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
PTR
-
Upper Saddle River, NJ Boston Indianapolis San Francisco New York * Toronto Montreal London a Munich Paris * Madrid Capelown a Sydney Tokyo * Singapore Mexico City
Many of the designations used by manufacturers and sellers to distinguish their product5 are claimed as tmdemarks. Where those designations appear in this book, and the publisher was aware of n trademark claim, the designations have been printed with ~nitialcapital letters or in all capitals. The author and publisher have taken care in the preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibil~tyfor errors or omissinns. No liability IS assumed for incidental or con=quential damages in connection with or arising uut of the use of the information or prognrns contained hereln.
The publisher offers excellent discount5 on this book when ordered in quantity for bulk purchases Dr s p ~ i n sales, l which may include electronic versions andlor custom covers and content pnrticular to your business. training goals. marketing focus, and branding interests. For more information, please contact:
U.S. Corporare and Government SnIes (800)387-3419
[email protected] For sales outside the U.S., please contact: International Sales international @ pearsoned.com
Visit us on the Web: www.phptr.com Library of Congress Ca~aloging-inPrtblication Data
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~l