www.sache.org).You will need n user name and password; both can be obt; from your department chair. Hit 2003 Tab.Go to K Problems. 2. The reactor lab developed by Professor Herz and discussed in Chapters 4 n could also be used here: www.macturlnb.net and also on the CD-ROM.
Catalysis and Catalytic Reactors
10
It isn't that they can't see the solution. It is that they can't see the problem. G , K. Chesterton
Overview. The objectives of this chapter are to deveIop an understanding of
catalysts, reaction mechanisms, and catalytic reactor desip. Specifically, after reading this chapter one should be abIe to (1) define a catalyst and describe its propwties, (2) describe the steps in a catalytic reaction and in chemical vapor deposition (CVD) and apply the concept of a rate-limiting step to derive a rate law, (3) develop a rate law and detwmine the rate-law parameters from a set of gas-solid reaction rate data, (4) describe the different types of catalyst deactivation. determine an equation for cataIytic activity from concentration-time data, define temperature-time trajectories to maintain a constant reaction fate, and (5) calculate the conversion or c a w Zyst weight for packed (fixed) M s . moving beds, well-mixed (CSTR) and straight-through (STTR] ff uid-bed react,ors for t ring and n iag catalysts. The various sections of this ch, $ 1 ~corn each of these objectives.
10.1 Catalysts Catalysts have been used by humankind for over 2000 years.' The first observed uses of catalysts were in the making af wine, cheese, and bread. It was found that it was always necessary to add small amounts of the previous batcb to make the current batch. However, it wasn't until 1835 that Benelius
' S.T- Oyama and G.A. Somojai, J. Chem. Educ,, 65,765($986).
646
Catalysis and Catalytic Reactors
Chap. 10
began to tie together observations of earlier chemists by suggesting that small
amounts of a foreign source couId greatly affect the course of chemical reactions. This mysterious force attributed to the substance was called catalytic. In 1894, OstwaId expanded Berzelius' explanation by stating that catalysts were substances that accelerate the rate of chemical reactions without being consumed. In over 150 years since Berzelius' wark, catalysts have come to play a major economic role in the world market. In the United States alone, sales of process catalysts in 2007 will be over $3.5 billion, the major uses being in petroleum refining and in chernica1 productjoti. 10.1.I Definitions
Summary hlotes
A cafalyst is a substance that affects the rate of a reaction but emerges from the process unchanged. A catalyst usually changes a reaction rate by pramoting a different molecu!ar path ("mechanism") for the reaction. For example, gaseous hydrogen and oxygen are virtuaIly inerl at room temperature, bur react rapidly when exposed to platinum. The reaction coordinate shown in Figure 10-1 is a measure of the progress abng the reaction path as Hz and 0: approach each other and pass over the activation energy bamer to form H20. A more exact comparison of the pathways, similar to the margin figure, is given in the Summary Notes for Chapter 10. Catal~eisis the occurrence, study, and use of catalysts and catalytic processes. Commercial: chemical catalysts are immensely important. Approximately one third of the material gross national product of the United States involves a catalytic process someuphere between raw material and finished p r ~ d u c t The . ~ development and use of catalysts is a major part of the constant search for new ways of increasing product yield and selectivity from chemical reactions. Because a catalyst makes ~t possible to obtain an end product by a different pathway with a Iower energy barrier. it can affect both the yield and the selectivity. Gas
HZ+ O2
'.
\,
H20
f
, ' .-...-catalyst
p z
r
Fast
Figure 10-1 Different reaction paths.
Normal1 y when we talk about a catalyst, we mean one that speeds up a reaction, although strictly speaking, a catalyst can either accelerate or slow?[fie
--
' V. Hacnsel and R. L. Burwell. Jr.. Sci. Am.. 22.5(10). 46.
Catalysts can acceImate the reactlon rate but cannot change the equilibrium.
f m a t i o n of a particular product species. A catalysr changes only the mle of o reaction; it does nor aflecr the equilibrium. Homogeneous catalysis concerns processes in which a catalyst is in solution with at least one of the reactants. An example of homogeneous catalysis is the industrial 0 x 0 process for manufacturing normal isobutylaldehyde. It has propylene, carbon monoxide, and hydrogen as the reactants and a liquid-phase cobalt complex as the catalyst.
Reactions carried out in supercritical fluids have been found to accelerate the reactino rate greatly.' By manipulating the properties of the solvent in which the reaction is taking place, interphase mass transfer limitations can be eliminated. Application of transition states theory discussed in the Professional Reference Shelfon the CD-ROMof Chapter 3 has proven useful in the analysis of these reactions. A heterogeneous catalytic prncpss involves more than one phase: usually the catalyst is a solid and the reactants and products are in liquid or gaseouy form. Much of the benzene produced in this country today is manufactured from the dehydrogenation of cyclohexane (obtained from the distillation of crude petroleum) using platinum-on-alumina as the catalyst:
Cyclohexane
Examples nf hetempeneon4 cata?ytic reactions
Benzene Hydrogen
Sometimes the reacting mixture is in both the liquid and gaseous forms, as in the hydrodesuEfurization of heavy petroleum fractions. Of these two types of catalysis, heterogeneous catalysis is the more common type. The simple and complete separation of the fluid product mixture from the solid catalyst makes heterogeneous calalysis economically attractive, especially because many catalysts are quire vaIuable and their reuse is demanded. Only heterogeneous catalysts will be considered in this chapter. A heterogeneous catalytic reaction occurs at or very near the fluid-solid interface. The principles that govern heterogeneous catalytic reactions can be applied ro both caralytic and noncatalytic fluid-solid reactions. The two other types of heterogeneous reactions invoIve gas-Iiquid and gas-liquid-solid systems. Reactions between gases and Iiquids are usually mass-transfer limited. -
-'
-
Savage, S. Gopalan. T. I. Mizan, C.J. Manino, and E. E. Brmk, AlChE 1.. 41 (7) 1723 (1995). B. Subramanian and M. A. McHugh, IEC Pn7ce.r~Desrgn and
P. E.
Dr~,elopmrrrt,25, I t 1486).
648
Catalysis and Catalytic Reactors
Chap
10.1.2 Catalyst Properties
Because a catalytic reactinn occurs at the fiuid-solid interface, a large inter cia1 area is almost always essential in attaining a significant reactian rate. Ten pram$ of this many catalysts, this area is provided by an inner porous structure {i.e., i cntalyv passes< solid contains many tine pores, and the surface of these pores suppties the a nlure surface area than a U.S. needed for the high rate of reaction), The area possessed by some porous ma fontball held rials is surprisingly large. A typical silica-alumina cracking catalyst has a pc volume of 0.6 crn3/g and an average pore radius of 4 nm. The correspondi surface area is 300 m2Jg. A catalyst that has a Large area resulting from pores is called a porn c a t ~ l y s t Examples . of these include the Raney nickel used in the hydrogenari~ of vegetable and animal oils, the platinum-on-alumina used in the reforming petroleum naphthas to obtain higher octane ratings, and the promoted in used in ammonia synthesis. Sometimes pores are so small that they will adn small molecules but prevent large ones from entering. Materials with this tyl Catalyst types: of pore are called molecular sieves, and they may be derived from natural su Porotls stances such as certain clays and zeolires, or be totally synthetic, such as son Molecular sievcs Monolithic crystalline aluminosili~ates(see Figure 10-2). These sieves can form the bas
- Supported Unsupponed
Typical zeolite catalvst
FaujasiteType Zeolite
0 12 Ring
(a)
Ibl
(a) Framework structures and (b) p r e cross sections of we types of zeolires. (a) Faujasite-type zeolite has a three-dimensional channel system wrth pores at least 7.4 A in diamerer. A pore i s formed by 12 oxygen atoms in a ring. (b) Schematic of reaction CH,and C6HICHy (Note the slze of the pore mouth and the interior of the zeolite sre nor to scale.) [(a) from N.Y.Chen and T. F. Degnan. Chem. Eng. Pmg., 8#(2), 33 (1988). Reproduced by permission of the Amencan Institute of Chemical Engineers. Copyright O I988 AIChE. All rights reserved.]
Figum 10-2
sec. 10.1
High setectivit~to para xylene
Catalysts
649
for quite selective catalyrts; the pores can control the residence time of various molecules near the catalytically active surfact to a degree that essentially allows only the desired molecules to react. One example of the high selectivity of zeolite cataIysts is the formation of xylene from toluene and methane shown in Figure 10-2(6).4 Here benzene and toluene enter through the pore of the zeolite and react on the interior surface to form a mixture of ortho. meta. and para xylenes. However, the size of the pore mouth is such that only para-xylene can exit thr~ughthe pore mouth as meta and onho xylene with their methyl group on the side cannot fit rhrough the pore mouth. There are interior sites that can isomerize ortho and meta to para-xylene. Hence we have a very high selectivity to form para-xylene. Another example OF zeolite specificity is controlled placement of the reacting molecules. After the motecules are inside the zeolite, the configuration of the reacting molecules may be abIe to be controlled by pPacement of the catalyst atoms at specific sites in the zeolite. This placement would facilitate cydization reactions, such as orienting ethane molecutes in a ring on the surface of the catalyst so that they Form
benzene:
Monolithic catalysts k~ either porous
or nonporous.
Not all catalysts need the extended surface provided by a porous structure, however. Some are sufftciently active so that the effort required to create a porous catalyst would be wasted. For such situations one type of catalyst is the monolithic catalyst. Monolithic catalysts are ~omnllyencountered i n processes where pressure drop and heat removal are major considerations. Typical examples include the platinum gauze reactor used in the ammonia oxidation portion of nitric acid manufacture and catalytic converters used to oxidize pollutants in automobile exhaust. They can be porous (honeycomb) or nonporous (wire gauze). A photograph of an automotive catalytic converter is shown in PRS Figure RIE.1-2. Platinum is a primary catalytic rnateriat in the monolith. In some cases a catalyst consists of minute particles of an active material dispersed over a less active substance called a slipport. The active materia[ is frequently a pure metal or metal alloy. Such catalysts are called supported cntalysts, as distinguished from unsupported catalysts. Catalysts can also have small amounts of active ingredients added called pmrnorers, which increase their activity. ExampIes of supported catalysts are the packed-bed catalytic converter automobile, the platinum-on-alumina catalyst used in petroleum reforming, and the vanadium pentoxide on silica used to oxidize sulfur dioxide in manufacturing suIfuric acid. On the other hand, the platinum gauze for ammonia oxidation, the promoted iron for ammonia synthesis, and the sitica-alumina dehydrogenation catalyst used in butadiene manufacture typify unsuppolted catalysts.
R. J. Masel, Chemical Kindics and Caralvsis (New York: Wiley Interscience, 2001). p. 741.
650
Deactivation by:
-- Aging Poisoning . Coking
Chemisorption on active sites rs what catalyzes the
reaction
Catalysis and Catalytic Readors
Chap 10
Most catalysts do not maintain their activities at the same levels for indefinite periods. They are subject to deactivation, which refers to the decline in a catalyst's activity as time progresses. Catalyst deactivation may be caused by (1) aging phenomenon, such as a gradual change in surface crystal structure; (2) by poisoning, which is the irreversible deposition of a substance on the active site; or (3) by fouling or coking, which is the deposit of carbonaceous or other material on the entire surface. Deactivation may occur very fast, as in the catalytic cracking of petroleum naphthas, where coking an the catalyst requires that the catalyst be removed after only a couple of minutes in the reaction zone. In other processes poisoning might be very slow, as in automotive exhausr catalysts, which gradually accumulate minute amounts of lead even if unleaded gasoline is used because of residual lead in the gas station storage tanks. For the moment, let us focus our attention on gas-phase reactions catalyzed by solid surfaces. For a catalytic reaction to occur, at least one and frequently all of the reactants must become attached to the surface. This attachment is known as adsorprion and takes place by two different processes: physical adsorption and chemisorption. Physical adsorption is similar to condensation. The process is exothermic, and the heat of adsorption is relatively small, being on the order of 1 to 15 kcallrnol. The forces of attraction between the gas molecules and the solid surface are weak. These van der Waals forces consist of interaction between permanent dipoles, between a permanent dipole and an induced dipole, andlor between neutral atoms and molecules. The amount of gas physically adsorbed decreases rapidly with increasing temperature, and above its criticai temperature only very small amounts of a substance are physically adsorbed. The type of adsorption that affects the rate of a chemical reaction is chernisorption. Here, the adsorbed atoms or molecules are held to the surface by valence forces of the same type as those thar occur between bonded atoms in rn01ecuIes. As a result the electronic structure of the chemisorbed molecule is perturbed significanrly, causing it to be extremely reactive. Interaction with the catalyst causes bonds of the adsorW reactant to be stretched. making them easier to break. Figure 10-3 shows the bonding from the adsorption of ethylene on a platinum surface to form chemisorbed ethylidyne. Like physical adsorption. chemisorption is an exothermic process, but the heats of adsorption are generally of the same magnitude as the heat of a chemical reaction ( i t . . 40 to 400 kTlmol). If a catalytic reaction involves chemisorption, it must be carried out within the temperature r a n g whek chemisorpfion of the reactants is appreciable+ In a landmark contribution to catalylic theory, Taylofl suggested that a reaction is not catalyzed over the entire solid surface but only at ceriain actil'f sires or centers. He visualized these sites as unsaturated atoms in the solids that resulted from surface irregulatities, dislocations, edges of cqstals. and cracks along grain boundaries. Other investigalors have taken exception to this
' H. S . Taylor, Proc. R. Sor. k~trdon,A108.
105 (19281.
Sec. 10.1
Cetalysts
651
Figurc 10-3 Ethylidyne as chem~snrbedon platinum. (Adapted from G.A. Somojai. Intmduction to Sudoce Chemr.rtr?.and Catalysis, U'iley. New Ymk, 1994.)
definition, pointing out that other properties of the solid surface are also important. The active sites can also be thought of as places where highly reactive intermediates (i.e., chemisorbed species) are stabilized long enough to react. This stabilization of a reactive intermediate is key in the design of any catalyst. However, for our purposes we wilI define an active site as a poinr on rhe caralysst surface that can form strong chemical bonds wirh an ad.~nrbedorom or
nrolecule. One parameter used to quantify the activity of a catalyst is the rurnm!Pr frequency (TOR. f. It is the number of molecules reacting per active site per second at the conditions of the experiment. W e n a metal catalyst such as platinum is deposired on a support, the metal atoms are considered active sites. The dispersion, D,of the catalyst is the fraction of the metal atoms depoqited that are on the surface, Example I U-1
Turnover Frequency in Fisther-Trnpsch Synrhesig
The Fischer-Tropsch synthesis was studied using a commercial 0.5 wt 4r Ru on y-A120~.h The catalyst d~speisionpercentage uf atoms exposed. determined from qR. S. Dixit and L. L. Tavlaiides, Ind. Eng. Chem. Pmcvss DPS.Dei:, 22, 1 (1983).
652
Catatysis and Catalytic Reactors
Char
hydrogen chemisorption. w35 round to br 498. At a pressure of 988 kPa and a t ptrature of 475 K. a turnover frequency. I;., , of 0.044 s-1 wa5 reported for meth What i s the rate of formation of methane, r i , in rnoIls.g of catalyst (metal pluc ,
port)?
- 0.044 molecules
I mol CH, 6 . 0 2 10" ~ molecules
-
(surface atom Ru) s 0.49 surface atoms total atoms RII
6 . 0 2 t~0'' atornL Ru g atom (mul) Ru
g atoms Ru 0.005 g Ru 101.1gRu
I
= 1.07
x !O-%olls.g
gtotal catalyst
(EIO-
That is, 1.07 x 1W moles (6.1 x 10" molecules) o f methane are jumping one gram o f catalyst every second.
Figure 10-4 shows the range of turnover frequencies (rnolecules/site as a function of temperature and type of reaction. One: notes that the turno frequency in Example 10-1 is in the same range as the frequencies shown the box for hydrogenation camlysts. 10.1.3 Classification of Catalysts
While platinum can be used for some of the reactions shown in Figure 1C we shall also discuss several other classes of reactions and the catalystsn7
Alkylation and Dealkylation Reactions. Alkyletian is the addition of alkyl group to an organic compound. This type of reaction is commonly c ried out in the presence of the Friedel-Crafts catalysts. AlC13 along wit1 trace of HCI. One such reaction is
A similar alkylation is the formation of ethyt benzene from benzene and ethyle
' J. W. Sinfelt. Ind. Eng. Chem., 62(2),23 (2970); 62(10).66 (1970). Also, W. B.Inr
in P. H. Emmertt. Eds., Catalysis, Vol. 2 (New York: Reinhold. 1955). p. I, : R. Masel, Kincrics (New York: Wiley, 2003).
Sec. 10.1
Catalysts
Hydrogenation
7 /-
Dehydrogenation
200
400 Temperature
600
8 00
(Kl
Figure 10-4 Range o f turnover Frequencies as a function for different reactions and temperatures. (Adapted from 6.A Somorjai. Intmduc!ion lo Srrrfacc C l ~ e t n i and ~ I ~Catrrlysrs, Wiley, New York. 1994.)
The cracking of petrochemical products is probably the most common deaikylation reaction. Silica-alumina, silica-magnesia, and a clay (montmorillonite) are common deatkyIation catalysts,
Isomesization Reactions. In petrochemical production, the conversion of normal hydrocarbon chains to branched chains is important, since !he latter ha.$ a higher gasoIine octane number. When n-pentane is isomerized to i-pentane, the octane number increases from 62 to 90!! Acid-promoted AIIOBis a catalyst used in such isomerization reactions. AIthough this and other acid catalysts are used in isomerization reactions, it has been found that the conversion of normal paraffins to isoparaffins is easiest when both acid sites and hydrogenation sites are present, such as in the catalyst Pt on A120,. Hydrogenation and Dehydrogenation Reactions. The bonding strength between hydrogen and metal surfaces increases with an increase in vacant d-orbltals. Maximum catalytic activity will not be realized if the bonding is too strong and the products are not readily desorbed from the surface. Consequently, this maxjrnum in catalytic activity occurs when there is approximately one vacant d-orbital per atom. The most active metals for reactions involving hydrogen are generally Co, Ni, Rh, Ru, Os, Pd, Ir, and Pt. On the other hand, V, Cr, Nb, Mo, Ta, and W, each of which has a large number of vacant d-orbitltls, are relatively inactive as a result of the strong adsorption for the reactants or the products or both. However, the oxides of Mo (MoOz) and Cr (CrIQ2) are quite active far most reactions involving hydrogen. Dehydrogenation reactions are favored at high temperatures (at least 200"C), and hydrogenation reactions are favored at
654
Catalys~sand Catalylrc Reactors
Chap. 10
lower temperatures. Industrial butadiene, which has been used to produce synthetic rubber, can be obtained by the dehydrogenation of buteoes:
"*'"
CH3CH=CHCH3 > CCW2=CHCW=CH2+H2 (possible catalysts: calcium nickeI phosphate, Cr2Q3,etc.) The same catalysts could also be used in the dehydrogenation of ethyl benzene to form styrene:
An example of cyclization, which may be considered to be a special type of dehydrogenation, is the formation of cydohexane from ra-hexane. Oxidation Reactions. The transition group elements (group VIII) and subgroup I are used extensively in oxidation reactions. Ag, Cu, Pt, Fe, NI, and their oxides are generally g d oxidation catalysts. In addition, V20s and M n 4 are frequently used for oxidation reactions. A few of rhe principal types of catalytic oxidation reactions are: 1. Oxygen addition: 2C,H,c O2 ZSO,+O,
2co+o,
)
v:o3
2C2H40 2SO,
2C0,
2. Oxygenolysis of carbon-hydrogen bonds:
2C,H,OH +O, 2CH,CHO+ZH,O 2CN,OH + 0, A" 2HCW0 + 2H,O 3. Oxygenation of nitrogen-hydrogen bonds: 4. Complete combustion:
Platinum and nickel can be used for both oxidation reactions and hydrogenation reactions.
Hydration and Dehydration Reactions. Hydration and dehydration catalysts have a strong affinity for water. One such catalyst is AIZO3,which is used in the dehydration of alcohols to form olefins. In addition to alumina, silica-alumina gels, clays, phosphoric acid, and phosphoric acid salts on inert carriers have also been used for hydration4ehydration reactions. An example of an industrial catalytic hydration reaction is the synthesis of ethanol from ethylene:
CH2=CH2+ H 2 0
PCH3CH20H
Sec. 10.2
655
Steps In a Catalytic Reaction
HaIogenation and DehaIogenation Reactions. Usual1y, reactions of this type take place readily without utilizing catalysts. However, when selectivity of the desired product is low or it is necessary to run the reaction at a lower temperature, the use of a catalyst is desirable. Supported copper and silver halides can be used for the halogenation of hydrocarbons. Hydrochlorination reactions can be carried out with mercury copper or zinc halides. Summary. Table 10-1 gives a summary of the representative reactions and catalysts discussed previously. TABLE 10- 1.
TYFu OF REACTIONSAND B P R E S ~ A T ~ VCATALYSTS E
Rcacrion
C~fpI~srs
1. Walogenation-dehalogen~tion 2. Hydration-dehydration 3. Alkylation4ealkylat1on 4. Hydrogenation-dchydragenation 5. Oxidation 6. Isornerization
CUCI,, AgCI. Pd AI:O~,M ~ O AICI,. Pd, Zeolites Co. Pt, Cr,D3. NI Cu.Ag, Ni, V,OI
AIC13. Pt/A1201. Zeolites
If, for example, we were to form styrene from an equimolar mixture of ethylene and benzene. we could carry out an alkylation reaction to form ethyl benzene, which is then dehydrogenated to form styrene. We need both an alkylation catalyst and a dehydrogenation catalyst:
10.2 Steps in a Catalytic Reaction A photograph of different types and sizes of catalyst is shown in Figure 10-5fa). A schematic diagram of a tubular reactor packed with catalytic pellets is shown i n Figure 10-5(b). The overalI process by which heterogeneous catalytic reactions proceed can be broken down into the sequence of individual steps shown in Table 10-2 and pictured in Figure 10-6 for an isornerization reaction.
(a) Figure 10-5 (a] Different shapes and sizes or catalyst. (Counecy of the Engelhard Corporat~nn. I
Catalysis and C ~ f a W i cR e ~ r ! c r r
Pecked catalyst bed
Cataiyst pellet
Chi
Catalyst pellet surface
Pores
(bl Figure 10.5
( h ) Catalytic pecked-bed reactor-schemaric.
Each step in Table 10-2 is shown schematically in J3gure 10-6.
1. Mass transfer (difhqian) o f the reactantls) (e.g..PpecIes A) fmm the bulk fluid to the exte surface of the catalyst pellet 2. O~ffusionof the reactant from the pore mouth through the catalyst pores to the irnmedia vicinity of the internal catalytic surface 3. Adsorption nT reactant A onto the catalyst surface 4. Reaction on the surface of the catalyst (e.g., A d B) 5. Deqorption of the product5 (e.g., 8 ) from the surface 6. Dtffusron of the pmducts from the interior o f she pellet to the pore mouth at the external surface
7. Mass trancfer of the praducts from the external pellet surface to the bvlk Ruid
Figure 10.6
Steps in r heterogeneous catalytic reacrion.
Sec. 10.2
A reaction takes place on the surface, but the species involved i n rhe reaction must get ro and from the surface
In this chapter we fucus on: 3 Adsorption
4. Surface reaction 5 . Desorption
Steps ~na Catalytic Reaction
657
The overall rate of reaction is equal ra the sate of the slowest step in the mechanism. When the diffusion sreps ( 1 , 2 , 6 ,and 7 in Table 10-2) are very fast compared with the reaction sreps (3, 4. and S ) , the concentrations in the immediate vicinity of the active sites are indistinguishable from those in the bulk fluid. In this situation, the mnspon or diffusion steps do not affect the overall rate of the reaction. In other situations, if the reaction steps are very fast compared with the diffusion steps, mass transport does affect the reaction rate. In systems where diffusion from the bulk gas or liquid to the catalyst surface or to the mouths of catalyst pores affects the rate, changing the flow conditions past the catafyst should change the overall reaction rate. In porous catalysrs, on the other hand, diffusion within the catalyst pores may limit the rate of reaction. Under these circumstances, the overall rate will be unaffected by external flow conditions even though diffusion affects the overall reaction rate. There are many variations of the situation described in Table 10-2. Sometimes, of course, two reactants are necessary for a reaction to occur, and both of these may undergo the steps listed earlier. Other reactions between two substances may have only one of them adsorbed. With this introduction, we are ready to treat individually the steps involved in catalytic reactions, In this chapter only the steps of adsorption, surface reaction, and desorption will be considered [i.e.. it is assumed that the diffusion steps (1, 2, 6, and 7) are very fast, such that the overall reaction rate is not affected by mass transfer in any fashion]. Funher treatment of the effects involving diffusion Iimitations is provided in Chapters 1I and 12.
Where Are W e Heading? As we saw in Chapter 5, one of the tasks of a chemical reaction engineer is to analyze rate data and to deveIop a rate law that can be used in reactor design. Rate laws in heterogeneous catalysis seldom follow power law models and hence are inherently more difficult to formulate from the data. To develop an in-depth undemanding and insight as to how the rate laws are formed from heterogeneous catalytic data, we are going to proceed in somewhat of a reverse manner than what is normally done in industry when one is asked to develop a rate Iaw. That is, we will postulate catalytic mechanisms and then derive rate laws for the various mechanisms. The mechanism will typically have an adsorption step, a surface reaction step, and a desorption step, one of which is usually rate-limiting. Suggesting mechanisms and rate-limiting steps is not the first thing we normally do when presented with data. I-lowever, by deriving equations for different mechanisms, we will observe the various forms of the rare law one can have in heterogeneous catalysis. Knowing the different forms that catalytic rate equations can take, it will be easier to view the trends in the data and deduce the appropriate rate law. This deduction is usually what is done first in industry before a mechanism is proposed. Knowing the form of the rate law, one can then numerically evaluate the rate law parameters and postulate a reaction mechanism and rate-limiting step that is consistent with the rate data. Finally, we use the rate law to design catalytic reactors. This procedure is shown in Figure 10-7. The dashed lines represent feedback to obtain new data in specific regions le.g., concentrations,
Catalvsis and Catalytic Reactors
Chap. 10
Obtain deta horn labralory resclm
,'
I
\
An algorithm
Figure 10-7 Collectin,g infnrmarion for catalytic reactor design.
temperature) to evaluate the rate law parameters more precisely or to differentiate between reaction mechanisms. We will discuss each of the steps shown in Figure 10-6 and Table 10-2. As mentioned earlier. this chapter focuses on Sreps 3, 4, and 5 (the adsorption. surface reaction and desorption steps) by assuming Steps 1, 2, 6, and 7 are very rapid. Consequenrly to understand when this assumption is valid, we shall give a quick overview of Steps 1, 2, 6. and 7. Steps 1 and 2 involve diffusion of the reactants to and within the catalyst pellet. While these steps are covered in detail in Chapters 11 and 12, it is worthwhile to give a brief description of these two mass transfer steps to better understand the entire sequence of steps. 10.2.1 Step IOverview: Diffusion from tha Bulk to the External
Transport
For the moment let's assume the transport of A from the bulk fluid to the external surface of the catalyst is the slowest step in the sequence. We Pump all the resistance to transfer from the bulk fluid to the surface in the boundary layer surrounding the pellet. In this step the reactant A at a bulk concentration CAh must travel through the boundary layer of thickness 6 to the external surface of the pellet where the concentration is CAsas shown in Figure 10-8. The rate of transfer {and hence rate of reaction. -[:) for this slowest step is Rate A kc (C,, - C,,) where the mass transfer coefficient, k , is a function of the hydrodynamic conditions. namely fluid velocity, U,and the particle diameter, D,,, As we will see (Chapter I I I the mass transfer coefficsent is inversely proportional to the boundary layer thickness. 6,
Sec. 10.2
659
Steps in a Catalytic Reaction
Figure 10-8 Diffusion through the external boundary layer. [Also see Figure Ell-1.1.1
and direct1y proportional to the coefficient diffusion (i-e., the diffusivity DAB). At low velocities of fluid flow over the pellet, the boundary layer across which A and B must diffuse is thick, and it takes a long time for A to travel to the surface, resulting in a small mass transfer coefficjeat kc. As a result, mass transfer across the boundary layer is slow and limits the rate of the overalE reaction. As the velocity across the pellet is Increased, the boundary layer becomes smaller and the mass transfer rate is increased. At very high velocities the boundary layer is so smalI it no longer offers any resisfance to the diffusion across the boundary layer. As a result, externat mass transfer na longer limits the rate of reaction. This external resistance also decreases as the particle size is decreased. As the fluid veIocity increases andlor the particle diameter decreases, the mass transfer coefficient increases until a plateau is reached, as shown in Figure 10-9. On this plateau, CAb= CAs,and one of the other steps in the sequence is the slowest step and Iirnits the overall rate. Further details on external mass transfer are discussed in Chapter 1I .
Overall Rate
longer the slowest step
External diffusion is ths slowest step
Figure 10-9 Effect nf overall raze on particle size and Ruid velocity.
660
Catalysis and Catalytic Reactors
Char
10.2.2 Step 2 Overview: Internal Diffusion Now consider that we are operating at a fluid velocity where external diffuz is no longer the rate-limiting step and that internal diffusion is the slowest sl In Step 2 the reactant A diffuses from the external surface at a concentral C,, into the pellet interior where the concentration is C,. As A diffuses E the interior of the pellet, it reacts with catalyst deposited on the sides of
pore walls. For large pellets, it takes a long time for the reactant A to diffuse into interior compared to the time it takes for the reaction to occur on the inte~ pore surface. Under these circumstances, the reactant is only consumed n the exterior surface of the pellet and the calalyst near the center of the pelle wasted catalyst. On the other hand, for very ma11 pellets it takes very li time lo diffuse into and out of she pellet inrerior and, as a result, internal ( fusion no Longer Iirnits the rate of reaction. The rare of reaction can expressed as Rate = k, CA, where CAsis the concentration at the external surface and k, is an overall c constant and is a function of particle size. The overall rate constant, increases as the pelIet diameter decreases. In Chapter 12 we show tl Figure 12-5 can be combined with Equation (12-34) to arrive at the plot of as a function of Dp shown in Figure 10-lO(b). We see that at small particle sizes internal diffusion is no longer the slc step and hat the surface reaction sequence of adsorption, surface reaction. a desorption (Steps 3, 4, and 5) limit the overall rate of reaction. Consider nc one more point about internal diffusion and surface reaction. These steps through 6) are not at all affected by flow conditions external to the pellet. In the rnateriai that follows, we are going to choose our pelIet size a1 external fluid velocity such that neither external diffusion nor internal diffusid
I
Surface reaction sequence
internal diffusion
k
- -1 OD
Figre 10-10 Effect of particle size on overall reaction rate constant. (a) Branching of a single pore with deposited metal; (b) Decrease in rate constant with increasing particle diameter..
Sec. 10.2
Steps
in a Catalytic Reaction
661
is limiting. Inqtead. we assume that either Step 3 adsorption, Step 4 surface reaction, or Step 5 desorption, or a combination of these steps. limits the overall rate of reaction.
10.2.3 Adsorption Isotherms
Because chemisorption is usually a necessary part of a catalytic process, we shall discuss i t before tnaring catalytic reaction rates. The letter S will represent an active site; alone it will denote a vacant site, with no atom, molecule, as complex adsorbed on il. The combination of S with another Iettes (e.p., A . S ) will mean that one unit of species A will be adsorbed on the site S. Species A can be an atom, molecule, or some other atomic combination. depending on the circumstances. Consequently, the adsorption of A on a site S is represented by
The total rnotar concentration of active sites per unit mass of catalyst is equal to the number of active sites per unit mass divided by Avogadro's number and will be labeled C, (rnolig .cat). The molar concentr?ttion of vacant sites, C ,, is the number of vacant sites per unit mass of catalyst divided by Avogadro's number, In the absence of catalyst deactivation, we assume that the total concentration of active sites remains constant. Some further definitions include partial pressure of species i in the gas phase. atm or kPa p, surface concentration of sites occupied by species i, rnollg cat Cres A conceptual mode1 depicting species A and B on two sites is shown in Figure 10-11.
Figure 10-11 Vacant and occupied sites
For the system shown, the total concentration of sites is Site balance
Postulate mdels: then see which one(s, fit(s) the data.
ct
=
c, + c~.s + CR.s
(10-1)
This equation is referred to as a ~ i l balance. e Now consider the adsorption of a nooreacting gas onto the surface of a cataryst. Adsorption data are frequently reported in the form of adsorption isotherms. Isotherms portray the amount of a gas adsorbed On a solid at different pressures but at one temperature. First, a model system is proposed and then the isotherm obtairled from the model is compared with the experimental data shown on the curve. If the curve predicted by the model agrees with the experimental one, the model may reasonably describe what is occurring physically in the real system. If the predicted curve does not agree with data obtained experimentally. the model fails
662
Catalysis and Catalytic Reactors
Ghap. 71)
to match the physical situation in at least one irnportarrt characteristic and perhaps more. Two models will be postulated for the adsorption of carbon monoxide on metal surfaces. In one model, CO is adsorbed as molecules, CO,
as
is the case on nickel
In the other, carbon monoxide is adsorbed as oxygen and carbon atoms instead of molecular CO.
as i s rhe case on
-Fe-Fe-Fe-
Two models: I . Adsorption as
CO 2. Adsorption as C and 0
-Fe-Fe-Fe-
The former is caIled molecular or nondissociared adsorprion (e.g., CO) and the latter is called dissociative adsorption (e.g., C and 0).Whether a molecule adsorbs nondi ssociatively or dissociative1y depends on the surface. The adsorption of carbon monoxide molecules will he considered first. Because the carbon monoxide does not react further after being adsorbed, we need only to consider the adsorption process:
See
Reference Shelf For Adcorption
irong
CO+S (c0.s
(10-2)
In obtaining a rate law for the rate of adsorption, the reaction in Equation (10-2) can be treated as an e l e m e ~ ~ r ureacrion. q The rate of attachment of the carbob monoxide molecules to the active site on the surface i s proportional to the number of collisions that these molecules make with a surface active site per second. In other words, a specific fraction of the molecules that strike the surface become adsorbed. The collision rate is, in turn, direc~ly-proportional to the car. bon monoxide partial pressure, Pco Because carbon monoxide moiecules adsorb only an vacant sites and not on sites already occupied by other carbon monoxide molecufes. the rate of attachment is also directly proportional ro the concentration of vacant sites, C,. Combining these two facts means that the
v p , ,
R. L. Masel, Principles of Adsorption and Reaction on Solid Sutjures (New Y d : Wiley. 19963.
Set. 10.2
Steps in a Catalytic Reaction
663
rate of attachment of carbon monoxide molecules to the surface is directly proportional to the product of the partial pressure of CO and the concentration of vacant sites; that is,
A
Rate of attachment = kAPcoC,
A
The rate of detachment of molecules from the surface can be a first-order process; that is, the detachment of carbon monoxide molecules from the surface is usually directly proportional to the concentradon of sites occupied by the adsorbed molecules (e.g., rate of detachment = k-ACco.s The net rate of adsorption is equal to the rate of molecular attachment to the surface minus the rate of detachment from the surface. If k, and k - , are the constants of proportionality for the attachment and detachment processes. then
~ d ~ ~ , , , The ~ ~ratio ~ ~ K A = kA/k-, is the adsorprio~equilibrium constanr. A . S to rearrange Equation (10-3) gives
A+S
The adsorption rate constant, kA, for mofecular adsorption is virtually independent of temperature. while the desorption constant, k-*, increases exponentially with increasing temperature. The equilibrium adsorption constant K A decreases exponentially with increasing temperature. Because carbon monoxide is the only material adsorbed on the catalyst. the site balance gives
)
- =
k,=
[A.
C,= C[. -t Cco
-
P A =( a m )
Using it
<]
(1 0-5)
At equilibrium. the net rate of adsorption equals zero. Setling the left-hand side of Equation (10-4) equal to zero and solving for the concentration of CC, adsorbed on the surface. we get G o s = K* c,.pro
Using Equation (10-5) to give C, in terms of Cco., and the total number of sites C,. we can solve for Cm., in terms of constants and the pressure of carbon monoxide:
G o . s = kA C, PCO = KAPCO(CI- CCO. S) Rearranging gives us
Catalysis and Catalytic Reactors
Chal
This equation thus gives the concentration of carbon monoxide adsor the surface, C,-o.s, as a function of the partial pressure of carbon monox and is an equation for the adsorption isotherm. This particular type of isothc equation is called a Langrnuir iso~herm.~ Figure 10-12(a) shows a plot of amount of CO adsorbed per unit mass of catalyst as a function of the par pressure o f CO. One method of checking whether a model (e.g., molecular adsorpt versus dissociative adsorption) predicts the behavior of the experimental d i s to linearize the model's equation and then lot the indicated variab against one another. For exarnp<e, Equation (10-7)maybe arranged in the fc od
MolecuIar adwmtrnn
-0
+ 1
Cc0.s KAC!
Pm
-pco C,
(10
and the linearity of a plot of P,,ICco.sas a function of Po will determine the data conform to a Langrnuir single-site isotherm. Next, the isotherm for carbon monoxide adsorbing as atoms is derivec Dissmiative adsorption
co + 25
a Cc.s+ 0 . S
When the carbon monoxide molecule dissociates upon adsorption, it i s referr to as the dissociative adsorption of carbon monoxide. As In the case of mole ular adsorption, the rate of adsorption is proportional to the pressure of cxb, monoxide in the system because this rate is governed by the number of ge eous collisions with the surface. For a molecule to dissociate as it adsort however, two adjacent vacant active sites are required rather than the sing site needed when a substance adsorbs in its moIecular fom. The probability
Linear
Figure 10-12 Langrnuir isotherm for (a) molecular adsorption (b) dissociative adwrption of CO.
after Irving Langmuir ( t 881-1957). who first proposed it. He received th Noh1 Prize in 1932 for his discoveries in surface chemistry.
9 Named
5%. 10.2
665
Steps ~na Catalyt~cRsact~on
two vacant sites occurring adjacent to one another is proportional to the square of the concentration of vacant sites. These two observations mean that the rate of adsorption is proportional to the product of the carbon monoxide partial pressure and the square of the vacant-site concentration, P,,Cj. For desorption ro occur, two occupied sites must be adjacent, meaning that the rate of desorption is proportional ta the product of the occupied-site concentration. (C - S) X (0S). The net rate of adsorption can then be expressed as
-
Factoring out k,, the equation for dissociative adsorption is Rate o f diswiative adsorption
where
For dissociative adsoption both k, and k-, increase exponential1y with increasing temperature while KA decreases with increasing temperature. At equilibrium, r,,, = 0, and
kAPcoclf = k-ACc.5Co.s
FOSCc.S = CO.$, (
~
~
P
~
=~ CO. ~ s" Z ~ v
(10-IO)
Substituting for Cc.sand CoAsin a site balance Equation (10-I),
Solving for C,
I
C, = C, /( 1 + 2 ( ~ ~ $ ~ ~ ) ~ ~ )
This vaIue may be substituted into Equation (10-10) to give an expression that can be solved for Co.s. The resulting equation for the isotherm shown in Fig-
1
I
Langmuir isotherm for adsorpt~onas
ure IO-lZ(b} is
atomic carbon
monoxide Dissociative adsorption
Taking the inverse of both sides of the equation, then multiplying through by (PCo)I", yields
2 *' I p
-= (PC01
c~S
CF:" PZ
sus : :P
I ct(KA)113
+
2(P,o)"Z
(10-12)
cf
If dissociative adsorption is the correct model, a plot of (ft$C0.,) vershould be linear with slope (2/C,).
666
Catalvsis and Catalytic Reactors
Cheo. 10
When more than one substance is present, the adsorption isotherm equations somewhat more complex. The principles are the same, though, and the isotherm equations are easily derived. It is left as an exercise to show that the adsorp tion isotherm of A in the presence of adsofiate B is given by the relationship ate
Ct-Cv+CS ~B'+~
Note assumptions in rhe model and check their validity
When the adsorption of both A and B are first-order processes. the desorptions are also first order, and both A and B are adsorbed as molecules. The derivations of other Langrnuir isotherms are relatively easy and are left as an exercise, In obtaining the Langrnuir isotherm equations, several aspects of the adsorption system were presupposed in the derivations. The mast important of these, and the one that has been subject to the greatest doubt, is that a unjfoform surface is assumed. In other words, any active site has the same artsaction for an impinging molecule as does any orher site. Isotherms different from the Langmuir types, such as the Fseundlich isotherm, may be derived based on various assumptions concerning the adsorption system, including different types of nonuniform surfaces.
10.2.4 Surface Reaction
The rate of adrorption of species A onto a solid surface,
A+S
A-S
is given by
Surface reaction
After a reactant has been adsorbed onto the surface, it is capable of reacting in a number of ways to form the reaction product. Three of these ways are: 1. Single sire. The surface reaction may k a single-site mechanism in
which only the site on which the reactant is adsorbed is involved in the reaction. For example, an adsorbed molecule of A may isomerize (or perhaps decompose) directly on the site to which it is attached, such as
N = n.p@mene
I= I - p m m
.A
A.S
B-S
5tngle stte
Because in each step the reaction mechanism is elementary, the surface reaction rate law is
667
Steps in a Catalytic Reactbn
Sec. 10.2
where Ks is the surface reaction equilibrium constant Ks = kslk-s 2. Dual site. The surface reaction may be a dual-site mechanism in which the adsorbed reactant interacts with another site (either unoccupied or occupied) to form the product.
K, = (dimensionless)
Duel site
For example. adsorbed A may react with an adjacent vacant site to yield a vacant site and a site on which the product is adsorbed, such as the dehydration of butanol.
For the generic reaction
the corresponding surface reaction rate Eaw is
Dual Sire
(10-16)
Another example of a dual-site mechanism is the reaction between two adsorbed species, such as reaction CO with 0
K,= (dimensionleas)
For the generic reaction
--
;,. ....,,-A
{ 0.
,TC ,
/.D.,. /
AaS
+ B.S
C.S
+ D.S
Dual srte
the corresponding surface reaction rate law is
668
Catalysis and Catalytic Reactors
Cha
A third dual-site mechanism is the reaction of two species adso on different types of sites S and S'. such as reaction of CO with
For the generic reaction
Dual site
the corresponding surface reaction mte law is
Reactions involving either single- or dual-site mechanisms, w were described earlier are sometimes referred to as following L muir-Hirsshelwood kinetics. 3. Eiq-Rideal. A third mechanism is the reaction between an adso molecule and a molecule in the gas phase, such as the reaction of pylene and benzene
Langmu~rHinshelwocd kinetics
C3*.*
B
.A:
@.: .
,@-.: &4&
,'C:
&-&
For the generic reaction
Eley-Rideal rnechan~srn
A . S + B(g)
c-S
the corresponding surface reaction rate law is
(10
This type of mechanism is referred to as an Eley-Rideal mechan, 10.2.5 Desorption
KK = (am)
.
En each of the preceding cases, the products of the surface reaction adso on the surface are subsequently desorbed into the gas phase. For the desoq of a species (e.g., C), C-S the rate of desorption of C is
C+S
Sec. 10.2
Steps in
a Catalytic Reaction
669
where Koc is the desorption equilibrium constant. We note that the desorption step for C is just the reverse of the adsorption step for C and that the rate of desorption of C, r,, is just opposite in sign to the rate of adsorption of C. r,,:
In addition, we see that the desorption equilibrium constant Kw is just the reciprocal of the adsorption equilibrium constant for C,Kc:
In the material that follows, the form of the equation for the desorption step that we will use will be similar to Equation (10-2 I).
10.2.6 The Rate-Limiting Step
When heterogeneous reactions are carried out at steady state. the rates of each of the three reaction steps in series (adsorption. surface reactiorl. and desorption) ate equal to one another:
However, one particular step in the series is usually found to be rare-limiting or mte-controlling.That is, if we could make this particular step go faster, the entire reaction would proceed at an accelerated rate. Consider the analogy
to the electrical circuit shown in Figure 10-13. A given concentration of reactants is analogous to a given driving force or electromotive force (EMF). The current I (with units of Cwbmbs/s) is analogous to the rate of reaction, -ri (rnol/sWgcat), and a resistance R, is associated with each step in the series. Since the resistances are in series. the total resistance is just the sum of the Endividual resistances, for adsorption (RAD),surface reaction (Rs), add desorption (RD),and the current, I, for a given voltage, E, 1s
ifowing us
down?
Plgure 10-13 Electrical analog to heterogeneous renctionu.
Catalysis and Catalflic Reactors
he concept of a
nte-limiting step
An alrwithm to determine the rate-limtting step
Chap. 10
Since we observe only the total resistance, R,,,.it is our task to find which resistance is much larger (say, 100 R) than the other two (say. 0.1 0).Thus,if we could lower the largest resistance, the current I (i.e., -rA) , would be Iarger for a given voltage, E. Analogously, we want to h o w which step in the adsorption-reaction-desorption series is limiting the overall rate of reaction. The approach in determining catalyric and heterogeneous mechanisms is usually termed the Longrnuir-Hinskelwood approach, since it is derived from ideas proposed by Hinshelwood'O based on Langmuir's principles for adsorption. The Langmuir-Hinshelwood approach was popularized by Hougen and Watson" and occasionatly inellides their names. I t consists of first assuming a sequence of steps in the reaction. In writing this sequence. one must choose among such mechanisms as moiecular or atomic adsorption, and single- or dual-site reaction. Next, rate Iaws are written for the individual steps as shown in the preceding section, assuming that all steps are reversible. Finally, a rate-limiting step is postulated, md steps that are not rate-limiting are used to eliminate all coverage-dependent terms. The most questionable assumption in using this technique to obtain a rate law is the hypothesis that the activity of the surface toward adsorption, desorption, or susface reaction is independent of coverage; that is, the surface is essentiaIly uniform as far as the various steps in the reaction are concerned. An example of an adsorption-limited reaction is the synlhesis of aamrnonia from hydrogen and nitrogen,
over an iron catalyst that proceeds by the following mechanisrn.j2
H2+2S
+ 2H.S
Rapid
Dissociative adsorption
of N, limits
NH.S+H+S
i
HH,N-S+S
H , N . S + H . S (NNH,S+S
Rapid
I0C. N. H i n s h e l w d , The Kinetics of Chemical Chan~e(Oxford: Clarendon h e s s 1940). I'O.A. Hougen and K. M. Watson. Ind. E ~ R Chem., . 35.529 (1943). '?From the literature cited in G. A. Sornorjai. lnfmdurrion m S~trjhceC h e m i s t ~atld Cataljj.~is(NEWYork: W~ley,1994). p. 482.
Sec. 10.3
Synthesiz~nga Rate Law, Mechanism, and Rate-Limiting Step
67 1
The rate-limiting step is believed to be the adsorption of the NZmolecule as an N atom. An example of a surface-limited reaction is the reaction of two noxjous automobile exhaust products. CO and NO,
carried out over copper catalyst to form environmentally acceptable products, N2 and CO?:
Rapid Surface reaction limits
Rapid
Analysis of the rate law suggests that C 0 2 and Problem P 10-7,).
N, are weakly
adsorbed (see
10.3 Synthesizing a Rate Law, Mechanism, and Rate-Limiting Step We now wish to develop rate laws for catalytic reactions that are not diffusion-limited. In developing the procedure to obtain a mechanism. a rate-limiting step, and a rate law consistent with experimental observation, we shall discuss a particular catalytic reaction, the decomposition of cumene to form benzene and propylene. The overall reaction is
A conceptual model depicting the sequences of steps in this platinum-catalyzed reaction is shown in Figure 10-14. Figure 10-14 is only a schematic representation of the adsorption of cumene: a more realistic model is the formation of a complex of the rr orbitals of benrene with the cataIytic surface. as shown in Figure 10-15. Adsorption Suriace reaction
cab
necorption
Adsorption of cumens
-Surfoca react ion
Derwpl~onof
benzene
Catalysis and Catalytic Reacton
Cha
Figure 10-15 n-orbttal complex on surface.
The nomenclature in TabIe 10-3 will be used to denote the various cies in this reaction: C = cumene, B = benzene. and P = propylene. The I tion sequence for his decomposition is
C These three steps represent the mechanism for
+S
+& C.S k*
"
C'S
k4
B.S <
ka
)
Adsorption of cumene on the surface
B .S + P
cumene decomposition k,
Ideal Gas
L ~ W
)
B
+S
Surface reaction to form adsorbed benzene and propylene in the gas phase
Desorption of benzene from surface
Equations (10-223 through (10-24) represent the mechanism proposed for reaction. When writing rate laws for these steps, we treat each step as an eler tary reaction; the only difference is that the species concentrations in the phase are replaced by their respective partial pressures:
PC = CcRT
cc
-
PC
There is no theoretical reason for this replacement of the concentration. with the partial pressure, PC;it is just the convention initiated in the 1930s used ever since. Fortunately PC can be calculated directly from Cc usini ideal gas law.
The rate expression for the adsorption of cumene as given in Equi (10-22) is C+S
C*S k - ~
TAD
1-
= kAPcC, - k-ACc.s
Adsorption:
r,, = k, P&,--
Sec 10 3
Syntbesr-rmg a RaZe Law, Mechan~sm,and Rate-Llmrt~ngStep
673
,
I f I.,,,, ha\ unit\ of (tnollg c a t - c ) ;IL~ C, ha\ unit\ nt'(mol cumcnc i ~ c l ~ ~ ~ h c t l / g cat) typical unitc of k , , X and K( ~woultlhe
,.
[ A , I =(I\P.r.s)-
' or(atrn.h)-'
{ k - % ]= h - I ors-I
The rate Law for the surface reaction step producing adsorbed benzcne and prnpylene in the gas phase.
with the s u r f ~ ~ crenorion c ~quilibriiurrconsranr being
Typical units for ks and Ks are s-' and kPa. respectively. Propylene is not adsorbed on the surface. ConsequentIy, its concentration on the surface is zero. Cps=O
The rate o f benzene desorptian [see Equation (10-74)j i s
I
I+D =
kDCR.s- k-DPRCL,
Desarption:
Typical units of kD and XDBate desorprion of benzene.
rD = k,,
S-'
( )CB s-
1
( lo-27)
[ 10-28)
and kPa, respectively. By viewing the
From right to left. we see that desorption is just the reverse of the adsorption of benzene. Consequently, it is easily shown that the benzene adsorption equilihrium constant Ks is just the reciprocal of the benzene desorption constant K,,ll:
Catalysis and Catatytic Reactors
Chap. 10
-
K, = 1
KD,
and Equation (10-28) can be written as
Because there is no accvmulation of reacting species on the surface the rates of each step in the sequence are all equal:
1-
(10-30)
For the mechanism postulated in the sequence given by Equations (10-22) through (10-24), we wish to determine which step is rate-limiting. We first assume one of the steps to be rate-limiting (rate-con~olling)and then formulate the reaction rate law in terns of the ~artialpressures of rhe species present. From this expression we can determine the variation of the initial reaction rate with the initial total pressure. If the predicted rate varies with pressure in the same manner as the rate observed experimentally. the implication is that the assumed mechanism and rate-limiting step are correct. 10.3.1 Is the Adsorption of Cumene Rate-Limiting?
To answer this question we shall assume that the adsorption of cumene is indeed rate-limiting, derive the corresponding rate law, and then check to see if it is consistent with experimental observation. By assuming that this (or any other) step is rate-limiting, we are considering that the reaction rate constant of this step (in this case k,) is small with respect to the specific rates of the other steps (in this case k , and kD).I3 The rate of adsorption is Need to express C,, and Cc,s in terms of PC. P,, and Pp
Because we can measure neither C, or Cc.s,we must replace these variables in the rate law with measurable quantities for the equation to be meaningful. For steady-state operation we have (10-30)
-~.d==r~,,=r~=r,,
''Strictly speaking one should compare the product k4Pc with k, and k,.
s kg cat
Dividing by KhL,PCIce note the ratio,
R,>P,
Ls
atm
= @ . The reawn i s that in order to cornpaw terns. kg car
mol ] (-3). ):[ and):( mud ali havr the same units r8
~ADPC
result is the same however
;kg c a t
~ h .end
Sec. 10.3
Synthesizing a Rate Law Mechanism, and Rate-Limiting Step
675
For adsorption-limited reactions, k~ is small and ks and kD are large. Consequently. the ratios rslks and r ~ l kare ~ very small (approximately zero), whereas h e ratio rAD/kAis relatively large. That is, the prduct for h e adsorption step [k,Pc] (s-I) is small with respect te the other rate constants: k, (s-1) for the surface reaction step and kD (s-I) for the desorption step. The surface reaction rate law is
Again, for adsorption-limited reactions the surface specific reaction rate ks is large by comparison, and we can set
r= ks
'S=
fJ
and solve Equation (10-31) for Cc.s:
To be able to express CC.ssoleIy in terms of the partial pressures of the species present, we must evaluate CB.$.The rate of desorption of benzene 1s
's-(J"yD to find C, and Cc 3 in terms of partial pressures
Hawever, for adsorption-limited reactions, k~ is large by comparison. and we can set
m= k, 1 and then solve Equation (10-29) for CB.s: Ce,s=KepaCv After combining Equations f 10-33) and
(10-351,we have
Replacing Cc.sin the rate equation by Equation (10-36) and then factoring C,., we obtain
( )
K P P
r A ~ = k Ap , - L P
(
~ , = k ,pc--
(10-371
Observe that by setting r,, = 0,the term (KsK c / K B ) is simply the overall partial pressure equilibrium constant. Kp. for the reaction
Catalysis and Catalytic Reactors
Chal
The equilibrium constant can be determined from thermodynamic I and is d a t e d to the change in the Gibbs free energy, AGO, by the equation I Appendix C)
where R is the ideal gas constant and T is the absolute temperature. The concentration of vacant sites, C,, can now be eliminated from Ec tion (10-37) by utilizing the site balance to give the total concentration of si C,, which is assumed constant: l4
Total sites = Vacant sites+Occupied sites Because cumene and benzene are adsorbed on the surface, the concentratior occupied sites is (Cc., +- CBAS), and the total concentration of sites Is Site balance
C, = C,
+ Cc.s+ CB-S
(1@
Substituting Equations (10-35) and (10-36)into Equation (10-401, we h
Solving for C,, we have
Combining Equations (10-41) and (10-371, we find that the rate law for catalytic decompositon of curnene, assuming hat the adsorptioa of curnene the rate-limiting step, is Cumene reaction
sate law if adsorption were the limiting step
'JSome prefer to write the surface reaction rate in terms of the fraction of the sud of sites covered (i.e.. f,? rather than the number of sites CA.Scovered, the differel being the multiplication factor of the total site concentration. C,. In my event. final form of the rate law is the same because C,, KA,ks, and so on, are dI lurn~ into the reaction rate constant. k.
Sec 10.3
Synthesizrng a Rate Law. Mechanism, and Rate-Llrniting Step
677
We now wish to sketch a plot of the initial rate as a function of the partial pressure of cumene. PC,,. Initially, no products are present; consequently, P , = PB = 0. The initial rate is given by ( 10-43)
-rho = C, k,4Pco= kPco
If the cumene decornpostion is adsorption rate limited. then the initial rate will be linear with the initial parlial pressure of cumene as shown in Figure 10- 16.
If adsorption were ln~tialpartial pressure of cumene, PCD
rate-limiting. the data should show -ri increasing
linearly with
Figure 10-16
CJninhibited adsorption-timited reaction.
Pa,.
Before checking to see if Figure 10-16 is consistent with experimental observation, we shall derive the carresponding rate laws and initial rate plots when the surface reaction is rate-limiting and then when the desorption of benzene is rate-limiting.
10.3.2 Is the Surface Reaction Rate-Limiting? The rate of surface reaction is Single-site
mechanism
Since we cannot readily measure the concentrations of the adsorbed species, we must utiEize the adsorption and desorption steps to eliminate Cc and CB.S
,
from this equation. From the adsorption rate expression in Equation (10-25) and the condition that kA and k, are very large by comparison with ks when surface reaction is controlling (i.e., rm/kA = 01, we obtain a relationship for the surface concentration for adsorbed curnene:
cc s = KcPcCu In a similar manner, the surface concentration of adsorbed benzene can be evaluated from the desorptjon rate expression [Equation / 10-29)] together Using 1 with the approximation: to find CB.sand
Cc.s in terms of partial pressures
when
-E 0
P~
k,
then C B .=~ K B P B C ,
678
Catalysis and Cataly#c Reactow
Chap. 10
Substituting for Capsand Cc.sin Equation (10-26) gives us
The only variable left to eliminate is C, :
ct = c, + cB.S + cc.S
Site balance
Substituting for concentrations of the adsorbed species, CB.S,and Cc.s yields P
Cumene rate law for surfacereaction-limiting
The initial rare is
At low partial pressures of curnene
1
+ KcPm
and w e observe that !he initial rate will increase linearly with the initial partial pressure of cumene:
At high partial pressures KrP,
I
and Quation I 10-45) becomes
and the rate i s independent of the panial pressure of curnene. Figure 10-17 shows the initial rate of reaction. as a function of initial partial pressure of cumene for the case of surface reaction controlling.
10.3.3 Is the Desorption of Benzene Rate-Limiting? Thc rate expression for the desorption of benzene is D'
= k,
(Cg
s-KBf'BCL
1
Sac. I 0.3
Synthesrz~nga Rate Law, Mechanism, and Rate-Limiting Step
679
If surface reaction were rate-limiting, the data would show
this behavior. Initial partial pressure of cumene, PCO
Figure 10-17 For desorplionl~rnltedreactiuns. h t h LAD and I , are vet! large compared wrth AD.
Surface-reaction-limited.
From the rate expression for surface reaction, Equation (10-26). we
set
!Lo As
a hich i%small.
to obtain
Similarly. for the adsorption step. Equation (10-251, we set
r~
o,
k*
to obtain Cc s = KcPCC ,
then substitute for Cc.s in Equation (10-46):
K K P C CB-$= PP Combining Equations ( I 0-28) and (10-47) gives us
where Kc i s the cumene adsorption constant. K , is the surface reaction equilibrium constant, and Kp is the gas-phase equilibrium constant for the reaction. To obtain an expression for C,. we again perfon a site balance:
Site balance:
C, = Cc.s + C,.,
+ C,
After substituting for the respective surface concentrations. we salve the site bal nnce for C,,:
680
Cata'ysis and Catalytic Reactors
Cl?a
Replacing C,. in Ecluat~on( 10-48) by Equation ( 10-49) and multipl the nuinerator and denominator by Pp, we vhtain the rate expression decorption cr~ntrol: C~~mene decr>mpn<~t~un rille law !f deqorptlun Ncrc Iiniiting
tf derorprlon control\. he uliti.11 rdre ic indepcnrlrnt of p a f l i ~ plrcrltre l of ctlrnenr.
To determine the dependence of the initial rate on partial pressur set P p = PR = 0; and the rate law reduces to
cumene. we again
- I"&, = k,C,
-I+:,,
with the corre~pondingplot of shown i n Figure IO- 18. If desorpticln controlling, we would see that the initial rate would be independent of the rial partial pressure o f cumene.
ln~tialpartial pressure of cumene, PCo
Fipre 10.18
Dewrpt~on-limitedrencrlon.
10.3.4 Summary of the Cumene Decomposition Cumcnc decompu'itlon surface-reaction-
limited
The experimental observations of -r&, as a function of Pco are shown in Fi 10- 19. From the plot in Figure 10- 19. we can clearly see that neither adsoq nor desorption is rate-limiting. For the reaction and mechanism given by
C-S
I3.S
+P
the rate law derived by assuming that the surface reaction is rate-limiting ai with the data. The rate law for the case of no inerts adsorbing on the surface is
See. 10.3
Synthesrzlng a Rate Law. Mechanism, and Rate-Limiting Step
Surt~cvlimited ~ ; 1 ~ t i omechilni=.m 1\ i, consictcnt with experimeti1;11 data.
Init~aipartial pressure of cumane, PCo Figure 11)-I9 Actual in~tialrate as a funct~onof partial preqsuw of uunlcne.
The forward curnene decomposition reaction is a single-site mechanism involving onEy adsorbed curnene while the reverse reaction of propy lene in the gas phase reacting with adsorbed benzene is an Eley-Rideal mechanism. tf we were to have an adsorbing inert in the feed, the inert wouId not participate in the reaction but would occupy sites on the catalyst surface:
Our site balance is now Because the adsorption of the inert is at equilibrium. the concentration of sites occupied by the inert is Cl.s= KIPLC,
( t O-51,)
Substituting For the inert sites in the site balance, the rate law For surface reaction control when an adsorbing inert is present is
10.3.5 Reforming Catalysts
We now consider a dual-site mechanism. which is a reforming reaction Found in petroleum refining to upgrade the octane number of gasoline.
Side Note: Ocbne Number. Fuels with low octane numben can produce spontaneous combustion in the cylinder before the airlfueI mixture is compressed to its desired value and ignited by the spark plug. The following figure shows-the desired combustion wave front moving down from the spark plug and the unwanted spontaneous combustion wave in the lower right-hand corner. This spontaneous combustion produces detonation waves which constitute engine knock. The lower the octane number the greater the chance of engine knock.
Catalysis and Catalytic Readors
%-~pontsnwu~ Combustion
I
Calibration Curve
1
Chap 10
t
I
Unknown
100% heptane 100% iso Octane Number
The octane number of a gasdine is determined from a calibration curve relating knock intensity to the % iso-octane in a mixture of jso-octane and heptane, One way to calibrate the octane number is to place a transducer on the side o f the cylinder to measure the knock intensity (K.1,) (pressure pulse) for various mixtures of heptane and iso-octane. The octane number is the percentage of iso-mtane in this mixture. That is, pure iso-octanehas an octane number of 100, 808 iso-octanel20% heptane has an octane number of 80, and so on. The knock intensity is measured for this 80120 mixture and recorded. The relative percentages of iso-octane and heptane are changed (e.g., 901101, and the test is repeated. After a series of experiments, a calibration curve is constructed. The gasoline to be calibrated is then used in the test engine, where the standard knock intensity is measured. Knowing the knock intensity, the octane rating of the fuel is read off the calibration curve. A gasoline with an octane rating of 92 means that it matches the performance of a mixture of 92% iiso-octane and 870heptane. Another way to calibrate octane number is to set the knock intensity and increase the compression ratio. A fixed percentage o f iso-octane and heptane is placed in a test engine and the compression ralio (CR) is increased continually until spontaneous combustion occurs, producing an engine knock. The compression ratio and the corresponding composition of the mixture are then recorded, and the test is repeated to obtain the calibration curve of compression ratio as a 'function of % iso-octane. After the calibration curve is obtained, the unknown is placed in the cylinder, and the compression ratio (CR)is increased untiI the set knock intensity is exceeded. The CR is then matched to the calibration curve to find the octane number. The more compact the hydrocarbon molecule, the less likely it is to cause spontaneous combustion and engine knock. Consequently, it is desired to isomerize straight-chain hydrocarbon molecules into more compact molecules through a catalytic process called reforming.
Sec. 10.3
Catalyst
Synthesizing a Rate Law, Mechanism, and Rate-Limiting Step
683
One common reforming catalyst is platinum on alumina. Platinum on alumina (A1203)(see SEM photo below) is a bifunctional catalyst that can be prepared by exposing alumina pellets to a chloropSatinic acid solution, drying, and then heating in air 775 to 875 K for several hours. Next, the material is exposed to hydrogen at temperatures around 725 to 375 K to produce very snlaIl clusters of Pt on alumina. These clusters have sizes on the order of 10 A, while the alumina pore sizes on which the Pt is deposited are on the order of 100 to 10,000 (i.e., 10 ta 1000 nml.
Platinum on alumrna (Ftpuru Irom K I \l.:kc.l.
Ch~~lrrrtrl Kmeiics ntrd Cutaleris. N'lley. NCI~York. 2001. p. 7W)
As an example of cataIytic reforming we shall consider the isomerization of n-pentane to f-pentane:
n-pentane
0 7 5 "I" A1:0,
<
,i-pentane
Normal penlane has an octane number of 62. whiIe iso-pentane has an octane number of 90! The 17-pentane adsorbs onto the platinum, where it i s dehydrogenated to fonn I?-penlene. The n-pentene desorbs from the platinum and 20% 25rk adsorbs onto the alumina. where it i s isomerized to i-pentene, which then 20% desorbs and subsequentIy adsorbs onto platinum. where it is hydrogenated to 10% form i-pentane. That is,
Gasoline
I na In"
Cr Cb
C7
,-, c, C I ~ .
I
5%
-H2
n-pentane
'H:
AL-01
( 2 n-pentene (i-pentene PI
(
)
PI
i-pentane
We shaIl focus on the i~orneriza~ion step to develop the mechanism and the rate law: A[-0;
n-pentcne
i-pentene
684
Catalysrs and Catalyt~cReactors
Char
The procedure for formulating a mechanism, rate-limiting step. and co sponding sate law is given in Table 10-4. Table 10-5 gives the forms of wte laws For different reaction mechanir that are irreversible and surface-reaction-limited. ----
-
. .
Single site A-S
+B.S
Dual site
A.S
+ B*S
-
C.S
+S
-r;
=
~ P P B
( 1 +K,P,+KnPa+
&PC)'
Eley-Rideal
We need a word of caution at this point. Just because the mechanism : rate-limiting step may fit the rate data does not imply that the mechanisn co~rect.'~ Usually, spectroscopic measurements are needed to confirm a me anism absolutely. However, the development of various mechanisms , rate-limiting steps can provide insight into the best way to correIate the c and develop a rate law.
10.3.6 Rate Laws Derived from the Pseudo-SteadyState Hypothesis
In Section 7.1 we discussed the PSSH where the net rate of formation of re tive intermediates was assumed to be zero. An alternative way to derive a ( alytic rate law rather than setting
is t o assume that each species adsorbed on the surface is a reactive interne ate. Consequently, the net rate of formation of species i adsorbed on the s face will be zero: The PSSH should
be used when mre than one step is limiung
I5R. I. Masel, Principles of Adsorprion and Reaction on Solid Surfaces {New Yr WiEey, 1996). p. 506.http:llwww.uiuc.edu/ph/www/r-masel/
Sec. 10.3
TABLE10-4.
Synthesizing a Rate Law, Mechanism, and Rate-Limfting Step A~COR~THM FOR
685
D ~ R M T Y I REACTION NG MECHANISM AND RATE-LIMITIKG STEP
Isornerization of n-pentene IN) to i-pentene (1) over alumina
Refnrming reactlon to rncrease octane number OF gasoline
I . SeIacr n merhonism. (Mechanism Dual Site)
N S S
Adsorption:
N.3
N-S + S
Surfa~ereaction:
I S+ S I
Teat each reaction step as an elementary reaction when writing rate laws. 2. Assume a mre-Itmiring step. Choase the surface reaction first. since more than 75% ofoll heremgeneow rencttons rhar are not diesion-limited ow surface+reaction-1Emt1ed. The rate law for the surface reaction step is
3. Find the expwssron for concenlmfionof the adsorkd speci6.t CC; s. Use the other steps that are not limit~ngto soFw for C,.s(e.g.. CN.S and C1.5).For this reaction. From
9 = 0:
CN = PNKGC,
~ A P
Foliowngthe Algorithm
From
9
k,
5
,
Cl =
0:
P;P
= KlP1C,
4. Write a sire balance.
c,= cv+ CN5 f
CT.~
5 . Derive the mte law. Combine Steps 2. 3, and 4 to arrive at the rate law:
6. Cornpure with data. Compare the rate law derived in Step 5 with experimental data. If they agree, there is a good chance that you have found the correct mechanism and rate-limiting step. If your derived rate law (i.e., model) does not agree with the data: a. Assume a different rate-limiting step and repent Steps 2 through 6. b. If, after assumlng that each step is rate-limiting, none of the derived rate laws agree with the experimental data. select a different mechanism (e.g.. A Single Site Mechanism):
. .
N +S
a N-S
NhS
I*S
ITS 14- S and then pmeed through steps 2 through 6. The single-site mechanism turns out to be the correct one. For this mechanism the rate law is
c. If two or more models agree, the statistical tests discussed in Chapter 5 {e.g.. comparison of residuals\ should be used to discriminate between them (see the Supplemenmy Reading}.
686
Catalysis and Catalyfic Reaclors
Chap. 10
While this method works well for a single rate-limiting step, it also works well when two w more steps are rate-limiting (e.g., adsorption and surface reaction). To illustrate how the rate laws are derived using the PSSH,we shall consider the isornerization of normal pentene to isn-pentene by the following mechanism, shown as item 6 of Table 10-4:
The rate Jaw for the irreversible surface reaction is
The ner rate of the adsorbed species (i.e.. active
The net rates of generation of N - S sites and 1. S sites are
intermediates)
Solving for CN+S and C,.sgives
and substituting for CN.Sin the surface reaction rate law gives
From a site balance we obtain
Use the PSSH method when
.
After substituting for C?;.s and Ci.S.solving for C,, . which we then substitute in the rate law, we find
pw
Some steps are irrevers~hlc.
n o ar more steps are rateltrn~t~ng.
k-, + ks
( 1 0-62 1
Sec. 10.3
Synthesizing a Rate Law, Macbnlsm, and Rate-Limiting Step
687
The adsorption constants are just the ratio of their respective rate constants:
We have taken the surface reaction step to be rate-limiting; therefore, the surface specific reaction constant, ks, is much smaller than the rate constant for the desorption of normal pentene, LN; that is, In-
ks
and the rate law given by Equation (1 0-62) becomes
This rate law [Equation (10-63)] is identical to the one derived assuming that rADlkAD= 0 and rD/kAD 0. However, this technique is preferred if two or more steps are rate-limiting or if some of the steps are irreversible or if none of the steps are rate-limiting.
10.3.7 Temperature Dependence of the ~ a t ' eLaw Consider a surface-reaction-limited irreversible istlmerization
in which both A and B are adsorbed an the surface, the rate law is
The specific reaction rate, k. will usually follow an Arrhenius temperature dependence and increase exponentially with temperature. However. the adsorp tion of all species on the surface is exothermic. Consequently, the higher the temperature, the smaller the adsorption equilibrium constant. That is, as the temperature increases, K, and KB decrease resulting in less coverage of the surface by A and B. Therefore, at high temperatures, the denominator of catalytic rate laws approaches 1. That is. at high temperatures (low coverage)
688
C a t a ? ~ iand s Catajytic R ~ l a c M r s
ChaF
The rate Inw could then he npproximated nh Ze:lc.ctin:: the ,dcorhed cptu~esa l high leinpr.~l~ire~
-1:;
=kfA
or for a reversiblr isomerization we wouId have
stnttgier ror mudel bu~lduig
Alenrithm
Drrhrcr Rate law Ftrrrl Mcchanr.~m E~,of~rrrrc Rate law paranlttcrr WPYIS~ PBR
CSTR
The algorithm we cafl use as a ctast in postulating n reaction mechani and rate-limiting step is shown in Table 10-4. Again, we can never really prc n mechanism by comparing the derived rate law with experimental data. In' pendent spectroscopic experiments are usually needed to confirm the mec nisni. We can. however, prove that a proposed mechanism is inconsistent N the experimental data by following the algorithm in Table 10-3. Rather than t, in9 a11 the experitnental data and then trying to build a model from the d: Box et al.Ih descrrbe techniques of sequential datii ~rtkingand model buildir
10.4 Heterogeneous Data Analysis for Reactor Design In this section we focus on four operations that reaction engineers need to able to accomplish: ( I ) developing an algebraic rate law consistent with exp irnental observations. (2) analyzing the rare law in such a manner that the ri law parameters ( e . ~ .k,, K,) can readily be extracted from the experimenfal da (3) tinding a mechanism and rate-limiting step consistent with the experirnen data. and 14) designing a caralytic reactor to achieve a specified conversion. 1 shall use the hydrodemethylation of toluene to illustrate these Foi~roperatior Hydrogen and toluene are reacted over a solid mineral catalyst cantal ing clinoptilolite la crystalline silica-alumina) to yield methane and benzene
We wish so design a packed-bed reactor and a fluidized CSTR to proce a feed consisting af 30% toluene, 45% hydrogen, and 25% iinerts. Toluene fed at a rate of 50 moI/min at a temperature of 6;10°Cand a pressure o f 40 at (4057, kPa). To design the PBR. we must first determine dle rate law from t,
differential reactor data presented in Table 10-6. Fn this table we find the ra of reaction of toluene s a function of the partial pressures of hydrogen (H). tc uene (T), benzene (B), and methane (MI. In the first two runs. methane w introduced inta the feed together with hydrogen and toluene, while the 0th product. benzene, was fed to the reactor together with the reactants only in ru~ 3.4, and 6. In runs 5 and 16 both methane and benzene were introduced in tl feed. In the remaining runs, neither of the products was present in the feel E. P. Box, W. G. Hunter, and I. S. Hunter. Stnristir~for Engi~zeers(New Yor Wiley. 1978). l7 J. Papp. D.Kallo. and G.Schay- J. CnraL. 23, 168 ( 1 97 I).
I%.
Sec 10.4
Heterogeneous Data Analysrs lor Reactor Deslgn
689
T,\H[t: 11)-h. DATA FROLI r\ D T F ~ E R F ~ REA(.IIIK TIAL
Set A I
-1 Set B 3 4 5 h
Unwrdmhle the data to find the rare law
C 7 8 9 Set D 10 II 12 13 I1 15 16 Set
stream. Because the conversion was less than t % in the difTerential reactor. the partial pressures of the products. methane and benzene. in these runs were essentialiy zero, and the reaction rates were equivalent to initial rates of reaction. 10.4.1 Deducing a Rate Law from the Experimental Data Assuming that the reaction is essentially irreversible [which is reasonable after comparing runs 3 and 51, we ask what qualitative c~nclusionscan be drawn from the data about the dependence of the rate o f disappearance of toluene, - r ; , on the partial pressures of toluene. hydrogen, methane, and benzene.
I. Dependence on (he product methnne. If the methane were adsorbed on the surface, the partial pressure of methane would appear in the denominator of the rate expression and the rate would vnry inversely with methane concentration:
Catalysis and Cabtytic Reacton
Foltowv7?g the Algorithm
If i t is in the denominator. it i s pmhably on the surface.
Chap. 10
However, from runs I and 2 we observe that a fourfold increase in the pressure of methane has little effect on -r;. Consequently, we assume that methane is either very weakly adsorbed {i-e.,KMPM G 1 ) or g m s directly into the gas phase in a manner similar to propylene in the cumene decomposition previousIy discussed. 2. Dependence on he producr benzene. In runs 3 and 4, we observe that for fixed concentrations (panial pressures) of hydrogen and toluene the rate decreases with increasing concentration of benzene. A rate expression in which the benzene partial pressure appeass in the denominator could explain this dependency:
The type of dependence of -r$ on PB given by Equation (10-68) suggests that benzene is adsorbed on the clinoptiloIite surface. 3. Drpe~~dence on loluene. At low concentrations of toluene (runs 10 and 1 I), the rate increases with increasing partial pressure of toluene, while at high toluene concentrations (runs 14 and 151, the rate is essentially independent of the toluene partial pressure. A form of the rate expression that would describe this behavior is
A combination of Equations (10-68) and (10-69) suggests that the rate law may be of the form
4. Dependence on Itydrugen. When we examine runs 7, 8, and 9 in Table 10-6. we see that the rate increases linearly with increasing hydrogen concentration. and we conclude that the reaction is first order in H:. In lisht of this fact. hydrogen is either not adsorbed on the surface or its coverage of the surface is extremely low ( 1 >> K H 7 P H 2for ) the preqsums used. If it were adsorbed, -r; would have a dependence on PHI analogous to the dependence of -r+ on the partial pressure of toluene. PT [see Equation (1 0-6911. For first-order dependence on Hz. '-r;
- PH2
(10-71)
Combining Equations (10-63) through (10-71).we find that the rate law
is in qualitative agreement with the data shown in Table 10-6.
Sec. 10.4
Heterogeneous Data Analysts for Reactor Design
691
10.4.2 Finding a Mechanism Consistent with Experimental Observations
We now propose a mechanism for the hydrodemethylation of toluene. We assume that toluene is adsorbed on the surface and then reacts with hydrogen in the gas phase to produce benzene adsorbed on the surface and methane in ALTroximatel~ the gas phase. Benzene is then desorbed from the surface. Because approxi75" O r a l ' heterogeneous mately 75% of all heterogeneous reaction mechanisms are surface-reacrmction tion-limited rather than adsorption- or desorption-limited, we begin by mechanisms are assuming the =action bet ween adsorkd toluene and gaseous hydrogen to be surface-reactinnreaction-rate-limited. Symbolically, this mechanism and associated rate laws for each elementary step are
,,,
a T-S
Adsorption: T(g) f S
Prows&
Surface reaction: H2(g)
mechanism
+ T.S
B - S + M(g)
Desorptian: B S (BB(g) + S
r~ = k, ICB s -KBP, Cu) For surface-reaction-limitedmechanisms,
we see that we need to replace that we can measure.
CT.sand CR.S in Equation ( I 0-73) by quantities
For surface-reaction-limited mechanisms, we use the adsorption rate Equarion (10-72) to obtain CT.S:
Then and we use the desorptinn rate Equation (10-74) to obtain CB5 :
692
Catalysis and Catalytic Reactors
Chap.
Then
The total concentration of sites is PMom a site balance to obtain C,.
Substituting Equations (10-75) and (to-76) into Equation (10-77) and arranging, we obtain
I
Next, substitute for C,., and C,., and then substitute for C, in Equatic (10-73) to obtain the rate law for the case of surface-reactioncontrol: k
Neglecting the reverse reaction we have Rate law for surface-reaction-
limited mechanism
Again we note that the adsorption equilibrium constant of a given species , exactly the reciprocaI of the desorption equilibrium constant of that species.
10.4.3 Evaluation ol the Rate Law Parameters In the original work on this reaction by Papp et al.,'s over 25 models wer tested against experimental data, and it was concluded that the precedin mechanism w d rate-limiting step (i.e., the surface reaction between adsorbe toluene and H, gas} is the correct one. Assuming that the reaction is essential1 irreversible, the rate law for the reaction on dinoptilolite is
We now wish to determine how best to analyze the data to evaluate th rate law parameters, k, KT,and Kg.This analysis is referred to as pamrnere e s t i m a t i ~ n . 'We ~ now rearrange our rate law to obtain a linear rdationshil between our measured variables. For the rate law given by Equation (10-80)
bid. l9
See the Supplementary Reading for a variety
law pnrameters.
of techniques for estimating the rat4
I
Sec. 10.4
693
Heterogeneous Data Analysis lor Reactor Design
we see that if both sides of Eq~~arion (In-80) are divided by PH,P.I.and the equation is then inverted, L~nearizethe rate equation to extract the rate law panmeters.
A linear-leastI
Fquares analys~sof the data fhown in h b l e 10-6 is presented on the CD-ROM.
The regression techniques described in Chapter 5 could be used to determine the rate law parameters by using the equation (R5.1-3) One can use the linearized least-squares analysis to obtain initial estimates of the parameters k, KT, KB,in order to obtain convergence in nonlinear regrecsion. However, in many caaes i t is possible to use a nodinear regression analysis directly as described in Section 5.2.3 and in Example 10-4. Example 10-2
Regression Analysis tn Determine the lWodel Parameters k, K g , nnd KT
The data from Table 10-6 were entered into the Polymath nonlinear-least-squms program with the following modification. The rates of reaction in column 3 were multiplied by 10"'. so that each of the number< in column I was entered directly (1.e.. 71.0. 71.3, ...). The model equation was
Living Example Problca
Rate =
k P ~ P ~ I E +KRP,+K,rPT
Following the step-by-step regression procedure in Chapter 5 and the Summary Notes, we arrive at the following pamrneter valufi shown in Table E 10-2.I.
-
Model: WTE k'PT'PH2lll +KB'PB+WPr)
Nonlimr regrea*on s e l t i q Max # l t s m h s = E-l
Precbn R*2
E
0.9999509
O.gB8~0.1128555 variance r 0.2508084
RaZadj ~msd
r
694
I
Catalysis and Catafylic Reactors
Chap. 10
Converting the rate law to kilograms of catalyst and minutes,
we have
Ratio of sites occupied hy toluene to those occupied by benzene
After we have the adsorption constants, KT and K,, we can calcuIaze the ratio of sites occupied by the various adsorbed species. For example, the ratio of toluene sites to benzene sites at 40%conversion is
We see that at 40% conversion there are approximately 12% more sites occu'
pied by toluene than by benzene. 10.4.4 Reactor Design
Our next step is to express the partial pressures P,. PB,and PH1as a function of X, combine the partial pressures with the rate Iaw, - r i . as a function of conversion, and carry out the integration of the packed-bed design equation
I
Brample l&3
Fired (i. @,. Packed)-Bed Reactor Design
The hydrodemethylation of roluene is to he carried out in a packed-hed reacror. Plot the conversion. the pressure ratio, p and the panial pressures of toluene, hydrogen. and benzene as a function of catalyst weight. The molar feed rare of toluene to the reactor is 50 rnollmm, and the rt'actor is operated at 40 atm and 640°C. The feed consists of 30% toluene, 459 hydrogen, and 25% inerts. Hydrogen is used in excess to help prevent coking. The prewure drop parameter, a. is 9.8 x lov5 kg-'. Also determine the catalyst welghr tn a CSTR w ~ t ha bulk density of 400 kg/m7 10.4 p l c d ) .
Living Exsmplc Problc
1
1 . Design Equation:
Sec. 10.4
Heterogeneous Data Analysts for Reactor Design
Balance on loluene (T)
2. Rate Law. From Equation (EIO-2.1) we have
with k = 0.00087 moIJam21kgcatlmin md K, = 1.39 amrl and KT = 1.038 c~m-1. 3. Stoichiometw:
Relating
Toluene (T) Benzene
(B)
Hydrogen (H,)
Ps Because P,,
E
= 0.
=P r I , X ~
I E TO-3.5)
we can use the integrated fortn of the pressure drnp term.
tn!al pressure at the enlrance
=
Note that P,
designate? the inilia1 partial pressure of toluene. In t h ~ sexample ~ h c ~nit~al total pressure 1s designated Po to avojd any confi~sjon.The initial mole f r x tion of toluene is 0.3(i.e..! , = 0.3). so that the initial partial prewre of toluene i s
Pressure drnp in PBRn i$ disc'u~scdin Section 4.5
PI,, = 10..1)(40) = 12 atm The maximum catalyst weight we can hate and nch fall below I atm 1s found from Quation (4-33) for an entering pressure of 40 atm and an exit pressure of I atm.
Conwquentll;, we will sct our final \\eight 10.000 kg and deterniinc the cclntel \lc>n as a fi~nction of catalyst we~ght up to th!r ~altle.Equntions tElO-3.1) through
696 The c:~lculatiuns f ( ~ r nn 1 P are given url the CD-ROM.
~ e f e r e n c Shelf t
Catalysis and Ce!aly?lc Reactcrs
( E 10-3.5) are hewn in the Polymath program in Table E 10-3 I. The conversion
Differentialequations as entered by the user [ 1i d(X)ld(w) = - W T O Explicit equations as entered by the user [l] m o = 5 0 [ 2 ] k-.00087 [ I ] KTr1.038 I 4 1 KBz1.39 [ 5 1 alpha = 0.000098 I6; P0=40
Living Example Problcr
Chap.
PTo = 0.3"Po t 8 l y t (I -afpha*w)"0.5 [7 1
[ 9 J P=y'Po [ 1a 1 PH2 = FTo'(l.5-X)*y [ 11 1 PB = PTo'X'y
[I21 PT = PToa(1-X)'y [?3 1 ti = -k'PT'PHZ(l +KB'PB+KT'PT) [I41 R A T E = - I ~
Conversion pmtile
down the packed hed
w (kg) Figure E10-3.1
Conversion and pressure ratio profiles.
Heterogeneous Oata Analysis for Reactor Design
1
=
Note the partial prcbsure ot knzenc goes rhrouph a n~nximt~rn. Why?
I Fluidized
We note in Figure EiR3.2 that the partial pressure o l hcnzenc goes through a rnnximom rs a rewit of the dccma.r in m1.1 presiurc becrlrie of ihc pressure d l v y We wili now criculvre the fluidized CSTR catalyst we~ght necessary to achieve the rrme conveninn as in the packed-bed reactor at the sitme operating conditions. The bulk density in the fluidized reactor is 0.4 gicm'. The design equation is
CSTR
A
In
-
Out
+
Gen
=
Accum
Fra
-
FT
+
r;w
=
0
Rearranging
Writing Equation (E10-2.3) in terms of and P,, = I 2 atm. we have
-rf
conversion
and then substituting X = 0.65
8.7x104prp~4 8.7~10~P?m(l-~)(1.5-~ =2,3x10-~ mo1 I + I.39PBr i . 0 3 8 ~ ~ = I.39PwY+ i+ I038P,(l -R) kgcat min
W =d F
-
=
(50 mol .F/min)lO.65) 2 . 3 loLJ ~ mnl T/kg cat-min
W = 1.4x lo4 kg of catalyst
698
How can the
ueieht of ca'aly''
he reduced?
Catalysis and Catalytic Reactors
Chap. 10
These values of the catalyst weight and reactor volume are quite high, especially for the low feed rates given. Consequently, rhe temperature of rlre r~acrirtg17ri.xt1rreshould be increased to reduce rke catcslysr weighr, pru~dded rlrar side lracfions do nnt become a problem or higher remperofure.r. Example 10-3 illustrated the major activities pertinent to catalytic reactor design described earlier in Figure 10-7. In this example the rate law was extracted directly from the data and then a mechanism was found that was consistent with experimental observation. Conversely, developing a feasible mechanism may guide one in the synthesis of the rate law.
10.5 Reaction Engineering in Microelectronic Fabrication 10.5.1 Overview
We now extend the principles of the preceding sections to one of the emerging technologies in chemical engineering. Chemical engineers are now playins an imponant mle in the electronic< industry. Specifically. they are becoming more involved in the manufacture of electronic and photonic devices and recording materials, Surface reactions play an important role in the manufacture of microelectronics devices. One of the single most important developments of this century was the invention of the integrated circuit. Advances in the development of integrated circuitry have led to the production of circuits that can be placed on a single semicanductor chip the size of a pinhead and perform a wide variety of tasks by controlling the electron flow through a vast network of channels. These channel?. which are made from semiconductors such as silicon. gallium asrenide. indium phosphide. and germanium. have led to the development of a multitude of navel microelectronic devices. Examples of m~croelectronicsensing devices manufactured ucing chemical reaction engineering principle$ are shown in the lefr-hand margin. The manufacture of an integrated circuit requires the fabrication of a network of p a r h ~ a y sfor elearom, The principal reaction engineering steps c ~ the f fabrication process include depositing material on the surface of a material called a substrate (e.g.. by chemical vapor deposition), changing the conducri\,il!: of regions of the wrface (e.g., by boron doping or ion-inplantation). and rernrwing unwanted material le.g.. by etchrng). Ry applying these \rep? syctemalically. miniature electronic circuits can he fabricated on very small senriconductor c h i p . The fabricat~onof microelectronic de~cicesmay include d~ few ?O or as many as 2110 individual sieph ro produce chipc with up lo 10V elernc~~ls per chip. An ahhreriated schematic of the r;tep\ involved i n producing a typical mclal-oxide semiconductor field-effect tr:in~istor (MOSFET) d e ice ~ is \IJC>!V~ in F i p r c 10-10. S~artingfrom the upper left, u'c xee that single-cr>st;il hilrcnn inpol\ are grotvn in a C7c>chr;il\ki cryrtallizer. sliced into wafers. and chemically al~dph\.\icall> poli5hed. T1iec;e porished wafer\ cerve ac \tarting material5
Sec. 10.5
Reaction Engineering in Microelectronic Fabrication
699
for a variety of rnicmelectronic devices. A typical fabrication seguence is shown for processing the wafer beginning with the formation of an SiO, Iayer on top of [the silicon. The Si02 layer may be formed either by oxidizing a silicon layer or by laying down a SiOz layer by chemical vapor deposition (CVD). Next, the wafer is masked with a polymer photoresist, a template with the pattern to be etched onto the SiO, layer is placed over the photoresist. and the wafer is exposed to ultraviolet irradiatjon. If the mask is a positive photoresist, the light will cause the exposed areas of the polymer to dissolve when the wafer is placed in the developer. On the other hand, when a negative photoresist mask is exposed to ulrraviolet irradiation, cross-linking of the polymer chains occurs, and the unexposed areas dissolve in the developer. The undeveloped portion of the photoresist [in either case) will protect the covered areas from etching.
CVD {two Rlmrl
Mm&, Etzh, then Strip Mslk
E i l h %#Id4
Wiw
CW.Muk. Etch. S ~ m pMh.L
D
m bu Phmphmr m w h
CYO of Rnal Layer
Figure 10-20 Micrmltctronic fabrication steps.
700
Catalysis and Catalytic Reactors
Chap.
Afier the exposed areas o f SiO? are etched to form trenches [eirher wet etching (see Problem P5-12) or by plasma etching), the remaining pha resist i h removed. Next. the wafer i s placed in a furnace containing gas mc cules of the desired dopant. which then diffuse into the exposed silicon. Al diffusion of dopant to the desired depth in the wafer, the wafer is removed : covered with S i 0 2 by CVD. The sequence o f masking, etching, CVD, and rr allization continues until the desired device is formed. A schematic of a fi chip is shown in the lower right-hand corner of Figure 10-20. In Section 10. we discuss one of the key processing steps, CVD. 10.5.2
Etching
We have seen in Figure 10-20 that etching (i.e,. the dissolution or physical chemical removal of material} is also an important step in the fabrication F crss. Etching takes on a priority role in rnicrmlectronics manufactur because of the need to create well-defined structures from an essentialiy hor geneaus material. I n integrated circuits, etching is necess~uy to remi unwanted material that could provide alternative pathways For electrons :
Reference Shelf P1U.3
thus hinder opersltion of the circuit. Etching is also of vital importance in fabrication of micr~rnechanical and optoelectronic devices. By selectit etching semic~nducrossurfaces. it is possible to fabricate motors and v a l ~ ultrasmall diaphragms that can sense differences in pressure, or cantile beams that can sense acceleration. In each of these applications, proper etch is crucial to remove material that would either short out a circuit or hin movement of the rnicromechanicrtl device. There are two basic types of etching: wet etching and dry etching. ' wet etching process, as described in Problem P5-1ZB, uses liquids such as or KOH to dissolve the layered material that is unprotected by the photore mask. Wet etching is used primarily in the manufacture, of rnicrornechan devices. Dry etching involves gas-phase reactions, which form highly reac species, usually in plasmas. that impinge on the surface either to react with surface, erode the surface, or both. Dry etching is used almost exclusively the fabrication of optoelectronic devices. OptoeZectronic devices differ f~ microelectronic devices in that they use light and electrons to c a q out tl particular function. That function may be detecting light. transmitting light emitting light. Etching is used to create the pathways or regions where I can travel and interact to produce the desired effects. Appliances using s devices include remote controls for I T sets, LED displays on clocks microwave ovens, laser printers, and compact disc players. The material on CD-ROM gives examples of both dry etching and wet etching. For dty etch the reactive ion etching (RIE) of InP is described. Here the PSSH is usec arrive at a rate 3aw for [he sate of etching. which is compared with experime observation. In discussing wet etching, the idea of dissolution catalysis is in duced and rate laws are derived and compared with experimental obsewatic In the formation of microcircuits. electrically interconnected films laid down by chemical reactions (see Section 12.10). One method by wl these films are made is chemical vapor deposition.
Sec. 30.5
Reaction Engineering in Mic~oelectronicFabricallan
10.5.3 Chemical Vapor Deposition The mechanisms by which CVD occurs are very similar to those of heterogeneous catalysis discussed earlier in this chapter. The reactant(s1 adsorbs an the surface and then reacts on the surface to fom a new surface. This process may be followed by a desorption step. depending on the particutat reaction. The growth of a germanium epitaxial film as an interlayer besetween a gal-
lium arsenide layer and a silicon layer and as a contact layer is receiving Ge used in %Iar
increasing arteation in h e microelectronics industry.20Epitaxial germanium is also an important materia1 in the fabrication of tandem solar cetls. The growth
of germanium films can be accomplished by CVD.A proposed mechanism is
Gas-phase dissociation: GeCt4(g)
GeCl,(g) k*
+ Cl, (g)
GeCI, .S
Adsorption:
GeC12(gj + S
Adsorption:
Hz+ 2 S
Surface reaction:
GeCl, - S + 2HmS ----+ Ge(s) + 2HCl(gj + 25
2B.S ks
At first it may appear that a site has been lost when comparing the right- and left-hand sides of the surface reaction step. However, the newly formed gem* nium atom on the Fight-hand side is a site for the future adsorption of H2(g) or GeCt2(g), and there are three sites on both the right- and left-hand sides of the surface reaction step. These sites are shown schematically in Figure 10-2 1.
Figure 10-21
The surface reaction between adsorbed molecular hydrogen and germanium dichloride is believed to be rate-limiting: Rate law for rate-limiting step
where
r:,,
= deposition
rate, nmls
ks = surface specific reaction
rate, nm/s
W .Ishii and Y. Takahashi, J. EIectrnckem SM., 135, p. 1539.
702
Catalysis end Catalytic Reador$ fGCCI2 =fraction
fH
Chap, 10
on the surface occupied by germanium dichloride
=fraction of the surface covered by molecular hydrogen
The deposition rate (filmgrowth rate) is usually expressed in nanometers per second and is easily convened to a molar rate (mol/m2+s)by multiplying by the molar density of solid germanium. The difference between developing CVD rate laws and sate laws for catalysis is that the site concentration (e.g., C,) is replaced by the fractional surface area coverage (e.g., the fraction of the surface that is vacant,f,). The total fraction of surface available for adsorption should, of course, add up to 1.0. Fractional area balance:
Area balance
We will first focus our attention on the adsorption of GeC12.The rate of jumping on to the surface is proportional to the partial pressure of GeC12. P,,,,, and the fraction of the surface that is vacant, f,. The net rate of GeCI, adsorption is
Since the surface reaction is rate-limiting, in a manner analogous to catalysis reactions, we have for the adsorption of GeCIz Adsorption of
GeCI, not rate-limiting
Solving Equation ( 10-84) for the fractional surface coverage of GeCI2 gives
For the dissociative adsorption of hydrogen on the Ge surface, the equation analogous to (10-84) is
Since the surface reaction is rate-limiting, Adxrrption of A: is nor rate-limiting
Then
Reaction Engineering in Microelectronic Fabricatjon
See. 70.5
703
Recalling the rate of deposition of germanium, we substitute for f,,,,? and fH in Equation (10-82) to obtain
We solve forf, in an identical manner to that for C. in heterogeneous catalysis. Substituting Equations ( 1 0-85) and 10-87) into Equation ( I 0-83) gives
Rearranging yieIds
Finally. substituting for,f, in Equation ( 1 0-88).we find that
and lumping K,, KH, and k, into a specific reaction rate k' yields Rare of deposition of Ge
If we assume that the gas-phase reaction Equilibrium in gas phase
GeC&lg) -+ C12(g)
GeCl,(gI is in equilibrium, we have
and if hydrogen is weakly adsorbed. tion as
I"'
Dcp
= ,(<
1 ) we obtain the rate
of deoosi-
=
Ir should also he noted that it i s possible that GeCI? may also be formed by the reachon of GeClj and a Ge atom on rhe susface, in which case a different rate Inw would result.
v;
704
Catalysis and Catalvtic Reactors
Chaa
10.6 Model Discrimination
) ;:* .
$- : : : :
-iI
i' Rtgmqslun
'
We have seen thnr for each mechanism and each rate-limiting step w e derive n rslte law. Consequently. if we had three possible mechanisms and tl: rate-limiting steps for each mechanism, we would have nine possible rate 1: to fit to the experimental data. We win use the regression techniques discus in Chapter 5 to identify which model equation best firs the data by choos the one with the smaller sums of squares andor carrying out an F-test. could also compare the residual plots for each model, which not only show error associated with each data point but also show if the error is randomly I tributed or if there is a trend in the error. If the error is randomly distribu this result is an additional indication that the correct rate law has been cho: We need to raise a caution here about choosing the model with the sm est sums of squares. The cnution is that the model parameter values that the smallest sum must be realistic, In the case of heterogeneous caralysis. values of the adsorption equiribrium constant must be positive. In additior the temperature dependence is given. because adsorption is exothermic. adsorption equilibrium constant must decrease with increasing temperature, illustrate these principles, let's look at the following example. Example 1W Hydrogenation of Ethylene to Ethane The hydrogenation (HIof ethylene (E)of Form ethnne IEA).
i.; carried out over a cobalt molybdenum catalyst (Col/ccr.Czecl?.Cltrm. Cnmm 51. 2760 (1988)l.Carry our a nonlinear regression analysis on the data give Table E 10-4.1, and determine which rate law best descriks the data.
Rtrr~
Number
Reacrion Rate cur. s)
P, (ortnl
/mot&
PEA lam)
(n~~n)
I
1.04
1
I
I
2 3
3.13
I
I
3
5.21
1
I
5
4
3.82
3
1
3
5
4.I9
5
1
3
b
2.391
0.5
1
3
[~r~>calure E ~ ~ t d.~ta sr Enter rnodcl .\fake i n i [ i i ~ l
I
Detcrriii~~c whrch of thr f n l l o ~ i n frdtc law\ hcht dotribes fhr data.
t ' h ~ i n ~ : ~ ~01 e'c
parameters Run rrgrc+*ion Examine pararnelcrs and var~:u~ct. Observe ermr distri-
Id)
-r;k~;#"
bulion Choose ~ n d e l
Polymath was chohen as the saknarc package to qolve this probbn. The data in Table EIO-4.1 were typed into the >ystem. A screen-shot by screen-shot set of instruction( on how to carry out regrcssinn ix given on the CD-ROM and the web. After entering the dara and Folloulng the htcpby-step pmcedurc\ dewribed in the Summar!: Nntex o n the WeblCD-RO.11 nf Chapter 5. the rehults h w n in Table E E9-4.2 were ohtained. T~RI,EE IO-!.?.
RESVLTS OF TfIF. POLYMATHNONLINEAR REGK~SION
Model (b)
Model car
Model: RATE =
954 c L
3 3438825
.a
0.0428119 2 2:10797
KZ
haodd: RATE = k'Pe'PHZr(1 +KE'Pel
k'Pe'PHZ(l+KEA'Pea+KFPe)
.
m
.
k KE
3.2922510.0636262
Nanhnear regrewon swings Max r ~feretionsr 84 A'Z RA2adj
Rmsd Vdrtaxce
-
= 0.99B32? =
0 9979614
R-i
= 0 9935878
R"2adj
W l : RATE
I
0,0191217 0.W4938t
i?P'Pa'PHZ(I +KE'P9F2
Nonrmaar rqemlon wttings Max I deraiMs = 84
3.2636933
r
OW72547
k-sd z 0.022872 Varlacce = 0.0060534
Model Id)
Model (v)
Lying Example Problem
3.28799R
Nonltnear rsgression s e h g s Man # Rwatlon~z 84 Pruws~on
Pnets~ln
3.:867851 2.1913363
0 2395535
Mdsl: RATE = K'PWa'PHPb
I
N o n l i w r r e w ' c n sanings Max # aerat~ons 64
PrscWon
0.9831504 RA2adj :L,
s 0.97753%
0.0605757
Variance = 0.0495372
1
1
Catalysis and Catalytic Reactors
I
Chap. 10
Model (a)
From Table E10-4.2 data. we can obtain
We now examine the sums of squares (variance) and range of variables themselves. Whife the sums of squares is reasonable and in fact the smallest of all the mode15 at 0.0049.However, let's look at KEA. We note that from the values for the 9 5 8 confi. dence limit of s.0636 is greater lhan the nominal value of K- = 0.043 atm-I itself (i.e., K,, = 0.043 f 0.0636). The 95% rimes means that if the experiment were run 100 times then 95 times it would fall within the range (-0.021) c KEA ': (0.1066). Because KEAcan never be negative. we are going to reject this model. Consequently we set KEA= O and proceed to Model lb3.
Model (b) From Table E10-4.2 we can obtain
The value af the adsorption constant Kt = 2.1 atm-I is reasonable ayd is not negative w~thinthe 9 5 9 confidence Iirnit. Also the variance is small at a, = 0.0061.
Model (c)
From Table EIO-4.2 we can obtain
While KE is small. it never-goes negative within the 9 5 8 confidence inremar. The variance OF this model at 0;: = 0.0623 is much Eqer than the other models. Comparing the variance of model (c) with m d e l (b)
We see that the 0: i s an order of magnitude greater than a:, and we eliminate model (c).?'
Model (d)
Similarly for the power I a n model. we obtain from Table E104.2
"See G.F. Fromen1 and K. B Blshoff, Chem~culReacrion A n o l y ~ i lr,r~dDesign, ?r.d ed. (New York: Wlley, 19911). p. 96
Sec. 10.7
I
Catatyst Deactivation
As with m d e l (c) the variance is quite large compared to model (b)
Consequetltly, we see that for heterogeneous reactions, Larigmuir-Hinshelwoodrate laws are preferred over power law models.
(
Choose the Besf Model.
Because aII the parameter values are realistic for model (b) and the sums of squares are significantly smaller for model (b) than for the other models, we choose model (b). We note again that there is a caution we need to point out regarding the use of regression! One cannot simply cany out a regression and then choose the modet with the lowest value of the sums of squares. If this were h e case, we would have chosen model (a), which had the smallest sums of squares of aII the modefs with a: = 0.0049. However, one must consider h e physical realism of the parameters in the model. In model (a) the 95% confidence interval was greater than the parameter itself, thereby yielding negative values of the parameter, KAE,which is physically impossible.
10.7 Catalyst Deactivation In designing fixed and ideal fluidized-bed catalytic reactors, we have assumed up to now that the aclivity of the catalyst remains constant throughout the catalyst's life. That is, the total concentration of active sites, C,, accessible to the reaction does not change with time. Unfortunately. Mother Nature is not so kind as to allow this behavior to be the case in most industrially significant catalytic reactians. One of the most insidious problems in catalysis is the loss of catalytic activity that occurs as the reaction takes place on the catalyst. A wide variety of mechanisms have been proposed by Butt and PetersenV2? to explain and mcdel catalyst deactivation. Catalytic deactivation adds another level of complexity to sorting out the reaction rate law parameters and pathways. In addition. we need to make adjustments for the decay of the catalysts in the design of catalytic reactors. This adjustment is usually made by a quantitative specification of the catalyst's activity, a ( i ) . In analyzing reactions over decaying catalysts we divide the reactions into two categories: separable kinerics and nonsepamhle kinetics. I n separabfe kinetics. we separate the rate law and activity:
22
J. B. Butt and E. E. Petersen, Activa~inn,D~acrivarionand Poisoning of Corollars (New York: Academic Press, 1988). See also S. Szkpe and 0. Levenspiel. C h m . Eng. Sci.. 23. 88 1-894 ( 1968).
708
Catalysis and Gatalvtic Reactors
Chau
Separable kinetics: -r- = a(?ast history) X -r- (Fresh catalyst) When the kinetics and activity are separable, it is possible to study cata decay and reaction kinetics independently. However, nonseparability,
Nonseparable kinetics: -ri
= -rL
(Past history, Fresh catalyst)
must be accounted for by assuming the existence of a nonideal surface or describing deactivation by a mechanism composed of several elemenl stepsh2" In this section we shall consider only separable kinetics and define activity of the catalyst at time t, a ( t ) , as the ratio of the rate of reaction c catalyst that has been used for a time r to the rate of reaction on a fresh c lyst ( f = 0): a(]):
catalyst
n (r) =
activity
-ri(t)
-ri(t=Q)
Because of the catalyst decay, the activity decreases with time and a typ curve of the activity as a function of time is shown in Figure 10-22.
0
t
Figure 10-22 Activity as a function of time.
Combining Equations (10-92) and (3-2), the rate of disappearance of reac A on a catalyst that has been utilized for a time t is Reaction rate law accounting for catalyst activity
where a([)=catalytic activity, time-dependent k(T) = specific reaction rare, temperature-dependent C, = gas-phase concentration of reactants, products, or contamina
The rate of catalyst decay,
r,, can
be expressed in a rate law analol
to Equation (10-93): Catalyst decay rate law
D.T.Lynch and G. Emig, Chern. Eng. Sci.. 44(6). 1275-1280 (1989).
Sec. 10.7
709
Calalys! Deactivation
where p [ a ( t ) ]i s some function o f the activity, k, is the specific decay constant, and h ( C , ) is the functionality of r,, on the reacting species concentrations. For the cases presented in this chapter, this functionality either will be independent of concentration (i.e.. h = 1) or will be a Einear function of species concentration (i.e., h = Ci). The functionality of the activity term. p[a(rE], in the decay law can take a variety of forms. For example, for a first-order decay,
p(a) = a
( 10-95)
and for a second-order decay,
The particular function, p(al. will vary with the gas catalytic system being used and the reason or mechanism for catalytic decay. 10.7.1 Types of Catalyst Deactivation Sintering Fouling
,p i s o i g
There are three categories into which the loss of catalytic activity can tradirionally be divided: sintering or aging, fouling or caking, and poisoning. Deactivation by Sintering (Aging).Z4 Sintesing, also referred to as aging, is the loss of catalytic activity due to a loss of active sudace area resulting from the prolonged exposure to high gas-phase temperatures. The active surface area may be Eost either by crystal agglomeration and growth of the metals deposited on the support or by narrowing or closing of the pores inside the catalyst pellet. A change in the surface structure may also result from either surface recrystallizarion or the formation or elimination of surface defects (active sites). The reforming of heptane over platinum on alumina is an example of cataIysz deactivation as a result of sintering, Figure 10-23 shows the loss of surface m a resulting from the flow of the solid porous catalyst support at high temperatures to cause pore closure. Figure 10-24 shows the loss of surface by atomic migration and agglomeration of srnalr metal sites deposited on the surface into a larger site where the interior atoms we not accessible to the reaction. Sintering is usually negligibie at temperatures below 40% of the melting temperature of the solid.25
The catalyst suppon becomes soft and
pore closure.
Figure 10-23 Decay by sintering: p r e cIwure.
C. Kuczynski. Ed., Sinkring and Catalysis, Vol. 10 of Malerials Science Research (New York: Plenum Press. 1975). Z5R.Hughes, Deac~ivntionof Coraiyst.~(San Piego: Academic Press, 1984). a.i
See G.
Catalysis and Catalytic Reactors
Chap. 10
The atoms move along the surface and agglomerate.
1
Side View t=O
t=t
Figure 10-24 Decay by sintering: agglomeration of deposited metal sires.
Deactivation by sintering may in some cases be a function of the malnstre'am gas concentration. Although other fonns of the sintering decay rate Eaws exist, one of the most commonly used decay laws is second order with respect to the present activity:
Integrating, with a
=
1 at time r = 0,yields
Sintering: secondorder decay
The amount of sintering is usually measured in terns of the active surface area of the catalyst S,:
The sintering decay constant, kd,follows the Arrhenius equation
Minimlzine sintering
The decay activatiorr energy, Ed. for the refarming of heptane on Pt/A1,03 is on the order of 70 kcal/mot. which is rather high. As mentioned earlier sintering can be reduced by keeping the temperature below 0.3 to 0.4 times the meta1"s melting point. We will now stop and consider reactor design for a fluid-soIid system with decaying catalyst. To analyze these reactors, we only add one step to our a l p rithm, that is, determine the catalyst decay law. The sequence is shown here.
-
Mole balance
The algorithm
-
Stoichiornetry
Reaction rate law
-
+ Combine and solve
Decoy rure lnw
4
w Numerical techniques
Example 1 6 5 Cu!culding Convdssion with Catalyst Decoy in Batch Reactors
The first-order isomerization
is being carried our isothemally in a batch reactor on a catalyst that is decaying as a result of aging. Derive an equation for conversion as a function of time.
Solurion
I . Design equation:
One extra atep (number 3 I IS added to the alporithm.
3. Decay law For second-order decay by sintering: 1
a(r)=I +k,r 4. Stoichiometrg:
Folio;rf;ng the Algorithm
CA = CAO(I- X I =
iYA" (1 - X ) 'I
5. Combining gives us
k t k = k 'WIV Then, separatinp variables, we have
Substituting for 0 and inkgrating yields
, No Decay
XK
In-=-I I-X
kd
ln(l+bt)
6. Solving for the canversinn X at any time
I.
we find that
712
Catalysis and Catalyt~cReactors
Chap
T l i ~ \i< thc sollvcr\ri?li ~h:it w ~ l he l achiered In ;i hatch re;lckt)r I'or n hrht-ol reaction when the c:itaIy\~ ~lt'cayI:IW r < 5ecot1d order. The purpilke nT t h ~ \exam u L b k to d c r n o n \ t ~ ~the i c algcrrithni for isotherr~ialcati~lyticreactor h i p n for a dec ing catalyst In pr(,hlcm 1 0 - l ~ c )you are ncked to \kclch the tempcriltitrc-time jevtorizs for bariouk values of C and k,,.
Deactivation by Caking or Fouling. This mechanism o f decay (see F i y 10-15 and 10-26) is common to reactions involving hydrocnrbonr. It resL from a carbonaceous (coke) material being dcporited on the surhce of a ca
lyst.
Figure 10-25 Schematic oE decay hy coking.
cab Frcrh vnwltst
c h ~Srrnt c ; ~ t ~ [ y s t
Figure 10-16 Decay by voh~ng.(Photos courtefy o f Enpelh:~rtlcatalyrt. cnp?rlght b! M~ch:icl Gal'fney Photographer. ?vl.lendha~n.N.J.)
The amount of coke on the surface after a time r has been found to obey tl following empirical relationship:
where Cc is the concentration of carbon on the surt'ace ($/rn2) and n and A ar fouling parameters, which can be functions of the feed rate. This expressic was originally developed by VoorhiesZhand has been found to hoId for a wic variety of catalysts and feed streams. Representative values of A and n for t6 cracking of East Texas light gas oilz7yield Ib 27
A. Voorhies, Itid. Eng. Ci~ern..37, 3 18 1 1945). C. 0.Pnter and R. M.Lago, Adv. Cntal., 8, 293 (1956).
Sec. 10 7
713
Catalyst Deact~vat~on
Different functionalitier between the activity and amount of coke on the s~lrfacehave been observed. One commonly used form is
or, in terms of time,
*
For light Texas gas oil being cracked at 750°F over a syn~heticcatalyst for short times. the decay law is
where r i s in seconds. Activlty for deactrvatinn by rtlk~ng
Other commonIy used forms ace n = e-nlrc
and
A dimensionless fouling correlation has been developed by Pacheco and
Petersen.?$ hllnimizin c t 7 k i v
When possible. coking can be reduced by running at elevated pressures (3000 to 3000 kPa3 and hydrosen-rich streams. A number o f other techniques for minimizing fouling are discussed by B a r t h o l ~ r n e w Catalysts .~~ deactivated by coking can usually be regenerated by burning off the carbon. The use of the shrinking core model to describe repeneration is discussed tn Section I E .5.1.
Deactivation by Poisoning. Deactivation by this mechanism occurs when the poisoning molecules become irreversibly chemisorbed to active sites. thereby reducing the number uf sites available for the main reaction. The poisoning molecule, P, may be a rcactant andfor a pmduct in the main reaction. or it. may be an irnpusiry in the feed stream.
l 3 M.
A. Rcheco and E. E. Petersen, J. Ca~rrl.,86. 7 5 119841. Sept. 12. 1984. p. 96.
=TC. Rmtholornew, Chm~.Eng..
714
It.s going to c o s ~you.
Catalysis and Catalytic R~actors
Chap. 10
Side Note. One of the most significant examples of catalyst poisoning occurred at thk gasoline pump. Oil companies found that adding lead to the gasoline increased fie Dctane number. The television commercials said "We are going to enhance your gasoline, but it's going no cosr you for the. added tetra-ethyl lead." So for many years they used lead as an antiknock c m p nent. AS awareness grew about NO, HC,and CO emission, from the engine, it was decided to add a catalytic afterburner in the exhaust system to reduce these emissions. Unfortunately, it was found that the lead in the gasoline poisoned the reactjve catalytic sites. So, the televj sion commercials now said "We are going to take the Iead out of gasoline but to receive the same level of performance as without lead, but it's going to c o s ~you because of the added refining costs ta raise the octane number.'' Do you think that financially, the consumer would have been better off if they never put the lead in the gasoline in the first place?? Poison in the Feed. Many petroleum feed stocks contain trace impurities such as sulfur. lead, and other components which are too costly to remove, yet poison the catalyst slowly over time. For the case of an impurity, P,in the feed stream, such as sulfur, for example, in the reaction sequence A+S
Main reaction:
A-S
B*S Poisoning reaction:
p+S
i
(A-S)
kc,
(B-S+C(g)) - r i = a ( t ) I +KACA+KBCn (B+S)
,P,
r d = -- = k ' C m a q dr
I'
the surface sites would change with time as shown in Figure 10-17. .A
P
+.~r,j(Jc)& . ..()QQQQ
Frogresqinnoi
t
t2
1
F~RuIT 10-27
t3
Decay by poisoning.
,,
If we assume the rate of removal of the poison, r , from the reactant gas stream onro the catalyst sites is proportional to the number of sites ithat are unpoisoned (Cto - Cp S ) and the Loncentration of poison in the gas phase. Cp: r f "=~ k d ( c t f l
- CP.S)CP
Cp s is the concentration of poi\oned sites and C,, is the total number of sites initially available. Because every molecule that is adcorbed frnln the gas phase onto a site i s assumed to poison the srle. this rate is alsa equal to the rate of rernovar of total active sites (C,} from the surface: where
Sec. 10.7
Catalyst Deactivation
715
Dividing through by Cfl and letting f be the fraction of the total number of sites that have been poisoned yields
The fraction of sites available for adsorption (1 - f ) is essentially the activity o ( t ) . Consequently, Equation (1 0- 108) becomes
A number of examples o f catalysts with their camspondiag catalyst poisons are given by Bartholornew.30
Packed-Bed Reactors. In packed-bed reactors where the poison is removed from the gas phase by being adsorbed on the specific catalytic sites, the deactivation process can move through the packed bed as a wave front. Here. at the start of the operation, only those sites near the entrance to the reactor will be deactivated because the poison (which is usually present in trace amounts) is removed from the gas phase by the adsorption: consequently, the catalyst sizes fmher down the reactor will not be affected. However. as time continues, the sites near the entrance of the reactor become saturated, and the poison must travel farther downstream before being adsorbed (removed) from the gas phase and attaching to a site to deactivate it. Figure 10-28 shows the corresponding activity profile for this type of poisoning process. We see in Figure 10-28 that by time 1, the entire bed has become deactivated. The corresponding overall conversion at %heexit of the reactor might vary with time ar shown in Figure 10-29.
F I g u ~ e10-a Movement of activity front in a packed bed.
The partial differential equations that describe the movement of the reaction front shown in Figure 10-28 are derived and solved in an exampre in the CD-ROMIWeb Summary Notes for Chapter 10. Summarv Nmtes
R. J. Farrauta and C. H . Banholomew, F u ~ ~ d a r n p ~ ~off uIrrdrr~rr-ial ls Caralylic Pro(New York: Blackie Academic and Professional, 1997). Thk hook is one of
ces.cec
the definitive resources on catalyst decay.
Catalysis and Catalpic Resctors
Chap,
Figure 10-29 Exit conversion as a function of time
-
Poisoning By Either Reactants or Products. For the case where the ma reactant also acts as a poison, the rate laws are:
Main reaction:
A+S
Poisoning reaction:
A +S
B+S
- r: = kACi
A. S
rd = kiCFaq
An example where one of the reactants acts as a poison is in the reactic of CO and HZover ruthenium to form methane, with
Similar rate taws can be written for the case when the product B acts as a poisol For separable deactivation kinetics resulting from contacting a poison , a constant concentration Cpoand no spatial variation: Separable deactivation kinetics
The solution to this equation for the case of first-order decay, n = I
Key Resource for camlyst deactivation
Empirical Decay Laws. TabIe 10-7 gives a number of empiricaI decay la% along with the reaction systems to which they apply. One should also see Fundamentals of Industrial Catalytic Processes, b Farrauto and BarthoEomew?l which contains rate laws similar to those in Tabl 10-7, and aIsa gives a comprehensive treatment of catalyst deactivation.
Sec. 10.7
Catalyst Deactivation
DECAY RATELAWS
T A R I10-7. ~ Functional
Deca~
Form ojAc6viryReaction Q r d ~ r DiferenrinI Farm Linear
0
- da = Dq d? do dt
Integral Form a =1
- pol Conversion of pam-hydrogen on tungsten when poison& with oxygena
a r e-'jF
--=B,a
Examples
Ethylene hydrogenation on Cu poisoned with
CO" Parafin dehydmgenation on Cr- Altol'
Examples of reactions with decaying catalysts and their decay laws
Cracking of gas oild Vinyl chloride monomer
formatione Hyperbolic
2
-dadr = ~
$ 2
-1 = 1 +pz I a
Vinyt chloride monomer
formation Cyctohexanc dehydrogenation on Pt/AlzOJe lsobutylede hybgenation on Nih
Reciprocal power
k-! =y PJ
-da -= P,oq.4P dt
a =dot-')
Cracking of gas oil add
gasoline on clayi
"D.D. Eley and E.J. Ridenl, Pmc R. SOC. Lurtdon. A178, 429 (194I). bU. N. Pease and L. Y.Steward. J. Am. Chem. Soc., 47. 1235 (19251. cE F. K. Herington and E. J. RideaI, Pmc. R. Soc. Landon, AJW, 434 (1945). dV.W.Weekman, End. Eng C k m . Pmcsss Des. Dev.. 7,90 (19683. CA.F. Ogunye and W. H. Ray, Ind. Eng. Chem. Ptwccss Des. ku., 9.6t9 (1970). 'A. F. Ogunye and W H. Ray, Ind. b ~ g Chem. . Pmcess Des. Dev., 10.410 (t971). *H V. Maat and L. Moscou. Pmc. 3rd lnt. Gong,:Coral. (Amsterdam: Nh-Holland. 1965), p. 1277. hA. t. POZZ~ and H. E Rase, Ind. Eng. Chem.. 50, 3075 (1958). 'A. Voorhies, Jr., Ind. Eng. Chem,. 37. 3 I8 (19d5); E. B. Maxted, 4hr. Cutal.. 3, 129 (i95I). JC.G.Ruderhausen and C. C. Watson. Chem. Eng. Scr., 3, I10 (t954). Source: J. B. Butt, Chemical Reactor Engineering-Washington. Advances in Chemistnr Series 109 (Washington. D.C.: American Chemical Swiety. 1972). p. 259. Also see CES 23, 883(19683
Example 2 0 4 CutaIyst Decay in a Fluidized Bed Modeled as a CSTR The gas-phase cracking reaction'
I
Gas oil (g)
.----t
Products (g)
* For simplicity. gas oil is used to represent the reactive portion of the feed. In actuality, gas oil, distilled from crude, is made up of complex hydrocarbons, which can be cracked. and simple hydrocarbons, which will not crack and are therefore inart in this
application.
7f8
Catalysis and Catalytic Reactors
Chap. 10
is carried out in ayuidized CSTR reactw. The feed stream contains 80% crude (A) and 20%ineR I. The m d e oil contains sulfur compounds, which poison the catalyst. As a first approximation we will assume that the cracking reaction is first order in the crude oil concentration. The rate of catalyst decay is first order in the present activity, and first order in the reactant concentration. Assuming thar the bed can be modeled as a well-mixed CSTR, determine the reactant concentration. activity, and conve~ionas a function of time. The volumetric feed rate to the reactor is 5000 rn31h. There are 50,000kg of catalyst in the reactor and the bulk density is 500 kglml. Additional informarion: Living Example Problem
CAn= 0.8 m o l l d d
k = p,k' = 45 h-I
Cm=I.0moltdm3
kd=9dm31mol-h
Solrrrion 1. Mole Balance on reactant:
Recalling hr, = CAYand r,V =
ri W
.then for constant volume we have
2. Rate Law:
3. Decay taw:
4. Stoicbiometry (gas phase. P = Po. T = To). From Equation (3-41) we have
Sec. 10.7
Catalys! Deactivation
Solving for v yields
5. Combining gives us
Dividing both sides of Equation [[email protected])by the volume and writing the equation in terms of *r = V/o, ,we obta~n
As an approximation we assume the conversion to be
Calculation of reactor volume and space time yields
are solved using Polymath as the Equations (EIO-6.4). (El0-6.9). and (E10-6.10) ODE solver. The Polymath program is shown in Table E10-6.1. The solution is shown in Figure El 0-6.1. The conversion variable X does not have much meaning in Row systems not at steady state, owing to the accumnlation of reactant. However, here the space time is relalively short (7 = 0.02 h) in comparison with the time of decay r = 0.5 h. ConsequentIy. we can assume a quasi-steady state and consider the conversion as defined by Equat~on(E10-6.10) valid. Because the catalyst decays in less than an hour, a fluidized bed would not be a gmd choice to carry out this reaction.
720
Catalys~sand Catalyt~cReacrors
Chap.
Dderent~alequarhs as entered by the usur ' . d(a)/d.(t)= -kd'a'Ca r . d(Cal1dlt) = CaWtau-((l+yao)l(l+CalCIO)+lau+aBk~~au
i Z ! Cao=.a :: ; tau = .D2 1:- CtO-1. i4: k=45 [ 6 1 yao = CaOICr0 i" : X = I-(l+yao)/(l+WCiO)'WCaO
C,.X , and c i time trajt.ctories In a CSTR rmt at steady state
I (hi
Figure EIO-6.1
Variation of CA.o, and X with time in
:I
CSTR.
We wilI now consider three reaction systems that can be used to hand systems with decaying catalyst. We wiIl classify these systems as those h a v i slow. moderate, and rapid losses of catalytic activity. To offset the: decline chemical reactivity o f decaying catalysts in continuous flow reactors. the f c lowing three methods are commonly used: Matching the reactor type with sped of catalyst decay
Slow decay - Tempercrture-Time Trajecrorifs ( 10.7.2) Moderate decay - Moving-Bed Reactors ( 10.7.3) Rapid decay - Straight-Through Transport Recrctors ( 1 0.7.4)
Sec. 10.7
Catalyst Deactivation
10.7.2 Temperature-Time Trajectories I n many large-scale reactors, such as those used for hydrotreating, and reaction systems where deactivation by poiron~ngoccurs, the catalyst decay is wlatively slow. In these continuous Rot+ systems, constant conversion is usually necessary in order that subsequent processing steps (e.g.. separation) are not upset. One way to maintain a constant conversion with a decaying catalyst in a packed or fluidized bed is to increase the reaction rate by steadily incrrasing the feed temperature to the reactor. Operation of a "fluidized" bed in this manner is shown in Figure 10-30.
Figure 10-30 Reactor with preheater ro increase feed tempemlure.
We are going to increase the feed temperature in such a manner that the reaction rate remains constant with time: -r; ( 6 = 0" To) = - r i ( r , 1)= a((.T){-r,i (r = 0, T ) ]
For a first-order reaction we have
k(To)CA = a ( [ , ~ ~ ( T ) C A We will neglect any variations in concentration so that the product o f the activity (a) and specific reaction rate (k) is constant and equal to the specific reaction rate, ko at time E = 0 and temperature To;that is,
k ( T ) n ( t ,T ) = k,
(10-t 13)
G t a d ~ ~ lrals~ng ly The goal is to find how the temperature should be increased with time lie., the the tenlperature can temperature-time trajectory) to maintain constant conversion. Using the Arrheo,.fsessteffects equation to substitute fork in terms of the activation energy, EA, gives ~ n t a l y r td ~ ~ y nius .
Solving for 1JT yields
The decay law also follows an Arrhenius-type temperature dependence.
Catalysis and Catalytic Reactors
where k, =decay constant at temperature To, s-I EA = activation energy for the main reaction (e.g,, A Ed = activation energy for catalyst decay, ldlrnol
Chap. 10
+JB), kllmol
Substituting Equation (10-I f 5 ) into (10-116) and rearranging yields
Integrating with a = 1 at t = O for the case n + (I
+ ,EdlE,),
we obtain
Solving Equation ( 1 0- I 14) for a and substituting in (10-1 18) gives
Equation (10-1 19) tells us bow the temperature of the catalytic reactor should be increased with time in order for the reaction rate to remain constant. In many industrial reactions, the decay rate law changes as temperature increases. In hydrocracking, the temperaturetime trajectories are divided into three regimes. Initially. there is fouling of the acidic sites of the catalyst followed by a linear regime due to slow coking, and finally, accelerated coking characterized by an exponential increase in temperature. The temperature t' me trajectory for a deactivating hydrocracking catalyst is shown in Figure 10-31. For a first-order decay, Krishnaswamy and Kinrellk expression [Equation (1 0- l 19)Jfor the temperature-time trajectory reduces to
-
10.7.3
Moving-Bed Reactors
Reaction systems with significant catalyst decay require the continual regeneration andlor replacement of the catalyst. Two types of reactors currently in commercial use that accommodate production with decaying catalysts are the moving-kd and straight-through transport reactor. A schematic diagram of a moving-bed reactor (ured for catalytic cracking) is shown in Figure 10-31,.
Sec. 10.7
Catalyst Deactivation
723 O Run 3
A Run 4 Model Comparing h o r y arid exprirnent
100
0
200
300
400
T ~ n l e(h) Run
EA
A
Ed
Ad
Figure 10-31 Temperature-time hjectories for deactivating hydrocracking catalyst, runs 3 and 4. [Reprinted with permission from S. Krishnaswamy and J. R. Kirtrell. Ind. En%. Chem. P r r m ~ sDes. D m ,18, 3 9 (1979). Copyright 8 1979 American Chemical Society.]
Moving bed reactor: Used for reactions with moderate rate of catalyst decay.
Figure 10-32 Thennofor calalytic cracking (TCC)unit. [From V. Weekman. AlChE Monogr. Ser.. 7 3 1 11. 4 (1939), With permission of the AIChE. Copyright 8 1879 AIChE All riphts rererved.]
Catalysis and Catalytic Reactors
Chap.
The value of the catalyst canrained in
a reactor
. of this type
is approximately
S1 million
Figure 1633 Moving-bed reactor-schematic.
The freshly regenerated catalyst enters the top of the reactor and th moves through the reactor as a compact packed bed. The cataIyst is coked co tinually as it moves through the reactor until it exits the reactor into the kiI
where air is used to bum off the carbon. The regenerated catalyst is lifted fro the kiln by an airstream and then fed into a separator before it is returned the reactor. The catalyst pellets are typicaily between and in. in diameter The reactant feed srream enters at the top of the reactor and flows rapid through the reactor relative to the flow of the catalyst through the reactor (Fi;
a
ure 10-33). If the feed rates of the catalyst and the reactants do not vary wi time, the reactor operates at steady state: that is, conditions at any point in rt reactor do not change with time, The mole balance on reactant A over AW i:
Molar
Molar
Molar
Molar
Dividing by b W, letting AW approach zero, and expressing the flaw rate i terms of conversion gives
The rate of reaction at any time r is
The activity. as before. i s a function of the time the catalyst has been in contact with the reacting ga6 rtseam. The decay rate law is
We now need to relate the contact time to the weight of the catalyst. Consider a point z in the reactor, where the reactant gas has passed cocurrently through a catalyst weigh;LV Since the solid catalyst IS moving through the bed at a rate U,(mass per unit time), the time t that the catalyst has been in contact with the gas when the catalyst reaches a point :: is I
Tf we
W =-
(10-122)
ri
now differentiate Equation (10- 122)
-
sit = dW U T
and combine it with the decay rate taw, we obtain
The activity equation is combined with the mole balance: The design equation for moving-kd reactor%
Example 1 6 7 Catalytic Cracking in a Moving-Bed Reactor The catalytic cracking of a gas oil chaye. A, to Form Cj* (B)and to form coke and dry gas (C)ir to be carried out in a screw-type conveyor moving-bed reactor at 900°F:
Llulng Example Problem
Gas oi,
CCsDry gas, Coke
(
This
~ t i 0 I - can I
also be vritrm ss A
k~
---4
Products
While pure hydmarbons are known to crack according to a first-order rare law, the fact that the gas oil exhibits a wide spectrum of cmcking rates gives rise to the fact that the lumped cracking rate i s well represented by a second-order rate law (see Problem CDP5-HB}with the following specific reaction '?Estimated from V, W. Weekman and D. M. Nace. AIChE J.. 16, 397 (1970).
726
Catalysis and Catalytic Reactors
-rA = 0.60
(drn)6
(g cat) (rnol) (min)
Chap. 10
c2
The catalytic deactivation i s independent of gas-phase concentration and follows a first-order decay rare law, with a decay constant of 0.72 reciprocal minutes. The feed stream i s diluted with nitrogen so that as a first approximation, volume changes can be neglected with reaction. The reactor contains 22 kg of catalyst that moves through the reactor at a rate of 10 kglmin. The gas oil is fed at a rate of 30 moI/rnin at a concenBation of 0.075 moIfdm3. Determine the conversion that can be achieved i n this reactor. Solution
1. Design Equation:
2. Rate Law:
3. Decay Law. First-order decay
Moving Ws: moderate rate of catalyst decay
Using Equatlon (10-124), we obtain
I
Integrating
4. Stoichiomefry. If v e u , [see Problem PI 0-2(g)] then
C, = C,(I
I
5. Combining, we have
6. Separating and integrating yields
- X)
Sec. 10.7
1
Catalyst Deactivation
7. Numerical evaluation:
We wiIl now rearrange Equation (E10-7.8)to a form more commonly found in the literature. Let X be a dimensionless decay time:
and Daz he the Damkohler number for a second-order reaction (a reaction l n i e dl-ided by a flonsporl r a r ~for ) a packed-bed reactor:
w
--kC,,,W Da, = (kc:,)( F A ~ u, Through a series of manipulations we arrive at the equation for the conversion in a rnorging bed where a second-order reaction is taking place:33 Second-order reaction in a moving-bed reactor
Da, (E -eLA) L+Da,(l - e - i ) SirnEIar equations are given or can easily be obtained for other reaction orders or decay laws.
Heat Effects in Moving Beds. We shall consider two cases for modeling the temperature profile in the tnoving-bed reactor. In one case the temperature of the solid catalyst and the temperature of the gas are different and in the other case they are the same.
+
Cmc J (T T,). The rate of heat transfer between the 9a.c at temperature T and the solid catalyst particles at temperature Ts is
= heat transfer coefficient. kJ Jm2.s .K = solid catalyw surface area per Inass of catalyst in the hed. rn'/kg cal Ts = temperature of the qolid. K. Al~o. 7;, = temperature of hear exchange fluid, K
where_h UP
728
Catalysis and Catalytic Reactors
Chap.
The energy balance on the gas phase is Energy holunce
If Dp i s the pipe diameter (m), ps is the butk catalyst density (kglm3), and is the wall surface area per mass of catalyst (rn2/kg)
I
The energy balance on the solid catalyst is Heat exchange between catalyct particle and gas
where CPr(Jl_kg.K)is the heat capacity of the solids, U,(kg/s) the catal! loading, and u p is the external surface area of the catalyst pellet per unit mc of cataIyst bed:
where II, is the pellet diameter. Case 2 (T5= _T). If the pmduct of the heat transfer coefficient. h, and t sudace area. a p . is very Iarge, we can assume that the solid and gas temper tures are identical. Under these circumstances the energy baIance becomes
10.7.4 Straight-Through Transport Reactors (STTR)
This reactor is used for reaction systems in which the catalyst deactivates ve rapidly. CommercialIy. the STTR is used in the production of gasoline fro the cracking of heavier petroleum fractions where coking of the catalyst peIle occurs very rapidly. In rhe STTR. the catalyst pellets and the reactant fet enter together and are transported very rapidly through the reactor. The bu density of the catalyst particle in the STTR is significantly smaller than moving-bed reactors, and often the particles ate carried through at the: sao vefocity as the gas velocity. I n some places the S l T R is also referred to as circulating fluidized bed (CFB). A schematic diagram is shown in Figure 10-3
Chap 10
STTR: U%d when catalyst decay (usually coking) IS very rapid
Figure 20-34 Straight-through tmnspurt reactor.
A mole balance on the reactant A over the differential reactor volume
AV
= A,&-
is F*I,-FAI_+k+rAA~
=Q
Dividing by k and taking the Iirnit as k -+ 0 and recalling that
r, = p,
r;
,
we obtain
In terms of conversion and catalyst activity,
For a catalyst particle traveling through the reactor with a velocity Up, the time the catalyst pellet has been in the reactor when it reaches n height ,: is just
Substituting for time t in terns of distance z [i-e., n ( r ) = cr(rlC'p)]. the mole balance now becomes
730
Catatysls and Catalytic
Reactom
Chap. 10
The entering moIar flow rate, FA,,can be expressed in terns of the gas velocity U,,CAD,and Ac: FA, = U,A,C,o
Substituting for FAo,we have
Exampie J&8
Living Evample Problem
A typical cost of the calalyst In the hysiern i? $ 1 m~llion
Decay in a Stmight-Through Transpon Reactor
The vapor-phase cracking of a gas oil i s to be carried out in a straight-through transport reactor (ST7R)that IS EO m high and 1.5 m in diameter. Gas-oil is a m~xtureof normal and branched paraffins {C,:-Clo). naphthenes, and aromatics. all of which will he lumped as a singie species. A. We shall lump the prlrnary hydrocarbon products according to disrillate temperature into ~ w orespective groups. dry gas (C-C,) B and gasoline (C,-C,,) C. The reaction
Gas-oil (g) can be written symbolically as
A
-
4
Products (g)
+ Coke
B + C + Coke
Both B and C are adsorbed on the surface. The rate law for a pa$-oil cracking reaction on fresh catalyst cad he approximated by
with k' = O.(f014 krnoI/kg cat.s.atm. K, = 0.05 atm-I, KB = 0.15 atm-I, and Kc = 0.1 atm-! me catalyst decays by the deposition of coke, which is produced in most cracking reactions along with the reaction products. The decay taw i s a=-
I
I + A1 'fl
with A
= 7.6 s-'"
Pure gas-oil enters at a preqsure of 11, a\m and a temperature of 4C)O"C. The bulk density of catalyst in the STTR is 80 kg catlrn3. Plot the activity and conversion of gas 011up the reactor for entering gar velocity Uo = 2.5 mls.
Mole Balance:
Chap. 10
Rate Law: -'A
= PB (-4)
-4 = a [-r',(t = 0)] Following the Algorithm
I
On fresh catalyst
I
Combining Equations (EIO-8.2) through E10-8.4)gives
Decay law. Assuming that the catalyst particIe and gas travel up the reactor at the velocity Up = Ug= U,we obtain
where ~ = ~ / A c = u , ( l + ~and ) /A d ,~= O / 4 .
Stoichiometry (gas phase isothermal and no pressure drop):
Parameter Evaluation:
~ = ~ ~ ~ 6 = ( l + 13 - 1 ) =
cAO = PA0 = RT,
12 atm
(0.082 mLatdkmol- K)(673 K)
= 0.22 lanol -
m3
Equations (El0-8.I), IE10-8.5).(El 0-8.7).and (El 0-8.8)through (E10-8.10) are now combined and solved using an ODE solver. T h e Polymath program is shown in Table ElO-8.1, and the computer output is shown in Figure EIO-8. I .
7'32
-
Catalysis and Catalvtrc Reactors
TaA1.F
EIO-8.1
Chaa
Eouhnor\.s FOR THE STTR L ~ N G E ~ ~ I R - H I ~ ~ HKrvm~cs ELwOOO
DWfeHel equatmns as. entered [ ? : d(X)td(z)= - d / C a o
[';
T140Dt273
[a:
mo1ao
the user
lcpnme = 0.0014 [ L O : D = 1.5 UO-2.5 t:: Kc.O.1 (131 U r Uom(l+6ps'X) t .c ] Pa = Pao'(1 .x)r(l +epa'X) r:S! Pb = Pao'wtt +epsqX) [ 9:
[?61 VO= U0'3,1416'D'W4 :i71 Cao=Pac4W I L P 7 KCa= W R ' T
(131 Pc=Pb L Z O l a = lI(l+A'(zRIpO.S) L 2: 1 rapma I am(-ikprtme'Pd(l+Ka'Pa+WPb+KccPt)) :i2; ra = ho'rapmne
Figure ElO-8.1 Activity and conversion profiles.
733
Summary
Chap. 10
Closnm. After teading this chapter, the reader should be able to discuss the steps in a hetemgeneous reaction (adsorption, surface reaction, and desorption) and describe what is meant by a rate-limiting step. The differences between molecular adsorption and dissociated adsorption shouId be explained by the reader as should the diffesent types of surface reactions (single site, dual site, and Eley-Rided). Given heterogeneous reaction rate data, the reader shouId be abIe to analyze the d:1m and to develop aI rate law for Langrnuir-finshelu;ood kinetics. Applications of mE in the elecuur~lc~ 1ndustry Hrere discussed and readers should be able to describe the analog between Langmuic-Hinshelqetics and chemicaI vapor deposition (CVD)and to derive a rate wood ki~ law for C3'D mechanisms. Because under a harsh environment catalysts do ,:,, LLUL IrrWlLtain rheit original activity, the reader needs to define and describe the three basic types of catalyst decay (sintering, fouIing, coking, and poisoning). h addition, the reader shouId be able to carry out calculations that predict conversion in the types of reactors (namely rnwing bed and S'ITR) used to offset the deactivation of the catalyst. ,A+
SUMMARY 1. Types of adsorption:
a. Chernisorptioo b. Physical adsorption 2. The Langmuir isotherm relating the concentration of species A on the surface to the partial pressure of A in the gas phase is
3. The sequence of steps for the solid-catalyzed isomerizatian
is:
Mass transfer of A from the bulk Ruid to the external surface of the pellet b. DifPusion of A into the interior of the vtlet c. Adsorption o f A onto the catatytic surface d. Surface reaction of A to form B e. Desorptian of B from the surface f. Diffusion of B from the pellet interior to the externat surface g. Ma- transfer of B away from the solid surface to the bulk fluid a.
4. Assuming that mass transfer is not rare-limiting, the rate of adsorption is
Catalysis and Catalytic Reactors
Chap. 70
The rate of surface reaction is
The rate of desorptian is
At steady shte,
rm = rs = rn
(SlO-6)
If there are no inhibitors present, h e total cmcentration of sires is
CI= CV+CA.S+CB s
(SID-7)
5. If we assume that the surface reaction is rak-limiting, we set
and solve for CA-S and CB.S in terms of PA and PB.After substitdon of these quantities in Equation (S10-4), the concentration of vacant sites is eliminatd with the aid of Equation (SIO-7):
Recall that the equilibrium constant for desorption of species B is the reciprocal of the equilibrium constant for the adsorption of species B:
and
K p = KAKsIKB 6. Chemical vapor deposition:
Chap. 10
Summary
7. Catalyst deactivation. The catalyst activity is defined as
The: rate of reaction at any time t is
-6 = a ( t ) k ( T )fn[C,,
C,, ..., C,)
(SlO-l4)
The rate of catalyst decay is
For first-order decay:
For second-order decay:
8. For slow catalyst decay the idea of a temperature-time trajectory is to increase the temperature in such a way that the rate of reaction remains constant.
9. The coupled differential equations to be solved for a moving-bed reactor are
For nth-order activity decay and m order in a gas-phase concentration of species i ,
10.The coupled differential equations to be solved in a straight-through transport reactor for the case when the particle and gas velocities. U,are i d e n t i 4 are
For coking
736
Catalysis and Catalytic Reactors
Chaa
ODE SOLVER ALGORITHM The ihorncrization A -+ B is carried out over n decaying catalyst in a ruol.in#-her1 wucror. Pure A enters the reactor and the cataly>t How+ rhrol~gh the reactor at a rate of 2.0 kg/r.
& -=-
dry
-k,,a:P,
=- 0.75
x-
&
'I
s-atm
F,, = 1 O moF Is
PA,, = 20 at,
a = 0.0019 kg-'
EV, = 500 kg cat
CD-ROM MATERIAL
Summary Notes
0
i
Cornputcr Modules
Learning Resources I . Sutnmrir?. Norex for Chapter 10 2. Ir~teracriveComputer Morlules A. Heterogeneous Catalysis
Example CD I O- I A11al)c!q uf a He1ero_crneou\Reaction [Class Problenl Cnlv e r i t y r ~ Mich~gan f 1 Example CD 10-2 Len\? Sq11~rt.s .41131y\lsto De~erminethe Rate Law Pnramrters k, k,, and k, Example CD 10-3 Decay in o Straight-Through Reactnr ho!ucd Problems Exarnpte CDIO-4 Cataly.rt Pu~snningin ;t Batch Reactor Living Example Problems I . Eron~plr10-2 Rq~t,.vrimt~ i.trrir/~ .\ i r to Dr,terrrrinr MRCIP~ Prrrrrtr~ererc -7. E.urt)?j~le10-3 Fi.rrd-Rerl Rmrror Drridrlrd ns o CSTR ?l":ng Example p r ~ b ! ~ 3. ~ Earrrl/~lt,10-8 I)erri,v it? rr Srmi~ht-ThtnttghTronhport Rt.~cror * Professional Reference Shelf R I 0. I H,wli-qy~tiAtiwrptiot~ A. Molecular Adhorption B. Dissociative Adsorption
;fl;rg
I.----
.
Slop*.
-k,
Reference Shelf
6
(torr'nl
R I0.2.Ana1~si.rclf CarnI~.rtD e r q Lrn1.s A. I~tegralMethod B. D~ffeferent~al Method R 10.3.Erctling of Smriconrl~~cror~
A. Dry Etching
t
Catalysis and Catalytic Reactors
B.
Chap. T O
Wet Etching Liquid Phase
S~I-Q-
$--I&
I
y-
I:
-?-
:I
0I
Si-0-9 i
I
I
Solid Phase
C. Dissolution Catalysis
-+ H'WS
H++S
MF + H+. S
H Product
This Rook, Ask Yourself a Question About I h a t You Read Q U E S T I O N S AND P R O B L E M S The subscript to each of the problem numbers indicates the level of difficulty: A, least difficul~;D. most difficult. A = @
B=.
C=* D=*+
P10-IA Read over the problems at the end of this chapter. Make up an original pmblem rhat uses rhe concepts presented in this chapter, See Problem P4-1 for gu~deltnes.To obtain a solution: Hall of Famc
(a) Create your data and reaction. .(h) Use a real reaction and real data. The journals listed at the end of Chapter 1 may be useful for pan (b). (c) Choose an FAQ from Chapter 10 and say why it was mort helpful. (d) Listen €0 the audios on the CD and pick one and say why it was most helpful. P10-2R (a) Example SO-lmCombine Table 10-1, Figure 10-4, and Example 10-1 lo calculate the maximum and minimum rates of reaction in lmolig cads) for (1) rhe isornerization of n-pentane, (2) the oxidation of SO:, and (3) the hydration of ethylene. Assume the dispers~onis 50% in all c a m a5 is the amount of catalyst at I f . (h) Example 10-2. ( 1 ) Wh3t is the fraction of vacant sites ar 609 conversion? (2) At 8 0 8 and 1 atm, what is the fraction of toluene sites? (3) How would you linearize the rate Ian?to evaluate the paramerers k. &:8. and KT from various linear plots? Explain. (c) Example 10-3. ( 1 ) What if the entering pressure were increased 10 80 atm or reduced 1 arm. how would your answers change? (2)What if the molar flow rate were reduced by SO%, how would X and y change? What catalyst we~ghtwould be required for 60% conversion? (dl Example 30-4. ( I I HOWwould your answers change if the Following data for R u n 30 were incorporated in your regrecsion table?
-a
Questiwrs and Prohlems
Chao. 10
-4 = 0.8,P, = 0.5 stma
= 15 atm, P, = 2
(1) How do the rate laws (el and Ifl
compare with the other rate laws? (e) Example 10-5. 11) Sketch X vs. I for various values of kd and k. Pay par-
(0 Mall cf Fame
(g)
(h)
(i)
Cj)
ticular attention to the ratio k/kd. (2) Repeat ( 1 ) for this example (i.e., the plotting of X vs. t ) for a second-order reaction with (CAI)= I moI/dm3) and first-order deca). (3) Repeat (2) for this example for a tint-order reaction and first-order decay. (4) Repeat (1) for this example for a second-order reaction (CAI,= I moWdm3) and a smond-oder decay. Example 10-6. Whar if . . . ( I ) the space time were changed? How would the minimum reactant concentration change? Compare your results with the case when the reactor is full of inerts at time t = 0 instead of 80% reactant. Is your cataIyst lifetime longer or shorter? (2) What if the temperature were increased so rhar the specific rate constants increase to k = 120 and kd = 12? Would your catalyst lifetime bc lodger or shorter than at rhe lower temperature? (3) Descrik how the minimum in reactant concentration changes as the space time r changes? What is the minimum if r = 0.005 h? If r = 0.01 h? Example 10-7. (1) What if the solids and reactants entered from opposite ends of the reactor? How would your answers, change? ( 2 ) What if the decay in the movlng bed were second order? By how much must the catalyst charge, Lr,, be increased to obtain the same conversion? (3) What if E = 2 (e.g., A + 3R) instead of zero, how would the results l x affected? Example 10-8. ( I ) What rf you varied the parameters PAo, Ur,R a and k' in the STTR? What parameter has the greatest effect on either increasing or decreasing the conversion? Ask questions such as: What is the effect of varying the ratio of k to U,or of k to A on the conversion? Make a plot of conversion versus distance as is varied between 0.5 and 50 d s . Sketch the activity and conversion profiles for U, = 0.025, 0.25,2.5, and 25 rnls. What generalizations can you make? Plot the exit conversion and activity as a function of gas velocity between velocities of 0.02 and 50 m l s . What gas velocity do you suggest operating at? What is the corresponding entering woturnevic Row rate? What concerns do you have operating at the velocity you selected? Would you like to choose another velocity? If so, what is it? What if you were asked to sketch the temperature-time trajectories and to find the catalyst lifetimes for first- and for second-order decay when EA = 35 kcallrnol, Ed = 10 kcaUmol, k& = 0.01 day', and T, = 4M K? How would the trajectory of the catalyst lifetime change if EA = 10 kcnWrnol and Ed = 35 kcal/mol? At what values of ka and ratlos uf Ed to EA would temperature-time traje~torie~ not be effective? What would your temperature-time trajectory Imk like if n = 1 + E,p'EA? Write a qucstion for this problem that involves critical thinking and explain why it involves critical thinking.
740
P10-3
*
-
:?).A
Cornpuyer Moduler
Catalysis and Catatytic Reactors
Char
Load the Interactive Computer Module (ICM) from the CD-ROM. Run module and then record your performance n u m k r for the module, wl indicates your mastering o f the mater~al.Your professor has the key to dec your performance nilmbes. ICM Heterogeneous Catalysis Performs #
P10-4, 1-Butyl alcohol (TBA) is an imponant octane enhancer that is used to rep lead additives in gasoline [Ii~d.Eng. Chern. Res., 27. 2224 (1988)l. t-B alcohol was produ>ed by the liquid-phase hydration (W) of isobutene (1) ( an Amberlyst- 15 catalyst. The system i s normally a rnukiphase mixturc hydrocarbon. water. and solid catalysts. However, the use of costllventr excess TBA can achieve reasonable miscibility. The reaction mechanism i s betieved to be I+S
1,s
Derive a rate law assuming:
(a) The surface reaction is rate-limiting. (b) The adsorption of isohrene is limiting.
1 ~ )The reaction follows Elty-Rideal kinetics and that the surface reaction is limiting. (d) Isobutene (I)and water (W) are adsorbed on different sites
TBA is
~ l a an i
rhe surface, and the surface reaction is rate-limiting.
(e) What generalization can you make by comparing the rate laws derived parts (a) through (d)? The process flow sheet for the commercial production of TBA is shas
in Figure P IO-4.
(0 What can you learn from this problem and the process flow sheet? P10-5,
The rate law for the hydrogenation (H) of ethylene (E) to form ethane ( over a cobaIt-molybdenum catalyst [Collection Czech. Chcm. Commun.. 1 2160 (1988)j is
(a) Suggest a mechanism and rate-limiring step consistent with the rate la (b) What was the most dimcult part in finding the mechanism?
Chap. 10
Duestions and Problems
Raw TBWwaler prcduci
TBA water azeotrope product
Drj
TEA prduct
--$
TEA synthes~sand raw TBA recovery
TBA azeotmpe prorluction
Heavies
Dy TeA prodwtron
Figure P1W Hiils TEA synthesis prmess. R. reactor: Ca, C1column: C*, CB
TBA column. (Adapted from R. E. Mepn, Ed.. Honclbo[~kof Chcmrcal~Prod~~crrori Pmce.ues, Chemrco! P r o c e s ~Technology H~rndhookSeries. McCraw-Hill. New York, 1983. p. 1.19-3. ISBN 0-67-041 765-2.)
column: AC, azeotrope column; TC.
formation of proponal on a catalytic surface is believed to proceed by the following mechanism
(c) The
Suggest a rate-limiting step and derive a rate law. P10-6R The dehydration of n-butyl alcohol (buranol) over an alumina-silica catalyst was inve~tigatedby J. F. Maurer (Ph.D. thesis, University of Michigan). The data in Figure P10-6were obtained at 750°F in a mcdified differential reactor. The feed consisted of pure butanol. (a) Suggest a mechanism and tare-controlling step that is consistent with the experimental data. (b) Evaluate the rate law panmeters. (c) At the point where the initial rate is a maximum. what is the fmction of vacant sites? What is the fraction of occupied sites by both A and B? (dl What generalizations can you make from studying this problem? (el Write a question that requires criticaE thinking and then explain why your question requires critical thinking. [Hint: See Preface Section B.2.1
Catalysis and Catalytic Reactors
Chap. 10
P10-7B The catalytic dehydration of methanol (ME)
to form dimethyl ether (DME) and water was carried out over an ion exchange catalyst [K. Klusacek, Collection Czech. Cllem. Commun., 49, 170C19W)j. The packed bed was initially filled with nitrogen and at I = 0 a feed of pure methanol vapor entered the reactor at 413 K. 100 kPa, and 0.2 cmVs. The following panial pressures were recorded at the exit to the differential reactor containing 1.0 g of catalyst in 4.5 cm3 of reactor volume. t(s)
0
10
50
100
ESO
200
300
PN,IkPa)
100
Q 25
26
0 26
35
37
37
38
60
2 23 30 45
0
0 0 Q
SO 2 10
10
PM; l k h )
40
37
37
PH-,(m) Pnk (Wa)
P30-8,
15
IS
Discuss the implications of these data. In 1981 the U.S.government put fonh the following plan for automobile manufacturers to reduce emissions from automobiles over the next few years. Year
1981
199.2
2004 0.125 1.7 01
Hydrocarbons
0.41
0.25
CO
3.4
NO
1.0
3.4 0.4
All values are in grams per mile. An automobile emitt~ng3.74 Ih of CO and 0.37 Ib of NO on a journey of 1 OOO miles wnuld meet the current government requirements.
Chap. 10
743
Questions and Problems
To remove oxides of nitrogen (assumed to k NO) fmm automobile exhaust, a scheme has been.prwsed that uses unburned carbon monoxide (CO)in the exhaust to reduce the NO over a solid catalyst, according to the reaction
CO
+ NO
-
Products (N,,CO,)
Experimental data for a particular solid catalyst indicate that the reaction rate can be well represented over a large range of temperatures by
where
P, = gas-phase partial pressure of NO PC = gas-phase partial pressure of CO k, K,.K, = coefficients depending only on temvrature
(a) Based on your experience with other such syskms. you are asked to propose an adsorption-surface reaction-desorption mechanism that will explain the experimenrally observed kinetics. (b) A certain engineer thinks that it would be desirable to operate with a very large stoichiometric excess of CO to minimize catalytic reactor volume. Do you apree or disagree? Explain. (c) When this reaction i s carried out over a supported Rh catalyst IJ. Phys. Chem.. 92. 389 (1988)], the reaction mechanism 1s believed to be
When the ratio of Pco/PNois small. the rate law that is consistent with the experimental data is
What are the conditions for which the rate law and mechanism are consistent?
PIO-9, Methyl ethyi ketone (MEK) is an important Industrial solvent that can be produced from the dehydrogenation of butan-2-01 IBu) over a zinc oxide catalyst [Ind. Eng. Ckm. Res.. 27, 2050 (l988)I: Bu +MEK + H? The following data giving the reaction rate for MEK were obtained in a differentia[ reactor ar 490°C.
Chap. f 0
745
Questions and Problems
The f(dlowing cl;kra were obtained
It is suspected that the reaction may involve a dual-site mechanism, but it i s not known forcertain. It is believedthat the adsorption equilibrium constant forcyclohexand is around 1 andis roughly oneortwoordersof magnitude gl-raterthan the adsorptionequil~briumconstants for the othercornpouods. Using thesedata: la) Suggest a rate law and mechanrsm consistent wlth the data given here. ( h ) Determine the constant? needed for the rate law. [Ind EnL?.Chem. Res., 32. 2626-2632 (I993).J (c) Why do you think estimates of the rate law parameters were given? P10-12, A recent study of the chemical vapor d e p n ~ i t ~ oof n silica from sllane (SiH,) i s believed to proceed by the following irreversible two-step mechanism [I. Electmchcm. Soc., 139(9). 2659 ( I 992)]:
This mechanism is somewhat different in that while SiHl i< irreversibly adsorbed, i t is highly reactive. In FJCL adsorbed SiH, react\ as h s r as it ih formed [it., ri,,, = 0. i.e., PSSH (Chapter 711, xo that it can be assumed to behave .as an actTve intermediate. (a) Determine if this mechanism is consistent with the following data:
,
--
S i nr
-
-
er
n
I _( 7 4 0
60
(h) At what partial pressuFes of siIane would you ?ake the next two data points? P1Q-13,Vanadium oxides are of interest for various sensor appIications. owing to the sharp metal-insulator transitions they undergo as a function of temperature, pressure, or stress. Vanadium triisopropoxide (VTIPO) war, used to grow vanadium oxide films by chemical vnpor depnsirion [J. Electm~~hctta.Sot., 136. 897 (198B)I. The deposition rate as a function o f VTIPO pressure for two different temperatures follows:
:auaxaqopils put m c m uuoj a] lsdinlcs c mnn p a s ~ dscm ~oumxqopi(-~' I T - O I ~ .,m3/39.l q! I ~ I C I C Saql l o illls~tap ayl puc ~ (s! b uO!r3nJJ ploh a q l ;Uo!sJaAuO3 ana!yx 01 h s s a ~ a u s! lyzram lsillnl~slcym '3uol IJ SF sad!d 08 stnpayx . ~ r ! - j I u! pay3~d an
' 3 , ~ 1~ z pawadn 1 c 3 1 s t a ~I ~ ~ I I I I c~Eursn J ~ I pau!mqo ~ =am awlm-r uunj 01 sumo-! jo uo!cua:olpill ~ 1 JOJ 1 mpp 4u~mollq391 3 n ~ - ~ ~ d * l o ~ s ~ aaq1 r umnp l q E ! a ~~sn'li:ir>lo uo!lsunj r! rr! x pur? .i ,old -,-SyC ~ =O m piln do.lp u n c w d ~u!l~;ncn.,t:(p) md ,cadla 11) t'7.a uo!lsag aagaJd aaS :IYHJ.?tl!llurql I~:JI!I.I~ sa.l!nba~uo!~sanh m0h' hym urydxa uaql pup S U ~ ~ U (ns!l!Ja ! ~ I .;al!nllal 1l:yl unysanb r? ~ Y U M (a) 'ay s:]I/,TH) 'ME1 alt?J a q ~yl!m l u a l < ~ r ~d~ om> 41ill?urtl-a~c~ p w llI~!u1:~~~.~111 ttotlnraj r! 1~39311s( q ) clrp ~ P I U ~ I U I . Ilql ~~X ~ ~H M Illal\r~utr.~ nq 2 1 e ~c l~a?f!ns (e)
Chap. 10
Questionsand Problems
PM (atm)
.;(""'
toluene)
5.kp cat
P10-t6n In the prducfion of ammonia
the following side reaction occurs:
NO+H,
H,O+iN,
Ayen and Peters [ I d . Eng. Chern. Pmress Dcs. Dm!,1, 204 ( 1 962)] studied catalytic reaction of niric oxide with Girdler (3-50 catalyst in a differential reactor at atmospheric pressure. Table PEO-16 shows the reaction rate of the side reaction as a functlm of PHI and PNoat a tempemture of 375°C.
r,,,x 10' (g rnoVrnin,p cat)
746
Catalysis and Catalytic Reactors
Chap. 10
T = 120°C: Grnwth Rare (pmlh)
1
0.004 0.015 0.025 0.04 0.068 0.08 0.095 0.1
In light of rhe material presented in this chapter. analyze the data and descrik your results. Specify where additionaI data should be taken. P10-14, Titanium dioxide is a wide-bandgap semiconductor that is showing promise as an insulating dielectric in VLSI capacitors and for use in solar cells. Thin films of TiOz are to be prepared by chemical vapor deposrrion from gaseous titanium tetraisopropoxide (TTJP).The overall reaction is The reaction mechanism in a CVD reactor is believed to be [K. L.Siefering and G . L. Griffin. J. Electrochem. Sac., 137, 814 (199011
where I is an active intermediate and P, i s one set of reaction products ie.g., HZO.C3H6)and P2 is another set. Assuming the homogeneous gas-phase reaction for 'STIF is in equilibrium. derive a rate law for the deposition of TiOzP The experimental results show that at 200°C the reaction is second order at low partial pressures of TTIP and zero order at high partial pressures, while at 300°C the reaction IS second order in TTiP over the entire pressure range. Diqcuss these results in light of the rate law you derived. P10-15, The dehydrogenation of methylcyclohexans (M) to produce toluene (TI was carsied out over a 0.3% Pt/AI,O, catalyst in a differential catalytic reactor. The reaction i s carried out in the presence of hydrogcn ( H 2 ) to avoid coking [J. Phyr. Ch~tli..64. 1559 ( 1 960)J. (a) Determine the model parameters for each of the following rate laws.
Use the data in Table P10-15. Ih) Which rare law hcst describes the data? (Hint: Neither K H , or K,, can take on negative valuer.) (c) Where aould you pIace addittonat data points'
Chap. 10
Questions and Problems
PIO-19, The elementary imverrible ga5 phase catalytic reaction
is to be carried out in a moving-bed reactor at constant tempemure. The reactor contains 5 kg of catalyst. The feed is stoichiornetric in A and B. The entering concentrat~onof A is 0.2 moUdm3. The catalyst decay law is zero order with R, = 0.2 s-I $nd k = 1 .O drn6/(rnol . kg cat . s) and the voIumetric flow rate is u, = I dm31s. (a) What conversion will be achieved for a catalyst Feed rate of 0.5 kgls? (b3 Sketch the catalyst activity as a function of catalyst weight h e . , distance) down the reactor length for a catalyst feed rate of 0.5 kg/s. [c) Whar is the maximum contersion that could be achieved (f.e.. at infinite catalyst loading rate)? (d) What catalyst loading rate is necessary to achieve 40% conversion? (e) At what catalyst loading rate (kgls) wit1 the catalyst activity be exactly zero at the exit of the reactor? (0 What doer an activity of zero mean? Can catalyst activity he less than zero? (g) How would your answers change if k = 5 m o n g catas and the catalyst and reactant were fed at opposite ends? (h) Now consider the following economtcs: The economics
The product sells for $160 per gram mole. The cost of opnting the bed Is $10 per kilogram of catalyst exiting the bed. What is the Feed rate of sotids (kgtmin) that wilI give the maximum profit? (Ans.: U,= 4 kglmio.) (NOW: For the purpose of this calculation, ignore all other costs, such as the cost of the reactant, etc.) 4i) Qualitatively how will your answer change i f k, = 5 mollkg catpmin and the reactant and catalyst are fed to opposite ends of the bed. PIO-20, W~ththe increasing demand f ~ xylcne r in the petrochemical industry, the production of xylene from toluene dispropltionation has gained attention in recent years [Ind Eng. Chem. Res.. 26, IS54 (1987) 1. T h i s reaction,
*
1Toluene
-
Benzene
+ Xytene
was studied over a hydrogen mordenite catalyst that decays with time. As a first approximation, assume that the catalyst follows second-order decay. and the mte law for low conversions is
with kr = 20 g mot/h.kg catnatmand k, = 1.6 h-I at 735 K. (a) Compare the conversion time curves in a batch reactor containing 5 kg Cat at different initial pmiat pressures ( 1 atm, 10 a m , etc.). The reaction volume containing pure toluene initially is 1 dm3 and the temperature is 735 K-
Catalysis and Catalytic Reactors
Chap. 10
The following rate law5 for side reaction (2). hased an various c~talytic mechanisms. were suggested:
Find the parameter values of the different Fate laws and determine which rate law best represents the experimental data.
PIO-17, Rework Example 10-6 when (a) The reaction is carried out In a moving-hed macmr at a catalyst loading rate of 250,000 kglh. (b) The reaction is canied out in a packed-bed reactor modeled as five CSms in series. (c) Repeat (a) when the catalyst and feed enter at opposite ends of the bed. (d) Determine the temperaturetime trajectory to keep conversion constant in a CSTR if activation energies for reaction and decay are 30 kcallmol and 10 kcallmol, respectively.
(e) How would your answer to part (d) change if the activation energies were reversed? P10-18, Sketch qualitorively the reactant, product, and activity profiles as a function of length str various times for a pocked-bed renctor for each of the following cases. In additton, sketch the efRuent concentration of A as a function af time. The reaction is a simple isornerizat~on:
(a) Rate law:
Decay law: Case I:
(b) -r;
kd
-r;\ = kaC, rd = kdaCA
S Case 11: k, = k Case 111: k,
S-
k
= kaC, and r, = kda2
(c) -rl = kaC, and r, = kdaCB (d) Sketch similar profiles for the rate laws in parts (a) and (c) in a moving-bed reactor with the sdids entering at the same end of the reactor as
the reactant. (e) Repeat pan (d) for the case where the solids and the reactant enter at opposite ends.
Chap. 10
751
Questions and Problems
PlO-22, The vapor-phase cracking of ps-oil in Example 10-8 i s carried out w e r a different catalyst, for which the rate law is
can vary the entering pressure and gas velocity, what operating conditions would you recommend? (b) What could go wrong wlth the conditions you chase? Now assume tbe decay law 1s (a) Assuming that you
-da = u
Coke
rnol .s
with k , = 100 dm' at 400°C
where the concentration, CCa,, in rnoi/dm" can be determind from a stoichiometric table. (c) For a temperature of 400°C and a reactor height of 15 rn, what gas velocity do you recommend? Explain. What is the corresponding convetxion? (d) The reaction is now to be carried in an S T I T 15 rn high and 1.5 m in diameter. The gas velocity i s 2.5 mls. You can operate in the temperature range between 100 and 500°C.What temperature do you choose. and what is the corresponding conversion? (e) What would 'the temperature-time trajecrory look like for a CSTR? Additional information:
PXO-23c When enhe irnpuriry cumene hydmperoxide is present in trace amounts in a cumene feed stream,it can deactivate the silica-alumina catalyst over whlch cumene is being cracked to form benzene and propylene. The following data were taken at 1 arm and 420°C in a differential reactor. The feed consists ~f rurnene and a trace (0.08 mol %) of cumene hydroperoxide (CHP).
of decay and the decay constant. (Ans.: k, = 4.27 x 10-3 S - I . ) fb) As a first approximation (actually a rather good one). we shall neglect the denominator of the catalytic rate law and colisider the reaction lo be first order in curnme. Given that the specific reaction rate with respect to curnene is k = 3.8 X 10' mollkg fresh cat .s+atrn.the molar flow-rate of cumene (9992% cumene. 0.08% CHP) is 200 mollmtn, the entering concentration is 0.06kmol/rn3, the catalyst weight is 100 kg. the velocrty o f solids 1s I .O kgJrnin, what conversion of cumene will be achieved in a movirrg-bed reacror? P10-24, The decomposition of spartanol to wulfrene and CO, is often carried out at high temperatures [J. Tliwr: Exp.. 15, 15 (7014)). Consequently. rhe denominator of the catalytic rate law I < easily approximated as unity. and the reaction i s first order with an activation energy of 150 kJlrnnl. Fortunately, the reaction (a) Determine the order
Catalysis and Catalytic Reactors
Chap. 10
(b) What conversion can be schieved in a moving-kd rmcror containing 50 kg of catalyst wlth a catalyst feed rate of 2 kgth? Toluene is fed at a pressure of 2 atm and a rate of 10 mol/rnin. (c) Explore the effect of caralyst feed rate on conversion. (d) Suppose that ET = 25 kcallmol and Ed = 10 kcallmol. What would the rernperarure-time trajectory look like for a CSTR? What if ET = 10
kcal/mol and Ed = 25 kcallmol? more closely follows the equation
(e) The decay law
with Jid = 0.2 atmL2 h-'. Redo parts (b) and (c) for these conditions. P10-LIP The elementary irreversible gas-phase catalytic reaction
is canied out isotherrmcally in a batch reactor. The catalyst deactivation follmls a first-order decay law and i s independent of the concentrations of both A and B. (a) Determine a general expression for catalyst activity as a function of time, (b) Make a qualitative sketch of catalyst activity as a function of time. Does a(r) ever equal zero for a first-order decay law? (c) Write our the general algorithm and derive an expression for conversion as a function of time, the reactor parameters. and the catalyst parameters. Fill in the following algorithm Mole balance Rare law Decay law Stoichiornet y Combine Solve I. Separate 2. Integrate
Id) Calculate the conversion and catalyst activity in the reactor after 10 mlnutes at 300 K. (el How would you expect your results in pans (h) and (d) to change if the reaction were ntn at 400 K? Briefly describe the trends qual~tatively. ( f ) Calculate rhe conversinn and cataly~tactivity in the reactor after 10 mioutes if the reaction were run at 400 K instead of 300 K. Do your results match the predictions in part (e)? Addiiional informarion: CAo= 1 rnolldm' Vo = l dm3
W = l kg k, = 0.1 m i d at 300 K k, = 0.2 dm'l(kg cat min) at 300 K
Chap. 10
Journal Critique Problems
The cata1yqt decay is litst order in activity with
1
k,=0.01 exp 7000K
---
[4;0
;)J
s-l
There are 50 kg o f catalyq in the bed. (a) What catalyst charge sate (kg/s) will give the highest conversion? (b) What is the correspondlnp conversionr! (c) Redu parts (h) and (b) €or Case 3 ( T # T q ) of heat effects in-moving beds. Use realistic values of the parameter values h and a, and a,, . Vary the entering temperatures of Ts and I:
JOURNAL CRITIQUE PROBLEMS PlOC-1 See NChE 1,7 (4), 658 (1%1). Determine if the following mechanism can &so be u d to explain the dam in this paper.
NO+ NO-S (N N 2 + Q 2 . S P10C-2 In J. CnmL. 63, 456 (1980). a rate expression is derived by assuming a reaction that is first order with respect to the pressure of hydrogen and first order with respect to the pressure of pyridine [Equation (IO)]. Would another reaction order describe the data just as well? Explain and justify. Is the rate law expression derived by the authors correct? P10C-3 See "The decomposition of nitrous axide on neodymium oxide, dysprosium oxide and erb~umoxide.'' J. Catnf., 28, 428 (1973). Some investigators have reporred the rate of this reaction to be independent of oxygen concentration md first order in nitrous oxide concentration, while othen have reported the reaction to be fint order in nitrous oxide concentration and negative one-half order in oxygen concentration. Can you propose a mechanism that is consistent with both observations? PIOC-4 The kinetics of seIf-poisoning of Pd/AI2O3 catalysis in the hydrogenolysis o f cpclopentane is discussed in J, Coral.. 54. 397 (1978). Is the effective diffusivity that is used realistic? Is the decay homographic? The authors claim that the deactivation of the catalyst is independent of metal dispersion. If one were to determine the specific reaction rate as a function of percent dispersion, would this information suppon or reject the authorr;' hypotheses?
*)1/ln3 3qlr 001 s! ~sLlclasp!ps ayl j o IC1pd83 I B ~ Qa u q u g u ! L l ~ n u ! ns! s ~ p3 u abp1c3 ~ aqr uaamlq aarsy,jaos ~ajsuul]cay aq, put 'g Q G 30 ~ arumadura~E IE J O I ~ R X ayi Jalua ~ m ~ s o a r aql put! ls61e~rapfios atp yloa .,utp,qoru ~2.0 JO u a ! i ~ . 1 1 u a m o o 3pr ~ t sj~our ! ~9.g 10 alc~B 18 ~ 0 v ~ aayl - 1 sJalua v and '3,0512IU'"ISUO> S! a~nlc~adwal 3uaIqm ayl p q ) p%!q f < 1 ~ ~ a ! 3 ~s!n JaZuaqsxa s ayi u! lwloos ayl do nm moH aqL
3+B
'#
u o n s r a ~fii~uautalaa!ruJaqloxa a s ~ q d - ~ caSl k (fl=5-8dpag!poln;) "gz-01d *alq!~scdamqm a.ylc~!~ucnhag Amap rshlra3 a41 ssnsva -01 = g ~ g f 10 i 0!1m ~ ~ ~ l ucupun l t d 6~IE Jno parue3 SEM u o I i s n a J ayL
-dap 01uc8aq uolrJanuos auatuaqlkpa aql 'malsAs 341 paJalua uo!l -cauaJuo:, auaqdoql u d d 00 1 t! u ; r q ~ '(4- ~ X ~ ] F JO I C imp)/auazuqlXy~a ~ lorn 8's = Y '3ESS Y 'i(686 I) 092 '(5) B t ".rW '1llJziJ ,8113 ./?1111 3 5 JO ~ uo!uan - u o ~a r ~ a z u a q l d q l auc 01 dn xluPlsval q i q u! JapJo maz si I S , { I B $ C ~~ ~ ! U ~ P J O U I - p p u 1: nzo auruayoph1Aq1a 01 auazuaqliqla l o uo1rsuaEo~p6qaqJ " p o ~ d Laur!laj!I 1sL1rlss 3q1 q! I ~ L ~(4) M LIDJJJ,~IIA~ J~IJ!~-JJ~~III do a ~ n l c ~ a d u IF!I!U! ~ a l ay1 31 (GI
.uo!va.\uoa ~ U P I S I I O ~tf
-.tadittar ayl a t l l w . l n a p 'y ggy c!
OL 'de43
U!CIU!~UI 01
I F < ~ B ) Cay3 ~
ZSL
sJol3esu ~ 1 1 6 1 apue l~~ s!s6lele3
Chap. 10
Octane Number
755
Supplementary Reading
Catalyst Decay CDP10-MB California problem. Isomerization reaction with catalyst decay. [3d ed. P10-51. Review the data and make a recommendation. CDP10-NR Fluidized bed reactor with catalyst decay as measured by decline in octane number. Real data. [3rd Ed. PIO-201 CDPIO-OB Catalyst decay in a batch reactor. I3rd Ed. PIO-231 CDP10-PB Deactivation by coking in a differential reactor. [3rd Ed. P10-241 2B IS carried out in a CDPIQ-QB The autocatalytic reaction A + B moving-bed reactor. The decay law is first order in B. Plot the activity and concentration of A and B as a function of catalyst weight. CDP10-R, The decompositon of cumene is carried out over a Lay zeolite camlyst, and deactivation is found to occur by coking. Dererrnine the decay law and rate law and use these to deslgn a S n R . CDPlO-S, Analyze a second-order reaction over a decaying catalyst takes place in a moving-bed reactor. [Final Exam, Winter 19941 Plot the conversion and actfvity profiles. CDPlO-TR Analyze a first-order reaction A -+ B + C takes place in a moving-kd reactor. Plot the profiles of conversion and activity. CDPIO-U, For the cracking of normal paraffins (P,). the rare has been found to increase with increasing temperature up to a carbon number of 15 (i.e., n 5 15) and to decrease w ~ t hincreasing temperature for a carbon number greater than 16. [J. Wei, Ckem. Eltg. Sci.. 51. 2995 (199633 Provide an explanarion, CDPIO-VR The reacbon A t. B + C + D is carried out in a movingbed reactor. Plot conversion and activity profiles. CDPlO-RIB Second-order reaction and zero-order decay in a batch reactor Plot X and a as a function of time. CDFIO-X, Analyze first-order decay in a moving-bed reactor for the series reaction A + B +JC. CDPIO-Y New Problem on CD-ROM.
SUPPLEMENTARY READING 1. A terrific discussion of heterogeneous catalytic mechanisms and rate-controIling steps may be found i n
E.. and R. J. DAVIS. Fundamentul~of Cl~etnicalRcacrinn Engineeriug. New York: McGraw-Hill, 2003. MASEL,R. I., Principlc.~of Aci.rorptian and Reacriun an Solid S~ctfa~aces.New D A V I S , M.
York: Wiley, 1996. SOMOWAI, G. A,, Infmduc~fon to Slrtfaface Chernistn and Ca~olysis.New York:
U'iley, 1994.
7. A truly excellent discussion of the types and rates of adsorption together
wtth
techniques used in measuring catalytic surface areas is presented rn
MASEL.R. I.. Principles York: Wiley. 1996.
of Adsorp~ionand Rcnctio~i on
SnIrd Sutfaces. New
754
Catalysis and Catalytic Reactors
Chap. 10
PlOC-5 A packed-bed reactor was used to study the reduction of nitric oxide with ethylene on a copper-silica catalyst [Ind. Eng. Chem. Pmcess Des. Dev., 9 (31, 455 (1970)]. Devetop the integral design equation in terms of the conversion and the initial pressure using the author's proposed rate law. If this equation is sorved for conversion at vanous inlrial pressures and temperatures, is there a significant discrepancy between the experimental results shown in Figures 2 and 3 and the calculated results based on the proposed rate law? What i s the possible source of this deviation? PIOC-6 The thermal degradation of rubber wastes was studied [lnt. J. Cllem, Eng., 23 (4j, 635 1 1983)]. and ir was shown that a sigrnoidal-shaped plot of conversion versus time would he obtained for the degradation reaction. Propose a model with physical significance that can explain this sigmoidar-shaped curve rather than merely curve fitting as the authors do. Also, what effect might the particle size distrjbution of ?he waste have on these curves? [Hint: See 0. Levenspiel. The Chenlical Rcacror Omnibook (Consallis. Ore.: Oregon State University Press. 1979) regarding gas-solid reactions.] Additional Homework Problems
Mechanisms
CDPIQ-A,
Suggest a rate law and mechanism for the catalytic oxidation of ethanol over tantalum oxide when adsorption of ethanol and oxygen take place on different sites. [2nd ed. P6-171 CDP10-BN Analyze the data for the vapor-phase esterification o f acetic acid over a resin catalyst ar 1 IB°C. CDPJO-CR Silicon dioxide is grown by CVD according no the reaction Use the rate data to determine the rate law. reaction mechanism, and rate law parameters. [2nd Ed. P6-131 Determine the rate law and rate law parameters for the wet etching of an aluminum silicate. Titanium films are used in decorative coatings as well as wear-resistant tools because of their thermal stability and low electrical resistivity. TiN is produced by CVD from a mixture of TiCI, and NH>TiN. Develop a raw law. mechanism, and rate-limiting step and evaluate the sale law parameters. CDPl 0-Fn The dehydrogenation of ethylbenzene is carried out over a shell catalyst. From the data provided, find the cost of the catalyst to produce a specified amount of styrene. [2nd Ed. P6-201 CDPl 0-Gh The formarlon of CH, from CO of Hz is audied in a differential reactor. Determine the rate law and mechanism for the reaction A + B C. Determ~nethe rate law from data where the pressures are varied in such a way that the rate is constant. [2nd Ed. P6-181 De~errninethe rate law and mechanism for the vapor phase dehydra. tion of ethanol. [?nd Ed. Ph-211 CDPXQ-KH Analyze the rate data. CDP 1O-IdR Carbonauon of ally1 chloride whereby the complexes PdoCQ and Pd-CO*NaOHAare formed. 13rd Ed. P10-l I]. Find the rate-limiting
'
rtep.
756
Catalysis and Catalyt~cReactors
Chap.
3. A discussion of the types of catalysis, methods of catalyst selection, methods prepararion, and chsses of catalysts can be found in
'
I
ENGELHARD CORPORATION, Engelhard Caralvsts and Pwciuus Metal Chemica
Cntalag. Newark, N.J.: Engelhard Corp.. 1985. GATES,BRUCEC., Catalytic Chemistry. New York: Wiley. 1992. SCHMIDT. 6.D..The Engineering of Chemical Reactions. New York: Oxfo Press, 1998. VAN S A ~ R. . A., and .! W. NIEMANTSVERDRIET, Chemical Kinetics ar, Catalvsis. New York: Plenum Press, 3945. 4. Heterogeneous catalysis and catalytic reactors can be found in
HARRIDIT.P., Chemical Rwcior Design. New York:Marcel fekker. 2003. SATERFIELD, C . N., Heterogeneous Caralysis in Industrial Pmcrice, 2nd el New York: McGnw-Hill, 1991.
and in the following journals: Advances in Caralysis, Journal of Catalysis, an
Catalysis Reviews.
5. Techniques for discriminating between mechanisms and models can be found in BOX, G.E. I?, W. G.H m and J. New York:Wiley. 1938.
S. HUNTER. Statistics for Enperimenrer
FROMENT. G.F.. and K. B. BISCHOFF, Ckemicol Reactor Analysis and Desigj New York: Wiley, 1979, SBC. 2.3. 6. A reasonably complete listing of the different decay laws coupled with vario types of reactors i s given by
BUTT,I. B., and E. E. PETERSEN. Activalion, Deactivalian, and Poisoning c Catalysts. San Diego, Calif.: Academic Press, 1988. FARRAUTO, R. J., and C. H.BARTHOLOMEW, Fundamentals of Industrial Carl lytic Processes. New York: Blackie Academic and Professiond. 1997.
7. Examples of applications OF catalytic principles to microelectronic manufacturin can be found in
DOBKIN,D. M., and M. K. ZURAW.The Netherlands: KIuwer Academic Puk lishers, 2003. HESS,D. W., and K. F. JENSEN, Microelecttvnics Pmcessing. Washington. D.C American Chemical Society, 1989. JENSEN, K. F.. "Modeling of chemical vapor deposition reactots for the fabrics tion of micmlectronic devices," in Chemical an$ Catalytic Reactor Mor. eling. Washington, D.C.:American Chemical Society, 1984, LEE, H. H., Fundnrnentals af Micmclecctronics Processing. New Tori McGraw-AiII, 1990.
External Dtffusion Effects on Heterogeneous Reactions
11
Giving up is the ultimate tragedy.
Robert 3. Donovan
or It ain't over 'ti1 it's over.
Yogi Berrn
Overview In many industria1 reactions. the overall rate of reaction is limited by the wte of mass transfer of reactants between the bulk fluid and the catalytic surface. By mass transfer, we mean any process in which diffiision plays a tole. In the rate Iaws and catalytic reaction steps described in Chapter 10 (diffusion, adsorption, surface reaction, desorption, and diffusion), we neglected the diffusion steps by saying we were operating under conditions where these steps are fast when compared to the orher steps and thus could be neglected. We now examine the assumption that diffusion can be neglected. In this chapter we consider the external resistance to diffitsion, and in the next chapter we consider internal resistance to diffusion. We begin with presentation of the fundamentals of diffusion, molar flux, and then write the mole balance in terms of the mole fluxes for rectangular and cylindrical coordinates. Using Fick's law. we write the full equations describing flow, reaction, and diffusion. We consider a few simple geometries and solve h e rnms flux equations to obtain the concentration gradients and rate of mass transfer. We then discuss mass 'transfer razes in terns of mass transfer coefficients and correlations for the mass transfer coefficients. Here we include two examples that ask "What I f . . ." questions about the system variables. We close the chapter with a discussion on dissolving solids and the shrinking core model. which has applications in drug delivery.
758
External Diffusion Effects on Metewgeneous Reactions
Chap. 11
-1 1.1 Diffusion Fundamentals The Algorithm Rate law
~
~
Combine Evaluate
~
The first step in our CRE algorithm is the mole balance, which we now need to extend to include the molar flux, W,, and diffusional effects. The molar direction, i flow ~rate of h A ini a given ~ ~ ~ such ~ as ~the z ~direction down the length of a tubular reactor, is just the product of the flux, W, (moVm2 s), and the cross-sectional area, A, (rn2), that is,
In the previous chaprers we have only considered plug flow in which case
We now will extend this concept to consider diffusion superimposed on the molar average velocity. 11.I .I Definitions
DiffuSion is the spontaneous intermingling or mixing of atoms or molecules by random thermal motion. It gives rise to motion af the species relative to motion of the mixture. In the absence of other gradients (such as temperature, electric potential, or gravihtional potentiai). molecules of a given species within a single phase will always diffuse from regions of higher concentrations to regions of lower concentrations, This gradient results in a molar flux of the species (e.g., A), W, {moles/area.time), in the direction of the concentration gradient. The flux of A, WA, is relative to a fixed coordinate (e.g., the lab bench) and is a vector quantity with typical units of rnolJrn2-s.In rectangular coordinates We now apply the mole balance to species A, which flows and reacts in an element of volume AV = drAyAz to obtain the variation of the molar fluxes in three dimensions.
Sm. 1 F .f
DiHus~onFundamentals
Molar
Molar
Molar
Molar
l
Mole Balance
,
-
AxA!.W,-I--Molar
hxApWA:J_+3;+ 4xA?.WA,I, - Axb:W,,I,
Molar +
[ Rate of ] [ =
generation
_
+dl
, +
Rate of accumulation
]
rAAx
Dividing by AxAyAz and taking the limit as they go to zero. we obtain the molar flux balance in sectanguiar coordinates ~ W A-,--$WA> --
dx
~ W A+: r A
a:
- ac,
ar
The corresponding balance in cylindrical coordinates wirh no variation in he rotation about the z-axis is FEMLAB
We will now evaluate the flux terms W,. We have taken the time to derive the molar fiux equations in this form because they are now in a form that is consistent with the partial differential equation (PDE) solver FEMLAR. which is included on the CD with this textbook. l f . 1 . 2 Molar Flux
The molar flux of A. W,, is the result o f two contributions: J,. the molecular diffusion flux relative to the bulk motion of the fluid produced by a concenlration gradient. and B,. the flux resulting from the bulk motion of the fluid: k ~ n Rux l diffusion
=
hulk mo~ion +
The bulk flow term for species A i s the total flux of a l l molecules relative io a fixed coordinate times the mole fraction of A, g,: i.e.. B, = y, 2 W,. The bulk f OW term B, can also be exprex~edin terms of ~ h cconcentration of A and the molar average veloci~yV:
External Diffusion Ef!ecls cn Heterogeneous Reactions
R , = CAV
-mol - -m2.s
Chap.
(I I-:
moi . rn
m3 s
where the molar average velocity is Molar average
velocity
Here V, is the particle velocity of species i, and yi is the mole fraction of spc cies i. 3 y particle wlocities, we mean the vector-average velocities of rnilIior of A molecules at a point. For a binary mixture of species A and B, we let h and Y, be the pnnicle velocities of species A and B. respectively. The flux ( A with respect to a f xed coordinate system le.g.. the lab bench), 1%. Is ju the product of the concentration of A and the particle velocity of A:
The molar average velocity for a b i n a ~ ysystem is (1 I-; V = vAVA+ ynVs The total molar ffux of A is given by Equation ( 1 1-4). BA can b expressed either in terms of the concentration of A, in which case
or in terms of the mole fraction of A: Binary system ot' AanJ B
We now need to evaluate the molar flux of A,
JA,
that is superimposed on th
molar average velocity V.
11.1.3 Fick's First Law Our discussion on diffusion will be restricted primarily to binary systems con taining onty species A and B. We now wish to determine how the molar diffu Expcriment~tion sive Rux of a species (i.e.. J,) is related to its concentration gradient. As an ail with frog "ps led lu in the discussion of the transport law that is ordinarily used to describe diffu F~ck's% r ~law t sion, recall similar laws from other transport processes. For example. in con ductive heat transfer the constitutive equation relating the heat flux q and thl
temperature gradient is Fourier's law:
where k, is the thermal conductivity. In rectangular coordinates, the gradient is in the form
Sec. 11.2
milks
rrmclur
762
Binary Diffus~on
The one-dimensional form of Equation ( 1 1 - 10) i 5
Heat Transfer
4: = -k,
dT dz
-
( 1 1-12]
In momentum transfer, the constitutive relationship between shear stress, t,and shear rare for simple planar shear Raw is given by Newton's Inw of viscosity:
r=
Momentum Transfer
-,J
-
du d:
The mass transfer flux law is analogous to the laws for heat and rnornentuln transport. i.e., for constant total concentration J,=
Mass Transfer
-D A B-
c/C, dz
The general 3-dimensional constitutive equation for JA, the diffusional flux of A resulting from a concentration difference. is related to the mote fraction gradient by Fick's first law:
JA = - c D ~ ~ V ~ ~ (1 1-13) where c is the total concentration (rnol/dm3), D,, is the diffusivity of A in B (drn2/s), and y,, is the mole fraction of A. Combining Equations (I 1-9) and I1 1-13]. we obtain an expression for the molar flux of A:
[WA = 4DABv~,l + .vA(WA+ wB)/
Molar flux equation
(1 1-14)
In terms of concentration for constant total concentration Molar Hux equation
11-2 Binary Diffusion Although many syrtems involve more than fwo components, the diffusion of each species can be treated as if it were diffusing thmugh another sin@ species rather than through a mixture by defining an effective diffusivity. Methods and examples for calculating this effective diffusivity can be Found in Hill.'
I
11.2.1 Evaluating the Molar Flux The task is ta now the bulk flow term.
We now consider four typical conditions that arise in mass transfer problems and show how the molar flux is evaluated in each instance. The first two
' C. G . Hi!l.
Chemical Engineering Kirletics nrtd Reocror Design (New York: Wiley.
19771. p. 480.
762
External Diffusion Effects on Heterogeneous Reactions
Chap. 11
conditions, equal molar counter diffusion (EMCD) and dilute concentration give the same equation for WA,that is. Summery tJotes
WA= -DABVCA
The third condition. diffusion through a stagnant film,does not occur as often and is discussed in the summary notes and the solved problems on the CD. The fourth condition is the one we have been discussing up to now for plug flow and the PFR, that is, FA= vc*
We will first consider equimolar counter diffusion (EMCD). colvcd problems
11-2.1A Fduimolar Counter Diffusion. In quimolar counter diffusion (EMCD), for everv molt of A that diffuses in a given direction, one mole of B diffuses in the opposite direction. For exarnplc,consider a species A that is diffusing at steady stare from the bulk fluid to a catalyst surface, where it isomerizes to form B. Species B [hen diffuses back into the bulk (see Figure 1 I - I). For every mole of A that diffuses to the surface, 1 rnol of the isomer B diffuses away from the surface. The fluxes of A and B are equal in magnitude and flow counter to each other. Stated mathematically,
Figure 11-1 EMCD in isomerization reaction.
An expression for 1% in (ems of the concentration of A. CA.for the case of EMCD can be found by first subaituting Equation (1 1-16) into Equation ( 1 1-91:
For constant total concentrarion EMCD flux equation
11.2.1R Dilute Concentrations. When the mole fraction of the diffusing solute and the bulk motion In the direction of the diffusion are small, the second term on the rrght-hand side of Equation ( 1 1-14) [i.e.. y,(W, + Wn)] can usually he neglected compared with the f rst term, J A . Under these condition<.
Sec. 11.2
Binary Dinusion
763
together with the condition of canstant total concentration, the flux of A is identical to that in Equation (1 1-161, that is, Flux at dilute concenttations
This approximation is almost always used for molecules diffusing w i h n aqueous systems when the convective motion is small. For example, the mole fraction of a 1 M solution of a solute A diffusing in water whose molar concenwation, Cw, is
would be
Consequently, in most liquid systems the concentration of the diffusing solute is small, and Equation ( 1 1-18) is used to relate W, and the concentration gradient within the boundary layer. Equation (I 1-14) also reduces to Equation ( 1 1-1 7) for porous catalyst systems in which the pore radii are very smalI. Diffusion under these conditions. known as K~ludscndifiaion, occurs when the mean free path of the molecule is greater than the diameter of the catalyst pore. Here the reacting motecules collide more often with pore walls than with each other, and molecules of different species do not affect each other. The flux of species A for Knudsen diffusion (where bulk flow is neglected) is
where D, is the Knudsen diff~sivity.~
Summarv Notes
11.2.1C Diffusion Through a Stagnant Gas (WB= 0). Because this condition only effects mass transfer in a limited number of situations, we will discuss this condition in the S u ~ n m aNores ~ of the CD-ROM. 11.2.1D Forced Convection, In systems where the flux of A results primarily from forced convec~ion.we assume that the diffusion in the direction of the flow (e.g., axial z direction). J,, is small in comparison with the bulkflow contribution in that direction. B,(V, = W ) ,
?
C. N. Sanerfield. Mass Transfer in Herei7)geneotl.s C a t a l ~ ~(Cambridge: is MIT Press, 1970). pp. 4 1 4 2 . discusses Knudsen Bow in catalysis and gives the expression for calculating D X .
E~ternalDiffusion Effects on Hoterousneous Reactions
w, = 0,: = c*v_= C,U Molar flux uf *pevirs A when axial difttts~on ettsch rlcgllgi blr:
Chao
-
U = c* = F A A,. A,
where A, is the cross-sectional area and v i s the vofumetric flow rate. Althou< the component of the diffusional flux vector of A in the direction of flow. J, is neglected. the component of the flux of A in the x direction, Jh. which normal to the direction of ff ow. may not necessarily be neglected (see Figure 1 1-: 'AI
Figure 11-3 Forced axial convection with diffusion to surface.
When diffusional effects can be neglected. F4 can be written as the pro uct of the volumetric flow rate and concentration:
F4 = vCA
Plug flow
11.2,lE Diffusion and Convective Transport. When accounting for diff stonal effects. the molar flow rate OF species A. FA,in a specific direction z, the product of molar flux in that direction, W,,, and the cross-sectional an normal to the direction of ff ow, A, :
In terms of concentration the flux is
The molar flow rate is
Similar expressioas follow for W,, and Wk Substituting for the flux W,,, and W, into Equation ( 1 1-2), we obtain Flow. diffusion. and reaction
W,
Sec. 11.2
FEULAB
765
Binary Diffusion
Equation ( I 1-21) is in a user-friendly form to apply to the PDE solver. FEMLAB. For one-dimension at steady state, Equation ( 1 1-21) reduces to
In order to solve Equation ( 1 1-22) we need to specify the boundary conditions. In this chapter we will consider some of the simple boundary canditions, and in Chapter 14 we will consider the more complicated boundar), conditions, such as the Danckwerts' boundary conditions. We wiiI now use this form of the molar flow rate in our mole baIance in the z direction of a tubular flow reactor
However, we first have to discuss the boundary conditions in solving the equation. 11.2.2 Boundary Conditions
The most common boundaty conditions ate presented in Table 11-1. Taaw I l - I .
T m
OF BOUNDARYCOND~~IONS
-
1 . Specify a: concentration at a boundary (t.g., 2 0,CA = CAD).For an instantaneous reaction at a boundmy, the concenmtion of the reactants at the boundary is taken to be zero (e.g.. Ch = 0).See Chapter 14 for the more exact and cornpticated Danckwens' boundary condztions at z = 0 and z = L 2. Specify a flux at a boundary. a. No mass transfer to a boundary, For ample, at the watt of a mnmcting pipe.
That is, because rhe diffusivity is finite. the onIy way the flux can be 2em is if the concentratron gradient is zero. b. Set the molar fluto t k surface equal to the rate of reaction on the surface, W~(surFace)=
- r; (surface)
f 11-24)
c. Set the molar flux to the boundary equal to convective transport ncmss a boundary layer.
where k, is the mass transfer cotficient and CA,and CAbare the surface and bulk concentrations, respectively. 3. Planes of symmetry. When the concentntion profile is symmetrical about t plane, the cowennation gradient is zem in that plane of symmetry. For example, in the case of radial diffusion in a pipe, at the center of the pipe
766
External Diffusion Effects on Heterweneous Reaction$
Chap. 11
11.2.3 Modeling DiffusionWithout Reaction
In developing mathematical models for chernicaIly reacting systems in which diffusional effects are important, h e first steps are:
Steps in modeling mass transfer
Step I: Step 2: Step 3: Step 4: Srep 5: Step 6:
Perform a differential moIe balance on a particular species A. Substitute for F, in terms of Wk. Replace W, by the apprnpriare e x p s i o n for iAe concenMon gradient State the boundary conditions. Solve for the concentration profile. Solve for the molar flux.
We are now going to apply this algorithm to one of the most important cases, diffusion through a boundary layer. Here we consider the boundary layer to be a hypothetical "stagnant film" in which all the resistance to mass transfer is lumped. Example 11-1 Diflusion Through a Film Yo a Catalyst Particle
Species A. which is present in dilute concentrations. is diffusing at steady state from the bulk fluid through a Stagnant film of B of thickness S to the external surface of the catalyst (Figure El 1 - 1 . 1 ). The concentration of A at the external boundary is CAb and at the external catalyst surface is C,,, wrth CAh> CAr.Because the thickness o f the "hypothet~calstagnant film" next to the surface is smaII with respect ro the diameter of the panlcle (i.e., 6 << d,), we can negrect curvature and represent the diffusion in rectilinear coordinates as shown in Figure El 1 - 1.2. Determine the concentration profile and the flux of A to the surface using (a) shell balances and (b) the general balance equations.
/
,I,
c*, ,=,
\
External M a f s
Transfer
figure EII-1.1
Transport to a cphere
Figure E1I-1.2 Boundary layer.
%c. 1 1.2
Binary Diffusion
(a) Shell Balances Our first step is to perfom a mole balance on species A over a differenrial element of width bz and cross-sectionel area A, and Uren arrive at a first-order diffmntia] equation in W, [i.e.. Equation (El 1-i .2)].
_1
Step 1: The general mole balance equation i s + -
An algorirhrn Mole balance * Bulk Row = ? Differential equauon
-
R ][;
-
J
Rate
generation
]
Rate of accumulation
Dividing by - igives ~ us
Boundary condi~~ons Concentration
and taking the limit as k
profile
4
00,we obtain
Molar flux
Step 2: Next, substitute for FA in terms of W, and A,. Fk = WkAc
Divide by A, to get Follovving the Algorithm
I
dUk2 d:
*
(Ell-1.2)
Step 3: To evaIuate the bulk flow term, we now must relate W, to the concentration gradient utiIizing the specification of the probIem statement. For d~ffusionof almost all solutes through a liquid, the concentration of the diffusing species is considered dilute. For a dilute concenttation of the diffusing soiute, we have for constant total concentration,
Differentiating Equation (EJ 1- 1.3) for comtant diffusivit. yields
However. Equation (El I- 1.2) yielded
Therefore, the differential equation describing diffusion through a liquid film reduces to
(El 1-1.5)
768
1
External Diffusion Effects on Heterogeneous Reactions
Chap. -
Step 4: The houndary conditions are:
Step 5: Solve For the concentration profile. Equation (ElI - 1.5) is an el mentary differential equation that can be solved directly by integnting twlc w ~ t hrespect to z. The first integration ylelds
1
and the second yieids
where K l and K2 are arbitrary constants of integration. We 11ow u%e11 boundary conditions to evaluate the constants K , and K!. At z = 0, C, = CAh;therefore,
Eliminating Kt and rearranging gives the following concentration profile: (Ell-I.' I
Rearnnging (El 1- I.7),we get the concentration profile shown in Figure-E I 1 - I .
Concentration profile
FIgure E l l - 1 3 Concentmtion profile.
Step 6: The next step is to determine the molar flux of A diffusing throug the stagnant film. For dilute solute concentrations and constant total concel tntion.
Sw.11.2
769
Binary Diffusion
To determine the flux. difyerentiote Equation (El 1-1.8) with respect to :and rhen mult~plyby -DAB:
Equation (El 1- 1.9) could also be written in terms of mote fracfions (El [-1.10)
I f DAB= 10-h m2/s. Cm = 0.1 kmollm?, and 6 = 1 0-6 m. y,, = 0.9, and v,, = 0.2, are substituted into Equation (E 1 1 1.9). Then
-
EMCD or dilute cancentntion
/b)General Balance Equations (1 1-2) and (11-21) Another method u ~ e dto arrive at the eqi~ationdescribing flow, reaction, and diffusion for a particular geornerry and set of conditions i s ro use the general balance equations L 1-21 and ( 1 1-211. I n this method, we examine each term and then cross out terms that do not apply. In this example, there is no reacrion. -r, = 0. no flux in the .+direction. (W,,= 0) or the >direction (W,,= 0). and Qe are at steady state so that Equation (1 1-21 seduces to
which i s the wrne as Equation (E I 1 - l .?I. Similarly one could apply Equation ( I 1-21 1 to this example realizing we are at steady state, no reaction, and there is no variation in concentration in either the x-direction or the -direction
so that Equation ( 11-2t ) reduces to
770
External Diffusion E m s on Heterogenws Reactions
Chap. 11
After dividing both sides by the diffusivity, we realize this equation is the same as Equation (El 1-15), This problem is reworked for diffusion through a stagnant film in the solved example problems on the CD-ROMlweb solved problems. Colved Problems
11.2.4 Temperature and Pressure Dependence of DAB
Before closing this brief discussion on mass transfer fuodamentaIs, further mention should be made of the diffusion coefficienr."quations for predicting gas diffusivities are given by Fuller' and are aEso given in Perry's I-/andbnnk." The orders of magnitude of the diffusivities for gases. liquids: and solids and the manner in which they vary with temperature and pressure are given in Table 11-2. We note that the Knudsen, liquid, and solid diffusivities are independent of total pressure.
lr is importan1 to know the magnitude and the T and P dependence of the diffusivity
Oi-dcr of Mugr~itude
Phare
cm21s
m p Ik. . liquid viscosit~esat
m71r
T~mperatrrreand Pressacre Drpnde~rcesP
temperarures T,and T2, sespcctively: ED. d~ffusionactivat~onenerg).
' For funher discussion of mass transfer fundamentals,
see R. B. Bird. \V, E. Stewan. {New York: Wiley. 1960). '' E. N.FuIIer. P.D.Schettler, and J. C. Giddings, Ind Eng. Chem.. 5815). 19 (f966). Several other equat~onsfor predicting diffusion coefficientscan he found in B. E. Polling, J. M . Prausnitz, and I. P. O'ConneII. Tile Properires ofcases unn' Liqrrids, 5rh ed. ( New York: McGraw-Hill. 21K)I ). R. H. Perry and D.N'. Green, Cl~~nriral Ensrneerb tirindbook. 7rh ed. (New Y d and E.N. L~ghtfoot,Tr~nsporrPhinorn~na,2nd ed.
McGrdw-HiIE. 1999). To es~imate llqufd drffusivittes fur binary syslerns, see Doraiswamy, h d En#. Cl~enl.F Z I I Ia,~ 77 , , (1967).
K. A. Reddy iind L. K.
Sec. 11.3
External Resistance to Mass Transfer
ni
f 1.2.5 Modeling Diffusion with Chemical Reaction
The method used in solving diffusion problems similar to Example 11-1 is shown in Table 1 1-3. Also see Cussler? TABLE 1 1-3.
Expanding the previous s,x
mdeling steps jusr a bit
STEPSIN MODELFNG CHEMICAL SYSTEMSuwi D I ~ S S O AMIN REnm03
I. Define the problem and state the assumptions. (See Problem Solving on the CD.) 2. &fine the system on which the balances are to be madc. 3. Perfom a differential mole bakance on a panicular species. 4. Obtain a differentinl equatioli In W, by rearranping your balance equation properly and ukIng thelimit as the volume of the elernen1 goes lo zero 5. Substirure the appropriate expression involving the concentration gradient for W, fmm Section E 1 2 lo abrain a second-order differential equation for the conccnwation of A,& 6 Express the reaction rate r, ( ~ any) f In terns of concentration and substitute into h e differ1. R 9. 10 I 1.
ential equation. Stale the appropriate boundary and initial condi~ions. PUI the differential equations and boundary condrtions. in dimensionless Tom. Solve the resultmg differential equation for the concenrraiion profile. Differentiate thrs concentration profile to obtain an expression for the molar flux of A. Suhstllute numerical values for symbols.
"In some instances it may be ea~ierto ~ntegratethe resulting differential equation in Sfep 4 before substituting for WA.
The purpose of presenting algorithms (e.g., Table 11-31 to solve reaction engineering probIems is to give the readers a starting point or framework with In ;---3 of the algorithm which to work if they were to get stuck. It is expected that once-readers are 6, familiar and comforrable using the algorithdframework, they will be able to to generate creati~esnlutions~-move in and out of the framework as they develop creative solutions to nonstandard chemicaI reaction engineering problems. Move
+
11.3 External Resistance to Mass Transfer
'
11.3.1 The Mass Transfer Coefficient
To hegin our discussion on the diffusion of reactants frum the bulk fluid to the external surface of a catalyst. we shalI focus attention on the flow past a single catalyst pellet. Reaction takes place only on the external catalyst surface and not in the fuid surrounding it. The fluid velocity in the vicinity of the spherical pellet will vary with position around the sphere. The hydrodynamic boundary layer is usually defined as the dismnce from a solid object to where the fluid velocity is 99% of the bulk velocity U U .Similarly. the mass transfer boundary layer thickness, 6, is defined as the distance from a solid object to where the concentration of the diffusing species reaches 99% of' the bulk concentration. A reasonable representation of the concentration profile for a reactant A diffusing to the external surface is shown in Figure 1 1-3. As illustrated, the
E.L. Cussler, Drffusion Mu1.7
Tmnsfer in Flltid S~rte~ns, 2nd ed. (New
bridge University Press. 1997).
York: Carn-
772
External Drftus~onEffects on Heterogeneous React~ons
Chap.
Side Note: Transdermal Drug Deliver)r The principles of steady state diffusion have been used in a number of drug delivery systems. Specifically, medicated patches are commonly used tc attach to the skin to deliver dmgs for nicotine withdrawal, birth control, and motion sickness, to name a few. The U.S. transdermal drug deIivq markel was $1.2 billion in 2001. Equations similar to Equation f 1-26 have been used to model the release, diffusion, and absorption of the drug from the patch into the body. Figure SN11.1 shows a drug delivery vehicle (patch) along with the concentration gradient in the epidermis and dermis skin layers. Skin Layers
Figure SN 11.1 Tmnsderrnal dryg delivery schematic.
As a first approximation, the delivery sate can be written as
where
R=R
6 + ' + A DAB,
8
DAB^
Where A, is the area of the patch; CAP, the concentration of drug in the patch; R, the overall resistance; and R,, the resistance to reIease from the patch. There are a number of situations one can consider, such zts the patch resistance limits the delivery, diffusion through the epidermis limits deIivery, or the concentra€ion of the drug is kept constant in the patch by using solid hydrogels. When difision through the epidermis layer limits, the nte of drug delivery rate is
Other mwers include the use of a quasi-steaay anruysls t~ coupIe the diffusio!n equation dance on the drug in the patcl3 or the zrm order (iissoluti.on of the n the patch.t Roblem 11-210 explores these situalions.
Fw t u r n infomutm see:Y. H.K d a and R Guy, Advmced Unrg Uellvery Km'ews 48,159 (2001); B.MuHer. M.Kasper, C.Surber. and G.Jinanidis, Ei4mpem J o u d of PhnmtaceuticaI Science 20. 18 t (2003); www.dmgdeLiverytech.codcgi-bid articIes,cgi?irlim'cLe=143: w w w . p h a m q ,umoiyland,ediu$%culry/rdalb?r//dul~/ Teaching%ZOWeb%2OPuge/Teaching.khn
Sac. 1 t .3
773
E~fernalR ~ 9 i s t a m eto Mass Transfer
chanze in concentration o f A from CAhto CA, takes place in a very narrow fluid Layer next to the surfslce of the sphere. Nearly all of the resistance to mass trancfer i s found in this I:iyer.
Ph
Figt~re11-3 Boundary layer around the surface of a cat;ilyst pellet.
11.3.2 Mass Transfer Coefficient
The concept or a b~~thrticnl stagnanr tilm within which the wsistance D I
external mas$ transfer exlsts
A useful way of modeling diffusive transport i s to treat the: fluid layer next to a solid boundary as a stagnant film of thickness 8. We say that all the resistance to mass transfer is found [i.e.. lumped) within this hypothetical stagnant film of thickness 6. and the properties (i.e., concentration, temperature) of the fluid at the outer edge of the film m identical to those of the bulk fluid. This model can readily be used so solve the differential equation for diffusion through a stagnant film, The dashed line in Figure l I-3b represents the concentration profile predicted by the hypothetical stagnant film model. while the solid line gives the actual profile. If the film thickness is much smalIer than the radius of the pellet, curvature effects can be neglected. As a resutt. only the one-dimensional diffusion equation must be solved. as was shown in Section 1 1. L (see also Figure L 1-4).
c A, Figure 11-4 Concentration protilr for E.VCD in stagnant film model.
For either EMCD or dilute concentrations, the solution was shown in Example El 1-1 to be in the form
While the boundary layer thickness will vary around the sphere, we will take it to have a mean film thickness 6. The ratio of the diffusivity DAB to the film thickness F is the mass transfer coefficient, kc, that is,
External Diffusion Effects on Heterogeneous Reactions
Chap. 71
The mass transfer cwfhcient
Combining Equations (1 1-26) and (1 1-27), we obtain the average molar flux from the bulk fluid to the surface Molar flux of A to the surface
In this stagnant film model. we consider all the resistance to mass transfer to be lumped into the thickness 6. The reciprocal of the mass transfer coefficient can be thought of as this resistance W, = Flux = Driving force - Cnh- C A ~
Resistance
-
-
(w)
11-3.3 Correlations for the Mass Transfer Coeflicient
The mass transfer coefficient kc is analogous to the heat transfer coeficient h. The heat flux q from the bulk fluid at a temperature Toto a solid surface at T,is
For forced convection. the heat transfer coefficient is normally cotrelated in terms of three dimensionless groups: the Nusselt number, Nu; the Reynolds number, Re; and the Prandtl number, Pr. For the single spherical pellets discussed here, Nu and Re take the following forms:
The Prandtl number is not dependent on the geometry of rhe system. The Nusselt. handtl. and Reynolds numbers are used in forced convection heat
transfer
cornlalions
\
' *
'
where or, = k,/pC, = thermal diffusivity, m*/s v =
= kinemarfc visco'sity (momentum diffusivity). m2/s P
d, = diameter o f pellet. m U = free-strearn velocity, mls k, = thermal conductivity. J I K . m a s p = Fluid density. kglrn3 h = beat transfer coefficient, Jlrn2.s-K or Watts/m2 K The other symboIs are as defined previously.
Sec. 1 I.3
775
External Resistance to Mass Transfer
The heat transfer correlation relating the Nusselt number to the Prandtl and Reynolds numbers for flow around a sphere is8 Nu = 2 +
( I 1-33) Although this correlation can be used over a wide range of Reynolds numbers, it can be shown theoretically that if a sphere is immersed in a sragnant fluid (Re = 0), then
and that at higher Reynolds numbers in which the boundary layer remains laminar, the Nusselt number becomes Although further discussion of heat transfer correlations is no doubt worthwhile, i t will not help us to determine the mass transfer coefficient and the mass flux from the bulk fluid to the external pellet surface. However, the preceding discussion on heat transfer was not entirely futile, because, for similar geometries, rhe hear and mass transfer cor~lationsare analogous. If a heat transfer correlation for the Nusselt number exists, the mass transfer coefficient can be estimated by replacing the Nusselt and Prandtl numbers in this correlation by the Shemood and Schmidt numbers, respecrively: Convening n heat transfer correlation to a maw transfer
correlation
Sh SC
-
-- NU fi
The heat and mass transfer coefficients are analogous. The corresponding fluxes are 4: = h(T
- T,)
( I 1-36)
Wk = k,(CA - )C, The one-dimensiona1 differential forms of the mass flux for EMCD and the heat flux are. respectively. For EMCD the hear and molar flux equations are analvpous.
If we replace h by k, and k, by DABin Equation (1 1-30), i.e..
a \hl. E. Ran? and W. R. Marshall. J r , C l ~ ~ rEng. n . Pmg.. 48, 141-146, 173-180 (19521.
776
External Diffusion Effects on Heterogeneous Reactions
Chao.
we obtain the mass transfer Nussel t number (j.e., the S herwood number):
The Prandtl number is the ratio of the kinematic viscosity (i.e., the momentu dihsivity) to [he thermal diffusivity. Because the Schmidt number is anal gous to the Prandtl number. one would expect that Sc is the ratio of tl momentum diffusivity (i.e.. the kinematic viscosity), v, to the mass diffusivi DAB.Indeed, this is true:
a, + D A B The Schmidt number is
s C = -Y - -m2/s
Schmidt number
The Sherwood. Reynnlds. and Sehmldt number< are used in forced convection
maw tmnGer correlation\.
D,, ..-
m2/s
dimensionless
( 1 1-31
Consequently, the correlation for mass transfer for flow around a spherical pe let is analogous to that given for heat transfer [Equation ( 1 1-33)]. that is. Sh = 2 -6 0.6ReT'2Sc"3 111-41 This relationship is often referred to as the Friisslirrg correl~~rion.~ 11.3.4 Mass Transfer to a Single Particle
In this section we consider two limiting cases of diffusion and reaction on catalyst particle.") In the first case the reaction is so rapid that the rate of di fusion of the reactant to the surface limits the reaction rate. In the second casl the reaction is so slow that virtually no concentration gradient exists in the g~ phase ( i t . , rapid diffusion with respect to surface reaction). Example 11-2 IF rapid reaction, then diffusion timits the overall rate.
Rapid Reactinn on a Catalyst Surface
Calculate the mass flux of reactant A to a single catalyst pellet I cm in diameter su! pended rn n Imge body of liquid. The reactant is present in dilute concentration: and the reaction is considered to take place instantrtoeousIy at the external peHt surface (i.e.. C,, = 0). The bulk concentration of the reactant is I.0 M. and th free-system liquid velocity is 0. I mls. The kinematic viscosity is 0.5 centistoke (cI I centistoke = 10AhmZls), and the liquid diffusivity of A is to-" rn2/b. SoI~rtion For dilute concentrations of the solute the radial flux is W4r =
k,(C~h- Ckr)
( 1 1-28
N. Frossfing, Gerlnndx Beit!: Geophy., 52, 170 (1938). "A comprehensive list of correlations for mass transfer to panicles is given by G. A Hughrnark, Ind Eng. Chem. fund.. 19I2), 198 (1980).
I
Set. 11.3
777
Exiernal Rasistance to Mass Transfer
Becauqe reaction i\ dcsumed to wcur in\tantanenuqly un the extem:~l c ~ ~ r f i l cnt' e the pellet. C4, = 0. A l w . C',,,, i\ given sc 1 molldrn7.The n1nh.c tr;~ri\trr ct~tficienlfor siugfe yheres i q calculated froom the Frridinp corrzlat~on:
Substituting these value!, into Equation I 11-40) gives us Sh = 2
+ 0.611Wo)0~5(500R)1'~ = 360.7
(El l-Zbl)
Substituting for k, and CAhin Equation ( 1 1-28), the molar Rux to the surface is LVn, = (4.61 X
.
m/s (1.0.'
- 0) mul/rn3 = 4.61 X lo--' mol(m2.s
Because W,,= - r,", thisr;lteisalsethente of~eactionper unit surface areaofcatalyst.
In Example 1 1-2, the surface reaction was extremely rapid and the rate of mass transfer to the surface dictated the overall rate of reaction. We now consider a more general case. The isornerizatian is taking place on the surface of a solid sphere (Figure 1 1-51. The surface reaction folIows a Langmuir-HinsheLwood single-site mechanism forwhich the rate law is
Figure 11-5 Diffusion to, and reaction on. external surface of pellet.
External Diffusion E M S on Heterogeneous Reactions
Chap, 1 7
The temperature is sufficiently high that we need only consider the case of very weak adsorption (i.e., low surface coverage) of A and B: thus (KBCBT+ K~Ckrl 1
Therefore,
Udng boundary conditions 2b and 2c in Table I I - I , we obtain
The concentration CAI is not as easily measured as the bulk concentration. Consequent?y. we need to eliminate CA,from the equalion for the flux and rate of reaction. Solving Equation (1 1-44) for C, yields ~,CA c*, = kr+k,
and the rate of reaction on the surface becomes Molar flux of A to lhe surface is equal to the rate o f consumption of A on the surface
One will often find the flux to or from the surface as written in termt of an efecrive transport coefficient kt,:
where
Rapid Reaction. We first consider how the o~erallrate of reaction ma!, he increased when the rate of mars tiansfer to the surface limits the overall rate of reaction. Under these circumstances the specific reaction rate constant is much greater than the mass transfer coefficient k, % k,
and k,. -41
k,
Sec. 11.3
M e r n a l Res~stancsto Mass Transfer
779
To increase the rate of reaction per unit surface area of solid sphere, one must increase C, andlor kc. In this gas-phase catalytic reaction example, and for most Iiquids, the Schmidt number is sufficiently large that the number 2 jn Equaljon (1 1-40) is negligible with respect to the second term when the ReynoIds number is greater than 25. As a result, Equation (1 1-40) gives It i s important to know how the mass transfer cwffic~ent vanes with fluid velocity, pangcle siw, and physrcal proprtles.
kc = 0.6 X (Term I ) X (Term2) Mass Transfer
-;/ r
Llrnlled
U
Reaction Rate Limited
Term I is a function of the physical properties DAB and v, which depend on temperature and pressure only. The diffusivity always increases with increasing temperature for both gas and liquid systems. However, the kinematic viscosity v increases with temperature (v w T p 2 ) for gases and decreases exponentialiy with temperature for liquids. Term 2 is a function of flow conditions and particle size. Consequently, to increase k, and thus the overall rate of reaction per unit surface area, one may either decrease the particle size or increase the velocity of the fluid flowing past the particle. For this particular case of flow past a single sphere. we see that if the velocity is doubled, the mass transfer coefficient and consequently the rate of reaction is increased by a factor of
Slow Reaction. Here rhe specific reaction rate constant is small with respect to the mass transfer coefficient:
1 u
The specific reaction rate i s independent of the velocity of fluid and for considered here. independent of particle size. Howet8er. for porous catalyst pellets, k, may depend on particle size for certain situations, as shown in Chapter 11. Figure 11-6 shows the variation in reaction rate with Term 2 in Equation ( 1 1-49), the ratio of velocity to particle size. At low velocities the mass
Mars tran~fer the solid sphere
effectr are no1
important when rhe reaction rate 1 5 Iimi~ing.
780
External Diffusion Effects on Heterogeneous Reaclions
Chap :
transfer boundary layer thickness is large and diffusion limits the reaction, A the veltxity past the rphere i s increased, the boundary layer thickne? decreases. and the mass transfer across the boundary layer no Ionger limits th rate of reaction. One also notes that for a given velocity, reaction-limiting cot ditions can be achieved by using very small particies. However, the smaller ~h particle size, the greater the pressure drop in a packed bed. When one I obtaining reaction rate data in the laboratory, one must operate at sufficient1 high velocities or sufficiently small particle sizes to ensure that the reaction I not mass transfer-limited.
Limited
When collecting rate law data, operate in the reaction-limited region.
Figre 11.6
Regions of mass transfer-limited and reaction-limited reactions.
1 1.3.5 Mass Transfer-Limited Reactions in Packed Beds
A number of industrial reactions are potentially mass transfer-limited becaus they may be carried out at high temperatures without the occurrence of unde sirable side reactions. In mass transfer-dominated reactions, the surface reac tion is so rapid that the rate of transfer of reactant from the bulk gas or liqui phase to the surface limits the overall rate of reaction. Consequentty. mas transfer-limited reactions respond quite differently to changes in ternpentur and flow conditions than do the rate-limited reactions discussed in previou chapters. In this section the basic equations describing the variation of conver sion with the various reactor design parameters (catalyst weight, Row condi [ions) will be developed. To achieve this goal, we begin by carrying out a mol balance on the following mass transfer-limited reaction:
carried out in a packed-bed reactor (Figure 11-7). A steady-state mole balanc on reactant A in the reactor segment between z and r + ht is
Sec. 1 1.3
781
External Resistance to Mass Transfer
rate out FA=!: - F
A
+
I-;cI,(A~Az)=
0
( 1 1-51]
Z+AZ Figure 11-7 Packed-bed reactor.
where r i = rate of generation o f A per unit catalytic surface area. mol/s.m? a, = external surface area of catalyst per volume of catalytic bed. m2/m" = 6( 1 - +)Id, for packed beds, m21m-' 4 = porosity of the bed (i,e.. void fraction)" d,, = particle diameter. m A, = cross-sectional area of tube containing the catalyst, rn2 Dividing Equation ( I I-51) by A,&- and taking the limit as k ,-0,we have
We now deed to express F, and r(; in terms of concentration. The molar Aow rate of A in the axial direction is
FA-.= A, Wk = (Jk
+ B,)A,
( l t -53)
I n aImost all situations involving flow in packed-bed reactors. the amount of material transported by diffusion or dispersion in the axial direction is negligineelwted. ble compared with that transported by convection h e . , bulk flow):
Axial diffusion is,
Jk < Bk (In Chapter 14 we consider the case when dispersive effects must be taken into account.) Neglecting dispersion. Equation ( 1 1-20) becomes Fk = A, W, = R,Bk = UC, A, f 1 1-54] where U is the superficial rnoIar average velocity through the bed Imls). Substituting for Fk in Equadon (I I-52) gives us
For the case of constant superficial velocity U .
"In the nomenclature for Chapter 4, for Ergun Equation for pressure dmp.
782
External Dlffusron Effects nn Heterogeneous Reactions
Chap. I t
Differential equation describing Row and reaction in a packed bed
For reactions at steady state, the molar flux of A to the particle surface, WA, (mol/rn2~s)(see Figure 11-8), is equal to the rate of disappearance of A on the surface -ri (mol /m2s); that is,
Boundary
I;------+--'
Layer
Figure 11-8 Diffusion across stagnant film s u m n d i n g catalyst pellet.
?A"
From Table 11-1, the boundary condition at the external surface is
-ri = Whr = kr(CA- CArl where
kc = mass transfer coefficient = 1DAB/8(s-1) CA = bulk concentration (moUm"> C,, = concentration of A at the catalytic surface (molJm")
Substituting for
I n mact~onsthat c o m ~ l e t c 1mass ~ transfer-ltmired, it IF not necewary to knoa the rate law.
ri in Quation
( 1 1-56), we have
In most mass transfer-limited reactions. the surface concentration is begligible with respect to the bulk concentration (i.e.. CAs CAT): ( 1 1-60)
integrating with the limit, at z = 0. C, = CAo:
Sec. 1 1 3
783
External Resistance lo Mass Transfer
me corresponding variation of reaction rate along the length of the reactor is
The concentration and conversion profiles down a reactor of length L are shown in Figure 1t-9. Reactor concentration profile for a mass
fransfer-limited reaction
1 .o
C~
X
,c
+
0 0
rlL
1.0
slL
0
1.0
Figure 11-9 Axial concentration (a) and conversion (b) profiles i n a packed bed.
To determine the reacto; length L necessary
to
achieve a conversion X,
we combine the definition of conversion,
with the evaluation of Equation (1 1-61) at z
=
L to obtain
I t .3.6 Robert the Worrier Robert is an engineer who is always worried. He thinks something bad will happen if we change an operating condition such as flow rate or temperature or an equipment parameter such as particle size. Robert's motto is "If it ain't broke, don't fix it." We can help Robert be a little more adventuresome by analyzing how the important parameters vary as we change operating conditions in order to predict the outcome of such a change. We first look at Equation I1 1-64} and see that conversion depends upon kc, a, U. and L. We now examine how each of these parameters will change as we change operating conditions. We first consider the effects of temperature and flow rate on conversion. To l e m the effect of flow rate and temperature on conversion, we need to know how these parameters affect the mass transfer coefficient. That is. we must determine the c o r n Iation for the mass transfer coefficient for the particular geometry and flow field. For flow through a packed bed. the correlation
784
E ~ t e ~ n D~%lsion al Effects on Weterogewaus Reac!io~s
given by Thoenes and KrarnersI2 For 0.25 I < Sc < 4000 is
Cbap 1
< + < 0.5. 40 < Re' < 4000, an
Sh' = I.O(Rc')"?Sc"?
( 1 1-65
Thurne+Kramer?;
corrrlatio~~ for Ruw through packed bedr
where Re' =
Re ( I -b)w
GI,, = particle diameter (equivaient diameter of sphere o f the sanl
voIume). m (volume of peElet)l1/
rb y
U y p
= kinematic viscosity. m21s
v = P
DAB= gas-phase diffusivity, m2/s For constant fluid propenies and panicle diameter:
For diffuqinnlimited reacttons. reactlvn rate depends on panicle w e and Buid veiwlty.
We see that the mass transfer coefficient increases with the square root of thl superficial velocity through the bed. Therefore, for n B e d concenfmtmn, CA such as that found in a differential reactor, the rate of reaction should vary wit1
ur/::
-
i-;:
x
k,.CAx U"?
However, if the gas velocity is continually increased, a point is reached when the reaction becomes reaction rate-limited and consequently, is independent o the superficial gas velocity, as shown in Figure 11-6. Most mass transfer correlations in the Iiterature are reported in terms o the Colburn J factor (i.e., I, )as a function of the Reynolds number. The rela tionship between JD and the numbers we have been discussing i s
Colburn J factor
Figure I L- 10 shows data from a number of investigations for the J fact01 as a function of the Reynolds number for a wide range of particle shapes ant I2D. Thoenes, Jr. and H.Kramers. Chctn. Eng. Sci., 8, 27 1 (1958).
Sec. 11.3
785
External Resistance to Mass Transfar
gas flow conditions. Note: Thcre are serious deviations from the Colburn a n d ogy when the concentration gradient and temperature gndient are coupled as shown by Venkatesnn and Fogler.I3
FEgure 11-10 Mass transfer correlntion Tor packed beds. $b=@ [Reprinted with pennivsion fmm P. N. Dwidevi and S. S. Upadhyuy, h ~ fEtrg . C/rertt. Pmcesr Dr! Dai:, 16, 157 11977). Copyright O 1977 American Chemical Society.]
A correlarion for Row through packed beds in terms o f the Collwrn J factor
Dwidevi and Upadhyayl-' review a number of mass transfer correIations for both fixed and fluidized beds and arrive at the following correlation, which is vntid for both gases (Re > 10) and liquids (Re > 0.01) in either fixed or fluidized beds:
0,365 mJ,=-+- ~0765 ~ 0 . 8 - R ~ 3~O
( 1 1-69)
For nonsphericill particies, the equivalent diameter used in the Reynolds and Sherwood numbers i s d, = where A, is the external = 0.564 surface area of the pellet. To obtain correIations for mass transfer coefficients for a variety of systems and geometries, see either D.Kunii and 0.Levenspiel, Fluidi;~tiotmEngineering, 2nd ed. (Butterworth-Heinemann. t991). Chap. 7, or W. L. McCabe. J. C. Smith, and P. Han-iott, Unit Operations in Chemical Engineering. 6th ed. (New York: McGraw-Hill, 2000). For other correlarions for packed beds with different packing arrangements, see I. Colguhoun-Lee and J. Stepanek, Chemical Engineer, 108 (Feb. 1974).
A,
Actual caw history and currellt
application
am
--
I
Example 11-3
Maneuvering a Space Satellife
Hydrazine has been studied extensively for use in monopropellant thrusters for space flights of tong duntion. Thrusters are used for altitude control of communication satellites. Here h e decomposition of hydrazinc over a packad k d of alumina-supported I3R. Venkatesan and H. S. Fogler. AICkE J.. 50. 1623 (July 2004). I T . N.Dwidevi and S. N. Upadhyay, ind. Eng Cl~em. Process Des. Dcv.. 16, 157 (1977).
786
External Diiusim Effects on Heterogeneous Reactions
Chap. 7 7
iridium catalyst is of intere~t.'~ In a proposed study, a 2%hydrazine in 98% helium mixture is to be passed over a packed bed of cylindrical particles 0.25 cm in diam. eter and 0.5 cm in length ar a gas-phase velocity of 15 m/s and a temperature of 750 K. The kinematic viscosity of helium at this temperature i s 4.5 X lo-' mVs. The hydrazine decomposition reacrion 1s believed to be externally mass transfer-limited under these conditions. If the packed bed as 0.05 m in length, what conversion can be expected? Assume isothermal operation.
m2/s a1 298 K DAB= 0.69 X Bed porosity: 30% Bed fluidicity: 95.7%
Rearranging Equation ( 1 1-64) g'~vesus (EI 1-3.1) ( a ) Thoenes-Kramers correlation I. First ule find the volume-average particle diameter:
(El 1-3.2)
2. Surface area per voiurne of bed:
(
3. Mass transfer coeflicient:
I
For cylindrical pellets.
Representative MIUCS
Re' = 143
5c = 1.3
Sh' = 13.0 k,
= 3.5 ml.r
I {'(
Correcting the diffurivi~yto 750 K using Table 11-2 gives
h. I. Smith and W. C. Solomon. Ind. En$ Chm~.F~rrrd..21.
I I ~
374 11982).
Sac, lI.3
787
External Resistance to Mass Transfer
(El1-3.5) Sc-
v D,,
-
4.Sx10-4m2Js = 1.30
3.47 X 10-"2/s
Substituting Re' and Sc into Equation (11-65) yields Sh' = (143.2)"2(1.3)1'3= (11.97)(1.09) = 13.05
X
[I .2)(13.05) = 3.52 m/s
(El 1-3.6)
(El 1-3.3)
The conversion is
=I
- 1.1 8 X
= 1.00
virtually complete conversion
(b) Colburn J , factor. Calculate the surface-area-averageparticle diameter For cylindrical pellets the external surface area is A=ndL,+Znl]
(El 1-3.9)
(El 1-3.1 1 )
(EI 1-3.12)
788
Extsma! E~fusionEf!ec!s on Hetarogenwus Reactions
I
I I
Sh = Sc"*e(Jf))
( E l 1-3.i
Then
=I
again, virtualty complete: conversion
It' there were such a thing as the bed fiuidicity, given in the problem stal ment, it would be a vseiess plece of information. Make sure that you know wt information you need to solve problems, and go after it. Do not let additional dr confuse you or l e d you astray with useless information or facts that represe someone else's bias, which are probably not well-founded.
11.4 What If J.
Chap.
. . . ? (Parameter Sensitivity)
One of the most important skills of an engineer is to be able to predict the effec system variables on the operation of a process. The engineer nee( to determine these effects quickiy through approximate but reasonably close ca culations, which are sometimes referred ta as "back-of-the-envelope calcul, tior~s."'~ This type of calculation is used to answer such questions as "What wi happen if I decrease the particle size?" "What if I triple the Row rate throug ~ a of~thek the reactor?" Envelope We will now proceed lo show how this type of question can be answere using the packed-bed, mass transfer-limited reactors as a model or example sy; tern. Here we want to Ieam the effect of changes of the various parameters (e.g temperature. particle size, superficia! velocity1 on the conversion. We begin wil a re,mangement of the mass transfer correlation, Equation ( 1 1-49), to yield
D.~
of a d d ~ ~ofd changes * ~
Find out how the
cmficient vanes with changes in physical properties p mpenie.~
The first term on the right-hand side is dependent on physical properties (ten peraturc and pressure), whereas the second term is dependent on system prol erties (flow rate and particle size). One observes from this equation that th mass transfer coefficient increases as the particle size decreases. The use c sufficiently small particles offers another technique to escape from the mas transfer-limited regime into the reaction rate-limited regime. 16Prof. J. D.Goddard. University of Michigan. 1963-1476. Currently at University ( California, San Diego.
Sec. 11.4
What If
Example 11-4
...?
(Parameter Sensitivity)
789
Tile Case nJDivide und Re Conquered
A m:\cs transfer-limited rcactlon is being carried o i ~ tin two reactors af equal vnTume and pack~ng.connected in series a\ shown tn F~gureE I 1-4.1. Currently. 86 5% convenion is being ach~etedwith th15 oii-angzmtnt. I t is \ugge.;ted that the reacton be separated and the flow rate he di~idederlually among each of the two reactors (Figure El 1-42) to decrease the prewre drop and hence the pumping requirements. I n term5 of achieving a higher conve:er?ion, Robert i s wondering if this i s a good idea'?
0)-(7)-
X=O,BBS
Figure El 14.1
Series arrangement
Reactors in series versus reactors in
parallel
2
Figure Ell-J.1
Panlfel arr:inpement.
As a first approximation, we negfect the effects of $mall changcc in ternpemture and pressure on mass tranqfer. We recall Equat~onE l 1-64). wfiluh gives conversinn as a function of reactor length. For a mafh transfer-Irmited reaction
I n 1= L Lk . 4L . I-X L! For case 1. the undivided syrjtem:
X, = 0.865 For case 2 , the divided system: (EI 1-42}
X: = ? We now take the ratio of case 2 (divided system) to case I (undivided system):
790
Enternnl Diffusion Effects on Heterogeneous
Reac!ions
Chap. $ 1
The surface area per unit volume a, is the same for both systems. From the conditions of the problem statement we know that
However, we must also consider the effezt of the division on the mass t r a d e r coefficient. From Equation 11 1-70) we know that
%, U1'2 Then
Multiplying by the ratio of superficial velocities yields
Solving for X? gives us
Bad idea!! Rokst worry.
was righr to
Consequently, we see that although the divided arrangement will have the advantage of a smaller pressure drop across the bed, it is a bad idea in terns of conversion. Recall thar if the reaction were reaction rate-limited. both arrangements would give the same conversion.
Example 11-5
The Case of the Overenffa~siaxticEngineers
The same reaction as that in Example 11-4 i s being carried out In the same tau reactors in series. A new engineer suggests that the rate of reaction could be increased hy a factor of 2'' by increasing the reacrion temperature from 4(X)"C to 500°C, reasoning that the reaction rare doubles for every 10°C increase lo temperature. Another engineer arrives on the scene and beratcs the new engineer with quotations from Chapter 3 concerning thir; rule of thumh. She points out that it is valid only for a specific afrivarion energy within a ~ p e c l f i ctemperature range. She then
SBC.11.4
1 Robcn worrres ~f this temptrarure lncreace will k wonh the trouble
.
What If . . ? (Parameter Sensitrvity)
741
suggests that he go ahead with the proposed temperature increase ht should only m p a an increase on the ads of or T.What do you think? Who is correct?
Because almost all surface reaction rates increase more rapidly w ~ t htemp-ature than do diffusion rates. increasing the temperature w ~ l lonfy increase the degree to w h ~ c hthe reaction is mass transfer-limited, We now consider the fr~llowingfwo cases: Case 1 : T = 400°C
X
= 0.865
T = 500°C
X
=?
Case ?+
Taking the ratio of case ? to case 1 and noting that the reactor length is the Fame for both cases (L1= L2). we obtain
(El 1-5.1 1
The molar feed rate FTo remains unchanged: =
..,(a) = I,,.
(p.) R TO?
(El 1-5.2)
Because v = A,.U, the superficial velocity at temperature T2 is
We now wish
learn the dependence of the mass transfer coefficient an temperature:
10
(El 1-5.4) Taking the ratio US cnTe 2 to caTe I and realizing that the particle diameter is the same for both cases gives us
The temperature dependence of the gas-phase diffusivity is (froni Table 11-2) DAB= T1.'5
(Ell-5.6)
For most gases. viscosity increa~esm ith incrcasingtemperatureaccording to ihe relatioil p " TI'?
From the ideal gar law, pxT-I
792
External Diffusion Effects on Heterogeneous Reactions
Chap
(El 1-5 It's re:tlly i n r p r t a n t 10 know huu to do !hi\ rypc of analysih.
I In
I -
I'h
(El1-5 I -X,
In-
Bad &a!! Robert war right to worry.
1 1
-X,
= I
I
I
n =2
- 0.865
Consequently, we see that increaring the temperature from 400°C 10 500°C lncreaues the conversion by only 1.74. Bath engineers would have benefited from nrore thorough study of this chapter.
For a packed catalyst bed, the temperature-dependence part of the mas: transfer coefficient for a gas-phase reacrion can be written as .-
kc r
uUZTI 1/12
( 1 1-72)
Depending on how one fixes or changes the molar feed rare, F,, U may also depend on the feed temperature. As an engineeq it LYextremely important that yorc reason out the eflects of changing conditions, as illustrated in the precedImportant concept
ing two examples.
11.5 The Shrinking Core Model The shrinking core model is used to describe siruarions in which solid particles are being consumed either by dissolution or reaction and, as a result, the
Sec. 11.5
0
'
Stomach acrd
793
The Shrrnkiog Core Model
amount of the material being conwned is "shrinking." This mode[ applies ra areas ranging from pharrnacokineticf (e.g., dissolution of pills in the stomach) ro the formation of an ash layer around a burning coal particle. to catalyst regeneration. To design the time release of drugs inrn the body's system, one must focus on the rate of dirsolution of capsules and solid pills injected into the stomach. See PRSl 1.4. In this section we focus primarily on catalyr;: regeneration and leave other applications such as drug delivery as exercises at the end of the chapter. 11.5.1 Catalyst Regeneration
Many situations arise in heterogeneous reactions where a 93s-phase reactant reacts with a species contained in an inert solid matrix. One of the most cammon examples is the removal of carbon from catalyst particles that have been deactivated by fouling (see Section 10.7.1 ). The catalyst reseneration process to reactivate the catalyst by burning off she carbon is shown in Figures 11-1 1 through 1 I - 13. Figure 1 1 - I 1 shows a schematic diagram of the removal of carbon from a single porous catalyst pellet as a function of time. Carbon i s first removed from Fhe outer edge of the pellet and then in the final stages of the regeneration from the center core of the pellet.
0
100
2W
Time. Mins
Progressive regeneration of buled pellet
Figure I 1 - l l Shell pmyressive regeneration or fouI& pellet. (Reprinted with permission from S. T.Richard~on.Ind. EIIc Clrrm. Pruccrs Deq. D r r , I ! ( 1 ). 8 (1972): copyr~ghtAmerican Chemlcal Soc1ety.l
As the carbon continues to be removed from the porous catalyst pellet. the reactant gas must diffuse farther into the material as the reaction proceeds to reach the unreacted solid phase. Note that approximately 3 hours was required to remove all of the carbon from the pellets at these conditions. The regeneration time can be reduced by increasing the gas-phase oxygen concen-
tration and temperature. To illustrate the principles of the shrinking core model, we shaIl consider the removal of caban from the catalyst particle just discussed. In Figure 1 I-17,
External Diffusion Effects on Heterogeneaus Reactions
Chap, 7 1
Fipre 11-12 Panially regenerated catalyst pellet.
a core of unreacted carbon is contained between r = 0 and r = R. Carbon has been removed from the porous matrix between r = R and r = Ro. Oxygen diffuses from the outer radius Ro to the radius R, where it reacts with carbon to form carbon dioxide, which then diffuses out of the porous matrix. The reaction
Qss~ U\e ~tead>-state prth Ier
at the solid surface is very rapid. so the rate of oxygen diffusion to the surface controls the rate of carbon removal from the core. Although !he core of carban is shrinking with time ran uns~eady-stateprocess). we assume the concentralion profiles at any lnstant i n time to be the steady-state profiles over the disrance (R, - R). This assumption is referred to as the quad-sfead~stute asslmlption {QSSA).
O x p e n must d i f f u ~ ethrough the porouc pelIel mstrlv u n ~ i l11 rcaches the unleaclcd
cnrhrln core
Figure 11-13 Sphere wilh unteac~edcarbon core of radius R.
TO study how the radius of unreacted carbon changes with time. we must first find the rate of diffusion of oxygen to the carbon surface. Next. we perform a mole balance on the elemental carbon and equate the rate of con+umPtron o f carbon to the rate of diffusion of oxygen to the gas carhon interface. In applying a differenrial oxygen mole balance over the inuremeill 3~ located somewhere between R,,and R, we recognize that O2 does not react in
Sec, 17.5
The Shrlnklng Core Model
795
is region and reacts only when it reaches the solid carbon inleiface lwated at r = R. We shall let species A represent 02.
Step I : The mole balance on O2lie., A) between r and r
[TI+
[z +
+ Ar is
Rate of [generation] = [accumulatim Itate Of
I
Dividing through by - 4 ~ A rand taking the limit gives Mole balance on oxygen
lim
W~rr'lr+ar-
Ar4O
w*rrlIr = ~ ( W A .=~ ~ )
Ar
dr
Step 2: For every mole of 0, that diffuses into the spherical pellet, 1 mu1 of CO, diffuses out (Wco2 = - W 0 2 ) ,that is, EMCD. The constirutive equation for constant total concenwation becomes
@ @
@&@@ Follclwrng the Algorithm
where D, is an effective difSuusivi@ in the porous catalyst. In Chapter 12 we present an expanded discussion of effective diffusivities in a porous catalyst [cf. Equation ( 1 2-1 )I.
Step 3: Combining Equations (11-73) and (1 1-74] yields
Dividing by (-D,)gives
Step 4: The boundary conditions are: At the outer surface of the particle, r = A,,: CA = CAO At the fresh carbodgas interface, r = R(r): C, = 0
(rapid reaction) Step 5: Integrating twice yields
External Diffusion Effects on Heterogeneous Reacttons
Chap. 1
Using the boundary conditions to eliminate K , and K2,the concentntio profile is given by
A schematic representation of the profile of Q1 is shown in Figure 1 1-1 dt a time when the inner core is receded to a radius R. The zero on the axis corresponds to the center of the sphere.
Concentration
pfofile at a given tlme, I li.e., cure mdius. RF
-
R
0.0
Increasing r
Figure 11-14 Oxygen concentr&tionprofile shown from the external n d i u of ~ the plket (R,,) to the pellet center. The gas-carbon rnterfxe is Iocated at R
Step 6: The molar flux of 0, to the gas-carbon interface is
Step 7: We now carry out an overall balance on elemental carbon. Ele mental carbon does not enter or leave the particle.
[ Y ] - [Z]+ [ generation] [ aceumu~ationI Rate of
=
Mole balance on shrinking core
where p, is the molar density of the carbon and +c is the volume fractioi of carbon in the porous catalyst. Simplifying gives
dR - --Lit
r"
+ ~ P C
Step 8: The rate of disappearance of carbon is equal to the flux of O?tr the gas-zarbon interface:
Sec. 11.5
797
The Shrinking Core Model
The minus sign arises with respect to WAF in Equation ( I 1-79) because 0: is diffusing in an inward direction [i.e., opposite to the increasing coordinate ( r ) direcBon]:
Step 9: Integrating with limits R = Ro at r = 0, the time necessary for the solid carbon interface to recede inward to a radius R is
We see that as the reaction proceeds, the reacting gassolid moves closer to the center of the core, The corresponding oxygen concentration profiles at three different times are shown in Figure 2 1-15.
Concentration
profiles at different times at inner core radii
FIgurt 11-15 Oxygen concentrationprofile at various times. At I , , the gas-carbon tnterface is Iwated at R ( r l ) ;at s2 it is located at Rfs2).
The time necessary to consume al1 the carbon in
the catalyst pellet is
Rme to complete regeneration of the particle
For a I-em diameter pellet with a 0.04 voiume fraction of carbon. the regeneration time is the order of 10 s. Variations on the simpIe system we have chosen here can be found on page 360 of Levenspiel" and in the problems at the end of this chapter.
-
-
"0. Levenspiel, Chemical Reactor Engineering, 2nd ed. (New York: Wley, 1972).
798
Exlernal Diffusion Effects on Heterogeneous Reactions
11-5.2 Pharmacokinetics-Diss01ution Particles
Chap 11
of Monodispersed Solid
We now consider the case where the total particle is being completely consumed. We choose as am example the case where species A must diffuse to the surface to react with solid 3 at the liquid-solid interface. Reactions of this type are typically zero order in B and first order in A. The rate of mass transfer to the surface is equal to the rate of surface reaction. WAF= kr(CA- Ch) = -ris = krCAs
(Diffusion)
(Surface reaction)
Eliminating CAI, we arrive at an equation identical to Equation (1 1-46) for the radial flux:
For the case of small particles and negligible shear stress at the fluid boundary, the Friissling equation. Equation ( 1 1 -40), is approximated by Sh = 2
or
where D is the diameter of the dissolving particle. Substituting Equation (I 1-82) into ( 1 1-46) and rearranging yields Diameter at which mass transfer and reaction rale reqistances are equal 1s
D'
where D* = 2D,/kr is the diameter at which the resistances to mass transfer and reaction rate are equal.
A mole balance on the solid particle yields Balance on the
In - Out +Generation =Accumulation
where p is the molar density of species B. I f 1 rnol of A dissolves 1 rnol of Ba then - r i , = - r [ , , and after differentiation and rearrangement uJe obtain
Sec. f I.5
1
The Shrinking Core Model
Ibuprofen
where a = 2k,C, P
At time r = 0. the initial diameter is D = Di. Integrating Equation (11-84) for the case of excess concentration of reactant A, we obtain the following diameter-time relationship:
Di+i?+
Excess A
I 2 D*
- ( ~ j - ~ 2 )
=
( 1 1 -85)
The time to complete dissolution of the solid particle is
a Reference Shelf
(I 1-86)
'She dissolution of polydispersz particle sizes is analyzed using population balances and is discussed on the CD-ROM.
Clasare. After completing this chapter, the reader should be abk to define and descrik rnoIecular diffusion and how it varies with temperature and pressure, the molar flux, bulk flow, the mass transfer coefficient, the Sherwood and Schmidt numbers, and the correlations for the mass transfer coefficient. The reader should be able to choose the appropriate correlation and calculate the mass transfer coefficient, the molar flux, and the rate of reaction. The reader should be able to describe the regimes and conditions under which mass transfer-limited reactions occur and when reaction rate limited reactions occur md to make calculations of the rates of reaction and mass transfer for each case. One of the most important areas for the reader apply the knowledge of this (and other chapters) is in their abifity to ask and answer '%at if . . ." questions. Finally, the reader should be able to descrjbe the shrinking core model and apply it to catalyst regeneration and phamacokinetics.
External DMvsron Effects on Heterogeneous Reactions
Chap.
SUMMARY I. The molar flux of A in a binary mixture of A and B i s
a. For equimolar counterdiffusion of the solute.
b. For diffusion through
IEMCD) or for dilute concentrati,
a stagnant gas,
c. For negligible diffusion,
w,, = y*w = c,v Representative Values
(S11-
2. The rate of mass transfer from the twlk fluid to a boundary at concentrati CAais
. .
h1bPhnse
- 5000 4000 Sh - 500
Re Sc
-
where k . is she mass transfer coefficient. 3. The Shenvood and Schmidt numbers are, respectively,
k, = 10-2 mls
!zLEb&
Re
- 500
Sc- l
Sh- lo
k, = 5 mls
4. If a heat transfer correlation exists for a given system and geometry, the ma transfer correlation may be found by replacing the Nusselr number by t S h e r w d number and the Prandtl number by the Sclitnidt number in ti
existing heat transfer correlation. 5. Increasing the gas-phase velocity and decreasing the panicle size will increa the overall rate of reaction for reactions that are externally ma tnnsfer-limited.
I/
External diffusion limited
Chao. 11
801
CD-ROM Material
6. The conversion for externally mass transfer-limited Rnctfuns can he round from the equation
In--- I
I-X
-k,OLL
u
7. Back-of-the-envelope calcuIatinnc, should be camed out to deternine the magnitude and directton that changes in process variables urill have on conversion. What i f . . .? 8. The shrinking core model states that the time to regenerate a coked catalyst panicle is
CD-ROM MATERIAL
Summarv Notes
$j!j -
Solved Problcms
Learning Resources I . Satmmuo Notes Diffusion Through a Stagnant Film 4. Sclhled Pmhlems Example CDI I - I Calculating Steady State Diffusion Example C D l l - 2 Relative Fluxes W,, 8,. and J4 Example CD11-3 Diffusion Through a Stagnant Gas Example CD I I-4 Measuring Gas-Phase Diffusivities
r
1
I
Paous DIskJk-cL
Example CDI 1-5 Measuring Liquid-Phase DifFusivities Professional RePerence Shelf R I 1. I . Mass Tmmfer-Lim~redRcncriorrs on Metallic Strrjkre.~
A. CataIyst Monolith? B. Wire Gauze Reactors
802
External Diffusion Effects on Heterogeneous Reactions
Chap. T 1
R 1 1.2. MerAods to Experimntally Measure Difusiviries Pun Lquld
Gas B
--+
Reference Shelf
in M u s ~ o n
Pure llqu~dA
A. Gas-phase diffusivities
-t
€3.
C. v,
Liquid-phase difiusivities
R 1 1.3. Facilifated Hear Transfer
R 1 3 -4.Dissoiution of Polydisperse Solids (e.g., pills D
in the sromach)
D*M
Jbuprofen
QUESTIONS
A N D PROBLEMS
The subscript to each of the problem numbers indicates the level of difficulty: A. least difficult; IT, most difficult.
PII-I,
Creatlvc Thinking
P11-2,
Read over the problem? at the end of this chapter. Make up an original pmblem that uses che concepts presented in this chapter. See Problem 4-1 for the guidelines. To obtain a soIution: la) Make up >our data and reaction. (b) Use a real reaction and real data. The journals Ilsted at the edd of Chapter I may hc useful for part (b). (a) Example 1-1. Consider the mass transfer-limited reaction A
-
?B
What would your concentration (mole fraction) profile look like? Using the same values for D,, .and so on,in Example 1 1-1, what i s the flux of A? (bl Example 11 -2. How would your answers change ifthe temperacure u'as ~ncreasedb!. 50°C. the panicle diameter was doubled, and Ruid velocit!' wab cut in half? Assurne propenies of water can be used for this %)..stem.
803
Questions and Problems
Chap. 11
Example 11-3. HOW would your answers change if you had a 5 6 5 0 mixture of hydrazlne and helium? If you increase d b a factor of 5 ? p , : Id) Emample 11-4. What jf you were: asked for represenrauve values for Re, Sc. Sh, and kc for both liquid- and gas-phase systems for a velocity of 10 cm/s and a pipe diameter of 5 cm (or a packed-bed diameter of 0.2 cm)? What numbers would you give? (e) Example 23-5. How would your answers change if the reaction were carried out in the liquid phase where kinetic viscosity varied as (c)
(0 Side Note. Derive eguatinns (SNl 1-1.1 ) and (SNJ 1.2). Next consider there are no gradients inside the patch and that the equilibrium solubility in the skin immediately adjacent to the skin is CAo= HCApwhere H is a f m of Henry's law constant. Write the flux as a function of H, 6,, D,,,, DAB!.and CAPFinally cart?, out a quasi-steady analysis, i.e.,
to predict the drug delivery as a function of time. Compare this result with that where the drug in the parch is in a dissolving solid and a hydro-gel and therefore constant with time. Explore this problem using different models and parameter values. Additional information cmvs. A, = 5 cm2, V = 1 ern3 H = 0.1, DAB1 = 1W6 crn2h,DAB? = and CAP= 10 mg/dm3 Pll-3B Pure oxygen is being absorbed by xylene in a cataIyzed reaction in the experimental apparatus sketched in Figure P13-3. Under constant conditions of temperature and Iiquid composition the folErnving dara were obtained:
k l l l m
CD
Xylem
Figure PI13
Externat D~ffus~on Effects on Heterogeneous React~ons Rurt
Srirrrr S[~ccd (rpm)
I$
Up~ukeUJ 0: im Ll h ) j ) r S,vsrern Pr@rsrtw(absolute)
1.2 atm
1.6 ntm
400
15
8Ull
20 21 21
31 59 62 61
1200 I600
Chap. 1
2.13 atrn
3.0 atm
75 105
152 205 208
1M
207
IM
No gaseous products were formed by the chemical reaction. What would yo^ conclude about the relat~veimportance of Iiquid-phase di8usion and about the order of the kinetics of this reaction? (California Pmkssionrtl Engineem
1
Exam) Pll-4c Ln a diving-chamber experiment, n human subject breathed a mixture of 0; and He while small areas of his skin were exposed to nitrogen gas. Aftel awhile the exposed areas became blotchy, with small blisters forming on the skin. Model the skin as consisting of two adjacent layers, one of thickness 6, and the orher of 6?. IF counterdiffusion of He out through the skin occurs at the same time as N? diffuses into the skin, at what point in the skin layers is the sum of the partial pressures a maximum? If the saturation partial pressure for the sum of the gases is 101 k h , can the blisters be a result of the sum of the gas panial pressures exceeding the saturation partial pressure and the gas coming out of the solution (i.e., the skin)? Before answering any of these questions. derive the concentration profiles for N, and He in the skin layers. Hint: See Side Note. Diffusivity of He and N? in the inner skin layer = 5 X lo-' cm2/s and 1.5 X lo-' cm2/s, respectively Diffusivity of He and N, in the outer skin layer = cm?/s and 3.3 X lo-' cm2/s, respectively E.rrernui Skin Bnundug Purriul Pres.rure
\'
20 )~.m
80 pm
Stratum corneum Epidermis
Pll-5, The decilrnposition of cyclohexane to benzene and hydrogen is mass tmsfer-limited at high tempentures. The reaction is canied out in a 5-cm-ID pipe 20 rn in length packed with cyhljddrical pellets 0.5 crn in diameter and 0.5 cm in length. The pellets are coated with the catalyst only on the outside. T h e bed porosiry i s 40%. The entering volumetric How rate is 60 dm3/min. (a) Catculnte the number of pipes necessary to achieve 99.9%conversion of cyclohexane from an entering gas stream of 5 8 cyclohexme and 95% Hz at 2 atrn and 500°C. (b) Plot conversion a s a function of length.
il"+, Green engineering
81
b
Inrernal Skin Bo~tndury Parttnl PW.TCI;TP
Chap. 11
PI1-6
Plt-7,
805
Questions and Problems
(c) How much would your answer change if the pelIet diameter and length were each cut In half? (dl How would your answer to part (a) change if the feed were pure cvclohexane? (el What do you believe is the point of this problem? Assume the minimum respintion rate of a chipmunk is 0.5mmat of q l m i n The corresponding vuIumetric rate OF gas intake is 0.5 dm31rnin at STP. (a) What is the deepest a chipmunk can burrow a 3-cm diameter hole beneath the s u f i c e in Pasadena, California? (b) In Boulder, Colorado? (c) How would your answers to (a) and (b) change in the dead of winter when T = O0F? (d) Critique and extend this problem (e.g.. COz poisoning). Carbon disulphide (A) is evaporating into air (8)at 35°CP, = 510 mrn Hg and 1 atm From the bottom of a 1.0 cm diameter vertical tube. The distance fmrn the CS2 surface to the upen end is 20.0 cm, and rhis is held constant by continuous addition of liquid CS2 from below. The experiment is arranged so that the vapor concentration of CSt ar the open end is zero. (a) Calculate the molecular diffusivity of CS2 in air (D,) and its vapor pressure at 35°C. (Ans.: DAB= 0.12 crn2/s.) (h) Find the molar and mass Ruxes (W, and n, of CS?)in the tube. (c) Calculate the following properties at 0.9, 5.0, LO.0, 15.0, 18.0, and 20.0 cm from the CS2 surface. Arrange columns in the following order oo one sheet of paper. (AddifionaI columns may be included for computational purposes if desired.) OR a separate sheet give the rerations used to obtain each quantity. Try to put each relation into a form involving the minimum computation and the highest accuracy: (1) yA and ys (mole fractions), C, (2) V,. VB, V*, V (rnacr velocity)
,,
(3) Jm JB (dl Plot each of the groups of quantities in (c)(t), (2)- and (3) on separate graphs. Name all variables and show un~ts.Do noi plot those parameters in parentheses. (el What is the rate of evaporation of CS, in cdday? (f) Discuss the physicst meaning of the value of VA and JA at the open end of the tube. (g) Is mnkcuSur diffus~onof air taking place? Pll-gB A device for measuring the diffusion coefficient of a gas mixture (Figure PI 1d)consists of two chambers connected by a smal! tube. Initially the chamben contain different proponions of two gases, A and B. The total pressure 1s the same in each chamber.
Figure P1I-% Diffusion cell.
806
Ewternal Diffusion Effects on Heterogeneous Reactions
Chap. 11
(a) Assuming that diffusion may be described by Fick's law, that the concentration in each flask is uniform, and that the concentration gradient in the rube is linear show that
State any other assumptions needed.
(b) B. G.Bray (Ph-D. Thesis, University of Michigan. 1%0) used a similar device. The concentration of hydrogen in hydrogen-argon mixtures was determined from measurements of an ionizing current in each chamber. The ionizing current is proportional to concentration. The difference in ionizing currenrs &tween chambers one and two is measured (AIC). Compute the diffusion coefficient,DAB,for the following data. 769 psia, T = 35°C.CT = 2.033 motldm3, cell constant#
Time, rnln AIC
1 10 1 20 1 33 1 50 1 1 36.M ( 32.82 ( 28.46 1 23.75 1
66 19.83
1
83
(
1660
)
1
IUO
)
117
13.89
[
11.63
1
1
133 9.79
PI1-9* A spherical particle i s dissolving in a liquid. The rate of dissolutton is first order in the solvent concentration, C. Assuming that the solvent is in excess, show that the following conversion time relationships hold. Rare-Limiting Regime Surface reaction
Conversion Erne Rebtionship 1
- (1 - X ) I / ~
=
Q[ 5i
Mass bansfm
5i [ I - (1 - x):/>]=
Mixed
[I
2D'
-
PIX-XOcA powder is to he completely
(1
!?!
D,
- ~ ) " 3 , + 3 11 - ( 1 2D'
- x)?'?]=
n,
dissolved in an aqueous solution in a large, well-mixed tank. An acid must be added to the solution to render the spherical particle soluble. The particles are sufficiently small that they are unaffected by fluid velocity effects in the hak. For the case of excess acid. C, = 2 M ,deme an equation for the diameter of the particle as a function of rime when (a) Mass transfer Iirn~tsthe dissolution: -W, = k,C,, (b) Reaction l~mitsthe dissolution: - r i = k,CAn What is the time for complete dissolution in each case? (c) Now assume that rhe acid is not in excess and thar mass transfer is limiting the dissolution. One mole of acid i s required to dissolve 1 rnol of solid. The molar coocentrdtion of a c ~ dis 0.1 M. the tank is 1(M t,and 9.8 rnol of rolid is added to the tank at time I = 0. Derive an expression
Chap. 7 1
807
Questions and Problems
for the radius of the particles as a function of time and catculate the time for the particles to dissolve completely. (d) How could you make the powder dissolve faster? Slower? Additional infomalion:
D, = lo-" m2/s,
k = lO-'"ls
initial diameter =
rn
P11-11, The irreversible gas-phase reaction
is carried out adiabaticafly over a packed bed of solid catalyst particles. The reaction is first order in the concentration of A on the catalyst surface:
The f e d consists of 50% (mole) A and 50% inens and enters the bed at a temperature of 300 K. The entering volumetric flow rate is 10 dm3/s ( i t . , lO,OW crn31s).The relationship berween the Sherwood number and the Reyoolds number is
As a firs! approximation, one may neglect pressure dmp. T h e entering concentration of A is 1.0 M. Calculate the catalyst weight necessary to achieve 60%conversion of A for (a) Isothermal operation. (b) Adiabatic operation. (c) What generalizations can you make after comparing parts (a) and (b)? Additional informafiott:
Kinematic viscosity: p / p = 0.02 crn2/s Particle diameter: d, = 0.1 cm Superficial velocity: V = 10 cmls Catalyst surface areaimass of catalyst bed: a = 60 cm2/g cat. Diffusivity of A: D, = 10-?crn?ls Heat of reaction: AHRx= - 10,000 calJg mol A Hear capacities: C,, = C,, = 25 callg rno1.K C,, (solvent) = 75 cal/g mol .K
k' (300 K) = 0.01 cm3/s.g cat with E = 4000 callmo1 PIX-XZL-(Pills) An antibiotic drug is contained in a solid inner core and is surrounded by an outer coating that makes it palatable. The outer coating and the drug are dissolved at d~fferentrates in the stomach, owing to their differences in equilibrium solubilities. (a) If D, = 4 mm and D ,= 3 mm,calculate the time necessary for the pill
to dissolve completely.
808
External D~ffusionEffects on Heterogeneous Rsact~ons
Chap,
-'
p; 1
(b) Acsuming firct-order kinetics Ik4 = 1 O h 1 for the ub
D, = 3 m m
PilI 2: D, = 4 mrn,
D ,= 3 mm
Pi11 3: D, = 3.5 rnm,
D , = 3 rnm
Pill I :
Stomach acid
(c) Discuss how you would maintain the drug level in the blood at a consfa,
level using different-size p~lIs?
(d) How could you arrange a distribution of pi11 sizes so that the concentr; tion in the blood w;is constant over a period (e.g.. 3 br) of time?
-
Amount of drug in inner core = 500 mg Solubility of outer layer at stomach conditions 1.0 mg/cm3 Solubility of inner layer at stomach condrtions = 0.4 mg/crn3 Volume o f fluid in stomach = 1.2 L Typical body weight = 75 kg Sh = 2., DAB= 6 X cm2/min
P11.13, If disposal of industrial liquid wastes by incineration is to be a feasible pro cess. ~t is important that the toxic chemicals be comptetely decomposed i n t ~ harmless substances. One study carried out concerned the atomization ant burning of a l ~ q u ~stream d of "principal" organic hazardous constituent {POHCs) [ E ~ ~ v i r oPmg., n. 8. 152 ( 198Q)I.The following data give the burninj droplet diameter as a function of time (both diameter and time are given it Green engineering
arbitrary units):
What can you learn from these data? Pll-148 (Estimating glacirtl ages) The following oxygen-I8 data were obtained fm soil samples taken at different depths in Ontano. Canada. Assuming that all the I3O was Iald down during the last glacial age and that the transport of IsO to the surface takes $ace by molecular diffusion. estimatc the number ol yean since the last glacial age from the following data. Independent measurements give the d~ffusivityof '$0 in soil as 2.64 X 10-lo mqs.
Figure P l l - l l Glaciers.
Chap. 11
809
Questions and Probfams
C, is the concentration of
at
25 m.
J O U R ~ ~ AA LR T I C L E P R O B L E M PI1J-2 After reading the anicle "Designing gas-spnrged vessels for ma*\ transfer" [Chrm. Eng.. 89(23).p. 61 (1982)l. deign a gas-spaged vessel to c;ltrlrate 0.6 m31s af water up to an oxygen celltent of 3 X 1 0-3 kg/& at ?(I"<'. A liquid holding ttme of 80 s 1s requ~rrd.
JOURNAL CRITIQUE PROBLEMS P11C-E The decomposition of nitric oxide on a heated platinum wire is diqcussrd in Chem. Eng. Sci.. 30. 781 ( 1975J. After making some assumptions about the d e n d y and the temperatures of the wire and atmosphere, and usrrlg a comelatmn For convective heat transfer. determine if Inass transfer limitations are a problem in this reaction. PZSC-2 Given the proposed rate equation on page 296 of the article in ltltt. Eng. Chern. Proces.~Dcs. Dev., 19, 294 (1980). determine whether or nut the concentration dependence on sulfur, C,, i s really second-order. Also, determine if the intrinsic kinet~crate constant. K2", is indeed only a function of temperature and panial pressure of oxygen and not of some other varinhles its well. P l l C - 3 Read through the art~cleon the oxidation kinetic.; of oil shaft char in Irtd Eng. Chem. Pmcess Dm. Der:. 18. 66 1 ( 19741. Are the uni ts for the mass transfer coefficient k, in Equation (6) consistent wrth the rate law9 Is the mass transfer coefficient dependent on sample size? Would the shr~nkingcore model tit the authors' data as well as the model proposed by the authon? 4
Additional Homework Problems
CDP11-A, CDP11-B,
An isomerization reaction that follows Langmuir-Hinshelwood kinetics is carried out on monol~thcatalyst. [2nd Ed. PIO- I I] A parameter sensitivity analysis is required for this problem in which an isomerization i s carried our over a 20-mesh gaum screen. [2nd Ed,
P10-121 CDP11-Cc This problem examines the effect on temperature in a catalyst monolith. [2nd Ed. P10-131 CDF11-DO A second-order catalytic reaction is camed out in a catalyst monolith. [2nd Ed. PI 0-141 CDPI1-Ec Fracture acidizing is a technique to increase the productivity of oil weI1s. Here acid is injected at high pressures 10 fracture the rock and form a channel that extends out from the well bore. As the acid flows through the channe! it etches the sides of the channel 10 make it larger and thus less resistant to the flow of oil. Derive an equation for the
810
External Diftus~onEffects on Heterogeneous Reactions
Chap. If
concentration profile of acid and the channel width as a function of distance from the well bore, CDP11-F, The solid-gas reaction of silicon to form Si@ is an important process In microelectronics fabrication. The oxidation occurs at the Si-SiO, interface. Derive an equation for the thickness of the Si02 layer as a function of time. [2nd Ed. PEO-171 CDF11-G, Mass mnsfer limitations in CVD processing to product material with fermelecrric md piezoelectric properties. [2nd Ed. PEO- 171 CDPI1-W, Calculate multicomponent diflwsivities. [2nd Ed. PID-91 FDPII-IB Application of she shrinking core mode1 to FeS? rock samples in acid mine drainage. [2nd Ed. PIO- 181 CDP11-JB Removal of chlorine by adsorption from a packed bed reactor: Vary system parameters to predict their effect on the conversion, [3rd Ed. P 1 1 -5,j CDPIl-KB The reversible reaction A B is carried out in a PBR. The sate law is 1 Kc
- r i = kA(cA- -cB) Green Engineering Pmblem [3rd Ed. PI 1-6,J CDP11-LB Oxidanon of ammonia over wire screens. [3rd Ed. PI 1-7,] CDPIl-M, Sgt. Arnbercromby on the scene again to investigate foul play. [3rd Ed. Pll-13,l CDPll-N, Additional Green Engineering prohlems can be found on the web at WWM: ~l~an.edu/Rreenengin~ering.
SUPPLEMENTARY
READING
1 . T h e fundamentals of diffusional mass transfer can be found in E. STEWART. and E. N . LIGHTFOOT, Tmnspsrr Phenomena. 2nd ed. New York: Wiley, 2003, Chaps. 17 and 18. CUSSLER, E. L.. Drfision Mass Tmnsfer in FIuid Systems, 2nd ed. New York: Cambridge Universrty Press. 1997. FAHIEN,R. W.. Furrdamentul.~of Transporf Phenomena. New York: McGrawBill. 1983. Chap. 7. GEANKOPUS, C. J., Trunspon Processes and Unir Operations. Upper Saddle River, N.J.: Prentice Hal!, 2003. B l s ~ sA. . L..and R. N . MADDOX, MUSSTransfer: Fundamentals and Applicar i m s . Upper Saddle River, N.J.: Prentice Hall, 1984. ZEVTCH, V. G., PFysiochemical H\pdrodynarnics. Upper Saddle River, N.J.. Prentice Hall, 1962. Chaps. 1 and 4. BIRD, R. B., W.
Supplementary Reading
Chap. 11
811
2. Equations for predicting gas diffusivities are given in Appendix D. Ex@rnental values of the diffusivity can be found in a number of sources, two of which are
PERRY,R. H., D.W. G m , and J. 0.MALONEY,Chemical Engineers' Handbook, 7th ed. New Ymk: McGraw-Hill, 1897. SHERWOOD, T.K., R. L. PIGFORD,and C.R. WLKE,Mass Tranxfer. New York: McGraw-Hill, 1975.
3. A number of correlations for the mass transfer coefficient can be found in
LYDERSEN, A. L.,MUSS Transfpr in Engineering Practice. New York WileyInterscience, 1983, Chap. 1. MCCABE,W, L.,1. C. SMITH, and P. HAR~IOTT,Uni; Operations of Chemical Engineering, 6th ed. New York: McGraw-Rill. 2000. Chap. 17. TREYBAL, R. E.,Mass Trensfer Opemrions, 3rd ed. New York: McGraw-Hill, '-
1980.
Diffusion and Reaction
12
Research is to see what everybody else sees, and
to think what nobody else has thought. Albert Szent-Gyergyi
concentntion in the internal surface of the pellet is less
than lhatOf external surface
Overview This chapter presents the principles of diffusion and reaction. Wile the fmus is primarily on catalyst pellets. examples iUustrating these principles are also drawn from biomaterials engineering and mimlecttonics. in our discussion of cataIytic reactions in Chapter 10, we assumed each point an the interior of catalyst surface was accessible to the same concentration. However, we b o w there are many, many situations where this equal accessibility will not be h e . For exmple, whan the reacrants must diffuse inside the catalyst pellet in order to react, we krlow the crmcenlration at the pore mouth must be higher than that inside the pime. C o wquently, the entire catalytic surface is not accessible to the same concentration; therefore, the rate of reaction throughout the pellet will vary. To account for variations in reaction rate throughout the pellet, we introduce a parameter known as the effectiveness factor, which is the ratio of the overall reaction rate in the pellet to the reaction rate at the external surface of the pellet. In this chapter we wilI develop mdeIs for diffusion and readan in two-phase systems, which include catalyst pellets, tissue generation, and chemical vapor deposition (CVD). The types of reactors discussed in, this chapter will include packed beds, bubbling fluidized beds, slurry reactors, trickle bed reactors,and C W boat reactors. After studying this chapter, you wilt be able to describe diffusion and reaction in two- and three-phase,syskms,determine when internal diffusion Iimits the overall rate of reaction, describe how to go aboi nating this limitation, and develop models for s:ysterns in which bo sion and reaction pIay a role (e-g,. tissue growth1, m 1 .
814
Dittusion and Reaction
Chap. 12
In a heterogeneous reaction sequence, mass transfer of reactants first sakes place from the bulk fluid to the external surface of the pellet. The reactants then diffuse from the external surface into and through the pores within the pellet, with reaction taking place only on the catalytic surface of the pores. A schematic representation of this two-step diffusion process is shown in Figures 10-6and 12-1.
surface
Figure 12-1 Mass transfer and reaction steps for a catalyst pellet.
12.1 Diffusion and Reaction in Spherical Catalyst Pellets In this section we will develop the internal effectiveness factor for spherical cataIyst pellets. The development of models that treat individual pores and pellets of different shapes is undertaken in the problems at the end of this chapter. We will first look at the internal mass transfer resistance to either the products or reactants that occurs between the external pelIet surface and the interior of the pellet. TQ iJIusmate the salient principles of this model, we consider the irreversible j sorneriza~ion A-B that occurs on the surface of the pore walls within the spherical pellet of mdius R. 12.1-1 Effective Diffusivity
The pores in .the pellet are not straight and cyljndsical; rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas. It would not be fruitful to describe diffusion within each and every one of the tortuous pathways individually; consequently, we chaIl define an effective diffusion cwfficient so as to describe the average diffusion taking place at any position r in tHe pellet. We shall consider only radial variations in the concentration; the radial flux W,, will be based on the total area (voids and soIid) normal to diffusion transport c.e., 4i~r2)rather than void area alone. This basis for W,, is made possible by proper definition of the effective diffusivity 5), The effective diffusivity accounts for the fact that:
.
1. Not all of the area normal to the direction of the flux is available (i.e.. the area occupied by solids) for the molecules to diffuse.
Sw. 12.1
825
Diffusion and Reaction in Spherical Catalyst Pellets
2. The paths are tortuous. 3. The pores are of varying cross-sectional areas. An equation that relates D, to either the bulk or the Knudsen djffusivit~is The effective diffusivity
where ? = tortuosityl =
Actual distance a moIecule travels between two points Shortest distance between those two points
$, = pellet porosity =
Volume of void space Total volume (voids and solids)
u, = Constriction factor The constriction factor, u,, accounts for the variation in the cross-sectional area that is normal to diffusion.* It is a function of the ratio of maximum to minimum pore areas (Figure 12-2(a)). When the two areas, A , and A * , are equal, the consrrjction factor is udty, and when P = 10, the constriction factor is approximately 0.5.
p = area A;, area A,
(a1
Figure 1L2
Example 12-1
(b) (a) Pore consrriction; (b) pore tonuosity.
Findittg the TorluesiQ
Calculate the tonuosity for the hypotheticar pore of length. L (Figure 12-2(b)). from
the definition of ?. 1
?
Some investigators lump consuicfion and tortuosity into one factor. called the tortuosity factor, and set it equal to 3nC.C. N. Safterfield. Muss Trun
816
Dtffus~onand React~on
Chap.
7 = Actual distance n~oleculetravels from A to B Shortest distance between I-I and B The %honestdistance between potnts A and B is ecule travels from A to B is 2L.
,E . The actual distance the mc
Although this value ih rcasonablc for i. values for ? = 6 to IO are not unknow~ Typical values of the constriction fac~or,the tortuosity, and the pellet porosity an respxtively, crc = 0.8, i= 3.0. and 0, = 9.10.
12.1.2 Derivation of the Differential Equation Describing Diffusion and Reaction
we will derive the cuncentration profile oFre,ctant 4 In the pellet.
We now perform a steady-state mole balance on species A as it enters. leaves and reacts in a spherical sheIl of inner radius r and outer radius r + Ar of tb pellet (Figure 12-3). Note that even though A is diffusing inward toward th center of the pel jet, the convention of our shell balance dictates that the ff ux bl in the direction of increasing r. We choose the flux of A to be positive in thl direction of increasing r (i.e.. the outward direction). Because A is actuall] diffusing inward. the Aux of A will have some negative value, such a: - 10 mol/m2.s. indicating that the flux i s acttlaIly in the direction of decreas ing r.
-
Figure 13-3 Shell balance on a catniyst pellet.
We now proceed to perform our shell balance on A. The area that appears in the balance equation is the total area (voids and solids) normal to the ditec-
tion of the molar Rux: Rate of A in at r = W,,;Area = W,, X 4ar21,
IrfAr
Rate of A out at ( r + Ar) = WAr.Area = WArX 4 ~ r Z
( 12-2) ( 1 2-3)
Sec. 12.1
817
Diffusion and React~on~nSpher~calCatalyst Pellets
1
Rate of generation Rate of reaerion] A withina = [Mass of cnaiyrr sheIl thickness of"
1
+
reaction jnsldc the cat:~Iy+lpellet
btole balnnce
Volume
x [volume of sheli]
1
-
Mole balance for diffusion and
[~ars
.
PC
x
x
4~lri~r
where r, is some mean radius between r and r + br that is used to approximate the volume A Y of the shelt and p, is the density of the pellet. The mole balance over the shell thickness L\r is
- (Out at r + b ) + (Generation within b)= 0 ( 1 2-5) - ( W A , x 4 ~ r 2 ] , + b+) ( x A ) =o After dividing by ( - 4 r A r ) and taking the limit as Ar -+ 0,we obtain
(In at r) ( WArx4nr2Ir)
the following differential equation:
Because 1 mol of A reacts under conditions of constant temperature and pressure to form 1 moE of 3,we have Equal Molar Counter Diffusion (EMCD) at constant total concentration (Section 11.2.1A), and. therefore. The
B t ~ xequation
where C , is the number of moles of A per dm3 of open pore volume (i-e.,volto (rnol/vol of gas and solids).-In systems where we do not have EMCD in catalyst pores. it may still be possible to use Equation ( 12-7) if the reactant gases are present in dilute concentrations. After substituting Equation ( t 2-7) into Equation / 12-6$, we arrive at the following differential equation describing diffusion with reaction in a catalyst ume of gas) as opposed
pellet:
We now need to incorporate the rate law. In the past we have based the rate of reaction in terms of either per unit volume, -
-
or per unit mass of cataIyst,
-r,[=](moVg cat. s )
818
Dtffuskm and Reaction
Chap. 12
we study reactions on the internal surface area of catalysts, the rate of reaction and rate law ax often based on per unit surface area.
When
As a result, the surface area af the catalyst per unit mass of catalyst,
is an important property of the catalyst. The rate of reaction per unit mass of catrtlyst, -rl , and the rate of reaction per unit surface area of catalyst are related through the equation
gm,
catalyst may cover as much surface area as a football field The rate law
-6 -rzsB 5
A typical value of S, might be 150 m2/gof catalyst. As mentioned previously, at high temperatures, the denominator of the catalytic rate law approaches 1 . Consequently, for the moment, it is reasonable to assume that the surface reaction is of nth-order in the gas-phase concentration of A within the pellet.
where '
L
-
Similarly, per unit Surgace Area: per unit Mass of Catalyst: k; = k
per unit Volume: k
p ~=[m ,
( k g , s )1
=~~S,~,[S-]] kn
Differential
equation and
=pcsakRpckn151
(gr
Substituting the rate law equation (12-9) into Equation (12-8) gives
boundary condition^
dercrib~ngdiffusion and reaction in a
catalys~pellet
d [r2(-De dC,/dr)] dr
+r2 ~'LS,P,'C~
=o
(12-101
By differentiating the first term and dividing through by -r2D,, Equation ( 1 2- 3 0) becomes
See. 12.1
-
Diffusion arid Reaction in Spherical Catalyst Pellets
The boundap conditions are:
1. The concentration remains finite at the center of the pellet:
C, is finite
at r = 0
2. At the extemaI surface of the catalyst pellet. the concentration is CAs:
12.1.3 Writing the Equation in Dimensionless Form We now introduce dimensionless variables and A so that we may arrive at a parameter that is frequently discussed in cataIytic reactions, the Thiele modulus. Let
With the transformation of variables, the boundary condition CA = CAa
at r = R
and the boundary condition C, is finite
at r = 0
becomes yf is finite
at h = 0
We now rewrite the differential equation for the molar flux in terms of our dimensionless variables. Starting with CA w*,= -n, ddr we use the chain rule to write
820
Diffusion and Reaction
Ch:
Then differentiate Equation (12-12) with respect to \y and Equation ( 1 2 with respect to r, md substitute the resulting expressions.
-
~ C =ACAs dyr
and
& - = -1 dr
R
into the equation for the concentration gradient to obtain
The flux of A in terns of the dimensionless variables,
and X, is
The total rate of consumption of A ~nsidethe pellet. IM, (mollsl
At steady state, the net flow of species A that eaters into the @let at external pellet surface reacts completely within the pellet. The overall rate reaction is therefore equal to the total molar flow of A into the catalyst pel The over311 rate of reaction, MA,can be obtained by rnuItiplying the molar B into the pellet at the outer surface by the external surface area of the pel1 47r R2: All the reactant that d~ffusesinto the pellet is consumed (a black hole)
MA=-4icR'W,,l
r=8
=4xRDeC,&1
=+4nR2Desl
dr
r=R
(12-1 k=I
Consequently, to determine the overall rate of reaction, which is given 1 ~ ~ u a t i o( 12n 13), we first solve Equation (12-I 1) for CA, wi respect to r, and then substitute the resulting expression into Equation (12-1; Differentiating the concentration gradient, Equation (12-13, yields
differentiate^^
After dividing by CAsIR2.the dimensionless form of Equation f 12-11 ) is wri ten as
Then Dimensionless form of equations describing diffusion and reaction
Sec, 12.1
821
Diffusion and React~onin Spher~calCatalyst Pellets
( 12-20)
Th~eIemodulus
The square root of the coefficient of v*. (i.e., h,) is called the Thiele moduius. The Thiele modulus, 4,. will always contain a subscript (e.g., n), which will distinguish this symbol from the symbol for porosity, 4, defined in Chapter 4. which has no subscript. The quantity &! i s a measure of the M I ~ O of "a" surface reaction rate to "a" rate of diffusion through the catalyst pellet:
-
$; =
k,R2C?;' knRCXq - "a" surface reaction rate D, D,[{CAC,, -O)/R 1 "a" diffusion rate
When the Thiele rnodrrlus is large, inwrnnl diffzuion ~ ~ s t t a litnits l l ~ rl~e overall rate of reaefion; when +,, is stnnfl, [he ~sttface reacrion is t t ~ t ~ n l i y mte-limiri~zg.If for the reaction
the surface reaction were rate-limiting with respect to the adsorption of A and the desorption of 13, and if species A and B are weakly adsorbed (i.e., low coverage) and present in very dilute concentrations, we can write the apparent first-order rate law
The units of K; are rn3/rn's (= mJs). For a first-order reaction. Equation {I 2- 19) becomes
where
822
Diilusion and Reaction
Chap. 12
The boundary conditions are
32-1.4 Solution to the Differential Equation for a First-Order Reaction
Differential equation (12-22) is readjIy solved with the aid of the transfonnation y = yA:
With these transformations, Equation (1 2-22) reduces to
This differentia1 equation has the following solution (Appendix A.3): = A , cash+, )l + B , sinhb, h
The arbitrary constants A* and El can easily be evaluated with the aid of the b o u n d a ~conditions. At h = 0;cosh4, h I, ( I l k ) + =. and sinhblA + 0. Because the second boundary condition requires to be finite at the center (i.e.. h = 0). therefore A , must be zero. The constant B, is evaluated from B.C. 1 (i.e., iy = I. h = 1) and the dimensionless concentration profile is Cancentralion profile
Figure 12-4 shows the concentration profile for three different values of the Thiele modulus, b , . Small values of the Thiele modulus indicate surface
reaction controls and a significant amount of the reactant diffuses well in to the pellet interior without reacting. Large values of the Thiele modulus indicate that the surface reaction is sapid and thal the reactant is consumed very close to the external pellet surface and very little penetrates into the interior of the
Sec. 12.1
I
Dfttusionand Reaction ~n Spherical Catalyst Pelbets
823
pellet. Consequently, if the porous pellet is to be plated with a precious metal: catalyst (e-g., Pt). it should only be plated in the imrnedia~evicinity of the external surface when large values of &,, characterize the diffusion and xaction. That is, it would be a waste of the precious metal to plate the entire pellet when internal diffusion is limiting because the reacting gases are consumed near the outer surface. Consequently, the reacting gases would never contact the center ponion of the pellet.
For large values of the Thiele modulus, ~nternaldiffusion
lirnln the rate of reaction.
Figure 12-4
Example 12-2 I
Concentration profile in a spherical cataly.rt pcllc~
application.^ of DifSusion arrd Reactio~;ia ESSIIP Engirreen'ng
The equations describing diffuusion and reaction In pornus cataly\t!, also can he uqed rates of tiswe growth. One important area of tissue growth I S in cartilage tissue in joints such as the knee. Over 200.NK)patients per year receive knee joint replacements. Alternative stratep~es include rhe growth af cartilage to repair the damaged knee.? One approach cumntly being researched by Professor Krifti Anqeth at the University of Colorado is to deliver cartilage forming cells in a hydrogel to the damaged area such as the one shown in Figure E12-7.1. to denve
I
Figure Ell-2.1 Damaged c;in~lugr, Flpvrc c o u n r i y ol
\'i,i~ iil
r ~ iScplrmhur .
3. 1IXl I . i
824
Biffuslon and React~on
Chap.
Here the patient's own cells are obtained from a biopsy and embedded if hydro gel. wh~ch1s a cross-l~nkedpolymer network that is swollen in water. In on for the celEs to riurvive and grow new tissue, many propenies of the gel must tuned to allow diffusion of important specles in and out le.g., nutrients in a cell-secreted extracellular molecules our. such as collagen). Because there is blood flow through the cartilage, oxygen transport to the canilage cells is prirnar by diffusion. Consequently, the design rnuqt be that the gel can maintain the necr aary nre'; of diffusion of nutrients le.g., O?) into the hydrogel. These rates exchange in the gel depend on the geometry and the thickness of the gel. To ill1 trate the application of chemical reaction engineering principles to tissue engine, ing, we wiH examine the diffusion and consumption of one of the nutrients, oxygr
Our examination of diffusion and reaction in catalyst pellets showed that many cases the reactant concentration near the center of the particle was virtual zero. Ifthis condition were to occur in a hydrogel. the cells at the center would dl Consequently, the gel thickness needs to be designed to allow rapid transport oxygen. Let's consider rhe simple gel geometry shown in Figure El?-2.2,
02
Figure El2-2.2 Schematic of cnrtilage cell system. We want to find the gel thickness at which the minimum oxygen consumption rat is 10-l3 moticell/h. The cell density in the gel is loL0celisidrn3, and the bulk con centration of oxygen = 0)is 2 x 1W rnoYdm3, and the diffusivity is IPScm2/s. Solldlion
A mole balance on oxygen, A. in the volume A V = A&
F*I=-F*I:+,+r*A,~ = 0 Dividing by Az and taking the limit as d* + 0 gives
is
Sec. 12.1
D~ffuSlOnand
Reactron rn Spherical Catalyst Pellets
For dilute concentrations we neglect tE 12-2.3) to obtain
ffc,
825
and combine Equatims (E12-2.2) and
If we assume the 0:consumption rate is zero order. then
I
Purring our equation in dimcnrionlcu form using
= C,.,fCAfjand :/L7 vc obtain
Recognizing the second term is just the ratio of a reaction rate to a diffusion rate for a zero order reaction, we call this mrio the Thiele modulus, @,., We divide and multiply by two to facilitate the integration:
The boundary conditions are At
h=U
v=I
At
L=l
ko A.
CA= CAI)
(E 12-2.9)
Symmetry condition (E12-2.10)
Recall that at the midplane (= : L, h = 1) we have symmetry so that there is no diffusion across the midplane so the gradient is zero at A = I . tntegrating Equation (El2-2.8) once yields
Using the symmetry condition that there is no gradient across the midplane, Equation (E12-2.101, gives k; = - 3$0:
integrating a second time gives
yr = $ , x ~ - ~ ~ , x + K ~
826
Diffusion and Reaction
Using the boundary condition concentration profile is
\y
Chap. 12
= I at h = 0, we find K2 = I . The dimensionless
Nore: The dimensionless concentration profile given by Equation (E12-2.13) is only valid for values of the Thiele modulus less than or equal to 1 . This restriction can he easily seen if we set & = 10 and then calculate yr at Ir = 0.1 to find y~ = -4.9, which 1s a negative concentrarion!! This condition i s explored further in Problem PI?-10,.
Evaluating the zem-order rate constant. k. yields -13
k
l ~=' ~ c e. 10 t ~ s mole 01= 10-~moie,dm3, dm3 cell . h
,,
and then the ratio
The Thiele module is
(a)
Consider the gel to be completely effective such tha! the concentration of oxygen I S reduced to zero hy the time it reaches the center of the pet. That is, ~f y = O at 7L = I. we solve Equation (El 2-2.13) to find that bo = 1
Solving for the gel half thickness L yields
(hl
Lel's critiqi~ethis answer. We said the oxygen concentration was zero st the center. and the cells can't furvive w~tboutoxygen. Consequently, we need to redcsign , is not zero at the center. Now confider ;he case whet;: the minimum oxygen concentration for the cells to survrve is 0.1 mmolldm', ~ h i c hi s one half that at the surface (i.e., y = 0.5 at = l .O). Then Equnt~on[El 2-2. I31 give<
cm' Solving Eqi~arion(EI2-2.171 for L gives
Sec. 12.2
(c)
Internal Effectiveness Factor
827
Consequently, we see that the maximum thickness of the cartilage gel (2L) is the order of 1 mm. and engineering a t h i c k tissue is challenging. One can consider orher perturbations to the preceding analysis by considering the reaction kinetics to foliow a first-order rate law, -r, = k,C,, or Monod kinetics,
The author notes the similarities to this problem wirh his research on wax build-up in subsea pipeline gels! Here as the parafin diffuses into the gel to form and grow wax particles. these particles cause paraffin molecules to take a longer diffusion path, and as a consequence the diffusivity is reduced. An analogous diffusion pathway for oxygen in the hydrogel containing collagen is shown for in Figure E12-2.3.
Figure EX2-2.3 Diffusion of Ol amund collagen.
where a and F, are predetermined parameters that account for diffusion around particles. Specifically, for cotlagen. a is the aspect ratio of the collagen panicle and F , is weight fraction of 'kolid" coliagen obstructing the diffusion.' A similar rnodification could be made for cartilage growth. These situations are left as an exercise in the end-of-the-chapter problems. e.g., PI 2-ZIb).
12.2 Internal Effectiveness Factor The magnitude of the effectiveness factor (ranging from 0 to 1 ) indicates the relative importance of diffusion and reaction limitations. The internal effectiveness factor is defined as
-a
P. Singh. R. Venkatesan, N. Nagarajan, and H,S, Fogler, AIChE J., 46, 1054 (2000).
Diffusion and Reaction
828 7 1s a measure of huw hr the reactant
diffuws
into the pellet before reacttne
Cbap
ActuaI overail rate of reaction = Rate of reaction that would result if entire interior surface were exposed to tlie external peIlet surface conditions C,,
'
The overall rate, -r;, is also referred to as the observed rate of react] [-r,(ubs)]. ln terms of symbols. the effectiveness factor is
To derive the effectiveness factor for a first-order reaction, It is easiest to wc in reaction rates o f (moles per unit time). M,. rather than in moles per UI time per volume of catalyst (i.e.. -r, )
- -r,xVolume of catalyst particle =>11.1 q=-- -r,, - I . , , x Volume of catalyst particle MM4, First we shall consider the denominator, MA,. If the entire surface we exposed to the concentration at the external surface of the pellet, CAs.the ra for n fitst-order reactinn wcluld be MA,=
Rate at external surfacexYolume of carslyrl Volume
The subscript s indicates that the rate -rAs is evaluated at the condition present at the external surface of the pellet (i.e., h = 1). The actual rate of reaction is the rate at which the reactant diffuses int the peltet at the outer surface. We recall Equation (12-17) for the actual rate c reaction, The actual rate of reactton
Differentiating Equation (12-77) and then evaluating the result at X = 1 yield:
-
Substituting Equation (1 2-30) into (12- 17) gives us MA = 45iRDeCA,(mlcothib1 - I ) We now substitute Equations (12-293 and (12-31) into Equation
obtain an expression for the effectiveness factor:
(12-31;
(12-28) to
Sec. 12.2
829
Internal Effectiveness Factor
Internal effectiveness factor
for a first-order reaction in a spherical catalyst
( 12-32)
gellet
A plot of the effectiveness factor as a function of the Thiele modulus is shown in Figure 12-5. Figure 12-5(a) shows q as a function of the Thiele modulus 4, for a spherical catalyst pellet for reactions of zero, first, and second order. Figure 12-5(b) corresponds to a first-order reaction uccurring in three differently shaped pellets of volume V, and external surface area A,, and the Thiele modulus for a first-order reaction, &,, is defined differently for each shape. When volume change accompanies a reaction (i.e., & + b ) the corrections shown in Figure 12-6 apply to the effectiveness factor for a first-order reaction. We observe that as the particle diameter kcomes very small, b, decreases, If $ , 7 2 s o that the effectiveness factor approaches 1 md the reaction is surface-reacthen ~ ( = - [ + ~ - l ] tion-limited, On the other hand, when the Tbiele modulus 4, is large (-301, the internal effectiveness factor q is small ( i t . , 7 4 1). and the reaction is diffusion-limited within the pellet. Consequently, factors inf uencing the rate of exterthen ?-nal mass urnsport will have a negligible effect on the overall reaction rate. For @I large values of the Thiele modulus,the effectiveness factor can be written as
To express the overall rate of reaction in terms of the Thiefe modulus, we aearrange Equation (12-28) and use the rate law for a first-order reaction in Equation (12-29) -rA =
1
rate reaction rate at C,,) Reaction rate at C,
=7(~ICAJ (12-34) Combining Equations (12-33) and (12-341, the overall rate of reaction for a first-order, internal-diffusion-limitedreaction is
Diffusion and Reaction
Chap. 12
Thiele modulus,
-4
Zero order
= ~jw~
+,o=~
+ , = ~ J k i : =
Second order
1 4 ~
Reaction
Internal effectiveness factor for different reaction orders and
catalyst shapes
0.1
I
1
I
I
0.4 0.6
I
1
I
I
I
1
2
4
6
I
10
m Sphere
$1
Cylinder
$1
Stab
( n ~ ~ h z= $lm i x = ( ~ 2 ) J =m $hz =
+ , = L J ~ =
Figure 12-5 (a) Effectiveness factor plot for nth-order kinetics spherical catalysl particles (from Mass T~ansfer m Heterogeneous Caralysrs, by C. N. Salkrtield. 1970; repnnt editron: Roben E. Krieger Publikhing Co., 1981; reprinted by permiss~onof the au~hor).Ib) First-order reactlon in differem pellet peometrics (from R. Aris. Inrmducr~on!o rhe Analwir of Cltemicrrl Rtacrors. 1965, p 13 I : reprtnted by permi.rf~onof Pren~ice-Hall.Englewood Cliffs. N.J.).
Sec. 12.2
Internal Effecttveness Factor
Comction for volume change with reaction (ire., c+O )
T' --
?
- Factor in the presence of volume chmnqe Foctor in obsence of volume change
Figure 12-6 Effectiveness factor satins for first-order kine~icson spherical catalyst pellets for various values of the Thiele modulus of a sphere. 6,.as a function of volume change. [From V. W.Weekman and R. t.Gor~ng,J. Carat. 4, 260 (1965j.j can the rate of reaction be increased?
HOW
Therefore, to increase the overall rate of reaction, -ri: ( 1 ) decrease the radius R (makepellets smaller); (2) increase the tempemure; (3) increase the concentration; and (4) increase the internal surface area. For reactions of order n, we have, from Equation (12-20),
For large values of the Thiele modulus, the effectiveness factor i s
Consequently, for reaction orders greater than 1 , the effectiveness factor decreases with increasing concentration at the external pellet surface. The preceding discussion of effectiveness factors is valid only far isothermal conditions. When a reaction is exothermic and nonisothemal, the effectiveness factor can be significantly greater than 1 as shown in Figure 1 2-7. Values of q greater than 1 occur because the external surface temperature of the pellet is less than the temperature inside the pellet where the exothermic reaction is taking place. Therefore, the rate of reaction inride the pellet is greater than the rate at the surfzce. Thus, because the effectiveness hcror is the
Oitfusion and Reaction
Chap,
Can you find regions where multiple wlulions (MSS) exist?
4'1
Figure 12-7 Nonirothermal effectiveness Factor.
ratio of the actual reaction rase to the rate at surface conditions, the effecdvt ness factor can be greater than 1, depending on the magnitude of the paramt ters p and y . The parameter y is sometimes referred to as the Arrhenit number, and the parameter p represents the maximum temperature differenc that could exist in the pellet relative to the surface temperature T,.
-
y = Arrhenius number = t
RT,
(See Problem P12-138 for the derivation of P.) The Thiele modulus for first-order reaction, is evaluated at the externaI surface temperature. Typicr values of y for industrial processes range from a value of y = 6.5 (P = 0.022 b, = 0.22) for the synthesis of vinyl chloride from HC1 and acetone to a valu of y = 29.4 (P = 6 X = 1.2) for the synthesis of amrnonian5Th lower the ?henna1 conductivity k, and the higher the heat of reaction, the greate the temperature difference (see Problems P 12- 13, and P12-14,). We observ
+,,
TypicaI parameter values
+,
H. V. Hlavacek,
N. Kubicek, and M. Marek, J, C~ataL,15, 17 ( 1969).
Ssc. 12.3 I
Ctiterton for no I\lSSs in thc pellet
Falsified Kinetics
from Figure 12-7 that multiple stead!: states can exist for values of the Thjele modulus less than I and when P is greater than approximately 0.2. There will be no multiple steady stares when the criterion developed by Lussh is fulfiIled.
/ 12-36]
12.3 Falsified Kinetics YOU may not be measuring what you th~nkyc~uare.
There are circumstances under which the measured reaction order and activation energy are not the true values. Consider the case in which we obtain reaction rate data in a differenrial reactor. where precautions are taken to virtually eIiminate external mass transfer resistance (i.e., C, = CAb}.From these data we construct a log-log plot of the measured rate of reaction -r, as a function of the gas-phase concentration, CAs(Figure 12-81, The slope of this plot is the apparent reaction order n' and the rate law takes the form
Measured rate with apparent reaction order n'
,c
log
Figure 12-8 Determining the apparent reaction order I-r, = p, I-r;)).
We will now proceed to relate this measured reaction order n' to the true reaction order n. Using the definition of the effectiveness factor. note that the actual rate -r,, is the product of r( and the rate of reaction evaluated at the external surface, k , C l , , i.e.,
For large values of the Tbiele modulus d,, we can use Equation ( 17-35) to substitute into Equation (12-38) to obtain
D. LUSS. Ckem. Eng. Sci.,23, 1 249 ( 1968).
Diffusion and Reactinn
Chap 72
We equate the true reaction rate, Equation (12-39), to the measured reaction rate, Equation (12-371, to get
We now compare Eqations (12-39) and (12-40). Because the overall exponent
of the concenlration, CAq,must be the same for both the analytical and measured rates of reaction, the apparent reaction order n' is related to the true reaction order n by The true and the apparent reaction
order
In addition to an apparent reaction order, there is also an apparent activation energy, EApp. This value is the activation energy we would calculate using the experimental data, from the slope of a plot of In (-r,) as a function of 1IT at a fixed concentration of A. Substituting for the measured and true specific reaction rates in terms of the activation energy gives
measured into Equation (12-401, we find that
true
Taking the natural log of both sides gives us
where ET is the true activation energy. Comparing the temperature-dependent terms on the right- and left-hand sides of Equation (12-42), we see that the true activation energy is equal to twice the apparent activation energy. The true activation energy
Sec 12.4
Overall Effectiveness Factor
835
This measurement of the apparent reaction order and activation energy results primarily when internal diffusion limitations are present and is referred to as disguised or falsified kinetics. Serious consequences could occur if h e laboratory data were taken in the disguised regime and the reactor were operated in Imponant industrial a different regime. For example, what if the particle size were reduced so that consquence Of jnternaI diffusion limitations became negligible? The higher activation energy, falsified kinetic ET,would cause the reaction to be much more temperature-sensitive, and there is the possibility for runoway reaciion conditions to occur.
12.4 Overall Effectiveness Factor For firstorder reactions we can use an overall effectiveness factor to help us analyze diffusion, flow, and reaction in packed beds. We now consider a situation where external and internal resistance to mass transfer to and within the pellet are of the same order of magnitude (Figure I 2-9). At steady state, the rransport of the reactant.(s)from the bulk Ruid to the external surface of the catalyst is equal to the net rate of reaction of the reactant within and an the pellet.
Here, borh internaI and external diffusion are important.
Figurn 12-9
Mass transfer and reaction steps.
The molar rate of mass transfer from the bulk fluid to the external surface is
Molar rate = (Molar flux) - {External surface area) MA = W,, (Surface area/Volume) (Reactor volume) = WA,-a,dV
112-44)
where oa,is the external surface area per unit reactor volume (cf. Chapter 11) and AV is the voIume. This molar rate of mass transfer to the surface. MA, is equal ro the net
(total) rate of reaction on and n-irhin the pellet:
Diffusion and Reaction
836
( External area =
iM, = -ri (External area + Internal area j
Mass of catalyst
= P C 11 - 4 ) = Purosity
Mass of catalyst Vol~~rne of catalyst
) MA=+;[a,
See nomenclature note
1
area x Reaclor v o l u m ~ Reactor volume
area = Internal area
$I
Chap
volume of catalystxReactor valum, Reactor volume
AV+S,pb A V ]
I
( 12-4
Combining Equations ( 1 2-44) and (12-45) and canceling the voIurne P one obtains
For most catalysts the internal surface area is much greater than the extern surface area (i.e., Sapb 4 a,). in which case we have
where -r," is the overall rate af reaction within and on the pellet per unit su face area. The relationship for the rate of mass transport is
where k, is the external mass transfer coefficient ( d s ) . Because internal diffi sion resistance is also significant, not all of the interior surface of the pellet accessible to the concentration at the external surface of the pellet, C,,. M have already learned that the effectiveness factor is a measure of this surfar accessibility [see Equation (12-3811:
Assuming that the surface reaction is first order with respect to A, we can ut lize the internal effectiveness factor ta write
Sec. 12.4
Overalk Effect~venessFactor
837
We need to eliminate the surface concentration from any equation involving the rate of reaction or rate of mass transfer, because CAscannot be measured by standard techniques. To accomplish this elimination. first substitute Equation (12-48) into Equation ( 12-46):
Then substitute for WA,a, using Equation (12-47)
I 1
Solving for CAs,we obtain Concentration at
the pellet surface as a function of bulk gas concentndon
Substituting for CAsin Equation (12-48) gives
In discussing the surface accessibility, we defined the internal effectiveness factor q with respect to the concenwation at the external surface of the pellet, CAs:
'= Two different effectivenes~factors
Actual overall rate of reaction Rate of reaction that would result if entire interior surface were exposed to the external pellet surface conditions, C,,, T,
We now define an overall effectiveness factor that is based on the bulk concen~ratjon:
Rate that would result if the entire surface were exposed to the bulk conditions, CAb,Tb Dividing the numerator and denominator of Equation (12-51) by kc@,, we obtain the net rate of reaction (total molar flow of A to the surface in t m s of the bulk fluid concentration), which is a measurable quantity:
838
Diffusion and Reaction
Chap. ? 2
Consequently, the overall rate of reaction in terms of the bulk concentration CAbis
where Overall effectiveness factor for a first-order reac~ion
The rates of reaction based on surface and bulk concentrations are related by
where
The actual rate of reaction is related to the reaction rate evaluated at the bulk concentrations. The actual rate can be expressed in terms of the rate per unit volume, -r,, the rate per unit mass, -ri, and the rate per unit surface area, -r,", which are relazed by the equation
In terms of the overall effectiveness factor for a first-order reaction and the reactant concentration in the bulk -r,=rA,1;2=r~,p,Q=-r~,S,pb~=k;CAbSap,fl
(12-57)
where again Overall effectiveness
factor
Recall that k;' is given in terms of the catalyst surface area (rn3/m2-s).
12.5 Estimation of Diffusion- and Reaction-Limited Regimes In many instances it is of interest to obtain "quick and dirty" estimates to learn which is the rate-limiting srep in a heterogeneous reaction.
Set. 12.5
Estimation of DMusion- and Reaction-Limited Regimes
12.5.1 Weisz-Prater Criterion for Internal Diffusion
The Weisz-Prater criterion uses measured values of the rate of reaction, -r; (obs), to determine if internal diffusion is limiting the reaction. This criterion can be developed intuitively by first rearranging Equation (12-32) in the fom
Showing where the Welsz-Prater
comes from
The left-hand side is the Weisz-Prater parameter:
cWP =q~b:
- Observed (actual) reaction rate:, Reaction rate evaluated at C,,
Reaction rate evaluated at CA, A diffusion rate
- Actual reaction rate A d~ffusionrate
Substituting for
q=- -rA(obs)
and
-
3, P, R2 - 4, R2 D,C*, D,C*,
$:=
-.as
PC
in Equation (12-59) we have
AR there any internal diffusion
Iirnitations
All the terms in Equation (12-61) are either measured or known. Consequently, we can calculate Cw Howwer, if
indicated from the
0 C,,
Weisz-Prater criterion?
c<
1
there are no diffusion limitations and consequently no concentration gradient exists within the pellet. However, if
internal diffusion limits the reaction severely. Ouch! Example 12-3
Estimating Thiele Modulus and Eflecctiveness Factor
The first-order reaction
840
1
Diffusion and Rpac!jon
was carried out over two different-sized pellets. The- pellets were contained j r spinning basket reactor that was operated at sufficteatly high rotation speeds tl external mass transfer resistance was negligible. The results of two experimen nlns made under identical conditions are as given in Table E13-3. I . (a) Estimate t Thiele modulus and effectiveness factor for each peliet. (b) How small should t pellers be made to virtually eliminate all internal diffusion resistance? TABLE E12-3.1 DATA mow
A
SP~NN~NGBASKET RWCTIO?~'
Menstwed Rare (obsl cat . s ) X I@
The.% two experiments yield an enomlous amount of information,
Chap.
[mollg Run 1 Run 2
Allet Raditt.~ (m)
3.0 15.0
0.01 0.00 t
See Figure 5- II(c),
(a) Combining Equations (11-58) and ( 1 2-6 t ), we obtain
Letting the subscripts 1 and 2 refer to runs 1 and 2, we apply Equation (E12-3.1) t~ runs 1 and 2 and then take the ratio 50 obtain
The terms p, D,,and C ,, cancel because the runs were carried out unda identicai conditions. The Thiclc modulus is
Taking the ratio of the Thielc moduli for runs 1 and 2, we obtain
Substituting for h , , in Equation (E12-3.2) and evaluating -ri and R for nrns 1 and 2 gives us
Sec. 12.5
1
Estimation of D ~ k s i o n and Reaction-Limited Regimes
We now hove one equation and one unknvwn. Solving Equation i E 1 2-3.7) we find that
61I= rod,?= 16.5 The corresponding effectiveness factors mentill polnts. one can predict the
particle cize where internal macs transfer does not limit the rate of reaction.
For R,:
for R l = 0.01
n
rn
q2=
ForR,: q, = 3(16.5 coth 16.5-1) (b) Next we calcuIate the panicle radius needed to virtually eliminate internal diffusion control (say. a = 0.95):
Solution to Equation (E12-2.&)yields
= 0.9:
(i.e.. q = 0.95).
12.5.2 Mears' Criterion lor External Diffusion The Mears7 criterion, like the Weisz-PMter criterion, uses the measured rate of reacrian, - r i . (kmoilkg catas) to learn if mass transfer from the bulk gas phase to the catalyst surface can be neglected. Mears proposed that when I s external diffusion limiting?
external mass transfer effects can be neg2ected.
'D. E. Mem, Ind
Eng. Chem Pmcess Des Der. 10. 541 (1971). Other interphase transport-limitingcriteria can be found in AlChE S ~ m p Sez . 143 (S. W. Weller, ed.). 70 ( 1974).
842
where
Diffusion and Readion
Chap. 12
n = reaction order R = catalyst particle radius, m ph = bulk density of catalyst bed, kg/rn3 = (1 +)p, (4 = porosity) p, = bulk density of catalyst bed, kg/m3 CAb= solid catalyst density, kg/m3 kc = mass transfer coefficient, m/s
-
The mass transfer coefficient can be calcuIated from the appropriate correlation, such as that of Thoenes-Garners, for the flow conditions through the bed. When Equation (12-62) is satisfied, no concentration gradients exist between the bulk gas and external surface of the catalyst pellet. Mears also proposed that the bulk fluid temperature, T, will be virtually the same as the temperature at the external surface of the pellet when Is there a temperature gradient?
where
h = heat transfer coefficient between gas and pellet, kJ/m2.s-K R, = gas constant, Wmol. K AHR, = heat of reaction, !d/mol E = activation energy, kYkmol
and the other symbols are as in Equation (12-62).
12.6 Mass Transfer and Reaction in a Packed Bed We now consider the same isomerization taking place in a packed bed of catalyst pellets rather than on one single pellet (see Figure 17-10). The concentration CAis the bulk pas-phase concentration of A at any point along the length of the bed.
z=o
z
Z*AZ
r=L
Rgure 12-10 Packed-bed reacior.
We shall perform a balance on species A over the volume element 11' neglecting any radial variations in concentration and assuming that the bed is
Sec. 12.6
Mass Transfer and Reaction in
B
Packed Bed
843
operated at steady state. The following symbols will be used in developing our model:
A, = cross-sectional area of the tube, dm1 CAb= bulk gas concentration of A, molldm3
pb = bulk density of the catalyst bed, g/dm3
v, = volumetric flow rate, dmVs
U = superficial velocity = v, /A,, dmls Mole Balance
A mole balance on the volume element (A,&) yields [Rate in]
-
Ac'Lnlr
- AcWkl:+a;
[Rate out]
$.
[Rate of formation of A] = o
+
Dividing by A, hz and taking the limit as bz
6 pbAc
=0
+ 0 yields
Assuming that the total concentration c is constant, Equation (11-14] can be expressed as
Also. writing the bulk flow term in the form B A= ~Y A ~ ( W+ A ~W B ~ = ) ?IAN = UCA~
Equation (12-64) can be written in the form
Now we will see how to use . Iand to calculate conversion in a nacked hed.
The term DAR(d2CAbJd~') is used to represent either diffusion andtor dispersion in the axial direction. Consequently. we shall use the symbol D,for the dispersion coefficient to represent either or both of these cases, We wiIl come back to this form of the diffusion equation when we discuss dispersion in Chapter 14. The overall reaction sate within the pellet. -I-;. is the overalI rate of reaction within and on the catalyst per unit mass of catalyst. It is a function of the reactant concentration within the cataIyst. This overall rate can be related ro the rate of reaction of A that would exist if the entire surface were exposed to the bulk concentration CAhthrough the overail effectiveness factor fl:
For the first-order reaction considered here. -rAb = -rib Sa= k3'SaCAh
844
Diffus~onand React~on
Chap
Substituting Equation ( 1 2-66) into Equation ( 12-57). we obtain the overall of reaction per unit mass of catalyst in terms of the bulk concentration C,,
Substituting this equation for -ri into Equarion ( 12-65). we form the c ferential equation describing diffusion with a first-order reaction in a cat$ bed: Flow and firstorder reaction in a packed bed
I
As an example. we shall solve this equation for the case in which the Row ra through the bed is very large and the axial diffusion can be neglecte Finlaysons has shown that axial dispersion can be neglected when Criterion For neglecting axial dispersion{diffusion
where W,, is the superficial velocity, d, the panicle diameter. and W, is tt effective axial dispersion coefficient. In Chapter 14 we will consider solutior to the complete form of Equation (12-67). Neglecting axial dispersion with respect to forced aria1 convection,
Equation (12-67) can be arranged in the form
With the aid of the boundary condition at the entrance of the reactor. C,, = C,4wl
at t = 0
Equation (12-693 can be integrated to give -(~Ksaibvc!
C A=~C A ~e O
The conversion at the reactor's exit. z = L. is Conversion in a
pack&-kd -tor
L. C.Young and B. A. Finlayson, Ind Eng. Chem, Fund. 12, 412 (1973).
Sec. 12.6
Mass Transfer and Reaction ~n a Packed Bed
845
Example 12-4 Reducing Niiroc~sOxides in a Phnl Efluent I n Section 7.1.4 we saw the role that nitric oxide plays in smog formation and the incentive we would have for reducing its concentration in the atmosphere. It is proposed to reduce the concentration of NO in an effluentstream from a plant by passing it through a packed bed o f spherical porous carbonaceous solid pellets. A 2% NO-98% air mixture flows at a rate of 1 X 1W6m3/s (0.001 drn3/s) through a 2-in.-ID tube packed wi!h porous solid at a temperature of F I73 K and a pressure of
101.3 kFa. The reaction
is first order in NO,that is.
-.j, = VSC ,, and occurs primarily in the pores inside the pellet, where Green
S,= Internal surface area = 530 m21g
chemical reaction
engineering
Calculate the weight of porous solid necessary to reduce the NO concentration to a level of 0.004%, which is below the Environmental Protection Agency limit. Additional information:
At I173 K. the fluid properties are
v = Kinematic viscosity = 1.53 X
lWsm2/s
DAB = Gas-phase diffusivity = 2.0 X IOV8m2/s
D, = Effective diffusivity The properties of the catalyst and bed
Also X e Web site ww.mwan.edu/
= 1.82 X
m2ts
are
p, = Density of catalyst particre = 2.8 g/cm3 = 2.8 x
pmengincering
106 g/m'
b = Bed porosity = 0.5
- 4) =
pb = Bulk density of bed = p,(t
R = Pellet radius = 3 X
1.4
x 106 glm'
m
y = Sphericity = 1.0 Solulion
It is desired to reduce the NO concentntion from 2.0% to 0.0048. Neglecting my volume change at these low concentrations gives us
1
where A represents NO.
846
Diffusion and Reaction
Chap. 12
The variation of NO down the length of the reactor is given by Equation (12-69)
Multiplying she numerator and denominator on the right-hand side of Equation (12-69) by the cross-sectional ma, A,, and realizing that the weight of solids up to a p i n t z i n the bed is (Mole baIancej
+ +
W = pbArz
(Rate law)
the variation of NO concentration with solids is
(Overall effectiveness
factor) Because NO i s present in dilute concentrations. we shall take E 6 I and set o = v , . We integrate Equation (EF2-4.1) using the boundary condition that when W = 0,then CAC= CAM:
(El 2-4.2) where
1
Rearranging. we have
1. Calculating fhc internal effectiveness facror for spherical pellets in which a first-order reaction is wcurring. we obtained
As a first approximation, we shall negIect any changes in the pellet size resulting from the reactions of NO with the porous carbon. The Thiele modulus for t h ~ ssystem isg .
where R = pellet radius = 3 x 10-3 m D, = effective diffusivity = 1.82 x
'L. K
m21s
p, = 2.8 glcm3 = 2.8 X IF pJm'
.ha", A. F. Sarohm, and J.
M. Beer, Comht~sr,Flame, 52, 37 (1983).
Set. 12.6
I
Mass Transfer and Reaction ~ne Packed Bed
847
k = specific reaction rate = 4.42 x 10-lo rn3tm2-s
4 , = I8 Because. L, is large,
2. To calculate the external mass fransfer cneficient. the Thwnes-Kramers correlation i s used. From Chapter 1 1 we recall
Sh' = ( ~ ~ ' ) ! / 2 $ ~ 1 / 3
( 1 1-65)
For a 2-in.-ID pipe, A, = 2.03 X lo-' m2. The supefidal velocity is
Re' =
Prdum Calculate Re ' Sc
Ud, ( 1 -0)v
-- (4.93x 1 OL4 m / S ) ( 6 ~10-3 m)= 386.7 ( 1 -0.5)(1.53x m2/s)
Nomenclature note: 4 with subscript 1. 4, = Thiele moduFus & without subscript. d~ = porosity
Then
I"? I Sh'
Sh'
= (386.T)117(0.765)v3 = (19.7)(0.915)= 18.0
3. Calcubtin~the extcrnal area per unit reactor volume, we obtain
I
= soo m2/s3 4. Evaluating the m~emlleff~ctivencssfactnr.Substituting into Equation (12-55). w e have
Dlffus~onand React~on
848
Chap.
In this example we see that both the external and internal resistances to mn transfer are significant. 5. Calruluti~lgthe weighr of . d i d necessnty to nchievc 99.8CTc cmtr~ersion.Su stituting into Equntion (E 12-4.3). we obtain
I
6. The reactor length is
12.7 Determination of Limiting Situations from Reaction Data For externa1 mass transfer-limited reactions in packed beds. the rate of reactio at a point in the bed is
Variation of reaction with system v:iriahleo
for the mass transfer coefficient. Equation 11 1-66), shows th kc is directly proportional to the square root of the veracity and inversely pro portional to the square root of the particle diameter: The correlation
We recaIl from Equation (EI2-4.5), a, = 6(1 - @)Ed,,that the variation of exter nal surface area with catalyst particle size is
Consequently, for external mass transfer-limited reactions, the rate is inversel: proportional to the particle diameter to the three-halves power:
Sec, 12.8
Many heterogeneou' reuction\ are diffilsion limited,
From Equation ( 1 1.-72) we see that for gas-phase externat mass trinsfer-limited reactions, the rate increases approximately linearly with temperature. When internal diffusion limits the rate of reaction, we observe from Equation (12-39) that the rate of reaction varies inversely with partick diameter, is independent of veiocity, and exhibits an exponential temperature dependence which is not as strong as that for surface-reaction-controlling reactions. For surface-reaction-limited reactions the rate is independent of particle size and is a strong function of temperature (exponential). Table f2-I summarizes the dependence of the rate of reaction on the velocity through the bed, particle diameter, and temperature for the three types of limitations that we have been discussing.
1 Very Important Table
Mulliphase Reactors
, I
Variation of Rcnction Rate w i h : Type of
Limitation
Velocip
Pnrticle Size
Tmpemture , 1
External diffusion
Internal diffusion Surface reaction
~ l n
Independent Independent
(dP)-312
(dp)-' Independent
=Lineat Exponential Exponential
i ,
The exponential temperature dependence for internal diffusion lirnitatians is usually not as suong a function of temperature as is the dependence for surface reaction limitations. IF we would calculate an activation e n e s y between 8 and 24 WJmol, chances are that the reaction is strongly diffusion-limited. An activation energy a€ 200 kT/moE, however, suggests that the reaction is reaction-rate-limited.
12.8 Multiphase Reactors Multiphase reactors are reactors in which two or more phases are necessary to carry out the reaction. The majority of multiphase reactors involve gas and liquid phases that contact a sotid. In the case of the sturry and trickle bed reac-
'Reference Shelf
tors, the reaction between the gas and the liquid takes place on a solid catalyst surface (see Table 12-21. However, in some reactors the liquid phase is an inert medium for the gas to contact the solid catalyst. The latter situation arises when a l q e heat sink is required for highly exothermic reactions. In many cases the cataIy st life is extended by these milder operating conditions. The multiphase reactors discussed in this edition of the book are the durry reactor, fluidized bed. and the trickle bed reactor. The trickle bed reactor, which has reaction and transport steps sirniIar to the slurry reactor, is discussed in the first edition of this book and on the CD-ROM along with the bubbIin8
850
Diffusion and Asaction TABLE 12-2.
Chap. 72
A m i c ~ n o ~OFsTHREE-PHASE REACTORS
I. S b r reactor ~ A. Hydrogenation
1. of fatty ackds over a suppwted nickel catalyst 2. of 2-butyne-1,4-dl01 wer a Pd-CeC03 catalyst 3. of glucose over a Raney nickel catalyst B. Oxidation I . of CzH4in an inen liquid over a PdC1:-ca&n catalyst 2. of SO2 in inen water over an activated carbon catalyst C. Hydmformation of CO w~thhigh-molecular-weight olefins on either a cubah or ruthenium complex bound to polymers D. Ethynylation React~onof acetylene with formaldehyde over a CaC12-supported catalyst 11. Trickle hed reartr8rs A. Hydnndesulfurization Removal of sulfur compounds from crude oil by reaction with hydrogen on Co-Mo on alumina
R. Hydrogenation 1. of anikine over a Ni-ctay catalyst 2. of 2-htyne-1.4-dm1 over a supponed Cu-Ni catalyst 3. of benzene, a-CH3 styrene. and crotonaldehyde 4. of aromatics in napthenic lube oil distillate C. Hydrodeni~ropenalion 1 . of lube oil dist~llate 2. of cracked Iight furnace oil D Oxidation 1. of cumene wer activated carbon 2. of SO, over carbon
Sourre: C.N. Satterfield. AIChE J., 21, 209 (1975): F.A. Ramachandmn and R. V. Chaudhari. Chcm. Ens.,87c241. 74 (19801: R. '4. Chaudhari and P. A. Rarnachandran. AEChE I . , 16, 177 1 1980).
Reference
fluidized bed. Tn slurry reactors, the catalyst is suspended in the liquid, and gas is bubbled through the liquid. A slurry reactor may be operated in either a semibatch or continuous mode.
($-%;~z
12.B.f
Slurry Reactors
A complete description of the slurry reactor and the transport and reaction steps are given on the CD-ROM,along with the design equations and a number of examples. Methods to det~rminewhich of the transport and reaction steps are rate limiting are included. See Professional Reference Shelf R12.1.
12.8.2 Trickie Bed Reactors The CD-ROM includes all the material on trickle bed reactors from the first edition of this book. A comprehensive example problem for trickle bed reactor design is included. See Professional Reference Shelf R 1 2.2.
Chap. 12
851
Summary
12.9 Fluidized Bed Reactors The Kunii-kvenspiel model for fluidization is given on the CD-ROM along with a comprehensive example problem. The rate limiting transport steps are also discussed. See Professional Reference Shelf R 1 2.3.
12.10 Chemical Vapor Deposition (CVD)
, VU ,
'1 Chemical Vapor Deposition in boat reactors is discussed and modeled. The equations and parameters which affect wafer thickness and shape are derived and analyzed. This material i s taken directly from the second edition of this book. See Professional Reference Shelf R12.4. Closure A h r completing this chapter, the reader should be able to derive
differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and mction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as bjomaterid tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-bed reactors, and CVD boat reactors and to make calculations for each reactor.
SUMMARY 1. The concentration profile for a first-order reaction occumng in a spherical cataJyst pellet i s
where bl is the Thiele modulus. For a first-order reaction
2. The effectiveness factors are
Internal effectiveness = 11 = factor
Actual rate o f reaction Reaction rate if entire interior sirrfacc I S exposed to concentration at the external pellet surface
"l
ool
b,
,, .a
Diffusion and React~on
Chap. 1
Overall Actual sate of reaction effectiveness = !2 = Reaction rate if entire surface area factor is exposed to bulk concentration
3. For large values of the Thzele modulus for an nh order reaction,
4.
For internal diffusion control, the m e reaction order i s related to the measured reaction order by
Ttte me and apparent activation energies are related by Errue
= ZEapp
(S12-51
5. A. The Weisz-Prater Parameter
Cw,=
9:rl =
-6 (observed) p, RZ D~CA~
The Weisz-Prater criterion dictates that
If CwpQ 1
no internal diffusion limitations present
If C, B 1
internat diffusion limitations present
B. Mears Criteria for Neglecting External Diffusion and Heat Transfer
and
CD-ROM MATERIAL * Learning Resaucees 1. Summary Notes Summary Notcr
Chap. f 2
CD-ROM Material
Professional Reference Shelf R 12. I .S l u r y Rencrors Trancpon Step%and Re~istanoes
mmh
mmIICI1
Reference Shclf
A. Description of Use of Slurry Reactors
Example R 12- I Industrial Sluny Reactor B. Reaction and Trampon Steps in a Step in Slurry Reactor
C. Determining the Rate-Limiting Step 1. Effect of Loading. Particle Size and Gas Adsorption 2. Effect of Shear Example RI?-2 Detennin~ngthe Controlling Resistance D. Slurry Reactor Design
Example R12-3 S l u q Reactor Design
R 12.2. Trickle Bed
Renctors
A. Fundamentals
B. Limiting Situations C. Evaluating the Transport Coefficients
h -.,
Dlftusion and Reacfbn
Chao. 12
R12.3. Fluidized-Bed Reactors A. Descriptive Behavior of the Kunii-LRvenspitl Bubbling Bed Modd
B. Mechanics of Fluidized Beds Example R12-4 Maximum Solids Hold-Up C. Mass Transfer in Fluidized Beds D. Reaction in a Fluidized Bed E. Solution to the Balance Equations for a First Order Reaction
Example R12-5 Catalytic Oxidation of Ammonia
F+
limiting Situations
Example R 17-6 Calculation of the Resistances Example R12-7 Effect of Particle Size on Catalyst Weight for a Slow Reaction Example R 12-8 Effect of Catalyst Weight for a Rapld Reaction R11.4.Chemical Vapor D ~ p o s ~ i i oRroctor.r n Refereace Shelf
a*-
C*LW"?I 6YS11u
-
>
: u
=/ \-,-A
Chap. 12
Questions and Problems
A. Chemical Reaction Engineering in Microelectronic b e s s i n g B. Fundamentals of CVD C, Effectiveness Factors for Boar Reactors
Example R12-9 Diffusion Between Wafers Example R 12-10 CVD Boat Reactor
~efcrcnceShelf
QUESTIONS AND PROBLEMS The subscript to each of the problem numbers indicates the level of difficulty: A. least dificult: 1). most difficult.
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Fl2-lc Make up an original problem using the concepts presented in Section (your instructor will specify the section). Extra credit will be gtven if you obtain and use real data from the literature. (See Prohrem P4-1 for thc guidelines.) P12-2R (a) Example 12-1. Efferril?~Drffusivin: Make a sketch of a diffus~onpath for n h ~ c hthe tortuosity is 5. How ~ ~ o u your l d effective gas-phase diffuslv~ty change if the ahsolute pressure were tnpled and the temperature were increased by SO%? How would your answers change if (b) Example 12-2. 7issr1eEtlginrerir~~. the reaction kinetics were ( I ) first order In O? concentration with k , = lW2 h-I? ( 2 ) Monod kinetics with p,,,, = 1.33 x h-I and K, = 0.3 rnolldm'. (3) zero-order kinet~cscarry out a quasi steady state analysis uslnf Equations (E12-2.19) along with the overall balance
to predict the 0?Aux and collagen build-up as a function of time. ,V01r: V = A,L. Assume o: = 10 and the ~toichiorne~ric coefficient for oxygen In collagen. v,. is 0.05 mass liaction of cclllmol 0:. (c) ExampFe 15-3. ( I ) What is the percent of the total reri? (3I ~f the Fac vclnc~tywere
D~Rus~on ano Rsact~on
Chap.
tripled? (4) if the particle v i ~ ewere decreased by a factor of 27 would the reactor length change in each case? (5)What length would required to achieve 99.99% conversion of the pollutant NO? What it.. (e) you app11c-dthe Mean and Weisz-Prater criteria to Examples E 1-4 a 12-4'? What would you find? What would you learn if AHR, = kcal/moI. h = 100 Btulh-ft:."F and E = 20 k callmol? (f) we let y = 30, = 0.4, and d~ = 0.4 in Figure 12-7? What wot~ldcau you to go from the upper steady state to the lower steady state and vi versa? (g) your internal surface area decrrnsed with time because of sintering. Hr would your effectiveness factor change and the rate of reaction chan with time if k, = 0.01 h-1 and q = 0.01 at r = O? Explain. [h) someone had used the False kinetics (it..wrong E. wrong n)'? Wou their catalyst weight be overdesigned or underdesigned? What are 0th positive or negative etficts that occur? (i) you were asked to compare the conditions (e.g.. catalyst charge, convt sions) and s~zesof the reactors in CDROM Example R l 2 . l . What diffc ences would you find? Are there any fundamental discrepancies betwe1 the two? If so, what are they, and what are some reasons for them? (j) you were to assume the ressstance to gas abwrption in CDROM Examp Rl2.1 were the same as in Example RZZ.3 and that the liquid phase rea tor volume in Example R12.3 was 50% o f the total, could you estima the controlling resistance? If so, what is it? What other things could yc calcuIat~in Example R12.1 (e.g., selectivity. conversion, molar flow rat, in and out)? Hint: Some of the other reactions that occur include
a
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(k) the temperature in CUROM Example R12.2 were increased? How woul the relative resistances in the slurry reactor change? (I) you were asked for alI the things that could go wrong in the operation ( a slurry reactor, what would you say? P12-3R The catalytic reaction
takes place within a fixed bed containing spherical porous catalyst X22. Fil ure P12-3 shows the overalf rates of reaction at a p i n t in the reactor as function of temperature For various entering total molar flow rates. Fm. (a) Is the reaction limited by external diffusion? (b) If your answer to part (a3 was "yes." under what conditions [nf thos shown (i.e.. T, F,)] is the reaction limited by external diffusion? (c) IS the reaction "reaction-rate-limited"? (d) If your answer to part (c) was "yes," under what conditions [of those show (i-e., T, Fm)] is the reaction limited by the rate of the surface reactions' (e) Is the reaction limited by internal diffusion? (fl If your answer to part (e) was "yes;' under what conditions [of thos shown (i.e., 7; FTu)]is the reaction limited by the rate of internal diffusion (g) For a flow ratate of ID g molth. deterrmne (if possible) the overall effec tiveness factor, R, at 360 K. Ih) Estimate (if possible) the internal effectiveness factor, r(. at 367 K.
Chap. 12
Quest~onsand Problems
I
0 350
1
I
I
I
I
360
370
380
390
400
T(k) Figure P12-3 Reaction ntes in a catalyst bed.
i
If the concentration at the external catalyst surface is 0.01 mol/dm3, calculate (if possible) the concentration at r = R12 inside the porous catalyst at 337 K. (Assume a first-order reaction.)
Addirionnl information:
Bed properties:
Gas properties:
Biffusivity: 0.I cmzls Density: 0.001 g/cm3 Viscosity: 0.0001 g l c m - s P12-.la The reaction
Tortuosity of pellet: 1.415 Bed permeability: 1 millidarcy Porosity = 0.3
A - B
is carried out in a differential packed-bed reactor at different temperatuses, flow rates, and particle sizes, The results shown in Figure P12-4 were obtained. Mall c f Fame
t
Figure P I 2 4 Reaction rates in a camlyst hzd.
8 58
P12-5,
Diffusion and Readion
Chap. 12
ta) What regions (i.e.. conditions d,, 7, F,) are external mass transfer-limited? (b) What regions are reaction-rate-limited? lc) What region is internal-diffusion-controlled? (d) What i s the internal effectiveness factor at T = 400 and d, = 0.8 cm? Curves A, B. and C in Figure P12-5show the variations in reaction rate for three different reactions catalyzed by solid catalyst pellets. What can you say ahout each reactlon?
! . I '
Figure P12-5 Temperature dependence of three reactions
PI2-6,
A first-order heterogeneous irreversible reaction is ttking place within a spherical catalyst pellet which is plated with platlnum throughout the pellet (see Figure 12-3). The reactant concentration halfway between the external surface and the center of the pellet f i x . . r = RI?) is equal to one-tenth the
concentratlon of the pellet's external surface. The concentratlon at the external surface is 0 001 g molfdrn3.the diameter ( 2 R ) is 2 X 30-hcm. and the dtffuslon c r ~ f f i c i c nis~ 0.1 cm=ls.
crn in (a) What is the concentration of reactant at a distance of 3 X from the external pellet surface? {Ans.: C, = 2.36 X 10-%ol/dm4.) (bk To wwh atdiameter should the pellet be reduced if the effectiveness factor . = 6.8 X I F 4 cm. Critique this answer!) is lo be 0.8? I A I I ~ .dp (c) Ifthe catalyst suppon were not yet plated with platinum. how would you suggest that the catalyst support be plated afrer it had been reduced by Applioaflon Pending rut Pmblem Hall of
grinding? PIZ-7,, The SMimming rate of a small organism fJ. Theorpi. Biol., 26, 1 1 (I970)] is related to the energy released by the hydrolysis of adenosine triphosphate (ATPI to adenosine diphosphate (ADP). The rate of hydroly~isis equal to the rate of diffuvon of ATP from the rnidp~eceto the tail (see Figure P 12-71. The diffusion coefficient of ATP in the midpiece and tail is 3.6 X cmZls.AQP is converted to ATP ~n the midsection. where its concentrafion is 4.36 X loc-' mollcm'. The cross-rcctional area of the tail is 3 X IO-IU cm2.
Figure P12-7 Swirnming of an organicrn. (a1 Derive an equation for diffusion and reactlon in the tail.
Ihl Derirc an equation Tor the effectiveness Factor in the tail
Chap, t 2
Questions and Problems
859
(c) Taking the reaction in rfie tail lo be of zero order. calcufate the lmgth of the tail. The rate of reaction in the tail is 23 X 10- I R molts. (d) Compare your answer with the average tail length of 41 pm. What
possible sources of error? P12.BB A first-order, heterogeneous, irreversible reaction is taking place within a catalyst pore which i s plated with platinum entirely along the length of the pore (Figure P12-8). The reactant concentration at the plane of symmetry (i.e., equal distance from the pore mouths) of the pore is equal to one-tenth the concentmtion of the pore mouth. n e concenmlion at the pore mouth is 0.001 g moIldm3, the pore length (2L) i s 2 x cm. and the diffusion coefficient is 0.1 cmYs.
Figum P12-8 Single catalyst pore.
(a) Derive an equation for the effectiveness factor. (b) What is the concentration of reactant at LIZ? (c) To what length should the pore length be reduced if the effectiveness factor is to be 0.8? (d) If the catafyst suppon were not yet plated with platinum, bow would you suggest the catalyst support be plated after the pore length, L, had been
reduced by grinding? P12-9, A first-order reaction is taking place inside a porous catalyst. Assume dilute concentrations and neglect any variations in the axial (x) direction. (a) Derive an equation for both the internal and overall effectiveness factors for the rectangular porous slab shown in Figure PI 2-9, (b) Repear part (a) for a cylindrical catalyst pellet where the reactants diffuse inward in the radial direct~on.
Figure P12-9 Flow over porous catalyst slab.
Pf2-10, The irreversible reaction
is taking place in the porous catalyst disk shown in Figure P12-9. The reaction is zero order in A. (a) Show that the concentration profile using the symmetry B.C.is
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Diffusion and Reaction
Char
where
(b) For a Thiele modulus of 1.0, at what point in the disk is the ioncenml zero? For = 4? (c) What i s the concentration you calculate at z = 0.1 L and = 10 us Equation (P12-10.I)'? What do you conclude about using this equatio, (d) Plot the dimensionless concentration profile y = CAiCAIas a function h = z/L for Q,, = 0.5. 1, 5 , and 10. H i ~ r there : are regions where the c~ centration i s zero. Show that h, = 1 - I/@, is the start of this reg where the gradient and coocedtntion are both zero. [L. K. Jang, R. York. J. Ctln. and L. R. Hile, Inst. Chem. Engr., 34, 319 (2003).]Sh that v=O; X'-2@,,($,)- 1 ) A+(&I ) = f o r & 5 h c 1. (e) The effectiveness factor can be written ns
@,,
where z , (&)is the point where both the concentration gradients and flux , zero and A, is the crors-sectional area of the disk. Show for a rem-ord
to
reaction that for ib,, 5 1.0
q = { i - x c = ~ t-orhzl $0
(f) Make a sketch for versus dosimiEar to the one shown in Figure 12-5 (g) Repeat parts (a) to ( f ) for a spherical catalyst pelIet. (h) Repeat parts {a) to (0 for a cylindrical catalyst petlet. ti) What do you believe to be the point of thls probiem? P12.11c The second-order decomposition reaction
is carried out in a tubular reactor packed with catalyst pellets 0.4 cm in diarnt ter. The reaction is inkmaldifft~sion-limited. Pure A enters the reactor at superficial velocity of 3 mls, n temperature of 25O"C, and a pressure of 500 kPi Experiments carried out on smaller gellets where surfilca reaction is lirmtin, yielded a specific reaction rate of 0.05 mblrnoE g cat -s. Calculate the length o bed necessary to achieve 80% conversion. Critique the numerical answer.
-
Effective diffusivity: 2.66 X LO-8 m2ls lneffectlve diffusivity: 0.00 m2/s Bed porosity: 0.4 Pellet density: 2 X 1P g/m3 Internal surface area: 400 m2/g
Chap. 72
861
Questions and Promems
P12-lZc Derive the concentration profile and effectiveness factor for cylindrical pellets 0,2 crn in diameter and 1.5 crn in length. Neglect diffusion through the ends of the pellet. (a) Assume that the reaction is a first-order isomerization. (Hint: Look for a Bessel function.) (b? Rework Problem 812- I1 for these pAlets. P12-13c Reconsider diffusion and reaction in a sphericaIcatalyst pellet forthe case where the reaction i s dot isothermal. Show rhat the energy balance can be written as
where k, is the effective thermal conductivity, calls - cm - K of the pellet with dTldr = 0 at r = 0 and T = T, at s = R. (a) Evaluate Equation (12-11) for a first-order reaction and combine with Equation (P12-15.1) to arrive at an equation giving the maximum temperature in the peIIet.
w e . ' At Tm,, c, = 0. (b) Choose representative values of the parameters and use a software package to solve Equations (12-11) and (P12-13.1) simultaneously for T(r) and CA(r)when the reaction is carried out adiabatically. Show that the resuiting solution agrees qualitatively with Figure 12-7. P12-14cDetermine the effecrrveness factor for a nonisothermal spherical catalyst pellet in which a tirst-order isomerisation is taking place.
Additional infomtion:
A, = 100 m2/m3
hH, = -800,000 Jlmol D, = 8.0 X m2/s CAs= 0.01 kmol/m3 External surface temperature of pellet, T, = 400 K
E = 120.000 Jlrnol ThermaE conductivity of pellet = 0.004 JErn .s
.K
d, = 0.005 m Specific reaction rate = 10-I m/s at 400 K Density of calf's liver = 1.1 gJdm3
How would your answer change if the pelIets were I t 2 , 1 P , and l P 5 m in diameter? What are typical temperature gradients in catalyst pellets? P12-15, Extension of Problem P12-8. The elementary isomerization reaction
is taking place on the walls of a cylindrical catalyst pore. (See Figure P12-8.) In one run a catalyst poison P entered the reactor together with the reactant A. To estimate the effect of poisoning, we assume that the poison renders the catdyst pore walls near the pore mouth ineffective up to a distance z,, so that no reaction takes place on the walls in this entry region.
862
DifIuslon and Reaction
C h a ~ 12 .
(a) Show that before poisoning of the pore occurred, the effectiveness factor was given by
where
with
k = reaction rate constant (lengthltirne) r = pore radius (length) I), = effective molecular diffusivity larealtime)
(b) Derive an expression for the concentration profile and also for the molar flux of A in the ~neffecriveregion 0 < x < 2,. in iems of:,. DAD,,. C,,, and CAs.Without solving any funher differential equations. obtain the new effectiveness factor T ' far the poisoned pare. P12-16B Fals$ed Kit~erics.The irreversible gas-phase dirnerization
Is carried out at 8.2 atm in a stirred contained-solids reactor to which only pure A is fed. There is 40 g of catalyst In each of the four spinning baskets. The following runs were carried out at 227°C: Toral Molar Feed Rote, F , (g rnollmin)
1
2
4
6
11
20
The following experiment was carried out at 237°C:
(a) What are the apparenl reaction order and the apparent activation energy? (b) Determine the true reaction order, specific reaction rate. and activation energy. (c) Calculate the Thiele modulus and effectiveness factor.
(d) What diameter of pellets should be used to make the catalyst more effective? (el Calculate the rate of reaction on a rotating disk made of the catalytic matenal when the gas-phase reactant concentration 1s 0.01 g mallL and the temperature is 527°C. The disk i s flat, nonporous, and 5 crn in diameter.
Effective diffusivity: 0.23 cm7s Surface area of porous catalyst: 49 m31g cat Density of caraly\t pellets: 1.3 g/crn3 Radius of catalyst pellet$: 1 cm Color of pellets: blushrng peach
Chap. 12
863
Journal Article Problems
P12-17, Derive Equation (12-35). Hint: Multiply both sides of Quation (12-25) for nth order reaction, that is.
by 2&I&
rearrange to get
and solve using h e boundary conditions d p l d = O at h = 0.
JOURNAL ARTICLE
PROBLEMS
P12J-I The article in Tmns. ini. Chcrn. Eng.. 60, 131 (1982) may be advantageous in answering the fotlowing questions. (a) Describe the various types of gas-liquid-solid reactors. (b) Sketch the concentration profiles for gas absorption with: (3) An instantaneous reaction (2) A very slow reaction (3) An intermediate reaction rate P12J-2 After reading the journal review by Y.T. Shah et al. [AIClrE J., 28. 353 (1982)j. design the following bubbie column reactor. One percent carbon dioxidd in aiT is to be removed by bubbling through a solution of sodlum hydroxide. The reaction i s mass-uansfer-limited. Calculate the reactor size (length and diameter) necessary to remove 99.9% of the COz. Also specify a type of sparger. The reactor is to operate in the bubbly flow regime and still process 0.5 m"ls of gas. The liquid flow rate through the column is lo-' rn3/s.
JOURNAL CRITIQUE PROBLEMS P12C-1 Use the Weisz-Praler criterion to determine if the reaction discussed in AIChE J., 10. 568 (1964) is diffusiun-rate-limited. P12C-2 Use the references given in Ind. Eng. Chem. Pmd. Res. Dei:, 14. 226 (1975) to define the iodine value, saponification number. acid number, and experimental setup. Use the slurry reactor analysis to evaluate the effects of mass uansfer and determine If there are any mass transfer I~rnitations. Additional Homework Pmblems CDPlZ-AB Determine the catalyst size that gives the highest conversion in a packed bed reactor. CDP12-BB Determine importance of concentration and temperature gradients in a packed bed reactor. CDP12-C, Determine concentration profile and effecttveness factor for the first order gas phax reaction
864
Diffusion and Reaction
Cha~.
Slurry Reactors
CDPIZ-D,
Hydrogenation of methyl linoleate-comparing cataiyst. 13rd 1 PI2-IYJ CDPlZ-E, Hydrogenation of methyl linolente. Find the rate-limiting step. 13rd Ed. PI2-201 CDP12-FR Hydrogenation of 2-butyne-1.4-diol to butenediol. Calculate perc, resistance of total of each for each step and the conversion. [3rd E PI?-211 %
CVD Boat Reactors CDPIZ-GI, Derennine the temperature profile to ach~eve\>nuniform thickne \>
CDPXt-H, CDPl2-IR CDP12-.Ic CDPIZ-Kc
P Mcrcbcr
Mall of Fame
Green Engineering
, [Znd Ed. P1 1-18) Explain how varying a number of the parameters in the CVD bc rerrctor will affect the wafer shape. [Znd Ed. P11-141j Determine the wafer shape in a CVD boat reactor for a series of opt ating condttions. [Znd Ed. P 1 3 -201 Model the build-up of a silicon wafer on parallel sheets. [2nd E PI I-21J Rework CVD boat reactor accounting for the reaction
SiH4 SiH2 + HI [Znd ed. Pll-221 Trickle Bed Reactors CDP12-LB Hydrogenation of an unsaturated organic is carried out in a trick.klebt reactor. [2nd Ed. PI2-71 CDPlZ-M, The oxidation of ethanol is camed out in a trickle bed recrcror. [2r Ed. P12-91 CDPlt-Nc Hydrogenation of aromatics in a srickle brd wauror ['nd Ed. P12-R: Fluidized Red Reactors CDPIZ-0, Open-ended Ruidizrttion problem that requires critical thinking 1 compare the two-phase fluid models with the three-phase bubblin bed model. CDP12-P, Calculate reaction ntes at the top and the bottom of the bed fc Example R12.3-3. CDP12-Q, Calculate the conversion for A + I3 in a bubbling fluidized bed. CDP12-Rk Calculate the effect of operating parameters on conversion for th reaction limited and transport limited operation. CDP12-Se Excellent Problem Calculate all the parameters in Example R 12-3. for a different reaction and different bed. CDPf2-Ta Plot conversion and concentration as a function of bed height in bubbling fluidized bed. CDP12-UB Use RTD studies to compare bubblingkd with a fluidized bed. CDP12-VB New problems on web and CD-ROM. CDPIZ-WB Green Engineering, www.rowan.edu/greenengineering.
Ghap. 12
Supptemefltary Reading
865
SUPPLEMENTARY READING I. There are a number of hooks that d ~ ~ internat c u ~ diffusion In catalyst peliets: however. one of the first b ~ k that s should be consulted on this and other topics on heterogeneous catnlyhis is
L ~ P m u s L.. , and N.R. AMU~DSOY. Chernicnl ReachtorTheor?.:A Review Upper Saddle River, N.J.:Prentice Hall. I977. In addition, see ARIT.R.. E f e m e t t ~ nChemicnl Rrrrcror Ann(v3i.r. Upper Saddle River. N.1.: Prentice Hall. 1969. Chap. 6. One should f n d the references listed at the end of this reading particularly useful. Lvss. D., "Diffusion-Reaction Interactions in Catalyst Pellets," p. 239 in ChernicnI Rerrcrion and Rerrcror Err,yineeritrg. New York: Marcel Dekker. 1987. The effects o f mass transfer on reactor performance are also discussed in DENBIGH, K.. and J. C. R. TURNER, Chemical Reactor Theov. 3rd ed. Cambridge: Cambridge Univer~ityPress. 1984. Chap. 7. SATERFIELD, C . N . , Heremgcneous Catnivsis in Industrinl Pmcrice, 2nd ed. New York: McGraw-Hill, 199 1 .
2. Diffusion with homogeneous reaction is discussed in ASTARITA, G.,and R. OCONE,Speci~rlTopics in Transport Phenomena. New York: Elsevier, 2002.
DANCKWERTS. P.V.. Ens-Liqrtid Rencrions. New York: McGraw-Hill. 1970. Gas-liquid reactor design is also discussed in CHARPG~E J. RC., , review article. Trans. lnst. Chem. Eng,60, 131 (L982). S H A HY . .T., Gas-Liquid-Solid Rencror Design. New York: McGmw-Hill. 1979.
3. Modeling of CVD reactors is discussed in HESS, D.W.. K. F. JENSEA,and T. I. ANDERSON,"Chemical Vapor Deposition: A Chemicat Engineering Perspective." Rev. Chrm. Eng.. 3, 97, 1985. JENSEN, K. E, "Modeling of Chemical Vapor Deposition Reactors for the Fabrication of Microelectronic Devices," Clternicul and Cntal,vric Rencror Modeling. ACS Syrnp. Ser. 237. M. F. Dudokavic, P.L. Mills. eds., Washington, D.C.: American Chemical Society. 1984. p. 197. LEE, Id. H.. fundamm~nis of ,Wicroelectronics Pmcessing . New York: McGnw-Hill. 1990. 4. Multiphase reactors are discussed in
R~UACHANDRAN, P.A.. and R. V. CHAUDHARI.Three-Plrase Caraiyfic Reactors. New York: Gordon and Breach, 1983. RODRIG~~ES, A. E..J. M. COLO,and N. H. SWEED. eds.. Mulriphnse R~actorx, Vol. I : F~ndDme??~~ljs. Alphen aan den Rijn, The Netherlands: Sitjhoff and Noordhoff, 198 1. RODRIGUES. A. E., J. M. COLO.and N. H . SWEED, eds.. Muftiphase Renctors, Val. 1: Design Metitoris. Alphen aan den Rijn. The Netherlands: Sitjhoff and Nmrdhoff. 198 1.
866
Diffusion and Reaction
Chap. 12
SHAH. Y. T.. 8 . G. KELKAR, S. P. GODBOLE,and W. D. DECKWEK"Design Parameters Estimations for Bubble Column Reactors" (journal review), AIChE J.. 28, 353 (1982). TARHAN. M.O.,Catalytic Reactor Design. New York: McGraw-Hill, 1983. YATES,J. G.,Fundamenrals of Fluidized-Bed Chemical Processes, 3rd ed. London: Butternorth, 1983. The following Advances in Chemistry Series volume discusses a number of mul-
tiphase reactors:
F'OGLERH, S., ed., Chernical Reactors. ACS Syrnp. Ser. 168. Washington. D.C.: Arnencan Chemical Society, 198 1, pp. 3-255. 5. Fluidization
In addition to Kunii and Levenspiel's book, many comlations can be found in DAV~DSON, J. F., R. CLIFF,and D. HARRISON, Fluidization, 2nd ed. Orlando: Academic Press, 1985. A discussion of the different models can be found in
YATES.I. G.. Fluidizad Bed Chemical Processes. London: Butterworth-Heinemann, 1983. Also qee GELDART, D. ed. Gas Fluidi:ation Technolop. Chichester: Wiley-Interscience, 1986.
Distributions of Residence Times for Chemical Reactors
13
Nothing in life is to be feared. It is only to be understood. Marie Curie
In this chapter we 'learn about nonideal reactors, that is, reactors that do not follow the models we have developed for ideal CSTRs, PFRs, and PBRs. I n Parl 1 we describe how to characterize these nonideal reactors using the residence time distribution function E(t), the mean residence time t,, the cumulative distribution function F(t), and the variance a2.Next we evaluate Elf), E(r), t,, and aZfor ideal reactors, so that we have a reference point as to how far our real (i-e., nonideal) reactor is off the norm from an ideal reactor. The functions E(t) and F ( t ) will be developed for ideal PPRs, CSTRs and lamjnar flow reactors. Examples are given for diagnosing problems with reaI reactors by comparing t, and E ( f ) with ideal reactors. We will then use these ideal curves to help diagnose and troubleshoot bypassing and dead volume in reaI reactors. In Part 2 we will leam how to use the residence time data and functions to make predictions of conversion and exit concenmtions. Because the sesidence time distribution is not unique for a given reaction system, we must use new mcdels if we want to predict the conversion in om nonjdeal reactor. We present the five most common models to predict conversion and then close the chapter by applying two of these models, the segregation model and the maximum mixedness model, to single and to multiple reactions. After studying this chapter the reader will be able to describe the cumulative F(t) and external age i 3 t ) and residence-time distrihtion functions, and to recognize these functions for PFR, CSTR,and laminar flow reactors. The reader will also be able to apply these functions to calculate the conversion and concentrations exiting a reactor using the segregation model and the maximum mixedness model for both single and multiple reactions. Overview
Distributions of Residenc$Times !or Chemical Reactor7
868
Chzp. 1
13.1 General Characteristics
We want to analyze and characterize nonrdeal reactor behavior.
The reactors treated in the book thus far-the perfectly mixed batch. th plug-flow tubular, the packed bed, and the perfect1y mixed continuous tan reactors-have been modeled as fdeal reactors. Unfortunately, in the real worl we often observe behavior very diflerent from that expected from the exem plar: this behavior is true of students. engineers. college professors. and chem ical reactors. Jusl as we mug[ learn to work with people who are not perfec~ so the reactor analyst must learn to diagnose and handle chemical reactor whose performance deviates from the ideal. Nonideal reactors and the princi ptes behind their analysis form the subject of this chapter and the next. J
Part 1 The basic ideas s
characterization and Diagnostics
d
t ;ue used in the distribution of residence times to charac terize and moddnonideal reactions are really few in number. The two majo,
uses of the residence time distribution to characterize nonideal reactors are
I. To diagnose problems of reactors in operation 2. To predict conversion or effluent concentrations in existinglavaiIab1~ reactors when a new reaction is used in the reactor
In a gas-liquid continuous-stirred tank reactor (Figure 13-I), the gaseous reactant was bubbled into the reactor while the Iiquid reactant was fed through an inlet tube in the reactor's side. The reaction took place at the gas-liquid interface of the bubbles, and the product was a liquid. The continuous liquid phase couId be regarded as perfectly mixed, and the reaction rate was proportional to the total bubble surface area. The surface area of a particular bubble depended on the time it had spent in the reactor. Because of their different sizes, some gas bubbles escaped from the reactor almost immediately, while others spent so much time in the reactor that they were almost comSystem 1
I Gas AFigure 13-1 Gas-liquid reacror.
Sec. 13.1
~ o at1t molecutes vending the same ttme in the
,,
General Characteristics
869
pletely consumed. The time the bubble spends in the reactor is termed the bltbble reside~zcetitne. What was important in the analysis of this reactor was not the average residence time of the bubbles but rather the residence time of each bubble (i.e., the residence time distribution). The total reaction rate was found by summing over all the bubbles in the reactor. For this sum. the distribution of residence times of the bubbles leaving the reactor was required. An understanding of residence-time distributions (RTDsf aad their effects on chemical reactor performance is tlws one of the necessities o f the technically competent reactor analyst,
System 2 A packed-bed reactor is shown in Figure 13-2. When a reactor is packed with catalyst, the reacting fluid usually does not flow through the reactor uniformly. Rather, there may be sections in the packed bed that offer little resistance to flow, and as a result a major portion of the fluid may channel through thrs pathway. Consequenrly, the molecuIes following this pathway do not spend as much time in the reactor as those flowing through the regions of high resistance to flow. We see that there is a distribution of times that molecules spend in the reactor in contact with the catalyst.
Figure 13-2 Packed-tred reactor.
System 3 In many continuous-stirred tank reactors, the inIet and outlet pipes are close together (Figure 13-3). In one operation it was desired to scale up pitar plant results to a much larger system. It was realized that some short circuiting wcurred, so the tanks were modeled as perfectly mixed CSTRs with a bypass stream. In addition to short circuiting, stagnant regions (dead zones) are often encountered. In these regions there is little or no exchange of material with the well-mixed regions. and. consequently, virtually no reaction occurs
We want to find ways of
determining the dead volume and amount of bypassing.
Dead zone
Figure 13-3 CSTR.
870
Tfie three concepts * RTD * Mixing
Model
D~str~but~ons of R e s l d e m Tomes tor Chernml Reaclors
Chap 13
there. Experiments were carried out to determine the amount of the material effectively bypassed and the volume of the dead zone. A simple modification of an ideal reactor successfully modeled the essential physical characteristics of the system and the equations were readily solvable. Three concepts were used to describe nonideal reactors in these examples: the disrribution uf residence time$ in the sysrem, rhe quolip of mixing, and the model used to describe the system. All three of these concepts are considered when describing deviations from the mixing patterns assumed in ideal reactors. The three concepts can be regarded as characteristics of the mixing in nonideal reactors. One way to order our thinking on nonideal reactors is to consider modeling the flow patterns in our reactors as either CSTRs or PFRs as afirsr approximation. In real reactors, however, nonideal flow patterns exist. resulting in ineffective contacting and lower conversions than in the case of ideal reactors. WE must have a method of accounting for this nonideality. and ro achieve this goal we use the next-higher level of approximation, which involves the use of macromixing information (RTD) (Sections 13.1 to 13.4). The next level uses microscale (micromixi~tg)information to make: predictions about the conversion in nonideal reaclors. We address this third level of approximation in Sections 13.6 to 13.9 and in Chapter 14. 13.1.1 Residence-Time Distribution (RTD) Function
The idea of using the distribution of residence times in the analysis of chemical reactor performance was apparently first proposed in a pioneering paper by MacMulIia and Weber.' However, the concept did not appear to be used extensively until the early 1950s. when Prof. P. V. Danckwerts2 gave organizational structure to the subject of RTD by defining most of the distributions of interest. The ever-increasing amount of literature on this topic since then has generally foIlowed the nomenclature of Danckwerts, and this will be done here as well. In an ideal plug-flow reactor. all the atoms of material leaving the reactor have been inside i t for exactly the same amount of time. Similarly. in an ideal batch reactor, all the atoms of materials within the reactor have becn inside it for an identical lenph o f time. The time the atoms have spent in the reactor is called the residence tirne of the atoms in the reactor. The idealized plug-flow and batch reactors are the only two classes of reactors in which all the atoms in the reactors have the same residence time. In all other reactor types, the various arorns in the feed spend different times inside the reactor; that is, there is a distribution of residence times of the marerial within the reactor. For example. consider the CSTR; the feed introduced into a CSTR at any given lime becomes completely mixed with the material already in the reactor. En other words. some of the atoms entering the CSTR
I
R. &. MacMullin and M. Wehcr. Jr.. Trullr. Am. Insr.
?
P.y. Danck~ens,Cl~rtli.En8. Sri.. 2. 1 4 1953).
Cltrrr~ D t g . . 31. 309 ( 1935).
Sec. 13.2
Measurement of itre RTD
871
leave it almost immediately because material is being continuousIy withdrawn from the reactor; other atoms remain in the reactor almost forever because all the material is never removed from the reactor at one time. Many of the atoms. of course, leave the reaclor after spending a period of time somewhere in the welcome. vicinity of the mean residence time. In any reactor, the distribution of e s i dence times can significantly affect its performance. The residence-tinre distribution (RTD) of a reactor is a characteristic of the mixing that occurs in the chemical reactor. There is no axial mixing in a We will use the plug-Row reactor. and this omission is reflected in the RTD. The CSTR is thorRTD to oughly mixed and possesses a far different kind of RTD than the plug-flow characteri~e ~ ares unique to a particular reacnonideal reacton. reactor. As will be illustrated Inter. not all R tor type: markedly different reactors can display identical RTDs. Nevertheless. the RTD exhihited by a given reactor yields distinctive clues to the type of mixing occurring within il wd is one of the most informative characterizations of the reactor.
The "RTD": Some molecules Fwve quickly, orhcrs oversmy their
13.2 Measurement of the RTD The RTD is determined experimentally by injecting an inert chemical, molecule. or atom. calIed a rr-acer. into the reactor a1 some time f = 0 and then ineasuting rhe rracer concentration. C, in the effluent stream as a function of time. In addition to being a nonreactive species that is easily detectable, the tracer should have physical properties similar 10 those of the reacting mixture and be completely soluble in the mixture. Et also should not adsorb on the walls or other surfaces in the reactor. The latter requirements are needed so that the tracer's behavior will honestly reflect that of the material flowing through the reactor. Colored and radioactive materials along with inen gases Use of tracers 10 determine the R m are the rnost common types of tracers. The two most used methods of injection are prrlsc ilrpu? and .TIP]?i ~ ~ p u f . 13.2.1 Pulse Input Experiment
~h~
c cllr\e
In a pulse input, an amount of tracer A', is suddenly injected in one shor into the feedstream entering the reactor In as short a time as possible. The outlet concentration is then measured as a funct~enof time. Typical concentmtion-time curves at the inlet and outlet of an arbitrary reactor are shown in Figure 13-4. The effluent concentration-time cun7e is referred to as the C curve In RTD analysis. We shall analyze the injection of a tracer pulse for a single-input and single-output system in which only f70w (i.e., no dispersion) carries the tracer material acrosq system boundaries. First, we choose an increment of time Ar sufficiently small that the concentration of tracer. C(r). exiting between time 1 and ! + A1 is ersentially the rame. The amount of trnccr matei i u l . Ah'. leaving the reactor between time r and r + Ar is then
872
Distributions of Res~denceTimesfor Ghernicai Reactors
Chap 1:
where u is the effluent volumetric flow rate. In other words. AN is the anoun of material exiting the reactor that has spent an amount of time between t an[ r At in the reactor. If we now divide by the total amount of material that was
+
injected into the reactor. No.we obtain
which represents the fraction of material that has a residence time in the reactor between time t and r + At.
For pulse injection we define
so that
The quantity E(t) is called the residence-time distri'brifioonfunction. It is the function that describes in a quantitative manner how much time different fluid elements have spent in the reactor. The quantity E{t)(it is the fraction of fluid exiting rhe reactor that has spent between time f and r + db inside the reactor.
+@--
Reactor
Pulse injection
Pulse response I
I
Step i!jection
/
I
Step response
Figure 13-4 RTD measurements.
Sec. 13.2
The C curve
If No is not known directly, it can be obtained from the outlet concentration measurements by summing up all the amounts of materials, AN.between time equal to zero and infinity. Writing Equation ( 1 3- 1 ) in differential form yieIds
t
djV = uC(t)dr
/13-5)
lx
(1 3-61
CUI
=lo
h
C(f)dt
~rea
COI
873
Measuremenl of the RTD
t
We find the RTD
and then integrating.
we
obtain N, =
u ~ ( id)
0
The volumetric flow rate v is usuaIly constant, so we can define E ( t ) as
function. E f t ) . from bhe tracer concentration
( 1 3-7)
C(1)
The E curve
t
The iniegral in the denorinrtor is the area under the C curve, An alternative way of interpreting the residence-time function is in its integral form:
Fraction of material leaving the that has resided in the reactor for times between r , and t2 We know that the fraction of a13 the material that ha.. resided for a time r in ithe reactor between t = 0 and t = TJ is 1: therefore, Eventually aIl must leave
The following example will show haw we can calculate and interpret E(t) from the efRuent concentrations from the response to a pulse tMCer input to a reaI (i.e., nonideaI) reactor. Example 13-1
Conshucting the C(t) and E(t) CUNCS
A sample of the tracer hytane at 320 K was injected as a pulse to a reactor, and the efRuent concentration was measured as a function of time. ~ s u l t i n gin the data shown in Table E13-t -1.
Pulse Input
The measurements represent the exact concentrations at the times listed and not average values between the various sampling tests. ( a } Construct figures showing C(r) and E(t) as functions o f time. (b) Determine both the fraction of rnatenal leaving
874
Distriberiions of ResidenceTimes lor Chemical Reactors
Chap. 13
the reactor that has spent between 3 and 6 min in the reactor and the fraction of material leav~ngthat bas spent between 7.75 and 8.25 min In the reactor, and (c) determine the fraction of material leaving the reactor that has spent 3 min or less in zhc reactor. Solurion
(a) By plotting C as a function of time. using the daza in Table El 3-1 .l. the curve shown in Figure El>-1.1 is obtained. To obtain the E ( r ) curve from the C(r) curve. we just divide
C ( r )by rhe integral
-0
C ( i ) dt .
The C curve
Figum E13-1.1 The C curve.
which is just the area under the C curve. Becaufe one quadrature (integration) Formula will not suffice over the entire range of data in Table E 13- I . I . we break the data inlo two regions. 0-10 minutes and 10 ta 14 minutes. The area under the C curve can now be found using the numerical integratron forn~ulas(A-31 ) and (A-25) in Appendix A.4:
+2(10>+41R)+2(6) +4(4)+2(3O)+412.2lr = 47.4 g - min !m7
l(1.513
Sec. 13.2
Measurement of the RTD
I We now calculate
with the folFowing results: TAHLE E13-1.2 C(t) AND E(t) I
(min)
1
2
1
3
4
5
6
7
8
9
10
1 2 1 4
(b) These data are plotted in Figure E13-1.2. The shaded area represents the fraction of material leaving the reactor that has resided in the reactor between 3 and 6 min.
Tne E curve
Figure E13-1.2 Analyzing the
E curve.
Using Equation (A-22) in Appendix A.4:
E (f) dr
= shaded area
3
=
]AfV,+ 3 , 1 ; + 3 S , + f , j
Evaluating this area. we find that 51% of the material leaving the reactor spends between 3 and 6 rnin in the reactor. Because the time between 7.75 and 8.25 rnin is very smalf relative to a time scale of 14 min. we shall use an alternative technique to determine this fraction to reinforce the inrerprelation of the quantity Elr)dt. The averaye value of E ( r ) between these times 1s 0.06 min- :
The tail
Consequently. 3.0%of the fluid leaving the reactor has been in the reactor between 7.75 and 8.25 min. The long-time portion of the Elt) curve is called the toil. In this example the tail IS that ponion of the curve between sny 10 and 14 min.
876
DIstrrbutians of Resrdence T!r-es tor Chemical aeaflors
Chap 1:
(c) Finally, we shall consider the fraction of mater~aithat has been in the rent tor for a time !or less. hat is. the fraction that has y e n t between O and I minute> rn the reactor. Thiq fraction i s just the rhaded area under the curve up to I = I min. uter. This area is qhomn in Figure E13-1.3 for r = 3 mln. Calculating the area undel the cune, we see that 20%-of the material has spent 3 min or 1e.r.~in the reactor.
0
1 2 3 4 5 6 7 8 91011121314 ! (min)
I
Drawbacks ur the Puix injection '0 obtain the RTD
Figure E13-1.3 Analyzing the E curve.
Tbe principal difficulties with the pulse technique lie in the problem: connected with obtaining a reasonable pulse at a reactor's entrance. The injec tion must take place over a period which is very short compared with residenc~ times in various segments of the reactor or reactor system, and there must be : negligible amount of dispersion between the point of injection and the entranc~ to the reactor system. If these conditions can be fulfilled. this technique repre sents a simple and direct way of obtaining the RTD. There are probiems when the concentration-time curve has a long tai because the analysis can be subject to larse inaccuracies. This probIem princi pally affects the denominator of the right-hand side of Equation (13-7) 1i.e the integration of the C(t) curve]. It is desirable to extrapolate the tail and ana lytically continue the calculation. The tail of the curve may sometimes b approximated as an exponential decay. The inaccuracies introduced by thi assumption are very likely to be much less than those resulting from eithe truncation or numerical imprecision in this region. Methods of fitting the tal are described in the Professional Reference Shetf -13R. 1. 13.2.2 Step Tracer Experiment Now that we have an understanding of the meaning of the RTD curve from pulse input, we will formulate a more general relationship between time-varying tracer injection and the corresponding concentration in the e H t ent. We shall state without development that the output concentration from vessel is related to the input concentration by the convolution integral:3 A development can be found in 0,Levenspiel, Chemical Reaction Engineering, 2n ed. (New York: Wiley, 1972). p. 263.
Sec. '3.2
Step input =I"
Ir
The inlet concentration most oAen takes the form of either a perfect pulse irtpitl (Dirac detta function). impetjfectp~ilwitjecrion (see Figure 13-41. or a step inplit. Just as rhe RTD function E(t1 can be determined directly fmm a pulse input, the cumulative distribution F(t) can be determined directly from a step input. We will now anaryze a srcp itlplit in the tracer concentration for a system with a constant volumetric flow rate. Consider u constant race of tracer addition to a feed that is initiated at time r = 0. Before this time no tracer was added to the feed. Stated symbolically, we have CoIO =
-- ----
[IJ
t
877
Measurement of the RTD
t
(C,) constant
r20
The concentration of tracer in the feed to the reactor is kept at this level until the concentration in the effluent is indistinguishable: from that in the feed; the test may then be discontinued. A typical outlet concentration curve for this type of input is shown in Figure 13-4. Because the inlet concentration is a constant with time, Cn.we can take it outside the integral sign, that is,
Dividing by Co yieIds =
E(r') dt' = F ( r ) (13-10)
I
We differentiate this expression to obtain the RTD function E ( t ) :
Advantages and drawbacks to the step injection
The positive step is usually easier to carry out experimentally than the pulse test, and it has the additional advantage that the total amount of tracer in the feed over the period of the test does not have to be known as it does in the pulse test. One possible drawback in this technique is that it is sometimes difficult to maintain a constant tracer concentration in the feed. Obtaining the RTD from this test also involves differentiation of the data and presents an additional and probably more serious drawback to the technique, because differentiation of data can, on occasion, lead to large errors. A third problem lies with the large amount of tracer required for this test. If the tracer is Very expensive. a pulse tesr is almost always used to minimize the cost.
878
Dis!ribv?ions of Residerrce Times !or Chemlca?Reac!ors
Chap. 13
Other tracer techniques exist, such as negative step ( i t . , elution), frequency-response methods, and methods that use inputs other than steps or pulses. These methods are usually much more difficult to carry out than the ones presented and are not encountered as often. For this reason [hey will not be treated here, and the literature should be consulted for their virtues, defects. and the details of implementing them and analyzing the results. A good source for this information is Wen and
53.3 Characteristics of the RTD From E ( r ) we can learn how long d~tferentmolecules in the have reaclnr.
Sornetinles E(r) is called the exif-a~e disrriburion .functiu~~. If we regard the "age*' of an atom as the time it has resided in the reaction environment. then E { t ) concerns the age distribution of the effluent stream. I t is the most used of the distribution functions connected with reactor analysis because it characterizes Ihe lengrhs of rime various atoms spend at reaction conditions.
13.3.1 Integral Relationships The fraction of the exit stream that has resided in the reactor for a period of time shorter than a given value r is equal to the sum over all times less than r of E ( r ) A!. or expressed continuously. Fraction of effluent that has been in for less than time r
The cumulative RTD funcuon Flr )
(13-12)
Analogous1y, we have
1
Fraction of effluent that has been in reactor = I - F ( r ) for longer than time t
(13-13)
Because r appears in the integration limits of these two expressions, Equations (13-1 2) and ( 13-1 3 ) are both functions of time. Danckwerts' defined Equation (13-12) as a curnularive dirtribution function and called if F ( I ) . We can calculate F ( t ) at various times r from the area under the curve of an E(r) versus r plot. For example, in Figure El 3-1.3 we saw that F(r j at 3 rnin a a s 0.20, meaning that 20% of the molecules spent 3 rnin or less in the teanor. The shape of the F ( i ) curve is sh6wn. Similarly, using Figure El 3-1.3 we calculate F(r) = 0.4 a1 4 minutes. We can continue in this manner to construct Fir). in Figure 13-5. One notes from this curve that 80% [ F ( r ) ]of the molecules spend 8 rnin or less in the reactor, and 20% of the molecules 11 - F(t)l spend longer than 8 rnin i n the reaaor.
C.Y. Wen
and L. T. Fan. Mr~riEls,fir F!onq .T~.I.IP~II,Tand Chen~irnlRt~artor.~ (Ficu York: Marcel Dekkcr. 1975). P V. Danckuens. Cltem. Eng. Sci., 2. 1 ( 1953).
Sec. 13.3
Characteristics of the RTD
The F curve
Vt)
a
t
(min)
Figure 13-5 Cumulative distribution curve. F ( t )
The F curve is another function that has been defined as the normalized response to a particular input. Alternatively, Equation (1 3-12) has been used as a definition of TI[),and it has been stated that as a resutt it can be obtained as the response to a positive-step zracer test. Sometimes the F curve is used in the same manner as the RTD in the modeling of chemical reactors. An excellent example is the study of Wolf and White.6 who investjgated the behavior of screw extruders in polymerization processes. 13.3.2 Mean Residence Time
In previous chapters treating ideal reactors, a parameter frequently used was the space time or average residence time 2. which was defined as being equal to VJU.It will be shown that, in the absence of dispersion, and for constant volumetric flow (D = vo) no matter what RTD exists for a particular reactor, ideal or nonideal, this nominal space time, T, is equal to the mean residence time. t,. As is the case with other variables described by distribution functions, h e mean value of the variabIe is equal to the first moment of the R T D function, E ( t ) . Thus the mean residence time is The first moment gives the average time the effluent molecules spent i n Ihe reactor.
We now wish to show how we can determine the total reactor volume using the cumulative distribution function.
6
D. Wolf and D.H.While, AlChE J.. 22, 122 (1976).
Distributtons of Res~dencef ~ m e sfor Chem~mlReactors
880
Chaa 1:
What we are going to do now is prove t, = T for constant volumetric flow u = vo. You can skip what ~01lowsand go directly to Equation (13-21) i
you can accept this result.
Consider the following situation: We have a reactor completely filled witi maize molecules. At time t = 0 we start to inject blue molecules to replace thr maize molecules that currently f i I I the reactor. IniriaIly, the reactor volume I/ i: equal to the volume occupied by the maize molecules. Now. in a time dr. tht volume of rnolecuIes that will leave the reactor is Ivdt). The fraction of thesr molecules that have been in the reactor a time t or greater is [ I - F ( t ) ] Because only the maize molecules have been in the reactor a time t or greater the volume of maize molecules, d V , leaving the reactor in a time dr is
If we now sum up all of the maize molecules that have left the reactor in time All we are doing here is proving thm the space time and mean residence time are equal.
h
0<
I
<
=,we have
v=
1' n
u [ l -F(r)]dr
(L3-16)
Because 'the volumetric flow rate is constant.*
Using the integration-by-parts relationship gives j
~
d
~
I=v d~r -
and dividing by the volumetric Aow rate gives
t
- F ( r ) ] = 0.The first term on the right-hand side is zero, and the second term becomes
At t = 0, F ( t ) = 0;and as t -+ x , then El
However, dF = E ( I ) dt; therefore,
The right-hand side is just the mean residence time, and residence time is just the space time t:
we see that the
mean
'Note: For gas-phase reactions at constant temperature and no pressure drop r, = T/(l EX).
+
Se. 13.3
End ot' p m f '
Characteristics of the FIT[?
and no change in volumetric flow rate. For gas-phase reactions, this means no pressure drop. isothermal operation, and no change in the total number of moles ( i t . , E = 0, as a result of reaction).
This result is true only for a closed sytenl (i.e., no disprsion across boundaries; see Chapter 14). The exact reactor volume is determined from the equation
13.3.3 Other Moments of the RTD It i s very common to compare RTDs by using their rnomen ts instead of trying to compare their entire distributions (e.g., Wen and Fan7). For this purpose, three moments are normally used. The tirst is the mean residence time. The second moment cornmonty used is taken about the mean and is called the variance, or square of the standard deviation. It is defined by The second mt~mentabout the mean is the variance.
The magnitude of this moment is an indication of the "spread" of the distribution: the greater the value of this moment is, the greater a distribution's spread will be. The third moment is also taken about the mean and is related to the skewness. The skewness is defined by The two parameters most commonly
used
to
characterize the
RTD art: r and v L
The magnitude of this moment measures the extent that a distribution is skewed in one direction or another in reference to the mean. Rigorously. for complete description of a distribution, a11 moments must be determined. Practically. these three are usually sufficient far a reasonable characterization of an RTD. Example 13-2
Mean Residence Erne and Vuriunce CalcuEations
Calculate the mean residence time and the variance for the reactor characterized in Example 13-1 by the RTD obtained from a pulse input at 320 K.
I
First, the mean residence time will be calculated from Equation (13-14):
C.Y. Wen and L. T. Fan, Models for Flow Sysrems and ChemicaI Rmctors {New York Decker. 1975). Chap. 1 1.
882
Distributions of Residence Times for Chemical Reatdors
Chap. 13
The area under the cum of a plot of sE(t) as a function oft wilI yield I,. Once the mean residence time is determined, the variance can be calculated from Equation ( I 3-23):
To calculate t,,, and u2,Table E13-2.1was constructed from rhe data given and interpreted in Example 13-1. One quadrature formula will not suffice aver the entire range. Therefore. we break the integral up into two regions, 0 to 10 min and 10 to 14 (minutes), i.e., infinity (.I).
Starting with Table E13-1.2in Example 13-1, we can proceed to calculate tE(1). (r-~)and(r-lm)2flr)sndZE(,)showninTablleE13-21.
1
1
o 0.02
2 3
5
0.10
0 0.02 0.20
8 10
0.16
0.48
0.20
R
0.16 0.12 0.08
0.80 0.80
O
4
5 6 7
f
l
6 4
4
3 2.2
10 12 I4
1.5 0.6 0
8
0.72 0.56
-5.15
0
-4.15
0.34 0.991 0.74 0.265
0 0.02 0.4 1 .44 3.2
0.004
4.0
0.087
4.32 3.92
-3.l~ -2.15 -1.15 -0.15 0.85 1.85 2.85
0.06 0.044 0.03 0.012
0.48 0 40 0.30 0.14
6.85
0
0
8.85
3.85 4.85
0.274 0.487 0.652 0.706 0.563 0
=These two columns are completed after the mean residence time
3.84
3.56 30 1.73 0 (I,)
is found.
Again, using the numerical integration formula$ (A-25) and (A-2 1) in Appndix A.4. we have
Yumerical integration to find the mean residence time. r, = 4.58
+ 0.573 = 5.15 min
Sec. 13.3
Calculating the mean residence T
=
,j,' =
ume,
883
Chamctertstics of the RTD
Note: One could also use the spreadsheets in Polymath or Excel to formulate Table E13-2.1 and to calculate the mean residence time t, and variance o.
rE{t)dr
t (rnin)
Figure E13-2.1 Calculating the mean residence time.
Platting t E ( t ) versus t we obtain Figure E13-2.1. The area under the curve i s 5.15 rnin. It,,, =
Calculating the variance.
5.15 mi111
Now that the mean residence time has been determined. we can calculate the variance by calculating the area under the curve of a plot of (t - !,,J2E{t)as a function of I (Figure El 3-2.2[a]). The area under the curve{s) is 6.11 min?.
One could also use Polymath or Excel lo make these
calculations. Fimre E13-2.2 Czlculating the variance.
Expanding quare term
We will use quadrature formula5 to evaluate the integral using the data (columns 1 and 7) in Tahk El3-2.1. Integrating ktween 1 and 10 minutes and 10 and 14 minutes using the same form as Equation (E13-2.3)
884
Distributions of Residence Times for Chemical Reactors
C h a ~r
= 32.7 1 rnin2 This value is atso the shaded area under the curve in Figure E13-2,2(b).
t ~ ( r ) d-t t: = 32.7 1 min2- (5.15 rnin)' = 6.19 minI
o2= 0
(
The square of the standard deviation is n2= 6.19 mi$, so cr = 2.49 rnin.
13.3.4 Normalized RTD Function, E(O) Frequently, a normalized RTD is used instead of the function E(r). If the parameter B is defined as
a dimensionless function
W h y we use a norrna1izea RTD
E ( I ) for n CSTR
E(O) can be defined as
and plotted as a function of @. The quantity @ represents the number of reactor volumes of fluid based on entrance conditions that have flowed through the reactor in time r. The purpose of creating this normalized distribution function is that the flow performance inside reactors of different sizes can be compared directly. For example, if the normalized function E(O) is used. all perfectly mixed CSTRs have numerically the same RTD. If the simple function E ( t ) is used, numerical values of E ( t ) can differ substantialIy for different CSTRs. As will be shown later, for a perfectly mixed CSIX.
-
From these equations it can be seen that the value of E ( t ) at identical times can be quite different for two different volumetric flow rates, say u , and v2. But for
Sec. 13.4
E(R)
885
RTD in Ideal 9eactc:s
the same value of @, the value of E(5.)) is [he same irrespective of the a perfectly mixed CSTR. It is a relatively easy exercise to show that
site
of
and is recommended as a 93-s divertissement. 13.3.5 Internal-Age Distribution,
Tomhstr)nr jail How long have you
k e n here? Ira) When do you expect to pet out?
41)
Although this section is not a prerequisite to the remaining sections, the internal-age distribufion is introduced here because of its close analogy to the external-age distribution. We shall let a represent the age of a molecule inside the reactor. The internal-age distribution function [(a)is a function such that I(a)dt-ris the fraction of material inside the rerrctor that has been inside the reactor for a period of time between tr and cr + dol. It may be contrasted with E(a)Aat, which is used to represent the material leaving the reucror that has spent a time between cc and a -t bor in the reaction zone; I(a) characterizes the time the material has been (and still is) in the reactor at a partici!lar time. The function E(a) is viewed outside: rhe reactor and [(a)is viewed inside the reactor. In unsteadystate problems it can be important to know what the particular state of a reaction mixture is, and I(a) supplies this information. For example, in a catalytic reaction using a catalyst whose activity decays with time. the internal age distribution of the catalyst in the reactor I(a)is of impottance and can be of use in rnodeting the reactor. The internal-age distribution is discussed further on the Professional Reference Shelf where the foIlowing relationships between the cumuiative internal age distribution !(a)and the cumuEative external age distribution F(a)
Reference Shelf
and between E(t) and I(t)
are derived. For a CSTR it is shown that.the internal ase distribution function is
13.4 RTD in Ideal Reactors 13.4.1 RTDs in Batch-and Plug-Flow Reactors
The RTDs in plug-flow reactors and ideal batch reactors are the simplest to consider. AIl the atoms leaving such reactors have spent precisely the same
886
Distributions of Residence nmes for Chemical Reactors
Chap. 13
amount of time within the reactors. The distribution function in such a case is a spike of infinite height and zero width, whose area is qua1 to 1 : the spike occurs at f = Vlv = T , or @ = 1. Mathematically, this spike is represented by the Dirac delta function:
1-1
E(r ) for a plugRow reactor
(1 3-32)
The Dirac delta function ha5 the following properties:
when x # 0 when x = O
s (x) = 0 x
Properties of the
Dimc delta function
To calculate z the mean residence time. we set g(x) =
,,, =
)
d
=
t
ts(r - r ) d t = i
0
But we already knew this result. To calculate the variance we set, g(t) = ( t - zI2, and the variance, 02,is
All material spends exactly a time T in the reactor, there is no variance! The cumulative distribution function F(1) is
F(r)=
jt E ( t ) d r = 1' 0
S(r -rMr
0
The E(t) function is shown in Figure 13-6(a), and F(r) is shown in Figure 13-S(b).
Fig~re13-6 Ideal plug-flow response tn a pulse tracer input
Sec. 13.4
887
RTD m ideal Reactors
13.4.2 Single-CSTR RTD From
a tracer halance we can
In an ideal CSTR he concentration of any substance in the effluent stream is identical to the concentration throughout the reactor. Consequently. it is possible obtain the RTD fmrn conceptual considerations in a fairly straighrforward manner. A material balance on an inen tracer that has been injected as a pulse at time r = 0 into a CSTR yields for t > 0
fn - Out
= Accumulation
Because the reactor i s perfectly mixed, C in this equnlion i q the concentration of the tracer either in the effluenr or within the reaclor. Separating the variables and integrating with C = Co at r = U yield< Cfr) = C , , P - " ~
(
13-39)
This relationship gives the concentration of tracer in the effluent at any time r. To find E ( r ) for an ideal CSTR, apefirst recall Equalion (13-7) and then substitute for C ( r ) uring Equation (1.1-39).That is.
Evaluatlne the integral in the denominator completes rhe denivationof he RTD for an ideal CSTR given by Equations ( 1 3-27) and ( 13-28):
E ( I ) and E[O) Ibr n CSTR
Recall fi = r / r and E ( @ ) = 'rE(r1. I
Reymnse of an ideal CSTR
/
7°C
'---- : -.'@
g
F
o
10
I-I
..-..........------------------
o
r 4.1
ra!
~ h l
ct
Distributions of Residence Times for Chemical Reactors
888
Chap+
The cumulative distribution F(O)is
1 -
It
F(@) =
E((0).10 =I-P-.
rl
The Ere) and F(O)functions for an Ideal CSTR are shown in Figure 13-7 (a) a Ib). respectively. Earlier it was shown that For a constant volumetric flow rate. the mean R dence time in a reactor is equal to Mu , or T. This relationship can be shown in a sil pler fashion for h e CSTR. Applying the delinition of a mean residence time to t liTD for a CSTR. we obtain
Thus the nominal holding time (space time) r = V l v is also the mean re: dence time that the material spends in the reactor. The second moment about the mean is a measure of the spread of tl distribution about the mean. The variance of residence times in a perfect mixed tank reactor is (let x = t/z) For n perfectly mixed CSTR: r, = r end u = T.
Then v = 2. The standard deviation is the square root of the variance. For CSTR. the standard deviation of the residence-time distribution is as large : the mean itself!!
13.4.3 Laminar Flow Reactor (LFR)
Before proceeding to show how the RTD can be used to estimate conversion i a reactor. we shalt derive E ( t ) for a laminar Bow reactor. For Iamina~flow in tuhular reactor, the velocity profile is parabolic, with the fluid in the center c the tube spending the shortest time in the reactor. A schematic diagram of th fluid movement after a time r is shown jn Figure 13-8. The figure at the let shows how far down the reactor each concentric ff uid elemeot has travelel after a time r. MoIecules near the center spend a shorter time in the reactor than those cIo~eto the wall. I
-
Figure 13-8 Schematic diagram of Ruid elements in a laminar flow reactor.
:u
The velocity profile in a pipe of outer radius R is
u
U
Parabolic Velocity Proflle
Ssc. 13.4
889
RTD ~nIdeal Reactors
where U,,, is the centerline velocity and U,,,is the average velocity through the tube. II,,, i s just the volumetric flow rate divided by the crou-sectional
area.
The time of passage of an element of Ruid at a radius r is
The volumetric Row rate of fluid out between r and ( r + dr), du, is du = U(r)Z ~ r d r
The fraction of total fluid passing between r and ( r + dr) is dv/uo. i.e.
The fraction of fluid between r and ( r + dr) that has a flow rare between IJ and ( v + du) spends a time between t and (r + dr) in the reactor is
We now need to relate the fluid fraction [Equation (1345)J to the fraction of fluid spending between time t and r -k dt in the reactor. First we differentiate Equation (13-43):
and then substirute for t using Equation (13-431 to yield
Combining Equations (€3-44) and (13-46). and then using Equation (13-43) for U(r), we now have the fraction of fluid spending between time r and t + dt in the reactor:
The minimum time the fluid may spend in the reactor is
D~stributronsot Resdsnce Times
for Chernlcal Reactors
Chap. 73
Consequently. the complete RTD function for a laminar flow reactor is
EC1 j for A 1:11ninar Row rc:lurt*r
The curnularive distribution function for
The mean residence time
i,,,
r
t / 2 is
is
For LFR I,,, = r
This recult u ; ~ sshown previou4y to be true for any reactor. The mean residence rlme is j u \ t the space time r. The dimensionles~form of the RTD function is Na~rn>nli~ed RTD
func~innfur ;I 1;11uinarflou red
and
1 5 plot~edin
Fipiirc. 13-9. The dimensionlr>+cumulative distribution. F(O) for 8 2 112. is
Sec.13.5
-
Diagnostics and Troubleshmtlng
891
Figure 13-9 (a) E(Q) for an LFR;fb) 40)for a PFR, CSTR, and LFR.
Figure 13-9(a) shows E(@) for a laminar flow reactor (LER),while Figure 9- 13tb) compares F(O) for a PFR, CSTR, and LFR. Experimentally injecting and measuring the tracer in a laminar flow reactor can be a difficult task if not a nightmare. For example, if one uses as a tracer chemicals that are photo-activated as they enter the reactor, the analysis and interpretation of E(t) from the data become much more i n ~ o f v e d . ~
13.5 Diagnostics and Troubleshooting 13.5.1 General Comments As discussed in Section 13.1, the RTD can be used to diagnose problems in existing reactors. As we will see in further derail in Chapter 14, the RTD functions E(r) and F(t) can be used to model the real reactor as combinations of
ideal reactors. Figure 13-10 illustrates typical RTDs resulting from different nonideal reactor situations. Figures 13-10(a) and (b) correspond to nearly ideal PFRs and CSTRs, respectively. In Figure 13-10(d) one observes that a principal peak occurs at a time smaller than the space time (T= V/v,) (i.e., early exit of fluid) and also that some fluid exits at a time greater than space-time r. This curve could be representative of the RTD for a packed-bed reactor with channeling and dead zones. A schematic of this situation is shown in Figure 13-10rc). Figure 13-10(f)shows the RTD for the nonideal CSTR in Figure 13-10(e), which has dead zones and bypassing. The dead zone serves to reduce the effective reactor volume, so the active reactor volume is smaller than expected.
8
D.Levenspiel. Chpmiral R~arrionEngineerfng. 3rd ad. (New York: Wlley. 1W ) ,p. 342.
892
Distributions of Residence T i r e s !or Chemfcal Reactors
1
j
Chap.
-
Ideal
RTDs that are uommonty observed
Gi O
\I
Dead Zones
r=L
-
Bypassing Long tail
/
Dead Zones
dead zone t
(1
I0 Figure 13-1U (a) RTD for near plug-Row reactor: (bl !CD for near perfectly mixed CSTR ( c ) Packed-bed reactor w ~ r hdead zones and channeling: (d) RTD for packed-bed reactor in (c); (e) tank reactur with short-c~rcuit~ng flow Ibypassl; (f) RTD for tank reactor with channeling (bypasskng or shon circuiting) and a dead zone in which the tracer slowly diRu%s in and out
13.5.2 Simple Diagnostics and Troubleshooting Using the RTD for ldeal Reactors 13.5.2A The CSTR
We will first consider a CSTR that operates (a) oormally, (b) with bypassin; and (c) with a dead volume. For a well-mixed CSTR, the moIe (mass) balanc on the tracer is
Rearranging, we have
Sec. 13.5
Diagnostics and TrouMeshootlng
We saw the response to a pulse tracer is
Concentration:
RTD Function: Cumulative Funcdon:
C(t) = ~ ~ ~ e - ' "
E [ ~=I !e-f'r T
F[t) = 1 - e-It'
where c' is the space time-the case of perfect operation. a. Perfect Operation (P) Here we will measure our reactor with a yardstick to find V and our Row rate with a flow meter to find uo in order to calculate T = Vlwo. We can then compare the curves shown below for the perfect operation in Figure 13-1 1 with the subsequent cases, which are for imperftct operation.
Figure 13-11 Perfect operation of a CSTR.
If r is large, there wiIl be a slow decay of the output transient. C(f), and ECtj for a pulse input. If t is small, there will be rapid decay of the transient, C(t). and E(r) for a pulse input. b. Bypassing (BP) A volumezric flow rate u,, bypasses the reactor whiIe a volumetric flow rate us* enters the system volume and (vo = us, + v,). The reactor system volume V, is the weI1-mixed portion of the reactor, and the volumetric flow rate entering the system volume is v,,. The subscript SB denotes that part of the flow has bypassed and only uss enters the system. Because some of the ff uid bypasses, the flow passing though the system will be less an the total volumetric rate, vs, < vg, consequently z , ~> T. Let's say the volumetric flow rate that bypasses the
Distributions of Residence Times
br Chemical Reacfors
Chap. 13
reactor, ub, is 25% of the total (e.g., ub = 0.25 v0). The volumetric flow rate entering the reactor system, us, is 75% of the total (vSB = 0.75 vo) and the corresponding true space time ( T ~ for ~ ) the system volume with bypassing is
The space time, KSB, wiIl be greater than that if there were no bypassing. Because TS, is greater than T there will be a slower decay of the transients C(r)and E(r) than that of pedect operation. An example of a corresponding Etl) curve for the case of bypassing is
The CSTR with bypassing will have RTD curves similar to those in Figure 13-1 2.
Figure 13-12 Ideal CSTR with bypass.
We see from the F(f) curve t h a ~we have an initial jump equal to the
fraction by-passed. c. Dead Volume (DV)
Consider the CSTR in Figure 13-13 without bypassing but instead with a stagnant or dead volume.
Figure 13-13 !deal CSTR with dead volume.
The total volume. V, i s the same as that for perfect operation.
v = v, + v,,.
Sec. 13.5
Diagnostics and Troubleshwting
895
We see that because there is a dead volume which the fluid does not enter, there is less system volume, Ifs, than in the case of perfect operation, VsD< K Consequently, the fluid will pass through the reactor with the dead volume more quickly than that of perfect operation, j.e., T s <~ 5.
-
Also as a result. the transients C(t) and E(t) will decay more rapidly than that for perfect operation, because there is a smaller system volume.
Summary A summary fw ideal CSTR mixing volume is shown in Figure 13-14,
Flgure 13-14 Comparison of E(r) and F(r) for CSTR under perfect operarion, bypassing, and dead volume. (BP=bypassing, P = perfect, and DV = dead volume).
Knowing the voIume V measured with a yardstick and the flow rate oo entering the reactor measured with a flow meter, one can calculate and plot Err) and F(r) for the ideal case (P) and then compare with the measured RTD E(r) to see if the RTB suggests either bypassing (BP) or dead zones (DV).
X3.5.2B 'lhbular Reactor A similar analysis to that for a CSTR can he carried out on a tubular reactor. a. Perfect Operation of PFR (P) We again measure the volume V with a yardstick and oo with a flow meter. The E(t) and F(t) curves are shown in Figure 13-15. The space time for a perfect PER i s h. PFR with Channeling (Bypassing, BP) Let's consider channeling (bypassing), as shown in Figure 13-16, similar to that shown in Figures 13-2 and 13-IO(d). The space time for the reactor system with bypassing {channeling) Tse is
896
Oistr~butbonsof ResrdenceTimes for Chernrcal Reactors
Chap 7
I
Yardstick
0
T
Figure 13-15 Perfect operation of a PFR
ubTj-;EL -
vo
0 Figure 13-16
7
=SO
'
=SB
PFR with bypassing similar to the CSTR.
Because us, c a,,, the space time for the case of bypassing is greate when compared to perfect operation, i.e., =sl3
'
2
If 25% is bypassing (i.e., ub = 0.25 uo) and 75% is entering the reac tor system (i.e.. U S B = 0.75 uo), then -csB = V l ( O . 7 5 ~=~ )1.33~.Th fluid that does enter the reactor system flows in plug flow. Here w have two spikes in the E(rE curve. One spike at the origin and on spike at TSB that comes after z for perfect operation. Because the vol umetric flow rate is reduced, the time of the second spike will b greater than 2 for perfect operation. c. PFR with Dead Volume (DV) The dead volume. V., could be manifested by internai circulation a the entrance to 'the reactor as shown in Figure 13-17.
5=2i+
Dead
zones
Figure 13-17 PFR with dead volume.
Sec I 3 5
D~agnosticsand Tmublsshoottng
897
The system I/,, is where the reaction takes place and the total reactor volume is (V = VsD + VD).The space time, T , for the reactor system with only dead volume is
Compared to perfect operation, the space time TsD is smaller and the tracer spike will occur before r for perfect operation. Here again, the dead volume takes up space that is not accessible. As a result, the tracer will exit eady because the system volume, Vs,. through which it must pass is smaller than the perfect operation case. Summary
Figure 13-18 is a summary of these three cases.
Figure 13-18 Comparison of PFR under perfect operation, bypassing, and dead volume (DV = dead volume, P = perfect PFR, BP = bypassing)
In addition to its use in diagnosis, the RTD can be used to predict conversion in existing reactors when a new reaction is tried in an old reactor. However. as we will see in Section 13.5.3. the RTD is not unique for a given system, and we need to develop models for the RTD to predict conversion. 13.5.3 PFR/CSTR Series RTD Modeling the real reactor as a CSTR and a PFR in senes
In some stirred tank reactors. there is a highly agitated zone in the vicinity of the impeller that can be modeled as a perfectty mixed CSTR. Depending on the location of the inlet and outlet pipes, the reacting mixture may follow a somewhat tortuous path either before entering or after leaving the perfectly mixed zone--ar even both. This tortuous path may be modeled as a plug-Row reactor. Thus this type of tank reactor may be modeled as a CSTR in series with a plug-Row reactor, and the PFR may either precede or follow the CSTR. In this section we develop the RTD for this type of reactor arrangement. Fint consider the CSTR foIlowed by the PFR (Figure 13-19). The residence time in the CSTR will be denoted by T~ and the residence time in the PFR by z., If a pulse of tracer is injected into the entrance of the CSTR,the CSTR output concentration as a function of time will be
898
Distributions of Residpnce Times for Chevica! Fleactn!~
.
C h p . 13
.+
Side Note: 1MedTcal Uses of RTD me application of RTD analysis i n bi& medical engineering is hing used at an &-&easing rate. For example, Professor Bob Langer's" group at MIT used RTD analysis for a novel Taylor-Couette flow device for blood detoxification while Lee et a1.t used an RTD analysis to study arterial blood Row-in the eye, In this later smdy, sodium fluorescein was injected hto the anticubical vein. The cumulative distribution function F(t) is shown schematically in Figure 13.5.N1, Figure 13.5N-2 shows a laser ophthalmost:ope imagce after injection of the sodium fluorescein. The mean residence time can be calculated for each m r y to or . estimate the mean circulation rime I..\ba. L.aJ sl. Chanees in the retinal blood flow m a y provide importa informati on for sickle-ceI1 discase and retinitis pigmen1
.,
-I
Figurn US&-I Cumulative RTD functicIn for arterial blwd flow in the eye. Courtesy 4sf Med. f i g . Phyxnt
Figure 13J.N-2 Image of eye after tracer inja:tion. Courtesy o f Med. fig. Phys.+
' G.A. A m , E.A. Gmven I:and R. Langer, AIChE adwic, C. J. 45, 633 (1999). E. T. Lee, R. G. Rehkopf. J. W,Warnick, T. F I I - ~ ~ "., N. Finegold, and E. G. Cape, Med. Eng. Phys. 19, 125 (1 997).
'
-
Figure 13-19 Real reacror modeled
as a
CSTR and PFR in
series.
This output will be delayed by a time T,, at the outlet o f the plug-flow section of the reactor system. Thus the RTD of the reactor system 1s
Dagnostics and TrouMeshmting
Sm. 13.5
See Figure 13-20.
The RTD is not unique to a particular reacm sequence.
fEgure 13-20 RTD curves EF!) and F(t) for a CSTR and a PFR In series. Next the reactor system in which the CSTR is preceded by the PFR will be treated. If the pulse of tracer is introduced into the entrance of the plug-flow section, then the same pulse will appear at the entrance of the perfectly mixed section z, seconds later. meaning that the RTD of the reactor system will be
E ( ! ) i s the same no matter which reactor comes first.
which is exactly the same as when the CSTR was followed by the PFR. It turns out that no matter where the CSTR occurs within the PFRICSTR reactor sequence, the same RTD results. Nevenheless, this is not the entire story as we will see in Example 13-3. Example 13-3
Comparing Second-Drdcr Reaction Systems
Consider a second-order reaction being carried out in a real CSTR that can be modExamples of furl! and late mixing for a given RTD
eled as two different reactor systems: In the first system an ideal CSTR i
Again. consider first the CSTR foIlowed hv the plug-Row section (Figure EI 3-3.1 ). A mole balance on the CSTR sectlon gwes
900
Distributions of ResidenceTimes for Chemical Reactors
Chap.
Figure EI3-3.1 Early mixing scheme.
Rearranging. we have
Solving for CAi gives
Then
This concentration will be fed into the
Pm.The PFR mole balance
Substituting CAi= 0.618. r = I , and k = 1 in Equation (E13-3.5) yields
Solving for C, gives
as the concentration of reactant in the effluent from the reaction system. Thus, thc conversion is 61.8%-i.e.. X = ([1 - 0.382]/1) = 0.618. When the perfectly mixed section is preceded by the plug-flow section (Fig ure E13-3.2) the outlet of the PFR is the inlet to the CSTR, C,,:
CA,= 0.5kmol/mJ and a material balance on the perfectly mixed section (CSTR) gives
Sec. 13.5
Diagnostics and Troubleshooting
Figure E13-3.2 Late mixing scheme.
X = 0.634
Early Mixing X = 0.618 Late Mixing X = 0.633
While ~ ( t was ) the same for both reaction systems, the conversion was not,
The Question
as the concentration of reactant in the effluent from
the reaction system. The comsponding conversion i s 63.44. I n the one configuration, a conversion of 61.8% was obtained; in the other, 63.4%. While the difference in the conversions is smslt for the parameter values chosen, the point is that there is a diffemnce.
The conclusion from this example is of extreme importance in reactor analysis: The RTD is not a complete description of structure for a particular reactor or system of reactors. The RTD is unique for a particular reactor. However, the reactor or reaction system is not unique for a particular RTD. When analyzing nonideaI reactors, the RTD alone is not sufficient to determine its performance, and more information is needed. It will be shown that in addition to the RTD, an adequate model of the nonideal reactor flow pattern and knowledge of the quality of mixing or "degree of segregation" are both required to characterize a reactor properly. There are many situations where rhe fluid in a reactor neither is well mixed nor approximates plug flow. The idea is this: We have seen that the RTD can be used to diagnose or interpret the type of mixing, bypassing, etc., that occurs in an existing reactor that is currently on stream and is not yielding the conversion predicted by the ideal reactor models. Now let's envision another use of the RTD.Suppose we have a nonideal reactor either on line or sitting in storage. We have characterized this reactor and obtained the RTD function. What will k the conversion of a reaction with a known rate law that is carried out in a reactor with a known RTD?
How can we use the RTD to predict conversion in a real reacror? In Part 2 we show how this question can be answered in a number of ways.
902
Dlstrlbutions of ResidenceTimes for Chern!cal Reacbz
Chap. 13
Predicting Conversion and Exit concentration
Part 2
13.6 Reactor Modeling Using the RTD Now that we have characterized our reactor and have gone to the lab no take data to determine the reaction kinetics, we need to choose a model to predict conversion in our real reactor. The Answer
RTD
+ MODEL c KINETIC DATA
{EXIT CONVERSlON and EXIT CONCENTIL.\TION
3
We now present the five models shown in Table 13-1. We shall classify each model according to the number of adjustable parameters. We will discuss the first two in this chapter and the other three in Chapter 14. TABLE 13-1. Ways we use the
RTD data to predict conversion in nonideal reactors
MODELSFDR P R E D I ~ NCONVER:J~SION G FROM RTD DATA
1. Zero adjustable parameters
a. Segregation model b. Maximum rnixedness model
2. One adjustable parameter a. Tanks-in-series model b. Dispersion mode! 3. Two adjustable parameters Real reactors modeled as combinations of ideal reactors
The RTD tells us how long the various fluid elements have been in the reactor. but it does not tell us anything about the exchange of matter between the fluid elements (i-e.. fke mixing). The rn~xingof reacting species is one of the major factors controlling the behavior of chemical reactors. Fortunately for first-order reactions. knowledge of the length of time each molecule spends in the reactor is all that is needed to predict conversion. For first-order reactions the conversion is independent of concentration (recall Equation E9-I: .3):
Consequently. mixing with the surrounding molecules is not imponant. Therefore. once the RTD is determined. we can predict the conversion that will be achieved in the real reactor provided that the specific reaction rate for the fisst-order react1011 is known. However, for reactions other than first order. knowledge of the RTD is not sufficient to predict conversion. In ther;e cases the degree of mixing of molecules must be known in addition to how long each molecule spends in the reactor. Consequently, we must develop models that account for the mixing of molecules inside the reactor.
Sec. 13.6
Referenceshelf
Reactor Modeling Using ihs RTD
903
The more complex models of nonideal reactors necessary to describe reactions other rhan first order must contain information about micromixing in addition to thar of rnacromixing. Macromixing produces a distribution of residence times u-dtl~nut,however, specifying haw molecules of different ages encounter one another in the reactor. Micromixing, on the other hand, describes how molecules of different ages encounter one another in the reactor. There are two extremes of r?~icmrnimng: (1) all molecules of the same age group remain together as they travel through the reactor and are not mixed with any other age until they exit the reactor (it., complete segregation): ( 2 ) molecules of different age groups are compIetely mixed at the molecular level ar soon as they enter the reactor (complete micromixing). For a given state of macromixing line.,a given RTD), these two extremes of micromixing will give the upper and lower limits on conversion in a nonideal reactor. For reaction orders greater than one or less than zero, the segregation model will predict the highest conversion, For reaction orders between zero and one, the maximum mixedness model will predict the highest conversion. This concept is discussed further in Section 1 3.7.3. We shall define a globule as a fluid particle containing millions of rnolecules all of the same age. A fluid in which the ,olobules of a given age do not mix with other globules is called a macrofluid. A macrofluid could be visualized as a noncoalescent globules where all the molecules in a given globule have the same age. A fluid in which molecules are not constrained to remain in the globule and are free to move everywhere is called a nricrqfl~id.~ There are two exiremes of mixing of the macrofluid globules to form a microfluid we shall study-arly mixing and late rnixlng. These two extremes of late and early mixing are shown in Figure 13-21 (a) and (b). respectively. These extremes can also be seen by comparing Figures 13-23 (a) and 13-24 (a). The extremes of late and early mixing are referred to as cnlnplefe segregario~and mn.rimunr rnixedne.r.7.respective1y.
Figure 13-21
-.
-
(a) MacroRuid:
and (h) microfluid mixing on the mnlecuIar level.
" I. Villerrnaux, Cl~nr~irrrl H P U ( Y ~Dt>ri,qn Irrrld Ecl~nr)lo~,v (Boston. Martinus XijIloff# 1986).
.
964
Distr~butionsof Res~dsnceTimes for Chem~mlReactors
Chao
1:
13.7 Zero-Parameter Models 13.7.1 Segregation Model
In a "perfectly mixed" CSTR. the entering fluid is assumed to be distribute( immediately and evenly throughout the reacting mixture. This mixing i! assumed to take place even on the microscale, and elements of different ages mix together thoroughly to form a completety micromixed Fluid. [f fluid elements of different aees do not mix together at all. the elements remain segregated from each other. and the Ruid is termed complefeiy segregared The extremes of complete micromixing and complete segregation are the limits ol the micromixing of a reacting mixture. In developing the segregated mixing model, we first consider a CSTR because the application of the concepts of mixing quality are illustrated most easily using this reactor type. In the segregated flow model we visuaIize the flow through the reactor to consist of a continuous series of globules (Figure 13-22).
In the segreptlon model globuIe~
&have
ah
batch
reactors operated for d~fferenttimes
Fimre 13-22 Little batch reactors (globules) inside a CSTR.
The segregation model has mixing at the latest po~riblepoint.
These globules retain their identity; that is, they do not interchange material with other globules in the Ruid during their period of residence in the reaction environment. i.e., they remain segregated. In addition, each globule spends a different amount of time in the reactor. In essence, what we are doing i s lumping all the molecules that have exactly the same residence rime in the reactor into the same globule. The principles of m m r performance in the presence of completely segregated mixing were first described by D a n c k ~ e s t s and Zwietering.11 Adother way of looking at the segregation model for a continuous flow system is the PFR shown in Figures 13-23(a) and (b). Because the Ruid flows down the reactor in plug flow, each exit stream corresponds to a specific residence time in the reactor. Batches of molecules are removed from the reactor at different locations along the reactor in such a manner as to dupIicate the RTD function. E ( t ) . The molecules removed near [he entrance to the reactor correspond to those molecules having short residence times in the reactor. I0P. V. Danckwerrs, Chcm. Eng. Sci.. 8, 93 (1958). 'IT.N. Zwiotering, Chcm. Eng. Sci., 11, 1 (1959).
Ssc. 13.7
Zero-Parameter Models
E(tl matches the emoval of the hatch reactors
.
**
a
Little hatch reactors
Ib)
Figure 13-23 Mixing at the latest possible point.
Physically, this effluent would correspond to the molecules that channel rapidly through the reactor. The farther the molecules travel along the reactor before being removed, !he longer their residence time. The points at which the various groups or batches of molecules are removed mmrpond lo the RTTD function for the reactor. Because there is no molecular interchange between globules. each acts essentially as its own batch reactor. The reaction time in a 6 one of these tiny batch reactors is equal to the time that the particular globule spends in the reaction environment. The distribution of residence times among the globules is given by the RTD of the particular reactor.
RTD + MODEL + KINETIC DATA
3
EXIT CONVERSION and
EXIT CONCENTRATION
To determine the mean conversion in the effluent stream. we must average the conversions of alI of the various globuIes ifl the exit stream: Mean conversion
rime I and t + dt in the reactor
Conversion
in the reactor
then
8 = XCt) X E ( t ) dt
1
Fraction
1
1 of ~iobvlcrthat 1 spend between t and t t.dt in the L reactor
1
906
Distributions ol Residence Times for Chemical Reactors
Chap. t 3
Summing over all globules, the mean conversion is Mean conversion for the segregation model
Summary Note:
Cnnsequently, if we have the batch reactor equation for X ( r ) and measure the RTD experimentally. we can find the mean conversion in the exit stream. T h q if we have rhe RTD, the reacfion rnlc expres.~ion,then ,for a segregaredflow siruarion (i.e., model), n7ehhave suficient information lo calculate tlre conversion. An example that may help give additionai physical insight to the segregation model is given in the Summary Notes on the CD-ROM, click the Back button just before section 4A.2. Consider the fallowing first-order reaction: A
--%
products
For a batch reactor we have
For constant volume and with hr, = NA,(l
- X),
Solving for X(t), we have
Mean conversion for a first-order reaction
P=
X ( f ) E ( l ) d i=
1
n
(I
-
C k lE)( r ) dt =
1 o
E ( t ) dr -
1
* n
p F L I E ( f 1 dt
( 1 3-55)
We will now determine the mean conversion predicted by the segregation model for an ideal PFR. a CSTR. and a laminar flow reactor.
Sec. 13.7
907
Zero-Parameter Mdels
Exanrpb 1 3 4 M~QIIConvemwn IS an Ideal PFR, an Ideal CSTR, and a Lominar Flow Reactor Derive the equation of a first-order reaction using the segregation mode1 when the RTD i s equivalent to (a) an ideaf PFR, (b) an ideal CSTR,and (c) a laminar flow reactor. Compare these conversions with those obtained from the design equation. Solution (a) For the PFR,the RTD function was given by Eguation (13-32)
E ( I ) = 6 ( t - T)
(23-32)
Recalling Equation (13-55)
Substituting for the RTD function for a PER gives
Using the integral properties of the Dirac delta function, Quation (13-35)we obtain
where for a first-order reaction the Damkohler number is Da = ~ kRecalI . that for a PFR after combining the mole balance. rate lam: and stoichiometric relationships (cf. Chapter 4), we had
Integrating yields
which is identical to the conversion predicted by the segregation model (b) For the CSTR, the RTD function is
2.
Recalling Equation ( 1 3-56), the mean conversion for a first-order reaction is
908
F,,X = - r,V
the
m d e l gives a mean conversion X identical to that obtained by using the a l p nthm In Ch 4.
Chap. I
Combining the CSTR mote balance. the rate law, and stoichiornetrg., we have
As expected, using
E(r) for an ideal PFR and CSTR wlth the segregation
Distributions of Residence firnes br Chemical Reactors
I
I
VD CAoX=
kCAo(i- X ) V
which is identical to the convenion predicted by the segregation model (c) For a laminar Row reactor the RTD function is
1.
=
for (tad2) The dimensionless form is
From Equation (13-15). we have
Integrating twice by parts
The last integral is !he exponential integral and can be evaluated from tabulated values. Fortunately, HjlderI2 developed an approximate formula ( ~ =k Da).
Sec. 13.7
I
909
Zero-Parameter Models
A comparison of the exact value along with Hilder's approximation is shown in Table E 13-4.1 for variour values of the Damktlhler number. Tk, along with the conversion i n an ideal PFR and an ideal CSTR. OF COXVERSION IN PFR. CSTlt, AND LAMINAR T.%EILE E 13-41. COMPARISON ROW REAC~ORFOR DIFFERENT. DAMK~HLER NUMBERS FOR A FIRST-ORDER REACTION
,,
where XLF Exec, = exact solution to Equation (B134.9) and X,, = Equation (E 13-4.10). For large vdues of the Damkiihler number then, there is complete conversion along the smamlines off the center streamline so that the conversion is determined along the pipe axis such that
I
Figure E13-4.1 shows a comparison of the mew conversion of the conversion in an LFR,PFR, and CSTR as a funsiian of the DnmLohleier numbcr for a Rnt-order reaction.
Figre E134.t Conversion in a PFR. LFR, and CSTR as a function of a Damkhhler number (Wa)for a finr-order reaction /Da = fkl.
910
I
Distr~butionsof Res~CmTIWS lor Chernal Reactors
Chap. f 3
We have just shown for a first-order reaction that whether you assume complete micromixing [Equation (El 3-46)] or complete segregation [Equation Important Point: (E13-4.5)] in a CSTR, the same conversion results. This phenomenon occurs For a first-order because the rate of change: of conversion for a first-order reaction does not depend on the concentration of the reacting molecules [Equatiw (13-54)j; it what r n o W 1 does ~ ~ not ~ matter ~ ~ ~ ~ kind of molecule is mxt to it or colliding with it. Thus the extent of micromixing does not affect a first-order reaction, so the segregated flow model can be used to calculate the conversion. As a result, only rhe RTD is necessaT fo calculafe the conversion for a first-order reaclion in any ppe of reactor (see Problem P13-3,). Knowledge: of neither the degree of micromixing nor the reactor flow pattern is necessary. We now proceed to calculate conversion in a real reactor using RTD dab.
I
Example 13-5 Mean Conversion Colcuhtions in n Real Reactor
Calculate the mean conversion in the reactor we have characterized by RTD measurements In Examples 13-1 and 13-2 for a first-order, liquid-phase. irreversible reaction in a completely segregated fluid: The specific reaction rate is 0.1 min-bat 320 K.
Because each globule acts as a batch reactor of constant volume, we use the batch reactordesign quation to arrive at the equation giving conversion as a function of time:
are easily csmed out with the aid of a spreadsheet such as Excel or
Pol) math.
To calculate the mean conversion we need to evaluate the integral:
8=
o
X ( t ) E ( f )dl
(13-53)
The RTXl Function for this reactor was determined previously and given inTable E13-2.1 and is repeated in Tahle El 3-5.1. To evatuate the integral we make a plot of X ( r ) E l r ) as a function of r as shown in Flgure E13-5.1 and determine the area under the curve.
Zero-Parameter Models
Sec. 13.7
0.07 0.m 0.05
E
For a given RTD, the segregation mode1 gives the upper bound on convers~on for reaction orders less than zero or greater than I .
W
X
0.04
0.03 0.02
0.01
0
2
d
6
8
10
72
14
15
t
Figure E13-5.1 Plot of columns 1 and 4 from the data in Table El 3-5.1 TABLE E13-5.1 I
(rnin)
PROCESSED DATA TO m~THE MEAN CONVERSIOY f E(r) (min-')
XO)
X ( i ) E ( t ) [rn~n-I)
--
Using the quadature formulas in Appendix A.4
I
I
=
(0.350)f (0.035)= 0.385
X = area = 0.385 The mean conversion is 38.59.Polymath or Excel will easily give 1 after setting up columns 1 and 4 in-Table El 3-5.I. The area under the cune in Figure E 13-5.1 i s the mean conversion X .
As discussed previously, because the reaction is jr.rr order. rhe conversion calculated in Example 13-5 w w l d be valid for a reaclor with cnmplete
912
Distributrons of Res~aencsTimes for Chem~calReactors
Chap. 1:
mixing, complete segregation. or any degree of mixing between the two Although early or late mixing does not affect a first-order reaction, micromix ing or complete segregation can modify the results of a second-order system significantly. Example 13-6
Mean Conversionfor a Second-Order Reaction in a Laminar
Flow Reactor The liquid-phase reaction between cytidine and acetic anhydride
cyiidine
acetic anhydride
IA)
(W
IC)
(0)
is carried out isothermally in an inert solution of N-metbyl-2-pyrrolidone (NMP with a ,,, = 28.9. The reaction follows an elementary rate law. The feed is equa molar in R and B with CAO = 0.75 mol/dmJ, a volumetric Row rate of 0.1 dm3/s ant a reactor volume of 100 dm3. Calculate the conversion in (a) a PFR. (b) a batcl reactor, and (c) a laminar flow reactor.
Additional information: l3
*
k = 4.93 x lo-' dm'lmol . s at 50°C with E = 13.3 kcallmol, AHRX = - 10.5 kcaVmol Heat o f mixing for 0,,, =
FA,
= 28.9. AHmkx= 4 . 4 4 kcallmol
Sohrion
The reaction will be carried out isothermalIy at 50'C. The space time is 2 = - - =V- -
uo
100dm3 - Z O O O s 0.1 dm'ls
-
For a PFR Mole BaIancc (a)
"1. 1. Shatynski and 13. Hanesian, Ind. Eng. Chem. Res., 32. 594 (1993).
Sec. 13.7
Zer~mParameterModels
c, = c* Combining
PFR
Calculation
I
Solving with T = Vlv, and X = O for V = 0 gives
I
where Do2 is the Darnkbhiei number for a second-order reaction
(b)
Batch Reactor
Batch Calculation
If the batch reaction time is the same time as the space time the batch conversion is the same as the BFR conversion X = 0.787. {c) Laminar Flow Reactor The diffmntid form for the mean conversion is obtained from Equation (13-52)
We use Equation (E13-6.9) to substitute forX(t) in Equation (13-52). Because E(t) for the LFR consists of two parts, we need to incorporate the IF statement in our ODE solver program. For the laminar flow reaction. we write
914
D~stributionsot Residence Times lor Chemical Reactors
T~ for r 1,= -
Chao. 13
2 212
2t3
Let
r, = f l
so that
the IF statement now becomes
LFR Calculation
E = If (I c t , ) then ( E l )eIse (E,)
-
(E13-6.12)
One other thing is that the ODE solver will recognize that El = at r = 0 and refuse to run. So we must add a very small number to the denominator such ar (0.001); for
exmpTe,
The integration time should be carried out to 10 or more times the reactor space time t. The Polymath Program for this example is shown below.
.- -
Living Example Problerr
k
0.00493
0.00493
0.D O 1 9 3
0.00493
Cao X
0.75 0
0.75
0.75 0.9866570
0.75 0.9666578
tau
1000
1000
t1
500
1000 500
BS E
5.0E*10 0
6,253-08 0
5.01+10 0.0034671
1000 500 6.25E-00 6.25E-08
0
500
[a1 CaoeO.75 [ 3 1 X = L'Caov(l+k-Cao~) [dl tauelm9 I31 H - h a
[sl E2 t a ~ . [ 7 I E = H(lct1)then (0)slee (E2)
~
~
We see that the mean conversion Xbar
)
(F)for the LFR is 7 4 . 1 9
In summary.
Compare this resalt with the exact analytical with a second+orderreaction Analytical Solution
IT= Dall - ( D U E )I~(I+~/D~)IE where Da =~C,,,T.For D a = 3.70 a e get
= 0.742
for the laminar flow reactor
Sec. 13.7
91 5
Zero-Parameter Mdels
13.7.2 Maximum Mixedness Model Segregation mudel mixing occurs aI the latest possible point.
In a reactor with a segregated fluid, mixing between panicles of fluid does not occur until the fluid leaves the reactor. T h e reactor exit is, of course, h e [arest possible point that mixing can occur, and any effect of mixing is postpond until after all reaction has taken place as shown in Figure 13-23. We can also think of completely segregated flow as being in a state of minimum rnixedness. We now want to consider the other extreme, that of maximum rnixedness consistent with a given residence-time distribution. We return again to the plug-flow reactor with side entrances. only this time the fluid enters the reactor along its length (Figure 13-24). As soon as the Ruid enters the reactor, it is c~mplctelymixed radially (but not longitudinally) with the other fluid already in the reactor. The entering fluid is fed into the reactor through the side entrances in such a manner that the RTD of the plug-flow reactor with side entrances is identical to the RTD of the real reactor.
Maximum mixedness: mixing accurs at the earliest passible point
0-
t (b)
Figure 13-24 Mixing
at
the earliest possible point.
The globules at the far left of Figure 13-24 correspond to the molecules that spend a long time in the reactor while those at the far righr correspond EO the molecules that channel through the reactor. In the reactor with side enmnces, mixing occurs at the earkst possible moment consistent with the RTD. Thus !he effect of mixing occurs as early as possible throughout the reactor, and this situation is termed the condition of maximum r n i x e d n e s ~ . ~ ~ The approach to calculating conversion for a reactor in a condition of maximum rnixedness will now be developed. In a reactor with side entrances, let X be the time it takes for the fluid to move from a particular point to the end of the reactor. ln other words, A is the life expectancy of the fluid in the reactor at that point (Figure 13-25).
I5T.N. Zwietering, Chcrn. Eng. Sci., 11, 1 (1959).
Distributions of Residence Times tor Chemical Reactors
Chap.
Figure 13-25 Modeling maximum mixedness by a plug-flow reactor with slde entrances
Moving down the reactor from left to right, A decreases and becomes zero at the exit. At the left end of the reactor. A approaches infinity or the maximum residence time if it is other than infinite. Consider the fluid that enters the reactor through the sides of volume AI/ in Figure 13-25. The fluid that enters here will have a life expectancy between h and k+dh. The fraction of fluid that will have this life expectancy between the product of the total volumetric flow rate. v,, and the fraction of the total that has life expectancy between h and h+Ah is E(h) hh. That is the volumetric rate of fluid entering through the sides of volume AV is uo E(h) Ah.
The volumetric flow rate at h. u i , is the flow rate that entered at b A k . plus what entered through the sides vo E(h)M,i.e.,
v, = v, +*, + v&(h)AX Rearranging and taking the limit as bh
0
Ssc. 13.7
917
Zero-Parameter Models
The volumetric Row rate uo at the entrance to the reactor ( X = 0) is zero because the fluid wly enters through the sides along the length. Integrating equation (13-57) with limits v h = 0 at h = m and uh = vh at h = X, we obtain
The volume of Auid with a life expectancy between h and X
AY
=
L~,[I
+ Ah is
- F ( X ) ]AX
( 13-59)
The rate of generation of the substance A in this volume is
We can
r , A V = r,u,[l -FIX)] Ah
( 13-60)
now carry out a mole balance an substance A between A and h
4- AX:
Mole balance
[at
t: .A 4
+
[through In
.
uo [l
- F ( ~ ~ ~ A I A + +A A
~0
ride] -I:?[
'[
~ ~ ~ z ~ =
O
C A O E (Ah ~)
b+ - V ~ [ I - F ( ~ ~ ] C ~ ~ ~ + ~ , , V ~ [ (13-61) ~ - F ( ~ ) ]
Dividing Equation (13-61) by v, AX and taking the limit as Ah E.(X)CAo +
Taking the derivative of
d ( [ l - F ( h ) l C ~ ( h+ )l dA the term
4
- F(h)]= 0
in brackets
We can rewrite Equation (13-62) in terms o f conversion as
0 gives
The boundary condition is as h + 0 3 , then CA= CAOfor Equation (13-62) [or X = 0 for Equation (13-64)]. Ta obtain a solution, the equation is integrated backwards numerically, starting at a very large value of h and ending with the rn gives the lower final conversion at A = 0.For a given RTD and reaction orders greater than one, 'bound On X. the maximum mixedness model gives the lower bound on conversion.
1
1
Example 13-7
Conversion Bounds for a Nonitfeu! Reactor
The liquid-phase. second-order dimerimtion
for which k = 0.01 dmVrno1 nmin is carried out at a reaction temperature of 320 K. The feed is pure A with CAO= 8 molldm3. The reactor is nonideal and perhaps could be modeled as two CS'17ts with interchange. The reactor volume is 1000 dm3. and the feed rate for our d~merizationis going to be 25 ddlmin. We have run a tracer test on this reactor, and the results are given in columns I and 2 of Table El3-7.1. We wish to know the bounds on the conversion for different possibIe degrees of micromixing for the RTD of this reactor. What are these bounds? Tracer test on tank reactor: No = 1 0 g, u = 25 dm3/min. TABLEE13-7.1
tlvrng Example Problem
f
(min)
0 5 10
Columns 3 through 5 are calculated from columns 1 and 2.
15 20 30 40 50 70 100
C (rngJdml)
112 95.8 82.2 70.6 60.9 45.6 34.5
26.3 15.7 7 67
150
2.55
200
0.90
1
2
E
m
R A W AND
i
0.0280 0.0240 0.0206 0.017'7 0.0152 001 I4 0.00863 0.00658 0.00393 0.00192 0 000h38 0.000225 3
I
PROCESSED DATA
- P(r) l.Oo0 0.871 0.760 0.643
E(i)l[l
- F(r)] (rnin-')
d (rnin)
0.353 0.278 0.174 0.087 0.024 0.003
0.0280 0 0276 0.0271 0.0267 0.0260 0.0242 0.0244 0.0237 0 0226 0.022 1 0.0266 0.075
200
4
5
6
0.584 0.472
0 5 I0 15 20 30 40 50
70 100 150
Solution
The bounds on the conversion are found by calculating conversions under conditions of complete segregation and maximum mixedness. Conversion {fflufdi s compl;te!s segmgared. The batch reactor equation for a second-order reaction of this type is
920
we need to take an
Segregation 6 1 % M a x . mix
56%
Oistrrbutions of Residence firres for Chemical Reactors
average of E / ( I
Chap.
- F) between h = 200 and h = 150.
Running down the values of X along the right-hand side of the preceding equatio shows that the oscillations have now damped out. Carrying out the remaining calcu lations down to the end of the reactor completes Table El3-7.3. The conversion fo
a condition of maximum mixedness in this reactor is 0.56 or 56%. It ts interestin! to note that there is little difference in the conversrons for the two conditions o complete segregation (61%) and maximum mixedness (56%). With b u n d s this nar mw, there may not be much point in mcdeling the reactor to improve the predict ability of conversion. For comparison it is Ieft for the reader to show that the conversion for a PFF of this size would be 0.76. and the conversion in a perfectly mixed CSTR with corn plete micromixing would be 0.58.
SM. 13.7
Zero-Parameter Models
Calculate backwards to reactor exit.
The Intensity Function, A(t) can be thought of as the probability of a particle escaping the system between a time r and ( t + dt) provided the particle is still in the sysrem. Equations (13-62) and (13-64)can be written in a slightly mare compact form by making use of the intensity function.I6The intensity function A(A) is the fraction of fluid in the vessel with age A that will leave between h and X + $A. We can relate A(h) to I(h) and E ( X ) in the following manner:
r
Volume of 7
_[
between times h and A+&
[ v , E(X1 A1 = [VI(Ml
Fraction of
1
time A and h + dh
1
(1 3-65)
Then
Combining Equations (13-64) and (13-66) gives
We also note that the exit age, r. is just the sum of the internal age. a,and the life expectancy, h: t=a+X (13-68) I6D. M. Nimmelblau md K.3.Bischoff, Process Analysis and Simltlntian (New York: Wiley. 1968).
922
Disiritwfions of Reshdence Times for Chemical Reactors
Chap. f 3
In addition to defining maximum rnixedness discussed above, ZwieteringI7 also generalized a measure of micromixing proposed by Danckwerts18 and defined the degree of segregation, J , as J = variance of ages between fluid '"points" variance of ages of all molecules in system A fluid "point" contains many molecules but is small compared to the scale of mixing. The two extremes of the degree of srgregation are J = 1: complete segregation J = 0: maximum rnixedness Equations for the variance and J for the intermediate: cases can be found in Zwietering.
13.7.3 Comparing Segregatron and Maximum Mixedness Predictions
Reference Chef
In the previous example we saw that the conversion predicted by the segregation model, X,, was grearer than that by the maximum rnixedness model X,,. Will this always be the case? No. To learn the answer we take the second derivative of the rate law as shown in the Professional Reference Shelf R13.3 on the CD-IWM.
a-c-'~)>~
ac', a?( -r*)
Comparing and X m m
ad,
Xw,
d2C-~*l -O
ac',
then
X,,, > X,,
then
X,, > X,,
then
X,, = X,,
For example, if the rate law is a power law model
- rA = kCf4
From the product [{n)(n -
!)I.
we see
Z
If n < 0.
then
"0:o
ac',
and
'7.N.Zwietering. Chrm. Eng. Sci., 11. 1 (1959). IMP.V. Danckwerts. C ~ P I IEng. I . Sri.. X. 93 (1958).
X..,>X,,
Sec. 13.8
then
Important point
923
Using Software Peckages
and
We note that in some cases X, is not too different from Xm. However, when one is considering the destruction of toxic waste where X > 0.99 is desired, then even a small difference is significant!! In this section we have addressed the case where all we have is the RTD and no other knowledge about the flow pattern exists. Perhaps the flow pattern cannot be assumed because of a lack of information or other possible causes. Perhaps-we wish to know h e extent of possible ersor from assuming an incorrect flow pattern. We have shown how to obrain the conversion, using only the RTD. for two limiting mixing situations: the earliest possible mixing consistent with the RTD,or maximum mixedness, and mixing only at the reactor exit, or complete segregation. Calculating conversions for these two cases gives bounds on the conversions that might be expected for different flow paths consistent with she observed RTD.
13.8 Using Software Packages Example 13-7 could have been solved with an ODE solver after fitting E(t) to a potynornial.
Fitting the E(t) Curve to s Polynomial Some forms of the equation for the conversion as a function of time multiplied by E(t) wilE not be easily integrated analytically. Consequently, it may be easiest to use ODE software packages. The procedure is straightforward. We recall Equation (1 3-53
where is the mean conversion and X(t) is the batch reactor conversion a t time I. The mean conversion is found by integrating between r = 0 and t = w or a very large time. Next we obtain the mole balance on X ( I ) from a batch reactor
and would write the rate law in terms of conversion, e.g., -5 = k d O ( 1-q2 The ODE solver will combine these equations to obtain X ( I ) which will be used in Equation (13-56). Finally we have to specify E(r). This equation can be an analytical funcrion such as those for an idea! CSTR,
924
Distributions of Residence Times for Chemical Reactors
or it can be polynomial
Chao. 1
or a combination of polynomials that have been use
to fit the experimental RTD data
We now simply combine Equations ( 13-52), (13-69), and (13-70) and use a, ODE sotves. There are three cautions one must be aware of when fitting E(t) ttl
a poIynomia1, First. you use one poIynomia1 E l ( { ) as E(rj increases with tim to the top of the curve shown in Figure 13-27. A second polynomial E2(1)i used from the top as E(r) decreases with time. One: needs to match the tWr curves at the top.
Match
E
h
t
Figure 13-27 Matching E,(r) and E,(r).
Summary Motes polymath~~~~~~l
Second. one should be certain that the polynomial used for E2(t) does no become negative when extrapolated to Iong times. If it does, then constraint: must be placed on the fit using IF statements in the fitting program. Finally one should check that the area under the E(t) curve is virtually one and that thr cumulative distribution F(t) at long times is never greater than 1. A tutorial or how to fit the C(r1 and EIt) data to a polynomial is given in the Summary Note: for Chapter 5 on the CD-ROM and on the web. Segregation Model
Here we simply use the coupled set af differential equations for the mean exit conversion, 2 ,and the conversion X(t) inside-a globule at any time, t.
The rate of reaction is expressed as a function of conversion: for example,
01
Sec. 13.8
Using S o h a r e Packages
and the equations are then solved numericaIIy with an ODE solver.
Maximum Mixedness Model Because most software packages won't integrate backwards. we need to change the variable suchthat the integration proceeds forward as h decreases from some large value to zero. We do this by forming a new variable, s, which is the difference between the longest time measured in the E ( t ) curve. T , and 1.In the case of Example 13-7, the longest time at which the tracer concentradon was measured was 200 minutes (Table E13-7.13. Therefore we will set T = zoo.
-
X = T -z=200-z Then,
One now integrates between the limit z = O and 2 = 200 to find the exit conversion at z = 200 which corresponds to h = 0. In fitting E ( t ) to a polynomial. one has to make sure that the polynomial does not become negative at large times. Another concern in the maximum mixedoess calculations is that the tern 1 - F ( X ) does not go to zero. Setting the maximum value of F ( t ) at 0.999 rather than 1.0 will eliminate this problem. It can also be circumvented by integrating the polynomial for E ( I )to get F ( t ) and then setting the maximum value of F ( I ) at 0.999. If F(r) is ever greater than one when fitting a polynomial. the solution will blow up when integrating Equation 113-72) numericaIIy. Example 13-8
Using Software to Make Maximum Miredrpess Model Calc1dat3ons
Use an ODE solver to determine the conversion predicted by the maximum mixedm e s s model for the E(r) curve given in Example E13-7.
Because of the nature of the E ( r ) curve, it i s necessary to use two polynomjals, a third order and a fourth order, each for a different part of the curve to express the RTD, E ( t ) , as a function of time. The resulting E t t ) curve is shown in Figure
E13-8.1. To use Polymath to cany out the integration, we change our variable from X to z using the largest time measurements that were taken fmm E(r) in Table E13-7.1,which i s 200 min:
926
Distributions of ResidenceTmes for Chernlcal Reactors
Chap 13
First, we fit E ( I ) .
Figure E13-8.1 Polynomiel fit of E ( f ) .
z = 200 - A The equations to be solved are A = 200 - 2 Maximum rnixedness modti
For values of h less than 70, we use [he polynomial
EI(~)=4.M7c-10h4-1.I80e-7h3+1.353eL5h2-8.657e-4k+0.028{E13-8.3) For values of A greater than 70,we use the polynomial
E2()c)= -2.640e-qh3
+ 1.361S C - ~ A-~ 2.407e-jh + 0.U15
E13-8.4)
(El 3-8.5) with z = 0
[I at
[
-F(h)]-
= 200). X = 0, F = 1 [i.e., F(A) = 0,9993.Caution: Because tends za infinity at+ = 1, (z = 0).we set the maximum vatue of F
0.999 at z = 0. The Polymath equations are shown in Table E13-8.1. The solution is
The conversion predicted by the maximum rnixedness model i s 56.3%.
Sec. 13.9
RfC)
snd Munipte Reactions
ExplH equalbs as mmwd
927
uw usor
[11
LfvingExample Problem
121 k t . 0 1 131 lam=2#z I 4 l cs r uto'(l-x) 1 S l E l = 4 ME€.bl[nswP+f . 1 8 ( t 2 e ~ ~ 1 3 6 3 ~ h ~ * . ~ 5 5 5 2 ' I ~ ~ 1 . 0 2 B D W I 6 1 E2 m -2.We-91em*a+1,381~7emh2-.0M12*08~gI~m+.o~ 5011 171 F1 = 4 ~ 5 8 a l [ Y 5 ? a ~ 1 . 1 8 0 2 b ? / 4 % m F d + 1 . 3 5 3 ~ R:'-FZ +9 3 0 7 6 ~ - B ' b ~ t i . 0 2 ~ 5 5 1 a ~ 2 - ."tarnc.618nl-2) 0W1 191 r a = - P W 2 110 1 E il (lam*~70)Fo) ((El)e h (E2) [ll l F = tf (lamc=7Q)ihen (Fl) s$s (F2) I121 EF = B(1-Q
I
-
Polynomiafs used to fir Err) and F(r) - I
uanaMe v a n W name : t lnn~alvalue O f~nalvalue : 200
.
13.8.1 Heat Effects If traces tests are carried out isothermally and then used to predict nonisotherma1 conditions, one must couple the segregation and maximum mixedness models with the energy balance to account for variations in the specific reaction rate. This approach will only be valid for liquid phase reactions because the vo1umetrjc flow rate remains constant. For adiabatic operation and
A?~=o,
As before, the specific reaction rate is
-
,
I."[;IkJj r
k=k,exp -
I
---
Assuming that E(r) is unaffected by temperature variations in the reactor, one simply soIves the segregation and maximum mixedness models, accounting for the variation of k w ~ t htemperature [i.e., conversion; see Problem P13-2(j)].
13.9 RTD and Multiple Reactions As discussed in Chapter 6.when multiple reactions occur in reacting systems, it is best to work in concentrations,moles, or molar flow rates rather than conversion. 13.9.1 Segregation Model
In the segregation model we consider each of the globules in the reactor to have different concentrations of reactants, CA,and products, Cp.These globules
928
Distnbutions of Residence Ttmes for Chemical Reactors
Chap.
are mixed together immediately upon exiting to yield the exit concentration I , which is the average of all the globules exiting: A,
The concentrations of the individual species, CA(t)and C B ( t ) ,in the difTere globules are determined from batch reactor calculations. For a constant-volun batch reactor. where q reactions are taking place, the coupled mole balan equations are
These equations are solved simultantousIy with
to give the exit concentration. The RTDs. E(t), in Equations (13-77) and (13-7 are determined from experimental measurements and then fit to a polynomial.
13.9.2 Maximum Mixedness For the maximum rnixedness model, we write Equation (13-62) for each sp cies and replace r, by the net rate of formation
After substitutionfor the rate laws for each reaction (e.g., r i =~k,CA),these equ tions are solved numericalIy by starting at a very large value of X, say T = 201 and integrating backwards to A = 0 to yield the exit concentrations CA,CB,.. We will now show how different RTDs with the same mean residenl time can produce different product distributions for multiple reactions.
Sec. $3.9
RTD and Multiple Reactions
Example 13-9
RTD and Complex Reactions
Consider the following set of liquid-phare reactions: Living Exarcple Problc
A+B
k'--, C
which are occurring in two different reacton with the same mean residence time r, = 1.26 min. However, the RTD is very different for each of the reactors, as can k seen in Figures E13-9.1 and E l 3-9.2.
t
Figure EI3-9.1
El (t): asymmetric distrihation.
t
Figure E13-9.2 E2(t): bimdal distribut~on
930
Distributions af ResidenmTlmes far Chernbl Reamts
Chap. 13
(a) Fit a polynomial to the m s .
(b) Determine the prduct distribution (e.g., S,. 1. The segregation model 2. The maximum mixedness model
)S,
for
Additional lnforrnat~on
k , = k2 = k3 = 1 in appropriate units at 350K. Solution
Segregation Model
Combining the mole balance and rate jaws for a constant-volume batch reactor he., globules), we have
and the concentration for each species exiting the reactor is found by integrating the equation
over the life of the E(r) curve. For this example the life of the E, I r ) is 2.42 mlnutes (F~gureE13-9.1f. and the life o f is 6 minutes (Figure El 3-9.2). The initial conditions are r = 0, C, = C, = 1, and C, = C, = C, = Q. The Pnlyrnath program used to solve these equations is shown in Table El 3-9.1 for the asymmetric RTD,J, ( r ) . With the exception of the polynomial for E&), an idenlical program to that in Table E13-9.1 for the bimodal distribution is given on the CD-ROM.A comparison of the exit concentration and selectlvities of the two RTD curves is shown in Table E 13-9.2.
See. 53.9
931
RTO and Multiple Reactions
Ltvlng Example Problem
Aqmmctric Disri-ibwtion
The solution
for E , ( t ) is:
-
C, = 0.E51
= 0.178
X
Bimodal Disrrihrian The solution for E2(r) is:
C, -
=0245
C, = 0.357
Sm = 1.18
-
C,
SD, = 1.70
= 0.265
= 0.454
-
-
0.303
=
84.94
Maximum Mixwines Model The equations for each s p i e s are
C, =0.5!0
C, =0.162
X=75.5%
0.321 ,3
Sm
= 1.21 = 1.63
932
Solved .Prohlcn?r
Oistribut~onsof Resrdence Trrnes for Chemical Reactors
C h a ~1:
The Polymath program for the bimodal distribution, Eltt), i s shown in Tab11 E13-9.3.The Polymath program for the asymmetric distribution is identical with thl exception of the polynomial fit for E, ( I ) and is given on the CD-ROM. A compari son of the exit concentration and select~vit~es of the two RTD distributions is show1 in Table E13-9.4.
FOR
TABLE E13-9 3. FOLYMATH PR~RAY MAXIMUM MIXEDYESSMODEL WITH BIMODAI.DISTRIBCTION (MULTIPLE RBA~ONS)
As!mmernc Disrribrrrion
BSmotIuI Dirtriburion
The solution hr E l ( 0 ( I 1 is:
The wlulion for & ( t ) (21 is:
Catcutations similar to those in Example 13-9 are given in an example on
*
Solved Problemr
kt
B
k?
>
c
I n addition, the effect of the: variance of the RTD on the parallel reactions in Example 13-9 and on the series reaction in the CD-ROM is shown on the
CD-ROM.
Chap. :3
Summary
Closure After completing h i s chapter the reader will use the tracer concentration time data to calculate the external age distribution function E(r), the cumuIative distribution function F(r), the mean residence time,r, and the variance, rsZ. The reader will be able to sketch E(t) for ideal reactors, and by comparing E(t) ftom experiment with E(t) for ideal reactors (PFR,PBR, CSTR, laminar flow reactor) the reader will be able to diagnose problems in red reactm. The reader will also be able to couple RTD data with reaction kinetics ta predict the conversion and exit concentrations using the segregation and the maximum rnixedness models without using any adjustable parameters. By analyzing the second derivative of the reaction rate with respect to concentration, the reader wiIl be able to deternine whether the segregation model or maimurn rnixedness model wiIl give the greater conversion. SUMMARY I. The quantity Err) dt is the fraction of material exiting the reactor that has spent between time t and r * dr in the reactor. 2. The mean residence time
:1
=
d t r ) dr = r
(~13-1)
is equal to the space time T for constant volumetric flow, v = vU. 3. The variance about the mean residence time is
4. The cumulative distribution function Fit) gives the fraction of emuent material that has been In the reactor a time r or less:
I
- F ( t ) = fraction of effluent material that has been in
( 5 13-3)
the reactor a time t or longer
5 . The RTD functions for an ideal reactor are
CSTR: Laminar flow:
E(t) = 0
!< f 2
(513-61
934
Dishibrrfim of Residence T i m e br Chemical Reactors
Chap. 13
6. The dimensionless residence time is
7. The internal-age distribution, [I(&) da],gives the fraction of material inside the reactor that has been inside between a time ol and a time (u + da). 8. Segregation model: The convwsion is
and for multiple reactions
9. Maximum fixedness: Conversion can be calculated by soiving the following equations:
and for multiple reactions
from h,,, to h = 0. To use an ODE solver let z = h,,,
-k
CD-ROM MATERIAL
Cummary tJotcs
& Links
Learning Resources I . Summaw Now5 2. Web ~ a i e r i a lLinks A, The Attainable Region Analysis www,engin.umich.edu/-cdChapters/ARpgcs/lntm/intm, h rm and w~~:wi~~.u~.:a/fac/engine~ring/pmmar/ar~gion 4. Solved Problems A. Example CD 13-1 Calculate the exit concentrations series reaction A-El-C
8 . Example CD 13-2 Deteminalion of the effect of variance on rhe exit concentrations for the series reaction Sobed Problems
A---+B-C
Chap. 13
Livrng Example Problem
CD-ROM Materiat
Living Example Problems I . Exansple I 3 4 Laminar Flow Reactor 2. Example 13-8 Using Sofware 10 Make Maximum Mixrdness Model Cnlculations 3. Example 13-9 RTD and Complex Reactions 4, ~ x m p l eCD13-J A + B + C Effect of RTD 5. Example CD13-2 A +JB + C Effect of Variance Professional Reference Shelf 13R.1. Frrrirrg the Toil Whenever there are dead zones into which the material diffuses in and out, - the C and E curves may exhibit long tails. This section shows how to analytjcaliy describe fitring these tails to the curves. E(I) =
b = slope of In E vs. r MI
a = be [ l - F ( t , ) ]
Reference Shelf
13R.2.h1temal-AgeDistribution The internal-age distribution currently in the reactor is given by the distribution of ages with respect to how long the molecules have been in the reactor. The equation for the internal-age distribution is derived along and an example is given showing how it i s applied to catalyst deactivation in a "fluidized CSTR."
Example 13R2.1 Mean Catalyst Activity In a Fluidized Bed Reactor. 1 3R.3.Con~paringX,,g with X,, The derivation of equations using the second derivative criteria
i s carried out.
@jstriSutio?sof Reside~ceTirnec
$7
Chemical Reactors
Chap. 13
QUESTIONS AND PROBLEMS The subscript to each of the problem numbers indicates the level of difficulty: A, least
-
A=.
B=.
C = *
D=**
Clomehmork Problems
P13-1, Read over the problems of this chapter. Make up
an original problem that
uses the concepts presented in this chapter The guidelines are given in hoblem P4-I. RTDs from real reactors can be Found i n Ind. Eng. Chem., 49, LW ( 1957); Ind. Enx. Chern. Pmce.rr Des. Den, 3, 38 1 (1964). Can. J. Chsm. Eng., 37, l OT 11 959 ): It~dEn p. Chcm.. 41, 2 18 ( 1 952); Cliem. Et1.p. Sci.. 3. 26 (1954): and Inri. Eng. Ckertz., 53. 38 I I196 I ) . P13-2, What if ... (a) Example 13-1. What fractlon of the Ruid spends nine minutes or longer in the reactor? (hj The combinations of ideal reactors modeled the following real reactors, given E(B),F(Q), or I - F(Q).
(c) Example 13-3. How would the E(r) change if ,T, as reduced by 50% and q , w a increased by 50%? (d) Example 13-4. For 75% conversion. what are the relative sizes of the CSTR. PFR. and LFR? (e) Example 13-5. How does the mean conversion compare with the conversion calculated with the same, I, applied to an ideal PFR and CSTR? Can you give examples of E(t) where this calculation would and would not be n good estimate of X? (fi Example 13-6. Load the Living Erample Problem. How would your results change if T = 40'C? How would your answer change i f the reaction was pseudo first order with kcAa = 4 x lP3ks? What if the reaction were carried out adiabatically where C,, = C,, = 20 cnVmoUK. 6 = -10 kcaVmal k = 0.01 dm3/ml/min at 25'C with E = -8 kcaUml
Chap. 13
937
Ouestrons and Problems
(g) Example 13-7. Load the Living ExampIe Pmblem. How does the X,,, and X,$, compare with the conversion calculated for a BFR and n CSTR at the mean residence time'' (h) Example 13-8. t o a d the Lirina Eramplc Probletrt. I-low would your results change if the reaction was pseudo first order with k, = CA&= 0.08 mind? If the reaction was third order with kdAo = 0.08 min-I? The reaction was half order with = 0.08 min-I. Describe any trends. (i) Example 13-9. Load the Living Example Pmbletn. If the activation energtes in callm~rland E , = 5.000.E: = 1,OIK). and E3 = 9,000. how would the selectives and conversion of A change as the temperature was raised or lowered around 350 KQ G) Heat Effects. Redo Lit'E~rgEmmple PmbSems 13-7 and 13-8 for the case when the reaction is carried out adlnbatically with (1) Exothermic reaction with
with k given at 320 K and E = 10.000 catlrnol. (2) Endothermic reaction with
and E = 45 Id/mol. How will your answers change? (k) you were asked to compare the results from Example 13-9 for the asymmetric and bimodal distributions in Tables E13-9.2 and E13-9-4. What similarities and differences do you observe? What generalizations can you make? (I) Repeat 13-2(h) above using the RTD in Polymath program E E 3-8 to predict and compare conversions predicted by the segregation m d e l . (m) the reaction in Example 13-5 was half order with = 0.08 min-I? How would your answers change? Hint: Modify the Living Example 13-8 program. (n) you were asked to vary the specific reaction rates k, and in the series
k,
Heat effects
reaction A B --bC given on the Solved ProbIems CD-ROM? What would you find? ( 0 ) you were asked to vary the isothermal temperature in Example 13-9 from 3 0 K, at which the rate constants are given, up to a temperature of 509 KT The activation energies in callmol are El = 5000. E, = 7000,and E, = 9000. How would the selectivity change for each RTD curve? (p) the reaction in Example 13-7 were carried out adiabatically with the same panmeters as those in Equation [PIS-2(j).ljqHow would your answers change? (q) If the reaction in Examples 13-8 and 13-5 were endothermic and carried out adiabatically with
TCK) = 320
- IOOX
and
E = 45 Hlmol
[?13-2u). I ]
how would your answers change? What generalizations can you make about the effect of temperature on the results (e.g., conversion) predicted from the RTD? (r) If the reaction in Example 8-12 were carried our in the reactor described by the RTD in Example 13-9 with the exception that RTD is in seconds rather than minutes (i.e., r, = 1.26 5). how would your answers change'?
938
Dtstr~b~rt~ons of Res~denceT~mesfor Chem~calReactors
Chap. 13
P13-3c Show that for a first-order reaction A-I3 the exit concentration maximum rnixedness equation
dC1 d)l
E("
1- F ( h )
(C*-C*,)
is the same as the exit concentration given by the segregation mode!
[Hinc Verify
is a solution to Equation (P13-3.I).]
P13-4, The first-order reaction with 1: = 0.8 min-I is carried out in a real reactor with the following RTD function
0
r
22
t, min
For then E(t) = circle) For t > 22 then E(r) = 0 (a) What 1s the mean residence rime? (b) What is the variance? (c) What is the conversion predicted by the segregation model? (d) What is the conversion predicted by the maximum mixedness model? P13-5, A step tracer input was used on a real reactor with the following results: For I S 10 min. then C, = 0 For 10 5 t 5 30 rnin. then C, = 10 g/drn3 For 1 2 3Q min, then c;= 40 gldrn) The second-order react~onA -+ B with k = 0.1 drn'lmol . min is ro be carried out in the real reactor with an entering concentration of A of 1.25 molldm' at a volumetric flow rate of 10 dm3/mln. Were k is given at 325 K. (a) What is the mean residence lime r,? (b) What is the variance a?'? (t) What conversions do you e x v c t from an ideal PFR and an ideal CSTR in a real reactor w ~ f hr,?
Chap. 13
Question&and ProMems
(d) What i s the conversion predicted by (1) the segregation model? (2) the maximum mixedness model? (e) What conversion is predicted by an ideal laminar flow reactor? (0 Calculate the conversion using the segregation model assuming T(K) = 325 500X. P13-6# The following E(f) curve was obtained from a Vawr test on a tubular reactor in which dispersion is believed to occur.
-
t
(min)
F i g n ~P13-5 RTD.
A second-order reaction A
4 B with
kc,,,= 0.2 min-I
is to be carried out in this reactor. There is no dispersion wcurring eather upstream or dowosaeam of the reactor, but there is dispersion inside the reactor. Find the quantities asked for in parts (a) through (e) in pmblem PI 3-SB? P13-7, T h e irreversible $quid phase reaction is half order in A. The reaction is carried out in a nonideal CSTR,which can be modeled using the segregation model. RTD measurements on the reactor gave values of t = 5 min and u = 3 rnin. For an entering concentmtion of pure A of 1.0 molldm" the mean exit conversion was 108. Estimate the specific reaction rate constant, k , . Hinc Assume a Gaussian distribution. P13-SB The third-order liquid-phase reaction
a reactor that has the following RTD E(r) = 0 for 1 2 min The entering concentration of A is 2 moUdm3. (a) For isothemaF operation, whac is the conversion predicted by 1 ) a CSTR, a PFR. an LFR. and the segregation model, X,,. 2) the maximum mixedness model, XMM.Plot X VS. z (or X) and explain why the curve looks the way it does. (b) For isothermal operation, at what temperature is the discrepancy between X,, and XMMthe greatest in the range 300 < T < 350 K. was carried out in
940
D~strrbutionsof Resrdence Tirnes for C3emical Reactors
Chap. 13
Suppose the reaction is carried out adiabatically with an entering temperamre of 305 K. Calculate X,,,. Aclrliriui~alI@ilrmation k = 0.3 drnh/rnol?/rninat 300 K EIR = 20.000 AHRx = -40.000 calJmol C, = C, = 25 callrnollK (c)
P13-9,
Consider again the nonideal reactor characterized by the RTD data in Example 13-5.The irreversible gas-phase nonelementaq reaction
is first order in A and second order in B and is to be carried out isothermally. Calculate the conversion for: (a) A PFR, a laminar flow reactor wirh complete segregation. and a CSTR. (b) The cases of complete segregation and maximum mixedness.
Also (c) Plot I(a) and A(X) as a function of time and then determine the mean age E and the mean Iife expectancy (d) Mow wouId your answers change if the reaction is carried out adiabatically with parameter values given by Equation [P13+2(h).l I?
x.
CAo= CB, = 0.0313 molldrn" V = 1000 dm3, v , = 10 dm3/s, k = 175 dmh/rno12~s at 320 K. PI3-10, An irreversible first-order reaction rakes place in a long cylindrical reactor. There is no change in volume, temperature, or viscosity. The use of the simplifying assumption that there is plug flow in the tube leads to an estimated degree of conversion of 86.5%. Whnt would be the actually attained degree of conversion if the reaI state of Bow is laminar. wath negligible diffusion? P13-11, Consider a PFR, CSTR. and LFR. (a) Evaluate the first movement about the mean in, = ( I - T ) E(t)dt for a In PFR, a CSTR,and a laminar flow reactor. (h) Calculate the conversion in each of these ideal reactors for a second-order liquid-phase reaction with Da = f .O ( r = 2 min and kcAo= 0.5 min-'1. P13-12BFor the catillyric reaction I~
the rate law can be written as
Which will predict the highest conversion. the maximum mixedness model or the segregation mode!? Hint: Are the different ranges of the conversion where one mode! will dominate over the other.
942
Distributrons of Res~dence T~mesfor Chemical Reactors
Chap. 13
P13-f78 The reactions descrikd in Problem P6-12, are to be camed out in the reactor whose R'TD is described in Example 13-9. (a) Determine the exit selechvities using the segregation mode!. (b) Determine the exit selectivities using the maximum mixedness model. [c) Compare the selectivities in parts (a) and (b) with those that would be found in an ideal PFR and ideal CSTR in which the space time is equal to the mean residence time. P13-1Sc The reactions described in Problem P6-11, are to be carried out in the reactor whose RTD is desctlkd in Problem CDP13-I, with CAD= 0.8 rnol/dm3and CBo= 0.6 mof/drnf. (a) Determine the exit selectivities using the segregation model. (b) Determine the exit selectivities using the maximum mixedness model. (c) Compare the selecriv~tiesin parts and (b) with those that woutd be found in an ideal PFR and ideal CSTR rn which the space time is equal to the mean residence time. (d) How would your answers to (a) and (b) change if the reactor in PsobIem PI 3-ldC were used? P13-19BThe flow through a reactor is 10 dmvmin. A pulse test gave the following concentration measurements at the outlet:
6)
(a) Plot the external age distribution E l f ) as a function of time. (b) Plol the external age cumulative distribution F ( t ) as a function of time. (c) What are lhe mean residence time t, and the variance. u2? (d) What fraction of the material spends between 2 and 4 min in the reactor7 (el What fraction of the material spends longer than h min in the reactor? ( What fraction of the, material s ~ n d sless than 3 rnin in the reactor? (g) Plot the normalized distributions E(O) and F(O)as a function of 8. (h) What i s the reactor volume? (i) Plot internal age distribhtion I ( ( ) as a function of time. (j) What is the mean intemaI age (Y,? (k) Plot the intensity function, A(t). as a function of time.
Chap. 13
943
Questions end Problems
(I)The activity of a "flujdized" CSTR is maintained constaof by feeding fresh catalyst and removing spent catalyst at constant rate. Using the preceding RTD data, what is the mean catalytic activity if the catalyst decays according to the rate law
Problem P13- 1 9B will be continued in Chapter 14. P14-13,.
with
k, = 0.1 sml? (m) What conversion would be achieved in an ideal PFR for a second-order reaction with kc,, = 0.1 min-I and CAO= I moVdm3? In) Repeat (m) for a laminar flow reactor. (o) Repeat (m) for an ideal CSm. (p) What would be the conversion for a second-order reaction with kc,, = 0.1 min-I and ,C , = 1 moUdrn3 using the segregation model? (q) What would be the conversion for a second-order reaction with kc,, = 0.1 min-band CAD= 1 rnolldml using the maximum rnixedness model? Additional Homework Problems .
CDP13-&
Aftex showing that El?) for two CSTRs In series having different volumes is
L ( f) =
CDP13-B,
-{
t exp ~ ( 2 m 1- )
(3
- - exp [( 1
1 : ) r]}
you are asked to make a number of calculations. [2nd Ed. P19-111 Determine E(r) and from data taken from a pulse test in which the pulse is not perfect and the inlet concentration msies with time. [2n6
M.P13-151 CDPl3-C,
Derive the E ( t ) curve for a Bingham plastic flowing in a cylindrical t u k . [2nd Ed. P13-161 CDP13-D, The order of a CSTR and PFR in series is investigated for a third-order reaction. [2nd Ed. P13- 101 a P 1 3 - E B Review the Murpfiree pirot plant data when a second-order reaction occurs in the reactor. [ I st Ed. PI 3- 153 CDP13-FA Calculate the mean waiting time for gasoline at a service station and in a parking garage. [2nd Ed. P13-31 CDP13-G, Apply the RTD given by
to Examples 13-6 through 13-8. [2nd Ed.PI 3-&1 The multiple reactions in Problem 6-27 are carried out in a reactor whose RTD is described in Example 13-7. CDP1.1-IR Real RTD data from an industrial packed bed reactor operating under poor operation. [3rd Ed. P13-51 CDP13-JB Real RTD data from dlstribwtion in n stirred tank. [3rd Ed. P13-7,] CDP13-K, Triangle RTD with second-order reaction. [3rd Ed. P13-8,] CDP13-L, Derive E(i)for a turbulent flow mctor with IJ7th power law. CDPl3-MB Good probrem-must use numerical techniques. [3rd Ed. P13-12,] CDP13-H,
944
Dlstr~htronsof Residence Times for Chemical ileaclors
Chap. 13
CDP13-N, Internal age distribution for n catalyst. [Id Ed. P 13- 13 ], CDPl3-DQEA U of M, Doctoral Qualifying Exam (DQE). May. ZOO0 CDP13-DQEB U of M, Doctoral Qualifying Exam (DQE). April, 1999 CDP13-DQEC U of M, Doctoral Qualifying Exam (DQE), January, 1999 CDP13-DQED U of M, Doctoral Qualifying Exam (DQE), January, 1999 CDP13-DQEE U of M, Doctoral Qualifying Exam (DQE), January- I998 CDP23-DQFF U of M, Doctoral Qualifying Exam (DQE).January, 1998 CDP13-ExG U of M. Graduare Class FinaI Exam CDP13-New New Problems will be inserted from time to time on the web.
SUPPLEMENTARY READING
1. Discussions of the measurement and analysis of residence-time distribution can lx found in
CURL,R. L., and M. L. McMr~~ru. "Accuracies In re~idencetime measurements," AlChE J., 12, 8 19-822 (1966). LEVENSPIEL, 0..Chemical Reaction Digineering, 3rd ed. New York Wiley, 1999. Chaps. 1 1-16.
2. An excellent discussion of segregation can be found in
DOUGLAS,J. M.. "The effect of mixing on reactor design," AlChE Symp. S ~ K 48. Vol. 60,p. 1 (1964). 3. Also see
DUDUKOVIC, M., and R. FELDER.in CHEMI Modules on Chemical R e u ~ ~ i o Engineering, Vol. 4, ed. B. Crynes and H. S. Fogler. New York: AIChE. 3 985.
NAUMAN,E. B.,"Residence time distributions and rnicrornixing." Chem. Eng. Commun., 8, 53 (1981). NAUMAN, E. B.,and B.A. BUFFHAM.Miring in Conrinttous Flow Systems. New York: Wiley. 1983. R o ~ m s o B. ~ ,A,, and J. W. T ~ RChem. , Eng. Sci., 41(3), 469483 (1986). VILLERMAUX, J., "Mixing in chemical reactors:' in Chemical Reaction Engineering-Plenaly hcrures, ACS Symposium Series 226. Washington. D.C.: American Chemical Society, 3982. -
Models for Nonideal Reactors
14
Success is a journey, not a destination.
Ben Sweetland
Overview Not a11 tank reactors ase perfectly mixed nor do a11 tubular reactors exhibit plug-flow behavior. In these situations, some means must be used to allow for deviations from ideal behavior. Chapter 13 showed how the RTU was sufficient if h e reaction was first order or if the fluid was either in a state of complete segregation or maximum mixedness. We use the segregation and maximum mixedness modeL to bound the conversion when no adjustable parameters are used. For non-frst-order reactions in a fluid with good micromixing, more than just the RTD is needed. These situations compose a great majority of reactor analysis problems and cannot be ignored. For example, we may have an existing reactor and want to carry use,he RTD to evaluate out a new reaction in that reactor. To predict conversions and product distriPameters butions for such systems, a model of reactor flow patterns is necessary. To model these patterns, we use combinations and/or modifications of ideal reactors to represent real reactors. With this technique, we classify a model as being either a one-parameter model (e.g., tanks-in-saies model or djspersion madel) m a two-parameter mode1 (e.g., reactor with bypassing and dead volume). The RX3 is then used to evaluate the pwameter(s) in the modet Mer completing this chapter, the reader will be able to apply the tnnks-in-series model and the dispersion modet to tubular reactors. In addition, the reader will ?m able to suggest combinations of ideal reactors to model a teal reactor.
946
Models for Nonideal Reactors
Chap. 14
14.1 Some Guidelines The overall goal is to use the following equation
1 RTD Data + Kinetics + Model = Prediction I Conflicting goals
A Model Must Fit the data
Be ahle to extrapolate theory and
experiment Have realistic pararnerers
The choice of the particular model to be used depends largely on the engineering judgment of the person carrying out the analysis. It is this person's job to choose the model that best combines the conflicting goals of mathematical simpficizy and physical reaIisrn. There is a certain amount of art in the deveIopment of a mode! for a particular reactor, and the examples presented here can onIy point toward a direction that an engineer's thinking might follow. For a given real reactor. it is not uncommon to use all the models discussed previously to predict conversion and then make a comparison. Usually, the real conversion will be hounded by the model calculations. The following guidelines art! suggested when developing models for nonideal reactors: 1. The nzodel musf be mathenraticall~tractable. The equations used to describe a chemical reactor should be able to be solved without an inordinate expenditure of human or computer time. 2. The model rnusr realistically describe she claaracrerisfics of f h p nonrdea/ reacror. The phenomena occurring in the nonideal reactor must be reasonably described physicaIly, chemically, and mathematically. 3 . The model must nor have more sham rwo adjustable parameters. This constraint is used because an expression with more than two adjustable parameters can be fined to a great variety of experimental data, and the modeling process in this circumstance is nothing more than an exercise in curve fitting. The statement "Give me four adjustable parameters and I can fit an elephant; give me five and I can include his tail!'"^ one that I have heard from many calleargues. Unless one is into modem art, a substantially larger number of adjustable paameters is necessar?, to draw a reasonable-looking elephant.' A one-parameter model is. of course, superior to a two-parameter model if the one-parameter model is sufficiently realistic. To be fair, however, in complex systems (e.g., internal diffusion and conduction. mass transfer limitations) where other parameters may be measured independently. then more than two parameters are quite acceptable. Table 14-1 gives some guidelines that will help your analysis and model building of nonideal reaction systems.
Sec. 14.1
Some Guldeiines
TO
The Guidelines
947
TABLE14-1. A PROCEDURE mu CHOOSING A MODEL. PRED~CT THE OLTLET CONCENTRATION AHD CONVERSION
I . Cook ot the rmcfo,: a. Where are the inlet and outlet smams to and from the reactors? (Is try-passing a possibility?) b. LDok at the mixing system. How many impellers are there' (Could thee multiple mixing zones in the reactor?) c. Look at the configuration. (Is internal recirculation possible? Is the packing of the catalyst particles loose so channeling could occur?) 2. Look a/ t h ~rrorer . darn. Plot the Ett) and FttE curves. a. b. Plot and analyze the shapes of the E(O) and A O ) curves. Is the shape of the curve such that the curve or parts of the curve can he 61 by an ideal reactor model? Does rhe curve have a long tail suggesting a stagnant zone? Daes the curve have an earty spike indicating bypassing? c. CaIculate the mean residence time. t,, and variance. 02. How does the t, determined from the RTD data compare with r as measured with a yardqrick and flow rne1er9 Hnw large 1s the variance; xr it laqcr or smaller than r:? 3, C h o a o~ model ~ or perhops m a or three rnod~ls. 4. Use !he tracer data lo determine rAr model pnrametcr.r le~g.,n. D,, vb). 5. U,ve [Ire CRE algorithm in Chapter 4. Calcuiate the exit concentrations and conversron for the model system you have selected.
14.j. 1 One-Parameter Models
Here we use a single parameter to account for he nonideality of our reactor. This parameter is most always evaluated by analyzing the RTD determined from a tracer test. Examples of one-parameter models for nonideal CSTRs include a reactor dead volume 1/,,where no reaction takes place. or a fraction f of fluid bypassing the reactor, thereby exiting unreacted. Examples of one-parameter models for tubular reactors include the tanks-in-series model and the dispersion model. For the tanks-in-series model, the parameter is the number of tanks, n, and for the dispersion model, it is the dispersion coefficient. D,. Knowing the parameter values, we then proceed to determine the conversion and/or emuent concentrations for the reactor. We first consider nonideal tubular reactors. Tubular reactors may be Nonideal tubular empty, or they may be packed with same material that acts as a catayysl, heat-transfer medium, or means of promoting interphase contact. Until now when analyzing ideal tubular reactors. it usually has been assumed that the fluid moved through the reactor in pisron-like flow (PFR), and every atom spends an identical length of time in the reaction environment. Here, the velociv profile is-pot, and there is no axial mixing. Both of these assumptions are false to some extent in every tubular reactor; frequently, they are sufficiently false to warrant some modification. Most popular tubular reactor models need to have means to allow for failure of the plug-flow model and insignificant axial mixing assumplions; examples include the unpacked laminar flow tubular reactor, the unpacked turbulent flow, and packed-bed reactors. One of two approaches is usually taken to compensate for failure of either or both of the ideal assumptions. One approach involves modeling the nonideaI tubular reactor as a series
948
Mcdels tcr Nonideal Reactors
Chap. 1 4
of identically sized CSTRs. The other approach [the dispersion mode[) involves a modification of the ideal reactor by imposing axial dispersion on plug flow. 14.1.2 Two-Parameter Models
The premise for the two-parameter model is that we can use a combination of ideal reactors to model the real reactor. For example, consider a packed bed reactor with channeling. Here the response to a pulse tracer input would show two dispersed pulses in the output as shown in Figure 13-10 and Figure 14-1.
Channeling
=LO
uh
If
Dead Zones
Figure 14-1 (a) Real system; (b) outlet for a pu1.w input; (c) model system.
Here we could mode1 the real reactor as two ideal PBRs in parallel with the two parameters being the fluid that channels, ub. and the reactor dead volume, V,. The real I-eaCtQCvoume is V = V, + V, with v, = oh + us.
14.2 Tanks-in-Series (T-I-S) Model
n=?
In this section we discuss the use of the tanks-in-series (T-I-S) model to describe nonideal reactors and calculate conversion. The T-I-S model is a one-parameter model. We will analyze the RTD to determine the number of ideal tanks, n, in series that will give approximately the same RTD as the nonideal reactor. Next we will apply the reaction engineering algorithm developed in Chapters I through 4 to calculate conversion. We are first going to develop the RTD equation for three tanks in series (Figure 14-2) and then generalize to n reactors in series to derive an equation that gives the number of tanks in series that best fits the RTD data. Pulse,
I n Figure 2-9, we saw how tanks in series could approxi-
mate a PFR,
Figure 14-2
Tanks in series: (a) rent system. (b) model system
The RTD will be analyzed from a tracer pulse injected into the first reactor of three equally sizetI CSTRs in series. Using the definition of the RTD presented in Section 13.2. the fraction of material leaving the system OF three reactors li.e., leaving'the third reactor) that has been in rhe system between time I and I + df is
Then
In this expression, C3(f) is the concentration of tracer in the effluent from the third reactor and the other terms are as defined previously. It is now necessary to obtain the outlet concentration of tracer. C,(t), as a function of time. As in a single CSTR,a material balance on the first reactor gives
We perfurn LI tracer balance on each reactor to obtain Cdt)
Integrating gives the expression far the tracer concentration in the effluent from the first reactor: C, = C,e
-cr/Y~
= Coe-f/~~
(14-3)
The volurnetri~flow rate is constant ( u = v,) and all the reactor volumes are identical ( V , = V j = V,): therefore. all the space times of the individual reactors are identical (T,= r2 = z,). Because Vi is the volume of a single reactor in the series, .ti here is the residence time in one of the reactors. not in the entire reactor system (i.e., .ti = 'r/n). A material balance on the tracer in the second reactor gives
Using Equation (14-3) to substitute for C,,we obtain the first-order ordinary differential equation
950
Models for Nonideal Reactors
Chap. 14
This equation is readily solved using an integrating factor e"" along with the initial condition C2 = 0 at t = 0,to give
The same procedure used for the third reactor gives the expression for the concentration of tracer in the effluent from the third mctor (and therefore from the reactor system),
Substituting Equation (14-5) into Equation (14-1), we find that
Generalizing this method to a series of n CSTRs gives the RTD for n CSTRs in series, ECr): RTD for equal-size tanks in series
Because the total reactor volume is nV,, then T , = zln, where T represents the total reactor volume divided by the flow rate, v:
where @
= tlz. Figure 14-3 illustrates the RTDs of various numbers of CSTRs in series in ;I two-dimensional plot (a) and in a three-dimensional plot (b). As the number becomes very large. the behavior of the system approaches that of a plug-flow reactor.
See. 14.2
Tanks-In-Series (T-I-S) Model
(b)
Figure
Tanks-in-seriesresponse to a pulse tracer input for different numbers of tanks.
We can determine the number of tanks in series by calculating the dimensionless variance r r i from a tracer experiment. =
2
=
"
(@- l)2E(0)d0
As the number of tanks increases, the
variance decreases.
The number of tanks in series is
This expression represents the number of tanks necessary to model the real reactor as n ideal tanks in series. If the number of reactors, n, turns out to be
kIo&ls !or Nonideal Reactors
952
Chap. 1
small, the reactor characteristics turn out to be those of a single CSTR or per haps two CSTRs in series. At the other extreme, when n turns out to be large we recalf from Chapter 2 the reactor characteristics approach those of a PFR Ifthe reaction is first order. we can use Equation (4-1 1) to calculate th conversion,
where
(t i s acceptable (and usual) for the value of n calculated from Equation (14-12 to be a noninteger in Equation (4-1I ) to calculate the conversion. For reaction other than first order, an integer number of reactors must be used and sequen tial mole baIances on each reactor must be carried out. If, for example, n : 2.53, then one could calculate the conversion for two tanks and also for thre tanks to bound the conversion. The conversion and effluent concentration would be solved sequentially using the algorithm developed in Chapter 4. ThE is, after solving for the effluent from the first tank, it would be used as th input to the second tank and so on as shown in Table 14-2.
TANKS-IN-SERIES SECOND-ORDER REACTON
TAALE 14-2.
1
Two-Reactor Sy stern
Three-Reactor System
For three equally sized reacton
For two squally r i ~ e dreactors
For a smond-order reaction. the combined mole balance. rate law, and stoichiometry for the first reactor gives =
C A ~C w t
k, Solving for
I/
CZ,,
C,,,,
Two-Reactor System:
'c2
=Z
2
Three-Reactor System: 7 , = 5 3
Solving for exit ooncenhtion from remtor I for each reactor system giver
The exit concentration from the second reactor for each reactor system gives
Two-Reactor System c42
=
-I
Three-Reactor System
+Jm
C'I? =
Zt2k
-1
-JX 2zJk
Balancing on the third reactor for the three-reactor system and solv~ngfor its oulIet concentragives
tlon. C,,
.
C'4 3 = - 1 JJ-
2e,k
The corresponding conveston For the two- and three-reactor Eysterns are
x,- C~~- C ~ 2
#
XI
C~~
For n = 2.53. tX, c
X < X;
CA~P -G 3 -c~~
)
L
chclf
Tanks-in-Series Versus Segregation for a First-Order Reaction We have already stated that the segregation and maximum mixedness models are equivalent for a first-order reaction, The proof of this statement was left as an exercise in Problem Pi3-38. We now show the tanks-in-series model and the segregation models are equivalent for a first-order reaction. Exurnpie 14-1 Eq~livulerrcyof Modelsfor a First-Order Reaction
Show that X-r-Is = XMM for a first-order reaction A
A
B
Sulutiofl
For a first-order reaction, we already showed i n Problem P13-3 that
Therefore we only need to show Xseg = XT.I.S. For a first-order reaction in a batch reactor the conversion is
Segregation Model
Using Maclaurin's series expansion gives
954
Modals for Nonideal Reactors
Chap. 14
neglecting the error term
-
x = rk -
51
2 r
h(r)dt
(E14-1.6)
0
To evaluate the second term, we first recall Equation (El 3-2.5) for the variance
Rearranging Equation (E14-1.8)
Combining Equations (E 14- 1.6) and (E 14-1.9). we find the mean conversion for the segregation model for a first-order reaction i s
Tanks in Series
Recall From Chaprer 4. for n tanks in series for a first-order reaction. the conversion is
Rearranging yields
We now expand in a binomial series
I
Neglecting the error gives
1
Rearranging Equation (14-12) in he fam
and substituting in Equation (E14-1.14) the mean conversion far the T-I-S model is
We see: that Equations (E14-1.lo) and (E14- 1.15) are identical; thus, the conversions are identical, and for a first-order reaction we have Important Result
But this is true only for a first-order reaction.
14.3 Dispersion Model The dispersion model is also used to describe nonideal tubular reactors. In this model, there is en axial dispersion of the material, which is governed by an analogy to Fick's law of diffusion, superimposed on the flow as shown in Figure 14-4. So in addition to transport by bulk ffow, UA,C, every component in the mixture- is transported through any cross section of the reactor at a rate equal to I-D&,(dCldz)] resulting from molecuEar and convective diffusion. By convective diffusion ( i t . , dispersion) we mean either Aris-Taylor dispersion in laminar flow reactors or turbulent diffusion resulting from turbulent eddies. Radial concentration profiles for plug flow (a) and a representative axial and radial profile for dispersive flow (b) are shown in Figure 14-4. Some molecuIes will diffuse forward ahead of molar average velocity while others will lag behind.
e4Zr Dispersion
Figure 14-4 Concentration profiles [a) without and (b) with dispersion.
956
Models for Nonideal Reactors
Chap. 1 4
To illustrate how dispersion affects the concentration profile in a tubular reactor we consider the injection of a perfect tracer pulse. Figure 14-5 shows how dispersion causes the pulse to broaden as it moves down the reactor and becomes less concentrated.
Tracer pulse with
tmcerpulse mth
dispenion
Figure 14-5 Dispersion in a tt~bularreactor. (From 0. Levenspiel. Cltrmirul Reuctron Engineering, 2nd ed. Copyright D 1972 John Wiley & Sons, Inc. Rephnted by permission of John Wiley & Sons. Inc. All sights reserved.)
Recall Equation ( 1 1-20), The molar flow rate of tracer IFT) by both convection and dispersion is
In this expression D ,is the effective dispersion coefficient (rnYs) and U (mls) is the superficial velocity. To better understand how the pulse broadens. we refer to the concentration peaks q and t3 in Figure 14-6. We see that there is a concentration gradient on both sides of the peak causing moIecules to diffuse away from the peak and thus broaden the pulse. The puise broadens as it moves through the reactor.
FIgure 14-6 Symetric concentration gradients causing the spreading by dispersion of a puIse input.
Correlations for the dispersion coefficients in both liquid and gas system! may be found in Levelspiel.' Some of these correlations are given in Sectior 14.4.5. 0.Levenspiel. Chemical Rmcrion Engineering (New York: Wiley, 14623, pp 29b293.
Sec. 14.4
Flow, Readion. and D~spars~on
A mole baiance on the inert tracer
T gives
ijF, -. = A c -JC,
ar Substituting for FT and dividing by the cross-sectional area A,, we have rj:
Pulse tracer balance dicpersion
D oa2cC, - - - - -a(uc,) 3%'
a;
ac, ar
Once we know the boundary conditions, the solution
The Plan
to Equation (14-13) will give the outlet tracer concentration-time curves. Consequently, we will have to wait to obtain this soIution until we discuss the boundary conditions in Section 14.4.2. We are now going to proceed in the following manner. First, we will write the balance equations for dispersion with reaction. We will discuss the two types of boundary conditions: closed-closed and open-open. We will then
obtain an anaiytical solution for the closed-closed system for the conversion for a first-order reaction in terms of the Peclet number (dispersion coefficient) and the Damkohler number. We then will discuss how the dispersion coeffrcient can be obtained either from correlations or from the analysis of the RTD curve.
14.4 Flow, Reaction, and Dispersion Now that we have an intuitive feel for how dispersion affects the transport of molecules in a tubular reactor, we shalI consider two types of dispersion in a tubular reactor, laminar and turbulent. 14.4.1 Balance Equations
A mole balance is taken on a particular component of the mixture (say, species A) over a short length Az of a tubular reactor of cross section A, in a manner identical to that in Chapter 1, to arrive at
Combining Equations (14-14) and the equation for the molar flux FA. we cad rearrange Equation (1 1-21,) in Chapter I1 as
This equation is a second-order ordinary differential equation. It is nonlinear when r+,is other than zero or first order. When the reaction rate r, i s first order. r, = -kc,. then Equation (14-16)
Models for Nonidesl Reactors
Chap. !4
Flow, reaction, and dispersion
is amenable to an analfizical solution. However, before obtaining a solution, we put our Equation (14-16) describing dispersion and reaction in dimensionless form by letting $ = C,/C,, and A = z / L : D, = Dispwsion coefficient h = Damklibler
Damktlhler number for a iirst-order reaction
The quantity Do appearing in Equation (14-17) is called the DarnkGhler number for convection and physically represents the ratio
Da
Per
-- 106. 'I
- lo'
For packed bed<
PC!
Rate of consumption of A by reaction = Rate of transport of A by convect~on
(14-18)
The other dimensionless term is the Pecler number Pe, Pe =
For open tubes
=
Rate of transport by convection Rate of transport by difhsion or dispersion
-
D,
(14-19)
in which I is the characteristic length term. There are two different types of Peclet numbers in common use. We can call Per the reactor Peclet number;it uses the reactor length, L, for the characteristic length, so Per = ULID,. It is Per that appears in Equation (14-17). The reactor Peclet number, Per, for mass dispersion is often referred to in reacting systems as the Bodenstein number, Bo, rather than the Peclet number. The other type of Peclet number can be called the fluid Peclet number, Pef: it uses the characteristic length that determines the fluid's mechanical behavior. In a packed bed this length is the particle diameter d,, and Pe, = Ud,/i$D,. (The term U is the empty tube or superficial velocity. For packed beds we often wish to use the average interstitial velocity. and thus U/+ is commonly used for the packed-bed velocity term.) In an empty tube, the fluid behavior is determined by the tube diameter d,, and ?ef = Ud,lD,. The fluid Peclel number. Pep is given in all correlations relating the Peclet number to the Reynolds number because both are directly related to the fluid mechanical behavior. It is, of course, very simple to convert Pef to Pe,: Multiply by the ratio Lld, or Llrl,. The reciprmal of Pe,, D,IUL. is sometimes called 'the vessel dispersion number. 14.4.2 Boundary Conditions
There are two cases that we need to consider: boundary conditions for closed vessels and open vessels. In the case of closed-closed vessels, we assume that there is no dispersion or radial variation in concentration either upstream (closed) or downstream (closed) of the reaction section; hence this is a closed-closed vessel. In an open vessel. dispersion occurs both upstream (open) and downstream (open) of the reaction section: hence this is an
Sec. 14 4
Flw, Reaction, anr! Dispersion
959
open-open vessel. These two cases are shown in Figure 14-7, where fluctuations in concentration due to dispersion are superimposed on the plug-flow velocity profile. A closed-open vessel boundary cond~tionis one in which there js no dispersion in the entrance section but there is dispersion in the reaction and exit sections.
I
Open-open vessel
Closed-closed vessel
Figure 14-7 Types of boundary conditions.
I
14.4.26 Closed-Closed Vessel Boundary Condition
I
~ d arclosed-closed vessel.
we have plug flow (no dispision) m the immediate left\of the entrance line (z = V )(closed) and to the immediate right of the exit t = L (Z = L+) (closed). However, between 2 = 0 ' and z = L-, we have dispersion and reaction. The corresponding entrance boundary condition is
++
Atz=b:
F,,
+
3
z =0
FA(@)= FA(@)
Substituting for FA yields
Solving for the entering concentration CA(O-) = C,,: Concentration hundary conditions at the entrance
At the exit to the reaction section. h e concentration is continuous, and there is no gradient in !racer concentration. Concentration boundary conditions ar the exit
At := L:
960
Models b r Nonideal Reactors
Chap. 14
These two boundary conditions, Equations ( 14-191 and ( 14-20), first stated by Dan~kwerts,~ have become known as the famous Dunrkwerts boundcondirions. Bischoff has given a rigorous derivation of them. sofving the differential equations governing the dispersion of component A in the entrance and exit sections and taking the limit as Do in entrance and exit sections approaches zero, from the solutions he obtained boundary conditions on the reaction section identical with those Danckwerts proposed. The closed-closed concentration boundary condition at the entrance is shown schematically in Figure 14-8. One shourd not be uncomfortable with the discontinuity in concentration at z = 0 because if you recall for an ideal CSTR the concentration drops immediately on entering from CAoto.,,C ,, For the other boundary condition at the exit z = L we see the concentration gradient has gone to zero. At steady slate, it can be shown that this Danckwerts boundary condition at := L also applies to the open-open system at steady state.
Prof. P V. Danckwens, Cumbridge University. U.K.
Cm
pb(z)qCA (L*)
CA (L-)
02
o+
t- L+ z =L
=0
(a)
(b)
Figuw 14-8 Schematic of Danckwerts b o u n d q conditions. (a) Entnnce (b) Exit
14.4.28 Open-Open System
For an open-open system there is continuity of flux at the boundaries at z = 0, F~CO-)= F ~ ( O + ) Open-open boundary condition
P.V. Danckwerts, Chem. Eng. ScL, 2, 1 (1953). 'K.B. Bischoff, Chem. Eng, Sci., 16, 13 1 (1961).
-
Sec 14 A
Flow, Reaction and Dtspersion
At z = L, we have continuity of concentration and
14.4.2C Back to the Solution for a Closed-Closed System
We now shall solve the dispersion reaction balance for a first-order reaction
For the closed-closed system, the Danckwerts boundary conditions in dimensionless form are At h = 0 rhen 1=-
At X = 1 then At the end of the reactor, where
dh
--
+ @(o*) ( I 4-25)
=0
A = 1. the solution
to
Equation (14- 17) is
- C A L =1 -X
+L--
'40
-
4qexpIPei2J2) ( I f q ) 2 cxp (Pe,q/2)- ( 1 - q)Z exp (- Per@)
where q = This solution was first obtained by Danckwerts5 and has been published many places (e.g., Levenspie16).With a sIight rearrangement of Equation (14-26), we obtain the conversion as a function of Da and Pe,.
X=l-
4q exp /PeJ2)
/ 1 + q)* exp (PedR) - ( 1 - q ) 2 exp (- Pe,q/2)
Outside the limited case of a fisst-order reaction, a numerical solution of the equation is required, and because this is a split-boundary-value problem, an iterative technique is required. To evaluate the exit concentration given by Equation (14-26) or the conversion given by (14-27), we need to know the Darnkohler and Peclet numbers. The DamkBhler number for a first-order reaction, Da = ~ k can , be found using the techniques in Chapter 5. In the next section, we discuss methods to determine Do, the Peclet number.
P. V. Danckwens, Chem, Eng. Sci., 2, 1 (1953). Levenspiel. Ckemiccll Reaction Engineering, 3rd ed. (NewYork: Wiley. 1999).
962
Models tor Nonideal Reactors
Chap 14
14.4.3 Finding Da and the Peclet Number Three ways to
find D,
There are three ways we can use to find D, and hence Per 1. Laminar flow with radial and axia3 molecular diffusion theory 2. Correlations from the literature for pipes and packed beds 3. Experimental tracer data At first sight, simple models described by Equation (14-13) appear to have the capability of accounting only for axial mixing effects. It will be shown, however, that this approach can compensate not only for problems caused by axiaI mixing, bur also for hose cawed by radial mixing and orher nonflat v e l o c i ~pm$lesn7 These fluctuations in concentration can result from different flow velocities and pathways and from molecular and turbulent diffusion. 14.4.4 Dispersion in a Tubular Reactor with Laminar Flow
In a laminar flow reactor, we know
that the axial velocity varies in the radial direction according to the I-Iagen-Poiseuille equation:
where U is the average velocity. For laminar flow, we saw that the RTQ function .€(I)was given by
In arriving at this distribution E(r). it was assumed thar there was no transfer of molecules in the radial direction between streamlines. Consequently. with the aid of Equation (13-431, we know that the molecules on the center streamline ( r = 0)exited the reactor at a time f = r12, and molecules traveling on the streamline, at r = 3R14 exited the reactor at time
Sec 14 4
Flow. Reaction, and Dispersion
963
The question now arises: What would happen if some of the molecules traveling on the streamline at r = 3W4 jumped (i.e., diffused) onro the streamline at r = O? The answer is that they would exit sooner than if they had stayed on the streamline at r = 3Rf4. Analogously, if some of the molecules from the faster streamline at r = 0 jumped (i.e., diffused) on to the streamline at r = 3W4, they would take a longer time to exit (Figure 14-9). In addition to the molecules diffusing between streamlines, they can also move forward or backward reIative to the average fluid velocity by molecular diffusion (Fick's law). With both axial and radial diffusion occurring, the question arises as to what will be the distribution of residence times when molecules a e transponed between and along streamlines by diffusion. To answer this question we will derive an equation for the axial dispersion coefficient, D,,that accounts for the axial and d i a l diffusion mechanisms. In deriving D,. which is referred to as the ARS-TapIor dispersion'cwfficient, we closely follow the development given by Btenner and E d w a r d ~ . ~
Molecules diffusing between streaml~nes
and hack and forth along a streamliiic
z=o
,'
1
zFigure 14-9 Radial diffusion in laminar Row
The eonvdctivedi~usionequation for solute (e.g., uacer) nsnspwt in both the axial and radial direction can be obtained by combining Equations (1 1-31 and (I 1-15),
where c is the salute concentration at a particular r. z, and I. We are going to change the variable in the axial direction z ro z', which corresponds to an observer moving with the fluid
A value of z* = 0 corresponds to an observer moving with the fluid on the center streamIine. Using the chain rule, we obtain
' H. Brenner and D. A. Edwards. Mocmtmnspar! Pmcesses (Boston: 8 utterworthHeinernann. 1993).
Models for Nonideal Reactors
Chap 74
Because we want to know the concentrations and conversions at the exit to the reactor, we are really only interested in the average axial concentration ?, which is given by
@ ,
I' 3 , ~ >
Consequently, we are going to solve Equation ( 14-30) for the solution concentration as a function of r and then substitute the solution c (r, z, t) into Equation (14-31) to find ,;( 0. All the intermediate steps are given on the CD-ROM R14.1. and the partial differential equation describing the variation of the average axial concentration with time and distance is
Reference Shelf
ac "C+uz=,*a*
a=*
f 14-32)
ar*l
where D* is the Aris-Taylor dispersion coefficient: Aris-Taylor disprsion
coefficient
That is, for laminar flow in a pipe
Figure 14-10 shows the dispersion coefficient. D' in terms of the ratio D' lU(2R) = D*lUdf as a function of the product of the Reynolds and Schmidt numbers. 14.4.5 Correlations for
D,
14.4.5A Dispersion for Laminar and Turbulent Flow in Pipes
An estimate of the dispersion coefticient, D,, can be determined from Figure 14-11. Here d, is the tube diameter and Sc is the Schmidt number discussed in Chapter 11. The Bow is laminar (streamline) below 2,100. md we see the ration {DoU/d,) increases with increasing Schmidt and ReynoIds numbers. Between Reynolds numbers of 2,100 and 30,000, one can put bounds on D, by calculating the maximum and minimum values at the top and bottom of the shaded region.
Sec
Id4
965
Flow, React!on, and Dispersion
Figure 14-10 Correlation for dispersion for streamline flow in pipes. (From 0.Levenspiel, Chemical Reaction Engineering, 2nd ed. Copyright O 1972 John Wiley & Sons. Inc. Repnntd by permission of John Wlley Sr Sons, Inc. All nghtr reserved.) [Note: D = D, and D =DAB]
Once the Reynolds number is calm-
Iated. D, can be found.
ail lol
I
! ~rlltrl
103
I ! ]
111rr8l tQ4
I
I 1111111
106
I
I I I I I ~
10"
Figure 14-11 Comefation for dispersion of fluids flowing in pipes. (From 0. Levenspiel. Chemical Reucrion En~irl~ering, 2nd pd. Copyright Q 1972 John Wiley & Sons, Inc. Reprinted by p e m i s ~ l o nof John Wiley & Sons. rnc. AH rights reserved.) [Nnie: D = D m ]
966
Models h r Nonideat Reac!ors
Chap. f 4
14.4.50 Dispersion in Packed Beds
For the case of gas-solid catalytic reactions that take place in packed-bed reactors, the dispersion coefficient, D,, can be estimated by using Figure 14-12. Here d, is the particle diameter and E is the porosity.
Figure 14-12 Experimental findings on dispersion of fluids flowing wirh mean axial velmity u m packed beds. (From 0.L-evenspiel, Chemical Rcacrion Engtneerins, 2nd ed. Copyright Q 1972 John Wiley & Sons, Inc. Reprinted by perm~ssionof John Wiley & Sons, Inc. All rights reserved.) [Note: D = Do]
14.4,6 Experimental Determination of DD,
The dispersion coefficient can be determined from a pulse tracer experiment. Here, we will use I, and u2to solve for the dispersion coefficient D, and then the Peclet number, Pe, Here the effluent concentration of the reactor is measured as a function of time. From the effluent concentration data, the mean resi d e time, ~ ~ r,, and variance, u2,are calculated, and these values are then used to determine Do. To show how this is accompljshed, we wiIl write
in dimensionless form, discuss the different types of boundary conditions at the reactor entrance and exit, solve for the exit concentration as a function of dimensionless time (O= r l z ) , and then relare Do, u2. and T. 14.4.6A The Unsteady-State Tracer Balance
The first step is to put Equation (14-13) in dimensionless form to arrive dimensionless group(s) that characterize the process. Let
at the
Sec. 14.4
967
Flow, Reaction, and Dispersion
For a pulse input, Cm is defined as the mass of tracer injected, M,divided by the vessel volume. K Then
The initial condition is At t = 0, 2 > 0, CdO",O) = 0,
Initial condition
${a+>= 0
(14-35)
The mass of tracer injected, M is
14.4.68 Solution for a Closed-Closed System
In dimensionless form, the Danckwerrs b o u n d q conditions a+e /-
Equation (14-34) has been solved numericaIly for a pulse injection, and the resulting dimensionless effluent tracer concentration. JJ,,;,, is shown as a function of the dimensionless time O in Figure 14-13 for various Peclet numbers. Although analytical solutions for $ can be found, the result i s an infinite series. The corresponding equations for the mean residence time, 1,. and the variance. . ' u areg
0 t,, = T
and
which can be used with the solution to Equation (14-34) to obtain
See K. Bischoff and 0.Levenspiel. Ad\: Chent. Eng . 4, 95 ( 1963).
'
Models for Nonideat Reactors
Cham 14
Intermediate amount d dispemon
Effects of dispersion on the efffuent tracer
concentration rga amounl ol d~spersmn.
Figure 14-13 C curves in closed vessels for various extents of back-mixing as predicted by the dispersion model. (From O Levenspiel. Cltemicflf Reactton Eng~necring,2nd ed. Copyright 0 1972 John Wiley & Sons. lnc. Reprinted by p e m i s ~ i o nof John Wley & Sons, Inc. All rights m e w e d ) [Note: D D,]I0
Calculating Pe, using t, and rr? determined from RTD data for a closed-closed system
Consequently, we see that the Peclet number, Pe, (and hence D,).can be found experimentally by determining t, and uZ from the RTD data and then solving Equation ( 14-39) for Pe,. 14.4.6C Open-Open Vessel Boundary Conditions
When a tracer is injected into a packed bed at a location more than two or three particle diameters downstrenm from the: entrance and measured some distance upstream from the exit, the open-open vessel boundaty conditions apply. For an open-open system, an analytical solution to- Equation ( 14- 13) can be obtained for a pulse tracer input. For an open-open system, the boundsry conditions at the entrance are
'"0. Levenspiel, Chemical Reaction Engineering, 2nd ed. (New York: Wlley. 1972).
pp. 282-284.
Sec. 14.4
Flow. Readion, and D~spers~on
969
Then for the case when the dispersion coefficient is the same in the entrance and reaction sections:
Eecause there are no discontinuities across the boundary at z = 0
As the exit Open at the exit
There are a number of perturbations of these boundary conditions that can be applied. The dispersion coefficient can take on different values in each of the three regions (z < 0. 0 2 2 5 L,and z > L), and the tracer can also be injected at some point z, rather than at the boundary, z = 0. These cases and others can be found in the supplementary readings cited at the end of the chapter. We shall consider the case when there is no variation in the dispersion coefficient for aIl ;and an impulse of tracer is injected at z = 0 at t = 0. For tong tubes (Pe > 100) in which the concentration gradient at k .s will be zero, and the solution to Equation (14-34) at the exit is1' Vnlid For Pe, > 100
The mean residence time for an open-open system is Calculate r for an open-opcn system
where T is based on the volume between z = 0 and z = L (i.e., reactor voIume measured with a yardstick). We note that the mean residence time for an open system is greater than that for a closed system. The mason i s that the molecules can diffuse back into she reactor after they exit. The variance for an
open-open system is Calculate Per for en
open-pen
system.
"W. Jost. Dgusion in Solids, Liquids gad Gases (New York: Academic Press. 1960). pp. 17, 47.
970
Mcdels for Nonideal Reactors
Chap. 14
We n o w consider two cases for which we can use Equations (14-39) and (14-46) to determjne the system parameters: Case J . The space time s is known. That is, V and vo are measured independently, Here we can determine the Peclet number by determining t,,, and u2 from the concentration-time data and then using Equation (14-46) to calculate Pe, We can also caiculate r, and then use Equation (14-45) as a check, but this is usually less accurate. Case 2. The space time T is unknown. This situation arises when there are dead ar stagnant pockets that exist in the reactor along with the dispersion effects. To analyze this situation we firsf calculate r,, and cr' from the data as in case 1. Then use Equation (14-45) to eliminate .c2 from Equation (14-46) to arrive at
Finding the effective reactor voume
We now can solve for the Peclet number in terms of our experimentally determined variables cr? and t i . Knowing Pe,, we can solve Equation (14-45) for z. and hence V. The dead volume is the difference between the measured volume (i-e., with
a yardstick) and the effective volume calculated from the RTD. 14.4.7 Sloppy Tracer Inputs It is not always possible to inject a tracer pulse cleanly as an input to a system because it takes a finite time to inject the tracer. When the injection does not approach a perfect pulse input (Figure 14-14), the differences in the variances between the input and output tracer measurements are used to calculate the Peclet number:
where rr?" is the variance of the tracer measured at some point upstream (near the entrance) and o:, is the variance measured at some point downstream (near the exit).
z =L Measure Figure 14-14 Imperfect tracer input.
971
Flow, Reaction, and Dispersion
Sec. 14.4
For an opeaspen system,it has been shownll h a t the Peclet number can be calculated from the equation
Now let's put all the material in Section 14.4 together to determine the conversion in tubular reactor for a first-order reactor. Exampic 1 4 2 Conversion Using Disgersion and Tanks-in-Series Models The first-order reaction
is carried out in a EO-cm-diameter tubular reactor 6.36 m in Ienpth. The s p i f i c reaction rate is 0.25 min-I. The results of a tracer test carried out on this reactor are shown in Table E14-2.1.
Calculate conversion using (a) the closed vessel dispersion model, (b) PFR, (c) the tanks-in-series model, and (d) a single CSTR. Solution (a) We will use Equation (14-27) to calculate the conversion
X=l-
4q exp (Per/2) (I + q I 2 exp{Pe,q/Z)- ( I - q ) 2 exp(-Pe,q/2)
( I 4-27)
JwT Do = ~ k and , Pe, = ULID,. We can calculate Per from
where g = Equation (14-39):
2 o'=--tZ
First cafculate I, and u2 Fmm RTD data.
2 (,-e-Pt') Pe, Pe:
However, we must find T~ and 02 fmm the tracer concentration data first.
'2R.
Aris, Chem. Eng. Sci.,
9,266 (1959).
972
Models tor Nonrdeal Reactors
Chap. 14
Conrider the data listed in Table E14-2.2. TO DETERUIWF I , AND TABLE E14-2.2. CALCCLATIOYS
U'
Here again spreadsheets can be used to calculate r2 and rr:.
To find E l f ) and then
t,
w e first find the area under the
1-
C curve, which is
C ( r ) di = 50 g m i n
0
Then
Calculating the first term on the right-hand side of Equation (E14-2.2). we find
Substituting these values to Equation (E14-2.2). we obtain the variance, ul.
Most people, including the author. would use Polymath or Excet to form Table E14-2.2 and to calculate t,,, and rr2. Dispersion in a closed vessel is represented by
Calcuinte Pe, from r, and d.
Solving for Pe, either by trial and error or using Polymath, we obtain Next. calculate Do, q. and X.
Next we catcvlate Do to be
Sec. 14.4
I
Two-Parameter Models
Using the equations for y and X give\
Then
Substitution into Equation (14-40) yields X= l -
Dispersion M d e l
4( 1.30)e''S1?'
(5.3)' exp (3 87) - (-0.3)' exp (-4.87) 68% conversion for the dispersion model
When dispersion effects am present in this tubular reactor, 68% conversion is achieved. (bl If the reactor were operating ideally as a plug-flow reactor. the conversion would be X = 1 - e-Tk = 1 - p-Da = 1
PFR Tanks-in-series model
-
29
= 0.725
That is, 72.5% conversion wouId be achieved In an ideal plug-flow reactor. (c) Conversion using the tanks-in-series model: We recall Equation (14-12) to calculate the number of tanks in series:
To calculate the conversion. we recaIE Equation (4-11). For a first-order reaction for n tanks in series. the conversion is 1 - 1I = 1[ I + ( t / n )kIn (1 ( I + t,kp X = 67.7% for the tanks-in-wries model
x= I--
1
+ 1 .29/4.35)435
(d) For a single CSTR,
CSTR
So 5 6 2 % conversion would be achieved in a singte ideal tank. Summary:
Summary
Dispersion: X = 68.0% Tanks in series:X = 67.7%
In this example. correction for finite dispersion, whether by a dispersion m d e t ar a tanks-in-series model, is significant when c o m p a ~ dwith a PFR.
974
Moclels for Nonideal Reactors
Chap. 14
14.5 Tanks-in-Series Model Versus Dispersion Model We have seen that we can apply both of these one-parameter models to tubular reactors using the variance of the RTD.For first-order reactions, the two models can be applied with equal ease. However, the tanks-in-series model is mathematically easier to use to ob~brainthe effluent concentration and conversion for reaction orders other than one and for multiple reactions. However, we need ta ask what would be the accuracy of using the tanks-in-series model over the dispersion model. These two models are equivalent when the Peclet-Bodenstein number is related to the number of tanks in series, n, by the equation1"
Equivalency
or
between models of
tan ks-in-senes and dispersion
where Bo = U r n " , where U is the superficial velocity, L the reactor length. and D, the dispersion coefficient. For the conditions in Example 14-2, we see that the number of tanks calculated from the Badenstein number, 330 (i.e., Pe,), Equation (14-SO), is 4.75, which is very close to the value of 4.35 caIculated from Equation (14-12). Consequently, for reactions other than first order, one would solve successively for the exit concentration and conversion from each tank in series for both a battery of four tanks in series and of five tanks in series in order to bound the expected values. In addition to the one-parameter models of tanks-in-series and dispersion, many other one-parameter models exist when a combination of ideal: reactors is to model the reai reactor as shown in Section 13.5 for reactors with bypassing and dead volume. Another example of a one-parameter model would be to model the real reaclor as a PER and a CSTR in series with the one parameter being the fraction of the total volume that behaves as a CSTR. We can dream up many other situations that would alter the behavior of ideal reactors in a way that adequately describes a real reactor. However. it may be that one parameter is not sufficient to yield an adequate comparison between theory and practice. We explore these situations with combinations of ideal reactors in the section on two-parameter models. The reaction rate parameters are usually known (i.e., Da), but the Peclet number is usually not known because it depends on the Raw and the vessel. Consequently. we need to find Pe, using one of the three techniques discussed earlier in the chapter.
I3K. Elgcti, Chm. Eng. Sci.,51, 5077 (1996).
Sec. 14.6
Numerical Solutions to Flows with Dispersion and Reaetion
975
14.6 Numerical Solutions to Flows with Dispersion and Reaction We now consider dispersion and reaction. We first write our mole balance on species A by recalling Equation (14-28) and including the rate of formation of A, r,, At steady state we obtain
AnaIytical solutions to dispersion with reaction can only be obtained for isothermal zero- and first-order reactions. We are now going to use FEMLAB to solve the flow with reaction and dispersion with reaction. A FEMLAB CD-ROMis included with the text. We are going to compare two soIurions: one which uses the Aris-Taylor approach and one in which we numerically solve for both the axial and radial concentration using FEMLAB. Case A. Aris-T~ylsrAnalysis for hrninar
Flow
For the case of an nth-order reaction, Equation (14-15) is
If we use the Aris-Taylor analysis, we can use Equation (14-15) with a caveat that $ = CA/cAO where ?A is the average concentration from r = 0 to r = R as given by
where
P+=
and Da
=r
kqi
Do
For the closed-closed boundary conditions we have
Danckwens boundary conditions
For the open-open boundary condition we have
Models !or Nonideal Reac!ors
Chap. 14
Equation (14-531 is a nonlinear second order ODE that is solved on the FEMLAB CD-ROM. Case
B. Full
Numerical Solution
To obtain axial and radial profile we now solve Equation (14-5 1)
First we wiIl put the equations in dimensionless form by letting $ = C,/C,\, , k = rtL, and 4 = r/R. Following our earlier transformation of variables. Equation ( l 4-5 l ) becomes
Example 1 4 3 Dispersion wifh Reactinn
(a)
(b) (c)
First, use E M L A B to solve the dispersion part of Example 14-2 again. How does the FEMLAB result compare with the solution to Example 14-2? Repeat (a) for a second-order reaction with k = O,5 dm3/mol mln. Repeat (a) but assume laminar flow and consider radial gndients in concentration. Use D,, for both the radial and axial diffusion coefhcients. PIor the axial and radial profile%.Compare your results with p;ln (a).
Additional information:
U, = UT = 1.24 mlmin, D, = UorJPe, = 1.05 ml/min. DAB= 7.6E-5rn2/rnin. Nore: For part (a), the two-dimensional model with no r;ldial gndients (plug flow) becomes a one-dimensional model. The inlet boundary condition for part (a) and part (b) is a closed-closed vessel (flux[; = 0-1 = flux[i = 0'1 or U;CA0 = flux) at the inlet boundary in EEMLab format is: -Nim = UORCAO.The boundary condition for laminar flow in FEMLAB format for part (c) is: -N;n = 2*UO*(I-(r/Ra)?}*CAO. CAn= 0.5 moVdm3.
The different types of E M L A B Boundary Conditions are given in Problem P 14- 2 9,
Solution
(a)
Equation (14-52) was used in the FEMLAB program along with the nte law
977
Nurnerlcal Solutions to Flows wlih 51saem1onand React~on
Sec. 14.6
We see that we get the bame result\ nh the: analytical solution in Example 14-1. W ~ t hthe Ari+Taylor n n a l y ~ i sthe two-dimensional profile becomes one-dimensional plug Row t e l r m ty profile. Figure E l4-3. I(a) shows a uniform concenrration surface and shows the plug flow k h a t lor OF the reactor. Figure E 14-3. I(b) shows the corresponding crosh-section plots at the inlet, half axial location, nnd outlet. The average outlet conversion ij 67.99.
The average outlet concentration at an axial distance z is foond by integrating across the radtus as hhown below
From the avenge concentrations at the inlet and outlet we can calculate the average conversion as
See FEMLAB 'rutorinls with w e e n shots
Concentration Surface
Radial Concentration Profiles
Load enclosed FEMLAB CD
Llulng Example Prcblem 015r
,
0
Radial Location (m)
3
001
OD1
DO3
OM
OD6
Radial Location (m)
FEgure E14-3.1 FEULAB results for a plug Row reactor w ~ t hfirst-order rcactlon. (Concen~mtionsin moVdm3.)
(b)
Now we expand our resu!rs to cpnsider the case when the reaction is second order (-7, = kc,=kCA,$) with k = O . S dm3/moI-min and CAo=0.5 moI/drn3. Let's assume the radial dispersion coefficient is equaI to the molecu3ar diffusivity. Keeping everything else constant, the average outlet conversion is 52.3%. However. because the flow inside the reactor i s modeled as plug flow the concentration profiles are still flat, as shown in Figure Ell-3.2.
P78
(
Models for Nonideal Reactors
Concentration Surface
Chap. 14
Radial Concentration Profiles
Radial Locat~on{m)
Radial Location (m)
(h)
Figure E14-3.2 E h 1 L A B result$ for a plug flow reactor with second-order reaction. (Concentmilons in moWdm3.)
{c)
Now. we will change the flow assumption from plug flow ta laminar flow and solve Equation (14-5 1 ) for a first-order reaction.
Radial Concentration Profiles
Concentration Surface 7
I
D5
0 6 ~
rmrt
1
oe. Awragc Outlel Converssn = 68 8%
401
.Dl
en!
c
-
3 5
C i
Radial Lucat~on(m)
C1-
0
OPi
002
003
ow
DO5
Radial Location {m)
Figure EI4-33 FEMLAB ou~putfor lan~inarflow in the reacror. (Concentratton< in mol/dmf 1
(dl
The average outlet converrion becomer 68 8 9 , not much d~fferentfrom the one i n part ( a ) In agreement w ~ t hthe Aris-Taylor analyris. Hnuever. due to the laminar iron7 assumption i n the reactor. the radial concenlrat~onprofiler are very d~fferentthroufhoui the reactor, As a homexnrk exercise, repeat pan (c) for the second-order reaction given in pad (b).
Sec. 14.7
Two-Parameter Models
14.7 Two-Parameter Models-Modeling Real Reactors with Combinations of Ideal Reactors We now will see how a real reactor might be modeled by one of two different combinations of ideal reactors. These are but two of an almost unlimited number of combinations that could be made. However, if we limit the number of adjustabIe parameters to two (e.g., bypass flow rate, uh, and dead volume, VD). the situation becomes much more tractable. After reviewing the steps in Table 14-1, choose a model and determine if it is reasonable by qualitatively comparA tracer ing it with the RTD. and if it is, determine the model parameters. Usual13 the expenmenl is used simplest means of obtaining the necessary data is some form of tracer test. 10 evaluate the model parameters. These tests have been described in Chapter 13. together with their uses in determining the RTD of a reactor system. Tracer tests can be used to defermine the RTD, which can then be used in a similar manner to determine the suitability of h e model and the value of its parameters. In determining the suitability of a particular reactor model and the parameter values from tracer tests, it may not be necessan, to calculate the RTD function Eli). The model parameters (e.g., V,) may be acquired directly from measurements of efffuent concen~ationin a tracer test. The theoretical prediction of the particular tracer test in the chosen model system is compared with the tracer measurements from the rea3 reactor. The parameters in the model are- chosen so as to obtain the closest possible agreement between the model and experiment. If the agreement is then sufficiently close, the model is deemed reasonable. If not, another model must be chosen. The quality of the agreement necessary to fulfill the criterion "sufficiently close" again depends on creatively in developing the m d e l and on engineering judgment.The most extreme demands are that the maximum error in the prediction not exceed the estimated error in the tracer test and that there be ne observable trends with time in the difference between prediction (the model) and observation (the seal reactor). To illustrate bow the modeling is carried out, we will now consider two different models for a CSTR. Creativity and
engineering judgment am necessary for model formulation
f4.7.2 Real CSTR Modeled Using Bypassing and Dead Space
A real CSTR is believed to be modeled as a combination of an ideal CSTR of volume V,, a dead zone of volume V,, and a bypass with a voFurnetric flow rate v, (Figure 34- 15). U% have used a tracer experiment to evaluate the parameters of the model Vs and u , . Because the total volume and volumetric Row rate are known, once V , and v , are found, u, and Vd can readily be calculated. 14.7.jP, Solving the Model System for C, and X
We $hall calculate the conversion for this model for the first-order reacfion
980
Models for Nonideal Reactors
Chap. 74
The model system
Deod zone
(01
Figure 14-15 (a) Real system: Ib) model system. A
Bnlnnce at junction
The bypass stream and effluent stream from the reaction volume are mixed at point 2. From a balance on species A around this point, [It11 = [Out]
We can solve for the concentration of A leaving the reactor,
For a first-order reaction, a mole balance on V, gives Mole balance on CSTR
U, CAD- V ,
or, in terms of ol and
, C - kcA, V, = 0
(14-59)
p.
Substituting Equation (14-60) into (14-53) gives the effluent concentration of species A: Conversion as a function of model
panmeters
We have used the ideal reactor system shown in Figure 14-15 to predict the conversion in the real reactor. The model has two parameters. a and P. If these parameten are known,we can readily predict the conversion. In the following section, we shall see how we can use tracer experiments and RTD data to evaIuate the model parameters.
Sec. 14.7
981
Two-Parameter Models
14.7.1B Using a Tracer to Determine the Model Parameters En CSTR-with-Dead-Space-and-BypassModel
In Section 14.7.1A, we ured the system shown in Figure 14-1 6, with bypass flowrate ub and dead volume y,. to rnodei our real reactor system. We shall inject our tracer, T, as a positive-step input. The unsteady-state balance on the noareacting tracer T i n the reactor volume V, is I n - out = accumulation Tracer balance for step input
Mdel system
Figure 14-16 Model system: CSTR with dead volume and bypaqsing.
The conditions for the positive-step input are
mejunction A balance around junctiofl balance
AttCO
C,=O
Atf2O
C,= C ,
point
2 gives
As before, = aV Vh
= Puo
t=-
v
uo
Integrating Equation (14-62) and substituting in terms of a and P gives
982
Models for Nonideal Reactos
Chap. 14
Combining Equations (14-63) and (14-641, the effluent tracer concentration is CT CiT -=
i l - p l e X P [
I).(
-aI
(14-65)
We now need to remange this equation to extiact the model parameters, a and
p, either by regression (Polyrnath/MATLAB/Excel)or from the proper plot of the effluent uacer concentrations as a function of time. Rearranging yields Evaluating the m d e l parameters
Consequently, we plot In[Cm/(Cm - CT)]as a function of t. If our model is correct, a straight line should result with a slope of (1 P ) / z a and art intercept of In[l/(l P)].
-
-
-
Example 1 4 4 CSTR with h a d Space a d Bypass
The elementary reaction A+B
C+D
is to be carried out io the CSTR shown schematically in Figure 14-15. There is both bypassing and a stagnant region i n this reactor. The tracer output lor this reactor is shown in Table E14-4.1. The measured reactor volume is 1.Q m3 and the flow rate to the reactor is 0.1 m3/min. The reaction rate constant i s 0.28 m31kmol.min. The feed is equjmolar in A and B with an entering concentration of A equal to 2.0 kmollm-'. Calculate the conversion that can be expected in this reactor (Figure E14-4.1).
Tkro-parameter mdel
Fipm E14-4.1 Schematic of real rcactor modeled with dead space ( V d ) and bypass ( v , ) .
Sec. 14.7
Two-Parameter Models
Solution Recalling Equation ( 14-66)
Equation (14-66) suggests that we construct Table E14-4.2from Table E14-4.1 and plot Cml(Cm - C), as a function of time on semilog paper. Using this table we get Figure E14-4.2.
-
TABLE E144.2.
PROCESSEu DATA
We can find a and 0 from either a semilog plot as shown in Figure E14-4.2or by regression using Polymath, MATLAB. or Excel.
Evaluating the parameters a and p
Figure E14-4.2 Response ta a step input.
The volumeuic flow rate to the well-mixed ponion of the reactor. u s , can be determined from the intercept, I:
The volume of the well-mixed region, V,, can be calculated from the slope:
I-a= s = , . 1 l 5 m i , - ~ 017
UT
=
1 - 0.2 = 7 min 0.115
Mwrers for Nonrdeal Aaaclors
984
Chap. fq
We now proceed to determine the conversion corresponding to these model parameters I. Balance on reactor volume V , :
[In]- Out + Generation = Accumulation v,CAO--U,C*, +r,,l/,
=Q
2. Rate law: -r*s = RCJ,CB, Equimolar feed
:. C,
= C, -r,,
=
3. Combining Equnt~ons(E14-4. I) and
(E14-4.7) gives
Rearranging. we have
T~RC:, + CAT- CAb= O The Duck Tupc Cormcil would like to point out the new wri~~kle: The Junction Balance.
Solving for C,,yields
4. Balance around junction point 2:
[In] = [Outl
Rearranging Equation (E14-46] g'lves us
Sec. 14.7
(
Two-Parameter Modeis
5. Parameter evaluation:
r, =
5 = 8.7 min v,r
(
Substituting into Equatian (ELJ-4.7) yields
Finding the
conversion
I f the reat reactor were acting as an ideal CSTR. the conversion would be -
I
(E13-1.9)
Other ModeIs. I n Section 14.7.1 it was shown bow we formulated a model consisting of ideal reactors to represent a real reactor. First, we soived for the exit concentration and conversion for our model system in terms of two parameters a and p. We next evaluated these parameters from data of tracer concentration as a function of time. Finally. we substituted these parameter values into the mole baIance, rate law, and stoichiomewic equations to predict the cortvecsian in our real reactor. To reinforce this concept, we will use one more example. 14.7.2 Real CSTR Modeled as Two CSTRs with Interchange I n this particular model there is a highly agitated region in the vicinity of the
impeller: outside this region, there is a region with less agitation (Figure 14-17). There is considerable material transfer between the two regions. Both inlet and outlet flow channels connect Pa the highly agitated region. We shall
986
Models for Nonideal Reactors
Chap. 14
model the highly agitated region as one CSTR, the quieter region as another CSTR. with material transfer between the two.
The model system
Figure 14-17 (a) Real reaction sysrem; (b) model reaction system.
14.7.214 Solving the Model System
Let
for C, and X
p represent that fraction of the total flow that is exchanged between reac-
tors 1 and 2, that is,
ol represent that fraction of the total volume V occupied by the highly agitated region:
and let
Two parameters: a and
p
Then
The space time i s VQ
As shown on the CD-ROM 1 4 ~ ~for 2 , a first-order reaction, the exit concentration and conversion are ~eferenccShelf
988
Models
14.8 Use of Software Packages to
!or Nonideal Reactors
Chap. ? 4
Determine
the Model Parameters If analytical solutions to the model equations are not available to obtain the parameters from RTD data, one could use ODE solvers. Here, the RTD data would first be fit to a polynomial to the effluent concentration-time data and then compared with the model predictions for different parameter values.
I
Example 14-5
CSTR with Bypass and Dead Volume
(a) Determine parameters a and p that can be used to model rwo CSTRs with interchange using the tracer concentrntion data Iisted in Table E 14-5.1.
TABLE E14-5.1. RTD DATA -
r (min)
0.0
Cr, (plm') 2000
I
-
20
40
60
80
120
160
200
240
1050
52fl
280
160
61
29
16.4
10.0
(b)Determine theconversion offirst-order reaction with k = 0.03 min-I and r = 40rnin
First we will use Polymath to At the RTD to a polynomial. Because of the steepnes! of the curve, we shall use two polynomials. Far f 5 80 rnin,
Trial and error using sortware packages
where CT, is the exit concentration of tracer determined experimentally. Next ws would enter the tracer mole (mass) balances Equations (14-7 1) and (14-72) into a ODE solver. The Polymath program is shown in Table E14-5.2.Finally. we vary th parameten a aod j 3 and then compare the calculated effluent concentration Cn wit the experimental effluent tracer concentration CT,.Affer a few trials we converge o the values a = 0.8 and P = 0. I.We see from Figure E14-5.1and Table E14-5.3 thz the agreement between the RTD data and the calculated data are quite good, indical ing the validity of our values of a and P. The graphical solution to this problem i given on the CD-ROM and in the 2nd Edition. We now substitute these values i Equation (14-68). and as shown in the CD-ROM, the comsponding conversion j SIR for the model system of two CSTRs with interchange:
Sec 14 8
Use of Software Packages ?a Determine the Mode! Parameters
Comparing models, we find
Livtng Example Problem
Differentla1 aquation8 as entered by the user I II d(CTl)ld(t) m fbeta'CT2-(l+beZa)*CTl)lalphdtau 12 J d(CT2)ld(t) = (betafCT1-beta*CT2)/(1-alpha)/tau
Expllc# equations as entered by the user 111 beta-0.1 [ 2 1 alpha = 0.8 [ 3 1 tau=40 [dl Cfel = 2MH)-69.B~+0.g4*t"2-O-000146"P3-f .047'IP(-5)W [ 5 1 CTe2 = 921 17.3't+O. 129'P2-0.000438T2"3+5.&'10"(-7)'th4 L S I tl =t-a0 I7 1 CTe = if(tc8Q)thsn(CTel)e1se~CTe2)
-
2.0
Scale:
Y: 10m3 KEY:
- m1
- CTe
(a/m3)
f .2
0.8
0.4
0.0 0.0
40.0
80.0 120.0 160.0mo.0
Figure E14-5.1 Cornparison of model and elrperimenrnl exit tracer co~centrations.
Models for Nunideal Reactors
Chap. 14
14.9 Other Models of Nonideal Reactors Using CSTRs and PFRs
A Ease hislor?, for terephthalic acid
Several reactor models have been discussed in the preceding pages. A11 are based on the physical observation that in almost all agitated tank reactors, there is a well-mixed zone in the vicinity of the agitator. This zone is usually represented by a CSTR. The region outside this well-mixed zone may then be modeled in various fashions. We have already considered the simplest models, which have the main CSTR combined with a dead-space volume: if some short-circuiting of the feed to the outlet is suspected, a bypass stream can be added. The next step is to look at all possible combinations that we can use to model a nonjdeal reactor using only CSTRs, PFRs, dead volume, and bypassing. The rate of transfer between the two reactors is one of the model parameters. The positions of the inlet and outlet to the model reactor system depend on the physical layout of the real reactor. Figure 14-18(a) describes a real PFR or PBR with channeling that is modeled as two PERsJPBRs in parallel. The: two parameters are the fraction of flow to the reactors li.e.. p and (1 - P)] and the fractional voIume [i.e., ol and (1 - a ) ] of each reactor. Figure 14-I 8(b) describes a real PFRlPBR that has a backmix region and is modeled as a PFRJPBR in parallel with a CSTR.Figures 14-191a) and (b) show a reaI CSTR modeled as two CSTRs with interchange. I n one case, the fluid exits from the top CSTR (a) and in the other case the fluid exits from the bottom CSTR. The parameter P represents the interchange volumetric flow rate and'cr the fractional volume of the top reactor, where the fluid exits the reaction system. We note that the reactor in model 14- 19(b) was found to describe extremely well a real reactor used En the production of terephrhalic acid." A number of other combinations of ideal Wactions can be found i n Levenspiel.
'"
'Trot. Indian Insi. Chem. Eng. Golden Jubilee, a Congress, Delhi, 1997, p. 323 ''Lcvenspiel. 0 Chrmicnl Reacriotl Ertgit~rrring,3rd ed. (New York: Wilcy. 19991, pp. 284-292.
Sec. 14.10
Appllcafions to Pharmacokinetic Modeling
Real System
Model S y s t m
Real System
Model System
Ih)
Figure 14-18 Combinations of ideal reactors used to m d e l real tubular reactors. (a) two ideal PFRs in pmllel (b) ideal PFR and Ideal CSTR in parallel
14.10 Applications to Pharmacokinetic Modeling The use combinations of idea1 reactors to model metabolism and dmg distribution in the human body is becoming commonplace. For example. one of the simplest models for drug adsorption and elimination is similar ro that shown in Figure 14-9(a). The drug is injected intravenously into a central companment containing the blood (the top reactor). The blood distributes the drug back and forth to the tissue compartment (the bottom reactor) before being eliminated (top reactor). This model will give the farnitjar linear semi-log plot found in pharmacokinerics textbooks. As can be seen in the figure for P~~fe.rsionul Refererice Shelf R7.5 on pharmacokinetics on page 453 there are two different slopes. one for the drug distribution phase and one for the elmination phase. More elaborate models using combinations of ideal reactors to model a real system are descrikd in section 7.5 where alcohol metabolism is discussed.
998
Models Icr Nonldeal Reacto~s
Chap. 14
P I 4 4 The elementary liquid-phase reaction
is carried out in a packed bed reactor in which dispersion is present. What is the conversion? Additional informution
Porosity = 50% Particle size = 0.1 cm Kinematic viscosity = 0.01 cm2/s
Reactor length = 0.1 rn Mean velocity = 1 cmls
P14-5, A gas-phase reaction is being carried out in a 5-cm-diameter tubular reactor that is 2 m in length. The velocity inside the pipe is 2 c d s . As a very first appmximation, the gas properties can be taken as those of air (kinematic viscosity = 0.01 cm21s), and the diffusivities of the reacting species are approximately O.QO5 cmr/s. ( a ) How many tanks in series would you suggest to model this reactor? (b) If the second-order reaction A + B d C + D is carried out for the case of equal molar feed and with CAO= 0.01moWdm3, what conversion can be expected at a temperature for which k = 25 drn3/molns? (c) How would your answers to parts (a) and (b) change if the fluid velwity were reduced to 0.1 c d s ? ~n&easedto 1 m/s? Id) How would your answers to parts (a) and @) change if the superficial velocity was 4 cm/s thmugh a packed bed of 0.2-cm-diameter spheres? (e) Wow would your answers to parts (a) to (d) change if the fluid were a liquid with properties similar to water instead of a gas, and the diffusivity was 5 X cm2/s? Plrdd, Use the data in Example 13-2 to make the following determinations. {The volumetric feed rate to this reactor was M) drn3/snin.) (a) Calculate the Peclet numbers for both open and closed systems. (b) For an open system, determine the space-time T and then calculate the 8 dead volume in a reactor for which the manufacturer's specifications give a volume of 420 dm3. (c) Using the dispersion and tanks-in-series models, calculate the conversion for a closed vessel for the first-order isorncrizatian
A - B with k = 0.18 min-I. Id) Compare your results in part (c) with the conversion calculated from the tanks-in-series model, a PFR, arid a CSTR. P14-7, A tubular reactor has beensized to okain 98%conversion and to prMess 0.03 m3/s. The reaction is a first-order irreversible isomerization. The reactor is 3 m long, with a cross-sectionaI area of 25 cm2. After being built. a pulse tracer test on the reactor gave the following data: t, = 10 s and u2 = 65 s2. What conversion can be expected in the real reactor? P14-BB The following E(t) curve was obtained from a tracer test on a reactor.
E in minutes
Chap. 14
999
Ouesllons and Problems
The conversion predicted by the tanks-in-series model for the isothermal elementary reaction
was 50% at 300 K. (a) If the temperature is to be mised 10°C (E = 25.000 callrnol} wd the reaction carr~edout isothermally, what will be the conversion predicted by the maximum mixedness model? The T-I-S model?
(b) The elementary reactions
k, = k, = k, = 0.1 min
-I
at
300 K, ,C , = 1 rnolldm3
were carried out isothermally at 300 K in the same reactor. What is the concentration of B in the exrl stream predicted by the maxlmum rnixedness model? (c) For the multiple reactions giveen in part (b). what is the conversion of A predicted by the dispersion model in an isothermal closed-closed system? P14-9B Revisit Problem P13-4, where the RTD function is a hemicircle. What is the conversion predicted by (a) the tanks-in-series model? (b) the dispersion model? P14-10, Revisit Problem P13-5B. (a) What combination of ideal reacton would you use to model the RTD? (b) What are the model parameters? (c) What is the conversion predicted for your model? P14-llR Revisit Problem PI 3-6,. (a) What conversion is predicted by the tanks-in-series model? (b) What is the Peclet number? (c) What conversion i s predicted by the dispersion model? P14-12, Consider a real tubular reactor In which dispersion is occurring. (a) For small deviations from plug flow. show that the conversion for a first-order reaction is ~ i v e napproximately as
(b) Show that to achieve the same conversion. the relationship between the volume of a plug-flow reactor. I'p and volume of a real reactor, K in which dispersion occurs is
(c) For a Peclet number of 0,1 based on the Pm length, how much bigger than a PFR must the real reactor be to achieve the 999 conversion pred~cledby the PFR?
.
996
Models br Nanideal Reactors
Chan. 14
Q U E S T I O N S AND P R O B L E M S The subscript to each of the problem numbers indicates the level OF difficulty: A. least difficult; D. most difficult.
\I
-\'Q Crea tivc TninXlng
P14-1, Make up and soIve an original problem. The guidelines arr given in Pmblem P4-1. However, make up a problem in reverse by first choosing a model system such as a CSTR in parallel with a CSTR and PFR [with the PFR modeled as four small CSTRs in series; Figure P14.lla)l or a CSTR with recycle and bypass [Figure P14- lo)].Write tracer m a s balances and use an ODE solver to predict the effluent concentrations. l o fact, you could build up an arsenal of tracer curves for different model systems to compare itgqinst real reactor RTD data. In t h s way you could deduce which model best describes the real reactor.
(b)
Figure PIQ1.1 Model systems.
? Lultrnne,
Hall of R m e
P14*2f, (a) Example 14-1. How lsrge would the ermr term be in Equation E14-1.4 i F t k = 0.1? ~ k l? = .tk= 10? (b) Example 14-2. Vary Do, k, U,and L. To what parameters or groups of parameters (e.g.. kL2/D,) would the conversion be most sensitive? What i f the first-order reaction were carried out in tubular reactors of different diameters, but with the space time, t, remaining constant? The diameters would range from a diameter of 0.1 dm to a diameter of f rn for v = ClJp = 0.01 cm2/s. U = 0.1 c d s . and DAB = 1W5cm2/s. How would your conversion change? Is there a diameter that would maximize or minimize conversion in this range?
Chap 14
997
Ouest~onsand Problems
(c) Example 14-3. ( 1 ) Load the reaction and dispersion program from the FEMLAB CD-ROM. V ~ r ythe Darnkbhler number for a second-order reaction using the Anr-Taylor approximation (part (b) in Example 14-31. 12) Vary the Pecler and Darnk6hler numbers for a second-order reaction in laminar flow. What values of the Peclet number affect the conversion significantly? (d) Example 14-4. How would your answers change if the slope was 4 min-I and the inrercepr was 2 in Figure E14-4.2? (el Example 14-5. Load the Livzng Example Pol~rrmthPmgmm. Vary a and p and describe what you find. What would be the conversion if a = 0.75 and p = 0.15? (i? Whnt if you were asked to design a tubular vessel that would minimize dispersion? What would be your guidehnes? How would you maximize the dispersion? How would your design change for a packed bed? (g) What if someone suggested you could use the solution to the flow-dispcrsion-reactor equation, Equation (14-27). for a second-order equation by s (kCAd2)CA= A'C, ? linearizing the rate law by lettering -r, = Under what circumstances might this be a good approximation? Would you divide CAoby something ofher than 2? What do you think of linearizing other non-first-order reactions and using Equation (14-27)? How could you test your results to learn if the approximation is justified? (h) What if you were asked to explain why physicaIly the shapes of the curves in Figure 14-3 Look the way they do. what would you say? What if the first pulse in Figure 14.l(b) broke through at @ = 0.5 and the second pulse broke through at @ = 1.5 in n tubular reactor in which a second-order liquid-phase reaction
k'e
P14-3,
was occurring. What would the conversion be if T = 5 min, CAo= mol/drn3. and k = 0.1 drn31mol*min? The second-order liquid-phase reaction
2
is to be &ed out isothermally. The entwing concentmuon of A is 1.0 mol/dm3. The spec~ficreaction rate is 1.0 drn3lrnol-min. A number of used reactom (shown klow) are available, each of which has been characterized by an RTD. There are hvo crimson and white reactors and three maize and blue reactors available.
Maize and blue Green and white Scarlet and gray Onnge nnd blue Purple and white Silver and black Crimson and white
2
2
4 3.05 2.3 1 5.17
4 4
2.5 25
f 25,000 50,000
4
50,000 50.003
4
50.000
4
50,W
2
25.N
(a) You have $50,000 available to spend. What is the greatest conversion you can achieve with the available money and reactors? (b) How would your answer to (a) change if you had $75,000available to spend? (c) From which cities do you think the various used reactors came from?
Modets for Nonideal Reactors
Chap. 14
For an open-open system. use
5. If a real reactor is modeled as a combination of ideal reactors, the m d e l should have at most two parameters.
CSJR w i t h bypass ond deod volume
Two CSf R s with inlerchonqe
6. The RTD is used to exmct model parameters. 7. Comparison of conversions for a PFR and CSTR with the zero-parameter and two-parameter models. X, symbolizes the conversion obtained from the segregation model and X,, that from the maximum mixedoess model for reaction orders greater than one.
Cautions: For rate Iaws with unusual concentration functionaljties or for nonisothemal operation, these bounds may not be accurate for certain types of rate laws.
CD-ROM MATERIAL
Summary Notes
RJ
A
PLE
tlvtng Example Probltm
Learning Resources I. Slrrnrnay Notes 2. Web Material FEMLAB CD-ROM Living Example Problems 1. Example 14-3 Dispersion wirh Reaction-FEMUB
2. Example 14-5 CSTH wirh Rjpass and Dead Volume
Chap. 14
CD-ROM Materiel
995
FEMLAB results
Radial Location (m)
Professional Reference Shelf R14.1 Derivation of Equation for Toylor-Aris Dispersion
t-a -
r-?U
Refcrcncc S ~ C B
C+$2=DddiC d*
z
R 14.2 Real Reuczor Modeled in on Ideal CSTR with Exchange Voluma Example R14-1 l b o CSTRs with interchange.
a ~ * ~
Models for Non~UealReactors
(a)
Chap. 14
(b)
Figure 14-19 Combinations of ideal reactor
Closure RTD Data + Kinetics + Model =Prediction
In this chapter, models w a e developed for existing reactors to obtain a more precise estimate of the exit conversion and concentration than estimates of the examples given by the zero-order parameter modeb of segregation and maximum mixedness. After completing this chapter, the reader will use the RTD data and kinetic rate law and reactor model to make predictions of the conversion and exit concentrations using the tank-in-series and dispersion one-parameter models. Tn addition, the reader should be able to create combinations of ideal reactors that mimic the RTD data and to solve for the exit conversions and concentrations. The choice of a proper model is almost pure art requiring creativi? and engineering judgment. The flow pattern of the model must. possess the most important characteristics of that in the real reactor. Standard models are available that have been used with some success, and these can be used as starting points. Models of real reactors usuallj consist OF combinations of PFRs, perfectly mixed CSTRs, and dead spaces ir n configurationthat matches the flow patterns in the reactor, For tubular reac ton, the simple dispersion model has proven most popular. The parameters in the model, which witknre exception should no exceed two in number, are obtained from the RTD data. Once the parnrne zers are evaluated, the conversion in the model, and thus in the real react01 can be calculated. For typical tank-reactormodels, this is the conversim i a series-paralIel reactor system. For the dispersion model, the second-orde differential equation must be solved, usually numericaH1y. Analytical sol1 tions exist fmfirst-order reactions, but as pointed out previously, no modr has to be assumed for the first-order system if the IUD is available. Correlations exist for the amount of dispersion that might be ezpecte in common packed-bed reactors, so these systems can be designed using tt dispersion model without obtaining or estimating the RTD. This situation perhaps the only one where an RlD is not necessary for designing a no
Chap. 14
Summary
SUMMARY
The rnodeIs
1. The models for predicting conversion from a. Zero adjustable parameters (1 ) Segregatition model (2) Maximum rnixedness model b. One adjustable parameter
RTD data are:
Tan ks-in-series model (2) Dispersion model c. Two adjustable parameters: real reactor modeled as combinations of ideal ( 1)
Teactors
2. Tanks-in-series model: Use RTD data to estimate the number of tanks in series,
For a first-order reaction
3. Dispersion model: For a first-order reaction. use the Danckwerts boundary conditions
X= 1-
4q exp (Per/2) ( I + q)* exp ( P e , q / 2 ) (1 - q)2 exp (-Pe,q/2)
-
(S 14-2)
where
Da = zk
Pe,
=
UL Dn
Ud Pe - P I-
D,E
(Si4-4)
4, Determine D, a. Far laminar flow the dispersion coeficient i s
b. Correlations. Use Figures 14-10 through 14-22. c. Experiment in RTD analysis to find (, and d. For a closed-closed system use Equation (S14-6) to calculate Per from the RTD data:
u2 - - -2 Lil-e-Per) 't Per Pef
1000
Models for Nnnideaf Reactors
(d) For an nth-order reaction, the ratio
o l exit
Chap. I d
concentration for reactors of
the same length has been suggested as
What do you think of this suggestion? (e) What is the effect of dispersion on zero-order reactions? P14-H8Let's continue Problem PI3-IgB. (a) What would be the conversion for a second-order reaction with kcAn = 0.1 min-I and CAfl= 1 mol/dm4 using the seeregatlon model ! (h) What would be the conversion for a second-order reaction ~ i t kcAo h = 0.1 mln-I and C ,,, = 1 molldm-' using the maximum rnixedness mode!? (c) If the reactor is modeled as tanks in series, how many tanks art: needed to represent this reactor? What i s the conversion for a first-order reaction with k = 0.1 min-I? (d) IF the reactor i s modeled by a dispersion model, what are the Peclet numbers for an open system and for a closed system'? What is the conversion for a first-order reaction with k = 0.1 min-I for each case? (e) Use the dispersion model to estimate the conversion for a second-order reaction with k = 0.1 drn.'tmol+sand CAD= 1 molldm3. (fl It is suspected that the reactor might be behaving as shown in Figure P14-13. with peri~ups(?) Y, = V,. What i s the "hnckflow" from the second to the first vessel. a< a multiple of u,, ?
Figure P14-I3 Proposed model system
Appkoetion
(g) U the model above i s correct, what would be the conversion for a second-order reaction with k = 0. I drn3Smol .min if C,,)= 1.0 molldm"? (h) Prepare a table comparing the conversion predicted by each of the models described above. (1) Bow wouId your answer to (a) change if the reaction were c m e d out adiabatically with the parameter values given in Problem P13-Yj)? P14-14, It is proposed to use the elementary reactions
Pewding Tor Problrm hdl of
to characterize mixing in a real reactor by monitoring the product distribution at different temperatures. The ratio of specific reaction rates (k,/k,) at temperatures T I .T,,T3. and Tdis 5.0, 2.0. 0.5, and 0.1. respectively. The comsponding values of tklCAo are 0.2, 2, 20, and 200. (I) (a)Calculate the pmduct distribution for the CSTR and PFR in series described in Example 13-3 for TcSTR = TPFR = 0 . 5 ~ . (b) Compare the product distribution at two temperatures using the RTD shown in Examples 13-1 and 13-2 for the complete segregation model and the maximum rnixedness model.
Chap. 14
Questions and Problems
loof
(c) Explain how you could use the product distribution as s function of temperature (and perhaps flo* nte) to characterize your reactor. For example,
could you use the test reactions to determine whether the early mixing scheme or the late mixing scheme in Example 13-3 is more representative of a real rractor? Recall that both schemes hnve the same RTD. (d) How should the reactions be carried out li.e., at high or low temperatures) for the product distribution to best characterize the micromixing in the reactor? P14-15, The second-order kaction is to be carried out in a real reactor which gives the following outlet concentration for a step input. F o r O S r i 10 min then C T = I O ( I - P - I ' ) For 10 5 r then CT=5+ 10 ( 1 -6-.If) (a) What model do you propose and what are your model parmeten, o!and Pq (b) What conversion can be expected in the reaE reactor? (c) How would your model change and conversiun change if your outlet tracer concentration was For r < 10 min, then CT = 0 Fort? !Omin, t h e n C t = 5 + 1 0 ( l - e 2 ~ f - ' o ~ )
vo = I dm'lmin. k = 0.1 drn3/rnol- min, CAD= 1.25 mol/dm3 P14-16RSuggest combinations of idear reactors to model real reactors given in Problem 13-2(b) for either E(B),E(r). F(0),F(r), or (I- E(0)). P14-17, Below are two FEMLAB simulations for a laminar flew reactor with heat effects: Run 1 and Run 2. The figures below show the cross-section plot of cuncentration for species A at the middle of the reactor. Run 2 shows a minimum on the cross-section plot. This minimum could be the result of (cisde ail that apply and explain your reasoning for each suggestion (a) through (e)) Es) the thermal conductivtty of reaction mixture decreases (b) overall heat tnnsfer coefficient increases (c): overall heat tnnsfer coefficient decreases ( d ) the ccolant flow rate increases re) the coolant flaw rate decreases Hint: Explore "fu"onisothermal Reactor''IT on the E M L A B
Figure Pf4-17 E M L A B screen shots
CD-ROM.
1002
Models for Non~aealReactors
Chap. 14
P14-18, Load the laminar flow with dispersion example on the FEMLAB CD-ROM. Keep Da and L R constant and vary the reaction order ti, (0.5 < n 5 5) for different Peclet numbers. Are there any combinations of n and Pe where dispersion is more important or less important on the exit concentration? What generalizations can you make? Hint: for n < 1 use r, = -k . (Rbs(C1)) P14-19, Revisit the FEMLAB Example 14-3 f o laminar ~ flow with dispersions. la) Plot the radial concentrahon profiles for zlL = 0.5 and 1.0 for a second-order react~onwith CAU= 0.5 moVdrn2 and kc,, = 0.7 mln-I using both the closed-vessel and the laminar flow open-vessel boundary conditions at the inlet. Is the average outlet conversion for the open-vessel boundaq condition lower lhan that which uses the clod-vessel boundary condition? In what situation. ~f any. will the two boundary conditions result I n significantly different outIet concentrations? Vary Pe and Da and describe what you find.
(b) Repeat (a) for both
a
third order with
k ~ = i0.7 min-I and
a
''
half-order reaction with k = 0.495 (mol/dm3)' min-'. Compare rhe radial convenion profiles for n first-. a second-, s third-, and a half-order .reaction at different locations down the reactor. Note in FEMLAB: Open-vessel Boundary (Laminar Flow): -Ni.n = 2*U0*( 1 -(rlRa)."2)*CAO Clase-vesqel Boundary: 4 i . n = UO*CAO Concentration Boundary Condition CA = CAO SymmetryRnsulat~onCondition n.N = 0 P14-20 The E curves for two tubular reactors are shown here. for a closed~losedsystem.
Flgure P14-20 F Curves
(a) Which curve: has the higher Peclet number? Explain. (h) Which curve has the higher dispersion coefficient? Explain. (c) If this F cuwe is for the tanks-in-senes model applied to two differen1 reactor<, which curve hqs the largest number of T-I-S ( 1 1 or (?j? U of M . ChES28 Fall 2000 Exam II P14-21 Consider the following system used to model a real reactor: Vh
v,=b v, V, = a
Figure P11-21 Model \!\!ern
V
Questions and Problems
Chap. 14
1003
Descrjbe how you would evaluate the paramefen a md B. (a) Draw the F and E curves for this system of ideal reactors used to model a real reactor using = 0.2 and a = 0.4. Identify the numerical values of the points on the F curve (e.g.. t,) as they relate lo r. 0)If h e reaction A 4 B is second order with kc,, = 0.5 rain-', what is the conversjon assuming the space time for the real reactor is 2 min? U of M, ChE528 Fall 2000 Final Exam P14-22 Thwe i s a 2 m3 reactor in storage that is to be used to carry out h e liquid-phase second-order reaction
and B are to be fed in equal molar amounts at a volumetric rate of I mVmin. The entering concentration of A is 2 molar, and the specific reaction rate is 1.5 mYkrnol min. A mcer experiment was carried out and reported in terms of F as a function of time i n minutes. A
Figure Pf4-22 F curve for a nonideal reactor
Suggest a two-parameter model consistent with the data; evaluate the m d e l parameters and the expected conversion. U of M,ChE528 Fall 2001 Firial Exam P14-23 The following E curve was obtained from a tracer test:
F i ~ r P14-23 e E curve for a nonideal reactor
(a) What is the mean residence time? (h) What is the Peciet number? How many tanks in series are necessarq. to model this non-idat reactor? U of M, Doctoral Qualifying Exam (DQE), May 2001
2 004
Models tor Nonrdeal Reactors
Chap. 14
P14-24 The A first-order reaction is to be carried out in the reactor with k = 0.1 min-I. Fill in the following table with the conversion predicted by each type of modellreactor. Ideal
laminar
Ideal PFR
Ideal CSTR flow reactor Segregation
Maximum mixedness
Tanks in Di~persion
series
The following outlet concentration tmjectory was obtained frorn a step Input to a nonideal reactor. The entering concentration was 10 millimolar of tracer.
Figure Pl4-24 C curve for a nonideal reactor Suggest a model using the collecrion of ideal reactors to model the nonideal reactor. U of M. Doctoral Qualifying Exam (DQE), May 7001
Additional Homework Problems CDPI4-A,
A real reactor is modeled as a combination of ideal
PFRs and
CSTRs. [2nd Ed. P14-51 CDP14-3, CDP14-Cc CDPICDB CDP14-E,
CDP14-FR CDPIJ-Gc CDPl4-BB CDP14-I, CDPlCJ, CDPl4New
A real batch reactor is modeled as a combination of two ideal reactors. [2nd Ed. P14-131 Develop a m d e l for a real reactor For RTD obtained frorn a step input. [2nd Ed. P14- 101 Calculate Do and X from sIoppy tracer data. [2nd Ed. P14-6,] Use RTD data from Oak Ridge National Labontory to calculate the conversion from the tanks-in-series and the dispersion models. [Znd Ed. P14-7R] RTD data from a s l u reactor. ~ [3rd Ed. Pi4-81 RTD data to calculate conversion for a second-order reaction for all models. [3rd Ed. P14-41 RTD data from barge spiIl on Mississippi River. /3rd Ed. P14-101 F!TD data to calculate conversion using all models. [3rd Ed. P14-1 I] Apply two-parameter modef to multiple reactions. [3rd Ed. P14-153 New problems wit1 be insened from time to time on the web.
Chap. 14
Supplementary Reading
t 005
SUPPLEMENTARY READING I. Excellent discussions of maximum mixedness can be found in DOCGLAS.I. M., "The effect of mixing on reactor design:' AIChE Sjmp. Ser 48, Vol. 60, p. 1 ( 1964). ZWIETERISG, TH.N.. C h ~ m E1.t.g. . Sci., 11, 1 (1959).
2. Modeling real reactoa with a combination of ideal reactors is discussed together with axial dispersion in
LEVEKSPIEL, Q., Chemical Reaction Engineering. 3rd ed. New York: WiTey, 1999.
WEN, C. Y.. and L. T.FAV.Modcls for Flow Systems and Chernicul Rencmrs. New York: Marcel Dekker, 1975. 3. Mixing and its effects an chemical reactor design have been receiving increasrngly sophistiw~edtreatment. See, for example: B~SCHOFF.K.
B.. "Mixing and contacting in chemical reactors," Ind. Eng. Chem., Sf?( I I),18 (1966). NAUMAN, E. B.."Residence time distributions and micromixirtg," Chern. Eng. Comrnun.. 8. 53 (1981). NAUMAN, E. B., and B. A. BUFFHAM,Mixing in Continuot~sF h w System. New York: WiIey, 1983. PA'TTERSON, G.K., "Applications of turbulence fundamentals to reactor modeling and scaleup." Chem. Eng. Commun..8, 25 (1981). 4. See also
Dunu~ov!c,M.. and R. FELDER.in CHEMi Modules on Chemical Reacrion Engineering, Vol. 4, ed. B. Crynes and H. S. Fogler. New York: AIChE, 1985.
5. Dispersion. A discussion of the boundary conditions for closed-closed, open-own. closed-open,and open-closed vessels can be found i n ARIS, R., Chem. Eng. Sci., 9.266 (1959). LEVENSPIEL, 0.. and K. B. Brsc~ow, Adv. in Ckem. Eng.. 4.95 (1963). NALVIAN, E.B..Chem. E q . Commun., 8. 53 (1981).
Modets for Nonldeal Reactors
Chap 14
This is not the end. It is not even the beginning of the end. But it is the end of the beginning. Winston Churchill
Appendices
Text Appendices Numerical Techniques Ideal Gas Constanr a d Conversion Factors Thermodynamic Relationships involving rhe Equilibrium Constant Measurement of Slopes on Semilog Paper Software Packages Nomenclarure Rate Law Data Open-Ended Problems How to Use the CD-ROM Use of Computational Chemistry Software Packages
CD-ROM Appendices Equal-Area Graphical Differen tiarion 23. Using Semilog Plots in Rate Data Analysis E. M A T U B H. Open-Ended Problems Use of Computational Chemistry Sofrware Pachges J K. A~lnlyticdSolutions ro Ordinary Diferenrial Equarions
A.
A
-
Numerical Techniques
A
App. A
Numerical Techniques
-+ 2-b
dx 2 - 2ax.t.b ax:+bx+c
for b2 = 4oc
(A- 10)
where p and q are the roots of the equation.
(A- 12) C
A.2 Equal-Area Graphical Differentiation There are many ways of differentiating numerical and graphical data. We shall confine our discussions to the technique of equal-area differentiation. In the procedure delineated here, we want to find the derivative of with respect to x. I. Tabulate the (T,, x i ) observations as shown in Table A-1. 2. For each intenla], calculate hx, = x, - x,-, and Ay, = y, - y
This method finds use in Chapter 5 .
,-,.
See. A.2
1011
Equal-Area GaphiEal DHerentiation
3. Calculate Ay,lAxn as an estimate of the average slope in an interval to X n . 4. Plot these values as a histogram versus x i . The value between xt and x 3 , for example, is ( y 3 kz)l(.r3- ~ 2 ) Refer . to Figure A-1.
-
Figure A-1 Equal-area differentiation.
5. Next draw in the smooth cusve that best approximates the area under the histogram. That is, attempt in each interval to balance areas such as those labeled A and B, but when this approximation is not possibIe, baIance out over several intervals (as for the areas labeled C and D). From our definitions of Ax and Ay we know that Y"-YI =
' 1
Ax,
(A- 13)
i=2
The equal-area method attempts to estimate dyldx so that See CD-ROM,
Appendix A, for
worked example
that is, so that the area under A y l h x is the same as &at under dyldx, everywhm possible. 6. Read estimates of dyr'dr from this curve at the data points x, , x,, .. . and complete the table.
Unkr
An example illustrating the technique is given in L c CD-ROM. Appendix A. Differentiation is. at best, less accurate than integration. This method also clearly indicates bad data and allows for compensation of such data. Differentiation is only valid, however, when the data are presumed to differentiate smoothly, as in rate-data analysis and the interpretation of transient diffusion data.
1012
Numerical Techniques
App. A
A.3 Solutions to Differential Equations First-Order Ordinary Differential Equation
A.3.A
d Links
See www.ucbac.uk/Mathematr'cs/geomath/EevcI2/deqpt/de8 htmI and CD-ROM
Appendix K.
2+ m y =,df) II).
Using integrating factor = exp fdt
the solution is
dt
y=e
Exampb A-I
(A- 15)
+ K,e
(A- 16)
Integrating Factor for Series Reactions -k,t
*+k$=k,e d!
%'
Integrating factor = exp k r
=
-
dye)
py= klI:k2y=-e
fk2 -11b kl e
d!
kl
k k,
-kt!
k2 - 4 r=O
(kz-kl)r l
- k*
+li;
-k21
+K,e
y=o
A.3.B
Coupled Differential Equations
Techniques to soIve coupled first order linear ODES such as
are given in Appendix K in the CD-ROM.
Numerical Evaluation of Integrals
Sec. A.4
A.3.C Second-Order Ordinary Differsntial Equation Methods of solving differential equations of the type (A- 17)
can be found in such tests as Applied Diflerential Equations by M.R. Spiegel (Upper Saddle River, N.J.: Prentice Hall, 1958, Chapter 4; a great book even though it's old) or in Diferentinl Equations by F. Ayres (New Yosk: Schaum Outline Series, McGraw-Hill, 1952). One method of solution is to determine the characteristic roots of
which are M
=
The solution to the differential equation is type are required
, =A,
+B , e t f i . ~
(A- 38)
where A! and B, are arbitrary constants of integration. It can be verified that Equation (A- 18) can be arranged in the form = A sinhfix
+ ~cosh,@x
(A- 19)
Equation (A-19) is the more useful form of the solution when it comes to evaluating the constants -4 and B because sinh(0) .= 0 and cosh (0) = 1 .O.
A.4 Numerical Evaluation of Integrals I n this section, we discuss techniques for numerically evaluating integrals for
solving first-order differentia1 equations. 1. Trapezoidal rule (two-point) (Figure A-2). This method i s one of the
sirnpIest and most approximate, as it uses the integrand evaluated at the limits of integration to evaluate the integrat:
when h = X I
- Xo.
1014
Numerical Techniques
App. A
2. Simpson S one-third rule (three-pint) (Figure A-3). A more accurate evaluation of the integral can be found with the application of Simp son's rule:
where
Methods to solve
in Chapters 2,4, 8, and
-E -
t
jbxtnnna in Chapter 13
I I
I
h&hYt
xo
xo
Xl
X
X, X
X2
Figure A-3 Simpson's three-point rule illustration.
Figure A-2 Trapezoidal rule iltustration.
3. Simpson 's three-eighths mIe (four-point) (Figem A-4). An improved version of Simpson's one-third rule can be made by applying Simpson's three-eighths rule: x3
1%
[ f N o ) + 3f (Xi 1+ 3f (Xz1 +f(X3) I
f
where
x,-x,
h=-
3
Figure A-4
XI
= Xo + h
X2 = Xo -I-2h
Simpson's four-point rule illusbation.
4. Five-point qwdratuw formula.
(A-22)
Sec. A.5
Software Packages
where
5. For N
+ 1 points, where (N13) is an integer,
where h =
6. Wr N
x,-x,, N
+ 1 points, where N is even,
where These formulas are useful in illustrating how the reaction engineering integrals and coupled ODES (ordinary differential equation(s)) can be solved and also when there is an ODE solver power failure or some other malfunction.
A.5 Software Packages LInks
Instructions on how to use Polymath, MATLAB, FEMLAB, and Aspen can be found on h e CD-ROM. For the ordinary differential equation solver (ODE solver), contact: PoIymath
CACHE Corporation P.O.Box 7939 Austin, Texas 78713-7379 Web site: w~w.pnlj~mathsofrware.comfogIer MATLAB The Math Works, Inc. 20 Nonh Main Street, Suite 250 Sherbom, Massachusetts 01770
Aspen Technology, Inc. 10 Canal Park Cambridge, Massachusetts 02141 -2201 Email: info @aspentech.corn W e b site: /hrw~:aspenrech.com
FErvlLAB COMSOL, Inc. 8 New England Executive Park. Suite 3 10 Burlington. Massachusetts 01803 Tel: 78 1-273-3322; Fax: 7 8 1-273-6603 Email: [email protected] Web site: MWW cor~~snl.com
A critique: of some of these software packages (and others) can be found in Chenr. Eng. Educ., XXV (Winter) 54 (199 1).
and Conversion Factors See www.onlineconversion.com. Unks
Ideal Gas Constant R = 0.73 ft3. atm Ib mol " R
R = 8.3 14 kPa. dm3
R = -8.3144 J mol .K
mol .K
.
R = 0.082 dm3.atm = 0.082 rn3.atm mol K kmol - K Boltzmann" ccostant kB = I .38 1 x 10-z3
- 1.987cal rno1.K
J molecule * K
Vokume of Idea1 Gas 1 lb-mo?of an ideal gas at 32°F and t a h occupies 359 fi3 (0.00279 lbmoWft31. 1 rnol of an ideal gas at VC and 1 am occupies 22.4 dm3 (0.0446 rnoWdrn3).
where
CA = concentration oFA, mol/dm3 T = temperature, K P = pressure, kPa y~ = mole fraction of A R = ideal gas constant, 8.314 kPa.dm3/mol.K
1018
Ideel Gas Constant and Conversion Factors
APP 0
Volume
1 cm3 1 in3 I fluid oz 1 ft3 1 m3 I U.S. gallon 1 liter (L)
1 A = 10-gcrn 1 dm = 10cm
= 0.001 dm3 = 0.0164 dm = 0.0296 dm3
1 prn = loL4cm 1 in. = 2.54 cm 1 ft = 30.48 crn I m = I00cm
28.32 dm3 = 1009 dm3 = 3.785 dm3 = 1 dm3 =
3 ft3 = 28.32 dm3 X
gal
3.785 dm3
=
7.482 gal
1
Pressure
Energy (Work)
1 torr (I m H g ) = 0.I3333 kPa 1 in. HzO = 0.24886 kPa 1 in. Hg = 3.3843 kPa 1 atm = 101.33 kPa 1 psi = 6.8943 P a 1 megadyne/crn2 = 100 kPa
1 kg.rn2/s2 = 1 J 1 Btu = 1055.06 J 1 cal = 4.1868 5 1L.at1-n =101.34J 1hp.h =2.6806X106J = 3.6 x 106J f kwh
Temperutwre
Mass
= 1.8 X "C + 32 =T 459.69 K = "C f 273.16 "R = 1,8 X K 'Rkamur = 1.25 X OC
OF
= 454 g 1 kg = 1OOOg 1 grain = 0.0648 g I oz (avoird.) = 28.35 g 1 ton = 908,000 g
1 Ib
+
R
1 poise = I g l c d s = 0.1 kglmls
1 centipoise = 1 cp = 0.01 poise
Force 1 dyne = I g cm/s2 +
1 Newton = 1 kg - m/s2
Pressure
Work
A. Work
=
Force x Distance
1 Joule = 1 Newton . meter = 1 kg m 2 / s L 1 Pa m' +
B. Pressure x Volume = Work 1 Newton/
m 2 .m"
1 Newton . rn = 1 Joule
App. 8
Ideal Gas Constant and Corwersinn Factors
Time Rate of Change of Energy with ZTme 1 watt = 1 J/s 1 hp = 746 J/s
GravMond Conversion FacLer Gravitational constant g = 32.2 ft/s2
American Engineering System
SI/cgs System g, = 1 (Dimensionless)
Gas (air, 7f°C, 10 1 Wa)
Liquid (water)
Solid
Density Concentration Diffisivity
1Cn-X) k g f d
1.0 kgfm'
3000 kglm"
55.5 moUdm3
0.04molldm3
-
1V m21s
Viscosity
I W 3kg/&
I t 5 m2/s 1.82 2(
-
Heat capacity
4.3 I JlglK
40 JlmollK
0.45 JfgX
Thermal conductivity Kinematic viscosity
1.Q JIslmflC
1 IF2lld& 1.8 x ICF5m2/s
I V" m2/s
Iw k g / d s
Prandtl number
1W m21s 7
0.3
Schmidt number
200
2
Liquid Heat Transfer Caefficient, h M s s Transfer Coefficient, kc
100 JldrrJK
Gas
1OOO Wlrn21K
65 W/m2fK
10-1 mls
3d
s
Thermodynamic Relationships Involving the Equilibrium Constant' For the gas-phase reaction
1. The true (dimensionless) equilibrium constant
RTlnK = -AG
where ai is the activity of spcies i
where f; = fugacity of species i fP = fugacity of the standard state. For gases, the standard state is 1 am.
For the limitations and for further explanation of these relationships, see, for example, K Denbigh, The Principles of Chemical Equilibrium, 3rd ed. (Cambridge: Cambridge University Press, 1971), p. 138.
Thermodynamic Relationships Involving the Equilibrium Constant
App. C
where: yi is the activity coefficient K
True equilibrium conslant K =Activity Guilibrium constant = Pregsure 2uilihriun-i msmt I=
Kc = Concentration equiribrium constant
-(dlp+cla-
K, has units of [atm]
k-
1
0
- bK, has units of [atm] " For ideal gases KT = 1.0 atm4
1
) = ~~m1-8
) = [am]'
2. The pressure equilibrium constant Kp is p;"g
KP = p*e0 P, = partial pressure of species i, atm, kPa. P,= C,RT
IC- 1)
3. The concentration equilibrium constant Kc is It is important to he able to relate
K,K,, Kc,and Kv 4.
For ideal gases, Kc and Kp are related by Kp = Kc(RT)8
5 . K p is a function of temperature only, and the temperature dependence of K p is given by van't Hoff's equation: Van't HoWs equation
6. Integrating, we have
K , and Kc are reIated by
App. C
Thermodynamic Relationships InvOlving the Equilibrium Constant1
1023
when
then KP = Kc 7. Kpneglecting ACp Given the equilibrium constant at one temperature T,, Kp (T,) and the heat of reaction AHR,, the partial pressure equilibrium constant at any temperature T is
I
I
8. From Le ChAtelier's principle we know that for exothermic reactions, the equilibrium shifts to the left (i.e.,K and X, decrease) as the temperature increases. Fjgures C-1 and C-2 show how the equilibsjum constant varies with temperature for an exothermic reaction and for an endothermic reaction, respectively.
Variation of
equilibrium constant with temperature
T
T
Figurn C-1 Exothermic reactton.
Figure C-2 Endothermic reaction.
9. The equilibrium constant at ternperamre T can be calculated from the change in the Gibbs free energy using -RT ln[K(T)I = AG:,(T)
(6-1 0)
10. Tables that list the standard Gibbs free energy of formation of a given species Gf are available in the literature.
1) w~n'/Ui~.~du:8O/-man~w~%rJl~mnd~nami~~Dcrta.and.Pmp~rr~~h~
& Llnks
2) w,ebbmknisr.gov 11. The relationship between the change in Gibbs free energy and enthalpy, H. and entropy, S, is AG=AH-TAS (C12) See hilbo chm u r i edw'CHMI lulecruresJI~crunS.l.hnn.
1024
[
Thormmlynamic Relationships Involving the Equilibdum Constant
Example G I
App. C
Water-Gas Shifi Reaction
The water-gas shift reaction to produce hydrogen,
is to be carried out at 1OOO K and 10 atm. For an equal-molar mixture of water and carbon monoxide, calculate the equilibrium conversion and concentration of each species.
Dnra: At IOOO K and 10 atm the Gibbs free energies of formation are Gc, = -47,860 cal/mol; GZo2 = -94,630 callmol; Gk0 = -46,040 callmol; G& = 0 SoIution
We first calculate the equilibrium constant. The first step in calculating K is to calculatt the change in Gibbs free energy for the reaction. Applying Equation fC-10) gives u: Calculate
AG:,
A%,
= G"H~+GEO~-EO~~-~%~O
(EC-1.1:
= -730 call rnol
-RT lnK = heR,(TJ
Calculate
K
1
b e (7-1 = l n K = -R"
RT
=
-(- 730 callrnol) 1.987calJmol~K(lOOOK)
(EC-1.2
0.367
then
K = L.44 Expressing the equilibrium constant first in terms of activities and then finally ir terms of concentration, we have
where ai is the activity,f,is the fugaciv, yr,is the activity coefficient (which we shal take to be 1.0 owing to high temperature and low pressure), and y, is the mole frac tion of species L2 Substituting for the mole fractions in terms of partial pressure
(EC-1.4
See Chapter 9 in J. M. Smith, In#mduction to Chemical Engineering T h e m d y m m Scs, 3rd ed. (New York: McGraw-Hill, 19591, and Chapter 9 in S. I. Sandler, Chemica and Engineering Themdynamics, 2nd ed. (NewYork:Wiley, 1989), for a discussiol of chemical equilibrium including nonideal effects.
Apo. C
I
Thermodvnamic Relationships Involving !he Equilibrium Constant1
1025
I n terms of conversion for an equal molar feed. we have
Relnre K and X,
(EC-1.7)
TEMPERATURE. DEGREES KELVIN
Figure EC-1.1 From M. Modclt and R. Reid. Thermodynamics and Its Applications, 0 1983. Reprinted by permission of Prentice Hall, Inc., Upper Saddle River, N.J.
1026
Thermodpamlc Relafionships I&ing
the Equilibrium Consfant
App. C
From Figure EC-1.1we read
at I000 K that log K, = 0.15; therefore, Kp = 1.41, which is close to !he calculated value. We note that there is no net change in the numkr of mdes for this reaction (i.e., S = 0); therefore,
K
= Kp = Kc (dimensionless)
Taking the square mt ofEquation (EC-1.7) yidds Calculate X, the Equilibrium Conversion
Solving for X,. we obtain
Then
Ca;lculate, ,,C EquiIi b r i m Conversion of CO
Figure EC-1.1gives the equilibrium constant as a function of temperature for a number of reactions. Reactions i n which the lines increase from left to Sight are exothermic.
The foIlowing links give themochemical dam. (Heats of Formation. Cp etc.) 1)
2) Links
w w .uic.edu/-mansoon%rAem~namic. Data.andPmpem-hzrrnl wtbbook.nist.gov
Also see Chem. Tech., 28 (3) (March), 19 (1998).
Measurement of Slopes on Semilog Paper
D
B y plo~ingdata directly on the appropriate log-Iog at semilog graph paper, a
Unts
great deaI of time may be saved over computing the logs of the data and then plotting them on linear graph paper. In the CD-ROM, we review the various techniques for plotting data and measuring slopes on semilog paper. Also see Physics Department web site at the University of Guelph, Ontario, Canada. www.physics.uoguelphca/ru~on'aldGLP.
Software Packages
E
Polymath 5.1 and a special version of FEhLAE3 are the software packages included with this book, along with extensive tutorials using screen shots.
E.1 Polymath E.l .A
About Polymath
Polymath is the primary software package used in this textbook, Polymath is an easy-to-use numerical compu~tionpackage that allows students and professionals to use personal computers to solve the following types of problems: Simultaneous linear algebraic equations Simultaneous nonlinear algebraic equations Simuttaneous ordinary differential equations Data Regressions (including the following) Curve fitting by polynomials and splines Multiple linear regression with statistics NanIinear regression with smtistics
Polymath is unique in hat the probIems are entered just like their mathematical equations, and there is a minimal learning curve. Problem solutions are easily found with robust algorithms. This allows very convenient problem solving to be used in chemical reaction engineering and other areas of chemical engineering, leading to an enhanced educational experience for students. The foIlowing special Polymath web site for software use and updating will be maintained for users of this textbook www.po1ymath-sofM,afi.com/fgler Lfn k
1030
Software Packages
E.1 .B
ADD. E
Polymath Tutorials
Polymath tutorials are given in the Summary Notes. Here screen shots of the various steps are shown for each of the Polymath programs.
Summary Notes Chapter 1 Summary htotes
A. Ordinary Differential Equation (ODE)Tutorial B. Nonlinear (NLE) Solver Tutorial Chapter 5 A. Fitting a Polynomial Tutorial B. Nonlinear Regression Tutorial
Note: The copy of Polymath supplied in the CD-ROM can be "opened" up to 200 times. Once Polymath is ''opened," any of the options (e.g., ODE solver, regression) can be run as many times as desired. That IS,you could open Polymath once, leave it open. and make an infinite number of runs and it would only count as one "'opening."After Polymath has been opened 200 times, directions on how to renew Polymath (with all recent updates and revisions and for a small fee) may be faund on the web at ~ ~ ~ ! p o l ~ r n a r h - , ~ ocorn fhuar@. Links
To obtain a discounted fee for Polymath, have: the following information available when you sign on the above web site to order Polymath! The ISBN number of the text (0-13-047394-4) and [he printing number which can be found on the back of the title page.
ITigurcE-l
I r l 9 t ~ l lPul!~nnth and [he Interactive computer
modules.
Sample MATLAB programs are given in CD-ROM,Appendix E. The disadvantage of the MATLAR ODE solver is t h a ~ it is nor particularIy user-friendly when trying to determine the variation of secondary parameter values. MATLAB can be used for the same four types of pmgrams as Polymath,
E.3 Aspen
Aspen
Aspen is a process simulator that is used primarily in senior design courses. It has the steepest learning curve of the software packages used in this text. It has a built-in database of the physical properties of reactants and products. Consequently, one has only to type in the chemicals and the rate Saw parameters. It is really too powerful to be used for the types of home problems given here. Example on The Pyrlosis of Benzene using Aspen is given on the CD-ROM using Aspen. CD-ROM Perhaps one home assignment shouId be devoted to using Aspen to solve a problem with heat effects in order to help familiarize the student with Aspen. An Aspen tutorial can be accessed directly from the CD-ROM home page.
E.4 FEMLAB FEMLAB is a partial differential equation solver (PDEI available commercially from COMSOL. Inc. Included with this text is a special version of FEMLAB that has been prepared to solve problems invofving tubular reactors. Specifically, one can solve CRE problems wirh heat effects involving both axial and radial gradients in concenrration and temperature simply hy loading the FEMLAB CD on one's computer and running the program. One can also use it to solve isasherma1 CRE probIems with reaction and diffusion. A step-by-step FEMLAB tutorial with screen show is given on the CRE CD-ROM web modules. There are two sections on the FEMLAB ECRE CD. The firrt one is "Heat effects in tubular reactors." and the second section i s "Tuhular reactors with dispersion.'" In the first section, the four examples focus on the effects of radial velocity profile and external cooling on the performances of isothermal and nonfsnthermal tubular reactors. I n the second section, ~ w oexamples examine the dispersion effects In a tubular reactor. Heat Effects I . Isothermal reactor. This example concerns an elenient'q, exothermic. second-order reversible liquid-phase reaction in a tubular reactor with a parabolic velocity distribution. OnIy the more, rate law. and stoichiometry balance in the tubular reactor are required in this FEMLAB chemical engineering module.
1032
Software Packages
App E
2. Nonisothermal adiabatic reactor. The isothermal reactor is extended to include heat effects whereby the tubular reactor is treated as an adiabatic reactor. In this model, malterial and energy balances i n the reactor are solved simultaneously in the FEMLAB module. 3. Nonisothermal reactor with isothermal cooling jacket. A coolant at constant temperature cooling jacket is added to the previous example to examine the performance of a nonisothermal reactor, In this model, the boundary condition for the energy balance at the radial boundary i s changed from the thermal insulation boundary condition to a heat flux boundary condition. 4. Nonisothermal reactor with variable coolant temperature. This example extends the third example by including the energy balance on the coolant in the cooling jacket as the temperature of the coolant varies along the reactor. Dispersion and Reaction
I . One-dimensional model with Danckwerts boundary conditions. In this example, the mass balance of an arbitrary reaction in a tubular reactor is described by a dimensioniess ordinary differential equation in terms of the Pkclet number and the Damktihler number. This model uses the Danckwerts boundary conditions when dispersion and reaction take place simultaneously. 2. One-dimensional model with upstream and downstream sections. This example uses the open-vessel boundary conditions where an inlet (upstream) section and an outlet (downstream) section are added to a tubular reactor where dispersion occurs but no reaction. It is suggested that one first use and play with the FEMLAB program for solutions to Examples 8-12 and 14-4, before making any changes. One should also review the web module for Chapter 8 on Axial and radial gradients in tubular reactors before running the program,
Nomenclature
ChemicaI species Cross-sectional area (mZ) Total external surface area of panicle Im2) External surface area of cataEyst per unit bed volume (mz/m3) Area of heat exchange per unit volume of reactor (m-I) External surface: area per volume of catalyst pellets (mz/rn3) Chemical species Flux of A resulting from bulk flow (rnollm2.s) Bodenstein number Chemical species Concentration of species i (mol/dm3) Heat capacity of species i at temperature T (callgmol K) Meao heat capacity of species i between temperature To and temperamre T (callrnol K) Mean heat capacity of species i between temperature TR and temperature T (callmol K) Total concentration (rnol/dm31 (Chapter 11) Chemical species Damkahler Number (dimensionless) Binary diffusion coefficient of A in B (dm2/s) Dispersion coefficient (cm21s) Effective diffusivity (dm21s) Knudsen diffusivity (drn2/s) Taylor dispersion coefficient Activation energy (caljgmol) Concentration of free (unbound) enzyme (rnol/dm3) Molar flow rate of species i (molls) Entering molar flow of species i (molls) Superficial mass velocity (g/dm2.s)
-
7 024
Nomenclature
App. F
Rate of generation of species i (molls) Gibbs free enegy of species i at temperature T (caISgrnol~K) Hi (TI Enthalpy of species i at temperature T (cal/mol i ) H,o(T> Enthalpy of species i at temperature To(callmol i ) Enthalpy of formation of species i at temperature TR(calJgmo1 i ) h Heat transfer coefficient (calErn2.s K) Molecular diffusive flux of species A (mollrn2.s) JA Adsorption equilibrium constan! KA Concentration equilibrium constant Kr Equilibrium constant (dimensionless) Kc Partial pressure equilibrium constant KP ASpecific reaction rate Mass transfer coefficient (m/s) A, Molecular weight of species i (g/mol) M, Ill, Mass of species i (g) Number of moles of species i (moI) N, n Overall reaction order Nu Nusselt Number (dimensionless) Pe Peclet number (Chapter 14) Panial pressure of species i (atm) p, Pr Prandtl Number (dimensionless) Hear flow from the surroundings to the s>lsiem IcalJs) Q R Ideal gas constant Reynolds number Re r Radial distance (m) Rate of generation of species A per unit volume (gmol A/s.dd) r~ Rate of disappearance of species A per unit volume -r, (rnoi A/s dm7) -ri Rafe of disappearance of species A per mass of catalyst (mol Ale-s) Rate of disappearance of A per unit area of caraiytic surface (mol k/rn2.s) An active sire (Chapter 10) Substrate concentration (grnol /dm3)(Chapter 7) Surface area per unit mass of cataIyst (m'tg) Selectil i ty parameter (instan~aneousselectivity) (Chapter 6 ) Overall selectivity of D to U Schmidt number (dimensionless) (Chapter 10) Shenvood number (dimensionless) (Chapter 10) Space velocity (5-I) Temperature ( K ) Time ( s ) Overall heat transfer coefficient (cal lm?. s . K ) Volume of reactor (dm'} Initial reactor volume (drns) Irolurneaic flow rate (dm7/s) G,
GPIn
HP
-
-
App. F
Nomenclature
Entering volumetric flow rate (drn3/s) Weight of catalyst (kg) Molar flux of species A (rnol/m2~s) Conversion of key reactant,A Instantaneous yield of species i Overall yield of species i Pressure ratio PIPo MoIe fraction of s p i e s i Initial mole fraction of species i CornpressibiIity factor Linear distance (crn) Subscripts 0 b c e
P
Entering or initial condition Bed [bulk) Catalyst Equilibrium Pellet
Greek Symbols a Reaction order (Chapter 3) 01 Pressure drop parameter (Chapter 4) Parameter in heat capacity (Chapter 8) a, Pi Parameter in heat capacity 3 I Reaction order YI Parameter in heat capacity 8 Change in the total number of moles per moIe of A reacted E Fraction change in voIume per mole of A reacted resulting from the change in total number of moles rl Internal effectiveness factor Ratio of the number of moles of species i initially (entering) to Oi the number of moles of A initially (entering) h Dimensionless distance ( z / L )(Chapter 12) h Life expectancy (s) (Chapter 13) P Viscosity (glcm.s) P Density Cg/cm3) Density of catalyst pellet (g/crd of pellet) PC Bulk density of catalyst (g/cm3 of reactor bed) Pb f Space time (5) 4 Void fraction (Porosity) bn Thiele modulus Ilr Dimensionless concentration (CAICAF) fl External (overall) effectiveness factor
Rate Law Data
Links
@ Unks
d Links
& Llnks
d Clnk
G
Reaction rate laws and data can be obtained from the following web sites: 1. National Institute of Standards and TechnoIogy (NIST) Chemical Kinetics Database on the Web Standard Reference Database 17, Version 7.0 (Web Version), Release 1.2 This web site provides a compilation of kinetics data on gas-phase reactions. kinetics.nist.gov/index.php 2. International Union of Pure and Applied Chemistry (IUPAC) This web site provides hnetic and photochemical data for gas kinetic data evaluation. www.iupac-kinetic.ch.cans.ac~uk/ 3, NASUJPL (Jet Propulsion Laboratory: California Insta'tute of Technology) This web site provides chemical kinetics and photochemical data for use in atmospheric studies. jpldataeval.jplnasa.gov/&wnload.html 4. BRENDA: University of Cologne This web site provides enzyme data and metabolic information. BRENDA is maintained and developed at the Institute of Biochemistry at the University of Cologne. www-bmnda.uni-koeln.de 5. NDRL Radiation Chemistry Data Center: Notre Dame Radiation Laboratory This web site provides the reaction rate data for transient radicals, radical ions, and excited states in solution. www.rcdc.ndedu
Open-Ended Problems
Links
H
The following are summaries for open-ended problems that have been used as term pmblcms at the University of Michigan. The complete pmblem statement of the problems can be found in the CD-ROM, Appendix H.
H.1 Design of Reaction Engineering Experiment The experiment is to be used in the undergraduate laboratory and cost less than $500 to buiId. Tbe judging criteria are the same as the criteria for the National AlChE Student Chapter Competition. The design is to be displayed on a poster board and explained to a panel of judges. Guidelines for the poster board display are provided by Jack Fishman and are given on the CD-ROM.
H.2 Effective Lubricant Design Lubricants used in car engines are formuIated by blending a base oil with additives to yield a mixture with the desirable physical attributes. In this problem, students examine the degradation of lubricants by oxidation and design an improved lubricant system. The design should include the lubricant sysrem's physical and chemical characteristics, as well as an explanation as to how i t is applied to automobiles. Focus: automotive industry, petroleum industry.
H , 3 Peach Bottom Nuclear Reactor The radioactive effluent stream from a newly constructed nuclear power plant must be made to conform with Nuclear Regulatory Commission standards. Students use chemical reaction engineering and creative pmblem solving to
f 040
Open-Ended ProMsrns
App. H
propose solutions for the treatment o f the reactor effluent. Focus: problem analysis, safety, ethics.
H.4 Underground Wet Oxidation You work for a specialty chemicals company, which produces large amounts of aqueous waste. Your chief executive officer (CEO)read in a journal about an emerging technology for reducing hazardous waste, and you must evaluate the system and its feasibility. Focus: waste processing, environmental issues, ethics.
H.5 Hydrodesulfurizatjon Reactor Design Your supervisor at Kleen Petrochemical wishes to use a hydrodesulfurization reaction to produce ethylbenzene from a process waste stream. You have been assigned the task of designing a reactor for the hydrodesulfurization reaction. Focus: reactor design.
H.6 Continuous Bioprocessing Most commercial bioreactions are carried out in batch reactors. The design of The complete a continuous bioreactor is desired since it may prove to be more economically pmbiem rewarding than batch processes. Most desirable is a reactor that can sustain &re found in the C D - ~ ~cells ~ , that are suspended in the reactor while growth medium is fed in, without Appendix H allowing the cells to exit the reactor. Focus: mixing modeling, separations. bia-
process kinetics, reactor design.
H.7 Methanol Synthesis Kinetic models based on experimental data are being used increasingly in the chemical industry for the design of catalytic reactors. However, the modeling process itself can influence the final reactor design and its ultimate perfomance by incorporating different interpretations of experimental design into the basic kinetic models. En this problem, students are asked to develop krnetic modeling meth~dslapproachesand apply them in the development of a model for the production of methanol from experimental data. Focus: kinetic modeling, reactor design.
H.8 Cajun Seafood Gumbo Most gourmet foods are prepared by batch processes. In this problem, students are challenged to design a continuous process for the production of gourmet-quality Cajun seafood gumbo from an old family recipe, Focus: reactor
design. Most gourmet foods are prepared by a batch process (actually in a batch reactor). Some of the most difficult gourmet foods to prepare are Louisiana
Sec. H.9
1041
Alcohol Metabot~sm
special ties, owing to the delicate baIance between spices (hotness) and subtle flavors that must be achieved. In preparing Creole and Cajun food, certain flavors are released only b!, cooking some o f the ingredients in hot oil for a period of time. We shall focus on one specialty. Cajun seafood gumbo. Develop a continuous-flow resctor system that wouId produce 5 gallh of a gourmet-quality seafood gumbo. Frepare a Row sheet of the entire operation. Outline certain experiments and areas of research that would be needed ro ensure the success of your project. Discuss how you would begin lo research these problems. Make a plan for any experiments to be carried out (see Section 5.6). Following is an old family fornula for Cajun seafood gumbo for batch operation (10 quarts, serves 40): 1 cup flour
i cups olive oil I cup chopped celery 2 large red onions (diced) 5 qt fish stock 6 Ib fish (combination of cod, red snapper, monk fish, and halibut) 12 oz crabrneat I qt medium oysters I lb medium to large shrimp
4 bay leaves, crushed
cup chopped parsley
3 large Idaho potatoes (diced) 1 tablespoon ground pepper
1 tabiespwn tomato paste 5 cloves garlic (diced) 1 tablespoon Tabasco sauce 2 1 bottle dry white wine 1 lb scallops
I. Make a roux (i.e., add 1 cup flour to 1 cup of boiling olive oil). Cook until dark brown. Add roux to fish stock. 2. Cook chopped celery and onion in boiling olive oil until onion is translucent. Drain and add to fish stock. 3. Add Iof the fish (1 lb) and of the crabmeat, liquor from oysters. bay ?eaves, parsley, potatoes, black pepper, tomato paste. garlic, Tabasco, and $ cup of the olive oil. Bring to a slow boil and cook 4 h, stirring intermittently. 4. Add 1 qt cold water. remove from the stove, and refrigerate (at least 12 h) until 2 ; h before serving.
5. Remove from refrigerator, add cup of the olive oil, wine, and scallops. Bring to a light boil. then simmer for 2 h. Add remaining fish (cut to bite size), crabmeat, and water to bring total volume to 10 qt. Simmer for 2 h, add shrimp, then 10 minutes later, add oysters and serve immediately.
H.9
Alcohol Metabolism The purpose of this open-ended problem is For the students to apply their knowledge of reaction kinetics to the problem of modeling the metabolism of atcohol in humans. In addition. the studenrs wiIl present their findings in a poster session. The poster presentations will be designed to bring a greater awareness to the university community of the dangers associated with alcohol
Living Example Problcmconsumptionm
1042
Open-Ended Problems
App. H
Students should choose one of the following four major topics to fwther investigate: I . Death caused by acute alcohol overdose 2. Long-term effects of alcohol 3. Interactions of alcohal with common medications 4. Factors affecting metahlism of alcohol General information regarding each of these topics can be found on the CD-ROM. The metabolism and model equations are given in section 7.5. One can load the Living Example problem for alcohol metabolism directIy from the
a
a pLE o n o w .
Lrulng Etample Probttrn
H.10 Methanol Poisoning The emergency room treatment for methanol poisoning is to inject ethanol intravenously ro tie up the aIcohol dehydrogenase enzyme so that methanol will not be converted to formic acid and formate, which causes blindness. The goal of this open-ended problem is to build on the physiological-based model for ethanol metabolism to predict the ethanol injection rate for methanol poisoning. One can find a start on this problem by reading problem P7-25c.
How to Use the CD-ROM
I
The primary purpose of the CD-ROMis ~o serve as an enrichment resource. The benefits of using the CD-ROMare fourfold: Link
1. To facilitate different student learning styles www engin. umich.edu/-cre/a~~Leaditresources. htm. 2. To provide the student with the optionlopportunity for further study or clarification of a particular concept or topic 3, To provide the opportunity to practice critical thinking,creative thinking, and problem-solving skills 4. To provide additional technical material for the practicing engineer 5. To provide other tutorial information such as additional homework problems and instnrctions on using computational software in chemi-
cal engineering
1.1 CD-ROM Components There are two types of information on this CD-ROM: information [hat is organized by chapter and information organized by concept. Material in the by chapter section on the CD-ROM corresponds to the material found in this book and i s further divided into five sections. Objectives. The objective$ page lists what the students wiIi learn from the chapter. When students are finished working on a chapter, h e y can come back to the objectives to see if you have covered everything in that chapter Or if students need additional heip on a specific topic, they can see if that topic i s covered in a chapter from the objectives page. Learning Resources. These resources give an overview of the material in each chapter and provide extra explanations, examples. and appEication5 to reinforce the basic concepts of chemical ~eacjjonengineering Summary Notes serve as an overview of each chapter and contain a logical flow of derived equations and
1044
6
!$LE
Llvlng ExsrnPlePr~blem
.
@
Homcwart problem^
App. I
additional examples. Web Modules and Interactive Computer Modules (ICM) show how the principles from the text can be applied to nonstandard problems. Solved Problems provide more examples for student5 to use the knowledge
A
Reference Shelf
How to Use the CD-ROM
gained from each chapter. Living Example Problems. These problems are usually examples from the text that require computational software to solve. Students can "play" with the problem and ask "what i f . . ?" questions to practice critical and creative thinking skills. Students can change parameter values, such as the reaction rate constants. to Ieam to deduce trends or predict the behavior of a given reaction system. Proi'e'essional Reference Shelf. The Professional Reference Sheif contains two types of information. First. i t 8ncIudes material that is ~rnportatitto the practicing engineer but that is typically not included in thl: majonty of chemical reaction engineering courses. Second, it ~ncludcsmaterial that gives a more detailed explanation of derivations that were abbreviated in the rexr. The intermediate steps to these derivations are shown on the CD-ROM. Additional Homework Problems. New problems were developed for this edition. They provide a greater opponunity to use today's computing power to solve walistlc problems. Instead of omitting some of the more traditional. yet excellent problems of previous editions, these probEems were placed on the CD-ROM and can serve as practice problems along with those unassigned problems in the text.
.
The materials in Learning Resources are further divided into Summary Notes, Web Modules, Interactive Computer Modules. and Solved Problems. Table I- 1 shows which enrichment resources can be found in each chapter. TABLE I- 1.
CD-ROM ENRICHME~T RFSOURCES
Interactive Computer Solved Problems
Living Example Pmhlems
Note: The ICMs are high-memory-use programs. Because of the memory
intensive nature of the ICMs, there have been intermiftent problems (lO-l5% of Windows computers) with the modules. You can usually solve the problem by trying the ICM on a different computer. Tn the Heatfx 2 ICM. only the first three reactors can be solved, and users cannot continue on to part 2 because of a bug currently in the program. The information that can be accessed in the by concept sections is not speci.fic to a single chapter. Although the material can be accessed from the by
Sec. 1.2
1045
Navigation
chapter sections, the by concept section allows you to access certain material quickly without browsing through chapters. lnteract~ve
4 Cornputs. Modules
Interactive Modules. The CD-ROM incrudes both Web Modules and ICitls. The Web Modules use a web browses for an interface and give examples nf how chemical reaction engineering principles can be applied to a wide nnge of situations such as modeling rvbra bites and cooking a potato. rCMs are modules that use a Windows or DOS-based pro,prn for an interface. They test knowledge on different aspects of ch'ernlcal reaction engineering through a variety of games mch as basketball and jeopardy. Problem Solving. Here students ran learn different stratcgles for prohIem so1villg in both dosed- and open-ended problems. See the ten diHerent types of home problems and suggesttom for approaching them. Extensive information on critical and creatlve thinking can also be found in this section. Frequently Asked Questions (EAQs). Over the years that I have taught this course. I have collected a number of auestions that the students have asked oker and over for years and years. The questions usually ask for clarification or for a different way of explaining the material or for another example of the principle being discussed. The FAQs and answers are arranged by chapter. Syllabi. Three representative syllabi have been included on the CD-ROM. See a 3-credit course schedule from the University of Illinois, or a couple of 4-credit course schcduEes from the Gniversity of Michigan. You can also practice for midterms, finals, or Doctoral Qualifying Exams with the actual exams given at the University of Michigan In previous years. Credits. See who was responsible for putting thiq CD-ROM together.
1.2 Navigation Students can use the CD-ROM in conjunction with the text in a number of different ways. T h e CD-ROM provides enrichment re.~urrw~s. It is up to each student to determine how to use these resources to generate the greatest benefit. Table 1-2 shows some of the clickable buttons found in the Summary rjotes within the Learning Resources and a brief description of what the students will see when they click on rhe butzons. TAWEI-?.
HOTBLTTONSIN SUMMARY NOTES
Solved example problem Genenl material that may not be related to the chapter
Hints and 6ps for solving problems A test on the material in a section with soIutions
Denvations of equations when not shown in the note<
--
1046
TABLE 1-2.
4+dw
How to U s e the CD-ROM HOT
B ~ N IN SSUMMARYN
App. C
m
Critical Thinking Question dated to the chapter
Wcb Module re1ated to the chaplei
Chapter objectives
Polymath solution of a problem from the Summary Notes Biography of the person who developed an equation or principle Chapter insen with more information on a lopic
Detailed solution of a problem
Plot of an equation or solution Extra information on a specific topic Audio clip
The creators of the CD-ROM tried to make navigating through the resources as easy and Iogical as possibIe. A more comprehensive guide to usage and navigation can be found on the CD-ROM. Figure 1-1 shows how to access the installation files for Polymath and the ICMs on the CD-ROM. The upper left window (CRE04)in the figure is what appears when the disc is inserted and "Explore the C D is chosen from the auto-run pop-up window.
1.3 How the CD-ROMMeb Can Help Learning Style
&
1.3.1
Global vs. Sequential Learners
See wwrw.engin.umich.edu/-cre/as~Leam~itreso~~~es. Arm.
Untc
Global
Use the summan, Iecture notes to pet an CD-ROM and see the big picture
werview of each chapter on the
Review real-world examples and pictures on the CD-ROM Look at concepts outlined in the ICMs
Sequential Use the Derive hot buttons to go through derivations In lecture notes on the web FolIow all derivations in the JCMs step by step Do all self-tests, audios, and examples in the CD-ROM tecture notes step by step
Sac. 1.3
How the CD-ROWeb
1.3.2
Can Help Learnln~Style
Active vs. Reflective Learners
Active *
Use all the hot buttons to interact witb the material to keep active Use self-tests as a good source of practice problems Use Living Example Problems to change seaings/parameters and see the result Review for exams using the ICMs
Reflective Self-tests allow you to consider the answer kfore seeing it Use Living Learning Problems to think about topics independently
1.3.3
Sensing vs. Intuitive Learners
Sensing
Use Web Modules (cobra, hippo, nanoparticIes) to see how material is applied to real-world topics Relate how Living Example Probfems are linked to real-world topics
Intuitive Vary parameters in supplied Polymath problems and understand their influence on a problem * Use the trial-and-error portions of some ICMs to understand "what i f . . . " style questions
1.3.4
Visual vs. Verbal Learners
Visual Study the examples and self-tests on the CD-ROM summary notes that have graphs and figures showing trends Do ICMs to see how each step of a derivation.problem leads to the next Use the graphical ourput from Living Example ProblemsPolyrnath code to obtain a visual understanding of how various parameters affect a system Use the Professional Reference Shelf to view ptctures of real reactors
Listen to audios on the web to hear information in another way Work with a panner to answer questions on the ICMs
Use of Computational Chemistry Software Packages
J
J.l Computational Chemical Engineering
L~nks
As a prologue to the future, our profession is evolving to one of molecular chemical engineering. For chemical reaction engineers. computation chemistry and molecular modeling. this could well be our future. Thermodynamic properties of molecular species that are used in reactor design problems can be readily estimated from thermodynamic data tabulated in standard reference sources such as Pelry"s Handbook or the JANAF Tables. Thermochemical properties of molecular species not tabulated can usually be estimated using group contribution methods. Estimation of activation energies is, however, much more difficult owing to the lack of reliable information on transition state structures, and the data required to carry out these catculatfons is not readily available. Recent advances in computational chemistry and the advent of powerful easy-to-use software tools have made it possible to estimate important reaction rate quantities {such as activation energy) with sufficient accuracy to permit incorporation of these new methods into the reactor design process. Computational chemistry programs are based on theories and equations from quantum mechanics. which until recently, could only be solved for the simplest systems such as the hydrogen atom. With the advent of inexpensive high-speed deskfop computers, the use of these programs in both engineering research and industrial practice is increasing rapidly. Molecular properties such as bond length, bond angle, net dipole moment, and electrostatic charge distribution can be calculated. AdditionalEy, reaction energetics can be accurately determined by using quantum chemistry to estimate heats of formation of reactants, products, and atso for transition state structures. Examples of commerciaIly availabte computational chemistry programs include Spartan developed by Wavefunction. Ine. (wwwWwovefi~n.corn). and Cerius2 from Molecular SimuIations, Inc. (www.accelrys.com). The following
7050
Use of Computational Chemistry Software Packages
App. J
example utilizes Spartan 4.0 to estimate the activation energy for a nucleophilic substitution reaction (SN2). The following calculations were performed on an IBM 43-P RS-6000W X workstation. An example using Spartan to calculate the activation energy for the reaction
C2H5CI + OW-
+ C2W50H + C1-
is given on the CD-ROM. CDPAppJ-I,
Redo Example Appendix J.1 on the CD-ROM. (a) Choose different methods of calculat~on,such as using a value of 2.0 A to constrain the C-Cl and C-0 bonds. (b) Choose different methods to calculate the potential energy surface. Compare the Ab Initio to the semiempirical method. (c) Within the semiempirical methd. compare the AM1 and PM3 models.
Index
PSSH in, 379-383 reaction pathways in, 391-394 Abia, S., 425 Accumulation in bioreactors, 435 Acetaldehyde in alcohol metabolism, 4 4 1 4 5 decomposition of, 378 from ethanol, 32 1-324 ethanol from, 41 8 pyrolysis of, 456 Acetic acid production. 608-614 Acetic anhydride cytidine reaction with. 912-914 production of, 504-505 adiabatic operation, 505-508 PFR with heat exchange for, 508-5 10 variable ambient temperature for, 510-51 1 dcetylation reactions, 2 16 Acetylene production, 370 Acmlein formation. 392 Activation energies in acetic acid production, 61 1 4 1 2 ARSST for, 605 bamer height for, 93 and bond strength, 97-98 determination of. 95-97 in rate laws, 85 and reaction coordinates, 92 Active intermediaks, 377-379 chain reactions in, 386391 for enzymatic reactionr. 394 mechanism searches in.
383-386
summary,4 4 7 4 9 Active l m r s , 1047 Active sites in catalysts. 65M51,661462 in enzymes. 395-396 Adenosene diphosphate (ADP) from adenosine mphosphate, 858-859 in biosynrhesis, 420 Adenosine tripbosphate ( A n ) adenosine diphosphate from, 858-859 in biosyathesis, 420 ARH {alcnhol dehydmgenase), 412, 41 8,466-467 Adiabatic operations in acetic anhydride production,
505-SOP, batch reactors, 594-598 butane liquid-phase isomerizatlon. 6 2 4 5 CSTRs, 526-531
eneqy baiance of, 4 7 8 4 7 9 hatch reactors, 594-598 equilibrium temperature. 514-515 in steady-state nonisothermal design. 48-87 tubular reactors. 488 exothermic irreversible gasphase reactions, 75-76 in nirroanaIine production, 603 propylene glycol production in, 526-53 1,595-598 1051
temperature and equilibrium conversion, 5 12-520 tubular reactors, 487-495 Adjustable parameters for nonideal reactors, 946 ADP (adenosene diphosphate) from adenosrne tnphosphate, 858-859 in biosynthesis. 420 Adswption. 650 of rumene, 672477
in CVD,701-703 dissociative, 287, 662, 664465, 702 equilibrium canstant, 663 isotherms, 661-666 rate constant, 663 in toluene hydrdemerhylarion, 69 1 Advanced Raaction System Screening Tool (ARSST). 605408 in acetic acid production, hGM14 in reaction orders, 638-640 Aerobic organism growth, 42 1 Aerosol reactors, 232-233, 235 Afiinity constant in MichaelisMenten equation, 400 Aging in catalyst deactivation. 650, 709-7 1 2 Air bags, 133-134 Air pallutiod. 32-33.392 AlcohoI dehydrogenase (ADH), 412, 418. 466-467 Alcohol metabolism. 393-394.441
Alcohul metabolism ( ~ n r ~ r . ) uenlrnl cornpantncnt in. 443 G 1. tract compnnent in. 443 lives compartment in, 44-445 model system for, 44 1 4 4 2 problem for. 1041-1012 stomach In. 4 3 2 4 3 Algorithmr complex reaction^. 327 mole balances, 327-37-8 net rates of reaction, 329-324 stoichiometry. 33.1-335 CRE problems. 116 data analysis. 254-255 ethylene glycol production. 154 vs. memorizing, 144 Aliphatic alcohol. 385 Alkencs, ozone reactions with, 298 Alkylation reactions, 652-653 Alpha order reactions. 83 Alumina-silica catalyst, 741-742 Amino acids in chymottypsin enzyme, 395 synthesis of. 420 Ammonia from hydrogen and nitrogen, 6 7 M 71 nitroanaline from, 599405 oxidation of, 35 1-355 producrion of. 747-748 from urea. 406-107 Ammonolysis. 216 Amylase, 395 Analytical solution Tor pressure drop, 181, 185-195 Anseth, Kristi. 823 Antib~otlcs coating on. 807-808 productton of, 419.423 An tifreeze from ethylene glycol. 163 from ethylene oxide, 191 Antithrombin. 376 Apoenzymes. 4 18 Apparent reactions In azornethane decomposition, 38 1 in falsified kinetics, 833-834 in kinetic rate law. 87 Approx~rnations in ethylene glycol pnaduction. 153 in segregation model, 909 Aqueous bromine, photochemical decay of, 297-298 A U {attainable region analysis), 360-36 1
Area balance in CVD. 702 Atts-Taylor andlysis for laminar flow. 975-978
A r k - b y l o r dispersion in dlhpersion model, 955 in tubular reactors, 963-964 Arrheniuc equation, 92. 95, YX Arrhen~ustemperature dependence, 407 ARSST (Advanced Reaction System Screening Tool L
605608 in acetic acid production. 608-6 14 in reaction orders. 638-6.U) Arterial blood in capillaries. 295 ln eyes, 898 Rniticial kidneys. 397 -ase suffix, 395 Aspen program, I94 explanation of, 1031 instructions for, 1015 Aspirin, 409 Assumptions in ethylene glycol production. 153 in tubular reactor radial and ax~alvariations, 559 Asymmetric distribution In maximum mixedness model. 932 in segregation model. 931 Atoms in diffusion, 758 in rtdctlon. 80 ATP (adenosine triphosphite) adenosine diphosphatc from, 858-859 in biofy nthesis. 520 Attainable region analysis (ARA), 360-361 Autocatalytic growth, 74-75 Autocatnlytic reactions, 421 Automobiles emissions nitrogen oxides in. 298-2W. 742-743 in smog formation, 32-33,
392-393 green gasolines, 584 Average molar flux in diffusion, 774 Axial diffusion. 781 Axial dispersion in dispersion model, 955 in packed beds, 844
Axial variations in tubular reactors, 55 1-56 1 Ayen. R. E.. 747 Azornethane decomposition. ethane and nitrogen from, 379-383
Back-of-the-envelope calculations, 788 Backmix reactors. See Contiouousst~rredrank reactors (CSTRr) Bacteria, 418. See also Cells in batch reactors, 4 3 2 4 3 4 in cell growth. 421-423 in enzyme production, 394 in ethanol production. 240 predictor prey relationships in, 464 in yogurt postacidification. 459 Bailey. J. E..421 Balance dispersion, 457 Balance equations in diffusion, 769-770 Balance in CSTRs, 980 Balance on A in semibatch reactors, 220 Balance on coolant in tubular reactors. 499-5 11 Balance on hydrogen in membrane reacton, 209-2 10, 2 12 Balance on reactor volume in CSTR parameter modeling, 984 BarthoIomew, C.,7 L3. 715 Basls of calculation ia conversions. 38 Batch reactors adiabatic operation of, 594-598 bactena growth in, 432434 bioreactors. 43 1 catalyst sintering in, 71 1-7 12 concentration equations for, 102-103 constant-volume. 103-106 cylindncal. 138 data analysis methods. 25&257 differentia!. 257-266 integnl. 267-271 nunlinear regression, 271-277 design equations for, 3 8 4 0 , 99 eneryy baiance of, 477.594-598 enzymatic reaction calculations, 404407
evaluation of. 29Q-291 with interrupted isothermal opration. 599-605 journal crit~queproblems. 250 mean conversion in, 9 10. 913 mole balances on, 10-12 in design equations. 39 enzymatic reactions. 404 gas phase, 200-202 integral data analysis, 267 liquid phase. 200 RTDs in, R85-886 space
time in, 67
stoichiometry in. 1M)-106 with variable volume, 109-1 11 Batch-type experiments. 6 Bed Ruldicity in hydrazine decomposit~an,788 Benzene adsorption of, 676 catalysts in production of, 647 from cumene, S from cyclohexane. 804-805 desorption of, 675, 6 7 8 4 8 0 ethylbenzene from,652-653 in Langmuir-Hinshelwood kinetics, 672 in reversible reactions, 89-90 from toluene, 87, 6 8 8 4 9 8 Benzene diazoniurn chloride, decomposition of, 95 Berra. Yogi on observation, 253 on questions, 29 on termination, 757
Bmelius. J.. 445646 Best estimates of parameter values in nonlinear regression.
273 Beta order reactions, 83 Bifurcation problems. 567, 588 Bimodal distributions, 932 Bimolecular reactions, 80 Binary diffusion, 7ML765 Bioconversions. 419 Biomass reactions in biosynthesis. 419 in nonisothermal reactor design. 571-578. 637-638 in reaction rate law, 86 Biomedical engineering, RTDs for, 898
Biopmessing design problem, 1040 Bioreactors. 4 1W20 autocadytic growth in, 74-75 cell growth In, 422423
chemostats, 137.434-435 design equations for. 435436 journal critique problems. 250, 469 mass balances in, 437434 oxygen-limited growth in. 438439 rate laws in. 423-425 scale-up in, 439 stoichiornetr): in, 426-430 summary,44749 supplementary reading for, 470 wash-out in, 436-438 Biosynthesis, 4 1 8 4 1 9 Binh control patches. 772 BischoSF, K. B., 960 Blindness fmm methanol, 4 11. 46E-467 Blood coagulation. 325-326, 362,
370 Blood detoxification, 898 Blood Rows in alcohol metabolism. 442 in capillaries, 295 in phamacokinetic mcdels. 440 Blowout velocity in multiple steady states, 539 Bodenstein number in dispersion and T-I-S models. 974 in tubular reacton, 958 Boltzmann's constant, 1017 Bomb cnlorimeter reactors constant-volume reactors. 103 for reaction rate data, 40 Bond distortions in reaction systems, 93-94 Bonding for enzyme-substrate c ~ m p l e x ,
spherical catalyst pellets, 8 19.
822 tubular reacton. 557, 560. 958-962 Bounded canve~ionsin nonideal reactors, 946 Brenner, H., 963 Bripgs-Haldane Equation. 404 Bromine cyanide in methyl bmmide prduction, 220-223 Bubble residence time, 869
Bulk concentration in d~ffusionand reaction, 838 in mass transfer to single particles. 778 Bulk density in packed bed flow, 179 in pressure drop. 188 Bulk diffusivity. 8 15 Bulk flow term in diffusion, 767 Bulk phase, difision in, 770
Bums,Mark. 408 Butadiene from
ethanol, 306
for synthetic rubkr. 654
Butane butene from, 21 1 from cyclobutane. 379 from ethane. 387 isomenzation of, 6 2 4 , 49w95,497499 Butanol dehydration, 741-742 Butene from butane. 21 1 Butt. J. B.. 707 Butyl alcohol P A ) , 740 Bypassing in CSTRs, 893-894,979-985.
988-990 in tubular reactors, 895-896
396 in intentage cooling, 519 Bonding strength and activatiun energies, 97-98 catalysts in. 653 Boundary conditions can~lageforming cells, 825-816 catalyst carbon removal, 795 diffusion. 659. 765. 768, 771, 773,819, 822 dispersion coefficient detemination. 967-970 dispersion models, 1032 mass kansfer to single particles,
778,780 maximum mixedness m d e l . 9 18-922
C curves in pulse input experiment, 87 1-876 in single-CSTR RTDs, 887 Cajun seafood gumbo, 1040-1041 Calcium magnesium carbonate, dissolution in hydrochloric
acid, 278 CaIcalations back-of-the-envelope, 788 for enzymatic reactions. 404407
for propylene glycol production, 528
Calorimeters ARSST, 605-614 bomb, 40, 103 Cape, E. G., 898
Capillaries. merial b l o d in. 295 Carbon dioxide in acetaldehyde formation. 321-324 from spartanol, 751-752 from urea, 4DWU7 Carbon disulph~de,805 Carbon monoxide adsorptfon of, 662664 methane from. 286288 Carbon removal in catalyst regeneration, 793-797 Cartilage forming celIs, 823-827 Cascades of CSTRs,23 Catalysts and catalytic reactors. 645 adsorption isotherms, 661-666 benzene rate-l~mit~ng, 67R-680 carbon removal in. 793-797 catalysis, 6 4 6 6 4 7 CD-ROM material. 7 3 6 7 3 8 classification of, 6 5 2 6 5 5 cracking, 725-728 deactivation of, 650. 707-709 by cok~ngand foultng, 712-713 empirical decay laws in, 716720
in moving-bed reactors, 722-728 by poisoning, 713-716 by sintenng, 709-712 in STTRs, 728-732 temperature-time trajectories in, 721-722 definitions, 6.16447 desorprion. 668-669 for differentla1 reactors. 281-282 diffuslon in from bulk to external transport, 658-659 differentla1 equatlon for, 81 6 819, 822-823 dimensionless form, 81 9-822 effective diffusivity in, 814-816 through film, 76G770 internal, 660461 for (IsSue enpineerinp, 823-827 in ethytene oxide production, 195
heterogeneous data analysis for. 688489
mechanisms in. 691-692 rate laws in, 6 8 9 4 9 3 ,
692-694 reactor design. 693698 in heterogeneous reactions, 88 journal critique problems, 753-755
membrane Eacmrs, 207-208 in mcrwlectronic fabrication,
698-700 Chemical Vapor Deposition in. 701-704 etching in, TOO model discrimination in, 704-707 f n moving-ted reactors. 712-728 properties of, 64&652 questions and problems. 738-753 rapid reactions on, 776-780 rate laws, 67 1-674 deducing, 689-690 denved from PSSH, 684487 waluating, 692494 temperature dependence of, 687-688 rate-lim~ting,669474, 677-680 reforming. 6 8 1 6 8 5 regeneration uf, 7'93-797 sheIl balance on, 81 6 steps in. 655-671 summary, 733-736 supplementary reading. 755-756 surface reaction in, 66-68 welght ethylene oxide, 19 1 heterogeneous reactions. 6 membrane reactors, 209 PER, 45, 179 w~thpressure drop, 18&190 Catalytic afterburners. 714 Catalytic dehydration of methanoI. 742 CD-ROM material active intermediates, enzymatic reactions. pharmacokinetic models, and bioreactors, 119453 catalysts, 7 3 6 7 3 8 components, 1043-1045 con\ ersion and reactor sizing, 71-72
d~ffusion.801-802, 852-855
isothermal reactor design, 23 1-234 for learning styles, 1 0 4 6 1047 mole balances, 26-29 multiple reactions, 359-36 1 navigating, 1 045-1 046 nonideal reactors, 994995 nonisothemal reactor design steady-state, 5 6 5 8 8 unsteady-state. 630-632 open-ended problems, 1039-1042 rate data coIlection and analysis, 293-294 rate laws and stoichiometry, 126-130, 139-140 RTDs, 934-935 Cells cartilage forming, 823-827 growth and division, 42M23 chemostats for. 137, 4 3 4 4 3 5 design equations for, 435-436 and dilution rate, 437 Luedeking-P~retequation,
429 mass balances in, 43 1 4 3 4
oxygen-limited, 438-439 rate laws in, 423-425, 428 stoichiometry in, 426430 wash-out in, 43W38 reactions in. 4 1 9 4 2 0 as reactors. 31 Centers in catalysts, 650451 Centrdl companment in alcohol metahlism. 443 Cereals, nufrients in. 245 Cerius program, 379. 1049 Chain reactrons. 386391 Chain rule for diffiision and reaction. 819 Chain transfer step, 386 Channels and channeling in mlcroreactors. 201 in tubular reactors. 895-896 Characteristic reactlon times in barch operation, 15 1 Chemical prodoction statistics, 31-32 Chemical reaction engineering (CRE).1-3 Chemical species, 4-5 Chemical h p o r Deposition (CVDJ in diffuslon, 84g-852 In microelectronic fabrication. 701-704 professional reference shelf for. 854-855
Chemisorption. 650, 661 Chernostats, 137. 434435 Chestenon. G. K.,645 Chipmunk respiration rate, 805 Chirping frequency of crickets,
132 Chloral in DDT. 6 Chlorination in membrane reactors. 347 in semibatch reactors. 216 Chlwobenzene from benzene diazoniurn c hloride, 95 in DDT. 6 Churchill, Winston, 1006 Chyrnotrypsin enzyme,395 Classes of cell reactions, 420 Clinoptilolite in toluene hydrodemethylation, 688698 Closed-cIosed boundary cond~tions in dispersion cwfficient determinat~on,967 in tubular reactors. 959-961 Closed systems, first law of thermodynamlcs for, 473 Clotting of b l d , 325-326, 362.
3711 CMRs (catalytic membrane reactors), 207-20s Co-current Row in tublar reactors,
5W50 1 Coagulatron of blood. 325-326, 362, 370 Coal ~asification,103 Iiquefact~on,369-370 Cobalt-molybdenum catalyst, 740-74 1 Cobra. bites. 363 Cocci. growth of. 421 Foenzy mes. 4 1 8 Cofactors, enzyme, 41 8 Coking in catalyst deactivation, 650. 712-713 Colburn J factor in h ydraz~nedecomposition. 787-788 In mass transfer correlations, 784-785 Collision rate in adsorption, 662 Collision theory, 84 active rntermediates in, 378 professional reference shelf for. 128 in reacrion systems. 43 Combtnarion step ticetaldehyde formatron. 322
acetic acid production, 61 0 ammonia oxidation, 354 butane isomerfzation. 491) catalyst decay, 7 19 catalyst stntering, 71 1 complex reacrion algorithms. 328 CSTRs batch operation, 149-150 cataIysr decay, 719 with cooling coils, 5.1 I second-order reaction in. 162 sinele. 157 ethylene glycol production, 154, 165
ethylene oxide prduction. 192-193
gas phase. 202 glucose-to-ethanol fennentatton, 433 laminar flow reactors mean conversion, 91 3 membrane reactors Row and reaction in, 2 13 tn multiple reactions. 350 mesitylene hydrodealkylation, 342 mole balance design, 199 moving-bed reactor catalytic cracking, 726 net rates of reaction. 332-334 nltroanaline production, 601 nitrogen oxide produn~on.205 nonisothermal reactor design. 472 PFR reactor volume. 146147 pressure drop in isothermal reactor design. 176 in tubular reactors, 186 propylene glycol production, 526. 596 triphenyl methyl chloride-methannl reaction. 262 tuhu1a.r reactors adiabatic. 488 flow In. 164-170, 173 Combinations in CSTR parameter modeling. 984 CSTRs and PFRs in, HI-64 and species identity, 5 Company chemical production statistrc~.32 Competing reactions. 305-3Rh Competitive inhihit~on.410-412
Complete segregation in RTD reactor modeling, 903 Completely segregated Rulds in segregation model. 904 Complex reactions, 305-3M algorithms for, 327 mole balanas, 327-328 net rates of reaction, 329-334 stoichiometry, 334-335 ammonra midation, 351-355 Compressibility factors in barch reactor state equation. 109
in Row systems. 111 Compression of ultrasonic waves, 384-386 Compression ratio end =tang number, 682 Concentration surfaces for laminar Row, 978
Concentrat~an-timedata in hatch reactors, 256 in nonlinear regression, 273-275 Concentrations and concentration profiles active site balances. 661 ammonia oxidation. 353-354 batch reactor data analysis,
167-268 catalyst carbon removal, 796 complex reacdon algorithms, 334-335 CSTRs. 13 differential reactors, 283 diffusion. 762-764, 768, 816. 820. 822 dilution, 437 dispersion model, 956 enzyme, 398-399 flow systems, 107, 1 1 1-1 12 key reactants, 115-1 17 liquid-phase, 108 specie<. 1 12-1 13 iqothermal reactor design. 198-1 99 laminar flow. 978 mass transfer correlat~ons.784 to s~ngIepartrcles, 778 methane pmduclion. 285-286 with pressure drop, 185 rate data analysis, 254-255 sern~batchreactors, 224 cpherical catalyst pellets, 81 2. 8 16. 820, 822 T-1-S model, 952
Concentrations and concentration proti les (cor!l.1 toluene hydrodemethylation, 692
tubular reactors, 959 Condensation reactions. 130 Confidence limits in nonlinear regression. 273 Configuration in chemical species. 4 Consecutive reactions. 306 Constant heat capacities in enthalpy, 483 Constant-volume batch systems. 40. 103-106. 257 Con6rant-volume decompositron of dimethyl ether, 297 Constant volumetric Row for differential reactors. 283 Conctriction factor in effective diffusivity, 815 Continuous-Bow systems, 106 design equation applications,
15-54 in mole balance. 12-21 reactor time in, 40 RTDs. 868-870 Continuous-stirred tank reactors (CSTRS),12-14 in butane i~omerization,494 bypassing in, 893-894. 979-985,988-990 cascades of. 23 catalyst decay in. 717-720 conversion in, 973 with cooling coils, 531-532 dead space
in, 979-985,
988-990 design, 14, 156-157 design equations for, 43-44. 99
parallel, 160-162 scale-up in, 148-156 second-order reaction in, 162-1 68 series, 158-160 single, 157-158 diagnostics and troubleshooting, 892-895 energy balance of, 47&477. 523, 531-532. 548-55 1, 620 in equilibrium equation. 123 for ethyIene gIycal. 152-156 evaluation of, 29 1 fluidized-bed reactors as, 24 with heat effects, 522-532
for l iquid-phase reactions, 32-13. 21-22 mass balnnces in. 43 1 mean conversion in, 907-910 mesitylene hydsodealkylarion in. 344-347 rnudeled as two CSTRs with znterchange. 985-987 mole balances on, 43, 2W202 multiple reactions, 343-347, 548-55 1 nonideal reactors using, 9'9tL99 1 parallel reactions, 160-162, 165-167. 3 1 4 3 1 7 propylcne glycol production in. 526-53 1 RTDs in. 868-870.887-888. 892-895. 897-90 1 runaway reactions in. S4G543 in series. 55-58 design, 158-160, 166-167 with PFRs, M M 4 RTDs for, 897-901 sequencing, 6 4 6 6 siztng, 48-50, 53-54 space ume In, 67 unsteady-state operation of energy balance of, 477 stanup. 216-217, 619-622 steady state falloff in. 622-624 Convection in diffusion, 763-765 mass transfer coefficient in. 774-776 id tubular reactors, 958,963 Convective-diffusion equation. 963 Conversion and reactor sizing, 37-38 batch reactors. 3&40 CD-ROMmaterial, 71-72 continuous-flow reactors. 35-54 conversion definition. 38 equilibrium. See Equilibrium conversions flaw reactors, 4045. 112, 1I7-123 mean conversion in laminar flow
reacton.
912-914
in real reactors. 916-9 LZ in segregation model, 9 0 6 9 I0 PBRs. 20 wlrh pressure drop, 185, 187 questions and problems. 72-77
rate law5 in, 98-99 reactors in senes. 54-65 space tlme. 66-67 space velmity, 6 8 4 9 summary. 69-7 1 supplementary reading. 77 Conversion bounds in maxlmum rntxrdness model. 918-92 Conversion factors for units, 1018-1019 Cooking potatoes, (34 seafood gumbo. 104% 1041 spagherti. 236 Coolant halance in steady-state tubular reacrors, 499-5 11 in tuhular reactors, 556 Coolant flow rate in interstage cooling, 5 18 Coolant temperature in energy balance. 478 in semibatch reactors, 614-619 in steady-state tubuiar reactors, 50 1 Cooling coils in CSTRs. 531-532 Cooling jackets in tubulnr reactors 560 Coordinates, reaction. 92 Corn starch. 395 Corn fations dispersion coefficient, 964-966 mass transfer, 774-776, 783-785 Corrosion of high-nickel stainless steel plates, 132-133 Costs
in ethyfene glycol production, 196198 of Pfaudler CSTRslbatch
reactors, 22 Counter current flow in tubular reactors, 50 1-502 Counterdiffusion. equimolar. 762 Cracking catalytic, 725-728 thermal, 387-392 CRE (chemical reaction engineering). 1-3 Creativity in reactor selection, 992 Cricket chirping frequency, 132 Critiquing journal articles. See Journal critique problems Crystals in microelectronic fabrication, 698 CSTRs. See Continuous-stirred tank reactors (CSTRs)
Diffeferentiat~on,equal-area graphical, 263, 1010-I01 1 Diffusion, 757, 813-8 Id blnsry, 760-765 boundary cond~tionsin, 659, 765. 768. 771, 773. 819. 822 from bulk to external transport. 658-659
w~thcatalysts. 658-661. 76.I-770. 814 differential equation for. 8 16-8 19. 822-82.7 d~mensionle>sform. 8 19-82? effective dtffunivity in. RI&Xlh for tissue engineering. 823-857 CD-ROM material, 801-802. 852-855
Chemical Vapor Deposition in. 849-852 convectton In. 763-765 definitions. 758-159 dtffvs~on-and reaction-limited regime estimation in. 838-R42
model. 955 externa! reqiszance to mtlrs transfer in example. 783-788 mass ~ransfercoefficient In. 771-776 mass transfer-limited reactions, 780-783 mass tranrfer to single panicles. 776-780 falsified kineucs In. 833-815 Fick's first law in. 7AC7hl through him. 7 6 6 7 7 0 f o r ~ r dconvec~innin. 763-764 fundamentals. 7.58 il~trrnal.660461 I nrernal effectiuenc~sfac~clor. 827433 journal urticlc problems. 863 journal cril~queproblemr. 809-8 10. 863-864 limiting siruiltionc for. 848-849 Inaw tranller in packed beds. 111 dispersion
842-K48
Mean' crrterlon fur. 84 1-84! modeling with. 77 1 mnlilr flux in. 75X-760. 763-765 nlultiphase reactors ill. 849-850
operating condition changes in.
783-788 overall effectiveness factor in,
835-838 in pharrnacokinetics, 798-799 questions and problems. 802-8 10, 855-863 shrinking core model. 792-799 through stagnant film, 762, 774 summary. 800-801, 851-852 supplcmentaty reading, 810-811. 865-866 l c m ~ r a l u r eand pressure dependence in, 770 for transdermal drug deli\wy. 772 DigttaI-age problems. multiple reactions for. 356-557 Dilute concentrations iri diffusiul~, 762-763
Dilution rate in hioreactors. 4 3 5 4 3 6 In chernostat5. 434 in wash-out. 437 Dimens~onlesscumulative distributlons. 890-891 D~mensionlessgroups in ma\s transfer cwfficient, 774 Dirnerize propylene, 6 M 1 Dimethyl ether (DME) decomposition of. 297 from methanol. 742 Diphenyl in reversible reactions. 89-90
Dmc delta function in PFR RTD.886 in step tracer experiment, 877 Di~appearanceof ~uhwrate.100. 431-432 Dl\appe:rrance rate. 5-6 Dlqguised kinetu. 873-835 Dtsk rupture In nrtmanalinc production. 605 Dispers~nn of catalysts. 631 FEMLAB lor. 975;978. 1032 une-parameter model\. 947 in packed bed<. 966 \ \. T-1-5mudel\, '3 74 in tubular reactors, 953-957. Yhi-9lrh. 971-973 Dikperriun coefficient erperlmental dererln~n;lhonof. 466-970 tubular seautorc, Lj(ll-Ybh Dl\\r~crntireadcorpr~(rn.X 7 , hh2. 664-b65. 7112
Distortions in reaction systems. 93-94 Divide and be conquered case. 789 D~ving-chamberexperiment. 804 Dlvismn of celIs. 4 2 0 4 2 3 DME (dimethyl erherj decomposition of, 297 from methanol, 742 DNA (deoxyribonucleic actdl identiticat~on,408-409 in protein pmduct~on.419420 Dolomiie dissoluriun. 278 Dunov:in. Rrrbert 3.. 757 Doubling times in growth rates. 125 Dr~nkingand driving. 364 Drug therapy sumpeti~ive~nhibi~ion in. 410 rran~JermaFdelnery syslems. 772 Dry etching. 700 Dual sites lrrevenible surface-reuction-limited rate lawr In. 684 surface reactionr, In catalysts. 667-668 Dwidevi. P. 3.. 785
E. cnli production. Ibh E curves in normalized RTD funcdm. XR+X85 in pulse input experiment. 871-873. X75 it1 RTD nloments. 882 Eadie-Hnf~~ee plots. 403 Easl Texas liphr pas nil. 7 12-7 1.3 Econom~cdeuiunns and incentireu irr irreversible ~ , a >phirse catal y ~ i crcac~rotlh.749 Tor p;~raIlelrea~,~iclns, ?I h for separations hycrelns. 307 Edwards. D. A,. 963 Effective diffusivity. 855 Ln catalyst carbon retnoval. 795 in spherical ci~talyctprllets. XI4816 EtYecti~ctranspon ct~tlicient.77s Efieutlb-enerc l'autrlr in diffusion internal. K27- X33. N3Y-811 ort.l.;~ll.X35-XqX i l l nitrous oxide it'dl~clit~~lc. XIfbXIK
Index
Eficient paralleI reactor schemes. 310 Electrical heating rate with calorimeters, 607 Electronics industry. micmelectronic fabrrcation. 299-300, 698 Chemical Vapor Deposition in. 701-704 etching in. 700 Elemenrary rate laws, 82-86 Elementary reactions, M. 662 Eley-Rideal mechanism. 68 1 irre\~erriblesurface-reaction-11mired rate laws in. 684 in surface reactions in catalysts. 668 Elution, 878 EMCD (equimoltlr vounterdiffusion) in binnn, diffuqion, 762 in spherrcal catalyst pellets, 81 7 Emissions. autornohjle nitrogen oxides in. 298-299. 742-743 in smog formation. 32-33, 392-393 Empirical decay laws. 71 6-720 Endotheliutn in blood clotting. 335 Endothermic reacrions, equilib rium ronrersron in. 5 1 1. 5 16-510 Energy, cnn\,emion factors for. 1018 Energy balances in acerlc acid productiol~.610 in acetic anhydride production. 505-506. 508-509 adiabatic operations. 478-479 batch reactors. 394-548 equilibrium temperature. 514-515 i n rtead!'-state nonI\othermal derign. 486-4X7 tubular reacrorq, J X R in butane isomer~zation,391 in roolanx balance. 503 CSTRs. 3 7 W 7 7 uith conllng cotfs. 531-532 hear exchanger in, 573 ill multiple reec~ionr, 538-15 1 un
first law of thermodynamics, 473-474 hear of reaction in. 483486 in movinpbed reactor catalytic cracking. 728 in nitroanaline production. 601 overview of. 476-479 PBRs. 477.576
PFRs. 477 with heat exchange, 495499 multiple reaction?. 5 6 . 5 4 7 parallel reactions. 546 in propylene glycol production. 528. 596. 620 in sern~bstchreactors, 477 w~thheat exchangers. 615 multiple reactions, 626 sleady-state molar flow rates, 47948 1 in tuhlar reactors. 495499. 554-556. 559 in un5ready-state nonisothermal renctorr. 591-593 work term in, 474476 Energy barriers, 92-93 Energy d~slributionfunction. 94 Energy economy h ydrogen-based, 247 membrane reactor<,21 1 Energy flux, 554 Enerey flux vectors. 554 ~ n e i - i yrate change with time. 1019
Enpine knock, mane number in. 681 -684 Engine oil. 457458 Engineering experiment design problem. 11139 Engineering lutlgrnent in reaclur \election. YYZ Entertng cuncentretions I n flow reactor design, 42 Enthalpies in enerpy balance. 475. 48 1 4 8 3
Enzxmat~creactions. 86. 394-395 hatch reactor calculations for. 404407
Br~~ps-Haldane Equation. 403 Eadie-Hnfstee plots. 403 enzyme-suhstrote complex. 395-397 111ducedf i t model, 396 inhrhit~ono f See Inhlhirion ot' e n q m c reactions luck and k y nrodeI. 396 rneclmnismh. 397-399
Michaelis-Menten equation, 399401 summary. 447449 supplementary reading for, 470 temperature in. 407 Enzyme-cam1yzed polymerization of nudeotides. 4 0 8 4 9 Epidemiology, PSSH for, 458 Epi~axinlgermanium, 701 Epoxydation of ethylene. 370-351 Equal-area differentiation, 163. 1010101 1 Equally spaced data points in data analys~s.258 Equation of state in batch reactors. 109 Equations for batch concentratlonq. I 02-1 03 for concentrations in flow syrfems. 107
differential. See Differentlei forms and equations Equ~librlum in adiabatic equilibrium temperature. 5 13 catalysts In. (747 in CVD, 703 Equili hrium constant in adiabatic equil~briumtemperature.
51 4
adrarptron, 663 in thermodynamic relationship. lU21-1026 Equillbrlum conversions. 5 1 I and adiabatic temperature. 5 12-520 tn butane isomerization. 492 in endothermic reaction%.5 1 1. 5 Ih-E;?O in exothemlc reactions. 512-5th feed temperature in, 520-521 in aerniha~chreactors. ?lr1?25 ~ i t variable h volumetr~cflow rak. 1 I R 123 Equimolar cnunterdiffu~inn (EMCD) tn binary difft~sion.762 !n spherical catalyst pellels. 817 Ergun equntion. 177-1 80, 1 RY Esterificalion reaction<. 2 I h Erl) cumrs, fittinp to pol!nornial\. 923-924 Etch~ng.semicnnductnr. 249-300. 700
PFR rcnclor volume. 146-147 pmpylene glycol pmdr~cllon. 5Yh-.51)7. h2 1 in tubular reactor Jc5ign. k73-174 Excel for ncllvation energy. 95-97 for CSTR parametcr ~nodel~ng. 982-983 for RTD moment\. 883 for tr~tyl-methanolre:stiun. 269-270 fur tubular reactor conversion. 972 Excexs method in batch reactors. 256 Exchange colurnes in CSTR parameter modeling. 987 Exhaust slreams, automobile nitrogen oxldes in. 298-299, 742-743 in smog formation, 32-33. 392-393 Exit-age disrribution funcct~un,$78 Exit temperature in interstage cooling, 5 17 Exothermic xactionr. 485. 5 1 1 equilibrium conversion in. 5 11-516 liquid-phase, 579 safety in, 599-605 Experirnentnl observation and measurements for diffusion. 802 for dispersion coefficient. 966-970 for rate laws, 84 ExperimentaI planning professional reference shelf for. 293-294 in n t e data collection and anelysis. 289 Explosicins Munsanto plant. 599605 nltrous oxide plant. 6 3 4 6 3 5 Explosive intermediates, microreacton for, 203 Exponential cell growth. 423124 Expnent~aldecay in catalyst deactivation. 7 17 Exponential integrals, 908 External diffusion effects. Scrc Difof
Elhdne fmm atomrthane. 379-383 ethylene from. 387-389.
740-74 1 in ethylene glycol product~on, 597-198 ethylene hydrogenation to. 70.1-707 thermal cracking of. 387-392 Ethanol from acetaldehyde. 41 8 acetaldehyde from. 32 1-324 AUH with, 412.466467 hrtadieoe
from.306
glucose-to-ethanul fennentntion. 4 3 2 4 3 4 in wine-making, 424 from Zymononas bacteria. 240 Ethoxylation reactrons. 347 Ethyl acetate saponiRcatit)n, 6l6-619 Ethylbenzene from benzene and ethylene. 652-653 tu ethylcydohexane. 752 styrene from. 2 1 1.585-586 Ethylvyclohex~ne.732 Ethylene adsorption of. 650 epuxydation of, 370-371 from ethane. 387-389, 740-71 1 ethane from. 704-507 ethyl benzene from. 652453 PB Rs for. 1 7 1 - 175 Ethylene chlorohydrin, 246 Ethylene glycol (EG) CSTRs for. 152-/56 from ethylene chlorohydrin and sodium bicarbonate. 246 from ethylene oxide. 191 product~onof. 163-1 68 synthesizing chemic:11 plant design for, 196-198 Ethytene clrlde. 306 in ethylene glycol prduction. In
196
pmduction of. 135-1 36. 191-195 Eukaqoter. doubling times for, 425 Euler method. 919 Evaluation
in CSTR scale-up. 149-1 50 in erhylene glycol productron, 154. 165
of laboratory reactors, ?8%291 in nitrogen oxide production. 206
In
fusion
External mnss transfer in diffusion, 766 in nlfrous oxide reductioris. &47 External resistance to mass transfer
example. 783-788 macc transfer cncfficient i n .
77 1-776 Inash trilmfer-limited re:lcrlollr. 780-783 mass trmsfer to single panicle$. 776-780 Exlinution temprature in multiple steady states. 537 Eyes ~rterialblood Aott in. 898 bl~ndnes~ fmrn methanol. 4 12, 466467 irri~int~ 392 , Eying equation, 129
F curves, 879 Fabrication, microelectronic, 299-300, 69R-700 Chemical Vapor Deporition in. 701-704 etching in. 7fl(l Facilitated heat transfer in diffusion, 802 Factor novoseven, 326 Falxified kinetics in d i t h i o n and reaction. 833-835 exercise. 862-863 Fan, L. T. on RTD moments. 88 1 on tracer rechniques, 878 Fanning frictlon factor. 182 FAQ (SrequentIy dsked questions\ convcrslon and reactor suing. 72 mole balance. 17 multiple reactions, 360 rate data collection and analysis.
293 Fast mange formation, 136 Fat compartment in alcohol metabolism. 444 Fed batch reactors. L e Semibatch reactors Feed stocks. poison in, 714-7 15 Feed tempraturn in equilibrium conversion, 520-52 1 FEMLAB program diffusion, 759. 765 dispersion, 975-978, 1032 explanat~onof. 1031-1032 instructions for, 10 15 PFR unsteady operation, 628
radial and axral variations. 551, 557. 560-5h l Femtoqecond \pectrnscnpy. 379 Fermentation plucose-to-crhanol. 4 3 ? 4 3 4 i n wine-rnaking. 42-25 Fermi. Enrico. 34-35 Fibers. temphthalic acid For. 367 Fibrin, 325 Fibrinogen, 37-5 Ftck's law in diffusion. 760-76 1 i n disper~ion.955, 96.1 Film. diffusion through. 762.
766-770. 774 Finegold, D.N.. 89% Finite difference method. 263-264 Finlayson. 3. A,. 844 Firefly flashing frequency. 132 First law uf thermodynamics. 473174 First-order dependence in CFRs. 45 First-order rate laws, 8.5 First-order reactions. 83 in CSTR design batch operations, 150
series. F 59 single. 158 stanup in unsteady-state operatlon. 2 I7 differentrnl equations for d~ffr~sion in spherical catelyst pellets. 822-823 soIution5. 1012 irreversible. 579 in multiple steady states. 535 PFR reactor volume I'or. 146-143
reversible, 512 Fischer-Tropsch reactors. 28-29 Fisch~r-Tropschsynthesis. turnover frequency in. 65 1-452 Fitting tail. 935 Five-point quadrature formula in PFR srzing. 51 solutions. 1014-1015 Fixed-bed reactors. See Packedbed mac~oxs(PBRs) Fixed concentration in mass transfer correlations, 784 Flame retardants. 455-456 Flashing frequency af fireflies, 132 flat veioc~typrofiles, 947 Flow in energy balance, 471
numericirl solutionc to. 975-978 through packed bed\. 177-1 8 I in pipe? d~\persionin, $64-965 pressure drop in. I 81-185 Flow n t e r ammonia oxidation. 355 CSTR parameter nrodeling. 983 interstage cooling. 5 I8 mass 1r;ln'rfer and reaction. 854 mass tran\fer correlat~ons.783 membrane reactor$. 349. 35 E
multiple reactions, 308 space tme, 66 Flow reactor% 106- t 07. See nlsu sperrfic J ~ I W rencrnrr hc
name concentrations in, 107-1 OX design equations, 4 0 4 3
CSTR. 4 3 4 t PBRs. 45 tubular, 44 journal critique problems. 250 moles changes in, 108-1 23 with variable volumetric flow mte. 111-173 Fluid fraction i n laminar Row reactors, 1189 Fluidized-bed reactors. 22. 24, 85 1 catalyst decay in, 7 17-720 in hetcropeneour reacttons. 88 professional reference shelf fur.
854 Flux equation For diffusion in spherical catalyst pellets. 8 17 In eqr~irnolarcar~nterdtffu~ion.
Fouling in catalyst deactivation. 650. 7 12-7 13 Four-point rule in integral cwluation. 1014 Fourier's law, 760 Frictional area bafance in CVD.
702 Free radical< as active intermediates. 378 in bimoleculdr reactions. 80 in smug formation. 393 Frequency factors in acetic acid piduction. 6 12 In activation energy. 95 ARSST for. 605 in nte l a ~ s 85 , Frequently asked questions (FAQ) conversion and reactor sizing. 72 mole kali~nce.27 multiple reactions, 360 rate h t a collection and nnalyqis,
293 Freudllch aotherms. 666 Fribeg, T . 998 Frict~onfactor i n pipe pressure drop, 182 Frtedel-Crafts catalysts. 652 fro^ leg< experiments, 760-76 1 Frossling correlation i n mass transfer coefficient. 776-777
in pharm;lcokinetics, 798 F(t) function i n lntegnl relationships, 878 Fuel cells. 247 Fuller. E. N.. 770 Furusawa. T.. 527
762 Fogler. H. 5., 785 Force. conversion facton for, lOF 8 Forced convection in diffuqiun, 763-76.1 mass transfer coefficient in.
774-376 Fomaldehyde formation of. 392 from methanol. 112.166467 oxidation of. 369 Formate from methanol. 412 Formation enthalpies, 4 8 1 4 8 3 Formation rates in ammonia oxidation. 354 in azomethane decomposition, 380 i n net reaction rates, 33 1
G.I.(gastruintesttnal)
tract component i n alcohol rnetsboli
443
Gallium amnide layers, 701 Gns-hourly spxe velocity. 68 Gas-liquid CSTRs, RTDr in,
868-870 Gas-oil. vapor-phase cracking of, 75 1 Gas phase and gas-phase reactions.
23-?4 adiabatic exothermic irrevenible. 75-76 batch systems. 40. 103, 1 10, 149 In CVD. 70 1
Gas phase and gas-phase reactions {con!.)
diffusion in. 770 in dimethyl ether decomposition, 297 elementary and reversible, 89 equilibrium constant in, 1021-1023 in flow reactors. 41. 1 14-1 18 in I~quid-phaseconcentrations. I08 in micrnreactors, 2 W ? O T mole balances on, 200-202 mole changes in. 108-1 23 in packed beds. 146 PFR reactor volume in. 146148 pressure drop in. 175-177 profe%slonalreference shelf for. 130 in tubular reactors. 14. 23. 169-171 Gas-solid heierogeneous reactions, 254 Gag sparpen. 438 Gas volurnerric flow rate in space velocity. 68 Gasoline catalyst poisoning of. 7 14 green. 584 octane number of i-pentanes in. 653 interstage beat transfer in. 516-517 Gastrointestinal 03.1.) tracr component in atcohol rnesaboli5m. 413 Gau~sianprogram. 379 Gel geometq in cartilage form~ng cell<, 874 General mole h~lanceequation. 8-10
For CSTRs. I3 for tubular reactors. 15 Generation heat in multiple steady Ftates. 534-536 Genenc power law rate laws in pas phase. 201 Germanturn epitaxial fiIm. 70 1 GHSV space ~elocity.68 Glbhs free energy in curnene nduorprion, 676 in equilibr~urnconslant. 1021. I024 Glacial age< estimation. 808-8M 'Slow sticks. 3R6 Glucose oxida~ionuf. 4 17
in wine-making, 424 Glucose oxidase, 417 Glucose-to-ethanol fermentation, 432434 Goals for nonideal reactors, 946 Goodness of fit In rate data analysis, 255 Gradientless differential reactors.
382 Graphical rnethds for activation energy determinations, 96-97 for batch reactor data analysis. 258 for equal-area differentiation. 1 0 1 e 1 0 t1 for triphenyl methyl chloridemethanol resctlon, 162-263 Gravitational conversion factor, 1019 Greek symbols. 1035 Green engineering. 457 Green gasolines. 584 Growth of microorganisms. See Bioreactors Gumbo. 104SI04 1
Heat of reactions in acetic acid production, 61 1
ARSST for, 605 in energy balance. 4 8 3 4 8 6 molar flow rates for, 479481 Heat terms in multiple steady states, 533-536 Heat transfer In broreactors, 438 to CSTRs. 523 in diffusion, 761. 802 in mash transfer coefficient, 775 in octane number, 516-517 in pressure drop, 190 Height, energy barrier. 93 Hellurn mixture in monopropellant thrusters, 786 Hemoglobin. deoxygenat~onof.
295 Hemostatis process, 325 Heptane, 682 Heterogeneous catalytic processes rn metl~aneproduction, 287 phases in, 647 Heterogeneous data analysis. 688489 mechanisms in. 69 1-692 rate laws in. 689690. 692694 reactor design. 693-698 Heterogeneous reactions, 6, 80, 87-88
Hagen-Poiseuille equation, 962 Half-live? rnethcd In rate data anal!.sis, 280-28 1 Halogenation reactions. 655 Hanes-Woolf model for M~chaeIis-Mentenequation. 403 of Monod equation, 430 Heat capacities in enthalpy. 483 Heat effect<.See ul.ro Temperature In catalytic cracking, 727-728 CSTRs u,ith. 522-532 FEMI-AB for, 103I in RTDs. 927. 937 in semibatch resaorx. 6 1 b 5 1 9 In rteady-ctate nonisotherrnal reactors. Sec Steady-stale nonisorhennnl reactors Heal exchdngers energy balance in. 523 i n lnter\tngc cooling, 5 1 8-5 19 i n rnlcroreaclors. 203 in semibatch reactors, 6 1 4 6 1 9 Heat load in inrer~tilgecuoling, 518
Heat nf fornlatlon for reactants. 1.79
data for. 254 external diffusion effects on.See Diffusion mass transfer of reactants in. 814
High-fructose corn syrup (HFCS).
395 Aigh-molecular-weighI olefin formation. 643 Hlph-nickel sra~nle~s steel plates. corrosion of, 132-133 High temperature In multlple ~ t e a d ystate%.535 Hilder, M. H.. 908 Hilder's approximation. 909 Hill, C. G.. 761 Holding time in space rime, 66 Holoenzymes. 41 8 Homogeneous catalysis. 647 Homogeneous liquid-phase flow reactors, 23 Homogeneous reac~ionq,80.8687 data for. 254 rate Ian parameters fur. 256 Hot XJK~IS in niicmreiictnrr. 203 Hr~ugen.0. A,. 670
Humphrey, A. E.. 425 Hydration reactions. 654 Hydrazine for space fl~ghts. 785-788 Hvdracarbons in catalyst coking and fouling, 712 partial oxidation of. membrane reactors for. 347 Hydrochloric acid. dolnmite dls~olution in, 278 Hydrocracktng. 722 H!tdrodealk!.lation o f mesitylene tn CSTRs. 344-347 rn PFR\. 34&343 Hydrcdemethylation of toluene. 87. 688698 Hydrodewlfurizatlon reactor design prohlenl, 1040 Hydmdyna~n~c boundary layer in diffusion, 77 1 Hydrogen ammonia from. 6 7 M l I from cycluhexane. 647. 804-805 discwietire adwrption of, 702 in enzyme-cubstrate complex. 396 from ethane. 387
in ethane thermal cracking. 388 in membrane reactorc. 109-2 10 in methane producttun. 284-288 In reversible reactions, 89-90 from waker-gas shift rescrlon. 1024-1 026 Hydmyen-based energy economy. 247 Hydrogen peroxide. 4 17 Hydrogenation reaction<. 653-654 of ethylene tu ethane. 704-7177 of i-nctene to i-octane. 744 membrane reactors for. 347 Hydrolases enzymes. 397 Hydrnlyris in semihatch reactor operation, 216 of starch. 461462 Hydrophobic forces for enzymesuh$trate colnplex. 396 Hyperbolic catalyst decay. 7 17 Hypotherlcal Ltapnant film in diffusion. 773
I-pentanes. 653 Ideal gas constant, 1017 ldeal Gas Law. 42-43 ldeal reactors RTD for, 892-901 batch and plug-flow, 885-886 laminar flow. 888-891 single-CSTR RTD,887-888 in two-parameter models. 979-987 Identity in chemical specres. 4 in reactbons. 5 Ignition-extinction cur\.e<.
536-540 Ignition temperature in equilibrium conversion, 521 in multiple steady slates. 537 Impellers 11n bioreactors. 438 Imperfect pulse inject~onin step tracer experiment. 877 IMRCFs (inert membrane reactors with catel!.st pellets on the feed ride), 207-208 Independent reaction<. 306307.
5 4 Induced fit model for enzyrne-substrate complex. 396 Industrial hutadiene, A54 Industrial reactors dimeri7e prnpylene into iwhexanes. 61 in mole balance, 21-24 space time in. 67 Industrial waste reaction, 245-246 Inert membrane reactors wirh catalqst peIlet~on the feed side I IMRUF5). 207-208 Inhihition nf enzyme reactions. 4 0 9 4 lo
comycr~titr.410-412 rnull!ple enfyme and substrate ytelns. 417418 nuncornpetitire. 414-41 h substrate. 4 16-41 7 uncornpetitive, 4 1 2 4 13 lnhihitnr molecules. 414 Inhibirors. 409 Inittnl rate< for dkfferential renclnrs. 281 in rate dara collectlrm and anal?sis. 277-279 In~ti:~tion step in chain reactinl~s.
386 I-octnnc and i-oc~ene.734
Inlet conditions in differential rraciorq. 283 ill equilibrium con\er\inn, 52 1
Instanianeous selectivity in multiple reactions. 307-308 In parallel reactions. 3 I8 in ~em~batch reactors, 21 8 Instantltneous yield in multiple reacllons, 3 10 Insulin production, 4 19 Inte~raldata analysis method, 267-27 1 Integral reactors evnlualion of. 19b291 PBRs, 19.45 Integral relationships in RTDs. 878-879 Integrals numerical evaluation of. 1013-1015 In reactor dehign. 1009- 1.010 lntegrarerl circuit fabrication.
698-700 Chern~calVapor Deposition in. 70 1-704 etching in. 700 Intcgrat~ngfaclor in ncetsldehl.de formation. 322 Intensiry filnc~ionin maximum mrxedness model. 921-922 Interchange in CSTR modeling. Y K5-987 Interfacial area for catalytic reactions, 648 Intermediate produc~yield. 32 1-324 Intermedlater, active. See Active ~ntrrmed~ate~ Internal-age RTDs. 885. 935 En~ernald~ffurran overview. 6660-6hI We1<7-Pratercriter~nnfnr, 839-841
Internid eliecr lvene\c factor In d ~ f f u ~ l o827-833 l~. In nltrntit oxldr reducr!nn<. 846-848
Interpha\e d~ffuvonrcacton. 369. 849-850. 853-854 Intempted isothermal opention<. 599-605 Intcr
577
Iunic forces for en7yme-suh\tr.11r: compleu. 3%
Irrevers~blereactions, 80 endothermic, 575-576 exothermic. 76-76 half-lives method5 in. 280 isomeritation. 657-688 order in. 256 Irse+erslblesurface-reaction-limited Mte laws, 6M [ha-octane. 682 Isu-pentene, 6 8 M 8 8 lsobutane production, 49n-395, 49 7 4 9 9 Iscrhexanes from dimerize propy-
que\tion%and problems. 234-249 structure for. 144- 118 sunimery. 226-230 supplementary reading. 252 synthesizing chemical pIanr design. 19&198 tubular reactors, 168-1 35 unsteady-state operation of stirred reactors. 2 15-11 6 semibatch reactors, 2 17-226
Kinetic?. solid-liquid dissolution. 378-279 Knee joint replacements. 823-827 K n u d e n ditt'i~siun rn d~luteconcentrations, 763 In spherical catalyst pellets, 815 Knudsen phase, diffusion in. 770 Kramers. H.. 784 Krishnasuaamy and Kittrell's exprersion. 722 Kunii-Levenspiel fluidization motlel. 85 1
Iene. 60-61 Isurnerases enzymes. 397
Isurnerizatlon. 653 in batch reactors. 1 1-1 2 of butane. 62-65, 490395, 497-499 irrevemble, 687-688 isothermal gas-phasc. 46 in reactions. 5 isopropyl isocyanate decomposition, 301 Isotherm equatlon in adsorption,
663 I~othermaloperations in Row reacton. 1 16. 1 I8 !as-phase isomerization. 46 rntermpted. 599-606 in nitroanaline production, 602 isothermal reactors. I43 CSTRs. 156-1 57 design equations for. 43-44, 99 pnnllel, 160-162 scale-up In. 148-156 second-order reaction in. 162- I 68 serres, 158-160 single, 157-158 FEMLAB for, 1032 journal critique problems, 349-252 learning resources for, 7,3&234 membrane. 207-2 I5 mtcroreactors, 201. 203-207 practical side. 226-227 pressure drop in, 175. 196 analytical solution for. 185-1 95 flow through packed beds in, 177-181 in pipes, 182-185 rate law in, 175-177 spherical PBRs in, T96
Jeffreys, G. V.. 50rl Johnson. Samuel. 377 Jones, A. W., 446 Journal critique problems actwe intermediates. enzymatic reactions. pharmacokinetic morlels. and bioreactorq.
468469 catalysts, 753-755 ditYusion, 809-810. 863-864 isothermal reactor design. 249-252 multiple reactions. 372-373 rate data collection and analysis. 302-303 steady-state nonisothemal reartor design, 589 techniques for, 233-234 Junction balance in CSTR panmeter modeling. 98 1. 984
Kargi, E. 41042 1 Kentucky Coal No. 9 liquefaction. 369-370 Key reactant concentrations. 115-1 17 Kidneys, artificial, 397 Klnd In chem~calspecies, 4 Kinematic v~scosity of helrum. 786 in mass transfer coefficient, 776 Kinetic Challeng~module. 131-132 Kinetic energy in energy balance, 475
Kinetic rate law. 82. 8687
Laboratory reactors bomb calorimeter, 103 evaluation of. 39-29 1 Labs-on-a-chip for DNA ~dentification.MUG9 microreactors for. 203 LaCourse, W. C.. 397. 403 Lactic acid production. 463 Lag phase in cell growth. 422 Lnm~narflow Aris-Taylor analysis for, 975-978 dispersion for in pipes, 964-965 In tubular reactors. 962-964 Laminar flow reactors (LERs) mean converston in, 908-906. 912-914 RTDs in. 888-891 in tubular reactors, 556 Langmuir-Hinshelwd kinetics in catalyst surface rertctians, 668 For heterogeneous reactions. 88. 254 nonl~nearregression for, 271 In rate limiting, 670 single-site mechanisms, 777 steps in. 672 for STTRs. 732 Cangmuir isotherm,
664465
Langmuir plots. 400 Large molecules. synthesis of. 420 Le Chnteiier's pnnciple. 1022 Leaded gnsoIine, 7 I4 Learning resources active intermedrates. enzymatic reactions. pharmacokinetic models. and bioreactors,
449453 catalysts. 736-737
cunversion and reactor si7ing. 71 dithion. 801. 852 explanation of. I (w? i\othermih reactor decign. 23G234 mole balances. 2 6 2 7 multlple reactions. 359-360 nonldeal reacton. 995 nonisothermal reactor design steady-state. 566 unsteady-state. 630 rate data collection nnd nnaltc~s. 293-294 rate taws and ~toich~ometry. 126-1 27 RTDs. 934 Learning styles. 1046-103-7 Lra~t-squaresanalysis for batch reactors, 27 1-273 rn multlple reaction analysk.
356 professional reference shelf for. 293 LeBlanc. Steve, 59 1 Lee. E. T..898 Length, conversion hctors for, 101s Levenspiel, 0. on dispersion coefficient determination. 968 on dispersion model. 956 on reaction combinations. 990 un tubular reactor boundary conditions, 961 Levenspiel plots for adiabatic isomerization. 64-65
for butane isomerization. 493 for Row reactors, 122 Lwlne, N.. 397. 403 LFRs (laminar flow reactors) mean conversion in, 908-W, 912-914 RTDs in. 388-89 I in tubular reactors, 556 LHSV space- velocity, 68 Ligases enzymes, 397 tight from ultrasonic waves. 384-386 Limiting reactants in batch systems. 105 in conversion. 38 in semibatch ceactors. 223 Limiting situations for diffusion. 548-849
Lindertnann. F. A,. 378
Linear dccay in catalyst deacrtvatlon. 717 Linear Iedst squares. 373 Linear plots In batch reactor data nnalysi\, 267-169 Linear regntssiun, 270-27 1 L~nenrizcdstability t h e o ~ , 631-632 Lineweaver-Burk plots for inhibiiion competitive. 41 t--112 noncompetitive. JIS uncompztitive, 413 for Michaelic-Menten equation. 402-403 Liquefaction of Kentucky Coal Kn 9.369-370 Liquid-hourly space velocity. 4X Liquld phase and liqutd-phase reactions. 21-23 in batch systems, 40, 103-105,
Log-log paper tor batch reactor analysi~. 257-258 slopes on, I027 for tnphenyl methyl chloridemethdnol reaction. 265-266 Logic vs, memorizing. 143, 116 Logistic growth law. 462 London van der Waals forces. 396 Los Angeles basin ~chemntlcdiagrams. 32-53 smog formatton in. 392 Low temperature in rnuItiple steadv states, 535 Lubricant dtstgn problem, 1039 Luedeking-Piwt equation. 419 Luminescence from 11ltrasonic waves. 384-386 L y a m enzymes. 397
148-156 in butane isomerizntion. 62-65.
490-495 concentrations in, I08 CSms for. 12-13. 21-22 diffusion In. 770 i n ethylene glycol productton. 155 in Row reactors, 41. 108 in methanol-tripheny1 reaction, 796-297 mole balances in. 200 pressure drop In. 175 scale-up of, 148-156 selectivity in, 217-218 in tubular reactors, 169 Liver cornpltnmenr in alcohol metahlam. 441,444-445 Living example prublems active intermediates. enzymatic reactrons, pharmacokinetic models, and bioreacrors, 450 catalysts, 737 explanation of. 1044 isothermal reactors. 232 multipie reactions, 360 nontdeal reactors. 994-995 nonisothermal reactors steady-state. 566 unsteady-state, 630 rate data collection and analysis. 293
RTDs, 935 Local steady-state values. 539 Lock and key model, 396
.
MacMullin, R. B 870 Macrofluid in RTDs,903 Macromixing. 870. 933 Maintenancc in cell growth, 427 M;~rx,Gro~lcho.I43 Mass, conversion factors for. 1018 Mass balances. Sea also Mole bal-
ances in cell growth, 43 143.1 in glucose-to-ethanol fermenta[;on. 433 Mass Row rate through packed k d s . I78 Mass tronsfer bundary layers, 77 1 Mass transfer coefficients correlattons for, 7 7 W 7 6 m diffusion, 771-774 in hydrazine decomposition, 786 In nitruus oxide: reductions, 847 operating condition changes in. 783-788 in PBRs. 848 what if conditions for, 788-792 Mass transfer-Iimited reactions, 789-790 on rnetailic surfacer, SO I in PBRs, 78G783 Mass transfers in diffusion, 761, 766 external resistance KO example, 783-788 mass transfer coefficient in.
77 1-776
Mas? transfers (rant) mass txansfer-limited reactions. 786783 mars transfer to single panicles. 776780 in heterogeneous reactions. 814 in microreac~ors,203 in packed beds, 78&783, 842-848 in pharmacokinetics. 798 tn single particles. 776-780 Mass telacity in ethylene oxide production, 193 Material Safety Dara Sheets IMSDS), 168 Mathematical definlliun of reaction rate, &R Mathema~icnllytmctahle nontdeat reactor models, 946 MATLAR program adiabatic tubular reactors. 488 CSTR parameter modeling, 982-983 ethylene oxldc producrion. 191 explanation of. 1031 instruct~onsfor. 1015 i~nthermalreactors, 230 membrane reactors. 2 13-214 nnn-adiabatic PFR energy bnl-
ance. 179 nonlinear represqion. 274
PFRIPBR mulriple reactions. 3.76-339 semibatch reactor multiple reactions. h?h Maximum mlxedness mndel, 9 1 5-922 rnuhiple reactionc in. 928-932 ODE qolver for. 925-927 in RTDF for reactor modeling. 903 vs. segregation model. 922-923 Mean conversion In laminar flow reaclors. 912-9 14 in real reactors. 9 10-4 I?. in segregation model. 9M-9 10 Mean residence time in disperrion cmfficlcnt delerrnlnarion, 969 In RTDs. 879-88 1 . KIJO in qpdce time. 66 Mean. D. E..83 1 Ifcars' cri~erlrm.84 I-XJZ Measured uarisbteq in rate &dta analysk. 254
in triphenyl methyl chloridemethanol reaction. 261 Mechanism searches, 383-386 Medical applications pharrnacokinerics See Pharmacokinet~c? of RTDs. 8Y8 transdermal drug delivery, 772 Medicated parches. 772 MEK (methyl ethyl ketone) production. 743-744 Membrane rcaclors des~pn.207-215 mole halances for. 198 for multrple reactions. 317-351 packed bed, 179. 576 Mrrnor~zationvs. Inpic, 111. 146 Mea~tylcnchydrirdeslLyla~ion In CSTRs. 3-l-k.317 in PFRc. 3401343 Me~abolisrnof alcohol. 393-394. 4s 1 central compnmenr in. 443 G.1. tract component in. 143 liver cnrnpannlent in. 444445 niadel system for. 341342 problem fur. 1041-104? slomach in, 4 4 2 4 1 3 Metallrc surface<.mass transferlim~tedreactions on, 801 Metaxylene rsomerization. 241
Methane from carbon monoxide and h! dropen. 284-288 from ethane. 387 from toluene. 87. 688-698 xylene frnm. 649 Methanol ADH u i t l ~ .4 12. 466-467 dirneth!l ether from. 717 poisoning hj. -11 7. 466467, 1042 syntltesir problem. 1 MO in mphenyl methyl chloride reaction. 310-266 in r i t ! I-niethanul rwction. 169-27 1 ~4e1hanol-tr1phenyl reaction. 296-297 Method of half-her. 181k281 M z t h r d uf initial rakes. '177-279 Metllyl alrl~nc.?3&213 Methyl Irrnln~depmduction. 13L223 \ l r ~ h ) ! cth!l ke~nnr1hlEK1produc~ion.713-744
Methylcyclohexane. toluene from, 746-747 Michaelis constant, 399-401. 41 1 MichaeIis-Menten kinetics and equations in alcohol metabolism. 44 1 In compelitive inhlbhion.
41M11 enzymatic reactions. 399404 i n oxyperi-limited growth. 439 cub\trilre concenuatron in. 305 in uncomperitive Inhibition. ill
412-413
Mscrobial gnwth. See Bioreactors Micruelectronlc fabrication, 2Y!L303. 698-703 Chemical Vapor Deposition in. 701-704
etching in, 700 Microfluids in DNA identrficalion, 408309 ~nRTDs. 903 Micromechanical fiibrrcation, 700 Micrumixin~,R70, 903 Microclrganlrrns growth of. Ser Bioreactors scale-up for, 439 Microplants. 148 Microreacrors designing, 201-207 for phosgene. 243-244 Mlld reaction conditions in bloconvenions. 4 19 Mills, K. F.,425 hl~nirnummixednew. 9 1.5 Mintmum sums of squares, 272 M~ssingrnformalion In ethylene glycol production, 153 hI~xedrnhlbition. 4 1 4 4 1 6 Mixer\ in microresctors, 203 M~xing in nonideal reactors. 870 In KTDs. 871. YO2 in segregn~ionmodel. 903-9fJ.T Mr)auchi, T.. 527 Model d~scrimlnationrn catal) F ~ S . 70G707 Molar average veloc~tyin d~ffuaion, 760 Molar feed rate in Rr~wreactors. -1 1 Molar flow. 107 alntnonia nxidatmn. 355 binary ditfu\ion. 76 1-765 cataly~tcarbon removal. 796 coolant bal:lnce. 503-501 CSTR5. I4
diffusion, 758-76 1, 764-765 ethylene glycol pmduction. 164.
196 flow reactors, 42, 1 1 1-1 13. 121 gas phase, 201 heat of reaction, 479481 isothermal reactors, 19&199 membrane reactors, 348-349 microreactors. 104-207 multiple reactions. 309. 34&349, 35 1
PBRs, 336739 PFRs. 17.336334 tracer. 956 tubular reactors, 552 Molar rate of mags transfer in diffusion. 835-827 Mole balances. 1-4 acetaldehyde formation, 321-324 acetic acid production. h 1 0 acetic onhydnde productson. 505-5M.508 adiabatic tubular reactors, 483488
ammonia oxidation. 352 batch reactors. 10-12 in design equations. 39 enzymatic reactions. 404 ~ntegraldata analysis, 267 butane isomerizstion. 490. 498 catalysts carbon removal. 795-796 decay. 7 18 sintering, 7 I0 CD-ROM material, 26-29 cms. 12-21 complex reactiuns, 327-328 crroInnt balance. 502-503 CSTRs. 43, 980 batch operations, 149-1 50 u ith cooling coils. 531 in design. 157 mulriple reactions. 344. 549 in series. 55-56 unsteady-~tateoperation, 620 diffusion. 758-759. 767. 8 17 ethane thermal cracking. 389 ethyl acetate saponification. 61 7 ethylene glycol production. 153-154 FAQ for. 27 gas phase. 2 W 2 0 2 generdl mole halance equatron, 8-10 industrial reactnrs. 2 1-24
inen tracer in disprsion model. 957
isothermal reactors, 176, 198-199 learning resources for, 26-27 liquid phase. 200 mass transfer and reaction in packed beds, 843 maximum mixedness modeI. 917 mean conversion in laminar flow reactors. 9 12 membrane reactors. 209. 2 12. 349 mesielene hydrodealkylation, 341, 345 multiple reactions, 309. 336-339, 344,349.549. 626 nilroanaline production, 601 nitrogen oxide producfion, 205 nitrous oxide reductions. 846 non~sothermalreactor design steady-state, 472 unsteady-state. 593 pmllel reactions, 3 16, 545 PBRs. 18-19. 2W203.
336-339 PFRs first-order gas-phase reaction. 146-147 with heat effects. 545 multiple reac~ions,336-339 in unsteady operation. 628 phannacokinetics, 798 pressure drop. 176. 186 professional reference shelf for. 27-29 propytene glycol productwn. 52E-530, 620 queshons and problems for. 3-35 rate data analysis. 254 reaction rate. 4-8 remibarch reactors, 2 19-220. 124. 626 SITRs. 730 summary,23 ~upplementaryreading. 3-36 T-I-S model, 952 triphenyl methyl chloside-methannl reaction. 261 t~lbrllarreactors. 44 adiabatic, 487488 deqipn. 172 radial and axial variatinns in. 553.559
Molecular adsorption, 662, 664 Molecular dynamics. professional reference shelf for. 129 Molecular sleves, 648
Molecularity of reactions, 80 Molecules lo diffusion, 758 Moles In hatch systems, 1W102 in gas phase, 108-1 23 in reactors in series, 54 Moments of Rms, 881-884 Momentum transfer in diffusion. 36 1 Monod equallon in b~oreacmrs.41 8. 435 for exponentral growth. 423424 Hanes-Wnolf form of. 430 ~n oxygen-limited growth. 4.79 Monodisprsed solid particle dissolution, 79g799 Monoethanolamine formation. 306 Monolithic catalysts. 449 Monopropellant thrusters. 785-786 Monsanto plant accident. 59960'; Moser equation. 425 hlOSEET devices. 698699 Motion sickness patches, 772 Mrtrnr oil, 457-458 Mo! ing-bed reactors. catalyst deactivation in, 722-728 MSDS (Material Safety Data Sheets). 168 Mukesh. D.,226-227 Multiphase reactors in difiuqion. 849-850 Multtple enzyme and ~ubstrate Fystems. 4 1 7 4 1R reactions, 305. 327 anal ysic for, 356 CD-ROM material for. 359-361
Multiple
complex. See Complex reactlnns in CSTRs. 343-347.548-55 1 for digital-age problems. 336-357 journal crttlque pmblemr for. 172-373 membrane reactors fnr. 347-35 I nonisothermal. 543 energy balance in. 544-55 1 unqteadystate. 625-627 In packed hed Bow. 179 parallel. See Parallel reactions In PBRs. 335-j43 In PFRs. 335-343. 477. 544-547 questions and problems for. 361-375
Mnltrple wactions (con!.) RTD In. 917-932 herles, 305-307 in bluod clotting. 325-326 desired product in. 320-326 In mas5 transfer-limited reactions. 789 wrnlnnry. 357-359 supplementary read~ngfor, 375 types of, 305-3 10 Mult~pleregres\ion technique\.
693 Multiple cteady Stdter. 533 heat of gtnerarion in. 53d-536 heat-rzlilo\ed tcrms in. 333-533 kgnitirl~l-extinction curves in.
536-54 rilllaway reactions In CSTRs.
S4rM43 Multiple substrate Fystelns, 417418, 453 Multtplication, cell. 421 Muscle/bt compartment in alcohol metabolism. 444 Mystery Theater module. 236
N-butyl alcohol, dehydration of. 74 1-742 N-pentanes in octane number. 653 Navrgatrng CD-ROM. 1045-1046 Negative steps in step tncer experiment. 878 Neoplastic diseases, 409 Net rates in complex reactions combination. 332-333 for each species. 319 rate laws for. 329-330 sto~chiornet~, 33&33 1 of formation in ammonia oxidation, 354 in complex reactions.
327-328 in membrane reactors. 349 in rnesltylene hydrodeaikylatlon i n CSTRs, 315 in PFRs, 341 in parallel reactions. 546 Newton's law of viscosity, 761 Nickel catalysts. 284-288 Nickelmordenite catalysts. 752 Nicotine patches. 772 Nlcotine species, 3 Nishimura. H.. 527
Nitration reactions. 347 N~troanal~nc frc~nlammonia and ONCB. 136. 59940 1 :idiabatic opemtlon tn, 603 hatch operation with heat
exchange, 60.7-6M disk rupture in. 605 isotherma1 operation in. 601 Nlrrobenzene exan~pleproblem, 18 Nitrogn ammonia fm111.670 A7 I from azornethane, 379-383 from benzene drazotiium chluride, 95 sklr~exposure to, 8(W i n smog forrna~inn.3112 Nitmgen dioxide from nitrogen oxide. 456 frnln reversible gas-phase decompositions, 1 1 RI 23 in smug format~on.392 Nitrogen oxides, 393 in automobile emissions. 298-299. 742-713 nirrogen dioxide from. 456 productson of, 204-207 In smog formation. 392-393 Nitrogen tetroxide decornpnqition. 118-113 Nitrous ox~des in plant effluents, 845-848 plant explosion. 634-635 NLES solution in flow reactors, 12CL12 1 i n mesitylene hydrodenlkylation,
346 Nomenclature. 1033-1035 Non-adiabatic energy balance, 479 Non-enzymatic lipoprotern. 325 Noncompetitive inhibition. 414416 Nondissociated adsorptian. 662 Nonelernentary rate laws, 86-88. 377-379 chain reactions in. 3815-39 I mechanism searches in,
283-386
PSSH in, 379-383 reaction pathways in, 391-394 summary, W 4 4 9 Nonflat veloc~typrofiles in dispersion. 962 Mongrowth associated product formation, 426.428429 Nonideal reactors. 945 CD-ROM material. 994-995 characteristics of, 867-87 I
using CSTRs and PFRs, 990-99 1 dispers~onRow solutions.
975-978 guidelines for. 946-947 one-parameter nlcdels. 947-948 questions and problems.
996- 1004 RTDs for, 991-993 summary. 993-994 suppIenientar!, reading. IOOS
T-1-5, 948-955. 974 trrbular balance eiluations in, 957 boundary condillr~~is in,
958-963 dispersion in. 955-957. Oh?-9M dlspenion coefficient correlati011 in, 944-966 d~spersiuncoefficient determination in, 966-970 sloppy tracer inputs in,
970-973 two-parameter models, 948,
979-987 Nonisothermal reactions, 543 FEMLAB for, 1032 internal effectiveness factor in. 83 1-83: 5teady-state. See Steady-state nonisnthermaE reactors unsteady-state. See Vnsteadystate nonisothemal
reactors Nonlinear least-5quaresb356 Nonlrnear regressiun for batch reactor data analysis, 27 1-777 For cell growth. 430 for ethylene hydrogenation to ethane, 705-707 for MichaeIis-Menten equation.
404 Nonpornus monolithic catalysts,
649 Nonreactive trajectory i n moIecular dynamics, 129 Nonseprtrable kinetics i n catalyst deacrivat~on,707-708 Normal pentme, octane number
of. 683 Normal plots for activation energy determinations, 96 Normalized RTD function. 884-885
Nuclc
3+35 Nuclear renctur problem. 1039 Ntlcledr recion in cells. 419 Sucleotideh, polytneriratlnn of. d t IX-IOT) Number in chemical specles. 4 Numerical technrques for adiabatic tubular reactors. 4x9 for batch reactors. 258-159 drfferentlal equations. See Differential form? and equations equal-am graphical different~atinn. 1010-101 E for flows ~ i t hdlspenron and reaction. 975-978 integrals numerisnI evaluation of,
1013-1015
in reactor design, 100%1010 For membrane reactors. 2 I3 software packages. 1015 Nusselt number, 774-776 Nutrients
In cell growth, 428
in ready-to-eat cereals. 245
Octane
butyl nlcohoI for, 740 TAME fur, 584 Octane number intewtage heat tnnsfer in. 516517 in petroleum i-efin~ng.681 4 8 4 ODE (ordinary differential equ:ttion) solvers. See olso FEMLAB program: MATLAB progrim: Polymath program adiabatic tubular reactors, 389 ammonia oxidation. 353 complex reactions. 327 fitting Eft) curves to polynomials. 923-921 gas phase. 202 mov~ng-hedreacton. 736 multiple reactions. 336-339.
356 segregation model, 924-925 steady-state nonrsothermal reacturs. 565
O!! Eact Texas l~ght$a< oil. 712-713 engine. 4 3 7 4 5 8 Olelins fornitltion. €43 Ollis. D. F.. 411 ONCB. nitmanaline from. 136. 59'4-60 I Ons-parameter models for nonidenrreacturs, 935 charicteristrcx ot', 9li-94% T-I-S, 948-955 One-third rule. 1014 Onset temperatures with calorimeter% 606-607 Open-ended problems, 3039-1042 Open-open boundary conditions in d~spersioncoefficient determinntinn, 968-970 in tubular reactors, 960-96 1 Open systems. Arst law of thermodynamics for, 473 Operating conditions in maqs transfer roefticients. 783-788 in paraElel reactions. 3 17-320 Operating cnsrs in ethylene glycol production, 196-197 Optimum feed temperature in equllibrfum convenion. 520-51 I Opt~murnyield in acetaldehyde formation. 3?2-323 Opruelectronic device fabrication.
700 Orbital distortions, 93-94 Order. reaction, 83 Onler of mngnitude of time In scale-up. 15 1 Ordinary differential equation solvers. See FEhitAB program: MATLAB program; ODE (ordinary differential equation) solvers: Polymath program Organ compartments in pharmacokinetic models, 440 Organic reactions, liquid-phase, 104 Oscillating reactions, 361. 372 OstwaId. Wilhelm, 646 Other work term in energy balance. 474 OTR (oxygen transfer rate) in biorzacton. 438 Outlet concentration in nortideal renctos. 937
of tracer i n T-1-S model. 949 Overall conversion in mesitylzne hydmdenlkyliition, 345 Overall effecfiveness factor in diffu~lunand reaction. 835-83R.855 in nitrous oxide reductionr, 846-848 Overall mass balance, 1 19 Overall mass transfer coefficient. 210 Overall selectiv~ty in membrane reactors, 345 In mesitylene hydwdealkyl:~tion. 3-17 in multiple reactions, 308-3 10. 348; Overall yield in multiple reactions. 310
Overenthusiastic engineers. 7%79:! Oxidation in acetaldehyde formation. 32 1 of ammonia, 35 1-355 in catalysts. 654 of formaldehyde, 389 of glucose, 4 17 membrane reactors for, 347 Oxidation problem. 1040 Oridoreductases enzymes, 397 Oxygen in cartilage forming cells
concentration. 826-827 consumption, 824-825 in smog formation. 392 Oxygen- 18 data, 808-809 Oxygen diffusion in catalyst pellet carbon removal. 794-797 Oxygen-limited grnwth. 338439 Oxygen tnnsfer rate (QTR) in bioreactors. 438 Ozone nlkene reactions. 298 formation of. 392-393 In green engineering, 457
Pacheco. M. A.. 713 Packed-&d reactors (PBRs). 12, 17-2 1 adiabatic. 488495 cataly5t poisoning of, 715 design equ;itions for. 99 dispainn in. 966 energy balance for. 477. 576
Packed-bed reactors (PBRs) (c.ont.1 flow reactor design equations, 45 gas-phase reactions in, E46 with heat exchange. 477, 502-503 journal critique problems. 25 1 mass transfer In. 780-783. M?-848
mole balances on, 18-1 9. 200-103. 3.76-339 mul~iplereactions in. 335-343 ODE solvers elgorithmr for. 230 pressure drop in, 177-1 8 1. 183-185. 196
RTDs,Rd9 in s~eady-statetubular reactors. 502-504 structure of. 13-24 for toluene hydrodemethylntion. 6 9 4 4 9 8 transfer-limrted reactions in. 848-849
Parallel reactions. 305-307. 310 CSTRL, 160-1 62, 165-167, 316.117 desired prcducts in. 3 1 1-31 7 i n mass transfer-lirn~tedreactions, 789 in PFRs with heat effects, 545-547 reactor selection and operating cond~tiunsin, 31 7-320 Parameters acetic anhydride production. 508-509 buttine iromerization, 491 CSTR modeling, 98 1-985. 987 ethylene oxide production. 192 mass transfer cnefficients. 788-792 membrane reactors. 2 13 mef~tylenehydrodeal kylation, 342-343 nonideal reactors. 945-946 nonlinear regression. 273 propylene glycol pmduction. 596597. 62 1 STTRs. 7.7 1
toluene hydrodemethylation. 692 tubular reactor de\ign. 173-1 74 Finial differentiaunn {PDE) \oIYers
for
diffusion. 759, 165
for tubular reactors. 551
Panial oxidation, membrane reactors for, 347 Panial pressure profiles. 697 Panicle size in internal diffusion, 660 in preswre drop, 1 89-1 90 PBPK (physiologically based pharrnncokinetic) modelr, 439446 PBR?. See Packed-bed reacton (PBRs) Peach Bmom nuclear reactor problem. 1039 Peclet number In dispersion and T-I-S models. 974
in dispersion coefficient deter~nination.966968. 970 in tubuiar reactors. 958. 961-962 Peclet-Bodenstein number. 974 PeIlets carbon removal from. 794-797 in internal diffusion. dm spherical. 81 4 differential equation for,, 81 6 8 19, 822-823 d~rnens~onless form. 8 19-822 effective diffusivity in. 814-816 for tissue engineering, 823-827 Penicill~umchrysogenum formation of. 323 as reacturs. 3 1 Pentene. iso-pentene from, 686488 Perfect mlxinp in CSTRs. 13. 13-44. 3 I l Perfect operalion In CSTRs. SLb3 in tubular reactors, 897 Peroxide radicals. 392-393 Peters. M. 5 23. 747 Pelerren. E. E.. 707. 7 E 3 Plaudler CSTRs/batch.reactors. 12 PFRs. See Plug-flow reactors IPFRs) Phnrmncnkinetics compe!irn e inhibi~ionin. -110312 diffuwon In. 748-799 m d n n k t p and driving. 3hJ
.
modelr. .I.:C)44h summary. 447349 Tdrjlon. 364-365
Phases cell growth, 422 enthalpy, 482 heterogeneous caialytic processes. 647 heterogeneous reactions. 6 miorgene productian, 243-244 Pho~ochemicaldecay of aqueous bromine, 297-298 Photos of real reactors, 27 Phthalic anhydride, 1-2 Physical adsorption. 650 Physiologically based pharrnacokinetic IPBPK) models. 43946 Picasso's reactor, 16-1 7 P~pes d i ~ p m ~ oin,n 9h4-965 pressure drop in. 182-1 RS Plant effluents, nitrous oxides in. 845-848
Platinum on alumina in benzene production. 647 as reforming catalvst, 68.7 Plug fiou in diffusion. 764 in tubular reactor design. 169. 172 Plug-Row reactors tPFRs). 12,
14-17 for acetic anhydride production. 508-5 10 adiaba~~c, 4K7dYS in butane isomenzation. 492494 conversion zn. 973 CSTR5 in ~ e r ~ as c s approxlmation of. 57-58, 611-64 design equatrons for. 44 energy balance of. 477 wth heat exchange. 495499 multiple reactions. 544-547 parallel reactrons. 536 ethylene production in. 171-1 75 for gn\-pha~ereactions. 13. 14til48 with heat ehchange. 502-504, 508-5 10
mean conversion in. 907. 9K-910.911-913 rne\it! lcne hydrodealkylat~on111. 340-343 mole balances on. ?(X)-201 lnultiple reaction\. 335-3.1.7. 553-537
nr~nidenlreacton using. Y%C991
numerical solutions to, 978 p~rallelreactions. 31 5-3 17. 545-547 reactor volume for, 146-148 RTDs for, 885-886. 897-901 runaway in. 567 in serieq. 5840 w~thCSTRs,6RTDF fox. 897-903 sequencing, 64.46 sizing, 5+54 steady-state tubular reactors, 502-501 unsteady operation of. 628 Point of no return in nitroanoline production. 603 Po~son~np
i n catalyri deactivation. 650. 71.7-736
methanol. 4 12, 46-467, 1042 Pnlanyl-Sernenav equation, 97 Pollshed wafers in miurmlectronic fabr~car~on. 698 PoFydisperse solidr, diffusion of.
802 Pnlyesters
ethylene gI!'ctll for. 163 from erhylene oxide, 191 Pol!, math propwrn acettc acid production. 61 3 4 14 acetlc anhydride production. 506-5(17. 5 ICf;l I adlabntlc reactor<. 479. 487 r~lcnhr~l metahlisrn. 445446 ammonia oxrdat~on.354355 blond clotting. 3 l h butane isomeritation. 493-494. 498-499 ua!:~lyst decay, 71 9-720 ceIl yroufh. 430 CSTR5
w~thbypass and dead voIirme. 988-989
W I I I I cooling coal<.532 x ith multiple reacttons. 550-55 1 parameter model~ng.982-983 unsteady5tate operation. 61 1-62? energy balance. 479 ethane thermal cracking. 190-39 1 tth!.l :Icetale snpon~lication. hlX-hl9 ethylene hvdrogenation to elllane. 705-707
ethylene oxide production, 193-195
explanation of, 1029-1 030 glucose-to-ethanol fermentation, 433 heat effects, 547 instructions for. 1015 isothermal reactor algurithms For, 129-230 maximt~mmixedness model.
925-927 mean conversion. 9 14 membrane reactors, 21 3-214, 350 mecir) lene hydrodealkylatinn, 342-343, 345-346 methane prduc~ion.287-288 methyl bromide prtrductinn,
22 1-222 Michael~s-hlentenequation. 404 multiple reactions, 3 16-3 17, 336-339. 350, 55C551, hn4, 61M27, 93 1-932 nitrnanal~neprduction, 604 nitrogen oxide piduction. 106207 nonlinear regression. 274-2.77. 430 PBR\, 336-339 PFRF,336-339, 547 prupylene glycol production. 53&53 1. 597-598. 62 1 4 2 2
RTD moments. 883 wm~batchreactors. 626 S T R s . 53 1-732 toluene hydrdemethylation, 693, 696 triphenyl methyl chloride-methGnol reaction, 264-266 trit! 1-methannl reaction. 269-27U rubular reacton. 487. 977 \ariatile volunietric flow rate. 120-121 Pnly~nerrzariun in batch sysrerns. 104. 151 in brorenctors, 420 journal cr~tiqueproblelns. 169 of nucleo~ides,408409 profess~onalreference shelf for. 45 I 4 5 2 screw extruders in. 879 l'olymers produclion. 4 19 Potyno~niallit for batch reactors. 25'6-260
for Elt) curves, 923-924 Polynomral method in triphenyl methyl chlox~de-methanol reaction, 764-27 1 Pomus catalyst systems. 648 carbon removal in. 793-79'7 diffusion in. 763 monolithic, 649 Pustacidification in yogun. 459 Potatcws, cnok~ng.134 Potential energy in energy halance. 475 Power law and elementary rate laws. 82-86 in gas phase. 201 for homogeneous reactions, 2.54 Pract~cals~ahil~ty rate in CSTR unsteady-state operatian. 619 in propylene pIycol prduction. 622 Prandtl number. 774-776 Predictor prey relationships. 464 Pressure convcrslon factors for, 101 8 in d~ffusion,770 In energy balance. 474 in
Pmfe~s~onnl ~Fcrenceshelf (con/.
cataly~t5,737-738 convenion and reactor si~ing. 71 diffwiun, 801-502. 853-855 explanation of, 1043 isothermal renctor design. 232 mole balance. 27-29 multiple reactions, 36g36I non ideal reactors. 995 nonisothermal reactor de~ign s~eady-state,566 unsteady-state, 53 1-63? rate data collection and analysir. 29.7-294 rate laus and stuichiometry. 128-130 RTDs. 955 Promoters, 649
Propagation step in cham reactions. 3Xb Propane, dehydrogenation for, 2 1 1 Propy lene adsorption ~ f 573 . from cumene. 5 In Langrnuir-HinsheZwood kinetics, 672 Propylene glycol prduction in adiabatic reactors, 526-53 I,
Q term rn CSTRh with heat effect%. 521
Quardrrer. G.C.. 7-89 Qunfi-steady state assumption (QSSA 1.794 Questions and problems active intermediates, enzymatic mactrons, phnrmacokinetic model\, and bioreactors. 454467
catalysts. 738-753 conversion and reactor sizing. 72-77 diffusion. 802-8 10. 855-863 isothermal reactor design. 13G2JY mole balances. 29-35 multiple reautionr. 36I-375 nonideal reactors. 996-1034 nonisothermal reactors steady-state, 5 6 6 5 8 8 unsteady-state, 633-643 rate data collection and analysis. 294-30 1 rate laws and sroichiomerry. 131-140 RTDs. 936944
595-598 in CSTR unsteady-state opention, 61 9424 multiple steady states in. 533 Propylene oxide, pmpylene glycol from. 52&53 1 Prostaglandin. inhibiting production of, 409 Protease hydrolpze~,394 Prothrornhin. 326 Pseudo-steady-state-hypothesis (PSSH). 377 for active ~ntermediates, 379-383 For epidemiology, 358 for ethane thermal cracking, 3 87-391 rate laws derived from,
'
684-687 Pulse injection, 872 Pulse input experiment for RTDs, 87 1-876 Pulse reactor evaluation. 29 1 Pulse tracer inputs, 873, 886-887, 948. 968 hrsley, I. A,, 284 Pyridfne hydrochloride, 260-266
Radial concentration profiles. 978 Radial mlxing in tubular reactors,
962 Radial whations in tubular reactors. 55 1-56] Radioactive decay, 80 Rapid reactions on catalyst surfaces, 576-780 CSTR startup in unsteady-state optration, 117 Rate constant adsorption, M3 reactton. 82, 9 1-98 Rate data collection and analysis,
253 batch reactor data. 156-257
differential rnechod. 257-266 integral method. 267-27 1 nonlinear regression, 27 1-277 data analys~salgorithm,
254-255 differential reacton, 28 1-288 experimental planning in, 289
I~alClives method. 2X(?-28 1 initlnl rater method. 277-279 journal critique problems. 302-303 laboratory reactors. 289-291 learning resourcex, 293-294 questions and problems for. 294-30 1 summary. 291-292 supplementary reading. 303-305 Rate equation. 7 Rate laws, 79 acetaldehyde formation.
321-322 acetic acid pmduction. 610 auetlc anhydride pmductton. 505 adiabatic equilibrium tempernm e . 513 adsorptron. 662 alcobof metabolism. 441 ammonia oxidation. 353 azomethane decomposition. 380-38 E hatch reactors. 260-267 butane isomerizdtion, 490, 498 catalyst decay, 7 18 catalyt ~ creactions. 671674 deducing. 68Y690 derived from PSSH, 68-87 evaluating. 692494 ternpenture dependence of, 687-688 CD-ROM material for, 126-1 30. 139-140 cell growth. 4 2 3 4 2 5 , 428 complex reacttons. 328 coolant balance, SO?-503 CSTRs batch operations. 149-150 with cooling coils. 531 multiple reactions, 549 parameter modeling. 984 second-order reaction in. 162 single. 157 unsteady-state operation, 620 cumene decornpositlon, 680 CVD,701 deducing. 383-384 definitions, 80-82 elementary, 82-86 ethane thermal cracking. 387.
389 ethyl acetate saponification. 617 ethylene glycol production, 153-154. 156, 164
Reactions Iconr.) temperature effects on, 97-98 Reactiee dlsrillation in semibatch reactor operalion. 216 i'nr thermodynnmically Iimr~cd leverxi ble reactlnnh, 225-226 Reuctive ion etchiilg (RIE), 700 Reactive trajzcwry sn moleculiir dynamics. 130 Rextur des~gofor toluene hydrodemetliyla~ion,094-698 Reactor lab for isothermal reactor design, 2.1 1-23? for mulriple reactions. 360 for rate data collect~on,293. 295 Reilctc~rlength in nitrous oxide reductions, 838 Rwcror modeling, RTDs for. 902-903 Reactor htaging wilh inrerqtate cooling of heating. 515-536 Reactor volume hutane isnmeri~ation.63 uatalysl decay, 771'6 cnntinunus-flou reactor<, 4M7 uonvenlun factors fur, 1018 CSTRs. 43. 56-58. 61-62. 6446, I61 ideal p e s . 1017 membrane reectorr. ?# PBRs, 18. 1&2E PFKs, 17. 21. 59. 644% in \pace time. 66 t ~ h u l i ~reactors. r 170 vi~~iahle, 109-) 1 1 Reactci<.354 In scrles. 5&55 USTRs. 55-58 CSTRs and PFRs cornhinalion. Albh3 CSTRc and PFRs compari5ona. h 4 4 h
PFRs. 58-h0 siling. See Conversion and redcror slzlng Ready-lo-eat cereal,. 245 Real reactors lnenn converaione in, 1)10-9 1 2
in tuo-parameter models. 979-987 Reallqtic models for nnnideltl reactors, 946 Reciprocal concenr~ationr.268 Reciprocal power decay, 7 I7 Reclrculat~nptransport reactors, 190-29 1 Recycle reactors journal critique problems, 25 1 prilfesrtonal reference shelf for. 232-233 Recycle strcam in parallel reacuons, 3 19 Reflect~xelearners. 1047 Reforming catsly~rs.681-685 Regeneration catalyc~.793-797 entymc. -1 17 Re~i-e+r~nn In ac~ivauonenergy deteminaLlnns. 95 for batch reactor dats analysis. 27 1-277 fnr cell growth. 430 for erhlene hydrogenation tn ethnne. 705-707 for methane production. 287-288 for Miuhnelis-Menten equation. 401 for tolucl>ehydrodemethy la11on.693 for tr~pheoylmethyl chloriden~elhat~ol reaction. 266 fclr intgl-methanol reaction. 27(L27 I Rehkopf. R. G.. 898 Related rn,ltrr~alin e ~ h !lene ~Iycol prc>tluctian. 155 Rel.lrl\e r;ltc\ 01' reactton .Immonia uxlda~lc~n. 353 me51!t l e ~ ~hydro~lealkyla~ion. c 34 1 m u l ~ ~ p lredctionk. e 330-3.1 I parntlel reaction\. 516 PFRlPBR multiple renclions. -:.7q ctriich~ome~ric caefticrcn~
H72
Rc',~Jrncr-tin~rdr\~rlbutirm\ r RTD\ I. 867. XhY-X7 1
CD-ROM material. 934-935 diagnostics and troubleshooting CSTRs, 891-895 P M S T R series, 897-90 1 tubular reactors. 89,5497 pas-liquid reactors, 868-870 ideal reactors batch and plug-Row. 885-886 laminar Row reactors. 888-89 2 single-CSTR. 887-888 integral relationqhips in. R78-879
internal-age distriburion. 885 mean residence time in. 879-88 1. 890 measurement of. 871-878 medical use< of. 893 microreactc~rs.70.7 moments of, 88 1-884 multiple reactions. 927-932 nonideal reactors. 991-942 normalized func~inn.884-885
PBRs. 869 pulse input experiment for. 87 1-876 questions and problems. 936-944 for reaccos modeling. 902-903 soflwai-e packages for. 923-927 step tracer experiment. 876-R7X summary. 933-934 supplementary reading. 944 T-I-S model, 950 two-paramerer models. 979 lijr zero-par:~metermodels maxilnu~nmixedness model. 9 15-022 maximt~mmixcdness predictwns. 922-913 <egmpation model. W4-9 I4 Hr\pira~innrdte of chipmunks. 8 0 Ketin;~lblood Row. 898 Rctinilis pigrnenrosa. 898 Rexerse re:ictrons in darn analysis. 277 Revenible gas-phase dcconipo
Rhizobium trifollic, 425 Ribonucleic acid (RNA), 419 Ribosomes. 419 RIE (reactive ion etching). 700 RNA (Ribonucleic acid). 419 Robert the Worrier, 783-788 Rotation in transition state theory. 129 RTDs. See Reridence-time distributions (RTDs) Runaway reactlons in CSTRs, 540-543 from falrified kinetics. 835 In PFRs. 567
Saccharomyces cerev~siae for glucose-to-ethanol ferrnentatlon. 432-434 production of, 300 Safety nf ethylene glycol, 167-168 1n exothem~creactions, 599-605 In unsteady-state nonisothemal reactor design, 605-614 Santa Ann winds, 33, 392 Saponification, 104-105 SatelIite maneuvenng. 785-788 Scale-up in broreactors, 439 of I~qu~d-phase batch reactor darn. 148-156 Scavengers with active ~ntermediaries, 385 Schmidt number in d~ffurinn.776. 779 in diqpenion. 964. 9hg Schmrtz, R. A.. 538-539 Seafood gumbo, 1040- 104I Searchrng
Fnr mechan~sms.383-386 in nonlinear regression. 271-273 Second-order decay. 7 10 Sccond-order ODE solutions. 1013 Second-order rate laws. 85 Second-order rcactlons, 83 tn batch reactor data annlyxis. 268 in CSTR design. 158, lhl-IhR irrcvc.r\~hle.158 iwthermnl. 150. 2?&223 In la~n~nar flow reactors. 912-914
mean conversion in, 91 2-914 in moving-bed reactor catalytic cracking, 727 in multiple steady states, 535 RTDs for, 899-901 Second reactors in interstag cooling, 5 19-520 Segregation model in maximum m~xednessmodel, 915,919, 922 multiple reactions in. 927-932 ODE solver for, 9 2 4 9 2 5 RTDs for. 904-9 14 vs. T-I-Smodel. 95S955 Segregation vs. maximum rnixedness predictions, 922-923 Sertz. Nick. 305 Sefectivity in liquid-phase reactions. 217-218 membrane reactors for. 215. 347-35 1
In multsple reac~ions.307-309. 347-351 In parallel reactions. 318
temperature effects on, 312-31 4 for Tramhouze reactions, 312-317 Self-heating rnte with calorimeters.
607 Semlbatch reactors. 215-216 energy balance of, 477,815,626 with heat exchangers. 61 4-619 for Tiquid-phaqe reactlons,
2 E -22 multiple reactions in. 625-627 ODE solver for. 730 substrate inhihttion in. 4E 7 unsteady-state operation in, 2 17-726 Semiconductor fabrication, 299-300. 698 Chemical Vapor Deposition in. 701-704 etching in. 700 Semilog plots, 96. 1027 Sensors. rnicmreactors for. 203 Separable kinetics, 707-708. 716 Separating variables u,ith pressure drop, 186 Separar~ons y t e n l s . economlc lncenlivc for. 307 Sequencing of reaclorr, 64-66 Serres. reaclors In. 54-55 cornhinntion~.HL44
CSTRS,55-58 design, 158-160. 166167 PFRs, 5&60 RTDq for. 897-901 Ser~esreactions, 305-307 blood clotting. 325-326 desired product rn. 320-326 In mass transfer-limited Waclions, 789 Severe eye irritants. 392 Shaft work in enegy balance. 471 Shell halnnces en catalyst pellets. 816 III drffusion, 767 Shenvood number< in mass transfer cneficieni. 775-776 In mars tnnsfer correlatinnf. 785
Shrinking core model. 392-793 regeneration in, 793-797 phar~namk~netics. 798-799 Shuler. M.L., 4 2 0 3 2 I Sickle-cell disettse, 898 Silicon and silicon dioxtde design problems For. 588 for mkcroelectronic devices. 299-300. 699 Sirnplificat~ons in rnte data analysis. 254 In tr~phenylmethyl chlondemethanol reaction, 26 1 Simpson's nne-th~xdmle. 1014 Sitnpson's three-eighths rule. 1014 Sin~pson'sthree-point formula for isomerlzation of butane. 64 lor PFRc in qerics. hO Single panicles. mass Transfer ro. 776780 Single-sue rnechanivns rate-limit~ngIn, 6 7 7 4 7 8 . 683 ~urfaccrcacrions in. 666-667. 684 S ~ n ~ e r ~706-7 n y . 12 Site halance I n adsorptron irotherms. A01 Si7ins heat exchangers. 5 l Y lnrerstage heat. 5 18 redctnrs See Converq~onand reactor siring relief valve<. 605614 Skewness In RTD moment,, 881 Skin. nitrogen gas exposure 10. 804 Slopes nn wnlilop p a p r . 1017 catalyr
Stoppy tracer input<.970-97.7 Slow reaciion~ CSTR startup in unrteady-state operorion. 2 17 in mas%transfer to ~inglepumties. 77!, S l u q reactors, 369, 849-850, 853 Small rnolecule synthesis. 420 Small-scale operations. 10 Smog formation. 32-33. 392-393 Soap. snpnn~ticat ion for. 1 0 4 - 105 Sodium bicarbonate, ethylene glycol from. 246 Sodium hydroxide in raponihcation. 105 Software packages, 1049-1050. See also specific snfo~.nn: packages b?: nrrrnr A~pen,1031 FEMLAB. 1031-1032 instructions for. 1015 ,MATLAB. 1031 Polymath. 1029-1 030 Solid catalysts in PBRs. 17-1 R Solids in CSTRs. 290-291 Solvents from ethylene oxide. 19 1 Space flights. 785-788 Space satelii te maneuvering, 785-7RR Space t ~ m e in catalyst decay, 719 in CSTR modeling, 986 definttion. 6&67 in dicpersion coefficient determination. 970 Space velocity, 68-69 Spaghetti, cooking. 236 Spanan program. 379 Spartanol, wulfrene and carbon dioxide from, 751-752 Speclnlty chemicals, 203 Spzcies, 4-5 mole balances on. 8-9 net rates of reaction for. 379-330 and variable volumetric ffow rate, 112-1 13 Specific reaction rate. 82. 91-98 Speciftcat~onsin ethylene glycol production, 153 Spectroscopic measurements. 684 Spherer. mass transfer coefficient for. 777 Spherical bacteria growth. 42 1 Spherical catalyst pellets, 514 differentla1equation for, S 16-8 19. 8 2 2 4 2 3
dirnen~ionles\form. X 19-872 effective d~Au\ivityIn. 814-81 h for tissue engineer~ng.823-8?7 Spherical renctnrs in pressilre drop. 196 profchsional reference shelf for, 232 Spread of distribution<.88 l Square of the srandnrd deviation. 88 1 Squares of difference. 275 Stab~litydiagrams. 542-543 Stability rates in CSTR unsteady-state operation, 619 in linearized srlibility theory. 631432 in propylcne glycol production. 622 Stagnant film, diffusion thrnugh. 762, 773 Stagnant gases, diffwion through, 763 Standard deviation. 88 t Standard temperature and preqsure (STP)in rpace velocity, 68 Srarch. hydrolysis of, 461-462 Startup of CSTRs. 2 16-2 17, 6 19-62' State equation In batch reactors, 1 OY
Stationap phase in cell growth, 423 substrate balance in. 432 Steady-state bifurcation. 567. 588 Steady Ftate in CSTRs. 13 Steady-sratc molar Aow rates, 479-48 1 Steady-state nonisothemal reactors, 47 1 adiabat~coperation. See Adiabatic operations CD-ROMmaterial. 5 6 6 5 6 8 CSTRs with heat effects. 522-532 energy balance. See Enerzy balances equilibrium conversion. See Equil~briumconversions infomation required for. 472-473 journal critique problems, 589 multiple chemical reactions. 5d3 in CSTRs. 548-55 1 in PFRs, 544-547 multiple steady states, 533 heat af generation in.
534-536
hcat-removed terms in. 533-534 ignrtino-extinction curves in. 536-540 nlllnwny react~on?In CSTRs. 54c543 practical hide. 562-562 questions and problems. 566-58K summary. (33-565 supplemenmn, reading. 589-590 tubular reactors mdral and axial variations, 55 1-56 1 tubular reactors ~ i t hheal exchange, 495. 502-504 balance on cul>ldnt In.
-1(1"65 1 1
PFR energy b:ilancr in. 495-499 Step tracer experiment. 876-878 Stcro-X'nlmer Equation, 384-386 St~rredrenctos batch. 190-291 CSTRT. SPP C~titi~!tlottf-stirred tnrtk QflCi0r.Y (CSTRx) unsteady-srate operation of. 215-216 semibatch reactors. 217-226 rtanup of CSTRs. 2 16-2 17 Stoichiometric coefficients in converuion. 38 in relative ntes of reaction, 81 Stnichiometry. 79. 94 acetaldehyde formation, 322 acettc acid product~on.610 acetlc anhydride prduction,
505 adiabatic equiltbrir~rntempenture, 313 ammonia ox~dation.353 batch systems. 101b106 butane isomerizmion. 490, 498 catalyst decay, 718-719 cafi1ys.t sintering. 7 10-7 1 1 CD-ROM materbal. 12Gt 30. 139-140
cell growth. 32-30 curnp1e.t reactions, 328. 334-335 cootant balance. 502-503 CSTRs batch operations, 139-150 with cooking ~wils,5 3 1 single. 157 ethyl acetate saponiticnlion, 617 ethylene glycol production. 154, 164
ethylene clxide prt>duction. 191-192 Llolc system\. 106-123 ga\ phase. 20 1-20? glucose-to-cthnnol fementation. 433 ibothermal reactor%,199 mean conversion. 9 13 membrane reactors. 112. 350 mesi tylene hydrodealky lation In CSTRs. 345 in PFRs, 341-342 moving-bed reactor catalytic cracking, 726 multiple reactions, 336-339. 350. 616 net rate< of reaction. 330-33 I nitroanaline production, 60 1 nitrogen oxide production. 205 parallel reactions, 546 PBRs, 336-339
PFRs w~thheat effects. 546 multiple reactions. 336-33'6 reactor volume. 146-147 pressure drop. 136, IRh propylene glycol production, 53-529, 5'36,620 questions and problems, 131-140 semibatch reactors, 626 steady-state non~sothermaIreactars. 472 STTRs. 73 1 summary. 125-126 supplementary reading, 141 toluene hydmdemethylation reactor?. 695 triphenyl methyl chloride-methanot reaction, Zh l tvbuIar reactors adiabatic. 188 design. 169. 173 r d r a l and axial variations in, 559 Stomach in alcohol metabolism, 442-443 STP (standard tempenture and preswrej in space ve!ocity. 68 Straight-through transport reactors
(STTRs) catalyst deactivation in,
728-732 evaluation of, 29G29L Streptomyces aureofaciens, 337438
Stunn Prowrr factor. 326 St!renc From eih~lhenzzne.21 I. 585-5Xh
Subscrtpt\. l lE35 Sub~trutes in cell growth. 421. 427-430 and dilutton rate, 437 disappearance of. 400.43 1-43: in en~yme-subltratecomplex.
395-397 inhibition by. 417. 414. JI6417. J6M67 maw balances, 13 3 4 3 2 in Michilelis-Menten equation, 400. 405 in niicroelzctron~cfabriuntion, 69tl rnult~plesystems. 41741 8. 453 Sulfunation reactions. 347 51lhric acid in DDT production, 6 in ethylene glycol production. 197-198 professiona1 reference shelf for. 568
Sums of squares for ethylene hydrogenar~onto ethane. 706707 in multiple reaction analysis, 356 in nonlinear repression. 273. 274, 277 Supesficial mass velocity. 196 Supplementary rending active intermed~ates.enzymatic reactions, pharmacok~netic models. and bioreactors, 469470 catalysts. 755-756 conversion and retrctor s~zing.77 diffus~on.8 I s 8 I I, 865-866 ~sothemaEreactor design. 252 mole balances. 35-36 multiple reactions. 375 nonideal reactors, 1003 nonisothemal reactors steady-state, 589-5W unsteady-state. 644 rate data collection and analysis.
303-304 rate
laws and stoichiomrtry, 141
RTDs, 944 Supported catalysts, 649
Surface area in catalyst sintering. 709 in hydnaine decomposition. 786788
in mas., tmnsfer-limited reuctlunh. 790 In
membranc reactor%,2 10
in mtcroreacklr\. 301 of splier~ci~l catalyst pellets, 8 18 Surfucc-catalyzed renctlon?, 203 Surbce-reaction-l~m tted operation., cumene decomposition. 680 ~mversibleisornertzation, 687-688 irrevers~blerate laws. 684 Surface reactions in catalysts, 66-8 in CVD. 70 1 In microelectronic fabrication.
698 in packed k d transfer-limited reactions, 8.19 rapid. 777-779 rate laws for. 673, 686 rare-limiting. 677-678 in toluene hydrodzmethylation. 69 I Surfaces catalyst, 776-JBU In effectiveness factor. 837 in mass transfer tu single pani. cles. 778 metallic, 801 Surfactants from ethylene oxide. I91 Sweerland, Ben, 945 Swimming rate of small organisms. 858-859 Synthesizing chemical plant design, 1 9 6 198 Synthetic rubber pmductlon. 654 System voluine in mole balance equation. R Szent-Gyorgyi. A l k a . 8 13
T-Anlyl Methyl Ether (TAME). 584 TaIs fitting, 935 in pulse input experiment. 875 Tanks-in-wries IT-I-S) models conversion in, 97 1-973 v5. dispetsion models, 974 nonideal. 948-955 one-parameter, 947 vs. segregation model, 953-955 Tartlon, 364-365 Taylor. H. S.. 650
Taylor-Couette flow device, 898 Taylor series for energy balance,
523 TBA (butyl alcohol). 740 TCC (therrnofor cetalytic cracking) units. 723 Temperature, 47 1 . See also Hear effects
and activation energy. 97-98 in adsorption. 663 in catalyst deactivation, 72 1-722 in cell growth, 425 co~lversionfactors for, 1018 in CSTRs, 13. 22 in diffusion, 770 in enzymatic reactions, 407 in ethylene oxide production. 193 in fludized-bed reactors, 24 tn internal effectiveness factor, 83 1-832 in mass transfer correlarions, 783 in mass transfer to single panicles. 778 non~sothemalreacton steady-state. See Steady-state nonisothemal reactors unsteady-stare. See Unsteadystate nonisothemal reactors in rate laws, 87. 6874588 in runaway reactions, 540-543 seltctiv~tyaffected by. 3 12-314 in state equations, 1 U9 Temperature-concentration phase planes. 619
Terephthalic acid (TPA). 367 Terminallon step in chain reactions, 386 Termolecular reactions, 80 Tessier equation. 42442.5 Testing new processes, batch reactors for, 10 Thermal conductivity, 554. 558 Thermal cracking of ethane. 387-392 Thermal decomposition of isoprcpyl Isocyanate, 301 Thermal diffusivlty in mass transfer coefficient. 776 Thermodynamic equilibrium constant. 89. 102 1-1026 ThermodynarnicaIly Iimlted reactions, 207. 225-226
Thermodynam~cs
equilibrium conversion from. 514
first law of. 473474 in reversible reactions, 91 T?m-mofor catalytic cracking (TCC)units, 723 Thiele modulus in cartilage forming cells, 825-826 in fals~fiedkinetics. 833 internal, 839-841 In internal effectiveness factor, 829. 832-833 in spherical catalyst pellets, 81 9-823 Third-order reactions, 84 Thoenes, D., Jr., 784 Thoenes-Kramess correlation In Row t t u ~ u p hpacked beds, 784
in hydrazine decomposition, 786 in Mears' criterion, 842 In nitrous oxide reductions, 847 Three-eighths rule, 1014 Three-phase reactors, 849-850 Three-pint differentiation formulas, 258-259 Three-point rule, 1014 Thrombin in bid clotting. 326 Thrusten, monopropellant. 785-786
Tic Tac module, 235
'lime in batch reactors. 15 1 concentration. 256 reactant, 39-10 in energy rate change, 1019 in growth rates. 425 in half-lives methnds. 280-28 I T~mefunction in semibatch reaclor%.2 19 Time order of magnitude In scale-
up, 151 Timmerlraus, K. D,,23 Tissue enpineenng, 823-827. 855 %sue factor in blood clotting. 325-326 Tissue water volume (TWV). 4-40-44 1 Titanium dioxide. 7'46 TOF (turnover frequency). 651 Toluene hydrodernethylatlon of, 87. 688-698 from methylcyclohexane, 746747 xylene from. 649.749-750
Tonuosity in effective dirusivity, 815-816 Total collective mass, 5 Total concentrations in flow reactors, 116-1 17 Total cycle time in scale-up, 151 Total energy in first Iaw of therrnodynamics, 473 Total enzyme concentration,
398-399 Total mass, 5 Total molar flow rate in Row reactws, 11 1-1 12 in gas phase. 201 in PFRmBR multiple reactions.
33b-339 Total volume CSTRs in series. 57-58 PFRs in series, 59 Toxic intermediates. 203 TPA (terephthalic acid), 367
Tracers
CSTR parameter modeling, 981-985, 987 dispersion coefficient deterrnination, 966-9157 dispersion model, 956957 puke input experiment, 874 RTDs. 87 1. 887 step tracer experiment, 876-878 T-I-S model, 949 tubuIar reactors. 97&973 two-parameter models. 979 Trains of reactors with interstage heating. 577 Trambnuze reactions. 312-3 17 Transdermal drug delivery. 772 Transfer. See Mass transfers Transferases enzymes. 397 Tranc~tionstate theory. 128-1 29 T n n ~ ~ t i ostates n and energ? barriers. 92
Transldlion in transition state theow, 129 Transport with catalysts, 657 In membrane reactors. 2 10, 21 1. 349 Trapeznrdal rule. E013 Trial and error method. 988 Trickle k d reactan. 849650, 853-854
Tr~ethanolarnineformation. 306 TriphenjI methvl chloride, methanol r e a c h with, 260-266 Tri~yl-methanolreaction, 269-27 1. 277
Troubleshooting, 891-892 corrosion, 132-1 33 CSTRS. 892-895 isothermal reactors. 239-240 olefin formation, 643 reactor systems. 580-581 tubular reactors. 895-897 Tmnlan, Harry S. 471
Tubes in microreacrors. 201 In pressure drop. 190 Tubular reactors, 1 4 - 1 7. 241 d e s ~ g nequations for. 44. 99 designing. 168-1 71 for ethylene production. 171-175 :as-phase reactions, 14, 23. IAPI71 hemoglobin deoxygenation in.
295 nonideal. 947-948 balance equations in, 957 boundary cond~tionsin. 9.58-952 disper~ionin. 955-957. 962-964 disper~ioncoefficient correlatlon in, 964-966 disperfion coefhc~entdetermination in, 966-950 sloppy tracer inputs In, 976973 plug-flow. See Plug-flow reactors (PFRqI radial and axial variations In. 551-561
RTDs for. 895-893 space time lo, 67 Turhulenr diflurion. 955 Turbulent flow In packed hed pressure drop, 184 in pipes. 964-96.5 Turnover frequency ITOF). 65 1 Turnover number In MichaelisMenten equation. 399 Two-parameter models for nnnidea! reactors. 945, 945. 979-987 Two-point rule. 1013 T W V (tissue water volume). 3 1 M 41 Tyrosinasc. 395
Ultrd\onic waie5, lrght fmm.
3U4-386
Unhund enzyme concentration, 398
Uncompetitive inhibition, 41241 3 Underground wet oxidation prob )em. 1040 Undesired products in multipie reactions, 307-309 Uniform surfaces. adsorption in, 666 Unirnolecular reactions, 80 Vnits. conversion factors far. 1018-1019 Unsteady-state nonisothermal reac-
tors. ,541 batch reactors adiabatic operation, 594-598 with ~nterruptedisothermal ovration. 599-605 CD-ROMmaterid. 6 3 6 6 3 2 CSTR operation energy balance of. 477 stanup. 2 16-2 17.619-622. stead! state falloff in, 6 2 2 4 2 4 energy balance In, 477, 591-594 mole balances for, 198, 593 multiple reactions. 625627 PFRF. 628 questions and problems, 633443 safety in. 605-6 14 hemihatch reactors, 614-619 wmrnaty. 6 2 9 4 3 0 supplementary reading. 644 Unsteady-state operation o f stirred reactors. ? 15-2 16 semibatch, 21 7-226 Ftanup. 316-717. 619-622
Used reactors. 2.1
Vacant sites in cunrene adsorp~inn, h76
Valves In rnicmreactors. 203 rel~ef.6 0 5 4 13 Van de Vusse kinetics. 360-361. 371 Vanadium oxides. 345-746 Vanadium triisopropaxide (VTPO), 745-746 Van't Hoff'a equation. 1022 Vapor-phase craclung. 75 1 Vapor-phase reacrlons, irrever~lbre endothermic. 575-576 Variable heat capacities. 567-568 Variable temperature for acetic anhydnde. 510-51 1 in energy balance. 478 Variable volume batch reactors wi~h.109-1 1 1 in pas phase flow systems, 10% Variable iolumetric Row rate, 11 1-123 Variance in ethylene hydrogenation to ethane. 706-707 in RTDs. 883-883, 886 in T-I-S model. 950-95 1 Vat reactors. See Cantinuousstirred tank reactors (CSTRs) Vejtasa. S. A,. 538-539 VeIcxity of fluid in Inass transfer to slnfle particle5, 779 space. 6869 Velocity profile< in tubular reacinrs. 947 dhperqion. 962 r a d ~ a land axrsl variations. 556 Venkatesan. R.. 785 Verbal learner?. 11147 Vermont Safely Information on the En~emet(Vermont SERI). 167 Vessel boundary conditions in dispersion coeffic~entdeterminatzon, 968-970 in tuhular reactors, 959-960 Vecqel d~spersionnumber, 958 Vibration In tran\ition state t h e o ~ . 129 Vibrational degrees of freedom. 378 Vinyl allyl ether. 3 Viqcosity conversion factors for. 1018 in diffuqion, 761 of helium. 786
Viscosity [mnt.) In mass transfer coefficient. 776 Visual Encyclopedia of Equipment, 27 Visual learners, I047 Volume. See Reactor volume Volume-average particle diameter in hydrazine decomposition. 786 Votumetric feed rate in chemostats, 434
Volumetric flow, t 07 CSTR parameter modeling. 983 differential reactors. 282-283 ethylene oxide production. 194 flow reactors, 1 14 methane production, 284-285 pulse input experiment, 873 RTDs, 880-88 1.889 T-I-Smodels, 949 tubular reactors, 171 variable, 1 1 1- 123 Voorhies, A,. 712 VTIPO (vanadium trii~opropoxide), 745-746
Wash-out in cell growth. 43&438
from methane. H9
Washington, Booker T., 79 Water, light from, 384-386 Water-gas shift reaction in coal gasification. 103 equilibrium constant in. 1024-1026 Watson, K. M.. 470 Web srtes for rate law data, 1037 Wekr, M., Er., 870 Weekman, V. W. , 289 Weighted least squares analysis. 293 U'eisz-Prater criterion, 839-84 1 Wen. C. Y. on RTD moments. 88 1 o n tracer techn~ques.878 Wet etching, 700 Wet oxidation problem, 1040 What if conditions for mass transfer coefficients, 788-792 White, D. H., 879 Wilkinson, P. K., 446 Wine-m&ing, 424425 Wolf, D.. 879 Wooden, John, 37 Work, conversion facton for, 1018 Work term in energy balance. 474476 Wulfrene from spartanol. 751-752
from toluene. 649. 749-750
Wafer fabrication, 298-300. 698699
Chemical Vapor Deposition in.
701-704 etching in, 7 0 0 Warnicki, J . W., 898
Xylene
isomers in, 584-585
Yeasts doubling times for. 425 gmwth of. 421 sacchmmyces rerevisiae
production, 300 Yields in bioconversions. 4 19 in cell growth, 426, 429-430 Yogur~,postacidification in, 459
Zeolite catalysts, M9
Zero-order reactions. 83 in batch reactor data analysis.
267 exothermic liquid-phase, 579 Zem-parameter models. RTDs for maximum mixedness model, 9 15-922 maximum mixedness predictions. 922-923 segregation model,904-914 Zewail, Ahrned, 379 Zwietering. T N. on maxlmum mixedness, 922 on segregation model. 904 Zyrnononas bacteria, 240
