Chemical Thermodynamics for Industry

Chemical Thermodynamics for Industry Chemical Thermodynamics for Zndustry Edited by Trevor Letcher University of Nata...

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Chemical Thermodynamics for Industry

Chemical Thermodynamics for Zndustry Edited by Trevor Letcher University of Natal, Durban, South Africa

RSC advancing the chemical sciences

ISBN 0-85404-591-0 A catalogue record for this book is available from the British Library 0 The Royal Society of Chemistry 2004

All rights reserved Apart from fair dealing for the purposes of researchfor non-commercial purposes or for private study, criticism or review, as permitted under the Copyright, Designs and Patents Act 1988 and the Copyright and Related Rights Regulations 2003, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in writing of The Royal Society of Chemistry, or in the case of reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency in the UK, or in accordunce with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page. Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 OWF, UK Registered Charity Number 207890 For further information see our web site at www.rsc.org Typeset by Macmillan India Ltd, London, UK Printed by Athaneum Press Ltd, Gateshead, Tyne and Wear, UK

Foreword The International Association of Chemical Thermodynamics (IACT) was established in 2002 to continue the activities of the former IUPAC Commission 1.2 on Thermodynamics. The challenge ahead for IACT is to provide leadership in academia and industry by maintaining the highest standards in research, teaching and application in all areas of chemical thermodynamics. At the crossroad of quite different areas, the communication among such a diversified audience of scientists, teachers and engineers includes publication of reference data collections, recommendations for using and applying thermodynamics and related techniques, publications of selected reviews in refereed journals as well as monographs and collective works. In each case, the responsibility of IACT is the identification and selection of the best authors/contributors. In this context, I would like to point out that IACT continues the successful series “Chemistry for the 21st Century” with contributions from international experts, which may be profitably used even by readers not specialized in the fields covered. Following the volume “Chemical Thermodynamics”, the present one on “Chemical Thermodynamics for Industry” deals with a large spectrum of applications of thermodynamics in modern chemical industry. It is a collection of contributions showing the importance of thermodynamics for pushing frontiers of efficiency and economics towards increasingly better-quality products and services for the benefit of mankind. The book assembles 23 chapters authored by leading thermodynamicists.The initiative for this work originated with Professor Trevor Letcher, the editor, who invested considerable energy and patience to select, collect and organize these timely highquality contributions. Undoubtedly, the book will find its place within IUPAC’smajor publications. As President of IACT, it is my honour and pleasure to thank Professor Letcher for his guidance in such an enterprise. My thanks also extend to the authors for making their knowledge available and for the time they devoted to this work. Jean-Pierre E. Grolier January 2004

Preface The history of chemical thermodynamics goes back 150 years and the subject has, for over a century, been the foundation for much of chemistry. This historical importance is often associated with an attitude that thermodynamics has little relevance to modernday chemistry and will have little importance in the future development of chemistry. To counteract this, the W A C Commission 1.2 on Chemical Thermodynamics, published a volume entitled “Chemical Thermodynamics for the 21st Century” in 1999. It consisted of 27 chapters, all focusing on the applications of thermodynamics to very recent developments in chemistry. The aim was to highlight the role that thermodynamics was playing at the forefront of chemical research at the dawn of the 21st century. Three years later, in 2002, the International Association of Chemical Thermodynamics, IACT, the successor to Commission 1.2, decided to publish a collection of essays on applied chemical thermodynamic topics under the same editorship. This new collection is aimed not only at those working in a specific area of Chemical Thermodynamics but also at the general chemist, the prospective researcher and those involved in funding chemical research. The new volume is dedicated to the chemical industry and will show that thermodynamics is not only helping us to understand the world we live in but is also helping us to create a better world. The topics for this volume have been chosen to include the very latest thermodynamic development in not only the traditional areas but also in newly established areas and potentially important new areas of chemistry. The traditional areas covered include aspects of: calorimetry microcalorimetry transport properties crystallization adsorption electrolyte systems transport fuels. The newly established areas include: multiphase modelling reactive distillation

...

Preface

Vlll

non-equilibrium thermodynamics spectro-calorimetry. Topics covered, which are considered potentially important to the chemical industry in the future, include: ionic liquids 0 new materials ab initia quantum chemistry nano particles polymer recycling clathrates the economic value of applied thermodynamics. 0

0

The topics have all been written by specialists in the field and include some of the most important names in chemical engineering, thermodynamics and industrial applications of applied thermodynamics. I wish to record my thanks to authors and to the publishers in producing a book of this breadth and importance. T.M. Letcher Durban, January 2004

Contents Chapter 1

Non-EquilibriumThermodynamicsfor Industry S. Kjelstrup, A. R4sjorde and E. Johannessen 1 What Can Non-Equilibrium Thermodynamics Offer?

Developments and Status of NET NET Applied to Phase Transitions 4 NET Equations for Distillation Columns 5 The Second Law Optimal Path of Operation 6 Second Law Optimal Distillation Columns 7 Second Law Optimal Chemical Reactors 8 Conclusions Acknowledgement References 2 3

Chapter 2

Chapter 3

1 1 2 2 5 6 7 8 10 10 10

A Modelling Technique for Non-equilibrium Metallurgical Processes Applied to the LD Converter M. Modigell, A. Traebert, I? Monheim and K. Hack

12

1 Introduction 2 Process Model Development 3 Modelling Tool 4 Simulation Results 5 Conclusions 6 List of Symbols References

12 12 16 17 21 21 22

Multiphase Thermodynamicsof Pulp Suspensions I? Koukbri, R. Pajarre and E. Rasanen

23

1 Introduction 2 Fibres in Aqueous Solution 3 Fibre-Ion Interactions 3.1 Donnan Equilibrium 3.2 Specific Interactions 4 Solid and Gaseous Phases

23 24 25 25 26 27

Contents

X

Chapter 4

5 General Gibbs Energy Model 6 Discussion References

27 31 32

Reactive Distillation

33

H.G. Schoenmakers and W Arlt

1 Introduction 2 Thermodynamics 2.1 Phase Equilibrium 2.2 Chemical Equilibrium 3 Technical Application of Reactive Distillation 3.1 Introduction 3.2 Process Synthesis 3.3 Process Design and Optimization 3.4 Limitations of the Methods for Synthesis and Design, the Scale-up Problem 3.5 Choice of Equipment 4 Conclusions References Chapter 5

Chapter 6

33 34 34 36 38 38 38 39 39 40 41 42

Thermodynamic Properties from Quantum Chemistry S.I. Sandler

43

1 Introduction 2 Quantum Mechanical Computation of Force Fields for Molecular Simulation 3 Approximate Quantum Mechanical Calculation of Thermodynamic Properties 4 Conclusions References

43

Thermodynamics of Natural Gas Clathrate Hydrates E.D. Sloan

57

1 Introduction 2 What are Natural Gas Clathrate Hydrates? 3 Development of Hydrate Thermodynamics 3.1 Discovery and Academic Interest 3.2 Thermodynamics Driven by Discovery in Flowlines 3.3 Thermodynamics Driven by Hydrates Outside Flowlines 4 Hydrate Thermodynamics 4.1 The Hydrate Phase Diagram 4.2 The Most Important Advance: The Statistical Thermodynamic Model

57 58 59 60

44

53 55

55

60

62 65 65 67

xi

Contents

Hydrate Challenges for the Future: Reversible and Irreversible Thermodynamics References

5

Chapter 7

Ionic Liquids in Separation Processes J. Gmehling

76

1 2

76

Introduction Thermal Separation Processes Using a Selective Solvent as Separating Agent 2.1 Extractive Distillation 2.2 Liquid-Liquid Extraction 2.3 Absorption 3 Thermodynamic Fundamentals 3.1 Vapor-Liquid Equilibrium (VLE) 3.2 Liquid-Liquid Equilibrium (LLE) 3.3 Gas-Liquid Equilibrium (GLE) 4 Selection of Selective Solvents 5 Thermophysical Properties Required for Selective Solvents 6 Ionic Liquids 7 Results 8 Conclusion Acknowledgement References Chapter 8

Chapter 9

72 72

Spectrocalorimetric Screening for Complex Process Optimization E Dan and J-RE. Grolier

77 77 78 79 79 79 80 80 80 81 82 83 86 87 87 88 88 89

Reaction Calorimetry: General Aspects Instrument Particularities of Anionic Polymerization of Lactams in Organic Media Spectro-Calorimetric Investigation Under Isothermal Conditions 4.1 The Manner of Contacting the Catalytic Species 4.2 Effect of the Catalyst Nature 4.3 Stirring Level Non-Isothermal Reaction Calorimetry Conclusion References

92 92 98 100 101 102 103

Microcalorimetry for the Pharmaceutical Industry

104

91

A.E. Beezer

1 Introduction 2 Stability Determinations

104 105

xii

Chapter 10

Chapter 11

Contents

3 Characterisation of Physical FodAmorphous Content 4 Drug/Receptor Studies by Titration Calorimetry (ITC) 5 Multiple Miniature Calorimeters and their Potential 6 Conclusions References

107 108 108 109 109

Isothermal Flow-Microcalorimetry:Principles and Applications for Industry M.A.A. 0 'neill

110

1 Introduction 2 Theory 3 Flow Calorimeter Design 3.1 Solution-Phase Flow Calorimeters 3.2 Gaseous-Phase Flow Calorimeters 4 Flow Calorimetric Equations 5 Validation of Flow-Through Microcalorimeters 6 A Simple Flow Calorimetric Experiment 6.1 Experimental Protocol 6.2 Determination of Thermodynamic and Kinetic Parameters from Calorimetric Data 7 Further Applications of Flow Microcalorimetry 8 Conclusions References

110 111 111 112 113 113 115 116 117

Transport Properties and Industry WA. Wakeham and M.J. Assael 1 Introduction 2 Measurements 2.1 Viscosity 2.2 Thermal Conductivity 2.3 Diffusion Coefficient 3 Theoretical Predictions 4 International Dimension

122

5 Databases 6 Future References Chapter 12

Micro- and Nano-particlesProduction Using Supercritical Fluids E. Reverchon and I. De Marc0 1 Introduction 2 Experimental Apparatus for SAS Processing

3

2.1 Experimental Procedure SAS Fluidodynamics, Mass Transfer and Thermodynamics

118 119 119 120

122 124 125 126 127 127 129 129 130 131 132 132 133 133 134

xiii

Contents

Relationship between Particles Morphology and High-Pressure VLEs 4.1 Experimental Evidences 5 System Modifications Induced by the Presence of Solute 6 Conclusions References

4

Chapter 13

Chapter 14

Chapter 15

135 135 140 142 142

CalorimetricMeasurements of Thermophysical Properties for Industry J-RE. Grolier and E Dan

144

1 Introduction 2 Heat Capacities 2.1 Heat Capacities of Gases 2.2 Heat Capacities of Liquids 3 Bulk Thermomechanical Properties 3.1 In Extended Ranges of T and p 3.2 In the Vicinity of the Critical Point 4 Phase Transition Thermal Properties 4.1 Fusion/Crystallization 4.2 Glass Transition Temperature 5 Conclusion References

144 145 145 147 149 149 151 152 152 154 157 158

Plastic Recycling W Arlt

159

1 Introduction 2 Cost of Recycling 3 Thermodynamic Considerations 4 Polymer Production 5 Recycling Schemes 6 Energetic Considerations References

159 160 160 160 162 165 165

Industry Perspective on the Economic Value of Applied Thermodynamics and the Unmet Needs of AspenTech Clients C-C. Chen, S. Watanasiri, 19 Mathias and V V De Leeuw

166

1 Economic Impact of Applied Thermodynamics 2 Applied Thermodynamics System 3 Elements of an Applied Thermodynamics System 3.1 Calculation Engine 3.2 Analysis Tools 3.3 Programmable Interfaces 3.4 Deployment Tools

166 167 167 168 169 169 169

Contents

xiv

Industrial Applications 4.1 Phase Equilibrium Modeling for Nylon-6 Process Simulation 4.2 Quickly Screen Solvents for Organic Solids 5 Quantitative Cost-Benefit Assessments 6 Unmet Needs of AspenTech Clients 6.1 Estimation Methods for Pure Compound Properties 6.2 Cubic Equations-of-State 6.3 Activity Coefficient Models 6.4 Modeling Electrolyte Systems 6.5 Modeling Polymer Systems 6.6 Data Packages 6.7 Equilibrium Calculations 6.8 Transport and Interfacial Properties 6.9 Model Deployment 7 Conclusion References

170

Thermodynamics of New Materials M. Schroeder and M. Martin

180

1 Introduction 2 Ceramic Membranes for Oxygen Separation and Catalytic Partial Oxidation of Hydrocarbons 2.1 Transport and Defect Thermodynamics in Membrane Materials 2.2 Oxygen Permeation 2.3 Stability of Membrane Materials 2.4 Conclusion 3 Solid Oxide Fuel Cells 3.1 Fuel Cells 3.2 Electrolytes for Solid Oxide Fuel Cells 3.3 Lanthanum Gallates 3 Conclusion References

180

4

Chapter 16

Chapter 17

Thermodynamic Prediction of the Formation and Composition Ranges of Metastable Coating Structures in PVD Processes f?J. Spencer

170 170 171 172 172 173 174 174 175 175 176 177 177 177 178

180 181 184 186 187 187 187 189 191 193 194

197

197 1 Introduction 198 2 (Ti,Al)N Coatings 2.1 Metastable + Stable Structural Transformation 199 Energies for Nitride and Carbide Phases 3 Al,O,-A1N Coatings 206

xv

Contents

Chapter 18

Chapter 19

Chapter 20

Chapter 21

4 summary References

208 208

Thermodynamics of the Nano-Sized Particles I: Tanaka, J. Lee and N. Hirai 1 Introduction 2 Gibbs Energy in Small Particle Systems 3 Calculation of Nano-Sized Phase Diagrams of Binary Alloys 4 Concluding Remarks References

209

Thermodynamics of Electrolyte Systems of Industry K. Thomsen 1 Standard States and Chemical Potentials 1.1 Ionic Mean Properties 2 Act: rity Coefficient Models 3 Equ librium Calculations 3.1 Speciation Equilibrium 3.2 Vapor-Liquid Equilibrium 3.3 Solid-Liquid Equilibrium 3.4 Liquid-Liquid Equilibrium 4 Pha ;e Diagrams 5 Density 6 Viscosity References

209 210 213 217 218 219 219 220 221 222 222 223 223 224 224 226 228 228

Thermodynamics of Crystallization A.S. Teja and R.W Rousseau

230

1 Introduction 2 Pure Component Phase Behavior 3 Binary Solid-Liquid Equilibria 3.1 Systems that Form Solid Solutions 3.2 Systems that Exhibit Eutectic Behavior 4 Ternary Solid-Liquid Equilibria 4.1 Systems that Form Solid Solutions 4.2 Eutectic Systems 5 Solid-Fluid Equilibria 6 Kinetics of Dissolution and Crystal Growth 7 Summary References

230 230 23 1 232 233 236 236 238 239 24 1 242 242

Thermodynamics of Adsorption A.L. Myers

243

1 Introduction 2 Adsorption Isotherm and Equation of State

243 244

xvi

Chapter 22

Contents

3 Thermodynamic Functions 4 Mixtures 5 Example 6 Summary References

246 248 249 253 253

Mesoscopic Non-equilibrium Thermodynamicsof Polymer Crystallization D. Reguera and J.M. Rubi

254

1 Non-Equilibrium Thermodynamics and the Mesoscopic Level of Description 254 1.1 Meso-Structures 254 1.2 Mesoscopic Non-Equilibrium Thermodynamics (MNET) 255 2 Polymer Crystallization 256 3 The Mesoscopic Non-Equilibrium Thermodynamics Approach to Polymer Crystallization 257 3.1 Mesoscopic Non-Equilibrium Thermodynamics 258 of Activated Processes 259 3.2 Polymer Crystallization 261 4 Conclusions 261 References Chapter 23

Applied Thermodynamicsfor Petroleum Fluids in the Refining Industry D. Ramjugemath and R. S h a m a

1 Introduction 2 Composition of Petroleum 3 Characterization of Petroleum and its Fractions 3.1 Measured Data 3.2 Calculated Parameters 3.3 Pseudo-Components Characterization 4 Thermo-Physical Property Prediction Methods 4.1 Enthalpy 4.2 P-V-T Relationship 4.3 Vapour-Liquid Equilibria (Including Vapour Pressure) 5 Codes and Data Sources 6 Future Trends References Subject Index

262 262 263 263 263 265 267 267 267 269 270 27 1 27 1 272 274

Contributors

xvii

Contributors W. Arlt Professor, Dr, Technische Universitat Berlin, Thennodynamik und Thennische Verfahrenstechnik,Building TK7, 0-10623 Berlin, Germany; Tel.: +49 30 314 22755; e-mail: [email protected]. MJ. Assael Professor, Dr, Chemical Engineering Department, Aristotle University, Thessaloniki 54124, Greece; Tel.: +30 6937 158044; e-mail: [email protected] A.E. Beezer Professor, Dr, Medway Sciences, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Kent ME4 4TB; Tel.: +44 1634 883362; e-mail: [email protected]. C-C. Chen Dr, Aspen Technology Inc., I0 Canal Park, Cambridge, MA 02141 USA; Tel.: +I 617 941 1202; e-mail: [email protected]. F. Dan Dr, Macromolecular Chemistry Department, “Gh. Asachi” Technical University of lasi, 71A, Mangeron Avn., P.O. 10, P.B. 2007, 66OO-lasi,Romania; Tel.: +40 232 27 86 83/ 2332; e-mil: [email protected]. V.V. De Leeuw Dr, STS Associks, 16 rue Jacques Tati, 91042 Evry, France. I. De Marco Dr, Dipartimento di Ingegneria Chimica ed Alimentare, Universita degli studi di Salemo, Via Ponte Don Melillo, 84084, Fisciano (SA),Italy. J. Gmehling Professor, Dr, Carl von Ossietzky Universitat Oldenburg,Fakultat V Technische Chemie, 0-2611I Oldenburg, Germany; Tel.: +49 441 7983831; e-mail: gmehling @ tech.chem.uni-oldenburg.de. J-P.E. Grolier Dr, Laboratoire de Thermodynamiques des Solutions et des Polymdres, Universite‘ Blaise Pascal, Clermont-Ferrand, 24, Avenue des Landais, 63177 Aubidre, France; e-mail:j-pierre.grolier@ univ-bpclermont.f x K. Hack Dr, Managing Director, GTT-Technologies, Kaiserstrasse 100, 0-52134 Herzogenrath, Germuny; Tel.: +49 2407 59533; e-mail: [email protected]. N. Hirai Dr, Department of Materials Science and Processing, Graduate School of Engineering, Osaka University, Osaka, 565-0871, Japan. E. Johannessen Dr, Department of Chemistry, Norwegian University of Science and Technology, N-7491 Trondheim,Norway. S. Kjelstrup Professor, Dr, Department of Chemistry, Norwegian University of Science and Technology, N- 7491 Trondheim, Norway; Tel.: +47 73594179; e-mail: [email protected]. P. Koukkai Dr, VTT Processes, PO.Box, FIN-02044 VlT, Finland; Tel.: +358 9 456 6366; e-mail: [email protected]. J. Lee Dr, Department of Materials Science and Processing, Graduate School of Engineering, Osaka University, Osaka, 565-0871, Japan. M. Martin Professor, Dr, Institute of Physical Chemistry,Aachen University of Technology, Templergraben 59, 0-52056 Aachen, Germany; Tel.: +49 241 80 94712; e-mail: [email protected].

xviii

Contributors

P. Mathias Dr, Aspen Technology Inc., 10 Canal Park, Cambridge, MA 02141 USA. M. Modigell Dr, lnstitut fur Verfahrenstechnik,RWTH Aachen, Germany. P. Monheim Dr, SMS Demag AG, Duisburg, Germany. A.L. Myers Professor, Dr, Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, PA, I9104, USA; e-mail: amyers @ seas.upenn.edu. M.A.A. O'neill Dr, Medway Sciences, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Kent, ME4 4TB, UK; Tel.: +44 1634 883362; e-mail: [email protected]. uk. R. Pajarre Dr, VTT Processes, P.0. Box, FIN-02044 VI?: Finland; Tel.: +358 9 456 6366. D. Ramjugernath Professor, Dr, School of Chemical Engineering, University of Natal, Durban 4041 South Afr?ca; Tel.: +27 2603128; e-mail: [email protected]. E. Rasiinen Dr, VTT Processes, €!O.Box, FIN-02044 Finland; Tel.: +358-9-456 6366. A. Rmjorde Dr, Department of Chemistry, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. D. Reguera Dr, Department de Fisica Fonamental, Facultat de Fisica, Universitat de Barcelona Diagonal 647, 08028 Barcelona, Spain. E. Reverchon Professor, Dr, Dipartimento di Ingegneria Chimica ed Alimentare, Universita degli studi di Salerno, Via Ponte Don Melillo, 84084, Fisciano (SA), Italy; e-mail: ereverchon@ unisa.it. R.W. Rousseau Dr, School of Chemical & Biomolecular Engineering, Georgia Institute of Technology,Atlanta, GA 30332-0100, USA. J.M. Rubi Professor, Dr, Department of Chemistry and Biochemistry, UCLA 607 Charles E. Young East Drive, Los Angeles, CA 90095-1569, USA; e-mail: [email protected]. S.I. Sandler Professor, Dr, Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, DE I971 6 USA; Tel.: +I 302-831-2945; e-mail: [email protected]. H.G. Schoenmakers Dr, BASF AG, Process Engineering, Building L540 0-67056 Ludwigshafen, Germany; Tel.: +49 621 60 56047; e-mail: [email protected]. M. Schroeder Dr, Institute of Physical Chemistry Aachen University of Technology Templergraben 59, 0-52056 Aachen, G e m n y . R. Sharma Dr, School of Chemical Engineering, University of Natal, Durban, 4041, South Africa. E.D. Sloan Professor, Dr, Gaylord and Phyllis Weaver Distinguished Professol; Center for Hydrate Research, Colorado School of Mines, Golden, Colorado 80401, USA; Tel.: +I 303 273 3723; e-mail: [email protected]. PJ. Spencer Dr, The Spencer Group, P.O.Box 393, Trumansburg, NY 14886, USA; e-mail: ps @spensergroupintl.com. T. Tanaka Professor, Dr, Department of Materials Science and Processing, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan; Tel.: +81 6 6879 7504; e-mail: [email protected]. A. Traebert Dr, Institut fur Verfahrenstechnik,RWTH Aachen, G e m n y . A.S. Teja Professor, Dr, School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA; Tel: +I 404 894 3098; e-mail: amyn.teja @ chbe.gatech.edu. K. Thomsen Associate Professor Dr, Department of Chemical Engineering, Technical University of Denmark, Building 229, DK 2800 Kongens Lyngby, Denmark. Tel.: +45 4525 2860, e-mail: kth @ kt.dtu.dk. W.A. Wakeham Professor, Dr, University of Southampton, Highfield, Southampton SO1 7 IBJ, UK; Tel.: +44 2380 592901; e-mail: [email protected]. S. Watanasini Dr, Aspen Technology Inc., I0 Canal Park, Cambridge, MA 02141 USA; Tel.: + I 617 949 1202.

CHAPTER 1

Non=Equilibrium Thermodynamicsfor Zndustry SIGNE KJELSTRUP, AUDUN RDSJORDE AND EIVIND JOHANNESSEN

1 What Can Non-Equilibrium Thermodynamics Offer? Non-equilibrium thermodynamics (NET) offers a systematic way to derive the local entropy production rate, 0, of a system. The total entropy production rate is the integral of the local entropy production rate over the volume, V, of the system, but, in a stationary state, it is also equal to the entropy flux out, J:, minus the entropy flux into the system, J;:

The entropy flux difference and the integral over CJ can be calculated independently, and they must give the same answer. The entropy production rate governs the transport processes that take place in the system. We have o = z JiXi > 0 1

where Ji and Xiare conjugate flux-force pairs. Each flux is a linear combination of all forces: J i = C L r- XJ I.

(3)

i

This means that NET gives flux equations in agreement with the second law of thermodynamics, and that the theory offers a possibility, through Equation (l), to check for consistency in the models that are used. The usefulness of NET in describing industrial problems has been questioned, because these problems are frequently non-linear. It is then important to know that the flux-force relations in Equation (3) also describe non-linear phenomena. The phenomenological coefficients Lii can, for instance, be functions of the state variables. By

2

Chapter I

including internal variables in the thermodynamic description, one can extend the application of NET to activated processes; see Chapter 2. For this reason, NET appears today as a non-linear and versatile theory that applies to many practical condition^.'-^ It is a misunderstanding that flux equations need to be linear on the global level.lo The total entropy production rate times the temperature of the environment is equal to the exergy destruction rate in a process. Processes with small losses in exergy have a high second law efficiency. A high second law efficiency, or exergy efficiency, is seldom a specific aim in process design. An increasing worldwide concern with CO, emission may change this. Multiobjective optimisation, with small entropy production as one target, may then be interesting in chemical engineering design.

2 Developments and Status of NET Non-equilibrium thermodynamics was founded by Onsager.11J2The theory was further elaborated by de Groot and Mazuri3and Prigogine.14 The theory is based on the hypothesis of local equilibrium: a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the Onsager assumed that the variables and the rate laws were the same on the macroscopic and the microscopic level; this is the so-called regression hypothesis. Also using the assumption of microscopic reversibility, he proved the reciprocal relations:

These assumptions restrict the validity of NET, but as stated above, they have a wide range of validity. It has long been known that the Navier-Stokes equations are contained in NET.13 More recently, NET has been extended to deal with transport across surfaces: quantum mechanical system^,^ and mesoscopic systems;a see Chapter 2. We have chosen to illustrate NET with cases of transport through surfaces in the following sections.

3 NET Applied to Phase Transitions Phase transitions are central in many industrial problems, for instance, in distillation, absorption, condensation, manufacture of liquid natural gas, and multiphase flow. We shall use the liquidhapour transition to illustrate the basic hypotheses and usefulness of NET for surfaces, a relatively new application. For applications to homogeneous phases, we refer the reader to the basic literature; see Section 2. R@sjordeet aZ? studied the phase transition in a pure fluid using non-equilibrium molecular dynamics simulations (NEMD). The NEMD method solves Newton's equations of motion for several thousand particles in an imaginary box; see Hafskjold16 for a review. The particles interacted with a Lennard-Jones-type pair

Non-Equilibrium Thermodynamicsfor Industry

3

potential. The phase diagram and the surface tension as a function of temperature of the system were first determined.2The ends of the box were then thermostatted to give an enormous temperature gradient: for argon-like particles, the gradient was lo8K-' m. The temperatures of the ends were also chosen so that a liquid-vapour interface established itself. The position of the surface was then determined according to Figure 1. The surface started where the equation of state for the vapour was no longer valid and ended where the density in the liquid phase started to vary in a linear manner. It was established that the surface was in local equilibrium: the surface tension calculated for the system with a gradient was indeed the same as for the system in global equilibrium at a given temperature; see Figure 2. The fluxes for heat and mass across the surface can be written as

J=-Ip,A - -I,,Rln-

P

with 1, being the surface transfer coefficients. The coefficients obey Equation (4). Equation (6) indicates that evaporation or condensation takes place when there is a thermal force (a difference in inverse temperature) and/or when there is a chemical potential difference (the pressure of the vapour differs from the saturation pressure at the temperature of the liquid). Equation (5) can be rewritten as

Here, A is the stationary-state thermal conductivity and q* is the heat of transfer. The results of the simulations for A and q* are shown as a function of surface tension in Figures 3 and 4. The transfer coefficients decrease in magnitude as we move from the critical point in the phase diagram (zero surface tension) to the triple point (maximum surface tension). It is reasonable that the surface becomes more resistive at high surface

-0

10

20

30

40

50

60

70

Figure 1 Density variation across a non-isotheml NEMD cell with Lennard-Jones spline particles. The extension of the su@ace is indicated by vertical bars

4

Chapter I

2 1 -

Figure 2 Surface tension as afunction of surface temperature in equilibrium (whole line) and in a temperature gradient (points)for Lennard-Jones spline particles

t .

.,-loo0

-1500

0

0

2

y/

Kinetic theory

N/m

4

6

Figure 3 Heat of transfer for the Lennard-Jones spline surface and from kinetic theory according to R#sjorde et aL9

tensions. In the same figures, the results are compared to computations from kinetic theory. Kinetic theory predicts the properties of hard spheres and dilute gases. Indeed, we observe from the figures that the closer one comes to the triple point, the closer the agreement between theory and simulations. The heat of transfer can be significant; its minimum value here is 20%of the heat of evaporation. The last term is frequently neglected in thermal modelling of interface transport. In this case, it was found that the flux equations were linear in the forces, even for large gradients. It was thus not necessary to invoke a nonlinear dependence on the forces, as is done in statistical rate tl1e0ry.l~ Little systematic information is available on surface transfer coefficients. There are indications that the surface transfer coefficients may depend on the range of particle interaction.18The entropy production rate per m2 of area can be high for the surface, implying that the surface conductivity per m is smaller than the conductivity of the homogeneous phases. But since the homogeneous phases are thicker than the surface, these phases will still give the dominating total contribution to the entropy production of the system.*9

Non-Equilibrium Thermodynamics for Industry

OCP

0

6 -

4

cu Y E 4 -

0

5

NEMD simulations Kinetic theory

0

0 0 .

$

8

\

i 3 x 2-

v)

o0 8

't 0

2

4 y / N/m

6

a

Figure 4 Thermal conductivity of a knnard-Jones spline surface and from kinetic theory according to R#sjorde et aL9

4 NET Equations for Distillation Columns Distillation columns are made to separate at least two different components. There are several different types of columns. The assumption of equilibrium between the liquid and vapour that leave each tray is still in common use to model tray distillation. The work of Krishna and Wesselingh20shows, however, that non-equilibrium models give results that are very different from those obtained with the equilibrium assumption. A description consistent with NET is the Maxwell-Stefan equations.20 In the application of Maxwell-Stefans equations, one frequently neglects the heat of transfer, q*. This assumption may be good for gases. It is not so good for liquid mixtures.16According to Section 3, it is likely that the heat of transfer plays an important role in flux equations of interface transport. Olivier21showed that failure to include the heat of transfer leads to an error of up to 20%in the heat flux calculated for some typical phase transition conditions. The entropy production rate and the complete set of equations that follows can be most conveniently written for the liquid film, the interface, and the vapour film in series.22 Film layers of thicknesses aL and 6" in the liquid and the vapour are illustrated in Figure 5. With constant fluxes (in stationary states), the integration is easy. The approach was called the integrated interface approach.22 For the three layers, the integrated overall force is the sum of the integrated force across each layer:

6

Chapter 1

Figure 5 Variationin chemicalpotential and temperature across the liquid luyel; the intelface, and the vapour layer during evaporatiodcondensation in a two-component system

With constant fluxes of energy and mass through the interface and adjacent films, the overall resistance coefficients were defined through the parentheses in the following equations:

With these relations, the combined films and interface is regarded as an “effective interface”. There is no need to assume phase equilibrium between liquid and vapour. The entropy production rate can alternatively be expressed by the measurable heat flux in the vapour and fluxes of mass.18p22This set of flux equations was used to explain the entropy production in tray distillation columns. However, it has not yet been used for predictive purposes. Much work remains to be done to include these equations in a software that is useful for industrial purposes.

5 The Second Law Optimal Path of Operation A process, natural or industrial, follows a path between states. In industry, this is called “the path of operation”. PrigogineI4investigated the dissipation of energy in a system out of equilibrium, and found that the entropy production rate was at a minimum in the stationary state close to global equilibrium. Also S ~ h e c h t e rgave ~~ a variational formulation of the problem by solving the partial differential equations that resulted from substituting the flux equations into the conservation equations, and confirmed that the entropy production was minimum in the stationary state, provided the phenomenological coefficients were constant. This may have strengthened the belief that processes need to be globally l i n e d for NET to be useful, an assumption that would limit the applicability of the theory seriously.

Non-Equilibrium Thermodynamicsfor Industry

7

Rather than asking for the general nature of a system that dissipates energy at the stationary state, like Prigogine and Schechter did, we find it more appropriate, at least for industrial purposes, to ask about the nature of the operating path, given certain constraints. A typical example would be: how can we minimise the entropy production of a system while maintaining a given production (or heat exchange)? The solution to this minimisation problem can be found for conditions far from global equilibrium, and is different from the solution found by Prigogine14or S ~ h e c h t e r . ~ ~ The question of constrained optimisation is answered in a standard manner by Euler-Lagrange optimisations. By formulating the problem with optimal control theory, Johannessen and K j e l ~ t r u explained p~~ that the “Hamiltonian” of the problem was constant in a study of chemical reactors. The total entropy production for a plug flow reactor was written as a function of a position-dependent state variable vector x(z) and the control variable u(z):

The Hamiltonian for the problem was constructed as

The solution to the problem is given by 2(rn+2) differential equations for the ternperature, pressure, degrees of conversion, and for Lagrange multipliers (T, P,ki, ;iT, Ap, and Ak), with partial derivatives of H , where rn is the number of reactions between the component^.^^ The constant Hamiltonian of this problem was reduced to a solution with constant entropy production, 0,in the case of a heat exchange process. Using NET, it was also found that this solution was approximated by a solution with constant driving force, Xi.25 How to realise this in practice, remains to be solved.

6 Second Law Optimal Distillation Columns For an adiabatic distillation column separating a binary mixture, the path of operation is shown through a McCabe-Thiele diagram in Figure 6. The compositions in the liquid and vapour streams at different trays are shown. Linear operating lines result when the heats of vaporisation of the two components are equal. For a diabatic distillation column,26however, the operating lines are curved and tend to follow the equilibrium line, as shown in the right part of Figure 6. This kind of operation is possible if heat is added or withdrawn on each tray. As a consequence, the entropy production rate in the column plus heat exchangers decreases. A reduction of 38% in the entropy production rate was obtained for a binary separation of benzene and toluene in an equimolar mixture. The heat added on each tray in the optimal diabatic column and the corresponding vapour flows are shown in Figure 7. The results for the corresponding adiabatic column are also shown for comparison. The most important trays for distributed heating were the trays closest to the reboiler and condenser.

8

Chapter I

X

1

Figure 6 Examples of operating lines of an adiabatic and a diabatic column

Figure 7 The vapourjlow (mol-'s) and heat exchanger load (kW)in an optimised diabatic column and an adiabatic column. Both columns separate an equimolar mixture of toluene and benzene

A new design that takes the varying vapour flows into account was proposed.27The effect of the changing hydrodynamic conditions has not yet been explored. It is still too early to conclude on the precise outcome of these optinisation studies, since the assumption of equilibrium on each tray was also used in the model. Progress in the methods of Section 4 may lead to improvements in the future. It is documented that the lost work can be reduced, but the increased investment costs are not yet clarified. Nevertheless, it is important to understand the thermodynamic conditions for optimal performance, independent of technical-economic considerations.

7 Second Law Optimal Chemical Reactors The entropy production of a sulfur dioxide oxidation (exothermic) reactor with heat exchangers was minimised in two different case^.^^,^^ Case 1 was a four-bed reactor with intermediate heat exchangers of a given total area,2gsee Figure 8. The entropy production rate was calculated from the entropy balance over the system. All inlet and outlet flow conditions were kept constant, except the pressure at the outlet. The

9

Non-Equilibrium Thermodynamics for Industry

problem was to find the inlet and outlet temperatures of the intermediate heat exchangers. These were found, and shown to give a saving of 16.7%in lost exergy. Case 2 was a typical plug flow reactor with temperature profiles as shown in Figure 9.29The entropy production rate was calculated from NET. The coolant temperature profile that gave the minimum entropy production rate was determined. Inlet and outlet conditions were kept constant, and so was the area of heat exchange. The bold lines in Figure 10 represent a case with 11% smaller exergy loss. In the next case, three constant boiling liquids were used as coolant. The length and temperature of the heat exchangers were set free to vary. The result is indicated by the grey lines in Figure 10. The lengths of the coolers are given by the vertical lines in the figure, and their temperature is shown on the ordinate axis. The lost work was now reduced by about 7%.29

Z

A T5

Product

Figure 8 The four-bed reactor with five intermediate heat exchangers according to de Koeijer et a1.28

-T

_ _ _ T,

I

I

900

600I 0

1

2

3

4

5

6

Position/m

Figure 9 Temperature of the reaction mixture, I: and of the coolant, Ta, in the plug flow reactor for sulfur dioxide oxidation according to Johannessen and K j e l ~ t r u p ~ ~

10

Chapter 1 1000

5

-T

___

600 I 0

1

2

3 4 Positiodm

5

Ta

6

Figure 10 Temperatures T and T, in the reactor with minimum entropy production. The cases of a continuous and a discrete coolant proBle are shown according to Johannessen and K j e l ~ t r q ? ~

8 Conclusions By referring to a few selected examples, we have discussed the following: Entropy production is important for a proper definition of fluxes and forces in non-equilibrium systems that are relevant to the industry. A higher accuracy in the fluxes can be obtained using NET. NET can be used to describe transport of heat and mass in the presence of substantial gradients in systems far from global equilibrium. Transport of heat and mass across interfaces can be described without using assumptions of equilibrium at the phase boundary. Expressions for the entropy production rate can serve as meaningful object functions in optimisations. NET offers an insight into the nature of the solution by characterising the system's H , 0,or Xi. These properties of NET may prove useful for modelling and in the design of more energy-efficient chemical processes in the future.

Acknowledgement The Research Council of Norway and the Statoil Vista Program are thanked for financial support.

References 1. B. Hafskjold and S. Kjelstrup Ratkje, J. Stat. Phys., 1995,78,463. 2 . A. Rosjorde, D. W. Fossmo, S. Kjelstrup, D. Bedeaux and B. Hafskjold, J. Colloid Inter$ Sci., 2000, 232, 178. 3. S. Kjelstrup and D. Bedeaux, Elements of Irreversible Thermodynamics for Engineers, International Centre of Thermodynamics, ITU, Istanbul, Turkey, 200 1.

Non-Equilibrium Thermodynamics for Industry

4. 5. 6. 7. 8. 9.

11

D. Bedeaux, Adv. Chem. Phys., 1986,64,47. D. Bedeaux and P.Mazw, Physica A, 2001,298,81. D. Reguera and J.M.Rubi, Phys. Rev. E, 2001,64,61106. A. Perez-Madrid, J. M. Rubi and P. Mazw,Physica A, 1994,212,231. J. M. G. Villar and J. M. Rubi, PNAS, 2001,98, 11081. A. Rbsjorde, D. W.Fossmo, S. Kjelstrup, D. Bedeaux and B. Hafskjold, J. Colloid Inte$ Sci., 2001,240, 355. 10. Y.Demirel, Nonequilibrium Thermodynamics. Transport and Rate Processes in Physical and Biological Systems, Elsevier, Amsterdam, 2002. 1 1. L. Onsager, Phys. Rev., 193 1,37,405. 12. L. Onsager, Phys. Rev., 1931,38,2279. 13. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover, London, 1984. 14. I. Prigogine, Thermodynamics of Zrreversible Processes, Charles C.Thomas, Springfield, 1955. 15. B. D. Coleman and C. Truesdell, J. Chem. Phys., 1960,53,28. 16. B. Hafskjold, in Thermal Nonequilibrium Phenomena in Fluid Mechanics, W. Kohler and S. Wiegand (eds), Springer, Berlin, 2002. 17. G. Fang and C. A. Ward, Phys. Rev. E, 1999,59,441. 18. S. Kjelstrup, T. Tsuruta and D. Bedeaux, J. Colloid Znte@ Sci., 2003, 256,451. 19. G. M. de Koeijer and S. Kjelstrup, Chem. Eng. Sci., 2003,58, 1147. 20. R. Krishna and J. Wesselingh, Chem. Eng. Sci., 1997,52, 861. 21. M.-L. Olivier, Proceedings of ECOS 2003, Copenhagen, Denmark, 2003. 22. D. Bedeaux and S. Kjelstrup, Chem. Eng. Sci., 2003,59, 109. 23. R. S. Schechter, The Variational Method in Engineering, McGraw-Hill, New York, 1967. 24. E. Johannessen and S. Kjelstrup, Proceedings of ECOS 2002, Berlin, Germany, 2002. 25. E. Johannessen, L. Nummedal and S. Kjelstrup, J. Heat Mass Trans$, 2002,45,2649. 26. R. Rivero, Energy, 2001,26,561. 27. G. M. de Koeijer, Energy efficient operation of distillation columns and a reactor applying irreversible thermodynamics, Ph.D. Thesis, Department of Chemistry, Norwegian University of Science and Technology, Trondheim, 2002. 28. G. M. de Koeijer, E. Johannessen and S. Kjelstrup, Energy, 2004,29, 525. 29. E. Johannessen and S. Kjelstrup, Energy, 2004, in print.

CHAPTER 2

A Modelling Technique for Non-equilibrium Metallurgical Processes Applied to the LD Converter M. MODIGELL, A. TRAEBERT, P. MONHEIM AND K. HACK

1 Introduction In the LD converter process, pure oxygen is blown on a molten iron bath for refining purposes. Elements dissolved in the molten iron, e.g. C, Si, Mn, P, etc., but also part of the molten iron, are oxidised. They either form a slag phase covering the hot metal or, in the case of C, gas bubbles containing CO and CO,. Several reaction zones can be identified in Figure 1. In the hot spot, the oxygen directly reacts with iron and dissolved elements. Due to the impact of the oxygen jet, iron droplets are dispersed in the slag phase as well as slag droplets in the metal bath. The metul-slug dispersion is mixed further by CO and CO, bubbles and serves as the main reaction zone. A third zone contains the hot metal that is not dispersed in the slag, but forms the bath underneath. Droplets from the dispersion fall back into this bath.

2 Process Model Development The task of developing a suitable converter model will be discussed based on the decarburisation reaction of the iron melt. Obviously, the rate of the decarburisation reaction depends on the reaction rate of

and the transport conditions in the converter. Thereby, these are determined by oxygen blowing conditions and CO formation. On the other hand, CO formation is influenced by the decarburisation reactions.

A Modelling Technique for Non-equilibrium Metallurgical Processes

13

Slab

\

Figure 1 LD converterprocess

Intermediary formation of FeO in the hot spot is the main oxygen source for the decarburisation. FeO is dispersed in the slag and the metal bath as well as Fe-C droplets, which are accelerated by the oxygen jet. The subsequent reaction of FeO with carbon dissolved in iron occurs in both phases. In the slag phase, dissolved FeO reacts with dispersed Fe-C droplets, and in the metal bath FeO droplets form the disperse phase. For the decarburisation, several reaction routes can be formulated: (FeO) +[C]+ [Fe]+ { CO}

(2)

(FeO) + { CO} + [Fel+ { CO,}

(3)

Reaction (2) is kinetically limited.' The time for the complete reduction of an FeO droplet with a diameter of 1 mm (equalling 4.7X1Op5 mol) in an Fe-C melt is between 30 and 175 s. The same amount of FeO reacts with pure CO according to reaction (3) in only 0.2 s. The consecutive reaction (4) takes place in only 2x10-3 s. From these data, it can be concluded that the direct reaction between FeO and C contributes little to the decarburisation due to its kinetic limitation. However, reactions (3) and (4) can only occur spontaneously when CO, bubbles come in contact with Fe droplets in the slag phase or CO bubbles with FeO droplets in the metal phase, respectively. The probability for such contacts depends on the transport conditions in the slag and metal bath, respectively.

Chapter 2

14

In principle, the conditions close to the phase boundary of the droplets can be modelled by transport and reaction equations. In a simple two-layer model, a transport equation according to Fick’s law can be stated for every component x: nx=Kx,

elTA(cx,

metal-'^, phase boundary)

(5)

The transport coefficients Kx,effare functions of the macroscopic and the local microscopic fluid movement, which is caused by the stirring of the gas bubbles. Additionally, the increase of the diffusion boundary layer during the reaction due to the formation of a pure Fe phase has to be taken into consideration for all three reaction routes (2)-(4). For the reaction rate, the following holds:

However, there is no reliable information on transport coefficients, reaction rate constants and concentration ratios that are needed for solving Equations (5) and (6). Hence, a process model based on these equations is not promising. Instead, the process will be modelled on a less detailed level using a cell model, which is based on the concept of local equilibrium. As shown in Figure 2, according to the main reaction zones, the converter is divided into three sections. These are treated assuming complete (local) equilibrium: the hot spot, where the reaction between oxygen and iron melt takes place, the metal-slag zone, where the conversion of the FeO from the hot spot with melt droplets takes place, and the metal bath. The fourth zone, the slag, as shown in Figure 2, is needed for the temporal discretisation of the model. It is assumed that

+

Bath reactor

Figure 2 Cell model for the LD converter process

A Modelling Techniquefor Non-equilibrium Metallurgical Processes

15

the total amount of FeO formed in the hot spot is transported to the metal-slag zone, where it reacts with carbon-containing iron melt and already available slag. This assumption is reasonable as reactions taking place in the bath zone can be transferred to the metal-slag zone. The individual ideal reaction zones are interlinked by material streams in a way that a circular flow through these zones results. This flow models the circulation of the material in the converter and the stream of metal droplets through the slag phase as they have been observed in experimental studies. With a given oxygen blow rate and lance height, the circulation rate between bath reaction zone and hot spot reaction zone can be defined. This also determines the mass flow between the assumed reaction zones, for which a boundary condition can be stated. Assuming the experimentally proven full conversion of oxygen in the hot spot leads to

2nO25

mFe, Hot Spot

(7)

MFe

Between the bath and the metal-slag zone, mass transfer of Fe droplets takes place. During a major part of the process, the decarburisation reaction is limited by the oxygen supply. After reaching the critical point, it is limited by the carbon transport to the reaction zone. Therefore, at the critical point, we have

Nv

5 -co=hVco=

2n0

m ~ eM . etal-slag

CO

P Fe These mass flows are a function of the oxygen blow rate and conditions as well as gas production, which are constant over the main process time. Towards the end of the process, gas formation depends on carbon concentration and, therefore, the mass transfer conditions change. As the kinetic and transport limitations of the process are modelled by the interlink of the reaction zones, the zones themselves can be treated assuming thermochemical equilibrium. Although derived with regard to the decarburisation reaction, the present model is also valid for other chemical components with similar behaviour present in hot metal and slag. Processes determined by other kinetic phenomena, such as the melting of scrap and the dissolution of lime, need to be modelled separately. The dissolution of fluxes, especially lime, is limited by the formation of a dicalcium silicate layer around the particles for a major part of the process. This behaviour is mainly dependent on the composition and temperature of the slag and mixing between metal and slag.2 A dissolution function was incorporated into the model regulating lime participation in reactions leading to the formation of slag. With this function, the heating up of the lime prior to its dissolution and the withdrawal of energy from bath and slag due to this heating up are also treated. A similar problem exists for the addition of steel scrap. Scrap melting not only depends on temperature but also on carbon concentration. Diffusive carbon transport to the scrap/metal bath interface is therefore an important factor for melting behaviour. A function that treats the melting as well as the heating up of the scrap prior to melting was implemented. Not only material streams to and from ideal reaction zones are defined in the model, but also energy streams. Energy losses are calculated and appropriate 2

t’

16

Chapter 2

amounts of enthalpy are withdrawn from the ideal reaction zones throughout the simulation. Thus, radiation losses from the converter mouth as well as radiation and convection losses through the converter walls are considered. Furthermore, enthalpy losses due to the hot off-gas stream are directly taken into account in the reaction zone model as the hot off-gas stream is removed throughout the process. A temporal discretisation of the described model is needed for simulation purposes. According to a chosen time increment At and the determined mass flows between the ideal reaction zones, in every time step the mass exchange between the reaction zones and the reaction progress is calculated. Hence, a step-by-step transport and conversion of different mass streams results.

3 Modelling Tool The process modelling tool SimuSage is a combination of the thermochemical programmers library ChemApp3 and a library of additional graphical components for Borland’s programming environment Delphi. Initially developed by Mannesmann Demag Metallurgy as an in-house tool under the name ProMoSys, it is now codeveloped by GTT-Technologies and SMS-DEMAG under the new name SimuSage. It consists of flow-sheeting components, which add to Borland Delphi’s powerful programming language the capability to easily generate stand-alone flowsheeting models with a sound thermodynamic basis. The elements of the reaction zone model shown in Figure 2, for example material streams and equilibrium reactors, are readily available in SimuSage as graphic components. This has made it possible to set up the converter model and to test modifications and additions to the model very quickly. As indicated above, the ChemApp programmers library3 (in the form of a Dynamic Link Library, DLL) has been used to form the thermodynamic backbone of SimuSage. The general concept is represented in Figure 3. The ChemApp library permits the easy use of full complex equilibrium calculations within a software by way of a set of interface routines. These interface routines are used to define the conditions for a “local” equilibrium calculation, execute the equilibrium calculation and extract information from the calculated equilibrium state (such as phase amounts or phase internal concentrations) that is needed for the process model to proceed. The equilibrium calculations in ChemApp are performed by the same Gibbs energy minimisation code as in the well-known interactive software ChemSage,4 now F a ~ t S a g eand , ~ are thus of proven reliability. The thermodynamic data used in the present modelling have been taken from the extensive file store of GTT-Technologies. One hundred and nine phases and altogether 202 species have been included in the calculations. The gas phase (60 species) has been treated as ideal, while the liquid Fe phase (dilute solution approach, 14 species) and the liquid slag (Gaye-Kapoor-Frohberg model, eight species) have been treated as non-ideal chemical solutions.

17

A Modelling Technique for Non-equilibrium Metallurgical Processes

SimuSage program Process parameters

Results

I I I I I I I I

E

E

l

I

I I I

I I

I I I I

I I I I I I

Data handling and phase equilibrium calculation module

I I I I

Figure 3 Integration of ChemApp into SimuSage

4 Simulation Results For the simulation calculations, the following process conditions are assumed according to Asai and Muchi?

4

150 t converter oxygen blow rate of 10.87 kg O,/s for 0 to the 15th minute, and 7.99 kg O,/s from the 15th minute 5 t scrap input 10 t CaO input

From the experimental results6 shown in Figure 4 and the corresponding process conditions stated above, model parameters can be calculated. The lower limit for theexchange ratio between the bath and the hot spot reaction zone defined as mHotSpot/mtotdis calculated using Equation (7); it amounts to 0.023. The exact value for this exchange ratio is derived by comparing the measured and calculated decarburisation results. Other reactions, for example of Si, and the temperature development also have to be considered. From the comparison of experimental and calculated results, the exchange ratio between the bath and hot spot reaction zone is found to be 0.04. In Figure 4, the critical point, where the change from a constant decarburisation rate to a rate controlled by carbon concentration takes place, is found to be at 13.5 min at a carbon concentration of 0.006. From this, the exchange ratio between the bath and the metal-slag reaction zone defined as mMetalSlag/mtotal is calculated to be 0.7. The results of the calculation with these parameters are shown in Figure 5. The decarburisation reaction, the critical point and carbon content are modelled correctly,

Chapter 2

18 0.045 0.04 5 0.035

0

2

4

6

8 10 Time [min]

12

14

16

18

Figure 4 Carbon content in hot metal 0.0035 0.003

0 C-content of hot metal

Figure 5 Decarburisation rate

as can be seen from the calculation of the decarburisation rate. With the calculated exchange ratio between bath and metal-slag reaction zone, a critical carbon content of 0.006 results, as expected. As discussed above, Equation (8) is valid for process conditions up to the critical point. The decrease in decarburisation rate from the critical point onwards results in a decrease in CO production and, therefore, less stirring is caused by the gas bubbles. Hence, mass transfer in the form of metal droplets in the slag decreases. In Figure 6, this effect is taken into account and modelled by a decrease of the bath/metal-slag reaction zone exchange ratio from the critical point onwards. After reaching the critical point, the exchange ratio is set to 0.35. Obviously, the decrease in decarburisation rate can be modelled quite well with this set of parameters. The calculated carbon content of the metal phase shows good agreement with measured values.6 At the beginning of the process, the oxidation of Si from the hot spot is limited by the transport of Si to the phase boundary. Hence, decarburisation starts, although

19

A Modelling Technique for Non-equilibrium Metallurgical Processes 0.0005

0.045

0.00045

0.04 C-content calculated -0- Si-content calculated 4 Mn-content calculated

0.035 Y

g

4

0.0004

-A-

0.03

0.00035 0.0003

0.025

0.00025

0.02

0.0002

I

5 0.015 w

2

’g

0

0.00015

0.01

o.Ooo1

0.005

O.ooOo5

0

0

0

2

4

6

8 10 Time [min]

12

14

16

18

Figure 6 Components in hot metal

-

Simulation Results Components in Slag Phase 25000

1 -

CaO calculated Si02 calculated 4 MnO calculated 4-

0

5

10

15

Time [min]

Figure 7 Components in slag phase

normally Si is oxidised preferentially to carbon. The limitation of transport to the phase boundary is modelled by a limited mass flow through the hot spot reaction zone and, therefore, by a limited amount of Si in this reaction zone. The resulting excess oxygen reacts to form CO and FeO-rich slag. This slag facilitates the dissolution of lime and forms the main reaction zone for further reactions. The oxidation of Si, therefore, not only takes place in the hot spot but also in the metal-slag reaction zone. The amount of Si calculated in the metal phase shows good agreement with measured value^.^ Additionally, the content of Mn and 0 in the metal phase was calculated. Both components show qualitatively good agreement with measured valu e ~Mn . ~ is oxidised towards the end of the process, and the oxygen content of the metal phase increases. Figure 7 shows the composition of the slag phase, which shows qualitatively good agreement with measured values7 The FeO content rises towards the end of the process, resulting in iron losses.

Chapter 2

20

The calculated gas production is shown in Figure 8. The shift of the main reaction zone from the hot spot to the metal-slag dispersion in the course of the process is modelled well, as can be seen from the gas production decreasing in the hot spot while increasing in the metal-slag reaction zone. The decreasing gas production at the end of the process is caused by the change of the oxygen blow rate and the slower decarburisation. As discussed above, the ideal reaction zones in the cell model are not only coupled by defined mass transfer but also by heat transfer. Energy losses from the converter are calculated at every time step and enthalpy is withdrawn from various reaction zones, mainly from the metal-slag reaction zone. In Figure 9, the evolution of the temperature in the hot spot and of the mean temperature of the hot metal bath 1600

I

I

Hot spot

1400 1200

800 600 400 200

0 0

1.5

3

4.5

6

7.5

9

10.5 12 13.5 15 16.5 18

Time [min]

Figure 8 Gas production

2600 2400

-

2200

E -5 2000 2

8 1800 b

1600 1400 1200 0

5

10 Time [min]

Figure 9 Temperature evolution

15

A Modelling Techniquefor Non-equilibrium Metallurgical Processes

21

are shown. The temperature in the hot spot is well described. Measurements give values of 2300-2400 "Cfor the first 8 min of the process.8 The mean temperature shows very good quantitative agreement with the experimenk6

5 Conclusions The aim of the research presented in this paper is the evaluation of a technique for modelling metallurgical processes. In the long term, the process models developed using this technique will be used to improve process design. Special focus is set on modelling non-equilibrium phenomena that are caused e.g. by transport limitations or dissolution processes. The objective is to model the process with a relatively simple model structure and only a few model-specific parameters. The LD converter process was modelled according to this intention. Based on experimental results for the decarburisation reaction and possible reaction routes, a simple cell model that well describes the transport limitations of the decarburisation was developed. Simple relationships that define the mass transfer between the cells (reaction zones) and thus the model parameters were deduced. Only a slight further adjustment was needed. Energy losses from the process are calculated throughout the simulation. The resulting temperatures of the hot spot show good agreement with real process data. With respect to the reactions, the model not only describes the decarburisation but also the behaviour of other elements dissolved in metal and slag. The temperature of the metal bath could also be reproduced by the model. Altogether, these facts show that, although only a simple model structure was employed, the physical and chemical conditions in the converter process are handled correctly.

6 List of Symbols A Cx, Ci CO kX

Kx,efi mtotal mtotal mHot Spot mMetal-Slag

mBath m X

M

X

nx N N RX

area of phase boundary concentration of component x, i critical carbon concentration reaction rate constant transport coefficient -

mHot Spot

-k

mMetal-Slag

-

mBath

total mass flow rate through the converter mass flow rate through the hot spot mass flow rate through the metal-slag reaction zone mass flow rate through the bath mass flow rate of component x molar mass of component x molar flow rate of component x number of droplets droplet flow rate reaction rate

22

t’ V

[I 0 {I At

P

Chapter 2

droplet residence time in the slag volume of droplet dissolved in liquid iron dissolved in slag in gas phase time increment density

References 1. I. Barin, M. Modigell and F. Sauert, Metall, Trans. B, 1987, ISB, 347-354. 2. V. I. Baptizmanskii et al., Steel in the USSR, 1973,8, 634-638. 3. G. Eriksson, K. Hack and S. Petersen, ‘ChemApp - A Programmable Thermodynamic Calculation Interface’, in Werkstomoche ’96, Symposium 8: Simulation, Modellierung, Informationssysteme, J. Hirsch (ed), DGM InformationsgesellschaftVerlag, 1997. 4. G. Eriksson and K. Hack, Metall. Trans. B, 1990, 21B, 1013-1023. 5. C. W. Bale, P. Chartrand, S. A. Degterov, G. Eriksson, K. Hack, R. Ben Mahfoud, J. Melancon, A. D. Pelton and S . Petersen, CALPHAD, 2002,26(2), 189-228. 6. S . Asai and I. Muchi, Trans. ISU, 1970,10, 250-263. 7. A. I. van Hoorn, J. T. van Konyenburg and P.J. Kreyger, in Role of Slag in Basic Oxygen Steelmaking, W.-K. Lu (ed), McMaster University Press, Hamilton, Ontario, 1976. 8. K. Koch, W. Fix and P. Valentin, Archivfur das EisenhUttenwesen, 1976,47(10),583-588.

CHAPTER 3

Multiphase Thermodynamics of Pulp Suspensions PERTTI KOUKKARI, RISTO PAJARRE AND ERKKI e S m N

1 Introduction By the year 2000, the global production of paper and cardboard products was exceeding 300 Mt annually. In the 1990s, the average growth rate of the use of printing paper and related commodities was 3.2%. Although the emergence of new information and communication technology is expected to replace some of the traditional uses of paper, the development is likely to lead to new improved paper products, such as qualities suitable for digital printing. Trees and other fibrous plants also provide a renewable resource with a long industrial tradition of utilisation, which has also prepared the ground for future development of new fibre-based materials. Pulp fibres made of wood are the basic raw material of the present pulp and paper industry. In order to obtain a final product with desired properties, various inorganic and organic chemicals are added to pulp at different stages of the manufacturing process. The interactions between the pulp fibres and these added chemicals determine the properties and quality of the resulting product. They also have a major influence on the stability and economics of the industrial process and have a prevalent effect on the maintenance of the equipment by ruling the conditions of their wear and corrosion. Therefore, the chemistry of pulp suspensions has gained considerable interest during the past decades. The importance of a proper understanding of the chemical state of pulp suspensions has been further increased by the often environmentally driven changes in pulp- and paper-making processes since the 1990s: Reducing the amount of water used in processes has increased the amount of metal ions circulating with process water. Transition metals like manganese, iron and copper catalyse the decomposition of oxygen-containing bleaching chemicals that have replaced chlorine gas, while magnesium has been noted to have a beneficial effect during alkaline oxygen or hydrogen peroxide pulp bleaching. Thus, the control of metal distribution in bleaching suspensions has become an important

24

Chapter 3

industrial practice. Similarly, the increasing use of CaCO, as a filler and coating material in paper-making has made the proper understanding and controlling of calcium chemistry in fibre suspensions more important. A great deal of the relevant knowledge of fibre-ion interactions has been on a purely empirical basis; this paper discusses work based on general thermodynamic principles aimed at a better understanding of the chemical interactions on a more fundamental level and using that knowledge to improve the current pulp-and paper-making process.

2 Fibres in Aqueous Solution Pulp fibres are composed of cellulose microfibrils, and gel-like hemicelluloses and lignin. A schematic picture of pulp fibres in aqueous solution is shown in Figure 1. The water in the system is divided between two sub-volumes: the volume of water surrounding the fibres and the volume of water inside the fibre walls. The amount of water enclosed by the fibre walls depends on the type of pulp and to some extent on the chemical composition of the suspension liquid, typical values being around 1-1.5 Vkg-l of dry weight. The cellulose microfibrils form the backbone of the fibre, whereas hemicelluloses and lignin contain acidic groups that can dissociate giving the fibre an anionic charge. The chemical nature and amounts of these functional groups depend on the wood raw material and the processes used to extract the fibres from the wood and to bleach the fibres,' typical total amounts of these acids varying in the range of about 50-300 mmoVkg-'. In order to neutralise the negative charge, mobile cations are enriched in the part of solution inside the fibre walls. The difference between ionic concentrations between the fibre wall exterior and interior increases with increasing pH, as more anionic groups are formed from dissociating organic acids, and decreases with increasing ionic concentrations, when a smaller fraction of the total mobile ion amounts is needed to balance the charge. In addition to simple ionic solutes, the system contains organic macromolecules and colloidal substances (thermochemistry of which will be mostly beyond the scope of this discussion) and often inorganic or organic metal salts.

c1-

Na'

Ca2+

cr Na'

CIFiber wall

CI-

External suspension solution

Figure 1 Schematic presentation of a pulp fibre in an aqueous electrolyte solution

Multiphase Thermodynamics of Pulp Suspensions

25

3 Fibre-Ion Interactions The interactions between the charged fibres can be divided into ion non-specific and ion-specific parts. Ion non-specific part is the electrostatic interaction or repulsion between the ions and the charged matrix of the fibre wall. The strength of this interaction is assumed to depend only on the charge of the ionic species. The ionspecific part of the interaction contains the specific complexation of metal ions with the bound acidic groups and the adsoqtion of ions into polarised surfaces. By its very nature, the ion-specific interactiom can be used to describe a more complex pattern of observed phenomena than non-specific interactions, but they also require a larger number of individually fitted numerical parameters in a model.

3.1 Donnan Equilibrium Electrochemical systems consisting of two parts where one or more charged species is restricted to one part of the system by a semipermeable membrane were first investigated by Donnan in the early twentieth century.2In such systems, the equilibrium condition for all solute species is that their electrochemical potentials are equal on both sides of the membrane: p;+RT In a[+z,Fy' = p;+RT In a;.'+z,Fy"

(1)

From Equation (1) the relation for the activities of ionic solute species on both sides is

where A is called the distribution coefficient. If the immobile ions are located on the side marked with superscript" and are negatively charged, the potential difference in Equation (2) will be positive and the coefficient k l . Mobile cations will be enriched on the side of the bound ions, whereas mobile anions will show the opposite behaviour. Shortly after its initial publication, the Donnan theory was applied to describe polyelectrolyte systems where anionic groups were confined to one part of an aqueous solution by chemical bonds instead of a semipermeable membrane.3 The theory was used by Neale ( 1929)4and Farrar and Neale ( 1952)5to characterise electrolyte interactions with cellulose fibres. In 1996 Towers and Scallan6published their mathematical model, based on the Donnan theory, which could be used to calculate the ionic distribution of mixtures of mono and divalent cations and monovalent anions in pulp suspensions. In their model equation (2) (assuming ideal solutions, so that the activites could be equated with corresponding concentrations), together with charge neutrality conditions for the two aqueous phases, equilibrium coefficients for the bound acidic groups and mass balances were used to derive an equation for A at a given pH, cation and acid amounts and water volumes. It was assumed that anionic groups formed from the bound acids did not form any specific complexes with any cations except the hydrogen ion. A good agreement could be found between the experimental and calculated distributions for Na', Ca2+,Mg2+ and Mn2+ ions in

Chapter 3 0.9 0.8 0.7

0.0

;

I

3

4

5

6

1

7

8

9

1

0

pH of external solution

Figure 2 Fraction of the total metal content present in the external solution in a pulp sample treated with chelating agent DTPA, redrawn based on the data in ref. 1

pulp suspensions of 1%consistency over a pH range of 2 to 10. Later, the model was extended to describe multivalent anions, hydroxyl complexes and chelating agents. l v 7 An example of modelling results in such systems is given in Figure 2. The formation of complexes of metal ions with hydroxide ions and chelating agents can strongly affect the cation distribution as negatively charged complexes will be repelled by charged fibres. The Donnan theory has also been used to determine the distribution of dissolved Cu2+and A13+ and it has been shown to be useful in describing the removal of anionic macromolecular components during pulp washing. lo. It has also been combined with models for displacement flow of aqueous solution through an immobile fibre network and dissolution and ion exchange kinetics to construct a model for pulp washing and chelation processes.' While most of the experimental work in the area has been carried out with pulps with 1-2% consistency, the Donnan theory has been successfully applied to suspensions up to 10%consistency.loJ1It is still to be expected that in industrial applications with increasing consistency of the pulp, for example in pulp washing and bleaching operations, the increasing inhomogeneity of the suspension will make the combination of the equilibrium model with suitable mass transfer kinetics' more and more important.

3.2 Specific Interactions Because of its simplicity, the concept of Donnan equilibrium is an attractive approach for the modelling of fibre-ion interactions. However, despite its general success in many cases, the Donnan theory alone is not sufficient to describe the experimentally observed distributions. In those cases, it is necessary to use ionspecific complexation equilibria with the acidic groups as a part of the model to describe the observed phenomena satisfactorily. Such behaviour has been shown to exist, for example, with chemically modified highly charged fibres.l 2 An apparent systematic deviation in the ionic distribution from the predictions of the Donnan theory has also been noted with common oxygen-delignified haft pulps,13 partly, but

Multiphase Thermodynamics of Pulp Suspensions

27

not entirely, explicable with the assumption of ideality of the aqueous solutions used with most Donnan-based models.

4 Solid and Gaseous Phases Not all the metal ions present in pulp suspensions are fully dissolved. Experiments with oxygen delignified haft pulps have demonstrated that Ca2+and Mg2+ are present in them predominantly as carbonate precipitates." CaCO, can also be used in paper-making as a filler or pigment. Under other conditions, when the amounts of carbonates in the system are less, the hydroxides are expected to dominate the species.14 Other solids that are known to form in various parts of pulp-and papermaking processes include calcium sulfates and oxalates, various silicates and aluminium salts. In addition to simple stoichiometric salts, Mn2+ and Fe2+ are known to co-precipitate with Mg2+ in pulp suspensions, the phenomenon having practical importance because of the detrimental effect of dissolved divalent manganese and iron on oxygen-based bleaching chemicals. Mn2+has been shown to form solid solution hydroxides with Mg2+ and to co-precipitate on MgCO, surfaces. l4 Dissolution kinetics of any solid precipitate should also be taken into account when modelling the chemical state of a pulp suspension system;' the stability of solid precipitates making it harder to quantitatively model the chemistry of ions such as Cu2+ and Fe3+. Of the various possible gaseous species, the most important is CO,. While the effects of its dissolution or release are difficult to model under actual process conditions, the gas-aqueous equilibrium can be shown to influence significantly the observed pH in laboratory experiment^.^^

5 General Gibbs Energy Model Computer programs that calculate the equilibrium state of a system for given initial composition based on the minimisation of the overall Gibbs energy of the system have been in use for several decades. They offer the possibility of easily carrying out calculations with fast and robust codes with chemical systems containing a large selection of possible phases, including non-ideal mixtures. A thermodynamic description of a pulp suspension system, including the Donnan equilibrium, that is compatible with an existing commercial thermodynamic software has been developed.16A description of the structure of the multiphase model of a pulp suspension is given in Table 1. As in a thermodynamic system description used for a normal solubility equilibrium calculation, the system contains a gas phase, if considered relevant for the problem at hand, an aqueous solution phase (external to the fibres), and a number of solid phases, which appear either with fixed stoichiometry or as solid solutions. The fibres are described as a separate aqueous phase. The thermodynamic data and stoichiometry for the solute species inside the fibre phase are identical to those describing the species in the external solution volume, with the exception that the charge of the species in the two aqueous phases must be defined separately. This will ensure that, given valid input values, charge neutrality will apply to both aqueous phases individually in the equilibrium composition calculated by Gibbs energy

28

Chapter 3

Table 1 Structure of an example thermodynamic multiphase system 0 H N C Na Cl Ca Mn EDTA H,Ojb, Acid,

e-

e,&

Gas

External solution H2O 1 2 H+ 1 OH1 1 Na+

-1 1 1

c1-

-1 1

c0;-

3

HCO,

3 2

co2

Mn2+ H4EDTA H,EDTAH2EDTA2H 1 EDTA3EDTA4MnEDTA2-

^

^

1

1 1

1 1

^

4 3

2 1 1

Fibres H2O H+ OHNa+

-2 1 1 1 1 1 1

1 2 3 4 2 1

1

1 1

-1

1 1

Ca2+

1

c0:-

3 3

HCO;

co2

1

2

Mn2+ H,EDTA H,EDTAH,EDTA2HlEDTA3EDTA4MnEDTA2Acid,Acid,H "

1 1 1 1

4 3 2

1 1

"

1 1

3

2

2

1 1 1 1 1 1 1 1

1 3

-1 1

c1-

Solids CaCO, MnCO, Mn(OH),

1 -2 2 1

1

Ca2+

1

1 1

1

-1

-1

1 -2 2 1

1 -2 2 1

-2

-2

1

1

2 3

2 3

4

4

2 1

2 1

Multiphuse Thermodynamics of Pulp Suspensions

29

minimisation. It should also be noted that the program used to run the calculations does not need to be coded to use electrochemical potentials (Equation (1)) or to explicitly consider the Donnan distribution (Equation ( 2 ) )while carrying out the calculations. Rather, since the Donnan equilibrium theory can be derived from the principle of charge neutrality of any macroscopic part of a thermodynamic system, the correct equilibrium composition that is consistent with the Donnan theory will be achieved using normal formulas for chemical potential without the electrostatic term. In a calculated equilibrium state, the following equality is valid for any solute species (superscript s denotes the aqueous solution external to the fibres and superscript f denotes the solution inside the fibre walls): p; +RT In u: =,up +RT In af-zjnc

(3)

where n stands for the chemical potential of the component differentiating the unit charge in the fibre phase from the charge in the external solution. Comparing Equations (1) and (2) with Equation (3) the following two equalities can be obtained:

A=..pi -+j Distribution coefficient A is also readily evaluated from any calculated equilibrium composition using Equation ( 2 ) with the calculated activities of any mobile ion in the system. Another necessary modification of the fibre phase data is related to the fact that the amount of water inside the fibres cannot be allowed to vary freely during calculations. The simplest solution is to keep its amount constant. This is not precisely true as the swelling of fibres is affected both by pH and the ionic strength of the solution. However, the calculated ionic distribution is not particularly sensitive to moderate changes in fibre phase volume and the same assumption of a fixed amount of fibre phase water has been commonly made with other Donnan theory-based models as well. Unlike the mobile ions and neutral solutes, the bound acidic groups are included only in the fibre phase. As no reactions that would change the total amount of these groups are assumed to take place, the chemical potential of the undissociated forms of these groups can be set to zero, while the chemical potential of the anionic forms can be calculated based on the thermodynamic relation

AGj=-RT In K j where the acidic dissociation constants and the corresponding molar amounts can be determined experimentally by potentiometric or conductometric titration. In a similar manner, species corresponding to explicit association between bound anionic groups and metal cations can be included in the model if experimental results show that the non-specific Donnan equilibrium alone is not sufficient to explain the ionic distribution with the fibre type in question. The multiphase model can be used to cover various fibre types or mixtures, once their acid-base characteristics (dissociation constants Ki)have been determined by titration. The model is independent of the valences of both anions and cations in the

Chapter 3

30

system. All such solute species, whose chemical potentials in terms of their mutual concentrations are known, can be included. Figure 3 shows the pH-dependence of Ca2+concentration in bleached eucalyptus sulfate pulp (ESA) and thermomechanical softwood pulp (TMP) suspensions. The pH adjustment has been performed with sulfuric acid and potassium hydroxide (ESA) and with carbon dioxide (TMP). The respective data input is shown in Table 2. The multiphase thermodynamic model described above has been implemented as a computational tool using the ChemSheet thermodynamic software18 (part of the widely used Solgasmix/ChemSage/ChemAppprogram family) that supports the use of the Pitzer formalism in describing ionic interactions in solutions and a number non-ideal solution models for solid phases. Despite the fact that the thermodynamic ESA/CaC03/H2S04 0.6

1

0.5

-

E 0.4

-

m

2 0.3

A

-

E

3

0.2 0.1

0

2

6

4

8

10

12

TMP/Ca2+/C02

0.8 I 0.7 0.6 -

0.5 0.4 0.3

-

0.1 -

0.2

0 : 0

'.*. ,,,',--.

I

I

I

I

I

2

4

6

8

I

10

12

PH

A [Cals measured

Figure 3 Calcium concentrations in the suspension of eucalyptus sulfate (ESA) and t h e m mechanical sofrwood (TMP) pulps. Experimental data are @om re$ I7

Multiphase Thermodynamicsof Pulp Suspensions

31

Table 2 Data input for the ESA and TMP models Pulp consistency

ESA

TMP

(%) 1 1

Water Acidic group content CaCO, Ca(OH), content of the jibre wall pK=3.6 pK=4.1 pK=9.1 (dm3kg-') (moVkg-') (moVkg-l) (moVkg-') (moVkg-') (rnoL4kg-l) 1 0.067 0.05 1.5 0.162 0.068 0.1 15

data used in calculations could still be refined in many ways and that the current knowledge of the relevant kinetically controlled processes is rather limited, the model has already been found to be useful in process design in the paper industry.19

6 Discussion The present thermochemical model describes the acid-base, ion exchange and solubility characteristics of a homogeneous pulp suspension. The important feature of the thermodynamic multiphase approach is that it provides the possibility to incorporate specific interactions of practically unlimited number of constituents into the system. Due to its general thermodynamic basis, the multiphase method can be applied both in the fibre line processes in pulp production and in the wet end chemistry of paper-making. The thermodynamic method is used to control the formation or dissolution of precipitates in different process stages, to model pulp washing efficiencies as a function of pH and to determine pH and alkalinity in the wet end suspensions of paper machines. An illustrative application is the utilisation of carbon dioxide in pH-control of both fibre line and paper machine processes. The practical paper machine stock system consists of the aqueous solution, a mixed fibre phase of different pulps together with some recycled suspension, which contains both fibres and calcium compounds. Solid CaCO, is introduced as a filler for opacifying and callipering the sheet. Here, the conventional pH-control with sulfuric acid has been extensively replaced with the more 'gentle' treatment with gaseous carbon dioxide or with a nonstoichiometric mixture of CO, and sodium hydroxide solution.20The latter forms an effective pH-buffer, and prevents the unwanted dissolution of solid calcium carbonate. The dosage of the buffering mixture is determined with the multiphase calculations to maintain the process pH within k0.2 pH-units. Other paper machine applications include the calculation of the impact of fresh water reduction on pH an free calcium levels of the stock suspension and estimation of wet end process conditions in systems including both sulfate and carbonate species. The multiphase method provides a practical 'screening tool' for industrial process research and development, even though under many circumstances the nonequilibrium effects such as supersaturation of solutions, retarded mass transfer or reaction kinetics and inhomogeneity of suspensions limit the applicability of the thermodynamic calculations. When the thermodynamic multiphase models are developed towards process simulation tools, one should incorporate such methods that include the effects of these non-equilibrium factors. They must be based on

32

Chapter 3

laboratory and process information and by necessity involve data from kinetic measurements. The approach towards the equilibrium state still works as a guiding principle, and in several cases the equilibrium state can be considered to be at least a useful approximation of the true state of the system. The thermodynamic method provides a quantitative basis for the acid-base, ion-exchange and solubility behaviour of the fibre stock suspensions. Many important measurable process indicators, such as concentrations, pH, alkalinity and Cahardness, can be directly calculated using the thermodynamic treatment, while additional information, concerning, for example, conductivity and colloidal stability, can be deduced from the simulated chemical state with the help of additional physicochemical and empirical relations.

References 1. E. Rasanen, Modelling Ion Exchange and Flow in Pulp Suspensions, Doctoral Thesis, Helsinki University of Technology,V?T Publications 495, Espoo, Finland, 2003. 2. F. G . Donnan and A. B. Harris, J. Chem. SOC., 191 1,99, 1554. 3. H. R. Proctor and J. A. Wilson, J. Chem. SOC., 1916, 109, 307. 4. S. M. Neale, J. Textile Znst., 1929, 20, 373. 5. J. Farrar and S. M. Neale, J. Colloid. Sci., 1952, 7 , 186. 6. M. Towers and A. M. Scallan, J. Pulp Pap. Sci., 1996,22, 332. 7. E. Rasanen, P. Stenius and P. Tervola, Nord. Pulp Pap. Res. J., 2001, 16, 130. 8. J. Lindgren, L. Wiklund and L.-0. Ohman, Nord. Pulp Pap. Res. J., 2001, 16, 24. 9. J. Lindgren, Experimental Studies of the Acimase Properties and Metal Ion Affinities of Wood Fibres, Doctoral Thesis, Umet University, Dept. Chemistry, Inorganic Chemistry, Umei, Sweden, 2001. 10. R. Anderson, J. LidCn and L.-0. Ohman, 7th International Conference on New Available Technologies, Stockholm, Sweden, 2002,42. 11. C. Sjolander, J. LidCn and L.-0. Ohman, International Pulp Bleaching Conference, Halifax, Canada, Oral presentations, 2000, 273. 12. J. Lindgren, P. Person and L.-0. Ohman, Nord. Pulp Pap. Res. J., 2001,16, 225. 13. K. Athley and P Ulmgren, Nord. Pulp Pap. Res. J., 2001, 16,204. 14. J. Liden, L. Wiklund and L.-0. Ohman, 11th International Symposium on Wood and Pulping Chemistry, Vol 1, Nice, France, 2001, 1, 193. 15. P. Koukkari, R. Pajarre and E. Rasiinen, 12th International Symposium on Wood and Pulping Chemistry, Madison, Wisconsin, 2003,3, 59. 16. P. Koukkari, R. Pajarre and H. Pakarinen, H., J. Solution Chem., 2002,31,627. 17. H. Maenpaa, ‘Paperikoneen kuituseosten ioninvaihtomallit’ (‘Ion-exchange models of the pulp mixtures at paper machines’, in Finnish), Masters Thesis, Helsinki University of Technology, Laboratory of Forest Products Chemistry, Espoo, Finland, 2001. 18. P. Koukkari, K. Penttila, K. Hack and S . Petersen, in Microstructures, Mechanical Properties and Processes, Euromat 99, Vol 3. Y. Brechet (ed), Wiley-VCH Publishers, Weinheim, 2000,323. 19. K. Viitikko and A. Kalliola, Wet End Technology, Scandinavian Paper Symposium, Helsinki, Finland, 2002. 20. A. Weaver, A. Kalliola and P. Koukkari, Third Major Pira International Conference of Scient$c & Technical Advances in Wet End Chemistry, Vienna, Austria, 2002.

CHAPTER 4

Reactive Distillation HARTMUT G. SCHOENMAKERS AND WOLFGANG ARLT

1 Introduction Reactive distillation, as the name implies, refers to a distillation process that incorporates a reaction and a separation step within a distillation column. The technique offers a key opportunity for improving the structure of a process.'V2 It is a so-called hybrid process, i.e. it merges two different unit operations in a single apparatus, namely reaction and distillation. But the combination of distillation and reactions is possible only if the conditions of both unit operations can be combined. This means that the reactions have to show reasonable data for conversions at pressure and temperature levels that are compatible with distillation conditions. Because of the limited hold-up in distillation column, those reactions having a conversion half-time of 10-30 min are preferred. So, the judicious use of the chemical equilibrium constant is the basis for the design of reactive distillation processes. The advantages of reactive distillation as compared to a reaction plus a separate distillation process are: a reduction of the investment, a simpler process, the use of the heat of reaction (if present) in situ, ease of control of the reaction temperature (evaporating system), and the possibility of overcoming azeotropes. A typical reactive distillation set-up is depicted in Figure 1. The high boiling reactant is fed as feed 1 and the low boiling reactant as feed 2. Between the two feeds, there is the reaction zone. As a special application, feed 1 can serve as an extractive agent, e.g. in the case of the production of methyl acetate, acetic acid serves as an entrainer for the binary azeotropic mixture methanol and methylacetate. The ensemble is then a reactive extractive distillation column. The low boiling product leaves the column at the top and the section between feed 1 and the reflux serves as a rectifying section. The high boiling product is withdrawn

34

Chapter 4

1

Feed 1 Catalyst

e distillation ZOI

Feed 2

Product 2

Figure 1 The basic elements of a reactive distillation column L

ca

g

40 35

1

2 30 '=0 8

25 20

2 15

.c

O

ti z

10

2 i13m.lllllllllllllllllllllllln 1965

1970

1975

1980

1985

1990

1995

year

Figure 2 A summary of the publications in the field of extractive distillation between I965 and 1995

at the bottom; the section between feed 2 and the evaporator is known as the stripping section. Reactive distillation is becoming increasingly popular and in the three decades from 1965 to 1995, there was a significant increase in the number of papers published in the literat~re.~ The figures of BeBling3are graphed in Figure 2.

2 Thermodynamics As in most separation processes, thermodynamics play a key role in designing reactive distillation equipment. The process is governed, on the one hand, by the thermodynamic properties related to separation by distillation and on the other hand, by the equilibrium constants related to the reactions.

2.1 Phase Equilibrium The general condition for phase equilibrium is the equality of chemical potentials. For practical reasons this is replaced by the equality of the fugacity f of each of the

35

Reactive Distillation

compounds in the liquid (L) and vapour (V) phase:

One of the two ways to describe the fugacity is by using activity coefficients yin the liquid phase and fugacity coefficients q in the vapour phase. Activity coefficients require the definition of a standard state. In distillation, the two terms are related by XiXJ+

= yiqivp

where + refers to the standard state, xi and yj are the mole fractions of the liquid and vapour phases, P is the pressure of the system and q is the fugacity coefficient. The standard state is chosen to be the pure substance at the temperature and pressure of the mixture. Pkv is the vapour pressure of the pure compound (subscript ‘0’)and is usually given in the form of an Antoine-type equation. The exponential expression relates to the change in going from the vapour pressure to the pressure of the system as a function of the pure liquid volume. In many cases, the difference of pressures is small enough such that the exponential expression tends to be unity:

The fugacity coefficients of the pure substance in Equation (3) and of the mixture in Equation (2) are of similar value for many compounds, so they often cancel. However, for organic and inorganic acids (very often feed 2 in Figure 1) and some other compounds that involve the formation of dimer and higher n-mers formation in the vapour phase, the fugacity coefficient must be taken into account. Activity coefficients can be derived from databank^,^ correlation model^,^ structure interpolation methods6 or a priori Modern methods allow the calculation of multicomponent behaviour on the basis of binary systems. In this case, only binary data are needed. Very often, however, experimental data for the binary mixture that forms the reactants are unknown. As an example let us consider the equation: A+B

H

C+D

(4)

described in Figure 3. Phase equilibrium data of the binary pair A,B will be rare. This is because the reaction makes it most difficult to measure the phase equilibrium. A comprehensive discussion is given by Ti~chmeyer.~ Only in the case when the reaction rate is slow is a fast determination of the phase equilibrium possible with a standard apparatus. In many other cases, special equipment must be used. A flow apparatus enables to resolve the reaction time as a reaction locus. So, an appropriate flow will make it possible to make the measurements. Roederer et aZ.I0 investigated an isomeric reaction (alpha- and beta-isophorone) at 388,423 and 493 K. They experimented in a stirred batch reactor in the form of

36

Chapter 4 D

A C

B

Figure 3 Concentration space of the reaction A +B H C+ D

a closed-loop reactor. For VLE experiments it was possible to use the experimental set-up as a flow-type cell. The compositions of the phases are obtained on-line with a Fourier-transform-infrared spectrometer. Alsmeyer et a2.l have set up a flowthrough apparatus for VLE measurements in reactive systems that also allows determination of reaction kinetics in the liquid phase. The reactive mixture is forced into a steady state that does not represent its chemical equilibrium but reaches phase equilibrium between vapour and liquid. Both phases are then analysed in situ by mid-infrared (MIR) spectroscopy. While the gas phase is measured by transmission, attenuated total reflection (ATR) is used for the liquid phase measurements. Another flow-type apparatus has been operated by Reichl et a2.12It was set up as a non-recycle flow still and, due to the short residence time, it was used to determine isobaric VLE data of thermally unstable components and of reactive mixtures. Two esterification systems (methyl formate and ethyl acetate) and one etherification system (tert-amyl methyl ether) were investigated. Arlt13 proposed a new type of static still. The temperature and pressure are recorded at a given binary start mixture over time. By extrapolation to the zero time, pressure and temperature for this point can be derived. The method works if the reaction half-times are longer than 10 min, which is a prerequisite for the use of reactive distillation columns (see above). The binary data of the mentioned apparatus are used to determine binary interaction parameters of @-models or of equations of state.

2.2 Chemical Equilibrium In formulating and understanding the problems inherent in reactive distillation, it is necessary to take the chemical equilibrium into account. Starting from the fundamental equation of thermodynamics for one component:

Equation (6) can derived. The free enthalpy G, which should reach a minimum within the field of the fundamental variables P,T and compositions, is related to the

37

Reactive Distillation

chemical potentials p of the species i. Here, the moles n of the compounds in the reaction are replaced by the variable 1that uses the information of the reaction kinetics, namely the stoichiometric coefficients v:

dnA - dnB - dnC - &D dL= vA

vB

'C

D '

The expression in Equation (6) can be calculated using the model of activity coefficients yor of the fugacityf. In terms of activity coefficients, the following equation can be derived: xixf&

vi

K=l$$)vi=n(,,)

i

The product of activity coefficients yand the mole fraction x is often called activity a . It is noteworthy that the chemical equilibrium constant can only be calculated if the acitivity coefficients are known. In the case of reactive distillation, this information is available. It should be mentioned that the equilibrium constant can be calculated from the pure free enthalpies of formation that have to be corrected to the reaction state depending on the model (activity coefficients or fugacity) used. It is further important to know that the kinetics must also be formulated in activity coefficients. too. The following equation gives the scheme for the above-mentioned equilibrium reaction:

The k's refer to rate constants. If the reaction kinetics are expressed in concentrations only, an ideal solution is assumed. Furthermore, under chemical equilibrium conditions the two reaction rates (to products and from products) should be the same. Because the chemical equilibrium constant K must be expressed in activity coefficients, an inconsistency might occur. In general, the illustration in Figure 3 is not suitable to present data in a plane area. Several proposals have been made by (a) Zharov, (b) Barbosa and Doherty and (c) Ung and Doherty14and Espinosa et aZ.15The latter will be used here. For a given concentration y the following rule is applied to calculate the new coordinate Y in reference to the component k:

38

Chapter 4

3 Technical Application of Reactive Distillation 3.1 Introduction In reactive distillation, the type of the catalysis is important. Homogeneous catalysis is possible in most cases but needs a separation step to recycle the catalyst. This can be avoided in heterogeneous catalysis, but here special constructions are necessary to fix the catalyst in the reaction zone. If everything harmonizes, considerable advantages arise as can be seen with reference to the Eastman-Kodak Chemicals process for the production of methyl acetate. As can be seen in Figure 4 only one column is needed if reactive distillation is used as opposed to nine and a reactor if it is not used. Is this an exception or does it stand for a type of process with a comparable potential for improvement? The questions are:

- how can processes be identified that exhibit potential for reactive distillation, - how can such a process be developed and - how can the equipment be designed?

Some answers to this question will be offered from the viewpoint of an industrial process development by simulation, experimental validation and the choice and design of suitable equipment.

3.2 Process Synthesis A vital aim of process synthesis is to reduce the complexity of the problem in order to enable simple solutions to be recognized quickly. In order to systematically analyse processes involving reversible reactions, a comprehensive process synthesis strategy has

-TI-

er

1

SOlVenI

I

water

catalyst

Figure 4 Production of methyl acetate at Eastmun-Kodakwithhithout reactive distillation

Reactive Distillation

39

been developed. One element of this strategy is the analysis of reactive distillation lines that enable the feasibility of reactive distillation processes to be examined simply. To achieve this, various authors have developed transformation m e t h o d ~ . ~ J ~ ~ ~ These transformation methods enable reactive distillation to be described by a system of equations that is known from conventional distillation; see Equation (9). The method is limited by the fact that it is possible only for equilibrium reactions and is more an estimation than a design.

3.3 Process Design and Optimization While process synthesis gives qualitative reference points, for industrial implementation we need quantitative results. Therefore, tools for rigorous process simulation including all effects are needed. In practice, the application of staged models with increasing complexity can be recommended but tools with this complexity are not yet on the market, so that in many cases reactive distillation cannot be simulated so far that a process design is possible without experiments.

3.4 Limitations of the Methods for Synthesis and Design, the Scale-Up Problem Another important question is how main and secondary reactions are described. In practice, it is frequently the case, for time and cost reasons, that a complete kinetic model cannot be developed and experiments in a reaction column are needed to verify the achievable purities and yields. With the results of these experiments, a scaleup problem exists. This is because in these experiments, reaction and mass transfer interact with one another. The product removal capability has to be designed with respect to separation performance and the property of converting the starting materials that show the characteristics of a reactor performance with different scale-up rules. If an engineer has to design an industrial column purely on the basis of mini-plant experiments, he has to maintain not only the separation performance but also the ratio of separation performance/reactor performance so that main and secondary reactions proceed to a comparable extent in the industrial-scale reaction column. One way in which this can be achieved is in terms of construction by separating reaction and product separation from one another both in mini-plant tests and on an industrial scale. This is possible, for example, using a heterogeneous catalyst in the downcomer or with side reactors on the column. An alternative is to use a structured packing with well-defined paths for the liquid flow. In general, the problem has not yet been solved, the main reason being the lack of reference columns on an industrial scale. The way we see is either: - to develop scale-up methods or - to carry out reactive distillation experiments on an industrial scale.

The latter possibility, which represents a step back into the process engineering stone age in the design of columns, will in many cases mean the end of the line for

Chapter 4

40

reactive distillation, since a company will only seldom be prepared, for time and cost reasons, to build integrated experimental plants on an industrial or semi-industrial scale. Thus, developing an appropriate scale-up procedure for reactive column internals is a key task.

3.5 Choice of Equipment Up to now it was assumed that reaction and distillation can favourably be combined in a column - in a normal distillation column in the case of homogeneous catalysis, and in a column with special internals or an additional exterior volume in the case of heterogeneous catalysis. This was discussed in the previous chapter under the aspect of scale-up in connection with separation and reaction performance. However, columns are an appropriate solution only for reactions that are so fast as to achieve considerable conversions in the residence time range of such columns. The question is whether the full potential for combining reaction and distillation can be found and industrially implemented using columns only. Thus two operating conditions can be distinguished: The range in which the conversion is influenced mainly by the concentration of the component to be separated; this range is called ‘controlled by distillation’. - The range in which the conversion is influenced mainly by the residence time and the reaction constant; this range is called ‘controlled by kinetics’. -

Industrial process design should aim at operating conditions within these two ranges: just the sufficient residence time and only the necessary expenditure for the distillation. So we have a second scale-up problem: what is the suitable equipment for complying with these demands? This means more precisely: how can the reactor performance be achieved over a broad range of reaction velocities? Different equipment may be chosen to combine reaction and distillation within the limiting conditions of reaction velocity, relative volatility and catalysis. Equipment in question are: - vessels stirred or with fixed beds, - cascades of vessels, both with or without columns and -

reaction columns.

Additional volume can be provided for all of them. The next considerations concentrate on homogeneous catalysis. Similar considerations apply to heterogeneous catalysis, and this will be commented on later. At first, a slow reaction is considered. ‘Slow’ means that the reaction rate is slow, compared with the residence times typical for separation equipment such as distillation trays. For residence time reasons a stirred vessel or, better, a cascade of stirred vessels is needed. Each vessel is supplied with energy to evaporate the component to be separated. If the relative volatility of this component is very high, a one-stage evaporation is sufficient. At a lower relative volatility, the separation

Reactive Distillation

41

Figure 5 Principles of the choice of equipment (homogeneous catalysis)

requires more stages, i.e. a column has to be placed on top of the vessel. If the separation is even more difficult, a stripping section must be added to the column and a reboiler is necessary. In the limiting case of a very low relative volatility, each stage of the cascade can be operated as a countercurrent stage. The first stage is additionally provided with a fractionating column to enrich the component to be separated. Such a set-up is equivalent to a reaction column with a large holdup on each reaction stage. Fast reactions do not demand long residence times. In this context ‘fast’ means that the reaction reaches equilibrium in the residence time range that is typical for column internals. The equipment therefore may be selected under the aspect of separation efficiency. The procedure to select the equipment is comparable to the procedure described above and leads to a list of possible options. As a result of such considerations a chart of process alternatives can be drawn2I for homogeneous catalysis - see Figure 5. As mentioned earlier, heterogeneous catalysis can also be treated in a similar manner. There is, however, one main difference and that is that additional, separate reaction volumes are necessary to retain the catalyst. These volumes can be arranged either within the equipment or in a side position,22coupled by recycle systems.

4 Conclusions In the case of conventional distillation, a sufficient body of knowledge has, thanks to intensive development, been acquired to enable many tasks in process synthesis, design and scale-up to be solved quickly.23For reactive distillation, the methods of design and synthesis have been developed largely with the aid of many analogies to A major focus of research and development in future years should be the scale-up of reaction columns. This is where great deficiencies still remain. In addition, the methodology in choosing the best equipment will have to be improved. In the expert community, columns are often seen as the only possible choice. Work will have to be carried out to ensure that the benefit of combining reaction and distillation can be enjoyed to the full by employing the most suitable equipment.

42

Chapter 4

References 1. M. Doherty, F. Michael and G. Buzad, Trans. Znst. Chem. Eng.,1992,70,448. 2. J. L. DeGarmo, V. N. Parulekar, V. Pinjala, Chem. Eng. Progr., March 1992, 88 (3), 43. 3. Befiling, B., Ph.D.Thesis (Dissertation), Universitat Dortmund, 1998. 4. J. Gmehling, U. Onken, W. Arlt, R. Grenzheuser, B. Kolbe, U. Weidlich, and J. R. Rarey, Vapor-Liquid Equilibrium Data Collection, Vol. I, DECHEMA Chemistry Data Series, Frankfurt, 1977. 5. H. Renon and J. M. Prausnitz, and A.1.C.h.E. J., 1968, 14 (l), 135-144. 6. J. Gmehling, R. Wittig, J. Lohmann and R. Joh, Znd.Eng.Chem.Res., 2001, 40, 5831-5838. 7. A. Klamt and G. Schiiiirmann, J. Chem. SOC.Perkin Trans., 1993,2,799-805. 8. I. Clausen, W. Arlt, Chem.-1ng.-Tech., 2000,72,727. 9. M. Tischmeyer, Thesis, Erlangen, 2004. 10. A. Roederer, Ph.D. Thesis, ETH Zurich, No. 13169, 1999. 11. F. Alsmeyer, Ph.D. Thesis, Aachen 2003. F. Alsmeyer, W. Marquardt and G. Olf, Fluid Phase Equilibr., 2002, 203, 3 1-5 1. 12. A. Reichl, U. Daiminger, A. Schmidt, M. Davies, U. Hoffmann, C. Brinkmeier, C. Reder and W. Marquardt, Fluid Phase Equilibr., 1998, 153 (l), 113-134. 13. W. Arlt, Fluid Phase Equilibr., 1999, 158-160, 973-977. 14. S. Ung and M.F. Doherty, Chem. Eng. Sci., 1995,50,2348. 15. J. Espinosa,; P. A. Aguirre and G. A. Perez, Chem.-hg.-Tech., 1995, 43, 541-550. 16. B. Bessling, J.M. Lohning, A. Ohligschlager, G. Schembeckerand K. Sundmacher,Chem. Eng. Technol., 1998, 21, 393400. 17. D. Barbosa and M. Doherty, Proc. Roy. SOC. Lond., 1987, A 413,459. 18. J. Stichlmair, ChemJng. Tech., 1988,60, 747-754. 19. J. Stichlmair, H. Offers and R.W. Potthoff, Znd. Eng. Chem. Res., 1993,32, 2438-2445. 20. D. Barbosa, Ph.D. Thesis, University of Massachusetts, Order Number 8727018. 21. U. Block, Chem. Zng. Tech., 1977,49, MS 4-44/77. 22. H. Schoenmakers and W, Buehler, Gel: Chem. Eng., 1982,5,292. 23. G. Ondrey, Chem Eng. 1996,6, Newsfront. 24. Jones, European Patent 461855, 1991. 25. L. A. Smith and M. N. Huddleston, Hydrocarbon Process, 1982,3,121. 26, J. Krafczyk and J. Gmehling, Chem. Zng. Tech., 1994,66, 1372. 27. U. Hoffman and K. Sundmacher, Chem Zng. Tech., 1997,69,613. 28. A. Gorak and L. Kreul, German Patent DE 197 01 045 Al, 1997.

CHAPTER 5

Thermodynamic Properties from Quantum Chemistry STANLEY I. SANDLER

1 Introduction The basis of computational quantum mechanics is the equation posed by Erwin Schrodinger in 1925 that bears his name. Solving this equation for multielectron systems remains as the central problem of computational quantum mechanics. The difficulty is that because of the interactions, the wave function of each electron in a molecule is affected by, and coupled to, the wave functions of all other electrons, requiring a computationally intense self-consistent iterative calculation. As computational equipment and methods have improved, quantum chemical calculations have become more accurate, and the molecules to which they have been applied more complex, now even including proteins and other biomolecules. However, ab initio quantum mechanical calculations increase dramatically in scale with N , the number of basis functions (e.g.,Gaussian functions) used to represent the wave functions of all electrons in a system (a molecule or a collection of molecules).' Hartree-Fock methods (which do not adequately account for the correlation between electrons) scale between N2 and P,and it is higher for levels of theory that introduce increasing extents of correlation: Moller-Plesset (MP) second-order perturbation theory scales as P to N4, more accurate coupled cluster (i.e., CCSD(T)) methods scale as fl to fl,and full configuration interaction (Full CI) methods, which should be almost exact, scale as N-' to I@. Because of this highorder scaling (referred to as the O(N) problem), the size of the molecules and the number of molecules that can be considered simultaneously, using high-accuracy quantum calculations are limited. The earliest and continuing successful application of quantum chemical methods has been to properties of a single molecule, such as the ideal gas heats of formation, bond energies, spectral frequencies, heat capacities and other energy properties. Heats of formation can now be calculated to what is referred to as a chemical accuracy of 2 2 kcal mol-', but with considerable effort.' However, much of the

44

Chapter 5

chemistry occurs in dense phases, and the accurate calculation of thermodynamic properties in a liquid is much more difficult, which is the subject of this review. For such a calculation of an ensemble average over many states, a sufficient number of interacting molecules to represent a bulk phase are required, and this is carried out by molecular simulation. This requires the energy of interaction of each configuration of a collection of molecules, which is difficult to calculate accurately due to the O(N) scaling. For example, a single full CI calculation for 100 molecules requires approximately 1007 greater computer resources than a single molecule calculation. This problem will be discussed here, along with the parameterized density functional theory and quantum mechanical polarizable continuum models. Semi-empirical methods will not be reviewed as they do not yield accurate interaction energies. A thorough discussion of the various quantum mechanical methods and the computer packages available can be found e l ~ e w h e r e . ~ * ~ One method to deal with the O(N) problem, although at the price of decreased accuracy, is to use the Density Functional Theory (DFT). In ab initio methods, the goal is to solve for the wave functions of the electrons, while in DFT, which is also based on single-electron wave functions, one calculates the total electron density distribution, as the ground state electronic energy and other system properties can be determined from this distribution function. The advantages of this method are its low cost and reasonable reliability, and consequently, it is widely used in molecular modeling to calculate ground-state energies and related properties. However, DFT is semi-empirical in that it contains parameters that have been adjusted to experimental data; unlike ab initio methods, there is no systematic way of improving predictions (e.g., by increasing the level of theory), and it is not of sufficient accuracy for the calculation of van der Waals forces.

2 Quantum Mechanical Computation of Force Fields for Molecular Simulation In molecular level (Monte Car10 or molecular dynamics) ~imulation,~ as is also discussed elsewhere in this book, typically hundreds or thousands of molecules are considered. Because of the O(N) scaling and the number of molecules and configurations that must be considered, high-level quantum mechanics calculations cannot be carried out. At present, there are two ways to proceed. The first method is that of Car and par in ell^,^ in which one performs molecular dynamics simulations for a large number (hundreds) of molecules, calculating the forces and energy from quantum mechanics for the whole assembly of molecules at each step. However, in order to be tractable, the calculations are carried out with DFT, but even then only for small molecules and requiring greater computational resources than are usually available. Also, the use of DFT restricts this method to systems in which electrostatic forces dominate; ab initio methods could be used instead, but only at very large supercomputer centers. The second more common and somewhat less computationally intensive way of using quantum mechanics is the temporal hybrid method in which interaction energies are first computed for pairs (and rarely triplets) of molecules at a large number

Thermodynamic Properties from Quantum Chemistry

45

of relative positions and orientations. Such calculations are usually carried out using the perturbation theory (such as MPn, where n=2, 3 or 4, indicates the level of theory) and a large basis set. When only weak interaction forces (van der Waals forces) are present, high-level calculations are needed to obtain a reasonable accuracy, as is also the case for very specific interactions, such as hydrogen bonding. This information is thtn fit to an analytic interaction potential or force field by adjusting parameters in site-site potentials using either atom-based or united-atom functional group-based (i.e., -CH3 or -CH,) sites, and perhaps off-atom sites (usually along bonds). While the two-parameter Lennard-Jones 6 - 1 2 potential has been used frequently, other more complicated potentials are preferred for better accuracy. Also, it can be useful to add site-site Coulomb interactions as atoms and/or functional groups in neutral molecules can have a nonzero charge. These charges are computed as part of the quantum mechanics calculation. The force field is subsequently used in simulations to compute thermodynamic properties. As the interaction energy in quantum mechanics is usually computed as the difference in energies of the dimer and the two separated molecules, two complications arise. First is that since the interaction energy is orders of magnitude less than the total energy of each molecule, each energy must be calculated to high accuracy to obtain a good value for this difference. Second, since the calculated energies depend on the number of wave functions, a Basis Set Superposition Error (BSSE)6 arises if different basis sets are used for each of the separated molecules and the dimer. This error is reduced by computing the energy of each molecule and of the dimer with the basis set of the dimer. The second virial coefficient (B,) can be exactly calculated from the ab initiobased pair potential using multidimensional integration, and such results provide a test of its quality. Some results obtained in this way are shown in Figures 1 and 2 for methyl f l ~ o r i d ehydrogen ,~ chloride* and methan01,~respectively. From these figures, we can see that the predicted second virial coefficients (B,) for CH3F and HCl are remarkably good for complete predictions (there are no adjustable parameters fit to the experimental data), but not for methanol. The B, values suggest that the interaction potential for methanol is not sufficiently attractive. This is probably because a higher level of quantum mechanics calculations is needed for this hydrogen-bonding fluid than could be done with the computational resources available. To calculate dense phase properties, molecular simulation must be used, and pairwise additivity is usually assumed, i.e., that the total interaction energy of a collection of molecules can be computed as the sum of the interactions between all pairs of molecules as if each pair of molecules was isolated from all others. [Note that this assumption of pairwise additivity in not used in the Car-Parinello method as the energy is calculated for the entire collection of molecules.] Using the painvise additivity assumption in Gibbs ensemble Monte Car10 (GEMC) simulation,1° we obtained the results shown in Figures 3 to 5 , where we can see that the combination of a quantum mechanics-based potential and simulation leads to excellent predictions of the vapor pressure, phase boundaries and densities for CH3F and HC1, but not for methanol. The results suggest that the pairwise additivity assumption is reasonable for nonhydrogenbonding molecular fluids, and that for such fluids, it is possible to use a temporalhybrid method to predict thermodynamic properties of reasonable accuracy.

46

Chapter 5

-100 n F

1

E

-a00

c3

E $ -a0

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300

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Temperature (K)

Figure 1 The predicted second virial coeficient for methyl fluoride and hydrogen chloride (lines) compared to experimental data (points)

There are two sources of error for hydrogen-bonding fluids: (1) that a higher level of theory and/or larger basis sets may be needed to obtain an accurate pair potential than for nonhydrogen-bonding fluids; and (2) that the assumption of painvise additivity may not be accurate. We can examine each of these separately. First, we can assume that because of the use of too low a level of theory and/or a basis set of insufficient size, the interaction potential for methanol is inaccurate in a systematic manner, and we can multiply the potential obtained by a single scale factor adjusted to reproduce the second virial coefficient. From Figure 2, we see that using a multiplicative factor of 1.2 gives excellent agreement for B, of methanol over the entire temperature range. The use of this scaled potential in simulation gives more accurate

Thermodynamic Properties from Quantum Chemistry

47

0 unscaled

:

-500

--

-1000

6

-1500

E

m

v

mi -2000

:

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r,

: L 6 I

Tl=

1.20

I

c

I

-2500 - 1 , I I

c -3000 250

I

I

350

450

550

Temperature (K)

Figure 2 The predicted second virial coeficient for methanol without and with scaling (as described in the text) compared to experimental data (points)

vapor pressures and phase boundary for methanol than obtained previously (Figure 5); however, the agreement is still not of engineering accuracy, nor as good as that for the nonhydrogen-bonding fluids. The simplest method to correct for the inadequacy of the pairwise-additivity assumption is to use a classical polarization model" to allow for the induction of one or more dipoles in each molecule due to the electric field of the surrounding molecules. The details of such calculations are given elsewhere. Because of the multibody nature of such a calculation, and the need for iteration over all the molecules in the system, this adds a factor of 3-10 to the computational load of an MC simulation. We have used a single-site dipole polarization model in some of our simulations and found that polarization (i. e., departures from pairwise additivity) had relatively little effect on the predicted thermodynamic behavior of methyl fluoride and hydrogen chloride, as expected. For hydrogen-bonding methanol, the simulation results with polarization (but not scaling) are significantly improved, although the results are still not satisfactory. However, the combination of both point polarization (to account for multibody effects) and scaling (to account for the use of an incomplete basis set) yields accurate vapor pressures, although the liquid density and critical point property predictions are still not as good as with nonhydrogen-bonding fluids. This suggests that for accurate properties predictions for hydrogen-bonding and other associating fluids, a high level of quantum theory and large basis sets must be used to compute the pair potential, and that corrections must be made for the assumption of painvise additivity. The extension of the temporal-hybrid method to mixtures is conceptually straightforward. What must be done for a binary mixture is to separately compute and fit the 1-2, 2-2 and 1-2 interactions, and use the potentials obtained in simulation.

Chapter 5

48 100

10 h

n 5 Y

2 u)

2

a

1

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I

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$

ez-

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dI

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(b)

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Figure 3 Vapor pressure ( a ) and vapor-liquid coexistence diagram ( b )for methyl fluoride from GEMC NVT simulations. The solid line indicates the experimental data and the symbols and dashed line are simulation results based on the ab initio pair potential. The closed symbol is the estimated critical point

The results of one such example are shown in Figure 6 using the scaled methanol potential and quantum-mechanically derived potentials for acetonitrile and their cross interaction. These results are true predictions from the combination of quantum mechanics and molecular simulation. No parameters have been adjusted to

49

Thermodynamic Properties from Quantum Chemistry 100

10

1

0.1

6

4

2

8

1OOO/Temperature (K-’)

350

r 300

x

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E

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200

t

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AL

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w I

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Density (g ~ m - ~ )

Figure 4 Vapor pressure and vapor-liquid phase boundary of hydrogen chloride as a function of temperature from experiment (line or .) and simulation A

mixture behavior (only a scaling of the pure methanol force field to experimental second virial coefficient data.) Indeed, there are no mixture parameters to adjust as all the interaction information comes from quantum mechanics. This example shows the subtlety of phase behavior that can be predicted with a completely “first principles” approach of quantum mechanics and molecular simulation. The use of quantum mechanics for large molecules, polymers, proteins or repeating small molecule structures (such as zeolites, hydrates or graphite sheets) is more difficult because of the O(N) problem. There are two typical approaches to deal with this problem. The first is to “divide and conquer”, i.e., to split the molecule into segments (usually capping broken covalent bonds with hydrogen atoms), and then

Chapter 5

50

100

1

no scalin no polarizagion

\

\

10 r n

n z

polarization// no scaling

Y

72

n

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c

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f

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350

25 0

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0.4

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Density (g ~ r n - ~ )

Figure 5 Vapor pressure and vapor-liquid phase boundary for methanol as a function of temperature from experiment (lines) and from simulation with combinations of scaling and polarization

adding up the contributions. The second is to recognize (e.g., in protein docking) that there may be a few interaction sites that are especially important, while distant parts of the molecule are less so. The important regions of the molecule are then treated with a high level of theory, and the distant regions with less accurate methods, or

51

Thermodynamic Properties from Quantum Chemistry I

0.0

0.2

0.4

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0.0 (b)

I

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I

0.6

I

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1.0

Xrnethanolp Yrnethanol

Figure 6 Vapor-liquid equilibrium for the methanol + acetonitrile system. The unfilled points indicate experimental data, and the$lled points indicate the predictionsfrom simulation

with the density functional theory, and the results are combined. Examples of this approach include the ONIOM method l2,I3 and others. Computational packages such as Gaussian, l 4 Jaguar15and others are available for the quantum mechanical calculation of the interactions energies, but it is very intensive, and requires the advice of an expert, as the results obtained depend upon the level of theory and basis set used, and it is not always obvious as to which set of

52

Chapter 5

choices is an optimal compromise between the computational resources required and accuracy. Although there are some molecular simulation packages available commercially,16 most simulators use programs that they have written. An alternative to the computationally intensive method of developing force fields from quantum mechanics has been to use empirical potentials, either transferable potentials that are meant to be used for many compounds (e.g., 0PLS,l7AMBER,'* TraPPE,19 etc.) or specialized potentials for specific compounds (e.g., SPC,20 TIP4P2I and later models for water). Such empirical potentials have been fit (frequently using simulation) to some experimental data. The results in Figure 7 for methanethiol illustrate the potential inaccuracies of using transferable potentials.22 There, we see that using a potential function fit to the quantum-mechanically

l!

3.4

3.2

3.0 2.8

2.6

2.4

2.2

2.0

1OOOflemperature (K-') I

0.0

0.2

0.4

0.6

0.8

Density (g ~ r n - ~ )

Figure 7 The vapor pressure and liquid-vapor coexistence curves (including estimated critical points)for methanethiol from simulation and experiment. The square symbols are obtained using the quantum-mechanical-based potential in simulation, the triangles from using the OPLS-UA potential, and the solid line shows the smoothed experimental data. The filled symbols are measured and estimated critical points (re$ 22)

Thermodynamic Properties from Quantum Chemistry

53

calculated energies yields quite good predictions of the phase behavior and vapor pressure of this fluid, while the results using the generalized OPLS potential in simulation are significantly in error. When such a potential fails for a system, there is no recourse except to use additional experimental data and refit, perhaps adding more terms to the potential.

3 Approximate Quantum Mechanical Calculation of Thermodynamic Properties The completely “first principles” way of computing thermodynamic properties described in the preceding section results in the O(N) scaling problem if a pair of large molecules are considered. A way around this is to use theories or models that require only a single molecule quantum mechanics calculation. [A high-level energy calculation for a single molecule takes less than one-hundredth (actually lD7) the time of a two-molecule calculation, and hundreds of two-molecule calculations are needed to develop a pair potential; thus, the computational load is increased by a factor of 10,000 or more.] One of the most well developed of the class of theories that require only a singlemolecule quantum mechanics calculation is the polarizable continuum model (PCM).23-25The goal of these models is to calculate the free energy of solvation (sol), i.e., the free energy change of bringing a molecule at fixed orientation from vacuum into a solvent that is treated as a polarizable continuum. This free energy change is computed as the sum of three terms:

where AGcavis the free energy required to create a cavity in the solvent to accept the solute, AGvdwis the contribution from the repulsion and dispersion forces between the solute and the continuum solvent, and AGelecis the electrostatic contribution that is computed from quantum mechanics. The last term is small for nonpolarizable molecules ( i e . , alkanes), but very important for polar solutes; it depends on the dielectric constant of the solvent, and is typically computed using the density functional theory. Once AG,,, is known, the vapor pressure and some other pure component thermodynamic properties can be computed directly; also, if AG,,, is known for inserting the solute both into its own pure liquid and into a (pure or mixed) solvent, mixture properties such as the activity coefficient of the solute can be obtained, from which the phase behavior can be computed. While the quantum mechanical calculation of the AGelecis reasonably well established, the term with the greatest uncertainty, which therefore limits the accuracy of the method, is the cavity formation term AGcav.Various approximations have been made for this term, including the use of spheres and nonspherical shapes to represent the solute with atomic radii empirically adjusted or based on crystal structure. However, as this term is less important for mixture properties (as AGcavfor a solute in its pure liquid and in the solvent to some extent cancel), PCM has been more useful for the calculation of activity coefficients, octanol-water partition coefficients

54

Chapter 5

T= 323.15k

- -- ---. .-.------------

41

A

Expt 323.15K

30 -COSMO-SAC

350

I

T=308.15k

1

1

h

2

200

E

E

150 100

50 0 0

0.2

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a8

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x1 J1

Figure 8 ( a ) Comparison of vapor-liquid equilibrium predictions from COSMO-SAC, UNIFAC and mod$ed UNIFAC models for water(1) + 1,4-dioxane(2)mixtures at temperatures 308.15 and 323.15 K, and (b) vapor-liquid equilibrium prediction from COSMO-SAC for benzene(l)/n-methylformamide(2)at temperatures of 31 8.15 and 328.15 K

and other mixture properties than for the calculation of pure component properties. The most successful PCM model is COSMO-RS (Conductor-like screening model for real solvents) of Klamt and co-workers26-28and others,29in which AGeI, is calculated from DFT for transferring a molecule from vacuum to a perfect conductor (infinite dielectric constant), and then an analytic correction is made for the dielectric constant of the actual solvent. A significant advantage of this approach is that the quantum calculation is for the solute in a perfect conductor regardless of the solvent,

Thermodynamic Properties from Quantum Chemistry

55

so that it needs to be done only once for each solute, and stored for later use with any pure or mixed solvent. Two examples of phase behavior predictions based on this method are given in Figure 8. This method is new and is still being developed, and will probably replace group contribution methods such as UNIFAC30 and modified UNIFAC3' in the future; a commercial product is already available.32Most importantly, one does not have to be an expert in quantum mechanics to use this software, and presently it is the most useful and efficient method of using quantum mechanics to directly compute phase behavior.

4 Conclusions Computational quantum mechanics continues to be a rapidly developing field, and its range of application, and especially the size of the molecules that can be studied, progresses with improvements in computer hardware. At present, ideal gas properties can be computed quite well, even for moderately sized molecules. Complete two-body force fields can also be developed from quantum mechanics, although generally only for small molecules, and this requires the study of pairs of molecules in a large number of separations and orientations. Once developed, such a force field can be used to compute the second virial coefficient, which can be used as a test of its accuracy, and in simulation to compute phase behavior, perhaps with corrections for multibody effects. However, this requires major computational effort and expert advice. At present, a much easier, more approximate method of obtaining condensed phase thermodynamic properties from quantum mechanics is by the use of polarizable continuum models based on COSMO calculations.

References 1. K. Morokuma, in Applications of Molecular and Materials Modeling. P. R. Westmoreland et al. (eds), Kluwer Academic, New York, 2002. 2. P. v. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III and P. R. Schreiner, The Encyclopedia of Computational Chemistry, Wiley, New York, 1998. 3. A. R. Leach, Molecular Modelling: Principles and Applications, 2nd edn, Prentice Hall, New York, 200 1. 4. D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd edn, Academic Press, New York, 2002. 5. R. Car and M. Paninello, Phys. Rev. Lett., 1985, 55, 2471. 6. N. R. Kestner and J. E. Combariza, in Reviews in Computational Chemistry, Vol 13, K. B. Lipkowitz and D. B. Boyd (eds), Wiley-VHC, New York, 1999. 7. S. I. Sandler, S. T. Lin and A. K. Sum, Fluid Phase Equalibr., 2002, 194,61. 8. P. K. Naicker, A. K. Sum and S. I. Sandler, J. Chem. Phys., 2003,118,4086. 9. A. K. Sum, S. I. Sandler, R. Bukowski and K. Szalewicz, J. Chem. Phys., 2002,116,7627. 10. A. Z. Panagiotopoulos, Molec. Phys., 1987,61, 813. 11. C. J. F. Bottcher, Theory of Electric Polarization 2nd edn, Elsevier, Amsterdam, 1973. 12. F. Maseras and K. Morokuma, J. Comp. Chem., 1995, 16, 1170, 13. T. Vreven and K. Morokuma, J. Chem. Phys., 1999,111, 8799. 14. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, R. E. Stratmann, J. C. Burant, S.Dapprich, J. M.

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Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, 0. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S . Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. G . Johnson, W. Chen, M. W. Wong, J. L. Andres, M. Head-Gordon, E. S. Replogle and J. A. Pople, Gaussian 98, Revision A. 9; Gaussian, Inc: Pittsburgh, PA, 1998. 15. Jaguar 5.0 User Manual, available at http://www.schrodinger.com/docs/jaguar5.0/pdf/ j 50-user-manual.pdf 16. Accelrys Materials Design Studio, information available at http://www.accelrys. com/mstudio/ 17. W. L. Jorgensen and J. Tiradorives, J. Am. Chem. SOC., 1988, 110, 1666. 18. W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz, D. M. Ferguson, D. C. Spellmeyer, T. Fox, J. W. Caldwell and P. A. Kollman, J. Am. Chem. Soc., 1996, 118, 2309; and later papers by Kollman et al. 19. B. Chen, J. J. Potoff and J. I. Siepmann, J. Phys. Chem. B., 2001, 105, 3093; and previous papers by J. I. Siepmann et al. 20. H. J. C. Berendsen, J. P. M. Potsama, W. F. van Gunseren and J. Hermans, in Intermolecular Forces, B. Pullman Reidel (ed), Dordrecht, 1981. 21. W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey and M. L. Klein, J. Chem. Phys. 1983, 79,926; and later papers by Jorgensen et al. 22. S. Garrison, private communication, 2003. 23. J. Tomasi and M. Persico, Chem. Rev. 1994, 94, 2027. 24. C. Amovilli and B. Mennucci, J. Phys. Chem. B, 1997,101,1051. 25. C. J. Cramer and D. G. Truhlar, J. Computer-Aided Molecular Design, 1992,6,629; and later papers by the same authors. 26. A. Klamt, J. Phys. Chem., 1995,99,2224. 27. A. Klamt, V. Jonas, T. Briiger and J. C. Lohrenz, J. Phys. Chem. A, 1998,102,5074. 28. A. Klamt and F. Eckert, Fluid Phase Equalibr, 2000, 172, 43; and later papers by these authors. 29. S.-T. Lin and S.I. Sandler, Znd. Eng. Chem. Res., 2002,41, 899. 30. A. Fredenslund, R. L. Jones and J. M. Prausnitz, A.Z.Ch.E. J., 1975, 21, 1086; and the many publications that followed. 31. J. Gmehling, J. Li and M. Schiller, Znd. Eng. Chem. Res., 1993, 32, 178; and the many publications that followed. 32. http://www.cosmologic.de/products.html

CHAPTER 6

Thermodynamics of Natural Gas Clathrate Hydrates E. DENDY SLOAN, JR.

1 Introduction Natural gas will be the premium fuel for this century for three reasons: (1) while we wish to use hydrogen as a fuel, we are economically restricted from obtaining it by reforming light hydrocarbons (with attendant inefficiencies) until we learn to efficiently electrolyze water, (2) gas bums cleanly, causes few pollution problems and, relative to oil or coal, produces less carbon dioxide, and (3) liquid fuels are better used as feedstocks for petrochemicals. Yet when we consider natural gas as a resource, we invariably find it associated with water. For example, when natural gas is produced from a Gulf of Mexico deepwater well, flowline design is driven by concurrent water production to prevent gas+ water from combining into a solid clathrate hydrate, which blocks the pipeline. As a second example, when dead plant and animal matter decompose on the bottom of the ocean, they form methane, which combines with water to form a clathrate solid. The amount of enclathrated methane in the bottom of the ocean is between 1015 and 1017 m3 (STP) - two orders of magnitude greater than the conventional worldwide gas reserve. Important application questions arise due to the profusion of natural gas hydrates:

1 . What defines economic flow assurance in deepwater energy production? 2. Is it possible that in situ gas hydrate reserves can be an economic energy resource? 3. Can subsea stability be affected by hydrate dissociation, affecting not only subsea structures but also climate change, if hydrated methane evolves to the atmosphere? 4. Is it possible to storehransport hydrated gas to onshore gas distribution, from a gas reserve that is too small to justify liquefaction, or too far from a pipeline? 5. What role do hydrates play in the environments of other planets?

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Answers to such difficult questions can be found in applied thermodynamics - in terms of measured, macroscopic values of pressures, temperatures, compositions, volumes, enthalpies, etc. This chapter provides an overview of natural gas clathrate hydrates - structures, phase diagrams, and thermodynamic predictions/measurements that guide our understanding in dealing with such questions. The hydrate historical perspective provides an example of how knowledge advances in a technical field. At the conclusion of the chapter, future thermodynamic challenges are presented.

2 What are Natural Gas Clathrate Hydrates? Clathrate hydrates form when small (<0.9 nm) non-polar molecules contact water at ambient temperatures (typically <300 K) and moderate pressures (typically >0.6 MPa). On a molecular scale, single small guest molecules are encaged (enclathrated) by hydrogen-bonded water cavities in these non-stoichiometric hydrates. Guest repulsions prop open different sizes of water cages, which combine to form three well-defined unit crystals shown in Figure 1. Cubic structure I predominates in the earth's natural environments with small (0.4-0.55 nm) guests and cubic structure I1 generally occurs with larger (0.6-0.7 nm) guests in mostly man-made environments. Hexagonal structure H may occur in either environment, but only with mixtures of both small and the largest (0.8-0.9 nm) molecules. The smallest hydrated molecules (Ar, Kr, 0, and N2) with diameters

46 HZO

/ -

Structure I

51262

8

16

512

\ +

'

3

5 264

Structure I

- 1

2

435663

136 H2O

___)

34 H2O

51268

Structure H

Figure 1 The three common clathrate hydrate structures, including the constituent cavities. Nomenclature: 5'268 indicates 12 pentagonal and 8 hexagonal sides in a cavity; numbers along lines indicate the number of cavities in each unit crystal structure. The rightmost numbers indicate the water molecules per crystal structure

Thermodynamics of Natural Gas Clathrate Hydrates

59

Table 1 Geometry of Cages in Three Hydrate Crystal Structures in Figure 1 Hydrate Crystal Structure Z zz H Cavity Description Number of cavitiedunit cell Average cavity radius, A Coordination numbe9 Number of waterdunit cell

Small Large 512 51262 2 6 3.95 4.33 20 24 46

Small 512 16 3.91 20 136

Large 51264 8 4.73 28

Small 512 3 3.91" 20 34

MediumLarge 435663 51268 2 1 4.06" 5.71" 20 36

aEstimates of structure H cavities from geometric models. bNumber of oxygens at the periphery of each cavity.

less than 0.4 nm form structure 11. Most of hydrate science, and thus most applications, concentrate on structure I (sI) and structure I1 (sII), with structure H (sH) in anecdotal occurance. While this work will emphasize sI and sII applications, many analogs occur for sH hydrates. Table 1 presents properties of the three common unit crystals shown in Figure 1. Water molecules hydrogen bond in a basic building block for both SI and sII, called the 512 (pentagonal dodecahedra) because there are 12 faces of pentagonally bonded water molecules in that cavity. Within the cavity, small guest molecules are enclathrated, with limited translation motion, but substantially more rotation and vibration, almost that of the gas. The 512 building blocks are joined to other 512cavities either through the vertices (sI) or via the 512 faces (sII). Yet, because the 512 basic building blocks cannot fill space without severe hydrogen bond strain and breakage, interstices between the 512 cages are filled with other cavities that relieve the strain by incorporating additional hexagonal faces with the 12 pentagonal faces - two hexagonal faces in the 512ti2cavity of SI and four hexagonal faces in the 51264of sII, in addition to the 12 pentagonal faces in each cavity. Thus, the SI and sII cages can contain larger guest molecules in decreasing order: 51264(in sII), 51262(in sI), and 512 (almost the same size in both SI and sII). The cages form with 46 H 2 0 molecules in basic repeating unit crystals with ratios of 2 X 512+6X 51262 sI, and 16X512+8X51264and 136 H 2 0 molecules in the sII unit crystal. In all three common structures, typically only one guest molecule forms within each cage. When all hydrate cavities are filled, the three crystal types (I, 11, and H) have remarkably high and similar concentrations of components: 85 mol % water and 15 mol % guest(s). Hydrate formation is most probable at the interface of the bulk guest and aqueous phases because the hydrate component concentrations greatly exceed the mutual fluid solubilities. The solid hydrate film at the interface acts as a barrier to further contact of the bulk fluid phases, and fluid interface renewal is required for continued, rapid clathrate formation. Three phase interfacial hydrate formation occurs with equilibrium of gas, liquid, and hydrate phases in artificial situations such as laboratories or in man-made processes.

3 Development of Hydrate Thermodynamics After their discovery and attendant interest, hydrate thermodynamic advances have been driven by practical concerns. In this section, motivating factors are shown with

60

Chapter 6

historical advances in three periods: (1) initial discovery and academic interest, (2) discovery of hydrates in pipelines, and (3) discovery of hydrates in nature. The applications of each of the three periods still exist as foundations for hydrate thermodynamic advances.

3.1 Discovery and Academic Interest While the clathrate hydrate discovery was announced in Davy’s Bakerian Lecture,’ Priestley2 apparently discovered SO, hydrates some years before, but failed to repeat, or pursue the original experiment. Davy’s hydrates of oxymuriatic acid (now known as chlorine) were succeeded by hydrate discoveries of methane, ethane, and propane through 40 years of efforts by Villard and deF~rcrand.~ These early researchers were intrigued by two aspects of these unusual hydrates: (a) the ratio of guest-to-water molecules (as high as 15:85) was very high relative to most other combinations of hydrocarbons and water, and (b) the fact that the ratio varied with each experiment - the hydration number changed from fractional values between 6 and 8, or higher for larger molecules such as propane. In addition, this era concentrated on finding a comprehensive list of chemicals that formed these unusual nonstoichiometric hydrates. More recently, academic hydrate research has moved from gentlemanly pursuits of science to being pragmatically driven. The rare research work stresses hydrate interest due to insights into water potential^.^ Currently, academic research is driven by funding vehicles, to provide answers to applications such as the five mentioned in the introduction. Fortunately, while pursuing single-minded goals, intellectually challenging research discoveries occur, usually providing the best driving force for hydrate advances. Consider the notable example of hydrate researcher D.N. Glew, who first published on ethylene oxide hydrates5 with Dow of Canada in 1959. Upon his retirement, his interest increased, as did his productivity in hydrates. In 2003, Glew published his 19th hydrate article6 in a series that provided the foundation for many new advances, including the new methane +water phase diagram discussed in Section 4.

3.2 Thermodynamics Driven by Discovery in Flowlines The discovery in 1934 of hydrates blocking energy flowlines led to a much more intensive period of hydrate research. Without knowledge of the hydrate structures, researchers generated methods to predict and to inhibit the conditions of hydrate formation in flowlines and to prevent them. Six major efforts, driven by pipeline discovery, hallmark this period. 1. In 1934, Hammer~chmidt,~ having discovered hydrate formation in pipelines, applied colligative physical chemistry to quantify prevention chemical effects. His discovery of alcohols, glycols, and other inhibitors and his

Thermodynamics of Natural Gas Clathrate Hydrates

2.

3.

4.

5.

6.

61

equation for hydrate formation temperature suppression still serve as industrial standards. The first systematic phase equilibrium study of hydrates was carried out at the US Bureau of Mines by Deaton and Frosts for a number of natural gas hydrate systems. Although the separation methods were somewhat crude (e.g.,a distinction could not be made between butane isomers), their hydrate formation pressure and temperature data continue to be foundations for prediction comparisons. In the 1940s, Katz had the insight that hydrates were ideal solid solutions rather than stoichiometric compounds. With the ideal solid solution concept, Katz and his students generated two experimentally based prediction m e t h o d ~ ~for t'~ hydrate formation conditions, without the knowledge of different hydrate structures. The first phase diagrams of hydrocarbons and water that contained hydrates were generated by Kobayashi and Katz for methane+water," and for propane +water. l 2 These diagrams were substantial advances in our understanding of water +hydrocarbon systems. Yet, the diagrams specified single, stoichiometric hydrate concentrations because the non-stoichiometric nature of hydrate structure was unknown. The intuitive insights from such diagrams were accompanied by a pragmatic, useful thermodynamic summary of hydrocarbon+water systems in Chapter 5 of the Handbook of Natural Gas Engineering by Katz et al.13 In a heroic effort after World War 11, von Stackelberg and his students at the University of Bonn determined powder X-ray diffraction patterns for many hydrates, particularly sI and sII. Using von Stackelberg's diffraction patt e r n ~ the ,~~ unit crystals of sI and sII were rapidly determined in the early 1950s by C l a u s ~ e n ' ~and > ' ~by Pauling and Marsh.17 It is interesting that, von Stackelberg's hydrate classification forced all of his data into the sI and sII paradigms, yet his data contained indications of hydrate structure H, discounted because they did not conform to the sI or sII formats. The idea of a fixed crystal structure in which single cages contained at most one guest proved irresistible to statistical thermodynamicists. After an initial effort by Powell,l8 Royal Dutch Shell workers van der Waals and PlatteeuwI9 generated a method that still stands today as a principal, regular industrial use of statistical thermodynamics. However, the model was not suitable for manual calculations (as were the methods of Katz in item 3 above), but required access to then-scarce computers, which limited its application to large companies or major universities. Widespread adoption of the model awaited the proliferation of personal computers.

The van der Waals and Platteeuw model represents the largest major advance in the thermodynamic history of hydrate technology. With this model, single hydrate formation data could not only be interpolated and extrapolated but also they could be extended to accurate mixture formation condition predictions. In Section 4, the details of this model are considered in light of modern hydrate phase measurements and modeling.

62

Chapter 6

In retrospect, the discovery of hydrates blocking energy flowlines motivated significant advances in clathrate hydrate thermodynamics, which may be typical of knowledge advances for many new compounds, with the following stages: 1. observation of initial phenomena with methods of prevention, 2. measurement of macroscopic formation temperatures and pressures, 3. generation of macroscopic prediction methods to interpolate and extrapolate data, 4. placement in the thermodynamic knowledge base via construction of phase diagrams, 5 . measurement and determination of microscopic crystal structure, and 6. construction of a statistical thermodynamics method to connect the microscopic and macroscopic domains.

The discovery of hydrates inside flowlines provided insights into hydrate physics and chemistry, which allowed application outside the pipelines. Even today, questions in flow assurance provide the most constant vehicle for hydrate research. However, the discovery of hydrates outside of flowlines has provided the most recent motivations for advances in hydrate science.

3.3 Thermodynamics Driven by Hydrates Outside Flowlines The Russian discovery of hydrates in nature spawned the third wave of advances in hydrate research. Ten major applications and basic research advances occurred in the modern time. 1. Makogon20 discovered in situ hydrates in porous media, leading to an instance of Siberian gas production from hydrates associated with normal gas reservoirs. Beginning in 1980, Kvenvolden and colleagues*' published surveys of in situ hydrates, independent of Makogon, but arriving at a consensus figure of 1015m3 (STP) of methane in hydrates. The intellectual heirs of such early workers are Ginsburg and Soloviev22in Russia, Dillon and C01lett~~ at the United States Geological Survey (USGS), and Hyndman and D a l l i m ~ r e ~ ~ at the Canadian Geological Survey. 2. Kobayashi and c o - ~ o r k e r began s ~ ~ to use the van der Waals and Platteeuw theory in the 1960s to predict hydrate formation in ternary systems. Parrish and Prausnitz26extended the method to prediction of hydrate incipient formation in natural gas systems causing the widespread industrial adoption of the van der Waals and the Platteeuw statistical method. Many academic (e.g., Holder and c o - ~ o r k e r s ~and ~ ) commercial programs (e.g., D.B. Robinson, and Associates2*)enabled the gas and oil industry to predict thermodynamic conditions at the incipient formation point, and thereby to prevent hydrate formation in industrial processes. All of the errors in the van der Waals and Platteeuw theory were placed in the solid phase. It may be argued that the theory's unusual success in prediction inhibited motivations for advances in hydrate phase measurements.

Thermodynamics of Natural Gas Clathrate Hydrates

63

3. Yet, beginning in the 1960s at the Canadian National Research Council (NRC), the group initiated by Davidson and extended by R i p m e e ~ t e rhas ~~ made the best modern measurements in hydrate science, with particular emphasis on nuclear magnetic resonance (NMR) spectroscopy, neutron diffraction, calorimetry, and molecular modeling. An abbreviated overview of the major advances in hydrate science at NRC is presented30in Table 2. 4. NMR success motivated other spectroscopic studies to measure the hydrate phase directly. This work represented an experimental departure, because previously only the fluid phases (vapor and liquid(s)) were measured, and any experimental error was incorporated in the solid-phase model of van der Waals and Platteeuw. However, with modern solid-phase measurements, the errors in the van der Waals and Platteeuw model could be clarified and corrected. Raman spectro~copy~~ and diffraction (X-ray and neutron, supplemented by Rietveld analysis32) have been successful: the first method to measure the relative occupation of single guest cages, and the second to extend the work to hydrate isothermal, adiabatic, and isobaric compressibilities. As shown in Section 4, these measurements combine with spectroscopic hydrate phase measurements to enable improvements of the model. 5. For computer simulations, in 1972, using early super-computing power at the s~~ USA Los Alamos National Laboratory, Tester and c o - ~ o r k e r performed the first Monte Car10 hydrate simulation, followed in 1983 by the first molecular dynamics simulation by Tse at NRC,34as computing power became more readily available. The most thorough, systematic simulations of hydrates have been performed by Tanaka and colleagues, who published 40 computer simulation articles on hydrates since 1984. Building on a foundation of thermodynamic simulations, since 1989 Rodger and c o - ~ o r k e r shas ~ ~extended the hydrate simulation studies to time-dependent and inhibitor-development studies. Recently, Klauda and Sandler36 and Trout and c o - ~ o r k e r shave ~~ extended ab initio studies to hydrates.

Table 2 Major Advances in Hydrate Science at NRC Canada NMR measurements of hydrate guest and host dynamics; many new hydrate sI and sII guests identified 1982 - Diffraction studies identify small guests as sII occupants 1983 - 016/018isotope ratio measurements for laboratory hydrate 1983 - First molecular and lattice dynamic calculations of hydrates 1983 - Confirmation of low thermal conductivity values of hydrates 1986 - First measurements of hydrate index of refraction 1987 - First reports of sH hydrate from powder diffraction and N M R 1988 - 13CMAS N M R determined as means of sI and sII hydrate identification 1990 - First published inventory of sH hydrate formers 1992 - First studies of phase equilibria in porous media 1992 - First inelastic neutron scattering of methane hydrate 1999 - Discovery of new hydrate structure - complex polytype of sII and sH, suggesting that interpenetrating lattices may exist naturally 2000 - New hydrate structure discovered for dimethyl ether hydrate 200 1 - First single-crystal X-ray diffraction for experimental composition measurement 1960-1980

- Dielectric and

64

Chapter 6

6. Of paramount importance is the reproducible generation of high-quality hydrate samples. In 1996, Stern et ~ 1 at USGS . ~ independently ~ generated a reproducible sample preparation method from melting ice, previously determined by Handa,39 that had gone unnoticed. This was a major advance because previously, when hydrates formed at the vapor-liquid interface, the solid hydrate barrier prevented further contact, so that the hydration number varied between samples, a sample variability that had plagued researchers since before Villard and deF~rcrand.~ 7. A major question arises from in situ hydrates detection, “Is energy recovery from hydrates viable?’ Two recent drillings - the 2002 Mallik well in Canadian permafrost (Dallimore et uL40) and the Hydrate Ridge ocean drilling (Trehu et ~ 1 . ~-’ document ) the technical viability of hydrate recovery. However, except in special circumstances (e.g., national security of very high gas prices), energy recovery from hydrates alone is not economical from an energy company’s per~pective,~~ principally due to the dispersed, lowenergy density of the resource. However, energy recovery is viable in special circumstances, as when hydrates overly a gas reservoir and can replenish the gas reservoir when the pressure is below the hydrate decomposition condition. A decade of such production was reported by M a k ~ g o in n ~the ~ Siberian Messoyaka Field in the 1970s. The Japanese hydrate program led by Japanese National Oil Companyu will attempt to produce energy from hydrates in the Nankai Trough by 2015. 8. The discovery of large amounts of in situ hydrates has led to two further research motivators. First, the community is concerned about seafloor stability, and the possibility of endangering the foundations of pipelines and subsea structures, as discussed in the monograph by Paul1 and Dil10n.~~ One major component of USA Department of Energy’s hydrate research program concerns hydrate developments in the Gulf of Mexico, whose results can be extended to other giant oil field sites such as offshore Africa or Brazil. 9. The most recent major application concerns hydrates and climate change. Using isotopic record data by Kennett et aZ., Dickens and colleagues proposed that the Late Paleocene Thermal Maximum (LPTM) was caused by gas evolution from hydrates. Most recently, Kennett et aZ.46 have proposed the “hydrate gun hypothesis,” which suggests that as late as 15,000 years ago, methane evolution from hydrates may have contributed to global warming. Such hypotheses, extended to more recent times, have motivated research programs in hydrate-induced climate change. Much more hydrate research is needed before such an hypothesis can advance to the stage of being an acceptable theory. 10. The discovery of hydrates in the natural environments initiated other hydrate monographs, with diverse topics of carbon dioxide seq~estration,~~ permafrost and oceanic and seafloor stability and climate change.49However, hydrate research proliferation has led to state-of-the-art records of the international hydrate community at three-year intervals at New York50in 1993, T o u l o ~ s in e ~1996, ~ Salt Lake City52in 1999, a n d Y ~ k o h a m a ~ ~ in 2002. The fifth international meeting is scheduled on June 12-16, 2005 in

Thermodynamics of Natural Gas Clathrate Hydrates

65

Trondheim, Norway. The most recent hydrate science is recorded in the four hydrate sessions of the Tenth International Conference on Physics and Ice.54 Any hydrate researcher will benefit from a review of such literature. The modem era of hydrate research is marked by the industrial adoption of the van der Waals and Platteeuw statistical thermodynamics model for the hydrate phase. The spectroscopic measurement of the hydrate phase, abetted with molecular simulation, led to accuracy improvements, and industrial applications to energy, seafloor stability, and climate change. With the above historical advances, consider modem thermodynamics of the hydrate phase itself.

4 Hydrate Thermodynamics Due to limitations of space, this section emphasizes only two major aspects of hydrate thermodynamics - namely the phase diagram and the hydrate prediction method. For examples of several counter-intuitive spectroscopic measurements results that impact the thermodynamic perspective, the reader is referred to a recent review,55presented with an overview of the 2002 Fourth International Hydrate Conference.

4.1 The Hydrate Phase Diagram One might think that a review of a single hydrocarbon+water phase diagram would be simplistic. Yet, discussion of such a diagram leads to both new and essential results to understanding of industrial gas mixture hydrate equilibria. In industry, the hydrocarbon or gas phase is almost always present in large excess for economic reasons, so that the hydrocarbon gas composition does not change on hydrate formation, but maintains a fixed composition, adding credibility to the study simpler diagrams. Here, we discuss the methane+ water diagram56for structure I. The reader may wish to supplement this discussion with more phase diagram details.57-58 Kobayashi and Katz" provided the first methane+water phase diagram, as a major advance in fundamental thermodynamics understanding. However, because these two researchers knew neither the hydrate structure nor their non-stoichiometric nature, their diagram did not indicate a variable composition for hydrates. This early advance was in violation of the phase rule because it suggests that only the specification of temperature or pressure is required to fix the hydrate composition in a two-phase condition. Recently, Raman spectroscopy was used to probe the methane+ water phase diagram further. A 4% decrease in methane hydrate concentration was measured by Huo et U Z . , ~ ~at the two-phase (H-V) condition, relative to two-phase (LurH) hydrates formed with dissolved methane in water at the same temperature and pressure. The result is a non-stoichiometric hydrate area in the methane+water isobaric T-y-z diagram shown in Figure 2, rather than the previously thought single hydrate composition, shown as a vertical line. Such solid solution areas are common in metallurgical phase diagrams. A similar non-stoichiometric hydrate region has been

66

Chapter 6

proposed59for the phase diagram of CO,+H,O, in which CO, also occupies both cavities of the sI hydrate lattice. The system is constrained to obey the Gibbs Phase Rule? In the most common application, with methane (or a fixed gas composition) and water, the two components will form the three phases of hydrates, vapor, and liquid with one degree of freedom. Fixing the temperature (or the pressure) determines the pressure (or the temperature) at the three-phase condition. For example, in Figure 2 the horizontal line connecting the three phase V+L,+ H (vapor+liquid water+ hydrate) phases indicates a single equilibrium temperature for the three phases in the isobaric diagram. This univariance allows one to apply the Clapeyron equation to obtain the useful relation of heat of hydrate dissociation from phase measurements of pressure and temperature, as verified by ~alorirnetry.~~ In contrast, with two-phase formation, without a free water (or a free vapor) phase, an additional degree of freedom, such as temperature, pressure, or phase composition, must be applied. In Figure 2, the two-phase areas (V+H, &+H, &+H, M + H , or H+I) indicate that two variables (temperature of the single-phase composition, in addition to the isobar of the diagram) must be specified to fix the system state. Hydrate formation from a water-saturated vapor counters the common myth that a free water phase must be present to have hydrates. That there is no thermodynamic necessity for a free water phase is perhaps shown most clearly by a study of the methane+water phase diagram.

~

~~

V vapor Lw: liquid water H: hydrate M: solid methane LM:liquid methane I: ice

v-L, Previous hydrate line

H,O

CH4.?H,0

CH4

Figure 2 The phase diagram of CH, +H,O. Note that the non-stoichiometric hydrate region replaces the previously proposed vertical line for stoichiometric hydrate concentration

Thermodynamics of Natural Gas Clathrate Hydrates

67

In Figure 2, there are no scales on either axis, and the diagram is only qualitatively shown. In the figure, the smaller concentration regions have been magnified for the sake of clarity. For example, at 298 K, the concentration of methane dissolved in water is typically 8.3 X and the concentration of water vaporized in the gaseous methane is 1X In a quantitatively correct Figure 2, both solubility lines would be vertical, at either extreme of the abscissa. So the qualitative Figure 2 is presented to illustrate the principles, trusting that the reader will turn to a reliable phase method as shown in the following subsection to quantify the diagram. Also, in Figure 2, hydrate formation from the two-phase regions is clearly permitted, and in fact comprises a large portion of the phase diagram. However, due to the small concentrations of water in the methane phase, or even smaller concentrations of methane in the water phase, it may be unlikely that kinetics will allow sufficient hydrate formation, for example, to plug a flow channel. Therefore, the restriction on a free water phase to form a hydrate plug is a kinetic restriction rather than a thermodynamic one. The phase diagram in Figure 2 has implications for mesoscopic structures. Kuhs61 has recently reported a crystalline morphology difference; porous hydrates formed with excess methane to the right of the non-stoichiometric region, while non-porous hydrates formed to the left.

4.2 The Most Important Advance: The Statistical Thermodynamic Model The major advance in hydrate thermodynamics was the generation of the van der Waals and Platteeuw model bridging the normal macroscopic and microscopic domains. Only a brief overview is given here to provide a basis for model improvements; the reader interested in more details should refer to another source.62The essence of the van der Waals and Platteeuw model is the equation for the chemical potential of water in the hydrate phase:

where pwH the chemical potential of water in hydrate gwpthe molar Gibbs free energy of water in the empty hydrate, k the Boltzmann’s constant vi the number of type i cavities per water molecule for sI: v,= 1/23, vL1=3/23; sII:vs=2/17, vL2=1/17; and e,. the occupancy of a hydrate cavity of type i with a molecule of type m. Equation (1) indicates that when a guest fills the cavities of a hydrate, the chemical potential of water in the cage is lowered, thereby stabilizing the hydrate phase. In principle, Equation (1) solves the hydrate prediction dilemma - that is, the hydrate formation conditions are determined by the pressures and temperatures that cause equality between the hydrate chemical potential of water in Equation (l), and the chemical potential of water in the other phase(s), as determined by separate equations of state. There are two important terms on the right: gwB the molar Gibbs free energy of water in the empty hydrate, and Qmi the filling of the cavity as a function of temperature and pressure. These details are not vital to the subsequent discussion, but are detailed elsewhere.62 I

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The van der Waals and Platteeuw principal equation (1) has five assumptions, which enable refinement of the theory:

1. No quantum effects are needed; classical statistics are valid. 2. Each cavity encages at most one guest molecule. 3. The internal partition function of guest molecules is the same as that of an ideal gas. 4. There are no interactions of the solute molecules, i.e., the energy of each encaged guest molecule is shielded by the surrounding water cage, and is therefore independent of the number and types of other solute molecules. 5. The host molecules’ free energy contribution is independent of cavity occupation. This assumption implies that encaged molecules do not distort the cavity. The first major success of the theory is that Equation (1) can be used to fit single guest hydrate dissociation pressures and temperatures. This fit of cavity occupation (yki) provides temperature- and pressure-independent Kihara potential parameters ( E and 0)for interactions between the guests and host cages. The fit enables interpolation between (and extrapolation beyond) single guest data for the eight common hydrate formers in gas and oil systems (methane, ethane, propane, iso- and normal butane, nitrogen, carbon dioxide, and hydrogen sulfide). The second major, more important success of the theory is that, using the fits to single component data, the theory allows predictions of mixture conditions for any combination of pure components. Due to the ideal solution, solid nature (assumptions 4 and 5 ) fits of pure component data provide acceptable predictions of all mixture conditions. Since laboratory data are costly both in terms of time and funds (e.g., US$2,000 per data point), it is very efficient to predict rather than to measure each hydrate condition. It is important to emphasize that, in contrast to vapor-liquid equilibria predictions, no interaction parameters are needed for mixture prediction. The above two successes have served the energy industry well for the last 40 years. Before the modifications indicated in the next section, the hydrate formation temperature prediction error to within 0.7 K (average absolute error for uninhibited hydrates) was acceptable, approximately twice the experimental error. With modern modifications of the van der Waals and Platteeuw theory, the energy industry feels confident in making multimillion dollar hydrate formation decisions based on the predictions, without obtaining data for important applications, such as: 1. Prediction of the temperatures and pressures of hydrate formation for an infinite variety of mixtures of the above eight components. This enables specification of flowline insulation or heating, to prevent fluids from entering the hydrate formation region. 2. Quantification of the effect of thermodynamic inhibitors (such as methanol, glycol, or salts) on the operation of processes. This determines the amount of inhibitor to be inserted into the free water phase to prevent hydrate formation from vapor and free water. 3. Specification of the water content to which a gas or liquid hydrocarbon must be dried to prevent hydrate formation, without a free water phase.

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4.2.1 Improvements to the Original Statistical Thermodynamics Theory The van der Waals and Platteeuw theory’s success is due to the general applicability of its five fundamental assumptions. Because the program was used mostly at low pressures in gas transportation and processing applications, it was fairly accurate at those pressures. However, the need for improvements was apparent at pressures above 20 MPa. Theoretical improvements have arisen by correcting the five assumptions, considered here in order. Modem experimentalists wish to supplement the normal macroscopic measurements with microscopic measurements, typically involving diffraction, Raman, and NMR spectroscopy. With the three spectroscopic methods, measurements of the solid phase not only provide hybrid e~perirnents,~~ but they also suggest improvements to microscopic models, and mitigate the earlier error of measuring the fluid phases and inserting all of the error in the hydrate phase. With the theoretical corrections in this section, prediction inaccuracies are reduced to values approaching experimental inaccuracies. The first assumption neglecting quantum effects stands unassailed. For the second assumption, under unusual conditions such as very high pressure, it is possible to have multiple-cage occupancy with unusual guests. For example, Mao et aL6, recently showed that hydrogen can form hydrates at a very high pressure with as many as two occupants in the small cage and four in the large cage of hydrate structure 11. However, these small guests and multiple occupancies are considered an aberration rather than the norm. With respect to the third assumption, for a number of years, Tanaka65has shown that the internal particle partition function of the guest molecules differ significantly in the cages, with restricted rotation and vibration, particularly for those molecules larger than methane. Indeed, some restriction on such motions is the basis, for example, in Raman spectroscopic determination of differing environments of the methane molecule, in the gas, in solution, and in the hydrate cages.66 Corrections to the last two assumptions have been the most beneficial in improving the theory. Together, the two assumptions (4. no interactions between guests, and 5. host molecules’ energy being independent of cavity occupation) comprise the ideal solid solution theory, which has been satisfactory for almost 40 years. However, a recent showed that a 0.5% change in the lattice parameter can change the pressure prediction (at P-100 bar) by 15%. The three ways in which the ideal solid solution assumptions are in error are based upon spectroscopic measurements of effects of pressure, temperature, and guest composition on hydrates: 1. Pressure. Using high-pressure neutron diffraction, Kuhs and colleagues32measured the isothermal compressibilities of CH4+H20 and CH4+D20hydrates to be 9.11 and 8.21 GPa, respectively, at 271 and 273 K. While these values conflict slightly with Helgerud’s68measured adiabatic compressibility (9.01 GPa at 273 K for CH, and H20), they provide quantitative evidence that hydrates are compressible. 2. Temperature and composition. Universal thermal expansivities of hydrate structures I and I1 have been measured for many common g ~ e s t sto~ within ~,~~

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Chapter 6

0.05% error. They showed that the hydrate lattice parameter increases significantly in proportion to both guest size and temperature. 3. Composition. Mooijer-van den Heuvel et aL7* have confirmed that when the larger cages (e.g., 512 64) are occupied by larger guests, the unit crystal lattice parameter increases. This lattice size increase also increases the small 512cage size, causing a decrease in Raman frequency - as much as 4 cm- for methane in the 512 when the largest molecules occupy the 512 64.In addition to addressing the ideal solution limit of how one guest molecule size affects the environment of another, this work suggests a limit to Raman spectroscopy, so that binary guest compositions cannot be determined a priori via integration of guest peaks. Instead, a method like NMR spectroscopy should be used to determine the relative cage occupation for >1 type of guest molecule. The above three errors can be seen as second-order corrections to a theory that has served the hydrate thermodynamics community well. To account for these secondorder changes, Equation (1) must be slightly modified as follows:

where the final term (yw) is an activity coefficient of water in the solid phase, which accounts for the change in the volume of the hydrate cavity. The best hydrate lattice models, incorporating the above three volume changes enable the prediction of hydrate unit cell volume to within 0.1%. The concept of this change is indicated in Figure 3. Figure 3(a) indicates that the free energy of water in the empty hydrate lattice is assumed to be known at a given temperature and volume. The volume of the empty hydrate lattice (V,"') must be equal to the volume of the filled lattice (V,), so that the only energy change in Figure 3(a) is due to occupation of the hydrate cavities. Yet, in Figure 3(b), the volume of the hydrate lattice is allowed to change due to filling of the lattice with guest molecules. As noted above, very small lattice changes (e.g., 0.5%) due to filling by guest can result in large (e.g., 15%) errors in high-pressure predictions. This non-ideality correction results in reducing the prediction inaccuracy substantially, For example, in comparison to all published uninhibited hydrate data (1685 data points), the average absolute temperature error can be reduced from 0.65 to 0.4 K, and the pressure decreases from 8 to 6% average absolute error. These hydrate prediction errors approach those of measurements, and may indicate a limit to fruitful improvements of the model. For more details, the reader is referred to the overview7' and to the four detailed publications of the model

4.2.2 Method Improvements in the use of the Statistical Thermodynamics Theory It should be noted that until the last decade, hydrate predictions using Equation (1) were restricted to incipient hydrate formation - i.e., specifying the temperature and pressure conditions at which the hydrate phase was initiated. The vapor-liquid equilibria analog of this is the determination of the dew point. However, Gupta et ~ l . , ' ~

Thermodynamics of Natural Gas Clathrate Hydrates

71

Figure 3 Concepts of the van der Waals and Platteeuw model and improvement. (a) original model that does not allow for distortion by guests (v,=vfT); (b) corrected model allowing guest distortion of lattice (vH# vHMT)

in Bishnoi’s laboratory, indicated how Equation (1) could be extended to predictions beyond the incipient hydrate formation condition, to determine the phase fractions, in the vapor-liquid analog of a flash calculation. Today, all modern hydrate prediction programs are carried out using this new technique, which is based on Gibbs energy minimization. It is extremely useful to determine phase fraction; in the aforementioned cases of two-phase hydrate formation (without a free water phase), one can use the Gibbs energy minimization to estimate whether there will be a sufficient hydrate amount to form a blockage in a process.

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5 Hydrate Challenges for the Future: Reversible and Irreversible Thermodynamics As we deplete the readily accessible natural gas reserves, we will encounter conditions that are both more remote and more severe. We will be challenged to explore deep ocean environments with higher pressures, permafrost environments with lower temperatures, and gases that were previously considered non-economical, such as those containing non-combustible components of nitrogen, carbon dioxide, and hydrogen sulfide. Such unusual conditions also stretch the applicability limits of hydrate phase equilibrium thermodynamics. In addition to their terrestrial applications, hydrates are of concern in outer space. Miller77suggested hydrates’ presence in comets, and on Jupiter, Mars, Neptune, Saturn, Uranus, and Venus. Recently, Loveday et aZ.78suggested that methane hydrate may be the dominant methane-containing phase in the nebula from which Saturn, Uranus, Neptune and their major moons, including an important role in formation models of Titan, Saturn’s largest moon. The temperature, pressure, composition, and enthalpy ranges for space applications encompass and usually exceed those encountered on earth. However, for normal conditions, hydrate thermodynamic predictions now rest on a comparatively sound foundation, with an accuracy approaching that of measurements. Yet, the largest future challenge goes beyond time-independent descriptions, to irreversible thermodynamics, or kinetics. We know very little about a kinetic mechanism founded on hydrate measurements. Due to the stochastic nature of nucleation, experimentalists have dealt with the deterministic growth process. Kinetic measurements are changing from macroscopic to microscopic scales. Initially, kinetics consisted of macroscopic measurements of the fluid phases associated with hydrates - such as gas consumption rates or liquid turbidity, fundamentally in Bishnoi’s l a b ~ r a t o r y .Subsequently, ~~ mesoscopic measurements of hydrate cry84 and film growth rates are Micros t a l ~ , 8 &particle ~ ~ size distribution,83+ scopic kinetic hydrate phase measurements are emerging.83.88-92 A review of microscopic hydrate science for both kinetics and thermodynamics is presented by ~oh.93 To borrow Churchill’s phrase, proving such hydrate phase kinetics measurement principles represents “the end of the beginning” for time-dependent experiments. The systematic kinetic study to generate a fundamental mechanism of hydrate nucleation and growth constitutes the major remaining challenge in hydrate physico-chemical science.

References 1. H. Davy, Phil. Trans. Roy. SOC. (London), 1811, 101, 1. 2. J. Priestley, Experiments and Observations on Different Kinds of Aid and Other Natural Branches of Natural Philosophy, Thos. Pearson, London, in 3 volumes, 1778, pp. 358ff. 3. R. deForcrand and P. Villard, Compt. Rendu., 1888, 106, 849. 4. J. S. Loveday, R. J. Nelmes, D. D. Klug, J. S. Tse and S . Desgreniers, Can. J. Phys., 2003, 81, 539. 5. D. N. Glew, Nature, 1959, 184, 545.

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42. K. Bil, Natural Gas Hydrate in Oceanic and Pennufiost Environments, M. Max (ed), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, 349. 43. Y F. Makogon, Natural Gas Hydrates: The State of Study in the USSR and Perspectivesfor Its Use,presented at the Third Chemical Congress of North America, Toronto, Canada, June 1988. 44. R. Matsumoto, Proceedings of the Fourth International Conference on Gas Hydrates, Yokohama, May 2002, 1. 45. C. K. Paul1 and W. P, Dillon, (eds), Natural Gas Hydrates: Occurrence, Distribution, and Detection, American Geophysical Union, Monograph 124, Washington, DC, 2001. 46. J. P. Kennett, K. G. Cannariato, I. L. Hendy and R. J. Buhl, Methane Hydrates in Quaternary Climate Change, American Geophysical Union, Washington, DC, 2003. 47. N. Handa and T. Ohsumi, (eds), Direct Ocean Disposal of Carbon Dioxide, Terra Scientific Publication Company, Tokyo, 1995. 48. M. D. Max, Natural Gas Hydrate in Oceanic and Permafrost Environments, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. 49. J.-P. Henriet and J. Mienert, Gas Hydrates: Relevance to World Margin Stability and Climatic Change, Geological Society, London, Special Publication 137, 1998. 50. E. D. Sloan, J. Happel and M. A. Hnatow (eds), (First) International Conference on Natural Gas Hydrates, Vol715, Annals NY Acad Sci., New York, 1994. 51. J. P. Monfort, (ed), Proceedings of Second International Conference on Natural Gas Hydrates, Toulouse, France, 1996. 52. G. D. Holder and P. R. Bishnoi (eds), Gas Hydrates Challenges for the Future (Third International Conference on Natural Gas Hydrates: Salt Lake City),Vol912, Annals NY Acad Sci., New York, 2000. 53. Y. H. Mori, (ed.), Proceedings of the Fourth International Conference on Gas Hydrates, Yokohama, Japan, 2002. 54. Tenth International Conference on Physics and Ice, Can. J. Phys., 81, (2003). 55. E. D. Sloan, J. Chem. Thermo., 2003,3541. 56. Z. Huo, K. Hester, K. T. Miller and E. D. Sloan, A.I.Ch. E. J,, 2003,49, 1300. 57. E. D. Sloan, Clathrate Hydrates of Natural Gases, 1st edn, Chapter 4, Marcel Dekker, Inc., New York, 1989. 58. E. D. Sloan, Clathrate Hydrates of Natural Gases, 2nd edn, Chapter 4, Marcel Dekker, Inc., New York, 1998. 59. F. Kiyono, A. Yamasaki and K. Ogasawara, Proceedings of the Fourth International Conference on Gas Hydrates, Y. H. Mori (ed), Yokohama, Japan, 2002,272. 60. J. W. Gibbs, Collected Works, Vol I, Longmans, Green and Company, New York, 1928. 61. W. F. Kuhs, Personal Communication, St. Etienne, France, June 17, 2003. 62. E. D. Sloan, Clathrate Hydrates of Natural Gases, 2nd edn, Chapter 5, Marcel Dekker, Inc., New York, 1998. 63. M. D. Jager, C. J. Peters and E. D. Sloan, Fluid Phase EquiZibx, 2002, 193, 17. 64. W. L. Mao, H. K. Mao, A. F. Goncharov, V .V. Struzhkin, Q. Z. Guo, J. Z. Hu, J. F. Shu, R. J. Hemley, M. Somayazulu and Y. S. Zhao, Science, 2002,297, 2247. 65. H. Tanaka, Can. J. Phys., 2003, 81, 55. 66. T. Uchida, S. Takeya, C. A. Wilson, C. A. Tulk, J. A. Ripmeester, J. Nagao, T. Ebinuma and H. Narita, Can. J. Phys., 2003, 81, 351. 67. R. A. Kini, Z. Huo, M. D. Jager, P. Bollavaram, A. L. Ballad, S. F. Dec and E. D. Sloan, Proceedings of the Fourth International Conference on Gas Hydrates, Y. H. Mori (ed), Yokohama, Japan, 2002,867. 68. M. B. Helgerud, Ph.D. Thesis, Standford University, 2001. 69. C. J. Rawn, A. H. Rondinone, B. C. Chakoumakos, S. Circone, L. A. Stern, S. H. Kirby and Y. Ishii, Can. J. Phys., 2003, 81, 431.

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70. M. M. Mooijer-van den Heuvel, R. Kini, 2. Huo, C. J. Peters and E. D. Sloan, Proceedings of Fourth International Hydrate Conference, Yokohama, May 2002,619. 7 1. A. L. Ballard and E. D. Sloan, Proceedings of Fourth International Hydrate Conference, Yokohama, May 2002,307. 72. A. L. Ballard and E. D. Sloan, Fluid Phase Equilibr., 2002, 194-197, 371. 73. M. D. Jager and A. L. Ballard, Sloan, Fluid Phase Equilibl:, 2003,211, 85. 74. A. L. Ballard and E. D. Sloan, Fluid Phase Equilibl:, 2004,218, 15. 75. A. L. Ballard and E. D. Sloan, Fluid Phase Equilibr., 2004, (in press). 76. A. K. Gupta, P. Bishnoi and N. Kalogerakis, Fluid Phase Equilibl:, 1991,63,65. 77. S . L. Miller, Proc. USA Nutl. Acad. Sci., 1961,47, 1798. 78. J. S. Loveday, R. J. Nelmes, M. Guthrie, S. A. Belmonte, D. R. Allan, D. D. Hug, J. S. Tse andY.P. Handa, Nature, 2001,410, 661. 79. P. Englezos, N. Kalogerakis, P. D. Dholabhai and P. R. Bishnoi, Chem. Eng. Sci., 1987, 42, 2647. 80. E. A. Smelik and H. E. King, Amel: Minerologist, 1997, 82, 88. 81. R. Larsen, C. A. Knight and E. D. Sloan, Fluid Phase Equilbr., 1998, 150-151, 353. 82. P. Bollavaram, S. Devarakonda, M. S. Selim and E. D. Sloan, Gas Hydrates: Challenges for the Future, Annls. NY Acad. Sci, Vol912, G. D. Holder and P, R. Bishnoi (eds), 2000, 533. 83. M. A. Clarke, and P. R. Bishnoi, Chem. Eng. Sci., 2000,55,4869. 84. A. Fiedel-Dufour and J.-M. Heni, Proceedings of the Fourth International Hydrates Conference, Yokohama Japan, May 2002,917. 85. D. K. Staykova, T. Hansen, A. N. Salamatin and W. F. Kuhs, Proceedings of the Fourth International Hydrate Conference, Yokohama, Japan, May 2002,537. 86. L. A. Stem, S. Circone, S. H. Kirby and W. B. Durham, Proceedings of the Fourth International Hydrate Conference, Yokohama, Japan, May 2002,673. 87. E. M. Freer, M. S. Selim and E. D. Sloan, Fluid Phase Equilibl:, 2001, 185, 65. 88. X. Wang, A. J. Schultz and Y. Halpern, Proceedings of the Fourth International Hydrate Conference, Yokohama, Japan, May 2002,455. 89. B. M. Freifeld, T. J. Kneafsey, L. Tomutsa, L. A. Stem and S. H. Kirby, Proceedings of the Fourth International Hydrate Conference, Yokohama, Japan, May 2002, 750. 90. S . Subramanian and E. D. Sloan, Fluid Phase Equilbr., 1999, 158-160, 813. 91. T. Pietrass, H. C. Gaede, A. Bifone, A. Pines and J. A. Ripmeester, J. Am. Chem. SOC., 1995,117,7520. 92. I . L. Moudrakovski, C. I. Ratcliffe and J. A. Ripmeester, Proceedings of the Fourth International Hydrate Conference, Yokohama, Japan, May 2002,444. 93. C. A. Koh, Chem. Soc. Rev., 2002,31, 157.

CHAPTER 7

Ionic Liquids in Separation Processes JURGEN GMEHLING

1 Introduction Separation processes are not only of great importance in refineries, but also in the chemical, petrochemical, gas processing, and pharmaceutical industries. Although the reactor can be regarded as the heart of a chemical plant, in most cases, 60-80% of the total cost is taken up by the separation step. This step involves one or more thermal separation processes such as distillation, extraction, absorption, crystallization, adsorption, membrane processes, etc., which are used to obtain the products at the required purity. In the various separation processes, concentration differences between two different streams (in most cases, two different phases) are employed to carry out the separation. The second stream is generated with the help of energy or a mass separating agent (solvent, membrane, adsorbent, etc.). Because of the various advantages of the distillation process (energy as separating agent, large density difference of the coexisting fluid phases (vapor, liquid)), 90% of all separations involve distillation. Other separation processes can become advantageous, when separation problems such as unfavorable separation factors (0.95 < aI24 . 0 5 ) or azeotropic points occur. In these cases, a special distillation process (extractive distillation) may be used. Extraction processes do not depend on a difference of vapor pressure between the compounds to be separated but on the relative magnitudes of the activity coefficients of the compounds. As a result, extraction processes are particularly useful in separating the different aromatic compounds (C, to CI2)from the different aliphatic compounds (C, to C12).Absorption processes are ideally suited for the removal of undesired compounds from gas streams, e.g., sour gases (H2S, CO,) from natural gas. For extractive distillation, extraction and absorption processes, highly selective solvents are required. The economic importance of extraction and extractive distillation processes can be recognized from the fact that the worldwide production of the BTX aromatics (benzene, toluene, xylenes, ethylbenzene) as important primary petrochemical products for the industrial manufacturing of many chemical products

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Ionic Liquids in Separation Processes

is approximately 60X lo6 t/a2. Aromatic compounds are mainly produced by steam cracking and reforming processes, together with extractive distillation and extraction processes used to separate the hydrocarbons. In all cases, the selective solvents (entrainers) have the task of altering the partition coefficients in a way that high separation factors and selectivities for the different phase equilibria (extractive distillation: vapor-liquid equilibrium (VLE), extraction: liquid-liquid equilibrium (LLE), absorption: gas-liquid equilibrium (GLE)) are achieved, resulting in a separation of compounds. The required partition coefficients, separation factors and selectivities can be calculated with the help of thermodynamic models ( gE-models, equations of state). As the total cost of these separation processes is strongly influenced by the selectivity of the solvent, the search for new and better solvents is important and has been a priority for many decades.

2 Thermal Separation Processes Using a Selective Solvent as Separating Agent 2.1 Extractive Distillation Extractive distillation is a suitable distillation process for the separation of azeotropic systems or systems with separation factors a,,close to unity. A typical extractive distillation process for the separation of aliphatics from aromatics is shown in Figure 1. In extractive distillation processes, the high boiling selective solvent (entrainer), introduced not far from the top of the extractive distillation column, has to alter the volatilities in such a way that the separation factor a12 attains a value very different from unity. Typical entrainers for the separation of aliphatics from aromatics are N-Methylpyrrolidone (NMP) or N-Fonnylmorpholine (NFM). In the presence of NMP or NFM, aliphatics

7

aromatics

7

Figure 1 Schematic diagram of an extractive distillation process (example: separation of aliphatics from aromatics)

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Chapter 7

but also in the case of most other entrainers, the aliphatic hydrocarbons become more volatile. This means that they appear at the top, while the aromatic hydrocarbons, together with the entrainer, leave at the bottom of the column. In the second distillation column, the high boiling entrainer can be separated easily from the aromatics and recycled to the first column. To achieve selectivities that are high enough, entrainer concentrations of 50-80% are required. The purity of the products is mainly influenced by the number of theoretical stages realized and the selectivity of the entrainer. Depending on the saturation vapor pressure of the entrainer and the feed location, traces of the entrainer may be lost with the distillate in the first column.

2.2 Liquid-Liquid Extraction For liquid-liquid extraction, a selective solvent is required, which shows only a partial miscibility with the liquid stream to be separated. In Figure 2, a typical multistage counter-current liquid-liquid extraction process is shown for the separation of aliphatics from aromatics. In this process, with the help of the selective solvent (extractant) the desired components (aromatics) are extracted from the feed stream. Distillation is typically used for the recovery of the selective solvent from the extract and raffinate stream leaving the extraction column. The selectivity, which strongly influences the number of separation stages, together with the capacity of the solvent, strongly influences the investment cost of a separation process. The objective is to find entrainers with a high selectivity and a sufficient capacity but unfortunately, in most cases, an increase in selectivity is linked to a decrease in capacity. The very low capacity of water, for example, is the reason that in spite of its high selectivity, it is not used for the separation of aliphatics from aromatics. distillation

extraction column ractant raffinate stream

Turn*\ I

aliphatics I

1

aromatics

A

T

‘ I hydrocarbons

Figure 2 Schematic diagram of a typical extraction process (example: separation of aliphatics from aromatics)

79

Ionic Liquids in Separation Processes purified gas (CH,)

desorbed gas ( C 0 2 , H2S)

---D

$4

P

c

natural gas

-+

f + -

absorbent

Figure 3 Schematic diagram of a typical absorption process (example: removal of sour gases from natural gas)

2.3 Absorption The main objective of absorption processes is the removal of one or more components from a gas stream using selective solvents. Figure 3 shows a typical absorption process in which, with the help of a selective solvent (absorbent), the undesired compounds (in this case, H,S, CO,) are removed from the raw gas (in this case, natural gas) in a multistage countercurrent process. While the purified gas leaves the absorber (saturated with the selective solvent), the absorbent is regenerated in a second column (desorber) and is recycled to the absorber. In the case of absorption, one can distinguish between physical and chemical absorption processes. In the case of physical absorption, the absorber is operated at high pressures and low temperatures while for the desorber the opposite conditions are used.

3 Thermodynamic Fundamentals For the right choice of the selective solvent and for the development and design of separation processes, a reliable knowledge of the phase equilibrium behavior (extractive distillation: vapor-liquid equilibrium (VLE); extraction: liquid-liquid equilibrium (LLE), absorption: gas-liquid equilibrium (GLE)) is required. This information is available from phase equilibrium thermodynamics.

3.1 Vapor-Liquid Equilibrium (VLE) The VLE behavior for weakly associating systems can be described with the required accuracy by the following simplified e q ~ a t i o n : ~

Chapter 7

80

where xi refers to the mole fraction of component i in the liquid phase, yi to the mole fraction of component i in the vapor phase, to the activity coefficient of component i, P is the total pressure and is Pi”the vapor pressure of pure i. Using Equation (l), the relationships for the partition coefficient, Ki, and the separation factor, aii,can be derived:

x

For the separation, only the difference, aii- 1, is used. In the case of azeotropic systems (a,=l),no separation can be achieved. For separation factors close to unity (e.g., 0.95 < a,, < 1.05), a large number of theoretical stages are required and a special distillation process has to be used.

3.2 Liquid-Liquid Equilibrium (LLE) Starting from the isoactivity criterion, the partition coefficients, Ki, and selectivities, Si,j,for extraction processes can be described using the following relations:

where E refers to the extract stream and R to the raffnate stream. For the calculation, a reliable knowledge of the activity coefficients as a function of composition and temperature is required.

3.3 Gas-Liquid Equilibrium (GLE) Gas solubilities can be calculated using Henry coefficients, Hi,j,In contrast to VLE, the value of the activity coefficient approaches unity, when xi becomes zero. For small solubilities and low gas pressure, the activity coefficient and the fugacity coefficient can be neglected and Equation (7) can be simplified:

x* x

xi *Hi,j=yiqiP

(7)

4 Selection of Selective Solvents The selectivities, Si,j,at infinite dilution are usually used4 for the selection of selective solvents. These values can be calculated from activity coefficients at infinite dilution (extractive distillation, extraction) or from Henry coefficients (absorption).

81

Ionic Liquids in Separation Processes

The required activity coefficients can either be calculated (predicted) with the help of thermodynamic models (group contribution methods) or obtained from factual data banks. The procedure for the selection of selective solvents for extractive distillation processes is given in refs. 4 and 5. The capacity Cj of extractants can be estimated using activity coefficients at infinite dilution.

r 3;.-

S"= lJ

(9)

For absorption processes, the partition coefficients and the separation factors at infinite dilution can be calculated directly using Henry coefficients.

where 1 and 2 refer to the gases to be separated and 3 refers to the solvent (absorbent).

5 Thermophysical Properties Required for Selective Solvents Typically, organic compounds are used as selective solvents. For a safe and efficient operation, all selective solvents for separation processes should have the following properties in common: 0 0 0

0 0 0 0

0 0

high selectivity high capacity no separation problems in the required regeneration step high thermal and chemical stability low melting point non-toxic low viscosity negligible corrosivity low price simple purification

Additionally, in order to have little or no impurities in the distillate of the extractive distillation column and also to avoid separation problems in the regeneration column, selective solvents for extractive distillation should have a boiling point that is at least 40 "C higher than the boiling points of the components to be separated. In the case of extraction, of course, it is most important that the solvent shows a miscibility gap with the feed and the raffinate stream. At the same time, a high

Chapter 7

82

capacity Ci for the extract is desired (see Equation (10)). Also, it is desirable that the extractant is almost insoluble in the raffinate phase. Furthermore, a sufficient density difference between the two liquid phases and suitable values for the surface tension are desired. In the case of absorption processes, the vapor pressure of the absorbent should be negligible in order to avoid solvent losses. Finally, it is desirable that the compounds to be removed should have a reasonable solubility. A large number of organic compounds are currently used for the various separation processes.' One of the most popular ones is N-Methylpyrrolidone (NMP), used not only for extractive distillation and for extraction but also for the various absorption processes. Since the cost of the separation step is greatly influenced by the selectivity, new and better extractive solvents are continually being investigated.

6 Ionic Liquids Over the past few years, ionic liquids have become very popular. These are a class of novel solvents, which show very interesting properties such as low melting point ( 4 0 0 "C), suitable viscosity, and in particular, negligible vapor pressure and therefore non-flammability. The first ionic liquid, ethylammoniumnitrate with a melting point of 12 "C, was synthesized by Walden in 1914.6Ionic liquids are good solvents for polar and non-polar organic and also for inorganic substances. Many are liquids at room temperature. These properties make ionic liquids interesting solvents for industrial applications and they may replace toxic, flammable and volatile organic solvents in future. Typically, an ionic liquid is composed of a large organic cation and an inorganic, polyatomic anion. The various cations and anions investigated are presented in refs. 7 and 8. Since there are a large number of cations and anions to choose from, there is virtually no limit to the number of ionic liquids that can be made. As a result, ionic liquids may be considered as designer solvents for bi-phasic reactions or as selective solvents (entrainers) for separation processes. In Figure 4, typical cations and anions are shown. Before 2001, only a few pure component properties (densities, viscosities, etc.) and mixture properties (phase equilibrium behavior, excess properties) involving ionic liquids were available. For a better understanding of their behavior and for the Cations

Anions

F O

,N\ @,N

R

"\

1-R-3-methylimidazolium R = methyl, ethyl, butyl, hexyl, ..

F O

Ethylpyridinium

Figure 4 Structural form of ionic liquids

O F

his( trifluoromethylsu1fonyl)imide

R-0-S-0' I

O F

F+{-A+F

0 II

II 0

R-sulfate

R = methyl, ethyl, methoxyethyl

83

Ionic Liquids in Separation Processes

development of thermodynamic models, reliable experimental data are required. As a result, various research groups started systematically measuring these properties and over the past 2-3 years, pure component properties (viscosities, heat capacities, densities, melting points, etc.), and, in particular, phase equilibrium data, such as VLE, LLE, SLE, gas solubilities and excess enthalpies HE have been measured for a variety of ionic liquids at different temperatures. To date, more than 6250 data points involving ionic liquids are a~ailable.~

r,

7 Results

r

Several research groups have measured infinite dilution activity coefficients for a variety of solutes (alkanes, alkenes, cyclic hydrocarbons, ketones, alcohols, water, etc.) in various ionic From the selectivities, it appears as though ionic liquids may replace the typical entrainers (extractants) currently used for extractive distillation or extraction processes. In Figures 5(a) and 5(b) the selectivities observed for the separation of cyclohexane from benzene (acting as a test system for the separation of aliphatics from aromatics) for different ionic liquids are shown as a function of temperature, together with the values obtained for NMP.9 A comparison indicates that the selectivities for ionic liquids are significantly higher than in the case of NMP. For all solvents, the selectivity decreases with increasing temperature. For the different imidazolium bis-trifluorosulfonylimides(R,R,IM BTI), it can be seen that the selectivities decrease with the length of the alkyl chain from methyl to butyl. The selectivities for 1,2-dirnethyl-3-ethylimidazolium BTI are lower than the values for 1-methyl-3-methylimidazolium BTI (MMIM BTI ), but larger than for EMIM BTI. For the imidazolium compounds with BF, or PF; as the anion, only the results for the hexyl compounds are available and these also show higher selectivities than does NMP. The highest selectivity is observed for 1-ethyl-3-methylimidazolium ethylsulfate. In comparison to NMP, the selectivities for the latter compound are approximately 2.5 times higher than the values for NMP. The second largest selectivities are observed for 4-methyl-N-butylpyridinium tetrafluoroborate. Similar promising results for ionic liquids were also obtained for the separation of systems such as hexane-benzene and heptane-toluene. l4 In Figure 6, P-x data for benzene or cyclohexane with 1-ethyl-3-methylimidazolium bis-trifluorosulfonylimide (EMIM BTI) at 353 K, measured with a static a~paratus,'~ together with the VLE data for the mixtures NMP+cyclohexane or +benzene, are shown.16 From these VLE measurements, it can be seen that high selectivities are achieved for the separation of aliphatics from aromatics using EMIM BTI. Furthermore, it can be seen that the ionic liquid EMIM BTI forms a large miscibility gap with cyclohexane. The system with benzene also shows a miscibility gap. The VLE behavior for the system EMlM BTI-benzene is particularly interesting. A miscibility gap is observed in spite of the system showing only a small positive deviation from Raoult's law. But surprisingly, over a wide composition range ( 0-80 mol%), the activity coeflicients remain nearly constant. While a value of 1.28 for the activity coefficient of benzene at infinite dilution is obtained, a similar value leads to the miscibility gap at about 80 mol% of benzene. From Figure 6, it can also be seen that the ionic liquid is nearly insoluble in cyclohexane (raffmate phase in the case of liquid-liquid extraction).

84

Chapter 7 20. 18.

$ 16.

.g 14. .*

-2

12. 10. 8.

6. 290.

300.

310. 320. 330. Temperature [K]

300.

310.

(a)

340.

350.

1

18. -

16. -

:

I

?? 12.

'S e e

v)

10. l4.I

290.

(b)

320.

330.

340.

350.

Temperature [K]

Figure 5 Comparison of the selectivity at injkite dilution Sz of various selective solvents for the separation problem cyclohexane(l)/benzene(2).(a) I -methyl-3-methylimidazolium [MMIM]' bis-trijluorosulfonylimide* [ I 01; I -ethyl-3-methylimidazolium[EMIM]+ bis-trtfluorosulfonylimide [BTI] A [ l o ] [I I ] ; 1-butyl-3-methylimidazolium [BMIM]' bis-trijluorosulfonylimide [BTI] [ I 01; 1 -ethyl-3-methylimidazolium [EMIM]' ethylsulfate + [lo] and NMP + [9]. (b) I,2-dimethyl-3-ethylimidazolium bis-tnj7uorosulfonylimide [BTI] [Ill; 4-methyl-N-butylpyridiniumtetrafluoroborate* [12]; I -Hexyl-3-methylimidazolium[HMIM] tetrafluoroborate A [14]; I Hexyl-3-methylimidazolium [HMIM] hemfluomphosphate [I31 and NMP + [ I 01

The large miscibility gap observed for an ionic liquid mixed with an aliphatic compound (without the addition of water as in the case of NMP) can be directly used for the separation of aromatic from aliphatic hydrocarbons by liquid-liquid extraction. Besides the miscibility gap, there are other requirements necessary for a successful extractant, such as high selectivity, high capacity, a low solubility of the extractant in the raffinate phase, a simple separation of the extract and the raffinate phase, low viscosity, high chemical and thermal stability and a sufficient density difference. Nearly all these requirements are met by the ionic liquids that have so far been investigated. However, our present knowledge of the thermal and chemical stability of ionic liquids is limited. For example, for some ionic liquids (largely dependent on the anion), hydrolysis does occur.

85

Ionic Liquids in Separation Processes I

100

I

4

-k

T = 354.15 K y;*= 4.81

-w'''-\

80

25 60

I

T=353.15K

r

100

2 2

2

40

L

20 0 0

0.2

0.6

0.4

0.8

1

X1

0

0.2

0.4

0.6

0.8

1

X1

Figure 6 VLE data for the systems at around 353 K: 0 Cyclohexane-1 -ethyl-3-methylimidazolium bis-trijluorosulfonylimide [EMIM BTI] [15] (left-hand side) and Cyclohexane-NMP [I51 (left-hand side). Benzene-I -ethyl-3-m bis-trifluorosulfonylimide[EMIM BTI] [I51 (right-hand side) and [16] (right-hand side)

The high solubility of aromatic compounds in ionic liquids can also be used for the removal of the different dibenzothiophenes using liquid-liquid extraction. l7 These are the compounds that are usually difficult to convert by hydro-treating in desulfurization processes. Because of their negligible vapor pressure, ionic liquids are uniquely suited as absorbents, since no contamination of the purified gas stream will occur. In Figure 7, the solubilities of carbon dioxide and methane in EMIM BTI at 50 "C are shown.'* Additionally, the Henry coefficients derived from these data are listed in Table 1 together with the values for NMP. It can be seen that the solubility of CO, is much higher than the solubility of methane. Furthermore, following Equation (12) similar separation factors (selectivities) are obtained for the solvents EMIM BTI and NMP, but a higher capacity is observed for EMIM BTI. This means that EMIM BTI may be used to remove CO, (one of the sour gases in natural gases') from methane.

Table 1 Henry coeficients for CO, and CH, in [EMIM][BTIIi8 and NMP at 323.I5 K8 Gas

co* CH4

Hi,3 (bar) [EMIM][BTI] 55.5 5 1.0 740 -+ 10

Hi,3(bar) NMP 105 -I 5 1120 ? 20

Table 2 Henry coeficients for CO, and CH, in [BMIh4][PF6]19andNMp8 at 298 K Gas

co* CH4

Hi,3 (bar) [BMIMI[PFJ

53.4 -I 0. 3 1690 2 180

'Gas solubility measurements for H,S are in progress.

Hi,3(bar) NMP 70 2 2 1040 2 20

86

Chapter 7 4 4 3

3 - 2

s2 1

1

0

0.2

0.4 0.6 x1[moVmol]

0.8

1

Figure 7 Solubility of CO,0 and CH, in 1 -ethyl-3-methyl-imidiazolium bis-trijluorosulfonylimide EMIM BTI at 323.15 K [ I 81

Gas solubility data have also been measured by other authors, e.g., Henry coefficients for CO, and CH, in 1-butyl-3-methylimidaoliumhexafluorophosphate [BMIM][PF,] have been determined at 298 K.19 In Table 2, the Henry coefficients for the two gases are given together with the values in NMP.9 As can be seen from Table 2, much better separation factors (selectivities) are obtained for [BMIM][PF,] than for NMP and in comparison to EMIM BTI, the capacity is higher. From the results shown in Figure 7 and listed in Tables 1 and 2, it can be concluded that ionic liquids may be suitable absorbents for absorption processes. The removal of the sour gas CO, from natural gas is only one example.

8 Conclusion Recent work has shown that ionic liquids are promising selective solvents for extractive distillation, liquid-liquid extraction and for separation by absorption. For these separation processes, high selectivities and capacities and negligible vapor pressure are important properties, but there are also other factors, such as cost, viscosity, difficulty of purification, etc., that must be taken into account. A negligible vapor pressure is particularly important for absorption processes and for the minimization of solvent emission, but it can also cause problems. Because of the high boiling point of an ionic liquid, their purification (e.g., the removal of high boiling compounds) will require new separation techniques (e.g.,extraction with supercritical fluids). For the industrial application of ionic liquids, as selective solvents for separation processes or other applications, a reliable knowledge of the pure component properties and mixture data (phase equilibria, excess properties) is required. Although

Ionic Liquids in Separation Processes

87

during the past 2-3 years, various research groups have begun measuring the thermophysical properties of ionic liquids, our knowledge is, today, still very limited. For example, we do not even know if the well-known gE-modelsor equations of state can be used to predict the behavior of multicomponent systems with ionic liquids, as is possible for the classical solvents (organic compounds, water). Because of the vast number of possible combinations of anions and cations, reliable predictive models would be most desirable. Predictive models would allow the search for the bestsuited ionic liquid for a given separation problem or as a solvent for chemical reactions for new processes. With the current tremendous research activity in this field there is a good chance that the required models can be developed within the next few years. Then, the future will indicate whether the promising properties of ionic liquids will result in them replacing the established organic solvents in separation processes and in other applications.

Acknowledgement The author would like to thank Dr. M. Krummen, Dr. R. Kato and Dr. J. Ahlers for their help in preparing the manuscript and Deutsche Forschungsgemeinschaft for financial support.

References 1. 2. 3. 4. 5. 6 7

J. Gmehling and A. Brehm, Grundoperationen, Wiley-VCH, Weinheim, 1996. K. Jiihnisch, Erdol Erdgas Kohle, 2001, 17, 232. J. Gmehling, B. Kolbe, Themodynamik, VCH-Verlag, Weinheim, 1992. J. Gmehling, C. Mollmann, Znd. Eng. Chem. Res., 1998,37, 31 12. B. Kolbe, J. Gmehling, U. Onken, Bel: Bunsenges. Phys. Chem., 1979, 83, 1133. P. Walden, Bull. Acad. Zmper. Sci. (St. Petersburg), 1914, 1800. P. Wasserscheid and T. Welton (eds), Ionic Liquids in Synthesis, Wiley-VCH, Weinheim 2003. 8 R. D. Rogers and K. R. Seddon (eds), Ionic Liquids - Industrial Applications for Green Chemistry, ACS Symposium Series 8 18, American Chemical Society, Washington, 2002. 9 Dortmund Data Bank 2003, www.ddbst.de. 10 M. Krummen, P. Wasserscheid and J. Gmehling, J. Chem. Eng. Data, 2002,47, 1411. 11 A. Heintz, D. V. Kulikov and S. P. Verevkin, J. Chem. Eng. Data, 2002, 47, 894. 12 A. Heintz, D. V. Kulikov and S. P. Verevkin, J. Chem. Eng. Data, 2001,46, 1526. 13 T. M. Letcher, B. Soko, D. Ramjugernath, N. Deenadayalu, A. Nevines and P.K. Naicker, J. Chem. Eng. Data, 2003,48, 708. 14 T. M. Letcher, B. Soko, P. Reddy and N. Deenadayalu, J. Chem. Eng. Data, 2003, 48, 1587. 15. M. Krummen, Ph.D. Thesis, Oldenburg 2002. 16. P. Gierycz, M.Rogalski and S . Malanowski S . , Fluid Phase Equilib., 1985, 22, 107. 17. A. Bosmann, L. Datsevich, A. Jess, A. Lauter, C. Schmitz and P. Wasserscheid, Chem. Commun., 2001,2494. 18. J. Gmehling, unpublished data. 19. J. L. Anthony, E. J. Maginn and J. F. Brenecke, J. Phys. Chem. B, 2002,106, 7315.

CHAPTER 8

Spectrocalorimetric Screening for Complex Process Optimization FLORIN DAN AND JEAN-PIERRE E. GROLIER

1 Reaction Calorimetry: General Aspects To identify optimal operating conditions of a chemical process, knowledge of kinetic and thermodynamic parameters for the most important main and side reactions is needed. A conventional method for investigating a reaction during process development is reaction calorimetry (RC).' RC is the technique accepted as the most leading method to study the process in near-to-the-industrial conditions. One of the objectives of RC is to simulate an industrial process at bench scale, allowing a wide spectrum of operation conditions and measurements. Areas of application of reaction calorimetry include determination of calorimetric data for reactions and process design, for the kinetic characterization of chemical reactions and of physical changes, for on-line monitoring of heat release and other analytical parameters needed in subsequent process development as well as for the development and optimization of chemical processes with the objective, for instance, to increase yield or profitability, control the morphology or degree of polymerization and/or index of polydispersity, etc. However, there is a remarkable disproportion between the three main areas occupied today by RC, namely hazards, process desigdoptimization and for monitoring the physico-chemical transformations. If there is an extensive activity in the field of hazards, often with little contribution to increased safety, probably less than 20% of the process development laboratories use calorimeters for process design and optimization; very little interest is shown for the use of heat released as a tracer for physico-chemical transformations since RC is still barely used in synthesis laboratories where reactions and process procedures are initiated. Basically, reaction calorimeters can operate in modes so that they closely approximate to isothermal, isoperibolic or adiabatic systems. Devices used to perform

Spectrocalorimetric Screening for Complex Process Optimization

89

RC measurements can be classified either as devices using jacketed vessels with control of the jacket temperature (heat balance calorimeters, heat flow calorimeters and temperature oscillation calorimeters) or as devices using a constant surrounding temperature, e.g., jacketed vessels with a constant jacket temperature, (isoperibolic calorimeters and power compensation calorimeters); such instruments may also feature single or double cells. It should be noted that the main driving force for developing process-oriented calorimetric instruments was the evolution of electronic hardware, which made measurements easily possible on a (non-micro) laboratory scale.

2 Instrument A detailed scheme of the instruments used in this study and constructed according to the principle of scanning transitiometry is presented in Figure 1. It consists of a calorimeter equipped with high-pressure vessels, a pVT system and a LabVIEW-based virtual instrument (VI) software. Two cylindrical calorimetric detectors (@ = 17 mm, 1 = 80 mm) made from 622 thermocouples chromel-alumel each are mounted differentially and connected to a nanovolt amplifier. The calorimetric detectors are placed in a calorimetric metallic block, whose temperature is directly controlled with an entirely digital feedback loop of 22-bit resolution K), being part of the transitiometer software. The calorimetric block is surrounded by a heating-cooling shield, and the temperature difference between the block and the heating-cooling shield is set constant (5,10,20 or 30 K) and controlled by an analog additional controller. The whole assembly is placed in a thermal insulation enclosed in a stainless-steel body and is placed on a stand, which permits to move the calorimeter up and down over the calorimetric vessels. A detailed description of the scanning transitiometer is given elsewhere.2

Figure 1 Detailed scheme of the transitiometric installation used as a reaction calorimeter

90

Chapter 8

In the case of polymer synthesis, the scanning transitiometer can be used as an isothermal reaction calorimeter, the advancement of a polymerization reaction being accurately monitored through the rigorous control of the thermodynamic parameter^.^ Two reaction cells - measuring and reference - are made as equal as possible to each other and they operate in a common surroundings in a so-called twin dzferential arrangement. The thermocouples composing the fluxmeters serve as a heat conduction path between cell walls and the block. The overall thermoelectric voltage, E, generated at the junctions provides an error-free indication of the temperature difference that causes the heat flux along the wires; the overall heat flux (thermal power), W, is summed up by series arrangement of the individual thermocouples. The thermoelectromotive force is related to the thermal power by the following relation:

where K is a calibration constant that depends on the thermoelectric capacity, E, and conductance, 'y, of the thermocouple wires. To gather additional information on a reaction, reaction calorimeters are often coupled with other analytical devices (e.g.,on-line FTIR, particle-sizing probes, turbidity probes, pH or other ion selective probes, etc.). Therefore, we developed a reaction cell that allows stirring, different dosing profiles for one or two reactants and can accommodate a small optical probe coupled to a miniaturized spectrometer, Figure 2.

IBI

Fiber Optic Spectrom

#I

-

-

Figure 2 Illustration of the reaction cell equipped with the spectrometric probe

Spectrocalorimetric Screeningfor Complex Process Optimization

91

The performances of this instrument will be illustrated in the case of processes for which reaction calorimetry was up to now difficult to apply, such as precipitant polymerization, particularly the ultrafine polyamide (PA) particles synthesis by anionic polymerization of lactams in organic media. For all experiments, the reaction mixture, consisting of the monomer, E-caprolactam (L6), the solvent (aliphatic hydrocarbon) and the catalyst (lactamate anion), as well as the synthesis of the catalyst (lactamate anion) were prepared in a separate reaction flask (see ref. 4); about 9 cm3 from it was transferred into the reaction cell at the reaction temperature with a preheated syringe. As soon as a stable base line was achieved, the chain initiators (CI) (a long-chain aliphatic isocyanate) was fed into the cell with the dosing device following a pre-established dosing profile.

3 Particularities of Anionic Polymerization of Lactams in Organic Media Anionic polymerization of lactams in organic solvents is a complex process that takes place mainly heterogeneously and allows to obtain powdered polyamide. The particles thus obtained are virtually spherical, possessing a porous structure, a narrow particle size distribution and a good solvent and heat resistance. This form is very handy for different processing procedures, such as flame spraying, electrostatic coating, paste production, dispersions, lacquer binders and cosmetic formulations.s*6 The most important steps in polyamide formation are the following : initiation and growth of macromolecules in a homogeneous medium, nucleation, phase separation and aggregation of the growing chains, solidification and, finally, polymer crystallization. All the above-mentioned events occur rapidly and partially over la^.^ The polymerization occurs by the activated monomer mechanism, which supposes a two-step mechanism involving the acylation of the lactam anion (NaL) by the N-acyllactam end-group followed by a fast proton-exchange with the monomer.* In bulk polymerization, the preformed N-acyllactams or their precursors (socalled CIS) are introduced into the system in order to avoid the slow initiation step due to the absence of N-acyllactam groups at the beginning of polymerization:

C.I.

NaL

Reaction Scheme In the presence of organic media, the reaction does not start in the absence of CI and, consequently, for calorimetric measurements the moment of adding the CI is considered the “zero” time of the polymerization. In a very strong basic medium, the

92

Chapter 8

CI disappears rapidly in side reactions (Claisen condensation) and the polymerization is prematurely ~ t o p p e dHowever, .~ as long as the monomer remains in the reaction medium, the polymerization can be continued by simply adding new amounts of chain initiation. This is a great challenge while using RC for process monitoring since in this way the process becomes a "controlled" one.

4 Spectro-CalorimetricInvestigation Under Isothermal Conditions The objective of this work is to simulate an industrial process on a bench scale, allowing a wide spectrum of operation conditions and measurements. Taking into account the complexity of the process as well as the fact that the calorimetric signal reflects the sum of physicochemical changes in the reaction system, an additional on-line mini-spectrometer (OceanOptics, S2000 spectrometer) for simultaneous in situ measurements as well as some off-line data furnished by TMDSC and gravimetric measurements have been used. The apparatus, a scanning spectro-transitiometer in such a configuration, is able to work either under isothermal or non-isothermal conditions; consequently, the first part deals with the isothermal RC on lactams polymerization and the second part deals with the non-isothermal polymerization, respectively.

4.1 The Manner of Contacting the Catalytic Species The main advantage of this process is the possibility of controlling the conversion and orienting it toward obtaining powders with desired morphologies through a finetuning of the synthesis parameters. There are many factors influencing the advancement of the process and the most important ones are as follows: temperature, solvent's quality and monomer/solvent ratio, chemical structure of catalytic species (catalyst and chain initiator), catalyst/monomer ratio, initiatodcatalyst ratio, the manner of contacting the catalytic species and hydrodynamic regime, e t ~In. what ~ follows, the effects of three important factors on the polymerization process are discussed: the manner of contacting the catalytic components since the polymerization starts as soon as the two species make contact with each other, the nature of the catalyst and the level of stirring.

4.1.1 One-shot Technique The simplest way of facilitating contact between the two catalytic species consists in adding the whole amount of CI just at the beginning of the process. This method is the so-called one-shot method and is well documented in the literature.5-7*10.11 The effect of temperature on the shape of heat released is illustrated in Figure 3. Clearly, the shape of the curves representing the heat evolution with polymerization changes from two peaks at 75 "C to a large shoulder at 90 "C that diminishes at 105 OC, and finally disappears at temperatures above 120 "C. The intensity of the first peak does not seem to be influenced by temperature, ranging from 25 to 35 mW per gram of L6 for all experiments.

Spectrocalorimetric Screening for Complex Process Optimization

300 1

t;i

0

1000

93

2000

3000

4000

5000

100

0

1000

2000

3000

4000

5000

0

1000

2000

3000

4000

5000 6000

Timels

Figure 3 Eflect of the temperature on the overall heat rate for the anionic solution polymerization of M using the one-shot method of contacting the catalytic species. In the insets, the reaction temperatures and gravimetric conversions are given. Reaction conditions: solvent, mixture of aliphatic hydrocarbons with bp. 140 - 160 "C; initial concentration of M, 3 mol-'L; catalyst, NaH = 3% mol mol-' M; Wcatalyst ratio = 1

On the contrary, the intensity of the second peak strongly increases with temperature, from about 33 to 273 mW per gram of L6 at 75 and 120 OC, respectively. The first peak corresponds to the initiation reaction between the two catalytic species (a very fast reaction, practically not influenced by temperature in the investigated range), while the second one, which is sensitive to temperature, corresponds to the propagation reaction. This assumption is sustained by the very sharp decline in the intensity of the spectrometric signal observed at the very moment when phase separation takes place, Figure 4. The drop in intensity corresponds to the occurrence of polymer precipitation; in the thermogram, this drop corresponds to the beginning of the second peak (see Figure 4(b)). Taking into account the crystalline character of the aliphatic polyamides, the second peak is given not only by the polymerization's heat but also contains the amount of heat corresponding to the crystallization of the separated polymer. It is very difficult to discern between heat of propagation and crystallization, respectively, since the two processes occur almost concomitantly. Consequently, additional off-line data, i.e., gravimetric and DSC were used in order to bring about the deconvolution of the overall heat, Figure 5. As Figure 5(a) shows, the overall heat contains tree terms, namely heat of initiation, heat of propagation (or polymerization) and heat of crystallization:

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Figure 4 On-line combination of calorimetry and spectroscopy: (a) 3 0 plots of the evolution of spectrometric signal during the anionic polymerization of M;(b)simultaneous calorimetric and spectroscopic signals recorded during polymerization at 90 "C (the intensity of spectrometric signal was taken at 600 nm). Note the stability of the spectroscopic signal before the phase separation and its sharp decrease at phase separation

Gravimetric data allowed the determination of L6 conversion into the high polymer, the DSC thermogram provided the quantification of crystallization heat (supposing that the heat of melting AHmelt for the first run is equal to the heat of crystallization during polymerization) and the heat of initiation was calculated by deconvolution of the heat rate using PeakFit v4.11 software (Systat Software 1nc.-SSI). In slurry polymerizations, changes in heat transfer properties are often neglected, as there are only slight changes in viscosity during the course of reaction; this is why the stirring heat was neglected in Equation (2). As shown in Table 1, the heat of melting did not change significantly with the polymerization temperature (it increases slightly), but the shape of the thermograms (Figure 5(b)) is strongly affected, suggesting crystalline and morphological changes of the PCA powders. As is shown in the insets of Figure 3, keeping all other synthesis conditions identical, the high polymer yield increases significantly with temperature. However, due

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Spectrocalorimetric Screening for Complex Process Optimization

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Table 1 Variation of the Thermal Properties of PCA Powders with the Reaction Temperature Reaction temperature ( "C) 1st 75 167.6 90 181.8 105 120 -

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to practical reasons the increase of the reaction temperature above 120 "C is not suitable due to the fouling at the reactor wall of the viscous precipitated particles. In addition, the particles size also increases, large agglomerates being obtained at high temperatures. An alternate route to increasing the polymer yield at lower temperatures is to divide all the CI into small portions and to feed each portion just after the baseline is reached.

96

Chapter 8

4.1.2 Successive Injections Technique The above-mentioned manner of contacting the catalytic species is based on the analysis of the heat released during the one-shot technique, i.e., the lifetime of the process is a little more than 30 min. In addition, it is well known that the catalyst (lactamate anion), in the absence of any impurities, is stable up to 48 h at the reaction temperature. Figure 6 illustrates the heat released at 90 "C when the same amount of CI, as that used in the one-shot method, was introduced by 12, 2 1 and 42 successive injections, respectively, In all cases, almost quantitative transformation of L6 was obtained in spite of the relatively low reaction temperature. For the greatest portion of CI (upper curve), the overall heat released resembles the heat observed for the one-shot technique, i.e., passing through a maximum with time. The peak of heat released is strongly tempered by a decrease in the amount of CI (lower curve in Figure 6). It should be stressed that the shape of each individual peak corresponding to the successive injections of CI is exactly similar to the one-shot shape at the same temperature. However, the amplitude of the individual peaks markedly decreases after a certain number of injections, i.e., at low monomer concentrations. Of course, the intensity of each peak and the corresponding heat are proportional to the amount of CI injected (see Figure 7(a)). Evidently, not only is the intensity of the main peak proportional to the amount of heat associated to chain propagation but also to the amount of heat corresponding to the initial shoulder, which reflects the initiation reaction. At the same temperature, the ratio between the intensities of the two heat effects is almost constant, In other experiments carried out at 83.5 OC, the injection rate of CI was progressively decreased for each injection; the heat rate evolution during the first 10 min is depicted in Figure 7(b). The slope of the heat rate corresponding to the initiation step is proportional to the injection rate until the first peak disappears at low injection rates.

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Spectrocalorimetric Screening for Complex Process Optimization

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The high polymer yield and the possibility of controlling the heat released during the successive injections method suggested another method of contacting the catalytic species, namely continuous injection of CI.

4.1.3 Continuous Injection Technique According to this method the entire amount of CI is delivered over a certain time period with a pre-established rate. Figure 8(a) shows the comparative heat evolution between the successive injection method and the continuous injection one. The whole amount of CI was delivered in 22 portions, 20 pl CI each, at a 20 min interval, or with a constant rate, 60 pl h-' over the same time period. A good agreement between the heat evolutions for the two processes can be noted. The effect of temperature on the heat rate is illustrated in Figure 8(b). In both cases, a stationary state was obtained but, for the same injection rate, its duration decreases with increasing temperature (3 h and about 7 h at 105 and 83.5 OC,respectively). The first small peak (or a shoulder) in thermograms was once again ascribed to the initiation step, i.e., it corresponds to the reactions occurring in the homogeneous phase. In the case of continuous addition of CI, three important indications are provided by the spectroscopic data, (Figure 9(a)). The first one (region I in Figure 9(a)) concerns the period between the start of reaction and the beginning of polymer

98

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Figure 8 Continuous injection method of contacting the catalytic species: (a) heat evolution versus time recorded during successive and continuous injection of the CI for the anionic polymerization of M at I05 "C;(b)eflect of temperature on the overall heat evolution (the CI was delivered 7 h at a rate of 60 pl h-l)

precipitation; it depends on temperature and the injection rate of CI. The second one (region I1 in Figure 9(a)) is related to the particle size evolution with time (this is only a qualitative information). Generally, after the phase separation the evolution of spectroscopic signal runs parallel to the conversion. As seen in Figure 9(a), when one approaches the end of the process, the intensity of spectrometric signal remains constant. Finally, the third piece of information (also qualitative) refers to the average size of particles. It was proved by SEM (see insets of Figure 9(a)) that the average particle size is directly proportional to the final intensity of the spectrometric signal (111 in Figure 9(a)).12 For the successive injections method, Figure 9(b), the spectrometric signal indicates the start of the phase separation. Quantitative information corresponds to a sharp decrease of the reflected light, i.e., the time period elapsed until the beginning of the polymer precipitation, and qualitative information, relating to the number of and size of PCA particles, is given by the absolute intensity of the reflected light at a pre-established wavelength (I1 in Figure 9(b)).

4.2 Effect of the Catalyst Nature The effect of the catalyst chemical nature is demonstrated using two catalysts, sodium hydride, NaH,and sodium bis(2-metoxy-ethoxy)aluminium hydride, RedAl@,which react with L6 giving rise to two lactamate anions with different nucleophilicity. The

Spectrocalorimetric Screening for Complex Process Optimization

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Figure 10 Heat rate of process versus time for the one-shot method of contacting the catalytic species using diferent catalysts. Reactions conditions are similar to those used in Figure 3

Chapter 8

100

second lactamate anion, the so-called "reduced lactamate salt," is characterized by a reduced nucleophilicity and a lower dissociation degree of the lactam salt.13As shown in Figure 10, the catalyst effect is quite similar to those of temperature, i.e., with decreasing its nucleophilicity the typical shoulder of the heat rate evolution at 105 "C is replaced by a two-peak curve, which characterizes the process below 90 "C.

4.3 Stirring level Considering the requirement of high turbulence to enhance the mixing of reactant streams to the desired degree of completeness on the molecular level,14 the heterogeneous character of the investigated process, and the presence of the intermediary viscous particles, where the polymerization mainly proceeds, one should expect a strong effect of the stirring level both on the high polymer yield and the particle size distribution. The stirring effect was investigated by the one-shot method at 120 O C , with stirring rates ranging from 150 to 750 rpm. As Figure 1l(a) shows, the shape of heat rate evolution was not affected by the stirring level, but the conversion strongly increased at the stirring level above 500 rpm, reaching a limit at about 750 rpm, s150 rpm

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Figure 11 Eflect of stirring level on the heat rate ( a )and overall heat (b)for the anionic solution polymerization of Mat 120 "C

101

Spectrocalorimetric Screening for Complex Process Optimization

Figure 1l(b). Further increase in stirring level above 1000 rpm (not given here) does not change the limit of conversion but negatively influences the particle size distribution due to the collision of the viscous particles.

5 Non-Isothermal Reaction Calorimetry As mentioned in the previous section, the apparatus is able to work in non-isotherma1 conditions. This is a useful feature since in the chemical industry very often the reactions are initiated on a temperature ramp. A first series of experiments was carried out by successive injections of CI with the temperature ranging from 75 to 130 "C and a scanning rate of 0.12 "C min-' (see Figure 12). The first portion of CI was injected just after the stable dynamic base line was reached; all other injections were initiated after the heat evolution of the previous step had ceased. As expected, the shape of the curves representing the heat evolution with polymerization changes from two peaks at low temperatures (up to 95 "C) to one peak with a small shoulder at the highest injection temperature (about 105 "C). The intensity of the first peak is not influenced by temperature, but the intensity of the second one strongly increases with temperature, and consequently the monomer conversion also increases. The second series of experiments deals with the continuous manner of contacting the catalytic species. Here, not only was the temperature scanned but also the injection rate was continuously changed. The temperature was scanned between 75 and 135 "C with a positive or negative ramp, and the CI profile was linearly and symmetrically varied on either side of the constant flow rate corresponding to the typical process. The delivered amount of CI was the same for all runs. Figure 13 illustrates the two limiting situations when both variables were concomitantly scanned in the same direction. Compared to the typical process (see Figure 8(b)), no stationary state was achieved with either positive ramps for temperature and injection rate (the upper thermogram) or negative ramps (the lower thermogram). The lower thermogram resembles that corresponding to the one-shot method (see Figure 3 at 105 and 120 "C) because the process temperature is the highest one and the amount of injected CI is sufficiently high just from the beginning. The opposite

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situation is encountered in the second case where the heat rate continuously increases and then rapidly declines to the base line after the total consumption of the monomer. It is well known that the reaction conditions favouring pre-oriented particle size and a narrow size distribution are not suitable from the viewpoint of the average molecular weight or average molecular weight homogeneity. This is why concomitant scanning of the two variables should be a very useful technique for controlling the particles size and the size distribution as well as the molecular weight and molecular homogeneity.

6 Conclusion Anionic polymerization of L6 in aliphatic solvents was monitored on-line using both isothermal and non-isothermal reaction calorimetry in combination with Vis-spectrometry. The temperature governs the "calorimetric signature" of this process and the analysis of the heat released, in combination with other off-line techniques, allows to discern the sequence of the events involved and to quantify the heat released in each event. This rapid process can be tempered either by successive or continuous addition of the chain initiator. In the first case, the overall heat evolution is split into a number of smaller distinct sequences and spread over the time scale. At the same temperature, each sequence presents the same shape of heat evolution with the corresponding overall evolution. In the second case, a steady state of heat evolution is achieved, whose amplitude and half-width are proportional to the feeding rate of CI.

Spectrocalorimetric Screening for Complex Process Optimization

103

Concomitantly scanning one or more variables should be an alternative route for controlling both the average molecular weight and the particle size distribution. This aspect is still under investigation.

References 1. J. Pastrk, A. Zogg, U. Fischer and K. Hungerbiihler, Org. Process Res. Develop., 2001,5, 158. 2. S . L. Randzio, Ch. Stachowiak and J-P. E. Grolier, J. Chem. Thermodynamics, 2003, 35, 639. 3. F. Dan, J-P. E. Grolier, Setaram News, 2002, 7, i3. 4. B. Stratula-Vahnoveanuand C1. Vasiliu-Oprea, Polym. Plast. Techn-Eng.,2002,41(5) 981. 5 . S. Chrzczonowicz,M. Wlodarczyk and B. Ostaszewski, Makromol. Chem., 1960,38,159. 6. C1. Vasiliu-Oprea and F. Dan, J. Appl. Polym. Sci., 1996, 62, 1517. 7. F. Dan and C. Vasiliu-Oprea, Colloid Polym. Sci., 1998,276,483. 8. H. Sekiguchi, Ring-Opening Polymerization, Vol 2, J. Iving and T. Saegusa, (eds), Elsevier, London, 1984, 833. 9. J. Stehlicek and J. Sebenda, EM Polym. J., 1986,22, 769. 10. C1. Vasiliu-Oprea and F. Dan, J. Appl. Polym. Sci., 1997, 64, 2575. 1 1 . F. Dan and C1. Vasiliu-Oprea, J, Appl. Polym. Sci., 1998, 67, 23 1. 12. J-P. E. Grolier, F. Dan, S. Boyer, M. Orlowska, and S. L. Randzio , Znt. J. Thermophys., 2004,25 (2), 297. 13. C. A. Veith and R. E. Cohen, ACS Polym. Preprints, 1989, 31, 42. 14. J. H. Rushton, E. W. Costish and H. J. Evert, Chem. Eng. Progr., 1950,46, 395.

CHAPTER 9

Microcalorimetry for the Pharmaceutical Zndustry ANTHONY E. BEEZER

1 Introduction Calorimetry and thermal methods have been practised in the Pharmaceutical Industry for some This chapter is, however, only concerned with applications of isothermal heat conduction microcalorimetry in the Pharmaceutical Industry (Chapter 10 describes Industrial Applications of Flow microcalorimetry, a topic not covered here). A feature observed in recent years is the growth (albeit small) in the number of Companies that offer broadly equivalent-in-performance microcalorimeters that operate over a range of modes (e.g., Thermometric AB, Jiirfalla, Sweden; Setaram, Caluire, France; Calorimetry Sciences Corporation, American Fork, UT, USA). Thus, these manufacturers all supply instruments with nanowatt detection limits complete with a range of specialised options for titration, humidity control, perfusion, etc. (for a general review of sensitivities of different instruments, see ref. 3). In addition, all supply solution calorimeters, high sensitivity scanning calorimeters and, more recently, multi-channel instruments (e.g., Thermometric offers a 48channel instrument). These multi-channel calorimeters generally sacrifice some of their sensitivity and detection limit capacity in the pursuit of higher throughput rates. This chapter will not describe in any detail the operation, design and techniques of microcalorimetric experiments; it will, however, concentrate on the applications to which microcalorimetry has been put. Calorimetry has several advantages that are of significance to pharmaceuticals. It is non-invasive, non-destructive and capable of dealing with all phases and with mixed phases. Thus, it can accommodate solids, liquids, gases (and their combinations), suspensions, emulsions, creams, ointments, etc. However it is essential, in an industrial context, to have traceable and validated bases for the determination of target parameters whatever the experimental technique used. Microcalorimetry is no exception and a standard test and reference reaction has been described4that fulfils this need. The imidazole-catalysed hydrolysis of triacetin has, as a result of an international multi-laboratory study, been selected as this test and

Microcalorimetryfor the Pharmaceutical Industry

105

reference system. The determined values for the enthalpy and (second order) rate constant are: A@= -91.723.0 kJ mol-’ and k=2.80+0.10X lov6dm3 mol-’ s-l. Calorimetric techniques have been applied across the range of issues concerned with drug development from lead discovery to formulated medicine and in aspects of this process - discovery, characterisation, formulation development, process development and scaleup - that are the subject of this chapter. As most studies have been concerned with stability and compatibility issues, it seems logical to consider these applications in almost reverse order.

2 Stability Determinations Conventionally, stability testing is performed5 in an accelerated reaction regime i. e. at elevated temperatures (>323 K) and at controlled humidity (say 70% RH). Derived reaction rate constants are used to predict, through application of the Arrhenius’ equation, the rate constant at the proposed storage conditions of the medicine, This extrapolation depends on the constancy of the reaction mechanism over the temperature range concerned. It would clearly be better to have direct determination of reaction rate constants under the storage environmental conditions. Until around 1995, it was unusual to find kinetic data determined from isothermal microcalorimetric studies - and this was especially true for long, slow reaction systems. However, the modern microcalorimeter has excellent long-term stability ( 1 pW or better over 24 h) and hence slow reactions can readily be investigated even those for which a complete reaction is not observed. Willson et aZ.”* showed that quantitative kinetic and thermodynamic data could be determined for reactions that had half lives of up to 2500 years. Commencing with a conventional (here simple for clarity) rate expression such as Rate=dxdt=k(a -x)n

(1)

where k is the rate constant, a the initial quantity (it is trivial to incorporate a volume term, however, as the enthalpy H is a molar term; it is more ~traightfonvard~*~ to outline the equations with quantity), x the extent of reaction to time t and, recognising that H=q/x where q is the number Joules associated with reaction to produce x mole of product and Q the number of Joules associated with complete reaction then

It is this calorimetric form of this equation (and the more complex equations that can be developed and used to describe sequential, parallel and more complex reacting systems) that can be analysed to yield values for the target parameters, n, k, H , K , G, S and Ea. References 6 and 7 provide details on the derivation of calorimetric kinetic equations and describe how to manipulate calorimetric data to determine equilibrium constants, Gibbs functions, entropies and activation energies. The approach described shows that, under normal storage conditions (“room temperature” i. e. 298 K and ambient humidity), it is possible, from only 50 h of power time (the calorimetric output is of W vs. t ) data, through these techniques, to distinguish between a

106

Chapter 9

reaction with a (first order) rate constant of 1 X lo-" s-l and a reaction with a rate constant of 2~ lo-" s-l. In the most detailed studies8*10published to date raw drug, benzoyl peroxide, its compatibility with selected excipients (see Table 1) and its presence at a range of strengths in formulated materials have been investigated (see Table 2). It is notable that it is possible to select, from a range of excipients from, here, only 12 h study of each system, those that contribute to greater drug stability and those that accelerate drug degradation. Moreover, Table 2 and Figure 1 show that, over the temperature range studied, there exist two regions exhibiting different activation energies. These observations emphasise the conclusion that a simple dependence on an Arrhenius' extrapolation from high-temperature studies can be unreliable. In the worst case, the derived rate constant for a 1% formulation in the absence of Carbopol is 9 orders of magnitude in error at 298 K. Table 1 Microcalorimetric kinetic stability studies of degradation of BPO in the presence of excipients in water at 313 K First-order rate constant, k (s-l )

Excipient -

2.00X 10-7+7.13X lop9 1.04X 10-7t2.64X 1.11x 1.51x lo-* 3.40X 1.62X 8.42X 10-956.76X lo-'' 3.41 X 10-723.75X lop9 7 . 2 4 1~0 - 9 ~ 5 . 0 010-lo x 5 mg ml-l in water cooled to 4 "C.

2% Poly trap 1% Sipernat 0.15% Monawet 5% Propylene glycol: 2.5% glycerine: 0.15% monawet 5% Propylene glycol: 2.5% glycerine: 2% polytrap Pluronic" P234 "(PO)-2250 M(EO)-1500. T, 33.43 and MW,,,,,,,=3750.

Table 2 Microcalorimetric studies of solid BPO and formulations (first-order rate constants: s - l ) Temperature

Placebo (reference talc) 5.03X10-8 15 52.30X lo-' 20 1.16X10-9 2.61X10-9 5.16X10-8 ? 1 . 0 1 ~ 1 0 - ~t1.01X10-8 +1.60x10-s 5.29x 1.36x 25 4.33~ ?3.59x10-9 +2.51X10-8 t8.73x10-9 5.65~ 2.34x 30 6.85~ 5 1.33X lo-* 5 1.64X 28.91 X lop8 6.94~ 5.27x lo-* 35 2 . 0 0 lo-* ~ +2.83x10-* 1+_6.36x10-9 +7.14X10-8 1.2OX lop7 3.43X 40 5.84X 27.70~ 52.53~ 22.1 1X 1.89X10-' 45 1.07X10-7 7.68X10-7 +8.49X1OP9 +1.94X10-7 +3.68X10-8 5 (52), E, (kJ mol-') 138 (?7) 172 ( t 1 4 ) 82 (23)

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Each of the reported experiments took 12 h observation and the whole study was completed in significantly less time (around 2 weeks) than a conventional high-temperature study would require. Moreover, the studies were performed on the “as is” formulated medicines without the need to develop assay systems (e.g. HPLC analyses). Further studies of this type on, for example, solid-state systems” and solutionphase systems1*can be found in the literature. The problems associated with freeze drying of peptides and proteins for therapeutic use have also received calorimetric attention recently - particularly, attempts to understand and interpret the dynamics of amorphous s01ids.l~Structural relaxation time is a measure of molecular mobility involved in enthalpy relaxation and thus is a measure of the dynamics of amorphous (glassy) solids. These dynamics are important in interpretation of the physicochemical properties and reactivities of drugs in amorphous formulations. The authors conclude that “microcalorimetry may provide data useful for rational development of stable peptide and protein formulations and for control of their processing”.

3 Characterisation of Physical FordAmorphous Content Recent review^'^.'^ have highlighted the capacity of microcalorimetry to provide sensitive and detailed information on the effects of processing and manipulation of drug substances on their physicochemical properties and the extent of amorphicity present in a sample. Using lactose as an example, the amorphous content of a sample has been determined to better than 0.5% (PXRD studies can usually only determine amorphous contents at 10%).The experimentation is simple and rapid. Surface

108

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energetics,16cry~tallisation,’~ dissolution, etc. are also discussed in detail in the cited reviews. More recently, it has been shown1*that microcalorimetry can be used successfully to study the kinetics and thermodynamics of a polymorphic transition. The object of the study was Seratrodast (a Takeda Chemical Industries product). The conclusions of the study were that: microcalorimetry can, with sensitivity discriminate between those outputs that are the sum of individual reactions associated with particular size fractions, significant saving in time can be made in such studies and no ancillary analytical method is required to facilitate the study.

4 Drug/Receptor Studies by Titration Calorimetry (ITC) Again, there are several review~’~-~l that treat this particular area in detail. Titration calorimetry is unique in that it allows direct determination of ARG (= -RT In K , where K is the equilibrium constant); A f l and A$ from one experiment. Thus, the characterisation of binding interactions that are important in improving understanding of biomolecular recognition and informing the process of rational drug design processes is made more amenable to study. For details, refer to these reviews but it is worth noting here that major issues remain - these are mostly associated with data interpretation and centre upon the role of structural changes and their impact upon the derived thermodynamic parameters. A further limitation associated with such isothermal titration calorimetric (ITC) studies is that the current equipment allows only one experiment to be conducted at a time - multi-channel instruments of normal design do not exist.

5 Multiple Miniature Calorimeters and their Potential As noted in Section 4 ITC, studies are limited in terms of throughput; thus, it is unlikely that these types of studies will contribute significantly to high-throughputscreening (HTS). The impact of combinatorial chemistry in drug synthesis makes the demand on HTS ever greater. However, modern technologies (nanotechnology) suggest that the landscape of calorimetry is changing. Whilst it is not possible to cite industrial applications of this new technology, it is clear that its potential in industrial contexts is enormous. To date, 3 Web sites have been located that offer high throughput calorimetric experimentation and indeed one of these sites explicitly offers HTS on a contract basis. Xensor (www.xensor.nl) offers “chip” calorimeters - four calorimetric units on a 20 mm square silicon chip. These units are simply sold to interested calorimetrists who can assemble them in any array and numbers required. The chips are cheap and relatively robust. Veeco (www.veeco.co.uk) can supply an eight bimetallic cantilever system that can be used to study, for example, protein ligand binding. The system is based on Atomic Force Microscopy technology and offers considerable potential for calorimetric development. Vivactiss (www.vivactiss.com) offer 96, 384 and 1536 well differential calorimeters for contract work in the area of “enzymekatalyst discovery and optimisation and physicochemical characterisation”. These instruments have very small heat capacities and hence require little in the way of equilibration. They work with nanolitre quantities

Microcalorimetry for the Pharmaceutical Industry

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of materials and, by design, will only permit the study of rapid reactions - they have no long-term stability properties.

6 Conclusions This overview has attempted to show that calorimetry has found applications in all areas of the pharmaceutical industry from discovery through receptor site binding to characterisation, on through compatibility to formulation and stability. The nanocalorimeters should find their real application in HTS whilst the more macro (but still micro!) calorimeters will still be needed, because of their better long-term stability to define the physicochemical properties of the selected pharmaceutical systems.

References 1. J. L. Ford and P. Timmins, Pharmaceutical Thermal Analysis, Techniques and Applications, Ellis Horwood, Chichester, 1989. 2. J. L. Ford and R. J. Willson, in Handbook of Thermal Analysis and Calorimetry, Vol 4, R. B. Kemp (ed), Elsevier Science, Amsterdam, 1999. 3. L. D. Hansen, Pharm. Technol., 1996, 20, 64. 4. A. E. Beezer, A. K. Hills, M. A. A. O’Neill, A. C. Morris, K. T. E. Kierstan, R. M. Deal, L. J. Waters, J. Hadgraft, J. C. Mitchell, J. A. Connor, J. E. Orchard, R. J. Willson, T. C. Hofelich, J. Beaudin, G. Wolf, F. Baitalow, S. Gaisford, R. A. Lane, G. Buckton, M. A. Phipps, R. A. Winneke, E. A. Schmitt, L. D. Hansen, D. O’Sullivan and M. K. Parmar, Thermochim. Acta, 2001,380, 13. 5. M. E. Aulton (ed), Pharmuceutics: The Science of Dosage form Design, Churchill Livingstone, Edinburgh, 1988. 6. R. J. Willson, Isothermal Microcalorimetry : Theoretical and Experimental Developments, Ph.D. Thesis, University of Kent, 1995. 7. R. J. Willson, A. E. Beezer, J. C. Mitchell and W. Loh, J.Phys.Chem., 1995, 99, 7108. 8. A. E. Beezer, A. C. Morris, M. A. A. O’Neill, R. J. Willson, A. K. Hills, J. C. Mitchell and J. A. Connor, J. Phys. Chem .B, 2001,105, 1212. 9. F. Zaman, A. E. Beezer, J. C. Mitchell, Q. Calrkson, J. Elliot, M. Nisbet and A. F. Davis, Znt. J. Pharm., 2001, 225, 135. 10. F. Zaman, A. E. Beezer, J. C. Mitchell, Q. Clarkson, J. Elliot, A. F. Davis and R. J. Willson, Int. J. Pharm., 2001, 227, 133. 11. M. A. A. O’Neill, A. E. Beezer, A. C. Moms, K. Urakami, R. J. Willson and J. A. Connor, J. Therm. Anal. Cal., 2003,73,709. 12. R. J. Willson, A. E. Beezer and J. C. Mitchell, Znt. J. Pharm., 1996, 132, 45. 13. J. Liu, D. R. Rigsbee, C. Stoltz and M. J. Pikal, J. Pharm. Sci., 2002,91, 1853. 14. M. A. Phipps and L. A. Mackin, Pharm. Sci. Technol. Today, 2000,3,9. 15. G. Buckton, Pharm. Tech. Europe, Nov. 2001,44. 16. G. Buckton and A. E. Beezer, Int. J. Pharm., 1988,41, 121. 17. V, P. Lehto and E. Laine, Pharm. Res., 1997, 14, 899. 18. K. Urakami and A. E. Beezer, Znt. J. Pharm., 2003, 257, 265. 19. J. E. Ladbury and B. Z. Chowdhry (eds), Biocalorimetry, Wiley, Chichester, 1998. 20. S. E. Harding and B. Z. Chowdhry (eds), Protein-Ligand Interactions, Oxford University Press, Oxford, 2001. 21. G. A. Holdgate, Bio Techniques, 2001,31, 164.

CHAPTER 10

Isotheimal Flow-Microcabrimetry: Principles and Applications for Industry MICHAEL A. A. O'NEILL

1 Introduction The use of isothermal microcalorimetry is now widely accepted as a powerful tool for the study of a vast array of systems within industry. Microcalorimetry is routinely used for compatibility studies' (active-excipient, excipient-excipient interactions), stability213(shelf-life), degradati~n,~ physical properties such as polymorphism5 and crystallinity,6 and can be used to probe surface characteristics of solid material^;^ some of these concepts are dealt with in more detail (with particular attention to pharmaceutical issues) by Beezer in Chapter 9. Generally, the information sought on a given system is related to the kinetics of that system (e.g. shelf-life). It is essential therefore that accurate/quantitative information can be derived for parameters such as rate and the rate constant. Also important is the quantity of material undergoing change in the system, which can be explored through the magnitude of the calorimetric signal and calculation of the enthalpy for that reaction. Knowledge of these parameters then allows a host of other thermodynamic and kinetic information to be derived from the system under 8-1 A large proportion of calorimetric studies are performed using statidbatch techniques, however, studies using batch mode techniques are generally unsuitable for the study of reactions that have a relatively short half-life such that the reaction is complete before the preparatory work required for static systems is carried out3 (approximately 1 h), such as some enzyme/substrate systems, and the activity of drugs and antibiotics on microbiological systems. Static techniques can be problematic for systems where sedimentation can be an issue e.g. cellular systems.'* These types of system are difficult to study using conventional techniques and usually require invasive sampling or study under non-relevant conditions (e.g, plating out of micro-organisms). Flow calorimetric techniques are also necessary for the study of systems in which the surface is to be

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probed by passing a gas or vapour over it, such as surface characterisation of solids for e ~ a m p l eFor . ~ studies on systems such as those described above, it can be advantageous to employ flow-through calorimetric techniques. Microcalorimetry provides a rapid, non-invasive approach to the study of such systems; moreover, flow calorimetry has a sensitivity that is superior to many other traditional techniques allowing accurate, quantitative information to be derived from the raw data.

2 Theory Calorimeters rely on the observation that all processes, chemical and physical, involve an exchange in heat energy to or from their surroundings. The calorimeter is non-specific in its operation, i.e., it will monitor and record all reactions that occur. There are several other advantages conferred by isothermal microcalorimetry. The isothermal Thermal Activity Monitor (TAM, Thermometric AB, Jarfalla, Sweden) has a combination of sensitivity and versatility that is superior over many other analytical techniques. It has been reported13 that the TAM has sufficient sensitivity to monitor slow reactions with lifetimes of up to 10,000 years. It has also been reported13that only 50 h of data are required to discriminate between a reaction that has a first-order rate constant of lXIO-lls-l and a reaction with a first-order rate constant of 2X 10-l s-l. The calorimeter is non-invasive to the sample under study and hence the reaction is not influenced by the calorimeter, nor does the reactant require special preparation before study. The sample can be in any physical state: solid, liquid, gas or any combination. The main advantage of the calorimeter is its ability to restore a wide variety of information that can be used to characterize a reaction system. Calorimetric data, with the aid of the relevant equations, allow a number of parameters to be determined. These parameters can then be used to characterize a reaction in terms of the reaction rate and mechanism. Each parameter contributes to the characterization by defining different aspects of the kinetic, thermodynamic and mechanistic properties of the reaction. By comparing this information between different types of reaction schemes and mechanisms, it might be possible to determine the trends or patterns that can be applied to unknown compounds or to facilitate the design of new compounds. Each of the parameters determined from the calorimeter makes a unique contribution to the understanding of the rate and mechanism of reaction. The mechanism can be deduced from reaction order, n, activation energy, Ea,change in enthalpy, H, change in entropy, AS, the quantity of material reacted and the experimental design. The order of reaction governs the dependency of rate on the reaction components. Determination of the reaction order with respect to individual components will yield the dependency of the rate with the individual reactants. Ea,H and AS may yield the information required to elucidate the process that is occurring.

3 Flow Calorimeter Design There are a number of commercially available microcalorimeters that are routinely employed within the chemical industry. Some are specifically designed to be operated

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solely as a flow calorimeter, such as the dedicated flow insert designed by Therm~metric'~and those manufactured by Micro~cal.'~Others, such as the Setaram16micro-DSC 111, have been designed such that they can be adapted for use as flow calorimeters through application of specially modified calorimetric ampoules. Calorimeters such as those developed by Thermometric and Setaram operate on a differential basis, i.e. they have a twin channel configuration incorporating a reaction (sample) channel with a reference channel. This arrangement gives, as the calorimetric output, a signal that is the differential of the sample and reference signals. This confers the advantage of reducing noise in the signal caused by external factors such as temperature fluctuations, electrical interference, etc. Moreover, if a suitable reference material is chosen, then the observed signal will be a direct reflection of the thermal process under investigation. Ideally, for flow calorimetric studies, one would use an inert reference solution flowing at a rate identical to the sample solution. In practice, this is difficult to achieve, therefore, one can either have the reference side filled with solvent and kept static or have it empty and seal it off. In this instance and for calorimeters that do not operate on the differential principle (such as those manufactured by Microscal), it becomes necessary to record a baseline before the experiment is conducted by flowing the solvent (or inert gas in the case of Microscal instruments) prior to the experiment. It should be noted that, in principle, all the flow calorimeters discussed here are capable of operating both in the gashapour phase and in the solution phase. However, the calorimeters manufactured by Setaram and Thermometric are generally used for solution-phase studies, whereas the Microscal instruments are designed specifically to facilitate gashapour-phase studies. For the purposes of this discussion, gaseous-phase flow calorimetry will centre around a consideration of the Microscal instruments, and solution-phase calorimetry will centre around the Setaram and Thermometric instruments.

3.1 Solution-Phase Flow Calorimeters There are two possible modes of operation for flow microcalorimeters used for solution-phase ~ t u d i e s . ~ J ~

3.1.1 Flow-Mix The flow-mix mode of operation allows a fast reaction (i.e. the reaction is rapid in comparison with the time constant of the instrument). Here, the reaction is initiated inside the calorimetric cell by the mixing of the pre-equilibrated reagents as they enter the cell, Figure 1. The effluent is not recycled but can be retained for further analysis. Note that it is not possible to distinguish kinetically between different rapid (with respect to the time constant of the instrument) reactions, i.e. the calorimetric signal appears zero order in nature regardless of the actual reaction order.

3.1.2 Flow-Through The flow-through cell is only suitable for relatively slow reactions with long half-lives (i.e. the reaction is slow in comparison with the time constant of the

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Figure 1 Schematic of mixing cell

Thermostat h

I

Calorimetric Ce 11 To Waste

React ion# Reaction

Vessel

Figure 2 Schematic for the flow through cell

instrument). In this mode of operation, the reactants are pre-equilibrated external to the calorimeter and the reaction initiated also external to the calorimeter. The reacting solution is either allowed to flow-through to waste (or further analysis) or is recycled to the reaction vessel, Figure 2. In both instances, the calorimeter records the change in heat through semi-conducting thermopiles positioned around the calorimetric cell. The data are recorded as a function of time and is displayed as J s-l.

3.2 Gaseous-Phase Flow Calorimeters The basic operation7 of the gaseous flow calorimeters is essentially identical to that of the flow-through solution-phase calorimeters with an external gas/vapour source that is passed, through a single calorimetric cell, across the solid of interest and the resulting heat change measured. For these instruments, the detectors are thermistors in direct contact with the solid under study. The form of the returned data is volts as a function of time. The signal can be converted to J s-l via a calibration constant.

4 Flow Calorimetric Equations As already mentioned all chemical and physical processes result in a change in heat. Depending on whether the process is endothermic or exothermic, heat will flow from, or to, the heat sink in order to maintain isothermal conditions. This heat

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flowlthermal power is measured as a change in heat energy (joules) per unit time (seconds) by the thermopiles and is expressed as dqldt (or $); it has units of J s-l (or watts). The isothermal microcalorimeter can therefore yield two types of data: heat flow (a kinetic term) and the time-independent reaction enthalpy change (a thermodynamic term). It is possible, in principle, then to derive thermodynamic and kinetic information from the raw calorimetric data. The equations discussed here are specifically for solution-phase flow calorimetric experiments. Equations that describe calorimetric output can be d e r i ~ e d ~ *from ' ~ J *basic kinetic expressions such as that seen in Equation ( 1 ) .

dx

-=k( C-xY

dt

where dxldt is the rate of reaction (mol dm-3 s-'), k is the appropriate rate constant, C is the initial concentration of the reagent (mol dm-'), x is the quantity of the reagent reacted to time t (mol dm-3), n is the reaction order and t is the time (s). Calorimetric equations that describe the output for flow calorimetric data were first derived by Beezer and Tyrell some 30 years ago.197 2o Equations (2), (3) and (4) describe first-, second- and zero-order reactions systems, respectively.

dq dt

-=

kC2HVc (1 + k t C + k z C ) ( l +ktC) d4 -=-kVcH dt

(3) (4)

where: dqldt is the calorimetric output (J s-'), F is the flow rate (dm3 s-l), H is the enthalpy of reaction (J mol-'), k is the appropriate rate constant, z is the residence time (s) and, Vc is the effective thermal volume (dm3). Equations (2)-(4) show that elucidation of the rate constant requires prior knowledge of the residence time, z (the time the reacting solution spends in the calorimetric vessel) and hence the thermal volume, Vc (i.e. the operational volume of the calorimeter). Determination of reliable values for rate constants and enthalpy changes from experimental data (power, time data) for reacting systems, studied by flow microcalorimetry; therefore, it requires an accurate and precise value for z at any given flow rate. This is determined from Equation (5) through knowledge of F and Vc.

Flowing reacting systems necessarily result in transport of heat from the detection area. Thus, whilst the physical volume of the cell can be known, this volume may not be the effective or thermal volume nor indeed the zero-flow-rate volume of the cell, Figure 3.

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OUT

Figure 3 Schematic for the flow-through calorimetric cell

5 Validation of Flow-Through Microcalorimeters The determination of an effective or thermal volume is a generic problem associated with all flow microcalorimeters operating in flow-through mode. The thermal volume, Vc,is the volume that the calorimeter detects as the medium flows through the vessel. It is affected by several different factors but the most important is the flow rate of the medium since, as the reacting solution flows through the vessel, a proportion of the heat will be carried out of the calorimetric cell retained in the medium, undetected21’22 (i.e. related to the heat capacity of the flowing medium). As the flow rate is increased, more heat will be carried out by the flowing medium and hence a smaller thermal volume will be observed. Conversely, as the flow rate is decreased, less heat will be lost such that as the flow rate approaches zero, the apparent thermal volume will approach a limiting value. It is clear therefore that, if quantitative thermodynamic and kinetic information is to be obtained from flow calorimetric data, there is some means of validating the calorimetric output in terms of the flow rate dependence of thermal volume. This has been achieved,21*22 for solution-phase flow calorimeters, through the use of a chemical test and reference reaction: the base-catalysed hydrolysis of methyl 4-hydroxy-benzoate (methyl paraben). Note that for reactions that are rapid (with respect to the time constant of the instrument), this validation is not necessary since the reaction is initiated inside the calorimetric cell, and has no measurable kinetic dependence (i. e. instantaneous) and the required thermodynamic parameters can be calculated through the use of a calibration constant. S t ~ d i e s using ~ , ~ ~the, ~base-catalysed ~ hydrolysis of methyl paraben (BCHMP) test and reference reaction have been conducted with a variety of different solutionphase flow calorimeters. The results obtained from these studies have shown that the flow rate dependency of the thermal volume is different for each of the instruments used and indeed for each experimental arrangement (e.g. sample and reference cell set-up). The determined value for Vc can differ by as much as 15% (over a range of experimental flow rates) from the nominal engineered volume (typically approximately lml). This effect can be minimised by careful design of the flow cell and also by careful consideration of the sample and reference cell arrangements (more details can be found in ref. 22 and references therein). Knowledge of how thermal volume varies with flow rate allows the physical characteristics of the flow cell to be investigated. For example, it is possible to simulate

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.l.O

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Figure 4 A 3 0 representation of the effect of varying theflow rate and the initial concentration, [A,], on the observed initial output for ajixed rate constant

data for a range of flow rates and investigate the interdependence of flow rate, rate constant and concentration. It should be noted that control is achievable over the flow rate, the concentration and to a lesser extent over the rate constant, the latter by changing the temperature. However, it is not possible to alter the enthalpy, assuming that it has no temperature dependence (it is possible, in some circumstances, to magnify the enthalpy by careful choice of an appropriate buffer system). The changes in the concentration and the rate constant must also satisfy the requirement that the mechanism of reaction is not affected as a consequence. Changing the flow rate will have no impact on the mechanism since it is purely a physical change to the system. Figure 4 demonstrates how the initial signal (Do (the maximum possible signal for a single reaction process) varies with the flow rate for fixed enthalpy and rate constant and varying initial concentration of the reagent. If the upper and lower detection limits for the calorimeter are known, then it is possible to choose the combination of Values for the concentration and the flow rate that yield an appropriate initial signal. It is apparent from these studies that all flow microcalorimeters should be validated in this way, if quantitative information is sought. To date,23similar studies have not been conducted for validation of gas-phase flow calorimeters. It is likely, however, that the problems associated with the removal of heat by the flowing medium will not be as significant in such systems since the associated heat capacity is much lower for gaseous/vapour systems and hence the amount of heat lost in this way is significantly reduced.

6 A Simple Flow Calorimetric Experiment It has been noted throughout this chapter that flow microcalorimetry can be used for a wide variety of reaction systems. This section will attempt to describe a simple,

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solution-phase, calorimetric experiment and discuss methods for the elucidation of relevant thermodynamic and kinetic parameters from the raw calorimetric data. In general, the object of most calorimetric experiments is the quantification of thermodynamic and kinetic parameters. Several recent developments now permit the recovery of such parameters through calculational methods. These are not discussed in detail here; for a more detailed discussion, see refs. 8-1 1. A good example of the range of parameters available from flow calorimetric data can be found from the study of enzymehubstrate systems. The kinetic nature of enzyme systems has been previously described by Michaelis and Menten.24In the treatment discussed here, the parameters sought are the enthalpy, rate constant, Michaelis constant and the enzyme activity. The following example describes a study on the well-known enzyme substrate system, uredurease.

6.1 Experimental Protocol A variety of pH 7.0 buffered, stock solutions of urea at various concentrations are prepared. The concentrations of urea chosen are such that the complete kinetic profile of the reaction can be observed (i.e. from first-order through mixed-order and finally to zero-order kinetics). A 50 ml aliquot of the urea solution is pre-thermostated to the operational temperature of the calorimeter; for these experiments the calorimeter is housed in a constant temperature environment and operated at 25 "C. The urea solution is then run in a continuous loop, at a known flow rate, until a stable baseline is achieved. This solution is then inoculated with 4.55 ml of a standard, fixed concentration, urease solution (also buffered to pH 7.0 and pre-thermostated) and the resulting calorimetric output recorded as a function of time. This is repeated for all concentrations of urea. Figure 5 shows a selection of typical calorimetric outputs for this enzyme system. Calorimetric data are recorded as a function of time and hence the rate of reaction can be determined from the calorimetric output. One of the distinguishing features of Michaelis-Menten type kinetics is the form of the plot of rate vs. substrate concentration. Figure 6 shows such a plot and it is clear that it has a form that is consistent with that predicted by the Michaelis-Menten kinetic theory.

7

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Figure 5 Calorimetric outputs forfirst,-mixed- and zero-order uredurease enzyme reactions

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Figure 6 Max rate vs. substrate concentrationfor the uredurease enzyme reaction

6.2 Determination of Thermodynamic and Kinetic Parameters from Calorimetric Data The form of the flow calorimetric equations described earlier can also be manipulated3, 25 to incorporate Michaelis-Menten parameters such as the Michaelis constant, KM,as well as yielding the first order rate constant and enthalpy for the overall reaction, Equation (6). 20+

where KMis the Michaelis constant and [atot is the total enzyme concentration (mol dm-3). Note that it is trivial to calculate the first-order rate constant from the calorimetric data; from Equation (6), it is simply the slope of the line for a plot of In dq/dt vs. t. However second-order rate constants require a more complicated approach and are accessible through a new calculational approach of data pairing, this is not considered further and more details can be found in refs. 11 and 26. Once the rate constant for reaction is known, it then becomes possible to calculate the enthalpy for the reaction for all time points of the reaction and obtain an average for the lifetime of the first-order reaction. Once the enthalpy is found, it is possible to calculate the value of k[EJ,,, from a consideration of the zero-order output and hence the Michaelis constant is now accessible from the slope of the first-order In dqldt vs. t plot (the slope of this plot is equal to -K[EJ,,JK,). It must be emphasized that this analysis is not possible without prior knowledge of the thermal volume of the calorimetric cell and therefore the validation routine described above must be conducted if accurate values for the enthalpy in particular are to be obtained. This new method also allows the effects of inhibitors of enzyme activity and their mechanism of action to be studied using flow calorimetry. More details of this study, and others, can be found in refs. 3, 11 and 26.

l

~

l

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119

7 Further Applications of Flow Microcalorimetry The examples here provide a ‘snapshot’ of the range of potential applications that flow microcalorimetry offers. Using the techniques described earlier, a number of systems have been explored through flow-calorimetric techniques. Early work in this area of calorimetry centred on the study of micro-organi~ms.~~~ 28 It was shown that, in principle, it is possible to distinguish between different microbial species by the shape of the calorimetric curve produced during organism growth. Quantification of the number of organisms present in the system is possible by comparison of the areas under the curve for an uncharacterized system with that of a known standard. Since calorimetric data are a measure of rate, it is possible to link heat output with the metabolic activity of human (and animal) cells. Both types of cell have been extensively studied by flow calorimetry; one example29is the study and optimization of Chinese hamster ovary cells producing recombinant interferon y Flow calorimetry has also been used to investigate dosage forms. Dissolution of tablets under various simulated conditions has been explored,30for example. Here, the tablet is presented with various solutions designed to mimic conditions in the gastro intestinal (GI) tract (pH 7 buffer) and stomach (pH 3 buffer for fasting and lipid solutions for fatty meals). The rate of dissolution can then be estimated for conditions in vivo. Gas-phase flow calorimetry has been used on a number of systems to investigate a variety of topics. Some examples are discussed below. The use of gas flow techniques are now widely recognized as important for the study of ad~orptiorddesorption~~~~ phenomena at solid interfaces. Such studies are particularly important for the characterization of catalytically active sites on the surfaces of solids. One particular example is a of the adsorption of ammonia onto activated carbon. It can be shown that adsorption of ammonia consists of reversible and irreversible steps and that these steps can be attributed to physisorption for the former and adsorption on particular chemical groups for the latter. The heats of adsorption for various sites can be measured and the data can be used to reveal the existence of a wide distribution of acid sites that are accessible to ammonia to varying degrees. Another example33of the use of this technique can be found in the determination of the heats of adsorption of water onto microporous carbons. Here, the effects of different gaseous carriers on the heat of adsorption and the amount of water adsorbed are investigated. It has been shown that adsorption of water on this type of surface is strongly influenced by the nature of the carrier gas from which the adsorption takes place.

8 Conclusions Isothermal flow microcalorimetry is now an indispensable tool for the study and characterization of a variety of reaction systems within the chemical industry. It permits the study of these systems as a function of time in a non-destructive, non-invasive manner and hence allows accurate, quantitative information for both the

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thermodynamic and kinetic character of the system under scrutiny and is only limited by the sensitivity of the instrument in question. This information is vital for the determination of essential parameters such as shelf-life and compatibility. Flow calorimetry has also been used to probe surface characteristics such as area, active sites, pore size, etc., information that is essential for the discovery of new catalysts and production of efficient separation systems for example. Calculational methods for the accurate determination of thermodynamic parameters (particularly for solution-phase calorimetry) have only recently become available through the BCHMP chemical test and reference reaction. It is likely therefore that values reported for enthalpy, for example, are likely to be significantly in error for some of the earlier work using flow calorimetry. These errors can be rectified, however, through the calculation of the thermal volume and relevant adjustment of the calorimetric data. Microcalorimetry is now firmly placed as an important tool within the chemical industry. As with all current technology, the trend is to produce ever smaller instruments that allow large numbers of parallel experiments to be conducted. This is evident through the d e v e l ~ p m e nof t ~microscale ~ ~ ~ ~ calorimeters (so-called ‘calorimeter on a chip’); such developments, however, do not preclude the need for flow instruments and indeed such microscale instruments can, and almost certainly will, be adapted for use as flow calorimeters.

References 1. F. Zaman, A. E. Beezer, J. C. Mitchell, Q. Clarkson, J. Elliot, M. Nisbet and A. F. Davis, Znt. J. Pharm., 2001, 225, 135. 2. S . Gaisford and G. Buckton, Thermochim. Acta, 2001,380, 185. 3. M. A. A. O’Neill, Ph.D. Thesis, University of Greenwich. 2002. 4. M. Riva, D. Fessas and A. Schiraldi, Thermochim. Acta, 2001, 370, 73. 5. K. Urakami and A. E. Beezer, Znt. J. Pharm., 2003,257,265, 6. D. Gao and J. H. Rytting, Znt. J. Phamz., 1997, 151, 183. 7. A. J. Groszek, Thermochim. Acta, 1998,312, 133. 8. R. J. Willson, A. E. Beezer, J. C. Mitchell and W. Loh, J. phys. chem., 1995, 99, 7108. 9. A. E. Beezer, A. C. Morris, M. A. A. O’Neill, R.J. Willson, A. K. Hills, J. C. Mitchell and J. A. Connor, J. Phys. Chem., 2001,105, 1212. 10. M. A. A. O’Neill, A. E. Beezer, A. C. Morris, K. Urakami, R. J. Willson and J. A. Connor, J. T h e m . Anal., 2003, 73, 709. 1 1 . R. J. Willson and A. E. Beezer, Thermochim. Acta, 2003,402, 75. 12. Y. H. Guan, P. C. Lloyd and R. B. Kemp, Thermochim. Acta, 1999,332,211. 13. R. J Willson, Ph.D. Thesis. University of Kent. 1995. 14. www.Thennometric.com 15. www.Microsca1.com 1 6. www.Setaram.com 17. LKB 10700-1 Flow Calorimeter Operation Manual. 18. A. K. Hills. Ph.D. Thesis, University of Kent at Canterbury, 2001. 19. A. E. Beezer, H. J. V. Tyrell, Sci. Tools, 1972, 19, 13. 20. A. E. Beezer, H. Tyrrell, and T. Steenson, Protides of the Biological Fluids, Vol 20, H. Peeters, (ed), Pergamon Press, New York, 1972, p. 563.

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21. M. A. A. O’Neill, A. E. Beezer, C. Labetoulle, L. Nicolaides, J. C. Mitchell, J. Connor, J. Orchard, R. B. Kemp and D. Olomolaiye, Thermochirn. Acta., 2003, 399, 63. 22. M. A. A. O’Neill, A. E. Beezer, G. Vine, C. Labetoulle, L. Nicolaides, R. B. Kemp, D. Olomolayie, P. L. 0. Volpe and D. Oliveira, Thermochim. Acta., 2003,413, 193. 23. A. J. Groszek, Personal communication, 2003. 24. L. Michaelis and H. Menten, Biochem Z., 1913, 49, 333 25. M. A. A. O’Neill, A. E. Beezer, J. C. Mitchell, J. Orchard and J. A. Connor, Thermochim. Acta, 2003, in press. 26. M. A. A. O’Neill, Thermochim. Acta, 2003, in prep. 27. A. M. James, Thermal and Energetic Studies of Cellular Biological Systems, Wright, Bristol, UK, 1987. 28. A. E. Beezer, R. D. Newell and H. J. V. Tyrrell, J. Appl. Bacteriol., 1976, 41, 197 29. R. B. Kemp, Thermochirn. Acta, 2001,380,229. 30. L. J. Ashby, A. E. Beezer and G. Buckton. Int. J. Pharm., 1989,51, 245. 31. G. J. Zajac and A. J. Groszek, Carbon, 1997,35, 1053 32. M. Domingo-Garcia, A. J. Groszek, F. J. Lopez-Garzon and M. Perez-Mendoza, Appl, Catal. A , 2002,233, 141. 33. A. J. Groszek, Carbon, 1997,39, 1857 34. C. Auguet, J. Lerchner, P. Marinelli, F. Martorell, M. Rodriguez de Rivera, V. Torra and G. Wolf, J. Therm. Anal., 2003, 71, 951. 35. C. Auguet, J. Lerchner, V. Torra and G. Wolf, J. Therm. Anal., 2003, 71,407.

CHAPTER 11

Transport Properties and Industry WILLIAM A. WAKEHAM AND MARK J. ASSAEL

1 Introduction The transport of mass, momentum and energy through a fluid are the consequences of molecular motion and molecular interaction. At the macroscopic level, associated with the transport of each dynamic variable is a transport coefficient or property, denoted by X, such that the flux, J, of each variable is proportional to the gradient of a thermodynamic state variable such as concentration or temperature. This notion leads to the simple phenomenological laws such as those of Fick, Newton and Fourier for mass, momentum or energy transport, respectively,

J=-XVY

(1)

Here, Y is the appropriate state variable conjugate to the flux J and X, and depends on the thermodynamic state of the system. These linear, phenomenological laws are fundamental to all processes involving the transfer of mass, momentum or energy but, in many practical circumstances encountered in industry, the fundamental transport mechanisms arise in parallel with other means of transport such as advection or natural convection. In those circumstances, the overall transport process is far from simple and linear. However, the description of such complex processes is often rendered tractable by the use of transfer equations, which are expressed in the form of linear laws such as J = - CAY

(2)

Here, the transport coefficient, C, is not simply a function of the thermodynamic state of the system but may depend on the geometric configuration of the system and the properties of surfaces for example. We are concerned here with the transport properties, X, of materials, which depend on thermodynamic state of the material only. In practical situations, the transport coefficients, C, will often have been expressed as correlations with a parametric dependence on the properties, X .

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The transport properties of a material determine the rate at which an initially nonuniform thermodynamic state evolves towards a uniform state over time. For this reason, from an industrial point of view, within the context of equipment or process design, the thermodynamic properties of a fluid often determine the feasibility of what is proposed, whereas the transport properties essentially determine how large the equipment or process unit must be or the time scale of the operation. This essential difference is responsible for the fact that the transport properties have, traditionally, been less emphasized in process design activities. However, as the needs for process integration and energy minimization grow, there is a tendency to examine more carefully the effects of the transport properties on the process design. In Figure 1, we show a schematic diagram of a catalytic reactor that could be used for the synthesis of methanol. The reaction takes place at 610 K and 30 MPa. The feed, hydrogen and carbon monoxide, enters at the bottom of the reactor vessel and is preheated to the required temperature by the hot product gases in the preheater. Subsequently, hydrogen and carbon monoxide react in two catalyst beds. In order to maintain the temperature for the exothermic reaction, a water-cooled interstage unit is employed between the two catalyst beds, from which the product emerges to heat the incoming gases. In the configuration shown, the entire reaction zone/cooler assembly is contained in a single high-pressure vessel. This type of configuration has the usual benefits of intensification, including a reduction in capital costs and energy efficiency. However, it follows that the heat exchanger design of both pre-heater and interstage cooler must be done very carefully so that they are able to perform their required tasks. Under-design will lead to inefficient reactor operation with little opportunity for remedial action, whereas over-design will unnecessarily increase the size of the overall pressure vessel. The area of the interstage heat exchanger depends on the viscosity, thermal conductivity and density of the gas mixture passing through it as well as the properties of the cooling water. Indeed, the design area of the exchanger, which may be taken to represent the design variable, is approximately Water

PRODUCT

Water

Preheater Catalyst bed lnterstage Unit Catalyst bed

610 K,30

FEED (H2 + CO)

Figure 1 Schematic diagram of a methanol catalytic reactor

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proportional to the ratio (viscosity/thermal conductivity) to the power 1/3.2,3Hence, and irrespective of any optimizing design procedure employed, if the gas viscosity is underestimated by 20% and its thermal conductivity is overestimated by an equal amount, then the area of the interstage heat exchanger required will be underestimated by 25% with a corresponding overestimate of the area from the reverse circumstances. In either case, the loss of efficiency or the unnecessary capital cost may have significant economic effects on the operation of the process. Uncertainties of as much as 20% in the properties of gas mixtures at high pressures and temperatures were relatively common until the latter part of the last century. Great improvements in methods of measurement and prediction have now been achieved for many systems of practical importance but not for all, so that the example given here remains relevant. Indeed, many other examples of the importance of transport properties to design can easily be found. One novel aspect of this kind of problem is revealed by the increasing focus on the very small systems. Miniaturized systems have many advantages for a wide range of purposes, for example, reducing the size of sensors reduces their thermal inertia and permits a more rapid response while the integration of sensors and actuators in single devices is essential if micro-reactor technology is to move from laboratory to prod~ction.~ In order to transport materials in such microscopic environments, fluid flow provides the simplest means. Indeed, nature is the best example of this since aqueous solutions are the main transport media in living organisms. Hence, nano-and micro-technologies are likely to drive fluid transport property research into relatively unexplored areas of size and materials. For example, in the fast-evolving area of nanotechnology, it is exceedingly important to be able to remove the heat generated in small components effectively. For this purpose, the heat transfer characteristics of conventional fluids have been found to be inadequate so that it has recently been proposed that small amounts of nanoparticles might be suspended in the fluid to enhance the thermal conductivity. There is now some experimental evidence that such an enhancement does take place because a 0.6% by weight addition of carbon, multi-walled nanotubes to water results in a 40% increase in its thermal cond~ctivity.~

2 Measurements The vast majority of accurate transport-property measurements have been performed on molecularly simple pure fluids under conditions close to ambient pressure and temperature. As one moves away from this set of circumstances, the amount of available information decays rather rapidly and its accuracy declines dramatically. The three transport properties of the greatest concern are the viscosity, thermal conductivity and mass-diffusion coefficients. In each case, although measurements had been conducted over a period of at least 150 years, it was not until around 1970 that techniques of an acceptable accuracy were developed for the relatively routine measurement of any of these properties. There is ample evidence in the literature69 of very large discrepancies among measurements made prior to that date. One reason for these discrepancies lies in the conflicting requirements that, to make a transport-property measurement, one must perturb an equilibrium state but, at the same

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time, make the perturbation as small as possible so that the property determined does refer to a well-defined thermodynamic state. The latter requirement implies that the signals to be measured in any such experiment are always small and up against the limits of resolution. A second reason for the discrepancies arises from the failure of some experimenters to develop rigorous working equations for their instruments using the full conservation equations of continuum mechanics. The last 25 years of the 20th Century witnessed a very considerable refinement in measurement resolution, and the theory of instruments has now obviated many of these difficulties. However, it is still necessary to consider whether it is possible to perform easily all transport-property measurements that industry requires. Let us assume that we need to measure only three properties at just 10 temperatures and 10 pressures, for 15 pure fluids and all their possible multicomponent mixtures, at five compositions in the liquid and gas phases. Then, the total number of measurements required is of the order of 1 O8 (3 X 10X 10X 32766 X 5 X 2). If one further assumes that one can perform three measurements per day, then it is obvious that even for the above program of measurements, 90.000 man-years will be required and it considered only 15 materials from among the set of several thousand involved routinely in industry. It is clear that measurement alone cannot serve the industrial appetite for transport property data. Nevertheless, accurate measurements will continue to be vital for the verification of theory and the validation of predication. In the following sections, the most important techniques employed at present for the measurement of the viscosity, thermal conductivity and diffusion coefficients will be briefly presented.

2.1 Viscosity Since 1970, two generic types of viscometer have received the greatest attention; the first makes use of the torsional oscillations of bodies of revolution and the second is based on the rather simpler concept of laminar flow through capillaries. Both reduce the measurement of viscosity to measurements of mass, length and time. In the case of torsional oscillating-body viscometers, an essentially exact description of the motion allows measurements of high accuracy. In such viscometers, the characteristics of the oscillator are affected by the presence of the fluid and its properties in a way that is readily measured. Thus, oscillating-disk viscometers have found the greatest application to both gases and liquids under relatively mild conditions of temperature and pressure. They have been especially important in the determination of gases at low density and high temperature in work pioneered by Kestin and his collaborators8 and continued by Vogel and his group.' For work on molten metals at elevated temperatures (up to 1500 "C), instruments using oscillating cylinders, with the fluid inside or outside, have been favored for operational reasons. The work of Oye and his grouplo is an excellent example of what can be achieved. Among the great advantages of oscillating viscometers, the fact that no bulk motion of the fluid is required and that the measurements can be made absolute are paramount. Although a knowledge of the density of a fluid is necessary to evaluate the viscosity from the measurements made, it need not be known with the same accuracy required of the viscosity.

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Capillary viscometers measure the time of efflux of a known volume of fluid through a circular section tube. They intrinsically determine the kinematic viscosity, (dynamic viscosity/density), so that evaluation of the dynamic viscosity requires a knowledge of the density with a comparably high accuracy. Such viscometers are most often employed for liquids at ambient conditions when a hydrostatic head provides the driving force, or for gases allowing the decay of a generated pressure difference. At elevated temperatures, the use of a fluid pump complicates the experimental installation. Although the theory of such viscometers is superficially simple, the details of some of the applicable corrections have only recently been resolved.6 Furthermore, owing to the difficulty of knowing the dimensions of the capillary tube and uniformity of bore with a very high precision, measurements are mostly performed on a relative basis. We note the excellent work of Smith and his collaborators9 in the dilute gas phase from 90 to 1500 K, as well as the work of Nagashima and his co-workers on water.l0 In the case of high pressures, different types of viscometer have been employed owing to the need to reduce the volume of fluid required. The most popular have been falling-body viscometers and torsional-crystal viscometers.8 Neither of these have, however, completely developed theories so that their accuracy is intrinsically l4 that limited.8 On the other hand, the newly developed vibrating-wire vi~cometer'~~ makes use of the damping of a transverse oscillation of a thin wire enjoys a complete theory. The application of optical techniques such as that involving the study of the frequency and decay of surface waves on fluids (ripplons)15has great potential for the future, especially since they require no contact with the fluid, but they are at an early stage of development now. It is possible to achieve an uncertainty in the measurement of the viscosity of a fluid of a few parts in a thousand under near-ambient conditions, which deteriorates to a few percent at extremely low and high temperatures for low densities, In very high-pressure gases and liquids, the uncertainty achieved is at best a few percent and frequently very much worse.

2.2 Thermal Conductivity In the case of the thermal-conductivity, there are three main techniques: those based on Equation (1) and those based on a transient application of it. Prior to about 1975, two forms of steady-state technique dominated the field: parallel-plate devices, in which the temperature difference between two parallel disks either side of a fluid is measured when heat is generated in one plate, and concentric cylinder devices that apply the same technique in an obviously different geometry. In both cases, early work ignored the effects of convection. In more recent work, exemplified by the careful work in Amsterdam with parallel plates,16and in Paris with concentric cylind e r ~ , ~the ' effects of convection have been investigated. Indeed, the parallel-plate cells employed in Amsterdam by van den Berg and his co-workers16have the unique feature that, because the temperature difference imposed can be very small and the horizontal fluid layer very thin, it is possible to approach the critical point in a fluid or fluid mixture very closely (mK).

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Only one transient technique has enjoyed success and that is the transient hot-wire method, in a form pioneered by H a m a n s and subsequently developed for a wide range of applications in gases and liquids for temperatures from 70 to 500 K and pressures up to 700 MPa. The essential features of this technique are that it measures the transient temperature increase of a thin metallic wire (a few microns) immersed in the test fluid following the initiation of electrical heating within it. Its principal advantages are that the temperature increases need last for no more than one second so that the inertia of the fluid inhibits the development of significant convective heat transfer. In addition, the small magnitude of the temperature difference applied (2 K), but the large temperature gradient (about lo6Krr-'), means that radiative heat transfer is also generally insignificant.8Although the technique is unsuitable for work close to the critical point, it has been successfully employed over a very wide range of conditions for different types of fluids. Generally, an uncertainty of a few parts in a thousand in the thermal conductivity around ambient conditions is possible for simple fluids, but this is degraded to several percent at extremes of temperature, although relatively unaltered by pressure. The technique has recently been applied to polar or electrically conducting fluids with considerable success, while a sensor made of a platinum wire embedded between two thin layers of alumina has recently been employed to measure the thermal conductivity of molten metals.18 Finally, we note that a variety of new techniques for measuring the thermal diffusivity of fluids have been introduced relying largely on light scattering.8These methods have distinct advantages in special fluids and regions of thermodynamic state, particularly close to the critical state. While they are not yet able to achieve the same level of accuracy for simple fluids as can be achieved by conventional techniques, the fact that they are contact free does give them great potential for the future.

2.3 Diffusion Coefficient The measurement of diffusion coefficients in either gases or liquids is a very slow process and many techniques require days to attain a single result. This is largely because of the intrinsic slowness of the process and the fact that most methods use equipment of large scale. For this reason alone, there are relatively few experimental results and the measurements do not extend over wide ranges of temperatures. On the other hand, there are many different techniques for the measurement of diffusion coefficients from optical interferometric methods to NMR measurements of spin relaxation and chromatographic flow-broadening.8 The range is too wide to treat here but it is worthwhile noting that the most accurate interferometric techniques yield diffusion coefficients with an uncertainty of about 0.1%, but only near-ambient conditions. Under different conditions, other techniques are usually employed and the uncertainty is then typically a few percent.8

3 Theoretical Predictions As noted above, the fluids of industrial concern only occasionally conform to those for which experiments have been performed. It is therefore both apparent and

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inevitable that the burden placed on prediction and estimation of transport property values is considerable. Hence, the need for theoretical input, in combination with a limited set of accurate data to enable the transport-properties estimation, is unavoidable. The extent to which rigorous theory is available depends greatly on the thermodynamic state and molecular complexity as the following material indicates. In the low-density gas, the mean-free path of the molecules is very much greater than the molecular diameter so that free molecular motion contributes most to the transport process and molecular collisions are relatively rare events involving only two molecules at any one time. Such molecular collisions modify the transport by deflecting molecules from their original course. Thus, the nature of the collision, which is determined by the forces exerted between a pair of molecules, necessarily contributes to the magnitude of the flux of a dynamical quantity brought about by a particular gradient. As the density is increased, the free motion of the molecules is progressively dominated by the transport process involving interactions among the molecules and especially groups of them. The mean-free path becomes smaller and the details of the interactions between the molecules therefore become less important compared with the fact that so many interactions take place. Thus, when the liquid state is attained, it seems that quite simple models of the interaction between molecules are adequate for a description of transport properties. In the extreme case of a fluid near its critical point, the specific intermolecular interactions become relatively unimportant since the properties are determined by the behavior of clusters of molecules and the size of these clusters rather than anything else. At present, a rather well-developed rigorous theory exists for transport properties in the low-density limit, which has been exploited to probe intermolecular potentials for spherically symmetric atoms and simple polyatomic molecules. As the density of the fluid is increased, the rigor of the theoretical description of transport properties decreases for a variety of reasons but rises again close to the critical point and then decays rapidly for liquid-like densities. In the moderately dense gas, the description of transport properties is based on the Enskog theory, which first deals with gases composed of rigid sphere molecules and assumes that the sole effect of increasing density is an increase in the collision frequency. It also allows for the transfer of momentum or energy across a plane without center of gravity motion of the molecules. This latter phenomenon, known as collision transfer, eventually becomes the dominant mode of transport for liquids and solids. For moderately dense gases where this collisional effect is modest, the Enskog theory turns out to provide a reliable basis for means of estimating transport properties even though it is unable to make predictions from first principles. In particular, for mixtures the theory has provided a remarkably effective means of predicting the properties of gaseous systems at high density given information only on the pure fluids under similar conditions. In the liquid phase also, the Enskog theory has also proved quite effective for the representation and prediction of the properties of pure and mixed substances. In this case, the theory has been modified using the results of computer simulations of hard spheres, which have indicated the limitations of the assumptions of entirely random motions at elevated densities. In combination, the Enskog theory corrected in this way provides a good description of the properties of some pure liquids. An even more general result of the Enskog theory is that the transport properties of a fluid or

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fluid mixture depend only on the ratio of the molar volume of the fluid to a molecular volume. For hard sphere molecules, this result is exact but for real molecules it now seems that it is also valid as long as a weak temperature dependence of the molecular volume is allowed. These observations have been made the basis of a scheme for the representation and prediction of a wide range of fluids over a large range of temperature and pres~ure.'~ From this analysis, it should be clear that for the dense fluids theory provides only a framework for description and prediction rather than an absolute basis and that this basis is most secure for spherical molecules. It follows that in the absence of experimental measurements and when the molecules deviate increasingly from a rigid spherical nature, the predictions of properties become less secure. It therefore remains rather difficult to make predictions or estimations of the properties of complex molecules for which, as was noted earlier, there are currently few measurements.

4 International Dimension The transport properties of fluids are an integral part of the design of process plant. For this reason, in commercial transactions across national borders involving process plant, the design data for transport properties among others are a part of the validation of the design. As a consequence, the validation of transport property data internationally has been of considerable significance. It was to provide such validation for transport properties that the International Union of Pure and Applied Chemistry (IUPAC) set up a Transport Properties Subcommittee under its former Commission on Thermodynamics. A number of representations of the transport properties of common fluids have been published by that subcommittee and its successor the International Association for Transport Properties.20Examples of this trade in design and the data associated with design include conventionally fuelled power plants where the properties of water are vital and the properties of carbon dioxide that feature in the safety calculations of certain kinds of gas-cooled nuclear reactors.21 A final point that also needs to be presented is related to quality assurance, i.e. the requirement to satisfy regulatory frameworks concerned with safety and environmental issues. Traditionally, only national regulatory bodies had to be satisfied. Now, however, international regulatory bodies have to be satisfied while national regulatory bodies must, in many cases, accept the regulations of other nations. In this context, quality assurance of a plant or a process may often require a demonstrable pedigree for each number in the calculation of some aspect of the plant.

5 Databases Individual companies can no longer afford to collect and maintain all the data for process modeling that they may need to carry through an increasingly diverse business. Hence, they rely increasingly on third-party software and databases. Thus, fluid property databases are becoming increasingly important, and with them the concept of standardized exchange protocols has evolved, so that data can easily flow from a number of different databases to a wide range of applications. For chemical

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species, such databases often combine the physical properties of the materials (including transport properties) with information on chemical stability, toxicity and flammability. For many of the reasons indicated earlier, the merits of a database do not lie simply in the wide range of compounds that it treats nor the ranges of thermodynamic state covered but also in the quality and pedigree of the data contained in it. The effort required to amass large quantities of validated data is now a major industry in its own right and the number of organizations engaged in it are rather few internationally. The American Institute of Chemical Engineers’ main data effort is the Design Institute for Physical Properties (DIPPR) program. DIPPR is funded by industrial sponsors and its best-known product is the database from Project 801, which contain evaluated and correlated pure-component data on specific key properties for over 1700 pure compounds. Another major source of evaluated fluid property data is the Standard Reference Data program of the National Institute of Standards and Technology (NIST). The Thermodynamics Research Center (TRC) (recently returned to NIST from Texas A&M University4) issues a variety of data products and maintains an enormous database, with information on approximately 100 properties for 16.000 pure components and numerous mixtures. A very important thermophysical properties database, DETHERM, has also been developed by DECHEMA in Germany. DETHERM incorporates 4.2 million data sets for more then 122.000pure fluids and mixtures. Finally, the National Engineering Laboratory in United Kingdom maintains the Physical Properties Data Service (PPDS). There can be little doubt that in the near future, access to GRID computing technologies can place these large databanks instantly on the desktop of every scientidengineer in academia and industry worldwide. Then, the premium to be attached to quality of the data and its certification then becomes ever more pressing as the data user and its source become increasingly disconnected.

6 Future Efficient design, energy savings and stricter environmental controls will produce a need for transport property data that are more accurate, internationally accepted and cover a wide range of conditions. The number of fluids for which such international accord is necessary is likely to be relatively small and determined by international agreement. For other materials, a small number of accurate measurements will be made in order to provide the basis for the development of prediction schemes of increasing sophistication. There is little doubt that in these prediction- schemes, direct-computer simulation of fluids will play an increasing role relative to analytical theory, but it is not likely that simulation can obviate completely the need for measurement in the next decade. It is therefore likely that the role of the experimentalist will need to shift from the study of simple molecules under convenient conditions to the study of unusual and complex materials under difficult conditions such as molten metals at high temperatures, molten polymers. Indeed, it is quite possible that the process of measurement will be integrated within devices or process plant so that in-situ measurements can be performed. This is most likely to be the case for devices constructed on the micro- or nano-scale where existing theories of fluid properties may be inadequate owing to the special conditions that pertain.

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References 1. R. B. Bird, W. E. Stewart and E. N. Lightfoot Transport Phenomena, John Wiley & Sons Inc., New York, 1960. 2. D. Q. Kern, Process Heat Transfer, McGraw-Hill International, Tokyo, 1950. 3. M. J. Assael, C. A. Nieto de Castro and W. A. Wakeham, Proceedings of CHEMPOR '78, 1978, 16.1-16.9. 4. A. H. Harvey and A. Laesecke, Chem. Engin. Prog., 2002, February, 34. 5. M. J. Assael, C. F. Chen., I. Metaxa and W. A. Wakeham, Proceedings of 15th Symposium on Thermophysical Properties, Boulder, Colorado, 2003. 6. J. Millat, J. H. Dymond and C. A. Nieto de Castro (eds), Transport Properties of Fluids. Their Correlation, Prediction and Determination, Cambridge University Press, London, 1995. 7. K. F. Jensen, Chem. Eng. Sci., 2001, 56,293. 8. W. A. Wakeham, A. Nagashima, and J. V. Sengers (eds), Experimental Thermodynamics Vol. III: Measurement of the Transport Properties of Fluids, Blackwell Scientific, Oxford, 1991. 9. E. Vogel, Wiss. Zeit. Rostock, 1972,21, 169. 10. H. A. Oye and K. Torklep, J. Phys. E: Sci. Instrum., 1979, 12, 875. 11. A. G. Clarke and E. B. Smith, J. Chem. Phys., 1968,48,3988. 12. K. Kobayashi and A. Nagashima, High Temp.-High Press., 1985, 17, 131. 13. M. J. Assael, M. Papadaki, S. M. Richardson, C. Oliveira and W. A. Wakeham, High Temp. - High Press, 1991, 23, 561. 14. M. J. Assael and W. A. Wakeham, Fluid Phase Equilibr:, 1992,75,269. 15. Y. Nagasaka, Proceedings of 16th European Conference on Thermophysical Properties, London, 1-6 September, 2002. 16. R. Mostert, H. R. van den Berg and P, S. van der Gulik, Rev. Sci. Instrum., 1989,60,3466. 17. R. Tufeu, Ph.D. Thesis, Paris University, 1971. 18. M. Dix, I. W. Drummond, M. Lesemann, V. Peralta-Martinez, W. A. Wakeham, M. J. Assael, L. Karagiannidis and H. R. van den Berg, Proceedings of 5th Asian Thermophysical Properties Conference, Seoul, 1998, 133-1 36. 19. M. J. Assael, J. H. Dymond, M. Papadaki and P. M. Patterson, Int. J. Thermophys., 1992, 13, 269. 20. S. Hendl, J. Millat, E. Vogel, V. Vesovic, W. A. Wakeham, J. Luettmer-Strathmann,J. V. Sengers and M. J. Assael, Int. J. Thermophys., 1992,15, 1. 21. M. J. Assael, The importance of thermophysical properties in optimum design and energy saving, in Energy and Environment: Technological Challengesfor the Future, SpringerVerlag, Tokyo, 2000.

CHAPTER 12

Micro- and Nano-particles Production Using Supercritical Fluids ERNEST0 REVERCHON AND IOLANDA DE MARC0

1 Introduction Micronization processes based on the use of supercritical fluids have been suggested during the last few years as alternatives to traditional techniques.' Indeed, one of the most intriguing challenges in the development of supercritical fluid (SCF)-based applications is the micronization of solid compounds that can be precipitated at mild temperatures and with reduced or no solvent residue. Moreover, SCF-based techniques guarantee the control of particle size (PS) and distribution (PSD) in the micrometric and nanometric range. These expectations are based on the peculiar characteristics of gases at supercritical conditions: very fast mass transfer and the fast and complete elimination of the SCF at the end of the micronization process. The first supercritical fluid-based micronization process has been the rapid expansion of supercritical solutions (RESS); it is based on the solubilization of the solid to be micronized in the SCF and its subsequent precipitation by fast depressurization of the solution. However, the use of this technique is largely limited by the low solubility in supercritical carbon dioxide (SC-CO,) of many of the solids of Particles generation from gas saturated solutions (PGSS) has also been proposed. In this case the product to be micronized is liquefied by heating and addition of SCCO,; then, the gas-liquid solution is sprayed in a low-pressure vessel, thus obtaining microparticles. This technique has been successfully used to process some polymers, but has limited applicability in the case of thennolabile compounds that can decompose during heating.9J0 The supercritical antisolvent (SAS) precipitation has also been proposed in various arrangements and under various acronyms.ll This technique is based on the use of SC-CO, as the antisolvent to precipitate the solid solute from a liquid solution. The prerequisites for the success of this technique are the complete solubility of the

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liquid in the SCF and the complete insolubility of the solid in it. In the batch mode (often indicated by the acronym GAS), the precipitation vessel is loaded with a given quantity of the liquid solution and, then, the supercritical antisolvent is added until the final pressure is reached. In the semi-continuous mode (often indicated by the acronym SAS), the liquid solution and the supercritical antisolvent are continuously delivered to the precipitation vessel in co-current or counter-current mode. Very recently, atomization processes assisted by supercritical CO, have been proposed by various authors. They are based on the mixing and/or solubilization of SCCO, in a liquid solution and the subsequent atomization of the mixture.12-16 At present, the most promising process seems to be the supercritical antisolvent precipitation, which is the most widely a ~ p l i e d ' ~and - ~has ~ been recently proposed on a semi-industrial scale.30.31 However, despite the fact that many works have been published on SAS precipitation, only a limited number of them have focused on the mechanisms controlling particle formation and on the role of the process parameters on the morphology and on the dimensions of the precipitated powders. This lack of information can be one of the main factors hampering the industrial application of this process. Therefore, the scope of this chapter is to describe the results obtained using SAS with particular emphasis on the production of particles with controlled PS and PSD in the micrometric as well as in the nanometric range. We will try to understand the role of high pressure vapor-liquid equilibria (VLEs) in determining the morphology and particle size of precipitates.

2 Experimental Apparatus for SAS Processing A semi-continuous SAS apparatus, as a rule, consists of two pumps used to deliver the liquid solution and supercritical CO,, respectively. A cylindrical vessel is used as the precipitation chamber. The liquid mixture is delivered to the precipitator through an injector. Different arrangements have been proposed in the literature like nozzles, capillaries, vibrating orifices and coaxial devices. Supercritical CO, is delivered through another inlet port that can be located on the top or the bottom of the vessel. The pressure in the precipitator is regulated by a micrometering valve located at the exit of the chamber. A stainless-steel frit located at the bottom of the precipitator is used to collect the produced powder. A second vessel located downstream the micrometering valve is used to recover the liquid solvent. Part of the experiments described in this chapter have been performed using a SAS transparent precipitator that has two quarts windows placed along the entire height of the vessel. This apparatus allows the visual observation of the macroscopic features of the precipitation process.

2.1 Experimental Procedure A SAS experiment begins by delivering supercritical CO, to the precipitator until the desired pressure is reached. An antisolvent steady flow is then established. Then, pure solvent is sent through the nozzle with the aim of obtaining steady-state composition conditions during the solute precipitation. At this point, the flow of the liquid solvent is stopped and the liquid solution is delivered through the nozzle. The

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experiment ends when the delivery of the liquid solution to the chamber is interrupted. However, supercritical CO, continues to flow to wash the chamber for the residual content of liquid solubilized in the supercritical antisolvent. If the final purge with pure CO, is not performed, solvent condenses during the depressurization and can solubilize the collected powder.

3 SAS Fluidodynamics, Mass Transfer and Thermodynamics Hydrodynamic requirements of SAS are related to jet break-up; i.e., the conditions under which small droplets are obtained to form a large surface between the liquid and supercritical fluid that favors mass transfer. The jet break-up is critical and may be problematic in the case of highly viscous solutions like polymer solutions. Mass transfer is also favored by the high diffusivity that characterizes supercritical fluids. The difference in density between the liquid and the antisolvent also plays a role in determining the direction of mass transfer. However, the SAS process does not seem to be substantially limited by mass transfer resistance^.^^ Thermodynamic constraints to the SAS process can be summarized in the required miscibility between the liquid solvent and the supercritical antisolvent and the insolubility of the solute in the antisolvent and in the solvent-antisolvent mixture. Data are available for various binary mixtures liquid-supercritical fluid 32 and can be described as type I using the classification of van-Konynenburg and If jet break-up is obtained and mass transfer is very fast, high-pressure VLEs of the ternary system liquid +solute+ supercritical fluid can control the precipitation process. Based on the previous considerations, some authors proposed thermodynamicbased approaches to SAS. De la Fuente Badilla et aZ.34attempted to develop a thermodynamic-based criterion for optimum batch antisolvent precipitation (GAS) using a definition of the volume expansion that takes into account the molar volume of the system studied. They analyzed various binary and ternary systems and concluded that the pressure corresponding to a minimum value of the liquid-phase volume expansion coincides with the pressure at which the solute precipitates. In a subsequent work,35 Shariati and Peters further highlighted the role of SC-CO, in GAS. It acts as a co-solvent (cosolvency effect) at lower concentrations, whereas at higher concentrations it acts as an antisolvent. M ~ k h o p a d h y a yproposed ~~ the relative partial volume reduction of the solvent as the key parameter for the selection of the solvent and of the optimum GAS process conditions. The solute solubility is proportional to the partial molar volume of the solvent due to CO, molecules clustering around solvent molecules, which lead to the loss of solvent power. Both the previous approaches apply to discontinuous GAS process (in which the pressure progressively increases) and indicate precipitation process pressures between 50 and 70 bar, i.e., under subcritical conditions (even when the critical pressure of pure CO, is considered), whereas the experimental evidences for semi-continuous SAS (steady-state pressure) indicate larger pressure values for the precipitation of micro- and nano-particles.

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Besides these thermodynamic criteria, the most common approach used in the literature is based on the operation at pressures above the binary (liquid - SC-CO,) mixture critical point, completely neglecting the influence of solute on VLEs of the system. But, the solubility behavior of a binary supercritical C0,-containing system is frequently changed by the addition of a low volatile third component as the solute to be precipitated. In particular, the so-called cosolvency effect can occur when a mixture of two components solvent+solute is better soluble in a supercritical solvent than each of the pure components alone. In contrast to this behavior, a ternary system can show poorer solubility compared with the binary systems antisolvent+solvent and antisolvent +solute; a system with these characteristics is called a non-cosolvency (antisolvent) s y ~ t e m .In ~ ~particular, !~~ in the case of the SAS process, they hypothesize that the solute does not induce cosolvency effects, because the scope of this process lies in the use of CO, as an antisolvent for the solute, inducing its precipitation. In the light of these considerations, a different approach based on ternary system thermodynamics could be considered. However, the phase behavior of ternary systems could be very complex and there is a considerable lack of data on ternary systems containing a component of low volatility; therefore, a possible compromise could be to consider that the solute addition can produce the shift of the mixture critical point (MCP) (i.e., the pressure at which the ternary mixture is supercritical) with respect to binary system VLEs and the modification of this kind of system that is formed according to the van-Konynenburg and Scott c l a s s i f i ~ a t i o n . ~ ~ . ~ ~ * ~ ~ Therefore, the approach followed in this chapter considers pseudo-binary diagrams, i.e., equilibria involving the third component are, however, neglected, but modifications due to the presence of the solute are considered on the binary system. We will observe in the analysis of the experimental results that this approach can provide interesting information regarding the evolution of the SAS process, and the morphology and dimension of the precipitated particles. A rationalization of the experimental results is also proposed.

4 Relationship between Particles Morphology and High-pressure VLEs In this part of the chapter, a summary of semi-continuous SAS experiments performed on various materials is proposed and an attempt at providing an interpretation of the different morphologies observed in relation to the position of the process operating point with respect to mixture critical point has been made.

4.1 Experimental Evidences Case a : In some cases, the solute was recovered in the liquid collection vessel, i.e., it was completely extracted by the solution formed by the solvent and SC-CO,. An example is represented by Medroxiprogesterone acetate (corticosteroid) processed in acetone at 120 bar, 40 "C. The visual observation across the windowed vessel showed that a single fluid phase occupied the entire precipitator and no precipitation was observed. Therefore,

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the solute was extracted due to an enhancement of its solubility in the solvent-antisolvent mixture with respect to its solubility in the SCF alone (the cosolvency effect described previously). Case b: In various cases, we observed the formation of large crystals at the bottom of the windowed precipitator. For example, we found large crystals of Prednisolone (corticosteroid) precipitated from Methanol (Figure 1 ) and of Prednisolone acetate (corticosteroid) precipitated from Tetrahydrofuran, in both cases operating at 150 bar, 40 "C.We also observed that a relative control of the crystal size down to some microns is also possible, in addition to decreasing the concentration in the liquid solution or increasing the ratio between the antisolvent and the liquid solution flow rates. In this case, the visual observation showed that two phases were formed in the precipitator. Crystals precipitated from a relatively small quantity of a liquid-rich phase located at the bottom of the vessel, whereas no precipitation was observed from the upper fluid phase (supercritical) that was separated from the lower one by a meniscus. Case c: The powder was formed by crystals and amorphous particles. These morphologies were observed, for example, in the case of Inulin (biopolymer) precipitation from Dimethylsulfoxide (DMSO) at 150 bar, 40 "C, 20 mg/ml and Amoxicillin (antibiotic) from DMSO under the same process conditions (Figure 2).41 The visual observation of the precipitation process showed that two phases formed in the precipitator. The precipitation took place from both phases; the upper one (supercritical) produced amorphous particles, whereas the lower one (liquid-rich) produced crystals. Case d Amorphous balloons, ix., hollow amorphous spherical particles several microns in diameter, were observed in some cases. They are characterized by various expansion degrees. We observed that the balloons diameter increases with CO, density. Examples of this morphology were observed during the study of the precipitation of Yttrium acetate (YAc) (superconductor precursors) from DMSO at 120 bar and 40 "C (Figure 3) 31 and of Cefonicid (antibiotic) from DMSO at 150 bar and 60 "C.40

Figure 1 Large needle-like crystals of Prednisolone precipitated from Methanol at 1.50 bal; 40 "C,20 mg/mL

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Figure 2 Amoxicillin precipitated from DMSO at 150 bal; 40 "C,20 mg/mL: ( a ) amorphous microparticles in the upper part of the precipitator; and (b) large crystals in the lower part of the precipitator

Visual inspection of the precipitation process showed that a single fluid phase was present in the precipitator. Cuse e: Nanoparticles formation was observed, for example, in the case of Yttrium, Samarium, N e ~ d i m i u mand ~ ~Gadolinium acetates21(superconductor precursors) and in the case of D e ~ t r a n(polymer), ~~ Tetra~ycline~~ (antibiotic) and other compounds. This morphology was obtained at CO, densities larger than those characteristic of case d. Nanoparticles down to a mean diameter of 50 nm were obtained in the case of Zinc acetate20 (catalyst precursor). Microparticles as well as nanoparticles were obtained for Amoxicillin, Rifampicin (antibiotic) (Figure 4) and other compounds. Visual inspection of the process confirmed that also in this case, the precipitation took place from a single fluid phase.

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Chapter I 2

Figure 3 Yttrium Acetate amorphous microparticles precipitated from DMSO at 120 bar; 40 "C,50 mg/mL

Figure 4 Rifampicin amorphous microparticles precipitated from DMSO at 90 bar; 40 "C, 40 mg/mL

The various cases discussed all refer to pressure and temperature conditions that are above the MCP of the binary mixture C0,-solvent; nevertheless, the presence of solutes has evidently modified the vapor-liquid behavior with respect to the one characteristic of the binary system at least in cases a-c: cosolvency and formation of two phases have been observed. Thus, visual observation and powder analysis confirmed our hypothesis of a shift of the MCP induced by the addition of the thirdcomponent. We can also suppose that the formation of crystals or of amorphous particles is controlled by the content of SCF in the liquid (and vice versa), i.e., a liquid containing small quantities of SCF has a liquid-like behavior and crystals are obtained; as the SCF content increases (SCF rich phase), we can observe the formation of amorphous particles.

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Xc0* Figure 5 Pseudo-binary diagram and the possible operating points for SAS

A pseudo-binary diagram is shown in Figure 5 and can be used to locate the various cases discussed with respect to VLEs. Case a can be explained in a trivial mode by the enhancement of solute solubility in SC-CO, because of the presence of the solvent and does not need to be reported in the diagram. Cases b and c, in our hypothesis, can be attributed to experiments performed at pressures lower than the MCP of the ternary system. A different solute partitioning between the two phases and their relative content in SCF can explain the formation of crystals and/or amorphous particles. In particular, point B in Figure 5 represents an operating point producing a large liquid-rich phase, whereas point C represents an operating point with a large gas-rich phase. We think that case d is relative at operating conditions below the ternary MCP, but located in the single gas phase region, in which mass transfer between the liquid droplet and the surrounding supercritical fluid is dominated by diffusion of CO, inside the droplet with the consequent formation of balloons of increasing dimensions (point D in Figure 5). This explanation is also supported by the modelling work performed by Werling and Debenedetti,44~45 which indicated the relative density of the two phases as the controlling mechanism for the expansion of the droplets. Their work confirms that droplets expansion prevails when the process is performed under sub-critical conditions with respect to the mixture studied. Case e is the result of precipitation under conditions above the ternary MCP and is the true supercritical antisolvent precipitation. The formation of very small particles indicates that characteristic times for surface tension disappearance are very short, droplets are not formed at the exit of the injector and the very small particles are released from the fluid phase.27These aspects have been studied in the previously cited works of Werling and Debenedetti,44t45which underline the interplay among the different consecutive or simultaneous process that characterize SAS, mainly mass transfer under sub-critical and supercritical conditions. The characteristic times

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of the various mechanism involved, including nucleation, have been postulated in a subsequent work.46

5 System Modifications Induced by the Presence of Solute Until now, an attempt at explaining the observed modifications of VLEs induced by the presence of the solute has been made. Some compounds do not interfere with the binary VLEs; therefore, the one-phase region is the same as in the binary system (at temperatures and pressures higher than MCP). But, we have seen that, in general, the

Figure 6 Cefonicid precipitated from DMSO at 120 bal; 50 "C, 10 mg/mL: (a) amorphous microparticles in the upper part of the precipitator; and (b) amorphous balloons in the lower part of the precipitator

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presence of the solute modifies the binary VLEs even at low solute concentrations. The simplest modification that can be presented by ternary non-cosolvency systems, with respect to the corresponding binary system, consists of the shift of the MCP towards higher pressures. This effect could also be a function of the concentration of solute in the liquid solution, i.e., the higher the concentration, the larger the pressure increase of t k MCP. However, the shift of the MCP towards higher pressures is only the simplest possible modification of the VLEs. More complex interactions among the three components may occur. For example, in the case of the Cefonicid precipitated from DMSO at 150 bar and 60 "C, we observed the presence of two different morphologies: in the upper part of the precipitator the powder was formed by sub-microparticles (Figure 6(a), whereas in the lower part it was formed by microparticles and balloonlike particles (expanded droplets) (Figure 6(b)). Both morphologies have been previously attributed to fluid phase precipitation. In this case, the question is: how is it possible to obtain two gas-like phases? When a two-phase equilibrium is obtained for type I systems, it is always characterized by a liquid- and gas-rich phase connected by a tie line. Indeed, these systems are characterized by an uninterrupted critical line that connects the critical points of the pure components .47 If we consider a type V system, instead, as the temperature increases, an azeotrope appears. The binary system DMSO-CO, is of type I and the presence of Cefonicid at 40 "C does not modify this characteristic. But when the temperature is increased at 60 OC, a type V situation is probably produced. It means that an azeotrope is formed; the critical line is not continuous, but is formed by two lines, which start

Figure 7 High-pressure vapor-liquid equilibrium diagram at 60 "C f o r the system DMSO-CO, in the presence of Cefonicid

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Chapter 12

from the critical point of the component A (supercritical fluid) and from the critical point of the component B (liquid solvent), respectively. Therefore, the precipitation occurs from two points in equilibrium corresponding to two C0,-rich phases (gaslike behavior) producing amorphous particles, but with different morphologies (Figure 7).

6 Conclusions In this chapter, we have highlighted that SAS performance is conditioned by thermodynamic constraints and it is possible to explain SAS products morphology through a thermodynamic-based approach. At this point, the question is: is the picture of SAS conditioning thermodynamic behaviors complete? Probably not. Indeed, very complex interactions are possible in ternary systems. The approach proposed is encouraging from another point of view. Even if a very complex behavior develops in the ternary systems, a systematic analysis of VLEs can allow the identification of the regions in which effective precipitation can be obtained.

References 1. J. Jung and M. Perrut, J. Supercrit. Fluids, 2001, 20(3), 179. 2. D. W. Matson, J. L. Fulton, R. C. Petersen and R. D. Smith, Znd. Eng. Chem. Res., 1987, 26(1l), 2298. 3. R. C. Petersen, D. W. Matson and R. D. Smith, Polym. Eng. Sci., 1987, 27, 1693. 4. R. S. Mohamed, P. G. Debenedetti and R. K. Prud’homme, A.Z.Ch.E. J., 1989, 35, 325. 5. A. K. Lele and A. D. Shine, A.Z.Ch.E. J., 1992,38(5), 742. 6. E. M. Phillips and V. J. Stella, Znt. J. Pharm., 1993, 94, 1. 7. E. Reverchon, G. Donsi and D. Gorgoglione, J, Supercrit. Fluids, 1994,6, 241. 8. E. Reverchon and P. Pallado, J. Supercrit. Fluids, 1996, 9, 216. 9. E. Weidner, Z. Knez and Z. Novak, European Patent EP 0 744 992, February 1995 PatentWO 95/21688, July 1995. 10. C. Xu, B. Watkins, R. E. Sievers, X. Jing, P. Trowga, C. S. Gibbons and A. Vecht, Appl. Phys. Lett., 1997,71(12), 1643. 11. E. Reverchon, J. Supercrit. Fluids, 1999, 15(l), 1. 12. E. Reverchon, Znd. Eng. Chem. Res., 2002, 41(10), 2405. 13. E. Reverchon and G. Della Porta, J. Supercrit. Fluids, 2003, 26(3), 243. 14. E. Reverchon and G. Della Porta, Znt. J. Pharm., 2003,258(1-2), 1. 15. R. E. Sievers, U. Karst, P. D. Milewski, S. P. Sellers, B. A. Miles, J. D. Schaefer, C. R. Stoldt and C. Y. Xu, Aerosol Sci. Tech., 1999,30, 3. 16. S. P. Sellers, G. S. Clark, R. E. Sievers and J. F. Carpenter, J. Pharm. Sci., 2001,90, 785. 17. S. D. Yeo, G. B. Lim, P. G. Debenedetti and H. Bernstein, Biotech. Bioeng., 1993,41,341. 18. W. J. Schimtt, M. C. Salada, G. G. Shook and S. M. Speaker, A.1.Ch.E. J., 1995,41(11), 2476. 19. E. Reverchon, G. Della Porta, S. Pace and A. Di Trolio, Ind. Eng. Chem. Res., 1998,37(3), 952. 20. E. Reverchon, G. Della Porta, D. Sannino and P. Ciambelli, Powder Technol., 1999, 102(2), 127.

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2 1. E. Reverchon, I. De Marco and G. Della Porta, J. Supercrit. Fluids, 2002, 23( l), 8 1. 22. G. J. Hutchings, J. A. Lopes-Sanchez, J. K. Bartley, J. M. Webster, A. Burrows, C. J. Kiely, A. F, Carley, C. Rhodes, M. Havecker, A. hop-Gericke, R. W. Mayer, R. Schlogl, J. C. Volta and M. Poliakoff, J. Catalysis, 2002,208(1), 197. 23. Y. Gao, T. K. Mulenda, Y. F. Shi and W. K. Yuan, J. Supercrit. Fluids, 1998, 13(1-3), 369. 24. P. M. Gallagher, M. P. Coffey, V. J. Krukonis and W. W. Hillstrom, J. Supercrit. Fluids, 1992, 5, 130. 25. R. Bodmeier, H. Wang, D. J. Dixon, S. Mawson and K. P. Johnston, Pharm. Res., 1995, 12(8), 1211. 26. D. J. Dixon, K. P. Johnston and R. A. Bodmeier, A.Z.Ch.E.J., 1993,39(1), 127. 27. T. W. Randolph, A. D. Randolph, M. Mebes and S. Yeung, Biotech. Progress, 1993,9(4), 429. 28. L. Benedetti, A. Bertucco and P. Pallado, Biotech. Bioeng., 1997, 53(2), 232. 29. S . Mawson, S . Kanakia and K. P. Johnston, J. Appl. Polym. Sci., 1997,64(1l), 2105. 30. E. Reverchon, I. De Marco, G. Caputo and G. Della Porta, J. Supercrit. Fluids, 2003, 26(1), 1. 31. E. Reverchon, G. Caputo and I. De Marco, Ind. Eng. Chem. Res., 2003,42, 6406. 32. S . Ohe, Vapor-liquid Equilibrium Data at High Pressure, Elsevier, Tokyo, 1990. 33. P. H. van Konynenburg and R. L. Scott, Philos. Trans. R. Soc., London 298 A, 1980,495. 34. J. C. de la Fuente Badilla, C. J. Peters and J. de Swaan Arons, J. Supercrit. Fluids, 2000, 17(1), 13. 35. A. Shariati and C. J. Peters, J. Supercrit. Fluids, 2002,23(3), 195. 36. M. Mukhopadhyay, J. Supercrit. Fluids, 2003,25(3), 213. 37. A. L. Scheidgen and G. M. Schneider, Fluid Phase Equilibr; 2002, 19k197, 1009. 38. F. Becker and P. Richter, Fluid Phase Equilibr; 1989,49, 157. 39. E. Reverchon, I. De Marco and G. Della Porta, Znt. J. Pharm., 2002,243(1-2), 83. 40. E. Reverchon and I. De Marco, J. Supercrit. Fluids, 2004, in press. 41. E. Reverchon, G. Della Porta and M. G. Falivene, J. Supercrit. Fluids, 2000, 17(3), 239. 42. E. Reverchon, G. Della Porta, I. De Rosa, P. Subra and D. Letourneur, J. Supercrit. Fluids, 2000, 18(3), 239. 43. E. Reverchon and G. Della Porta, Powder Tech., 1999, 106( 1-2), 23. 44. J. 0. Werling and P. G. Debenedetti, J. Supercrit. Fluids, 1999, 16(2), 167. 45. J. 0. Werling and P. G. Debenedetti, J. Supercrit. Fluids, 2000, 18(l), 11. 46. F. ChBvez, P. G. Debenedetti, J. J. Luo, R. N. Dave and R. Pfeffer, Ind. Eng. Chem. Res., 2003,42(13), 3156. 47. J. C. G. Calado and J. N. Canongia Lopes, Pure Appl. Chem., 1999,71(7), 1183.

CHAPTER 13

Calorimetric Meas urernents of Thermophysical Properties for Industry JEAN-PIERRE E. GROLIER AND FLORIN DAN

1 Introduction Materials science is currently witnessing an impressive acceleration of activities. New advanced materials are constantly emerging. The materials that will be used or needed in the next decade are not yet known. Since such materials are often used in special environments or under extreme conditions of temperature and pressure, their careful characterization must be done not only at the early stage of their development but also in their entire life cycle. Furthermore, their properties as a function of temperature and pressure must be well established for the optimal control of their processability. Undoubtedly, thermal and calorimetric techniques are essential in this respect. In relating thermal as well as mechanical behavior to materials’ structures, these techniques are perfectly adapted to provide accurate data in wide ranges of temperature and pressure. Typically, thermophysical properties are the most important when dealing with materials subjected to thermal variations and mechanical constraints. The purpose of this chapter is not to produce an exhaustive catalogue of such properties. The choice has been made to consider the most important or the key properties. Interestingly, materials considered here encompass gases, liquids and solids, polymers essentially in the latter case. The selected properties are of two types: bulk properties and phase transition properties. The bulk properties are either caloric properties, like heat capacities Cp, and mechanical properties, like isobaric thermal expansivities a,, isothermal compressibilities q-, and isochoric thermal pressure coefficients &. The two main thermal properties are related to first-order transitions, fusion and crystallization, and glass transitions. All these properties are now accessible thanks to recent progress in experimental techniques, which allow measurements in the three physical states over extended ranges of T and p , including in the vicinity of the

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critical point (at least in the case of gases and liquids). Measurements are also possible for polymers submitted to pressure in the presence or absence of gases. In what follows, some of the above properties in selected materials or systems serve to illustrate the state of the art in both available techniques and instrument performances. As a matter of fact, the development of the instrumentation parallels the need for data for new materials to be used in specific applications and under further extreme or particular conditions. For all selected examples, the industrial needs, which were very often the incentive factor for making the measurements, will be stressed.

2 Heat Capacities The heat capacities of pure substances or of their mixtures, irrespective of their aggregation states, are often used to correlate the main thermodynamic properties, including the phase equilibrium properties, over large domains of temperature. They are now recognized as key data for engineering calculations. Heat capacities are also used to verify the validity of existing equations of state (EOS) or even to develop new ones. Furthermore, direct measurements of heat capacities in the vicinity of the critical point are of practical importance, due to the spectacular changes generally observed under conditions around this point.

2.1 Heat Capacities of Gases Understanding the thermodynamics of natural gas is essential for optimizing the management of its resources. Correlation and prediction of thermodynamic and transport properties of natural gas are needed at different stages during the exploration and assessment of reserves as well as for its production, transport, and storage. In this context, the calorimetric technique originally developed for measuring heat capacities for liquids2 has been adapted to investigate gases.3 Basically, the method involves recording the differential heat flux between the sample cell containing the gas under study and the reference cell during a temperature increment AT from an initial temperature Tito a final temperature Tf at constant pressure. The total heat Q exchanged is obtained after integration of the heat flux between the two thermal equilibria reached at the initial and final temperatures, respectively. In order to calculate the isobaric specific heat capacities cp (J K - 1 g - l ) , two additional quantities are necessary; the molar densities have been calculated from semi-empirical equations of state, and the inner volume of the sample cell, Vs, was calculated by calibrating the calorimeter with gases for which the molar densities and heat capacities are known. The heat quantity 6Q obtained from the integration of the heat flux can be expressed as

6Q = mcpdT-m7($)

dP

where m refers to the mass and v refers to the specific volume of the gas in the sample cell. The two terms of Equation (1) have been calculated for some gases, such as nitrogen, helium, and carbon dioxide at temperatures ranging from 323.15 to 423.15 K

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and pressures ranging from 5 to 25 MPa, respectively. It was found that the influence of the pV-work (the second term on the right-hand side of Equation (1)) may not be negligible in the case of CO, and in this case it has been taken into account in the cp calculations. The same technique was then used to determine the specific isobaric heat capacities of two reference gases, carbon dioxide and argon, and of five synthetic gaseous mixtures representing different types of natural gas in the above-mentioned ranges of pressure and temperature. For the pure gases, the results have been compared with the calculated values obtained with the equations of state of Ely et aL4 and of Stewart and J a ~ o b s e nrespectively. ,~ Experimental and calculated values for CO, and Ar are given in Figures l(a) and l(b), respectively; they show the ability of the experimental technique to deliver results that can be obtained with a 2% confidence. The experimental heat capacities of the five synthetic gaseous mixtures having the molar compositions listed in Table 1 were used to compare the performance of various equations of state, namely: AGA 8,6 S ~ a v e Peng-Robinson,* ,~ Lee-Ke~ler,~ Simonet-B6har-Rauzy,l0 and chain-of-rotators. Numerical comparison between experiment and EOS is given in the right-hand part of Table 1. The absolute average deviations (in %) show that the different EOS, with the exception of the chain-ofrotors model, agree with the isobaric heat capacities of the investigated gases within the experimental uncertainty, i.e .=2%. 0.75

0.70 r

m

k

0.65

2

0" 0.60

0.55

5

10

15

20

25

Figure 1 Experimental heat capacities (symbols) compared with calculated values (lines) of carbon dioxide (a) and argon (b), respectively

Table 1 Molar composition (mole/ %) of the five gaseous mixtures and the absolute average deviations (in %) obtained using different equations of state (EOS), over the full pressure and temperature ranges investigated

GasA Gas B Gas C Gas D GasE

CH, 87.570 96.072 83.427 85.200 99.000

Composition (% /mole) C,H, C,H, N , CO, 9.380 2.155 0.895 2.077 1.433 0.418 3.584 0.871 10.62 1.535 14.80 1.000

EOS (% absolute average deviations) AGA8 Soave P-R L-K S-B-R C-0-R 1.8 2 2.3 2.0 2.5 3.8 0.6 2.6 0.6 1.3 0.5 1 0.7 1.1 0.6 0.9 0.5 5.8 - 0.5 - - 0.8 -

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2.2 Heat Capacities of Liquids The above-mentioned method was initially developed for measuring the isobaric heat capacities of aqueous salt solutions up to 573 K and 30 MPa.2 For a typical run, the sample cell was loaded with the sample solution and the reference cell was loaded with a reference fluid of known heat capacity (usually water). Then, the temperature was increased from Tito Tf, at constant pressure, and the difference Q in the transferred heat was corrected taking into account both the cell's volumetric dissymmetry and the differences between the densities and specific heat capacities of the measured sample and reference fluids, respectively. Such an experiment allows the measurement of the product pc, representing the isobaric heat capacity divided by volume. In order to obtain the desired isobaric heat capacity, c,, of the solution, it was necessary to know its density. For this purpose, the isobaric specific heat capacity and density were represented by polynomials in terms of temperature T: n

m

c , = C c i T i and p = c r j T J i=O

j=O

The parameters rj for the density polynomial were obtained from available corresponding densities. Knowing the calibration constant for each set of experiments and the densities of the measured fluids allowed the calculation of ci and finally the specific heat capacity. The performances of the calorimetric arrangement have been tested by measuring the heat capacity for liquid n-hexadecane as well as of NaCl (aquous) in the molality range 0.1-2.0 mol kg-'. Measurements taken at different temperatures and pressures compare well with the literature data. 12J3-15. Later, the pressure-scanning technique16was used to investigate the thermophysical properties, isobaric molar heat capacity C, (J K-'mol-'), and ap,over extended T and p of several fluids or their mixtures, such as quinoline, n-hexane, l-hexanamine, and its binary mixtures with 1-hexanol, rn-cresol, and its binary mixtures with quinoline, etc. As a rule, for simple liquids without strong intermolecular interactions, such as n-hexane, for example, both the C,-isotherms and the pressure effects (isotherms) on the isobaric heat capacity at pressures up to 700 MPa exhibit minima. It is worth recalling that the pressure effect on the C, is related to the isobaric thermal expansibility a, by the following equation (the effect of pressure on the a, is discussed in the next section):

When the a,-isotherms cross at a given pressure, (dad3T), changes its sign and this property is responsible for the above-mentioned minima. In addition, as the temperature increases, the minima are shifted to higher pressures. At higher temperatures, especially when approaching the critical temperature, the minimum becomes a shallow one. As an example for associated liquids, some selected C,-isotherms and the pressure effects at different temperatures on the isobaric heat capacity of rn-cresol are presented in Figures 2(al) and (a2),17respectively.

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Chapter 13 503K

. _._.-

.... . - . . _ --.

k

353 K 2.15

0

100

(al)

200

300

400

-0.124 0

(a2)

p IMPa

I

I

100

I

1

200 p MPa

I

I

300

I

I

400

0

1

-2

-3

0

0.2

0.4

0.6

0.8

Figure 2 Isobaric heat capacities ( a l ) and pressure efects on the isobaric heat capacities (a2)for m-cresol. Typical W-shapedexcess molar heat capacities (b)at 298.15 K of {1,4-dioxane+n-alkanes]mixtures; numbers refer to the n-alkane carbon numbers

At lower temperatures, where the associations are strong, there are no minima, because the isotherms of a, cross at a higher pressure and the change of sign of (daJdQ, does not sufficiently compensate the positive value of ap’ over the pressure range under investigation. At higher temperatures, where the association bonds are weakened or even broken, the minima appear and the behavior of m-cresol becomes similar to that of simple liquids. As regards normal-pressure heat-capacity calorimetry, during the two decades 1973-1993, Grolier and his co-workers produced the largest database on binary organic mixtures of different types; this appeared in more than 40 papers with a rough estimate of 150 systems.’*Among the systems investigated, the most striking result was obtained in collaboration with Inglese and Wilhelm; it concerned the unusual composition dependence of excess molar heat capacity of { 1,4-dioxane+ n-alkane} binary mixtures19 as shown in Figure 2(b). Typically, the CF -curve plotted against the mole fraction of the cyclic ether showed a double minimum separated by a marked maximum, yielding a W-shaped curve. Such a behavior was assigned to the existence of preferential orientations and aggregations of some “flexible molecules” in admixtures with similarly flexible alkanes, depending on the concentration range. Later, the W-shaped curves, including theoretical modeling, were obtained in several laboratories worldwide, and the number of systems known for showing W-shaped CpE -curves has increased steadily, and this behavior is now recognized as being of relatively wide occurrence in mixtures of a strongly polar substance and n-alkanes, although it is not restricted to these types of systems only.

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3 Bulk Thermomechanical Properties 3.1 In Extended Ranges of T andp Nowadays, for a thermodynamicist, p W-calorimetry (further referred to as scanning transitiometry, its patented and commercial name20) is the most accomplished experimental concept. It allows direct determinations of the most important thermodynamic derivatives; it shows how, in practice, the Maxwell relations can be used to fully satisfy the thermodynamic consistency of those derivatives. Of particular interest is the use of pressure as an independent variable; this is typically illustrated by the relatively newly established pressure-controlled scanning calorimeters (PCSC).16J7Basically, the isobaric expansibility a,@,7)=( l/v)(dv/d7), can be considered as the key quantity from which the molar volume, v, can be obtained and therefore all subsequent molar thermodynamic derivatives with respect to pressure. Knowing the molar volume as a function of p at the reference temperature, To,the determination of the foregoing pressure derivatives only requires the measurement of the isobaric expansibilities a, as a function of p and T. Thus, it is easy to compute the total variation of the above-mentioned thermodynamic function over extended p, T-surfaces. Many substances starting with gases, passing on to liquids and their mixtures, and ending with high-molecular-weight semi-crystalline polymers have been characterized from the viewpoint of a, over extended ranges of pressure and temperature using the PCSC technique; some selected examples are illustrated in Figure 3. Measurements of the isobaric thermal expansivity a, of methane in the pressure range from 50 to 165 MPa and along four isotherms (303,333,363, and 393 K) have been performed in decreasing the pressure at a low constant rate, a=0.02 MPa.s-', in such a way as to remain at thermodynamic equilibrium. Several hundreds of data points were collected and fitted, as a function of p along each isotherm, to the following empirical equation: A

a,= @+PIC

(4)

The 32-term Modified Benedict-Webb-Rubin (MBWR) equation of state proposed by Younglove and Ely21 was used to represent the pVT-behavior of supercritical methane and, in particular, to compute a,. The experimental data agree within 2% with these values at the temperature 333 K, as shown in Figure 3(a). The same technique was used to investigate liquid substances and their binary mixtures. A general conclusion for these investigations is that nonpolar simple liquids, without strong intermolecular interactions, like n-hexane, exhibit a single crossing point of a, isotherms at moderate low pressure [Figure 3(bl)]. For liquids with strong intermolecular interactions, the crossing of a, isotherms, which is pressure dependent, is shifted to a higher pressure and this dependence is related to the nature of the specific molecular interactions. This is typically shown by associated liquids, like water [Figure 3(b2)].22 Crystalline polymers, like polyethylene as an example of the stable coexistence of crystal and amorphous phases, have been characterized by a series of apmeasurements on polyethylenes with various crystallinities at several temperatures as a function of

Chapter 13

150

pressure, up to 300 MPa.23Clearly, all a,-isotherms of the different polyethylenes have a tendency to converge at high pressures, but they do not cross as observed for simple liquids. Exceptions are only for isotherms near or above the temperature of the fusion of crystalline polymers at atmospheric pressure as seen in Figure 3(cl). From these measurements, it was found that a, is a linear function of crystallinity over the entire p , T-surface under investigation, Figure 3(c2). The straight lines presented in Figure 3(c2) have been obtained by the least squares method. From such approximations, a, of both amorphous and crystal phases have been derived. Thus, in Figure 3(c3), not only the thermal expansivities of low-, medium-, and high-density polyethylenes, but also the estimated a, of polyethylene amorphous and crystal phases as a function of pressure up to 300 MPa at 302.6 K are presented.

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Figure 3 Transitiometric investigation of the pressure effect on the isobaric thermal expansivities apof gaseous, liquid, and solid substances: ( a ) comparison between calculated and experimental data of methane at 333 K; ( b l ) quinoline as a simple fluid; (b2) water as the associated liquid; ( c l ) efect of temperature on medium-density polyethylene (MDPE); (c2) evolution of apwith the crystallinity (points represent the experimental data and the lines were obtained by the least squares fitting of data); (c3) comparison between the experimental apfor three polyethylenes with dijferent degrees of crystallinity and the predicted values for crystal and amorphous phases obtained by extrapolationfrom linear fitting of the experimental data

151

Calorimetric Measurements of Thennophysical Properties for Industry

3.2 In the Vicinity of the Critical Point The above-presented pressure-scanning procedure allows direct determination of both a, and K~ of the investigated substance. However, when approaching the saturation line, especially in the vicinity of the critical point, where both apand KT tend to infinity, achievement of a measurable pressure variation requires a very large volume variation, and the pressure-scanning procedure becomes less precise or even unusable. Consequently, the isochoric thermal pressure coefficient pv= (dp/dT), =( o ! ~ / K ~ ) is hardly accessible. Thus, the volume-scanning procedure was used instead for measuring the thermomechanical coefficients of n-hexane in the vicinity of the critical point.24When the volume is varied as a linear function of time at isothermal conditions with a scanning rate c, the variation of heat effect with time may be written as

where PT(V)is the thermal power generated by such a process, and S is the entropy. The volume variations can also be expressed as a step function of time. Then, for each step, the integration of Equation (5) permits the direct determination of the thermal pressure coefficient, pv,as follows:

as

aP

Q

pV=(m)T=($F)v= m This simple relation holds, since the volume is an extensive variablz, under the condition that the measured heat effect, Q, corresponds to the same amount of substance under investigation as the change in volume AV (see ref. 24). In addition, apcan also be obtained from heat effects, Q, and pressure variations, Ap, and similarly, K~ can be obtained from volume and pressure variations as follows:

0.020

0.35 0.30

0.016 0.25

P

k

2

0.20

3

0.15

h

h -$4a

r

0.012

0.008

0.10 0.004

0.05

0.00

0

5

10

15

20

0.000

Figure 4 Thennomechanical coeficients a K and Pv of n-hexane at 507.2 K determined p’ .? by volume-controlled scanning transitlometry

Chapter 13

152

where Vin is the whole internal volume, AVe,c,, is the external volume variation corrected for compressibility of the hydraulic liquid, and V, is the volume of the investigated substance. However, when approaching the critical point, the pressure variations are too small to be measured and the derivation of both apand becomes impossible. This is illustrated (Figure 4) by the behavior of the three thermomechanical coefficients for n-hexane at 507.2 K as a function of pressure. The value of pv at 3.05 MPa (0.0406 MPa K-') is very close to the value obtained from the correlation equation for the saturation pressure of n-hexane, i.e. 0.040 MPa K-' (ref. 25). Although it is much more laborious than the pressure-scanning procedure, the volume scanning permits one to approach the critical point much closer and to determine & directly in the vicinity of the critical point with a reasonable precision.

4 Phase Transition Thermal Properties Scanning transitiometry is an extremely useful technique for studying phase transitions. Different transitions, mainly in polymer systems under various constraints of pressure, temperature, or chemical reagents, are reported hereafter.

4.1 Fusionkrystallization The simplest case is the first-order phase transition of semicrystalline polymers in the presence of a chemically inert fluid as a pressure-transmitting medium. Mercury was preferred due to its chemical inertness and due to its convenient and well-known thermomechanical coefficients (ap= 1.8OX K-' and K~ = 0.40X MPa-I). The polymer sample was always in intimate contact with the pressure-transmitting fluid. The pressure effect on the melting/crystallization temperature of an MDPE sample, in the pressure range from 50 to 200 MPa, is illustrated in Figure 5. The isobaric temperature scans were performed at a rate of 0.833 mK s-l either in ascending or 80 60 -

ym40 -

$ 20

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-80 390

.

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'

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i

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430 Te mperaturelK

450

470

Figure 5 Heatflow curves obtained on heating and cooling at a rate of 0.833 mK s-I for the mercury-pressurized MDPE. The base lines were shifted for the sake of clarity on the pressure efect on the meltinghystallization temperatures

Calorimetric Measurements of Thermophysical Properties for Industry

153

descending mode and simultaneous recording of the heat flow and volume variations. Both the melting temperature Tmand crystallization temperature Tcrhave been obtained at the onset of the transition by the conventional method applied to the respective peaks, i.e. from the intercept of the greatest slope of the measured signal with the base line. Taking the fusion at 200 MPa as an example, the rates of variation with temperature of both volume and heat flow yielded the same value of the melting temperature, i.e. 456.2 K. Details with respect to the rate of volume variations are given elsewhere.26 As expected, both fusion and crystallization are shifted toward higher temperatures with increasing pressure. The obtained value of 0.297 K MPa-’ for AT,,,/Ap (Clapeyron slope) is in fairly good agreement with the known literature values.27 Variations of volume and enthalpy in the same process were obtained by integration of the respective curves, the corresponding values being accordingly280.0573 cm3 g-I and -88.54 J g-]. The scanning transitiometric technique was also adapted to work under supercritical conditions. In this case, the pressure-transmitting fluid is the gas that is in a supercritical state. The investigated sample, polymer or paraffin, was placed in an open glass ampoule supported on a spring in the reaction cell, in order to be positioned in the central active part of the detector zone and in direct contact with the pressurization fluid. The reaction vessel was connected to a pressure gauge and to the high-pressure pump through a stainless-steel high-pressure capillary tube; it can also be connected to a supercritical fluid tank or compressor for filling the fluid into the pump. Such a setup can be easily used to investigate the pressure effect on the (solid +fluid)-to-fluid transitions. The tetracosane/methane (C,,H5dCH,), poly(vinylidenefluoride)/1,l -difluoroethylene (PVDF/C,H,F,), and poly(vinylidenefluoride)/nitrogen (PVDFN,) binary asymmetric systems have been selected to illustrate this particular It is well known that at elevated pressures, these systems show a pronounced nonideal behavior because of large differences in the molecular size of components. On the other hand, both CH, and C,H,F, are known as “good” solvents for the corresponding solids, tetracosane, and PVDF, respectively; even if C2H,F, dissolves PVDF at high pressures and temperatures (over 200 MPa and 500 K), whereas nitrogen is inert with respect to the polymers, a comparative investigation is of interest. Two series of calorimetric heat flow curves of melting of tetracosane and PVDF are presented in Figures 6(al) and (a2), respectively. The isobaric temperature scans are reported at a rate of 1 rnK.s-’. The experiments were performed first in ascending pressures. The initial pressure was automatically kept constant while heating; then, the apparatus was cooled down at the same pressure. Next, the pressure was increased and the heating and cooling procedure was repeated. As Figures 6(al) and (a2) show, despite the differences between the molecular weight of the two solids (for PVDF 113,000 and % = 330,000) as well as the physical properties of the fluids, the curves of fusion are very similar. The fusion peaks broaden with increasing pressure, especially above 40 MPa, while the amplitude of the peaks concomitantly decreases. One can observe that at a pressure of 180 MPa, the transition is only slightly visible, especially in the case of the tetracosane/CH, system.

==

154

Chapter 13

(81)310

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.

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.

310

315

330

335

336

340

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.

180 160

60 40 20 0315

320

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3.30

Figure 6 Pressure eflect on the temperature-inducedand gas-assisted melting of tetracosane and PVDF (for more details and explanations, see refs. 26 and 29): ( a l ) and (a2) heat rate evolution during fusion in the presence of supercritical CH, and C,H,F, (VF), respectively; ( b l )fluid phase equilibria in the tetracosane/methane system; and (b2)partial p-T phase diagramfor the PVDF- VF and PVDF-N, systems. Note the depression in the melting/crystallization temperatures in the pressure range up to 30 MPa for the tetracosane/CH, and PVDFNF systems

A detailed presentation of the temperature of the melting transition for the both systems as well as the temperature of crystallization for the PVDF/C,H,F, and PVDF/N, is given in Figures 6(bl) and (b2). Under the inert N, pressure, both melting and crystallization temperatures increase with increasing pressure, similar to the effect observed on the mercury-pressurized MDPE (Figure 5). In the investigated range of pressure (0.1 to 30 MPa), the slope of T,-pressure plot and T,,-pressure plots were 0.108 2 0.002 and 0.115 k0.002 K MPa-I, respectively. On the contrary, CH, and C,H,F, depress the melting/crystallization point by sorption of the gas in the polymer up to 30 MPa. The antiplasticization effect of the hydrostatic pressure of the two fluids is apparent above 30 MPa. Consequently, crystallization and melting temperatures are a compromise between the effect of hydrostatic pressure, which increases the transition temperatures of crystallization and fusion, and the solubility of the fluids in the solute-rich phase, which reduces these values.

4.2 Glass Transition Temperature

4.2.1 By Temperature-ModulatedDifferential Scanning Calorimetry Differential scanning calorimetry (DSC) has been used to successfully characterize structure and transitions (changes in structure) in plastics for more than 40 years. Plastics are becoming more complex in order to meet the demand for lower-cost materials and improved physical properties. Consequently, it is becoming increasingly more difficult to characterize the structure and resulting physical properties of

Calorimetric Measurements of Thermophysical Properties for Industry

155

plastics that are often polymer blends and composites. The result is that new approaches to using DSC or new techniques like Temperature Modulated DSC@ (TMDSC) are required. TMDSC employs a modulated or sinusoidal change in the heating rate in order to automatically separate the heat capacity-reversing contribution from the nonreversing kinetic contribution in the total heat flow signal. Apart from traditional DSC, with TMDSC, the two signals are generated in a single experiment so that each of the components to the total heat flow signal can be shown and analyzed independently.

Total = TMDSC Reversing + TMDSC Nonreversing. The high potentiality of TMDSC is illustrated here for the evaluation and selec31 The tion of the polymers used in the NanoImprint Lithography technique (NIL).30* nano-imprint process can be successful in a manufacturing environment only if a good critical dimension (CD) control can be achieved across the wafer. Usually, the nano-structures obtained by NIL [see Figure 7(al) for printing geometric parameters] are studied as a function of process parameters like the type of resist, pressure, temperature, and time, depending on the thermal properties of polymer. Figures 7(b) and (cl) illustrate the thermal signatures of two different polymers: mr-18030, Figure 7(b), is a polymer designed for nano-imprint lithography and NEB 22, Figure 7(cl), is a negative e-beam photoresist formulated by Sumimoto Chemical. The glass transition temperature, Tg,is observed on the reversing curve at

50

tb)

100 150 200 Temporituro/*C

-0 24 250

Figure 7 Illustration of the TMDSC technique for measuring Tg of the polymers designed for nanolithography: ( a l ) printing geometric parameters; (a2) exumple of apperiodic dense lines; (b)t h e m 1 signature of the polymer mr-18030; ( c l ) t h e m 1 signature of the polymer NEB 22; and (c2) variation of the thickness of the NEB22 resist as a function of the bake temperature, showing the domains of optimal deposition temperature

156

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around 125 and 80 "C for mr-I8030 and NEB 22, respectively. It is worth mentioning that under classical DSC, such glass transition temperatures cannot be observed. In the case of mr-18030, the decomposition of polymer is characterized by the endothermic peak observed on the non-reversible curve at around 230 "C, showing that the polymer has to be pressed in the temperature range 130-200 "C. In the case of NEB 22 photo-resist, several peaks are observed on the non-reversing part of the heat flow; the two endothermic peaks located at around 100 and 130 "C correspond to the evaporation of the Photo Acid Generator and residual solvent. The exothermic peak above 150 "C was attributed to the reaction between the polymer and crosslinker. Further, NEB 22 polymer has been printed under different time and temperature conditions and the results of the residual polymer thickness measurements allowed to identify the domains of optimal deposition temperatures, Figure 7(c2). Many other resists or their blends have been studied by TMDSC, including positive resists, which contain a protection group and no crosslinker. Typically, TMDSC experiments were conducted under a modulation amplitude of A, = 0.8 K, a period of p = 60 s, and an average heating rate of q = 2 K min-'. All samples analyzed had a mass of about 1-3 mg.

4.2.2 By Scanning Transitiometry The glass transition temperature is also affected by pressure since an increase in pressure causes a decrease in the total volume, and an increase in Tg is expected based on the prediction of decreased free volume. This result is important in engineering operations such as molding or extrusion, when operation too close to Tgcan result in a stiffening of the material. Investigation of the glass transition of polymers under pressure is not a simple problem, especially in the case of elastomers whose Tg are usually well below the ambient temperature. In this case, the traditional pressure-transmitting fluid, mercury, must be replaced since its crystallization temperature is relatively high, i.e. 235.45 K. The choice of the replacement fluid is a challenge because it should be chemically inert with respect to the investigated sample (with respect to all its constituents). Also, values of its thermomechanical coefficients, compressibility, K~ and thermal expansivity, ap,should be smaller than those of investigated samples. An additional difficulty in the investigation of second-order-type transformations is the relatively weak effect measured. It is well known that the amplitude of the heat flux at the glass transition Tgincreases with the temperature scanning rate, while the time constant of Tian-Calvet-type calorimeters imposes temperature scan rates that are slow compared to typical DSC scan rates. However, with the use of an ultra-cryostat coupled to the transitiometer, we have used a temperature program that allowed the accurate measurement of Tgat relatively high scanning rates.26In a typical run, the temperature of the cooling liquid was lower than that of the calorimetric block during the stabilization periods (isothermal segments) and it was higher during the dynamic segment. In such a way, the scanning rate could be increased up to 0.7 K min-I, always maintaining a minimal difference between the target and real temperatures of the calorimetric block. Since the temperature gradient between the "heating fluid" and the calorimetric block was kept constant (20 K), the power uptake of

Calorimetric Measurements of Thermophysical Properties for Industry

157

Figure 8 Illustration of the scanning transitiometry technique for low-temperature, highpressure investigation of polymers glass transition. The pressure coeficient of the glass transition temperature is given in the inset

the heating elements was quasi-constant; thus, the interference of sudden changes of power uptake on the calorimetric signal was avoided. Figure 8 shows the evolution of Tgof a poly(butadiene-co-styrene) vulcanized rubber during isobaric scans of T with the temperature ranging 218.15 to 278.15 K at 0.4 K.min-', in the pressure range from 0.25 to 90 MPa. As can be seen in the inset of this figure, the Tg increases linearly with pressure with a slope of 0.193 20.002 K MPa-'. It should be noted that Tg is expressed as the temperature corresponding to the peak of the first derivative of the heat flux (i.e.the inflexion point of the heat flux).

5 Conclusion Modern industrial activities are characterized by more and more sophisticated processes to exploit natural resources and produce high-quality products and services in the best economic conditions while preserving the natural environment and human health. The basis of these elementary rules relies on elaborated and pertinent chemical engineering calculations. A large amount of good-quality thermodynamic data and of thermophysical properties are a prerequisite for such calculations. In the past and for quite a long time, equations of state and theoretical models have very often been developed to predict or estimate thermophysical properties as well as phase equilibria of a variety of fluid systems. Very few and poor experimental data were available for validation of the models and most of the models concerned rather simple molecules, or simple fluid mixtures. New advanced materials including specialty polymers and processing operations, like extrusion, over extended temperature and pressure ranges, require accurately measured above-mentioned properties, very often under extreme conditions of T and p . Newly developed techniques and instruments, such as p VT-calorimetry, scanning transitiometry, and temperature-modulated differential calorimetry (TMDSC), to cite a few, show how important thermal

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and calorimetric methods are in this scientific endeavor. Technological progress has permitted these different techniques to be used to provide excellent, say unambiguous, reference values from low to high temperatures, over very large pressure ranges and in the critical region.

References L. Dordain,J.-Y. Coxam, J. R. Quint and J.-P. E. Grolier,J. Supercritical Fluids, 1995,8,228. J.-Y. Coxam, J. R. Quint and J.-P. E. Grolier, J. Chem. Thermodynam., 1991,23, 1075. L. Dordain, J.-Y. Coxam and J.-P. E. Grolier, Rev. Sci. Znstrum., 1994,65, 3265. J. F. Ely, J. W. Magee and W. M. Haynes, Research Report 110, Gas Processor Association, 1987. 5. R. B. Stewart and R. T. Jacobsen, J. Phys. Chem. Ret Data, 1989,18, 639. 6. American Gas Association, Transmission measurement Committee, N8, American Petroleum Institute MPMS, Chapter 14.2, 2nd edn, November 1992. 7. G. Soave, Chem. Eng. Sci., 1972,27, 1197. 8. D.-Y. Peng and D. B. Robinson, Znd. Eng. Chem. Fundam., 1976,15,59. 9. B. I. Lee andM. G. Kessler,A.Z.Ch.E.J., 1975, 21, 510. 10. E. BChar, R. Simonet and E. Rauzy, Fluid Phase Equilibr, 1986,21,237. 11. C. H. Chien, R. A. Greenkorn and K. C . Chao, A.Z.Ch.E. J., 1983,29,560, 12. V. Jr. Ruzicka, M. Zabransky and V. Majer, J. Phys. Chem. Ref., 1991,20,405. 13. J. A. Gates, D. H. Tillet, D. E. White and R. H. Wood, J. Chem. Thermodynam., 1987, 19, 131. 14. P. S. Z. Rogers and C. J. Duffy, J. Chem. Thermodynam., 1989, 21,595. 15. K. S. Pitzer, J. C. Pepier and R. H. Busey, J. Phys. Chem. Re$ Data, 1984, 13, 1. 16. S. L. Randzio, J.-P. E. Grolier, and J. Quint, Rev. Sci. Znstrum., 1994, 65, 960. 17. S. L. Randzio, E. A. Lewis, D. J. Eatough and L. D. Hansen, Znt. J. Thermophys., 1995, 16, 883. 18. J.-P. E. Grolier, Thermochim. Acta, 1997,300, 149. 19. J.-P. E. Grolier, A. Inglese and E. Wilhelm, J. Chem. Thermodyn., 1984, 16, 67. 20. Scanning Transitiometer ST-6VI from BGR TECH; S . L. Randzio, J.-P. E. Grolier, J. Zaslona and J. Quint, French Patent 09227, 1991; S. L. Randzio and J.-P. Grolier, French Patent 02612, 1998. 21. B. Younglove and J. E Ely, J. Phys. Chem. Re& Data, 1987, 16, 577. 22. S. L. Randzio, Thermochim. Actu, 1997, 300, 29. 23. L. Rodier-Renaud, S. L. Randzio, J.-P. E. Grolier, J. R. Quint and J. Jarrin, J. Polym. Sci. Part B: Polym. Phys., 1996,34, 1229. 24. S. L. Randzio, J.-P. E. Grolier and J. R. Quint, High Temp.-High Pressures, 1998,30,645. 25. S. L. Randzio, J.-P. E. Grolier, J. R. Quint, D. J. Eatough, E. A. Lewis and L. D. Hansen, Int. J. Thermophys., 1994, 15,415. 26. J.-P. E. Grolier, F. Dan, S. A. E. Boyer, M. Orlowska and S . L. Randzio, Znt. J. Thermophys., 2004,25 (2), 297. 27. S. L. Randzio and J.-P. E. Grolier, Anal. Chem., 1998,70, 2327. 28. S. L. Randzio, J. T h e m . Anal. Cul., 1999, 57, 165. 29. S. L. Randzio, Ch. Stachowiak and J.-P. E. Grolier, J. Chem. Thermodyn., 2003,35, 639. 30. C. Perret, C. Gourgon, G. Micouin and J.-P. Grolier, Jpn. J. Appl. Phys., 2002,41, 4203. 31. L. Pain, C. Higgins, B. Scarfoglikre, S. Tedesco, B. Dal’Zotto, C. Gourgon, M. Ribiero, T. Kusumoto, M. Suetsugu and R. Hanawa, J. Vac. Sci. Technol., 2000, B16(8), 3388. 1. 2. 3. 4.

CHAPTER 14

Plastic Recycling WOLFGANG ARLT

1 Introduction This chapter describes post-consumer plastic recycling, where “recycling” refers to plastic material and may involve material recycling, chemical recycling or thermal recycling. In material recycling, the material of the original application is restored but in another form. The recycling can involve a pure material or a mixture. Recycling of the pure material often occurs in the production of the merchandise, e.g. the production of insulating roofing materials. Various plastic manufacturers return rejected parts and the trim from fabrication processes back to the production process, thus contributing to the formation of new plastip Droducts. This procedure, known in the plastics industry as regrinding, can be repeated numerous times, as long as the percentage of regrinds remains low. Material recycling in mixtures occurs when plastic materials from the households are collected. Because most polymers, even of the same species, do not mix well in the melt, a demixed polymer is normally not useful in recycling. One example of mixed material recycling is the production of park benches from household plastics, which were first produced in Germany in the1990s. Chemical recycling relates to a chemical change and usually involves a crude mixture of polymeric material, e.g. for thermosetting materials; it may be of interest to chemically treat the plastic in order to return to the initial monomers or similar materials. From the thermodynamics point of view, it should be noted that the production of polymers from monomers is an exothermic process and as a result, the monomer production requires work and heat. Unfortunately, in many cases, the amounts involved are unreasonable and hence uneconomical. Thermal recycling involves, in most cases, the simple burning of collected material and the utilization of the resultant exothermic heat. In some cases, the material may have a chemical use such as in the chemical reduction of iron oxides in a furnace. In this example, the plastic replaces coal as the reducing agent.

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2 Cost of Recycling The purpose of plastic recycling is to save energy (crude oil) and to reduce the problems related to landfilling plastic material. When considering plastic recycling, it is important that at least the energy balance is fulfilled. The steps in the recycling process of mixed plastics include collection, separation and replacement. The collection of mixed packaging material from the public and consumers involves either curbside collection or collection from drop-off centres. Curbside collection involves consumers placing designated plastics in special containers outside their homes for pick-up by a public or private hauler, and drop-off centres refer to consumers bringing their plastics to a centrally located collection point. In contrast to the collection at a production site, these costs are estimated to reach 0.50 m g . ’ The cost of separating or sorting the collected recycling material has been estimated to add another 1-2 €/kg to the overall cost.’ Furthermore, the recycled material has to be cleaned and shaped for the future application. In some cases, the sorted plastics are converted into flakes or pellets, which can then be used to manufacture new products. The cost depends on the designed application. Recycled material cannot be used in every application, for example, strict rules for food packaging material often prevent the use of recycled plastics. Summing up, the recycling cost quickly adds up to over 1.5-2.5 a g . This often exceeds the price of virgin bulk plastics. One way to increase the amount of plastic recycling is to institute a fee for landfilling plastic material. However, in some countries in the European Union, plans are being made to eventually close landfill sites. This will boost the amount of plastics being recycled. It should be mentioned that most polymers are made from crude oil, so plastic recycling is one way to reduce our dependence on oil imports.

3 Thermodynamic Considerations From the point of view of thermodynamics, a polymer is a valuable low-entropy product that is produced by dissipating heat to the surrounding. Any breaking of the bonds within the recycled polymer will be costly in terms of energy. As a result, plastic recycling ideally should not involve the breaking of chemical bonds. Furthermore, the second law of thermodynamics (disorder is the preferred state) implies that the separation (creating order) of plastic materials from a mixture of plastics must involve supplying heat and work. Recycling plastics is, by nature, a costly business and in the extreme case, total recycling requires infinite work and heat.

4 Polymer Production Polymer production in the world is increasing and as an example, the polymer production in Germany is reported in Table 1 for the years 1999 and 2000. The data in Table 1 give a good impression of the magnitude of the plastics being produced and used in Germany and of the amounts of plastic that need to be recycled in the future as a result of long residence times e.g. in construction. A detailed breakdown of the collection and recycling of plastics has been obtained from the German trade organization, “Griiner Punkt” (“Green Dot”). The Packaging

Plastic Recycling

161

Table 1 Polymer production in Germany,2in units of 106 metric tonnes, Mt. In this table, PVC refers to poly-vinylchloride, PP to poly-propylene, PE to polyethylene (high density or low density), PS and EPS to poly-styrene and PET to poly-ethyleneterephthalate Year Production PVC PP PE-LD+LLD PE-HD PS +EPS PET Thermosetting Trade Export Import Balance

1999

2000

14.8 1.55 1.43 1.30

15.5

1.60 1.46 1.32 1.08 0.83 0.30 4.08

1.06 0.79

0.21 3.61 8.47

9.39

6.00 2.47

6.32 3.07

Ordinance is the legal basis for the work of the private company DSD (Duales System Deutschland). It searches and contracts companies involved in recycling plastics, pays them and collects a licence fee from every producer of goods. This is dependent on the weight of the total packaging material and the number of pieces produced. The producer of the original packaging material also has to pay a licence fee. In the year 2000,0.570 Mt, with 64% (0.365 Mt) being mixed plastics, were collected by the contractors of the Green Dot. The structure of the material is presented in Figure 1.

EPS/ cups

2%

bottles

foils

23%

mixed plastics 64%

Figure 1 Details of the packaging material collected in the year 2000 in Germany

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I

I

distillation

+

naphta, gas oil

.c

fi thermal cracking

f

ethylene

I

polymerisation

+

I

poly -eth ylene Figure 2 Pathway involved in the production of poly-ethylene from crude oil

Because the cups, bottles and foils are more or less of one species of polymers, the mixed plastics component is a more serious problem. It consists mainly of polyolefins. As a result, much of this remaining chapter will refer to poly-olefins. Figure 2 describes the production pathway from crude oil to poly-ethylene.

5 Recycling Schemes There are numerous schemes for separating plastics. In this chapter, the mechanical methods are excluded. Examples of these include a swim-float scheme and mechanical handling based on the identification of the type of plastics on a conveyor belt. A comprehensive summary of the various types of plastics is given by ref. 3. A thermodynamic method, more fitting to this chapter, has been proposed by Nauman et al. They claim a process for the separation of a physically mixed solid polymers by selective dissolution. They rely on the different polymer solubility characteristics. Tables of this property have been reported and are based on regular solution theory and Hildebrand solubility parameters. The core of the Nauman invention is to find suitable solvents to dissolve particular polymers under defined temperature and pressure conditions. A mixture of polymers is first added to one solvent, at a given temperature, in order to dissolve a particular polymer. The remaining polymer mixture is then treated at a higher temperature with the same solvent or with a different solvent. For clarity, two examples are taken from the patent4 A mixture of PS, PVC, LDPE, HDPE, PP and PET has to be separated. If the solvent tetrahydrofuran (THF) is used then: (i) PS and PVS dissolve at room temperature and form two liquid phases of 10% impurity in each;

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(ii) LDPE dissolves from the remaining polymer at 65 "C; (iii) HDPE and PP dissolve in a single liquid phase at 165 "C and (iv) PET at 190 "C. The four liquid mixtures have to be treated separately. The pressure over the solution is only slightly below the vapour pressure of the pure solvent, since a polymer concentration of up to 20 wt% does not really influence the vapour pressure. In the given case, the final pressure is about 17 bar. The same polymeric mixture can be treated with the solvent xylene. In this case, the LDPE dissolves at 75 "C,HDPE at 105 "C, PP at 118 "C and PVC at 138 "C. The PET remains undissolved. It should be noted that the fractions are not pure in the sense of there being only one polymer present. However, at each of the above temperatures, the fractions include one main polymer with small amounts of other polymers being present. Another proposal has been made by Arlt et aL5 The basic principle of the invention is that most polymer melts tend to form two more or less pure (in terms of polymers) phases. If a solvent is found that does not interfere with this, the same behaviour should be observed at higher temperatures. It was found that the solvent should be of a chemical nature similar to the polymers. There is, however, a mass transfer problem of demixing at lower temperatures caused by high viscosities. Concentrated polymer solutions tend to take hours to form two distinct liquid phases. A solution to this problem is the use of the lower critical solution temperature. Because of their thermodynamic nature, all polymer-solvent mixtures tend to form two liquid phases ("LL") with low viscosities, at higher temperatures (LCST) as depicted in Figure 3.

T

LCST-region

UCST-region

mass fraction PE

Figure 3 The composition-temperature diagramfor pure poly-olefins showing UCST (upper critical solution temperature) and LCST (lower critical solution temperature) behaviour (because of the nature of this mixture, the UCST cannot be observed and therefore is dotted)

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T=20 "C

solvent

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T=140 "C

+

unsoluble compounds

(111) filtration

T-140 "C

F (IV) shear crystallization

solvent

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(V) solid-liquid separation

T-100 "C

(VII) partial evaporation

T-100 "C

(VIII) liquid-liquid demixing

(IX) evaporation

I

(XI evaporation

TLLE- 140 "C

T-140 "C

Figure 4 Recycling scheme as invented by Arlt et al. and Lindner

Figure 3 shows a more or less pure solvent phase (left-hand side of the diagram) and a polymer phase existing at low and high temperatures. It was important to prove that the demixing of the melt is maintained in the LCST region, forming two liquid phases, with one phase involving one solvent containing one polymer and the other phase containing the solvent and the other polymer, Because only pure phases (in terms of polymers) are required, the liquid demixing was not applied to the separation of HDPE and LDPE. An alternate process of shear crystallization was incorporated. The full scheme of the process by Arlt et dllindner, as published in the cited patents (see refs 5-7), is shown in Figure 4. The shear crystallization process is given in step IV in Figure 4. The solvent-containing polymer phases are separated by evaporation (steps VI, VII, IX and X in Figure 4) and traces of solvent can be removed by vacuum extrusion.

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Table 2 Spec@ CO, emission in kg per kg HDPE Production of virgin polymer as depicted in Figure 2 1.064

Polymer recycling as depicted in Figure 4 0.402

Diference 0.662

6 Energetic Considerations As indicated in the chapter, the cost of recycling the plastic must be advantageous in terms of work and energy. Arlt et aZ.* estimated the total heat for 1 kg of recycled polymer (average for all types of polymers) following the scheme given in Figure 4 to be 3.459 MJ and the electrical energy to be 0.250 kWh. Arlt compared this result to the production scheme depicted in Figure 2. Table 2 shows that the proposed recycling scheme produces 40% of the carbon dioxide emissions of virgin material. This fact underlines that it is advantageous in terms of energy to recycle polymers and to use crude oil for energy production. The reason is inherent in the many energy-consuming steps leading to the virgin polymer.

References 1. J. Brandrup, M. Bittner, W. Michaeli and G. Menges, Die Wiedervenvertung von Kunststoffen, 1995. 2. Verband Kunststofferzeugende Industrie e.V.(German society for the polmer producing industry): Oktober 2001. 3. R. J. Ehrig (ed), Plastics Recycling - Products and Processes, 1992. 4. E . B. Nauman, Polymer recycling by selective dissolution, US-patent 5 198471, 1993. 5. W. Ark, B. Bungert, G. Sadowski and S. Behme, Thermisches Trennverfahren fur vermischte Polymere, German patent appl. 198 06 355.5; W. Arlt, G. Sadowski, M. Seiler and A. Thiele, Beschleunigung der Phasentrennung von Polymer-enthaltenden flussigen Phasen durch verzweigte Losungsmittel, German patent appl. DE 199 22 944, 14.5.99; W. Arlt, G. Sadowski, M. Seiler and S. Behme, Verfahren zur Trennung vermischter Polyolefine, German patent appl. DE 19905029, 29.1.99; W. Arlt, B. Bungert, G . Sadowski and S. Behme, Thermisches Trennverfahren fur vermischte Polymere, German patent appl. DE 19806355, 10.2.98, internat. Patent appl. PCT/EP99/05809, 6.8.99. 6. W. Lindner, Verfahren zur Trennung polyolefinischer Kunststoffmischungen, German patent application 19927523, 2000. 7. A. Keller and F. M. Willmouth, J. Macrom. Sci.-Phys., 1972, B6(3), 493-538. 8. W. Arlt and A. Korporal, unpublished data.

CHAPTER 15

Industry Perspective on the Economic Value of Applied Thermodynamics and the Unmet Needs of AspenTech Clients CHAU-CHYUN CHEN, SUPHAT WATANASIRI, PAUL MATHIAS AND VALENTIJN V. DE LEEUW

1 Economic Impact of Applied Thermodynamics Thermophysical properties, phase equilibria, and solution chemistries are the underlying physical and chemical phenomena of industrial chemical processes. Rigorous thermodynamic modeling of such phenomena establishes a sound scientific foundation for simulation of industrial chemical processes and subsequent process development, optimization, and control. The oil and gas, chemical, polymer, and pharmaceutical industries are striving to create new products, catalysts, recipes, and processes. At the same time, they are challenged to shorten the time-to-market, ensure process and plant safety, and achieve regulatory compliance. Worldwide, the industries spend an estimated $500 billion annually’ in product and process research and development, pilot plants, process engineering, detailed engineering, construction and startup, manufacturing operations, and plant maintenance. In these industries, chemists and engineers enhance the quality and efficiency of their value-added activities and decision-making processes by using engineering software that helps them model the complex physical and chemical phenomena of the chemical systems and processes they investigate. These enterprise-wide, first principles-based engineering software products are easy-to-use, effective, and accurate. Underlying these software systems are the integrated and validated applied thermodynamics systems that describe the thermophysical properties and chemistries of the relevant chemical systems. In short, the quality and efficiency of chemists’ and engineers’ pursuits in the process industry are profoundly impacted by

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the scientific rigor and engineering accuracy of the underlying applied thermodynamics models and tools.

2 Applied Thermodynamics System An applied thermodynamics system includes thermophysical property calculation methods, applied thermodynamic models, and model parameter databanks. It also includes phase equilibrium calculation algorithms, parameter estimation and regression, and a user interface. The system is an engineering tool for studying the physical and chemical behavior of chemical systems. When used along with other engineering programs, such as a process simulator or a heat exchanger design program, it provides rigorous and accurate thermophysical properties that are the foundation for all process calculations. Its economic impact is greatly magnified when the system is used consistently in all engineering calculations and programs, at every stage of life cycle modeling and throughout the enterprise. These benefits and values are summarized in Figure 1 and Table 1.

3 Elements of an Applied Thermodynamics System The elements for an enterprise-wide applied thermodynamics system can be divided into the following broad categories: Calculation engine Analysis tools Programmable interfaces Deployment tools Aspen Properties@Helps Drive Chemical Product 8 Process Innovations

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Figure 1 Applied thermodynamics system helps drive chemical product and process innovations

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Table 1 Applied thermodynamics system delivers value at every stage of life cycle modeling, and throughout the enterprise Value Support and improve decision making in various process R&D activities and tasks such as solvent selection, solvent recovery, product purification, etc. Develop high-fidelity models for process design, revamp, or Process engineering optimization Reuse and develop thermophysical property knowledge developed during the conceptual design stage Eliminate the use of unvalidated models and manual transfer of model parameters Apply high-fidelity models for equipment design Detailed engineering Apply consistent design by leveraging the same validated thermophysical properties knowledge and models used in preliminary design stages Eliminate the potential errors associated with manual transfer of data Plant start-up and operation Develop a high-fidelity model of the plant for operator training by using the same validated, rigorous thermophysical property models Reduce operating costs by debottlenecking or optimizing the process using either offline or online models developed with the same validated, rigorous thermophysical property models Improve capital and operating efficiencies by improving Throughout the enterprise communications and sharing physical property knowledge Ensure physical property data and model integrity across various engineering tools used at each stage of the process lifecycle

Stage Conceptual design

3.1 Calculation Engine The calculation engine is the heart of the property system. The specific contents of the engine are determined by the industries the system is designed to serve. In general, the engine should contain the following: A flexible thermophysical property calculation framework to support various calculation methods. A comprehensive library of thermophysical and transport property models. Extensive pure component and binary system parameter databanks, including high-fidelity data packages for important industrial processes. Proven parameter estimation capability to fill in the gaps of missing properties and for evaluation of new chemicals. Comprehensive data regression capability to fit experimental data to models. Robust and efficient mathematical algorithms for phase equilibrium calculations (e.g., VLE, LLE, VLLE, and solid precipitation). Verified analytical derivatives of properties with respect to temperature, pressure, and composition. This is a key requirement for equation-oriented process calculations and dynamic or transient process simulation. Robust implementation and continuity of results.

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The engine should also be able to: Extend easily to client property models and data to allow the company to use proprietary knowledge. Link to sources of experimental measurement data, such as the Dortmund Databank or NIST TRC database. Interface with third-party applications through CAPE-OPEN COM Thermodynamic, OLE automation, or FORTRAN interfaces. This allows third-party models and data to be plugged in and used.

3.2 Analysis Tools Thermophysical property analysis tools are needed to allow chemists and engineers to evaluate the performance of the models being used or developed for the chemical systems and processes at conditions of interest. These analysis tools include property tables, phase equilibrium calculations, phase envelopes, and plots of phase diagrams.

3.3 Programmable Interfaces Programmable interfaces, whether proprietary or public, allow the calculation engine to be used by other engineering programs. The interface must include: Configuration (setup) of compounds and property methods to be used. Property and phase equilibrium calculations. Query of model details, such as parameter requirements. Access (read) and change (write) of model parameters.

3.4 Deployment Tools In order to maximize the value of the applied thermodynamics system throughout the enterprise, it must be accessible to all process engineers and chemists who require accurate thermophysical property calculations in their daily work. Web applications, which do not require installation of the calculation engine on the user’s computer, facilitate easy access to the system. Web applications can be designed to provide pure component data such as normal boiling point and critical properties. They can also provide access to the most frequently carried out calculations, such as phase equilibrium calculations, tabulation, and plotting of pure component properties as a function of temperature and pressure, and mixture property calculations. Microsoft Excel is one of the most widely used engineering programs. When coupled with a rigorous applied thermodynamics calculation engine, it becomes a very powerful engineering tool for various engineering applications. This can be accomplished through the Excel Add-In feature, which facilitates access to the calculation engine for: Selection of compounds and property methods. Calculation of properties and phase equilibrium using an Excel Function. Such calculations can be performed on one or more cells. The input required for

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the calculations can be defined in other cells and referenced. Calculation results are updated and propagated automatically when these input cells are changed. As an alternative to the Add-In, users can also develop their own Excel Macros to call the programmable interfaces that have been published.

4 Industrial Applications Numerous industrial applications of applied thermodynamics have been reported in the literature for engineering analysis of wide varieties of chemical systems and processes. For example, Chen and Mathias' reported examples of physical property modeling for the high-density polyethylene process and for sulfuric acid plants. Here, we present two recent examples that are illustrative of numerous applications of applied thermodynamics models in the industry for various process and product development studies.

4.1 Phase Equilibrium Modeling for Nylon-6 Process Simulation A recent study by Seavey et aL2 presented a thermodynamically consistent model for the phase equilibrium of water/caprolactam/nylon-6 mixtures, based on the POLYNRTL (polymer non-random two-liquid) activity-coefficient model3 in Aspen Properties*, the commercially available applied thermodynamics system from AspenTech. The model predicts phase equilibrium for any binary or ternary mixture containing water, E-caprolactam, and nylon-6 at industrially relevant temperatures and pressures with an average error of 1%.The model was validated with operating data from commercial manufacturing process units. It was then used to simulate a melt train and a bubble-gas kettle train for the industrial production of nylon-6. The phase equilibrium model facilitates fundamental kinetic, thermodynamic, and mass transfer calculations for detailed predictions about reactor products over wide ranges of temperature and pressure conditions. The model represents a significant advancement toward the goal of having a unified, consistent model that describes the phase behavior of all commercially significant nylon-6 polymerization and depolymerization technologies. It forms the scientific basis for the enterprise efforts in new polymer grade development, evaluation of new catalysts, process development, and optimization and control, to name a few.

4.2 Quickly Screen Solvents for Organic Solids Frank et aL4reported examples of quickly screening solvents for organic solids. In one particular example, solubilities of aspirin in four different solvents (acetone, ethanol, chloroform, and cyclohexane) were used to regress the Hansen solubility parameters for the solute, aspirin. Once the Hansen solubility parameters are identified for aspirin, Frank et al. showed that one could quickly estimate the solubilities of aspirin in any solvent or solvent mixture as long as the Hansen solubility parameters are also available for the solvents. Figure 2 shows the Hansen model calculation results versus experimental data for aspirin solubility in various solvents at room temperature. The Hansen solubility

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Solubility of Aspirin (Wt%)at 25 "C 100

A UNIFAC Prediction

U .CI

8e

A Hansen Correlation

0.1

c1

0.01 0.001 0.001 0.01

0.1 1 10 Literature Data

loo

Figure 2 UNIFAC model and Hansen model results versus experimental data for aspirin solubility in various solvents at room temperature

parameters for solvents were taken from H a n ~ e nwhile , ~ the Hansen parameters for aspirin were regressed from the experimental data. Also shown are the UNIFAC predictions. Neither the Hansen model nor the UNIFAC model provides adequate quantitative results. Clearly, better models for solute solubility modeling are needed. This example shows one of the numerous opportunities for developing innovative models to address industrial product and process development needs. Since solvents facilitate organic synthesis, separation via crystallization or extraction, product formulation, and equipment cleaning,6 a rapid and reliable solvent selection model would benefit the industry in many different ways, including faster production, separation and purification, reduced solvent emission and waste, lower overall costs, and better production processes.

5 Quantitative Cost-Benefit Assessments AspenTech has developed and applied quantitative cost-benefit analysis for engineering systems, including applied thermodynamics. The analysis, provided by Aspen Value ProcessTM(AVP), is a collaborative process yielding multi-level, broad commitment for business process changes enabled by software solutions. A key outcome of AVP is the customers' validation of the estimated value. The financial information is highly sensitive and consequently there are no published cases available. Here, we present the basic approach and give typical quantitative assessment results. One could either carry out an extrapolation of this analysis to assess the industry-wide benefits (as is done below), or, alternatively, carry out a series of these assessments with specific producers. The mechanics of the value calculation involves using a company's income statement, in one form or another, (for example, the cost-of-goods-sold), and identifying the independent, major profit contributors that impact the different items in the income statement, e.g., average sales price increase, total production increase, engineering efficiency increases and cost savings, operations cost savings, and capital cost savings. For each, a complete list of specific mechanisms that generate quantifiable benefits is developed, by a business process. For example, having more accurate

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density estimations can improve the timing for ordering feed for a stock tank. Improved order timing can, in turn, increase the number of inventory turns and, therefore, lower working capital. Line items are totaled; estimations for total potential and a realistically reachable solution are made. For comparison, benefits of current work practices and systems are subtracted. Solution cost is taken into account to deliver net, additional recurrent added value. Additional public information is available at http://www.aspentech.com/brochures/AVPBrochure.pdf. A rough estimation for the process industries based on experiences with AVP would be, ‘state of the art engineering systems can deliver 1-10% of turnover in net, recurring additional benefits.’ The industry generally validates between 10 and 50% of this ‘claim.’ A conservative estimation is that 10%is directly attributed to applied thermodynamics. This gives a net, recurrent benefit range of 0.0145% of turnover of the process industries, being $600 M to $30 B for a $6 trillion industry.

6 Unmet Needs of AspenTech Clients While applied thermodynamics has contributed considerably to the work of chemists and engineers in the process industries, numerous unmet or emerging needs for innovative applied thermodynamic models remain. Some of these unmet needs, as we understand them, are summarized below.

6.1 Estimation Methods for Pure Compound Properties Many estimation methods are available for a wide variety of properties, such as normal boiling point, critical properties, vapor pressure, ideal gas heat capacity, and liquid These estimation methods are based largely on the group-contribution concept and were developed from databases consisting mainly of relatively small molecules and for properties mainly of interest to small molecules. We are, therefore, constantly forced to question the quality of the predictions, especially when dealing with large, complex molecules. Many of these large molecules exist as solids under normal conditions. Hence, properties for the solid phase are particularly important. As the industry shifts its focus to larger and more complex molecules, we find existing property data, database, and property estimation methods to be rather inadequate. For complex molecules, 3-D molecular configuration can have a significant impact on the properties. To our knowledge, no estimation methods have taken into account the effects of molecular configuration. The deficiencies with large, complex molecules exist for virtually the entire spectrum of physical properties, from heat of formation, Cp,and density, to vapor pressure, melting temperature, etc. We particularly welcome estimation methods that incorporate available experimental data, such as melting point. Engineers engage in a number of steps to quantify their confidence in an estimation method and to improve the accuracy of the prediction when working with new mole~ules.~ As shown in Table 2, a new molecule with the structure has a molecular weight of 163.606, a measured normal boiling point (Tb) of 237.6 OC,and a few vapor pressure data. Using an estimation method

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Table 2 Estimations of vapor pressure with or without Tb experimental data

T,, "C T,,"C Vapor pressure at 300 "C,bar

Experimental 237.6 3.445

Structure only 273.94 532.8 1.753

With Tb 237.6 479.2 3.697

that relies on the structure only (the Joback method), the estimated Tb is in error by 15%. This Tb is then used to estimate critical temperature, Tc. Critical pressure, Pc, is also estimated from the Joback method. Based on these estimated Tb, Tc,and Pc, the Riedel method is then used to estimate liquid vapor pressure at 300 OC, with an error of 49%. By using the experimental Tb, the estimated vapor pressure is significantly more accurate (7% error). Chemists and engineers have a healthy, professional skepticism of results from all prediction methods, whether ab initio or empirical. The use of experimental data not only improves the prediction results but also raises the confidence level associated with the prediction methods.

6.2 Cubic Equations-of-State A broad range of cubic equation of state models (EOS) are successfully used today. The EOS range from the standard Soave-Redlich-Kwong and Peng-Robinson, which is widely used in the hydrocarbon processing and related industries (oil & gas and petrochemicals), to a new class of models that extend the range of applications to chemic a l ~ . New ' ~ ~models ~ ~ are continually being developed and are too numerous to cite. For non-ideal chemical systems, given the large number of models/mixing ruledalpha functions available, clear choices for the preferred EOS have not yet emerged. We have found the PSRK model of Holderbaum and Gmehling'O to be useful to our users. The PSRK model's predictive capability is based on the UNIFAC model and is supplemented with special parameters for light gases. At present, it is our opinion that a model that is based on one of the CEOS/AE mixing rules, an appropriate alpha function, and an improved density prediction will adequately serve industry needs to model non-ideal chemical systems. 12-14 However, to be useful to the industry, a successful model must have an extensive set of pure component and binary interaction parameters available. Unfortunately, most published models are only accompanied by a small set of such parameters. Therefore, the deficiency in this particular area lies probably not in the model or mixing rule, but in identifying a satisfactory model and then devoting the resources toward developing a comprehensive library of pure component and binary interaction parameters to support it.

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6.3 Activity Coefficient Models Existing activity coefficient models such as NRTL and UNIQUAC have been very successful correlative tools that provide versatile, flexible thermodynamic frameworks to correlate available experimental data. Once properly parameterized, the models are then used to perform both interpolation and extrapolation. In the exploratory phases of product and process research and development, generally little or limited data are available for model parameterization. However, chemists and engineers need to evaluate a multitude of possible molecular or process variations. Rather than high accuracy, successful evaluation depends on the ability to discard the least viable options and select better options for further detailed studies. Estimation methods that give reliable results for new or unknown species are required. The well-known group-contribution methods, like UNIFAC, have demonstrated their value in cases where the molecules can be decomposed to functional groups for which parameters are already available. At present, molecular simulation consumes too much CPU time to be used directly as a means for general-purpose property estimation, including phase equilibrium. However, recent advances with the COSMO-RSl5 and COSMO-SAC16models allow prediction of activity coefficients from electron density profiles that are computed from quantum chemistry methods for molecules. If libraries of electron density profiles for molecules can be made public, these models can provide an alternative predictive technique to UNIFAC. However, the quality of the predictions still needs to be improved. We are currently exploring an interesting approach with the segment NRTL method,17where three common model segments with established interaction parameters (e.g. ethylene segment as a model segment for all non-polar, hydrophobic chains or groups, -OH segment as a model segment for all hydrophilic, hydrogen bonding groups, etc.) are used to characterize any known or unknown molecules. The model correlation parameters are the equivalent numbers of these common model segments, i.e., hydrophobicity, polarity, and hydrophilicity, in each molecule. Available experimental phase equilibrium data are used to determine these model parameters. Once all molecules are characterized this way, the model becomes a simple and flexible thermodynamic framework to provide first approximation predictions. This approach combines the correlative power of the NRTL model with the predictive power of the UNIFAC model. The model removes the rigidity associated with the UNIFAC model in which molecules must be defined a priori in terms of specific functional groups. The model seems to be particularly valuable in modeling systems with large, complex molecules. It also provides a versatile framework that can be easily extended to systems with electrolytes, polymers, and biologicals.

6.4 Modeling Electrolyte Systems We have seen many successful industrial applications of applied electrolyte thermodynamics models. In particular, the electrolyte NRTL activity coefficient model of Chen and Evans1*has proved to be the model of choice for various electrolyte systems, aqueous and mixed-solvent. However, there are unmet needs that require further development.

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First, we need a predictive activity coeffcient model for electrolyte systems. The electrolyte NRTL model is correlative, and it requires extensive experimental data sets from which NRTL binary interaction parameters can be identified. The OLI electrolyte model, with its extensive parameter database, has been serving as a pseudo-predictive model. However, use of the OLI electrolyte model is limited to dilute aqueous electrolytes, its parameter database is not open to the public, and its electrolyte speciation is not supported by experiments. Second, we need an equation-of-state for electrolyte solutions. Equations-of-state are needed for modeling high-pressure applications with electrolyte solutions. Significant advances are being made in this area.19-21Given that the electrolyte NRTL model has been widely applied for low-pressure applications, we are hopeful that, some day, there will be an equation-of-state for electrolytes that is compatible with the electrolyte NRTL activity coefficient model. Modeling mixed-solvent electrolyte systems proves to be a more challenging task. Based on our experiences in modeling mixed-solvent electrolytes, we believe progress needs to be made on two fronts: (1) better analytics to establish the correct solution chemistries and the resulting speciation, and (2) better activity coefficient models that provide higher resolution in accounting for the physical interactions between various species, and segments of species, in the liquid phase.

6.5 Modeling Polymer Systems Significant progress has been made with polymer thermodynamics recently. The most notable models useful for polymer process modeling are the polymer NRTL model3 and the PC-SAFT equation-of-state.22Going forward, we expect further evolution of the PC-SAFT EOS for modeling polar and aqueous phase polymers. It is interesting to note that, in industry, we have observed strong resistance to the use of the association term due to the much higher level of complexity with the EOS. We are also looking for models and methodologies that can be used to characterize polymer microstructures such as chemical composition distribution and polymer crystallinity.

6.6 Data Packages Data packages are those models that provide high accuracy for a particular industrial application or technology. Both high-fidelity data packages and engineering quality data packages are highly valuable and are much sought after for all major industrial chemical processes and related chemical systems. In fact, a great deal of industry resources have been spent to create and recreate data packages for various chemical systems. Examples of successful high-fidelity data packages are the NIST data packages for the thermophysical properties of fluids (e.g., NIST/ASME Steam Database and the NIST Thermophysical Properties of Hydrocarbon Mixtures; for further information, visit the NIST web site at http://www.nist.gov/srd/fluids.htm). Many engineering quality data packages have been developed and they remain confidential properties of the owner companies. However, at present, publicly available and proven data packages are limited to pure fluids and relatively simple mixtures like hydrocarbons. Publicly available data

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packages are particularly lacking for complex fluids like electrolytes and polymers. Specifically, the industry could benefit considerably from data packages for major electrolyte systems such as nitric acid, sulfuric acid, caustic, amines gas treating, etc. Ideally, such data packages should be developed from widely accepted, thermodynamically consistent, thermodynamic frameworks that allow industrial practitioners to further extend the data packages to cover additional compounds and for wider ranges of conditions.

6.7 Equilibrium Calculations Reliable and fast equilibrium calculations (or so-called “flash” calculations) are the mechanism by which thermodynamic properties are used in industry. This area has received much attention in the past. Algorithms include successive substitution with acceleration and stability a n a l y ~ i s , ~I n~~. i~d~e - O u t and ~ ~ Interval methods,26 Homotopy continuation methods27 with application to three-phase systems,28 and systems with simultaneous physical and chemical e q ~ i l i b r i u m An . ~ ~area of recent focus is the flash algorithm for mixtures containing polydisperse polymers.30 However, many challenging problems remain. The flash algorithm of Behme et aL30only addresses polydisperse homopolymers. While the algorithm works for two phases, vapor-liquid or liquid-liquid, the algorithm is not yet capable of performing vapor-liquid-liquid three-phase flash. Also, while the algorithm computes molecular weight distributions for polymers in different phases, the algorithm does not yet take into account the bivariate distribution of polymer molecular weight and copolymer composition. Further challenges include the effects of copolymer sequence structure, i.e., block vs. random vs. alternating, etc. High-pressure systems in the vicinity of critical points, such as synthesis gas and air separation systems, remain a challenge. Our flash algorithm has difficulty in identifying the correct phase state, or converging to the correct vapor-liquid solutions. This problem may be exacerbated by the difficulty in obtaining the equation of state volume root in the vicinity of the critical points.31Further work to improve the algorithm and the equation of state volume root determination is required. It is believed that the homotopy continuation methods are probably better suited for calculations near the critical points. Liquid-liquid equilibrium calculations remain a problem area especially for systems containing non-volatile species such as strong electrolytes or high polymers. These species have negligible or no fugacities, and, as a result, many flash algorithms cannot properly account for them. Many important industrial systems make extensive use of surfactants for various reasons. Surfactants may dissolve in the bulk liquid phases, form distinct micelles, or preferentially concentrate on the interfaces. Existing flash algorithms do not address micelles or interfaces as possible reservoirs for the surfactants. The flash results are simply not valid for systems with surfactants. The pharmaceutical and life science industries often deal with large, complex molecules, and separation via crystallization is an important practice. Robust flash algorithms for solid-liquid equilibrium, particularly systems with multiple polymorphs, are highly desirable.

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6.8 Transport and Interfacial Properties Transport and interfacial properties are often neglected in favor of research and development efforts directed to phase equilibrium properties. Even less attention has been devoted to such properties for electrolytes and polymers. In industrial practice, the needs for transport and interfacial properties are numerous, i e . , detailed design of heat exchangers, and distillation column tray and packing sizing calculations. Both predictive and correlative models are needed for liquid viscosity, thermal conductivity, surface tension, diffusion coefficients, etc. Some recent works on polymer v i s c ~ s i t yand ~ ~ *electrolyte ~~ viscosity34 are very promising. The simplicity and accuracy of these models make them particularly attractive for engineering applications. However, extensive successful industrial applications are essential if any of these models are to become accepted in the industry.

6.9 Model Deployment One of the impediments to realizing the full value of an applied thermodynamic model is the speed and cost associated with its deployment. For a published model to gain acceptance by practitioners, either industrial or academic, it must be used and validated in actual applications. The cost of deploying a model includes coding, testing, validation of results, model application, and continued maintenance. If a new model can be published in such a way as to facilitate its deployment in inhouse program or commercial process simulators, then it has a better chance of being used and accepted. This is a difficult and complex subject. Efforts in the past include model validation initiative35and CAPE-OPEN.36The model validation and deployment should work in concert to ensure that the model being deployed is sound from the point of view of thermodynamic consistency, reasonable behavior at extreme conditions, and ability to return derived or related properties (e.g., heat capacity or enthalpy departure). CAPE-OPEN defined a standard interface for physical property calculations. Major process simulators support this interface standard. A property model that has been developed conforming to this standard should be able to be used directly in these simulators. However, the standard, as currently published, is not sufficiently precise and leaves many implementation details to interpretation, causing potential interoperability problems (e.g., how to handle properties of a physically non-existent phase). Nevertheless, significant progress has been made during the past few years in reducing ambiguity, interoperability, and performance problems in the early versions of the standard. An organization called CO-LaN has been formed (http://www.colan.org) with the responsibility for maintaining, promoting, and extending the standard.

7 Conclusion Applied thermodynamics has a major and essential contributing role in the 6 trillion dollar process industry. It is a key enabling science and technology for chemists and engineers in their pursuit of new and better products and processes in the industry. Continuing vigorous innovations and advances in applied thermodynamics are

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essential to address many unmet or emerging industry needs. We have shared our personal perspectives in the applications and needs of the industry in applied thermodynamics from the privileged position of an industry software provider. Our views, which are inherently limited by our personal biases, have benefited from close collaboration with our colleagues and partners in industry and academia.

References 1. C.-C. Chen and P. M. Mathias, A.Z.Ch.E. J., 2002,48, 194. 2. K. C. Seavey, N. P. Khare, Y. A. Liu, T. N. Williams and C.-C. Chen, A New PhaseEquilibrium Model for Simulating Nylon-6 Polymerization Processes, Znd. Eng. Chem. Res., ASAP web release date: July 18, 2003. 3. C.-C. Chen, Fluid Phase Equilibr., 1993,83, 301. 4. T. C. Frank, J. R. Downey and S. K. Gupta, Chemical Engineering Progress, December, 1999, 41. 5. C. M. Hansen, Hansen Solubility Parameters: A User’s Handbook, CRC Press, 1999. 6. P. Kolar, J.-W. Shen, A. Tsuboi and T. Ishikawa, Fluid Phase Equilibr., 2002, 194-197, 771. 7. R. C. Reid, J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids, 4th edn., McGraw-Hill, New York, 1987. 8. B. E. Poling, J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th edn., McGraw-Hill, New York, 2000. 9. E. C. Carlson, Chem. Eng. Prog, October. 1996, 35. 10. T. Holderbaum and J. Gmehling, Fluid Phase Equilibr., 1991, 70, 25 1. 11. S. H. Wong and S. I. Sandler, A.Z.Ch.E. J., 1992, 38, 671. 12. C. H. Twu, W. D. Sim and V. Tassone, Znd. Eng. Chem. Res., 2002,41,931. 13. C. H. Twu, W. D. Sim and V. Tassone, Chem. Eng. Prog., Nov. 2002,58. 14. J. Ahlers and J. Gmehling, Fluid Phase Equilibr., 2001, 191, 177. 15. A. Klamt and F.Eckert, Fluid Phase Equilibr., 2000,43, 172. 16. S.-T. Lin and S. I. Sandler, Znd. Eng. Chem. Res., 2002,41, 899. 17. C.-C. Chen, unpublished. 18. C.-C. Chen, and L. B. Evans, A.Z.Ch.E. J., 1986,32, 444. 19. W. Fiirst and H. Renon, A.Z.Ch.E. J., 1993,39, 335. 20. J. Wu and J. M. Prausnitz, Ind. Eng. Chem. Res., 1998, 37, 1634. 21. J. A.Myers, S.I. Sandler and R. H. Wood, Znd. Eng. Chem. Res., 2002,41, 3282. 22. J. Gross and G. Sadowski, Znd. Eng. Chem. Res., 2002,41, 1084. 23 M. L. Michelsen, Fluid Phase Equilibl:, 1982, 9, 1. 24 M. L. Michelsen, Fluid Phase Equilibr., 1982, 9, 21. 25 J. F. Boston, and H. I. Britt, Comp. Chem. Eng., 1978, 2, 109. 26 J. Z. Hua, R. W. Maier, S. R. Tessier, J. F. Brennecke and M. A. Stadtherr, Fluid Phase Equilibr., 1999, 158, 607. 27 A.C. Sun and W. D. Seider, Fluid Phase Equilibr., 1995, 103, 213. 28. V. Gopal and L. T. Biegler, A.1.Ch.E. J., 1999, 45, 1535. 29. W. D. Seider and S. Widagdo, Fluid Phase Equilibr., 1996, 123, 283. 30. S. Behme, G. Sadowski, Y. Song and C.-C. Chen, A.l.Ch.E. J., 2003,49,258. 31. P. M. Mathias, J. F. Boston and S. Watanasiri, A.Z.Ch.E. J., 1984, 30, 182. 32. L. T. Novak, Znd. Eng. Chem. Res., 2003,42, 1824. 33. Y. Song, P. M. Mathias, D. A.Tremblay and C.-C. Chen, Znd. Eng. Chem. Res., 2003,42, 2415.

Industry Perspective on the Economic Value of Applied Thermodynamics

179

34. J. Jiang and S. I. Sandler, Ind. Eng. Chem. Res., 2003,42,6267. 35. R. Gani and J. P. O’Connell, Comp. Chem. Eng., 2001,25,3. 36. J-P. Belaud, B. Braunschweig, M. Halloran, K. Irons, D. Pinol, New Generation Simulation Environment: Technical Vision of the CAPE-OPEN Standard, paper presented at the AIChE annual meeting, Reno, Nevada, 2001.

CHAPTER 16

Thermodynamics of New Materials M. SCHROEDER AND M. MARTIN

1 Introduction Progress in all fields of our technological world depends on the development and optimization of materials. This applies to metallic materials, ceramics and polymers as well. Research and applications related to the thermodynamics and transport processes of new materials is therefore a vast field that cannot be covered in a single article. As a consequence, the focus of this article will be on ceramic materials, and within this broad field we will focus on oxide ceramics. In the following two sections, ceramic membranes that are mixed ionic and electronic conductors and oxygen ion conductors will be discussed. Both classes of materials exhibit interesting properties and can be used for important technical applications, such as oxygen separation and catalytic partial oxidation of hydrocarbons or conversion of chemical energy into electrical energy in solid oxide fuel cells.

2 Ceramic Membranes for Oxygen Separation and Catalytic Partial Oxidation of Hydrocarbons Today, huge amounts of pure oxygen are required for the production of chemicals, petrochemicals and primary metals, as well as for medical, military and other applications. The yearly consumption of oxygen adds up to more than 35 million tons worldwide.' Although ubiquitously available in air, the production of oxygen is dominated by cost-intensive processes, such as cryogenic distillation and pressure swing a d ~ o r p t i o nA. ~new ~ ~ technology to generate oxygen in a more economical way would seem to be very attractive for industrial gas-to-liquid (GLT) applications. Recently, it was demonstrated that ceramic membranes that selectively permeate oxygen at high temperatures (>700 "C) could be used for continuous oxygen separation from air.4.5 As various industrial oxidation processes are carried out in the same temperature range, an integration of oxygen generation and chemical oxidation into a single-step process would lead to a drastic reduction of investment costs in GLT processes.6The

Thermodynamicsof New Materials

181

most prominent example for such an integrated process is the conversion of natural gas to synthesis gas by catalytic partial oxidation, which could be carried out directly at the membrane surface. In the subsequent sections, some essential properties of high-temperature ceramic membranes and problems related to their application in oxygen generators and partial oxidation reactors will be discussed.

2.1 Transport and Defect Thermodynamics in Membrane Materials In contrast to conventional permeation membranes, where separation usually takes place by pore diffusion?J oxygen transport in high-temperature ceramic membranes occurs by an electrochemical process. Provided the membrane is dense enough to prevent any leakage of other feed gas components, an infinite oxygen separation factor can be achieved. The permeation is driven by a difference of the oxygen partial pressures in the gas phases in contact with the membrane surfaces. According to Figure 1, the overall permeation process may be divided into three steps: (a) reductive incorporation of oxygen from gas phase (') with the larger oxygen partial pressure, whereby oxygen ions are formed in the membrane; (b) transport of oxygen ions through the membrane bulk and (c) oxidation of oxygen ions and release to gas phase (") with the lower oxygen partial pressure. To maintain local charge neutrality, a simultaneous internal flow of electronic charge carriers (electrons or electron holes) is necessary. It is obvious that a highly permeable membrane material must exhibit large conductivies for both ionic and electronic charge carriers. Partial conductivities of various, so-called mixed ionic electronic conductors (MIEC), as calculated or directly obtained from Refs. 9-21, are presented in Figure 2.

Gas phase '

Membrane

Gas phase

"

'2)

Carrier gas 0 2

P" ( 0 2 )

Figure 1 Working principle of a mixed-conducting oxide membrane

182

Chapter 16

Ce0.8Gdo.202-8

L%.6Sr0.4C00.8Fe0.203-~~

0

9 L%.5sr0.5c003-8

YSZ (10 d o Ti02)g

:1E-5

c

.-

:1E-6 d,

5

1E-7 2 1E-8

L%.8sr0.2cro3-8 I

1 1 1 1 1 1 1

11111111,

1 1 1 1 1 1

'1

1111111

111,11111

*

1111111

1~-9

High-performance materials are expected in the upper right section of the plot, where mainly complex metal oxides of the general formula A, --x A: B,-, Bi,03-6, with the dopant elements A' and B', are abundant. The B-sublattice contains transition metal elements, which are a necessary prerequisite for electronic conduction as they are multivalent and provide electronic defects as mobile charge carriers. For this reason, MIEC oxides are predominantly electronic conductors; however, they can also exhibit extraordinarily large oxide-ion conductivities that are comparable to, or even exceed, those of the best-known solid oxide electrolytes. This is due to the perovskite structure of these oxides, which is known to accommodate large quantities of unoccupied oxygen lattice sites. These structural defects, known as oxygen vacancies, enable fast oxide-ion migration. In contrast, the number of cation defects is generally small. The deviation from stoichiometry, 6, is therefore a direct measure of the fraction of oxygen vacancies. Controlling 6 and the average transition metal valence provides a means to tailor the transport properties of the material and can be achieved by doping with aliovalent ions. This will be illustrated for the case of undoped, stoichiometric LaBO,: substitution of the acceptor ion Sr2+for La3+ requires charge compensation either by formation of B4+or by loss of oxygen. In general, both processes take place and 6 depends on both the dopant fraction x and the fraction v of B4+defects:

183

Thermodynamics of New Materials

On the other hand, the oxide can exchange oxygen with the gas phase. The exchange redox reaction is an equilibrium reaction: 1

2'2

(gas)

-k

'(Oxide)

+ 2B&ide)

o:6xide)

+ 2B:&ide)

From Equation (2), it is obvious that 6varies with the oxygen partial pressure of the gas phase, p ( 0 , ) . In terms of the dependence of 6 on p(O,), different regions may be d i s t i n g ~ i s h e d :at ~ ~high . ~ ~p(O,), charge compensation of the acceptor dopant occurs mainly by the formation of B4+ (region I). From Equation (l),it follows that v = x and 6=0.At medium p(O,), both B4+ and vacancy defects provide charge compensation (region 11) and at low p(O,), mainly vacancies are formed (v = 0 and 6 = x/2, region 111). Thus, by going from high to low p(O,), the conductivity is expected to change from predominantly electronic to predominantly ionic and only in region I1 the oxide should be a mixed conductor. Finally, at very low p(O,), B3+ gradually reduces to B2+ and 6 increases further (region IV). Figure 3 illustrates the dependence of 6 on p ( 0 , ) for acceptor-doped MIEC oxides with the transition metals cobalt and iron on B - s i t e ~ .The ~ ~ cobaltate ?~~ exists only in a narrow partial pressure range, which belongs to region 11, and is an excellent MIEC material. As expected from equation (l), the deviation from stoichiometry increases with the dopant fraction x. For the iron-containing oxide, all four regions are observed. By means of defect model calculations, a quantitative description of the dependence of 6 on p ( 0 , ) was obtained for this oxide, and the defect concentrations were calculated.26 In Figure 4, the variation of the defect concentrations with p ( 0 , ) is

(.o

cr)

2,8

-

0

La,$jro.lFe03-p 1273 K

-Calculated from defect model La,.,Sr,CoO3-~ 2,7 -

1173 K

0 x=0.2 A x=0.3 0

x=o.5 x=0.7

Figure 3 Deviation from stoichiometry as a function of oxygen partial pressure for La,,Sr,, FeO,-, (solid line: calculated from defect model), and as a function of p ( 0 , ) and compositionfor La,-x SrxCoO,-6 (solid lines are a guide to the eye). Data were taken and adapted from refs. 24,25

184

Chapter 16 0 1

I

re"

I

1

Fe2+

1

1

Figure 4 Oxygen partial pressure dependence of the defect concentrations of Lao,,Sro,,Fe0,-6 at I273 K,as obtained from model calculations6

illustrated. The model calculations revealed that the formation of B2+ by the disproportionation reaction

cannot be neglected, and that significant amounts of B2+ and B4+ defects exist even in region 111, providing for mixed conduction. Measurements of the total electrical cond~ctivity'~ support these results: the conductivity is largest at high p ( 0 , ) in region I and exhibits a minimum at a p ( 0 , ) belonging to region 111, but even at this minimum it is much higher than the ionic conductivity.

2.2 Oxygen Permeation As outlined in the previous section, oxygen permeation through an MIEC membrane relies on the migration of ionic and electronic defects. The transport theory of Wagner27may be adopted to express the oxygen flux j(0,) through a membrane exposed to a gradient of oxygen chemical potential Vp(O,),

where q and 0,are the partial ionic and electronic conductivities, respectively. Equation (4) is valid only if the overall permeation process is limited by the bulk transport of charge carriers, which implies that the oxygen exchange reaction with the gas phase at the membrane surface (see Equation (2)) must be fast enough to maintain equilibrium. If predominant electronic conduction is assumed, the conductivitydependent term in Equation (4) is approximately equal to 4. Furthermore, if a vacancy mechanism is assumed for the bulk transport of oxygen ions, is proportional to the vacancy mobility u, and to the molar fraction of vacancies, which is given by 6:

Themdynumics of New Materials

185

where Vm is the molar volume of the oxide. In MIEC oxides, the vacancy mobility is often nearly independent of oxygen partial pressure, but the deviation from stoichiometry varies with p ( 0 , ) . The oxygen flux is then obtained by the combination of Equations (4) and (5) and subsequent integration over the membrane thickness L:

Provided the dependence of 6 on p ( 0 , ) is known, the permeation flux can be calculated as a function of the partial pressures p ' ( 0 , ) and ~ " ( 0 , )on the oxygen-rich and the oxygen-lean side of the membrane, respectively. By evaluating the nonstoichiometry data of L~,_,S~,COO,-~ given in Figure 3, it was found that 6 is approximately proportional to ~(0,)"s. Integration of Equation (6) then yields:

with 4 being the deviation from stoichiometry at p(0,) = 1 bar. n6 is dependent on temperature and the molar fraction of the dopant x and assumes negative values in the range - 0 . 2 4 1 n 6 1 -0.07. Recently, the oxygen permeation through La, -,S~,COO,-~was investigated. Selected results are presented in Figure 5. Linear plots are obtained if Equation (7) is applied; however, only for membranes whose surfaces had been modified to enhance the kinetics of oxygen exchange with the gas phase, values of n were found to be negative and consistent with the n6 obtained from the nonstoichiometry data. For membranes without surface modification, n>O was found in all cases, which indicates that Equation (7) is not applicable. Apparently, sluggish surface exchange kinetics limit the permeation process through La, -xSr,Co0,-6. Surface exchange limited permeation has also been observed for 19728

1

I

I

I

1

Figure 5 Oxygen permeation flux through membranes of uncoated Lao~,Sro,,CoO,-sat 11 73 K and Luo,,Sro,,Co03~s coated with SrCoo,,Feo,,O,-s at 1183 K. The coating was deposited on the feed side. Data were taken from re5 28

186

Chapter 16

other MIEC oxides. A detailed analysis of the influence of the surface exchange kinetics on the permeation is beyond the scope of this paper, but can be found in several excellent discourse^.^^-^^ However, it is stressed that data on the surface exchange reaction kinetics are important for the design of permeation membranes. Equation (7) implies that reducing the membrane thickness L is one way to increase the permeation flux. This is because the bulk resistance of oxygen transport decreases with decreasing thickness. On the other hand, the resistance due to surface exchange does not depend on L, and below a critical membrane thickness, no gain in permeation flux can be obtained by further reducing the membrane thickness. The critical thickness depends on various factors (e.g. oxide composition and temperature) and ranges from several 10 pm up to several 100 pm.

2.3 Stability of Membrane Materials In an operating partial oxidation reactor, the membrane material must tolerate harsh conditions over long time periods. The large gradient of oxygen chemical potential across the membrane causes a gradient of oxygen vacancy concentration. The resulting spatial variation in lattice expansion may cause significant tress,^^.^^ and mechanical failure after short times of operation has been observed particularly with cobalt-containing materials.34 Furthermore, the reaction side of the membrane is exposed to reducing gases (CO and H,) at very low oxygen partial pressures ( 10-18-10-20bar). While A-site acceptor doping and high reducibility of the B-site transition metal ions are beneficial for the oxygen permeation performance, it diminishes the chemical stability of the membrane. The critical p ( 0 , ) below which reductive decomposition into a nonconductive multiphase mixture takes place is approx. bar for undoped LaFe03-6 and bar for Even higher critical pressures have been reported for the respective strontium-doped oxides.36 Reductive decomposition causes deterioration of the membrane permeability and is therefore a critical problem. The partial replacement of B-site transition metals by less reducible elements (e.g. gallium or chromium) improves the membrane stability to some e ~ t e n t ~ ~ however, . ~ * ; at the cost of lower permeability. Recently, a new concept to stabilize permeation membranes has been i n t r o d ~ c e d . ~It ~relies . ~ ~ on bilayer materials that consist of a substrate layer of an MIEC material, e.g. Lal-xSrxCo03-6, and a thin protective coating on the low-p(0,) side. The coating material is required to be stable towards reduction and should exhibit high oxide-ion conductivity but low electronic conductivity. Samarium-doped ceria meets these criteria. Figure 6 illustrates the effect of the coating. Because of its low electronic conductivity, the coating has a higher resistance to oxygen permeation than the substrate. If the membrane is exposed to an oxygen chemical potential gradient, a substantial drop of oxygen potential occurs in the coating. This causes the oxygen chemical potential ~ ~ ( at 0 the , ) interface of substrate and coating to rise to above the critical potential p(O,>,,, of the substrate. In this way, the reduction of the substrate can be avoided. Bilayer membranes have been successfully operated under partial oxidation conditions for several hundred hours without any significant deterioration of the permeation

187

Thermodynamicsof New Materials MIEC

Uncoated membrane

MIE3C

Coating

Bilayer membrane

Figure 6 Function of a bilayer membrane subjected to a large gradient of oxygen chemical potential. The protective coating diminishes the gradient acmss the delicate MIEC substrate, which would decompose below p(02)crir.

2.4 Conclusion In the previous sections, it has been demonstrated that, in order to improve the properties of known MIEC membrane materials and to develop new materials, a fundamental understanding of their defect chemistry and the transport of defects is crucial. This is achieved through interpretation of experimental thermochemical and transport data by means of defect models. While the introduction of a membrane separation step in catalytic partial oxidation processes is most attractive for technical and economical reasons, various problems related to the material and reactor design have yet to be solved. The mechanical and chemical stability of highly permeable MIEC oxides needs to be improved to ensure long-term membrane stability under operation conditions and during thermal cycling. But other issues also need to be addressed: Perovskite membrane surfaces often favour deep oxidation of hydrocarbon^.^^ Therefore, partial oxidation catalysts have to be used to improve the selectivity for partial oxidation. If the membrane surface serves as support for these catalysts, unwanted reactions between catalyst and membrane oxide must be avoided. The oxidation process may require additional components, such as H20, to be present in the hydrocarbon feed stream, which also might react with the membrane and alter its properties. On the other hand, the oxygen fluxes obtained with some known MIEC membrane materials already meet the minimum requirements for economic operation of a membrane reactor,43 which raises hope of a breakthrough in this technology in near future. It seems that hightemperature membrane reactor technology will continue to provide an ample field for both applied and fundamental research,

3 Solid Oxide Fuel Cells 3.1 Fuel Cells In a fuel cell, an electrolyte is used to divide the strongly exothermic chemical reaction H2+Y2O2 t)H,O into two electrochemical reactions: a reduction and an oxidation reaction. In this way, the conversion of the reaction Gibbs energy, A,G, into electrical energy is possible, and the only reaction product is water (if pure

Chapter I6

188

hydrogen is available as fuel). The voltage, E, of this battery is given by E= -A,GInF (where n is the number of electrons and F is the Faraday constant).The first fuel cell was described already in 1839 by Groveu using an aqueous acid as electrolyte. Nowadays, several fuel cell types exist for various application^,^^ and numerous efforts are being undertaken in basic research and development with the aim to optimize a sustainable energy conversion method. As shown in Figure 7, there are essentially two possibilities for the operation of a fuel cell. If protons, H+, are mobile in the electrolyte, the oxidation and reduction reactions are H, +2H' +2e- and 2H++2e- +%O, +H,O, respectively (see Figure 7(a)). If, on the other hand, an electrolyte is used where oxygen ions are mobile, the corresponding oxidation and reduction reactions are 02-+H, +H,O +2e- and $,+2e- -+0,-(see Figure 7(b)). As can be seen from Figure 7, it is necessary for the optimum operation of a fuel cell that the two half cell reactions are separated by a membrane that does not conduct electrons, but only ions. The most important fuel cells that are in use nowadays are the polymer electrolyte membrane fuel cell (PEMFC), the molten carbonate fuel cell (MCFC), and the solid oxide fuel cell (SOFC).45In a PEMFC, the electrolyte is a polymer membrane that conducts protons, H+, in an MCFC the electrolyte is a carbonate melt in which oxygen is conducted in the form of carbonate ions, Cog-, and in an SOFC the electrolyte is a solid oxide that conducts oxygen ions, 0,-. While a PEMFC can be operated at low temperatures of about 80 "C, an MCFC works at intermediate temperatures of about 65OoC, and an SOFC needs relatively high temperatures of 800-1000 "C (see next sections). In both PEMFCs and SOFCs, there is a strong need to improve the electrolyte materials and to find substitutes for the traditional electrolytes, which are Nafion for the PEMFC and yttria-stabilized zirconia (YSZ) for the SOFC. In this paper, we will focus on the discussion of new developments in the field of SOFC electrolytes. For recent developments concerning other fuel cell types, the reader is referred to review articles and books.4547

a

b

Figure 7 Schematic set-up of a fuel cell. ( a ) proton-conducting electrolyte, and (b) oxygen ion-conducting electrolyte

Thermodynamicsof New Materials

189

3.2 Electrolytes for Solid Oxide Fuel Cells The electrolyte in an SOFC is a solid, crystalline oxide in which conduction of oxygen ions is possible by means of oxygen vacancies, V,. The site exchange between oxygen ions and oxygen vacancies is a thermally activated process (the typical activation energy in YSZ48is 80-90 kJ mol-’). Since a sufficiently high electrical conductivity is required to minimize the internal resistance of the electrolyte, relatively high operating temperatures are needed in an SOFC. The traditional oxygen ion conductor is yttria-stabilized zirconia (YSZ). Doping ZrO, with Y203 has two effects: (i) It stabilizes a cubic structure, which is stable from room temperature to elevated temperatures. This is very important for the possibility to change the temperature of an SOFC without destroying the electrolyte as a result of phase transformations. (ii) Oxygen vacancies are produced, which are necessary for a high oxygen ion conductivity. As can be seen in Equation (8), doping with Y203results in two yttrium ions occupying zirconium sites, Y;:, three oxygen ions occupying oxygen sites, 0;-,and one oxygen vacancy, V,. The formation of the oxygen vacancy is necessary to preserve the lattice structure of zirconia with two oxygen sites per zirconium site.

Y20, + 2Y;:

+ 30;- + V,

As a consequence of Equation (8), the concentration of oxygen vacancies is fixed by the dopant concentration, cvo=cY,,,. Therefore, in a typical YSZ electrolyte, e.g. (Zr02)o~9(Y203)o~l, the oxygen vacancy fraction is as high as 10 mol%. Due to this high concentration and the high vacancy diffusion coefficient, D,, a high oxygen ion conductivity, a(02-)=4F2DVocvo/RT(Nernst-Einstein relation), is possible. A typical value49for (Zr02)o.9(Y203)o,l at 1000 “C is a(02-)= 10-’ S cm-l. It is interesting to note that the first use of doped zirconia and its high oxygen ion conductivity goes back to Nernst,” who used this material already in 1899 in his Nernst glower ( a lamp where the light-emitting filament is an oxygen ion conductor). It took, however, more than 40 years till the conduction mechanism was explained in detail by Wagner.51 As mentioned above, a perfect electrolyte has a vanishing electronic conductivity. In reality, however, an oxygen ion-conducting oxide always contains electrons, e’, and electron holes, h’, as minority charge carriers. Their contribution to the total conductivity, o,,= , o(02-) + fie’) +o(h’), is determined by the electronic transference number, tel=(o(e’)+ o(h’))/qot. A good oxygen ion conductor is characterized by tel< lop3.To determine the so-called ionic domain, one has to consider that the concentrations of electrons and electron holes depend on the external oxygen partial pressure p ( 0 , ) . This can be observed immediately from the incorporation reaction for oxygen from the gas phase (oxygen is reduced and occupies an oxygen vacancy)

1 2

- 0,(g)

+ 2e’ + V,

H

and the electronic equilibrium nil

++ e’ + h’

06-

(9)

190

Chapter 16 1

I

I

I

I

1

I

I

I

I

0

-6

-8

.I_ - -10 -50

- - _ _ _-_ .-

-,-T

ionic domain

I

I

I

I

I

I

I

I

I

-40

-30

-20

-10

0

10

20

30

40

log(p(0,) 1bar)

Figure 8 Conductivities of oxygen ions, electrons and electron holes in Zro,9Yo,101,95 as a function of oxygen partial pressure at diflerent temperatures (adopted from re$ 52). The ionic domains and typical p(0,)-ranges offuel cell operation are also shown

Assuming equilibrium for both reactions and considering that the concentration of oxygen vacancies is fixed by the high Y203fraction, cvo= cy203,we obtain:

The concentration of electrons increases with decreasing oxygen partial pressure, while the concentration of electron holes decreases. The above results are summarized in Figure 8, showing the partial conductivities of oxygen ions, electrons and electron holes in YSZ as a function of oxygen partial pressure.52 As can be seen in Figure 8, the ionic domain (tel<10-3) extends over more than 20 orders of magnitude. The p(0,)-range of an SOFC corresponding to a cell voltage of 0.8 V (air at the cathode and a H,/H,O-mixture at the anode, using the Nernst equation E=(R~/4~ln(p(0,)""fh/po""d)), is well within the ionic domain. In summary, the high operating temperatures of 900-1000 "C yield sufficiently high ionic conductivity of the electrolyte and small electronic transference. On the other hand, lower operating temperatures are desired, (i) for economic reasons, and (ii) to minimize problems that are due to the combination of different materials in a fuel cell. For example, problems arise from interdiffusion and chemical reactions49 between the electrodes and the electrolyte (typical electrodes in an SOFC consist of strontium-doped lanthanum manganate, (La,Sr)MnO,_&, at the cathode and a mixture of fine-grained nickel and YSZ at the anode). To obtain large enough voltages from an SOFC, cells have to be combined to a stack. In this stack, the individual cells are separated by the interconnector that acts as a current collector and separates the 0,- and H,-containing compartments of adjacent cells. At high temperatures, there are severe problems (sealing, reactivity, thermal mismatch, evaporation) using ceramic interconnectors (such as (La,Ca)CrO,), which could be avoided by going to intermediate temperatures of about 700 "C where conventional metallic components could be used as interconnector materials.

191

Thermodynamics of New Materials Temperature ("C)

900800 700 600

500

400

300

I

I

I

150 pm

15 Pn

1.5 pm

0.15 p

03

1

1 2

1,4

1,6

1,8

2

1000A'(K1)

Figure 9 Oxygen ion conductivities of three electrolytes as afunction of reciprocal temperature (adopted from re& 45)

However, to decrease the operating temperature of an SOFC from high temperatures of 900-1000 "C to intermediate temperatures (IT) of about 700 "C, it is necessary to decrease the cell resistance. This is possible by decreasing the thickness, L, of the electrolyte and/or by new electrolyte materials with higher oxygen ion conductivities. A characteristic number is the area r e ~ i s t a n c eR,=LIa, ~~, which should be smaller than 0.15 i2 cm2. Figure 9 shows the temperature dependence of the oxygen ion conductivities of YSZ and two new, recently discovered oxygen ion conductors, doped lanthanum gallate (LSGM) and gadolinium-doped ceria (GDC)45, both of which exhibit higher conductivities than YSZ. The electrolyte thickness on the right-hand side of the diagram was calculated from the oxygen conductivity on the left-hand side of the diagram for a desired area resistance R,=O.15 Rcm2. To remain below this value, the thickness of aYSZ electrolyte in an IT-SOFC at 700°C must be smaller than 15 pm.With such a small thickness, a self-supported electrolyte layer is no longer possible and one must go to anode-or cathode-supported cells. With the new materials, LSGM and GDC, thicker electrolytes can be used even at 500-700 "C. The ionic domain of LSGM53is comparable with YSZ. In contrast, the ionic window of GDC is much smaller (due to reduction of Ce4+ to Ce3+ at low oxygen partial pressures) causing problems in its use as an SOFC e l e c t r ~ l y t eTherefore, .~~ in the next section, we will discuss only the new oxygen ion conductor LSGM in more detail.

3.3 Lanthanum Gallates In the search for new oxygen ion conductors to be used as electrolytes in SOFCs, oxides have been extensively studied in recent years, which crystallize in the perovskite structure and in structures derived therefrom (e.g. the brownmillerite

192

Chapter 16

structure). Doped lanthanum gallate, La,-,+4xGa,-$,0,-,, has proved to be a particularly suitable candidate.5456In this material, the oxygen ion conductivity can be clearly increased in comparison to the classic SOFC electrolyte, YSZ, by doping at A- and B-sites. Testing comprised different compositions with partial substitution of La by Sr, Ca or Ba and of Ga by Mg, In or Al.54Other tested gallates had e.g. the composition (L~,&~~I)o~8Sro~2G~~,Mgo~,02~8 (Ln=Nd, Sm, Gd, Y, Yb).57The highest conductivities are achieved with Sr at A- and Mg at B-sites. Lanthanum gallate of the composition L~~,Sr0~,G~,,Mg,,O2,, shows conductivities between lop2and 10-1 R-' cm-' at temperatures between 500 and 800 "C (see Figure 9). The Gibbs energy of formation of LaGaO, has been determined using galvanic ~ e l l s ~and * -vapour ~ ~ pressure measurements combined with a defect chemical analysis60,61; the enthalpy of formation was determined by calorimetry.62 The phase diagram of the binary system La,O,-Ga,O, shows only two comp o u n d ~ , ~LaGaO,, , which is the basis for LSGM, and La4Ga,0,. The stability field of Sr- orland Mg-doped lanthanum gallate was studied in several investigat i o n ~However, . ~ ~ ~ phase diagram studies are rendered difficult due to the small cation diffusion coefficients in these oxides and the necessary long annealing times for complete equilibration. Recently, the stability field of LSGM at 1400°C has been determined: (i) the oxides were prepared using a modified Pechini method,68 where an aqueous solution of the metal nitrates, citric acid and ethylenglycole yields a polyester gel, in which the cations are statistically distributed in chelat complexes. (ii) After calcination, equilibration of the samples was ensured by longterm annealing and by considering diffusion of the cations,69which are the slowest moving species. Solid phase equilibria in the quasi-quaternary system La20,-Ga,O,-MgO-SrOnear the solubility lobe of the (La,Sr)(Ga,Mg)O, -6 perovskite phase were determined by XRD, SEM/EDX and electron microprobe analysis (EPMA) of annealed and quenched samples. The resulting phase diagram in Figure 10 shows the stability field of LSGM and the adjacent phase field^.^^,^^ The solubility of strontium in LaGaO, is only about 2 at% and much smaller than reported in the other studies. Co-doping with strontium and magnesium increases the Sr-solubility and yields a stability field, which also contains the composition with the highest ionic conductivity, L%.9Sr0.,G%,9 Mg0.1O2.9. It must be emphasized that this equilibrium phase diagram was measured at a temperature of 1400 "C,which is much higher than the operating temperature of an SOFC. The phase diagram and the solubilities of Sr and Mg in LSGM at lower temperatures are not known. Nevertheless, LSGM prepared at high temperatures is expected to be also stable at lower temperatures due to the very small cation mobilities, by which LSGM is kinetically stabilized. Cation self-diffusion coefficients of La, Sr and Mg in LSGM have been measured recently for temperatures between 900 and 1400°C using stable isotopes and Secondary Ion Mass Spectrometry (SIMS).72At 900 "C, the cation diffusion coefficients are 10 orders of magnitude smaller than the oxygen diffusion coefficients. Despite these small values, cation diffusion is also important in an oxygen ion conductor, since there are several processes that are determined by the slowest moving species, such as sintering, kinetic d e m i ~ i n g(due ~ ~ to different cation diffusion

Thermodynamics of New Materials

193

“LaMgO2.5’’

A

0.6

0.0 0.0

LaGaO,

0.2

0.4

x(W

0.6

0.8

“SrGaO,,“

Figure 10 Section of the phase diagram of the La,O,-SrO-Ga,O,-MgO system at 1673 K. Phase regions: ( I ) (La,Sr)(Ga,Mg)03-6perovskite phase (P), MgGaLa,O,, MgO; ( 2 ) R MgGaLa307, MgO, SrLuGaO,; (3) R SrLaGaO, MgO; (4) P, MgO, SrLaGaO, SrLaGa,07; and (5) R SrLaGaO, SrLaGa,O,

coefficients) or creep. The activation energy reported for creep of LSGM at temperatures between 1200 and 1300 “C is about 5 eV.74,75This value is in fair agreement with the activation energy of the self-diffusion coefficients of the cations,72which are the slowest moving species in LSGM and should determine the creep rate. In further investigations on LSGM, the thermodynamic stability under operating conditions76and the reactivity with electrode m a t e r i a l ~ ~were ~ . ~ *studied. Evidence of evaporation of Ga under operating conditions at the anode side of the SOFC was found in ref. 76 and confirmed by vapour pressure measurements and thermodynamic calculations.70~71 However, recent studies show that gallium evaporation at the anode can be suppressed successfully using a thin buffer layer consisting of samariumdoped ~ e r i aWith . ~ ~such a cell, a fairly high power density of 700 mWcm-2 could be obtained at 800 “C, indicating that an SOFC based on an LSGM electrolyte can operate at intermediate temperatures and would be competitive with an SOFC based on a YSZ electrolyte.

4 Conclusion Lanthanum gallates doped with strontium and magnesium (LSGM) are promising new oxygen ion conductors, which can be used as electrolyte material in an SOFC and may substitute the traditional electrolyte YSZ in IT-SOFCs. Further research is required (i) for a better understanding of the basic physical and chemical properties of these materials and their long-term behaviour as SOFC component, and (ii) for the search for new and better oxygen ion conductors.

194

Chapter I6

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31. S. Kim, S. Wang, X. Chen, Y. L. Yang, N. Wu, A. Ignatiev, A. J. Jacobson, and B. Abeles, J. Electrochem. SOC., 2000, 147, 2398. 32. P. V. Hendriksen, P. H. Larsen, M. Mogensen, F. W. Poulsen and K. Wiik, Catal. Today, 2000, 56, 283. 33. A. Atkinson, A. Selcuk, Solid State Zonics, 2000, 134, 59. 34. U. Balachandran, J. T. Dusek, P. S. Maiya, B. Ma, R. L. Mieville, M. S. Kleefisch and C. A. Udovich, Catal. Today, 1997,36, 265. 35. T. Nakamura, G. Petzow, and L. J. Gauckler, Matel: Res. Bull., 1979, 14,649. 36. B. C. H. Steele, Matel: Sci. Eng., 1992, B13, 79. 37. A. Atkinson, T. Ramos, Solid State Zonics, 2000, 129, 259. 38. S. Kim, A. J. Jacobson, B. Abeles, Materials Research Society Symposium Proceedings, Vol548, (Solid State Zonics V), 1999, 563. 39. M. Schroeder, K. Q. Huang, H. Y. Yin and J. B. Goodenough, Proceedings of the Znternational Symposium “Solid State Ionic Devices ”, in The Electrochemical Society Proceedings Series, PV 99-13, E. D. Wachsman, J. R. Akridge, M. Liu and N. Yamazoe (eds), Pennington, NJ, 1999, 121. 40. T. L. Clites, and E. D. Wachsman, Proceedings of the International Symposium “Solid State Ionic Devices”, in The Electrochemical Society Proceedings Series, PV 99-13, E. D. Wachsman, J. R. Akridge, M. Liu and N. Yamazoe (eds), Pennington, NJ, 1999, 110. 41. M. Schroeder, K. Q. Huang and J. B. Goodenough,Mass and Charge Transport in Inorganic Materials. Fundamentals to Devices, in Advances in Science and Technology, 29. Proceedings of the International Conference on “Mass and Charge Transport in Inorganic Materials - Fundamentals to Devices ”, P. Vincencini and V. Buscaglia (eds), Jesolo di Lido (Venice), Italy, 2000, 1507. 42. R. Doshi, C. B. Alcock, N. Gunasekaran and J. J. Carberry, J. Catal., 1993,140,557. 43. R. Bredesen and J. Sogge, SINTEF Report, S96017, 1996. 44. W. R. Grove, Phil. Mag., Ser.3, 1839,14, 127. 45. B. C. H. Steele, J. Mat. Sci., 2001,36, 1053. 46. L. J. N. Blomen and M. N. Mugerwa (eds), Fuel Cell Systems, Plenum, New York, 1993. 47 * K. Kordesch and G. Simander, Fuel Cells and their Applications ,VCH, Weinheim, 1996. 48. P. S. Manning, J. D. Sirman, R. A. De Souza and J. A. Kilner, Solid State Zonics, 1997,100, 1. 49. A. Hammou and J. Guindet, in The CRC Handbook of Solid State Electrochemistry, P. J. Gellings and H. J. M. Bouwmeester (eds), CRC Press, Boca Raton, 1997,407. 50. W. Nernst, Z. Elektrochem., 1899, 6, 41. 51. C. Wagner, Naturwissenschafien, 1943,31,265. 52. B. C. H. Steele, J. Power Sources, 1994, 49, 1. 53. H. Yokokawa, N. Sakai, T. Horita and K. Yamaji, Fuel Cells, 2001, 2, 117. 54. T. Ishihara, H. Matsuda and Y. Takita, J. Am. Chem. SOC., 1994, 116, 3801. 55. M. Feng and J. B. Goodenough, Eul: J. Solid State Inorg. Chem., 1994, t13,663. 56. K. Huang, M. Feng and J. B. Goodenough, J. Electrochem. SOC.,1997,144,3620. 57. T. Ishihara, H. Matsuda and Y. Takita, Solid State Ionics, 1995, 79, 147. 58. A. M. Azad, R. Sudha and 0. M. Sreedharan, Mat. Res. Bull., 1991,26,97. 59. K. T. Jacob, J. Matel: Res., 2000,15,2836. 60. W. Kuncewicz-Kupczyk, D. Kobertz, M. Miller, C. Chatillon, L. Singheiser and K. Hilpert, J. Am. Ceram. SOC.,2002, 85, 2299. 61. M. Matraszek, L. Singheiser, D. Kobertz, K. Hilpert, M. Miller and M. Martin, Phys. Chem. Chem. Phys., 2003,53042. 62. Y. Kanke, A. Navrotsky, J. Solid State Chem., 1998, 141,424. 63. M. Mizuno, T. Yamada and T. Ohtake, Ygyo Kyokaishi, 1985,93,295.

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CHAPTER 17

Thermodynamic Prediction of the Formation and Composition Ranges of Metastable Coating Structures in PVD Processes P. J. SPENCER

1 Introduction A large amount of systematic development work on new high-performance mixed carbide, nitride, boride and oxide coating materials produced by physical vapour deposition (PVD) techniques is being carried out, because the mixed coatings often display significantly improved properties (e.g. wear-, corrosion-, oxidation-resistance or certain physical property behaviour) compared to the properties of the individual constituents. Experiments with the deposition of metastable, multicomponent coatings using PVD methods have shown that the required coating compositions can be achieved relatively easily. Furthermore, variation of the substrate temperature as well as the coating composition sometimes allows the structure and hence the properties of the deposited coating to be changed. By carrying out thermodynamic calculations, it is possible to predict the conditions for the appearance of different metastable phases during PVD coating processes and thereby assist in the selection of coating parameters required to produce coatings with optimum desired properties. The general principle relating to thermodynamic calculation of the metastable phase ranges resulting from PVD of multicomponent coatings on a low-temperature substrate has been propounded by Saunders and Miodownik' as follows: If an alloy, that in equilibrium contains a multiphase structure, is co-deposited at low enough temperatures, the surface mobility is insufficient for the breakdown of the initially fully intermixed depositing atoms, and the film is therefore

198

Chapter 17

constrained to be a single-phase structure ... Nucleation and subsequent growth processes are controlled by the substrate temperature, and therefore, if the film is constrained to be a single phase, the structure of t h e j l m should reflect the most energetically stable single-phase form available to it at the temperature of the substrate. This phase can be directly found from the Gibbs energy versus composition (G/x)diagrams of the alloy system of interest.. . The above-defined principle has been used in a series of calculations relating to various possible carbide and nitride coating combinations formed from among transition metal nitrides and carbides (all with the cubic NaCl structure) on the one hand and A1N or S i c (hexagonal wurtzite structure) or TiB, (hexagonal structure) on the ~ t h e r . ~ - ~ Further calculations have been carried out for coatings in the A120,-A1N s y ~ t e m . ~ . ~ The present paper compares the results obtained from thermodynamic prediction of the metastable phase ranges in the AlN-TiN and A120,-A1N coating systems with results obtained from experimental investigations of the two systems.

2 (Ti,AI)N Coatings A1N (hexagonal wurtzite structure) and TiN (cubic NaCl (cF8) structure) show virtually no solubility in each other under equilibrium conditions at temperatures below about 1500°C (see Figure l), but when co-deposited on a cold-temperature substrate using PVD techniques, the metastable cubic NaCl structure can be retained to high concentrations of A1N in the deposited coating. Such coatings are already being produced commercially for cutting tool applications. They have been shown to exhibit superior performance as compared to TiN under conditions of wear at elevated temperatures due to their better oxidation resistance and hardness. The oxidation resistance of the coatings increases significantly with increasing A1 Content. However, the deposition of (Ti,Al)N films with TUA1 ratios below about 30/70 leads to a hexagonal coating structure that is not suitable for tribological coatings. Until more recently, relatively little was known about the location of the cubic to hexagonal phase transition and hence about the optimum composition of (Ti,Al)N hard coatings. 2500 /

1

1

gas + fcc

2300 \

'hexagonal P.

cubic NaCl

2100

G

G 1900

hexagonal+ cubic NaCl 1700

1500

Figure 1 Calculated AlN-7iN section of the Al-7i-N system

.

Thermodynamic Prediction of the Formation

199

The significance of the metastable structural phase transition in the AlN-TiN system on the properties of coatings produced by PVD allows the important potential of thermodynamic calculations in predicting the ranges of composition and temperature in which different metastable phases may form in a particular coating system to be demonstrated. The thermodynamic properties of hexagonal AlN and cubic transition metal nitrides and carbides are known,* but for the calculation of the ranges of metastability of the cubic and hexagonal phases in the AlN-MeN or AlN-MeC systems, the energy differences for the transitions from the stable to the metastable structures, e.g. MeN,,, 4 MeN,,, and AlN,,, +AIN,,, are of primary importance. In addition, the thermodynamic properties of mixing of the metastable solutions are required. Since virtually no experimental data are available for the energy differences referred to, a combination of calculation and estimation techniques must be used to obtain them. Methods available for this purpose are described below.

2.1 Metastable + Stable Structural Transformation Energies for Nitride and Carbide Phases A standard method of obtaining missing thermochemical values is to make plots of relevant properties in comparison with neighbouring substances in the Periodic Table, as shown in Figure 2 for enthalpies of transition fcc +bcc of transition metals from the 2nd long period. It is clear from Figure 2 that it is not a straightforward matter to predict a missing value from such an irregular curve. However, it has been found that the reorganization of the Periodic Table used by Pettifor’ (Figure 3), in which the elements are listed in a single string based on what Pettifor terms the Mendeleev Number, allows enthalpies of formation of particular types of compound with a given structure to be plotted on a smooth curve (see ref. 10). Figure 4 presents such a plot for the enthalpies of formation of nitrides with the cubic-NaC1 structure.

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Figure 2 Enthalpies of the fcc +bcc transition .for elements of the 2nd Long Period of the Periodic Table

Chapter 17

200

Figure 3 The Pettifor single string rearrangement of the Periodic Table in the sequence of the Mendeleev Number

Mendeleev number

Figure 4 Enthalpies of formation of nitrides with the cubic-NaCl structure plotted using the Pettifor arrangement of the Periodic Table

S t ~ l t e nhas ~ . ~developed a semi-empirical estimation method to obtain enthalpies of formation of carbides, nitrides and oxides in metastable crystallographic structures. The calculations involved are based on the classical alloying criteria of atomic size, valence and electronegativity, and incorporate a correction factor obtained

20 1

Thermodynamic Prediction of the Formation

using one or two reliable experimental enthalpy of formation values for the structure concerned. When plotted as a function of the Mendeleev number, the calculated enthalpies are in all cases essentially linear for compounds of transition metals with Mendeleev numbers from 49 (Zr) to around 57 (Cr). At higher Mendeleev numbers, a marked curvature is observed. Figure 5 presents the results obtained by Stolten for the enthalpies of formation of nitrides of the transition metals in the cubic-NaC1and hexagonal-ZnS structure. Fernandez Guillermet and Grimvall have also written a series of papers [ 1 1-17] devoted to deriving enthalpies of formation and standard entropies for a range of metastable carbide and nitride phases of the 3d, 4d and 5d transition metals. Thermodynamic values were obtained from an analysis of the cohesive properties and vibrational entropies of the compounds concerned, whereby a quantity with the dimensions of energy, E,, was defined as

where R is the average volume per atom and is obtained from experimental lattice parameters. The derived values were found to vary in a regular and related way when studied as a function of the average number of valence electrons per atom in the compound, n,. Figure 6 a and b, taken from Fernandez Guillermet and Grim Vall", shows the entropy-related characteristic energy, E,(Ry), and the enthalpy of formation (per mol of atoms) for the 3d transition metal carbides. Tables 1 and 2 present a summary of available data, obtained both by experiment8 and by calculation for nitrides and carbides with the cubic-NaC1 and hexagonal wurtzite structures.

7

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Zr Hf Ti Ta Nb V WMoCrRe TcMnFeRu OsCoRh Ir Ni Pi Pd 1 ~ . ~ , ~ . ~ . ~ . ~ . ~

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Figure 5 Experimental (solid squares) and c a l c ~ l a t e d(open ~ - ~ circles and stars) enthalpies of formation of nitrides of the transition metals in the cubic-NaCl and hexagonalZnS structures plotted as a function of the Mendeleev Number

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Figure 6(a) and (b) E,and AoH of carbides asfinctions of ne,The numbers refer to ( 1 ) ScC, (2) Tic,(3) VC, (4) Cr3C2, (5) Cr7C3,(61 Cr2& (7) Mn7c3, (8) Mn&, (9)Mn,C, (10)Mn,,C, ( 1 1 ) Fe3C, (12) C O ~ C(13) , Ni3C (14) Sc2C9(15) Sc,C3, with identical identijicationfor the E, values at the same n,. The dashed lines refer to the (cF8) structure

Comparison of the different calculated values with each other and with experimental data from Landolt-Bornstein Series8 is presented in Figures 7 and 8 for transition metal carbides and nitrides with the NaCl structure. The calorimetric study carried out by Stolten for three compositions in the metastable A1N-TiN system provides direct experimental data both for the enthalpy difference AlN,, +AlN,,,, NaCl (see Table 1) as well as the mixing enthalpy of AlN-TiN cubic NaCl ~olutions.~ These values, together with the calculated value for the Gibbs energy difference between the NaCl and wurtzite structural forms of pure TiN,3,4 allow a thermodynamic analysis of the TiN-AlN system to be made. Gibbs energy curves have been calculated for the hexagonal and cubic phases for various hypothetical (Ti,Al)N deposition temperatures (Figure 9). The point of intersection of these curves at each temperature defines the composition at which there is a transition from one structure to the other. It can be seen that this composition is nearly temperature independent and has a calculated value of around 0.7 mol fraction AlN. Experimental studies of the extent of the metastable cubic range in the section AlN-TiN have been carried out by Knotek and Leyendecker18and more recently, with

203

Thermodynamic Prediction of the Formation

Table 1 Enthalpies of formation of transition metal nitrides and of AlN in the cubicNaCl and hexagonal wurtzite structures Nitride

Af H(298 K)-cubic Wmol-' MeN

Re$

Af H(298 K)-hexag. W mol- MeN

Re$

ZrN HfN TiN TaN NbN VN WN MoN CrN ReN TcN MnN FeN RUN OsN CON RhN IrN NiN PtN PdN A1N

-365.3 -369.2 -337.6 -252.3 -236.4 -217.1 -111.8 -38.0 - 117.0 -43.0 -34.8; 90.0 - 185.8; -59.2 2.0; 6.4 137.8; 159.2 121.4 13.2; 37.4 117.6; 108.0 103.2 22.0; 43.2

8 8 8 8 8 8 394 394 8 3,4 3,4; 14 3,4; 17 3,4; 17 3,4; 14 5,6 3,4; 17 3,4; 14 334 3,4; 17

-310 -340 -240 -40 0 160 420 600 620

3,4 3,4 394 394 394 394 374 334 3,4

77.2 -278.0

17 334

-3 18.0

8

'

Table 2 Enthalpies of formation of transition metal carbides in the cubic-NaCl structure Carbide AfH(298K)-cubic Re$ Carbide AfH(298K)-cubic Re$ Wmol-' MeC Wmol-' MeC ZrC

HfC Tic TaC NbC vc0.88

wc

MoC CrC ReC TcC

- 196.6 -230.1 -184.1 - 144.1 - 140.6 - 101.7 -40.0 - 10.0 -28.4; 1.2 24.0 29.6; 64

8 8 8 8 8 8 8 8 3,4; 13 394 3,4; 14

MnC FeC RuC osc COC RhC Irc NiC PtC PdC

-86.4; 12.6 50.0; 42.0 140.6; 74 130.6 56.8; 51.2 124.0; 72 114.0 63.6; 62

3,4; 13 3,4; 13 3,4; 14 394 3,4; 13 3,4; 14 394 3,4; 13

64

14

greater accuracy, by Cremer et aZ.19using electron probe microanalysis, X-ray photoelectron spectroscopy and thin-film X-ray diffraction. The latter study showed that for co-deposition temperatures between 300 and 700"C, a two-phase region exists in the metastable AlN-TiN phase diagram. This expands with increasing temperature, whereas at 100°C substrate temperature, no two-phase region was observed (Figure 10).

204

Chapter I 7 150 125

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Mendeleev Number

Figure 7 A comparison of experimental and calculated enthalpies of formution for transition metal carbides with the cubic-NaCl structure. Experimental - solid squares (Ret 8); Calculated - open circles (refs. 3 and 4), crosses (refs. 13 and 14) 200 150 100

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-350

-400 Mendeleev Number

Figure 8 A comparison of experimental and calculated enthalpies offormation for transition metal nitrides with the cF8 structure. Experimental - solid squares (ReJ8); Calculated - open circles (Refs. 3 and 4), crosses (Refs. 14 and 17)

The composition of the metastable structural transition in the experimental studies lies between about 0.63 and 0.69 mol fraction AlN. The calculated transformation composition is therefore in good agreement with these observations. Based on the experimental work, a new Ti,-,Al,N hard coating for tribological applications with a composition of 0.62 mole fraction AlN has been developed, which shows an increase of oxidation resistance by a factor 4 over conventional TiN coatings. Because of the importance of the cubic to hexagonal transition in determining the properties of coatings for particular applications, a series of calculations have been carried out for a number of A1N-MeN systems (MeN is a transition metal nitride),

205

Thermodynamic Prediction of the Formation

mol fraction TiN

Figure 9 Gibbs energy of formation of the cubic and hexagonal phases in the A1N-TiN system at difierent temperatures 900

1

~

1

'

1

~

1

'

/

cubic + hexagonal

800 -

700 -

Y

600-

1500:

km:

F

-

300 -

(A1,Ti)N hexagonal

(Ti,Al)N cubic

-

A A I I ( A

200 100

0

1

'

1

'

1

'

1

.

Figure 10 Metastable TiN-AlN phase diagram according to experimental work from Ref 19. Experimental compositions with a large cross indicate a two-phase specimen

with a view to determining the extent of the metastable cubic and hexagonal ranges in each case. Energies of transition between the wurtzite and NaCl structures have been taken from Table 2, and the assumption has been made that the mixing properties are the same for the wurtzite and NaCl solutions and are equal to those in the system AlN-TiN in all cases. Figure 1 1 illustrates the calculated points of intersection of the Gibbs energy curves for the cubic and hexagonal solutions at 500 K, thereby defining the ranges of the two structures for the different, potential (Me,Al)N coatings. It can be seen that hafnium reduces the range of the cubic phase significantly compared with Ti, whereas Cr is effective in increasing the range further. Table 3 presents a summary of the calculated metastable hexagonal and cubic coating ranges for a number of MeN-AlN systems at 500 K.

206

Chapter 17 400

/

"

"

I

350 5- 300 0

c

.s 250 E

g

200

100 (3

50 0

0

01

02

03

04

05

06

07

08

09

10

rnol fraction MeN

Figure 11 Calculated ranges of the metastable cubic and hexagonal phases for a number of MeN-AlN systems at 500 K

Table 3 Calculated composition ranges for the cubic and hexagonal metastable coating phases in various MeN-AlN systems at 500 K MeN

HfJY ZrN TiN TaN NbN VN CrN

NaCl-cubic range mol fraction AlN 0 to 0.42 0 to 0.59 0 to 0.71 0 to 0.84 0 to 0.86 0 to 0.91 0 to 0.95

Hexagonal wurtzite range mol fraction AlN 0.42 to 1 0.59 to 1 0.71 to 1 0.84 to 1 0.86 to 1 0.91 to 1 0.95 to 1

3 AI,O,-AIN Coatings Al,O,-AlN coatings are of considerable interest because they offer the possibility of combining the advanced physical properties of electrovalent A1,03 with covalent AlN. Although for temperatures below 1627 K, there is no reciprocal solubility in the solid state, a metastable, homogeneous oxynitride phase could have potential applications in many areas because, e.g.,

-

the refractive index of a pure AlN film is high20, whereas the refractive index of A1,03 is low2' the thermal conductivity of A1N2, is high compared to that of A1,0,,, AlN is a semiconductor, A1,0, an insulator. If homogeneous, metastable Al-0-N phases with variable composition can be stabilized by vapour deposition, then it may be possible that the different properties of the binary compounds can be combined. Metastable A1-0-N layers have been produced by reactive magnetron sputtering ion plating using an A1 target and an Ar-0,-N, gas mixture.24For a substrate temperature of 190°C and variable partial pressures of 0, and N,, a variety of coatings with different 0 and N contents were prepared and analysed by means of X-ray photoelectron spectro-scopy, Auger electron spectroscopy, high-resolution transmission

207

Thermodynamic Prediction of the Formation

0.02

0.0

1.o

0.8

0.6 Mol fraction AlN

0.4

A1203

AlN

Figure 12 Gibbs energy of formation of the spinel, amorphous and wurtzite phases in the Al,O,-AlN system at 463 K

-

1

.

1

1

300-

1

0.0 A1203

I

I

0.2

1

'

Hexagonal

Spinel

50:

'

I

I

0.4

.

\

I

0.6

Mol fraction AIN

I

, ! . 0.8

II

1.o

AIN

Figure 13 Calculated metastable phase ranges in the Al,O,-AlN system with experimentally observed structures at 190°C from Van Richthofen and Dimnick 24

electron microscopy and high-resolution scanning electron microscopy. Oxygen-rich Al-0-N phases were found to be nanocrystalline with a cubic gamma-A1203(spinel) structure. Nitrogen-rich phases displayed a hexagonal AlN (wurtzite) structure. For O/N ratios around 1.5, the deposited phase was "EM amorphous (see Figure 13). Using Gibbs energy values provided by a thermodynamic evaluation of the A1,0,A1N system,25and incorporating thermodynamic descriptions for the wurtzite and amorphous phases for the entire composition range, Gibbs energy curves as a function of composition have been calculated for the spinel, wurtzite and amorphous phases for temperatures between room temperature and 600 K. The experimentally observed transition in structure from amorphous to hexagonal at a composition close to 70 mol% AlN was used to adjust the thermodynamic parameters of these two phases. The calculated curves for the deposition temperature (463 K) used in the experimental studies are illustrated in Figure 12.

Chapter I7

208

From the loci of the points of intersection of these curves as a function of temperature, the deposition ranges of the phases have been determined (Figure 13).

4 Summary The results presented above with respect to the AlN-TiN and Al,O,-AlN coating systems demonstrate that thermodynamic calculations are able to provide information of significant practical importance. In particular, the capability of calculating composition and temperature ranges of metastable coating phases produced by PVD processes offers significant time and cost-saving predictive support in the development of new, multi-component coatings with desired structures. Using the results of calculations to define selected experiments, much more rapid and reliable progress can be made than that associated with widely applied trial-and-error development methods.

References 1. N.Saunders and A.P.Miodownik,J. Muter. Res. 1986, 1, 38. 2. P.J.Spencer and H.Holleck, High Temp. Sci. 1990, 27, 295. 3. H.Stolten, Ph.D. Thesis, Lehrstuhl fur Theor. Hiittenkunde, RWTH Aachen, 1991. 4. H.Stolten, P.J.Spencer and D.Neuschutz, J. Chim. Phys. 1993, 90, 209. 5 . P.J.Spencer, G.Eriksson and A.von Richthofen, Proc. CODATA’94, Chambery, France, Sept. 1994, Springer, Berlin, 1998, 6. P.J.Spencer, in Ringberg workshop on computational thermodynamics, Group 5, new applications of thermodynamic calculations, CALPHAD 24 (2000) 72. 7. P.J.Spencer, 2. Metullkunde, 2001, 92, 1145-1150. 8. Thermodynamic Properties of Inorganic Materials compiled by SGTE, LandoltBornstein, New Series, Group lV,Vol 19, SubvolumeA - Pure Substances, Subvolume B Binary Alloys, Springer-Verlag, Berlin, 1999 onward. 9. D.Pettifor, New Scientist, 1986, 48. 10. P.J.Spencer, Thermochim. Actu, 1998, 314, 1. 11. A.Fernandez Guillermet and G.Grimval1, Phys. Rev., 1989, B40, 10582. 12. J.Haglund, G.Grimvul1, T.Jarlborg and A.Fernandez Guillermet, Phys.Rev., 1991, B43, 14400.

13. A.Fernandez Guillermet and G.Grimval1, J. Phys. Chem. Solids, 1992,53, 105. 14. A.Fernandez Guillermet, J.Haglund and G.Grimval1, PhyxRev., 1992, B45, 11557. 15. A.Fernandez Guillermet and K.Frisk, K., unpublished work. 16. A.Fernandez Guillermet, L.Haglund and G.Grimval1, Phys.Rev., 1993, B48, 11673. 17. A.Fernandez Guillermet and K.Frisk, J.Alloys Compds, 1994,203, 77. 18. 0.Knotek and T.Leyendecker,J.So1id State Chem., 1987, 70, 318. 19. R.Cremer, M.Witthaut and D.Neuschutz, in ‘Value Addition Metallurgy’, W.D.Cho and H.Y.Sohn and (eds), The Minerals, Metals & Materials Society, 1998 249. 20. D.H. Wang and L. Guo, Thin Solid Films, 1987, 158, L39. 21. D.H. Wang, in Proceeding of the 172nd Meeting of the Electrochemical Society, Hawaii, Oct. 1987; J.Electrochem.Soc., 1987, 2, 575. 22. N.K. Smit, G.A. Acker and C.J. van der Laan, Thin Solid Films, 1986, 138, 171. 23. H. Schule and W.-I. Ratzel, Technische Kerumik, lst.edn, Vulkan, Essen, 1998. 24. A. von Richthofen and R. Domnick, Thin Solid Films, 1996,283, 37. 25. M. Hillert and S. Jonsson, Z.Metullkde., 1992,83, 714.

CHAPTER 18

Thermodynamics of the Nano-Sized Particles TOSHIHIRO TANAKA, JOONHO LEE AND NOBUMITSU HIRAI

1 Introduction Various thermodynamic databases have been compiled to be mainly applied to the calculation of phase diagrams of alloys, salts and oxides.' The accumulation and assessment of thermodynamic data and phase equilibria information to establish those databases is sometimes called Computer Calculation of Phase Diagrams (CALPHAD) approach.2 The CALPHAD approach has been recognized to be useful in various aspects of materials science and engineering.'Y2 If it would be possible to use the thermodynamic databases to evaluate various phy sicochemical properties as well as phase diagrams, we could not only widen the applicability of those thermodynamic databases but also understand the physicochemical properties more deeply. The authors have applied those thermodynamic databases to the evaluation of the surface tension of liquid alloys, molten ionic mixtures and the interfacial tension between liquid steel and molten lag.^-'^ These works are aimed to comprehend the thermodynamic properties of a material system including its surface or interface as well as the bulk.13,14Since the effect of its surface on the total thermodynamic properties cannot be negligible in small particles of metals and alloys, the phase relations in these metals and alloys are dependent on the size of the particle and its surface property. The thermodynamics of solid-liquid-phase equilibira in a small particle system was firstly discussed by Pawlow in 1909,15followed by Hanzen16in 1948. Since Takagi17observed in 1957 that the melting points of some pure metals decrease with decreasing size of these metallic particles, several studies have been carried out on the effect of the particle size on the melting phenomena of pure metals and As described above, the authors have evaluated the surface properties as well as phase diagrams of alloys on the basis of thermodynamic databa~es.~-l~ Extending these techniques, a new system might be developed to estimate phase equilibria of materials in small particle systems as shown in Figure 1. The authors have tried to calculate some nano-particle binary alloy phase diagrams

210

Chapter I8

7

7 Alloys, Salts & Oxides etc.

Nano-sized Particles

(Surface Tension,

7

Figure 1 Concept of a new evaluation system for phase equilibria in nano-sized particle systems on the basis of thermodynamic databases

of Cu-Pb, Cu-Bi and Au-Si systems by using thermodynamic databases,13 and in addition, to evaluate the effect of the surface properties on those phase diagrams on the basis of a regular solution rn0de1.l~In the present paper, we describe an outline of our trials13>14 to evaluate phase diagrams of binary alloys in the nano-sized particle systems.

2 Gibbs Energy in Small Particle Systems When a pure solid phase is selected as the reference state of Gibbs energy, the total Gibbs energies in liquid and solid phases, Adrotdl Phase (Phase=Liq or Sol), of an A-B binary alloy system in a small particle with its radius r are described in the following equations (1)-(6):l39l4 AGTotal, Phase

=

A G B u l k , Phase

+

AGSurface, Phase

(1)

The Gibbs energy of the bulk of A-B binary alloy in liquid and solid phases, A G B u l k ? Liq and A G B u l k , in Equation (l),which corresponds to AGTotal*(P=Liq or Sol) with r=-, are expressed in Equations ( 2 ) and (3).

where AGkS and AGf are Gibbs energies of pure liquid phases relative to those of pure solid phases, in other words, the Gibbs energy of the melting. GExcess, L1q and GExcess,Sol are the excess Gibbs energies of liquid and solid phases in the A-B alloy. N A and N B are mole fractions of components A and B. Although detailed functions of compositions and temperature for excess Gibbs energies are generally stored in thermodynamic databases, the regular solution model is applied here to evaluate the excess Gibbs energy GExcess, Phase because various interaction energies can be selected easily as follows: GExcess, Phase

=

N

A

.N~ .

w,PPase

(4)

21 1

Thermodynamics of the Nano-Sized Particles

where WEae (Phase=Liq or Sol) is the interaction energies in liquid and solid phases. In addition, the Gibbs energy of the melting AGF is also compiled for various elements in thermodynamic databases, but here we use the following simple equation (5) as Pelton and Thompson32assumed this equation in their evaluation of phase diagrams:

where Tx,mp and Sx,mp are the melting point and the entropy of fusion for pure substance X(X=A or B). S , mp is roughly assumed to be 10JK-' mol-' according to Richard's rule.33In addition, we select here TA: mp= 1200K and TB,,,=600K as an example to evaluate a phase diagram of A-B binary system. AGsu~ace9 Phase in Equation (l), the effect of the surface on AGTota'* Phase, is assumed as f01lows:'~J~ AGsUrfaCe, Phase =

2

Phase

v Phase - 2 ( N A o r 1 V r 1 + N o

r

V

" ")

(Phase = Liq or Sol) (6)

where oLiqand oSo1 are surface tensions of liquid alloy and solid solution, VLiqand p' the molar volumes of liquid alloy and solid solution, r the radius of a particle, orland or1the surface tensions of pure solid A and B and Vrl and VSp' the molar volumes of pure solid A and B. The second term on the right-hand side of Equation (6) comes from the fact that a pure solid phase is selected as the reference state of Gibbs energy. On the surface tension of pure liquid metals at their melting points we use the following approximation:l4

oFmp,

o!&,~=

TXm (Nm-') 1000

(7)

In addition, the temperature coefficient of the surface tension of pure liquid has been reported to be about 0.0001 Nm-' K-1.34Thus, we assumed the following equation for okiq and o,L'4:

= 0.6 -

0.0001 (T - TB,,,)/Nm-'(T,,

mp

=

600K)

(9)

As shown in Equation (6), we need the value of the surface tension oF1of pure solid X, but the precise information on the value of OF'and its temperature dependence are i n ~ u f f i c i e n t . From ~ ~ - ~ the ~ data reported in some reference^,^^-^^ the value of oT1 of pure solid at the melting point is found to be 25% larger than the surface tension of pure liquid on the average. Equation (10) is, therefore, assumed to express the surface tension OF'of pure solid X in the present work:

a?'

=

1.250~~qm, +

aopq

7 (T - Tx, mp)

( X = A or B )

(10)

where oXLfgmis p the surface tension of pure liquid X at its melting point Tx,mp. The temperature dependence of o?' in Equation (10) is assumed to be the same as that of opq. In addition, the effects of crystal faces on o;c"lare ignored here. When

212

Chapter 18

Equation (10) is applied for the surface tension opl of pure solid X , the following relations are obtained from Equations (8) and (9):

or1= 1.25 X 1.2-0.0001 = 1.25 X 0.6 - 0.0001

(T - TA, mp)/Nm-l

(1 1)

(T - TB, mp)/Nm-l

(12)

On the molar volumes of pure components, we used the following values in Equation (13) by assuming that the temperature dependence of the molar volume and the volume change due to the melting are neglected because the effect of the excess Gibbs energy and the surface tension on the phase equilibria is focused.13*14

VzhaSe= VPhase B = 10 X lO-'/m3rnol-'

(Phase = Liq or Sol)

(13)

It is assumed that the molar volumes of liquid alloy VLiq and solid alloy VSo1in Equation (6) are obtained from the following simple additivities: VPhase

=

N

A

VPhase A

+N

VPhase B B

(14)

The authors have evaluated the surface tension oPhase in Equation (6) of A-B binary alloys on the basis of Butler's equation [38] as follow^:^-^^

In Equations (15a) and (15b), gurf and Ni& are mole fractions of element B in the surface and the bulk, respectively. A~ase=LN~b(VxM)2b (No Avogadro number: X=A or B, L= 1.091) is the molar surface area of pure X , and this is obtained from the molar volume Vxm Phase, Bulk (T, Ni"lk)and G F Bulk (T, @") are partial excess Gibbs energies of component X in the bulk and the surface, respectively, as functions of T and Ni& or @". Since the partial molar excess Gibbs energy of component X (X= A or B) in the bulk GFEss*phase, Bullc (T, Ni&) in Equations (15a) and (15b) can be obtained directly from GExcesst Phase in Equation (4) by using Equations (16a) and (16b), q

m

s

9

GF Phase, Bulk (T, N,Bulk) = G Excess, Phase G?,

Phase, Bulk (T, ~BBulk) =

G Excess, Phase -

a GExcess, Phase

NB

aNB

( W

a G Excess, Phase - NB)

aNB

( 16b)

For the excess Gibbs energy in the surface, we derived the following equation( 17)3-'4 based on the model proposed by Yeum et ~ 1 . ~ ~

213

Thermodynamics of the Nano-Sized Particles

where PMrX* Liq = 0.83:liquid alloys & pMIX* Liq =

= 0.75:solid solutions. 12 s"f(T,iVid),which has the same formula Equation (17) indicates that G F as GF Phase, Bulk (T, iVi"Lk), is obtained by replacing iVFkby NiUrfin G F Phase, s"f (T, N i d ) (X=A or B) and then multiplying P M I X y Phase with G F (T, N p - 9 . pMIX* Phase is a parameter corresponding to the ratio of the coordination number in the surface to that in the bulk considering the surface rela~ation.~-'~ In particular, for the solid solutions, here we use the value 0.75 as p MIX* by assuming that the coordination number in the bulk is 12 and that in the surface is 9 in a closed-packed structure. The surface tension crPhasein Equation (15) of liquid alloys or solid solutions can be calculated from Equations (15)-( 17) as follows:

(1) Setting temperature T and composition iVpkof a solution. (2) Inserting the values for surface tension cryase and molar volume V!hase of pure substances at the above temperature in Equations (15a) and (15b). (3) Determining excess Gibbs energies in the bulk phase at the above temperature and composition, and substituting them in to Equations (15a) and (15b). (4) Then, Equations (15a) and (15b) become the simultaneous equations with unknown Nid and cr Phase. These equations are solved for those unknown and CT Phase numerically.

3 Calculation of Nano-Sized Phase Diagrams of Binary Alloys From Equations (l), (2), ( 6 )and (lo), the Gibbs energy of pure component X including the surface is obtained as follows:

The temperature T, which yields A g o t a l = O in Equation (18), is the melting point of pure X at a given radius r of a particle. Figure2 shows the change in the melting point of pure Au with the radius of the particle calculated from Equation (18) with experimental data.20*21 The data34740necessary in this calculation are given in Table 1. Liquid-solid-phase equilibria in an alloy are obtained from the following thermodynamic conditions:

P p l , Liq

=

(20)

P P l ,

a AG TO^, Phase PpalPhase ,

=

& T o t a l . Phase -

NB

aN,

(Phase=Liq or Sol)

(21)

214

Chapter 18

li

J

5a 1250

Mdtlny polnt of pure Au

hp,

115B

10

20 30 40 50 Radius of particle / nm

f1

Figure 2 Change in melting point of pure Au with particle size. Expe.: Sambles,20Coombes.21

Table 1 Parameters used in the calculation of phase diagrams Melting point Entropy of IK fusion IJK- 'mol-

Substance A: 1200

10

Substance B: 600

10

Sur$ace Tension lNm-

OX IO-Wq. 1,2-0.0001( T- 12OO):Liq 1.2X 2.5 -0.000 1 (T- 1200):Sol. 10X 1O-? Sol. 0.6-0.0001(T- 600):Liq . OX 10-6:Liq. 0.6X2.5-0.0001(T-600):S01. lox 10-6:S~l.

30 w,;q/k.J

20 15 0 -20

Molar volume lm3mol-'

Figure 4(a) Figure 4(b) Figure 4(c) and Figure 3(c) Figure 3(d)

15

0

Figure 3(b)

Figure 3(a)

- 15

Figure 3(d)

aAGT0ta'.Phase

ppl,Phase = AGTota1. Phase + (1 - N ~ )

JNB

(Phase = Liq or Sol)

(22)

Figures 3 and 4l3,I4show the calculated results of the phase diagrams using the parameters in Table 2. In these figures, the solid curves indicate the phase equilibria in the bulk (r=m). On the other hand, the chain and the dotted curves are the calculated results for r= 10 and 5nm, respectively. Figure 3(a) indicates the phase diagrams for the ideal solutions in both solid and liquid phases, in other words, Wk2=WZ1=O. When W&' increases from 0 to 30 kJ with Wgq=O, the phase diagrams change as

215

Thermodynamics of the Nano-Sized Particles

I

I

Figure 3 Calculated results of the phase diagrams with systematic change of W;? and WAS': on the basis of the regular solution model

shown in Figures 3(b) and (c). Figure 3(d) shows a phase diagram for Whq= -20 kj and Wz1=-15kj as an example for Wzq>Oand W&'%O. These composition dependences of the surface tension affect the contribution of AGSurface* Phase to AGOd, Phase largely. In addition, when Wzl>O, the solid solutions do not appear in some of the phase diagrams of the bulk as shown in Figures 3(c), (d) and 4(d). However, when the particle size decreases, the contribution of AGsurfacev to AQod, in the solid phase cannot be ignored. Consequently, as shown in Figures 3 and 4, the solid solution appears in the small particle systems even when the bulk phase diagrams do not show the solid solutions. As metioned above, when an alloy has a large positive interaction among the components ( W P > O ) ,the phase diagram of the alloy changes remarkably with the size of a particle. Figure 5 shows one example of the calculated result of Cu-Pb alloy, which has a large positive excess Gibbs energy.13 Since this phase diagram consists of a liquid phase and pure solid phases, the solid solutions are not considered in the calculation. The calculation procedures and the data are given in detail in the authors' previous work. l 3

216

Chapter I8 ,

I

.

.

-

..------....--. -.....-....-. -,..-........

*...

: ; * . I : I; : ; :

i

Figure 4 Calculated results of the phase diagrams with systematic change of WkLq in the liquid phase and W;;'= 30 kJ on the basis of the regular solution model

As can be seen in Figures 2-5, the phase diagrams of binary alloys in the nanosized particle systems can be estimated from the information stored in thermodynamic databases with some fundamental physical properties, although the rough assumptions have been used here.I3J4As Howe pointed the atomic bonding near the surface of the bulk materials is weaker than that in the bulk, which causes the surface melting. In this phenomenon, the thin layers near the surface of metals are melted below the melting points of the bulk. For example, Kojima and S U S ~ ~ ~ investigated the surface melting of pure Cu by using the MD method. They reported that the surface melting occurs at 900 K in the face [ 1111, although the bulk Cu melts at 1356K. Thus, in the nano-sized particles, whose surface area per unit volume is

Table 2 Data necessary for calculation of the melting point of pure Au Surj4ace tension o;/Nm-' of pure liquid Au: Ref: 34

adz -/Nm-'T-' aT

0ku= 1.169-0.00025

-0.00025

(T- 1336.15) Tx, mp/K:Ref. 34 1.169 1336.15 Molar volume/ m3mol-' of pure liquid Au: Ref. 34 Vku=11.3X10-6{ 1.0+0.000069 (T-1336.15)) Gibbs energy change of fusion of Pure Au/ J mol-': Ref. 40 AGk:= 12552.0-9.385866 T

ok,,dNrn-': Ref. 33

Thermodynamics of the Nano-Sized Particles

217

Figure 5 Phase diagram of Cu-Pb alloy

larger than that of the bulk systems, the surface phenomenon may occur more remarkably than the bulk system. We should, therefore, consider the combination of those microscopic approaches with the above macroscopic thermodynamics for further development of the evaluation of the nano-sized phase diagrams.

4 Concluding Remarks The authors have described some fundamental procedures to evaluate the solidliquid phase relations of nano-sized particles in binary alloys from the calculation of the surface properties as well as the phase equilibira on the basis of thermodynamic databases, which are usually used for the calculation of phase diagrams of the bulk materials. In order to obtain quantitatively precise values of the melting points and liquidus temperatures in alloys, we should carry out further investigation as follows: More detail discussion on the thermodynamic criteria on the melting of nanosized particles. The reliability of the data used for the evaluation of the melting phenomena of the nano-sized particles, especially the surface energies of liquid and solid alloys. The change in the bonding of atoms near the surface from that of the bulk in relation to the surface melting phenomena. We can, however, evaluate the phase equilibria of small particles qualitatively from the macroscopic thermodynamic point of view as described in the present paper. Thus, the authors wish that enormous attention be paid to the application of thermodynamic databases to the calculation of the phase diagrams of the nano-sized particles, whose information will be in demand increasingly in the near future.

218

Chapter 18

References 1. C. W. Bale and G. Eriksson, Canad. Metall. Quart. 1990,29, 105. 2. T. Nishizawa, Matel: Trans. JIM, 1992,33,713. 3. T. Tanaka and T. Iida, Steel Research, 1994, 65, 21. 4. T. Tanaka, K. Hack, T. Iida and S. Hara, 2.Metallkd., 1996,87, 380. 5 . T. Tanaka, S. Hara, M. Ogawa and T. Ueda, 2. Metallkd., 1998,89, 368. 6. T. Tanaka, S. Hara, M. Ogawa and T. Ueda, Molten Salt Forum, 1998,5-6,213. 7. T. Tanaka, S. Hara and T. Ueda, in Proceedings of the 11th International Symposium on Molten Salts, San Diego, ECS, USA, P.C. Trulove, H.C. De Long, G.R. Stafford, S.Deki (eds), 1998,645. 8. T. Tanaka, K. Hack and S . Hara, MRS Bulletin, 1999, 24,45. 9. T. Tanaka and S. Hara, Electrochemistry, 1999,67, 573. 10. T. Tanaka and S. Hara, 2.Metallkd., 1999, 90, 348. 11. T. Ueda, T. Tanaka and S. Hara, 2.Metallkd., 1999, 90, 342. 12. T. Tanaka, K. Hack and S. Hara, Calphud, 2001,24,463. 13. T. Tanaka and S. Hara, 2.Metallkd., 2001, 92,467. 14. T, Tanaka and S. Hara, 2.Metallkd., 2001, 92, 1236. 15. P. Pawlow, 2.Phys. Chem., 1909,65, 1. 16. H. Reiss and I.B. Wilson, J. Colloid Sci., 1948, 3, 551. 17. M. Takagi, J. Phys. SOC.Jpn., 1954,9, 359. 18. K.J. Hanszen, 2.Phys., 1960, 157, 523. 19. C.R.M. Wronski, Brit. J. Appl. Phys., 1967,18, 1731. 20. J.R. Sambles, Proc. Roy. SOC.Lond. A., 1971,324, 339. 21. C.J. Coombes, J, Phys. F., 1972,2,441. 22. Ph. Buffat and J-P. Borel, Phys. Rev., 1976,13, 2287. 23, P.R. Couchman and W.A. Jesser, Nature, 1977,269,481. 24. P.R. Couchman and C.L. Ryan, Phil. Mag., 1978, A37,369. 25. G.L. Allen, W.W. Gile and W.A. Jesser, Acta Metall., 1980, 28, 1695. 26. J. Ross and R.P. Andres, Su@ Sci., 1981,106, 11. 27. J.P. Borel, Surf.Sci., 1981, 106, 1. 28. G.L. Allen, R.A Bayles, W.W. Gile and W.A. Jesser, Thin Solid Films, 1986,144,297. 29. H. Saka, Y. Nishikawa and T. Imura, Phil. Mag. A, 1988, 57, 895. 30. K. Sasaki and H. Saka,, Phil. Mag. A, 1991,63, 1207. 31. J.G. Lee, H. Mori and H. Yasuda, Phys. Rev. B, 2002,66,012105-1. 32. A.D. Pelton and W.T. Thompson, Prog. Solid State Chem., 1975, 10, 119. 33. D.R. Gaskell, Zntroduction to the Thermodynamics of Materials, 3rd edn, Taylor & Francis, Washington DC, USA, 1995 34. T. Iida and R.I.L. Guthrie, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1988. 35. B.C. Allen and W.D. Kingery, Trans. Metall. Sco. AIME, 1959,215, 30. 36. L.E. Murr, Interfacjal Phenomena in Metals and Alloys, Addison-Wesley,London, 1975. 37. V.K. Kumikov and Kh.B. Khokonov, J. Appl. Phys., 1983,54, 1346. 38, J.A.V. Butler, Proc. Roy, Soc., 1932, A135, 348. 39. K.S. Yeum, R. Speiser and D.R. Poirier, Metall. Trans., 1989, 20B, 693. 40. A.T. Dinsdale, CALPHAD, 1991,15, 317. 41. J.M. Howe, Interfaces in Materials, John Wiley & Sons, Inc., New York, 1997. 42. R. Kojima and M. Susa, High Temp. High Press., 2002,34.

CHAPTER 19

Thermodynamics of Electrolyte Systems of Industry KAJ THOMSEN

1 Standard States and Chemical Potentials The chemical potential pi of a species i is the partial molar derivative of the Gibbs energy G, enthalpy H , Helmholtz energy A, or internal energy U of substance i1

Pi =

(1)

In Equation (l),niis the amount of component i, T is the temperature, P is the pressure, S is the entropy, and V is the volume. Matter flows spontaneously from a region of high chemical potential to a region of low chemical potential just like electric current flows from a region of high electric potential to a region of low electric potential and mass flows from a position of high gravitational potential to a position of low gravitational potential. The chemical potential can therefore be used to determine whether or not a system is in equilibrium. When the system is in equilibrium, the chemical potential of each substance will be the same in all phases of the system. In an ideal solution, the chemical potential pjdeal of species i is calculated from the standard chemical potential PO, the gas constant R, the Kelvin temperature T, and the mole fraction xi as2

The chemical potential of component i will be equal to the standard chemical potential when the mole fraction of component i equals 1. The standard state is thus pure component i. If species i is an ion, the standard state is the hypothetical state of pure ion i. In a real solution, an extra term, RT In is added to the chemical potential in order to account for the deviation from ideality:

220

Chapter 19

x

where is the activity coefficient of component i. In the standard state, the activity coefficient equals 1. There is no way of directly measuring the standard chemical potential of pure ions. It has therefore been found practical to introduce a normalized activity coefficient equal to the ratio of the activity coefficient and the activity coefficient at infinite dilution:

‘

y* = - + 1

[ I T

for xi+ o

(4)

By introducing this unsymmetrical activity coefficient in Equation (3), the expression for the chemical potential of component i can be written as:

The term p;= p; + RT In YF does not refer to the chemical potential in a specific physical state, but has the advantage that it can be determined by experiment in the limit of infinite dilution. A molal activity coefficient f can be defined as the product of the water mole fraction and the unsymmetric activity coefficient:

By insertion of this expression into Equation (5), the following expression for the chemical potential of component i is obtained as a function of the molality mi(mol per kg water) of component i, and the molecular mass M , of water in kg per mol water:

In this equation, the following definition of the chemical potential of ion i in “the hypothetical state of the pure ions dissolved in pure water to give an ideal solution at unit mean molality, under the standard state pre~sure”~ has been used:

Like p*,p: can be measured and is the value that is most often tabulated for the standard chemical potential of ion^^-^. mo= 1.0 mol per kg water is inserted in order to make the expressions dimensionless. The molal activity coefficient f is the activity coefficient usually measured experimentally and reported, rather than the unsymmetric or symmetric mole fraction activity coefficients. Most activity coefficient models, however, yield values of the symmetric or the unsymmetric mole fraction activity coefficient. It is therefore often necessary to convert back and forth between activity coefficients and chemical potentials in the various systems.

1.1 Ionic Mean Properties Electroneutrality is required in all solutions; thus, it is not possible to measure the properties of a single ion without influence from an ion of opposite charge. By convention, the standard enthalpy of formation and the standard entropy of the

Thermodynamics of Electrolyte Systems of Industry

22 1

hydrogen ion in aqueous solution are given the value zero at 298.15 K6These values refer to the molal standard state. A value of zero is also assigned to the (molal) standard Gibbs energy of formation, heat capacity, and volume of the hydrogen ion. The corresponding values for all other ions can be calculated relative to the value for the hydrogen ion. In some tables, other conventions are used for the standard state heat capacity and volume of the hydrogen ion.5 Like the standard state properties, the activity coefficient of a single ion is immeasurable by any thermodynamically valid m e t h ~ dThe . ~ measured and calculated activity coefficients are therefore usually presented as the geometric mean of the cationic and the anionic activity coefficients weighted according to the amounts of each ion:

A similar expression can be obtained for the symmetric and the unsymmetric mole fraction activity coefficient. In a completely dissociated solution of n mol Na2S0,, the mean molal activity coefficient is

or

The mean molality is similarly defined by 1 lnm, = - x n j l n m j ions

A corresponding definition is used for the mean mole fraction. The mean ionic activity is the product of the mean concentration and the mean activity coefficient.

2 Activity Coefficient Models Activity coefficient models are equations representing the Gibbs or the Helmholtz energy of solutions. Activity coefficients and related properties are derived form these energy functions by proper differentiation (Equation ( 1)). In addition to the short-range interactions between species in all solutions, longrange electrostatic interactions are found in electrolyte solutions. The deviation from ideal solution behavior caused by these electrostatic forces is usually calculated by some variation of the Debye-Huckel theorys or the mean spherical approximation (MSA).9 These theories do not include terms for the short-range attractive and repulsive forces in the mixtures and are therefore usually combined with activity coefficient models or equations of state in order to describe the properties of electrolyte solutions. The Electrolyte NRTL modellO-lland the Extended UNIQUAC are examples of activity coefficient models derived by combining a deb ye-Huckel term with a local composition model. Equation of state models with electrostatic terms for

222

Chapter 19

electrolyte solutions have not yet been developed to a level where they can be of practical use in industry.Ig The Pitzer mode120 was developed by adding virial expansion terms to the Debye-Huckel equation. The Pitzer model is based on another definition of the ideal solution than the one given in Equation (2). A UNIQUAC model combined with a Debye-Huckel term and a virial expansion term was introduced by Wang et al. in 2002.,l

3 Equilibrium Calculations Equilibrium calculations for electrolyte solutions include speciation equilibrium, vapor-liquid equilibrium, solid-liquid equilibrium, and liquid-liquid equilibrium. As an example of the first three types of equilibria, we will consider the ternary H,O-NH,-CO, system. Chemical and phase equilibria are attained when the chemical potential of the reactants equals that of the products. This criterion for equilibrium can be expressed by the mass action law as

AGO is the change in standard state Gibbs energy by the process, calculated from the standard state chemical potentials that can be found t a b ~ l a t e d .R~ *is~the gas constant, T is the temperature in Kelvin, aiis the activity of species i, and vi is the stoichiometric coefficient of component i in the process under consideration. Components on the left-hand side in the process equation are given negative stoichiometric coefficients. Usually, only the values of the chemical potentials at 298.15K are found in tables. It is then necessary to calculate the value of AGO at the temperature of interest. This can be done by using the Gibbs-Helmholtz equation:

a(AGo/R7') -- --AHO aT

RT2 at constant pressure

(14)

The variation of AHO, the change in standard state enthalpy by the process, with temperature can be calculated from the heat capacity of the species involved in the process.22

3.1 Speciation Equilibrium The term speciation is used to describe the reactions that take place when an electrolyte is dissolved in water. Water dissociates, sour gases hydrolyze, some ions dissociate, and other ions associate until thermodynamic equilibrium is attained. The liquid phase of the ternary H20-NH,-C0, system contains at least the following nine species: H20, NH,(aq), CO,(aq), H', OH-, N H:, HCO;, CO:-, and NH,COO-. (aq) indicates that the species is in aqueous solution to avoid ambiguity. In order to adequately model this system, interaction parameters for the interaction between each pair of species need to be determined; thus, speciation calculations are performed simultaneously with the parameter estimation, and the calculated amount of each species is compared with experimental data. Some models20also require ternary parameters and consequently an additional amount of data to determine these parameters.

223

Thermodynamics of Electrolyte Systems of Industry

One of the speciation processes taking place in this solution is the formation of carbamate ions, NH2COO-, from CO, and NH,: H 2 0 + C02(aq) + NH,(aq)

H30f

+ NH2COO-

(15)

The AGO for this process to be used in Equation (13) is calculated by

In Equation (16), it has been assumed that the pure component standard state is used for all components and that the activity coefficients of the solutes are normalized according to Equation (4). The right-hand side of Equation (13) becomes ~ H , o~NH,COO+

) =In(

'H20 'C02 'NH,(aq)

xHIO+ 1/*H30+XNH,C00- lfNH2C00XH,OYH,O

x c o 2 ~ oXNH,&H, ,

When all independent speciation equilibria in this way are written in the form (13), the equations can be solved for the mole fractions. The equation set must be solved by an iterative procedure, as the activity coefficients are dependent on the mole fractions.

3.2 Vapor-Liquid Equilibrium The gas phase of the system will mainly consist of H,O(g), NH,(g), and C02(g). In order to perform vapor-liquid equilibrium calculations with an activity coefficient model at pressures higher than ambient pressure, the activity coefficient model can be combined with an equation of state for the gases.', Usually, there is no need for binary interaction parameters in the equation of state as gas phase fugacities are only slightly dependent on these binary interaction parameters. For the equilibrium between ammonia in the liquid phase and ammonia in the vapor phase, Equation (13) gives:

The mole fraction and the fugacity coefficient are y and #, and the total pressure is P . This is the so-called gammdphi formulation of vapor-liquid equilibrium.,

3.3 Solid-Liquid Equilibrium A number of different solid phases can precipitate from a solution of ammonia and carbon dioxide in water. As an example, the equilibrium of ammonium bicarbonate with an aqueous solution is written in the form (13):

NH,HCO, E NH:

+ HCO,

As a standard state for the solid salt, the state of pure, solid salt is used. For solid-liquid equilibrium calculations in mixed solvent systems, it is important to choose the proper standard states for the ions.15

224

Chapter 19

3 -4 Liquid-Liquid Equilibrium For liquid-liquid equilibrium to occur, it is required that the activity of each component is the same in both phases. For a system of NaCl, water, and isopropanol (IP) three equations of the form (13) can be written for the equilibrium between liquid phase I and liquid phase 11:

The left-hand side of each of these equations is zero when the same standard state for each species is used in both phases.23The mean ionic activity is being used for the salt. For solutions containing more salts, it will be necessary to add one equation for the mean activity of each independent salt.

4 Phase Diagrams Solid-liquid equilibrium phase diagrams play an important role in the design of industrial crystallization processes.24-26The calculation of phase diagrams can be used to validate the activity coefficient model used for process simulation. An example of a binary phase diagram is shown in Figure 1. Experimental data from the open literature are plotted in the diagram along with the calculated phase diagram. In the temperature range from 49 to 89 "C, magnesium nitrate has three different solubilities at each temperature corresponding to two different solubilities of the hexahydrate, and the solubility of the dihydrate. Figure 2 shows the 50°C solubility isotherm for the ternary H20-Na2S04-MgSO, system. This isotherm consists of three different branches. The first branch represents the solubility of sodium sulfate, the second branch represents the solubility of astrakanite (Na2S04*MgS04.4H20),and the third branch represents the solubility of hexahydrite, (MgS04.6H20). In Figure 3, the liquid-liquid phase diagram for the ternary H20-Na,$04-t-butanol system is shown. The units in this phase diagram are mass percent. Polytherms for ternary systems can also be presented in two-dimensional phase diagrams.17 Phase diagrams for reciprocal salt pairs can be presented by the method of J a n e ~ k eAccording .~~ to this method, a projection of the three-dimensional diagram is made, and the water content is indicated by contour lines. In the Janecke diagram in Figure 4, the water content is marked in mass percent at each grid intersection. The concentrations of the four ions, K', NH4+, C1-, and NO,-, are indicated as charge fractions. The lines in Figure 4 indicate compositions of liquids in equilibrium with two salts. Each field in the diagram represents a range of compositions that are saturated with one salt. The phase diagrams shown in Figures 1 4 were calculated with the Extended UNIQUAC model,12 and the experimental data marked with circles in the diagrams come from various sources.

225

Thermodynamics of Electrolyte Systems of Industry

0

20

40 60 Mass percent Mg(NO,),

80

100

Figure 1 The solubility of Mg(NO,), in water as afunction of temperature

T = 50.0"C 80

20

N

0 0

10

20

30

40

50

60

70

80

90

100

MgS04

Figure 2 The 50 "C solubility isotherm in the H,O-Na,SOcMgSO, the three solid phases. Concentrations are in mass 96

system. Tie lines indicate

For systems with more than four independent components, two-dimensional phase diagrams can be made by fixing the amount of each additional component. An isotherm for such a system will then consist of a series of phase diagrams with varying amounts of these components.

226

Chapter 19

6

Figure 3 Liquid-liquid equilibrium in the H,O-Na$Oct-butanol

system. Calculated and

experimental tie lines

I

0.00

T = 20.0"C

I

I

I

I

I

I

0.10

0.20

0.30

0.40

0.50

CI-

I

I

I

I

I

l

l

1

0.60

0.70

0.80

0.90

1.00

Anion Charge fradon NO;

Mass percent water given at grid intersections

Figure 4 Janecke diagram for the reciprocal salt pair KCl-NH,,NO, in water. Calculated two-salt saturation lines and experimental two-salt saturation points

5 Density The apparent molar volume Vp of a salt is

vp = nV-

'H20 '&O

ns

227

Thermodynamics of Electrolyte Systems of Industry 43.0 42.5 42.0 41.5 m

i

%

41.0

::: 39.5 39.0 38.5 38.0

0

10

20

30

40

50

60

sqrt (c, m0Ym3)

Figure 5 Apparent molar volume of potassium nitrate solutions calculated with Masson's rule (22), compared with experimental data

where V is the molar volume of the solution, n is the total amount in mol, nH,ois the amount of water, Viz0is the molar volume of pure water, and ns is the amount of salt. The apparent molar volumes of salts are often linear functions of the square root of the molar concentration c in the concentration range from infinite dilution to saturation of the salt:28

Equation (22) is known as Masson's rule. The apparent molar volume of potassium nitrate is shown in Figure 5 as a function of the square root of the concentration c in mol m-3 up to saturation at 25 "C.Experimental data are plotted, together with the line calculated from Equation (22) with a=38.13X m3 mol-' and b=7.45x (m3 m ~ l - ' ) ~ ' ~ . A similar relation, which can be derived theoretically from the deb ye-Hiickel theory, exists for the partial molar volume of electrolyte^.^^ Apparent molar volumes of ions are often considered additive. The a and b parameters from Equation (22) cannot be measured for single ions, but the parameter values for salts can be split appropriately between the anions and cations of the salt if the parameter values for one ion are fixed by convention. Unfortunately, several conventions are in use.5 As data based on different conventions are incompatible, it is important to check some reference values when using this type of data. A good approximation for the density of multicomponent electrolyte solutions can be obtained by adding the individual ionic contributions to the apparent molar volume:

228

Chapter 19

The density p of a multicomponent salt solution can then be calculated from

Temperature-dependent parameters for calculating the densities of binary and ternary aqueous solutions of salts have been compiled by Sohnel and N o v ~ t n L . ~ ~

6 Viscosity The viscosity of aqueous electrolyte solutions is usually described with an equation of the following form, as a function of the concentration c in m ~ l / m ~ : ~ l

'

-= 1 '0

+ Ac'" + Bc

(25)

The A parameter in this equation can be calculated t h e ~ r e t i c a l l y .The ~ ~ ,B~ ~parameter is determined from experimental data. 77 is the viscosity of the solution and qois the viscosity of the pure solvent. The B parameter is usually considered additive and an approximation of the viscosity of a multicomponent salt solution can be obtained by adding the contributions from all ions.

References 1. J. M. Prausnitz, R. N. Lichtentaler and E. G. Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd edn, Prentice-Hall PTR, Upper Saddle River NJ 07458, 1999. 2. J. M. Smith, H. C. Van Ness and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 6th edn, McGraw-Hill, New York, 2001. 3. D. D. Wagman, W. H. Evans, V. B. Parker, R. H. Schumm, I. Halow, S. M. Bailey, K. L. Churney and R. L. Nuttall, J. Phys. Chem. Re$ Data, 1982, 11, Suppl. 2. 4. NIST Chemical Thermodynamics Database Version 1.1, U.S. Department of Commerce, National Institute of Standards and Technology, Gaithesburg MD, 1990, 5. Y. Marcus, "Ion Properties", Marcel Dekker Inc. New York, 1997. 6. J. D. Cox, D. D. Wagman, V. A. Medvedev, CODATA Key Valuesfor Thermodynamics, Hemisphere Publishing Corporation, New York, 1989. 7. R. P. Buck, S. Rondinini, A. K. Covington, F. G. K. Baucke, C. M. A. Brett, M. F. Camdes, M. J. T. Milton, T. Mussini, R. Naumann, K. W. Pratt, P. Spitzer and G. S . Wilson, Pure Appl. Chem., 2002, 74 (1 l), 2169. 8. P. Debye and E. Hiickel, Phys. Z., 1923,24, 185. 9. A. H. Harvey, T. W. Copeman and J. M. Prausnitz, J. Phys. Chem., 1988,92,6432. 10. C. C. Chen, H. I. Britt, J. F. Boston and L. B. Evans, AIChE J., 1982,24,588. 11. C. C. Chen and L. B. Evans, A.Z.Ch.E. J., 1986,32,444. 12. K. Thomsen, P. Rasmussen and R. Gani, Chem. Eng. Sci., 1996,51,3675. 13. K. Thomsen and P. Rasmussen, Chem. Eng. Sci., 1999,54, 1787. 14. K. Thomsen and P. Rasmussen in Steam Water and Hydrothermal Systems: Physics and Chemistry Meeting the Needs ofhdustry, P. G. Hill, P. Tremaine, D. Irish and P. V. Balakrishnan (eds), NRC Press, Ottawa, 2000, 118. 15. M. Iliuta, K. Thomsen and P. Rasmussen, Chem. Eng. Sci., 2000, 55,2673. 16. S . Pereda, K. Thomsen and P. Rasmussen, Chem. Eng. Sci., 2000, 55,2663.

Thermodynamics of Electrolyte Systems of Industry

229

17. M. C. Iliuta, K. Thomsen and P. Rasmussen, A.Z. Ch.E J., 2002,48,2664. 18. S. G. Christensen and K. Thomsen, Ind. Eng. Chem. Res., 2003,42,4260. 19. J. A. Myers, S. I. Sandler, R. H. Wood, Znd. Eng. Chem. Res., 2002, 41,3282. 20. K. S. Pitzer, Activity Coeficients in Electrolyte Solutions, CRC Press, Boca Raton, FL, 1991,33431. 21. P. Wang, A. Anderko and R. D. Young, Fluid Phase Equilibr. 2002,203, 141. 22. K. Thomsen, Aqueous electrolytes: model parameters and process simulation, Ph.D. Thesis, Department of Chemical Engineering, Technical University of Denmark, 1997. 23. H. Zerres and J. M. Prausnitz, A.Z.Ch.E. J., 1994, 40,676. 24. B. Fitch, Ind. Eng. Chem., 1970, 62,6. 25. K. Thomsen, P. Rasmussen and R. Gani, Chem. Eng. Sci., 1998,53, 1551. 26. L. A. Cisternas and R. E. Swaney, Znd. Eng. Chem. Res., 1998,37, 2761. 27. Janecke, E., 2. Anorg. Chem., 1906,51, 132. 28. D. 0. Masson, Philos. Mag. and J. Sci., 1929, 8,218. 29. 0. Redlich and P. Rosenfeld, 2. physikal. Chem. Abt. A., 1931, 155,65. 30. 0. Sohnel and P, Novotny, in Densities of Aqueous Solutions of Inorganic Substances, Physical Sciences data, vol22, Elsevier, Amsterdam, 1985. 3 1. H. D. B. Jenkins and Y. Marcus, Chem. Rev., 1995,95,2695. 32. H . Falkenhagen and M. Dole, Phys. Z., 1929, 30,611.

CHAPTER 20

Thermodynamics of Crystallization AMYN S. TEJA AND RONALD W. ROUSSEAU

1 Introduction Crystallization generally involves the separation of a solute from a solution (mother liquor) via the creation of a second (solid) phase. The solid phase may be a pure compound, a nonstoichiometric compound, a “mixed crystal” phase, or a solid solution. The solid may be thermodynamically stable, or metastable due to slow rates of change to more stable forms. Metastable solid phases are quite common, and can result from surface heterogeneities as well as slow kinetics. The driving force for crystallization is supersaturation, which is defined as the difference between the chemical potential of the solute in the solution under the prevailing conditions, and that in the solution at equilibrium. Although a state of equilibrium may not be achieved in solid-liquid systems because of kinetic considerations, thermodynamics plays a key role in determining the driving force for the crystallization process via the determination of supersaturation. Phase diagrams provide information about the equilibrium characteristics of a particular system and are essential to an understanding of the crystallization process, The focus of what follows is therefore on phase diagrams of solid-liquid systems, and the representation of the solid-liquid equilibrium curves by means of thermodynamic relationships.

2 Pure Component Phase Behavior Phase diagrams of mixtures terminate at pure-component freezing points where the pure solid crystallizes. A proper understanding of the solid-liquid behavior of multicomponent systems therefore requires a description of the freezing points of the pure components. A typical pressure-temperature phase diagram for a single-component system is shown in Figure l(a). There are three single-phase regions on this diagram, denoted by S (solid), L (liquid), and G (gas). Equilibrium between any two phases is represented by curves such as AD, BD, and CD. AD represents the locus of

Thermodynamics of Crystallization

23 1

Figure 1 Phase diagrams of single-component systems

solid-vapor equilibria and is the sublimation curve; BD represents the locus of solid-liquid equilibria and is the melting/freezing curve; and finally, CD represents the locus of vapor-liquid equilibria and is the vapor pressure curve of the substance. The three loci meet at the triple point D, where the S, L, and G phases simultaneously coexist at equilibrium. Quite often, the sublimation curve is discontinuous, as shown in Figure l(b) where a system that exhibits two crystalline polymorphs (aand p ) is shown. AD' and D'D in the diagram represent the sublimation curves of the two solids, and the two curves intersect at a transition point D'. This is also a triple point because two solids and a vapor coexist simultaneously at this point. If the liquid is cooled slowly, the solid Sp crystallizes first and then changes to the solid S, on further cooling. Several such transitions are known to exist in some substances. It is also possible for one of the solids (Ss in Figure l(c)) to exist as a metastable polymorph. In Figure 1(c), A'D' represents the sublimation curve of the metastable solid S p . A hypothetical transition point between the two solids exists at D". However, D" is at a higher temperature than the melting point of the solid and, therefore, cannot exist in practice. The coexistence curves in Figure 1 can be represented by the Clapeyron equation

where (dP/dncoexrepresents the change in pressure with temperature along the coexistence curve, and AH, AV are the changes in enthalpy and volume due to the phase transition. Thus, for example, the change in sublimation pressure with temperature is related to the enthalpy of sublimation, and to the difference in volume between the vapor and solid phases. The Clapeyron equation can be used to determine the conditions for crystallization of pure substances either from the melt or via sublimation, using appropriate values of AH and AV.

3 Binary Solid-Liquid Equilibria Only the solid and liquid phases are of interest in most crystallization processes; therefore, the vapor phase will generally be ignored in the following discussion. Furthermore, pressure has little effect on the properties of condensed phases, and will also be ignored in most of what follows.

232

Chapter 20

3.1 Systems that Form Solid Solutions A system of two components can form solid solutions in which the atoms or molecules of one of the species occupy substitutional or interstitial sites in the (crystal) lattice of the other species, without substantially affecting the crystal structure of that species. Such a system is called isomorphous because of complete miscibility in both the liquid and solid phases. Figures 2(a) to 2(c) show the temperature-composition phase diagrams of such systems. Three different regions can be found in these diagrams: S, L, and S +L. The curve ACB separating the L and S +L regions is the liquidus or freezing point curve of the solution, and the curve ADB separating the S and S +L regions is the solidus or melting point curve. A homogeneous liquid phase is present at all temperatures and compositions above the liquidus, whereas a homogeneous solid phase is present at all temperatures and compositions below the solidus. When subjected to cooling, the two components are crystallized simultaneously and a homogeneous solid phase is obtained at equilibrium conditions below the melting point curve. Any point M in the region bounded by the liquidus and solidus curves represents a mixture of liquid C and solid solution D. Compositions and relative amounts of the phases may be determined by drawing horizontal tie-lines (since the temperature must be the same in both phases) and using the lever rule. A minimum can be exhibited by the liquidus and solidus curves as shown in Figure 2b or, in rare cases, a maximum as shown in Figure 2(c). However, the thermodynamic description is the same for all three types of phase behavior. The thermodynamic relationships for equilibrium between a binary solid solution and a binary liquid solution are given by

where zi,xi are the compositions, and ys, yk are the activity coefficients of component i in the solid and liquid phases, respectively. The fugacities of the pure solid

x or z

Figure 2 Phase diagrams of binary systems

233

Thermodynamics of Crystallization

fis and of the pure subcooled liquid fiL at the temperature and pressure of the system may be obtained using a thermodynamic path described by Prausnitz et aL1:

where Ahp is the enthalpy change upon melting of component i at its triple point temperature Tti,and ACpi is the difference between the heat capacities of the component i in the liquid and solid phases. The differences in the ACpi terms are often small in comparison with the Ahys term, and can generally be neglected.' The equation therefore simplifies to

If the solid undergoes polymorphic transitions, then one or more terms involving the transition enthalpies and temperatures must be added. Thus

where Ahqmsis the enthalpy change of transition of component i at the transition temperature Ti. Finally, if the pressure is large and its effects must be considered, then a term involving the volume change must also be included:

In Equation (6), AV is the difference in volume between the liquid and solid phases, and the integration is carried out from the triple point pressure to the pressure of the system. Equation (2) and any one of Equations ( 3 ) to (6) may be written for each of the two components and solved to obtain the freezing and melting curves shown in Figures 2(a) to 2(c), provided the pure-component properties (Mys, Ah,ms, ACpi,AV, Tti, and Ti) are available, and provided the activity coefficients f , can be estimated using methods such as UNIFAC.2

xL

3.2 Systems that Exhibit Eutectic Behavior Frequently, the solidus curve in Figure 2(b) is horizontal and gives rise to two regions in the phase diagram where a liquid solution is in equilibrium with a solid phase (Figure 2(d)). Note that in the system shown in Figure 2(d), there is negligible solubility of solid A (S,) in solid B (S,) and of S, in S,. Horizontal line CED in the figure represents the temperature at which solid mixtures of S, and S, melt or crystallize. AE and BE are liquidus curves, which bound regions AEC and BED, respectively. Solid is produced when processing (cooling or concentrating) moves a solution within one of these regions. In region SA+L, pure solid A is in equilibrium with liquid solutions whose compositions fall on the liquidus curve AE. Similarly, in region S,+L, pure solid B exists in equilibrium with liquid solutions on curve BE. Horizontal tie-lines again give the composition of the liquid in equilibrium with

234

Chapter 20

either S, or S,. Although both AE and BE are liquidus curves, AE is often called the “solubility curve”, and BE is called the “freezing point curve”, when A is considered to be the “solute” and B is considered to be the “solvent”. However, it should be clear from this discussion that the terms “solute” and “solvent” depend on which component is in excess in a solution. The liquidus curves meet at the eutectic point E, which represents the lowest temperature at which a liquid may coexist in equilibrium with one or more solid phases in the system. At point E, a liquid of composition xE exists in equilibrium with two pure solids S , and S,. The thermodynamic relationships along the liquidus curves are given by

Equation (7) is obtained from Equation ( 2 ) by noting that the solid phase is pure, and therefore the mole fraction and activity coefficient in the solid phase are both unity. The ratio of pure-component fugacities can be obtained from any one of Equations ((3) to (6)), and the activity coefficient in the liquid, must be estimated. The composition-temperature behavior along the liquidus curves may then be calculated. The eutectic point is found from the intersection of the two liquidus curves. If the solution is dilute (i.e. the liquidus curve exists over a narrow concentration range near the pure solvent axis), then the activity coefficient of the solute in the liquid solution can be replaced by its value at infinite dilution. Furthermore, a simple relationship may be employed for the temperature-dependence of the infinite-dilution activity coefficient as follows:

x,

Combining Equations (3), (7), and (8) leads to expressions of the type lnx = a

+ b/T

(9)

that are often quite satisfactory for correlating the solubility behavior of sparingly soluble solute^,^ as shown in Figure 3 for several amino acids. If the two components react with one another to form one or more compounds, then phase diagrams of the type shown in Figure 2(e) are obtained. Again, there is negligible solubility of S, in S,, or of S, in S,. However, the system incorporates a reaction A + B + AB. AB may be a solvate or a hydrate, for instance. In the range of compositions between C and E in Figure 2(e), pure solid A (S,) exists in equilibrium with a liquid solution and AE is the liquidus or solubility curve of AB in A. In the range of compositions between E and D’, solid AB (S,) exists in equilibrium with liquid solutions along the liquidus curve FE. FE is then the liquidus or freezing point curve for AB in the presence of A. AB can exist by itself only at the composition represented by the vertical line FF’. This vertical line is similar, in principle, to the vertical axes used to represent the two pure components. Pure AB melts at F, the congruent melting point of S,. Similarly, curves FE’ and BE’ are the solubility and freezing curves of AB and B, respectively. When system compositions are to the right of F’, C’E’D is the temperature below which solids B and AB crystallize as a eutectic

235

Thermodynamics of Crystallization

‘

-5.00 -fL-isoleucine

c -5.20

d

-5.40

-5.60 -5.80

0.0030 0.0031 0.0032 0.0033 0.0034 0.0035

1IT(K) Figure 3 Solubilities of a series of amino acids

mixture. The complete phase diagram may therefore be considered to be two simple eutectic diagrams (one for the A-AB system and one for the AB-B system) joined at the vertical line FF’. The liquidus curves can be obtained by solving the thermodynamic relationship already presented above, since pure solid A, pure solid B, or pure solid AB is in equilibrium with a liquid phase in each of the instances described. The fugacity of pure subcooled liquid AB may be obtained by considering the reaction equilibrium in the liquid phase, so that AG;

=

fib -RT In JAJB

The standard Gibbs energy may be obtained from the standard Gibbs energies of fusion and dissolution of solid AB. Partial miscibility in the solid phase is also possible, as shown in Figure 2 ( 0 . In this case, there are three single-phase regions (S,, Sp, and L), and three two-phase regions (S, + L, So+ L, and S, + S,) in the diagram. The S, phase is a solid solution rich in A, whereas the S, phase is a solid solution rich in B. The “solubility limit” of S, corresponds to the curve ACC’. It increases with temperature to a maximum at C (where all Sp has melted) and then decreases until the melting point of pure A is attained. Similarly, the curve BDD’ represents the “solubility limit” of Sp. S, and Sp coexist at all temperatures and compositions in the region S,+SB. AE and BE are the liquidus curves of S, and Sp and they meet at the eutectic point E. Phase equilibrium relationships in all regions except S, + Sphave already been presented in the previous discussion. In the region S, + S,, we have

xsazrf,as = f p ~ , B f , . ’ ~ or

(i = A, B)

( 1 1)

236

Chapter 20

These relationships may be solved at a given temperature for the phase compositions :z and z r , provided the activity coefficients can be estimated.

4 Ternary Solid-Liquid Equilibria 4.1 Systems that Form Solid Solutions Ternary systems of two solutes in a solvent can form solid solutions. In this case, phase diagrams of the type shown in Figure 4(a) are obtained. Points a and b on the diagram represent the solubilities of solute A and solute B in solvent C at a fixed C

e

/

0

0

0

0

XC

mole fraction

Figure 4 (a) Ternary systems that form solid solutions. (b) Ternary systems in which the three constituent binaries exhibit eutectic behavior. (c) Projection of liquidus curves in (b). (d) Ternary system in which two of the constituent binaries form eutectics. ( e ) Ternary system in which A and B combine to form AB

Thermodynamics of Crystallization

237 C

C

C

L

A

Figure 4 (continued)

B

238

Chapter 20

temperature (and pressure), and curve adb is an isotherm representing the solubility at a specific temperature. At point m,a liquid of composition d is in equilibrium with crystals of composition e. The broken lines are tie-lines at the given temperature (and pressure). The thermodynamics of ternary mixtures that form solid solutions may be obtained by writing Equation (2) for the two solutes A and B. Thus, for solute A:

The pure-component fugacities may again be obtained from enthalpies of fusion and triple point temperatures as described above. If Equation (3) is used for the fugacity ratio, then ZA

- r,” M p

-- -

-(T TtA

xA

RTtA

r,”

- 1)

Alternatively, in the absence of enthalpies of fusion and triple point temperatures, the solubility of pure A in solvent C may be used to obtain the fugacity ratio as described by Teja et d4Equation (14) then becomes

Similar expressions may be obtained for solute B. A relative-distribution ratio or “selectivity” may be defined as follows:

Note that the relative distribution of the solutes depends on nonidealities in both the liquid and solid phases, in addition to pure-component properties. Also, if one of the solutes is an impurity, then crystallization of the other solute from a solution may result in crystals containing significant levels of the impurity.

4.2 Eutectic Systems The phase diagram of a ternary system in which the three species do not form solid solutions with each other and the constituent binary systems form eutectics, is shown in Figure 4(b). The temperatures TA, TB, and Tc correspond to the melting points of A, B, and C, respectively. The vertical faces of the prism represent the temperature-concentration behavior of the three binaries. Note that the behavior of each binary system is that shown in Figure 2d. The solidus lines are not shown for the sake of clarity. Points EAB, EAC, and EB, are the eutectic points of the three binary systems, and point D is the ternary eutectic point. At D, a homogeneous liquid phase coexists in equilibrium with the three pure solids. The dashed lines from the binary eutectic points to D represent the lowering of the binary eutectic temperatures upon the addition of the third component. Projections of the dashed lines onto the triangular base are shown in Figure 4c, where the apexes of the triangle represent the pure solids at their melting points. The binary eutectic points are indicated on the three

Thermodynumics of Crystallization

239

sides of the triangle. The three liquidus curves DE,, DE,, and DEB, meet at the ternary eutectic point D. The temperature falls from the apexes and from the sides of the triangle toward the eutectic point D. A few isotherms in one region of the diagram are shown in Figure 4c (T,>T,>T,) in order to follow the cooling of a ternary solution. A ternary liquid solution of composition rn starts to deposit crystals of pure B at temperature T I .Upon further cooling, more B is deposited and the composition changes along the line BmD' until the liquidus curve is reached at D'. A second solid (pure C) now begins to crystallize along with B and the composition of the mother liquor moves along D'D toward D. At the ternary eutectic point D, the third solid (A) crystallizes along with B and C, and the system solidifies without further change in the composition of the residual liquid. The thermodynamic relationships for equilibrium between a pure solid and a (ternary) liquid solution have already been presented in equation (7). However, the activity coefficient is now a function of the mole fractions of the three components, as well as the temperature. Along the liquidus curve EBCD, pure B and pure C crystallize as the solution is cooled. We may therefore write

and

At any fixed temperature along the liquidus curve, we may simultaneously solve equations (17) and (18) for the unknown compositions xB,xc, and xA (= 1-xB-xC). The location of the ternary eutectic point (TD,xB,xc) requires the simultaneous solution of three equations such as equation (17) written for each of the three components. Many different types of phase diagrams are possible in a ternary system. Figure 4d shows a system in which there are two eutectics in systems A+C and B+C (a system of two salts and water, for example). The two solids (salts) are completely immiscible, and there are therefore four regions on the phase diagram, labeled L, L+SA, L+SB, and L+SA+SB. The ternary eutectic and the liquid curves are labeled as in previous figures. Figure 4e shows a system in which A and B react to form a new compound F, which is comprised of A and B in a specific stoichiometric ratio. Such a species is referred to as a double salt. This leads to six regions on the phase diagram labeled L, L+SA, L+SB, L+Sp L+SA+SF, and L+SF+S,. The fugacity of the compound F is obtained by considering its reaction equilibrium using relationships such as Equation (10). The other equilibrium equations are similar to those described above. Thus, many of the phase diagrams for ternary and higher mixtures can be calculated using the equations presented in this work.

5 Solid-Fluid Equilibria Crystallization from the gas phase, and in particular from a gas phase above the critical temperature and pressure of the solution, has attracted much interest in the literature for making particles of uniform size via a process known as supercritical

240

Chapter 20

P

Figure 5 The solubility of a nonvolatile solid in a supercritical fluid, as a function of pressure at two temperatures close to the critical temperature of the system

cry~tallization.~ Generally, a change in pressure is employed to bring about a large change in solubility, and hence in supersaturation, of a nonvolatile solute. Since pressure plays a significant role in these crystallizations, it is instructive to outline the thermodynamic relationships employed in the calculation of solid-supercritical fluid equilibria. Figure 5 shows a typical solubility vs. pressure curve for a nonvolatile solute such as naphthalene in a solvent such as carbon dioxide. As the pressure increases from a low value, the solubility first decreases and then dramatically increases before it attains an approximately constant value. The dramatic increase in solubility occurs near the critical point of carbon dioxide, so that a small change in pressure near the critical point can lead to large changes in solubility. Large supersaturations can therefore be generated by small changes in pressure near the critical point of the solvent. The thermodynamic relationships for a pure solid solute A (assuming that the gas does not dissolve in the solid) in equilibrium with a supercritical solution is given by

where $A is the fugacity coefficient of the solute in the supercritical solution, yA is its solubility, and P is the system pressure; fiSis the fugacity of the pure solid solute at the temperature and pressure of the system and may often be replaced by a product of its sublimation pressure and a Poynting correction for pressure. Thus,

The fugacity coefficient qAis generally calculated from an equation of state. However, many equations of state require a knowledge of the critical parameters of the solute, which may not always be available. Nevertheless, solubilities can be correlated, and sometimes extrapolated, using this approach. The addition of more solutes poses few problems from a thermodynamic viewpoint, as long as the appropriate solid-state fugacity is used in the calculations. This type of approach may also be used to study

24 1

Thermodynamics of Crystallization

the effect of co-solutes and co-solvent on the solubility behavior. Alternatively, an approach based on the dilute solution theory6 may be employed for this purpose.

6 Kinetics of Dissolution and Crystal Growth Crystal nucleation and growth are key factors in determining characteristics of crystalline products that may determine their usefulness or even the ability to separate them from their mother liquor. The kinetics of these phenomena and of dissolution are related to the difference between the values of the system variables and the Values of those variables at equilibrium. In crystallization, this difference is referred to as supersaturation and it is given by the expression

where p is the chemical potential of the crystallizing species at the prevailing system conditions, and p* is the equilibrium value at the system temperature and pressure. Common practice has resulted in the use of more easily expressed driving forces for the kinetic phenomena, often without a sound basis; among these are the following: concentration difference: Ac = c - c*

(21a)

temperature difference: AT = T* - T

(21b)

saturation or supersaturation ratio: S = c/c*

(2 1c)

relative supersaturation: CT = ( c - c*)/c*= S - 1

(21d)

where the asterisk (*) designates the value at equilibrium. The relative supersaturation has special utility because it approximates the rigorous definition of supersaturation in Equation (21), which is related to solute concentrations by the expression

AP YC RT =Iny*c* For ideal solutions, where both yand y* are approximately unity, or for conditions that lead to the approximation y= f ,

AP =In-;-c = l n S RT c

(23)

When the supersaturation ratio, S, is small (say less than l.l), it follows that

For this reason, then, kinetics of nucleation, growth, and dissolution are often expressed in terms of relative supersaturation. The actual relationships for these

242

Chapter 20

kinetic phenomena depend on the controlling mechanism and are beyond the scope of the present d i s c u s ~ i o nHowever, .~~~ the importance of the solubility is clear from the definition of supersaturation.

7 Summary This chapter outlines thermodynamic relationships and phase diagrams relevant to the crystallization of relatively simple molecules. In particular, electrolytes and mixed solvents have not been treated, although many features of their phase diagrams are similar to those described here. Specialized books and monographs should be consulted for further information on all t~pics.~-l*

References 1. J. M. Prausnitz, R. N. Lichtenthaler and E. G. Azavedo, Molecular Thermodynamics of Fluid Phase Equilibria, 3rd edn, Prentice-Hall, Upper Sadle River, NJ, 1999. 2. A. Fredenslund, J. Gmehling and P. Rasmussen, Vapour-Liquid Equilibria Using UNZFAC, Elsevier, Amsterdam, 1977. 3. J. C. Givand, A. S. Teja and R. W. Rousseau, A.Z.Ch.E. J., 2001,47,2705. 4. A. S. Teja, J. C. Givand and R. W. Rousseau, A.1.Ch.E. J., 2002,48, 2629. 5. A. S. Teja and T. Furuya, Encyclopedia of Separation Science, Vol 111, Academic Press, London, 2000. 6. J. Mendez-Santiago and A. S. Teja, Z & E C Res., 2000,39,4767. 7. J. W. Mullin. Crystallization, 4th edn, Butterworth-Heinemann,Oxford, 2001. 8. R. W. Rousseau, Kirk-Othmer Encyclopedia of Chemical Technology, 4th edn, Vol 7, Wiley, NJ, 1993. 9. W. C. Blasdale. Equilibria in Saturated Salt Solutions. ACS Monograph Series, Chemical Catalog Co., New York, 1927. 10. R. Haase and H. Schonert (translated by E. S. Halberstadt), Solid-liquid Equilibrium, Pergamon Press, London, 1969. 11. K. S . Pitzer (ed). Activity Coeficients in Electrolyte Solutions, 2nd edn, CRC Press, Boca Raton, FL, 1991. 12. G. T. Hefter and R. P. T. Tompkins (eds), The Experimental Determination of Solubilities, Wiley, Chichester, UK, 2003.

CHAPTER 21

Thermodynamics of Adsorption ALAN L. MYERS

1 Introduction The attachment of molecules to the surface of a solid by adsorption is a broad subject. This chapter is focused on the adsorption of gases in high-capacity solid adsorbents such as active carbon or zeolites.2These commercial adsorbents owe their enormous capacity to an extensive network of nanopores of various shapes (cylinders, slits) with specific volumes in the range from 100 to loo0 cm3 kg-'. Applications of adsorption exploit the ability of nanoporous materials to adsorb one component of a gas preferentially. For example, the preferential adsorption of nitrogen from air passed through an adsorption column packed with zeolite creates a product stream of nearly pure oxygen. Thermodynamics has the remarkable ability to connect seemingly unrelated properties. For example, the temperature coefficient of adsorption is directly proportional to the heat of immersion of the solid adsorbent in the gas. The most important application of thermodynamics to adsorption is the calculation of phase equilibrium between a gaseous mixture and a solid adsorbent. The basis for thermodynamic calculations is the adsorption isotherm, which gives the amount of gas adsorbed in the nanopores as a function of the external pressure. Adsorption isotherms are measured experimentally or calculated from theory using molecular simulation^.^ Potential functions are used to construct a detailed molecular model for atom-atom interactions and a distribution of point charges is used to reproduce the polarity of the solid material and the adsorbing molecules. Recently, ab initio quantum chemistry has been applied to the theoretical determination of these potentials, as discussed in another chapter of this book. Thermodynamics applies only to equilibrium adsorption isotherms. Equilibrium means that any point can be reached from either direction by raising (adsorption) or lowering (desorption) the pressure at constant temperature. If the desorption isotherm does not coincide with the adsorption isotherm, then equilibrium has not been achieved and the usual thermodynamic equations do not apply. The mismatch of adsorption and desorption, which is called hysteresis, does not occur in pores smaller than 2 nm but is observed' when the pores are large enough for the adsorbing molecules to condense to a liquid. For adsorption of supercritical gases or for

Chapter 21

244

adsorption of subcritical vapors in nanopores, most experiments and simulations yield equilibrium isotherms with no evidence of hysteresis. Molecular simulations yield absolute adsorption or the actual number of molecules in the nanopores. Experiments measure excess adsorption, which is the number of molecules in the nanopores in excess of the amount that would be present in the pore volume at the equilibrium density of the bulk gas. The difference between absolute and excess adsorption is negligible at the sub-atmospheric pressures of greatest interest. For supercritical gases adsorbed at high pressure (e.g. 100 bar), the difference between absolute and excess adsorption is too large to i g n ~ r e . ~

2 Adsorption Isotherm and Equation of State Whether the adsorption isotherm has been determined experimentally or theoretically from molecular simulation, the data points must be fitted with analytical equations for interpolation, extrapolation, and for the calculation of thermodynamic properties by numerical integration or differentiation. The adsorption isotherm for a pure gas is the relation between the specific amount adsorbed n (moles of gas per kilogram of solid) and P, the external pressure in the gas phase. For now, the discussion is restricted to adsorption of a pure gas; mixtures will be discussed later. A typical set of adsorption isotherms is shown in Figure 1. Most supercritical isotherms, including these, may be fit accurately by a modified virial e q ~ a t i o n . ~ r

3

2 ,-

h c (

g 1

0 0

40

80

120

PkPa

Figure 1 Adsorption isotherms of C,H, in NaX (zeolite structure FAU), where n is the-amount absorbed and P is the pressure. Points indicate experimental data.6Solid lines indicate Equation ( 1 )

Thermodynamics of Adsorption

245

where K is the Henry's constant, or the slope of adsorption isotherm dn/dP at the limit of zero pressure, rn is the saturation capacity (mol kg-I), and the Ciare virial coefficients, three of which usually suffice to fit the data within experimental error. If the virial coefficients are all zero, Equation (1) reduces to the well-known Langmuir equation.' Equation (1) has the form P(n) so that the inverse function n(P) is implicit. This slight inconvenience is offset by the fact that the implicit form can be integrated analytically for the thermodynamic functions (see below). The determination of an accurate value for Henry's constant ( K ) is essential for the calculation of thermodynamic properties and for mixture calculations. According to Equation (l), a plot of ln(P/n) as a function of n intersects the y-axis at 1/K. If the scatter of the data at low pressure is so large that an accurate value of K is impossible to determine, then Henry's constant should be measured at a higher temperature. It is difficult to obtain reliable values of Henry's constants from gravimetric measurements because the amount of gas adsorbed at low pressure is given by the difference of two weight measurements that differ by an infinitesimal amount. Volumetric measurements are preferred for measuring Henry's constants because the amount of gas adsorbed is determined by the large difference between the amount of gas dosed to the system and the amount of gas left in the system after adsorption. Interpolation of adsorption isotherms with respect to temperature is based on the thermodynamic e q ~ a t i o n : ~

where h is the differential enthalpy of adsorption, a negative quantity because adsorption is exothermic. The absolute value of h is called "isosteric heat". The partial differentiation is performed at constant n. In the rigorous equation, the pressure P is replaced by the fugacity of the gas. The differential enthalpy may be expressed as a polynomial: -

h(n) = D o + D,n -k D,n2

+ D3n3+

(3)

The constants Diare assumed to be independent of temperature. For wide variations in temperature over several hundred degrees Kelvin, this approximation can be corrected by introducing heat capacities. The integrated form of Equation (2) is

which provides the temperature dependence P(T) given a reference point P*(T") measured at the same value of n. Combination of Equations (1) and (4) yield an adsorption equation-of-state, which includes the temperature variable:

x exp[C,n

+ C2n2+ C3n3+ - -

*

J

(5)

where the constants K , m, and the Cjrefer to the reference isotherm at T * .The constants of Equation (5) for the adsorption isotherms in Figure 1 are: T*= 298.15 K, K= 1.9155

246

Chapter 21

0.1

10

1

100

PMa

Figure 2 Adsorption isotherms of C,H, in NaX (zeolite structure FAU), where n is the amount adsorbed and P is the pressure. Points indicate experimental data.6 Solid lines calculated from Equation (5)

mol kg-' kpa-', rn=2.9997 mol kg-', C,=0.841 kg mol-', C2= -0.0631 1 kg2 mol-2, C3= -0.009415 kg3m ~ l - Do= ~ , -39.5 W mol-', andD1=2.25 kJ kg molP2.The experimental data are compared with Eq. (5) in Figure 2, which includes interpolated and extrapolated isotherms. Logarithmic plots are useful for examining the accuracy of the equation-of-stateat low pressure. The calculation of enthalpy, free energy, and entropy from these constants is explained in the next section. Usually, the differential enthalpy is determined from Equation (2) using two or more adsorption isotherms. Alternatively, the differential enthalpy can be measured directly using a calorimeter.8In either case, a reference isotherm should be measured for the lowest temperature at which an accurate value of the Henry constant can be extracted. In the example shown in Figure 1, the reference isotherm is at 25 "C. For a particular gas and solid, the combination of a reference isotherm with the differential enthalpy provides complete thermodynamic information about the system.

3 Thermodynamic Functions The grand potential plays a central role in adsorption thermodynamics. The grand potential is defined by

where F is the Helmholtz free energy. The independent variables of the grand potential are temperature, volume, and chemical potential. These variables are precisely the ones needed to describe the amount adsorbed from a bulk gas at specified values of temperature and chemical potential in a solid adsorbent of fixed volume. For the same reason, molecule simulations of adsorption are conveniently performed in the grand canonical ensemble for which a=-kT In E,where Z is the grand canonical partition fun~tion.~

247

Thermodynamics of Adsorption

For adsorption of a pure gas, the grand potential is obtained from an isothermal integrati~n:~

R is expressed in J kg-' of solid adsorbent. Physically, the grand potential is the free energy change associated with isothermal immersion of fresh adsorbent in the bulk fluid. The absolute value of the grand potential is the minimum isothermal work necessary to clean the adsorbent. Since adsorption occurs spontaneously, the cleaning or regeneration of the adsorbent after it equilibrates with the feed stream is the main operating cost of an adsorptive separation process. Any extensive thermodynamic property of the system (free energy, enthalpy, entropy, or heat capacity) may be written as the sum of three terms for: 1. the value of the property for the adsorbate molecules at the state of the equilibrated bulk gas mixture at { T, P, yi}; 2. the value of the property for the clean solid adsorbent in vacuo at T; and 3. the change in the property associated with immersion of the clean adsorbent in the bulk gas at constant { T, P , y i } . The thermodynamic functions for items 1 and 2 are calculated using the standard equations for bulk gases and solids, respectively,1° so that the focus for adsorption thermodynamics is on item 3. It follows from Equations (5) and (7) that the grand potential (free energy of immersion) for each pure component is

2 3 + -C2n3 + -C3n4 + ... 3 4 +-

[f

- - -:](kDln2

2 3 + -D,n3 + -D3n4 + 3 4

- - a

The constants rn and Cirefer to the values for the reference isotherm at T";the constants Di refer to the polynomial for the differential enthalpy in Equation ( 3 ) . Note that the free energy is independent of the limiting value of the enthalpy at zero pressure, Do in Equation (3). The enthalpy of immersion (H) is the integral of the differential enthalpy (h):

H = /:%dn

(9)

The enthalpy of immersion, like R, has units of J kg-'. From Equations ( 3 ) and (9): H(n) = Don

1 1 1 + ... +Dln2 + -DD,n3 + -DD,n4 4 2 3

(10)

It is convenient to report the enthalpy of immersion as an integral molar enthalpy (J mol- ') using h= Hln: 1 h(n) = Do + -Dln 2

1 1 + -DD,n2 + -D3n3 + 3 4

Chapter 21

248

Given the free energy of immersion ( R ) and the enthalpy of immersion (H),the entropy of immersion is

s=

H-R

-

T

4 Mixtures The grand potential provides the standard state for the formation of adsorbed solutions from the pure components. Given the pressure ( P ) , temperature (0,and mole fraction of component 1 in the gas phase (yl) for a binary mixture, three equations are solved simultaneously7 to establish the amounts adsorbed (ny,ni) at the standard state:

Thus, the partial pressures on the left-hand side of Equations (13) and (14) are known and the three unknowns are ny, n;, and xl, where x2= 1 -xl. For mixtures containing more than two components, each additional component adds two equations and two unknowns (np and xi). In the rigorous form of Equations (3, (7), and (13)-( 1 9 , the pressure or partial pressure is replaced by the fugacity. Given the adsorbed-phase composition x 1 from the solution of Equations (13)-( 15), the selectivity of the adsorbent for component i relative to componentj is

The larger the selectivity, the easier the separation of component i from component j by adsorption. Zeolites with a selectivity as high as 10 for nitrogen relative to oxygen are used in pressure-swing adsorption processes” to produce oxygen from air. The specific amount of each component adsorbed for an ideal solution is given by ni=n,xi

where the total specific amount adsorbed from a mixture of gases is 1 -n,

-?$

(17)

(18)

In summary, the procedure for predicting the thermodynamic properties of an adsorbed mixture begins with the determination of the thermodynamic properties of each individual component as expressed by its equation of state, Equation (5). After fixing the independent variables { T, P , yi} for a system containing Nc components, the set of (2Nc- 1) Equations (13)-(15) is solved for the adsorbed-phase mole fractions xi and standard-state amounts adsorbed (np),with the constraint that x i x i = 1.

249

Thermodynamics of Adsorption

Knowledge of the standard states and the adsorbed-phase composition allows the calculation of the selectivity by Equations (16) and the amount of each species adsorbed by Equations (17) and (1 8). Finally, the entropy and enthalpy of immersion are given by the equations:

H; and S; are evaluated at the standard-state amount adsorbed (n;). It may seem at first glance that an entropy of mixing term is missing from Eq. (20), but S refers to the entropy of immersion of the solid in the gas mixture. The total entropy of the adsorbate mixture relative to its pure, perfect-gas reference state includes a separate term for mixing and compressing the adsorbate gas to its equilibrium state { T, P , y i }. The integral enthalpy H of the mixture divided by the total amount adsorbed is the integral molar enthalpy h, as in Equation (1 1) for adsorption of a single component.

5 Example The application of Equations (13)-(20) is illustrated for binary mixtures of ethylene (1) and ethane (2) adsorbed on NaX zeolite (faujasite). The constants for the singlegas adsorption equations of state5are given in Tables 1 and 2. The selectivity of NaX is a function of temperature, pressure, and the for ethylene relative to ethane composition of the gas. The selectivity at constant temperature (20°C) is shown in Figure 3. The selectivity at the limit of zero pressure is the ratio of Henry's constants (K,/K2=33.7). At constant mole fraction of ethylene in the gas, the selectivity decreases rapidly with increasing pressure. At constant pressure, the selectivity decreases with increasing mole fraction of ethylene in the gas. The selectivity at constant pressure and gas composition decreases with temperature, as shown in Figure 4. Decrease of the selectivity with temperature, pressure, and the mole fraction of the preferentially adsorbed species is typical behavior for binary adsorption.

Table 1 Constants of Eq. ( 1 ) for reference adsorption isotherms of gases in NaX zeolites at 293.15 K. Virial coeficients Ci in units of kg' mol-' Gas C2H, C2H6

K (mol kg-' kPa-') 5.2039 0.1545

m (mol kg-')

4.5341 3.8937

C, 0.385 -0.267

c 2

c.3

c 4

0.0075 -0.0499

0.0012 0.0192

0.0012 0.0

Table 2 Constants of Eq. (3)for diflerential enthalpy (isosteric heat) of adsorption of gases in NaX zeolite5 at 298.15 K. Mrial coeficients Di in units of W kgi mol- ( i + 1) Gas C2H4 C2H6

Do -41.836 -26.893

4 0.3215 -1.1719

D2 - 1.2203 0.0328

D3

D4

0.9452 -0.1195

-0.1576 0.0

250

Chapter 21

0

0.2

0.4

0.6

0.8

1

Y1

Figure 3 Selectivity (xIy2)/(x2y1) for adsorption of ethylene (1) relative to ethane (2) in NaX (zeolite structure FAU) at 20"C,plotted against y,, the mole fraction of C,H, in the gas. Isobars calculated from Equations (13)-(16)using the constants for pure gases in Table 1

0

20

60

40

80

100

TIT

Figure 4 Selectivity (xly2)/(xg,)for adsorption of ethylene (1)relative to ethane (2)in NaX (zeolite structure FAU) at 100 kPa and y,=O.l, plotted against the temperature. Calculated from Equations (13)-(16) using the constantsfor pure gases in Tables 1 and 2

The selectivities in Figures 3 and 4 were calculated from the single-gas isotherms using Equations (1 3) and (14), which are written for ideal adsorbed solutions (IAS) with activity coefficients ~ = 1 These . equations are rigorous at the limit of low pressure. At high pressure, mixtures adsorbed in nanopores display negative

25 1

Thermodynamics of Adsorption 5

4

7 3

3 4

2

;* 1

0 4

3 I

4

24 M

3 2 E \

I:

1

0

7

x

P=lOkPa

C2H6

2

I

24 w

“01

E. I:

a

P=lkPa

I

I

2

0.4

0.6

0.8

0

Y1

Figure 5 Individual and total isotherms at 20 “C for isobaric adsorption of mixtures of ethylene (1) and ethane (2) in NaX (zeolite structure FAU), where n is the amount adsorbed and y l is the mole fraction of ethylene in the gas. Dashed lines calculated from Equations (13)-(15), (17) and (18) using the constants for pure gases in Table 1. Solid lines indicate experimental datas

252

Chapter 21

Figure 6 Enthalpy for isobaric adsorption of mixtures of ethylene ( I ) and ethane (2) in NaX, where h is the integral enthalpy and x, is the mole fraction of ethylene in the nanopores. Dashed lines calculated from Equations (13)-(15) and (19) using constants for the pure gases in Tables 1 and 2. Solid lines indicate experimental data.’ At 1 kPa, the dashed and solid line coincide

(x<

deviations from Raoult’s law 1). These deviations are dominated by heterogeneity of the gas-solid energy and therefore cannot be estimated from the activity coefficients of the bulk fluids. The strongest deviations from ideality are observed for mixtures in zeolites such as NaX (faujasite), which has strong electric fields and electric field gradients in its nanopores that interact differently with quadrupolar (C,H,) and nonpolar molecules (C,H,). Mixtures adsorbed in materials with weak electric field gradients such as silicalite (MFI structure) or active carbon are more nearly ideal ( p l ) than zeolites like NaX, which contain exchangeable nonframework cations. Activity coefficients for nonideal mixtures have been r e p ~ r t e dThe . ~ error associated with the use of IAS theory is shown in Figure 5. The solid lines are the experimental data and the dashed lines were calculated from Equations (13)-( 18). The comparison of the IAS prediction with experimental data in Figure 5 raises the following question: is it possible to predict activity coefficients? Correlations of activity coefficients with single-gas adsorptive properties5 suggest that such predictions are possible, and reliable methods may be discovered in the future. The estimate of the integral enthalpy ( h )by Equation (19) is shown by the dashed lines in Figure 6. The solid lines are the experimental data determined by calorimetry.5 The error in the estimated enthalpy (dashed lines) increases with pressure but the largest error is 1.6%.The values at the two end points (yl=O and y,= 1) are the integral enthalpies for pure ethylene and ethane given by Equation (1 1).

Thermodynamics of Adsorption

253

6 Summary Equation (5) is an equation-of-state for the adsorption of a pure gas as a function of temperature and pressure. The constants of this equation are the Henry constant, the saturation capacity, and the virial coefficients at a reference temperature. The temperature variable is incorporated in Equation (5) by the virial coefficients for the differential enthalpy. This equation-of-state for adsorption of single gases provides an accurate basis for predicting the thermodynamic properties and phase equilibria for adsorption from gaseous mixtures.

References 1. D. M. Ruthven, Principles of Adsorption and Adsorption Processes, John Wiley & Sons, New York, 1984,7, 50, 56. 2. http://www.iza-structure.org/databases 3. D. Nicholson and N. G. Parsonage, Computer Simulation of the Statistical Mechanics of Adsorption, Academic Press, London, 1982 4. A. L. Myers, P. A. Monson, Adsorption in porous materials at high pressure: theory and experiment, Lungmuir, 2002,18, 10261-10273. 5. F. R. Siperstein and A. L. Myers, Mixed-gas adsorption, A.Z.Ch.E.J., 2001, 47, 1141-1 159. 6 . S. H. Hyun and R. P. Danner, J. Chem. Eng. Data, 1982,27, 196. 7. A. L. Myers, Thermodynamics of adsorption in porous materials, A.I.Ch.E. J., 2002, 48, 145-1 60. 8. J. A. Dunne, R. Mariwala, M. Rao, S. Sircar, R. J. Gorte and A. L. Myers, Calorimetric heats of adsorption and adsorption isotherms. 1. O,, N,, Ar,CO,, CH,, C&, and SF6 on silicalite, Lungmuir, 1996, 12, 5888-5895. 9. D. A. McQuarrie, Statistical Mechanics, Harper & Row, New York, 1976, p. 51. 10. J. M. Prausnitz, R. N. Lichtenthaler and E. G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd edn, Chapter 3, Prentice-Hall, Upper Saddle River, New Jersey, 1999. 11. D. M. Ruthven, S. Farooq and K. S. Knaebel, Pressure Swing Adsorption, John Wiley & Sons, New York, 1993.

CHAPTER 22

Mesoscopic Non-equilibrium Thermodynamics of Polymer Crystallization DAVID REGUERA AND J.M. RUB1

1 Non-Equilibrium Thermodynamics and the Mesoscopic Level of Description In 193 1, Onsager published two papers on simultaneous irreversible processes' in which he formulated the well-known reciprocal relations between phenomenological (Onsager) coefficients establishing in this way the basis for the formulation of a thermodynamical theory of irreversible processes, referred nowadays to as non-equilibrium thermodynamics.* The success of the theory is mainly due to the simplicity of its method, which runs parallel to that of equilibrium thermodynamics, to the many situations pertaining to the macroscopic world to which it applies, and to its solid connection with statistical mechanics and kinetic theory. Irreversible processes as for example heat conduction, mass diffusion and electrical phenomena, which constitute fundamental mechanisms of transport in bulk phases and at interfaces3 and are the basis of many industrial processes, can convincingly be explained under the framework of the theory. Non-equilibrium thermodynamics nowadays constitutes a powerful tool with many applications in physical, physicochemical and biological systems as well as in engineering processing.

1.1 Meso-Structures The scope of non-equilibrium thermodynamics is restricted to irreversible processes taking place in macroscopic systems, i.e. at length scales much larger than any molecular characteristic length. The continuum hypothesis, for which matter is assimilated to a continuum medium, invoked to set up a local description of the system, ignores its molecular nature. Although this approach may work out extremely well at long length scales, when reducing the observation scales focusing on regions

Mesoscopic Non-equilibrium Thermodynamics of Polymer Crystallization

25 5

whose energy is not drastically much larger than thermal energy, the molecular nature of the system becomes unavoidable manifest. The more experimental accessibility to shorter and shorter scales and the interest in elucidating the intimate nature of materials and the elementary mechanisms governing irreversible processes make the formulation of a non-equilibrium thermodynamics valid at the mesoscopic level a task of primary importance. Many interesting irreversible processes in nature as for example nucleation, growth and agglomeration phenomena, which constitute basic mechanisms in industrial applications in metallurgy, food industry and material science, occur at the mesoscale. In these processes, the starring entities (such as the emerging crystals, droplets or aggregates) are mesostructures, which are neither particles nor macroscopic objects. They undergo genuine meso-scale processes as assembling and impingement and they are affected by the fluctuations of the thermal bath in which they are embedded and by an external driving as is the case in industrial processing.

1.2 Mesoscopic Non-Equilibrium Thermodynamics (MNET) It has been shown that the scheme of non-equilibrium thermodynamics can be extended to describe irreversible processes occurring at the mes~-scale.~*~ This generalization entails the formulation of the local equilibrium hypothesis, the comerstone of non-equilibrium thermodynamics, in a more general context. The local state of the system has to incorporate the variables that have not yet relaxed toward equilibrium at mesoscopic scales. Starting from the corresponding Gibbs equation that incorporates those degrees of freedom, one then obtains the entropy production and from it the expression for the currents in terms of the thermodynamic forces. As a result, in general, one obtains expressions for the currents in the space of the mesoscopic degrees of freedom of the system and the kinetic equations describing the evolution of the probability density in that space. In this paper, we will illustrate the application of mesoscopic non-equilibrium thermodynamics to the dynamics of activated processes. Together with transport process, they constitute the two basic mechanisms for which a system can be removed from an equilibrium state. Although both processes act in the sense of yielding non-equilibrium situations, their natures are markedly different. Whereas transport processes always exhibit a linear regime that manifests at small values of the driving force, activated processes are genuinely non-linear since a certain amount of energy, not necessarily small, has to be given to the system in order that it may overcome the free energy barrier to change its state. Non-equilibrium thermodynamics cannot account for the kinetic of such non-linear processes. For a chemical reaction, for example, non-equilibrium thermodynamics formulates a linear relationship between the reaction rate and the affinity, which constitute only the first term in the development of the law of mass action. To obtain the full law, one has to take into account not only the initial and final states of the kinetics but all intermediate configurations, i.e. one has to introduce a mesoscopic degree of freedom accounting for the different molecular configurations. When this is done, in the framework of mesoscopic non-equilibrium thermodynamics, one arrives at the law of mass action governing the kinetics for arbitrary values of the thermodynamic

256

Chapter 22

force. To illustrate the use of the theory, we will focus, in particular, on polymer crystallization, a basic mechanism in many industrial applications.

2 Polymer Crystallization Crystallization of polymers is a process for which structural reorganization of a melted polymer leads to the emergence of an ordered phase. The transition from melted to crystalline polymer first occurs at the melting temperature. The crystalline phase manifests in the temperature range comprised between the melting temperature and the glass transition temperature. For temperatures higher than the melting temperature, thermal agitation impedes the formation of an ordered phase, whereas for temperatures smaller than the glass transition temperature, diffusion of the monomers to form a crystal is prevented. We can schematically distinguish between three stages in the entire crystallization process: nucleation, growth and secondary crystallization. At the earliest stages, crystals are formed from the melted phase via a nucleation process. The presence of density fluctuations may eventually create small aggregates of polymers having the same properties as the crystalline phase. The small crystals are continuously being created and destroyed by fluctuations. The appearance of a small crystal involves competition between two effects. One is the decrease of energy due to the fact that, below the melting temperature, the chemical potential of the crystal phase is lower than that of the melt, which competes with the energetic cost due to the surface tension associated with the creation of an interface. For clusters of small size, surface effects are dominant. As a consequence, their growth is energetically unfavorable and the small crystals tend to spontaneously dissolve. There exist, however, a critical size beyond which the volume effects become dominant, and the growth of the cluster is favored by a global reduction of the energy. In the second stage, or growth, once formed, crystals grow freely until they progressively begin to compete to fill the whole space. They may eventually hit each other, and therefore the growth is stopped at the contact surface. This phenomenon, called impingement, is relevant at later stages of the process and determines the final morphology of the system. In the final stage of crystallization, also known as secondary crystallization, amorphous non-crystallized matter that remains trapped among clusters can eventually selforganize, joining to the main crystal structure and therefore increasing the degree of crystallinity of the sample. In quiescent homogeneous melts, crystallization proceeds by the formation of isotropic quasi-spherical clusters known as spherulite. Figure 1 illustrates the final structure of a crystallized polymer (PHB), showing clearly the different spherulites originating from nucleation, whose growth competing to fill the space was stopped by impingement. The usual theoretical framework to describe this process is the Avrami- Kolmogoroff based on purely stochastic grounds, and aimed at describing the space filling in the crystallization kinetics. The underlying physical ingredients of this model are the nucleation and growth rates. This highlights the importance of providing an accurate theory to obtain nucleation and growth rates that ideally may incorporate the influences of the host melted phase and also the peculiarities of the processing of the melt.

Mesoscopic Non-equilibrium Thermodynamics of Polymer Crystallization

257

Figure 1 Crystal structure of PHB grown at 5.5" Cfrom Ref: 10)

3 The Mesoscopic Non-EquilibriumThermodynamics Approach to Polymer Crystallization Nucleation and growth are the most important processes in determining the final morphology of the crystallized polymer and consequently the properties of the material. Both processes can be quite conveniently described within the framework of mesoscopic non-equilibrium thermodynamics. Crystallization of polymers is an activated process that may be influenced by the presence of mechanical stresses and external gradients. Clusters are formed at the expense of the metastable phase due to the presence of thermal fluctuations. The process can be modeled by a particle crossing a free energy barrier, whose height depends on the interactions and the external drivings, upon the intervention of

Figure 2 Schematic representation of crystallization as a difSusion process over a free energy barrier

258

Chapter 22

fluctuations. The states at both sides of the barrier correspond to the metastable melting phase and to the crystallized phase (see Figure 2).

3.1 Mesoscopic Non-Equilibrium Thermodynamics of Activated Processes Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallizationproceeds by the formation of spherical clusters, the process can be characterized by a coordinate which may represent for instance the number of monomers in a cluster, its radius or even a globalorder parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. Since activation kinetics is affected by the presence of fluctuations in the metastable phase, we will provide a description of the system in terms of the probability distribution function P(xt). Our task will be then to derive the expression of the activation current, a quantity accessible to experimentsin many instances. To this end, our starting point is the expression for the Gibbs equation accounting for entropy variations due to the underlying diffusion process of the probability density

x

(1)

where P ( x t ) is the probability density, p(xt) is a non-equilibrium chemical potential defined in yspace and Tis the temperature of the system assumed to be constant. In the absence of impingement effects, the chemical potential is given through the expression

Here, H(33is the free energy barrier and k is Boltzmann’s constant. The presence of impingement effects can be taken into account by introducing an activity coefficient in the chemical potential as done for non-ideal mixtures. The evolution of the probability density is governed by the continuity equation

where J is the activation current defined in yspace, which has to be determined. Its value can be obtained from the entropy production CJ = dS/dt, which follows from the Gibss equation, by taking the time derivative and then using the continuity equation. Due to the fact that the definition domain of yis bounded, the current has to disappear at the initial and final states of the process. After performing a partial integration, one then arrives at

Mesoscopic Non-equilibrium Thermodynamicsof Polymer Crystallization

259

The integral consists of contributions of flux-force pairs and then has the habitual form of the entropy production in non-equilibrium thermodynamics. Also assuming linear dependence between fluxes and forces, one establishes the relations

where L is the corresponding Onsager coefficient, which may in general depend on the internal coordinate. To derive this equation, locality in the internal space has also been taken into account, for which only currents and forces at the same value of the coordinate become coupled. To obtain the expression of the current, we need to elaborate eq (5) more. Using Equation (2), it can be expressed as

where b=kUP is a diffusion coefficient assumed to be constant in first approximation. Activation processes are usually fast enough to favor the setup of a quasistationary state with a nearly constant current. Integration in yspace then leads to

which expresses the constant current J in terms of forward and backward contributions corresponding to the two exponentials and 1 is a kinetic coefficient that can be identified from the integration procedure. We have then shown that application of the scheme of non-equilibrium thermodynamics to a mesoscopic description of the system enables us to obtain the non-linear kinetic laws of activated processes. The derivation we have presented is general and therefore valid for polymer crystallization. In the next subsection, we will discuss the way in which MNET can also account for the peculiar characteristics of these processes.

3.2 Polymer Crystallization Crystallization of polymers takes place usually under inhomogeneous conditions as those related to the presence of temperature gradients or mechanical processing of the melt, such as extrusion, shearing or injection, which largely influence the kinetics. The formalism introduced in the previous subsections is able to incorporate the effect of these influences in the crystallization kinetics, thus providing a more realistic modeling of the process, which is mandatory for practical and industrial purposes. Due to the strong foundations of our mesoscopic formalism in the roots of standard non-equilibrium thermodynamics, it is easy to incorporate the influence of other transport processes (like heat conduction or diffusion) into the description of crystallization. In addition, our framework naturally accounts for the couplings between all these different influences. Contrary to the case of solidification of simple substances, crystallization of polymers occurs at a wide range of temperatures. Moreover, the low thermal conductivites and large heat capacities, characteristic of many polymer substances, imply that, even the latent heat naturally released in the phase transformation sets up important temperature gradients in the melt, able to alter the crystallization process significantly.

Chapter 22

260

Within the framework outlined in the previous section, the coupling between thermal effects, diffusion and crystallization can be easily worked out in a simple and unified framework, as described in ref. 8. Crystallization in a sheared melt8 can also be described by the formalism introduced in the previous subsection in which the internal coordinate represents the coordinates of the center of mass of the crystallites and their number of monomers. The main influence of the flow on crystallization kinetics is to modify the free energy barrier, which is given by

Q(n,u) = +(n)

1

+ -rn(n) 2

(u-vo)2

(8)

+

where is the free energy of formation of a cluster with n monomers at rest, rn its mass, u its velocity and vo the shear flow velocity. Proceeding along the lines indicated previously, one obtains two significant results. First, the diffusion coefficient of the crystallites in the direction of the velocity gradient is modified by the presence of the shear.9 Its new expression is given by

where Do is the diffusion coefficient in the absence of flow. This expression clearly shows that the correction depends on the shear rate p, the pressure p and the viscosity 77 through the dimensionless factor @/p. In practical applications, crystallization occurs at very large values of the shear rate, which originates through the injection process of the melt; moreover, the viscosity of the supercooled melt is also very large. This implies a significant correction to the diffusion mechanism of the crystallites due to the presence of an external flow that originates as a result of the presence of driving forces in the processing of the melt. The second important result is related to the change in the crystallization rate due to the shear flow. Since the rate of addition of polymer to a spherulite is roughly proportional to the diffusivity of the molecules, variations of the diffusion coefficient

Figure 3 Polarizing optical micrographs of poly(ethy1ene terephthalate) (PET) crystallized at 240°C in the absence (A) and in the presence (B) of a shearing. As a consequence of the shearing, nucleation becomes increasingly profuse, and the shape of the spherulites becomes elliptical. (From re$ 1 I )

Mesoscopic Non-equilibrium Thermodynamicsof Polymer Crystallization

26 1

affect the value of the forward current in Equation (7). Moreover, the presence of the flow destroys the isotropy of the system and leads to a distinction between growth rates (and diffusion) in different directions. The growth then proceeds by the formation of non-spherical crystallites, as shown in Figure 3.

4 Conclusions We have shown that mesoscopic non-equilibrium thermodynamics satisfactorily describes the dynamics of activated processes in general and that of polymer crystallization in particular. Identification of the different mesoscopic configurations of the system, when it irreversibly proceeds from the initial to the final phases, through a set of internal coordinates, and application of the scheme of non-equilibrium thermodynamics enable us to derive the non-linear kinetic laws governing the behavior of the system. This description has to be compared with that proposed by non-equilibrium thermodynamics in terms of only two states, corresponding to the melted and crystallized phases in the example we are discussing, from which only one may account for the linear domain, when the chemical potentials at the wells are not very different. This feature imposes serious limitations in the application of NET to activation processes since that condition is rarely encountered in experimental situations and has therefore restricted its use to only transport processes. The mesoscopic version of non-equilibrium thermodynamics, on the contrary, circumvents the difficulty offering a promising general scenario useful in the characterization of the wide class of activated processes, which appear frequently in systems outside equilibrium of different nature. We have applied the MNET theory to a particular activated process: polymer crystallization, in which the initial and final states of the system correspond to the melted and crystallized phases, respectively. The processing of the melt is carried out under out-of-equilibrium conditions due to the presence of driving forces and gradients usually of significant strength. Our conclusion is that MNET can also account for the kinetics in these more extreme situations.

References 1. L. Onsager. Phys. Rev., 1931,37,405;Phys. Rev., 1931,38,2265. 2. S . R. de Groot and P. Mazur, Non-equilibrium Thermodynamics,Dover, New York, 1984. 3. B. Hafskjold and S . Kjelstrup, J. Stat. Phys., 1995, 78, 463 ;A. Rosjorde, D.W. Fossmo, S. Kjelstrup, D. Bedeaux and B. Hafskjold, J. Colloid Znter$ Sci., 240, 355. 4. J. M. G. Vilar and J. M. Rubi, Proc. Nut. Acad. Sci.,2001, 98, 11081. 5. D. Reguera and J.M. Rubi, Phys. Rev. E 2001,64,061106. 6. M. Avrami, J. Chem. Phys. 1939,7, 1103; ibid 1940,8, 212; ibid 1941,9, 177. 7. A. N. Kolmogorov, Bull. Acad. Sci. USSR Mat. Ser., 1937, 1, 355. 8. D. Reguera, J.M. Rubi and L.L. Bonilla, Chapter 3 in Mathematical Modelling for Polymer Processing, V. Capasso (ed), Springer-Verlag, Berlin, 2003. 9. I. Santamm’a-Holek, D. Reguera and J.M. Rubi, Phys. Rev. E, 2001,63, 05 1106. 10. J. K. Hobbs, D. R. Binger, A. Keller and P. J. Barham, J. Polymer Sci., 2000, 38, 1575. 11. H. S. Myung, W. J. Yoon, E. S. Yoo, B. C. Kim and S. S. Im, J. Appl. Poly. Sci., 2001,80, 2640.

CHAPTER 23

Applied Thermodynamics for Petroleum Fluids in the Refining Industry DERESH RAMJUGERNATH AND RAJ SHARMA

1 Introduction Crude petroleum has been explored, extracted and processed for thousands of years.’ The modern petroleum industry as we know it, however, is just over a hundred years old. The processing of crude oil into various products, by-products, charge-stocks for petroleum-based finished products and petrochemicals has been determined by the ever-changing demand of the product slate and new products or processes. Historically, the petroleum industry is, perhaps, the single industry having had the greatest impact on the development of Chemical Engineering as a discipline. It is definitely a case where technology has preceded engineering science! The use and application of ‘thermodynamics of petroleum fluids’ encompasses all facets of the petroleum industry, ranging from production and refining of crude oil to processing of petrochemicals. This chapter is limited to a review of the most widely used thermodynamic methods in the petroleum refining industry, together with a discussion on a few recent developments. The chapter provides a single source of easily accessible relevant information. Perhaps the most important thermophysical properties required in petroleum refining for rating as well as process and equipment design are enthalpy and vapor-liquid equilibria (fugacities). Enthalpies and fugacities often provide sufficient information to calculate mass and energy balances across most unit operations in a refinery. Since crude oil is an undefined, wide-boiling mixture of many different types of hydrocarbons ranging in carbon numbers extending to 70 or beyond, conventional thermodynamic methods used for property prediction of pure components or well-defined mixtures are no longer applicable. The several direct products of crude petroleum such as gasoline, kerosene, diesel and others are also defined by their boiling point range cuts and the conventional methods cannot be used. Special

Applied Thermodynamics for Petroleum Fluids in the Re$ning Industry

263

methods and correlations have been developed over the past several decades to predict these, and other, properties of petroleum crude and its fractions. Most of these correlations or methods rely on bulk property characterization of crude oil and its cuts, such as specific gravity (SG or "API), various average boiling points, etc.

2 Composition of Petroleum In spite of different sources and origins of crude oils across the world, surprisingly the elementary chemical composition is relatively uniform (Table 1), even though the physical characteristics (specific gravity, boiling ranges, pour points and the like) vary ~ i d e l y . ~The 9 ~ physical characteristics are, however, generally affected by the small differences in composition, which also, consequently, have an impact on the value and processing of the crude oil. Crude petroleum is an undefined mixture of several hydrocarbons (with carbon numbers up to 70 or more) consisting of mainly paraffinic (including their isomers), naphthenic and aromatic-type molecules. The small quantities of non-hydrocarbon elements such as sulfur, nitrogen, vanadium, nickel, chromium and oxygen present in the crude oil are usually present as part of complex structures involving hydrocarbon^.^?^ While olefinic hydrocarbons are generally not present in the whole crude, various fractions of petroleum may contain a small percentage as a result of some dehydrogenation of paraffins and naphthenes that could occur during downstream processing.

3 Characterization of Petroleum and its Fractions 3.1 Measured Data Historically, and to date, perhaps the two most important bulk physical properties measured in the refining industry to characterize petroleum and its fractions are the distillation datdcurve [True Boiling Point (TBP) - US Bureau of Mines m e t h ~ dor, ;~ ASTM D 866] and the mid-percent gravity curve in terms of "API. The distillation datdcurve gives the boiling point profile and the mid-percent gravity curve as a function of liquid volume percent distilled (Figure 1). The functional relationship between "API and specific gravity (SG) is given as 141.5 OAP1

= SG6%00~ -

131.5

The distillation data are generally reported to be 572°F (-573 K) at 40 mm Hg (-5.33 kPa) U.S. Bureau of Mines method5), which corresponds to about 790°F

Table 1 Typical Elementary Composition of Petroleum Element Carbon Hydrogen Nitrogen Sulfur

Average percent

- 85 - 12 - 0.1

up to 5%

264

Chapter 23

1000 -

-- 80

3 800-

i$

-- 60

E

600-

c

i 400-

-- 40

200 -

-- 20

0

I

I

I

0

I

Figure 1

:

1111 1088

901

8.8 A T

711

u

Mfi

R E

50°

400

300

OM

0.05 1.2 0.5 1 2

5

10 21 30 40 50 60 70 811 90 LIQUID VOLUME PERGNT DISTILLED

95

91 99

99.899.9

99.99

Figure 2 Illustration of the use of probability graph paper - crude distillation data (ternperature vs liquid volume percent distilled)

(-694 K) at atmospheric pressure (760 mmHg; -101.32 kPa). However, the end points of the whole crude ordinarily extend beyond 1000"F (-81 1 K). Hence, often, the distillation data (temperature as a function of percentage distilled) are plotted on a probability graph paper and a 'best-estimate' straight line drawn extrapolating to about 1200 OF (-922 K) (Figure 2). The data points are then picked off from this line to obtain a smooth TBP curve over the entire range as in Figure 1. These two bulk properties (TBP data/ "API) are then used to calculate other 'constants' such as molecular weight (MW), the pseudo-critical temperature (T,) and pressure (P,),respectively, and the pseudo-acentric factor (a). The other properties generally measured are the kinematic viscosities at 100°F (-311 K) and 200°F (-366 K), respectively, and the Reid Vapor Pressure (RVP) (mainly for the gasoline range cut, defined as the vapor pressure exerted by the cut at 100 O F (-31 1 K) ). All of the above-measured properties and the calculated constants are generally

Applied Thermodynamics for Petroleum Fluids in the Refining Industry

265

Distillation OAPT

Estimate

Other Properties

Mass and Energy Balances

Figure!3 Hierarchy of property prediction for 'design'

adequate to completely characterize 'petroleum' and enable estimation of various thermo-physical properties. Borrowing from the concept of the onion diagram, the hierarchy of thermodynamic property prediction in the refining industry for design is perhaps best illustrated by Figure 3.

3.2 Calculated Parameters From the specific gravity (SG/"API) and the distillation curve/data, the mean average boiling point (MeABP) and the Watson characterization factor (K,) are calculated. The MeABP is the arithmetic average of the molal and cubic average boiling points and is mainly used for correlating hydrogen content, molecular weight (MW), pseudo-critical properties (T,, P,), pseudo-acentric factor (m), K,, and the like. For a petroleum fluid, MeABP may be estimated directly from the volume average boiling point (VABP) by subtracting a differential expressed as a function of VABP and the slope of the TBP curve as shown in Figure 4. Detailed correlations/curves are presented by M a ~ w e l l . ~ The Watson characterization factor (K,) is calculated as follows:

The Kw for crude oil and its fractions generally range between 10 (highly naphthenic crude) and 13 (highly paraffinic crude). For highly aromatic compounds to highly paraffinic compounds, the value of Kw usually ranges between 10 and 15, respectively, and therefore, can be said to be a qualitative measure of either the aromaticity or the paraffinicity of a crude oil. In addition to MeABP and Kw,the pseudo-critical properties (T,, P,), MW and the pseudo-acentric factor (m) are also required for estimating various thermo-physical properties of crude and its fractions. Several correlations/methods exist for calculating

266

Chapter 23

I

--

MeAB= VABPa'DIFFERENTIAL' + +90% (DISTILLEDTEMPS) VABP-096

0

2

4

6

8

10

12

Figure 4 Schematic of correlation between mean average boiling point and volumetric average boiling point (Maxwell)

Tc,Pc, w and MW, in terms of MeABP and SG (or K,.,).Some of the most widely used methods for estimating these parameters are listed in Table 2. Most of these methods use a simple 2-parameter correlation of the form

8 = f (MeABP,SG)

(3)

where 8is the property (Tc,Pc, w, MW) to be predicted.

Table 2 Some Widely Used Parameter Estimation Methods for Characterization of Petroleum and its Fractions Parameter Molecular weight, MW

Critical temperature, T,

Methods Hariu-Sage8 Kesler-Lee9 Riazi-Daubert Riazi-Daubert Kesler-Lee9 Riazi-Daubert lo l-bVU12

Critical pressure, P,

Critical volume, V,

Riazi-Daubert Kesler-Lee9 Riazi-Daubert l o Twu'* Riazi-Daubed Riedel' %U'2

Acentric factor, o

Kesler-Lee9

267

Applied Thermodynamicsfor Petroleum Fluids in the Refining Industry

i-

f

Liquid Vol % Distilled

Figure 5 Illustration of pseudo-component ‘characterization’ of petroleum crude into ‘narrow’ boiling range ‘compounds’.A similar approach is used for estimating spec@ gravities of ‘pseudo-components’

3.3 Pseudo-Components Characterization Petroleum crude, in recent times, is also on occasion defined in terms of pseudocomponents by breaking up the overall distillation curve (Figure 1) into several components as in Figure 5. These pseudo-components are not single components, but mixtures boiling over narrower temperature ranges as compared to the whole petroleum or its products. These pseudo-components are again characterized in terms of an average boiling point and SG for estimating other properties using the methods of Table 2 as outlined in Calculated Parameters above. Since the crude oil is now represented in terms of a defined mixture, the well-known multicomponent methods for rating and design of process equipment become available. It is generally accepted that the pseudo-multicomponent approach improves accuracy in thermodynamic property predictions over the conventional petroleum correlations leading to better designs. However, it has been the author’s experience that perhaps the pseudo-component characterization of crude oil leads to more elegant methods becoming available for calculations, but, with relatively no improvements in accuracy even at the cost of a significant increase in computation time and effort. There are several underlying reasons for this, starting with the assumptions about mixing rules to predict petroleum mixture properties.

4 Thermo-Physical Property Prediction Methods 4.1 Enthalpy Enthalpy is one of the most important thermo-physical properties required for calculating heat loads in process design. The most common models used for estimating enthalpies of petroleum and its fractions in the refining industry are based on the corresponding-states approach. Methods for calculating the necessary critical properties

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268

and an acentric factor have been discussed in Calculated Parameters above. The American Petroleum Institute Technical Data Book - Petroleum Refining14 is the most widely used reference manual in petroleum refining and presents enthalpy correlations in terms of the two standard bulk properties measured in refineries - the MeABP and SG (and, in turn, K.J. It is important to mention here that since it is not possible to estimate the absolute enthalpy of a compound, a reference temperature is generally chosen. Ideally, this reference temperature is such that all the enthalpies under consideration are zero at this temperature. In the absolute sense, unfortunately, different compounds have different molecular motions (and hence different enthalpies) at a given temperature. Hence, the solution lies in choosing a temperature that is sufficiently low so that different compounds may be assumed to have the same enthalpy at the reference point.15 Kesler-Lee9 defined the reference state for enthalpies of petroleum fluids as saturated liquid at - 200 O F (- 144 K), and it is reasonable to assume that enthalpies of all the petroleum fractions at this low temperature will be zero. In general, the estimation of enthalpy (departure) can be represented in the following manner:

AH

I

Tb

T

=ref

Tb

=

Cp,dT + AHv + ICp,dT

+ AHpc

(4)

where Cpland Cpvare the liquid and vapour heat capacities, respectively, AH,, is the latent heat of vaporization and AHx is the pressure correction term to compensate the effect of system pressure on enthalpy. Often, the effect of pressure on a liquid that exists below 1000 psia ( -6.9 MPa) pressure is neglected. The Kesler-Lee9 correlations for liquid and vapour phase heat capacities of petroleum fluids are used for estimating the respective enthalpies at temperatures of interest. The Lee-Kesler16 corresponding-states method is used for obtaining estimates of the heats of vaporization and for developing the saturation envelope enthalpies. This method uses the Curl and Pitzer17 approach and calculates various thermodynamic properties by representing the compressibility factor of any fluid in terms of a simple fluid and a reference fluid as follows:

where Z ( O ) and Z(') are given by

The enthalpy departure (and other thermodynamic properties) is then calculated using equations obtained from the compressibility factor via the usual thermodynamic relationships. In the case of reacting systems in petroleum refining, the required estimates of heats of reaction are obtained, as usual, by the difference between the enthalpy departures of the products and reactants. As an example, the endothermic heat of

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cracking reactions in a fluidized-catalytic-cracker (FCC) may be estimated as follows with excellent results.'* The heat of cracking reaction may be defined as

where m p r o d u c t s is the enthalpy of products, and AHrcactants is the enthalpy of reactants. As discussed above, in general, the enthalpy departure is given by Equation (4). Substituting Equation (4) into Equation (8) gives

where Cpl and Cpgare calculated using the Kesler-Lee9 correlation and AHvapis calculated using the generalized thermodynamic correlation reported by Lee and Kesler16 as before.

4.2 P-V-T Relationship Density or the pressure-volume-temperature (P-V-7') relationship is considered along with enthalpy and vapour-liquid equilibria as the three most essential thermophysical properties in the petroleum refining industry. The compressibility or density is commonly used in the petroleum industry to determine the volumetric properties of gases and liquids, information that is vital for transportation, safety and sale of petroleum fluids. The pressure-volume-temperature relationship is concisely represented by the compressibility factor, PV RT

Z=--

Depending on the conditions (temperature and pressure) and the complexity and accuracy required for the computations, generalized correlations for gases or analytical equations of state can be used to compute the gas compressibilities and the subsequent molar volumes. Generalized correlations for the compressibility factor, 2, as well as analytical expressions, based on the second virial coefficients, have been developed by Pitzer et aZ.19The correlation for 2 takes the form:

where Zo and 2' are functions of both reduced temperature (T,) and reduced pressure (P,). Of the Pitzer correlations, the one developed by Lee and Kesler16 has found greatest favour. The analytical equations of state used to compute the pressure-volume-temperature relationships are generally one of the following three classes, viz. virial equations of state, cubic equations of state and the Benedict-Webb-Rubin equation or modifications thereof.

270

Chapter 23

For the liquid phases, the compressibilities are most commonly computed using the generalized correlations for liquids and analytical equations of state. Reid et aL20 describe in detail the various correlations available for the computation of liquid phase densities for pure components, with the modified Rackett equation being the most popular. For mixtures, the use of an analytical equation of state is most preferable with cubic equations of state being the favourite type of equation of state.

4.3 Vapour-Liquid Equilibria (Including Vapour Pressure) The emphasis on vapour-liquid equilibria (including vapour pressure) is inherant in the petroleum industry due to the importance of distillation in separations. If separations by extraction are to be undertaken, then liquid-liquid equilibrium is equally important. Fugacities for thermodynamic equilibrium (flash calculations) are probably one of the most sought-after properties. This is because fugacities and enthalpies often provide sufficient information to calculate a mass and energy balance. The description of vapour-liquid equilibrium behaviour can be obtained from analytical equations and generalized correlations. The generalized correlations are generally for the equilibrium ratio, K, and the fugacity coefficients. For the analytical equations, there are two methods to compute the vapour-liquid equilibrium for systems. The equation of state method (also known as the direct or phi-phi method) uses an equation of state to describe both the liquid and vapour phase properties, whereas the activity coefficient method (also known as the gamma-phi approach) describes the liquid phase via an activity coefficient model and the vapour phase via an equation of state. Recently, there have also been modified equation of state methods that have an activity coefficient model built into the mixing rUles.21*22 These methods can be both correlative and predictive. The predictive methods rely on the use of group contribution methods for the activity coefficient models such as UNIFAC23and ASOG. Recently, there have also been attempts to develop group contribution methods for the equation of state method, e.g. PRSK.24For a detailed history on the development of equations of state and their applications, as well as activity coefficient models, refer to Wei and S a d u ~ , ~ ~ Sandler26and W a l a ~ . ~ ~ DePreister 28 produced nomographs for K-values of light hydrocarbons. The K-values represent the lightness of a compound or fraction, and are computed from the vapour and liquid equilibrium phase compositions as follows:

Yi K. = -

'

Xi

The K-values are correlated as functions of temperature and pressure. The DePreister correlations have approximate validity for mixtures of light hydrocarbons and other simple molecules. Vapour pressure is a key property in VLE calculations and is thus an important petroleum property. The most common method for prediction of vapour pressures is the corresponding states method. The method requires knowledge of the critical properties and the acentric factor. For petroleum fractions, the Ma~well-Bonnell~~ method is standard.

Applied Thermodynamics for Petroleum Fluids in the Rejhing Industry

27 1

The generation of pseudo-components for petroleum fluids has been discussed in Pseudo-Components Characterization. Once the various pseudo-components have been generated and classified, the conventional methods for multicomponent VLE apply and the methods discussed above for VLE computation can be used.

5 Codes and Data Sources The most widely used codes and data sources today are perhaps the following three:

1. API Technical Data Book - Petroleum Refining.14 2. Aspen Physical Property System - Physical Property Methods and Models.30 3. SUPERTRAPP - National Institute of Standards and Technology (NIST) Thermophysical Properties of Hydrocarbon Mixtures Databa~e.~' All the above three provide extensive data sources and simulators for estimating various thermodynamic properties of pure compounds, well-defined mixtures and petroleum fluids. The respective manuals describe in detail various methods used for estimating the properties and their limitations while at the same time providing the user with choices. The Aspen Physical Property System and SUPERTRAPP can also be used to estimate thermodynamic properties using the pseudo-component characterization of petroleum fluids. The reader is referred to the respective manualshimulators for details.

6 Future Trends Even though thermo-physical property prediction of petroleum fluids has come a long way since the early years of crude oil refining, the approach used by industry still mainly remains rooted in methods developed several decades ago. Most of these methods, as discussed earlier, rely on bulk property characterization deriving from them other parameters needed for property predictions. The current analyses of petroleum fluids are typically limited to identifying the front-end through to C,/C, (or C,) and then characterizing the C l / C i (or C;) fraction in terms of boiling points and specific gravities. In view of the complexity of composition, a detailed component-wise characterization of petroleum fluids is very difficult and is neither practical nor f e a ~ i b l e . ~However, ~ 3 ~ ~ with growing emphasis on improved designs, better characterization and property prediction methods are required. The pressure-volume-temperature (PVT) and density predictions are particularly sensitive to accurate characterizations of the heavier hydrocarbons. Recently, attempts have been made to better characterize the heavier hydrocarbons in petroleum fluids using a continuous model for C,+ fractions. The continuous model is based on the use of distribution functions for characterizing c,+ fraction^^^,^^, and very good agreement between predicted values and experimental data has been claimed.34The characterization parameters, including the pseudo-critical properties/acentric factors, obtained from the distribution function approach, are then used for predicting the phase-behaviour of petroleum f l ~ i d s .The ~ ~methods , ~ ~ developed are mostly applied

Chapter 23

272

to simple petroleum fluids with a high concentration of light hydrocarbons and a low concentration of C,’ fractions. Most recently, Fazlali et aZ.33have developed a new phase behaviour prediction method for complex petroleum fluids (consisting of high and concentrations of the C,’ fraction), using a combination of the Peng-R~binson~~ the Riazi-Mansoori3* equations of state, which requires a minimum of input characterization data such as an average value of molecular mass and the specific gravity of the C,+ fraction. Prediction of phase-behaviour of complex petroleum fluids using ‘continuous thermodynamics’ is a new area of research and will perhaps lead to important developments in the field. Further, molecular simulation and computational chemistry have evolved, and are evolving, into important tools for developing better characterization techniques where it is not possible to measure all data.32Even so, it is precisely the molecular complexity of petroleum fluids that seems to be an inhibiting factor in the use of these methods for developing better characterization methods. However, identification of important functional groups in petroleum fluids applying molecular simulation and/or computational chemistry for use with group contribution methods to predict thermo-physical properties may be an area for further research.

References 1. J. G. Speight, The Chemistry and Technology of Petroleum, 3rd edn, Marcel Dekker, New York, 1999. 2. J. H. Gary and G.E. Handwerk, Petroleum Refining - Technology and Economics, MarcelDekker, New York, 1975. 3. W. L. Nelson, Petroleum Refinery Engineering, 4th edn, McGraw-Hill, New York, 1958. 4. M. C. K. Jones and Hardy, R. L., Znd. Eng. Chem., 1952,44,2615. 5. N. A. C. Smith, H. M. Smith, 0. C. Blade and E. L. Garton, ‘The Bureau of Mines Routine Method for the Analysis of Crude Petroleum’, Bureau of Mines Bull., 1951, Number 490. 6. .ASTM Designation D 86, 1934. 7 J. B. Maxwell, Data Book on Hydrocarbons, D. Van Nostrand, New York, 1950. 8 I0. H. Hariu and R. C. Sage, Hydrocarbon Process., 1969,48(4), 143-148. 9. .M. G. Kesler and B. I. Lee, Hydrocarbon Process., 1976,55(3), 153-158. 10 M. R. Riazi and T. E. Daubert, Hydrocarbon Process., 1980,59(3), 115. 11. . M. R. Riazi and T. E. Daubert, Ind.Eng.Chem.Res.,1987,26,755-759. 12. ’C. H. Twu, Fluid Phase Equilibl:, 1984, 16, 137. 13. .Reidel, L. API Procedure 4A 1.15. 14. .American Petroleum Institute Technical Data Book - Petroleum Refining, 5th edn, T. E. Daubert and R. P. Danner (eds), American Petroleum Institute, Washington, DC, 1988. 15 V. K. Pareek, A. A. Adesina, A. Srivastava and R. Sharma, Chem. Eng. J., 2003,92(1-3), 101 - 109. 16. B. I. Lee and M. G. Kesler, A.I.Ch.E.J., 1975, 21,3, 510. 17. R. F. Curl and K.S. Pitzer, Ind. Eng. Chem., 1958, 50, 265. 18. V. K. Pareek, A.A. Adesina, A. Srivastava, and R. Sharma, J Mol. Cat, A: Chem., 2002, 181(1-2), 263-274. 19. K. S. Pitzer , Thermodynamics, 3rd edn, McGraw-Hill, New York, 1995. 20. C. R. Reid, J. M. Prausnitz and B. E. Poling, The Propeties of Gases and Liquids, 4th edn, McGraw-Hill, Singapore, 1988. .

I.

I

.

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21. M. J. Huron and J. Vidal, Fluid Phase Equilibl:, 1979, 3,255-271. 22. D. S. H. Wong and S. I. Sandler, A.I.Ch.E. J., 1992,38(5), 671-680. 23. A. Fredenslund, J. Gmehling and P. Rasmussen, Vapor-Liquid Equilibria using UNIFAC. A Group Contribution Method, Elsevier, Amsterdam, 1977. 24. T. Holderbaum and J. Gmehling, Fluid Phase Equilibl:, 1991, 70, 25 1. 25. Y. S. Wei and R. J. Sadus, A.I.Ch.E. J., 2000, 46, 169-196. 26. S. I. Sandler, Models for Thermodynamics and Phase Equilibria Calculations, Marcel Dekker, New York, 1994. 27. S. M. Walas, Phase Equilibria in Chemical Engineering, Butterworths, Stoneham, 1985. 28. C. L. DePriester, Chem. Eng. Progl: Symp. Sel:, 1953,49(7), 1-43. 29. Maxwell-Bonnell, API Procedure 5A1.15, 1955. 30. Aspen Physical Property System 11 . l , Aspen Tech Inc, Cambridge, MA, 2001. 3 1. NIST Thermophysical Properties of Hydrocarbon Mixtures Database (SUPERTRAPP), National Institute of Standards and Technology (US Department of Commerce), Gaithersburg, MD, 2003. 32. C. Tsonopolous and J. H. Heidman, Thermophysical property needs into the 21st century: petroleum refining and petrochemical industry view, 13th Symposium on Thermophysical Properties, Boulder, CO, USA, June 22027, 1997. 33. A. Fazlali, H. Modarress and G.A. Monsoori, Fluid Phase Equilibl:, 2001,179,297-317. 34. M. R. Riazi, Ind. Eng. Chem. Res., 1997,36,4299-4307. 35. L-S Wang and J. Gmehling, A.I.Ch.E. J., 1999,45, 5 , 1 125-1 134. 36. H. Manafi, G.A. Mansoori and S. Ghotbi, J.Petroleum Sci. Eng., 1999, 22, 67-93. 37. D. Y. Peng and D.B. Robinson, Ind. Eng. Chem. Fund., 1976,15,59-64. 38. M. R. Riazi and G. A. Mansoori, Oil Gas J., 1993,12, 108-1 11.

Subject Zndex Ab initio quantum Mechanical Calculations, 43 Activity, 37 Activity Coefficients, 35, 37 Activity Coefficient Model, 174 Activated Processes, 258 Adsorption, 79,243 Adsorption Isotherm, 244 Alloys, 213 Analysis Tools, 169 Anionic Polymerization, 91, 108 Applied Thermodynamics, 166, 167 Arrenhenius Plot, 107 Aspen, 166 AspenTech Clients, 172 Binary Alloys, 213 Binary Solid-Liquid Equilibria, 23 1 Bulk Thermodynamic Properties, 149 Calculation of Phase Diagrams, 213 Calculation Engine, 168 Calorimetric Data, 118 Calorimetric Design, 111 Calorimetric Measurement, 144 Catalyst Nature, 98 Catalytic Partial Oxidation, 180 Catalytic Reactor, 123 Catalytic Species, 92 Ceramic Membranes, 180 Chemical Equilibrium, 36 Chemical Potentials, 29,219 Chemical Reactors, 8 Chemical Recycling, 159 ChemApp, 16,30 ChemSage, 30 ChemSheet Thermodynamic Software, 30 Coatings (A1,0, - AlN), 206 Coatings (Ti, Al)N, 198 Codes, 271 Computational Quantum Mechanics, 43

Complex Process Optimization, 88 Composition of Petrol, 263 Continuous Injection, 97 COSMO-RS Model, 54 Cost of Recycling, 160 Cost-Benefit Analysis, 171 Critical Point, 151 Crystal Growth, 241 Crystallization, 230 Cubic Equation of State, 173 Data Bases, 129 Data Packages, 175 Data Sources, 271 Decarburisation rate, 18 Density, 3, 226 Deployment Tools, 169 Differential Scanning Calorimetry, 154 Diffusion Coefficients, 127 Dissolution of Crystal Growth, 241 Distillation, 33 Distillation Columns, 5, 33 Donnan Equilibrium, 25 Drug Receptor Studies, 108 Economic Impact, 166 Economic Value, 166 Electrolytes, 174, 189 Electrolytes for Fuel Cells, 189 Electrolyte Systems, 174 Energetic Considerations, 165 Energies of Transformation, 199 Enthalpy Predictions, 267 Entropy Production, 5, 41 Equation of State, 173, 244 Equilibrium Calculations, 176, 222 Equipment, 40 Estimation Methods, 172 Eutectic Behaviour, 233 Eutectic Systems, 238 Extractive Distillation, Process, 77 Extraction Process, 78

Subject Index

Fibres, 23, 24 Flow Calorimeter, 111 Flow Calorimetric Equations, 113 Flow Microcalorimeter, 110, 119 Flow-Through Microcalorimeters, 115 Force Fields, 44 Fuel Cells, 187 Fugacity Coefficients, 35 Fusion Crystallization, 152 Gaussian Comutational Package, 5 1 Gas Clathrate Hydrates, 57 Gas-Liquid Equilibrium, 80 Gas-Phase Flow Calorimetry, 113 Gas Solubility, 86 GEMC NVT simulations, 48 Gibbs Energy in Small Particle Systems, 210 Gibbs Energy Minimization, 16, 28, 71 Gibbs Energy Model, 27 Glass Transition Temperature, 154 Heat Capacities of Gases, 145 Heat Capacities of Liquids, 147 Heat of Transfer, 4 Henry Coefficients, 85 Heterogeneous Catalysis, 4 1 High Pressure VLE, 135 Hydrate Phase Diagram, 65 Hydrate Thermodynamics, 59 Infinite Dilution Activity Coefficients, 83 Industrial Applications, 170 International Dimension, 129 Interfacial Properties, 177 Ionic Liquids, 76, 82 Ionic Mean Properties, 220 Irreversible Thermodynamics, 72 Isothermal Conditions, 92 Isothermal Flow-Microcalorimetry, 110 Isothermal Microcalorimetry, 110 Kinetics of Dissolution of Crystal Growth, 24 1 Kinetic Parameters, 118 Kinetic Stability, 106 Lactams, 91 Lanthanum Gallates, 191 L-D Convertor, 12

275 Lennard Jones Potential, 45 Liquid-Liquid Equilibrium, 80,223 Liquid-Liquid Extraction, 78 Mass Transfer, 134 Material Recycling, 159 Membranes, 180, 186 Mesoscope Non-Equilibrium Thermodynamics,255,259 Meso-Structures,254 Metallurgical Processes, 12 Metastable Coatings, 197 Microcalorimetry, 104 Micro-Particle Production, 132 Mixtures, 248 Model Deployment, 177 Modelling, 16 Modelling Electrolyte Systems, 174 Modelling Polymer Systems, 175 Molecular Simulations, 44 Monte Car10 Simulations, 45 Multi-Phase Thermodynamics, 23 Multiple Miniature Calorimeter, 108 Nano-Particle Production, 132 Nano-Sized Particles, 132, 209 Nano-Sized Phase Diagrams, 213 Natural Gas, 57 Natural Gas Hydrates, 57 New Materials, 180 Non-Equilibrium Thermodynamics, 1, 254 Non-Equilibrium Thermodynamics of Polymer Crystallization, 254 Non-Isothermal Reaction Calorimetry, 101 Oxide Fuel Cells, 187, 189 Oxygen Permeation, 184 Oxygen Separation, 180 Particles Morphology, 135 Petrol Composition, 263 Petroleum Fluids in the Refining Industry, 262 pH, 31 Pharmaceutical Industry, 104 Phase Diagrams, 66, 224 Phase Equilibrium, 34, 170 Phase Equilibrium Modelling, 170 Phase Transitions, 2 Phase Transition Thermal Properties, 152

276 Plastic Recycling, 159 Polymer Crystallization, 256,259 Polymer Production, 160 Polymer Systems, 175 Prediction Methods, 267 Prediction Methods for PVT Relationships, 269 Prediction Methods for VLE, 270 Process Design, 39 Process Model, 12 Process Synthesis, 38 Programmable Interfaces, 169 Pseudo-Binary Diagrams, 139 Pseudo-Component Characterization, 267 Pulp Fibres, 23 Pulp-Ion Interaction, 25 Pulp Suspension, 23 Pure Component Behaviour, 230 PVD Processes, 197 PVT Predictions, 269 Quantitative Cost-Benefit Assessments, 171 Quantum Chemistry, 23 Quantum Mechanical Calculations, 43, 53 Quick Screen Solvents, 170 Raman Spectroscopy, 65 Reaction Calorimetry, 88 Reaction Kinetics, 37 Reactive Distillation, 33 Recycling, 160 Refining Industry, 262 Reversible Thermodynamics, 72 SAS Processing, 133 Scale-Up Problem, 39 Second Law, 6 , 9 Selective Solvent, 77, 80, 81 Selectivity at Infinite Dilution, 83 Separating Agent, 77 Separating Processes, 76 Simulation, 17 SimuSage, 16 Slag, 14 Solid-Fluid Equilibria, 239 Solid-Liquid Equilibrium, 223, 23 1, 236 Solid Oxide Fuel Cells, 187 Solid Solutions, 232 Solution-Phase Flow Calorimeters, 112

Solvent Screening, 170 Speciation Equilibrium, 222 Spectro-CalorimetricScreening, 88 Stability Determinations, 105 Stability of Membranes, 186 Standard States, 219 Statistical Thermodynamic Model, 67 Steel Melting, 15 Stirring Level, 100 Successive Injection Technique, 96 SupercriticalAnti-solvent Precipitation, 132 Supercritical Fluids, 132 Surface Tension, 4 Ternary Solid-Liquid Equilibrium, 236 Theoretical Predictions, 127 Thermal Conductivity, 126 Thermal Recycling, 159 Thermal Separation Processes, 77 Thermodynamic Considerations, 165 Thermodynamic Functions, 246 Thermodynamics of Adsorption, 243 Thermodynamics of Crystallization, 230 Thermodynamics of Electrolyte Systems, 219 Thermodynamics of Nano-Sized Particles, 209 Thermodynamics of New Materials, 180 Thermodynamic Prediction, 197 Thermo-MechanicalProperties, 149 Thermo-Physical Properties, 144, 267 Titration Calorimetry, 108 Transport and Interfacial Properties, 177 Transport Coeficients, 14 Transport Properties, 122 Transport Thermodynamics, 181 Unmet Needs of AspenTech Clients, 166, 172 UNIFAC Model, 5 5 , 171 Van der Waals and Platleeuw Model, 69, 71 Vapour-Liquid Equilibrium, 79, 85, 223 Virial Coefficient Prediction, 46 Viscosity, 125, 228 VLE Data, 85 VLE Measurements, 36 VLE Predictions, 270 VLE Simulation, 48, 49, 54