Separation of Molecules, Macromolecules and Particles: Principles, Phenomena and Processes 9780521895736

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Separation of Molecules, Macromolecules and Particles

Providing chemical engineering undergraduate and graduate students with a basic understanding of how the separation of a mixture of molecules, macromolecules or particles is achieved, this textbook is a comprehensive introduction to the engineering science of separation. • Students learn how to apply their knowledge to determine the separation achieved in a given device or process. • Real-world examples are taken from biotechnology, chemical, food, petrochemical, pharmaceutical and pollution control industries. • Worked examples, elementary separator designs and chapter-end problems give students a practical understanding of separation. The textbook systematically develops different separation processes by considering the forces causing the separation, and how this separation is influenced by the patterns of bulk flow in the separation device. Readers will be able to take this knowledge and apply it to their own future studies and research in separation and purification. Kamalesh K. Sirkar is a Distinguished Professor of Chemical Engineering and the Foundation Professor of Membrane Separations at New Jersey Institute of Technology (NJIT). His research areas are membranes and novel membrane based processes.

“The first comprehensive book that takes the fundamentals of separation on a molecular level as the starting point! The benefit of this approach is that it gives you a thorough insight in the mechanisms of separation, regardless of which separation is considered. This makes it remarkably easy to understand any separation process, and not only the classical ones. This textbook finally brings the walls down that divide separation processes in classical and non-classical.” Bart Van der Bruggen, University of Leuven, Belgium “This strong text organizes separation processes as batch vs continuous and as staged vs differential. It sensibly includes coupled separation and chemical reaction. Supported by strong examples and problems, this non-conventional organization reinforces the more conventional picture of unit operations.” Ed Cussler, University of Minnesota “This book fills the need by providing a very comprehensive approach to separation phenomena for both traditional and emerging fields. It is effectively organized and presents separations in a unique manner. This book presents the principles of a wide spectrum of separations from classical distillation to modern field-induced methods in a unifying way. This is an excellent book for academic use and as a professional resource.” C. Stewart Slater, Rowan University “This book is an excellent resource for the topic of chemical separations. The text starts by using examples to clarify concepts. Then throughout the text, examples from many different technology areas and separation approaches are given. The book is framed around various fundamental approaches to chemical separations. This allows one to use this knowledge for both current and future needs.” Richard D. Noble, University of Colorado

“This book provides a unique and in depth coverage of separation processes. It is an essential reference for the practicing engineer. Unlike more conventional textbooks that focus on rate and equilibrium based separations, Prof Sirkar focuses on how a given separation takes place and how this is used in practical separation devices. Thus the book is not limited by application e.g. chemical or petrochemical separations. “As chemical engineering becomes increasingly multidisciplinary, where the basic principles of separations are applied to new frontier areas, the book will become an essential guide for practitioners as well as students. “The unique layout of the text book allows the instructor to tailor the content covered to a particular course. Undergraduate courses will benefit from the comprehensive and systematic coverage of the basics of separation processes. Whether the focus of a graduate course is traditional chemical separations, bioseparations, or separation processes for production of renewable resources the book is an essential text.” Ranil Wickramasinghe, University of Arkansas “This advanced textbook provides students and professionals with a unique and thought-provoking approach to learning separation principles and processes. Prof. Sirkar has leveraged his years of experience as a separation scientist and membrane separation specialist, to provide the reader with a clearly written textbook full of multiple examples pulled from all applications of separations, including contemporary bioseparations. Compared to other separations textbooks, Prof. Sirkar’s textbook is holistically different in its approach to teaching separations, yet provides the reader with a rich learning experience. Chemical engineering students and practicing professionals will find much to learn by reading this textbook.” Daniel Lepek, The Cooper Union

Separation of Molecules, Macromolecules and Particles Principles, Phenomena and Processes

Kamalesh K. Sirkar New Jersey Institute of Technology

University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521895736 © K. Sirkar 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalog record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Sirkar, Kamalesh K., 1942- author. Separation of molecules, macromolecules and particles : principles, phenomena and processes / Kamalesh Sirkar, New Jersey Institute of Technology. pages cm. – (Cambridge series in chemical engineering) isbn 978-0-521-89573-6 (Hardback) 1. Separation (Technology)–Textbooks. 2. Molecules–Textbooks. I. Title. TP156.S45S57 2013 5410 .22–dc23 2012037018 ISBN 978-0-521-89573-6 Hardback Additional resources for this publication at www.cambridge.org/sirkar Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents Preface List of notation Introduction to the book Introduction to chapters Linked footprints of a separation process/technique Classification of separation processes Additional comments on using the book Textbooks, handbooks and major references on separation processes

page xi xiii 1 2 3 6 7 7

1

Description of separation in a closed system 1.1 Binary separation between two regions in a closed vessel 1.2 Multicomponent separation between two regions in a closed vessel 1.3 Definitions of composition for a binary system in a closed vessel 1.4 Indices of separation for binary systems 1.5 Comparison of indices of separation for a closed system 1.6 Indices for separation of multicomponent systems between two regions 1.7 Some specialized nomenclature Problems

19 19 22 24 25 29 33 35 35

2

Description of separation in open separators 2.1 Preliminary quantitative considerations 2.2 Binary separation in a single-entry separator with or without recycle 2.2.1 Examples of separation in single-entry separators 2.2.2 Single-entry separator with a product recycle 2.2.3 Separative power and value function 2.3 Binary separation in a double-entry separator 2.3.1 Examples of separation in double-entry separators 2.3.2 Separation in a double-entry separator with recycle/reflux 2.4 Multicomponent systems 2.4.1 Size-distributed particle population 2.4.2 Continuous chemical mixtures 2.4.3 Multicomponent chemical mixtures 2.5 Separation in an output stream with time-varying concentration Problems

39 39 42 45 48 49 50 51 53 53 54 62 64 66 70

3

Physicochemical basis for separation 3.1 Displacements, driving forces, velocities and fluxes 3.1.1 Nature of displacements 3.1.2 Forces on particles and molecules 3.1.3 Particle velocity, molecular migration velocity and chemical species flux 3.1.4 Integrated flux expressions for molecular diffusion and convection: single-phase systems

76 76 76 77 88 102

vi

Contents

3.2

3.3

3.4

4

5

3.1.5 Flux expressions in multicomponent systems 3.1.6 Additional topics Separation development and multicomponent separation capability 3.2.1 Separation development in a closed system 3.2.2 Multicomponent separation capability 3.2.3 Particulate systems Criteria for equilibrium separation in a closed separator 3.3.1 Phase equilibrium with equal pressure in all phases 3.3.2 Phase equilibrium where different phases have different pressures 3.3.3 Single-phase equilibrium in an external force field 3.3.4 Equilibrium between phases with electrical charges 3.3.5 Equilibrium between bulk and interfacial phases 3.3.6 Curved interfaces 3.3.7 Solute distribution between phases at equilibrium: some examples 3.3.8 Particle distribution between two immiscible phases Interphase transport: flux expressions 3.4.1 Interphase transport in two-phase systems 3.4.2 Interphase transport: membranes 3.4.3 Interphase transport in two-phase systems with phase barrier membranes Problems Appendix: Diffusion coefficients in different systems

112 117 118 119 124 127 128 129 130 132 132 133 136 137 159 159 160 170 189 192 202

Separation in a closed vessel 4.1 Equilibrium separation between two phases or two regions in a closed vessel 4.1.1 Gas–liquid systems 4.1.2 Vapor–liquid systems 4.1.3 Liquid–liquid systems 4.1.4 Liquid–solid systems 4.1.5 Interfacial adsorption systems 4.1.6 Liquid–ion exchanger systems 4.1.7 Supercritical fluid–bulk solid/liquid phase 4.1.8 Bulk fluid phase – mesophase systems 4.1.9 Partitioning between a bulk fluid phase and an individual molecule/macromolecule or a collection of molecules for noncovalent solute binding 4.1.10 Gas–solid particle–liquid system in mineral flotation 4.2 Equilibrium separation in a single phase in an external force field 4.2.1 Centrifugal force field 4.2.2 Electrical force field 4.2.3 Gravitational force field 4.2.4 Particle separation with acoustic forces 4.2.5 Externally imposed temperature gradient: thermal diffusion 4.3 Equilibrium separation between two regions in a closed vessel separated by a membrane 4.3.1 Separation by dialysis using neutral membranes 4.3.2 Separation between two counterions in two solutions separated by an ion exchange membrane: Donnan dialysis 4.3.3 Separation of a gas mixture by gas permeation 4.3.4 Separation of a pressurized liquid solution through a membrane Problems

205 205 206 208 217 222 223 228 231 231

Effect of chemical reactions on separation 5.1 Extent of separation in a closed vessel with a chemical reaction 5.2 Change in separation equilibria due to chemical reactions 5.2.1 Gas–liquid and vapor–liquid equilibria 5.2.2 Liquid–liquid equilibrium

280 280 281 281 289

234 245 245 246 253 256 260 261 262 262 264 266 272 273

Contents

5.3

5.4

vii 5.2.3 Stationary–mobile phase equilibria 5.2.4 Crystallization and precipitation equilibrium 5.2.5 Surface adsorption equilibrium 5.2.6 Complexation in Donnan dialysis 5.2.7 Enzymatic separation of isomers Rate-controlled equilibrium separation processes: role of chemical reactions 5.3.1 Absorption of a gas in a reactive liquid 5.3.2 Solvent extraction of a species with chemical reaction Rate-governed membrane processes: role of chemical reactions 5.4.1 Reverse osmosis: solute ionization 5.4.2 Ultrafiltration: complexation 5.4.3 Dialysis: reaction in dialysate 5.4.4 Chemical reactions in liquid membrane permeation–separation 5.4.5 Separation through solid nonporous membrane Problems

299 303 306 308 308 309 309 315 318 318 319 323 324 337 338

6

Open separators: bulk flow parallel to force and continuous stirred tank separators 6.1 Sources and nature of bulk flow 6.1.1 Hydrostatic pressure induced bulk flow 6.1.2 Gravity induced bulk flow 6.1.3 Free convection 6.1.4 Bulk motion due to capillarity 6.1.5 Electroosmotic flow 6.1.6 Centrifugal force driven flow 6.1.7 Surface tension gradient based flow 6.1.8 Drag flow 6.1.9 Feed introduction mode vs. bulk flow 6.2 Equations of change 6.2.1 Equations of change for species concentration in a mixture 6.2.2 Equation of motion of a particle in a fluid: trajectory equation 6.2.3 General equation of change for a particle population 6.3 Bulk flow parallel to force direction 6.3.1 External forces 6.3.2 Chemical potential gradient driven phase-equilibrium systems 6.3.3 Filtration and membrane separation processes 6.4 Continuous stirred tank separators 6.4.1 Well-mixed separators – CSTSs and batch separators 6.4.2 Well-mixed separators – membrane based devices Problems

346 348 348 350 350 352 353 354 354 355 356 358 358 368 368 372 373 390 412 445 445 469 475

7

Separation in bulk flow of feed-containing phase perpendicular to the direction of the force 7.1 Chemical potential gradient based force in phase equilibrium: fixed-bed processes 7.1.1 Fixed-bed adsorption/desorption processes 7.1.2 Pressure-swing adsorption process for gas separation 7.1.3 Potential-swing adsorption 7.1.4 Parametric pumping 7.1.5 Chromatographic processes 7.1.6 Expanded bed adsorption (EBA) from a broth/lysate aided by gravitational force 7.1.7 Counteracting chromatographic electrophoresis and electrochromatography 7.2 Crossflow membrane separations, crossflow filtration and granular filtration 7.2.1 Crossflow membrane separations, crossflow filtration 7.2.2 Granular filtration of hydrosols (and aerosols) 7.3 External force field based separation: bulk flow perpendicular to force 7.3.1 Electrical force field

485 487 487 511 519 520 527 550 551 555 555 586 596 596

viii

Contents 7.3.2 Centrifugal force field 7.3.3 Gravitational force field 7.3.4 Field-flow fractionation for colloids, macromolecules and particles 7.3.5 Magnetic force field 7.3.6 Radiation pressure – optical force Problems

8

9

10

618 634 640 648 654 655

Bulk flow of two phases/regions perpendicular to the direction(s) of the force(s) 8.1 Countercurrent bulk flow of two phases or regions perpendicular to the direction(s) of the force(s) driving species 8.1.1 Development of separation in countercurrent flow systems 8.1.2 Gas (vapor) absorption/stripping 8.1.3 Distillation 8.1.4 Countercurrent solvent extraction 8.1.5 Countercurrent melt crystallization in a column 8.1.6 Countercurrent adsorption and simulated moving bed system 8.1.7 Membrane processes of dialysis and electrodialysis 8.1.8 Countercurrent liquid membrane separation 8.1.9 Countercurrent gas permeation 8.1.10 Countercurrent gas centrifuge 8.1.11 Thermal diffusion and mass diffusion 8.2 Cocurrent bulk motion of two phases or regions perpendicular to the direction(s) of the force(s) driving species/particles 8.2.1 Cocurrent two-phase flow devices – general considerations 8.2.2 Chromatographic separations in cocurrent two-phase flow devices 8.2.3 Particle separation in cocurrent gas–liquid flow–Venturi scrubber 8.2.4 Cocurrent membrane separators 8.3 Crossflow of two bulk phases moving perpendicular to the direction(s) of the driving force(s) 8.3.1 Continuous chromatographic separation 8.3.2 Crossflow plate in a distillation column Problems

670

Cascades 9.1 Types of cascades 9.1.1 Countercurrent cascades, ideal cascade 9.1.2 Other cascade configurations 9.2 Cascades for multicomponent mixture separation via distillation 9.3 Cascades for multicomponent mixture separation involving other separation processes Problems

812 812 812 818 822 824 824

Energy required for separation 10.1 Minimum energy required for separation 10.1.1 Evaporation of water from sea water 10.1.2 Recovery of water by reverse osmosis 10.1.3 Net work consumption 10.1.4 Minimum energy required for membrane gas permeation, distillation, extraction, and other separation processes 10.2 Reducing energy required for separation 10.2.1 Evaporation of water for desalination 10.2.2 Distillation 10.2.3 Free energy of mixing 10.2.4 Dilute solutions Problems

827 827 828 828 830

670 670 683 709 736 751 754 761 767 771 775 781 782 783 785 788 789 794 794 799 804

831 836 836 839 843 844 845

Contents 11

Common separation sequences 11.1 Bioseparations 11.2 Separation sequences for water treatment 11.2.1 Sea-water and brackish-water desalination 11.2.2 Ultrapure water production 11.2.3 Pharmaceutical grade water 11.3 Chemical and petrochemical industries 11.4 Hydrometallurgical processes

Postface Appendix A Units, various constants and equivalent values of various quantities in different units Appendix B Constants Appendix C Various quantities expressed in different units References Index

ix 847 847 851 851 852 853 853 855 856 857 858 859 861 877

Preface This is an introductory textbook for studying separation. Primarily, this book covers the separation of mixtures of molecules; in addition, it provides a significant treatment of particle separation methods. Separation of macromolecules has also received some attention. The treatment and coverage of topics are suitable for chemical engineering students at undergraduate and graduate levels. There is enough material here to cover a variety of introductory courses on separation processes at different levels. This book is focused on developing a basic understanding of how separation takes place, and of how the resulting separation phenomenon is utilized in a separation device. The role of various forces driving molecules or particles from a feed mixture into separate phases/ fractions/regions is basic to such an approach to studying separation. The separation achieved is then amplified in an open separator via different patterns of bulk-phase velocities vis-à-vis the direction(s) of the force(s). The forces are generated by chemical potential gradient, electrical field, rotational motion, gravity, magnetic field, etc. The resulting separation is studied under three broad categories of separation processes. Separation processes driven by a negative chemical potential gradient are generally multiphase systems and are treated under the broad category of phase equilibrium driven processes. External force driven processes populate the second category, and include those operating under an electrical field, rotational motion, magnetic field or gravity; thermal diffusion processes are also briefly included here. The third category of membrane based processes studied is driven generally by a negative chemical potential gradient; however, electrical force is also relevant for some processes. The treatment of any external force driven processes will cover both separation of molecules and particle separations. These physical separation methods are often reinforced by chemical reactions, which are usually reversible. An elementary treatment of the role of chemical reactions in enhancing separation across a broad spectrum of phase equilibrium driven processes and membrane based processes has been included. The level of treatment in this book assumes familiarity with elementary principles of chemical engineering thermodynamics and traditional

undergraduate levels of knowledge of ordinary differential equations and elementary partial differential equations. Specific aspects of a given separation process are studied in the chapter devoted to those aspects for all separation processes. To study a particular separation process in great detail, one therefore has to go to different chapters. The footprints of a given separation process are provided at the beginning of the book (Tables 1–7); there are quite a few tables to cover a variety of separation processes. The list of processes is large; however, it is far from being all inclusive. The introductory chapter, which provides additional details about various chapters, as well as about the book, is preceded by a notation section. All references appear at the end of the book. The description of the extent of separation achieved in a closed vessel for a mixture of molecules is treated in Chapter 1. Chapter 2 illustrates how to describe the separation of molecules in open separators under steady and unsteady state operation; a description of separation for a size-distributed system of particles is also included. Chapter 3 introduces various forces developing species-specific velocities, fluxes and mass-transfer coefficients, and illustrates how the spatial variation of the potential of the force field can develop multicomponent separation ability. The criteria for chemical equilibrium are then specified for different types of multiphase separation systems, followed by an illustration of integrated flux expressions for twophase and membrane based systems. Chapter 4 develops the extent of separation achieved in a closed vessel to a variety of individual processes under each of the three broad categories of separation processes. Chapter 5 demonstrates how separation can be considerably enhanced by chemical reactions in phase equilibrium based and membrane based processes under both equilibrium- and rate-controlled conditions. For open separators having bulk flow in and out, including continuous stirred tank separators (CSTSs), Chapter 6 provides first the equations of change for molecular species concentration in single-phase and two-phase systems, the trajectory equation for a particle in a fluid and the general equation of change for a particle population. Chapter 6 then treats individual separation processes under each of the three

xii broad categories of separation processes when the bulk flow is parallel to the direction of the force and in CSTS mode. Chapter 7 follows this latter approach of treating individual separation processes under each of the three broad categories of separation processes when the bulk flow of feed-containing phase is perpendicular to the direction of the force. Chapter 8 follows the same approach when the bulk flows of two phases/regions in the separator are perpendicular to the direction(s) of the force(s). Chapter 9 briefly elaborates on cascades, which were already introduced in the countercurrent multistaged flow systems of Chapter 8. Chapter 10 introduces the energy required for a number of separation processes. Chapter 11 illustrates a few common separation sequences in a number of common industries involved in bioseparations, water treatment, chemical and petrochemical separations and hydrometallurgy. Conversion factors between various systems of units are provided in an Appendix. Virtually all separation processes taught to chemical engineering students in a variety of courses have been covered via the approach illustrated in Chapters 3, 4, 6, 7 and 8; in addition, many particle separation methods have been treated. The structural similarity in the separation method between apparently unrelated separation processes becomes quite clear. A few basic principles equip the students with the capability to understand a wide variety of separation processes and techniques, including emerging ones. To aid the student, there are 118 worked examples, 300 problems, 340 figures, 100 tables and 1011

Preface references. A website will provide guidance for computer simulations for a few selected problems. The introductory chapter provides references to articles and books which influenced the development of various aspects of this book. I have benefitted considerably from the comments on selected chapters of the book by reviewers, anonymous or otherwise. Comments by Professors C. Stewart Slater, of Rowan University, Steven Cramer, of Rensselaer Polytechnic Institute, and Ranil Wickramasinghe, of Colorado State University (now at University of Arkansas), were particularly useful. Many doctoral students and postdoctoral fellows were of invaluable help during the long gestation period of this book, either in formulating solutions of the problems or in developing illustrative drawings. I want to mention in particular Amit Sengupta, Theoharris Papadopoulos, Xiao-Ping Dai, Meredith Feins, Dimitrios Zarkadas, Quixi Fan, Praveen Kosaraju, Fei He, Atsawin Thongsukmak, Sagar Roy, Dhananjay Singh, John Tang and John Chau. The first two students helped me when we were at Stevens Institute of Technology. Sarah Matthews of Cambridge University Press patiently provided manuscript preparation guidelines and encouraging comments during an ever-shifting timetable. Irene Pizzie did an extraordinary job as the copy editor. Brenda Arthur of New Jersey Institute of Technology tirelessly typed the draft of the whole manuscript over a considerable length of time, while carrying out many other duties. I must also mention at the end my wife, Keka, without whose patience, help and understanding this book would never have been finished.

Notation Equation numbers identify where the symbols have been introduced or defined. The following styles have been adopted. Bold

vector quantity

asp av, avc

Overlines —

—= ^

quantity averaged over time or a specific coordinate direction, multicomponent system, Laplace transformed dependent variable averaged quantity quantity in a mixture, per unit mass of bulk phase

A(r)

Underlines _ ~, ¼

a A; A1, A2, A3

hypothetical binary system quantity (2.4.23), (2.4.24) vector quantity, tensor quantity

Ac Ahex; Aij; Am; Atm ; Ai ðTÞ; Aoi

Brackets 〈vtj〉

average value of vtj over surface area Sj A1ρ

a

a1; a2, a3,…

a1 aA aH ai; ai(T); aij, ail; 0 ^a i ; am ; amsℓ

ap as1, as2; asw

ellipsoid semiaxis dimension; see also (3.1.10a); constant in relations (3.1.49), (4.3.7) and (4.3.43a); interfacial area per unit volume, defined by (7.2.191), (7.3.25) constant in (3.3.105), (4.1.42b), (4.3.29) and (5.2.147); constants in (7.2.73), (7.3.50) and (7.3.139) constant in (7.2.198a) stoichiometric coefficient for species A Hamaker constant (3.1.16) activity of species i; equilibrium constant (7.1.63); atom fraction of ith isotope of an element in region j (1.3.6), value of ai in region j and liquid phase, respectively; amplitude (7.1.72b); constant in (4.3.43c); membrane surface per unit channel length (7.2.70) surface area of a particle activity of solvent s in regions 1 and 2, respectively; activity of salt in water

Ap

Aþ p ðxÞ AT

Amþ, Am−, Am± b

pore surface area per unit volume of the porous medium of porosity ε surface area of a particle per unit particle volume, value of av in a cake (6.3.135j) mean electrolyte activity (3.3.119d) amplitude (7.3.18), pure-water permeability constant in reverse osmosis and diffusive ultrafiltration; constants in (7.1.90b), three surface areas in control volume of Figure 7.2.6(b) cross-sectional area of a cone in centrifugal elutriation cross-sectional area of duct heat exchanger surface area; constant in equations for activity coefficients (4.1.34d); surface area of membrane; total membrane surface area; modified equilibrium constant (7.1.66); constant (7.1.72b) constant in crystal growth rate equation (6.4.27) projected area of a particle (3.1.64); transport coefficient in solutiondiffusion-imperfection model (3.4.60a) cumulative crystal surface area distribution fraction (6.4.17) total particle surface area per unit volume of total mixture (Example 2.4.2); total crystal surface area per unit liquid volume (6.4.16) three forms of amino acid ellipsoid semiaxis dimension; proportionality constant in osmotic pressure relation (3.4.61b); half of channel gap; width of region of gas flow completely cleaned up by a fiber (6.3.42a); constant in crystal growth rate expression (6.4.35); parameter (7.2.18); liquid envelope radius (7.2.208)

xiv

Notation

b1, b2, b3 b1 bf; bi; bif; biℓ; bi1

0

0

bi−k , bk 0

bm, bm B B; Bo; Boi ;Bp

Bi B1p, B2p

c; cC

C C1,C2; C2(x)

C2f C(2) C 0A , C δA CAℓb, CAℓi

0

CAwb, CAwi; C Awb

CBob, CBoi

0

Cc; C Cb ;CD

CFC 0

0

C H1 , C H2

constants in relations (3.1.143f) constant in (3.3.105), (5.2.147) membrane feed channel height (Figure 7.2.3(a)); constant in equilibrium relation (3.3.81); constant in (7.1.73), constant in Freundlich isotherm (3.3.112c); constant in Langmuir isotherm (3.3.112a); constant in equation (4.1.42a) proportionality constant in (5.2.154), equilibrium constant in (5.2.155) constants in (4.3.43a,c) magnetic induction vector (3.1.19) constant, density function of the birth rate of new particles (6.2.50g), value of B as rp ! 0; value of Bo for crystal growth rate gri; duct perimeter (7.3.41) Biot number (3.4.35) second virial coefficients for interaction between polymers 1 and 2 (4.1.34p) velocity of light; gap between plates at entrance (Figure 7.3.10), stoichiometric coefficient for species C clearance of a solute (8.1.390) integration constants (3.3.10b); molar concentration of species 2 at location x molar concentration of species 2 in initial mixture, mol/liter defined in (6.3.75) molar concentration of A at z = 0, δ (3.1.124) molar concentration of A in phase bulk and phase interface and in liquid at gas–liquid interface (3.4.1b) molar concentration of A in bulk water and in water at phase interface (3.4.45e); critical value of CAwb for maximum enhancement (5.3.53) molar concentration of species B in organic-phase bulk and interface, respectively slip (Cunningham) correction factor (3.1.215); critical value of bulk concentration of C for maximum enhancement (5.3.29b); particle drag coefficient (3.1.64), (6.3.4) molar concentration of fixed charges in ion exchange resin dual mode sorption constants for species 1 and 2 (3.3.82a,b)

Ci; C i ; C ti ; C i ; C ig ; C ii1 , C ii2

C i10 , C i20 C oi ðπÞ

C igel Cij

C tij ; C ijn

C 0ij , C ℓij , C δij

C iℓb , C iℓe , C iℓi

0

0

C ig , C igf ^ ik ; C it C ik , C ik , C

p

C dim , C H im ; C im

Cimi,Cimo,Ciob, Ciwi, Ciwb CpR, Cpw

molar concentration of species i; an average of the molar concentration of species i in the feed and the permeate (5.4.74), (6.3.158b); total molar concentration of species i in the porous medium per unit volume (3.1.118b); nondimensional species i concentration (5.3.35j), hypothetical gas-phase species i concentration (8.1.47); initial bed concentration of solute i in phases 1 and 2 values of C i1 , C i2 at z = 0 liquid-phase concentration of pure solute i at spreading pressure π and temperature T providing the same surface phase concentration of i as the mixture molar species i concentration in gel molar concentration of species i in region j or location j or stream j; j = b, bulk; j = E, extract; j = f, feed region; j = g, gas phase; j = k, kth phase; j = ℓ, 0 liquid; j = ℓ , adhering liquid phase on crystal; j = m, membrane; j = o, organic; j = p, permeate, product; j = r, raffinate; j = R, ion exchange resin phase, raffinate; j = s, solution, solid phase or pore surface; j = w, water, mol/liter total molar concentration of species i in region j including complexed or dissociated forms; value of Cij on nth plate/stage species i concentrations in phase j at locations 0 (or initial concentration), ℓ and δ molar concentrations of species i in bulk solution, at the end of concentration process and at the beginning of concentration process (6.3.173) molar species i concentration per unit gas phase volume in a pore intrinsic phase average, phase average and deviation in Cik for species i concentration in phase k (6.2.24a,b), (6.2.28); defined by (7.1.94) species i concentration in membrane: for Henry’s law and Langmuir species, respectively, in dual sorption model (3.3.81); membrane pore liquid molar concentrations of species i at various locations in Figure 3.4.11 molar protein concentrations in resin phase and aqueous solution

Notation

xv

Csf, Csm, Csp

Ct; Ctj; Cvp

d; dh; di; d ℓm

di; dgr; dimp

dion; dp; dw; dp1, dp2

d32 D

DA, DC; DB

Deff Dgr Dp; Dp(ϕ)

Dr Di,eff; Di,eff,r, Di,eff,z; Di,eff,1

Di; eff; k ; D

i; eff; k

Dij; Dil; Dis DTA , DTB ; DTis

DAB; DBR

DiD, DiH Die

molar solvent concentration in feed, membrane and permeate, respectively total molar concentration; Ct in a mixture in region j; volume of particles per unit fluid volume (7.2.176) diameter of tube/pipe/vessel; hydraulic diameter (Table 3.1.8); effective diameter of a molecule of gas species i (3.3.90a); logarithmic mean diameter (8.1.417) force-type term (3.1.178), (3.1.181); grain diameter; diameter of an impeller mean diameter of a molecular ion; mean diameter of a particle (6.1.4b); wire diameter (3.1.23); diameters of particles 1 and 2 Sauter mean diameter of a drop or particle (6.4.88), (6.4.89) diffusion coefficient of species in countertransport through liquid membrane diffusion coefficient of species A and C, respectively; dialysance in hemodialysis (8.1.389) effective diffusion coefficient (5.4.64a) crystal growth diffusivity (6.4.45) diffusion coefficient for particle (3.1.68), (6.2.52); shear-induced particle diffusivity (3.1.74), (7.2.126), (7.2.131a) desalination ratio (1.4.25), (2.2.1a) effective diffusion coefficient of i in liquid (6.2.18), (6.3.16b); value of Di,eff in r-and z-directions; value of Di,eff in phase/region 1 effective diffusion coefficient of i in phase k (6.2.33); dispersion tensor (6.2.31) diffusion coefficient of species i in region j; j = l, liquid; j = s, solvent thermal diffusion coefficient for species A and B (3.1.44); for species i in solution binary diffusion coefficient for mixture of gases A and B; diffusion coefficient of particles due to Brownian motion (7.2.216) diffusion coefficients in dual sorption– dual transport model (3.4.78) effective diffusion coefficient of i in a porous medium (3.1.112d)

Dim; Dimo; Dip

Dik , Dim

DiK DiM Dis ; D0is ; DN is

D12 De; (De)mv

Df; DF

e ei enb,enM,enMN, enMP,enp

E; E; E c ; E Di ; E e

Ei; Eo; EoG; Es; Ey; EBRS, EGrS

EIS; EIS

diffusion coefficient of i in membrane (6.3.149); value of Dim for Cim = 0 (3.4.67b), Di in a pore (3.4.89c), (6.3.145a); effective Di in the pores of a particle multicomponent diffusion coefficient of species pair (i, k) and (i, m) in Maxwell–Stefan approach Knudsen diffusion coefficient for species i (3.1.115c) effective binary diffusivity of species i in a mixture (3.1.184), (3.1.185) binary diffusion coefficient for solute i/ solvent s; value of Dis at infinite dilution; Nernst–Planck binary diffusion coefficient for species i/solvent s binary diffusion coefficient for species 1 and 2 density function of particles which disappear (die) (6.2.50h); see (7.2.170a) and (7.2.172) decontamination factor (2.2.1c); dilution factor (6.4.106), (7.2.91c) charge of an electron, 1.60210  10−19 coulomb constant in adsorption isotherm for solute i (3.3.113d) molecular energy in the bulk, due to intramolecular interactions, due to intermolecular interactions, due to interaction between molecules and pores and total energy for molecules in the pore (3.3.89d) electrical force field; its magnitude (3.1.8), (6.1.22), (6.3.8f), extraction factor (8.1.281), stage efficiency (6.4.72); electrical field strength Ec (7.3.32a); activation energy for diffusion of species i in polymer (4.3.46b); extraction factor for extraction section (8.1.303) enrichment of species i by pervaporation (6.3.193b), particle collection efficiency (7.2.200b); overall column efficiency (8.1.195); point efficiency (8.3.13); extraction factor for the scrubbing section; electrical field strength in y-direction (7.3.48); particle collection efficiency (7.2.219), (7.2.214) inertial impaction based single fiber capture efficiency (6.3.42a); particle capture efficiency by interception (7.2.224)

xvi

EME, EMR; EMV; EN

ET; E 1T ; E T i

Notation

Murphree extract stage efficiency, Murphree raffinate stage efficiency (6.4.70), (6.4.71); Murphree vapor efficiency (8.1.198); Newton particle separation efficiency (2.4.14a) total efficiency in solid–fluid separation (2.4.4a), overall filter efficiency (6.3.45), (7.2.201); reduced efficiency of Kelsall (2.4.16a); ET for ith solid–fluid separator (2.4.17c,d)

F; F^ , F^ p ;F

F(rp)

Fi; F ext i

Facrx f

f2 f(r); f(rp); f(ε)

ff(rp), f1(rp), f2(rp)

fA, fi; f 0i fg(gr) fil fm, f0 m f 0ij , f 0ig , f 0il

f di , f dp ^f , ^f , ^f ; ^f ijpl ig ij il

f dio f dim , f dsm f f ðMÞ, f l ðM Þ, f v ðM Þ; f∞, fλ

fM; fQm, fyo

friction factor (6.1.3a), fractional consumption of chemical adsorbent (5.2.19d) fraction of the solute in ionized form (i = 2) in RO (5.4.4) molar density function in a continuous/semi-continuous mixture with characteristic property r; particle size probability density function (2.4.1a), pore size distribution function in a membrane; defined by (7.2.222a,c) value of f(rp) for feed stream, overflow and underflow based on particle weight fraction in a given size range (2.4.1b) fugacity of species A, species i; standard state fugacity of species i probability density function of crystal growth rate (6.4.41a) fugacity of pure species i in liquid phase quantities characteristic of a membrane polymer (4.3.46a,d) standard state fugacity of species i in region j; j = g, gas phase; j = ℓ, liquid phase frictional coefficient for species i and spherical particle value of fugacity of i in a mixture in gas phase, phase j and liquid phase, respectively; value of ^f ij for a planar surface value of f di for a sphere of equivalent volume (3.1.91e) frictional coefficient for solute i and solvent s in a membrane value of f(r), where r = M, molecular weight, for the feed mixture, liquid fraction and the vapor fraction, respectively; defined by (7.1.59a), (7.2.187), respectively fraction of the total metal ion concentration in the aqueous phase present as Mnþ (5.2.97); probability density functions (7.3.79), (7.3.80)

Frad F BR F ELK i F ELS i F Lret i Fm k F ext net iner ext F drag p ; F p ; F pz

F ti , F ext ti ext F ext tp ; F tpx , ext F ext ; F tpz tpy

F TA F

g; gc; gm

ext g ext ; g ext x , gy , g ext z

gri G; Gg

G; Ga, Gb; Gi; Go

force on a particle; value of F per unit particle mass; degrees of freedom (4.1.22) probability distribution function corresponding to f(rp) (2.4.1c), crystal size distribution function (6.4.11) electrostatic force on 1 gmol of a charged species in solution (6.3.8a); magnitude of external force on 1 gmol of species i acoustic radiation force in x-direction (3.1.48) radiation pressure force (3.1.47), (7.3.267) force on very small particle due to random Brownian motion (3.1.43) electrokinetic force on particle in double layer (3.1.17) electrostatic force on particle i, Coulomb’s law (3.1.15) London attraction force (3.1.16) force on species k in mass flux ji, force relation (3.1.202) net external force; for gravity see (3.1.5) frictional force on a particle; inertial force on a particle (6.2.45); external force on a particle in z-direction total force and total external force on 1 gmol of species i (3.1.50) total external force on a particle (3.1.59), (6.2.45); components of F ext tp in x-, y-and z-directions force on 1 gmol of species A due to a temperature gradient (3.1.44) Faraday’s constant, 96 485 coulomb/ gm-equivalent acceleration due to gravity; conversion factor; a quantity characteristic of a membrane polymer (4.3.46a) external body force per unit mass; its components in x-, y-and z-directions intrinsic growth rate of ith crystal (6.4.41a) superficial mass average velocity based on empty flow cross section, G for gas phase growth rate of crystal (6.4.25), (6.4.3b); value of G under condition a, condition b; factor representing contribution of species i properties to Qim (4.3.56a); constant

Notation

Gc ; GDr ; G Dμ

Gi ;Gij

Gr(rp) Gtj; Gcrit

Gr; Gr(βr, σv); Gz

h; ho

hþ, h−, hG hmin H; Hf; Hi; 0 H Ci , ½HCi  0 ; HPi ; Hlf, Hvf

H oℓ , H oℓP Hm H A , H oA , H cA ; H1, H2 HD, HM, HS, HSM Hi HTU

i I; I; Ij; IðC þ isbL Þ

xvii

convective hindrance factor (3.1.113); drag factor reducing solute diffusion by hindrance (3.1.112e); function of particle volume fraction in hindered settling (4.2.61) partial molar Gibbs free energy of species i, ratio of solute i velocity to the averaged pore solvent velocity, convective hindrance factor (=Gc ) (3.1.113), (3.4.89b); value of Gi in region j grade efficiency function (2.4.4b) total Gibbs free energy of all molecules in region j (3.3.1), (4.2.23); defined by (7.1.58e) Grashof number (3.1.143e); function defined by (7.2.174); Graetz number (8.1.276) membrane flow channel height, distance between particle and collector (3.1.17), constant in (4.1.9a) for Henry’s law constant, (1/h) is a characteristic thickness of double layer (6.3.31a), height of liquid in a capillary at any time t; value of h as t ! ∞ (6.1.11) contributions of different species to h (4.1.9c) minimum value of h plate height, stack height; molar enthalpy of feed; value of H for species i (6.3.22), Henry’s law constants for species i in gas–liquid equilibrium; (3.3.59) (4.1.7); (3.4.1b), (5.2.6); (5.2.7); (8.1.49), (3.4.1a); molar enthalpy of liquid fraction and vapor fraction, respectively, of the feed height of a transfer unit defined by (8.1.96) magnetic field strength vector Henry’s law constants for species A (3.4.1a,b), (3.4.8); defined by (7.1.20b) components of plate height (7.1.107e–i) partial molar enthalpy of species i height of a transfer unit (6.4.85), (8.1.54b), (8.1.57b), (8.1.65e), (8.1.245b), (8.1.247a), (8.1.357b) unit vector in positive x-direction, current density (3.1.108c) ionic strength of the solution (3.1.10c), (4.1.9b); purity index (1.4.3b), current; value of purity index for region j (1.4.3b); integral (7.2.86)

j ji; jix, jiy, jiz

JD Ji, J i ; ðJ i Þk ; J 1 , J T1

Jiz, Jsz; J iy J iz , J jz , J sz , J Az J t Ay Jvz

k; k

kB ka, kd kb, kf kAo,kAw kc, kg, kxj, ky 0

0

0

0

k c, k g, k x , k y kd, ks; k ℓ ,

kigc, kigx, kigy

0

kT ; k T kgf, kgs

k ℓf , k ℓs ; k pρ

unit vector in positive y-direction mass flux vector of species i, Mi Ji (3.1.98), Tables 3.1.3A, 3.1.3B, (6.2.5n); components of ji in x-, y- and z-directions factor defined by (3.1.143g) molar flux vector of species i (3.1.98), (3.1.99), Tables 3.1.3A, 3.1.3B; value of Ji in region k; diffusive molar flux vector of species 1 (4.2.63); temperature gradient driven molar flux vector of species 1 (4.2.62) z-components of flux vectors Ji and Js; y-component of flux vector J i z-components of flux vectors J i , J j , J s and J A total molar flux of species A in ydirection (5.4.51) volume flux through membrane in z-direction (3.4.60c), (6.3.155a) unit vector in positive z-direction; region or phase, constant in (2.2.8a–c), (2π/λ) (3.1.48) Boltzmann’s constant (3.1.72), (3.3.90c) rate constants for adsorption and dissociation, respectively (4.1.77a) backward and forward reaction rate constants (5.4.42) mass-transfer coefficient of species A in organic or water phase mass-transfer coefficients for species i (3.1.139), (3.4.3) values of kc, kg, kx and ky for equimolar counterdiffusion (3.1.124) mass-transfer coefficients in crystallization (3.4.23a,b); liquid film mass transfer coefficient (5.3.3) mass-transfer coefficients for species i in gas phase when the concentration gradient is expressed in terms of C, molar concentration of species i in gas phase, x, mole fraction of species i in gas phase and similarly y, mole fraction in gas phase, respectively thermal diffusion ratio (3.1.45); thermal diffusion constant (4.2.64) gas film mass-transfer coefficient on feed side and strip side of a liquid membrane (5.4.97a), (5.4.99a) liquid film mass-transfer coefficient on feed side and strip side of a liquid membrane (5.4.97b), (5.4.99b); particle mass-transfer coefficient (7.2.217b)

xviii

Notation

k sþ , k s− 0

0 k il ; k_ i1

kcR, kcE

kgr, knu kil; kil3, kil4, kilo; kimo

k1 k 1m K

Kx; K* K1, K2, K3 KAo, KAw; KAB

Kc, Kg, Kx, Ky KcE K AC Kd

Kd1, Kd2 Ki; K1, K2, K3; K ai ; K ∞i

K iℓ ; Kℓ

rate constants for forward and backward interfacial reactions (5.3.40) distribution ratio of species i between regions 1 and 2, also called capacity factor (1.4.1); distribution ratio defined by (2.2.19) for species i between streams 1 and 2 mass-transfer coefficient in the continuous phase, raffinate based, extract based (6.4.97a,b) rate constant for crystal growth and nucleation, respectively (6.4.51) mass-transfer coefficients for species i in liquid phase; value of kil for condition 3, condition 4, channel inlet; species i mass-transfer coefficient through organic filled membrane pore first-order reaction rate constant (5.3.7) membrane mass-transfer coefficient for species 1 (4.3.1) equilibrium constant for a chemical reaction (3.3.68), or an ion exchange process (3.3.121i), a constant (6.3.49) mole fraction based K for a chemical reaction (5.2.35); defined by (5.4.100) constants in membrane transport (6.3.155a,b) overall mass-transfer coefficient of species A based on organic or water phase; equilibrium constant (7.1.42c) overall mass-transfer coefficients (3.4.5), (3.4.6) Kc based on extract phase (6.4.80) equilibrium constant for ion exchange reaction (5.2.122) equilibrium constant for protein–ion exchange resin binding (4.1.77c), ionization equilibrium constant (5.2.4) dissociation constant for solutes 1 and 2, respectively (5.2.61a), (6.3.29) equilibrium ratio of species i between regions 1 and 2 (1.4.1) or (3.3.61); value of Ki for species 1, 2 and 3; values of Ki in terms of activities (4.1.3); o ƒ il γ∞il =P in dilute solution stripping (4.1.19b) overall liquid-phase mass-transfer coefficient for species i (7.1.5a); reaction equilibrium constant in the liquid phase based on molar concentrations (5.2.52a)

0

Kijc; kijx; k ijx

Ko, Kw

Kis, Kps

KxE, KxR

l; ℓ

ℓik , ℓki ℓloc

ℓx , ℓy , ℓz L; Lf ; Lþ

Lii, Lis, Lss Lik,Lki,LiT Lp ; Lap ; Lbp

LT; Lmin; LMTZ; LUB m; mB; mi; m1; m0i

mij; moij ; mij ðnÞ; maij

miσ

molar concentration based overall mass-transfer coefficient for phase j (8.1.1c); j phase mass-transfer coefficient (8.1.60); value of kijx for equimolar counterdiffusion (8.1.62a) overall mass-transfer coefficient based on organic or aqueous phase, ionization product for water (5.4.41c) values of K for ion i/protein (p)–salt (s) exchange on an ion exchange resin (7.1.109d), (3.3.122b) overall mass-transfer coefficient Kx (3.4.5) based on extract phase and raffinate phase, respectively (6.4.77), (6.4.81) length of a device, length of molten zone in zone melting (6.3.109b), characteristic dimension of the separator; constant in (2.2.8a–c), length phenomenological coefficients (3.1.203) characteristic length of a local volume corresponding to a point in volume averaging Section 6.2.1.1 dimensions of a rectangular separator, Figure 3.2.1 length of a separator, dimension of length, characteristic crystal size; molar feed flow rate; nondimensional L (7.2.38) phenomenological coefficients for binary system (i, s) (3.1.208), (3.1.209) phenomenological coefficients (3.1.205) hydraulic transport parameter in Kedem–Katchalsky model (6.3.158a); value of Lp in perfect region; value of Lp in leaky region separator length (= L); (7.1.60); Figure 7.1.5(b); (7.1.21g) velocity profile constant (7.3.134); moles of B; moles of species i in separator; moles of species 1 in separator; total number of moles of species i in separator moles of species i in region j, total number of moles of i in region j at t = 0; moles of i in region j after nth contact; number of atoms of ith isotope of the element in region j moles of species i in interfacial region σ

Notation

xix

mi,j

mF,R mp; msl mtj

m11(t), m21(t) mr11 ðtÞ, mr21 ðtÞ mag

mop M; Mi, Ms; M sl

Mt Mw Mse MT; MTa, MTb

ðiÞ

Mo(n); Mof , ðiÞ ðiÞ Moℓ , Mov

j

j

Mof r , Mogr MWCO

n; nc

~ ðr p Þ nðr p Þ; no ; n

ni; nix,niy,niz; nij

nk

molality, moles of i per kilogram of solvent in region j: j = R, resin; j = w, aqueous phase molality of fixed charges in the resin phase mass of particle; solvent moles in stationary liquid phase (7.1.104b) total moles of all species in region j (j = f, feed; j = 1, vapor phase; j = 2, ℓ, liquid phase) moles of species 1 and 2 in region 1 at time t values of m11(t) and m21(t) in the case of a chemical reaction magnetophoretic mobility (7.3.251) molecular weight, a metal species, number of stages in stripping section of a cascade; value of M for species i, for solvent s; M for coating liquid in stationary phase average molecular weight of solution (3.1.56) magnetization of wire (3.1.23) seed mass density per unit liquid volume (6.4.40a) suspension density of a crystalcontaining solution (2.4.2f), (6.4.18); value of MT for cases a and b nth moment of the density function (2.4.1g); ith moment of molecular weight density function of feed, liquid and vapor, respectively (6.3.70) jth moment of crystal size density functions f(rp) and fg(gr) molecular weight cut off of a membrane number of species/components in a system, number of contacts, stage/ plate number, number of positive charges in a metal ion, number of unit bed elements; number of collectors, number of channels (7.3.109) population density function, particle number density function (2.4.2a); nucleation population density parameter (6.4.7); defined by (6.4.46b) mass flux vector of species i, MiNi; its components in x-, y- and z-directions; ni through surface area Sj outwardly directed unit normal to the k-phase surface (6.2.26b–d)

nmax; nmed, npar

np, np; npy; nt

N; N(rp); Nðr min , r p Þ, Nðr p max Þ

~ Ni N;

N A ; N rA , N tAy ; N dil

Ni; Nim, Njm; Nix, Niy, Niz; Nir; Nij; jN iy j p

N iz

Noj; NoℓP; Np

NR; NS Ni(rp); Nit

Nt NtoE, NtoR

Noi NTU

p p

peak capacity (3.2.32) (6.3.26a); refractive index of medium and particle particle number flux, (3.1.65), (3.1.66), (3.1.68); particle flux in y-direction; number of turns by gas in a cyclone (7.3.146b) total number of stages in a multistage device or in the enriching section of a cascade, anionic species in Donnan dialysis, a metal species, number concentration of molecules; numbers/ cm3, number of particles per unit fluid volume in the size range of rmin to rp; value of N(rmin, rp) for r p max Avogadro’s number (6.02  1023 molecules/gmol); plate number for i (6.3.27a) molar flux of species A in a fixed reference frame without and with reaction; total molar flux of species A in facilitated transport or countertransport or co-transport in y-direction; normality of diluate solution (8.1.404) species i flux; Ni through membrane, Nj through membrane; components of Ni in x-, y- and z-directions; radial component of Ni; Ni through surface area Sj; magnitude of Niy z-component of species i flux, Niz, based on unit pore cross-sectional area (3.1.112a) number of transfer units (8.1.92), (8.1.96); defined in (8.1.96); number of pores per unit area size rp (6.3.135d) flux ratio (3.1.129a); solvent flux number density of crystals having a size less than rp and growth rate gri; total number of crystals per unit volume having a growth rate of gri (6.4.41c) total number of particles per unit volume (6.4.10) number of transfer units based on extract and raffinate phases, respectively (6.4.86a), (6.4.83) number of particles of size rpi (2.4.2k) number of transfer units (8.1.54c), (8.1.57c), (8.1.66b), (8.1.67d), (8.1.338) dipole moment of a dielectric particle stoichiometric coefficient for product P, kinetic order in the dependence of nucleation rate (6.4.30a)

xx

pA; pB; pi,pj pAb; pAi; pB;ℓm

pif, pij, pip, piv pib , pii

pH; pI

pKi P; Po;Pc P; P f ; P j , 0 Pp ; Pp ; Pℓ

P 0i ðπÞ; P i

Patm; Pliq; P1, P2

Pe; Pei ; Pem i ; Pez; eff ; Pezj

sat P sat i , Pj ; sat P sat ; P iPℓ i, curved

P sat M P0 PR, Pw Po, Pr q

Notation

partial pressures of species A, species B; species i and species j value of pA in the gas bulk; value of pA at gas–liquid interface; logarithmic mean of pB (3.1.131b) value of pi in feed gas, region/stream j, permeate gas, vapor phase species i partial pressure in bulk and particle interface, averaged over bed cross section, Figures 3.4.4(a), (b) indicator of hydrogen ion concentration (5.2.65a); isoelectric point for a protein/amino acid; at pI = pH, net charge is zero −log10 Kdi, (5.2.65b) for i = 1, (5.2.74b) for i = 2, (6.3.29) total pressure, system pressure; standard state pressure; critical pressure local solute permeability coefficient (6.3.157b); feed pressure; total pressure of jth region and permeate, respectively; gas pressure (Figure 7.2.1(b)); gas pressure at the end of a capillary of length l (6.1.5d) equilibrium gas-phase pressure for pure i adsorption at spreading pressure π, which is the same for a mixture (3.3.111a); pressure at crossover point for solute i in supercritical extraction atmospheric pressure; pressure in the liquid (6.1.12); purification factors (7.2.97) Péclet number (3.1.143g), (7.3.34d); Pe number for dispersion of solute i (6.3.23a); pore Péclet number (6.3.145a); (z vz/Di,eff.z) (7.1.18h); j phase Pez (8.1.92) vapor pressure of pure i and pure j, respectively, at system temperature; value of P sat i on a plane surface; value of P sat i on a curved surface value of P sat i for pure species i of molecular weight M amplitude of pressure wave (3.1.48) pressure of resin phase and external aqueous solution, respectively power number (6.4.976), power (3.1.47) number of variables in a problem, stoichiometric coefficient for species Q, heat flux in a heat exchanger attached to a cooling crystallizer (6.4.47a), the power of MT in expression (6.4.39a) for B0, factor (8.1.150)

qfr qi(Ci2), qi1(Ci2), qi1 ; q0i1 , qsi1

qiR, qis

qmaxR q(r, ψ, λ)

Q

Qc; Qd Qf; Qg; Qh; Qi; Qℓ, Qo, Qp, QR, Q1, Q2

QAm, QBm, Qim, Qjm

Qij

Qov im

Qcrys, Qsub

Qim0 Qsm, Qsc

r

fraction of light reflected (3.1.47) moles of species i in solid phase 1 per unit mass of solid phase, crosssectional average of qi1, initial value of qi1; saturation value of qi1 moles of species i per unit mass of ion exchange resin (R) or solid adsorbent (s) maximum molar fixed charge density per unit resin mass probability that a molecule having configuration (r, ψ, λ) does not intersect pore wall (3.3.89f) volumetric fluid flow rate, product species in reaction (5.3.5), hydraulic permeability in Darcy’s law (6.1.4g,h), heat transfer rate (6.4.47a,d) electrical charge of a collector; volume flow rate of dialysate solution volumetric feed flow rate to separator; Darcy permeability for gas through packed bed; amount of heat supplied at a high temperature; electrical charge on 1 gmol of charged species i; amount of heat rejected at a low temperature; volumetric flow rate at membrane channel inlet; electrical charge of a particle; excess particle flux (7.2.123), volumetric flow rate of product stream from separator; heat supplied at the reboiler per mole of feed; volume flow rate of overflow; volumetric rate of underflow/concentrate permeabilities of species A, B, i and j, respectively, through membrane in gas permeation and pervaporation, respectively permeability of species i through region j (= A, B, C, D, 1, 2) of the membrane overall permeability of species in membrane pervaporation for a membrane of thickness δm heat transfer rate of a solution during crystallization and subcooling, respectively value of Qim for Cim = 0 solvent permeability through membrane and cake in cake filtration vector of molecular position, radius vector, unit vector in radially outward direction

Notation

r r1, r2; r1i, r2o; rc; rf

rg; rh; ri; rin

ro

rp; rp,a; rp1, rp2; r pi ; rs; rt; rw

rmax, rmin; rp,50 qffiffiffiffiffi r 2p

r p , r p1; 0 , r piþ1; i

r p3; 2 ra; re; re|flow R; R

^ cδ , R ^ cw Rc ; R

Rh; Ri; R0i; RL

Rm ; R s ; R ∞

RA, Ri

xxi

radial coordinate, any characterizing property of a continuous mixture radii of curvature of interface (3.3.47); liquid outlet radii in tubular centrifuge; critical size of a nucleus (3.3.100b), cyclone radius; free surface radius in tubular centrifuge radius of gyration of a macromolecule; hydrodynamic viscosity based radius (3.3.90f); radius of spherical solute molecule (3.3.90a); radius of liquid– liquid interface in tubular centrifuge radius of a sphere whose volume is equal to that of an ellipsoid (3.1.91g), radial location of the center of solute peak profile, radius of a cylindrical centrifuge radius of particle, pore radius; analytical cut size (2.4.18); particles of two different sizes; dimension of particle of a certain size; particle size for classification (2.4.8); cyclone exit pipe radius; radius of wire maximum and minimum radius of membrane pores or particle sizes; equiprobable size hydraulic mean pore radius (3.4.87) mean of particle size distribution based on f(rp) and n(rp), respectively (2.4.1e), (2.4.2g), (2.4.2h) Sauter mean radius of a drop, bubble or particle (6.4.89) rate of arrival of cells; fractional water recovery; value of re in a cell (6.3.172b) universal gas constant; radius of a tube or capillary, reflux ratio (2.3.5), (2.3.7), (8.1.137), solute rejection in reverse osmosis, solute retention in ultrafiltration cake resistance; specific cake resistance per unit cake thickness and unit cake mass, respectively (6.3.135l) hydraulic radius (6.1.4c); solute rejection/retention by membrane for species i, retention ratio for species i (7.3.211); fraction of solute i in mobile phase (7.1.16c); largest radius of a conical tube membrane resistance; resolution between neighboring peaks (2.5.7) in chromatography; value of Ri at large Pem i molar rate of production of A per unit volume (5.3.7), (6.2.2d), for species i

R1, R2 Rþ (R−)

Rij;Rim;Ri,reqd

Rik, Rki; Rmin

Robs, Rtrue; Re; ReL; Reimp

s

sm; smp, sms; sp

s1, s2 S; Sc

Sf, Sk, SM; Sj; SM, Sdp

Sij; Sij Sim; Sim0 S∞ Sσ ; Sσ ; S1 , S2

Stj; Smag

intrinsic RO rejection of the unionized species 1 and ionized species 2 ion exchange resin with fixed positive (negative) charge; cation (anion) of a surface active solute permeation resistance of region j (j = A, B, C, D) in the membrane to species i; membrane permeation resistance of species i; solute i rejection required in a RO membrane phenomenological coefficients (3.1.207); minimum reflux ratio (8.1.170) observed and true solute rejection in ultrafiltration and RO; Reynolds number; Re for a plate of length L (3.1.143a), Table 3.1.5; Re for an impeller (Table 3.1.7), (6.4.96) fractional supersaturation (3.3.98b), ^ cδ power of the ΔP dependence of R (6.3.138j), Laplace transform complex variable (5.4.35), solution volume fed to bed per unit empty bed cross section (7.1.17a), eluent/salt (7.1.109b), salinity (10.1.13) pore surface area/membrane volume (6.3.135e); value of sm for membrane pore volume or membrane solids volume (6.3.135f,g); sedimentation coefficient (4.2.16b) two different solvents solute transmission/sieving coefficient (6.3.141e), supersaturation ratio (3.3.98c), stripping factor (8.1.135), (8.1.189b); bed/column cross-sectional area cross-sectional area vector of feed entrance, kth feed exit and membrane surface area vector in a separator; cross-sectional area of flow for jth stream; selectivities (7.3.219a,b) molar entropy of species i; partial molar entropy of species i (3.3.17b) solubility coefficient of species i in membrane; value of Sim for Cim = 0 value of S when Pem i >>1 (6.3.145b) surface area of interfacial region (Figure 3.3.2A); molar surface area in gas–solid adsorption (3.3.107); sieving coefficients for species 1 and 2 total entropy for region j (3.3.3); magnetic field force strength (7.3.251)

xxii

Sobs; Strue Sc; Scc Sh; Shc, ShD; Shp; Shz

St

t; tbr; tc; tres; tσ

þ þ tþ i ; t1 , t2 ; t; t 1 , t 2 , t 3

0 t in i , ti

tim, tis t s1 , t s2 t Ri ; t RM ; t Ro

T; T1; T2; Tc

Tf; Tg; Ti; Tp

TC; TH, TL; TR

Tcf; Tci; Tmi

Tsat Tsol Th u uc

Notation

observed solute transmission/sieving coefficient; true value of S Schmidt number (3.1.143a); Sc for continuous phase Sherwood number (3.1.143a); Sh for continuous phase and dispersed phase, respectively; Sh in a packed bed (7.2.218a); Sh at location z (7.2.64) Stanton number (3.1.143g), Stokes’ number (6.3.41) time; breakthrough time; time for cut point in the chromatographic separator output (2.5.1); residence time; thickness of interfacial region (Figure 3.3.2A) nondimensional time variable for species i (6.3.12); value for t þ i for species 1, 2 (3.2.9), (3.2.20); breakthrough time (7.1.15c); value of t for species 1, 2 and 3 times when species i appears and disappears, respectively, from a chromatographic separator output transport number of i in membrane or solution (3.1.108d) time required for solvents, s1, s2 (3.2.24) retention time for species i in capillary electrophoresis (6.3.18a), (7.1.99d); retention time for the mobile phase; retention time based on vz,avg (7.3.207) absolute temperature; temperature of cooled plate; temperature of the heated plate; critical temperature feed temperature; glass transition temperature of a glassy polymer; value of absolute temperature T of region i; product temperature temperature of condenser; two temperatures in supercritical extraction; reboiler temperature temperature of cooling fluid, critical temperature of species i; melting temperature for species i temperature at which the solution is saturated temperature of the solution due to undergo crystallization dimensionless group (3.1.46b) number of fundamental dimensions (Section 3.1.4.1) mass fraction of solute in the crystallized solution on a solid-free basis

ui; uk; uij; uil; uio; uis; uilb

uo; urc; uro

um i uiEn, uiRn

urig, uris urij U i ; Up Ui new

Ui Uix, Uiy, Uiz

^ ik U ik ; hU ik ik ; U int int U int p ; U px i ; U pr p

Upt; Upr; Uprt; Upzt, Upyt

v, vt ; v , vt ; vþ , ref vt ; vtj , vtj

vi; vij

vk ; vkz ; vint k

vrp

mass fraction of impurity species i; mass fraction of species k; mass fraction of species i in region j (1.3.5) or jth stream (2.1.20); value of ui in melt; value of ui initially in the solid; value of ui in the recrystallized solid; value of uil in the bulk melt mass fraction of solute in the solution charged for crystallization; uc/(1 − uc); uo/(1 − uo) (6.4.47b) ionic mobility (3.1.108j) mass fraction of species i in extract and raffinate streams, respectively, from stage n defined by (8.1.349) and (8.1.350) weight of solute i per unit weight of phase j (9.1.32) migration velocity vector for species i (3.1.84b); particle velocity vector averaged velocity vector of ith molecules due to all forces defined by (3.1.103) components of migration velocity of species i in x-, y- and z-directions, (3.1.82) value of Ui in the kth phase/region; average of Uik in the kth phase/region; defined as a fluctuation by (6.2.28) internal particle velocity vector (6.2.50d); its component in the direction of the internal coordinate xi; internal particle velocity vector for all particles of size rp terminal velocity vector of particle (3.1.62); radial particle velocity; terminal value of Upr; value of Upt in z-direction (6.3.1), (7.2.211), y-direction (7.3.154) mass averaged velocity vector of a fluid (also vt); molar average velocity vector of a fluid (also vt ); nondimensional v (6.3.39); reference vt; values of vt and vt on surface area Sj (2.1.1), (2.1.2) averaged velocity vector of ith species; value of vi on surface area Sj (2.1.1) velocity of region k; z-component of velocity vk of region k; mass average velocity of the interface of two phases particle diffusion velocity relative to that of the fluid phase (3.1.43)

Notation

vci ; vCi ; vo; vo; vþ Ci ; vog

vr; vs; vso; vtw; vtz; vvz

vx, vy; vyw; vywf

vz ; vzmax ; vz; avg ; vz; avg; f

v∞ vf r ; vþ i, eff

vAA z ; vzH , vzL

vEOF; vEOF,z vr V ; V b; V d ; V f ; ^ a ; V_ ; V V h; V

~ c; V i; V i ; V j V

xxiii

average velocity of liquid zone carrying species i (7.3.208); concentration wave velocity of species i (7.1.12a); velocity of micelles, superficial velocity (3.1.176), in a packed bed (6.1.4a); amplitude of square-wave velocity (7.1.72a); nondimensional concentration wave velocity (7.1.105o); superficial velocity for gas phase radial velocity; shell-side velocity (3.1.175), volume flux of solvent/ permeate (7.2.71); value of vs at membrane channel inlet; tangential fluid velocity in wall region (7.3.134); same as vz; vapor velocity in z-direction (6.3.47a) local convective velocity in x-direction, y-direction; value of vy at wall; value of vyw at feed entrance (7.2.39) velocity of fluid in z-direction; maximum value of vz, averaged value of vz over flow cross section; value of vz,avg at feed location (6.1.5f) uniform gas velocity far away from object velocity of freezing interface in zone melting (6.3.110c); nondimensional velocity of species i in capillary electrophoresis (6.3.12) velocity of interface AA between suspension and clarified liquid in zdirection (4.2.52); interstitial gas velocity at high-pressure feed step and low-pressure purge step, respectively (7.1.53a,b) electroosmotic velocity (6.1.22); value in z-direction defined by (7.2.167) volume of a region, volume of separator, volume of feed solution/ eluent passed, volume of a sphere (3.3.52a), voltage between electrodes; volume of buffer; volume of the dialysate solution; volume of feed solution; hydrodynamic volume of a macromolecule/protein (3.3.90f); adsorbed monolayer phase volume/ weight of adsorbent (3.3.114b); radial volume flow rate of solution (Figure 7.1.6); defined by (7.1.18i) critical molar volume of a species; molar volume of pure species i; partial molar volume of species i; volume of region j: j = 1, stationary phase; j = 2, mobile phase

p

Vk; Vl; V ℓ ; Vo; V to ;Vo

VM; Vp; Vs; V Sℓ ; Vsp

V s; V t ; V w; V σ

Vfi; Vfe Vf0; VfR; V N i ; V Ri

V iex cap ; V eff ; V channel V AY V ic Vim V ij VCR

w; wcrys ; wi ; wso

wij; wtj; wtz; wstj ; wtE, wER

wE, wM, wR wl; ws; wsp (rp); wa

volume of region k; liquid volume; specific pore volume of a microporous adsorbent (4.1.64a); volume of organic solvent; total amount of Vo (6.3.99); sample volume (7.1.101b) mobile-phase volume in a column (7.1.99g); volume of a particle, cumulative permeate volume (7.2.87); stationary adsorbent-phase volume in a column; suspension volume; stationary liquid-phase volume; mobile-phase volume present inside the pores of a particle (7.1.110c) partial molar volume of solvent; averaged partial molar volume (3.1.56); volume of water; volume of interfacial region, Figure 3.3.2A initial volume of feed solution; final volume of solution volume of feed solution at time t = 0; volume of retentate; net retention volume for species i (7.1.99j), retention volume for species i (7.1.99e,f), (7.1.99h) volumetric ion exchange capacity of packed bed; liquid filled centrifuge volume; volume of channel partial molar volume of electrolyte AY (3.3.119b) value of V i for a crystal of i molar volume of species i at its normal boiling point value of V i in region j volume concentration ratio (Vf 0/Vf R) (6.4.98) mass of adsorbent, quantity defined by (5.2.54); mass of crystals formed at any time, mass of solute i charged; mass of solution charged to the crystallizer mass flow rate of species i through Sj (2.1.4); total mass flow rate through Sj (2.1.5); total mass flow rate in zdirection; total solids flow rate in jth stream (2.4.3a); wtj for j = extract, E, wtj for j = raffinate, R weight of mixture at E, M and R, respectively mass of stationary liquid phase used as coating; mass of particles per unit of fluid volume (2.4.2e); mass of a solid particle of size rp (2.4.2e); waist size of a light beam

xxiv

Notation

W, WA W Wbi, Wij, Wtj

WtAm, WtBm Wtf, Wtl, Wtv Wtlb, Wtld

Wt1, Wt2, Wtp W ttjn We x, xA x A

xAb, xAi xi, xij, x iσ , xi,j

x iE , x iE , x eiE

xilb, xild

x iR , x iR , x eiR

0

xjf, xjp, x ip

width of rectangular channel walls, molar transfer rate of species A total reversible work done (3.1.25a,b) band width of the chromatographic output of species i (Figure 2.5.2), molar flow rate of species i through Sj (2.1.7), total molar flow rate through Sj (2.1.8) molar rate of permeation of species A and B through membrane molar flow rate of feed, liquid fraction and vapor fraction, respectively total molar flow rate of bottoms product and distillate product from a distillation column total molar flow rate of streams j = 1, j = 2 and j = permeate total molar flow rate of stream j for stage n in enriching section of cascade Weber number, (ρc vc2 dp/γ) (6.4.91) coordinate direction, mole fraction of species A in liquid phase hypothetical liquid phase mole fraction of species A in equilibrium with yAb bulk liquid mole fraction of species A, value of xA at a two-phase interface mole fraction of species i, value of xi (i = A,B, 1, 2, s(solvent), etc.) in region j or stream j (j =1, 2, f(feed), g = gas, l = liquid, p = permeate, R = resin, s = solution, v = vapor, w = water), xij where j=σ, the surface adsorbed phase, mole fraction of species i in region j when molecular formula is substituted for species i mole fraction of species i in extract phase, value of xiE in equilibrium with xiR, value of xiE if both streams leave stage in equilibrium bulk value of xil, mole fraction of i in liquid product from reboiler in a distillation column, xil in liquid product from condenser at the top of a distillation column mole fraction of species i in raffinate phase, value of xiR in equilibrium with xiE, value of xiR if both streams leave stage in equilibrium mole fraction of species j in feed and permeate, respectively, mole fraction (Figure 7.2.1(b))

xs,lm X Xij, Xijn

X aij

y, yf

yi, yA, yH, yL

0

y A , y  , y ℓim

yAb, yAi Yij, YiR, Y2, Y2,f

Y11(t), Y21(t) Y aij , Y_ ij; Y_ rj

z, zcr (or zcrit), zH, zL, zAA, zBB

z HL , z HS zþ , z þ i

z α , zβ , z o ðC i2 Þ Zi, Zi,eff, Zp

defined by (3.1.137) molar density of fixed charges in electrically charged system mole ratio of species i in region j (1.3.2) or stream j (2.2.2d), Xij for state n abundance ratio of isotope i in region j (1.3.7) coordinate direction, normal to gas– liquid, liquid–liquid or membrane– fluid interface, feed gas mole fraction (Figure 7.1.14) mole fraction of species i and A in vapor/gas phase, respectively, fluid mole fraction (7.1.62b), gas-phase mole fractions defined by (7.1.54a) hypothetical gas-phase mole fraction of species A in equilibrium with xAb, nondimensional y-coordinate (5.3.5j), y-coordinate of limiting trajectory (7.3.263a) bulk gas mole fraction of species A, value of yA at a two-phase interface segregation fraction of species i in region j (1.3.8a), segregation fraction of solute i in retentate in a membrane process (6.4.107), defined by (7.2.19) and (7.2.20) segregation fraction of species 1 and 2 in region 1 at time t segregation fraction for an isotopic mixture (1.3.9a), of species i stream j (2.2.12), of particles of size r in stream j (2.4.5) coordinate direction, critical distance (7.2.133), characteristic locations at pressures PH and PL (7.1.55e), vertical coordinates of interfaces AA and BB, respectively height of the liquid level of a dilute suspension, height of sludge layer nondimensional z-coordinate (Figure 3.2.2), (6.3.12), value of zþ for center of mass of concentration profile of species i (3.2.22a,b) z-coordinate locations of regions α and β, respectively, defined by (7.1.17e,f) electrochemical valence of species i, effective charge on ion i due to the diffuse double layer (< Zi) (6.3.31a), value of Zi for a protein

Notation

xxv

Greek letters α

αf , αl , αv

hf

ft

αij ; αij , αij , αht ij ; αijn

αin 0

α αAB ; αevap ; αperm ij ij

β

βf , βℓ , βv

βi ; βp; βA, βB; β; βc

γ; γcr ; γA ; γR ; γR2

γ f , γℓ , γv γi; γif, γij; γjf

constant in (3.1.215), separation factor between two species 1 and 2 in capillary electrophoresis (6.3.27e) parameters for feed, liquid fraction, vapor fraction in a continuous chemical mixture described by a Γ distribution function (4.1.33f), (6.3.71) separation factor between two species i (i = 1, A, s (solvent)) and regions j (j = 2, B (species), i), value of αij between two streams (2.2.3), (2.2.4), (2.2.2a); value of αij for nth plate multicomponent separation factor between species i and n (1.6.6) separation factor of Sandell (1.4.15) value of αAB under ideal condition of zero pressure ratio (6.3.198), (8.1.426); value of αij for evaporation only (6.3.180); ideal separation factor in vapor permeation (6.3.180) defined by (2.2.8a,b), exponent in (3.3.89a), indicator of macromolecular shape for a given rg (3.3.90e), constant in (3.1.215), defined by (7.1.52b), (7.2.58), (7.2.138) compressibility of fluid (3.1.48), parameters for feed, liquid fraction and vapor fraction in a continuous chemical mixture (4.1.33f), (6.3.71) exponent in (3.3.112c), parameter (7.2.18); compressibility of particle (3.1.48); defined by (7.1.52a); coefficient of volume expansion (6.1.9); a parameter (6.4.129); coefficient of thermal expansion of solution density (7.3.233) surface tension, interfacial tension, pressure ratio (6.3.197), (6.4.123), (8.1.426), parameter (7.1.18f); critical surface tension for a polymer; Damkohler number (kbδ2m =DA ) (5.4.100); (DAk1/k 2ℓ ) (5.3.11); defined by (5.3.35j) parameters for feed, liquid and vapor fraction (4.1.33f), (6.3.69) activity coefficient for species i, dimensionless steric correction factor; value of γi in feed stream, in phase j; value of γj in feed stream (species j)

γ1, γ2, γ3, γ4, γ12; γSL, γLG, γSG

γ_ ; γ_ w γ∞il ; γ∞il, u

Γ Eis

Γ iσ ; Γ Eiσ

δ; δ(x), δ(z), δðz þ Þ; δc δg ; δi ; δℓ ; 0 δm ; δs ; δ ℓ

δA , δB , δC δ1 , δ2 , δn ΔC; ΔE s ; ΔG; ΔGt ; ΔH s ; ΔP; ΔS; ΔT

ΔH crys ΔG0 , ΔH0 , ΔS0

Δdp ; Δr p ; Δws

factors/parameters in plate height (7.1.107f–i); interfacial tension between two bulk phases 1 and 2; interfacial tension between phases S and L, L and G, S and G, respectively (S = solid, L = liquid, G = gaseous) shear rate (3.1.74), (6.1.31); wall shear rate infinite dilution activity coefficient for species i in the liquid; activity coefficient of species i in liquid phase on a mass fraction basis at infinite dilution (6.3.83) molar excess surface concentration of species i/cm2 of pore surface area (3.1.117a) surface concentration of species i (3.3.34); algebraic surface excess of species i (3.3.40b), (5.2.145a) falling film thickness; delta function in x-, z- and zþ-directions (3.2.14b); cake layer thickness thickness of gas film; solubility parameter of species i, characteristic thickness of concentration profile (7.3.202); thickness of liquid film; membrane thickness, pore length; sorbed surface phase thickness (3.1.118b); thickness of liquid film from phase interface to reaction interface (5.3.22) membrane thickness in regions A, B, C of a membrane (Figure 6.3.35(b)) thickness of layer 1, layer 2, layer n in a composite membrane molar supersaturation (3.3.98a); activation energy for crystal growth rate (3.4.29); Gibbs free energy change, e.g. for forming a crystal (3.3.100a); change in Gibbs free energy for all molecules; enthalpy of solution of a gas species in a polymer; pressure difference (Pf − Pp), etc.; selectivity index (7.2.93); extent of supercooling (3.3.98d), temperature difference (T2 − T1) heat of crystallization standard Gibbs free energy change, enthalpy change and entropy change for a reaction particle diameter difference between two consecutive sieves; (Δdp/2) or the net growth in size of all seed crystals (6.4.40i); mass of crystals retained on a sieve of given size

xxvi

Δx, Δy, Δz, ΔzL , ΔzH Δμ0i ; Δν1 , Δνþ , Δν− , Δπ

Δρi; Δρi,sat; Δϕ

ΔR ri

Notation

lengths of a small rectangular volume element in three coordinate directions, defined by (7.1.53a,b) μ0i1 −μ0i2 , number of water molecules, cations and anions, respectively released during binding; osmotic pressure difference (πf – πp) mass concentration change of species i in crystallizer (6.4.24); change in saturation concentration due to ΔT (6.4.51); potential difference between two phases defined by (8.1.328) del operator in internal coordinates xi, yi and zi

θ

θi

θid, θkd

κAo ; κd ;κi ε εa; εb εd; εi; εk

εm;εp;ε0;ε0

εB; εC

ε12

ζ

void volume fraction in a packed bed, electrical permittivity of the fluid (= ε0εd) phase angle; fractional cross-sectional area of membrane which is defective dielectric constant of a fluid, porosity of particle deposit on filter media; Lennard–Jones force constant for species i; volume fraction of phase k in a multiphase system porosity of membrane, pellet, bead or particle; dielectrical constant of a particle; electrical permittivity of vacuum; porosity of spacer in spiralwound module (3.1.170), porosity of clean filter fractional area of defects in glassy skin region B (Figure 6.3.35(b)); value of fractional membrane cross section in region C enrichment factor for species 1 and 2 (1.4.10), (2.2.2e) zeta (electrokinetic) potential (3.1.11a)

κi1 , κN i1

κif ; κij ; κim

0

00

κij , κij κio

κip κi; loc ðr Þ 0

κis , κis η

ηþ 0 ηi ; ηj; ηij; η j , ½η; ½ηi ; ηiF

hηi i;hη2i i

recycle ratio (2.2.22), nondimensional variable (3.2.11), (6.3.14b), (6.3.114), fraction of particles collected (7.3.37), (7.3.38), (7.3.46), intrinsic viscosity variable defined by (3.2.17) variable (2.5.5), η for species i (3.2.25); impurity ratio for region j (1.4.3a); impurity ratio for ith species in region j (1.6.4); defined by (1.4.4); intrinsic viscosity (3.3.90f); its value for polymer i (7.1.110f); current utilization factor (8.1.406) first moment of Ci(ηi , t þ i ) (3.2.26); defined by (3.2.29)

0

κis, eff κiE κiR κp1 ; κ0p1

κm salt

cut (2.2.10a), stage cut, fraction of adsorbent sites occupied (3.3.112a), contact angle, gas–liquid–solid system (Figure 3.3.16), contact angle between membrane surface and liquid (6.3.140) component cut for species i (2.2.10b), Langmuir isotherm (7.1.36a), nondimensional mobile phase concentration (7.1.18b) component cut for species i and k, respectively, in the distillation column top product (8.1.224) distribution coefficient of species A between aqueous and organic phases (3.4.16); reciprocal Debye length (3.1.17); distribution coefficient of species i distribution coefficient of species i between regions 1 and 2 (1.4.1), its Nernst limit (3.3.78), (3.3.79) partition coefficient of solute i between feed solution and feed side of the membrane; partition/distribution coefficient of solute i between two immiscible phases; κij for feed solution and the membrane (3.3.87), (3.3.89a) effective value of κij when there is no reaction or there is reaction partition coefficient of i between an organic and an aqueous phase (6.3.101a) κim at permeate–membrane interface (3.4.58) local equilibrium partition constant (3.3.89g) impurity distribution coefficient (6.3.109a) effective partition coefficient in zone melting for species i (6.3.119a) distribution (partition) coefficient for species i in solvent extraction distribution coefficient of i in ion exchange system (3.3.115) protein partition coefficient in aqueous two-phase extraction; protein distribution coefficient in the absence of a charge gradient at interface distribution coefficient of salt, molal basis (4.1.34o)

Notation

λ

λ; λx, λz; λi ; λþ ; λ− ; λoi

λ1 , λ2 Λ

μ; μ0 ; μdr μC , μD μm EOF μi , μ0i , μm i ; m μm ; μ i, eff ion, g

μij ; μijn ; μeℓ ij

μijPl μℓ ðMÞ, μv ðMÞ

m m μm 0 , μp , μs ; μ∞

xxvii

Debye length (3.1.10b), mean free path of a gas molecule (3.1.114), filter coefficient (7.2.187), parameter for a dialyzer (8.1.399), parameter for a distillation plate/stage (8.3.38), latent heat of vaporization/condensation molecular conformation coordinate (3.3.89c); electrode spacings (7.3.18); retention parameter for species i (7.3.213), ionic equivalent conductance of ion i (3.1.108r); value of λi for a cation; value of λi for an anion; value of λi at infinite dilution (Table 3.A.8) defined by (5.4.100) equivalent conductance of a salt (an electrolyte) (3.1.108s) dynamic fluid viscosity; value of μ at inlet; viscosity of drop fluid fluid viscosity of continuous and dispersed phases, respectively electroosmotic mobility (6.3.10c) species i chemical potential and its standard state value; ionic mobility of ion i (3.1.108m); effective value of μm i (6.3.8d,g); ionic mobility of ion in gas phase value of μi in region j; value of μij for plate/stage n; value of μij in a system with electrical charge (3.3.27) value of μij for a planar interface (3.3.50) chemical potential of species of molecular weight M in liquid and vapor phases, respectively, in continuous chemical mixtures magnetic permeabilities of vacuum, particle and solution, respectively; chemical potential of a crystal

π; π πf ; πi ; πp , πw

π1 , π2

ρ; ρb ; ρc ; ρe ; ρf ; ρf m ; ρi ; ρij

ρℓ ; ρp ; ρpg ; ρs ; ρt ; ρtj

ρv ; ρi1 , ρif

ρiℓ ; ρip ; ρp; rp

ρR ; ρ1 , ρ2

ρdR ; ρpR ρavg ; ρisat

ρik , ρik kinematic viscosity; υ at axial location z; υ at inlet stoichiometric coefficient for species i in any chemical reaction moles of ions A and Y produced by dissociation of 1 mole of electrolyte AY

υ; υðzÞ; υ0 υi υA , υY

ξ ξ 12 , ξ rp ; r p 1

2

extent of separation for a binary system (1.4.16), (2.2.11), (6.3.105) extent of separation for components 1 and 2 in particle classification (2.4.9), for particles of size r p1 and r p2 (2.4.6)

ρik; avgi hρi i

σ σi

constant (3.1416…); osmotic pressure, spreading pressure (3.3.41a) osmotic pressure of feed solution; spreading pressure in adsorption of solute i from a solution (4.1.63a); osmotic pressures of permeate solution and the solution at the membrane wall, respectively osmotic pressure of solution in regions 1 and 2 fluid density; bulk density of the packed bed; density of continuous phase (also fluid density at critical point); electric charge density per unit volume; density of feed fluid; moles/ fluid volume; mass concentration of species i (also density of solute i (3.3.90b)); value of ρi in region j (1.3.4) density of liquid (melt); mass density of particle material; particle mass concentration in gas; density of solid material; total density of fluid; total mass density of mixture in region j (1.3.5) density of vapor phase; mass concentration of species i in crystallizer outlet solution and feed solution, respectively mass density of species i in liquid (melt); mass density of crystals of species i; mass density contribution of particles of size rp to rp þ drp (6.2.55) density of resin particle, reduced density of fluid (¼ ρ=ρc ); density of liquids 1 and 2 fluid resistivity; particle resistivity average gas density (6.1.5e); mass density of solute i in solution at saturation intrinsic phase average and phase average, respectively, of ρik (6.2.24a,b) ρik (6.2.24a) mass density of solute i in solution averaged over flow cross section (6.2.16b) electrical conductivity of the solution (3.1.108q), particle sticking probability standard deviation of any profile (3.2.21a,b), Lennard–Jones parameter, ionic equivalent conductance (3.1.108r), solute i reflection coefficient through membrane (6.3.157a,b)

xxviii

σ p ; σ v , σ iv ; σ x , σ y

σ AB ; σ ix σþ i ; σm

σ ti , σ zi ; σ þ zi ; σ V i

τ; τ m ; τ w; τ wc

τ yx , τ yz

ϕ; ϕA, ϕB

ϕc ; ϕD; ϕs

ϕj ; ϕm ϕiℓ ; ϕim ; ϕm ; ϕp ; ϕpℓ

ϕN ϕ0 ϕw ϕmax

Notation

steric factor for protein SMA model (3.3.122d); specific volume based deposit in filter (7.2.172), (7.2.189); standard deviations (7.3.173) average of σ A and σ B (3.1.91b); standard deviations (7.3.12a) nondimensional standard deviation of a profile (3.2.21a,b); average electrical conductivity of the ionic and electronic species in a mixed conducting membrane standard deviation in the output profile of species i in t-coordinate and z-coordinate, respectively (2.5.3), (6.3.18c,b); nondimensionalized σ zi (6.3.16b); standard deviation in volume flow units (7.1.102b) tortuosity factor; value of τ for porous membrane; wall shear stress (7.2.118); τ w in cake region (7.2.136a) components of tangential stress τ y in x- and z-directions (6.1.24) potential of any external force field (also ϕext), angle in spherical polar coordinate system, enhancement factor due to reaction, solids volume fraction profile in suspension boundary layer (7.2.109); enhancement factors for species A and B volume fraction of solids in cake; volume fraction of continuous, dispersed phase (Table 3.1.7), (6.4.88); volume fraction of particles and solids in a suspension electrical potential of phase j; defined in (7.1.107h) volume fraction of species i in liquid phase; volume fraction of species i in membrane; polymer volume fraction in membrane; voltage drop over a cell pair in electrodialysis; volume fraction of polymer in liquid phase correction factor for nonequimolar counterdiffusion (3.1.136b) arbitrary value of centrifugal potential at r = 0 (3.1.6d) volume fraction of water in resin, particle volume fraction at wall maximum enhancement due to an instantaneous reaction, maximum particle volume fraction

ext ext ϕext i ; ϕiα , ϕiβ

^ ig ; Φsat Φ; Φ i

   ϕtot i ;ϕi ; 0ϕi max ; þ þ ϕi ; ϕ i ϕext ti

χ, χ p , χ s χ ij , χ ip

ψ;ψ

ψk ψs , ψv

ψ As ψ eℓ 〈ψ k 〉 〈ψ k 〉k

value of ϕ or ϕext for any species i; values of ϕext i in regions α and β fugacity coefficient (3.3.56); fugacity coefficient of species i in a mixture in ^ ig for pure i at P sat gas phase; value of Φ i at system temperature defined by (3.2.4); defined by (3.2.5b), defined by (3.2.6); defined by (3.2.9) and (6.3.14a) X ϕext i , summation over different external forces (3.2.4) volume suspectibility of fluid, particle and the solution Flory interaction parameter between species i and j and species i and polymer molecular orientation coordinate vector (3.3.89c); extent of facilitation of flux (5.4.58) (5.4.59a), selectivity (7.2.92) any property or characteristic of the kth phase shape factor for particle surface and volume (Example 2.4.1), (2.4.2e) (3.4.26), (6.4.15), (6.4.19) association factor (3.1.91a) electrical potential in the double layer phase average of ψ in the kth phase (6.2.25a) = ψk,avg intrinsic phase average of ψ in the kth phase (6.2.25b) = ψk,avgi

ω

angular velocity, solute permeation parameter (6.3.158b), sign of fixed charge in (3.3.30b)

ΩD; AB

quantity in diffusion coefficient expression (3.1.91b)

Superscripts a; b; Br d, drag eff; eph; ext ft; G hf ht i; iner; int

activity based; bottom/stripping section of a column; Brownian motion drag effective; electrophoretic; external between the feed stream and the tails stream; gas phase between the heads stream and the feed stream between the heads stream and the tails stream species i; inertial; internal

Notation

L m N 0, o oo, ov p P r

s

S t,T vℓ x, δ ∞ 0 00

þ

*

xxix

liquid phase magnetic, mobility, based on molal quantities, maximum number related to Nernst–Plank equation (3.1.106) location z = 0, standard state, infinite dilution, original quantity (7.1.101c), overall pore, pure species mole fraction based in Henry’s law quantity in a system with recycle or reflux, for the case of a chemical reaction related to solids only in solid–fluid separation, at saturation, scrubbing cascade solid phase total value, top/enriching section of a column, thermal diffusion vapor liquid mole fraction based, location z = δ or δm infinite dilution condition equimolar counterdiffusion case, first derivative, feed side of membrane permeate side of membrane nondimensional quantity hypothetical quantity

A,B,C c d D e eff; ex; ext f; fr g; gr H; hex i ij

imp; iner

k 0 l, liq; loc; ℓ ; ℓ; L

m; mc; max; min M ME, MR n, (n þ 1), (n − 1) N; nu o; obs; og; ol; out

p; pd; P r; r p1 , r p2 ; R

s

sþ, s−; se

Subscripts atm; b

in; ion j

atmospheric; bulk phase value, bottom product stream from a column, backward reaction species A, B and C based on molar concentration, continuous phase, cake, critical quantity diffusive, dialysate phase, dissociation, distillate drag related, Henry’s law species related, dispersed phase enriching section effective value; exit location; external feed side, feed based; during formation of drop, size based gas phase; growth based high; heat exchanger species i, where i = 1, 2, 3, A, B, M, s, phase interface, ith module/tube/stage ith species in jth stream, j = f for a single feed stream, j = f1, f2 for two feed streams entering separator, j = 1,2 for product streams rich in species 1 and species 2, respectively impeller; inertial

S t; tOE; tOR

tj

true; T v;w x y Y z 1, 2 3, 4 α, β σ, s

at inlet; ionic species region/phase j, where j = 1, 2, E, f, g, l, 0 ℓ m, o, p, R, s, t, v, w, T, σ species k liquid phase; local; adhering liquid phase on crystal; low; at end of separator of length L membrane phase; micellar; maximum; minimum mixture, molecular weight based, metallic species Murphree based on extract phase, raffinate phase stage number n, (n þ 1) and (n – 1), respectively metallic species; nucleation organic phase; observed; overall gas phase based; overall liquid phase based; at outlet permeate side, product side, particle; dominant particle; planar interface radial direction; related to particles of size r1 or r2; ion exchange resin phase, resistive, raffinate phase solvent, surface integration step in crystallization, surface adsorption site, salt/eluent counterion, stripping section, stage number forward and reverse surface reaction; seed crystal location S, stationary phase total, top product stream in a column; transfer units (overall) based on extract phase; transfer units (overall) based on raffinate phase total quantity in jth stream, j = f for a single feed stream, j = f1,f2 for two feed streams entering separator, j = 1,2 for product streams rich in species 1 and 2, respectively true value; thermal diffusive, total vapor phase, volumetric; aqueous phase, water liquid mole fraction based, coordinate direction gas phase mole fraction based, coordinate direction as in species AY coordinate direction species 1 and species 2, phase 1 and phase 2 species 3 and species 4, phases 3 and 4 phase α and phase β surface phase

xxx

Notation

Abbreviations and acronyms

HFCLM

Å AEM atm avg AVLIS BET bar barrer

HGH HGMS HK HPCE

angstrom anion exchange membrane atmosphere average atomic vapor laser isotope separation Brunauer–Emmet–Teller 105 pascal unit for permeability coefficient of gases through membrane, 1barrer ¼ 10−10

Btu CAC CACE CCEP CD(s) CDI CE CEC CEDI CEM CFD CFE CGC CHO CHOPs cm Hg CMC CMS CSC CSTR CSTS CV CZE ED ELM EOF ESA FACS FFE FFF FFM GAC GLC GPC HDPs HETP

cm3 ðSTPÞ-cm cm2 -s-cmHg

British thermal unit continuous annular chromatograph counteracting chromatographic electrophoresis countercurrent electrophoresis cyclodextrin(s) capacitive deionization capillary electrophoresis capillary electrochromatography continuous electrodeionization cation exchange membrane computational fluid dynamics continuous free-flow electrophoresis countercurrent gas centrifuge Chinese hamster ovary Chinese hamster ovary cell proteins pressure indicated in the height of a column of mercury critical micelle concentration carbon molecular sieve continuous-surface chromatography continuous stirred tank reactor continuous stirred tank separator control volume capillary zone electrophoresis electrodialysis emulsion liquid membrane electroosmotic flow energy-separating agent fluorescence-activated cell sorting free-flow electrophoresis field-flow fractionation free-flow magnetophoresis granular activated carbon gas–liquid chromatography gel permeation chromatography high-density particles height of an equivalent theoretical plate

HPLC HPTFF IEF ILM IMAC ITM ITP LDF LK LLC LM LRV LSC LTU mAB MBE ME MEKC MEUF MSA MSC MSF MSMPR MTZ NEA OEA PAC PBE PSA psia psig RO SCF SEC SLM SMA SMB SPs TDS TFF TOC

hollow fiber contained liquid membrane human growth hormone high-gradient magnetic separation heavy key high-performance capillary electrophoresis high-performance liquid chromatography high-performance tangential-flow filtration isoelectric focusing immobilized liquid membrane immobilized metal affinity chromatography ion transport membrane isotachophoresis linear driving force approximation light key liquid–liquid chromatography logarithmic mean log reduction value liquid–solid adsorption chromatography length of transfer unit monoclonal antibody moving boundary electrophoresis multiple effect micellar electrokinetic chromatography micellar enhanced ultrafiltration mass-separating agent molecular sieve carbon multistage flash mixed suspension, mixed product removal mass-transfer zone nitrogen-enriched air oxygen-enriched air powdered activated carbon population balance equation pressure-swing adsorption pound force per square inch absolute pound force per square inch gauge reverse osmosis supercritical fluid size exclusion chromatography supported liquid membrane steric mass action simulated moving bed structured packings total dissolved solids tangential-flow filtration total organic carbon

Notation

TSA UF UPW VCR VLE

xxxi

thermal-swing adsorption ultrafiltration ultrapure water volume concentration ratio vapor–liquid equilibrium

VOCs WFI WPU ZE

volatile organic compounds water for injection purified water zone electrophoresis

Introduction to the book

The basic objective of this chapter is to describe the organization of this book vis-à-vis separations from a chemical engineering perspective. Separation, sometimes identified as concentration, enrichment or purification, is employed widely in large industrial-scale as well as small laboratoryscale processes. Here we refer primarily to physical separation methods. However, chemical reactions, especially reversible ones, can enhance separation and have therefore received significant attention in this book. Further, we have considered not only separation of mixtures of molecules, but also mixtures of particles and macromolecules. The number of different separation processes, methods and techniques is very large. Further new techniques or variations of older techniques keep appearing in industries, old and new. The potential for the emergence of new techniques is very high. Therefore, the approach taken in this book is focused on understanding the basic concepts of separation. Such an approach is expected not only to help develop a better understanding of common separation processes, but also to lay the foundation for deciphering emerging separation processes/techniques. The level of treatment of an individual separation process is generally elementary. Traditional equilibrium based separation processes have received considerable but not overwhelming attention. Many other emerging processes, as well as established processes dealing with particles and external forces, are not usually taught to chemical engineering students; these are integral parts of this book. To facilitate the analysis of processes over such a broad canvas, a somewhat generalized structure has been provided. This includes a core set of equations of change for species concentration, particle population and particle trajectory. These equations are expected to be quite useful in general; however, separation systems are quite often very complicated, thereby limiting the direct utilization of such equations. Separation and purification are two core activities of chemical engineers. The first wave of textbooks on separation/mass transfer (until the early 1980s) concentrated on

distillation, absorption and extraction, with some attention to adsorption/ion exchange/chromatography. The second wave expanded the treatment of adsorption/ion exchange/ chromatography and incorporated an introduction to membrane processes. The overwhelming emphasis in these books was on chemical separations. Simultaneously, a series of textbooks emerged focusing on bioseparations. In these textbooks, the treatment of particle based separations appears briefly under mechanical separations, or under the equilibrium based process of crystallization, or as special operations under bioseparations. Fundamental principles that facilitate understanding of a variety of different separations have, however, been emerging in the literature for quite some time. It is useful to structure the learning of separation around these basic principles. Forces present in the separation system act on molecules, macromolecules or particles and make them migrate at different velocities and sometimes in different directions. When such velocities/forces interact with the bulk velocity of the individual phase(s)/region(s) present in the separation system, molecular species or particles follow different trajectories or concentrate in different phase(s)/region(s), leading to separation. This book will systematically develop this overall framework of a few important configurations of bulk flow direction vis-à-vis the direction of the force(s) for open separators. Chemical thermodynamics provides the local boundaries/limits in such configurations for chemical separations. The individual separation processes/techniques will then be illustrated in each such configuration of bulk flow direction vs. force direction in three categories of processes: phase equilibrium driven; external force driven; membrane processes. These three basic categories of separation processes rely on three different types of separation phenomena. These different types of separation phenomena achieve different extents of separation when coupled with particular configurations of bulk flow vs. force pattern. The description of each process/technique generally includes

2 its conventional treatment and often elementary process/ equipment design considerations. This illustrative framework is preceded by a few chapters that provide the basic tools for achieving this goal. In earlier literature, a broad category of separation techniques is identified as a mechanical separation process. These techniques are invariably restricted to the separation of particles in a fluid or drops in another fluid subjected generally to an external force. In this book, particle separations have been studied along with chemical separations when a particular external force is considered. Therefore the structure of this book is somewhat different. The following section provides a brief introduction to each chapter in the book.

Introduction to chapters What happens to a perfectly mixed binary mixture of two species in a closed vessel as separation takes place is introduced in Chapter 1. How one describes the extent of separation achieved in the closed vessel is illustrated via a few common separation indices. The separation indices are based on the notion of different species-specific regions in the separation system, and their differing compositions and capacities. Double subscript based notation, with the first subscript i referring to a component and the second subscript j referring to a region/phase/fraction, is introduced here. This notation has been used throughout the book as often as possible. Use of these separation indices is illustrated for three basic classes of separation systems without any particles: immiscible phases; membrane-containing systems; and systems having the same phase throughout the separation system. A description of separation in multicomponent systems has been included along with the notion of a separating agent required for separation. Chapter 2 presents the description of quantities needed to quantify separation in open systems with flow(s) in and out of single-entry and double-entry separators for binary, multicomponent and continuous chemical mixtures, as well as a size-distributed particle population. Separation indices useful for describing separation in open systems with or without recycle or reflux are illustrated for steady state operation (Sections 2.2 and 2.3); those for a particle population are provided in Section 2.4. At the end (Section 2.5), indices for description of separation in time-dependent systems, e.g. chromatography, have been introduced. The physicochemical basis for separation is the primary focus of Chapter 3. Separation happens via species-specific force driven relative displacement of molecules of one species in relation to other species into species-specific region in the separation system. Particles of different sizes/ properties similarly undergo relative displacements. To develop this perspective, Chapter 3 (Section 3.1) identifies various external forces and chemical potential gradient based

Introduction to the book force generating different migration/terminal velocities and fluxes for chemical species and particles. Integrated flux expressions for molecular diffusion and convection for single-phase systems, mass-transfer coefficients and empirical correlations for mass-transfer coefficients are introduced. Chapter 3 (Section 3.2) further points out the role of the spatial profile of the potential attributable to the force in developing a multicomponent separation capability. The criteria for achieving equilibrium between different phases and regions in the separation system with or without an external force and various types of phase equilibria are discussed in Section 3.3. The presentation of the partitioning of a species between two phases is at a phenomenological level. The molecular basis of this partitioning via intermolecular interactions has not been considered. This is followed by species flux expressions in interphase transport, including membrane transport (Section 3.4). The notion of an overall mass-transfer coefficient and its relation to single-phase mass-transfer coefficients are introduced here. Chapter 4 provides a quantitative exposition of how much separation is achieved at equilibrium in a closed vessel for three broad classes of separation systems: phase equilibrium between two phases (Section 4.1); single phase or a particle suspension in an external force field (Section 4.2); two regions separated by a membrane (Section 4.3). The phase equilibrium systems considered are: gas–liquid, vapor–liquid, liquid–liquid, liquid–solid, interfacial adsorption systems, liquid–ion exchanger and the supercritical fluid–solid/liquid phase. The external force fields and configurations studied are: centrifuges, isopycnic sedimentation, isoelectric focusing, gravity (sedimentation, inclined settlers), acoustic forces and thermal diffusion. In the case of a membrane based system of dialysis and gas permeation if separation is to be achieved, we come across the need for an open system. Chapter 5 focuses on the beneficial effects of chemical reactions in phase equilibrium and membrane based separation systems. A few common types of reactions, such as ionizations, acid–base reactions and different types of complexation equilibria, are found to influence strongly the separation achieved across the whole spectrum of separations involving molecules and macromolecules. The phase equilibrium systems studied are: gas–liquid, vapor–liquid, liquid–liquid, liquid–solid, surface adsorption and Donnan equilibrium. Reaction based enhancement in the rates of interphase transport as well as membrane transport has been illustrated for a variety of systems. Separation is most often implemented in open systems/devices with bulk flow(s) in and out. The treatment of separation achieved in such separators is carried out in Chapters 6, 7 and 8. Chapter 6 begins with the sources and the nature of bulk flow in separation systems in a multiscale context as well as the feed introduction mode vis-à-vis time (Section 6.1). The various equations of change for species concentration in a mixture, the equation of motion of a particle in a fluid and the general

Linked footprints of a separation process equation of change for a particle population, including that in a continuous stirred tank separator, are provided in Section 6.2. Section 6.3 covers the separation processes/ techniques in which the direction of the bulk flow is parallel to the direction of the force(s). Figure 6.3.1 illustrates the widespread use of this flow vs. force configuration for three basic classes of separation systems. External force based processes of elutriation, capillary electrophoresis, centrifugal elutriation, inertial impaction and electrostatic separation of fine particles are introduced first. Chemical potential gradient driven processes of flash/vaporization/ devolatilization, batch distillation, liquid–liquid extraction, zone melting, normal freezing and drying are studied next. The membrane based processes covered are: cake filtration/microfiltration, ultrafiltration, reverse osmosis, pervaporation and gas permeation. In the final section (Section 6.4), Chapter 6 considers the continuous stirred tank separator (CSTS) as a special category of bulk flow vs. force configurations; the separation processes studied are: crystallization (precipitation), solvent extraction, ultrafiltration and gas permeation. The nature and extent of separation achieved when the direction of flow of the feed-containing fluid phase is perpendicular to the direction of the force(s) are studied in Chapter 7. This treatment illustrates the basic separation mechanism clearly, even though the particulars vary widely, as in, for example, free-flow electrophoresis, electrostatic precipitators, electrostatic separation of plastic mixtures, laser excitation of isotopes and flow cytometry (all of them driven by an electrical force field perpendicular to the bulk flow). Figure 7.0.1 provides this broad perspective across all three classes of separation processes: phase equilibrium driven, membrane based, external force driven. This chapter begins (Section 7.1) with the treatment of fixed-bed adsorption processes, pressure-swing adsorption, parametric pumping and chromatography. Crossflow membrane processes considered next (Section 7.2) are: gas permeation, reverse osmosis, ultrafiltration, microfiltration; this has been followed by granular filtration. The external force field based processes studied in Section 7.3 involve electrical force (mentioned earlier), centrifugal force (centrifuges, cyclones), gravity (gravity based settlers), magnetic force field (highgradient magnetic separation) and optical force. Field-flow fractionation as a special case of a force perpendicular to bulk flow interacting with the velocity profile in a novel way has also been treated; a variety of forces may be used. Chapter 8 deals with the configuration of bulk flows of two phases/regions (one of which may be solid) perpendicular to the direction of force(s). The directions of motion of the two phases may be parallel to each other in either countercurrent or cocurrent fashion, or they may be in crossflow. Figures 8.1.1–8.1.4 illustrate the countercurrent flow vs. force configuration for all three classes of separation systems. Conventional countercurrent devices/ processes of gas absorption/stripping, column distillation

3 with a condenser and reboiler, solvent extraction in columns, melt crystallization, adsorption and simulated moving beds, dialysis and electrodialysis, liquid membrane separation, gas permeation, gas centrifuge, thermal diffusion and mass (sweep) diffusion are studied in Section 8.1. How cocurrent flow of the two phases/flows changes the separation achieved is considered vis-à-vis a few systems in Section 8.2. Local multicomponent feed injection in a crossflow format in fluid–solid systems leads to the achievement of continuous chromatography. Overall crossflow of two phases is exemplified by a crossflow distillation plate (Section 8.3). Although countercurrent multistaged processes of distillation, gas absorption, solvent extraction, etc. have been studied in some detail in Chapter 8, the subject of multistaging/cascades is considered briefly in Chapter 9. Ideal cascades and constant or variable cross-sectional area are introduced, as are cascades of multistage columns for nonbinary systems. Chapter 10 describes at an elementary level the minimum energy required for separation by different separation processes. Additional topics discussed in Chapter 10 include the consideration of various concepts that reduce the energy required for separation, recovering the free energy of mixing via a dialytic battery and additional deliberations for treating dilute solutions in the context of bioseparations. In many real-life applications, sequences of different separation processes are employed with and without reaction processes. Chapter 11 illustrates such sequences of separation processes for bioseparations, water treatment, chemical and petrochemical industries and hydrometallurgical separations. Each of these chapters provides a particular aspect/ perspective of the broad subject of separations. One is often interested, however, in a particular separation process/technique in all of its aspects, beginning with the basic concept and ending with devices designed to implement the separation. Tables are therefore provided at the end of this chapter that identify the essential and important components located in different chapters for a given separation process. Obviously it is not possible to provide a comprehensive treatment of every process, and a few commonly used separation processes have received much more attention than others. However, the treatment of each such commonly used separation process is at a level illustrative of the basic principles relevant to the particular chapter. Furthermore, the treatments are not exhaustive. Readers interested in greater detail are encouraged to go to major texts on such separation processes identified along with their treatments.

Linked footprints of a separation process/technique We provide in this section seven tables; they appear at the end of this chapter Each table has nine columns. The first column identifies the name of a particular separation

4 process in a particular row (e.g. ‘Absorption’ in row 1 of Table 1). The second column focuses on Chapters 1 and 2. Six more columns are identified progressively with each of the Chapters 3–8. The final ninth column covers the much smaller Chapters 9–11. Each row in the tables is dedicated to a particular separation process. The entry in a box for a given row and a given column identifies sections in the chapter where that particular separation process or fundamental material needed to understand the transport and thermodynamics for that process has been presented. In Chapters 1 and 2 and in Sections 3.1, 3.2, 3.3.1–3.3.6, 6.1 and 6.2, general features or fundamental relations valid for a variety of separation processes are presented. Therefore, entries for a given separation process under columns 2, 3 and 6, specifically Sections 3.1, 3.2, 3.3.1–3.3.6, 6.1 and 6.2, providing fundamental information on the description of separation species/particle transport, thermodynamics relations, balance/conservation equations and equations of change, respectively, for species/particles are not tied in general specifically to that separation process; however, any entry will be useful for understanding that separation process. Table 1 covers many of the common phase equilibrium based separation processes. The entries contain a few separation techniques/processes which are not employed on a large scale or illustrate important conceptual developments, e.g. cycling zone adsorption, foam fractionation, parametric pumping. Table 2 includes membrane separation processes, where different membrane transport rates of different species provide the selectivity in open systems. This table also includes membrane contactor based separation processes, where the basis for separation is the partitioning equilibrium between two fluid phases contacting each other at membrane pore mouths. Tables 3 and 4 identify separation processes driven by centrifugal force and electrical force, respectively. Table 5 is devoted to a few processes driven by magnetic force or gravity. A few separation processes/techniques driven by other forces, such as acoustic force, radiation pressure, inertial force and thermal gradient driven force, are listed in Table 6. Table 7 is devoted to additional separation processes such as field-flow fractionation and mass (sweep) diffusion. It is useful now to illustrate how the descriptive treatment of a particular separation process, e.g. distillation, has been implemented in an evolutionary fashion via the different chapters as identified in row 7 of Table 1. In Section 1.1, Example I of Figure 1.1.2 illustrates the result of heat addition to an equimolar liquid mixture of benzene–toulene: a benzene-rich vapor phase and a toluene-rich liquid phase. Using definitions of compositions etc. introduced in Section 1.3, separation indices such as the separation factor αij (also the equilibrium ratio Ki) describe the separation achieved in a closed vessel for the benzene–toluene system and a methanol–water system for various liquid-phase compositions. Section 1.5 illustrates via Example 1.5.1 and the values of various separation indices, α12 and ξ, the

Introduction to the book separation achieved in the benzene–toluene system in a closed vessel. Section 1.6 describes multicomponent mixtures and develops the relations between the compositions of two phases in equilibrium, a result useful for distillation in later chapters. Section 2.1 introduces various quantities describing flow rates and compositions in an open system; a sieve plate in a distillation column is used as one example, among others, of a double-entry separator. A flash distillation stage with liquid fraction recycle illustrates recycle in a single-entry separator (Section 2.2). Section 2.3 for double-entry separators provides a numerical example of benzene–toluene distillation in a countercurrent column without a condenser or reboiler. This and other examples provide a quantitative background on the separation achieved in a given device without discussing the separation mechanism. The same strategy of description of separation achieved via reflux to a column is pursued in this section to demonstrate that a higher reflux ratio leads to higher separation. Sections 2.4.2 and 2.4.3 introduce indices to describe continuous chemical mixtures and multicomponent mixtures vis-à-vis flash vaporization. The introductory Section 3.1.2.5 in Chapter 3 identifies the negative chemical potential gradient as the driver of targeted separation, and the relevant species flux expression is developed in Section 3.1.3.2 (see Example 3.1.9 also). Section 3.1.4 introduces molecular diffusion and convection and basic mass-transfer coefficient based flux expressions essential to studies of distillation and other phase equilibrium based separation processes. Section 3.1-5.1 introduces the Maxwell–Stefan equations forming the basis of the rate based approach of analyzing distillation column operation. After these fundamental transport considerations (which are also valid for other phase equilibrium based separation processes), we encounter Section 3.3.1, where the equality of chemical potential of a species in all phases at equilibrium is illustrated as the thermodynamic basis for phase equilibrium (i.e. μiv ¼ μil). Direct treatment of distillation then begins in Section 3.3.7.1, where Raoult’s law is introduced. It is followed by Section 3.4.1.1, where individual phase based mass-transfer coefficients are related to an overall mass-transfer coefficient based on either the vapor or liquid phase. Section 4.1 via Section 4.1.2 formally illustrates vapor– liquid equilibria vis-à-vis distillation in a closed vessel along with bubble-point and dew-point calculations for multicomponent systems. How vapor–liquid equilibrium is influenced by chemical reactions in the liquid phase is treated in Section 5.2.1.2, where two subsections, 5.2.1.2.1 and 5.2.1.2.2, deal with reactions influencing vapor–liquid equilibria in isotopic systems. We next encounter open systems in Chapter 6. The equations of change for any two-phase system (e.g. a vapor–liquid system) are provided in Section 6.2.1.1 based on the pseudo-continuum approach for the dependences of species concentrations

Linked footprints of a separation process on time and the main axial coordinate (i.e. z) direction. Section 6.3.2.1 starts with the simplest of open systems, a flash vaporizer, and illustrates isothermal flash calculations under the constraint of phase equilibrium and bulk flow parallel to (jj) the force direction for multicomponent systems and continuous chemical mixtures. Batch distillation without any reflux is then studied as a particular illustration of this flow vs. force configuration for a fixed amount of feed liquid as well as for constant-level batch distillation employed for solvent exchange. Residue curves are introduced here. Column distillation is the most common form of an open separation system in distillation. Here the two phases have, on an overall basis, bulk motions in parallel flow in the countercurrent direction with the forces causing separation being perpendicular (⊥) to the directions of bulk flows. The general characteristics of such a separation system are briefly identified in Section 8.1.1, specifically 8.1.1.1–8.1.1.3. We learn the structural consequences of this flow vs. force configuration, namely a distillation column cannot at steady state separate a ternary mixture; we need two columns for a ternary mixture. Further, the particular forms of equations of change for the two phases are obtained from the more general equations in Section 6.2.1.1 (as well as by a control volume analysis). Distillation columns with reflux and recycle are studied in detail in various parts of Section 8.1.3. The conventional approach of assuming ideal equilibrium stages (the stage may have crossflow in an overall countercurrent flow configuration) is adopted in the McCabe–Thiele graphical framework to study the following: operating lines in both sections of a column, q-line, total reflux, minimum reflux, partial/total reboiler, partial condenser, open steam introduction, Kremser equation and side stream. The deviation from ideal equilibrium stages is studied next via stage efficiency in Section 8.1.3.4. Vapor–liquid contacting on a plate/tray in a column is considered in Section 8.1.3.5 vis-à-vis estimation of column diameter (with reference to Section 6.3.2.1). Topics such as the rate based approach for modeling distillation and separation of a multicomponent mixture in a column are briefly introduced, the latter via the Fenske equation, the Underwood equation and the Gilliland correlation. Distillation in a packed tower and in a batch vessel with reflux are studied next. Section 8.2.1 briefly touches on distillation in a cocurrent two-phase flow device. Section 8.3.2 studies separation in a crossflow distillation plate employing general equations from Section 6.2.1.1, ultimately yielding the American Institute of Chemical Engineers (AIChE) tray efficiency expression. The total number of worked examples involving distillation in one form or another in Chapters 1–8 is 19. Various other aspects of distillation are considered further in Chapters 9–11. Chapter 9 (Section 9.2) introduces briefly the methodology for multicolumn distillation for separating a mixture containing more than two species. Chapter 10

5 covers the minimum energy required for distillation, and the concepts of net work consumption, multieffect distillation and heat pump vis-à-vis distillation. Chapter 11.3 introduces very briefly the important role of distillation in the chemical and petrochemical industries. If the treatment of distillation in a given section of the book needs certain building blocks, it is most likely that those concepts/methods/building blocks have been introduced in an earlier chapter or section of the book. Furthermore, in whichever section distillation appears, it is studied as part of a specific pattern followed by many other separation processes based on phase equilibrium. Such patterns have been emphasized often throughout particular chapters. A few pointers on phase equilibrium based separation processes are useful. Table 3.3.1 lists possible useful combinations of two bulk immiscible phases for separation such as gas–liquid (vapor–liquid included), gas–solid, liquid–liquid, etc. Quite a few of these combinations form the basis of existing separation processes. In this book, therefore, each chapter, from Section 3.3 onwards, focusing on a particular aspect of the subject of separation, has the subject of phase equilibrium driven separation processes organized along such two immiscible phase combinations. However, all such combinations in practical use do not appear in each chapter. The treatment of membrane separation processes in this book merits some deliberation. The most commonly used driving force in membrane separation processes is negative chemical potential gradient; a few processes also employ electrical force. Figure 3.4.5 identifies the variety of feed phase–membrane type combinations with variations due to the nature of the permeate phase when negative chemical potential gradient is imposed across the membrane. Section 3.4.2 illustrates the interphase membrane transport aspects of many such configurations. The developments in later chapters follow these feed phase–membrane type permeate phase combinations as often as possible, subject to space limitations. Electrodialysis as an example of an application of electrical force appears in Sections 3.4.2 and 8.1.7. Membrane contactors appear with their phase equilibrium process counterparts in Sections 8.1.2 and 8.1.4, whereas the basic transport considerations in such membrane devices appear much earlier in Sections 3.4.3.1 and 3.4.3.2. A most important item in membrane separation processes is that such devices in the absence of external forces achieve separation when operated as an open system – Sections 4.3.1 and 4.3.3 demonstrate this feature via the processes of dialysis and gas permeation. The descriptive treatment of the membrane process of reverse osmosis (RO) in the book as identified in Table 2 will be briefly illustrated here. Section 1.1 identifies the basic configuration of RO in Figure 1.1.3. Example 1.5.4 illustrates calculations of separation indices describing separation in RO shown in Figure 1.5.1. Sections 2.1 and 2.2 describe various quantities, as well as the separation indices relevant

6 for RO; Example 2.2.1(c) is directly applicable to RO. Sections 3.1.2.5, 3.1.3.2 and 3.1.5.1 provide a general transport background. Section 3.1.5.2 is directly relevant to an irreversible thermodynamics based solute and solvent transport through RO membranes. Section 3.3.7.4 provides a membrane–liquid equilibrium relation from an osmotic equilibrium point of view. Section 3.4.2.1 formally introduces transport rates in RO membranes and flux expressions, along with issues of concentration polarization in a feed solution. A closed vessel of Chapter 4 has very limited relevance for RO (Section 4.3.4). Sections 5.4.1 and 5.4.1.1 describe how chemical reactions, such as ionization, in the solutions influence separation in RO processes. Section 6.3.3.3 studies RO in bulk flow parallel to the force configuration and describes various membrane transport considerations and flux expressions. Practical RO membranes are employed in devices with bulk feed flow perpendicular to the force configuration, as illustrated in Section 7.2.1.2. A simplified solution for a spiral-wound RO membrane is developed: analytical expressions for the water flux as well as for salt rejection are obtained and illustrated through example problem solving. A total of six worked example problems have been provided up to Chapter 7. Chapter 9 (Figure 9.1.5) shows a RO cascade in a tapered configuration. Section 10.1.2 calculates the minimum energy required in reverse osmosis based desalination and compares it with that in evaporation. Section 11.2 covers the sequence of separation steps in a water treatment process for both desalination and ultrapure water production. The very important role played by RO in such plants is clearly illustrated. The evolution of separation through different chapters due to an external force needs some discussion as well. Whereas negative chemical potential gradient driven distillation is utilized to separate low molecular weight liquids having different volatilities, an external force, such as electrical force arising from a negative gradient of electrical potential, can be used to separate small charged molecules, charged macromolecules, charged cells, charged particles, etc.; the medium may be liquid or gaseous. The canvas is large, and the variety of separation processes/ techniques driven by electrical force is significant. Although there is considerable variety also in phase equilibrium processes resulting from a variety of two-phase systems, the separation systems are more often limited to smaller molecules. Separation of proteins/macromolecules via chromatography (Section 7.1.5.1) and biphasic/reverse micellar extraction (Sections 4.1.4 and 4.1.8) provide exceptions; flotation (Section 3.3.8) separates particles with the helping hand of an external force, gravity, as does a Venturi scrubber (Section 8.2.3) via inertial impaction. Consider the electrophoretic motion of charged molecules/macromolecules/proteins in an aqueous solution/ buffer subjected to an electrical force. Three separation techniques, isoelectric focusing, capillary electrophoresis

Introduction to the book and continuous free-flow electrophoresis, exploit, among others, electrophoretic transport under the constraints of a closed vessel, bulk flow parallel to force and bulk flow perpendicular to force, respectively. Correspondingly, in Table 4, isoelectric focusing does not appear in Chapters 6–8; capillary electrophoresis is absent from Chapter 8. However, each such technique benefits from relevant discussions in earlier chapters, even though the technique itself is treated in detail in a later chapter; therefore materials in Chapters 2 and (especially) Chapter 3 are identified for each of the three techniques. Capillary electrophoresis appears also in Section 7.1.7.1, where it has been coupled with chromatography where the bulk flow is perpendicular to the force.

Classification of separation processes This book has not adopted a comprehensive classification scheme for all separation processes. Readers should go to the references, especially Figure 30 and Table 7 of Lee et al. (1977a) and Table 1-1 of King (1980), to that end. What has been adopted here is apparent from the titles of Tables 1–7. Separation processes are classified into three categories based on the three basic types of physiocochemical phenomena: (1) phase equilibrium based separation processes; (2) membrane separation processes; (3) external force based separation processes. There are a few processes where there is an overlap. For example, electrodialysis is a membrane-separation process driven primarily by an external force, the electrical potential gradient; most membrane-separation processes are driven by negative chemical potential gradient. There are a few others, e.g. mass diffusion/sweep diffusion, which cannot be neatly put into these three categories; they possess characteristics of different categories. In this framework of three broad categories of separation processes, further separation development/classification comes about due to the nature of the interaction between the basic separation phenomena in each category and the directions of bulk flow vis-à-vis the direction of force(s) responsible for the basic separation mechanism. Considerable additional separation development is achieved by reflux, recycle, creation of an additional property gradient in an external force field, mode of feed introduction, etc. These aspects have been addressed in the following sections: reflux (Sections 2.3.2, 8.1.1, 8.1.4, 10.1.4.2, 10.2.2.1); recycle (Sections 2.2.2, 2.4.1, 7.2.1.1, 7.2.4, 8.1.1); development of an additional property gradient in an external force field (Sections 4.2.1.3, 4.2.2.1, 4.2.3.3, 7.1.7); mode of feed introduction (Sections 6.1.9, 7.1.5, 7.1.6, 8.1.1, 8.2.2.1, 8.2.2.2, 8.3.1). An additional classification approach considers the nature of the mixture to be separated: mixtures of small molecules and/or ions in solution or gas phase; mixtures of

Textbooks, handbooks and major references macromolecules in solution; mixtures of particles, where particles in this book include biological cells (Tables 4.2.1, 7.3.1), cell debris, colloidal material and inorganic and organic particles of varying dimensions (submicron to visible particles, Figure 2.4.1(b)). Of the numerous separation techniques involving different types of macromolecules, the following have received some attention here: separation of proteins from each other/ one another or solvent via isoelectric focusing, etc. (Sections 4.2.2.1, 4.2.2.2), ultrafiltration (Sections 6.3.3.2, 6.4.2.1, 7.2.1.3), chromatography (Sections 4.1.6, 4.1.9.4, 7.1.5.1.6, 7.1.5.1.7, 7.1.5.1.8, 7.1.6, 7.1.7), electrophoresis (Section 7.3.1.1), field-flow fractionation (Section 7.3.4), aqueous biphasic extraction (Section 4.1.3) and reverse micelles (Section 4.1.9); separation of nonbiological macromolecules via size exclusion chromatography (Section 7.1.5.1.7), flash devolatilization (Section 6.3.2.1), sol–gel separation (Section 2.4.2); DNA separation via isopycnic sedimentation (Section 4.2.1.3). It is useful to provide a list of the basic physical or physiochemical properties, each of which could be a basis for separation; it is also useful to list simultaneously the core phenomenon exploiting such a physical or physicochemical property for separation. It is to be noted that this list is not exhaustive; rather, it contains the more familiar properties. Table 8 identifies a variety of these basic properties and lists phenomena employing a particular basic property leading to separation. For each basic property and phenomenon in this table, there are three columns corresponding to three different types of basic separation processes: phaseequilibrium-based separation processes; membraneseparation processes; and external force based separation processes. An entry into these three columns identifies a separation process or processes where the particular basic property is key to separation. References to Tables 1–7, a section in the book or a separate reference have been provided to each entry in these three columns. There are some items of interest here. A few basic properties are the basis for separation in two different types of basic separation processes. For example, condensability of a vapor/gas species is useful for vapor absorption as well as for membrane gas separation; geometrical partitioning (or partitioning by other means between a pore and an external solution) is useful both in adsorption/chromatography as well as in the membrane processes of dialysis and ultrafiltration, etc. Further, there are many cases where chemical reactions are extraordinarily useful for separation; these are not identified here since chemical reactions can enhance separation only if the basic mechanism for separation exists, especially in phase equilibrium based separations. However, there are a few cases where chemical reactions, especially complexations, provide the fundamental basis for separation, as in affinity chromatography, metal extractions and isotope exchange reactions.

7 Additional comments on using the book This book has 118 separate numerical examples spread over Chapters 1–4 and 6–9. The numerical examples are not in finer print. Chapter 5 has sometimes employed numerical calculations to illustrate the effect of chemical reactions on separations without formal numerical examples. Chapter 10 follows this strategy as well to illustrate the amount of energy required for a particular separation. The total number of problems provided at the ends of all the chapters is 299. The specific separation process relevant for the problem is generally obvious from the introductory sentence in the problem. Further, the sequence of appearance of a problem on a given separation process reflects/follows the sequences of appearance of that separation process in the text. Footnotes have been employed occasionally. All references used appear at one location in alphabetical order at the end of the book. The symbols and notation employed throughout the book are consistent; any local deviation has been identified. In a few locations, advanced material or additional information has been provided.

Textbooks, handbooks and major references on separation processes There is an extraordinarily rich literature on separations. This book has freely drawn material from this literature consisting of textbooks, monographs or extended chapters in multiauthor edited volumes apart from numerous journal articles. Here we list these books and chapters (but no journal articles) under the following categories: separations; chemical separations; bioseparations; membrane separations; particle separations; other books. Such books and relevant journal articles have been cited through each section in each chapter. Occasionally some comments have been attached here to a given reference. Books devoted solely to a given separation process/ technique are not, in general, mentioned below. The following list is given in chronological order. At the end of each reference, its formal reference has been identified. Separations

(1) Karger, B.L., L.R. Snyder and C. Horvath, An Introduction to Separation Science, Wiley, New York (1973). Chapter 18 devotes 19 pages to particle separation; otherwise it covers primarily separations of chemicals and macromolecules. (Karger et al., 1973.) (2) Lee, H.L., E.N. Lightfoot, J.F.G. Reis and M.D. Waissbluth, “The systematic description and development of separation processes,” in Recent Developments in Separation Science, Vol. III, N.N. Li (ed.), Part A, CRC

8 Press, Cleveland, OH (1977), pp.1–70. An important contribution to structuring separations from a morphological perspective with a distinct transportbased input. (Lee et al., 1977a.) (3) Giddings, J.C., “Principles of chemical separations,” in Treatise on Analytical Chemistry, Part I. Theory and Practice, Vol. 5, P.J. Elving, E. Grushka and I.M. Kolthoff (eds.), Wiley-Interscience, New York (1982), chap.3. An early and useful contribution toward transport-based understanding of analytical separations. (Giddings, 1982.) (4) Giddings, J.C., Unified Separation Science, John Wiley, New York (1991). An important contribution to separation science with an emphasis on methods used in analytical chemistry, especially chromatography. (Giddings, 1991.) (5) Schweitzer, P.A., Handbook of Separation Techniques for Chemical Engineers, 3rd edn., McGraw-Hill, New York. (Schweitzer, 1997.) Chemical separations

(1) Benedict, M. and T.H. Pigford, Nuclear Chemical Engineering, McGraw-Hill, New York (1957). Introduces isotope separations and cascades in a chemical engineering context for the nuclear industry. The second edition (1981), with added author H.W. Levi, substantially expands the treatment and coverage. (Benedict et al., 1981.) (2) Pratt, H.R.C., Countercurrent Separation Processes, Elsevier, Amsterdam (1967). Contains, among others, an introduction to cascade analysis for chemical separations and isotope separations. (Pratt, 1967.) (3) Sherwood, T.K., R.L. Pigford and C.R. Wilke, Mass Transfer, McGraw-Hill, New York (1975). (Sherwood et al., 1975.) (4) King, C.J., Separation Processes, 2nd edn., McGrawHill, New York, (1980). An important textbook which analyzes chemical separations in a generalized framework with considerable emphasis on multistage separation processes. The first edition appeared in 1970. (King, 1980.) (5) Treybal, R.E., Mass-transfer Operations, 3rd edn., McGraw-Hill, New York (1980). A textbook focusing primarily on conventional mass transfer operations. (Treybal, 1980.) (6) Hines, A.L. and R.M. Maddox, Mass Transfer: Fundamentals and Applications, Prentice-Hall PTR, Upper Saddle River, NJ (1985). (Hines and Maddox, 1985.) (7) Wankat, P.C., Rate-Controlled Separations, Elsevier Applied Science, New York (1990). Textbook providing an extensive treatment of adsorption, chromatography, crystallization, ion exchange and membrane separations. (Wankat, 1990.)

Introduction to the book (8) Humphrey, J.L. and G.E. Keller II, Separation Process Technology, McGraw-Hill, New York (1997). Book oriented towards separation technology, useful for process design. (Humphrey and Keller, 1997.) (9) Seader, J.D. and E.J. Henley, Separation Process Principles, John Wiley, New York (1998). Textbook with broad coverage of chemical separation processes including adsorption, crystallization and membrane separations. A second edition was published in 2006. (Seader and Henley, 1998.) (10) Noble, R.D. and P.A. Terry, Principles of Chemical Separations with Environmental Applications, Cambridge University Press, Cambridge, UK (2004). Treats chemical separations in an environmental context. (Noble and Terry, 2004.) (11) Wankat, P.C., Separation Process Engineering, 2nd edn., Prentice Hall, Upper Saddle River, NJ (2007) (formerly published as Equilibrium Staged Separations, Elsevier, New York, 1987). (Wankat, 2007.) (12) Benitez, J., Principles and Modern Applications of Mass Transfer Operations, 2nd edn., John Wiley, Hoboken, NJ (2009). (Benitez, 2009.)

Bioseparations

(1) Belter, P.A., E.L. Cussler and W.-S. Hu, Bioseparations: Downstream Processing in Biotechnology, Wiley-Interscience, John Wiley, New York (1988). An introduction to bioseparations. (Belter et al., 1988.) (2) Garcia, A.A., M.R. Bonen, J. Ramirez-Vick, M. Sadaka and A. Vuppu, Bioseparation Process Science, Blackwell Science, Malden, MA (1999). (Garcia et al., 1999.) (3) Ladisch, M.R., Bioseparations Engineering: Principles, Practice and Economics, John Wiley, New York (2001). (Ladisch, 2001.) (4) Harrison, R.G., P. Todd, S.R. Rudge and D.P. Petrides, Bioseparations Science and Engineering, Oxford University Press, New York (2003). (Harrison et al., 2003.)

Membrane separations

(1) Hwang, S.T. and K. Kammermeyer, “Membranes in separations,” Vol. VII in Techniques of Chemistry, A. Weissberger (ed.), Wiley-Interscience, New York (1975). Reprinted, Kreiger Publishing, Malabar, FL (1984). (Hwang and Kammermeyer, 1984.) (2) Meares, P. (ed.), Membrane Separation Processes, Elsevier Scientific Publishing Co., Amsterdam (1976). (Meares, 1976.) (3) Belfort, G., Synthetic Membrane Processes, Academic Press, New York (1984). (Belfort, 1984.) (4) Ho, W.S.W. and N.N. Li, “Membrane processes,” in Perry's Chemical Engineers' Handbook, R.H. Perry

Textbooks, handbooks and major references

(5)

(6)

(7)

(8)

(9)

and D.W. Green (eds.), 6th edn., McGraw-Hill, New York, (1984), pp. 17.14–17.35. (Ho and Li, 1984a.) Rautenbach, R. and R. Albrecht, Membrane Processes, John Wiley, New York (1989). (Rautenbach and Albrecht, 1989.) Mulder, M., Basic Principles of Membrane Technology, 2nd edn., Kluwer Academic Publishers, Dordrecht (1991); a second edition followed in 1996. (Mulder, 1991.) Ho, W.S.W. and K.K. Sirkar (eds.), Membrane Handbook, Van Nostrand Reinhold (1992). Reprinted, Kluwer Academic Publishers, Boston (2001). (Ho and Sirkar, 2001.) Noble, R.D. and S.A. Stern (eds.), Membrane Separations Technology: Principles and Applications, Elsevier, Amsterdam (1995). (Noble and Stern, 1995.) Baker, R.W., Membrane Technology and Applications, 2nd edn., John Wiley, Hoboken, NJ (2004). (Baker, 2004.)

Particle separations

(1) Wark, K. and D.F. Warner, Air Pollution, Its Origin and Control, IEP – Dun-Donnelley, Harper & Row, New York (1976). (Wark and Warner, 1976.) (2) Friedlander, S.K., Smoke, Dust and Haze: Fundamentals of Aerosol Behavior, John Wiley, New York (1977). (3) Svarovsky, L. (ed.), Solid-Liquid Separation, Butterworths, London (1977). (Svarovsky, 1977.) (4) Svarovsky, L., Solid-Gas Separation, Elsevier Scientific Publishing, Amsterdam (1981). (Svarovsky, 1981.) (5) Flagan, R.C. and J.H Seinfeld, Fundamentals of Air Pollution Engineering, Prentice Hall, Englewood

9 Cliffs, NJ (1988). A useful book written from a fundamental perspective. (Flagan and Seinfeld, 1988.) (6) Randolph, A.D. and M.A. Larson, Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization, 2nd edn., Academic Press, New York (1988). (Randolph and Larson, 1988.) (7) Soo, S.L., Particulates and Continuum: Multiphase Fluid Dynamics, Hemisphere Publishing, New York (1989). (Soo, 1989.) (8) Tien, C., Granular Filtration of Aerosols and Hydrosols, Butterworths, Boston, MA (1989). (Tien, 1989.) Other books

(1) Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley, New York (1960); 2nd edn., John Wiley, New York (2002). A key book for transport phenomena. Bird et al. (2002.) (2) Foust, A.S., L.A. Wenzel, C.W. Clump, L. Maus and L. B. Anderson, Principles of Unit Operations, John Wiley, New York (1960). (Foust et al., 1960.) (3) McCabe, W.L. and J.C. Smith, Unit Operations of Chemical Engineering, 3rd edn., McGraw-Hill, New York (1976); 5th edn., (1993), with additional author P. Harriott. (McCabe et al., 1993.) (4) Perry, R.H. and D.W. Green (eds.), Perry's Chemical Engineers' Handbook, 6th edn., McGraw-Hill, New York (1984). (Perry and Green, 1984.) (5) Cussler, E.L., Diffusion: Mass Transfer in Fluid Systems, 2nd edn., Cambridge University Press, Cambridge, UK (1997); 3rd edn. (2009). (6) Geankoplis, C.J., Transport Processes and Separation Process Principles, 4th edn., Prentice-Hall PTR, Upper Saddle River, NJ (2003). (Geankoplis, 2003.)

Table 1. Relevant sections for each phase equilibrium based separation process

Chapter 3 Basis for separation

Chapter 4 Separation in a closed vessel

Chapter 5 Effect of chemical reaction

Chapter 6 Bulk flow jj to force(s) and CSTS

Chapter 7 Bulk flow ⊥ to force(s)

Chapter 8 Bulk flow of two phases ⊥ to force(s)

1.1–1.4, 2.1, 2.3

3.3.7.1,a 3.4.1.1, 3.4.3.1

4.1, 4.1.1.1

5.1, 5.2.1.1, 5.3.1

6.2.1.1

7.1.2.1

Adsorption (& simulated moving beds)

1.1–1.4, 2.1, 2.5

4.1, 4.1.5

5.2.3.2

6.2.1.1

7.1.1–7.1.7.1

Chromatographyc

1.1–1.4, 2.1, 2.5

6.2.1.1

7.1.1.1, 7.1.5– 7.1.7.1

1.1–1.4, 2.1, 2.4.1

4.1.3, 4.1.5– 4.1.8, 4.1.9.1 4.1.4, 4.1.9.1

5.2.3.2

Crystallizationd

3.1.3.2.3, 3.1.3.2.4,b 3.3.5, 3.3.7.4, 3.3.7.6, 3.4.1.4 3.1.3.2.3, 3.1.3.2.4, 3.2.1, 3.2.2, 3.3.7.4, 3.3.7.6, 3.4.1.4, 3.4.1.5 3.3.1, 3.3.7.5, 3.4.1.3

8.1.1, 8.1.1.1– 8.1.1.3, 8.1.2, 8.2.1.1, 8.2.2.2 8.1.1, 8.1.1.1– 8.1.1.3, 8.1.6

5.2.4

6.2.3, 6.4.1.1

5.2.1.2

6.3.2.1 6.2.1.1, 6.3.2.1

Chapters 1 & 2 Describe separation Absorption (and stripping)

Cycling zonee adsorption Devolatilization Distillation

3.1.5.1,a 3.3.7.1, 3.4.1.1, 3.4.3.1

3.1.4, 3.3.7.5

Extraction

1.1–1.4 1.1–1.4, 2.2, Ex. 2.2.2 1.1–1.6, 2.1, 2.3

Flotation Foam fractionation Ion exchange

Prob. 2.2.4 1.1–1.4, 2.1, 2.5

Leaching Melt crystallization Normal freezing Parametric pumping (see adsorption)

10.1.4.4

8.2.2, 8.3.1

11.1

8.1.1, 8.1.1.1– 8.1.1.3, 8.1.5

9.1.2.2, 9.1.2.3

8.1.1, 8.1.1.1– 8.1.1.3, 8.1.3, 8.2.1, 8.3.2

10.1.1, 10.1.3,f 10.1.4.2, 10.2.1.1, 10.2.1.2, 10.2.1.3, 10.2.2 11.1 10.1.1, 10.2.1

8.1.1, 8.1.1.1– 8.1.1.3, 8.1.4

10.1.4.3, 11.4

8.1.1, 8.1.6

11.2

7.1.4.4 1.1–1.7, 2.1–2.3, 2.4.2, 2.4.3

Drying (and freeze-drying) Evaporation

Chapters 9, 10 & 11 Other aspects

1.1–1.4, 2.1, 2.4.1 1.1–1.4, 2.1 1.1–1.4, 2.1, 2.5

3.2.2, 3.3.7.2, 3.3.7.9, 3.4.1.2, 3.4.3.2

4.1, 4.1.2 4.1, 4.1.2

6.1.4, 6.3.2.4

4.1.3, 4.1.7, 4.1.8

5.2.2, 5.3.2

4.1.5 4.1.6

5.2.5 5.2.3.2

6.2.1.1, 6.3.2.2, 6.4.1.2

3.3.8 3.3.5, 3.3.7.6 3.1.3.2, 3.3.7.7, 3.4.1.5, 3.4.2.5 3.3-7.4 3.3.7.5

4.1.4 4.1.4

6.2.1, 6.3.2.3

3.3.7.5 3.3.7.4, 3.3.7.6, 3.4.1.4

4.1.4 4.1.5

6.3.2.3 6.2.1.1

7.1.1.4, 7.1.5.1.6

8.1.1, 8.1.5

7.1.4, 7.1.4.1, 7.1.4.2, 7.1.4.3

Precipitation (see crystallization) Pressure-swing adsorption (see adsorption)

3.3.7.5

Solvent extraction (see extraction)

1.1–1.6, 2.1, 2.3

Supercritical fluid extraction

1.1–1.6, 2.1, 2.3

Zone melting (see crystallization)

1.1–1.4, 2.1

a

3.1.3.2.3, 3.1.3.2.4, 3.3.5, 3.3.7.4, 3.3.7.6, 3.4.1.4 3.3.7.2, 3.3.7.9, 3.4.1.2, 3.4.3.2

4.1, 4.1.5

6.2.1.1

4.1.3, 4.1.7, 4.1.8

5.2.2, 5.3.2

3.3.7.2, 3.3.7.9, 3.4.1.2, 3.4.1.6, 3.4.3.2

4.1.3, 4.1.7

5.2.2

3.3.7.5

4.1.4, 4.1.9.1

See also Sections 3.1.2.5, 3.1.3.2, 3.1.4, 3.3.1 and Example 3.1.9. See also Sections 3.1.2.5, 3.1.4, 3.2.1, 3.2.2, 3.3.1. c See also those listed under “Adsorption”. d See also adductive crystallization, clathration, Sections 4.1.9.1.1, 4.1.9.1.2. e See also those listed under “Adsorption” and “Parametric pumping”. f See also Chapter 9 and Sections 10.2.4, 11.3. b

6.2.3, 6.4.1.1

6.2.1.1, 6.3.2.2, 6.4.1.2 6.2.1.1, 6.3.2.2, 6.4.1.2 6.2.1, 6.3.2.3

7.1.1, 7.1.2

8.1.1, 8.1.1.1– 8.1.1.3, 8.1.4 8.1.1, 8.1.4

8.1.5

10.1.4.3, 11.3, 11.4

Table 2. Relevant sections for each membrane based separation process

Chapter 5 Effect of chemical reaction

Chapters 1 & 2 Describe separation

Chapter 3 Basis for separation

Chapter 4 Separation in a closed vessel

Cake filtration

2.1, 2.2, 2.4

3.4.2.3

4.2.1.2

Dialysis

1.1–1.4, 2.1, 2.3

4.3.1

5.4.3

Donnan dialysis Electrodialysis

2.1, 2.3 1.1–1.4, 2.1, 2.3

4.3.1, 4.3.2 4.3.2

5.2.6

Emulsion liquid membrane

1.1–1.4, 2.1, 2.3

3.1.3.2.3, 3.1.4, 3.3.7.4, 3.4.2.3.1 3.1.3, 3.3.7.7, 3.4.2.5 3.1.3.2, 3.3.7.7, 3.4.2.5 3.1.4, 3.3.7.2, 3.4.1.2

4.1.3, 4.1.8

5.2.2, 5.3.2, 5.4.4, 5.4.4.1–5.4.4.3, 5.4.4.5

Filtration (see cake filtration) Gas permeation (membrane gas separation) Gaseous diffusion Hollow fiber contained liquid membrane Immobilized liquid membrane

2.1, 2.2, 2.4

3.4.2.3

1.1–1.4, 2.1, 2.2

3.1.3.2, 3.3.1, 3.3.7.3, 3.4.2.2

4.3.3

2.1, 2.2 1.1–1.4, 2.1–2.3

3.1.3.2.4, 3.4.2.4 3.1.4, 3.3.7.2, 3.3.7.4, 3.4.3 3.1.4, 3.3.7.2, 3.3.7.4, 3.4.3

Prob. 4.3.9 4.1.3, 4.1.8

1.1–1.4, 2.1–2.3

3.1.4, 3.3.7.2, 3.3.7.4

4.1.3, 4.1.8

Membrane contactor gas absorption/ stripping See reverse Microfiltration

1.1–1.4, 2.1–2.3

3.1.2.5, 3.1.3.2.3, 3.1.3.2.4, 3.1.4, 3.3.7.1, 3.4.1.1, 3.4.3.1 3.1.2, 3.1.3.1, 3.3.6, 3.4.2.3

4.1.1.2, 4.1.7

Pervaporation

1.1–1.4, 2.1, 2.2

4.1.2, 4.3.3

Reverse osmosisa (Hyperfiltration)

1.1–1.4, 1.5, 1.7, 2.1, 2.2

Ultrafiltration

1.1–1.4, 1.5, 2.1, 2.2, 2.4.2, 2.5

3.1.3.2, 3.3.7.3, 3.3.7.4, 3.4.1.1, 3.4.2.1.1 3.1.2.5, 3.1.3.2, 3.1.5.1, 3.1.5.2, 3.3.7.4, 3.4.2.1 3.1.3.2, 3.1.3.2.3, 3.1.5, 3.3.7.4, 3.4.2.3

Ion transport membrane Liquid membrane processes

a

1.1–1.4, 2.1–2.3

2.1, 2.4

Forward osmosis, Prob. 4.3.12.

4.1.3, 4.1.8

Chapter 6 Bulk flow jj to force(s) and CSTS

Chapter 7 Bulk flow ⊥ to force(s)

6.1.6, 6.3.1.4, 6.3.3.1

7.2.1.4, 7.2.1.5

Chapter 8 Bulk flow of two phases ⊥ to force(s)

11.1, 11.4 8.1.1, 8.1.7, 8.2.4.1 8.1.1, 8.1.7

5.4.4.1, 5.4.4.5, 5.4.4.6

6.2.1.1, 6.3.2.2, 6.4.2.2 6.1.6, 6.3.1.4, 6.3.3.1 6.3.3.5, 6.4.2.2

Chapters 9, 10 & 11 Other aspects

11.1 11.4 10.2.3, 11.2

8.1.1, 8.1.8

7.2.1.4, 7.2.1.5 7.2.2 7.2.1.1

11.2, 11.4 8.1.1, 8.1.9, 8.2.4.2

10.1.4.1

9.1.1 5.2.2, 5.3.2, 5.4.4, 5.4.4.1–5.4.4.4 5.2.2, 5.3.2, 5.4.4, 5.4.4.1– 5.4.4.3, 5.4.4.5 5.4-5.2 5.2.2, 5.3.2, 5.4.4 5.4.4.1–5.4.4.3, 5.4.4.5 5.2.1, 5.3.1

4.2.3.1–4.2.3.3

6.2.1.1, 6.3.3.5, 6.4.2.2 6.2.1.1, 6.3.3.5, 6.4.2.2

8.1.8, 8.1.9 7.2.1.1

8.1.8, 8.1.9

6.4.1.2

8.1.8, 8.1.9

6.2.1.1

8.1.1.1, 8.1.1.2, 8.1.2.1, 8.1.2.2, 8.1.2.2.1

11.2

6.2.2, 6.2.3, 6.3.1.4, 6.3.3.1 6.3.3.4

7.2.1.4, 7.2.1.5

11.1, 11.2

4.3.4

5.4.1, 5.4.1.1

6.3.3.3

7.2.1.2

9.1.2.1, 10.1.2, 11.2

4.3.4

5.4.2, 5.4.2.1–5.4.2.3

6.3.3.2

7.2.1.3

9.1.1, 9.1.2.1, 11.1, 11.2

Table 3. Relevant sections for each centrifugal force driven separation process Chapters 1 & 2 Describe separation Centrifugal elutriation

2.1, 2.4, 2.5

Centrifugal separations (centrifugal filtration)

2.1, 2.4, 2.5

Cyclone dust separator

2.1, 2.4, 2.5

Gas centrifuge

2.1, 2.2, 2.3

Hydroclone

2.1, 2.2, 2.4

Isopycnic sedimentation

1.1–1.4, 2.1, 2.2, 2.4

Separation nozzle

1.1–1.4, 2.1 2.2, 2.4

Chapter 3 Basis for separation 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.1 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.1 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.1 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.1, 3.3.3 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.1, 3.3.3 3.1.2.2, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.1

Chapter 4 Separation in a closed vessel

Chapter 5 Effect of chemical reaction

Chapter 6 Bulk flow jj to force(s) and CSTS

Chapter 7 Bulk flow ⊥ to force(s)

Chapter 8 Bulk flow of two phases ⊥ to force(s)

6.2.2, 6.3.1.3

11.1

4.2.1.1, 4.2.1.2, 4.2.1.3

6.1.3, 6.1.6, 6.2.2, 6.3.1.3

7.3.2.1– 7.3.2.4

8.2.3

4.2.1.1, 4.2.1.3

6.2.2

7.3.2.3

8.2.3

4.2.1.1

6.1.3, 6.2.1

7.3.2.4

8.1.1, 8.1.10

4.2.1.2

6.2.1, 6.2.2

7.3.2.3.1

4.2.1.3

4.2.1.1

Chapters 9, 10 & 11 Other aspects

7.3.2.4

11.1

9.1.1

Table 4. Relevant sections for each electrical force driven separation process

Capillary electrophoresis Continuous free-flow electrophoresisa Corona-discharge reactor Dielectrophoresis Electrochemical membrane gas separation Electrochromatography

Chapters 1 & 2 Describe separation

Chapter 3 Basis for separation

1.1–1.4, 2.1, 2.4.2, 2.4.3, 2.5 1.1–1.4, 2.1, 2.2, 2.4, 2.5

3.1.2.2, 3.1.2.7, 3.1.3.2, 3.2.1 3.1.2.2, 3.1.2.7, 3.1.3.2, 3.2.1 3.1.6.1, 3.1.6.2 3.1.2.2, 3.1.2.7

Chapter 5 Effect of chemical reaction

4.2.2.1, 4.2.2.2

5.2.2.1, 5.2.2.2

4.2.2.1, 4.2.2.2

5.2.2.1, 5.2.2.2

Chapter 6 Bulk flow jj to force(s) and CSTS 6.1.5, 6.2.1 6.3.1.2 6.1.1, 6.1.5, 6.2.1, 6.3.1.2

Chapter 7 Bulk flow ⊥ to force(s)

Chapter 8 Bulk flow of two phases ⊥ to force(s)

Chapters 9, 10 & 11 Other aspects

7.1.7, 7.1.7.1

8.2.2.1, 8.2.2.3

11.1

7.3.1.1 7.3.1.2 7.3.1.1.1

5.4.4.6

Electrodialysis Electrosettler Electrostatic particle separator Electrostatic precipitator Flow cytometry Ion mobility spectrometry

2.1, 2.1, 2.1, 2.1,

Isoelectric focusing

1.1–1.4, 2.1, 2.5

Laser isotope separation

2.1, 2.2, 2.4

Micellar electro-kinetic chromatography Potential swing adsorption (Electrosorption, capacitive deionization)

1.1–1.4, 2.1, 2.4.2, 2.4.3, 2.5

a

Chapter 4 Separation in a closed vessel

2.2, 2.4 2.2, 2.4 2.2, 2.4 2.2, 2.4

3.1.2.2, 3.1.2.7, 3.1.3.2, 3.2.1 see Table 2 3.1.2.2, 3.1.2.7 3.1.2.2, 3.1.2.7, 3.1.3.1 3.1.2.2, 3.1.2.7, 3.1.3.1 3.1.2.2, 3.1.2.7, 3.1.3.1 3.1.2.2, 3.1.2.7, 3.1.6.1, 3.1.6.2 3.1.2.2, 3.1.2.7, 3.1.3.2, 3.2.1 3.1.2.2 3.1.2.7, 3.1.6.1/2 3.1.2.2, 3.1.2.7, 3.1.3.2, 3.2.1

Moving boundary electrophoresis, zone electrophoresis: Section 4.2.2.2.

11.1 8.1.8

4.2.2.1

7.1.7, 7.1.7.1

8.1.1, 8.1.6

4.2.2, 4.2.3.4.1 6.2.2, 6.3.1.5 6.2.2, 6.3.1.5 6.2.2

7.2.2, 7.3.1.4 7.2.2, 7.3.1.3 7.3.1.5 7.3.1.2

4.2.2.1, 4.2.2.2 7.3.1.2 4.2.2.1, 4.2.2.2

5.2.3.2

6.1.5, 6.2.1, 6.3.1.2

8.2.2.1 7.1.3

11.2

Table 5. Relevant sections for gravitational or magnetic force driven separation processes Chapters 1&2 Describe separation

Chapter 3 Basis for separation 3.1.2.1, 3.1.2.7, 3.1.3.1, 3.1.3.2, 3.2.3 3.1.2.4, 3.1.2.7, 3.1.3.1, 3.1.3.2 3.1.2.4, 3.1.2.7, 3.1.3.1, 3.1.3.2 3.1.2.4, 3.1.2.7, 3.1.3.1, 3.1.3.2

Gravity driven separation (elutriation, jigging) Magnetic force driven separation Magnetophoresis

High-gradient magnetic separation

Chapter 4 Separation in a closed vessel

Chapter 5 Effect of chemical reaction

4.1.10, 4.2.3, 4.2.3.1, 4.2.3.2– 4.2.3.4

Chapter 6 Bulk flow jj to force(s) and CSTS

Chapter 8 Bulk flow of Chapters 9, Chapter 7 10 & 11 two phases Bulk flow ⊥ to force(s) ⊥ to force(s) Other aspects

6.3.1.1, 6.3.1.5

7.2.2, 7.3.3, 7.3.3.1, 7.3.4

7.3.4, 7.3.5, 7.3.5.1, 7.3.5.2, 7.3.5.2.1 7.3.5, 7.3.5.1

7.3.5, 7.3.5.1, 7.3.5.2, 7.3.5.2.1

Table 6. Relevant sections for each separation process driven by other forces Chapters 1&2 Chapter 3 Describe Basis for separation separation Acoustic forces/waves Inertial impaction–depth filtration Radiation pressure– photophoretic separation Thermal diffusion 1.1–1.5 Thermophoresis

Chapter 5 Chapter 4 Effect of Separation in chemical a closed vessel reaction

3.1.2.6 3.1.2.6.1

Chapter 7 Bulk flow ⊥ to force(s)

6.2.2, 6.3.1.4

7.2.2

6.2.2

7.3.6

Chapter 8 Bulk flow of Chapters 9, two phases 10 & 11 ⊥ to force(s) Other aspects

4.2.4, 4.2.4.1

3.1.2.6, Ex. 3.1.4 3.1.2.6 3.1.2.6

Chapter 6 Bulk flow jj to force(s) and CSTS

4.2.5

6.2.1.1

11.2

8.1.1, 8.1.11

Table 7. Relevant sections for additional separation processes Chapters 1&2 Describe separation

Chapter 3 Basis for separation

Field-flow 1.1–1.4, 3.1.2.1– fractionation 2.1, 2.5 3.1.2.7, 3.1.3.1, 3.1.3.2 Mass diffusion 1.1–1.6, 2.1, 3.1.2.5, (sweep 2.3, 3.1.3.1, diffusion) 2.4.3 3.1.3.2, 3.1.4, 3.1.5.1

Chapter 5 Chapter 4 Effect of Separation in a chemical closed vessel reaction

Chapter 6 Chapter 8 Bulk flow jj Chapter 7 Bulk flow of to force(s) Bulk flow ⊥ two phases ⊥ and CSTS to force(s) to force(s) 6.1.1

7.3.4

8.1.11

Chapters 9, 10 & 11 Other aspects

Table 8. Basic properties and phenomena underlying separation processes for molecules, macromolecules and particles

Basic property

Phenomenon causing separation

Charge-to-mass ratio

different charge-to-mass ratios of different particles/molecules

Chelation of metal ions

selective chelation

Condensability of a vapor/gas species

different condensabilities of vapors/gases

Density of liquid

heavier liquid is located further from the center for an immiscible liquid mixture in a centrifugal force field particles/cells develop a dipole moment in a nonuniform electrical field different diffusivities for different species

Dielectric constant

Diffusivity

Electrical charge on an element, molecule, macromolecule, particle

Phase equilibrium based separation processes

Membrane separation processes

electrostatic separation of particles (Section 7.3.1.4), mass spectrometry (Karger et al., 1973) solvent extraction of metals (Section 5.2.2.4, Table 1) vapor absorption, vapor–liquid chromatography (Table 1, Section 4.1.1.1)

liquid membrane processes (Table 2, Section 8.1.8) membrane gas permeation (Section 4.3.3, Figure 4.3.4, Table 2) centrifugal separation (Section 7.3.2.1.1)

dieelectrophoresis (Table 4, Section 7.3.1.1.1) ultrafiltration, nanofiltration, dialysis, gas permeation, gaseous diffusion (Table 2) electrodialysis (Table 2)

attraction toward oppositely charged electrode

membrane surface has opposite charge size exclusion for different size solutes

thermal diffusion (Table 6), mass/ sweep diffusion (Table 7, Section 7.3.4) element, molecule, particle in gas phase: laser isotope separation, corona-discharge reactor, ion mobility spectrometry, electrostatic precipitator, flow cytometry (Table 4) capillary electrophoresis, continuous free-flow electrophoresis

different ionic/electrophoretic mobilities and attraction toward oppositely charged electrode

Geometrical partitioning of a solute between pore and a solution

External force based and other separation processes

gel permeation chromatography, size exclusion chromatography (Table 1, Sections 3.3.7.4, 3.4.2.3.1, 7.1.5.1.7)

nanofiltration separation of divalent anions, ultrafiltration of charged proteins (Table 2) dialysis, ultrafiltration (Table 2, Sections 3.3.7.4, 3.4.2.3, 4.3.1, 6.3.3.2)

Magnetic permeability or susceptibility Mass, density of particles

Molecular mass of species

Particle compressibility

pI, isoelectric point of a protein Reversible complexation of solute/macrosolute with a ligand Reversible electrostatic interaction between counterions in solution and porous charged solids Solubility in a solvent of a solute present in a solution or a solid

different magnetic permeability/ susceptibility between particles and the solution different terminal/settling velocities in gravitational/ centrifugal force field heavier gas/isotope concentrates further from the center in a centrifugal force field positive and negative values of f-factor in an acoustic force field proteins having different pI values equilibrate at different locations in a pH gradient selective complexation capability

magnetophoresis, high-gradient magnetic separation (Table 5) gravitational settlers, jigging, elutriational devices, centrifuges, cyclone separators (Tables 3, 5) gas centrifuge, separation nozzle (Table 3) acoustic force based separation (Table 6) isoelectric focusing (Table 4)

affinity chromatography (Section 7.1.5.1.8)

preferential interaction of a counterion in solution with porous charged solid

ion exchange, ion exchange chromatography (Table 1, Sections 3.3.7.7, 4.1.6)

different solubilities for different solutes

solvent extraction, liquid–liquid chromatography, leaching, supercritical extraction (Table 1, Sections 4.1.3, 4.1.7, 3.3.7.5) micellar extraction, reverse micellar extraction (Section 4.1.8)

Solubility in micellar core or selective complexation with micellar headgroups Solubility of a gas/vapor in a liquid

selective partitioning into micellar core or complexation with charged headgroups different solubilities of gases/ vapors in a liquid

Solubility of a gas/vapor in a polymer Species come out of solution or melt as a crystal Surface active nature of solute

different solubilities of gases/ vapors phase change leading to crystal formation excess accumulation of a species at the interface of phase 1 and phase 2: gas–liquid, liquid–liquid

absorption, gas–liquid chromatography (Table 1, Figure 3.3.3, Section 4.1.1.1)

membrane chromatography (Section 7.1.5.1.8), ultrafiltration (Section 5.4.2.1) electrodialysis, Donnan dialysis (Table 2, Sections 3.4.2.5, 4.3.2)

liquid membrane processes (Table 2, Section 8.1.8)

micellar-enhanced ultrafiltration (Section 5.4.2.2) membrane contactor (Table 2)

membrane gas permeation, pervaporation (Table 2) crystallization, zone melting (Table 1, Section 3.3.7.5) foam fractionation (Table 1, Sections 3.3.7.6, 4.1.5, 5.2.5)

micellar electrokinetic chromatography (Section 8.2.2.1)

Table 8. (cont.) Phase equilibrium based separation processes

Basic property

Phenomenon causing separation

Surface adsorption potential

excess accumulation of a species at the interface of phase 1 and phase 2: gas–solid, liquid–solid different volatilities of bulk liquids and solutes evaporation of water

distillation, stripping (Table 1, Sections 4.1.1, 4.1.2) evaporation, drying (Table 1)

sublimation of water

freeze-drying (Table 1)

Vapor pressure of a liquid, volatility of a solute Water removed by vaporization from a solution or moist solid Water removed from a solid by sublimation

adsorption, chromatography (gas–solid, liquid–solid) (Table 1, Sections 3.3.7.6, 4.1.5)

Membrane separation processes gas separation by surface diffusion (Section 3.4.2.4), preferential sorption and capillary transport in reverse osmosis (Table 2) pervaporation (Table 2, Section 6.3.3.4) (plus membrane permeability) membrane distillation (Song et al., 2008)

External force based and other separation processes

1

Description of separation in a closed system Separation is a major activity of chemical engineers and chemists. To separate a mixture of two or more substances, various operations called separation processes are utilized. Before we understand how a mixture can be separated using a given separation process, we should be able to describe the amount of separation obtained in any given operation. This chapter and Chapter 2 therefore deal with qualitative and quantitative descriptions of separation. Chapter 2 covers open systems; this chapter describes separations in a closed system. In Section 1.1, we briefly illustrate the meaning of separation between two regions for a system of two components in a closed vessel. Section 1.2 extends this to a multicomponent system. In Section 1.3, various definitions of compositions and concentrations are given for a twocomponent system. In Section 1.4, we are concerned with describing the various indices of separation and their interrelationships for a two-region, two-component separation system. A number of such indices are compared with regard to their capacity to describe separation in Section 1.5 for a binary system. Next, Section 1.6 briefly considers the definitions of compositions and indices of separation for the description of separation in a multicomponent system between two regions in a closed vessel. Finally, Section 1.7 briefly describes some terms that are frequently encountered. The separation of a mixture of chemical species 1 and 2 may involve separating either a mechanical mixture or a true solution of the two species. Examples of mechanical mixtures include water containing fine particles forming a slurry, air containing fine dust particles, etc. Here water or air comprise species 1, whereas the chemical substance present in the form of solid particles comprises species 2. (Although air is a mixture of various species, we consider it here as species 1.) Examples of true solutions of species 1 and 2 are: a solution of sugar in water, a liquid solution of benzene and toluene, a gas mixture of nitrogen and carbon dioxide, a solid solution of silver in gold, etc. In such true

solutions, the two different chemical species are intimately mixed at the molecular level. In a mechanical mixture, solid particles, comprise aggregates of molecules of species 2, so the two species are not intimately mixed at the molecular level. Colloidal solutions and macromolecular solutions provide a spectrum of behavior in between these two limits. In this book we will primarily treat separation of true solutions, the so-called chemical separations. Separation of mechanical mixtures, as well as colloidal and macromolecular solutions, will also receive significant coverage.

1.1 Binary separation between two regions in a closed vessel Consider a closed vessel A containing a true solution of species 1 and 2, as shown in Figure 1.1.1. Let the solution composition be uniform throughout the vessel. We will refer to this condition as the perfectly mixed state. The process of separation is now initiated somehow, either by the addition or extraction of some amount of energy without allowing any mass to enter or leave the system. The end result of any such process is shown by the condition existing in another vessel, B. Depending on the solution and the process employed, vessel B can have any number of conditions. If the condition is represented by I, we note that vessel B has two regions, each region being occupied by molecules of only one kind. Thus species 1 and 2 have been completely separated from each other. If the material in any one of the two regions is now removed from vessel B, we will have obtained the corresponding pure species. Such a condition is called perfect separation. It is the exact opposite of the perfectly mixed state present in vessel A. Further, the objective of all separation would be to achieve condition I in vessel B whatever the initial conditions. Note that, henceforth, we would like to identify region 1 with species 1 and region 2 with species 2. This

20

Description of separation in a closed system

Condition I Region 1

Perfect separation

Region 2 Vessel B Condition II

Uniform mixture

Condition IV

Region 1

Region 1

Region 2

Region 2 Vessel B

Vessel A

Vessel B

Condition III Region 1 Imperfect separation

Species 1 Species 2

Region 2 Vessel B

Figure 1.1.1. Binary separation between two regions in a closed vessel.

means that, for perfect separation, region 1 will have only molecules of species 1, and region 2 will have, only molecules of species 2. The condition represented by II in Figure 1.1.1 is, however, quite different. Vessel B still has two regions. But region 1, instead of having only species 1, has some molecules of species 2, while region 2, instead of having only species 2, has some molecules of species 1. If the material in any one of the two regions is removed from vessel B, we will obtain a solution of two species instead of an amount of pure species. This condition illustrates imperfect separation. All real separation processes almost always result in an imperfect separation. Although perfect separation is therefore never attained, the condition of perfect separation is a valuable yardstick for measuring the amount of separation achieved by an actual separation process. If we refer to the separated material in any one of the two regions in condition II as a fraction, then the purity level or the impurity level of the fraction would provide one such yardstick. By the convention adopted earlier, region 1 will have now species 2 as an impurity, whereas region 2 will have species 1 as an impurity. Another estimate of the separative efficiency of the process is provided by how far removed the compositions of the separated regions are from the original mixture in vessel A. Before we develop and use an index of separation, which serves as a criterion of merit to indicate the separative efficiency, we need to examine the quantities that describe the composition of a binary mixture. However, knowing the composition only can sometimes be misleading. For example, see the condition represented by III in

Figure 1.1.1. We have, as before, two regions, region 1 and region 2. Region 1 has only pure species 1, whereas region 2 has both species 1 and 2. If we consider region 1 in III and region 1 in I and consider only the compositions of this region, we may mistakenly conclude that perfect separation has been achieved in both cases, since region 1 has only pure component 1 under both conditions. What we should also consider is how much of species 1 present in the original mixture in vessel A has been recovered in region 1 of vessel B in condition III. Since not all of the species 1 has been recovered and put in region 1, condition III represents another case of imperfect separation. Similarly, if region 2 had only species 2 and region 1 had a mixture of species 1 and 2, we again have imperfect separation. If the previous paragraphs have led to the belief that a separation process always starts with a uniform mixture, as in vessel A of Figure 1.1.1, and ends up with any of the three types of conditions I, II or III in vessel B, this is not the case. One can start with a condition shown in II and carry out a separation process, perhaps by altering the physical state (e.g. temperature, pressure, etc.), and achieve greater separation. For example, consider condition IV in vessel B achieved by changing condition II. Obviously, region 1 in condition IV is purer in species 1, and similarly for species 2 in region 2. Thus a separation operation can be carried out not only on a uniform mixture, but also on two mixtures of different compositions existing in two contiguous regions of a vessel or a container. We have used the notion of a region in a separation system without much explanation; it is worthwhile to

1.1

Binary separation in a closed vessel

21

Feed mixture

Example I Add heat

Example II Extract heat

Example III Extract heat

Example IV Extract heat

Benzene toluene mixture (Liquid)

Saline water (Liquid)

Aniline hexane mixture (Liquid)

Carbon dioxide phthalic anhydride mixture (Gas)

Separated system

Region 1

Vapor phase

Benzenerich

Region 2

Liquid phase

Toluenerich Ice crystals

Region 1 Region 2

Region 1 Region 2

Region 1 Region 2

Concentrated brine

Salt-rich water

Hexane-rich liquid layer Aniline-rich liquid layer

Carbon dioxide (gas) Phthalic anhydride crystals

Temperature less than 126 ⬚C

Figure 1.1.2. Binary separation systems with immiscible phases or regions.

explain its physical meaning. A region is a certain volume in space enclosed within given boundaries such that its contents have a composition different from that of an adjoining region (or those of adjoining regions if the separated system has more than two regions). The words fraction or phase may also be used to denote a region in a separation system. When the separation system contains two immiscible phases in a vessel, e.g. vapor (or gas) and liquid, solid and liquid, solid and vapor (or gas) or liquid and liquid, the use of “phase” or “fraction” is common. Consider Figure 1.1.2, where we show examples of several practical separation processes. In example I, a liquid mixture containing the same number of moles of benzene and toluene is separated by the addition of heat into two immiscible phases – a vapor phase richer in benzene and a liquid phase richer in toluene. The space occupied by vapor is region 1, and the liquid phase is region 2. In example II of Figure 1.1.2, a solution of salt in water can be separated by cooling the solution (extraction of heat) to form two distinct phases: ice crystals floating at the top (since they are significantly lighter than water or brine) and concentrated brine at the bottom. If the ice crystals are collected together without any brine sticking to the ice crystals (something which is almost impossible to carry out), then the concentrated brine is phase 2 and the ice crystals make up phase 1. If ice crystals cannot be collected together, then each ice crystal is considered to be part of region 1. Similarly, examples III and IV have two regions in a separated system; example III has a liquid–

liquid system and example IV has a vapor–solid system1 (Lowenheim and Moran, 1975). In each of these four examples, the top phase, which is the lighter phase, is usually referred to as the light fraction, while the bottom phase is called the heavy fraction. Remember that for separation systems containing immiscible phases, phase or fraction is more commonly used than region. On the other hand, consider the separation of saline water by a semipermeable membrane, as shown in example I of Figure 1.1.3. The uniform saline solution can be separated by the application of pressure energy on the feed saline to yield almost pure water on the low-pressure side of the membrane and concentrated brine on the highpressure side. The membrane (or the barrier or partition) separates the two fractions in the two regions. However, both of these fractions are miscible with each other, unlike the immiscible phases of Figure 1.1.2. The words fraction or region are more appropriate here. Whereas in Figure 1.1.2, the two regions were separated only by the individual phase boundaries, in this case the membrane actually defines the boundaries of the two regions. Another example of a membrane providing the boundaries between two regions is obtained in the separation of compressed air by a silicone membrane (example II of 1

In the manufacture of phthalic anhydride by naphthalene oxidation, the gaseous products are CO2, water vapor and phthalic anhydride. For illustration here we use a CO2 and phthalic anhydride system, where phthalic anhydride crystals are formed from the vapor below 126
C.

22

Description of separation in a closed system

Atmospheric pressure

Region 2

Purified water Concentrated brine

Air 25 ⬚C

Region 1 Region 2

50% N2–50% H2 mixture

Semipermeable membrane Bulb 1

O2 - rich fraction N2 - rich fraction

Silicone membrane

H2 - rich region 1

Bulb 1 500 ⬚F

Pressure energy

Bulb 2 System at 50⬚F

High pressure

Pressure energy

Example II

Example III

energy

Saline water

Region 1

Thermal

Example I

N2 - rich region 2

Bulb 2 50 ⬚F

Figure 1.1.3. Binary separation systems with both regions miscible with each other.

Figure 1.1.3). Although separation in a closed vessel with membranes is often time-dependent, the conditions at any instant of time are sufficient for describing separation. There can also be a separated system in which the two regions are not separated from each other by a membrane nor are they immiscible with each other. Such a separation system is shown in the form of a two-bulb cell in example III of Figure 1.1.3. Originally, the two-bulb cell contained a uniform gas mixture of nitrogen and hydrogen (say, 50% N2–50% H2, mole percent) at, say, 50
F. If now the bulb 1 (region 1) has its temperature raised to and held at, say, 500
F, while the temperature of bulb 2 (region 2) is maintained at 50
F, we will observe that region 1 has become richer in the light component, hydrogen, whereas region 2 has become richer in the heavy component, nitrogen, due to the phenomenon of thermal diffusion. What is important here is to recognize that the two regions having different gas compositions at steady state are not separated by either a membrane or a phase boundary.2 In this case, the composition changes continuously through the capillary from one bulb to the other. However, since the capillary volume is very small compared to the volume of either of the bulbs, we may neglect it. So we have a separated system with two regions of different compositions such that not only are the materials in both regions completely miscible, but also there is no barrier separating the two regions. Although “light fraction” (“hydrogen-rich fraction”) or “heavy fraction” (“nitrogen-rich fraction”) are used to describe such a separation, the word region is more descriptive. A few separated systems with continuously varying compositions are described in Problems 1.4.3 and 1.4.4. These involve systems without any barriers or immiscible phases.

2

In any separated system without two immiscible phases or a barrier separating the two regions, the composition profile will be continuously varying.

1.2 Multicomponent separation between two regions in a closed vessel If the separation system contains more than two species, strictly we are dealing with multicomponent separation. Thus, if we have species 1, 2 and 3 or 1, 2, 3 and 4 or 1, 2,. . ., n, etc., the separated system will have, in general, different compositions with respect to each species in the two regions. All situations described in Figures 1.1.1–1.1.3 are also valid here. The only difference is that the uniform solution we start with in Figure 1.1.1 is a multicomponent mixture. Further, whereas in Figure 1.1.1 one region has only pure component 1 present and the other region has a binary mixture of species 1 and 2, for multicomponent separation, if region 1 has only pure species 1, region 2 will have a mixture of all species (with or without species 1). There is an additional point worth emphasizing. If there are only two regions in the closed vessel, it is not possible to have perfect separation since one requires as many regions as there are numbers of species present in the original mixture. Therefore, perfect separation for a three-component mixture requires three regions (or four regions for a four-component mixture). These requirements are illustrated by various alternative separation conditions in Figure 1.2.1 for a four-component mixture. For example, starting with a uniform mixture in vessel A, one can conceive of perfect separation in vessel B with four regions (conditions I or II). On the other hand, if two regions are available, only an imperfect separation is possible (conditions III or IV). The case of multicomponent separation between two immiscible phases requires further consideration if each phase is primarily made up of one species and the remaining components are present in these phases in small amounts. Consider a three-component system consisting of, say, water, benzene and picric acid, the latter being present in very small amounts. Water and benzene

1.2

Multicomponent separation in a closed vessel

23

Perfect separation Condition II

Condition I Region 1

Region 1

Region 3

Region 2

Region 4

Region 3 Region 4 Region 2 Vessel B

Vessel B Uniform mixture

Condition III Region 1 Region 2 Vessel B

Vessel A

Imperfect separation

Condition IV Species 1

Region 1

Species 2 Region 2

Species 3 Species 4 Vessel B

Figure 1.2.1. Separation of a four-component mixture in a closed vessel.

Uniform mixture

Add benzene and stir

Region 1 (Benzene phase)

Region 2 (Water phase) Picric acid in water Water

Picric acid

Benzene

Figure 1.2.2. Multicomponent separation between two immiscible liquid phases.

form two immiscible liquid phases, with the lighter benzene layer above the heavier water layer. It is known that picric acid at low concentrations will be distributed (Hougen et al., 1954) between the two phases such that its concentration in the benzene phase is higher than that in the water phase. Let the original mixture to be separated be picric acid in water. Benzene is added to this mixture, shown in Figure 1.2.2, since it is easier to recover picric acid from benzene than from its original mixture in water. Two fractions have been formed: a benzene based fraction, the extract, and the water based residue, the raffinate. Although this is a three-component system, the description of each phase can be given as if it were a binary system, i.e. a solution of picric acid in water or a solution of picric acid in benzene. Similarly, suppose two solutes, picric acid and benzoic acid, are distributed between two phases, one of

which is a benzene phase and the other one is a water phase. Each phase can be described as if it were a ternary system since benzene and water are immiscible (in reality, very small amounts of water are soluble in benzene and vice versa). In fact, one can go a step further by identifying the benzene extract phase as region 1 and the water raffinate phase as region 2 and working only with the concentrations of the two acids in each region. We then, in effect, have a binary system of benzoic acid and picric acid between two regions, 1 (benzene layer), and 2 (water layer), and one could proceed with the description in the manner of Section 1.1. Thus the description of a fourcomponent separation system may be reduced to that of a two-component separation system provided each of the immiscible phases is made up of essentially one species

24

Description of separation in a closed system

only and the two components, in whose separation we are interested, are present in small amounts. A variety of conditions are possible with multicomponent systems in general. When one needs to separate particles of one size from particles of all other sizes present in a suspension, or when particles of different sizes are to be separated from one another as well as from the medium of suspension, a multicomponent separation problem exists. If a macromolecular substance or solution is to be fractionated, the nature of the description of the separation problem is similar. Such problems are handled most conveniently by dealing with pseudo binary systems. We will come across such cases as we proceed further in this book, although binary systems will be encountered much more frequently.

ρij and the mass fraction uij of species i in region j are defined by ρij ¼ uij ¼

mij M i ¼ C ij M i , Vj ρij ρij ¼ , 2 X ρtj ρij

So far we have described separation qualitatively. In order to describe separation quantitatively, first we have to define compositions and then use such compositions to define indices of separation. Only with the help of indices of separation can one describe separation quantitatively. Refer to Figure 1.1.1 for a physical background behind the definitions of compositions. With both vessels A and B closed, we assume that the separation process which resulted in the condition of vessel B from that of vessel A conserved the total number of moles of each of the species 1 and 2. Assume further that within any region the composition is uniform everywhere (although this is not true for some of the representations actually used in Figure 1.1.1). Let mij denote the number of moles of the ith species in the jth region in vessel B under any particular condition. For a binary system, i ¼ 1, 2, since there are two regions only, j has values 1 and 2. The mole fraction xij of species i in region j is defined by x ij ¼

mij ; 2 X mij

i ¼ 1, 2;

j ¼ 1, 2:

ð1:3:1Þ

i¼1

The mole ratio Xij of species i in region j is given by X ij ¼

mij ; mkj

i ¼ 1, 2;

k ¼ 1 or 2:

where Mi is the molecular weight of the ith species and ρtj is the total mass density of the mixture in region j. In dealing with mixtures of two isotopes (e.g. H2 and D2), one comes across instead of mole fraction the atom fraction aij. The atom fraction aij of the ith isotope of an element in region j is obtained as

C ij ¼

mij : Vj

ð1:3:3Þ

It should be noted that C1j þ C2j ¼ Ctj, the total molar density of the mixture in region j. The mass concentration

maij 2 X

,

ð1:3:6Þ

maij

i¼1

maij

is the number of atoms of the where  ith isotope  of the element in the jth region. The sum ma1 j þ ma2 j represents the total number of atoms of the element in the jth region, with i ¼ 1, 2 being the only two isotopic forms under consideration. For example, in a gas mixture of H2 and D2, if there are 0.186 moles of D2 for 0.189 moles of H2 and D2, the atom fraction of deuterium in that gas mixture is (0.186  2/0.189  2) ¼ 0.984. For binary isotopic mixtures, the quantity similar to mole ratio Xij is the abundance ratio X aij , which is given by X aij ¼

aij aij ¼ ; akj 1− aij

i ¼ 1, 2;

k ¼ 1 or 2:

ð1:3:7Þ

Thus the abundance ratio represents the number of atoms of the desired ith isotope per atom of the other isotope, where a1 j þ a2 j ¼ 1. The quantities defined so far describe the composition of a given region, phase or fraction with respect to how much of one species is present amongst all the species in the given region, phase or fraction. It is also useful to know what fraction of the total amount of a species present in the total separation system is present in a given region. Such a quantity is the segregation fraction Yij :

ð1:3:2Þ

Note that (xij þ x2j) ¼ 1 but (Xij þ X2j) need not equal 1. If the volume of the jth region is Vj, then the molar concentration Cij of species i in region j of vessel B is given by

ð1:3:5Þ

i¼1

aij ¼

1.3 Definitions of composition for a binary system in a closed vessel

ð1:3:4Þ

Y ij ¼

mij mij ¼ 0: 2 X mi mij

ð1:3:8aÞ

j¼1

m0i

moles of the ith species are present in Here separation vessel B having two regions j ¼ 1 and 2. Further, Y i1 þ Y i2 ¼ 1. For a two-component system with j ¼ 1, 2, one may define a segregation matrix [Yij] as well as the corresponding matrix for mole numbers [mij]:

1.4

Indices of separation for binary systems



  Y Y ij ¼  11 Y 21

  Y 12     m11 ¼ ; m ij  m21 Y 22 

 m12  : m22 

ð1:3:8bÞ

See Problem 1.3.1 for related representations. Following the segregation fraction Yij for nonisotopic systems, one can define the segregation fraction Y aij of an isotopic mixture as Y aij ¼

maij maij ¼ a0 : 2 X mi maij

ð1:3:9aÞ

j¼1

All three quantities, the mole fraction xij, the mass fraction uij and the atom fraction aij, vary between the limits of 0 and 1. When their value is 1, we have pure species i in region j. The closer their value is to 1, the greater is the purity of the region (fraction or phase) in the ith species. On the other hand, the mole ratio Xij and the abundance ratio X aij vary between 0 and ∞, the latter value indicating only pure species i in region j. Thus, the upper limits of the two sets of quantities (xij, uij, aij and Xij, X aij ) have radically different values, although all of these quantities indicate the level of purity of the ith species in the given jth region in their own ways. The remaining quantities Yij and Y aij have limits similar to those of xij, uij and aij, namely 0 and 1. But the upper limit has a different meaning. For example, Y22 ¼ 1 signifies that region 2 has all of species 2 present in the separation system. But this does not mean that region 2 has pure species 2, since some species 1 may also be present in region 2. Thus if Y22 ¼ 1, Y12 need not be zero (compare: if x22 ¼ 1, x12 ¼ 0). Further, Y11 need not be equal to 1. Condition III of Figure 1.1.1 illustrates such a separation. So Yij provides an estimate of the extent of recovery of species i in region j with respect to the total amount of species i present in the whole separation system. The goal of perfect separation is to segregate all of each species in its designated region in a pure form. For a separation system with i ¼ 1, 2 and j ¼ 1, 2, perfect separation may be represented as follows:       ðY 11 ¼ 1Þ 0 ; Y ij ¼  ðY 22 ¼ 1Þ  0   h i   a  Y ¼ 1 0  : Y aij ¼  11 ð1:3:9bÞ a Y 22 ¼ 1  0 It should be noted at this stage that some mass balance relations are necessary for relating conditions in vessel A to those in vessel B due to the law of conservation of mass. If the ith species mole fraction in the initial uniform mixture in vessel A is xif , then, in the absence of chemical reactions, a balance of the total number of moles of both species leads to 2 X 2 X i¼1 j¼1

mij ¼ m01 þ m02 :

ð1:3:10Þ

25 A balance of the ith species only yields 2 X  0  m1 þ m02 x if ¼ mij :

ð1:3:11Þ

j¼1

Sometimes the initial binary mixture to be separated is not a uniform mixture as in vessel A, but instead has two regions whose compositions are characterized as x if 1 and x if 2 corresponding to j ¼ f1, f2. Note that these regions are such that region f1 usually has more of species 1, whereas region f2 has more of species 2. The total and ith component mass balances in such a case lead to m01 þ m02 ¼ 2 X i¼1

1.4

!

mif 1 x if 1 þ

2 X i¼1

2 X i¼1

mif 1 þ

2 X

mif 2 ;

i¼1

!

mif 2 x if 2 ¼ m0i :

ð1:3:12Þ

Indices of separation for binary systems

A number of indices of separation may now be developed using the quantities defined in Section 1.3. A separation index is needed to indicate how much separation is obtained in a given separation process. Such a quantity enables one to determine quickly the separation capabilities of any particular process with respect to a given separation task. Since none of the quantities defined in Section 1.3 are restricted to any particular material or separation process (except aij, X aij and Y aij , which are restricted to isotopic mixtures), it is expected that the indices of separation given below for a two-component system will likewise be sufficiently general in their application. We restrict ourselves here to describing only the separation that has been achieved in vessel B (Figure 1.1.1) between regions 1 and 2. Further, the list of indices given below is not exhaustive. The simplest separation indices are the distribution ratio (or capacity factor) ki10 , the distribution coefficient κi1 and the equilibrium ratio Ki:3 k i10 ¼

mi1 ; mi2

κi1 ¼

C i1 ; C i2

Ki ¼

x i1 : x i2

ð1:4:1Þ

In general, k i10 ¼

mi1 C i1 V 1 V1 ¼ ¼ κi1 , mi2 C i2 V 2 V2

ð1:4:2Þ

so that if V2 ¼ V1, k i10 ¼ κi1 . Note that the three indices in equation (1.4.1) are all defined with the light fraction, i.e. fraction 1, on the top. In this convention, then, for species 1 to concentrate more in region 1 implies that κi1 is greater than 1, since Ci1 > Ci2. This does not mean that k il0 > 1

3

In chemical engineering literature k il0 is rarely used. But κi1 is often used, especially in liquid extraction where the system, in general, is a multicomponent system (see Section 1.2). In distillation and flash separation processes, Ki is used frequently.

26

Description of separation in a closed system

since for that to be true V1  V2. However, the upper and lower limits for all three indices are, respectively, ∞ and 0. Further, an upper limit of ∞ for k il0 and κi1 as well as for Ki does not necessarily imply perfect separation. Similarly, the lower limit of zero does not necessarily imply either zero or perfect separation. Before we present some complex indices of separation, some more indices apparently similar to those of equation (1.4.1) require further consideration. These are as follows. Impurity ratios (Gleuckauf, 1955a): η1 ¼

m21 ; m11

m12 : m22

η2 ¼

ð1:4:3aÞ

Purity indices (de Clerk and Cloete, 1971): I j ¼ − log10 ηj ;

j ¼ 1, 2;

I¼

2 X j¼1

Ij:

ð1:4:3bÞ

The subscript on the impurity ratio ηj refers to the phase (fraction or region) under consideration. Since phase 1 is supposed to contain primarily species 1 as a valid separation goal, species 2 in phase 1 is an impurity. Perfect separation therefore requires η1 ¼ 0 and η2 ¼ 0. For imperfect separation ηj > 0. Note, however, that η1 ¼ 0 does not imply η2 ¼ 0 or vice versa. Since perfect separation requires ηj ¼ 0 for both j ¼ 1, 2 simultaneously, improved separation in a given problem implies decreasing values of ηj. If it is desired that the separation index value should increase as separation improves, one can define an enrichment ratio η0j by utilizing the definition of ηj (Boyde, 1971): η0j ¼ 1− ηj :

ð1:4:4Þ

For such a definition, the maximum value of η0j ¼ 1 implies no impurity in region j. One could add subscripts and superscripts to η0j to indicate whether the separation of the desirable species 1 is being sought from the undesirable species 2 or vice versa (Boyde, 1971). The most commonly used separation index in chemical engineering is the separation factor α12, where the subscripts refer to the two species 1 and 2. It is defined by α12 ¼

x 11 x 22 : x 21 x 12

ð1:4:5Þ

For a binary system of species 1 and 2, since (x1j þ x2j) ¼ 1, we can express it also as α12

x 11 ð1  x 12 Þ X 11 ¼ ¼ , ð1  x 11 Þ X 12 x 12

following two relations for the mole fractions of any species between the two regions: x 11 ¼ x 12 ¼

α12 x 12 , 1 þ x 12 ðα12  1Þ

x 11 : α12  x 11 ðα12  1Þ

ð1:4:7bÞ ð1:4:7cÞ

The relationship between x11 and x12 is often presented graphically. For example, in systems having a vapor phase ( j ¼ 1) and a liquid phase (example I, Figure 1.1.2), x11 and x12 are plotted as the ordinate and the abscissa, respectively. Figure 1.4.1 displays this for the systems benzene– toluene and methanol–water. The straight line in each figure represents x11 ¼ x12, a condition where no separation is possible, since x11 ¼ x12, x21 ¼ x22 and α12 ¼ 1. The figure for each system has a dashed line representing the value of α12 as a function of x12. In the benzene–toluene system, α12 is a constant; in the other system, α12 varies with composition. The nature of this variation can sometimes be very complex. By using equation (1.4.1) in definition (1.4.5), we get a few relations between α12 and some other indices:4  0  0 : α12 ¼ ðK 1 =K 2 Þ ¼ k 11 =k 21

ð1:4:8Þ

α12 ¼ ð1=η1 η2 Þ:

ð1:4:9Þ

Similarly, definitions (1.3.1) and (1.4.3a) substituted into the definition of α12 yield

Note that if region 1 or region 2 has pure component 1 or 2, respectively, α12 ¼ ∞, a case of infinite separation factor. However, as indicated in Figure 1.1.1 (conditions I and III), unless both η1 and η2 are zero simultaneously, perfect separation is not achieved. Thus α12 ¼ ∞ need not imply perfect separation, although at least one region has a pure component only. An example of such a separation is the evaporative desalination of brine, where the vapor generated is pure water. Thus, with water as component 1 in the vapor region designated as 1, salt is absent in the vapor so that η1 ¼ 0 or x21 ¼ 0 and α12 ¼ ∞. However, if during vapor generation some brine droplets are entrained by the rising vapor, x21 6¼ 0 and α12 6¼ ∞. (See Example 1.4.2 for another problem of this type.) Often instead of the separation factor α12, one encounters the enrichment factor5 ε12 defined by ε12 ¼ ðα12  1Þ ¼

ð1:4:6Þ

ðx 11  x 12 Þ : x 12 ð1  x 11 Þ

ð1:4:10Þ

where k ¼ 2 in (1.3.2) so that X 11 ¼ α12 X 12 :

ð1:4:7aÞ

The mole ratios of any species between two regions are therefore related linearly through the separation factor α12. Similarly, using relations (1.4.6) one can easily obtain the

4

 0  0 The ratio k 11 =k 21 has been called the separation quotient (Rony, 1968a). 5 In separation literature, subscripts 1 and 2 are usually dropped and only ε is used.

1.4

Indices of separation for binary systems

27

1.0 0.8

5

0.6

4

x11

a12

x11 = x12

0.4 0.2

0

0 0.1

0.3

0.5 x12

0.7

1.0

11

0.8

9

0.6

7

x11

a12

x11 = x12

3

0.4

5

2

0.2

3

0.9 1

0

0 0.1

i = 1 = benzene, C6 H6

0.3

0.5 x12

0.7

0.9 1

1

i = 1 = methanol, CH3 OH

j = 2 = toluene, C7H9

j = 2 = water

j = 1, vapor; j = 2, liquid

j = 1, vapor; j = 2, liquid

Figure 1.4.1. Relationships between x11 and x12 and α12 and x12 for benzene–toluene and methanol–water mixtures in a vapor–liquid system.

If ε12 is nonzero, α12 is greater than 1, indicating that separation is possible. If ε12  1, the situation corresponds to close separation. In such a case, the following approximation is valid (Pratt, 1967a, p. 10): ðα12  1Þ2 þ    ffi ðα12  1Þ ¼ ε12 : ℓnðα12 Þ ¼ ðα12  1Þ  2 ð1:4:11Þ Therefore α12 ¼ e ε 1 2 ¼ exp ðε12 Þ:

ð1:4:12Þ

Further, for close separation, ðx 11  x 12 Þ ¼ ε12 x 12 ð1−x 11 Þ, a small quantity so that x11 is quite close to x12. Therefore, often x12 can be interchanged with x11 in the term ε12 x 12 ð1−x 11 Þ and vice versa, leading to the following two relations: x 11 ¼ x 12 þ ε12 x 12 ð1  x 12 Þ;

ð1:4:13Þ

x 12 ¼ x 11  ε12 x 11 ð1  x 11 Þ:

ð1:4:14Þ

Some of the more recent indices of separation between two regions for a two-component system in a closed vessel are as follows. Separation factor α0 (Sandell, 1968): α0 ¼

Y 11 : Y 22

ξ ¼ abs jY 11 − Y 21 j ¼ abs j1 − Y 12 − Y 21 j

¼ abs jY 11 þ Y 22 − 1j; ξ ¼ abs jY 22 − Y 12 j

¼ abs jY 11 Y 22 − Y 12 Y 21 j ¼ jdet½Y ij j

ð1:4:17Þ

Therefore one may express ξ in general by ξ ¼ abs jY ij − Y kj j,

i, k ¼ 1, 2; j ¼ 1, 2; i 6¼ k: ð1:4:18Þ

Since the largest value of Yij is 1 and the smallest value is 0, ξ varies between 1 and 0, with the absolute sign taking care of any negative sign if it arises. In terms of perfect and imperfect separation, since the maximum value of Yij is 1, ξ ¼ 1 implies perfect separation (if Yij is 1 and Ykj is zero; if Ykj is 1 and Yij is zero). A uniform composition between the two regions is indicated by ξ ¼ 0, and there is no separation (see equation (1.4.22)). The relations between ξ and some of the other indices are going to have some use later, so we will obtain them now. By definition (1.4.16), m   11 m21  ξ ¼ abs jY 11 −Y 21 j ¼ abs  0 − 0 : ð1:4:19Þ m1 m2 Further, from equations (1.4.1)

ð1:4:15Þ

Extent of separation ξ (Rony, 1968a): ξ ¼ absjY 11 − Y 21 j:

Substituting definition (1.3.8a) of Yij, we can easily show that the following relations are valid:

1 þ k il0 ¼

m0i ; mi2

k i10 þ 1 m0i ¼ , mi1 k i10

ð1:4:20Þ

yielding ð1:4:16Þ

One should consider the index ξ, the extent of separation, in some detail due to its versatility (Rony, 1972).

  ξ ¼ abs 

   0 0 0 0  k 11   k 11 k 21 k 11     , 0 − 0  ¼ abs  0 − 0 1 þ k 11 1 þ k 21 1 þ k 11 α12 þ k 11

ð1:4:21Þ

28

Description of separation in a closed system

where for the final relation we have utilized equation (1.4.8). An alternative relation between ξ and α12 can be obtained by expressing the last of the equalities (1.4.17) as     Y 11 Y 22 ξ ¼ Y 12 Y 21 abs  − 1 ¼ Y 12 Y 21 abs jα12 − 1j: Y 12 Y 21 ð1:4:22Þ

Similarly, the other separation factor α0 is related to k i10 by   0 0 k 11 1 þ k 21 Y 11 m11 =m01 0   : ð1:4:23Þ α ¼ ¼ ¼ 0 m22 =m02 Y 22 1 þ k 11

There is no direct relation between ξ and I or Ij, the purity indices. But both are different functions of two quantities, as can be observed from Problem 1.4.1. Although indices of separation have been devised to indicate the nature of separation between two contiguous regions in a separated system, sometimes the nature of the composition difference between one of the separated regions and the initial mixture is of interest. The following indices, with their names originating from specialized separation processes, are of this type: decontamination factor, Df j j ¼ ðC if =C ij Þ;

ð1:4:24Þ

desalination ratio, Dr ¼ ðC if =C i1 Þ,

i ¼ 2:

ð1:4:25Þ

In the above definitions, Cif is the molar concentration of i in the initial uniform mixture. For the desalination ratio, i is usually 2, corresponding to salt, which concentrates in region 2, whereas water concentrates in region 1 (ice in example II, Figure 1.1.2; purified water in example I, Figure 1.1.3). For the decontamination factor used with radioactive or surfactant impurities in solvents (or particles in air filtration), i is the impurity and j is the region for the purified solvent. These definitions need not be restricted to salt or a radioactive or surfactant impurity. They are

applicable to any impurity to be removed from a solvent, especially in the case of a dilute solution. Note that, in a similar manner, one can define α12 and ε12 between the initial mixture and the separated region of interest. Example 1.4.1 Close separation in thermal diffusion of an isotopic mixture Isotopic mixtures are difficult to separate since isotopes are very similar to one another. One method sometimes adopted is thermal diffusion. Consider a two-bulb cell as shown in example III of Figure 1.1.3, with one bulb at 300
C and the other at 23
C. Initially, both bulbs at the same temperature, 23
C, contained an equimolar mixture of C12H4 and C13H4. After the temperature of bulb 1 was raised to 300
C, the mole fractions of C12H4 in bulbs 1 and 2 were found to be 0.5006 and 0.4994, respectively. The separation factor α12 for C12H4 as species 1, with the hot bulb as region 1, is given by

x 11 x 22 0:5006  0:5006 ¼ 1:00482; ¼ x 21 x 12 0:4994  0:4994 ¼ α12 −1 ¼ 0:00482:

α12 ¼ ε12

The expressions for some other indices of separation in twobulb thermal diffusion are given in Example 1.5.3. Such a small separation would render the process of thermal diffusion useless unless a way is found to increase α12 . Example 1.4.2 Description of freeze-concentration of fruit juices Large quantities of water are removed from freshly obtained fruit juices to produce fruit juice concentrates. This reduces, among others, the problem of shipping large amounts of liquid material from production centers to distribution centers. This concentration may be achieved by cooling the fruit juice to between −3
C and −15
C when highly pure ice crystals are formed as a suspension in the concentrated fruit juice (Figure 1.4.2). Next, the slurry can be separated in a filtering centrifuge or a filter press or wash column (Thijssen, 1979) to yield two products: almost pure ice and the fruit juice concentrate. Consider water as component 1 and the active constituents of fruit juice as component 2. Let the ice phase be region 1 and the concentrate be region 2. If the ice crystals formed were absolutely devoid of fruit juice active constituents, the separation factor α12 would be infinity since x21 would be zero. Actually when ice

Use filtering centrifuges etc.

Vessel A

Vessel B

Fruit Juice

Pure Ice (?)

Region 1

Concentrate

Region 2

Extract heat

Ice crystals in suspension in concentrated fruit juice

Figure 1.4.2. Freeze-concentration process for fruit juices.

1.5

Compare indices separation for a closed system

z=L

z=0

Molten zone

z

29 Since the impurity concentration is low, the density of the rod material may be assumed equal everywhere. Therefore taking two small and equal volumes at the two ends of the rod (effectively the same amount of material), we have

α12 ¼

I

But the impurity concentration being very low, C11 ffi C12. Therefore

Unmelted zone Direction of heater movement

Refrozen zone

α12 ffi

C 2f ½1 − ð1 − κ21 Þ exp ð−10 κ21 Þ C 22 ¼ C 2f ½1 − ð1 − κ21 Þ C 21 1 − 0:5 exp ð−5Þ ¼ ¼ 1:9932: 0:5

For κ21 ¼ 0.1,

C2f

α12 ¼

Impurity concentration profile in refrozen solid

C2(z)

0

L z

Figure 1.4.3. Zone refining of a rod.

crystals are formed, some fruit juice solids are frozen inside the crystals as impurities. In addition, some fruit juice concentrate will stick to the surfaces of ice crystals in the separator vessel B, which may be any one of the three types of equipment mentioned earlier. Therefore x21 6¼ 0 and α12 has a high value, which is, however, less than infinity. Example 1.4.3 Zone refining of solid materials Consider a solid rod of silicon of length L cm containing some impuity (species 2) at a low concentration level of C2f gmol/cm3. By a process called zone refining (Pfann, 1966) (see Section 6.3.2.3), a portion of the solid rod can be made substantially more pure by slowly moving a heater of length ℓ ( L) along the rod from one end to the other (Figure 1.4.3). The portion of the rod directly concentric with the heater remains molten, while that immediately to the left starts solidifying. This solidifying section rejects part of its impurity content into the molten zone. As the heater moves to the right, the molten zone also moves to the right. The impurity follows this movement so that the left end of the refrozen rod is substantially purer than the right end. Consider the whole rod as the separated system. The impurity concentration in the solid after the process is over is given as a function of distance z from the left end by (Pfann, 1966)   C 2 ðz Þ=C 2f ¼ 1 − ð1 − κ21 Þexp ½−κ21 z=ℓ :

During the melting and refreezing process, κ21 ¼ C 2 ðz Þ= C 12 ðz Þ, C 12 ðz Þ being the impurity concentration in the molten rod at a distance z from the left end, which is in contact with the refrozen rod at z, having a concentration of C2(z). Determine the separation factor for this process for κ21 ¼ 0.5, L ¼ 10ℓ if region 1 is z ¼ 0 and region 2 is z ¼ L. If κ21 ¼ 0.1, what happens to α12? Solution For κ21 ¼ 0.5, L ¼ 10ℓ:

α12 ¼

x 11 x 22 m11 m22 ¼ : x 21 x 12 m12 m21

m11 m22 C 11 C 22 ¼ : m12 m21 C 12 C 21

1 − 0:9 exp ð−1Þ ¼ 6:7: 0:1

Note here that κ21 is a distribution coefficient for species 2 during the melting and refreezing processes. Thus, the lower the distribution coefficient of the impurity between the solid and the melt, the greater is the separation factor. Further, such a separated system exists without a membrane (Example 1.5.4) or two immiscible phases.

1.5 Comparison of indices of separation for a closed system There are many desirable properties of a versatile index of separation. Some of these are listed below (Rony, 1972): (1) It should be dimensionless. (2) It should be usable at all levels of concentrations of any species in the mixture as well as with any separation process. (3) It should be easily calculable. (4) It should preferably be normalized, varying between one (indicating perfect separation) and zero (implying no separation at all), with increasing number indicating improved separation. (5) It should be unaffected if the component subscript (i ¼ 1, 2) and the region subscript ( j ¼ 1, 2) are interchanged for a binary system. (6) It should be sensitive. One can, no doubt, list additional desirable features. In Table 1.5.1, we have indicated some of these features for all indices introduced in Section 1.4. Note that indices k 0i1 , κi1, Ki , η1, η2, I1 and I2 are particularly inadequate when it comes to describing separation which involves both species and both regions, since each index either incorporates both regions or both components. It is also apparent that, of the remaining indices, e.g. I, α12, ε12, α0 and ξ, only the extent of separation, ξ, is normalized. Further, except for the purity index I and ξ, the maximum values of the others, namely α12, ε12 and α0 do not represent perfect separation.

30

Description of separation in a closed system

Table 1.5.1. Properties of various indices of separationa

Index of separation

Dimensionless

Applicable to both components

Applicable to both regions

k0il κi1 Ki1 η1 η2 I1 I2 I1 α12 ε12 α0 ξ

yes yes yes yes yes yes yes yes yes yes yes yes

no no no yes yes yes yes yes yes yes yes yes

yes yes yes no no no no yes yes yes yes yes

a b c

Maximum value (¼ (?) perfect separation)

0 (?) 0 (?) 0 (?) 0 (?) 0 (?)

∞ (?) ∞ (?) ∞ (?)

c

b

c

b

c

b

b

c

b

b

c

b

∞ (yes) ∞ (?) ∞ (?) ∞ (no) 1 (yes)

∞

1 (yes) 0 (yes) 0 (?) 0 (yes)

What value implies perfect separation? c c

c c c

1

For a binary system in a closed vessel with two regions only. Depends on further conventions about the limits of contents of each region. See Problem 1.4.1. No specific value is capable of describing perfect separation.

Next we give some examples and calculate the values of a selected few indices to obtain an estimate of their capabilities to describe the quality and amount of separation. Consider first the hypothetical cases of a binary separation given by the following four conditions.6 (a) (b) (c) (d)

Condition 1: k 011 ¼ 106 , k 021 ¼ 104 ; condition 2: k 011 ¼ 10−4 , k 021 ¼ 10−6 ; Condition 3: k 011 ¼ 10, k 021 ¼ 0:1; Condition 4: k 011 ¼ 20, k 021 ¼ 0:05:

In Table 1.5.2, we show the calculated values of Yij for all four conditions of separation. In condition 1, most of species 1 as well as species 2 are located in region 1, whereas in condition 2 they are almost totally located in region 2. Therefore, these two conditions represent poor separation. On the other hand, in condition 3, species 1 is located mostly in region 1, whereas species 2 is located mostly in region 2. Thus condition 3 represents much better separation than conditions 1 and 2. Condition 4 represents even better separation than condition 3 since the amounts of impurities in both regions are reduced considerably compared to those in condition 3. We can now judge which one of the four indices of separation α12, α0 , ξ or I accurately reflects the conditions of separation described above. Note that in Table 1.5.2 the two indices α12 and I do not discriminate between the three conditions of separation 1, 2 and 3. The index α0 incorrectly indicates condition 1 as a much better separation than either of conditions 2 or 3. The extent of separation, ξ,

6

Minimum value (¼ (?) zero separation)

The first three conditions are based on an example provided by Rony (1972) in his discussion on the desirable properties of indices of separation. Note, however, the differences in our definition,k 0i1 , and Rony’s ki2.

however, correctly demonstrates that condition 3 represents far better separation than conditions 1 or 2. Thus ξ is a much better index of separation. However, with regard to indicating how much more improved the separation in condition 4 is with respect to that in condition 3, note that the relative changes in both α12 and I are much more than that of ξ. Therefore, as perfect separation is approached, the sensitivity of ξ becomes limited compared to either I or α12. Example 1.5.1 Consider the binary system of benzene (1) and toluene (2) distributed between a liquid and a vapor region within a closed vessel. Given the benzene mole fraction in the liquid and the vapor phase to be 0.780 and 0.90, respectively, determine the values of α12 and ξ for the following two cases: (a) 1 gmol of liquid, vapor volume 0.293 liters; (b) 1 gmol of liquid, vapor volume 2.93 liters. The total pressure is 1 atmosphere and the temperature is 85
C. The vapor mixture may be assumed to be ideal. Solution Since benzene (species 1) is present more in the vapor phase, region 1 is vapor phase. Therefore

So

x 12 ¼ 0:78,

x 22 ¼ 0:22,

α12 ¼

x 11 ¼ 0:90,

x 21 ¼ 0:10:

x 11 x 22 0:90  0:22 ¼ 2:54: ¼ x 12 x 21 0:78  0:1

This is valid for both cases (a) and (b). For the calculation of ξ, the values of four quantities, m11, m12, m22 and m21, have to be determined. Case (a): 0.293 liters vapor volume. Assuming that the ideal gas law holds, the total number of gram moles present in the vapor is given by

ðm11 þ m21 Þ ¼

0:293 273  ¼ 0:01 gmol: 22:4 358

1.5

Compare indices separation for a closed system

31

Table 1.5.2. Comparative descriptions of separation by indices α12, α0 , ξ and I Description of separation

Yij

Y11 Y21 Y12 Y22

α12 α0 ξ I

Condition 1a

Condition 2b

Condition 3c

Condition 4d

0.999999 0.9999 0.000001 0.0001

0.0001 0.000001 0.9999 0.999999

0.909 0.091 0.091 0.909

0.9523 0.0477 0.0477 0.9523

100 104 0.000099 2

100 10–4 0.000099 2

100 1 0.818 1.9991

400 1 0.9046 2.601

a

k011 ¼ 106 , k021 ¼ 104 : k011 ¼ 10−4 , k 021 ¼ 10−6 : c 0 k11 ¼ 10, k 021 ¼ 0:1: d 0 k11 ¼ 20, k021 ¼ 0:05: b

Therefore m11 ¼ 0.01  0.9 ¼ 0.009 gmol; m21 ¼ 0.001 gmol; m12 ¼ 0.78  1 ¼ 0.78 gmol; m22 ¼ 0.22 gmol, so       0:009 0:001   m11 m21 ¼  ¼ 0:00688, ξ ¼  − − m11 þ m12 m21 þ m22   0:789 0:221 

a case of very poor separation.

Case (b): 2.93 liters vapor volume. Therefore m11 þ m21 ¼ 0.10 gmol ) m11 ¼ 0.1  0.9 ¼ 0.09 gmol; m21 ¼ 0.01 gmol; m12 ¼ 0.78 gmol; m22 ¼ 0.22 gmol, and     0:09 0:01  ¼ 0:0597: − ξ ¼  0:09 þ 0:78 0:01 þ 0:22 

In both cases, ξ indicates poor separation, although case (b) represents somewhat better separation than case (a). However, α12, based on mole fraction only, is insensitive to such changes. Further, α12 ¼ 2.54 mistakenly indicates reasonable separation since it is far away from α12 ¼ 1.0, corresponding to zero separation. Example 1.5.2 Aniline and hexane are completely miscible with each other at all temperatures higher than 59.6
C, below which they separate into two immiscible phases. A uniform mixture of aniline and hexane containing 52 mole percent aniline is cooled in a closed vessel from 60
C to 42
C. At 42
C, the hexane-rich layer has 83 mole percent hexane while the heavier and immiscible aniline-rich layer has 88 mole percent aniline. Determine the values of α12 and ξ to indicate the separation that has been achieved compared to the original feed mixture. Solution Basis: 100 gmol of feed mixture. Species 1 – hexane; species 2 – aniline; j ¼ 1, hexane-rich top layer; j ¼ 2, anilinerich bottom layer. (Note: aniline is heavier than hexane.) Given that m11 x 11 ¼ 0:83 ¼ , ð1:5:1Þ m11 þ m21 m22 ð1:5:2Þ x 22 ¼ 0:88 ¼ m22 þ m12 From equation (1.3.11),

100 x 1f ¼ 100  0:48 ¼ m11 þ m12 ¼ 48;

ð1:5:3Þ

100 x 2 f ¼ 52 ¼ m21 þ m22 :

ð1:5:4Þ

Solving the four equations (1.5.1)–(1.5.4) for m11, m12, m21 and m22, we get m11 ¼ 42.09, m12 ¼ 5.91, m21 ¼ 8.62 and m22 ¼ 43.38 gmol. The extent of separation is given by        42:09 8:62  m11 m21 ¼  ¼ 0:7122: − ξ ¼  −   m11 þ m12 m21 þ m22 48 52  The separation factor is given by

α12 ¼

x 11 x 22 0:83  0:88 ¼ 35:8: ¼ x 12 x 21 0:12  0:17

Although the value of α12 indicates an extremely effective separation, the value of ξ indicates that, since the extent of recoveries is not sufficient, the separation is far from perfect. Example 1.5.3 Consider the thermal diffusion separation of a gas mixture of H2 and N2 initially present with a hydrogen mole fraction x1f in a two-bulb cell. The volumes of the two bulbs are V1 and V2. Let the uniform temperature of the twobulb cell be changed so that the bulb of volume V1 now has a temperature T1 while the other one is at T2 (T1 > T2) (Figure 1.1.3, example III). Assuming that this is a case of close separation such that x11 ffi x1f, we obtain

α12 − 1 ¼ ε12 ffi

x 11 − x 22  : x 1f 1 − x 1f

ð1:5:5Þ

Given that (Pratt, 1967b, p. 404)

α12 − 1 ¼ ε12 ffi γ12 ln



T1 , T2

ð1:5:6Þ

where γ12 is a property of the H2–N2 system, obtain a relation between ξ and ε12. How can you increase separation in such a system for given T1 and T2? Assume the ideal gas law to be valid. Neglect the volume of connecting capillary. Solution Since the ideal gas law is valid, with the light component hydrogen (species 1) concentrating more in region 1 (higher temperature)

m11 ¼

PV 1 x 11 ; RT 1

m12 ¼

PV 2 x 12 ; RT 2

ð1:5:7Þ

32

Description of separation in a closed system

m22 ¼

PV 2 x 22 ; RT 2

m21 ¼

PV 1 x 21 , RT 1

where P is the total pressure of the gas mixture everywhere in the cell. Therefore     m11 m21  ξ ¼ abs  − m11 þ m12 m21 þ m22    ðx 11 V 1 =T 1 Þ ¼ abs  ðx 11 V 1 =T 1 Þ þ ðx 12 V 2 =T 2 Þ   ðx 21 V 1 =T 1 Þ : ð1:5:9Þ − ðx 21 V 1 =T 1 Þ þ ðx 22 V 2 =T 2 Þ  But

x 1f ¼

m11 þ m12 ðm11 þ m21 Þ þ ðm12 þ m22 Þ

ðV 1 x 11 =T 1 Þ þ ðV 2 x 12 =T 2 Þ : ðV 1 =T 1 Þ þ ðV 2 =T 2 Þ

ð1:5:10Þ

  ðV 1 x 21 =T 1 Þ þ ðV 2 x 22 =T 2 Þ : 1 − x 1f ¼ ðV 1 =T 1 Þ þ ðV 2 =T 2 Þ

ð1:5:11Þ

¼

Therefore

   x −x  V1 V2  11 12   ξ¼  abs  2  x 1f 1−x 1f  V1 V2 T 1T 2 T1 þ T2

ð1:5:12Þ

on introducing (1.5.10) and (1.5.11) in the expression for ξ obtained by simplifying expression (1.5.9). On rearrangement, the relation between ξ and ε12 is obtained as (Sirkar, 1977) 1 V 1 T2 V 2 T1

þ VV 21 TT 12 þ 2

  ε12 :

Suppose T1/T2 ¼ 10. Let us ðaÞ ðV 1 =V 2 Þ ¼ 1; ðbÞ ðV 1 =V 2 Þ ¼ 10:

Case (a)

ξ¼

ð1:5:13Þ take

two

cases:

ε12 ε12 ¼ ; ð0:1 þ 10 þ 2Þ 12:1

Case (b) ξ ¼ 10

10

Feed brine Semipermeable membrane Permeated water Figure 1.5.1. Separation of water from a salt solution under pressure through a membrane.

Similarly,

ξ¼

Piston

ð1:5:8Þ

ε12 ε  ¼ 12 : 4 þ 0:1  10 þ 2

Obviously case (b) has much better separation since by providing the hotter bulb with a much larger volume, much more hydrogen can now be segregated in the hotter bulb which provides the light fraction. This is not evident from α12 or ε12; both are independent of V1 or V2. Example 1.5.4 In the vessel shown in Figure 1.5.1, a membrane which allows water to pass through easily and prevents salt from going through has on one side 1000 cm3 of aqueous solution containing 0.1 gmol of salt per cm3 and atmospheric air on the other side. If the saline water is pressurized by means of a piston to a constant but high enough pressure, water permeates through the membrane but salt does not.

(a) After operation for some time, 500 cm3 of pure water has come to the atmospheric air side through the membrane. Determine the separation factor α12 and the extent of separation ξ if the density of brine in the concentration ranges encountered is 1 g/cm3. (b) Suppose some salt also leaks through the membrane to the permeate side so that the salt concentration in the solution on the atmospheric pressure side is 0.026 gmol/ cm3. If 500 cm3 of water has permeated through the membrane as before, what are the values of α12 and ξ ? Solution Region 1, permeated solution or water; region 2, high-pressure brine; component 1, water; component 2, salt. Part (a): α12 ¼ x 11 x 22 =x 12 x 21 . But x11 ¼ 1 (pure water), x21 ¼ 0 (no salt in permeate) and x12, x22 6¼ 0. Therefore α12 ¼ ∞, a case of infinite separation factor. By definition,     m11 m21 : ξ ¼  − m11 þ m12 m21 þ m22      Now m21 ¼ 0, m11 ¼ 500=V 1 and m12 ¼ 500=V 1 , where V 1 is the partial molar volume of water in the brine solution, assumed to be constant in the range of pressures and concentrations (unit, cm3/gmol). Therefore     1 500 ξ ¼  − 0 ¼ , 500 þ 500 2

definitely not a case of perfect separation, inspite of an infinite separation factor.

Part (b): m21 ¼ 0.026  500 ¼ 13 gmol; m22 ¼ (1000  0.1) – 13 ¼ 87 gmol;    500 13  ξ ¼  − ¼ 0:5−0:13 ¼ 0:37, 500 þ 500 100 

so separation is reduced from case (a).

1.6

Separation indices for multicomponent systems

α12 ¼

33

  500=V 1 ðm22 Þ 500  87 ðm11 Þ ðm22 Þ  ¼ ¼ ¼ 6:7 ðm12 Þ ðm21 Þ 500=V 1 ðm21 Þ 500  13

1 A 10 20

Thus the change in α12 between cases (a) and (b) is much more dramatic than that in ξ. From the above examples, it is clear that ξ, the extent of separation, often describes separation much better than α12, the separation factor. This is due to the ability of ξ to take into account the extent of recovery of a given species in a given region. On the other hand, when one deals with two regions with small amounts of impurity, any change in the impurity levels are better recognized through the indices indicating the composition of any region, e.g. α12, ηj, I, Ij, etc. Therefore, no single existing index describes all aspects of separation efficiently (see Problem 1.5.3).

1.6 Indices for separation of multicomponent systems between two regions In Section 1.2, we introduced a brief description of separation for multicomponent systems. Although we learnt there that perfect separation in such a system requires as many regions as there are components, we will restrict ourselves here to separation systems with only two regions in a closed vessel. Thus perfect separation is, in general, ruled out from our considerations. The familiar quantities used to define the composition of a mixture in a region have to be redefined since i ¼ 1, 2, 3,. . ., n for the n-component system: x ij ¼

mij n X mij

i ¼ 1, 2, . . . , n;

j ¼ 1, 2;

ð1:6:1aÞ

i¼1

mij X ij ¼ mkj

k, i ¼ 1, 2, . . . , n;

j ¼ 1, 2: ð1:6:1bÞ

Note n X i¼1

n X i¼1

C ij ¼ C tj ;

30 C 40

wt % 2

x ij ¼ 1

aij ¼

for

maij ; n X maij i¼1

j ¼ 1, 2; uij ¼

ð1:6:1cÞ ρij : ð1:6:1dÞ n X ρij i¼1

All other basic definitions of concentrations remain the same, except for i ¼ 1, . . . , n. For multicomponent systems, it is worthwhile illustrating the value of aij by way of an example. Consider an isotopic mixture of water having H2O, D2O and HDO present in the amounts 0.6, 0.1, 0.3, respectively (in mole fractions, i.e. if H2O is component 1, then x1f ¼ 0.6). The atom fraction of hydrogen is then [(0.6  2 þ 0.3  1)/2] ¼ 0.75. A graphical method of describing compositions is often adopted for three-component systems. Consider an equilateral triangle with apexes identified as 1, 2 and 3

MC 80 70 B

MB

60

M

wt % 1 50

50 60

40 E

70 F

80

30

90

MA

20

R

10 L

2

10

20

30

60 40 A 50 wt % 3

70

80

90

3

Figure 1.6.1. Compositions of a ternary system in equilateral triangular diagram.

(Figure 1.6.1) representing pure components 1, 2 and 3, respectively. From any point M inside the triangle representing a ternary mixture, draw perpendiculars MA, MB and MC to the sides 23, 31 and 12. The distances MA, MB and MC represent weight percentages of species 1, 2 and 3, respectively, in the mixture. The sum of the distances MA, MB and MC is equal to the altitude of the triangle. Since the perpendicular distance from any one of the apexes to the opposing base is the same for an equilateral triangle, this distance represents 100 percent of the mixture. Therefore the weight fractions of species 1, 2 and 3 in the mixture are given by u1 j ¼

k 6¼ i;

90

MA MB MC , u2 j ¼ , u3 j ¼ , AL AL AL

ð1:6:1eÞ

where AL is the altitude of the triangle. For easy determination of these fractions, triangle 123 has percentages marked on each side. Thus point M in Figure 1.6.1 has weight fractions of 0.50 of 1, 0.30 of 2 and 0.20 of 3. Two different points inside the triangle represent two different mixtures. These two mixtures may or may not be immiscible. If these two mixtures are brought together, the composition of the overall mixture (subscript j ¼ M) will be represented by the point F on the straight line joining the two points E (extract) and R (raffinate). The location of point F is determined by the lever rule: line EF uiE − uiM wR ¼ ¼ , line RF uiM − uiR wE

ð1:6:2Þ

where wR and wE are the total weights of mixtures at R and E, respectively. This relation follows from a total mass balance, wR þ wE ¼ w M ,

ð1:6:3aÞ

and an ith-component balance, wR uiR þ wE uiE ¼ wM uim :

ð1:6:3bÞ

34

Description of separation in a closed system

With regard to the various indices of separation, the definitions of k 0i1, κi1 and Ki remain unaffected, regardless of whether i ¼ 1, 2 or i ¼ 1, 2,. . ., n. The impurity ratio ηj defined earlier according to de Clerk and Cloete (1971) has to be modified for a multicomponent system to the impurity ratio for the ith species in the jth region: ηij ¼

mij mjj

j ¼ 1, 2;

i ¼ 1, 2, . . . , n,

i 6¼ j:

ð1:6:4Þ

One could define a purity index corresponding to such a definition of the impurity ratio as follows: 2 X

I ¼−

n X

log10 ηij ,

j¼1 i¼1

ð1:6:5Þ

following the suggestions of de Clerk and Cloete (1971). The separation factor α12 for two species 1 and 2, with species 1 being lighter than species 2, is usually changed for multicomponent separations to αin (Pratt, 1967b, p. 451), x i1 x n2 , αin ¼ x i2 x n1

ð1:6:6Þ

such that species n is the heaviest (with the highest molecular weight in general) and species 1 is the lightest with i ¼1, 2,. . ., n. Such a definition ensures that αin is greater than 1 if there is any separation. Species n is often referred to as the heavy key component. The corresponding index of enrichment factor is defined as εin ¼ ðαin − 1Þ:

ð1:6:7Þ

It is not necessary, however, to have the separation factor of species i always defined with respect to the heaviest species. One could define the separation factor αij as αij ¼

x i1 x j2 , x i2 x j1

ð1:6:8Þ

where the jth species is heavier than the ith species. This ensures that αij  1 since, by convention, the ith species concentrates more in region 1 and a species heavier than i will concentrate in region 2. The corresponding enrichment factor is defined as   εij ¼ αij − 1 :

ð1:6:9Þ

Note that if the separation factor is defined as x j1 x i2 αji ¼ , x j2 x i1

ð1:6:10Þ

where the jth component is heavier than the ith component, αji will have a value less than 1. As in equation (1.4.7b), the relation between the mole fractions xi1 and xi2 is of interest for the n-component system. Now Ki ¼

x i1 , x i2

i ¼ 1, 2, . . . , n:

ð1:6:11aÞ

But n X

x i1 ¼ 1:0,

n X

K i x i2 ¼ 1:

i¼1

and therefore

i¼1

ð1:6:11bÞ

If the separation factor αin is defined by (1.6.6) with respect to the heavy species n, then n n X X Ki 1 ¼ αin x i2 : x i2 ¼ Kn Kn i¼1 i¼1

ð1:6:11cÞ

But x i1 ¼ K i x i2 ¼ and therefore x i1 ¼



Ki Kn



K n ðx i2 Þ,

αin x i2 : n X ðαin x i2 Þ

ð1:6:12Þ

i¼1

The inverse relation is x i2 ¼

x i1 =αin : n X ðx i1 =αin Þ

ð1:6:13Þ

i¼1

None of the definitions (1.6.4) and (1.6.6)–(1.6.9) provide a single number to indicate the quality of separation, as was the case for a binary system. If a single number is required to describe the separation of a particular ith component from the rest, one can lump all the other species together and treat them as the other component of a binary system. With this rearrangement, all the indices of separation defined earlier for binary systems become useful. This is especially true for ξ, the extent of separation, since Rony (1972) has shown that the automatic extension of   ξ ¼ jdet Y ij j from a binary system to an n-component system distributed between n regions is not of much use:      Y 11 Y 12 Y 13 . . . Y 1n        Y 21 Y 22 Y 23 . . . Y 2n    : ð1:6:14Þ ξ ¼ det    Y 31 Y 32 Y 33 . . . Y 3n     Y n1 Y n2 Y n3 . . . Y nn  

Since a determinant is zero if two rows or columns of the determinant are identical element by element, ξ would be zero, even though (n − 2) components (say) are completely separated and segregated in their respective regions. (See Problem 1.6.1 for other descriptions of multicomponent systems.) As pointed out in Section 1.2, for a three- or fourcomponent system in which two of the components form two immiscible phases while the other components are distributed between these phases in small quantities, indices of separation defined for a two-component system

Problems

35

are quite useful. If we have a three-component system with species 1 distributing itself between phase 1 (essentially species 2, say) and phase 2 (essentially species 3, say) then the indices k 0i1 , κi and Ki are particularly useful. In fact, other indices, e.g. ηj, Ij, I, α12, ε, α’ and ξ are not useful at all in such a case since i ¼1 only. On the other hand, with a four-component system with species 1 and 2 distributing themselves between immiscible phases 1 and 2 (phase 1 is, say, essentially species 3 and phase 2 is essentially species 4), the indices ηj, Ij, I, α12, ɛ, α’ and ξ regain their usefulness. Here, the system is to be treated as if two components 1 and 2 are being separated between two regions. Therefore perfect separation is possible, with species 1 completely segregated in region 1 and species 2 in region 2. In a five-component system with two immiscible phases, the above argument would suggest that we are effectively dealing with a three-component system. The indices of separation (1.6.4)–(1.6.7) valid for multicomponent systems are to be used in such a case. There are systems where there may be thousands of species. Quantitative indices have been developed to describe composition and separation in such systems. Chapter 2 provides an introduction to this topic.

1.7

Some specialized nomenclature

Terms such as concentration, enrichment, purification and separation are commonly used in describing chemical separation processes. Based on the usage pattern for these terms, Rony (1972) has suggested that a specified range of composition should be associated with each term. For example, purification processes will increase the mole fraction of a species being purified, with the mole fraction of the species remaining above 0.90 always. In concentration processes, the species mole fraction being increased remains below 0.90. Enrichment processes, traditionally used for isotopic separations, rarely get the mole fraction of the species above 0.10. All processes achieving

purification, concentration or enrichment are, however, separation processes. When salt is removed from sea water by various desalination processes, we have purification of water. Germanium is purified to contain only 1 in 1010 electronically active impurities by the zone-refining process (Example 1.4.3). Cane juice containing 10 to 12 wt% sugar is concentrated to about 65 wt% sugar solution by multiple-effect evaporation (King, 1980). Orange juice solids are concentrated sometimes by the freeze-concentration process (Example 1.4.2). The amount of U235 isotope in natural uranium compounds (usually around 0.0075 atom fraction) are increased (with respect to the U238 isotope) by gaseous diffusion (Pratt, 1967b, p. 349), which becomes an enrichment process. Chemical separation processes are often described broadly by means of the external agents introduced into the feed mixture to effect separation. In Figures 1.1.2 and 1.1.3, the external agent is energy addition or energy extraction. The form of energy is either heat or pressure energy. Many other forms of energy addition are also practiced. Such separation processes are called energyseparating-agent (ESA) processes. On the other hand, consider Figure 1.2.2 where benzene is added from outside to the feed mixture to create a separated system. Such a process is characterized as a mass-separating-agent (MSA) process; benzene is the mass-separating agent. Different materials and different phases have been used in mass-separating-agent processes. A comprehensive characterization of different separation processes in terms of the ESA process or the MSA process is available in King (1980). It is wrong to assume that no energy is required in separation processes using a mass-separating agent. In the example of Figure 1.2.2, the extract phase containing the solute picric acid in benzene has to be subjected to an ESA process to separate benzene from picric acid. Only then is picric acid recovered in purer form, and the two initial species, water and picric acid, are separated.

Problems 1.3.1

One can characterize a binary separation system with two regions in terms of a 2  2 matrix of elements indicating the number of moles of species i in region j (Rony, 1972). Write the matrix S1 for ideal separation. (a) Obtain the matrix for the actual separation achieved, S. (b) Determine the matrix Sf for the initial condition before the separation was started in terms of m0i and Y 0j . Here Y 0j is that fraction of the total number of moles present in the system which is present in region j before the separation was initiated at time t ¼ 0.

1.4.1

(a) If the purity index I ¼

2 X

j¼1

I j may be expressed as I ¼ log10 A − log10 B, show that the extent of separation, ξ,

is given by ξ ¼ abs [A − B] for a binary system distributed between two regions in a closed vessel. Indicate the expressions for each of A and B. (b) Determine the minimum value of I corresponding to maximum impurity in both regions. For this purpose, recognize that since each region is identified with a species, the composition of any region may not be

36

Description of separation in a closed system such that the region-specific component i has less than half of m0i in its designated region. While determining the condition for minimum I with respect to a species, note that you have to assume that the distribution of the other species is unaffected. Avoid differentiation. (Ans. 0.)

1.4.2

Two enantiomers, species 1 and species 2, are present in a solution. By a process called resolution, two fractions are obtained. The optical purity of one of the fractions ( j ¼ 1) highly purified in species 1 is generally expressed in terms of an enantiomeric excess eeð1Þ ¼

C 11 − C 21 : C 11 þ C 21

Relate this quantity to an appropriate index identifying the purity of the fraction in terms of species 1. 1.4.3

If a gas or gas mixture is enclosed within a cylindrical vessel of radius r2 and height h, and the closed vessel is rotated about its axis at an angular velocity of ω radian/s, then, due to centrifugal forces, the partial pressure pi(r) of species i at any radial location at a distance r from the center is given by  M i ω2 r 2 , pi ðr Þ ¼ pi ð0Þ exp 2 RT where Mi is the molecular weight of species i. (a) Assuming the ideal gas law to be valid, obtain the following expression for the separation factor α12 between gas species 1 and 2 at radial locations r1 and r2, where r1 < r2 and M1 < M2:    ðM 2 − M 1 Þ ω2 r 22 − r 21 x 11 x 12 , ¼ exp α12 ¼ 2RT x 12 x 21 where region 1 is located at radius r1, etc. Remember that the total pressure varies with r in such a gas centrifuge. (b) Consider now the regions 1 and 2 to be thin cylindrical shells of height h and thickness dr at radial locations r1 and r2, respectively. Obtain the following relation: 2

3 2

7 6 1 6   7 ξ¼6  2 2 2 7 5 4 M ω r − r r1 1 1 2 1þ exp r2 2RT 1.4.4

3

7 6  1 6   7 α12 − 1:  6 2 2 2 7 5 4 M ω r − r r2 2 2 1 1þ exp r1 2RT

For the thermal diffusion separation of solute 2 present in solvent 1, located between two flat plates as shown in Figure 1.P.1, it is known that the solute will concentrate near the cold plate (region 2) while the solution near the hot plate at the top gets depleted in solute. With y coordinate values of 0 at the top plate and ℓ at the bottom plate, it is known that the mole fraction distribution of the solute x2(y) is given by the equation  dx 2 α T 1 − T 2 x 2 ð1 − x 2 Þ: ¼ dy T ℓ Initially, the uniform solute mole fraction in the solution was x2f , where the relation

x 2f ¼ ð1=ℓÞ

ðℓ

x 2 dy

0

is valid due to solute conservation. (a) Show that the solute mole fraction profile is given by (Powers, 1962, p. 29) x 2 ðy Þ ¼ ½1 þ ψ 0 exp ð−2ΑϕÞ −1 , where   A ¼ αðT 1 −T 2 Þ=2T , ϕ ¼ ðy=ℓÞ and     exp f2A 1−x 2f g−1   : ψ0 ¼ 1−exp f−2A x 2f g

Problems

37

Hot Plate y=0

Solution

y=ℓ

Cold Plate Figure 1.P.1. Static thermal diffusion cell with two plates for a solution.

(b) Obtain an expression for the separation factor α12, with y ¼ 0 being region 1 and y ¼ ℓ being region 2. (Ans. α12 ¼ exp(2A).) (c) Obtain the expression for ɛ12 if this is a case of close separation. (Ans. ɛ12 ¼ 2A.) 1.5.1

(a) In Example 1.5.3, the index ξ was found to depend on V2 and V1, both of which can be varied to achieve an increased value of ξ. Thus, if (V2/V1) is considered to be a variable, ξ can be maximized with respect to (V2/V1). In general, the maximum value of ξ, ξmax, with respect to a variable γ can be obtained for the separation of a two-component system between two regions from ∂ξ ∂ ½abs ðY 11 − Y 21 Þ ¼ 0, ¼ ∂γ ∂γ whose solution will yield a value of γ, which, when substituted in ξ, will give ξmax. Consider a twocomponent system distributed between two immiscible phases such that α12 is constant. Show that for a value of the variable k 011 ¼ ðα12 Þ1=2 ,    α1=2 −1   12  ξ max ¼ abs  1=2 : α þ 1 12

(b) If the above system is such that ɛ12 ! 0 (i.e. close separation), show that ξmax is given by ξ max ¼ ε12 =4. (c) Consider relation (1.5.13) between ξ and ɛ12 for binary thermal diffusion separation. Show that, for constant values of T1 and T2 for a given two-component system (equation (1.5.6)), ξ max ¼ ðε12 =4Þ and that the corresponding value of (V2/V1) ¼ (T2/T1). 1.5.2

(a) Consider the separation of a hexane–aniline liquid mixture as described in Example 1.5.2. Obtain the following relation between ξ, α12 and κ11:    C 11 C 21 1 ξ ¼ V 2V 1 abs α12 −1: m01 m02 κ11 (b) For the separation of picric and benzoic acids between two immiscible layers of benzene and water, show that α12 ¼ κ11/κ21 if both liquid phases may be considered as dilute solutions in benzene and water. (c) For the problem posed in part (b), it is known that both κ11 and κ21 are substantially independent of concentrations of either solutes in either phases. Utilizing the result in (a), suggest ways to increase the separation at constant temperature (this means κ11 and κ21 are constants).

1.5.3

Consider two different separators indicated by supercripts A and B. Each has a uniform binary mixture to start with. After separation of each mixture into two different regions, both separators have k 011 ¼ 106 and k 021 ¼ 10−6 . In separator A, mo1 ¼ mo2 ¼ 1 gmol. But in separator B, mo1 ¼ 10−6 gmol and mo2 ¼ 106 gmol. (a) Obtain the value of the extent of separation for each vessel and show that they are equal. (Ans. ξ A ¼ ξ B ¼ 0.9999981.)

38

Description of separation in a closed system (b) Define an extent of purification ξ p by the absolute value of the determinant of the matrix whose elements are x ij . (c) Show that the extent of purification for separator A is much greater than that of separator B.   This problem illustrates the role of initial mole ratio m01 =m02 in developing a comparison between two different separation systems behaving identically with respect to distribution ratios (Rony, 1968b). Conversely, it illustrates the relative insensitivity of the extent of separation ξ to considerable degrees of purification (as pointed out in Section 1.5).

1.6.1

For a qualitative description of separation of a single multicomponent feed mixture of n species into k product regions or fractions, Lee et al. (1977a) have defined a relative molar fraction of species i in product region j by πij ¼ xij/xif , where each of the mole fractions refer to its averaged value in a given region. Further, r j is defined as rj ¼

mj k X

,

mj

j¼1

where mj is the total number of moles in the

jth product or region.  k  X π ij r j : Ans:Y ij ¼ π ij r j =

(a) Develop a relation between Yij and πij.

j¼1

(b) Obtain a matrix of product compositions in terms of relative molar fractions. (c) What criterion must be satisfied to indicate that there is some separation between two species in one of the product regions?

2

Description of separation in open separators Separations for preparative or analytical purposes are often carried out batchwise in a closed vessel. On the other hand, industrial-scale separations are commonly achieved in a continuous manner with open vessels into which feed streams enter and from which product streams leave. In this chapter, we consider the available methods of describing separation in such open separators. The quantities, fluxes and mass balances necessary for such descriptions are presented first in Section 2.1. Section 2.2 describes the available indices of separation and their interrelationships for binary separation with a single feed stream entering the separator. In Section 2.3, we briefly introduce indices for binary separation with two feed streams entering a separator. The complications encountered in describing multicomponent separations with a single-entry or double-entry separator are presented in Section 2.4. This section provides also an introduction to the description of systems of continuous chemical mixtures and size-distributed population of particles. Separation by any of the separators considered in these sections presupposes that the output streams have different compositions. There are separation processes, e.g. chromatography, in which the separator has only one output stream, but that has a time-varying composition. The description of separation in such a separator with the help of various indices has been considered in Section 2.5. Triple-entry separators etc. have not been dealt with here. Further, except for Section 2.5, steady state operation is assumed throughout.

2.1

Preliminary quantitative considerations

Open separators can be broadly classified into single-entry and double-entry separators. A single feed stream enters the single-entry separator. Two feed streams are necessary for a double-entry separator (Figure 2.1.1). A feed stream enters a separator through a defined fraction of its surface area, just as the product streams (at least two in number in

general) leave through some other defined surface areas of the separator. The two product streams may be in contact with each other or they may be separated from each other by a barrier. Next we provide some examples of the nature of contact between various feed and product streams in open separators. Figure 2.1.2(a) shows a nitrogen-rich gas fraction leaving the single-entry separator on one side of the silicone membrane, whereas the oxygen-rich gas fraction is withdrawn from the low-pressure side of the silicone membrane permeator used for separating air. Figure 2.1.2(b) shows a single plate in a distillation column in which bubbles of vapor leave the plate (i.e. a double-entry separator) after contacting a liquid stream which flows along the plate. Only a small part of the vapor product stream is in contact with part of the liquid product stream leaving the plate separator. A product stream may also be in contact with a feed stream as it leaves the separator. Figure 2.1.2(c) shows how a gas stream leaves the top of a spray scrubber where a feed liquid stream is introduced to absorb an undesirable species from a gas mixture. Only part of the cross-sectional area of the tower is utilized by the falling drops of liquid feed, with the rest being utilized by the product gas. While the specification of the flow crosssectional area of the separator for a given stream is quite difficult in this case, it is straightforward in Figure 2.1.2(a). We now consider these separators to be fixed in space and focus attention on a particular surface area Sj of any separator shown in Figure 2.1.1. Let vij be the local average velocity vector of species i with respect to coordinate axes fixed on the surface area Sj. If the mass concentration of species i at this location on such a surface area is indicated by ρij, with ρtj being the local value of the total mass concentration, we define the local mass average velocity vector1 vtj by

1

See Bird et al. (2002), pp. 533–535 for more details.

40

Description of separation in open separators S1

S1

Sf 2

Sf

Actual velocity profile

Actual velocity profile

Sf 1

S2

S2 Single-entry separator

Double-entry separator

Figure 2.1.1. Single-entry and double-entry separator. The velocity profiles shown at the inlet(s) and exits of the separators need not be parabolic.

Silicone membrane O2-rich fraction

Air

N2-rich fraction Compressor (a)

Benzene-rich vapor product

Scrubbed gas

Feed liquid

Liquid feed

Vapour feed

Taluene-rich liquid product

Gas (b)

(c)

Figure 2.1.2. Some examples of the nature of contact between various feed and product streams. (a) Membrane separates air into two product streams in a single-entry separator, which is a silicone membrane permeator for air separation. (b) Limited contact between vapor and liquid products in a sieve plate (the separator) in a distillation column. (c) Complete contact between a feed scrubbing liquid stream introduced as drops in a spray and the product gas stream after scrubbing.

vtj ¼

n X

ρij vij

i¼1 n X

ρij

¼

n X

ρij vij

i¼1

ρtj

ð2:1:1Þ

i¼1

for a system of n components. The magnitude of this velocity, vtj, is, for example, measurable for liquids by use

of a suitable pitot tube. The local molar average velocity vector vtj is defined by

vtj ¼

n X

C ij vij

i¼1 n X i¼1

C ij

¼

n X

C ij vij

i¼1

C tj

,

ð2:1:2Þ

2.1

Preliminary quantitative considerations

41

where Cij is the local molar concentration of species i. In general, the quantities ρij, ρtj, Cij, Ctj, vij, vtj, vtj , will vary with location on the surface area Sj. The local mass flux vector nij of species i relative to the stationary surface area Sj is defined as nij ¼ ρij vij :

ð2:1:3Þ

For these special cases, we have assumed that vij is always locally perpendicular to Sj. Further, the quantities Sj, vij, vtj and vtj are the magnitudes of the respective vector quantities. If now vij and therefore vtj and vtj vary in magnitude only along Sj, and vij is locally perpendicular to Sj, the following average values hviji, hvtji and hvtj i may be defined such that

The mass rate of inflow or outflow of species i through surface area Sj is given by ð wij ¼ nij  dSj : ð2:1:4Þ Sj

The rate at which mass is entering or leaving the separator through Sj is then given by 2 3 ð n n X 6 7 X wtj ¼ wij : ð2:1:5Þ 4 nij  dSj 5 ¼ i¼1

i¼1

Sj

We adopt the convention that when material is entering a separator, it is to be considered positive for mass balance purposes, whereas when material is leaving the separator, we consider it to be negative. Further, diffusive contributions to the mass rate of inflow or outflow of species i through Sj are not considered in this chapter. Quite often the molar rate of inflow or outflow of mass with respect to the surface area Sj of a separator is of greater interest. In such a case, the molar concentration Cij is used. The quantities corresponding to nij, wij and wtj are then Nij, Wij and Wtj, the local molar flux vector of species i, the molar rate of inflow or outflow of species i and the total molar rate of inflow or outflow of all species, respectively: N ij ¼ C ij vij ; ð W ij ¼ N ij  dSj ;

ð2:1:6Þ ð2:1:7Þ

Sj

W tj ¼

n X i¼1

2 3 ð n X 6 7 W ij ¼ 4 N ij  dSj 5: i¼1

ð2:1:8Þ

Sj

If the quantities ρij, Cij, vij,, vtj and vtj do not vary across the surface area Sj, we obtain the following relations from (2.1.4), (2.1.5), (2.1.7) and (2.1.8), respectively: wij ¼ ρij vij Sj ; wtj ¼

n X i¼1

ð2:1:9Þ !

ρij vij Sj ¼ ρtj vtj Sj ;

W ij ¼ C ij vij Sj ; W tj ¼

n X i¼1

ð2:1:10Þ ð2:1:11Þ

!

C ij vij Sj ¼ C tj vij Sj :

ð2:1:12Þ

wij ¼ ρij hvij iSj ;

ð2:1:13Þ

wtj ¼ ρtj hvtj iSj ;

ð2:1:14Þ

W ij ¼ C ij hvij iSj ;

ð2:1:15Þ

W tj ¼ C tj hvtj iSj :

ð2:1:16Þ

ρij =M i ¼ C ij :

ð2:1:17Þ

Note that

If the concentrations also vary across the surface area Sj, we can define the various mass and molar flow rates with respect to averaged concentrations and averaged velocities. The mole fraction xij of species i in the stream entering or leaving the separator through Sj is defined by x ij ¼ ðW ij =W tj Þ:

ð2:1:18Þ

If vij ¼ for the uniform velocity cases, (2.1.11) and (2.1.12), then only v∗ tj

x ij ¼ C ij =C tj :

ð2:1:19Þ

The mass fraction uij of the ith species in the stream passing through area Sj is defined for an open separator by uij ¼ ðwij =wtj Þ:

ð2:1:20Þ

When uniform velocity profiles exist and vij ¼ vtj, this relation reduces to uij ¼ ðρij =ρtj Þ:

ð2:1:21Þ

An open separator will have at least three streams coming in and out through three different surface areas Sj (see Section 2.5 for a different case). We adopt the convention that for a single-entry separator the subscript j will have values 1, 2, . . ., k corresponding to k product streams, but j ¼ f for the single feed stream. For a double-entry separator, the two feed streams are to be denoted by j ¼ f1 and f2, respectively, while the k product streams will continue to have j ¼ 1, 2, . . ., k. Consider the steady state operation of the single-entry separator shown in Figure 2.1.1. Since there is no accumulation of mass inside the separator, rate of mass input ¼ rate of mass outflow ) wtf ¼

k X j¼1

wtj :

ð2:1:22Þ

42

Description of separation in open separators

Assuming further no accumulation of species i in the separator and no chemical reaction, we have the following mass balance for species i: uif wtf ¼

k X

uij wtj ,

j¼1

ð2:1:23aÞ

where we know that n X i¼1

uij ¼ 1,

n X i¼1

uif ¼ 1:

ð2:1:23bÞ

For a single-entry separator with two output streams, (2.1.22) and (2.1.23a) reduce to wtf ¼ wt 1 þ wt 2

ð2:1:24Þ

uif wtf ¼ uil wt1 þ ui2 wt2 :

ð2:1:25Þ

and

The corresponding molar balances in the absence of a chemical reaction are: W tf ¼ x if W tf ¼

k X

W tj ;

ð2:1:26Þ

x ij W tj ;

ð2:1:27Þ

j¼1

k X j¼1

W tf ¼ W t1 þ W t2 ;

ð2:1:28Þ

x if W tf ¼ x i1 W t1 þ x i2 W t2 :

ð2:1:29aÞ

Further, n X i¼1

x if ¼ 1,

n X i¼1

x ij ¼ 1:

ð2:1:29bÞ

With identical assumptions, the total mass balance and the ith species balance equations for a double-entry separator (see Figure 2.1.1) are: wtf 1 þ wtf 2 ¼ uif 1 wtf 1 þ uif 2 wtf 2 ¼

k X

wtj ;

j¼1

k X

uij wtj :

j¼1

ð2:1:30Þ ð2:1:31aÞ

Note that in this case n X i¼1

uij ¼ 1,

n X i¼1

uif 1 ¼ 1,

n X i¼1

uif 2 ¼ 1:

ð2:1:31bÞ

Like (2.1.23b) and (2.1.29b), these equations (2.1.31b) are needed as such to solve for the variables, the mass fractions and the mass flow rates. However, only (n − 1) of equations (2.1.31a) are to be used. The corresponding steady state molar balances, W tf 1 þ W tf 2 ¼

k X j¼1

W tj

ð2:1:32Þ

and x if 1 W tf 1 þ x if 2 W tf 2 ¼

k X

x ij W tj ,

j¼1

ð2:1:33aÞ

are valid in the absence of any chemical reaction in the separator. Further, n X i¼1

x if 1 ¼ 1;

n X i¼1

n X

x if 2 ¼ 1;

i¼1

x ij ¼ 1:

ð2:1:33bÞ

2.2 Binary separation in a single-entry separator with or without recycle We consider now the description of binary separation in a single-entry separator with only two product streams. Assume steady state operation without any chemical reaction. As in Chapter 1 with region 1 and species 1, we assume here arbitrarily that species 1 is lighter than species 2 and that j ¼ 1 refers to that product stream which is richer in the lighter species 1. If we had perfect separation, product stream j ¼ 1 will have only species 1 and product stream j ¼ 2 will have only species 2. To start with, consider only a nonrecycle separator. Such a separator is sometimes called a splitter (Figure 2.2.1a). If xi1 6¼ xi2 and both are different from xif, the singleentry separator has achieved some separation. How much separation has been attained can be estimated by suitable descriptors, indices of separation. The simplest of these indices bear the name of specific separation processes where they are extensively used: the desalination ratio Dr the solute rejection R and the decontamination factor Df. Desalination ratio: Dr ¼

C 2f : C 21

ð2:2:1aÞ

Solute rejection: R¼



1

 C 21 : C 2f

ð2:2:1bÞ

Decontamination factor: Df ¼

C 2f : C 2l

ð2:2:1cÞ

Note that all three definitions involve elimination of solute from a solvent (species 1). Further, the solvent is supposed to concentrate in exiting stream 1 and solute (species 2) in exiting stream 2. If there is separation, both Dr and Df will have values greater than 1, whereas R  1. It is also clear that     1 1 R¼ 1 ¼ 1 : ð2:2:1dÞ Dr Df For dilute solutions, the following simplifications are valid:   x 2f x 2f x 21 ; Df ¼ ; R ffi 1− : ð2:2:1eÞ Dr ffi x 21 x 21 x 2f

2.2

Binary separation in a single-entry separator

xi 1

43

qWtf

hf α12

r

ht α12

x ri1

xi 2

Wtf

W rtf

xif

x if

ft α12

(1–q)W rtf

r

r

x i2 (b)

(a)

Wtf, xif

qW rtf

x ri1

(1−q)Wtf

Wtf xif

(1–h)qW rtf

xi1

hqW rtf

x ri1

Pressure-reducing valve

Vapor fraction

r

Fresh feed

W tf Flash separator x rif

Butane (1) Decane (2)

Pump

Liquid fraction r

x i2

Heater (c)

Figure 2.2.1. (a) Nonrecycle single-entry separator. (b) Single-entry separator with part of light fraction recycled to feed stream. (c) Singleentry separator with part of heavy fraction recycled to the feed.

All such indices therefore describe separation by indicating how different the exit stream 1 composition is from the feed stream with respect to the solute species. A somewhat similar index is the equilibrium ratio of species i (obtained under phase-equilibrium conditions), K i ¼ ðx i1 =x i2 Þ,

ð2:2:1fÞ

which relates the mole fraction of species 1 in product stream 1 to that in product stream 2. The most common index of separation for the single-entry separator with two product streams is the separation factor αht 12 for species 1 and 2 between the product streams j ¼ 1 (the heads or light fraction, indicated by the superscript h) and j ¼ 2 (the tails or heavy fraction, indicated by the superscript t): αht 12

x 11 x 22 ¼ : x 12 x 21

ð2:2:2aÞ

It is sometimes called the stage separation factor. If αht 12 > 1, species 1 is preferentially present in the heads stream j ¼ 1 and species 2 concentrates in the tails stream j ¼ 2. Commonly conditions are chosen such that αht 12 > 1. A few common and useful relations involving the separation factor and the compositions of the exiting streams are given below:

x 11 ¼

x 12

"

# αht x 12  12  ; 1 þ x 12 αht 12  1

"

# x 11   : ¼ ht α12  x 11 αht 12  1

ð2:2:2bÞ

ð2:2:2cÞ

Further, if one defines the mole ratio Xij ¼ (xij/(1 − xij)) for a binary system, then X 11 ¼ αht 12 X 12 :

ð2:2:2dÞ

The enrichment factor εht 12 , defined by ht εht 12 ¼ ðα12  1Þ ¼

ðx 11  x 12 Þ , x 12 ð1  x 11 Þ

ð2:2:2eÞ

takes on very small values for close separations (εht 12 x21 but θ > 1. But the separation is poor, since very little of species 1 present in the feed has been recovered in the heads fraction j ¼ 1. Most of species 1 is present along with most of species 2 in the stream j ¼ 2. Note, however, that θ does not directly indicate the extent of recovery in the light fraction of species 1 present in the feed stream. That is indicated by the component cut, θi, for the ith species, defined by

ft

hf

Thus if α12 > 1 and α12 > 1, the stage separation factor αht 12 hf ft is greater than both α12 and α12 . This need not be true if hf ft either α12 or α12 is less than one. The single-entry separator is said to be symmetric if hf

ft

α12 ¼ α12 :

ð2:2:6Þ

 2  2 hf ft αht ¼ α12 : 12 ¼ α12

ð2:2:7Þ

In such a case

When equation (2.2.6) is not valid, we have an asymmetric hf ft separator. If β is the highest common root of α12 and α12 such that (Wolf et al., 1976) hf

α12 ¼ βk

ft

α12 ¼ βℓ ,

and

ð2:2:8aÞ

we get kþℓ , αht 12 ¼ β hf

ft

k=kþℓ α12 ¼ ðαht ; 12 Þ

ℓ=kþℓ α12 ¼ ðαht : 12 Þ

ð2:2:8bÞ ð2:2:8cÞ

Although αht 12 has been defined in terms of xij (j ¼ 1, 2) only, it may be defined in terms of Wij also. Use (2.1.18) in (2.2.2a) to obtain αht 12 ¼

W 11 W 22 : W 12 W 21

ð2:2:9aÞ

Similarly, hf α12

W 11 W 2f ¼ : W 1f W 21

ft

W 1f W 22 : W 2f W 12

θi ¼

W t1 x i1 , W tf x if

ð2:2:10bÞ

the value of which ranges between 0 and 1. The index ξ, the extent of separation, was shown in Chapter 1 to reflect both the quality of the separated regions in terms of composition and the amount of recovery of a species in its designated region. For a single-entry separator, an appropriate definition of ξ is as follows (see Rony (1970) and the efficiency formula 9 in Rietema (1957)): ξ ¼ jΥ˙ 11  Υ˙ 21 j ¼ jθ1  θ2 j,

ð2:2:11Þ

where the quantities Υ˙ 11 and Υ˙ 21 are obtained from the following general definition of the segregation fraction Υ˙ ij of the ith species in the jth stream: Υ˙ ij ¼

W ij x ij W tj x ij W tj ¼ : ¼ k X W if x if W tf W ij

ð2:2:12Þ

j¼1

Note that Υ˙ 11 ¼ θ1 and Υ˙ 21 ¼ θ2 , where θi is the ith component cut. For perfect separation in a single-entry separator and two product streams, x 11 W t1 ¼1 Υ˙ 11 ¼ x 1f W tf

and

x 22 W t2 Υ˙ 22 ¼ ¼ 1: ð2:2:13aÞ x 2f W tf

If there is no separation at all (i.e. x1f ¼ x11 ¼ x12), then ð2:2:9bÞ

and α12 ¼

ð2:2:10aÞ

ð2:2:9cÞ

Υ˙ 11 ¼ θ

and

Υ˙ 22 ¼ 1−θ,

ð2:2:13bÞ

since, for a binary system, x22 ¼ x2f ¼ x21 for no separation. We will now obtain a more useful form of ξ for a singleentry separator. Substitution of definition (2.2.12) in (2.2.11) leads, after simplification, to (Sirkar, 1977)

2.2

Binary separation in a single-entry separator

   x 21  x 11 x 2f ;  1  x 2f  x 21 x 1f     x 1f x 21  W t1 x 11  1 ξ¼ ;  W x x x  ξ¼



W t1 W tf



W i1 0 : ¼ k˙ i1 W i2



tf

ξ¼θ

1f

ð2:2:14Þ

11 2f

       x 21  hf  ¼ θ2 αhf  1 ¼ θ2 εhf ; α  1 12 12 12    x 2f 

"

ξ ¼ θ1 1 

# 1 : αhf 12

Example 2.2.1 Obtain simplified expressions of ξ for (a) close separation, (b) a close-separation symmetrical separator and (c) a dilute solution of species 2 in solvent 1. Solution (a) Close separation In such a case, and x21 < x2f, but x11 – x1f ffi 0(∊), where ∊ small quantity W tf . Similarly, the fresh feed composition is given by x1f, whereas the actual feed composition to the recycle separator is x r1f (>x1f since x11 > x1f for hf α12 > 1). Since we have assumed that the actual values of hf ft θ, α12 and α12 for the recycle separator are the same as those of the simple separator, we obtain, by a total molar balance at steady state for a recycle ratio of η, W rtf ¼

Further

0:685  0:694 ¼ ¼ 4:83: 0:315  0:306

  0:97  0:315  0:685  0:502  W t1 x 21  hf   − 1 ξ¼ α12 − 1 ¼  1:91  0:502 0:315  0:498 W tf x 2f ¼ 0:318j2:19−1j ¼ 0:378:

The cut θ ¼ Wt1/Wtf ¼ 0.506. Note that ft α12 ¼ ð4:83=2:19Þ ¼ 2:20; it is behaving almost as a symmetric separator.

It would appear from the preceding treatments that for a hf single-entry separator, the cut θ is independent of αht 12 , α12 or ft α12 . For any given separation process, the relation between a separation factor and the cut, θ, can be derived only by detailed considerations of the separation mechanism operative in the separator. In general, the separation factor is likely to depend on θ. These dependencies will be considered when individual separation processes are discussed in later chapters.

hf

α12 ¼ ft

α12 ¼

x r11 x r2f x r21

x r1f

x r1f x r22 x r2f

x r12

ffi ffi

x r1f x r12

;

hf

α12 ¼ ft

α12 ¼

x 11 x 2f x 11 ffi ; x 1f x 21 x 1f

ð2:2:23Þ

x 1f x 22 x 1f ffi : x 2f x 12 x 12

ð2:2:24Þ

W tf x 1f þ ηθW rtf x r11 ¼ x r1f W rtf ,

ð2:2:25Þ

which on rearrangement using (2.2.22) and (2.2.23) yields x r1f ¼ x 1f

ð1 − ηθÞ

hf

½1 − ηθα12 

:

ð2:2:26Þ

hf

Since α12 > 1, obviously x r1f > x 1f , and therefore hf x r11 ð¼ α12 x r1f Þ is greater than x11. The apparent cut and the apparent heads separation factor for the recycle separator are given by

Single-entry separator with a product recycle

Sometimes, to achieve a better separation, a fraction of one of the product streams is recycled to the feed end of the single entry separator. Consider Figure 2.2.1 where we show a nonrecycle traditional separator with an actual cut hf ft θ and actual heads and tails separation factors α12 and α12 . This figure also shows the schematic of a separator, where a fraction η of the actual light fraction molar output from the separator is recycled to the feed stream (Figure 2.2.1 (b)). Thus the actual molar feed rate to the recycle separator is higher than that to the nonrecycle separator. Let us now consider a special case of a binary mixture in which x1f α12 : ¼ α12 x 1f W tf

x r11 x 1f

¼

ð2:2:27Þ ð2:2:28Þ

Thus, the apparent values indicate a better quality of separation accompanied by a lesser amount of product stream 1. For a better picture, let us compute the extent of separation with recycle and without recycle:     x r ð1 − ηÞθW r x r ð1 − ηÞθW r   r r   11  21 tf tf  ξ r ¼ Υ˙ 11 − Υ˙ 21  ¼  − ;  W tf x 1f W tf ð1 − x 1f Þ  ξr ¼

 x r x 1f ð1 − ηÞθx r11  1 − 21 r ð1 − ηθÞx 1f  x 2f x 11

ξ ¼ jY_ 11 − Y_ 21 j ¼ Therefore,

   ; 

ð2:2:29Þ

  x 21 x 1f  θx 11  1 − : x 2f x 11  x 1f 

r

ξ ð1 − ηÞ x r11 ¼ ξ ð1 − ηθÞ x 11

 xr21 x1f   1− r   x2f x11   x x :  1 − 21 1f  x 2f x 11

ð2:2:30Þ

ð2:2:31aÞ

2.2

Binary separation in a single-entry separator

Since all the mixtures are dilute in species 1, we may assume that x r21 x 1f x 1f ffi ; x 2f x r11 x r11

x 21 x 1f x 1f ffi : x 2f x 11 x 11

ð2:2:31bÞ

Utilizing relations (2.2.23) and (2.2.26) in (2.2.31a), we get, after various manipulations, r

ξ =ξ ¼

hf ð1 − ηÞ=½ð1 − ηθα12 Þð1 − ηθÞ,

ð2:2:31cÞ

from which we can conclude that if ξ r > ξ, then the following inequality must hold:2 hf

α12 >

1−θ : θð1 − ηθÞ

ð2:2:31dÞ

For a binary mixture dilute in one species and being separated in a single-entry separator, there will therefore be a better hf separation with recycle of light fraction for constant θ, α12 and ft α12 if condition (2.2.31d) is satisfied. Obviously close-separation cases will not be helped by recycle unless θ is unrealistically close to 1. The description of separation in a recycle single-entry separator is now less ambiguous compared to that based on earlier results (2.2.27) and (2.2.28).3 Note that ξ r has been calculated by using the fresh feed and the net separated output streams to estimate the actual separation achieved. A recycle single-entry separator may have part of its tails stream recycled to the feed (flash distillation, Figure 2.2.1(c)). Examples of radioactive rare gas separation with various recycle arrangements are given in Ohno et al. (1977, 1978). See Problem 2.2.7. One aspect to be noted while comparing separators with or without recycle is that since θ is constant, but the actual feed flow rate to the separator changes with recycle from that without recycle, the dimensions of the two separators will be different. See, however, Problem 2.2.8. 2.2.3

Separative power and value function

An additional index, often used for isotope separation plants and originally introduced by P.A.M Dirac (Cohen, 1951), is the separative power δU of a single-entry separator. Assuming that the value of a particular process stream increases with the increase in mole fraction of a given species, the separative power of a given separating unit could be determined from the net increase in the value of the streams coming into and out of the separator. Such an index may be defined, following Dirac, as δU ¼ W t1 V aðx 11 Þ þ W t2 V aðx 12 Þ−W tf V aðx 1f Þ, ð2:2:32aÞ

49 where Va(xij), the value function, is the value of one mole of the jth stream of composition xij. For the separator of cut θ, the following alternative expression is more useful:   δU ¼ θV aðx 11 Þ þ ð1 − θÞV aðx 12 Þ − V a x 1f : W tf

ð2:2:32bÞ

One demands Va(xij), the value function, to be such that δU is independent of feed and product compositions. The quantity δU then indicates the net increase in the value of products over that of the feed stream, and is therefore a measure of the separator capability or separation achieved. There exists no general solution for Va(xij) to make δU independent of feed and product compositions. For a few special cases, solutions for Va(xij) satisfying such a requirement are available. One such case is close separation, for which both xi1 and xi2 are sufficiently close to xif to permit a Taylor series expansion of Va(xij) around Va(xif):  dV aðx i1 Þ  V aðx i1 Þ ¼ V aðx if Þ þ ðx i1 − x if Þ dx i1 x i1 ¼x if  ðx i1 −x if Þ2 d2 V aðx i1 Þ  þ . . . ; ð2:2:33Þ þ  2 2 dx i1 x i1 ¼x if  dV aðx i2 Þ  ðx i2 − x if Þ dx i2 x i2 ¼x if  ðx i2 − x if Þ2 d2 V aðx i2 Þ  þ þ . . . : ð2:2:34Þ  2 dx i2 2 x i2 ¼x if

V aðx i2 Þ ¼ V aðx if Þ þ

For a continuous and well-behaved function Va(xij),   dV aðx i1 Þ  dV aðx i2 Þ  ¼ and dx i1 x i1 ¼x if dx i2 xi2 ¼xif   d2 V aðx i1 Þ  d2 V aðx i2 Þ  ¼ : ð2:2:35Þ dx 2 i1 x i1 ¼x if dx 2 i2 x i2 ¼x if Recognizing from equations (2.1.29a) and (2.2.10a) that x if ¼ θx i1 þ ð1 − θÞx i2 ,

ð2:2:36Þ

we obtain from definition (2.2.32b)  δU 1 d2 V aðx i1 Þ  ¼ ½θðx i1 − x if Þ2 þ ð1 − θÞðx i2 − x if Þ2  W tf 2 dx 2i1 x i1 ¼x if

ð2:2:37aÞ

when relations (2.2.33), (2.2.34) and (2.2.35) are used. To simplify the above result further, rewrite the ith species balance equation (2.2.36) in the following two forms:     θ x i1 − x if ¼ ð1 − θÞ x if − x i2 ð2:2:37bÞ

and 2 3

  hf This analysis is valid only if 1 − ηθα12 is nonzero and positive.

K.K. Sirkar and S. Teslik, “Description of separation in separators with or without recycle,” unpublished (1982).

  θðx i1 − x i2 Þ ¼ x if − x i 2 :

ð2:2:37cÞ

With these two relations, expression (2.2.37a) for (δU/Wtf) is simplified:

50

Description of separation in open separators The value function is plotted in Figure 2.2.4 against xif. As has been pointed out (Pratt, 1967), a zero value for Va(xif ¼ 0.5) is reasonable, especially for nonisotopic mixtures. This is because the mixture is in the lowest energy state and will require the largest energy for separation (see Chapter 10). See Cohen (1951), Benedict et al. (1981, chap. 12) and Pratt (1967) for detailed treatments on separative power, value functions, etc. for isotope separation plants.

7 6

(2xif –1)ln[xif /(1–xif)]

5 4 3

2.3

2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

xif Mole fraction Figure 2.2.4. Value function of equation (2.2.42).

δU ¼

 W tf d2 V aðx i1 Þ  : θð1 − θÞðx il −x i2 Þ2 2 2 dx i1 x i1 ¼ x if

ð2:2:37dÞ

For close separations, use relation (2.2.2h) with xil ffi xi2 ffi xif on the right-hand side and substitute in (2.2.37d): δU ¼

 2 d2 V aðx if Þ  2   W tf θð1 − θÞ αht : x if 1 − x if 12 − 1 dx 2if 2

ð2:2:38Þ

To make δU independent of compositions of various streams, assume 2

Therefore,

d V aðx if Þ 1 ¼   2 : dx 2if x if 1 − x if δU ¼

 2 W tf θð1 − θÞ αht 12 − 1 2

ð2:2:39Þ

and

if

x if ¼0:5

¼0

Unlike for a single-entry separator, two feed streams enter a double-entry separator. The number of product streams leaving such a separator in general is two. As shown in Figure 2.3.1, the relative orientations of the two feed streams with respect to each other as well as the separator may vary, resulting in crosscurrent, cocurrent or countercurrent configurations. Separation with such an arrangement is achieved when xil 6¼ xi2, and either xif1 6¼ xi1 or xif 2 6¼ xi2 or both. The most frequent situation encountered is xif 1 6¼ xi1, xif 2 6¼ xi2, xil 6¼ xi2, with all three inequalities being valid simultaneously. Further, product stream 1, usually termed the heads fraction, the light fraction or the vapor fraction, will have only species 1 when pure, whereas product stream 2, usually identified as the tails fraction, the heavy fraction or the liquid fraction, will have only species 2 when pure. The double-entry separator performance is often characterized for a two-component system by the equilibrium ratio Ki (definition (2.2.1f)) as well as by the separation factor αht 12 (definition (2.2.2a)). Both definitions relate the two product stream compositions and are related to each other by (2.2.2i). The compositional differences between the light fraction and feed stream 1 may be indicated by a heads separation factor hf

α12 ¼

ð2:2:41bÞ

for Va(xif) satisfying (2.2.39). The solution is simply        ð2:2:42Þ V a x if ¼ 2x if − 1 ln x if = 1 − x if :

x 1l x 2f 1 : x 1f 1 x 21

ð2:3:1aÞ

Similarly, the tails separation factor between feed stream 2 and the heavy fraction is ft

ð2:2:40Þ

for close separations. The corresponding value function can be obtained by specifying the following conditions:   ð2:2:41aÞ V a x if ¼ 0:5 ¼ 0  dV aðx if Þ   dx

Binary separation in a double-entry separator

α12 ¼

x 1f 2 x 22 : x 2f 2 x 12

Unlike that for a single-entry hf ft αht 12 ¼ α12  α12 by definition, here, hf

ft

αht 12 6¼ α12  α12

ð2:3:1bÞ separator,

where

ð2:3:2Þ

unless the two feed streams have the same composition, i.e. x if 1 ¼ x if 2 . However, relations (2.2.2b)–(2.2.2d) are valid for a double-entry separator. One may also define an enrichment factor εht 12 in exactly the same way as in (2.2.2e). In addition, for close separations (εht 12 R2)? We opt to determine the relative values of the extents of separation for the two cases. But we recognize that, in effect, the separator of Figure 2.3.4 has one feed stream and two product streams. Therefore we use ξ defined by (2.2.14) to obtain the following:

1

xif

1

r

W t2

Wt 2 xi 2

or

x ri2

Figure 2.3.4. Double-entry separator with or without reflux.

ξ r jR¼R1 ξ r jR¼R2

ð2:3:7Þ

To simplify the problem further, let us assume that W tf 1 ¼ W rt1 and W rtf 2 ¼ W rt2 (the so-called constant total molar overflow assumption to be encountered in Sections 8.1-3.1/8.1-1.1). Further, let the following relation be valid regardless of the value of R: x if 1 ¼ K i x ri2 :

R+1

ð2:3:5Þ

In general, x ri1 6¼ x rif 2 (for example, in Figure 2.3.2(b), where the vapor from the top of the column is only partially condensed, the composition of the product vapor and the product liquid will be different; the partial condenser is also a single-entry separator. However, with a total condenser, all vapor from the column top is condensed to liquid: x rif 2 ¼ x ril ). But with a total condenser on the column, Wtf2¼ 0, x if 2 ¼ 0, x rif 2 ¼ x ri1 and R is the reflux (instead of the recycle) ratio (so that (R/R þ 1) fraction of W rt1 is recycled to the top of the separator). Equations (2.3.5) and (2.3.6) may now be simplified to W tf 1 ¼

R+1

R

W rt1

 r      Wr   W t1  r   t1 x r11 R1  R1 þ1 R1 þ1 x 21 R1    −  W x W tf 1 x 2f 1  tf 1 1f 1   r   : ¼  r   W t1  W t1 x r  r    R2 þ1 11 R2 R2 þ1 x 21 R2    −  W x W tf 1 x 2f 1  tf 1 1f 1 

ð2:3:12Þ

On simplification, it yields (Sirkar and Teslik, 1982; see footnote 3, p. 49) for R1, R2 6¼ 0 ξ r jR¼R1 ð1 þ 1=R2 Þ : ¼ ξ r jR¼R2 ð1 þ 1=R1 Þ

ð2:3:13Þ

Thus ξ r jR¼R1 > ξ r jR¼R2 if R1 > R2. That a higher reflux ratio achieves a better separation (which is a well-known fact, as we shall see in Section 8.1-3.2) can therefore be described with the help of the separation index ξ.

2.4

Multicomponent systems

Up to this point, we have considered only the description of separation of a binary mixture in a single- or doubleentry separator. We now briefly turn to multicomponent mixtures. In general, a multicomponent mixture will have, say, n different chemical species so that i ¼ 1, 2,. . .,n. An analogous problem is encountered when solid particles of the same material having a wide distribution of sizes are to be separated from one another or from a fluid in which they are suspended. If solid particles of a certain size are to be considered as one species or one component, then such a

54

Description of separation in open separators

mixture, in general, will have an infinite number of species or components since the particle sizes are, in general, continuously distributed over a range. Similarly, a macromolecular mixture which is not monodisperse will have a molecular weight distribution and therefore an infinite number of species in general. We should recognize now that whether the system is of n components or an infinite number of components, no special indices are available to accommodate this complexity. The indices developed for describing binary separation have to be adopted for this task. This is done by lumping a large number of components or a range of particle sizes (or molecular weights) into one component of a hypothetical binary system whose other component embraces the remaining components or the remaining range of particle sizes (or molecular weights). An alternative approach is to work with the heavy key component and the light key component, as pointed out in Section 1.6. We will study first the description of separation of a particle population and then a chemical solution containing an infinite number of species. At the end of the section, we describe n-component chemical solutions using the approach of a heavy key component and a light key component. 2.4.1

The composition of the particle population is usually indicated by the particle size density function f (rp), where rp is the characteristic particle dimension of importance. We denote the values of this function for the feed stream and the product streams by ff (rp), f1(rp) and f2(rp), respectively. Since the fraction of particles in the size range rp to rp þ drp is given by4 f (rp) drp when the particle size density function is f (rp), such that r max ð

r min

ð2:4:1aÞ

the following relations are also valid: r max ð

r min

f f ðr p Þdr p ¼

r max ð

r min

f 1 ðr p Þdr p ¼

r max ð

r min

f 2 ðr p Þdr p ¼ 1: ð2:4:1bÞ

Here, rmax and rmin refer to the maximum and minimum particle sizes in the feed stream. The nature of such a density function is illustrated in Figure 2.4.3(a). The particle size distribution function F(rp) is defined by dF ¼ f ðr p Þdr p )

Size-distributed particle population

Consider a single-entry separator used either for separating solid particles from a fluid or separating solid particles having sizes above a particular value from those having sizes below the particular value. Let wstf be the total mass flow rate of solids in the feed fluid whose total volumetric flow rate is Qf. In such a separator, there are only two product streams, the overflow (j ¼ 1) and the underflow (j ¼ 2). The total mass flow rate of solids in the overflow and the underflow are, respectively, wst1 and wst2 . The overflow is identified with essentially the carrier fluid and the finer particles, whereas the underflow is assumed to have most of the coarser particles and small amounts of carrier fluid. Figure 2.4.1 illustrates this for a hydrocyclone (Talbot, 1980). Sizes of various natural and industrial particles are shown in Figure 2.4.2. If the objective is to obtain a particle-free fluid, then perfect separation means no solid particles in the overflow (equivalent to the heads stream of Section 2.2) and no carrier fluid in the underflow (the tails stream of Section 2.2). If particle classification is the goal, then perfect separation requires all particles above a given size to be in the underflow and all particles smaller than the given size in the overflow. In an imperfect separation, some particles are always present in the overflow (when the goal is to have a particle-free carrier fluid). Similarly, due to imperfections in the separator, some particles coarser than the given size are in the overflow, just as some finer particles are in the underflow from the separator functioning as a classifier.

f ðr p Þdr p ¼ 1,

rðp

r min

f ðr p Þdr p ¼

rðp

r min

dFðr p Þ ¼ Fðr p Þ: ð2:4:1cÞ

The maximum value of F(rp), namely 1, is achieved when the upper limit of integration is rmax. A variety of particle size distributions are used in practice. The most widely used example is the Normal distribution (Gaussian distribution), "  2 # rp − rp 1 f ðr p Þ ¼ pffiffiffiffiffi exp − , ð2:4:1dÞ 2σ 2r 2π σ r where r p is the mean of the distribution defined by rp ¼

r max ð

r min

r p f ðr p Þdr p

ð2:4:1eÞ

and σ 2r , the variance, is defined by σ 2r

¼

r max ð

r min

ðr p − r p Þ2 f ðr p Þdr p :

ð2:4:1fÞ

Other distributions employed are the log-normal distribution, the gamma distribution, the Rosin–Rammler distribution, etc. We will encounter some of them later in the book.

4

One can also think of f (rp) drp as the probability of finding a particle in the size range rp to rp þ drp.

Multicomponent systems

55

100

% Weight less than diameter

% Weight less than diameter

2.4

Slurry feed

80 60 40 20 0

0.2

0.5 1 2 3 45 Diameter (µm)

100 Overflow 80 60 40 20 0

10

0.2

0.5

1 2

3 45

Diameter (µm)

40 20 0

40

Feed

60

rflow

60

80

Unde

Underflow 80

w

100

Over flo

% Weight less than diameter

% Weight less than diameter

100

20 0

0.2

0.5

1 2 3 45

0.2

10

0.5

1

2 3 45

10

Diameter (µm)

Diameter (µm)

Figure 2.4.1 Typical particle size distribution curves for removal and separation of solids from an aqueous kaolin solution by a hydrocyclone. (After Talbot (1980).)

Often, moments of distribution/density functions are used in particulate systems. The nth moment of the particle size density function is defined by MoðnÞ ¼

r max ð

r min

r np f ðr p Þdr p

ð2:4:1gÞ

particle number density function, n(rp), (also called the population density function) by     ð2:4:2aÞ dN r p ¼ n r p dr p , rðp

r min

From definition (2.4.1e), we see that the mean of the particle size distribution,r p , is the first moment of f(rp), Mo(1). The zeroth moment of f(rp), Mo(0), is F(rmax) ¼ 1. The variance σ 2r of the Gaussian distribution from (2.4.1f) may be related to σ 2r

¼

r max ð

r min

r 2p

f ðr p Þdr p − 2r p

r max ð

r min

2

r p f ðr p Þdr p þ ðr p Þ

r max ð

r min

f ðr p Þdr p

¼ Moð2Þ − 2r p Moð1Þ þ ðr p Þ2 ¼ Moð2Þ − 2r 2p þ ðr p Þ2 ,

Moð2Þ ¼ r 2p þ σ 2r :

ð2:4:1hÞ

Thus the variance, σ 2r , is easily related to the second moment Mo(2) and the first moment Mo(1). An item of significant interest is the actual number of particles N (rmin, rp) in the size range rmin to rp per unit fluid volume; the corresponding number is dN(rp) in the size range rp to rp þ drp. This quantity is related to the

nðr p Þdr p ¼ Nðr min , r p Þ:

ð2:4:2bÞ

The dimensions of N(rmin, rp) are number/volume, i.e. number/[L]3; the dimensions of n(rp) are number /[L]4. The total number of particles per unit fluid volume, Nt, is obtained from ð∞ ð∞ ð∞     N t ¼ dNðrmin , r p Þ ¼ dN r p ¼ n r p dr p , 0

0

0

ð2:4:2cÞ

by integrating over all possible dimensions of the particles, where we have assumed rmin ¼ 0. For such a case, we write N (rp) instead of N(0, rp). Figure 2.4.3(b) illustrates the population density function n(rp) as well as the cumulative number of particles N(rmin, rp) per unit volume against the particle dimension, rp. See Randolph and Larson (1988) for more details. Problem 2.4.1 is a useful exercise for determining n(rp). One can now relate the particle size distribution function F(rp) to the number density function n(rp) as follows:

56

Description of separation in open separators

10–3

10–2

10–1

µm

dp

1

102

10

urban aerosols smoke

smog

dust

mist, fog

colloidal silica

spray silt

clay

nanoparticles

sand

paint pigment

carbon black

pulverized coal

flexible long-chain macromolecule (MW≈106) coiled

extended

viruses

bacteria synthetic polymer spheres red blood cells yeast cells

mfp of air

Figure 2.4.2. Sizes of various natural and industrial particles: dp ¼ particle diameter or length; 1 μm ¼ 10−4 cm; 1 nm ¼ 10 Å ¼ 10−3 μm; mfp ¼ mean free path; MW ¼ molecular weight. (After Davis (2001).)

(b)

Distribution function 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

f(rp)

F(rp)

rmin

Nt n(rp), population density function of particles

Density function 1.0

N(rp), cumulative numbers of particles

(a)

rmax Particle size, rp

rmin

rp

Particle size, rp

rmax

Figure 2.4.3. (a) Particle size density function and distribution function. (b) Typical plots of population density function and cumulative particle numbers against the particle size. rðp

ð2:4:2dÞ

          dws ¼ f r p dr p ws ¼ n r p dr p wsp r p ¼ n r p dr p ρs ψ v r 3p ,

If, in lieu of f(rp) based on mass fraction, we use n(rp), where the number of particles present per unit volume between the limits rp and rp þ drp is n(rp)drp, then the differential change in mass of the particle population ws per unit fluid volume for a differential change in rp is given by

where wsp (rp) is the mass of a particle of size rp, ρs is the density of the particle material and ψν is a shape factor (Foust et al., 1960; Svarovsky, 1977) for particle volume assumed independent of particle size and having the value of (4π/3) for a sphere (for example). The total particle mass per unit volume, ws (also referred to as the suspension density, MT) is

Fðr p Þ ¼

rmin

nðr p Þdr p =N t :

ð2:4:2eÞ

2.4

Multicomponent systems

rmax ð

rmin

s

s

dw ¼ w ¼ M T ¼ ρs ψ v

rmax ð

rmin

57

  r 3p n r p dr p ¼ ρs ψ v Moð3Þ : ð2:4:2fÞ

The number-based mean radius r p1, 0 of the particle population is obtained by using the zeroth and the first moment, Moð0Þ and Moð1Þ , of n(rp): 2r 3 , rmax max ð ð     4 n r p dr p 5 r p n r p dr p r p1, 0 ¼ rmin

rmin

¼

ð∞   Moð1Þ ¼ r p n r p dr p =N t , Moð0Þ 0

ð2:4:2gÞ

where we have replaced the limits rmin and rmax by 0 and ∞, respectively. In general, r piþ1, i ¼

rmax ð

rmin

  r iþ1 p n r p dr p

, rmax ð rmin

  r ip n r p dr p :

¼ 50  2 ðg=cm3 Þ  ψ v  ð0:04Þ3 cm3 ,

so

4  10−2  106 4  102 ¼ ¼ 6:25: 100  64 64   3 If ψ v were defined via dws ¼ dN r p ρs ψ v d p , then ψv ¼

ψv ¼

4  10−2  106 4  102 ¼ ¼ 0:78: 100  64  8 64  8

(c) For a spherical particle,

ð2:4:2hÞ

Another commonly used radius is based on the mass of the particle (which varies with r 3p ; see (2.4.2f)), r p4, 3: , ð∞ ð∞     4 r p4, 3 ¼ r p n r p dr p ð2:4:2iÞ r 3p n r p dr p : 0

Solution (a)(i) From equation (2.4.2e), for one spherical    particle (i.e. dN r p ¼ n r p dr p ¼ 1), the mass of one spherical particle is dws ¼ ρs 43 πr 3p ; by definition, dws ¼ ρs ψ v r 3p which implies that ψ v ¼ 43 π. If ψ v is defined as dws ¼ ρs ψ v d 3p this implies that ψ v ¼ π=6: (ii) The volume of a rectangular parallelepiped of dimensions L  W  W is LW2. By definition, dws ¼ ρs ψ v L3 ¼ ρs LW 2 . Since W ¼ L/3, we get ψ v ¼ W 2 =L2 ¼ 1=9: (b) From equation (2.4.2e),   dws ¼ dN r p ρs ψ v r 3p ) 4  10−2 g

0

ψs ¼

4πr 2p surface area ) ψ s ¼ π: 2 ¼ d 2p ðcharacteristic dimensionÞ

For a spherical particle, 3

ψv ¼

ðπ=6Þd p volume ¼ π=6: 3 ¼ d3p ðcharacteristic dimensionÞ

Therefore

Example 2.4.1 (a) We are going to consider the issue of volume shape factor ψv for different types of particles. (i) What is the value of ψv for spherical particles of radius rp? (ii) Consider crystals shaped as rectangular parallelepipeds with dimensions of L, W, W, where (L/W) ¼ 3. What is the value of ψv if ψv is defined with respect to L3? (b) When a mixture of different sized particles was fractionated by a sieve having openings of size 0.09 cm, the particles which passed through were fractionated next by a sieve of size 0.07 cm. Those particles which were retained by this sieve may be considered to have a characteristic dimension (2rp ¼ dp) of 0.08 cm. The number of particles retained on the sieve is 50; their weight is 4  10−2 g. The particle density is 2 g/cm3. What is the value of ψν ? If ψν were defined with respect to d 3p what would be its value? (c) A surface shape factor ψ s of particles may be defined as follows:

surface area : ψs ¼ ðcharacteristic dimensionÞ2 Show that the sphericity ψ, defined by

ð6ψ v =π Þ2=3 ψ¼ ψ s =π has the value of 1 for spherical particles when the characteristic dimension for ψν and ψs is the diameter.

ψ¼



 6ψ v 2=3 π ψ s =π

¼



6π π6

2=3

π=π

¼ 1:

Generally, for other particle shapes, 0  ψ < 1: Example 2.4.2 Consider a particle number   density function   n r p having the form n r p ¼ n0 exp  −ar  p :  Determine expressions for Nt, F r p , f r p , r p1, 0 , d32 , MT and AT. Here d 32 is the Sauter mean diameter equal to 2r p3, 2 ; AT is the total particle surface area per unit volume of the total mixture. Assume that rp varies between 0 and ∞. Solution From definition (2.4.2c),

ð∞ ð∞       N t ¼ n r p dr p ¼ n0 exp −ar p dr p ¼ n0 =a : 0

0

From definition (2.4.2d),

  F rp ¼

rðp

r min

  n r p dr p =N t :

Assume now that rmin ¼ 0. Then rðp

 

  exp −ar p expð−0Þ − n0 exp −ar p dr p ¼ a −a −a   0    ¼ −1 exp −ar p − 1 ¼ 1 − exp −ar p :

  a F rp ¼ 0 n

By definition (2.4.1c),

58

Description of separation in open separators

          dF r p ¼ n r p =N t dF r p ¼ f r p dr p ) f r p ¼ dr p

X

Noi r pi X , ¼ Noi i¼1

r p1, 0

(from definition (2.4.2d)), so

i¼1

n0 expð−ar p Þ f ðr p Þ ¼ ¼ a expð−ar p Þ: n0 =a By definition (2.4.2g), 0∞ 1 ð   r p1, 0 ¼ @ r p n r p dr p =N t A

0 ∞ ð   n0 a a1 1 ¼ 0 r p exp −ar p dr p ) r p1, 0 ¼ 2 ¼ : n a a 0

By definition (2.4.2h), d 32 ¼ Sauter mean diameter ¼ 2r p3, 2 : ð∞ 2 r 3p nðr p Þdr p 6=a4 6 d 32 ¼ ∞0 ¼ : ¼2 ð 2=a3 a 2 r p nðr p Þdr p 0

By definition (2.4.2f), ð∞ ð∞     3 0 M T ¼ ρs ψ v r p n r p dr p ¼ ρs ψ v n r 3p exp −ar p dr p 0

0

Γ ð4 Þ 3! ¼ ρs ψ v n0 4 ; M T ¼ 6ρs ψ v n0 =a4 : a4 a From the given definition of AT, if ap is the surface area of a particle of size rp, then, from the previous example, ap ¼ ψ s r 2p . Therefore ¼ ρs ψ v n0

ð∞ ð∞     AT ¼ ap n r p dr p ¼ ψ s n0 r 2p exp −ar p dr p 0

ψ n0 2! 2ψ n0 ¼ s 3 ¼ s3 : a a

ð2:4:2kÞ

0

The treatment of size-distributed particle populations has so far assumed that the number of particles in the population is quite large and there is almost a continuous distribution in particle sizes in the particle population under consideration. If the number of particles in the population is not large, then we have essentially a discrete distribution in particle sizes. Suppose the number of particles in the size range r pi to r pi þ Δr pi is ΔNi. Then the population density function ni of particles in this size range is ni ¼ ΔN i =Δr pi

ð2:4:2jÞ

(Note: we are dealing with a discrete function not a continuous function, hence the subscript i.) This approach is needed to treat experimental information on different particle size fractions collected over different size sieves: that is, particles collected on a particular size sieve have sizes larger than this sieve opening but smaller than the opening size of the sieve immediately above it through which the particles fell. If we can characterize the number of particles of a certain size r pi as Noi , then the mean particle size r p is 1, 0

where the denominator represents the total number of particles in the sample. In many particle based separation processes, particles break, coalesce or grow (e.g. crystal growth). Common expectations for the mean particle size in small-sized populations can often be misleading. Consider the example provided by Neumann et al. (2003). Let there be ten particles of volume equivalent size of 1 unit and one large particle of size 100 units. The number mean size r p1, 0 from definition (2.4.2k) r p1, 0 ¼

10  1 þ 1  100 ¼ 10: 11

ð2:4:2lÞ

If the large particle breaks into two small particles of similar shape, then since the particle volume/mass, ρs ψ v r 3p , is conserved, we get two new particles of size rpn:  3 ρs ψ v ð100Þ3 ¼ 2ρs ψ v r pn ;

r pn ¼

100 2

1= 3

¼ 79:37: ð2:4:2mÞ

The new value of r p1, 0 is r p1, 0 ¼

10  1 þ 2  79:37 ¼ 14:06, 12

which is larger than before. Since the breakage led to two smaller particles, one is tempted to think that the mean particle size will decrease. It did not because the sample size was small. Neumann et al. (2003) have suggested that the number mean size will increase due to halving breakage of the larger particles (each particle breaking into two equal fragments) provided the particles broken are more than 70% larger than the mean particle size of the initial distribution. We will now focus on describing separation in devices used for separation of particles. The fraction of particles in the range rp to rp þ drp could be based on the weight fraction of particles, or the number fraction of particles, etc. Suppose we assume that f (rp) drp gives the weight fraction (and therefore the mass fraction) of particles in the size range rp to rp þ drp. Then a mass balance of all particles, as well as of particles in this size range, yields at steady state for the single-entry separator: wstf ¼ wst1 þ wst2 ;

ð2:4:3aÞ

wstf f f ðr p Þ dr p ¼ wst1 f 1 ðr p Þ dr p þ wst2 f 2 ðr p Þ dr p :

ð2:4:3bÞ

Current practice utilizes two descriptors for such a separation problem, the total efficiency ET and the grade efficiency Gr for particles of size rp (Svarovsky, 1977, chap. 3): ET ¼

wst2 ; wstf

ð2:4:4aÞ

2.4

Multicomponent systems

Gr ¼

59

wst2 f 2 ðr p Þ f ðr p Þ ¼ ET 2 : wstf f f ðr p Þ f f ðr p Þ

ð2:4:4bÞ

The total efficiency ET is the fraction of particles of the feed which are exiting through the underflow or the tails stream. When ET equals 1, the separation between the solid and the overflow fluid is complete. If the carrier fluid stream is the preferred one containing finer particles, then obviously E T ¼ ð1−θs Þ, where θs is the cut defined by (2.2.10a) in terms of only the solid particle flow rates. The grade efficiency Gr of particles of size rp is the ratio of the mass of particles of size rp in the underflow to that in the feed. It depends on rp and the separator characteristics. If the particle classifier or the solid–liquid separator is of any use, then at least all particles of size rmax should be in the underflow. To generalize,Gr jr p ¼∞ ¼ 1: As r p ! 0, Gr tends to a limiting value defined by (Svarovsky, 1977, chap. 3)

to be component 2. We can define the segregation fractions Y_ 11 ð¼ Y˙ rp 1 Þ and Y_ 21 ð¼ Y_ rp 1 Þ as follows: 1

Y_ r p

1

1

ws f 1 ðr p1 Þ , ¼ t1 wstf f f ðr p1 Þ

2

Y_ rp

2

1

¼

wst1 f 1 ðr p2 Þ , wstf f f ðr p2 Þ

ð2:4:5Þ

where we have assumed that the mass fraction of the particles is equal to the mole fraction of the particles. Borrowing the definition of the extent of separation ξ from (2.2.11), we have for the separation of particles of two sizes rp1 and rp2      ws f 1 ðr p1 Þ wst1 f 1 ðr p2 Þ  : ð2:4:6Þ ξ r p rp ¼ Y_ r p 1 − Y_ rp 1  ¼  t1 − 2 1 2 1 wstf f f ðr p1 Þ wstf f f ðr p2 Þ 

ð2:4:4cÞ

Utilizing the relations (2.4.3b), (2.4.4a) and (2.4.4b) for two particle sizes rp1 and rp2, we can easily show that    f ðr p Þ f ðr p Þ    ξ rp rp ¼ Gr ðr p1 Þ−Gr ðr p2 Þ ¼ E T  2 1 − 2 2 : 1 2 f f ðr p1 Þ f f ðr p2 Þ

where Q2 is the total volumetric flow rate of the underflow. In other words, even if there is no separation per se of the finest solid by the separation mechanism from the fluid, the finest solids of the feed are split between exit streams 1 and 2 since some fluid inevitably leaves with the underflow. The behavior of such types of grade efficiency functions, along with those for which Gr jrp ¼0 ¼ 0, are shown in Figure 2.4.4. Such functions are also known as Tromp curves. We now turn our attention to the description of separation of particles of only two sizes, rp1 and rp2. Assume particles of size rp1 to be component 1 and those of size rp2

If r p1 ¼ r max and r p2 ¼ r min , then ξ rp r p ¼ 1 − ðQ2 =Qf Þ, the 1 2 maximum possible value. Since Q2 6¼ 0, ξ rp rp jmax is never 1 2 equal to 1. On the other hand, as r p1 ! r p2 , ξ rp1 rp2 ! 0. The familiar quantity grade efficiency may thus be easily related to the index, ξ, the extent of separation. If we are interested in describing the separation of all particles above and below a particular size rs in the overflow and the underflow, we may define component 1 to be all particles with rp < rs and component 2 to be all particles with rp > rs. Continuing with the assumption of the equivalence of the mass fractions of particles with mole fractions of particles, we have in such a case

Gr jr p ¼0 ¼

Q2 , Qf

ð2:4:7Þ

rmax

rmax

rmax

1.0 Grade efficiency function for Gr r π 0 p

0.8

0.6

Grade efficiency function for Gr rp = 0

0.4 Gr 0.2

0 Particle size, rp Figure 2.4.4. Nature of various grade efficiency functions.

60

Description of separation in open separators ðrs

wst1 Y_ 11 ¼

wstf

r min ðrs

r min

wst1

f 1 ðr p Þdr p

rs

Y_ 21 ¼

;

wstf

f f ðr p Þdr p

r max ð r max ð rs

wst1

f 1 ðr p Þdr p : f f ðr p Þdr p ð2:4:8Þ

The extent of separation for components 1 and 2 defined in this manner is then   r r max   ðs ð     f ðr Þdr f ðr Þdr p p 1 p 1 p   wst1  r min rs  ð2:4:9Þ −r ξ 12 ¼ s  rs : max  ð wtf  ð     f f ðr p Þdr p f f ðr p Þdr p     r min r s

If we define the particle size distribution function F(rp) (see Figure 2.4.3(a) for illustration) by Fðr p Þ ¼

rðp

r min

Y_ 21 ¼

f ðr p Þdr p ,

we may rewrite (2.4.9) as    F ðr Þ 1 − F ðr Þ   1 s 1 s  − ξ 12 ¼ ½1 − E T    F f ðr s Þ 1 − F f ðr s Þ 

ð2:4:10Þ

r min

¼ f f ðr p Þdr p

r min

ð2:4:11Þ

ð2:4:12aÞ

wt1 u11 Y_ 11 ¼ , wtf u1f

ð2:4:12bÞ

Note further that ET in such a case is given by ET ¼

F f ðr s Þ − F 1 ðr s Þ F 2 ðr s Þ − F 1 ðr s Þ

ð2:4:12cÞ

if one uses equations (2.4.3a) and (2.4.3b) and integrates over the size range.5 Instead of considering only the separation of particles from one another, consider now the separation of all particles (species 2) from the fluid (species 1) feed or the elimination of liquid from the solid-rich underflow (region 2). Instead of the segregation fractions defined by (2.4.8), we will have

ð∞ ð1 E T ¼ Gr f f ðr p Þdr p ¼ Gr dF f ðr p Þ: 0

0

ð2:4:12dÞ

ð2:4:13bÞ

ξ 12

   w u  t1 11 wt1 u21  − ¼ ,  wtf u1f wtf u2f 

ð2:4:13cÞ

where we recall that wtj is the total mass flow rate of both species 1 and 2 in the jth stream. In solid–liquid separation practice, one comes across the Newton efficiency and the reduced efficiency. The Newton efficiency has been defined as follows (Van Ebbenhorst Tengbergen and Rietema, 1961; Svarovsky, 1979): wt2 u22 wt2 u12 − : wtf u2f wtf u1f

ð2:4:14aÞ

Using mass balance relations (2.1.24) and (2.1.25), one can easily show that ξ 12 of (2.4.13c) is the same as EN. Note that if the mass fraction of solids recovered in any stream (e.g. wt2 u22/wtf u2f) can be replaced by the corresponding mole fraction (Wt2 x22/Wtf x2f) and similarly for the fluid, we have W t2 x 22 W t2 x 12 − , W tf x 2f W tf x 1f

ð2:4:14bÞ

which is the extent of separation ξ for components 1 and 2 being separated between product streams 1 (overflow) and 2 (underflow). If we rewrite (2.4.14a) as E 1N

  u12 u2f wt2 u22 ws 1− ¼ ¼ t2 wstf wtf u2f u22 u1f

  ! u22 u1f −1 1− u12 u2f ð2:4:15Þ

to illustrate the composite nature of EN, we see that ðwst2 =wstf Þ indicates how much of the solid in the feed is recovered in the underflow (and therefore is essentially separated from the feed liquid), whereas (u22 u1f/u12 u2f) indicates how free from liquid this solid in the underflow is. The reduced efficiency of Kelsall (1966) (Svarovsky, 1977, chap. 3) is given by

5

From (2.4.4b), the grade efficiency may be related to ET in general by (see Prob. 2.4.2b)

ð2:4:13aÞ

leading to

EN ffi f ðr p Þdr p ¼ Fðr max Þ ¼ 1:

wst1 wt1 u21 : ¼ wstf wtf u2f

Similarly,

and r max ð

f 1 ðr p Þdr p

r min r max ð

EN ¼

since

ðwst1 =wstf Þ ¼ ð1 − E T Þ

wstf

r max ð

E 1T ¼

E T − ðQ2 =Qf Þ ¼ 1 − ðQ2 =Qf Þ

Qf

wst2 wstf

!

− Q2

Qf − Q2

:

ð2:4:16aÞ

When wst2 ¼ wstf , E 1T ¼ 1, the maximum value, since wst2  wstf : Further, note that, for dilute slurries or suspensions,

2.4

Multicomponent systems

61

Dust-free air

Feed air with dust

Cyclone II Cyclone I Dust hopper Dust hopper Collected dust Collected dust Figure 2.4.5. Two dust-collecting cyclones in series.

  E N ffi E T − Q2 =Qf

ð2:4:16bÞ

Solution (a) Use the results of Problem 2.4.2 (b), and obtain, in general,

since the solid volume is small compared to the liquid volume so that E 1T ffi

EN : 1 − ðQ2 =Qf Þ

ð2:4:16cÞ

In solid–liquid separation practice, sometimes three or more phases are encountered. Each phase may be made up of only one chemical species or a number of chemical species. But the phases are immiscible. Such a system, if separated into two product streams, may be described by indices similar to EN; one such is available in Sheng (1977). Example 2.4.3. Consider the separation of dust particles from air by means of two cyclones connected in series (shown schematically in Figure 2.4.5). The particle size distribution function of the feed to cyclone 1 is given by (Van der Kolk, 1961)

F f 1 ðr p Þ ¼ 1 − e−arp :

ð2:4:17bÞ

(a) Show that the total efficiency ET1 for the first cyclone is given by

ET 1 ¼

b1 : b1 þ a

0

ð2:4:17cÞ

(b) Similarly, show that the total efficiency ET2 for the second cyclone, based on the quantity of dust entering cyclone 1, is

a a : ð2:4:17dÞ ET 2 ¼ − a þ b1 a þ b1 þ b2 (c) What is the total efficiency of the two-cyclone system based on the material entering the first cyclone?

ð1 Gr f f ðr p Þdr p ¼ E T ¼ Gr dF f ðr p Þ: 0

For the first cyclone,

ET 1 ¼

ð∞ 0



 1−e−b1 r p ð−Þð−aÞe−arp dr p ¼

b1 : a þ b1

(b) The air that comes out through the top of cyclone 1 enters cyclone 2 as feed. This air has some dust in it. The amount of dust entering cyclone 2 is the difference between the total quantity of dust entering cyclone 1 and the amount of dust collected in cyclone 1. If we adopt the basis that a total of 1 kg of dust is entering cyclone 1, then

ð1 ð∞ 1 ¼ dF f 1 ðr p Þ ¼ ae−arp dr p :

ð2:4:17aÞ

The types of cyclones being used are such that the grade efficiency function is given for the ith cyclone by

Gri ¼ 1 − e−b1 rp :

ð∞

0

0

Further, the quantity of dust collected by cyclone 1 and appearing in the underflow stream 2 is then equal to

ð1 0

Gr1 dF f 1 ðr p Þ ¼

ð∞ 0

Gr 1 f f 1 ðr p Þdr p ¼ E T 1 ¼

b1 a þ b1

ð∞

¼ ae−arp ð1−e−b1 rp Þdr p : 0

For the second cyclone, the total quantity of dust entering between size rp and rp þ drp is (based upon 1 kg of dust entering cyclone 1)    1  ae−arp −ae−arp 1−e−b1 rp dr p :

If the grade efficiency function for cyclone 2 is (1 − e−b2rp), the value of total efficiency ET2 in this case is

62

Description of separation in open separators

ET 2

ð∞ 0

 1 ae−arp − ae−arp ð1 − e−b1 r p Þð1 − e−b2 rp Þdr p ¼



 a a : − a þ b1 a þ b1 þ b2

(c) The total efficiency of the series connected system is ET1,2 ¼ ET1 þ ET2, where ET2 is based on the amount of dust entering cyclone 1. Therefore

E T 1, 2 ¼

b1 a a a − ¼ 1− : þ b1 þ a b 1 þ a b 1 þ b 2 þ a b1 þ b 2 þ a

Obviously ET1,2> ET1. Other types of connections between two cyclones are also possible. See Problems 2.4.4 and 2.4.7 for two such arrangements.

Description of separation in solid particle–fluid separation devices by means of the grade efficiency Gr or related functions is often avoided in practice. Instead, a single number is used to describe the separation characteristics. Such a single number is provided by the “cut size,” which is the size of the opening of a hypothetical and ideal screen achieving the same separation as the device under consideration. An ideal screen will generate a step function for Gr: all particles of size smaller than the screen opening will appear in the overflow, whereas all particles of size larger than the screen opening will appear in the underflow. Actual grade efficiency functions for separators in practice are hardly of the step function type; therefore a “cut size” approach cannot be a true substitute for Gr. However, since the “cut-size” concept is used frequently, we will indicate very briefly a few definitions of cut size. One of the most commonly used cut sizes is the equiprobable size rp,50, for which the value of the grade efficiency Gr is 0.50 (Figure 2.4.4). A particle having this size has an equal probability of appearing in both the overflow, as well as the underflow from the separator. A smaller particle will most likely be carried away by the fluid in the overflow whereas a larger particle will most likely be separated from the fluid and appear in the underflow (Svarovsky, 1979). Note that if Gr is plotted against a normalized particle radius (rp /rmax), the equiprobable size will be independent of the actual values of the particle sizes for a given problem. Since the grade efficiency function Gr is needed to know the equiprobable size rp,50, and substantial information is necessary before Gr is known (e.g. ET, f2(rp), and ff (rp) should be available according to equation (2.4.4b)), two other cut-size definitions are frequently used in industry: the analytical cut size rp,a and the cut size by curve intersection (Svarovsky, 1979). We will only touch upon the analytical cut size here. The analytical cut size rp,a is defined such that a hypothetical and ideal screen having this size opening will give from the feed solid mixture the same value of total efficiency ET as the actual separator. In terms of the feed particle size distribution function, Ff (rp),

  1 − F f r p,a ¼ E T :

ð2:4:18Þ

The analytical cut size rp,a is not equal to the equiprobable cut size rp,50. The relation between these two can be derived, for example, for a log-normal particle size distribution. 2.4.2

Continuous chemical mixtures

Coal-derived liquids, heavy petroleum fractions, vegetable oils and polymers are mixtures that have very large numbers of components. It is practically impossible to identify each component by ordinary chemical analysis. One can no longer use mole fractions of individual components. Traditionally, the pseudo-component approach, or the key component approach, has been used to handle such complex mixtures. In the pseudo-component approach, the complex mixture is represented by a discontinuous distribution of pseudo-components, where the mole fraction xi of pseudocomponent i is represented by a bar in Figure 2.4.6(a) (which shows ten components). In the approach of continuous chemical mixtures, discrete components are not identified. Instead, the mixture is described by a property density function f(r) of a single distribution variable r (Figure 2.4.6(b)). The variable r can be the normal boiling point, the molecular weight, the number of carbon atoms, the degree of polymerization, etc. (Cotterman et al., 1985; Kehlen et al., 1985). The fraction of molecules in the mixture characterized by the range r to r þ dr is given by f(r)dr such that r max ð

r min

f ðrÞdr ¼ 1:

ð2:4:19Þ

Figure 2.4.6(c) shows the cumulative distribution of components up to any component i. For continuous chemical mixtures, we have instead a property distribution function. The property distribution function F(r) is defined by (Figure 2.4.6(d)) ðr

r min

f ðrÞdr ¼ FðrÞ:

ð2:4:20Þ

These quantities are thus analogous to those we have already defined for a size distributed particle population. Instead of particle size, we have a distribution variable r, which is intrinsic to a given chemical species. For example, for a flash vaporizer single-entry separator (Figure 2.4.7), the material balance for 1 mole of feed having a molecular weight density function of ff (M) is f f ðM ÞdM ¼ θf v ðM ÞdM þ ð1 − θÞf l ðM ÞdM,

ð2:4:21aÞ

where θ is the fraction of feed vaporized, fv(M) is the molecular weight density function of the vapor and fl(M) is the molecular weight density function of the liquid product (Cotterman and Prausnitz, 1985). Again,

2.4

Multicomponent systems

Finite-component (discrete) mixture

63

∑ xi = 1.0

i

∑ xi

xi

all i

0

xi = mole fraction Component no. i (c)

Component no. i (a)

∞ 0

F (r = ∞) = ò f (r ) dr = 1.0 Infinite-component (continuous) mixture

f(r)

f (r ) = Density function

F(r)

r is a characterizing quantity, e.g., molecular weight. Distribution variable r

Distribution variable r

(b)

(d)

Moles

Figure 2.4.6. Discrete and continuous composition for a multicomponent mixture. Reprinted, with permission, from Ind. Eng. Chem. Proc. Des. Dev., 24(1) (1985), 194. Copyright 1985 American Chemical Society.

Moles

Vapor

M Constant T & P

Moles

Feed

M Liquid

M Figure 2.4.7. Flash vaporizer for a continuous chemical mixture. (After Cotterman and Prausnitz, 1985.)

Mðmax

M min

f v ðMÞdM ¼ 1;

Mðmax

M min

f l ðMÞdM ¼ 1:

ð2:4:21bÞ

When a particular quantity, such as the pressure or volume, of a vapor–liquid mixture is to be determined for a continuous chemical mixture, the method of moments is to be adopted. Suppose the molar volume V of a pure chemical liquid species of molecular weight M can be expressed as a function of temperature T, pressure P and M, i.e. V(M,T,P). If this mixture is an ideal mixture and we are dealing with a simple n-component system, then the molar volume V of the solution is given by

V ¼

n X i¼1

x i V i ðT, P, M i Þ:

ð2:4:22aÞ

In a continuous chemical mixture, we have instead

V ¼

Mðmax

M min

V ðT, P, MÞf 1 ðMÞdM:

ð2:4:22bÞ

Such a description of continuous chemical mixtures can also be adopted for polymer solutions containing a molecular weight distribution. Consider a solution of polydisperse polystyrene in cyclohexane. The mass fraction

64

Description of separation in open separators

Sol

Mean = 2.19 × 105 Variance = 24.6 × 109

6

Feed

4

4 2 0

1 2 3 4 5 6 Molecular weight × 10−5

7

T = 21.6 °C 8

2 0

Mean = 1.25 × 105 Variance = 4.28 × 109

6

0

0

1

2 3 4 5 6 Molecular weight × 10−5

7 Gel

Mass probability density × 106

Mass probability density × 106

8

Mass probability density × 106

8

Mean = 3.13 × 105 Variance = 27.1 × 109

6 4 2 0

0

1 2 3 4 5 6 Molecular weight × 10−5

7

Figure 2.4.8. Phase equilibria in the system (polydisperse) polystyrene/cyclohexane. Flory parameter χ 0.50–0.60. Reprinted, with permission, from Ind. Eng. Chem. Proc. Des. Dev., 24(1) (1985), 194. Copyright 1985 American Chemical Society.

of polymers having a molecular weight in the range M to M þ dM in the solution is shown in Figure 2.4.8. The fraction of polymers in the solution is 0.67. When this solution is chilled to 21.6 C, it forms two phases, a polymer-rich phase called the gel and a solvent-rich phase called the sol. Obviously, the density function of polymer molecular weight in the gel is different from that in the sol; the gel has much more of the higher molecular weight species than the sol as shown (Cotterman et al., 1985). Semi-continuous mixtures form a category in between the continuous chemical mixtures and the ordinary multicomponent mixtures. For example, solvents in a polymer solution or light hydrocarbons in a gas-condensate system can be described by discrete concentrations or mole fractions, whereas the continuous components are described by a density or distribution function approach as just outlined. For an introduction to such systems and the basics of calculation procedures needed to describe separation, consult Cotterman et al. (1985) and Cotterman and Prausnitz (1985).

separators, but are equally applicable here if regions 1 and 2 are substituted by product streams 1 and 2. Current practice regarding the definition of heavy key (Section 1.6) and light key is as follows. The component with the highest molecular weight appearing in the light fraction in significant amounts is termed the heavy key. The component with the lowest molecular weight appearing in the heavy fraction in significant amounts is the light key. Instead of the molecular weight, other criteria regarding the heaviness or lightness of a component may also be used. Example 2.4.4 illustrates the selection of heavy key and light key components for the distillation process. Note also that one can analyze the separator as though there are only two components, the heavy key (i ¼ 2) and the light key (i ¼ 1). Of course, in such a case the total molar feed flow rate is to be reduced to Wtf [x1f þ x2f] for a single-entry separator, where x1f and x2f refer, respectively, to the actual mole fractions of species 1 and species 2 in the actual feed. Thus x1f þ x2f < 1. But in the hypothetical binary system, define xlf and x2f such that (Hengstebeck, 1961) x 1f ¼

2.4.3

Multicomponent chemical mixtures

For a multicomponent chemical mixture being fed into a single-entry separator with only two product streams, existing methods of description involve amongst others separation factors for selected components. Some of these separation factors, namely αin (definition (1.6.6)), αij (definition (1.6.8)), etc., were introduced earlier with closed

x 1f x 1f þ x 2f

x 2f : x 1f þ x 2f

and

x 2f ¼

and

W tf x if ¼ W tf x if ;

ð2:4:23Þ

Furthermore, W tf ¼ W tf ½x lf þ x 2f 

i ¼ 1, 2,

ð2:4:24Þ where Wtf is the total molar feed flow rate in this hypothetical binary system.

2.4

Multicomponent systems

65

If a multicomponent feed stream is separated by a single-entry separator into more than two product streams, the description acquires much greater complexity. Very little is available in the literature for such cases. The best approach appears to be to concentrate on two specific species and the respective species-specific product streams. Perfect separation will then consist of each product stream containing only a specified species. The various indices defined for binary systems in this chapter may be used with due provisions for the distribution of each species in each product stream. Example 2.4.4 Consider a paraffinic hydrocarbon liquid feed having a composition in mole fractions as given in Table 2.4.1. This liquid feed at a pressure of 15 atm and 90 C is introduced into a tall distillation column, which produces a light and a heavy fraction having the compositions given in Table 2.4.2. The light key component in this case is n-C3H8 because C2H6 is absent in the heavy fraction. Since n-hexane does not appear in the overhead vapor, n-pentane is the heavy key. The component n in αin defined by (1.6.6) is then n-C5H12. In Table 2.4.3 we calculate αin values for all the six components; here i ¼ 1, 2, 3 and 4 correspond, respectively, to CH4, C2H6, n-C3H8 and n-C4H10, whereas i ¼ 6 corresponds to n-C6H14. Naturally αin > 1 for i ¼ 1, 2, 3 and 4, but αin < 1 for i ¼ 6. Another type of multicomponent separation involves one or more solutes distributed between two solvents that are immiscible and which form two different phases. A third type of multicomponent separation involves a gas phase and a liquid phase such that the solvent, which is the dominant Table 2.4.1. Composition in mole fractions for liquid feed in Example 2.4.4 n-C6H14

n-C5H12

n-C4H10

n-C3H8

C2H6

CH4

0.20

0.25

0.31

0.11

0.09

0.04

Table 2.4.2. Compositions of light and heavy fractions in Example 2.4.4 n-C6H14 n-C5H12 n-C4H10 n-C3H8 C2H6 CH4 Overhead vapor Bottoms liquid

0.35

0.01

0.32

0.33

0.22

0.12

0.362

0.281

0.007

–

–

constituent of the liquid phase, does not appear to a significant extent in the gas phase. Although there are many other types of multicomponent, multiphase systems, we will describe one of the preceding types of separation now since the pseudo-binary approach is valid for both cases. Example 2.4.5 Benzene is to be absorbed in a heavy wash oil from a nitrogen stream which enters a countercurrent absorber at a total molar flow rate of 10 gmol/s. The mole fraction of benzene in this stream is 0.020. The heavy wash oil entering the absorber at 1.0 gmol/s contains no benzene or nitrogen. It is known that benzene is distributed between the gas and the wash oil according to the relation (xbenzene)gas ¼ 0.125 (xbenzene)oil for the higher concentration levels near the gas inlet, but that nitrogen is not absorbed at all. Obtain values of the extent of separation and the separation factor. Consult the countercurrent schematic in Figure 2.3.1. Solution Product stream j ¼ 1 is the nitrogen stream stripped of benzene (almost). Product stream j ¼ 2 is the wash oil stream leaving the absorber after absorbing benzene. Component i ¼ 1 is nitrogen; component i ¼ 2 is benzene. In a pseudo-binary calculation scheme, we neglect the wash oil content in j ¼ 2. Similarly, feed stream f1 is the nitrogen stream, whereas feed stream f2 is the wash oil stream on a wash oil-free basis. The extent of separation ξ is     ξ ¼ Y_ 11 −Y_ 21      W t1 x 11 W t1 x 21   − ¼ ;  W tf 1 x 1f 1 þ W tf 2 x 1f 2 W tf 1 x 2f 1 þ W tf 2 x 2f 2 

but x1f2 ¼ 0, x2f2 ¼ 0, and also x12 ¼ 0. Furthermore, Wtf1 x1f1 ¼ Wt1 x11 þ Wt2 x12 ¼ Wt1 x11. Therefore    W x   t1 21  ξ ¼ 1− :  W tf 1 x 2f 1 

So, the species 2 balance is given by

   W x   t2 22  W tf 1 x 2f 1 ¼ W t1 x 21 þ W t 2 x 22 ) ξ ¼  :  W tf 1 x 2f 1 

Since some benzene always escapes with the nitrogen in product stream j ¼ 1, Wt2 x22 < Wtf1 x2f, so ξ < 1 unless we have complete absorption of benzene. Now,   gmol gmol benzene W tf 1 x 2f 1 ¼ 10  0:020 ¼ 0:20 : s s Using the distribution relation at the absorber gas inlet, we have

ðx benzene Þoil ¼ Table 2.4.3. Value of αin for the six components in Example 2.4.4

αin

n-C6H14

n-C5H12

n-C4H10

n-C3H8

C2H6

CH4

0

1

41.2

1710

∞

∞

ðx benzene Þgas 0:125

¼

0:020 ¼ 0:16: 0:125

Therefore

  moles benzene W t2 x 22 ¼ ðwash oil flow rateÞ moles wash oil

 ðx benzene Þoil 0:16 ¼ 1:0 ¼ 1:0  ffi 0:1905: 1 − ðx benzene Þoil 0:84

66

Description of separation in open separators

So,

   0:1905    ξ¼  ¼ 0:954,  0:20 

a reasonably good separation. But α12 ¼ x 11 x 22 =x 21 x 12 ¼ x 11 x 22 =x 21 0 since x12 ffi 0 (no nitrogen in wash oil product stream (on a wash oil-free basis or otherwise)). Therefore α12 ¼ ∞, which is not helpful in determining the changes in the performance of the absorber (if any) since nitrogen is always absent in the j ¼ 2 stream.

2.5 Separation in an output stream with time–varying concentration In some separation processes, only one product stream comes out of a single-entry separator. Further, the molar flow rates of various species in this product stream as measured by a detector (Figure 2.5.1) vary with time in a manner useful for separation. Consider the molar rates at which two chemical species 1 and 2 present in small amounts in a carrier liquid or gas stream emerge from the separator. Figure 2.5.1 shows two general types of output profile. In Figure 2.5.1 0 (a) between t in 1 and t 1 seconds, the exit gas or liquid stream 0 contains only species 1, whereas between t in 2 and t 2 seconds, the exit stream has only species 2 with 0 in t 02 > t in 2 > t 1 > t 1 . (Here the superscripts in and 0 refer, respectively, to the times when a particular species appears in the output for the first time and when it disappears completely.) Thus, if we assume the volumetric output of 0 the separator between t in 1 and t 1 seconds to constitute 0 region 1 and the volumetric output between t in 2 and t 2 to constitute region 2, we have a case of perfect separation of

species 1 and 2 between the two regions. On the other hand, consider Figure 2.5.1(b), where the output is such 0 in in 0 that t 02 > t in 2 ; t 1 > t 1 but t 2 < t 1 . In this case, the output 0 stream between t in and t will have both species 1 and 2. If 2 1 we now consider the purity of two regions obtained by collecting the outputs between, say, t in 1 and tc and tc and t 02 , it is obvious that we are dealing with an imperfect separation. Both the regions now have some impurity. It is now apparent that the description of separation between species 1 and 2 achieved with such an output stream can be carried out with the indices developed for a closed separator with two regions. There are also other indices which grew out of the practice of processes using such separators (e.g. in chromatographic separations). Before we introduce these indices, consider the variation in concentration Ci(t) of the ith solute species in the output stream as a function of time. This is necessary to estimate the total number of moles of the given species in a given region obtained by collecting the output between specified times. Let m0i be the total number of moles of species i introduced into the separator by the feed stream. Assume further that all of m0i has come out of the separator in the 0 output stream between times t in 1 and t i seconds. Let the output volumetric flow rate be constant at Qf cm3/s. It is also assumed to be equal to the input volumetric flow rate. The number of moles of species i in the output in between t in 1 and tc (for t c > t 1 ) is given by mii ¼

t in i

Qf C i ðtÞdt

ð2:5:1aÞ

in if region i for t c > t in 1 is t 1  t  t c . On the other hand, if in 0 region i for t c > t 1 is t i  t  t c , then

Output stream Ci (t )

Separator

Feed

ðt c

Detector

Species 2

gmol/s QFCi(t) 0

in

t1 (a)

Species 1

Species 2

gmol/s QFCi (t)

Species 1

0

t

t1

in

t2

in

in

t1

0

t2

t2

tc

0

t1

t (b)

Figure 2.5.1. Time-varying molar output rate of solute species in the single product stream from a single-entry separator.

0

t2

2.5

Separation in time–varying composition

67

Inflection point QF C1 (t)

Inflection point

A⬘

A

tR1 Wb1 = 4st 1

t Figure 2.5.2. Molar output rate of species 1 for a Gaussian profile. 0

mii ¼

ðt i

tc

Qf C i ðtÞdt:

ð2:5:1bÞ

Note that here Ci (t) is the number of moles of species i per unit volume of solution collected at any time t. Along with definition (2.5.1a), we have 0

m0i ¼

ðt i

t in i

Qf C i ðtÞdt,

ð2:5:2Þ

which is also valid for definition (2.5.1b). Sometimes it is difficult to specify a value of t 0i where Ci (t) ¼ 0; one then uses t 0i ¼ ∞. Similarly, one uses t in 1 ¼ −∞. At this time, we require a functional description of Ci (t) vs. t to make estimates of mii to be used in various indices. A Gaussian profile is often close to what is observed in many separator outputs (e.g. the so-called chromatograms, Figure 2.5.2, where tR1 is called the retention time for species 1): "  # Qf C i ðtÞ 1 1 t Ri −t 2 p ffiffiffiffiffi : ð2:5:3Þ ¼ exp − m0i 2 σ ti σ ti 2π

Note that with t 0i ¼ ∞ and t in 1 ¼ −∞, such a profile yields 92 3 2 8 ð∞ ð∞ t in i and t c  t  t i . For the second type of region i with t c > t in i and 0 t i  t  t c , we use definition (2.5.1b) to obtain "  # ð∞ m0i 1 t Ri − t 2 pffiffiffiffiffi exp − mii ¼ dt: ð2:5:14Þ σ ti 2 σ ti 2π tc

On using transformation (2.5.5) and splitting the integral, we get

ð0 ð∞ mii 1 1 2 p ffiffi ffi pffiffiffi exp½−η2i dηi ð2:5:15Þ ¼ exp½−η dη þ i m0i π π t c −t R 0 pffi i 2σ ti pffiffiffi since (t c − t Ri = 2σ ti ) is negative here. Changing the vari00 ables in the second integral to ηi ¼ −ηi , we get

ð mii 1 1 tpRiffi2σ−t c 1 1 t Ri −t c 00 2 00 ti exp½−η p ffiffi ffi p ffiffi ffi ¼ dη ¼ : þ þ erf i i 2 2 2 m0i π 2σ ti 0

ð2:5:16Þ

t in 1

t in 1 ,

With region 1 as t c > and t c  t  we can use relation (2.5.13) to determine Y11 and Y21. The index ξ, the extent of separation for such a system of two solute species 1 and 2 distributed between regions 1 and 2, is therefore given by (Rony, 1968b) 



1 t c −t R t c −t R ξ ¼ absjY 11 − Y 21 j ¼ abs erf pffiffiffi 1 − erf pffiffiffi 2 : 2 2σ t1 2σ t2 ð2:5:17Þ Here t R1 and σti are characteristic parameters for the solute species i and the separation system. If t R2 − t R1 ¼ ∞ and tc halfway between t R1 and t R2 , both error functions will have an upper limit whose magnitude is infinitely large. Therefore, from (2.5.17) we have 2 3 t c −t R1 t c −t R2   p ffiffi ffi p ffiffi ffi   ð 2σ 6 2 ð 2σ t1 2 7 t2 1 2 2  6 7 ξ ¼ abs6pffiffiffi e−η dη− pffiffiffi e−η dη7 4 π 5 π 2   0 0    2 3 −∞   ð∞ ð   2 1 2 2 2 ffi abs4pffiffiffi e−η dη− pffiffiffi e−η dη5 π π 2   0 0 2 3 ∞   ð  1  1 2 02 ffi abs41 þ pffiffiffi e−η dη0 5 ffi abs½1 þ 1 ¼ 1: π 2  2  0

ð2:5:18Þ

2.5

Separation in time–varying composition

The maximum extent of separation of ξ ¼ 1 is achieved even if tc is not halfway between the two tR values as long as each jt c − t Ri j is large. Obviously, if t R1 ¼ t R2 and σt1 ¼ σt2, ξ ¼ 0. Thus, there is no separation when the two concentration profiles have identical shape and come out at the same time. Separation under any other conditions then depends on the nature of the two profiles, their locations and the location of tc, the cut point. It is also possible to calculate the value of the separation factor α12 and some of the other types of indices defined in Chapter 1 for such an open system. The interrelationships been various separation indices (ξ and Rs, for example) may also be developed under certain conditions. See Example 2.5.1 and Problems 2.5.1 and 2.5.2. For a description of multicomponent separation in chromatographic outputs, the semi-quantitative approach by Stewart (1978) involving the concept of a resolution matrix is likely to be useful to interested readers. Example 2.5.1 Consider the time-varying molar outputs of two species 1 and 2 from a chromatographic system (Figure 2.5.1(b)). Assume both outputs, which are overlapping each other, to be Gaussian and σt1 ¼ σt2. If the cut point is located such that the impurity ratio in each region is the same, and if it is known that m01 ¼ m02 , develop a relation between this impurity ratio and the resolution. Calculate the values of this impurity ratio for Rs ¼ 0.2, 0.6, 1.0 and 1.4. Solution The impurity ratios for regions 1 and 2 are defined by relation (1.4.3), where region 1 is from, say, t in 1 to tc and region 2 is from tc to t 02 . We have η1 ¼ m21 =m11 and η2 ¼ m12 =m22 ,

m21 m21 =m02 m02 m21 =m02 Y 21 ¼ ¼ ¼ m11 m11 =m01 m01 m11 =m01 Y 11

η1 ¼

since m02 ¼ m01 . Similarly for η2: η2 ¼

m12 m12 =m01 m01 m12 =m01 Y 12 ¼ ¼ ¼ : m22 m22 =m02 m02 m22 =m02 Y 22

From (2.5.13),

Y 11 ¼ But

 

1 tc − tR erf pffiffiffi 1 þ 1 : 2 2σ t1

Y 12 ¼ 1 − Y 11 ¼

 

1 tc − tR 1 − erf pffiffiffi 1 : 2 2σ t1

This last result could also have been obtained from relation (2.5.16). Similarly,  

1 tR − tc Y 22 ¼ 1 þ erf p2ffiffiffi and 2 2σ t2  

1 tR − tc : Y 21 ¼ 1 − Y 22 ¼ 1 − erf p2ffiffiffi 2 2σ t2 Therefore, as required in Example 2.5.1,

69

 3  3 2 t −t t −t 1 − erf pR2ffiffi2σ c 1 − erf pc ffiffi2σR1 t2 t1 4 5 4   ¼ η2 ¼  5: η1 ¼ t −t t −t 1 þ erf pc ffiffi2σR 1 1 þ erf pR2ffiffi2σ c 2

t1

t2

This leads to

 

  2 tR − tc 2 tc − tR erf p2ffiffiffi ¼ erf pffiffiffi 1 : 2σ t2 2σ t1

Since σt1 ¼ σt2 ¼ σt (say), this means either tR − tc erf p2 ffiffiffi 2σ t 



! t c − t R1 ¼ erf pffiffiffi , 2σ t

which means t R2 − t c ¼ t c − t R1, i.e. t c ¼ ðt R1 þ t R2 Þ=2, or tR − tc −erf p2 ffiffiffi 2σ t 



! t c − t R1 p ffiffi ffi : ¼ þ erf 2σ t

Since erf (−x) ¼ −erf (x), this means t c − t R2 ¼ t c − t R1 i.e. t R1 ¼ t R2 , which is a trivial result since we know t R2 6¼ t R1. Therefore t c ¼ ððt R1 þ t R2 Þ=2Þ and 2

t R2 − t c  pffiffi   2σ t 5 t R1þt R2 t −t t ¼ erf cpffiffi2σR 1  c 2 t

1 − erf

η1 ¼ η2 ¼ 4 1þ



 3

t R2 − t R1 pffiffi  2 2σ t  : t R2 − t R1 erf 2pffiffi2σ t

1 − erf ¼

1þ





But the resolution in this case is Rs ¼ ðt R2 − t R1 Þ=4σ t . Thus η1 ¼ η2 ¼

pffiffiffi 1 − erfð 2Rs Þ pffiffiffi 1 þ erfð 2Rs Þ

is the required relation. This η has been indicated by Glueckauf (1955a) as η1:1. pffiffiffi The values of erf ( 2Rs ) for Rs ¼ 0.2, 0.6, 1.0 and 1.4 are, respectively, approximately 0.310, 0.7658, 0.9545 and 0.9949. The corresponding values of η1:1 are 0.526, 0.133, 0.0233 and 0.00256. Thus a resolution value of 1.4 implies excellent separation with very low impurities in each band. For a graphical relation between η1:1 and Rs over a wider range of Rs, see Said (1978). The preceding treatment considered separation in a timevarying output from a separator of the chromatographic type. Similar methods of describing separation, namely Rs, ξ, η1, η2, etc., may also be adopted for describing the separation between two concentration profiles in Figure 2.5.1 if the abscissa is a distance coordinate z instead of a time coordinate t. One then replaces t Ri by zi, σti by σzi, Ci(t) by Ci(z), tc by zc, etc. The mathematical formulae will be identical with the proper substitutions made, and we may conclude that no special treatment is needed to describe the separation between two concentration profiles spatially displaced but having some overlap.

70

Description of separation in open separators

Problems 2.2.1

The absorption factor, A, in the absorption of species i in a flowing gas stream by a flowing solvent stream is defined by



molar liquid flow rate liquid mole fraction  A¼ : molar gas flow rate gas mole fraction i, equilibrium The stripping factor, S, which describes the stripping of a volatile species i into a flowing gas stream from a flowing liquid stream is the inverse of A. The extraction factor for any species is defined in solvent extraction by



mole fraction in extract extract molar flow rate  E¼ : mole fraction in raffinate i, equilibrium raffinate molar flow rate Relate A, S and E to k_0 i1 :ðAns: S ¼ ð1=k_0 i1 Þ;E ¼ k_0 i1 :Þ

2.2.2

hf

ðα12 − 1Þ ¼ 2.2.3

hf

Obtain the following relation between the separation factor αht 12 , the heads separation factor α12 , the cut θ and x11 for a single-entry separator and a binary feed: ðαht 12 − 1Þð1 − θÞ : 1 þ θðαht 12 − 1Þð1 − x 11 Þ

Consider the separation of a binary mixture in two single-entry separators connected together as shown in Figure 2.P.1. The tails stream from separator 2 is recycled back to the feed stream to separator 1. The mole fraction of the ith species in the jth stream of separator n is indicated by xij(n) (for n ¼ 1,2). If these two separators are operated such that6

and if the values of both

hf α12

and

ft α12

x i2 ð2Þ ¼ x if ð1Þ are the same for both separators, show that hf

ft

1=2 : α12 ¼ α12 ¼ ðαht 12 Þ hf

(Hint: Use mole ratios to convert separation factor definitions to X 11 ðnÞ ¼ α12 X 1f ðnÞ.) 2.2.4

Consider the elimination of a surface active impurity from water by the foam fractionation process as shown in Figure 2.P.2. An inert gas is bubbled through the impure water generating a foam rich in the surface xi1(2) Light fraction product Separator 2 xif (2) xi2(2)

xi1(1) Separator 1

xi2 (1) Tails product stream Wtf (1) xif (1)

Feed

Figure 2.P.1. Two single-entry connected separators with tails recycle.

6

When two streams to be mixed together are such that they have the same composition, we encounter the so-called “no-mixing” condition.

Problems

71

Gas out

Foam breaker

Foam rising in the column Waste water feed

Surfactantrich water Purified water

Inert gas Figure 2.P.2. Water purification by foam fractionation removal of detergent impurities in a column.

active impurity. The foam rises up the column and is collapsed to obtain an impurity-rich water phase and the inert gas, which is recirculated to the column bottom for renewed foam making. The surfactant concentration in the feed is in the range 10−6 gmol/cm3. The decontaminated water has a surfactant concentration 0.01 times that of the feed. The molar water flow rate in the impurity-rich collapsed foam is 3% of the feed molar flow rate, Wtf. (a) Analyze this double-entry separator as a single-entry separator by drawing a suitable envelope in the figure and showing any other necessary arrangements. (b) What is the value of the extent of separation? (c) Show that the value obtained in (b) is very close to that obtained if the decontamination factor is 1000. (d) Although the extent of purification of water in (c) is much larger than that in (b), ξ does not reflect it. Define impurity ratios for the X flow system as η_ 1 ¼ ðW 21 =W l1 Þ and η_ 2 ¼ ðW12 =W22 Þ. Is the new impurity I j more sensitive for such a case? (Note: Ij or I is the purity index for the ratio η_ j or I j ¼ −log10 η_ j or I ¼ j flow system.) 2.2.5

Thorman et al. (1975) have used two permeators of the type described in Example 2.2.4 in series to obtain O2enriched air containing 0.310 mole fraction O2 from feed air (0.209 mole fraction O2). The feed air flow rate to the first permeator is 0.665 std. cm3/s. The O2-enriched permeated stream containing 0.250 mole fraction O2 and having a flow rate of 0.365 std. cm3/s is introduced as feed after compression into the second permeator, which has a permeate composition of 0.310 mole fraction O2. The permeate flow rate from the second permeator is 0.151 std. cm3/s. The second permeator is somewhat smaller (silicone capillary length ¼ 217 cm) than the first permeator (silicone capillary length ¼ 691 cm). Obtain an estimate of the overall separation using αht 12 , θ and ξ.

2.2.6

To recover as pure a C8 hydrocarbon liquid as possible from a feed containing small amounts of a C5 hydrocarbon, it is proposed to use a flash separator with a fraction of the bottoms liquids recycled to the feed. (See Figure 2.2.1(c) for a schematic; replace decane by C8 hydrocarbon and butane by C5 hydrocarbon.) A fraction η0 of the bottoms liquid fraction is compressed and heated before it is mixed with fresh feed. It is proposed to compare the purification ability of the recycle system with the no-recycle system. Assume that the equilibrium ratio Ki1 for the flash separator for the ith species is independent of recycle and depends only on the pressure and temperature of the flash, which do not change much with recycle. Assume further that the cut θ for the flash separator is independent of recycle (not strictly correct). Do not introduce any simplification based on the C5 concentration in the feed. (a) Develop a relation between xr22 and x2f containing only η0 , θ and K21. Ans. x r22 ¼

x 2f ½1−η0 ð1−θÞ : ½θK 21 þ ð1−η0 Þð1−θÞ

(b) Show that x r22 is always greater than x22 (corresponding to no recycle).

72

Description of separation in open separators (c) Show further that the fraction of the fresh feed recovered as the liquid fraction is lower due to the recycle. Note: C5 hydrocarbon goes much more into the vapor phase than C8 hydrocarbon, which has a very low volatility.

2.2.7

Consider a single-entry separator with a dilute feed solution of heavy species 2. If a fraction η0 of the tails stream hf ft is recycled to the feed stream and the separator operates with the same values of θ, α12 and α12 as does the nonrecycle separator, show that the following are true. (a) The apparent cut for the recycle separator based on the light fraction product flow rate from the system and the fresh feed flow rate is greater than θ, based on the light fraction actually leaving the separator and the feed rate actually entering the separator. ft ft (b) The apparent value of α12 ðffi x r22 =x 2f Þ is greater than the actual value of α12 for the separator feed and tails stream. (c) The condition under which the extent of separation based on the net product flow rates from the recycle system and the fresh feed is greater than that for a no-recycle system is (Sirkar and Teslik, 1982; see footnote 3, p. 49) " # ft α12 þ 1 1 0 η < − : ft ft ð1−θÞα12 ð1−θÞ2 α12 Comment in general on the utility of such systems.

2.2.8

Consider a single-entry separator for the separation of a binary mixture of species 1 and 2 (Figure 2.2.1(a)). We wish to compare its separation performance with that of a single-entry separator having a fraction η of its light fraction recycled to the feed stream. The operational conditions are as follows: hf ft (a) the heads and tails separation factors α12 and α12 have the same value for both the recycle and the nonrecycle separator; (b) the recycle separator operates such that its apparent cut (definition (2.2.27)) is equal to the cut of the nonrecycle separator. Show that under such conditions for any feed mixture ξ r > ξ holds and that the recycle separator is therefore a better separation device (Sirkar and Teslik, 1982; see footnote 3, p. 49).

2.2.9

The separative power δU of a single-entry separator for a binary feed stream is defined by Dirac in terms of a value function Va(xij) of composition xij as δU ¼ W t1 V aðx i1 Þ þ W t2 V aðx i2 Þ−W tf V aðx if Þ, where the value function Va(xij) indicates the value of one mole of a given fluid stream of composition xij, and it changes as the composition of the fluid stream changes. Show that the above relation may also be expressed as follows (Lee et al., 1977a): ! ! ! ! ! ! hf hf ft ft ft 1 þ α12 X 1f α12 X 1f α12 þ X 1f X 1f δU α12 −1 α12 −αht 12 ¼ −V aðx 1f Þ, þ V a V a ft hf ft ft W tf 1 þ X 1f αht 1−αht α12 þ X 1f 1 þ α12 X 1f α12 þ X 1f α12 12 12 −1   ft hf where the feed light component mole ratio X 1f is given by X 1f ¼ x 1f =ð1−x 1f Þ , and α12 , α12 and αht 12 are defined by (2.2.3)–(2.2.5).

2.2.10

Consider a single-entry separator separating two species g and h in the feed stream. The molar flow rates of species g and h in the feed stream are, respectively, G0 and H0. The corresponding molar flow rates of species g and h in stream 1 and stream 2 are, respectively, G1, H1 and G2, H2. Stream 1 is the light fraction. The following quantities are defined as follows: a ¼ mole percent of g in the feed; b ¼ mole percent of g in stream 1; j ¼ amount of stream 1 as a percentage of the feed. (1) Define the extent of separation ξ in terms of the quantities defined here (G0, H0, G1, H1, G2, H2, a, b, j). (2) A new separation efficiency η has been defined as η ¼ 100j Show that η ¼ 100ξ.

b−a : að100 − aÞ

Problems

73

2.4.1

Particle size distribution may be determined in practice for liquid suspensions by sieving, i.e. using screens/sieves having different openings and determining the weight of the particles retained by the sieve of a particular size. Part of the following information is adopted from Randolph and Larson (1988): ws ¼ MT ¼ 0.15 g/cm3 of particle free liquid volume; ρs¼ 2 g/cm3; particle volume shape factor ¼ 0.6. The sieving process yielded the following information for two sizes of particles: the fraction of particles (Δws =ws ) around the two sizes, 100–120 µm (av. 100 µm), 200–220 µm (av. 210 µm), are 0.15 and 0.04, respectively. Calculate the values of the population density n(rp) for these two sizes in units of numbers/cm3 · µm. (Hint: Employ (2.4.2e) over the size range.) (Ans. 704 numbers /cm3 · µm; 27 numbers/cm3 · µm.)

2.4.2

(a) Particle removal effectiveness of dead-end filters (Chapter 6.3-3.1) is determined by fractional penetration P¼

ϕout , ϕin

where ϕout and ϕin are the particle volume fractions at the outlet and the inlet of the filter, respectively. An additional index is the log reduction value (LRV) of the filter: LRV ¼ log10 ð1=P Þ: Identify the indices in Chapters 1 and 2 which are closest to these two indices. (b) Obtain the following relation for the total efficiency ET in a solid–liquid separator operating as a classifier as a function of the grade efficiency Gr of particles of size rp: ð1 E T ¼ Gr ðr p ÞdF f ðr p Þ, 0

where Ff(rp) is the particle size distribution function of the feed (0  Ff(rp)  1). Develop a relation to predict f1(rp) from a knowledge of Gr, ET and ff(rp). (Ans. f 1 ðr p Þ ¼ f f ðr p Þ  ðð1−Gr Þ=ð1−E T Þ:Þ 2.4.3

For classification of particles above size rs and below size rs, respectively, into the underflow and the overflow of a solid–liquid classifier, define component 1 to be the undersize particles and component 2 to be the oversize particles. Show that the extent of separation based on mass fractions in such a case is given by   ws wst2  u22 −u21  ξ ¼ t1 :  wstf wstf  u1f u2f 

Compare its numerical value with that of another efficiency E, defined in the literature (McCabe and Smith, 1976, p. 920) as

ws wst2 u21 u12 E ¼ t1 wstf wstf u1f u2f for the case where u2f ¼ 0.540, u21 ¼ 0.895 and u22 ¼ 0.275, obtained for a rs value equal to 0.75 mm. Assume that mass fractions of particles are equivalent to mole fractions. (Ans. ξ 12 ¼ 0:61; E ¼ 0.63.) 2.4.4

Consider two cyclones in a series for dust cleaning of air, such that the dust collected in the underflow of cyclone 1 is withdrawn, along with a small amount of air, and introduced as feed to the smaller cyclone 2, whose overflow gas stream containing some dust is recycled to the fresh feed air stream entering cyclone 1. Assume that the grade efficiency functions for cyclones 1 and 2 are given, respectively, by (Van der Kolk, 1961) Gr 1 ¼ 1−e−b1 r

and

Gr 2 ¼ 1−e−b2 r :

(a) Show that if the cleaned air from cyclone 1 and the dust from cyclone 2 are the net products, and if ηr kg of dust of certain size r is recycled from cyclone 2 for 1 kg of dust of the same size in the fresh air feed to cyclone 1, then

74

Description of separation in open separators

ηr ¼

Gr 1 − Gr 1 Gr 2 : 1 − Gr 1 þ Gr 1 Gr 2

(b) Show that the total efficiency E T 1, 2 . of this arrangement of two cyclones is given for a feed size distribution function Ff1(r) ¼ 1 − e–ar by E T 1, 2

ð∞

  ae−ar 1 − e−b1 r − e−b2 r þ e−ðb1 þb2 Þr ¼ dr: f1 − e−b2 r þ e−ðb1 þb2 Þr g 0

2.4.5

A disk centrifuge has the following grade efficiency function (Svarovsky, 1977, chap. 7) Gr ¼

r 2p r max 2

for r p  r max ;

Gr ¼ 1 for r p > r max:

It is known that the feed particle size density function is Gaussian with an average value r p and a standard deviation of σ r . (For the limits of integration, a value of infinity may be substituted for rmax. However, in the integrand, retain rmax.) Obtain an expression for a total efficiency ET as a function of r p , rmax and σ r : 2.4.6

The feed solids in a slurry to a hydrocyclone obey the following log-normal law: " # dFðr p Þ ðℓnr p − ℓnr g Þ2 1 pffiffiffiffiffi exp − ¼ , dðℓnr p Þ ℓnðσ g Þ 2π 2ðℓnσ g Þ2

where F(rp) is the particle size distribution function, with σg being the standard deviation and rg being the mean particle size. The feed volumetric flow rate is 6.35  10−4 m3/s. The feed solids concentration is 30 kg/m3. The underflow volumetric flow rate is 3.5  10−5 m3/s and the underflow solids concentration is 314.2 kg/m3. Determine (1) the total efficiency ET; (2) the reduced efficiency of Kelsall; (3) the particle size distribution function F(rp).

2.4.7

Consider the separation of dust particles from air by means of two cyclones connected in the following fashion. Feed air containing dust enters cyclone 1; the feed particle size distribution is F f 1 ðr p Þ ¼ 1−e−ar p . The underflow from cyclone 1 is introduced with a small amount of air as feed to a small cyclone 2. The underflow from cyclone 2 is collected as dust from the system. The overflow air from cyclone 2 is mixed with the overflow air from cyclone 1 to obtain the cleaned air. The grade efficiency of the ith cyclone is given by Gri ðr p Þ ¼ 1−e−bi r p : (1) Show by actual calculation the total efficiency of the first cyclone, E T 1. (2) Based on 1 kg of dust entering cyclone 1, determine step-by-step the total efficiency E T 1, 2 of the twocyclone system.

2.4.8

2.5.1

Consider the separation of a ternary gas mixture of species 1, 2 and 3 through a membrane separator with two different types of membranes M0 and M00 such that species 1 appears preferentially in product stream j ¼ 1 from membrane M0 , product stream j ¼ 2 from membrane M00 is enriched in species 2, while the tails stream j ¼ 3 is enriched in species 3. If the gas composition everywhere inside the separator is indicated by xi3, the mole fraction of species i in tails stream 3, define the following separation factors (Sirkar, 1980): (a) α0 1n and α00 1n for species 1 in the product streams 1 and 2, respectively, with respect to all other species n ¼ 2, 3 and the reject stream j ¼ 3; (b) α1n for species 1 and all other species (n ¼ 2, 3) with respect to product streams j ¼ 1 and j ¼ 2. Show that   α1n ¼ α0 1n =α00 1n . It has been pointed out in Section 2.5 that separation in the output stream from a chromatographic separator depends on the location of tc, the cut point. Assume σ t1 ¼ σ t2 ¼ σ t and Gaussian output streams for two species 1 and 2. Use the extent of separation to obtain the value of the optimum location of the cut point and the

Problems

75

corresponding optimum value of the extent of separation. (See Problem 1.5.1 prior to solving this problem.) With this optimum value of ξ, show that (Rony, 1968b) pffiffiffiffiffiffiffi ξ opt ¼ absjerf 2Rs j,

where Rs is the resolution for the given problem. 2.5.2

Obtain an expression for the separation factor α12 between two solute species coming out of a chromatographic separator.

3

Physicochemical basis for separation

The preceding chapters introduced first the notion of separation and then a variety of indices to describe separation. These indices were used to characterize quantitatively the amount of separation achieved in a closed or an open separation vessel. The quantitative description included systems at steady or unsteady state involving chemical or particulate systems. Systems studied were either binary or multicomponent or a continuous mixture. Not considered in these two chapters was the fundamental physicochemical basis for these separations; appropriately, this is the focus of our attention in this chapter. In Section 3.1, we distinguish between bulk and relative displacements and describe the external and internal forces that cause separation-inducing displacements. This section then identifies species migration velocities and the resulting fluxes as a function of various potential gradients. Section 3.2 is devoted to a quantitative analysis of separation phenomena and multicomponent separation ability in a closed vessel as influenced by two basic types of forces. The criteria for equilibrium separation in a closed separator vessel and individual species equilibrium between immiscible phases are covered in Section 3.3. Section 3.4 treats flux expressions containing mass-transfer coefficients in multiphase systems. Flux expressions for transport through membranes are also introduced here.

3.1 Displacements, driving forces, velocities and fluxes When a particular component in a mixture is displaced in a given direction, it moves with a certain velocity. This velocity leads to a flux of the species, which is the molar rate of species movement per unit area in any given frame of reference. The nature of the displacements and the forces that cause the displacements leading to species velocity and flux are considered first in this section. Expressions for species velocities and fluxes are then studied to provide the foundations for a quantitative analysis of separation later.

3.1.1

Nature of displacements

The separation of a mixture involves the setting apart of the mixture components present in a given region. When perfect separation is attained, each component occupies a separate region where no other component is present. The total volume of all such regions may or may not be equal to the volume of the region originally occupied by the mixture. To achieve this separation, each component must move selectively toward its own designated region. Therefore, molecules of each component undergo displacement toward their own region during a separation process. The initial mixture, as well as the final separated state, may consist of either a single phase or a collection of immiscible phases. If separation is desired in a feed consisting of two immiscible phases, then each component has to be selectively displaced toward its designated phase. Such component-specific displacement and the separation achieved thereby may or may not lead to pure phases. It is, however, a prerequisite to any separation. The direction and the rate of displacements of molecules of a component (or particles of a certain type) will in general depend on the nature of the chemical species (or particle), the type, magnitude and direction of forces acting on the chemical species (or particle) and the surrounding medium. Such species movements take place spontaneously at the microscopic level in response to conditions imposed on the separation system (Sweed, 1971, p. 175). In most separation processes, bulk displacements also take place simultaneously. Generally, bulk displacements move molecules of all components at the same speed and in the same direction. On the other hand, the spontaneous microscopic species-specific movement in response to forces acting on a particular species (mentioned in the preceding paragraph) is identified as relative displacement (Giddings, 1978). These two types of displacements may occur simultaneously or consecutively (Sweed, 1971).

3.1

Forces, displacements, velocities and fluxes

Lighter phase Vessel A

Heavier phase Vessel B Figure 3.1.1. Decantation of upper phase from vessel A to vessel B.

Bulk displacement may be caused by fluid flow or direct mechanical conveying of a complete phase or fraction from one location or vessel to another. Consider, for example, the mechanical decantation of the upper phase from vessel A to vessel B. In vessel B, additional separation takes place because of the redistribution of components of the upper phase in vessel A between itself and the lower phase in vessel B (Figure 3.1.1). But all the components in the upper phase of vessel A are displaced to the new vessel B at the same rate and in a nonselective fashion. Such a procedure is followed in laboratory decantations as well as in industrial decanters. Bulk displacement by fluid flow is much more common. When a multicomponent mixture moves in a vessel or in a region of a vessel at a certain bulk velocity, the fluid motion carries all the components, in general, at the same velocity. In the case of a flat velocity profile with convective transport being dominant over diffusive transport, the fluid flows like a “plug” (Froment and Bischoff, 1979), and every species has the same displacement and velocity. If no forces are present to impart different displacement rates to different components, all components will be non-selectively carried by bulk fluid motion and there would not be any separation.1 Relative displacement takes place when different forces acting on molecules of different species (or on different particles) cause molecules of one species to move relative to those of the other species (similarly for particles). Such motions lead, in general, to separation, whether the mixture as a whole has a bulk velocity or not. Examples of such

1

There can be separation processes in which bulk displacement could vary from location to location due to gradients in bulk velocity. Additional complexities and sometimes separation can be achieved by a time dependence in the bulk displacements. The effects of spatial and temporal dependence of bulk displacement will be treated in later chapters.

77 species-specific forces are: chemical potential gradient, electrostatic potential gradient, centrifugal force field, gravitational force field, magnetic force field, thermal gradient, etc. Of these, only the chemical potential gradient is referred to as an internal force; the others are caused by devices or phenomena external to the separation system. We have described above the notions of bulk displacement and relative displacement primarily for molecules. They are equally applicable to the separation of macromolecules, colloids or macroscopic particles from the continuous phase (or to fractionation of macromolecules, particles, etc.); however, the detailed mechanisms and methods of description may be different. For example, as we will see later in deep bed filtration and aerosol filtration, particles of different sizes and therefore different masses may have different inertial forces in an accelerating or decelerating flow field. With larger particles, certain types of bulk motion may, therefore, lead to a relative displacement. Even for gas molecules, the persistence of velocity phenomenon creates different displacements for molecules of differing masses moving at a given bulk velocity when an impingement on a target gas occurs (Anderson, 1980). 3.1.2

Forces on particles and molecules

A variety of external forces are of use in separation. Of course, the internal force of chemical potential gradient has great importance in chemical separations. We will begin, however, by treating external forces. Of these, the gravitational force due to the earth is perhaps the most familiar one and is therefore appropriate for consideration now. 3.1.2.1

Gravitational force

In the external gravitational force field, the work required to raise a particle of mass mp from a location of height z1 (where the gravitational potential is Φ1 ¼ gz1 ; Φ is a scalar quantity) to one of height z2 (Figure 3.1.2A) in free space is mp(gz2  gz1) for z2 > z1.2 Here the positive z-coordinate is vertically upward. For a differential displacement dz, the vertically downward gravitational force F is related to dW, the differential amount of work done on the particle and dΦ, the change in gravitational potential, by a force F in the positive z-direction acting over the distance dz: dW ¼ F  dz ¼ mp dΦ ¼ F dz ¼ mp g dz;

ð3:1:1Þ

where the magnitude of the force per unit mass of the particle, F^ , is

2

g is acceleration due to gravity, 980 cm s2. The unit for force is the newton or kg m/s2. The unit for work is the joule or newtonmeter (N-m).

78

Physicochemical basis for separation

Figure 3.1.2B. Centrifugal force of magnitude mprω2 (¼ mpvθ2/r) acting radially outward on a particle of mass mp rotating in a circle of radius r with a tangential velocity vθ ¼ rω.

Figure 3.1.2A. Gravitational force of magnitude mpg acting on a particle of mass mp vertically downward.

F dΦ ¼ g: F^ ¼ ¼ mp dz

ð3:1:2Þ

In vector notation, the vertically downward gravitational force per unit mass may be described as F F^ ¼ ¼ rΦ: mp

ð3:1:3aÞ

The relevant z-component in vector form is F^ z k ¼ ðdΦ=dzÞk:

ð3:1:3bÞ

The vector gravitational driving force per unit mass of the particle due to the external force field is simply the negative of the gradient of the scalar potential of that force field. Here Φ increases with positive z; nature, i.e. gravity, spontaneously drives the particle to a lower Φ. The movement of the particle of mass mp under the action of the gravitational force considered above, however, assumed free space. If this particle were instead immersed in a fluid of density ρ, a buoyancy force would act on the particle in the vertically upward direction. The net external driving force on the particle acting vertically downward would be ! ! mp ρ ext F net ¼ mp g  ρg ¼ mp g 1  ; ð3:1:4Þ ρp ρp where ρp is the mass density of the particle. This force acts whether the particle moves or not. If, instead of a single particle, we have one mole of molecules of the ith species, the total gravitational force on one mole is     ρ ρ ext F ext ; F k; ð3:1:5Þ ¼ M g 1  ¼ M g 1  i i net net ρi ρi where Mi is the molecular weight and ρi is the density of the ith species. (To develop this expression, follow the procedure illustrated later for a centrifuge in (3.1.51).)

For a particle in the gravitational field, the net external force, F ext net , will remain constant if mp, ρp and ρ are constant in the z-direction. In a separation system, one can create a condition such that the surrounding fluid composition, and therefore the fluid density ρ, change with the height z. The net external force on the particle will now depend on the vertical location of the particle. A vertical gradient in fluid density may then be considered as an additional source of an external driving force on a particle along with the gradient of gravitational potential. Clearly, such an additional external force requires the existence of the gravity force. Note that F ext net will be zero if ρ¼ρρ for a particle or ρ¼ρi for molecules of species i. Further, the net external force vector per unit particle mass is g(1  ρ/ρp)k. 3.1.2.2

Centrifugal force

If a particle rotates at an angular velocity of ω radian/ second (rad/s) in a circle of radius r (Figure 3.1.2B), the centrifugal force3 on the particle per unit mass in the outward radial direction is F^ ¼ ðF=mp Þ ¼ rω2 r ¼ ðdΦ=drÞr:

ð3:1:6aÞ

Here Φ is the scalar centrifugal force field potential and r is the unit vector along the outward radial direction. If a fluid particle of mass density ρ is rotated as a rigid body in a vessel such that ρ is constant, the centrifugal driving force on a unit mass of the fluid particle is also given by the above relation. The vector centrifugal force acting on one gram mole of molecules of species i in the radially outward direction is F ¼ M i rω2 r:

ð3:1:6bÞ

From relation (3.1.6a), we obtain

3 The tangential velocity vθ of the particle is rω. The magnitude of the centrifugal force is mρ v2θ =r ¼ mρ rω2 . The Coriolis force is also present under these conditions. However, it may be neglected in comparison to the rω2 term.

3.1

Forces, displacements, velocities and fluxes

y Electrode potential F2 + + + + + + + + Positively + charged + electrode + + + + + + + + + +

E

+

Electrode potential - F1 F1 < F2 - Negatively - charged - electrode Positively charged particle/ion z

Figure 3.1.2C. Ion/charged particle placed in an electric force field of constant strength E ¼ ðdϕ=dzÞ.

rω2 ¼ ðdΦ=drÞ:

ð3:1:6cÞ

Integrating, we can write Φ ¼ Φ0  ðr 2 ω2 =2Þ;

ð3:1:6dÞ

where Φ0 is an arbitrary value of the centrifugal potential at r ¼ 0. As r increases, Φ decreases. 3.1.2.3

Electrical force

Consider next an electrical force field of constant strength E (volt/m) whose electrostatic scalar potential Φ (volt) is such that the potential difference dΦ over a distance dz in the direction of the field is E dz (Figure 3.1.2C). If molecular ions of the ith species, each having a valence of Zi (the magnitude varies from 1 to 10), are exposed to this field in an aqueous solution in between the electrodes, the force exerted on 1 gmol of such charged molecules in the z-direction is F iz ¼ Z i F E ¼ Z i F

dΦ dz

ð3:1:7Þ

or F i ¼ Z i F rΦ and

E ¼ rΦ:

ð3:1:8Þ

Here F is Faraday’s constant having the value of 96 485 coulomb per g-equivalent (C/g-equiv.); the electrical field has the unit of newton/coulomb (N/C). The above expression for force assumes that the electrical field has not been disturbed by the ions and vice versa. The work done to move these charged molecular ions having a total charge of ZiF coulomb along the z-coordinate from z1 to z2 is given by W ¼ Z i F Eðz2  z1 Þ

ð3:1:9Þ

per gram mole or gram ion obtained from 1 gmol of the compound (the unit of work is the joule (¼volt-coulomb))

79 with Φ decreasing in the positive z-direction. The electrical ~ where N ~ is Avogadro’s charge on a single ion is (Z i F =N), number. (See Section 3.1.6.2 for ions in a gaseous mixture.) A binary electrolyte AZνþþ YZν when dissociated fully in a solvent will produce νþ ions of cation AZ þ (whose electrochemical valence is Zþ) and ν ions of anion YZ  (whose electrochemical valence is Z). For example, for Na2SO4 in water, νþ ¼ 2; Z þ ¼ 1; ν ¼ 1; Z  ¼ 2 since the ions are Naþ and SO 4 . Thus Zi  s are positive for cations and negative for anions. If there is an electrical field in the solution, each species, positive and negative, will experience a force due to it according to (3.1.7). In any electrolytic solution, any molecular ion of interest having a mean diameter dion and an algebraic valence Zion will attract ions of opposite charge, namely counterions. However, the centers of such counterions can approach the center of the ion of interest to a distance of “a” only due to short-range repulsive forces. For radial distances r (> a), the distribution of the net potential Φnet due to the ion of interest and the counterions is given as (see Newman, 1973)   ðr  aÞ Z ion e exp  λ Φnet ðrÞ ¼ ; 1 þ aλ 4πεr

ð3:1:10aÞ

where the potential due to the ion of interest only at location r is (Z ion e=4πεr) and e is the electronic charge, 1.6021  1019 coulomb. Thus the net potential Φnet ðrÞ decays very rapidly with r. The extent of rapidity in decay is determined by the parameter λ called the Debye length λ¼



ε RT F2 I

1=2

;

ð3:1:10bÞ

which is influenced by the ionic strength I defined by Ι¼

n 1X Z 2 C il : 2 i¼1 i

ð3:1:10cÞ

The higher the value of I, the smaller the value of λ. Here F is Faraday’s constant, R is the universal gas constant, T is the absolute temperature and ε is the electrical permittivity of the fluid. This quantity ε is the product of the relative dielectric constant of the medium εd (for water εd ¼ 78.54 at 25  C) and the electrical permittivity ε0 of vacuum (¼ 8.8542  1014 farad/cm or coulomb/volt-cm ¼ 8.854  1012 coulomb2/newton-m2, where recall that 1 newton-m ¼ 1 voltcoulomb ¼ 1 joule). In a uni-univalent electrolyte solution of 0.1 M strength (of, say, NaCl) the value of λ at 25  C is 9.6  108 cm, i.e. 0.96 nm (Newman, 1973). The ions of opposite charge shield the charge of the ion of interest, and the effect of the ion of interest decays very rapidly with distance. So the description of the electrical force on an ion in an applied field E by definition (3.1.8) is generally satisfactory. The ion of interest, however, has in an aqueous solution a solvation shell of water molecules. Similarly, the

80

Physicochemical basis for separation

counterions have water molecules around them. When the ion of interest moves in an applied electrical field E (as in electrophoresis), the water molecules solvating the counterions move in an opposite direction with the counterions whose charge is equal and opposite to the charge of interest. This motion of the solvent molecules located in a shell at a distance λ exerts a retardation force, F ret i , on the main force on the ion of interest, Z i F rΦ. The net driving force, F net i , on the ion of interest becomes (Wieme, 1975) n d o ion ; F i þ F ret ¼ Z i F rΦ  ðZ i F rΦÞ  i 2λ   d ion F net : ¼ Z F rΦ 1  i i 2λ

ð3:1:10dÞ

Often, this correction is ignored for small ionic species where the description of the species-specific force (3.1.8) is reasonably accurate. If the charged species are larger, for example proteins, or if we are dealing with colloidal particles in a solution, the retardation forces are regularly taken into account. Generally, small ions of charge opposite to that of the charged protein or particle will collect in a diffuse layer next to the protein or particle. An electrical double layer is created by the fixed charges of the protein or particle and the counterions collected from the solution (Figure 3.1.2D). The total charge in this double layer is zero. However, there is an electrical potential Ψel in this layer which decreases to zero with distance from the particle or protein surface. Note that the counterions are mobile and can be influenced by the external electrical field E. If the macroion, protein or particle moves in a given direction due to the external field E, resulting in electrophoretic motion, the large number of counterions move in an opposite direction. Since these ions carry some solvent Hydration shell

z Particle or protein

rp

Drp

y

ye1

0 Figure 3.1.2D. Charged particle with its hydration shell and double-layer potential profile.

with them by the phenomenon of electro-osmosis, the macroion motion is retarded by this solvent motion. The electrophoretic retarding force is given by F ret p ¼ ðεd ζ r p  Qp ÞE:

ð3:1:11aÞ

Here εd is the dielectric constant of the medium, rp is the effective particle/macroion radius, Qp is its charge and ζ is the zeta (or electrokinetic) potential. (This is the potential at the surface of shear around the particle; there are solvent molecules tightly bound to the particle of radius rp up to the radius rp þ Δrp defining the hydration shell, and they move with the particle defining the shear surface.) Thus the net force on the particle is F p þ F ret p ¼ Qp E þ ðεd ζ r p  Qp ÞE ¼ εd ζ r p E:

ð3:1:11bÞ

A more general analysis yields the following results (Wieme, 1975):   fðr p =λÞ ; F p þ F ret p ¼ εd ζ r p E ð1 þ r p =λÞ

ð3:1:11cÞ

where the function f(rp /λ) tends to 1 when (rp /λ) >> 1 (rp /λ ! 1000). For proteins the range is 1 < rp /λ < 300. When (rp /λ) ! 0.1, f(rp /λ) tends to 2/3. We assume the electric field strength E to be constant here, independent of the z-coordinate. Therefore, the electrical force on 1 gmol of the charged ith species is also independent of the z-coordinate location (we can generalize this to all three coordinate directions). However, if the charge on the ith molecular ion species, Zi, changes from one location to another, the electrical force will become location dependent. For example, it is known that the net charge on protein molecules in a solution depends on the solution pH. At the isoelectric pH (identified as pI), Zi ¼ 0, but Zi 6¼ 0 for all other pH values (see Figure 4.2.5(c)). The net charge is positive at sufficiently low pH and negative at sufficiently high pH. Thus, the electrical force on protein molecules can vary even if the field strength E is constant. It is possible to create a pH gradient in the solution of concern in the separation system by external means. At the location where pH has the isoelectric value for the ith protein species, Zi will be zero, leading to a zero external electrical force on the ith species. At other locations, the force will be nonzero. The forces on molecules or ions in solution due to an externally imposed primary field (i.e. electrical field) can then be suitably altered by the imposition of additional property gradients in the solution by external means. If we have macroscopic particles instead of molecules or macromolecules, the driving force per particle may be obtained from definition (3.1.8) simply by replacing ZiF by Qp, the net particle charge in coulomb.

3.1

Forces, displacements, velocities and fluxes

81

+Q +

r

F

a

Electric dipole

E is nonuniform

F

– -Q

Figure 3.1.2E A dielectric uncharged particle placed in a nonuniform electric field develops an induced dipole of moment p = Qa, where Q is the magnitude of the charge of each pole separted by a distance a where the vector coordinate of negative charge –Q is r.

If the electrical force field E is nonuniform and a dielectric particle4 is placed in such a field and develops a dipole moment p, then the net electrical force on the particle with the induced dipole is (Figure 3.1.2E)

the effective polarization of the spherical particle (Jones, 1995). When the particle is metallic (ρpR ¼ 0), with εp ! ∞,

F ¼ ðp  rÞ E ¼ p rE:

Note: r E2 is r (E  E) so that (E  E) is the local electrical field intensity. Both of these forces are proportional to the particle volume as well as gradient of the square of the electrical field (Von Hippel, 1954; Lin and Benguigui, 1977). The above results can be obtained from (3.1.12) by introducing an expression for p, the dipole moment, resulting from the polarization of the particle in the nonuniform electric field (p ¼ (polarizability)  (particle volume)  (E)). Note that the dielectric force F is directed along the gradient of the electric field intensity r E2. For the metallic particle, the force direction is always toward the direction of the largest field. On the other hand, the force on a nonmetallic particle will be toward the direction of the largest field only if εp > εd (positive dielectrophoresis); it will be toward the lowest field if εd > εp (negative dielectrophoresis). If εp >> εd or εd >> εp, the magnitude of the force is not influenced, but the direction is. Obviously if there are two particles with εp1 < εd < εp2 , the forces experienced by the two particles will be in opposite directions (Lin and Benguigui, 1977). The dielectrophoretic force expressions given above are proportional to the square of the field strength and are independent of the direction of the field. Therefore an alternating current (AC) field can be used here, unlike electrophoretic motions induced by a DC field (Pohl and Kaler, 1979). Such a time-varying nonuniform electrical field has been used to separate mixtures of whole cells. In the cases considered above, there was an applied electrostatic potential gradient E, uniform or nonuniform, acting on the particle with or without charge. If the particle had a charge, it was assumed (although we did not indicate it) that the field generated by this charge did not influence the external electrical force field generated by E. On the

ð3:1:12Þ

The motion of particles caused by polarization effects in the nonuniform electrical field is identified as dielectrophoresis (Pohl, 1978), whereas, as we have seen already, the motion of charged particles in a uniform electric field is termed electrophoresis. In the force expression (3.1.12), rE is the gradient of the nonuniform electric field at the particle center and p is the magnitude of the dipole moment given by the product of the charge and the distance between the charges (in coulomb-meter). The gradient of the electrical field is to be determined at the center of the particle (Halliday and Resnick, 1962). If an uncharged particle is placed in a dielectric fluid having a dielectric constant εd and an electrical resistivity ρdR, the particle experiences a force only if the electrical field E is nonuniform in space. Assume the particle to be spherical with radius rp. If the particle is not metallic and the fluid resistivity ρdR is very high, the force is given by (Jones, 1995)   εp  εd ð3:1:13Þ rE 2 ; F ¼ 2π r 3p εd εp þ 2 εd where εp is the dielectric constant of the particle having a resistivity ρpR. The quantity within the brackets is called the Clausius–Mossoti function, and it indicates the strength of

4

The small dipole is characterized by two equal and opposite charges, +Q and Q, located at a vector distance a apart (Jones, 1995). Since the electric field is nonuniform, the two charges are subjected to different values of the electric field, E(r + a) and E (r); here r is the location of –Q. The force F ¼ QE(r + a) – QE(r). However, since jaj  > = < 2a 1 H F Lret ¼  ih ; ð3:1:16Þ h  i2 3r p > ; : hmin þ 2 ðhmin =r p Þ2 > rp

where

aH ¼ Hamaker constant; hmin ¼ the minimum separation between particle and collector; ih ¼ unit vector in the direction hmin outward from the collector. The electrokinetic force in the double layer (Spielman and Cukor, 1973) is given by " εd r p κ d exp ðκd hÞ F ELK ¼ ðζ 1 þ ζ 2 Þ2 1 ∓ exp ðκd hÞ 4 # exp ðκd hÞ ih ; ð3:1:17Þ ðζ 1  ζ 2 Þ2 1  exp ðκd hÞ where h ¼ distance between particle and collector; εd ¼ dielectric constant for the fluid; κd ¼ reciprocal Debye length (= 1/λ); ζ1,ζ2 ¼ zeta potential of collector and particle, respectively.

The upper sign in this equation corresponds to approach at constant charge, while the lower sign corresponds to approach at constant potential. 3.1.2.4

Magnetic field

An ionic species i with a charge Zi moving with a velocity vi can essentially be considered to be a current flow with a current of magnitude Z i F jvi j for 1 gmol of ionic species. A conductor carrying current in a magnetic field of constant strength B experiences a force. Since the motion of ionic species constitutes a current, the force on 1 gmol of the ith ionic species in the magnetic field is given by mag

Fi

¼ Z i F ½vi  B:

ð3:1:18Þ

The vector magnetic potential Φ for the magnetic field B is given by B ¼ r  Φ:

ð3:1:19Þ

Vector B is also called the magnetic induction. The unit of B is (newton/coulomb)/(m/s) and is identified usually as weber/m2, or tesla. Consider now particles without charges in the magnetic field. Particles can be classified into three general classes depending on how much magnetization is induced in them in a magnetic field. Ferromagnetic materials, such as iron, cobalt and nickel, are strongly magnetized. In general, the induced magnetism in these materials become relatively independent of the applied magnetic field. The extent of magnetization induced in the second class of materials, paramagnetics, is far weaker than in ferromagnetics. However, 55 elements in the periodic table are paramagnetic, and a magnetic field is used to separate paramagnetic particles. The third class of materials, diamagnetics, have even weaker magnetization. The magnetic force exerted on a small magnetizable paramagnetic particle of volume (mp/ρp) in a magnetic field of strength Hm (unit, amp/m) and volume susceptibility χp in vacuo is ! 1 m mp ð3:1:20Þ χ p rðH m  H m Þ; F ¼ μ0 ρp 2 where μm 0 is the magnetic permeability of vacuum (Watson, 1973; Birss and Parker, 1981). Note that the force is proportional to the volume of the particle (i.e. proportional to d 3p ). For paramagnetic and diamagnetic particles, Hm is linearly proportional to the magnetic induction vector B. The magnitude of this magnetic force in the z-direction (say) is given by ! dH m m mp χp H m ; ð3:1:21Þ F z ¼ μ0 ρp dz so that increasing Hm or (dHm/dz) or both will increase Fz. If the particle is in a fluid of volume susceptibility χ, then the force on the particle is modified to (Birss and Parker, 1981)

3.1

Forces, displacements, velocities and fluxes

Ferromagnetic wire Particle H

Magnetic force

Figure 3.1.2F. Simplified cross-sectional view of the magnetic forces acting on a paramagnetic particle flowing past a magnetized ferromagnetic wire in a background field H. (After Dobby and Finch (1977).)

ð3:1:22Þ

The variation of Hm in the z-direction could be brought about in the following way even if a uniform Hm is created by a solenoid as a magnet. If the region where the magnetic field is applied has fine ferritic steel wires, the uniform field is grossly distorted near the wire (Figure 3.1.2F) and the particles are therefore forced toward the wire (Dobby and Finch, 1977) due to the positive gradient of the magnetic field. The background field Hm magnetizes the ferromagnetic wire of magnetic permeability μw. If the magnetization of the wire is Mw, the local field gradient becomes, approximately, dH m ðH m þ M w Þ  H m M w amp ¼ ¼ ; dz dw d w m2

the suspension medium and is directed along the gradient 2 of the magnetic field intensity rH m 0 . The quantity in brackets on the right-hand side of (3.1.24) is a Clausius– Mossotti function K of sorts, where ε has been replaced by μms. In positive magnetophoresis, K > 0, and particles are attracted to magnetic field intensity maxima and are repelled from the minima. In negative magnetophoresis, K < 0, and the phenomenon is reversed.

3.1.2.5

z

1 F ¼ μm vp ðχ p  χÞrðH m  H m Þ: 2 0

83

ð3:1:23Þ

where dw is the wire diameter. The magnetic force on a spherical paramagnetic (linearly polarizable) particle of radius rp in a paramagnetic solution exposed to a nonuniform magnetic field having a local field intensity vector of H m 0 in vacuo is given by (Jones, 1995) ! m μm 2 p  μs 3 ð3:1:24Þ r rH m F ¼ 2π μm s p 0 : m μm p þ 2μs m Here μm s and μp are the magnetic permeabilities of the solution and the particle, respectively. The permeabilities may be related to the susceptibilities via μm ¼ μm 0 ðχ þ 1Þ; 2 corresponds to that in vacuum. Further rH 0m where μm 0 is related to the magnetic flux density B0 in the absence 2 m 2 of matter via rH m 0 ¼ rðB0 =μ0 Þ . This phenomenon of particle motion is sometimes described as magnetophoresis. Note that the force is proportional to the particle volume through r 3p , the magnetic permeability μm of s

Chemical potential gradient

A number of external force fields have been described in the preceding subsections. Of these, the gravitational force field exists in nature regardless of our desire to require it or not. Other force fields, like centrifugal, electrical and magnetic, are created by engineers and scientists to achieve separation. Most of these external force fields are often described by the negative of the gradient of their respective scalar potentials, i.e. rΦ (exceptions are the magnetic and nonuniform electrical fields). But all such force fields originate outside the separation system. The force of chemical potential gradient, on the other hand, originates inside the separation system due to the escaping tendency of molecules. For example, suppose we introduce two species into a closed vessel at constant temperature (T) and pressure (P) as a vapor–liquid system which is not in chemical equilibrium. As time progresses, we will observe that the species with the lower boiling point will preferentially escape the liquid phase and accumulate in the vapor phase; simultaneously, the higher boiling species will preferentially escape into the liquid phase. This directionally oriented escaping tendency disappears when the chemical potential of each species becomes uniform throughout the two phases and chemical equilibrium is attained (see Section 3.3). No external agency is involved in this separation phenomenon. We shall soon see that a large number of separation processes are governed by the gradient of this chemical potential, rμi , where μi is the chemical potential of the ith species per gmol of the ith species. For convenience, this chemical potential is sometimes referred to as μint i , the internal chemical potential, the superscript indicating its origin inside the separation system (Giddings, 1982). It is known from chemical thermodynamics (Guggenheim, 1967) that, at constant T and P, the reversible work W done to transfer mi moles of species i from a state of partial molar Gibbs free energy Gi j1 to Gi j2 ð> Gi j1 Þ is given by W ¼ mi ðGi j2  Gi j1 Þ:

ð3:1:25aÞ

However, μi, the chemical potential, is simply Gi . If states 1 and 2 refer to spatial locations z1 and z2, where z2 > z1, then we immediately conclude that

84

Physicochemical basis for separation W ¼ mi ðμi j2  μi j1 Þ:

ð3:1:25bÞ

This work is needed to overcome a natural force which resists taking mi moles from μi j1 to a higher value μi j2 . This natural and spontaneous force on species i must then act in the opposite direction: W ¼ mi ðμi j2  μi j1 Þ ¼ F total iz ðz 2  z 1 Þ:

ð3:1:25cÞ

Per gram mole of species i, the force Fiz is F iz ¼

F total μ j  μi j 1 iz ¼ i2 : mi z2  z1

ð3:1:25dÞ

Vectorially, we can now describe this force due to this chemical potential per gmol of ith species as F i ¼ Fjgmol of ith species ¼ 

dGi dμ k ¼  i k: dz dz

ð3:1:26Þ

In vector notation, the force on 1 gmol of the ith species due to the chemical potential gradient is rμi. By analogy, it has often been indicated that all external driving forces per mole of the ith species may be represented as rμext i . The shortcomings of such an approach (from the point of view of the fundamental property relation in chemical thermodynamics) for external forces have been demonstrated by Martin (1972, 1983). We therefore express the total driving force on 1 gmol of the ith species at constant temperature as F ti ¼ F t jmole of ith species ¼ rμi þ F ext ti :

ð3:1:27Þ

Of course, when possible, F ext ti may be represented by ΣrΦext i so that F ti ¼ F t jmole of i ¼ rðμi þ

ΣΦext i Þ

¼ rðμi þ

Φext ti Þ;

ð3:1:28Þ

where the summation for the external force potentials indicates the sum of the available external forces representable as rΦ. The above representation of external forces combined with the chemical potential force for molecules or ions is quite useful for those external force fields representable by the negative of the gradient of their scalar potentials. We indicate in Table 3.1.1 the value of Φext i for 1 gmol of the ith species for a few cases. For magnetic and nonuniform Table 3.1.1. The value of external force field Φext i for 1 gmol of species i (1) Uniform electrical field of electrostatic potential Φ and constant Zi (2) Gravity; species i in a solution of density ρ with z-axis vertically upwards (3) Centrifugal force field

electrical force fields, such representations are not possible for a system of molecules or ions. For macroscopic particles, where no chemical potential exists, only external force fields are important. Just as we have an idea of how to calculate Φext and i therefore rΦext i , we need to know more details about rμi . Consider any region in the separation system where we assume equilibrium to exist (see Section 3.3). For a binary system of i ¼ 1, 2 at constant temperature, if the pressure P and the mole fraction xi in the region are changed by differential amounts, the corresponding change in μi is given by       ∂μi ∂μi  ð3:1:30Þ dμi  ¼ dP þ dx i : T ∂P T;x i ∂x i T;P

The partial molar volume V i of the ith species in this region is defined as   ∂μi : ð3:1:31Þ Vi ¼ ∂P T;xi Furthermore, 

∂μi ∂x i



T;P

  M i g 1  ρρ z i

1 Φ0  M i ω2 r 2 ð3:1:29Þ 2

d ℓn ai ; dx i

ð3:1:32Þ

where ai is the activity of species i in this region. One can therefore write dμi jT ¼ V i dP þ RT d ℓn ai :

ð3:1:33Þ

Correspondingly, the gradient of μi is given by rμi jT ¼ V i rP þ RT r ℓn ai

ð3:1:34Þ

for a binary system at constant temperature. The force due to a chemical potential gradient on species i at constant temperature, rμi jT , then arises due to the existence of a pressure gradient, an activity gradient, or both, in the separation system in a given region. For liquid phases at not too high a pressure, the activity ai may be related through an activity coefficient γi to the mole fraction xi as follows: ai ¼ γi x i :

ð3:1:35Þ

Thus rμi jT ¼ V i rP  RT r ℓn ðγi x i Þ; which can be expressed as rμi jT ¼ V i rP  RT

ZiF Φ

¼ RT



dℓn γi x i dℓn x i



T

r ℓn x i :

ð3:1:36Þ

This expression shows that, ultimately, the mole fraction gradient can be used instead of the activity gradient. For an ideal solution, we have a simpler expression (since γi ! 1): rμi jT ¼ V i rP  RT r ℓn x i : For gaseous mixtures, we can use

ð3:1:37Þ

3.1

Forces, displacements, velocities and fluxes

ai ¼

^f ig f 0ig

;

ð3:1:38Þ

where f 0ig is the standard state fugacity of the gas i at system temperature and specified pressure of 1 atmosphere and ^f is the fugacity of species i at system temperature and ig pressure. For an ideal gas mixture, ^f ig ¼ pig , the partial pressure of species i in the mixture; for such a case, rμi jT ¼ V i rP  RT r ℓn pig :

85 of different kinds. In a gas mixture subjected to a temperature gradient, it has been observed that the lighter gas species concentrates in the hot region and the heavier species concentrates in the cold region.5 This phenomenon, known as thermal diffusion, is also observed in liquid mixtures. The force exerted on gas species A in a binary mixture of A and B subject to a temperature gradient is given by6

ð3:1:39Þ

Fjgmol of A ¼ 

For isobaric systems, we find, for ideal solutions, liquid solutions : rμi jT;p ¼ RT r ℓn x i ; gaseous mixtures : rμi jT ¼ RT r ℓn pig :

ð3:1:40Þ

Thus, mole fraction gradient, or partial pressure gradient, is the basis of the chemical potential gradient in a binary system at constant T and P. Recognize, however, that there is no specific force leading to a mole fraction or partial pressure gradient (see Section 3.1.3 after equation (3.1.76)). It is important to qualify these characterizations of rμi by indicating that they are valid within a phase, liquid, gaseous or solid. When the binary separation system has two phases, the variation of standard state chemical potential between the two phases also needs to be considered. Suppose rP¼0 in the separation system. Integrate relation (3.1.33) at constant P and obtain, for any phase j, μij ¼ μ0ij þ RT j ℓn aij :

ð3:1:41Þ

Thus rμij jT;P ¼ rμ0ij  RT j r ℓn aij ;

ð3:1:42Þ

where μ0ij is the standard state chemical potential of the ith species in phase j at the standard state conditions used to calculate of aij. (In Section 3.3, Table 3.3.2 identifies standard states for various conditions.) In a two-phase system, μ0ij for species i is usually different for the two phases; thus rμ0ij is also a driving force for species movement and separation. The value of μ0ij in any phase may be calculated from standard thermodynamic relations (Denbigh, 1971). Very small particles in liquid or gas streams have a random Brownian motion due to the thermal energy of the continuous phase molecules. If there is a concentration gradient of particles due to a particle sink, then there is a Brownian motion force on these particles: vrp

F BR ¼ 6 π μ vrp r p ;

ð3:1:43Þ

where is the particle diffusion velocity relative to that of the liquid or gas phase, rp is the particle radius and μ is the fluid viscosity.

RT DTA r ℓn T ¼ F TA ; DAB C A M A

ð3:1:44Þ

where DTA is the thermal diffusion coefficient for species A in a binary mixture of species A and B and DAB is the ordinary diffusion coefficient. A thermal diffusion ratio kT has been defined (Bird et al., 1960) as kT ¼

ρt DTA : C 2t M A M B DAB

ð3:1:45Þ

Furthermore DTA ¼ DTB . Thus if species A goes toward the hotter region, species B moves to the colder region. No phenomenon analogous to thermal diffusion in chemical mixtures is encountered with particles. However, when placed in a temperature gradient, small particles in a stagnant liquid have been found to move in the direction of lower temperature. This phenomenon has been referred to as thermophoresis (Fuchs, 1964). It is said to arise in a gas because gas molecules originating in the hot regions impinge on the particles with greater momenta than molecules coming from the colder regions. The magnitude of the force due to the thermophoresis, FTP, for a spherical particle obeying Stokes’ law in the z-direction is   6 π μ2 r p dT Th F TPz ¼ ; ð3:1:46aÞ C c ρt T dz where Th is a dimensionless group defined for a gas phase of viscosity μ by Th ¼ 

U pzt ρt T : μ ðdT=dzÞ

ð3:1:46bÞ

The value of Th is said to range between 0.42 and 1.5 from theoretical predictions. Here Upzt is the steady particle velocity due to thermophoresis (see Section 3.1.3.2 for its definition), rp is the particle radius and Cc is a correction factor (see Section 3.1.6.1). For additional details, see Talbot et al. (1980). Explanations for the same phenomenon in a liquid are somewhat uncertain (McNab and Meisen, 1973). An alternative situation, in which particle motion is generated in a solute gradient and termed diffusiophoresis, is treated in Anderson et al. (1982).

3.1.2.6 Other forces: thermal gradient, radiation pressure, acoustic force 5

Until now, we have considered the temperature to be uniform in our separation system. The existence of a temperature gradient exerts unequal forces on molecules

This is true for gases observing the inverse power law of repulsion: (force) = (constant)  (intermolecular distance)υ and υ > 5. See Present (1958). 6 In a frame of reference where vt ¼ v∗ t (see Section 3.1.3).

86

Physicochemical basis for separation

Radiation pressure from continuous wavelength (cw) visible laser light is known to accelerate freely suspended particles in the direction of the light. The magnitude of the radiation pressure force, Frad, has been given as (Ashkin, 1970) F rad ¼

2 qfr Pr : c

ð3:1:47Þ

Here qfr is the fraction of light effectively reflected back (generally assumed to be ~0.1), Pr is the power of the laser light and c is the velocity of light. The medium of interest is a liquid. The acoustic radiation (acr) force Facrx in the x-direction due to an ultrasound of wavelength λ induced on a particle of volume Vp, density ρp and compressibility βp suspended in a fluid medium of density ρf and compressibility βf, is given by (Petersson et al., 2005) ) !( π P 20 V p βf 5 ρp  2ρf βp F acrx ¼   sinð2kxÞ; 2λ 2ρp þ ρf βf ð3:1:48Þ where k is defined as (2π/λ), x is the distance from the pressure node and P0 is the amplitude of the pressure wave. The direction of the force Facrx is dependent on the sign of the quantity within the brackets { }, which is sometimes called the ϕ-factor. Those particles which have a positive value for the ϕ-factor will move toward the pressure node; those with the reverse sign will move toward the pressure antinode. For an introduction, see Nyborg (1978), Ter Haar and Wyard (1978) and Weiser et al. (1984). 3.1.2.6.1 Inertial force So far, we have presented a variety of forces that can come into play and act on molecules of a given species and/or particles. When such forces act, the molecules or particles move in a given direction. At the beginning of this motion, the molecules/particles undergo acceleration. From Newton’s second law, we know that the molecules/particles are subjected to an inertial force, F iner, in the same direction during this period of acceleration. The magnitude of this inertial force is simply the product of the mass of the molecules/species and the magnitude of the acceleration (see the beginning materials in Sections 3.1.3.1 and 3.1.3.2). In most circumstances encountered in separations, one can assume that the acceleration ceases to exist after some time and a steady velocity comes about; therefore the inertial force ceases to exist. However, if the fluid flow field is such that the direction and magnitude of its velocity continues to change, inertial forces will always be present. This is particularly true of particles flowing in a medium with many obstacles, or if there is a change in flow direction and flow cross-sectional area. 3.1.2.6.2 Lift force on a particle in shear flow When a particle is flowing in a shear flow field, it experiences a lift force normal to the fluid flow direction. The magnitude of

this lift force normal to (i.e. in the y-direction) the flow direction (say, in the z-direction) is given by F lift ¼ a μ vz ðdvz =dyÞ1=2 r 2p =υ1=2

ð3:1:49Þ

for a spherical particle of radius rp in an axial fluid flow field (i.e. velocity vz) having a simple shear ((dvz/dy) 6¼ 0) (Saffman, 1965); the constant “a” has the value 6.46 and υ ¼ (μ/ρ) for the fluid in which the particle is suspended. 3.1.2.7

A generalized expression for all forces

In the preceding discussions, a large number of forces, both external and internal to the separation system, have been identified and described briefly. Note that any force so identified was, for example, specific to molecules of the ith species.7 However, it is known that forces specific to the jth species can also affect the motion of molecules of the ith species. For the immediate objectives in the paragraphs that follow, these effects are ignored by assuming uncoupled conditions:8 molecules of species i in a stagnant fluid move only due to forces specific to the ith species; similarly for the jth species. It is further assumed that the conditions are not too far removed from equilibrium (see the introduction to Section 3.3 and Sections 3.3.1–3.3.6 for descriptions of equilibrium conditions); therefore thermodynamic quantities (defined only for equilibrium conditions) can be used to described nonequilibrium conditions where a net transport of molecules of species i exist due to external and internal forces. For illustrative purposes, an expression for the total driving force Fti on 1 gmol of species i or 1 gion of ion i in a solution or mixture due to a variety of forces identified earlier is given below (force is positive in the direction of the positive axis). Obviously, only one or a few of these forces exist at any time in a given separation system: F ti ¼ rμi jT þF ext ti þF Ti ; mag þF Ti ; F ti ¼ rμi jT þΣ rΦext i þF i F ti ¼ RT r ℓnai V i rP rμ0i þM i ω2 r rM i g kZ i F rΦ þZ i F ðvi BÞ

RT DTi rℓnT: Dij C i M i

ð3:1:50Þ

This expression9 does not include the effect of a nonuniform electrical field (see (3.1.12)) and the electrophoretic retardation. Note that buoyancy forces do not appear in Fti as such. Lee et al. (1977a) have tabulated the magnitudes of each of these forces, i.e. gradients, which can create a value of rℓnai jT;P equal to 1 cm1. A clarification on the forces acting on solute molecules in a solvent undergoing rigid body rotation in a centrifuge 7

Particles will be considered next. For an illustration of coupling, see Section 3.1.5.2. 9 For alternate representations, see Lee et al. (1977a). 8

3.1

Forces, displacements, velocities and fluxes

is useful here. The (non-Brownian) force exerted on the solute molecules is due not only to the ω2rr term, but also to the pressure gradient in the solvent developed by rotation. To determine this pressure gradient, consider only the rotation of solvent species i ¼ s in the centrifuge (no solute present). At equilibrium, there is no net force acting on solvent molecules in the centrifuge: F ts ¼ 0 ¼ V s

dP r þ M s ω2 r r: dr

ð3:1:51Þ

This provides an expression for (dP/dr) due to rotation. Assume now that this pressure gradient generated by solvent rotation is unaffected by the presence of solute species i in the rotating centrifuge. The net force10 on solute species i in the radial direction is then obtained as (excluding the concentration gradient contribution) dP r þ M i ω2 r r dr 0 10 1 # " Vi Ms 2 ¼ M i ω r 1  @ A@ A r; Mi Vs

F ti ¼ V i

ð3:1:52Þ

where ðV i =M i Þ is the partial specific volume of solute i and ðM s =V s Þ is essentially the solvent density in any region. The assumption that the solute species will not affect the pressure gradient in a centrifuge is not generally valid. Consider the total force on any ith species in such a case. At equilibrium, Fti is zero, leading to  dμi  F ti ¼ 0 ¼   r þ M i ω2 r r; dr  T  ð3:1:53Þ dP dμi  2 þ M i ω r ¼ 0:  V i  dr  T;P dr

From the principles of chemical thermodynamics (Guggenheim, 1967), at constant T and P the Gibbs–Duhem equation is  n X  ¼ 0: ð3:1:54Þ x i dμi  T;P i¼1 Multiply equation (3.1.53) by xi, sum over n species and use the Gibbs–Duhem equation (3.1.54) to obtain 

n X i¼1

x i V i dP þ

n X i¼1

x i M i ω2 r dr ¼ 0:

ð3:1:55Þ

Define an average partial molar volume V t and an average molecular weight of the solution Mt as follows: n X i¼1

xi V i ¼ V t

and

Rewrite equation (3.1.55) as 10

n X i¼1

xi M i ¼ M t :

ð3:1:56Þ

One can similarly develop (3.1.5) for species i in a solvent under gravity.

87 V t dP þ ðM t ω2 rÞdr ¼ 0:

ð3:1:57Þ

With solution density ρt defined as (M t =V t ), we can now simplify equation (3.1.53):    ð3:1:58Þ dμi  þ ðM i  ρt V i Þ ω2 r dr ¼ 0:  T; P

Note that ρt and V t will, in general, depend on radial location r since the pressure and composition vary with r. We now focus on macroscopic particles and provide an expression for the total external force acting on a particle of mass mp, density ρp, radius rp, charge Qp, velocity vp and volume (mp/ρp). We have not included here the Brownian motion force F Br, nor any force due to thermophoresis, radiation pressure, acoustic force and the electrical force in a nonuniform electrical field given by (3.1.13). Although not generated by an external force field, coulombic types of interactions, London dispersion and electrokinetic forces in the double layer are included in the expression given below: 0 1 0 1 ρ ρ t t ext 2 F tp ¼ mp g @1  Ak þ mp rω @1  Ar  Qp rΦ ρp ρp þ ðεd ζ r p  Qp ÞE

0 1 1 m @mp A χ p r ðH m  H m Þ þ Qp ½vp  B þ μ0 ρp 2 þ F ELS þ F Lret þ F ELK : ELS

Lret

ELK

ð3:1:59Þ

and F may be obtained, The expressions for F , F respectively, from (3.1.15), (3.1.16) and (3.1.17). Note that the pressure gradient generated in a centrifugal field has been replaced by means of equation (3.1.58) as a centrifugal buoyancy term in which V i has been replaced by particle volume ðmp =ρp Þ and Mi by mp to provide ð1  ½ ρt =ρp Þ. Example 3.1.1 Calculate the gravitational force exerted on a particle for the following two cases: (1) particle diameter 10 μm, particle density 2 g/cm3, fluid density 1.3 g/cm3; (2) particle diameter 2 cm, particle density 2 g/cm3, fluid density 1.3 g/cm3. Solution (1) From the force expression (3.1.59), the z-direction gravitational force on the particle is given by

4 F tz ¼  π r 3p ρp gð1  ðρt =ρp ÞÞ 3 ¼

4πð5  104 Þ3  2  980ð1  ð1:3=2ÞÞ g cm 3 s2

¼

4  π  125  1012  2  980  0:35 3

¼

4  π  1:25  2  0:98  0:35  107 g cm 3 s2

¼ 3:59  107

g cm ¼ 3:59  1012 newton: s2

88

Physicochemical basis for separation

(2)

F tz

¼

Z i F E ¼ 4:5  96 500  30 coulomb-volt=cm

4  π r 3p ρp g ð1  ðρt =ρp ÞÞ 3

¼

4π ð1Þ3  2  980ð1  ð1:3=2ÞÞ g cm 3 s2

8π  980  0:35 g cm ¼ 2873:5 2 ¼ 3 s ¼ 2:873  102 newton: Example 3.1.2 Calculate the centrifugal force exerted for the following two cases. Case (1) 1 gmol of ovalbumin molecules (molecular weight ¼ 45 000) in an aqueous solution of density 1 g/cm3 rotating in a centrifuge at 7000 radians/s at a radial coordinate of 1 cm, given ovalbumin density in solution ¼ 1.34 g/cm3. Case (2) a particle of diameter 10 μm, density 2 g/cm3 in a solution of density 1.3 g/cm3 at a radial distance of 1 cm. Solution Case (1) From the force expression (3.1.52), the magnitude of the radial force experienced by 1 gmol of ovalbumin molecules (species i) is given by

¼ 4:5  96 500  3000 coulomb-volt=m ¼ 1:3  109 coulomb-volt=m:

But 1 coulomb-volt ¼ 1 joule ¼ 1 newton-m. Therefore, ZiF E ¼ 1.3  109 newton, a force much stronger than the centrifugal force considered in Example 3.1.2, case (1). Example 3.1.4 Calculate the magnitude of the force due to radiation pressure on a lossless dielectric spherical particle of radius 0.5145 μm and density ρp ¼ 1g/cm3 subjected to a cw argon laser light of power 1 watt at a wavelength λ ¼ 0.5145 μm. Calculate also the instantaneous acceleration experienced by the particle. Solution From equation (3.1.47),

F rad ¼

Assume that qfr ¼ 0.1; velocity of light, c ¼ 2.799  1010 cm/s; Pr ¼ 1 watt. Then

F rad

0 1 0 1# "   Vi Ms  ¼ M i 1  @ A @ A ω2 r F tir  mole M V i

1

g cm ¼ 45000½1  0:74649  10 2 s gmol 6

¼ 560 070  106

g cm s2 gmol

¼ 5:60  106 newton=gmol: Case (2) From the generalized force expression (3.1.59), we get F tr jparticle experienced by the particle as follows: 0 1 0 1   ρ 4 ρ F tr particle ¼ mp rω2 @1  t A ¼ πr 3p ρp rω2 @1  t A ρp ρp 3 0 1 4π 1:3 3 2 ¼  ð5  104 Þ  2  1  ð7000Þ  @1  A 3 2 ¼

4π  125  2  49  0:35  106 g cm=s2 3

¼ 0:179  101 g cm=s2 : Example 3.1.3 Calculate the force exerted on ovalbumin molecules exposed to an electrical field of constant strength 30 volts/cm, given Zi ¼ 4.5 for ovalbumin at the solution pH. Neglect the electrical double-layer effect. Solution The electrical force per mole of ovalbumin is obtained from (3.1.8) in the absence of other information (e.g. zeta potential, ionic strength, etc.) It is given by

¼ ¼

s

0

# 45 000 g 1 A ð7000Þ2 @ 1 ð1Þ 1 cm ¼ gmol 1:34 s2 "

2 qfr Pr : c

2  0:1  1 watt 0:2  1010 watt-s ¼ : 10 2:799 cm 2:799  10 cm=s

0:2  1010  107 erg 0:2  103 g-cm2 ¼ 2:799 cm 2:799 cm s2

¼ 7:145  105

g-cm s2

¼ 7:145  105 dyne:

If the instantaneous acceleration is d2z/dt2, then Frad ¼ mp d2z/dt2. Now, mp ¼ ð4=3Þπ r 3p ρp , so

mp

¼

4 4π πð0:5145  104 Þ3  1 g ¼  0:1362  1012 3 3

¼

0:57  1012 g;

d2 z 7:145  105 dyne ¼ acceleration ¼ ¼ 1:25  108 cm=s2 : dt 2 0:57  1012 g The magnitude of this acceleration is quite high – much larger than that due to gravity (Ashkin, 1970).

Note that the following forces acting on the particle are proportional to the particle volume (and therefore r 3p ): gravity force; centrifugal force; dielectrophoretic force; magnetic force on a paramagnetic particle. The electrophoretic retardation force is proportional to the particle radius.

3.1.3 Particle velocity, molecular migration velocity and chemical species flux It is now appropriate to calculate the velocities of particles or molecules resulting from the forces acting on them. For clarity, we assume first that velocities created by bulk flow are zero. Let the particle velocity vector due to external forces be represented by Up. Similarly, let U i be the

3.1

Forces, displacements, velocities and fluxes

average velocity vector of the ith chemical species in any region due to external as well as internal forces. 3.1.3.1

Particle velocity and particle flux

We focus on particle motion first. If a macroscopic particle (mass mp) moves in the z-direction (positive z, vertically upward) under the action of forces acting in the z-direction, then, from the principles of mechanics (Newton’s second law),   d2 z magnitude of force on the ¼ mp 2 : ð3:1:60Þ particle in the z-direction dt If the particle was moving in free space, this force would equal the external force or forces we have identified earlier in magnitude and direction. If, however, the particle moves in a gaseous, liquid or solid (rarely) medium, the force on the particle consists of the external force, (or forces, including the buoyancy force) and a frictional force, F drag p , which opposes the particle motion (and thus has a negative sign). This frictional force comes into play as soon as the particle moves. It is known that, at small values of particle velocity Upz (in the direction of the positive z-coordinate), the frictional resistive force is linearly proportional to the magnitude of the particle velocity. For example, according to Stokes’ law, the resistive or drag force vector on a spherical particle of radius rp moving at a velocity Upzk through a fluid of viscosity μ is 6π μ r p U pz k. If this particle is falling under gravity in a fluid of density ρt and viscosity μ, then ! d2 z ρt dz drag  6 πμr p  F ¼ m g 1  ; mp 2 ¼ F ext p pz pz ρp dt dt ð3:1:61Þ where Upz ¼ dz/dt. Due to a nonzero acceleration (d2z/dt2 6¼ 0), the velocity of the particle increases; the frictional resistive force (the drag force) also increases. In many systems, after some time the particle acceleration in the stagnant fluid becomes zero and the particle velocity becomes constant. This velocity is called the terminal velocity. We express the resistive force vector on the particle as f dp U p ; when Up is the terminal velocity, for the z-component, d F ext pz ¼ f p U pz ) U pz ¼

F ext pz f dp

F ext p f dp

;

Upt, for example in the z-direction as Upzt. (See Section 3.1.6.2 for corresponding quantities for ions in a gas phase.) Suppose the fluid medium is not stagnant but has a mass average velocity vt. If Stokes’ law still determines the resistive force, then the resistive or frictional (or drag) force on the spherical particle is to be determined using the velocity of the particle relative to the fluid: F drag ¼ 6πμr p ðU p  vt Þ: p

:

ð3:1:62Þ

where f dp is the frictional coefficient of the spherical particle moving slowly in a stagnant medium of viscosity μ. For a particle resistive force described by Stokes’ law, f dp is equal to 6πμr p . Often, this terminal velocity will be identified as

ð3:1:63Þ

These relations based on Stokes’ law are valid only for small particles whose particle Reynolds number, defined by ð2r p U p ρ=μÞ, is very small ( b),

Table 3.1.2. Estimation of Perrin factor for various proteins

Protein Bovine serum albumin Ovalbumin α-Lactalbumin Myoglobin

ð3:1:91dÞ

ð3:1:91eÞ

Molecular weight

pI

r0, radius, (nm)

f di =f di0

66 000

5.74

3.5

1.29

45 000 14 200 16 900

5.08 4.57 7.1

2.78 2.3 2.4

1.16 1.18 1.18

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðb2 =a2 Þ f di  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ 1þ 1ðb2 =a2 Þ f di0 b2=3 ℓn a b=a

ð3:1:91fÞ

where the volume of the prolate spheroid is equal to the volume of a sphere of radius r0, 4 4 π a b2 ¼ π r 30 : 3 3

ð3:1:91gÞ

For oblate ellipsoids (semiaxes a < b), correspondingly r 30 ¼ a2 b and f di ¼ f di0 a1=3 b

Estimates of ri, the hydrodynamic radius, may be obtained from expressions (3.3.90a–c). (The basic assumption here is that Stokes’ law holds reasonably well for the motion of solute molecules also. In fact, it is routinely used for proteins and macromolecules (Tanford, 1961).) Therefore for a given magnitude of the driving force, as solute size increases, the displacement or migration velocity Uiz decreases. If the solute shape is nonspherical, a relation other than Stokes’ law will apply. For the determination of the resistance of nonspherical macromolecules, the reader may consult pp. 356–364 of Tanford (1961). We will provide a very brief perspective on this effect here. Some cells, and especially many proteins, are ellipsoids of revolution. The drag force encountered by such an ellipsoid of revolution of species i is described in terms of the drag encountered by a sphere of equal volume whose radius is r0 via an appropriate correction factor ðf di =f di0 Þ which is always greater than 1. This factor is called the Perrin factor. The drag magnitude of F i is enhanced by this factor, i.e. Fi

93

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ða2 =b2 Þ #: "  1=2 2 b tan1 1  ab2 a

ð3:1:91hÞ

For a brief introduction, see Probstein (1989) for references to the work by Perrin; for greater details, see Happel and Brenner (1983). In Table 3.1.2, we list estimates of the Perrin factor for several proteins (information from Basak and Ladisch (1995)). The increase in the frictional coefficient is natural since the surface area for the spheroid is larger than that of a sphere of equal volume. In any given medium, when the solute size increases, the frictional coefficient will increase. For a given solute, as the medium viscosity increases, f di will increase, leading to a decrease in diffusion coefficients and therefore in the displacement velocity. For linear flexible macromolecules, especially charged ones such as polyelectrolytes, estimation of the drag force is complicated; see Viovy (2000) for an introduction. Note that Ui for a polyelectrolyte that is uniformly charged is likely to be independent of its length in free solution; the situation is different in gels. Example 3.1.5 Case (1) Calculate the displacement or migration velocity of ovalbumin molecules subjected to the conditions in a centrifuge described in Example 3.1.2, case (1), where the solution viscosity is 1 cp. Case (2) Calculate the terminal settling velocity of the particles in Example 3.1.2, case (2), where the solution viscosity in 1.5 cp. You are given that for ovalbumin r i ¼ 2:78 nm and ðf di =f di0 Þ ¼ 1:16.

94

Physicochemical basis for separation

Solution Case (1) If we assume Stokes’ law to be valid, then the radial displacement/migration velocity of ovalbumin molecules is given by

U ir ¼ ðF tir=gmol Þ=f dijgmol ¼

5:60  1011 g-cm=s2 ~ ð f di =f di0 Þ 6 π μ rp N

g-cm 5:60  1011 2 s ¼ g  27:8  108 cm 6 π  1  102 cm-s 1:16  6:02  1023 11

¼

13

5:60  10  10 cm ¼ 0:151  104 cm=s: 36:12  π  32:48 s

This migration velocity appears to be much larger than that generated in the centrifugal field of Example 3.1.5, case (1). Example 3.1.7 Determine the gravitational terminal settling velocity of particles in the following two systems. (1) Particle of diameter 10 μm, density 2 g/cm3 in a solution of density 1.3 g/cm3 and viscosity 1.5 cp. (2) E. coli cells of diameter 1 μm, particle density 1.09 g/cm3, solution density 1 g/cm3 and viscosity 1.5 cp. Solution Case (1)

U pzt ¼ 

Check now the validity of Stokes’ law:

¼

2 r 2p g ρp ð1  ð1:3=2ÞÞ cm s 9  1:5  102

¼

2  ð5  104 Þ2  980  2  0:35 13:5  102

¼

100  980  0:35  108  102 cm s 13:5

U ir r i ρ Reynolds no: ¼ μ ¼

g 8 0:151  104 cm s  27:8  10 cm  1 cm3 10

¼ 4:19  10

1  102 ðg=cm sÞ 1, (λ/rp) is the Knudsen number) the particle appears to other gas molecules (which are far apart) as if it were a gas molecule. On the other hand, when (λ/rp) 0; as molecules of the solute species move in the positive z-direction, the concentration of the solute remains uniform in the xyplane (at all z). How the concentration pulse of a given solute species moves as a function of z and t is best studied using the solution of the species conservation equation to be developed now. The species conservation equation can be obtained by a simple mass balance of solute i in a small volume element of cross-sectional area lxly and thickness Δz in Figure 3.2.1 at any z for 0 z l z . For species i ¼ 1, 

     rate of Species 1 rate of Species 1 rate of Species 1  ¼ going out coming in accumulation :   ∂C 1 ðl x ly ΔzÞ ¼ N 1z z lx ly  N 1z zþΔz l x l y ∂t

ð3:2:1Þ

Dividing both sides by lxly Δz and taking the limit Δz ! 0, we get

120

Physicochemical basis for separation

Uniform continuous potential profile

Discontinuous potential profile

(fi∗)max Uniform continuous Discontinuous

fi∗

z

0

lz

Figure 3.2.2. Profiles of ϕ i analyzed. (After Giddings (1978).)

∂C 1 ∂N 1z ¼ : ∂t ∂z

ð3:2:2Þ

Assume now that the movement of solute species 1 in the solvent does not affect the movement of solute species 2 and vice versa. Following the derivation of the species balance equation (3.2.2) for i ¼ 1, we may obtain a similar species conservation equation for i ¼ 2: ∂C 2 ∂N 2z ¼ : ∂t ∂z

ð3:2:3Þ

The solution of these two partial differential equations, subject to appropriate boundary and initial conditions, will yield the concentration profiles for solutes i =1,2 as a function of z and t. How each of these two concentration profiles develops along the z-coordinate with time is intimately connected with the nature of the species-specific forces, since the latter determine Niz. Recall from expressions (3.1.27) and (3.1.42), at constant temperature,  X ext F ti ¼ F ext ϕ þ μ  rμ ¼ r i i i ti ¼

rðϕext ti

þ

μ0i Þ

 RTr ℓn ai ¼

rϕtot i ;

where ext 0 ϕtot i ¼ ϕti þ μi þ RT ℓn ai

ð3:2:4Þ μ0i

is the total potential acting on species i (note: includes here any pressure gradient effects also; otherwise employ the expression from (3.1.84d). Further, the rϕext reprei sentation for any external force is insufficient for magnetic and some other forces). Rewrite ϕtot i as ϕtot i ¼ ϕi þ RTℓn ai ;

where 0 ϕ i ¼ ϕext ti þ μi ;

ϕext ti ¼

X

ϕext i :

ð3:2:5aÞ ð3:2:5bÞ

We will find out later that the nature of the profile ϕ i in the z-direction along the separator is crucial. It can be either continuous or discontinuous, with all sorts of variation with z in either category. We consider now a continuous profile of ϕ i , specifically a uniformly continuous profile of ϕ i (the discontinuous profile is studied in Section 3.2.2). In Figure 3.2.2, the uniform continuous profile of ϕ i is represented as a straight line varying from ðϕ i Þmax at z ¼ 0 to 0 at z ¼ lz. Two simple examples of such a profile are: a uniform electrical field of electrostatic potential; a gravitational potential along the vertical axis. A nondimensional representation ϕþ i ¼ ϕi =ðϕi Þmax

ð3:2:6Þ

will have a maximum value of 1 at z ¼ 0 and a minimum value of 0 at z ¼ lz. Since both ϕþ i and ϕi are functions þ of the z-coordinate only, we may write them as ϕþ i ðz Þ, where z þ ¼ z=l z :

ð3:2:7Þ

Substitute now the expression (3.1.81) for N 1z in the differential equation (3.2.2), under the conditions of v tz ¼ 0 and an ideal solution, and obtain for species 1 in solvent s the following: # " ∂C 1 1 ∂ ∂μ01 ∂ϕext RT ∂2 C 1 t1 ¼ d þ C1 : þ d C1 ∂z ∂z ∂t ∂z f1 f 1 ∂z2 Therefore # " ∂C 1 1 ∂ ∂ϕ 1 RT ∂2 C 1 þ d ¼ d ; C1 ∂t ∂z f 1 ∂z f 1 ∂z 2 where ðRT=f d1 Þ ¼ D01s .

ð3:2:8Þ

3.2

Separation development and multicomponent separation Nondimensionalize ð∂ϕ 1 =∂zÞ and the time t by ∂ϕ 1 ∂z

¼

ðϕ 1 Þmax lz

∂ϕþ 1 ∂zþ

¼

ðϕ 1 Þmax lz

0 þ ðϕþ 1 Þ ; t1 ¼

tðϕ 1 Þmax l 2z f d1

i.e. at time t ¼ 0, all of the solute species 1 (m1 moles) is contained in a very thin slab of cross-sectional area lxly at zþ ¼ 0. An alternative way of looking at this is as follows:

;

ð∞

ð3:2:9Þ 0 where we note that ðϕþ 1 Þ is a constant for the uniform continuous ϕ1 profile (Figure 3.2.2). Equation (3.2.8) may now be rewritten as

∂C 1 RT ∂2 C 1 þ 0 ∂C 1 ¼ ; þ  ðϕ1 Þ þ ðϕ1 Þmax ∂zþ2 ∂z ∂t 1

ð3:2:10Þ

where C1 depends on the independent variables zþ and t þ 1. To simplify equation (3.2.10) further, define a new set of independent variables η and t þ 1 , where þ

η¼z þ

0 þ ðϕþ 1 Þ t1 :

ð3:2:11Þ

In these two new independent variables, equation (3.2.10) is reduced to17 ∂C 1 RT ∂2 C 1 : ¼ 2 ðϕ ∂t þ 1 1 Þmax ∂η

ð3:2:12Þ

This equation is far simpler than the earlier ones; in fact, it is essentially the diffusion equation. We now need the solution of this equation for the given initial condition and suitable boundary conditions. Although the z-dimension of our separator is finite (= lz), we may assume that lz is sufficiently large for the values of t in our range of interest. Then the solution for z ! ∞, i.e. η ! ∞ will be usable for our purpose. Obviously, at all t þ 1, for zþ ¼ ∞ or η ¼ ∞; C 1 ¼ 0 and

∂C 1 ∂C 1 ¼0¼ ; ∂z þ ∂η ð3:2:14aÞ

0

ð∞ C 1 l x l y dz þ ¼ m1 δðzþ Þdzþ ¼ m1 :

The initial condition is:

ð3:2:14bÞ

0

ð3:2:15Þ

þ

is applicable here provided the z -coordinate varies between þ ∞ and  ∞. To determine the constant A, note that solute species 1 is conserved. Therefore, the total number of moles of species 1 in a separator of dimensions x ¼ 0, lx; y ¼ 0, ly; z ¼  ∞ to þ ∞ must be the sum of the number of moles in two separators, one from z ¼ 0 to þ ∞ and the other from z ¼  ∞ to 0. However, the latter reservoir does not have any solute in this problem. Therefore 3 2 m1 þ 0 ¼ m1

¼

    ∂C 1 ∂C 1 dt þ dη: 1 þ þ ∂η t þ ∂t 1 η

Therefore

ηþ ¼

1



tþ 1

,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4RTt þ 1 : ðϕ 1 Þmax

ð3:2:17Þ

Since dzþ ¼ dη, we get rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð∞ 4RT exp½  ðηþ Þ2  dηþ ðϕ 1 Þmax ∞

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð∞ RT ¼ Al x l y l z 4 exp½  ðηþ Þ2  dηþ : ðϕ 1 Þmax 0

1

ð3:1:13bÞ

Further, ∂C 1 ∂zþ 1

η

Therefore

          ∂C 1 ∂C 1 ∂C 1 ∂η ∂C 1 0 ∂C 1 þ þ ðϕþ ¼ : 1Þ þ þ þ þ ¼ ∂η t þ ∂t 1 ∂η t þ ∂t 1 zþ ∂t 1 η ∂t 1 η



7 6 2 7 6 1 6 η 7dzþ : pffiffiffiffiffi exp þ7 6 þ 4RTt t1 4 1 5 ∞ ðϕ 1 Þmax ð∞

Define

ð3:1:13aÞ

1

∞ 0 0

7 6 7 6 A η2 7dx dy dz pffiffiffiffiffi exp 6  7 6 þ 4RT þ t1 5 4 t ðϕ 1 Þmax 1 3 2

ð3:2:16Þ

þ þ For C 1 ¼ C 1 ðt þ 1 , z Þ ¼ C 1 ðt 1 , ηÞ,

dC 1 ¼

ð∞ ðly ðlx

¼ Al x l y lz

m1 ¼ Al x l y l z 17

ð3:2:14cÞ

A well-known solution of the diffusion equation (Carslaw and Jaeger, 1959),18 " # A η2 þ þ þ C 1 ðz, tÞ ¼ C 1 ðz ;t 1 Þ ¼ C 1 ðη;t 1 Þ ¼ pffiffiffiffiffi exp  4RT þ ; tþ ðϕ 1 Þmax t 1 1

i.e. the value of C1 as well as its gradient in z will be zero far away from the source of C1 at z ¼ 0. at t þ ¼ 0; C1 ¼ ðm1 =l x l y Þδðzþ Þ;

121

¼



∂C 1 ∂η



tþ 1

)

 2   2  ∂ C1 ∂ C1 ¼ : þ2 ∂z ∂η2 t þ tþ 1

1

ð3:1:13cÞ

18

The solution of the heat conduction equation given in sect. 10.3.II of this reference for an instantaneous plane source of strength Q parallel to the plane z = 0 can be used to solve the particular diffusion problem. The reader can verify it by substituting solution (3.2.15) into equation (3.2.12) and seeing that it is satisfied.

122

Physicochemical basis for separation

Species 2 Species 1

z2+

or

Ci

+

z1

z+

0

Figure 3.2.3. Concentration profiles of species 1 and 2 at any time t.

m1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:2:18Þ RT π 2l x l y lz ðϕ1 Þmax pffiffiffi since the value of the integral is π=2. The concentration profile for species 1 is therefore given by 2 n o2 3 þ 0 þ   þ þ ðϕ Þ t z 7 6 1 1 m1 1 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 6 C 1 ðzþ ;t þ 1Þ ¼ 4 RT 5: þ lx ly lz 4πRTt 1 tþ 4 1 ðϕ1 Þmax ðϕ 1 Þmax A¼

ð3:2:19Þ The concentration profile for species 2 may be similarly obtained as follows: 2 n o2 3 þ 0 þ   þ z þ ðϕ2 Þ t 2 7 6 m2 1 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 6 C 2 ðzþ ;t þ 2Þ ¼ 4 RT 5; þ lx ly lz þ 4πRTt 2 4 t ðϕ 2 Þmax 2 ðϕ 2 Þmax ð3:2:20Þ where tþ 2 ¼

tðϕ 2 Þmax : l 2z f d2

Here þ þ C max 1 ðz ;t 1 Þ ¼



m1 lx ly lz

ffi  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðϕþ 1 Þmax ; 4 π RT t þ 1

þ þ similarly for C max 2 ðz ;t 2 Þ: Figure 3.2.3 shows the instantaneous concentration profiles for solutes 1 and 2 at any time t (only the zþ-coordinate is shown for simplicity). The concentration profile of species i (i ¼ 1,2) is located around a þ 0 þ zþ i ¼ ðϕi Þ t i and has its maxima there. The profiles are Gaussian with a standard deviation σ þ i (nondimensional σi), where

o1=2 n þ σþ ¼ σ i =lz i ¼ 2RTt i =ðϕi Þmax

ð3:2:21aÞ

n o1=2 2 0 ; ðσ i =l z Þ ¼ σ þ i ¼ 2Dis t=l z

ð3:2:21bÞ

since ðRT=f di Þ ¼ D0is . Therefore, the larger the value of D0is or the longer the time allowed for separation, the higher will be the value of σi as the species i concentration profile moves along the z-axis. These results have several important implications. Consider first the instantaneous location of the center point of each of the two concentration profiles where C max ðzþ ; t þ i Þ occurs. i Species 1:   tðϕ 1 Þmax l z ∂ϕ 1 þ þ 0 zþ : 1 ¼ t 1 ðϕ1 Þ ¼  2 d l z f 1 ðϕ1 Þmax ∂z But ∂ϕ 1 ∂ϕext ∂μ0 ¼ t1 þ 1 ¼ f d1 U 1z ; ∂z ∂z ∂z so that zþ 1 ¼

tU 1z : lz

ð3:2:22aÞ

zþ 2 ¼

tU 2z : lz

ð3:2:22bÞ

Species 2: Similarly,

Note that U1z and U2z are the z-components of the displacement or migration velocities of species 1 and 2, respectively. Thus, as long as U 1z 6¼ U 2z , the two concentration profiles (of 1 and 2) will have different locations of their centers. Although at t ¼ 0, species 1 and 2 were present in a uniform mixture at z ¼ 0 (and x ¼ 0, lx; y ¼ 0, ly), different values of U1z and U2z have created a condition whereby species 1 and 2 are now spatially separated in the solvent present in the separator. From (3.2.21a,b) for σi, it is clear that, as the diffusion coefficient D0is of solute species i increases, the concentration profile becomes broader at any time t. Therefore, the larger the value of D0is , the greater the extent of overlap between two contiguous solute profiles. Thus, an increased value of D0is leads to reduced separation. Although molecular diffusion due to a concentration gradient could be interpreted in terms of a concentration driving force that is part of the overall driving force Fti , here it acts to reduce separation in the presence of external forces. Therefore it is useful to subtract it from Fti in developing an expression for displacement or migration velocity Ui of species i (as was done in definition (3.1.82)). The effect of time on the separation achieved is an additional feature that can be investigated. First, the

3.2

Separation development and multicomponent separation

zþ-coordinate of the center of mass of the concentration profile of the ith species, zþ i , increases linearly with time (see the developments after equation (3.2.25)). The distance between the centers of two contiguous profiles, þ i.e. z þ 1  z 2 , for species 1 and 2 therefore increases as ½tðU 1z  U 2z Þ=lz . Second, the standard deviation σi of each profile increases as t 1=2 . Thus, as time progresses, any given solute species disperses over a wider region of space around the center of the profile. This feature is often called band broadening or dispersion. The net effect of time on the separation between species 1 and 2 can be determined by using an index, e.g. the resolution Rs, where Rs ðtÞ ¼

þ 2ðz þ 2ðz 1  z2 Þ t 1  z2 Þ / 1=2 ¼ t 1=2 ¼ þ þ σ Þ 4ðσ Þ 4ðσ 1 þ σ þ t 1 2 2

ð3:2:23Þ

Thus separation between species 1 and 2 improves with increasing time, although there is increased dispersion in each profile. With time, then, we have better separation but increased dilution in each product region. Obviously, the finite boundaries of the closed separator impose a limit on the time during which such a behavior is feasible with a uniform continuous ϕ i potential profile. In practice, open separators are used and additional factors come into play. We have studied here the one-dimensional migration of solutes introduced as a δ-function at one end of the vessel. Lee et al. (1977a) have illustrated the threedimensional migration of a compact solute pulse. If a certain state of separation of two solutes or the development of an individual solute profile is under consideration, the time lapsed, as we have seen above, is quite important. This time will depend amongst other things on the solvent used. For example, to arrive at a given nondimensional position in the separator with two different solvents s1 and s2, the times t s1 and t s2 required for a given 0 ðϕþ i Þ value may be obtained from þ tþ s1 ¼ t s 2 )

ðl 2z Þs1 ð f di Þs1 ½ðϕ i Þmax s2

t s1 ¼ t s2 ðl 2z Þs ð f di Þs ½ðϕ i Þmax s 2 2 1

ð3:2:24Þ

for any solute i. Obviously, the frictional coefficient of solute i in the solvent and therefore solvent viscosity will be important for estimating the time required for achieving a specified state of solute profile development. The previous conclusions on separation development were based on two primary characteristics of any solute profile: (1) the center point of each concentration profile is þ 0 þ given by z þ i ¼ ðϕi Þ t i ; (2) the nondimensional standard þ deviation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ i of the Gaussian profile is equal to ð2D0is t=l 2z Þ. These characteristics of the concentration profiles (3.2.19) and (3.2.20) need to be justified. To justify our locating of the center point of each conþ 0 þ centration profile at zþ i ¼ ðϕi Þ t i , take the first moment of the concentration profile (see the definition in (2.4.1g)) with respect to the variable ηi defined from (3.2.11) as

123 þ 0 þ ηi ¼ z þ i þ ðϕi Þ t i :

ð3:2:25Þ

Then

hηi i ¼

ð∞

ηC i ðηi ;t þ i Þdηi

0 ð∞ 0

;

ð3:2:26Þ

C i ðηi ;t þ i Þdηi 0

ð∞

1

η2i

C B ηi ðmi =lx l y l z Þ C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expB @  RT  Adηi =ðϕ Þ 4πRTt þ þ i i max 4 t 0 ðϕ i Þmax i 0 1 : hηi i ¼ ð∞ 2 B C ðmi =l x ly l z Þ ηi C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expB @  RT  Adηi 4πRTt þ =ðϕ Þ þ i i max 4 t 0 ðϕ i Þmax i

ð3:2:27Þ

tþ i ,

the numerator has a zero value. Since hηi i At any given should provide the center of mass of the species i concentration profile, þ 0 þ hηi i ¼ 0 ¼ hzþ i þ ðϕi Þ t i i ¼ 0

)

zþ i

¼

0 þ ðϕþ i Þ ti :

ð3:2:28aÞ ð3:2:28bÞ

To determine the standard deviation of the concentration profile, we take the second moment of the concentration profile with respect to the variable ηi, since 2 ðσ þ i Þ

hη2i i

¼ hðηi  hηi iÞ2 i ¼ hη2i i with hηi i ¼ 0; ð∞ η2i C i ðηi ;t þ i Þdηi

¼

0

ð∞ 0

:

ð3:2:29Þ

C i ðηi ;t þ i Þdηi

Now, the integral in the numerator, 0

ð∞

1

C B η2i η2i

C ðmi =lx l y l z Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expB  Adηi ; @ þ RT 4πRTt i =ðϕi Þmax 4 tþ 0 i ðϕi Þmax

can be changed to the integral below if we define ηþ i ¼ (

ηi

RTt þ i 2 ðϕ i Þmax

)1=2 :

ð∞   ðmi =l x l y l z Þ 4RTt þ 2 þ 2 þ i pffiffiffi ðηþ i Þ exp ðηi Þ dηi ðϕi Þmax π 0 0 1 þ pffiffiffi π mi A 4RTt i pffiffiffi ¼@ lx ly lz πðϕi Þmax 4

124

Physicochemical basis for separation

at any t þ i . The value of the integral in the denominator of (3.2.29) is 

mi lx ly lz

ð∞ 0

!

1 η2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  RTi þ dηi : 4 4πRTt þ =ðϕ Þ ðϕi Þmax t i i i max

This is considered the zeroth moment of the concentration profile Ci. Using ηþ i , it can be simplified to 

   pffiffiffi ð∞ mi 1 mi π þ pffiffiffi expðηþ2 pffiffiffi : Þdη ¼ i i lx ly lz lx ly lz 2 π π 0

Therefore ðσ þ i Þ

2

ðmi =l x l y l z ÞfRTt þ 2RTt þ i =ðϕi Þmax g ¼ i 1 ðϕi Þmax ðmi =l x l y l z Þ 2

¼

hη2i i ¼

¼

2RTt þ 2RTtðϕ Þ 2D0 t i ¼ 2 d i max ¼ 2is ðϕ i Þmax lz lz f i ðϕi Þmax

at any time t. Example 3.2.1 A pulse of ovalbumin molecules is introduced at the xy-plane at z ¼ 0 at t ¼ 0 in a uniform electrical field (in the z-direction) of strength 30 volt/cm (see Figure 3.2.1). The length of the vessel lz ¼ 10 cm; the solution viscosity is 1 cp; the molecular weight of ovalbumin is 45 000; ri ¼ 2.78 nm; ( fi/fiw) ¼ 1.16 (see Examples 3.1.2, 3.1.5 and 3.1.6). Calculate (1) the time-dependent location of the center point of the ovalbumin concentration profile as it moves along the z-coordinate; (2) the standard deviation of this profile (D0iw ¼ 7:76 107 cm2 =s; see Table 3.A.5), given T ¼ 20  C. (3) How long would it take for the profile center point to reach the end of the vessel? Solution (1) From the solution of Example 3.1.6, the migration velocity of ovalbumin molecules under the above-mentioned conditions is 0.35  102 cm/s. From (3.2.22a),

U iz t ) zi ¼ U iz t ¼ 0:35  102 t cm: lz

(2) The dimensional standard deviation of the profile around the center point z þ i is equal to (from (3.2.21b)) 0 1=2 σþ =l z   l z ¼ ð2D0is tÞ1=2 i  l z ¼ ½ð2Dis tÞ 7 1=2

¼ σ i ¼ ð2  7:76  10 Þ ¼ 1:245  103 t 1=2 cm:

t

1=2

(3) z i ¼ l z ¼ 10 ¼ 0:35  102 t ) t ¼

103 cm ¼ 0:066 cm:

3.2.2

Multicomponent separation capability

What kind of separation processes or forces causing separation has the capacity to achieve multicomponent separation in a single separation vessel? To answer this question, consider the solute concentration profiles (3.2.19) and (3.2.20) obtained earlier. If we assume that the solute mixture pulse at z ¼ 0 and t ¼ 0 contained more than two solute species, and if the movement of each species does not influence those of any other species, then 2 n o2 3 þ 0 þ þ þ ðϕ Þ t z i i ðmi =l x l y l z Þ 6 7 o n ffiffiffiffiffiffiffiffiffiffiffiþffi exp4 C i ðz þ ;t þ 5 i Þ ¼ q4πRTt þ RT i t 4 i ðϕ Þ ðϕ Þ i max

i max

ð3:2:31Þ

ð3:2:30Þ

zþ i ¼

1:245  103  ð2857Þ1=2 cm ¼ 1:245  53:45

103 s; 0:35

t ¼ 2857 s ¼ 47.6 minutes. At this time, the standard deviation will be

for i ¼ 1,2. . .,n. Obviously if each solute species i has a unique value of Uiz, the center point of the instantaneous concentration profile for each solute will be at different locations in the separator. Further, if the standard deviations of the profiles are small compared to the distance between the center points of two neighboring concentration profiles, i.e. the zþ i , we have succeeded in separating a multicomponent mixture at any given instant of time. For practical separations, one has to isolate physically the solvent in each region characteristic of a given species at any instant of time. Regardless of the practical difficulties in achieving this, it can be concluded that uniform continuous profiles of ϕ i , where i ¼ 1,. . .,n, are inherently capable of separating an n-component solute mixture in a closed system with no convection since each i has a separate migration velocity Ui. The above analysis was carried out for a constant value 0 of ðϕþ i Þ found in systems with a uniform continuous ϕi þ 0 profile. As long as ðϕi Þ varies continuously along the separator and has a nonzero value, the multicomponent separation ability is retained. We will learn soon that multicomponent separation ability is absent in separation 0 systems where ðϕþ i Þ ¼ 0 and there is no bulk velocity. What is the maximum number of components that can be separated in a closed separator of length lz having uniform continuous profiles of ϕ i ? This number, called the peak capacity, is defined by nmax ¼

lz ; 4σ

ð3:2:32Þ

where we have assumed an averaged value of standard deviations for all profiles and indicated it by σ. The quantity nmax is thus the maximum number of Gaussian solute concentration profiles that can exist in the separator of length lz with contiguous profiles being separated, as shown in Figure 3.2.4, by a distance of 4σ (thus Rs ¼ 1 between each neighboring peak). To calculate σ, choose a

3.2

Separation development and multicomponent separation

4s

4s

125

4s

Ci

z

lz Figure 3.2.4. Maximum number of species separable in a separator of length lz.

particular ith species and use19 its ðϕ i Þmax and f di . Remember, however, that the expression for σi indicated earlier by (3.2.21a/b) is nondimensionalized by lz since η or zþ is nondimensionalized with respect to lz. Therefore nmax ¼

lz lz ¼ ( )1=2 : 4σ i 2RT tðϕ i Þmax 4 lz ðϕ i Þmax l2z f di

nmax ¼

for uniform continuous ϕ i profiles. This result based on definition (3.2.32) assumes that an average value of 4σ may be defined. However, in practical situations, when long separation length or times are involved, 4σ will not be a constant. Suppose one is interested in finding out the maxmum number of components that can be separated in a given time window Δt at any given location in a column/vessel. Since the basic migration of peaks remains unchanged, nmax may be defined as Δt 4σ

ð3:2:33bÞ

for Rs ¼ 1. Now 4σ is not a constant for different species/ peaks. Expressing 4σ as the base width of the peak, Wb, we can rewrite the expression for nmax: nmax ¼

Δt : Wb

dn ¼

dt : Wb

ð3:2:33dÞ

Integrating this over the time difference Δt ¼ t 2  t 1 , we get

For an estimate of the characteristic time t, use t ¼ ðl z =U iz Þ,where Uiz is the migration velocity of species i and is given by ð∂ϕ i =∂zÞ=f di . With these, and ð∂ϕ i =∂zÞ ¼ ðϕ i Þmax =l z , we obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðϕ i Þmax ð3:2:33aÞ nmax ¼ ( )1=2 ¼ 32RT d 2RT l z f i 4 2 d ðÞ l z f i ∂ϕi =∂z

nmax ¼

In a differential form, this expression may be written as (Giddings, 1969)

ð3:2:33cÞ

nð max 0

ðt 2

dn ¼ ðdt=W b Þ: t1

ð3:2:33eÞ

For a homologous series of compounds, Medina et al. (2001) have proposed that Wb may be related to the time t when the peak appears by W b ¼ a1 ða2 t þ a3 Þ;

ð3:2:33fÞ

where a1, a2, a3 are empirical constants. Substituting into (3.2.33e), we get nmax ¼

ðt 2

t1

dt 1 a2 t 2 þ a3 ¼ ℓn : a2 t 1 þ a3 a1 ða2 t þ a3 Þ a1 a2

ð3:2:33gÞ

Example 3.2.2 A pulse of ovalbumin molecules (molecular weight, 45 000) in a mixture of a variety of proteins is exposed to an aqueous solution (density, 1 g/cm3) subjected to a steady electrical field of strength 30 volt/cm. Given Zi ¼ 4.5, develop an estimate of the peak capacity when the electrodes are separated by a distance of 1 cm and the potential difference is 30 volt; ovalbumin may be used as the representative species. Solution From equation (3.2.33a), the expression for peak capacity nmax for Rs ¼ 1.0 is

Here,

 1=2 nmax ¼ ðϕ i Þmax =32RT : ðϕ i Þmax ¼ ðϕext i Þmax ¼ Z i ϕmax F ;

where ϕmax is the electrical potential, 30 volt. Therefore 19

As a representive value for obtaining σ.

Z i F ϕmax ¼ 4:5  96500  30 volt-coulomb=gmol;

126

Physicochemical basis for separation

R ¼ 8.317 joule/gmol-K. Assume the temperature to be 25  C, T ¼ 298 K. We have 0 11=2 joule jouleA @ nmax ¼ 4:5  96500  30 =32  8:317  298 gmol gmol ¼ ð164Þ1=2 ¼ 12:8:

1

2 +

About 12 peaks may be separated. Compare this with the number of peaks in the centrifugal separation of Problem 3.2.1(c).

Ci (z, t i ) 3

Example 3.2.3 The definition of peak capacity (3.2.32) was developed for the case of Rs ¼ 1 where the two neighboring peaks were separated by the distance 4σ (where σ is an assumed average value of the standard deviations of all profiles). If poorer separation is acceptable, i.e. Rs < 1, what is an estimate of the peak capacity in a given system?

Figure 3.2.5A. Concentration profiles of species 1, 2 and 3 at any time t for a discontinuous ϕ i profile in the vessel 0 z lz.

Solution From the definition of resolution,

Rs ¼

þ zþ 1  z2 : þ 2ðσ 1 þ σ þ 2Þ

þ þ þ Assuming ðσ þ 1 þ σ 2 Þ ¼ 2σ , where σ is an averaged standþ þ ard deviation, Rs ¼ ðzþ 1  z 2 Þ=4σ . Therefore for any value of þ þ Rs less than but close to 1, ðzþ 1  z 2 Þ ffi Rs 4σ . We can then write

nmax

¼

lz þ ðzþ 1  z 2 Þl z

)

nmax ¼

1

1 ; Rs 4σ þ

1 max

The maximum value, as well as the center of mass of this profile, is located at zþ¼0 (try calculating hz þ 1 i using the approach of (3.2.26) with z1 instead of η1). The nondimensional standard deviation of this profile will be given, as 2 1=2 0 before, by σ þ . In Figure 3.2.5A, the profile 1 ¼ ð2tD1s =l z Þ of C1 is shown at any time t.

lz : Rs 4σ

We will now study the multicomponent separation capability of a discontinuous ϕ i profile (shown in Figure 3.2.2 by the dashed line). The particular discontinuous profile we have chosen has a maximum value of ðϕ i Þmax at all values of 0 z l z , but at z ¼ lz, ϕ i abruptly drops to the value of zero and stays at that level beyond lz. It is thus a step function. Such a profile can be obtained in practice ext by having ϕext ti ¼ 0 or F ti ¼ 0 and maintaining two different phases or solvents in the two regions: 0 z l z and l z z 2l z (say). Recognize that, for all z < l z , the value of 0 ðϕþ i Þ (equation (3.2.9)) is zero. The governing equation for the concentration distribution of species 1 is then obtained from equation (3.2.10) as the simpler ∂C 1 RT ∂2 C 1 : ¼ þ2 ðϕ ∂t þ 1 i Þmax ∂z

thin slab of cross-sectional area lxly at z ¼ 0 for t ¼ 0. The solution is obtained as a special case of solution (3.2.19) as ! ðm1 =l x ly lz Þ z þ2 ð3:2:35Þ C 1 ðzþ ;t þ Þ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  RT þ : 4 ðϕ Þ t 1 4πRTt þ ðϕ 1 Þmax

where σþis nondimensional. For dimensional standard deviation σ, nmax ¼

z lz

ð3:2:34Þ

Note that this is merely a special case of equation (3.2.12) 0 þ obtained when ðϕþ i Þ is zero so that η¼z from definition (3.2.11). The initial and boundary conditions of (3.2.14b) and (3.2.14a) for solute 1 may still be used here; the initial condition merely states that all solute is contained in a very

Example 3.2.4 In the configuration shown in Figure 3.2.5A, a pulse of species i is injected in fluid j in the thin slab at x ¼ 0, lx; y ¼ 0, ly; z ¼ 0. Calculate the standard deviation of the profile as a function of time for the following systems: system 1, n-propanol (species i)–water (fluid s); system 2, CO2 (species i)–N2 (fluid s). The system temperature is 25  C. Obtain numerical values for t =100 s. Solution System 1: Propanol–water, ) D0is ¼ 1:1  105 cm2 =s (Table 3.A.3). From the formula for standard deviation (dimensional), σ i ¼ ð2  1:1  105 tÞ1=2 ¼ 4:69  103 t 1=2 cm. System 2: CO2–N2 ) D0is ¼ 0:165 cm2 =s ¼ DCO2 N2 ; σ ι ¼ ð2 0:165  tÞ1=2 ¼ 0:574  t 1=2 cm. For t ¼ 100 s we have (1) propanol–water, σ i ¼ 0:0469 cm; (2) CO2–N2, σ i ¼ 5:74 cm.

Assuming that the diffusion of each of the solute species i ¼ 1,2,3,. . .,n contained in the solute mixture pulse at z ¼ 0 takes place independently of one another, we have plotted the profiles of two more solutes i ¼ 2 and 3 in Figure 3.2.5A. Obviously, at zþ ¼ 0, the solute whose diffusion coefficient is smallest will have the lowest σ i and the highest peak

3.2

Separation development and multicomponent separation

+

Solute introduction slab, z = lz 1

Ci (z, t i )

1 2

3 2

3

0

z

2lz

lz

Figure 3.2.5B. Concentration profiles of species 1, 2 and 3 at any time t for a discontinuous ϕ i in the two adjoining vessels, 0 z lz and lz z 2lz, with solute slab introduced at z ¼ lz at time t ¼ 0.

for the same mi; suppose D01s < D02s < D03s and mi ¼ m1 (a constant) for all i, then ðm1 =l x l y l z Þ ðm1 =l x l y lz Þ C 1 ð0;t þ 1 Þ ¼ pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 2π 2D1s t=l z 2π 2D02s t=l 2z þ ¼ C 2 ð0;t þ 2 Þ > C 3 ð0;t 3 Þ:

ð3:2:36Þ

Therefore, solute 1 may be recovered at a high concentration around z ¼ 0 if D02s and D03s are much larger than D01s . However, this high concentration of species 1 will have impurities of species 2 and 3. On the other hand, solute 3 will have spread furthest; one can collect the solvent from this furthest region and get pure species 3 in the solvent. Obviously it will be a very dilute solution of pure species 3. We observe then that no more than two species may be 0 imperfectly separated when ðϕþ i Þ is zero, and that only one pure species may be obtained with a poor recovery. To determine what will happen in the other solvent in the region z > lz, it is better to start imagining what will happen if the initial solute mixture pulse were introduced at z ¼ lz (Figure 3.2.5B). If we can assume that a different sequence20 of profiles of species will be observed in each solvent vessel, then one can get at the most two pure species, one in each vessel from the furthest region from the solute introduction point. It is clear that this type of discontinuous potential profile can at the most separate two pure species and thus inherently lacks multicomponent separation capability. When the diffusion process ceases in such a system after an adequate time has elapsed, the species are going to distribute themselves between the two solvents in a

20 Different solutes will have different solubilities and partitioning properties in the two solvents; thus the concentration difference driving the diffusion will be different in the two solvents for any species.

127

manner governed by the criterion for chemical equilibrium. This aspect is treated in Section 3.3. It is important to understand why a continuous ϕ i profile can separate a multicomponent mixture when a discontinuous profile cannot. In the case of a continuous profile, the major feature is that the concentration profile of any species i appears to be translated bodily in the z-direction. In reality, the process may be understood by considering the Ci profile at any instant of time. There are two regions in this profile: the trailing edge, whose zþ-values are less than z þ i , the center-of-mass location, and where the ∂Ci/∂zþ term is positive, and the leading edge, whose zþ values are more than zþ i and where the ∂Ci/∂zþ term is negative. Consider now equation (3.2.10) without the diffusion term fRT=ðϕ 1 Þmax gð∂2 C 1 =∂z 2 Þ: ∂C 1 0 ∂C 1 ¼ ðϕþ : 1Þ ∂zþ ∂t þ 1

ð3:2:37Þ

0 The value of ðϕþ 1 Þ is negative for the continuous ϕi profile chosen. Therefore, in the trailing edge of the profile, C1 is decreasing with time: the solute species 1 is being picked up and taken ahead (to larger z-values) to make room for another species coming from behind. In the leading edge of the profile, C1 is increasing with time since solute is being transferred from the trailing edge into it (both 0 ∂C 1 =∂z þ and ðϕþ 1 Þ negative). Thus, the existence of a non0 zero ðϕþ Þ directly leads to the capacity of continuously 1 evacuating solute from the trailing edge and bringing it to the leading edge to make room for another species profile to move in. Meanwhile, all of the species are moving forward in the positive z-direction. A discontinuous ϕ i profile just does not have this capability. We have already observed that there are two compon0 ents in a ϕ i profile: ϕext ti and μi . In developing the result (3.2.35), we had ϕext ¼ 0 and nonzero μ0i ; but, as is true in ti two-phase systems (see Section 3.3), μ0i does not vary within a phase at equilibrium. Therefore ð∂μ0i =∂zÞ ¼ 0 within a phase. Thus, it appears that, unless we have 0 external forces, ðϕþ i Þ will be zero and there will be no multicomponent separation capability. We will learn in Section 7.1.5 (equation (7.1.100)) that this is not correct; the multicomponent separation capability can exist even though external forces are absent, e.g. in chromatographic processes in the presence of a nonzero bulk velocity.

3.2.3

Particulate systems

So far, we have studied the separation of chemical solutions including solutions of macromolecules, e.g. proteins. In macroscopic particulate systems without any Brownian forces of significance, for particles of size rp, mass mp, etc., F tp ¼ F ext tp

ð3:2:38Þ

since there is no chemical potential. We know that, in many cases, these external forces may be represented by

128

Physicochemical basis for separation

their scalar potentials. Such scalar potentials are usually continuous functions of the spatial coordinate, quite often uniformly continuous analogous to the cases studied in Section 3.2.1. Thus, it would appear that the external forces acting on particles have a multicomponent separation ability. Just as different chemical species had different migration velocities Ui, similarly particles with different sizes or different densities or different charges will have different terminal velocities Upt. For example, in the gravitational force field, the terminal particle velocity vector is given by   2 mp g 1  ρρt k F ext 2 rp g tp p U pt ¼ d ¼ ðρ  ρt Þk ¼ 6πμr p 9 μ p fp ð3:2:39Þ for spherical particles obeying Stokes’ law. Thus particles of different radii or different densities (or both) will have different terminal velocities, providing the possibility of separating different particles. However, different particle speeds, by themselves, are insufficient for separation since all of them are moving in the same direction. Some other technique or condition has to be created to collect these particles at different locations of the separator (as we shall see later in Chapters 6, 7, etc.). However, different values of Upt provide necessary conditions for separation. Note that identical comments can also be made about different molecular species i in a chemical mixture subjected to a uniform continuous potential profile.

3.3 Criteria for equilibrium separation in a closed separator If two or more immiscible phases are kept in a closed container for a sufficient length of time, isolated from their surroundings, the phases come to equilibrium with one another. The amount of separation achieved at equilibrium is of considerable interest. We need to know the thermodynamic criteria for equilibrium to determine this separation. In this section, such criteria are specified for a variety of equilibrium conditions encountered in separation processes, including those where a single phase is exposed to an external force field in a closed vessel. Chapter 4 covers the extent of separation achieved under equilibrium conditions in a closed container. Thermodynamic equilibrium between two or more phases or two or more regions requires the existence of thermal, mechanical and chemical equilibrium. The closed separator has thermal equilibrium if all phases and regions are at the same temperature T, which is constant. Mechanical equilibrium requires equality of pressure P in all phases or regions of the separator with plane phase interfaces. If the phase interfaces are curved, the pressures

in the two phases under mechanical equilibrium will be different. If the two regions are separated by a semipermeable membrane, or if there is a swelling pressure in one phase (e.g. ion exchange resin), the constant pressure in each phase or region will also be different under mechanical equilibrium. We are more interested in chemical equilibrium, achieved after transfer of species between two or more phases or regions. The criteria for equilibrium here will directly allow the calculation of different concentrations of a given species in different phases. This calculation presumes the existence of thermal and mechanical equilibrium. If the region is subjected to an external force field, the criterion for equilibrium separation is affected by the external potential field. This and other related criteria will be indicated in Section 3.3.1 without extensive and formal derivations (for which the reader should refer to different thermodynamics texts and references). The development of such criteria will be preceded by a brief illustration of the variety of two-phase systems encountered in separation processes. Our emphasis will be on two immiscible phase systems. Mixtures in a separation system can exist in a number of different bulk phases. The most common adjectives used to characterize different phases are: gaseous, liquid, solid, supercritical fluid, membrane and ion exchange material.21 The gaseous phase includes both gas and vapor, just as solid includes crystalline as well as amorphous materials. Some combinations, e.g. gas–gas, gas–supercritical fluid, are to be eliminated since they do not behave as two immiscible phases. From diffusional rate considerations, two-phase combinations of solid, membrane and ion exchange materials (a total of six, i.e. solid–membrane, solid–ion exchange material, solid–solid, membrane–membrane, membrane–ion exchange material, ion exchange material–ion exchange material) are not useful because separation would take forever!22 Thus one of the phases in any useful twophase combination should be a fluid; i.e. gaseous, liquid or supercritical fluid. This eliminates a total of eight combinations. (Giddings (1982) has suggested that, in theory, there can be [k(k þ 1)/2] combinations of two phases for k available phases, i.e. there are 21 possible combinations of six phases.) The remaining possible combinations are identified in Table 3.3.1. Of the 13 two-phase combinations identified, two combinations, gas–ion exchanger and supercritical fluid–ion

21

We ignore liquid crystals or mesophases, vesicles, caged molecules, etc. for the time being. They are, however, studied in Section 4.1.8. The membrane phase includes gels. Interfacial phases are ignored for now. 22 Echoes of this may have mistakenly led to the formulation “Corpora non agunt nisi fluida sive soluta” – substances do not react unless in a liquid or a dissolved state (Helfferich, 1995).

3.3

Equilibrium separation criteria closed vessel

Table 3.3.1 Possible useful combinations of two bulk immiscible phasesa Combinations having a particular phase Combinations having a gaseous phase Combinations having a liquid phase

Combinations having a supercritical fluid phase

Possible useful combinations gas–liquid; gas–solid; gas– membrane; gas–ion exchanger liquid–liquid; liquid–solid; liquid– membrane; liquid–ion exchanger; liquid– supercritical fluid supercritical fluid–solid; supercritical fluid–membrane; supercritical fluid– supercritical fluid; supercritical fluid–ion exchanger

a

Table 3.V in Giddings (1982) is based on seven phases, unlike the six phases used here.

exchanger, may not be fruitful since ions are not normally transferred between these phases. The rest are the basis of existing equilibrium based separation processes or have a significant potential for becoming practical techniques. These useful two immiscible phase combinations are to be kept in mind as we consider now the criteria for chemical equilibrium in such systems, as well as other related material. 3.3.1 Phase equilibrium with equal pressure in all phases We choose for this purpose first a two-region (or twophase) separator at constant and uniform T and P without any chemical reaction. If we focus on a particular region, it is obvious that it is open since molecules can be transferred from this to a contiguous region or vice versa. Impose now the restriction that a region can only have a single phase. For this particular open single-phase region chosen as a thermodynamic system, the total Gibbs free energy of all molecules in region j, Gtj, is a function of temperature T, pressure P and the mole numbers, mij, of all chemical species i in region j: Gtj ¼ Gtj ðT;P; m1j ; m2j ; m3j ;. . .Þ;

ð3:3:1Þ

for the n-component system i ¼ 1, 2. . ., n. From standard thermodynamics texts (Van Ness and Abbott, 1982), we know that the total differential of Gtj following a small change in T,P and mijs is dGtj ¼ Stj dT þ V j dP þ

n X i¼1

μij dmij :

ð3:3:2Þ

Here Stj is the total entropy of all molecules in region j, Vj is the volume of region j,

129 ð∂Gtj =∂TÞp;mij ;all i ¼ Stj ; ð∂Gtj =∂PÞT;mij ;all i ¼ V j ; ð∂Gtj =∂mij ÞT;P;mkj ;k6¼i ¼ μij :

ð3:3:3Þ

With the closed separator considered as a thermodynamic X system, the species mole numbers, m0i ¼ mij , are conj¼1

stant in the absence of any chemical reaction. The total Gibbs free energy of the whole separator as a system (obtained by summing Gtj over all j) is then only a function of T and P. At constant T and P, the total differential of this total Gibbs free energy is zero:   2 2 X n X X  ð3:3:4Þ dGtj T;P ¼ 0 ) μij dmij ¼ 0;  j¼1 j¼1 i¼1

where we have used (3.3.2) to obtain the second result. For a two-phase (or two-region) system, dmi1¼dmi2 due to conservation of species i. This simplifies (3.3.4) to n X i¼1

ðμi1  μi2 Þ dmil ¼ 0;

ð3:3:5Þ

which is valid for any arbitrary change dmi1. Therefore μi1 ¼ μi2 :

ð3:3:6Þ

This criterion for chemical equilibrium in a two-phase system then requires that the chemical potential of any species i should be uniform and constant through the separator. If the closed separator has more than two phases or two regions, a criterion for chemical equilibrium can be derived by considering any two of the phases or regions, say j 1 and j 2 , at a time. The relation μij1 ¼ μij2

ð3:3:7Þ

can be easily shown to be valid for the total system of regions j1 and j2 by following the earlier procedure. Since this result is valid for any (j1, j2) pair of phases or regions, the general criterion for chemical equilibrium in a closed separator containing a total of k phases or regions is μi1 ¼ μi2 ¼    ¼ μij ¼    ¼ μik

j ¼ 1;2;. . .;k: ð3:3:8Þ

A different form of the criterion for chemical equilibrium (3.3.6) or (3.3.8) is often quite useful since it uses the fugacities of a component i in solution instead of the chemical potential of species i. Let ^f ij be the fugacity of species i in a homogeneous mixture in region or phase j. At a constant temperature, we know from the thermodynamic properties of homogeneous mixtures (Van Ness and Abbott, 1982) that any small change in the chemical potential of species i in region or phase j is related to the change in ^f ij by dμij jT ¼ RT d ℓn ^f ij jT : Integration yields

ð3:3:9Þ

130

Physicochemical basis for separation

h i μij  μij ðinitialÞ ¼ RT ℓn ^f ij  RT ℓn ^f ij ðinitialÞ : ð3:3:10aÞ

We assume that

μij ðinitialÞ ¼ C 1 ða constantÞ; ^f ij ðinitialÞ ¼ C 2 ða constantÞ;

ð3:3:10bÞ

where the constants are dependent on the constant T and P of the separator and are assumed independent of j, the region or phase. Then, using the general criterion (3.3.8), we find ℓn ð ^f i1 =C 2 Þ ¼ ℓn ð ^f i2 =C 2 Þ ¼    ¼ ℓn ð ^f ij =C 2 Þ ¼    ¼ ℓn ð ^f ik =C 2 Þ;

ð3:3:10cÞ

3.3.2 Phase equilibrium where different phases have different pressures We consider next the requirements for chemical equilibrium between two regions or phases when the temperature is uniform and constant throughout the separator, but the pressure has different (but uniform) values in different phases (or regions). Assume the phase interface to be planar and assume no chemical reactions. Equation (3.3.2) is changed now to dGtj ¼  Stj dT þ V j dP j þ

ð∂Gtj =∂TÞPj ;mij ;all i ¼  Stj ; ð∂Gtj =∂P j ÞT;mij ;all i ¼ V j ; ð∂Gtj =∂mij ÞT;Pj ;mkj ;k6¼i ¼ μij :

for j ¼ 1;2 . . .;k;

ð3:3:11Þ

as another form of the criterion for chemical equilibrium in a chemically nonreactive separator at constant T and P. (Note: The ^ over a quantity indicates, among other things, “valid for a mixture”; here, the fugacity of species i in a mixture.) Equality of the chemical potential of a component/ species in two contiguous immiscible phases constituting two regions rarely implies equality of concentration. In fact, the relation between the different concentrations of different species in two phases at equilibrium in a separator will be developed for various types of separation phenomena in Section 4.1 using these relations. The relations given above are valid for any n-component system. As pointed out in Section 2.4, the mixture may be a continuous chemical mixture, where the composition is described by a molar density function f(r) whose independent variable r is some characterizing property, e.g. molecular weight, carbon number, etc. The criterion for chemical equilibrium in a two-phase (or two-region) system of j ¼ 1, 2 is, for all values of r, (Cotterman et al., 1985) μ1 ðrÞ ¼ μ2 ðrÞ;

ð3:3:12Þ

where μ1 ðrÞ and μ2 ðrÞ refer, respectively, to the chemical potentials of species of property r in phases 1 and 2, respectively. If the mixture is semicontinuous, i.e. the concentrations of some components (i ¼ 1,. . ., n) have discrete values in terms of mole fractions, whereas the concentrations of all others are described by a molar density function f(r), there are two criteria for chemical equilibrium (Cotterman and Prausnitz, 1985), μi1 ¼ μi2 ;

ð3:3:13aÞ

for each i ¼ 1,. . ., n; for the continuous mixture part, μ1 ðrÞ ¼ μ2 ðrÞ for all values of r.

μij dmij ;

i¼1

ð3:3:14Þ

where

which yields ^f ¼ ^f ¼    ¼ ^f ¼    ¼ ^f i1 i2 ij ik

n X

ð3:3:15Þ

The total Gibbs free energy of the system consisting of the two phases or regions of the separator as a whole depends only on T, P1 and P2 for j ¼ 1,2 and not on the conserved mole number m0i of any species i. At constant, T, P1 and P2, the total differential of the Gibbs total free energy of the two phases should be zero:  2 2 X n  X X  dGtj T;P1 ;P2 ¼ 0 ) μij dmij ¼ 0:  j¼1 j¼1 i¼1 Using the familiar argument of dmi1 ¼ –dmi2 and the fact that the magnitude of dmi1 or dmi2 is arbitrary, we obtain μi1 ¼ μi2 :

ð3:3:16Þ

The chemical potential of a species i should then be uniform and constant throughout the separator under conditions of equilibrium that include different pressures in different phases. In the osmotic equilibrium of two liquid solutions on either side of a semipermeable membrane (see Example 1.5.4), the semipermeable membrane, if perfect, prevents the exchange of the solute across the region boundary while the solvent passes through the membrane, the region boundary. In the development of the criteria for chemical equilibrium, it was always presumed that any species i could be exchanged between adjoining phases or regions. Therefore, the criteria for equality of chemical potential apply only to those species which are permeable through the membrane and not to impermeable species. We will digress now and describe the dependence of μij ðP j ;T ;x ij Þ on Pj, T and xij. A total differential of μij for any phase j can be written as follows (Denbigh, 1971): dμij ¼  Sij dT þ V ij dP j þ

ð3:3:13bÞ where

 n1  X ∂μij k¼1

∂x kj

dx kj ; T;Pj ;x l;l 6¼k

ð3:3:17aÞ

3.3

Equilibrium separation criteria closed vessel

Sij ¼ ð∂μij =∂TÞPj ;xnj ;all n ;

ð∂μij =∂P j ÞT;x nj ;all n ¼ V ij ð3:3:17bÞ

and i ¼ 1,2,. . ., n. Here, Sij is the partial molar entropy of species i in region j. Note that only ðn  1Þ x kj s are independent. At any fixed composition xij,all i, we get dμij ¼ Si dT þ V ij dPj :

ð3:3:18aÞ

Integrating from standard state pressure P0 to the pressure Pj of region j, we get, at constant T, 0

μij ðP j ;TÞx ij ¼ μij ðP ;T Þx ij þ

P ðj

V ij dPj :

P0

ð3:3:18bÞ

For liquid solutions, V ij is generally a weak function of pressure. Therefore 0

0

μij ðP j ;T Þxij ¼ μij ðP ;TÞxij þ V ij ðP j  P Þ:

ð3:3:19Þ

At any fixed pressure Pj, we can also integrate (3.3.17a) between the pure ith species and the composition xij to get, at constant T,   xðij   n1  X ∂μij  μij ðP j ;T;x ij Þ  μij ðP j ;TÞxij ¼1 ¼ dx kj :  ∂x kj T;Pj ;x l l6¼k; k¼1  x ij ¼1 ð3:3:20aÞ

But we know from relation (3.3.10a) that, for the initial state being pure i,

131

μij ðP j ;T;x ij Þ  μij ðP j ;TÞjx ij ¼1 ¼ RT ℓn

^f ij f 0ij

!

¼ RT ℓn aij ; ð3:3:20bÞ

where aij is the activity of species i in region j. Use relation (3.3.19) for xij ¼ 1 to obtain μij ðP j ;T ;x ij Þ ¼ μij ðP 0 ;TÞjx ij ¼1 þ V ij ðP j  P 0 Þ þ RT ℓn aij : ð3:3:20cÞ Usually, μij ðP 0;TÞ at standard state pressure P0 and system temperature T for a pure substance i is identified as the standard state chemical potential μ0i ðP 0;TÞ ¼ μ0i . Then μij ðP j ;T;x ij Þ ¼ μ0i þ V ij ðP j  P 0 Þ þ RT ℓn aij

ð3:3:21Þ

is the general relation linking μij, Pj, T and xij (through aij). A special result for a pure substance is μij ðP j ;TÞ ¼ μ0i ðP 0 ;TÞ þ V ij ðP j  P 0 Þ:

ð3:3:22Þ

One further obtains, at standard state conditions, dμ0ij ¼ V ij dP 0

ð3:3:23Þ

for a differential change in the standard state pressure P0. Since the standard state is important for many calculations, Table 3.3.2 summarizes the commonly employed standard states for the more frequently encountered conditions.

Table 3.3.2. Standard states State or phase of system and/or species under consideration

Standard state defined by pressure, temperature or composition

(1) Gas or vapor

pure component gas or vapor behaving ideally at 1 atm and system temperature pure liquid component at the same temperature and pressure as the solution pure solid component at the same temperature and pressure as the solid solution pure liquid or solid component at the same temperature and pressure as the solution

(2) Liquid acting as a solvent (3) Solid acting as a solvent (4) Solute in a liquid or solid solution such that the pure component solute is liquid or solid at the temperature and pressure of the solution (5) Pure component solute does not exist in the same phase as the solution at the temperature and pressure of the solution

Infinite dilution standard state whereby the standard state value of fugacity or activity of a component at the temperature and pressure of the solution is given by the ratio of the fugacity or activity to the mole fractiona under conditions of infinite dilution

a Sometimes instead of the mole fraction xij, the molality, (mi,j, moles of i/kg of solvent in region j) is used. Whether molality or mole fraction is used, this standard state is intimately connected with Henry’s law:

^ ¼ x il f 0 ¼ x il H i , il

lim f x il !0 il

where Hi is Henry’s law constant for species i in the solution and is the hypothetical fugacity fil0 of the species at the infinite dilution standard state (see relation (3.3.59)).

132 3.3.3

Physicochemical basis for separation Single-phase equilibrium in an external force field

We observed in Chapter 1 that separation is possible in a closed separator even if it contains only one phase (see Figure 1.1.3, example III; Examples 1.4.1 and 1.5.3; Problem 1.4.3; Section 3.2.1 analyses such a system in general). In such cases, there exists usually one (or more) external force field (or temperature gradient) which creates a difference in composition at different locations within the separator. These different locations then become different regions. A phase at equilibrium in our earlier results for criteria for equilibrium has the same properties everywhere. The presence of an external force field, however, imparts different values of the external potential ϕext to i different locations in a single-phase closed separator. Consequently, no two locations in such a separator are identical; one could presume the system to be composed of an infinite number of phases of differential thickness in the direction of the external force field. Assume now two locations α and β a differential distance apart. At equilibrium, the net total force Fti on species i should be zero. For the case of a single external force field F ext i , we get from the general expression (3.1.50)    ð3:3:24Þ rμi  ¼ F ext i : T For an external force field describable by rϕext i , the onedimensional representation of the above relation along the coordinate direction z is    dμi  dϕext  i ð3:3:25Þ k ) dμi  ¼  dϕext  k¼ i ; T dz  T dz relating the change in chemical potential of the ith species for a differential distance dz to the corresponding externalforce-created potential difference. For a finite distance zα  zβ between two regions α and β, we obtain, on integrating (3.3.25) (Denbigh, 1971),           ext  ext  ext μi zα  μi zβ ¼ ϕi zβ  ϕi zα ) μiα  μiβ ¼ ϕext iβ  ϕiα :    

ð3:3:26Þ

3.3.4

Equilibrium between phases with electrical charges

In systems using, say, ion exchange resins (see Section 3.3.7.7), the resin particles have fixed electrical charges. Similarly, ion exchange membranes (Section 3.4.2.5) have fixed electrical charges. If there is no externally applied electrical field, it is useful to enquire what criteria govern chemical equilibrium in two-phase systems containing such a phase. It is common practice to define an electrochemical potential μel ij in the jth region by μel ij ¼ μij þ Z i F ϕj ;

total work done in bringing the species i from a vacuum into phase j (Adamson, 1967). The criterion for equilibrium for all ith species (i ¼ 1,. . ., n) is that the electrochemical potential of a species must be the same in all phases: e1 μe1 i1 ¼ μi2 ;

j ¼ 1;2:

ð3:3:28Þ

Using definition (3.3.27) in this relation leads to μi1  μi2 ¼ Z i F ðϕ2  ϕ1 Þ;

ð3:3:29Þ

where the phase potentials ϕ2 and ϕ1 are independent of the ionic species under consideration. This potential difference (ϕ2ϕ1) has been identified as the Donnan potential. An additional relation has to be used in each phase containing charged species/ions. This is the electroneutrality condition, according to which there should be no net electrical charge at any bulk location (there may be deviations from it at phase boundaries; see Helfferich (1962, 1995)): X X Z i C i1 ¼  Z k C k1 ; ð3:3:30aÞ

where i ¼ positive ions, k ¼ negative ions, for phase j ¼ 1 without any fixed charges; X X ð3:3:30bÞ Z i C i2 ¼  Z k C k2  ωX ;

for phase j ¼ 2 with fixed charges, where X is the molar density of fixed charges and ω is the sign of fixed charges (þ for positive charges, – for negative charges). It is useful to consider an explanation regarding electroneutrality and the potential difference ϕ2ϕ1 in ion exchange systems provided by Helfferich (1962, 1995). Ion exchange resin particles have fixed positive or negative charges.23 When they are placed in a solution of electrolytes, counterions (i.e. ions with charges opposite to the fixed charges) diffuse into the porous ion exchange resin particles and maintain electroneutrality. However, there then develop large concentration differences between the two phases. For a cation exchange resin system, say (with fixed negative charges), the cations are counterions; their concentrations are larger in the ion exchange resin particle, whereas the concentrations of anions (coions) are larger in the solution. Therefore anions tend to migrate to the resin particle and cations from the resin tend to migrate to the solution. The migration of a few ions in both directions immediately builds up an electrical potential difference between the two phases. This is the Donnan potential. The positive charge in the solution prevents cations from leaving the resin, whereas the negative charge on the resin prevents the migration of anions from the solution to the resin. An equilibrium is established between the repulsive

ð3:3:27Þ

where ϕj is the electrical potential of the jth phase and Zi is the electrochemical valence of species i. It is defined as the

23

See Figures 3.3.9, 3.3.10 and 3.3.12 for schematics of such particles and their charges.

3.3

Equilibrium separation criteria closed vessel

133

PI

Solute concentration profile

Cib Gas phase Csb Solution phase Scales differ for solute and solvent concentration

Solvent concentration profile

t1

t2

Interfacial region Figure 3.3.1. Concentration profiles across a gas–liquid interface: PI = phase interface.

forces of similar charges and the so-called driving force of a concentration difference. An identical condition is obtained with an anion exchange resin, except the Donnan potential has an opposite sign. Regardless of the Donnan potential, however, electroneutrality is maintained in the ion exchange resin or the solution. Helfferich (1962) states that Migration of just a few ions is sufficient to build up so strong an electric field counteracting any further migration that deviations from electroneutrality remain far from the limit of accuracy of any method except for the measurement of the electric field itself.

3.3.5

Equilibrium between bulk and interfacial phases

In the absence of external force fields, we have so far assumed that, in a two-phase or multiphase system, the intensive properties of any phase such as pressure, temperature, composition, etc., were constant and uniform throughout the phase, including and up to the interface between the phase under consideration and the adjoining phase or phases. However, in many systems, the interfacial region demonstrates properties different from those of either of the bulk phases. For example, in a dilute aqueous solution of surfactants exposed to air, the surfactant concentration at the air–solution interface on the solution side is greater than that in the solution bulk. Skimming off the surface layer provides a way of obtaining a more concentrated surfactant solution, and is therefore a reasonable basis for a separation process or processes. A two-phase system has thus become a three-phase system: two bulk phases and one interfacial or surface phase. Figure 3.3.1 illustrates how the composition of a surface active solute i and the solvent s varies in the interfacial region of the surfactant solution (j ¼ 1) – air (j ¼ 2) system.

Since the interfacial phase is in addition to the two bulk phases, the criteria (3.3.8) for phase equilibrium should be applicable if there are no external force fields. Therefore μi1 ¼ μi2 ¼ μiσ ;

ð3:3:31Þ

where we have identified the surface phase by the subscript σ. In conventional phase equilibria, such a relation is generally sufficient to determine the equilibrium concentrations of any species in the two phases at equilibrium. But estimation of the surface phase concentration to determine equilibrium separation between a bulk phase and the surface phase requires consideration of surface forces. The molecules of any species i present in the bulk of a liquid phase experience a net force if they are at the surface of this phase (i.e. at the interface of two contiguous phases, liquid–liquid or liquid–air). This force creates a tendency for the bulk phase surface to contract and is termed the interfacial tension γ12 (unit, dyne/cm), where the superscripts refer to the two bulk phases 1 and 2. Consider a system with a planar interface between the phases. Let the volume, surface area and thickness of such a surface region shown in Figure 3.3.2A between two bulk phases 1 and 2 with planar interfaces 110 and 220 be, respectively, Vσ, Sσ and tσ (Guggenheim, 1967). Denote the uniform pressure in both bulk phases by P. The force in the direction parallel to planes 110 and 220 inside the surface phase is, however, given by Pt σ  γ12 . This force acts on an area of height tσ and unit width perpendicular to the plane of the paper in Figure 3.3.2A. Following Guggenheim (1967), it may be shown that, if the surface layer values Vσ, Sσ and tσ are changed by a differential amount to Vσ þ dVσ, Sσ þ dSσ and tσ þ dtσ through a

134

Physicochemical basis for separation dγ12 ¼ Γ 1σ dμ1σ þ Γ sσ dμsσ ¼ Γ 1σ dμ11 þ Γ sσ dμs1 ð3:3:36Þ where one of the bulk phases is j ¼ 1. The Gibbs–Duhem equation for bulk phase 1 at constant T and P is

2

x 11 dμ11 þ x s1 dμs1 ¼ 0:

2⬘

2

ts

s 1⬘

1 1

Figure 3.3.2A. Schematic of the interfacial region according to Guggenheim (1967).

ð3:3:37Þ

Substitution of this relation into (3.3.36) results in   x 11 Γ sσ dμ11 ¼ dγ12 : ð3:3:38Þ Γ 1σ  x s1 The coefficient in brackets on the left-hand side may be interpreted as follows. Since species 1 is the solute and s is the solvent, Γ sσ ðx 11 =x s1 Þ is the number of moles of solute 1 per unit area of the interface if the solvent and solute are present in the σ-phase in the same ratio as in bulk phase 1. The coefficient in brackets in (3.3.38), if positive, indicates the excess number of moles of solute 1 per unit interfacial area at the interface due to a deviation in behavior from the bulk phase. Denote this explicitly by rewriting (3.3.38) as Γ E1σ dμ11 ¼ dγ12

ð3:3:39aÞ

or reversible process, the reversible work done by the interfacial phase as a system is P dV σ  γ12 dSσ . This is in contrast to the reversible work P dV done by a bulk phase when there is a differential change dV in volume V. Incorporating such a departure in a total differential of the internal energy of the surface phase, and using Euler’s theorem on homogeneous functions, the following relation can be shown to be valid for the surface phase (Guggenheim, 1967): 12

Stσ dT  V σ dP þ Sσ dγ þ At constant T and P, Sσ dγ12

n X i¼1

miσ dμiσ ¼ 0:

Define the surface concentration Γiσ of species i (in gmol/ cm2) by miσ : Sσ

ð3:3:34Þ

The isotherm (3.3.33) may be rewritten as n X i¼1

Γ iσ dμiσ ¼

n X i¼1

Γ iσ dμi1 ¼

ð3:3:39bÞ

for the two-component (solute i ¼ 1) two-bulk-phase system with an interfacial region of distinct properties. (The superscript E denotes a surface excess quantity.) Generalizing such a procedure for an n-component solution with a solvent i ¼ s, we get, corresponding to (3.3.39a), n X i ¼1 i 6¼ s

ð3:3:32Þ

ð3:3:33Þ

i¼1

dγ12 ¼

dγ12 dμ11

Γ Eiσ dμi1 ¼

n X i¼1 i 6¼ s

Γ Eiσ dμiσ ¼ dγ12 ;

ð3:3:40aÞ

where

n X miσ dμiσ : ¼

Γ iσ ¼

Γ E1σ ¼ 

n X i¼1

¼ Γ iσ dμi2 ð3:3:35Þ

using relation (3.3.31). This result is, however, not used in this form. Consider two species i ¼ 1, the solute and i ¼ s, the solvent. Equation (3.3.35) can be written for i ¼ 1, s as

Γ Eiσ ¼



Γ iσ 

 x i1 Γ sσ ; x s1

i ¼ 1; 2 ;. . .;n i6¼s

ð3:3:40bÞ

Here Γ Eiσ is called the surface excess of species i. Note that when i ¼ s, Γ Esσ ¼ 0, i.e. the surface excess of the solvent is zero. For surface active solutes, 12 12 ðdγ12 =dμint i1 Þ ¼ ðai1 =RTÞ ðdγ =dai1 Þ is negative since γ E decreases with increasing solute concentration: thus Γ iσ is positive. For electrolytic solutes (e.g. salts) in an aqueous solution at an air–water interface, ΓEiσ could be negative since the electrolytic solute prefers the environment of the water molecules in the bulk compared to that of the air–water interface, raising the value of γ12 over that of pure water. Consider now equation (3.3.39b). It relates the interfacial concentration of solute 1 to the bulk concentration of solute 1 through the dependence of interfacial tension on the bulk chemical potential of solute 1. Remember, we need the equilibrium criterion so that we can relate the

3.3

Equilibrium separation criteria closed vessel

solute concentrations in the two phases. Both results (3.3.39a) and (3.3.40a) are known as the Gibbs equation or Gibbs adsorption isotherm. The procedure normally used to derive the Gibbs equation, however, is different and is based on the assumptions that Vσ ¼ 0 and the solvent surface excess is also zero (Davies and Rideal, 1963; Adamson, 1967). There is a sound reason for adopting the Gibbsian strategy of Vσ ¼ 0. The quantities Γ E1σ and Γ Esσ in equation (3.3.38) are arbitrary since their values are dependent on the locations of the boundaries 110 and 220 . However, the quantity ðΓ 1σ  fx 11 =x s1 g Γ sσ Þ is independent of the volume Vσ and the locations of the boundaries. Gibbs therefore replaced the real situation with an imaginary one where bulk properties continued all the way to a hypothetical phase interface, a twodimensional region which had the surface excess or surface deficiency of the solute only. For surface-active solutes lowering the value of γ12, a film pressure or spreading pressure π has been defined, for example for an air–water system, as 12 π ¼ γ12 solvent  γsolution :

ð3:3:41aÞ

This is a sort of two-dimensional pressure for the twodimensional interfacial region with no volume (Gibbsian formulation Vσ ¼ 0) but a surface area Sσ. Thus instead of pressure, volume and temperature (P, V, T) of a conventional three-dimensional phase, the hypothetical interfacial phase has π, area and temperature (π, Sσ, T). Correspondingly, the ideal gas equation of state for a pure gas, PV ¼ RT for 1 mole of gas, is replaced by π Sσ ¼ RT;

ð3:3:41bÞ

where Sσ is the surface area per mole of the gas. For qiσ moles of a gas i in the surface phase per unit adsorbent mass (see equation (3.3.43a) for a changed definition of Sσ) π Sσ ¼ qiσ RT:

X

i¼positive ions

k¼negative ions

Z i Γ Eiσ

¼

X 

k¼negative ions

same thermodynamic equations which led to the Gibbs equation. The Gibbs adsorption isotherm in this case has been shown to be (Hill, 1949) Sσ dπ ¼

Z k Γ Ekσ :

ð3:3:42bÞ

If an applied electrical field is present, the surface adsorption equilibrium relation (3.3.40a) is modified by the inclusion of an electrical potential term (Davis and Rideal, 1963). The criteria for equilibrium in gas adsorption on the surface of a solid adsorbent may also be obtained using the

n X i¼1

qiσ dμiσ ¼

n X

qiσ dμig ;

i¼1

ð3:3:43aÞ

where qiσ is the number of moles of gas i adsorbed per unit mass of adsorbent, Sσ is the surface area per unit mass of the adsorbent, and π, the spreading pressure, is defined to be the lowering of surface tension at the gas–solid interface due to adsorption. Unlike gas–liquid interfaces where the interfacial tension γ12 is measurable, the spreading pressure π of a gas–solid surface is not measurable. The interfacial region is often identified as the adsorbate. Expressing (3.3.43a) for a binary gas mixture of species 1 and 2 as Sσ dπ ¼ dπ ¼

2 X q i¼1

iσ

Sσ

2 X

qiσ dμig ;

i¼1

ð3:3:43bÞ

RT d ℓn ^f ig ¼ Γ 1σ RT d ℓn ^f 1g þ Γ 2σ RT d ℓn ^f 2g ;

dπ ¼ Γ 1σ d ℓn ^f 1g þ Γ 2σ d ℓn ^f 2g : RT

ð3:3:44Þ

For a pure gas species i dπ ¼ Γ iσ d ℓn fig RT

ð3:3:45Þ

d ℓn fig dπ ¼ RT Γ iσ : dΓ iσ dΓ iσ

ð3:3:46aÞ

or

Since ðd ℓn fig =dΓ iσ Þ is obtainable from an isotherm (experimental or otherwise), dπ=dΓ iσ can be determined (Lewis and Randall, 1961). Note that for an ideal gas behavior with the pure species at pressure P,

ð3:3:41cÞ

In the application of equation (3.3.40a), all independent ionic species are to be included along with nonionic solutes. Further, the electroneutrality condition requiring the absence of net charge anywhere suggests that the following should be satisfied: X X Z k C k1 ; ð3:3:42aÞ Z i C i1 ¼  i¼positive ions

135

dπ d ℓn P ¼ RT Γ iσ : dΓ iσ dΓ iσ

ð3:3:46bÞ

Expressing Γiσ as qiσ / Sσ, we obtain, by integrating from 0 to P as π increases from 0 to π, 1 RT

ðπ 0

Sσ dπ ¼

π Sσ ¼ RT

ðP 0

qiσ dP ; P

ð3:3:46cÞ

an alternative form of (3.3.46a) for a pure ideal gas being adsorbed. So far, the surface excess of a solute species was considered on the surface of a bulk phase, e.g. water, air, solid adsorbent, etc. The surfaces of macromolecules, especially proteins, have an interfacial region, sometimes called the local domain, which can have compositions different from the bulk domain, namely the bulk of the solution. For example, the bulk of an aqueous protein solution may have cosolvents (or cosolutes) such as urea, guanidine

136

Physicochemical basis for separation

r

P2

P1 P1 > P2

Figure 3.3.2B. Pressure in two regions (inside and outside) of a spherical bubble.

hydrochloride, sucrose, etc., added in large amounts (40– 50% by weight). As Tang and Bloomfield (2002) state: These cosolute molecules bathe and solvate the macromolecular solute as water does.

A question of some importance is: Does the curved surface of a drop or other geometries affect the chemical potential (or fugacity) of a species compared with that on a planar surface? For gas–liquid, vapor–liquid, solid–liquid or liquid–liquid systems, this subject has been considered by Lewis and Randall (1961), starting from the general equation of a differential change in total Gibbs free energy of a phase j (compare with relation (3.3.2)): X dGtj ¼ Stj dT þ V j dP j þ γ12 dSσ þ μijPl dmij ; ð3:3:49Þ

where γ12 dSσ appears due to interfacial tension between phases j ¼ 1 and j ¼ 2. Note that the quantity μijPl in (3.3.49) refers to any process of transfer of mij without any change in Sσ (planar interface, subscript Pl); to indicate this explicitly,   ∂Gtj μijPl ¼ ; ð3:3:50Þ ∂mij T;Pl;Sσ ;mkj ;k6¼i

The composition of the local domain is important from the point of view of how the macromolecular solute, for example protein, will behave. If the local domain is depleted in the cosolutes and enriched in water relative to that in the bulk solution, the macromolecule (protein) is said to undergo preferential hydration. If, however, the local domain, is depleted in water relative to the bulk, then either of two things can happen. The “cosolutes” may be preferentially accumulated in the local domain, or the local domains of neighboring macromolecules/proteins may interact with each other (for example via hydrophobic interaction; see Section 4.1.9.4) leading to, for example, precipitation (see Section 3.3.7.5).

where we have included a subscript Pl to refer to a planar interface. From Lewis and Randall (1961), for any curved interface system, we can express relation (3.3.49) in general as X dGtj ¼ Stj dT þ V j dP j þ μij dmij : ð3:3:51Þ

3.3.6

Correspondingly,

Curved interfaces

The interfaces between two bulk phases considered so far have been planar, and the pressures in the two bulk phases under equilibrium have been equal for gas–liquid and liquid–liquid systems. When the interface has a curvature, mechanical equilibrium requires different values of the pressures in the two phases. The general relation (the Young–Laplace equation) governing the pressure difference between bulk phases 1 and 2 is as follows (Adamson, 1967; Guggenheim, 1967):   1 1 ðP 1  P 2 Þ ¼ γ12 þ ; ð3:3:47Þ r1 r2 where γ12 is the interfacial tension and r1 and r2 are the principal radii of curvature of the interfaces. By convention, r1 and r2 are positive if they are in phase 1 and ΔP is positive, with the interface convex going from phase 1 to phase 2. For a true spherical surface, as in a spherical bubble of radius r (Figure 3.3.2B), ðP1  P 2 Þ ¼ 2γ12 =r

For a sphere, any small change in the surface area of the sphere dSσ due to a differential change dr in the radius will lead to the following change in volume dV of the sphere: dV ¼ 4πr 2 dr;

dSσ ¼ 4πðr þ drÞ2  4πr 2 ffi 8πr dr:

dSσ ¼ 2

dV : r

ð3:3:52aÞ

If the volume change is due to the addition of dmi moles of species i for all i, then X dV ¼ V ij dmi ; ð3:3:52bÞ

which leads to

γ12 dSσ ¼ 2γ12

dV 2γ12 X V ij dmij ¼ r r i¼1

) μij  μijPl ¼

2V ij γ12 r

ð3:3:52cÞ ð3:3:52dÞ

is a specific example for a spherical droplet of radius r (Figure 3.3.2B). Using the relation (3.3.9) between μij and ^f , we get24 ij

ð3:3:48Þ

since r1 ¼ r2 ¼ r. For both results, it has been assumed that γ12 is not affected by r1, r2 or r. For a planar interface, r1 ¼ r2 ¼ ∞ and P1 ¼ P2 in relation (3.3.47).

24 Other considerations may come into play to prevent ^f ij from increasing continually as r is reduced.

3.3

Equilibrium separation criteria closed vessel

ℓn

^f 2Vij γ12 ij : ¼ ^f RTr ijPl

ð3:3:53Þ

Here, ^f ij is the fugacity in a system with a spherical surface, with the interface convex going from the droplet phase to the surroundings, and ^f ijPl corresponds to that with a planar surface. For ideal gas behavior, replace ^f ij by pij. This result has implications for droplets, which have their surface convex toward the vapor. Consider a pure liquid drop of radius r. Then ^f ijPl is equal to the vapor pressure of the pure liquid (on a flat surface), P sat iPl , if the fugacity coefficient ϕsat i under the conditions of P and T is assumed to be unity. However, equation (3.3.53) implies that the vapor pressure of the pure liquid in the space above the convex curved surface of the drop, Psat i , is higher than P sat iPl at equilibrium (Figure 3.3.2B):   2 V ij γ12 sat P sat : ð3:3:54Þ ¼ P exp i iPl RTr Further, the smaller the drop diameter, the higher the observed vapor pressure of the liquid in the vapor space, leading to a higher rate of evaporation from the drop. Equation (3.3.54) is identified as the Kelvin equation and the phenomenon is called the Kelvin effect. This increased evaporation tendency appears because a molecule near the droplet surface is attracted to a lesser extent toward the interior by the surrounding molecules (since there are fewer of these surrounding molecules). However, this analysis breaks down as the drop size becomes too small and the number of molecules become much smaller. If the drop consists of a solution of a nonvolatile solute, the Kelvin equation applies to the solvent. Note that if the liquid surface is concave toward the vapor, the fugacity ^f at the surface is less than that on a planar surface. See ij Section 3.3.7.5 as well as Problem 3.3.2 for the application to a solid–liquid system.

3.3.7 Solute distribution between phases at equilibrium: some examples When two or more immiscible phases are at equilibrium, species generally distribute themselves between the phases such that the uniform and constant concentration of a species in one phase is different from those in other phases. Since there can be a large number of combinations of two-phase systems in equilibrium (see Table 3.3.1), the variety in distribution coefficient (κil) relations is enormous. Our objective here is somewhat limited. We pick a few common two-phase systems in equilibrium and illustrate the relations between the concentrations of any solute between the phases at equilibrium. This will not only allow us to develop integrated flux expressions in multiphase

137 systems in Section 3.4, but will also facilitate calculating the separation achieved in a closed vessel in many systems considered in Chapter 4. We consider separately equilibrium in gas–liquid, liquid–liquid, gas–membrane, liquid– membrane/porous sorbent, liquid–solid, interfacial adsorption systems, ion exchange systems, as well as in supercritical fluid–solid/liquid systems. Only elementary and essential aspects of each equilibrium solute distribution will be our concern. Although we describe the equilibrium distribution of a solute25 between two phases in general, there are cases here, e.g. vapor–liquid equilibrium, where the solute is a major constituent of the phases under consideration. The notion of a solute in a solvent is inappropriate in such cases. 3.3.7.1

Gas–liquid equilibrium

We discuss first the distribution of a solute i between a gas and a liquid at pressure P. At equilibrium, we obtain from the criterion (3.3.11) ^f ¼ ^f : ig il

ð3:3:55Þ

From standard thermodynamics texts, the fugacity of species i in each phase may be expressed as ^f ¼ x ig Φ ^ ig P; ig

ð3:3:56Þ

^f ¼ x il γ f 0 ; il il il

ð3:3:57Þ

^ ig is the fugacity coefficient of species i in the gas where Φ phase at P and system temperature T, and fil0 is the standard state fugacity of species i in the liquid phase at a standard temperature and pressure. Therefore, the mole fraction ratio of i between the two phases in equilibrium is given by ^ ig P x il Φ : ¼ x ig γil f il0

ð3:3:58Þ

If the gas phase behaves as an ideal gas and the liquid ^ ig ffi 1, phase behaves as an ideal solution, we know that Φ γil ffi 1 and pig ¼ x ig P. If the gas phase is distinctly in the nonvapor region (see item (5), Table 3.3.2), it is known from thermodynamics that lim ^f il ¼ x il f il0 ¼ x il H i ;

x il !0

ð3:3:59Þ

where Hi is Henry’s law constant. The relation (3.3.58) is then reduced to either x ig ¼ ðH i =PÞx il

ð3:3:60aÞ

or

25

Particle distribution between two immiscible phases has been considered separately at the end of this section.

Physicochemical basis for separation

pig, partial pressure of species in gas (mmHg)

138

2000 T °C

1500

SO2–water

20

NH3–water

20

O2–water

25

1000

500

0 0.000

0.100

0.200

0.300

xil, mole fraction of gas i in water Figure 3.3.3. Solubility behavior of three different gas species in water. The value of the Henry’s law constant may be calculated from (pig /xil) = Hi. For example, for the SO2–water system, at a pig = 450 mm Hg, xil 0.017. Therefore Hi = (450/0.017) mm Hg/mole fraction = 2.65  104 mm Hg/mole fraction = 35 atm/mole fraction.

pig ¼ H i x il ;

ð3:3:60bÞ

which is Henry’s law of solubility of gas species i in the liquid under consideration. Figure 3.3.3 illustrates the Henry’s law behavior of a number of gases in water according to relation (3.3.60b). These are based on values of Hi available in the literature for three gases, NH3, SO2 and O2 (Geankoplis, 1972; Perry et al., 1984). Note that the solubility of O2 is so low that its mole fraction in the liquid phase does not show up at the scale used. Further, the lower the solubility of a gas species, the higher the value of Hi. The liquid phase is often called absorbent. If the gaseous phase can be considered as a vapor, then the distribution of a species between the vapor and liquid phase is of interest. Obviously, the species can be a major constituent of each phase here. The equilibrium ratio Ki of a component i is defined as (equation (1.4.1)) Ki ¼

x i1 : x i2

Generally, phase 1 is the vapor phase ( j ¼ v) and phase 2 is the liquid phase ( j ¼ l). At equilibrium, use relations (3.3.56) and (3.3.57); then, Ki ¼

x iv γ f0 ¼ il il ; ^ iv P x il Φ

ð3:3:61Þ

since ^f il ¼ ^f iv . For low to moderate pressures, the standard state fugacity in the liquid phase is the fugacity of pure liquid at the temperature and pressure of the system (Smith and Van Ness, 1975):

sat f 0il ¼ f il ¼ P sat i Φi ;

ð3:3:62Þ

is the vapor pressure of pure component i at where P sat i the system temperature and Φsat i is the fugacity coefficient of pure i at P sat at the system temperature. For pressures i when the vapor phase is an ideal gas, Φsat i ffi 1. Further, ^ iv ffi 1. If the liquid phase is an ideal solution, then Φ γil ffi 1. Thus x iv P sat ¼ Ki ¼ i ; P x il

ð3:3:63Þ

where the system pressure is P, and both the vapor and the liquid phase behave ideally. When written as x iv ¼ ðP sat i x il =PÞ;

ð3:3:64Þ

we obtain the well-known Raoult’s law of vapor–liquid equilibrium. An alternative representation of Raoult’s law is useful for the case of continuous chemical mixtures. Rewrite relation (3.3.63) as Px iv ¼ piv ¼ P sat i x il ;

ð3:3:65Þ

where piv is the partial pressure of species i in the vapor phase behaving ideally. For a continuous chemical mixture of vapor and liquid phases at equilibrium, let fl(M) and fv(M) be the molecular weight density functions of the liquid and vapor phases, respectively. If the vapor pressure of a species of molecular weight M and T is indicated by Psat (T;M), then Raoult’s law for species in the molecular weight range M to M þ dM is Pf v ðMÞ dM ¼ P sat ðT;MÞf l ðMÞ dM;

3.3

Equilibrium separation criteria closed vessel

i.e. sat

Pf v ðMÞ ¼ P ðT;MÞf l ðMÞ:

ð3:3:66Þ

One can use density functions fv and fl as functions of quantities other than molecular weight, e.g. boiling point, carbon number. In the vapor–liquid or gas–liquid equilibrium studied above, the vapor or gas species existed as molecules of that species in the liquid. With diatomic gases, e.g. H2, O2, N2, etc., and liquid metal, it is, however, found that the gas is dissolved atomically in the molten metal. For example, in molten iron, the following equilibrium has been suggested for nitrogen gas (Darken and Gurry, 1953): N2 ðgÞ ¼ 2NðlÞ:

ð3:3:67Þ

139 illustration of each type of system.) In the general case, we use the equilibrium criterion (3.3.6) for solute i between two immiscible liquid phases j ¼ 1,2, namely μi1 ¼ μi2. But at any P, T, μil ðPÞ ¼ μ0il ðP 0 Þ þ RTℓn ail þ V ij ðP  P 0 Þ from (3.3.20c) and (3.3.21) for μij. Therefore V i1 ðP  P 0 Þ þ RTℓn ai1 þ μ0i1 ðP 0 Þ ¼ RT ℓn ai2 þ μ0i2 ðP 0 Þ þ V i2 ðP  P 0 Þ:

ð3:3:68Þ

Case 1 If pure solute i exists as a liquid at P and T, then the standard state is independent of phase 1 or phase 2 (see item (4), Table 3.3.2): μ0i1 ðP 0 Þ ¼ μ0i2 ðP 0 Þ:

ð3:3:73Þ

ðai1 =ai2 Þ ¼ 1;

ð3:3:74Þ

x i1 γi2 ¼ : x i2 γi1

ð3:3:75Þ

This implies

For an ideal gas, aN2 g ¼ x N2 g ¼ ðpN2 g =PÞ:

ð3:3:69aÞ

For the liquid phase, the activity of the atomic nitrogen species is related to its mole fraction: aNl ¼ x Nl γNl :

ð3:3:69bÞ

Substituting these into (3.3.68), we get

x Nl ¼

 1=2 ðpN2 g Þ1=2 K : P γNl

ð3:3:70Þ

Thus, the mole fraction of atomic nitrogen in the liquid metal phase is proportional to the square root of the partial pressure of molecular nitrogen in the gas phase. This square-root dependency is known as Sievert’s law and is found to be valid for diatomic gases like O2, H2, etc. The effect of a chemical reaction on solute distribution between two phases in equilibrium has been considered in great detail in Chapter 5. 3.3.7.2

Liquid–liquid equilibrium

Next, we consider the distribution of a solute in liquid– liquid equilibrium. Liquid–liquid systems can be of three main types: aqueous–organic, nonpolar organic–polar organic; aqueous–aqueous. Both phases in each case must be immiscible with each other. (See Section 4.1.3 for an

ð3:3:72Þ

Two cases arise depending on the solute, P and T if we assume V i1 ¼ V i2 .

The equilibrium constant for this chemical reaction may be written in terms of activities of each species as ðaNl Þ2 K¼ : aN2 g

ð3:3:71Þ

i.e.

In such a case, the mole fraction of solute i in the two liquid phases 1 and 2 will be different only if the solute behaves nonideally in both phases and γi2 6¼ γi1. Partitioning of, say, acetic acid at 25  C between water and isopropyl ether will fall into this category since pure acetic acid is a liquid. Case 2 Pure solute i does not exist as a liquid at P and T; an infinite dilution standard state is necessary (item (5), Table 3.3.2). In general, such standard state values μ0ij (P0) are dependent on phase j. Therefore, from relation (3.3.72),  0 0  0 ai1 μ ðP Þ  μ0i1 ðP 0 Þ Δμi ¼ exp ; ¼ exp i2 RT ai2 RT where Δμ0i ¼ μ0i2 ðP 0 Þ  μ0i1 ðP 0 Þ:

ð3:3:76Þ

  γi2 Δμ0 exp  i : RT γi1

ð3:3:77Þ

Correspondingly, x i1 ¼ x i2



If the solutions in each phase are dilute enough for γi2 ffi 1 and γi1 ffi 1, then (xi1/xi2) is a constant:

140

Physicochemical basis for separation

cm3 CH4 STP/cc. Polymer

10.0 9.0

25 °C

8.0

35 °C 45 °C

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0 0

2 4

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Pressure (atm. abs.)

Figure 3.3.4. Solubility of methane in oriented polystyrene. Reprinted from Journal of Membrane Science, Vol. 1, W. R. Vieth, J. M. Howell and J. H. Hsieh, “Dual sorption theory,” pp. 177–220. Copyright (1976), with permission from Elsevier.

ðx i1 =x i2 Þ ¼ κN i1 ;

ð3:3:78Þ

0 where κN i1 can be determined from μij  s. This behavior is identified as the Nernst distribution law. Even if the amount of solute i, and therefore its mole fraction, is varied in one phase, the ratio of the solute mole fractions in the two phases remains constant. An alternative form is used for dilute solutions:

ðC i1 =C i2 Þ ¼ κi1 ;

ð3:3:79Þ

where kil is the constant distribution coefficient. 3.3.7.3

Gas–membrane equilibrium

The dissolution of a gas in a nonporous organic polymeric membrane is an example of gas–membrane equilibrium. Organic polymeric membranes can be amorphous, semicrystalline or crystalline. Crystalline regions are virtually impenetrable. Therefore practical polymer membranes are either amorphous or semicrystalline. Semicrystalline polymers are a combination of amorphous and crystalline regions. The amorphous region/membrane may be rubbery or glassy. The gas molecules dissolve in the amorphous region of the membrane as if it were a liquid. For permanent gases at temperatures T > Tci (= the critical temperature of gas species i) and nonporous rubbery polymeric membranes, this behavior has been observed as long as there is no interaction with the membrane material. Further, under these conditions, Henry’s law is obeyed for gases such as H2, He, Ne, Ar, O2, N2, CH4 (Stern and Frisch, 1981). The form of Henry’s law used in gas– membrane equilibrium with the gas phase behaving as an ideal gas is C im ¼ Sim pig ;

ð3:3:80Þ

where Sim is the solubility coefficient of gas species i, Cim is the concentration of species i in the membrane exposed to a gas, with the species i partial pressure being pig. The commonly used units of the solubility coefficient Sim are (cm3(STP))/(cm3 ∙ cm Hg), with Cim and pig expressed, respectively, in cm3(STP)/cm3 and cm Hg. The gas species is often identified as the penetrant. Generally, the more condensible the gas, the higher the solubility coefficient. For a useful correlation of solubility coefficient with the normal boiling point, see Stannett et al. (1979). Sometimes, the solubility coefficient is also reported as the sorption coefficient. Figure 4.3.4 illustrates the values of the solubility coefficient for a number of gases and vapors in natural rubber membranes. Section 4.3.3, specifically, equations (4.3.43a–c) and (4.3.44a), illustrate how the solubility coefficient Sim depends on the polymer, the temperature as well as the critical temperature Tci of the gas/vapor species for amorphous polymers. When the organic polymer is glassy (i.e. the system temperature is lower than Tg, the polymer glass transition temperature), the solubility behavior of a pure gas species in the membrane is different from Henry’s law (Figure 3.3.4): C im

C 0 Hi bi P ½1 þ bi P 

¼

Sim P þ

¼

C dim þ C H im :

ð3:3:81Þ

At very low pressures, where bi P > 1, a linear behavior ðC im ¼ Sim P þ C 0 Hi Þ is again observed. Figure 3.3.4 displays how a low-pressure linear region is connected by a nonlinear curved behavior to the high-pressure linear region for dissolution of CH4 in a polystyrene film (Vieth et al., 1976). The first term in the above isotherm corresponds to species dissolving according to Henry’s law. The second term is due to species dissolving according to the Langmuir isotherm (see Section 3.3.7.6). Specifically, the Langmuir species are assumed to be sorbed in microvoids in the membrane (regions having 0.5–0.6 nm diameter). The overall behavior is described as the dual sorption26 mode for two different dissolution modes of gas molecules in the glassy membrane. When a binary gas mixture (i ¼ 1, 2) is in equilibrium with a glassy polymeric membrane, the following sorption behavior has been suggested (Koros, 1980):

26

It has been suggested that there is no sharp boundary between absorption and adsorption. The term “sorption” was introduced by J. W. McBain in 1909 to describe situations where both are important (Bikerman, 1970).

3.3

Equilibrium separation criteria closed vessel

C 1m ¼ S1m plg þ h

C 0 H1 b1 p1g 1 þ b1 p1g þ b2 p2g

C 2m ¼ S2m p2g þ h

C 0 H2 b2 p2g 1 þ b2 p2g þ b1 p1g

i;

ð3:3:82aÞ

i:

ð3:3:82bÞ

Here, S1m and S2m are the solubility coefficients of those molecules of species 1 and 2 which dissolve according to Henry’s law. The utility of such a mixture sorption isotherm model has been verified (see, for example, Sanders et al. (1984) for a mixture of CO2 and C2H4 in poly(methyl methacrylate)). The gas–membrane equilibrium is basically somewhat different if the membrane is made of a metal or an alloy and the gas is diatomic. We have already noted that a diatomic gas such as N2, H2 or O2 dissolves in a molten metal according to Sievert’s law, i.e. the mole fraction of the gas (in the atomic state) in the molten metal is proportional to the square root of the gas partial pressure. The same equilibrium behavior, relation (3.3.70), is also observed between a gas and a solid metallic membrane. It is therefore necessary to assume that the gas is present in the atomic state while developing any relation for membrane transport of diatomic gases. 3.3.7.4 Liquid–membrane equilibrium and liquid–porous sorbent equilibrium There are two aspects to liquid–membrane equilibrium. The first one is concerned with the osmotic equilibrium between two solutions on two sides of a semipermeable membrane permeable to the solvent and impermeable to the solute; the second one covers partitioning of the solute between the solution and the membrane. Both porous and nonporous membranes are of interest. The second aspect is also useful for porous sorbent/gel particles. In osmotic equilibrium between two regions 1 and 2 separated by a semipermeable membrane, the pressures P1 and P2 of the two regions are usually different (see Figure 1.5.1). For the solvent species i transferable between the two regions, we have, at equilibrium, μi1 ðP 1 ;T;x i1 Þ ¼ μi2 ðP 2 ;T;x i2 Þ

ð3:3:83Þ

from relation (3.3.16). Using expression (3.3.21) for μij (Pj,T, xij), we get μ0i1 ðP 0 ;TÞ þ V i1 ðP 1  P 0 Þ þ RTℓn ai1

¼ μ0i2 ðP 0 ;TÞ þ V i2 ðP 2  P 0 Þ þ RTℓn ai2 :

ð3:3:84Þ

If the pure ith species exists in the same physical state in both phases at P0 and T, then μ0i1 ðP 0 ;TÞ ¼ μ0i2 ðP 0 ;TÞ ) V i1 ðP 1  P 2 Þ ¼ RTℓn

ai2 ; ai1 ð3:3:85Þ

141 where we have assumed V i1 ¼ V i2 . This relation illustrates how the pressure difference between two phases under osmotic equilibrium can be tied to the composition difference between the two solutions. For a one solute–one solvent system with a semipermeable membrane impervious to solute i but permeable to solvent s, the above results can be reexpressed as follows: V s ðP 1  P 2 Þ ¼ RTℓn

as2 ¼ V s ðπ1  π2 Þ; as1

ð3:3:86aÞ

where π1 and π2 are the osmotic pressures of the solutions in region 1 and region 2, respectively. If π1 > π2, aS2 > aS1; further, P1 > P2 for osmotic equilibrium. The osmotic pressure of a dilute solution of small molecules of species i is often calculated from the van ’t Hoff equation πi ¼ C i RT:

ð3:3:86bÞ

Consider now the second type of equilibrium when a solute partitions between the solution and the membrane, which is no longer semipermeable to the solute: the solute distribution is somewhat similar to that in liquid–liquid equilibrium of the type (3.3.78) or (3.3.79). Solute partitions into the membrane, which is assumed to be nonporous, and is dissolved in it as if the membrane were a liquid. For low solute concentrations in the membrane and no swelling, the following distribution equilibria generally hold: i ¼ 1; . . . ;n:

ðC im =C il Þ ¼ κim ;

ð3:3:87Þ

If the membrane happens to be porous (or microporous) and uncharged, the nature of the liquid–membrane equilibrium will be determined by the relative size of the solute molecules with respect to the pore dimensions in the absence of any specific solute–pore wall interaction. Similar considerations are also valid for liquid–porous sorbent equilibria. If the solute dimensions are at least two orders of magnitude smaller, then the solute concentration in the solution in the pore should be essentially equal to that in the external solution. However, the solute concentration in the porous membrane/porous sorbent/gel will be less than that in the external solution due to the porosity effect. Assuming that the solute exists only in the pores of the membrane/porous sorbent with a porosity εm, the value of kim should be equal to the membrane or sorbent porosity εm if the solute characteristic dimensions are at least two orders of magnitude smaller than the radius of the pore. When the solute dimensions are larger and there are no specific solute–pore wall interactions, the partitioning effect is indicated by a geometrical partitioning factor for a cylindrical pore as p

C im ¼ C il



1

ri rp

2

;

ð3:3:88aÞ

142

Physicochemical basis for separation

Center line of cylindrical pore

Liquid-filled membrane pore

2rp Membrane wall

ri ri (rp − ri)

ri Molecules of radius ri

Feed liquid

Figure 3.3.5A. Geometrical partitioning effect: only a distance of r p  r i from the pore center is available for locating the center of the molecules of the solute. p

where C im is the concentration of solute i in the membrane/sorbent pore liquid, rp is the pore radius and ri is the radius of the solute molecule (assumed spherical). The center of the solute molecule cannot approach the wall beyond a radius of rp – ri (Figure 3.3.5A). Thus only the volume fraction πðr p  r i Þ2 =π r 2p of the pore can have solute molecules at the same concentration as the external solution, Cil; yet the solvent molecules exist effectively throughout the pore, making the pore liquid (total volume π r 2p  1 for unit length) leaner in solute molecules compared to the liquid phase concentration Cil outside the pores: p

C im ðπr 2p Þ  1 ¼ C il π ðr p  r i Þ2  1:

ð3:3:88bÞ

This was originally suggested by Ferry (1936) as a reduction in cross-sectional area for diffusion. Giddings et al. (1968) have developed the same result theoretically by considering the limitations in orientations and structural configurations of macromolecules to avoid sterical overlap with the pore wall. If the pore has the shape of a slit (β ¼ 1), cylinder (β ¼ 2) or a cone (β ¼ 3), the geometrical partitioning factor is given by   ri β p κim ¼ ðC im =C il Þ ¼ 1  ; ð3:3:89aÞ rp where rp is the characteristic pore dimension. See Colton et al. (1975) for a brief review of the relevant literature. In most porous media being modeled as a collection of cylindrical capillaries, most pores are unlikely to be truly cylindrical. Therefore, rp may be replaced by (2/s), where s is the surface area of the wall of the capillary per unit pore volume:  r i s 2 κim ¼ 1  : 2

ð3:3:89bÞ

We will illustrate very briefly now a fundamental approach to calculating the partitioning factor of (3.3.88a) as developed by Giddings et al. (1968). In the formalism of statistical mechanics, the equilibrium partition constant

kim is the ratio of partition functions for molecules within the pores and within the bulk liquid:   ∭exp enp ðr;Ψ ;λÞ=kB T ðdr dΨ dλÞ κim ¼ : ð3:3:89cÞ   ∭exp enb ðλÞ=kB T ðdr dΨ dλÞ

Here the coordinates r, Ψ and λ describe the molecular position, orientation and conformation, respectively. The energy enp in the porous configuration consists of the enM due to intramolecular interactions, an energy enMN due to intermolecular interactions and an energy enMp due to interactions between the macromolecule and the pore; therefore enp ¼ enM þ enMN þ enMP :

ð3:3:89dÞ

Correspondingly, for the bulk liquid, enb ¼ enM þ enMN :

ð3:3:89eÞ

If we choose the condition of infinite dilution, enMN will be zero since the macromolecules are far apart. We further assume that essentially there is only one conformation of the molecule/macromolecule (rigid) (alternatively all conformations have the same energy); therefore, enM ¼ 0. Further, we assume that there are no adsorptive forces between the macromolecule and the pore wall; in addition, the pore wall and macromolecule are distinct and discontinuous: consequently, exp(enMP/kBT) has the value of 1 for molecular configurations free from overlap with the wall and the value of 0 for configurations of overlap with the wall. In random-pore networks, an ensemble average of exp(enMP/kBT), q (r, Ψ, λ), can replace it: κim ¼

∭qðr;Ψ ;λÞ dr dΨ dλ ∭dr dΨ dλ

:

ð3:3:89fÞ

This quantity q(r, Ψ, λ) may be defined in general as the probability that a molecule having a given configuration does not intersect a pore wall. If we define a local equilibrium partition constant ki,loc(r) such that

3.3

Equilibrium separation criteria closed vessel

κi;loc ðrÞ ¼

∭qðr;Ψ ;λÞ dΨ dλ ∬dΨ dλ

;

ð3:3:89gÞ

where κi,loc(r) depends only on the position coordinate r in the pore, then ð κi;loc ðrÞ dr ð κim ¼ : ð3:3:89hÞ dr For rigid macromolecules/molecules, dependence on λ disappears, and we get ð κi;loc ðrÞ dr ∬qðr;Ψ Þ dr dΨ ð : ð3:3:89iÞ ¼ κim ¼ ∬dr dΨ dr To illustrate, consider spherical molecules of radius ri and pores of radius rp in a rigid medium. For pores of infinite length, it is obvious that, for radial locations r of the center for the molecule, 0 r ðr p  r i Þ; κim ¼ 1;

ð3:3:89jÞ

ðr p  r i Þ r r p ; κim ¼ 0:

ð3:3:89kÞ

Use now equation (3.3.89h); it is now the ratio of two circular areas, one with the diameter (2(rp  ri)) corresponding to (3.3.89j) in the numerator and the denominator corresponding to the pore area of diameter 2rp:   π4ðr p  r i Þ2 =4 ri 2 ; ð3:3:89lÞ κim ¼ ¼ 1  π 4 r 2p =4 rp which is the relation (3.3.88a) described earlier. The radius, ri cm, of the solute molecule (which is assumed spherical) needs to be known in liquid–porous membrane/porous sorbent equilibrium. For relatively small molecular species, the specific molar volume of the solute will lead to ri ¼



3M i ~ i 4πNρ

1=3

ð3:3:90aÞ

;

where Mi is the molecular weight, ρi is the solute density ~ is Avogadro’s number (6.022  1023). Spriggs and Li and N (1976) have therefore suggested that, for solids, 8

2r i ¼ 1:465  10 ðM i =ρi Þ

1=3

;

ð3:3:90bÞ

where ρi is in g/cm3 and ri is in cm, whereas, for liquids, 2r i ¼ 108 V im 1=3 ; where Vim is the molecular volume at the normal boiling point (MVNBP) in cm3 (Reid et al., 1977). Expression (3.3.90a) is also used for determining ri values for larger molecules, like globular proteins, where one usually assumes that ð1=ρi Þ ffi 0:75 cm3 =g is the partial specific volume.

143 Since the solute molecule is in a solution, an alternative estimate of ri is provided by the Stokes–Einstein equation: r i ¼ kB T=6πD0is μ;

ð3:3:90cÞ

B

~ D0is is the difwhere k is Boltzmann’s constant (¼ R=N), fusivity of solute i in the solvent and μ is the solvent viscosity. (This ri is the same hydrodynamic radius discussed in Section (3.1.3.2).) This solute radius based on the Stokes–Einstein equation and identified therefore as the hydrodynamic radius has been correlated with the molecular weight of globular proteins by (Tanford et al., 1974) logðr i ρi Þ ¼ log M i  ð0:147  0:041Þ;

ð3:3:90dÞ 3

where the recommended value of (1/ρi) is 0.74 cm /g. For a macromolecule in general, whether it can enter a given pore of certain size or not depends on its radius of gyration rg, which depends on the molecular weight Mi and the macromolecular shape β1 via r g / M i β1 :

ð3:3:90eÞ

For rod-shaped molecules, β1 ¼ 1, for spheres, β1 ¼ (1/3), whereas for flexible chains, β1 ¼ (1/2). The radius of gyration is proportional to the hydrodynamic viscosity based radius rh determined from the intrinsic viscosity of the macrosolute via the hydrodynamic volume Vh (Ladisch, 2001, pp. 557–561): rh ¼



3Vh 4π

1=3

;

Vh ¼

½η M i : ~ Ψ sh N

ð3:3:90fÞ

Here [η] is the intrinsic viscosity and Ψsh is a shape factor having the value of 2.5 for spheres; Ψsh is greater than 2.5 for ellipsoids. For compact globular proteins, r h ¼ 0:718M i 1=3 ;

ð3:3:90gÞ

which is very close to the estimate of ri by equation (3.3.90b) for ð1=ρi Þ ffi 0:75 cm3 =g. Hagel (1989) has indicated that the estimates of macromolecule/protein dimensions via the hydrodynamic radius, ri, and the hydrodynamic viscosity based radius, rh, follow the relation rh > ri :

ð3:3:90hÞ

The porous medium (sorbent, membrane, gel particle) can be made of soft spherical particles that swell quite a bit when immersed in a solvent; they are called gels. If they swell in water, they are called hydrogels; however, the polymers are crosslinked so that they retain their overall structure. Of course, the polymers are soft and compress under pressure; a typical example would be gels (which are crosslinked) based on agarose (Figure 3.3.5B). Such gel particles are widely used in a variety of separation techniques for separating macromolecules/proteins, etc.

144

Physicochemical basis for separation

50

Temperature, °C

40

Liquid e

g

30

in ez

lin

e Fr

e

g ltin

20

lin

e

M

Solid

10

0 0

0.2

Figure 3.3.5B. Microporous polymer in the form of a gel particle formed by crosslinking linear chains of monomer; crosslinks are shown by heavy lines.

Various empirical expressions have been developed that describe the partition coefficient kim of solute i as a function of the molecular weight Mi or hydrodynamic radius ri. A few are illustrated in the following: ðκim Þ1=3 ¼ a  b ðM i 1=2 Þ;

κim ¼ a þ b log M i ;

κim ¼ a þ br i :

ð3:3:90iÞ

(For more details, see Hagel (1989); for an introduction in a gel permeation chromatography context, see Ladisch (2001).) If the molecule cannot enter the pore, kim ¼ 0. From the relation (3.3.90e), for a given molecular weight Mi, the radius of gyration rg is the highest for a rod-shaped molecule and lowest for a sphere. Therefore, for a given molecular weight, a rod-shaped molecule has the lowest kim and a spherical molecule the highest kim.

(1) a solid solution in equilibrium with a molten mixture; (2) a solute being leached from a solid mixture by a solvent; (3) crystallization equilibria between a crystal and the mother liquor. When components 1 and 2 are miscible in the solid phase over the whole composition range between pure 1 and pure 2, we have a solid solution. For example, Figure 3.3.6A shows the temperature vs. composition diagram for a naphthalene/β-naphthol system. The lower curve is

0.6

0.8

1.0

xβ-naphthol Figure 3.3.6A. Phase diagram for the system naphthalene/βnaphthol. (J. M. Prausnitz, Molecular Thermodynamics of Fluid-Phase Equilibria, 1st edn. © 1969, pp. 404. Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ.)

known as the solidus and the upper curve is known as the liquidus. The system is completely liquid above the liquidus and is completely solid below the solidus. In between, both liquid and solid phases exist. At any temperature, the solid and liquid phases at equilibrium have different compositions. Not all solid solutions span the whole composition. For example, in iron–copper alloys around 1000  C, a homogeneous solid solution exists only below a composition of 10 wt% copper (Darken and Gurry, 1953). To determine the equilibrium ratio of species i between the melt (j ¼ 1 ¼ l) and the solid solution ( j ¼ 2 ¼ s), note that, at equilibrium,

3.3.7.5 Liquid–solid equilibria: leaching, crystallization, precipitation Liquid–solid equilibria in which a liquid and a solid phase (or solid phases) are coexisting can be of the following types:

0.4

^f ¼ ^f ) ^f ¼ ^f i1 i2 il is

ð3:3:91Þ

But ^f ¼ γ x il f 0 il il il

and ^f is ¼ γis x is f is0 ;

ð3:3:92Þ

x il x i1 γ f0 f0 ¼ ¼ K i ¼ is is0 ffi jideal is0 : x is x i2 γil f il f il

ð3:3:93Þ

leading to

The final simplification is valid only for ideal behavior. If experimental data on Ki are not available, the ratio f 0is =f 0il has to be calculated by procedures that are elaborate and complicated (Prausnitz, 1969). If the behavior is nonideal, the liquidus and solidus behavior shown in Figure 3.3.6A is changed to more complicated ones similar to maximum or minimum boiling azeotropes (see Section 4.1.2).

3.3

Equilibrium separation criteria closed vessel

Crystals present

C

p2 B

Supersaturated solution

A″ A′ A

Concentration

p3 Csat

Metastable region A′

B′ p1

Solubility curve f Unsaturated solution

T2

Tsat

T1

Temperature

Figure 3.3.6B. Concentration vs. temperature behavior for saturation, supersaturation and crystallization.

Consider now the leaching of a solid mixture (which can be treated as a collection of aggregates of pure components) by a solvent. In the solid phase, each component fugacity is then equal to its pure component solid fugacity f 0is . Let the leaching solvent be insoluble in the solid mixture. At equilibrium between the solid mixture and the solvent, for species i, f is ¼ f is0 ¼ ^f il ¼ γil x il f il0 :

ð3:3:94Þ

x il ¼ f 0is =γil f il0 :

ð3:3:95Þ

So

If the solution temperature is lower than the melting point of solute i (which is usually the case), and if vapor pressures could be substituted for f is0 and f il0 , we find x il ¼

P vap is γil P vap il

:

ð3:3:96Þ

If the system temperature is the normal melting temperature of the solid, then f is0 ¼ f il0 and x il ¼ ð1=γil Þ:

ð3:3:97Þ

Refer to Prausnitz et al. (1999) for a more comprehensive treatment on standard state fugacity calculations in solid– liquid systems. At any given temperature and pressure, a solvent has a certain capacity for a solute. When the limit is reached, the solution is said to be saturated. This limit generally increases with temperature and provides the solubility curve (Figure 3.3.6B). Conversely, if a solution is cooled to a temperature below the saturation temperature, there would be a tendency for the solute to come out of the solution. The solute may crystallize or precipitate. In reality, crystallization does not occur till the solution gets supersaturated to some extent, as shown in Figure 3.3.6B.

145 So there are three regions of importance in crystallization (Miers, 1927): an unsaturated region (with no crystals), a supersaturated region (with crystals) and an intermediate region, where crystal growth will occur if there are crystals, but there will be no nucleation (i.e. crystal formation without any crystals). This intermediate region is also called the metastable region. A feed solution represented by f in the unsaturated region can produce crystals if (1) it is cooled to the supersaturated region p1, (2) the solvent is evaporated to reach the supersaturated region p2 at constant T, or (3) a combination of cooling and evaporation is implemented to reach p3. (An additional method employed to attain supersaturation involves the addition of an antisolvent). It is now believed that there are a number of other factors such as solution cooling rate, mechanical shock or friction which influence the locations of the curves. True equilibrium is rarely attained in crystallization processes. Consider point A on the solubility curve (Figure 3.3.6B). The saturated solution concentration of solute at temperature Tsat is Csat. When this solution loses some solvent at the same temperature T and reaches the curve for the supersaturated solution (point B), the solution concentration becomes C > Csat. If the solution were saturated at this higher concentration C, its temperature would have to be higher, namely T1, corresponding to point A0 on the solubility curve. At temperature Tsat, the molar supersaturation ΔC is defined as ΔC ¼ C  C sat :

ð3:3:98aÞ

The fractional supersaturation, or relative supersaturation, s, is defined as s¼

ΔC C  C sat C ¼ ¼  1; C sat C sat C sat

ð3:3:98bÞ

whereas the supersaturation ratio S is defined as S¼

C ¼ 1 þ s: C sat

ð3:3:98cÞ

If instead of removing the solvent at Tsat, we lower the 0 solution temperature and reach point B at T2 ( 50 nm to micropores < 2 nm, to pores of molecular dimensions (0.3–1 nm and above) in zeolites, which are crystalline aluminosilicates. The pores are invariably tortuous. Gas/vapor molecules are adsorbed on the pore surfaces generally by physical adsorption, involving relatively weak intermolecular forces, and rarely via chemisorption, involving bond formation. The physical adsorption due to these weak forces is assumed to take place on “sites” strewn throughout the pore surface area. To understand the functional nature of gas–solid equilibria for physical adsorption processes, one may develop a relation between Γiσ and the gas phase composition using equation (3.3.43b) and the illustrations thereafter. Two alternate approaches will be illustrated here using the same basic equation (3.3.43b) and a binary system of species i ¼ 1,2. Recognize that Sσ is the total interfacial area per unitmass ofadsorbent and therefore may be P2 expressed as i¼1 qiσ Sσ , where Sσ is the molar surface area. Therefore (Van Ness, 1969)

149

Mole fraction benzene in gas phase

3.3

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Mole fraction benzene in adsorbed phase

ð3:3:107Þ

Figure 3.3.8A. Vapor-adsorbed phase equilibria of benzene and cyclohexane on activated charcoal at 30  C. : P = 0.27 kPa; •: P = 1.33 kPa; o: P = 13.3 kPa. Reprinted with permission from A. L. Myers et al., AIChE J., 28(1), 97 (1982). Copyright © [1982] American Institute of Chemical Engineers (AIChE).

x 1σ dx 1g x 2σ dx 2g Sσ Sσ dπ ¼ d ℓn P þ dπ ¼ 2 þ ; X RT x 1g x 2g qiσ RT

P observed data of 2i¼1 qiσ . The experimental and numerical procedures are described in Myers et al. (1982). The adsorption behavior of benzene in a benzene– cyclohexane mixture for the adsorbent, activated charcoal, is shown for a particular temperature in Figure 3.3.8A. P Note that 2i¼1 qiσ , i.e. the total number of moles adsorbed on the solid adsorbent surface per unit adsorbent mass, is obtained from the experimental data of mass of gas adsorbed per mass of adsorbent using

2 2 X X q Piσ d μig ¼ Sσ dπ ¼ x iσ d μig : qiσ i¼1 i¼1

For ideal gas behavior in the gas phase, Sσ dπ ¼

2 X

x iσ RT d ℓn x ig P;

i¼1

i¼1

ð3:3:108Þ

since x 1σ þ x 2σ ¼ 1. Further, x 1g þ x 2g ¼ 1. It follows, therefore, x 1σ  x 1g Sσ dπ dx 1g : ¼ d ℓn P þ 2 X x 1g ð1  x 1g Þ qiσ RT

ð3:3:109Þ

i¼1

For an ideal gas, this is a form of Gibbs adsorption isotherm for a binary mixture: it relates the species 1 mole fraction in gas phase, x1g, to species 1 mole fraction, x1σ, in the adsorbed phase, the adsorbate (Van Ness, 1969). At constant total gas pressure, we obtain !  2 X x 1σ  x 1g ∂ðπ Sσ =RTÞ ¼ qiσ : ð3:3:110aÞ ∂x 1g x 1g ð1  x 1g Þ P i¼1 At constant gas phase composition, x1g, we get !  2 X ∂ðπ Sσ =RTÞ ð3:3:110bÞ qiσ =P: ¼ ∂P x 1g i¼1 These two first-order partial differential equations can be solved numerically to express x1σ and ðπ Sσ =RTÞ as a function of total pressure P and gas phase composition x1g for

2 X i¼1

qiσ ¼

total mass of gas adsorbed=mass of adsorbent : M 1 x 1σ þ M 2 x 2σ ð3:3:110cÞ

For real gas mixtures, Ritter and Yang (1989) have developed simple numerical and graphical procedures to determine adsorbed phase compositions at elevated pressures. An alternative procedure suggested by Myers and Prausnitz (1965) based on the molar Gibbs free energy of mixing provides Px ig ¼ P 0i ðπÞ γiσ x iσ

ð3:3:111aÞ

as the relation between the gas phase mole fraction, xig, and the adsorbate phase mole fraction, xiσ, for a total gas pressure P. Here P 0i ðπÞ is the equilibrium gas phase pressure for pure i adsorption at spreading pressure π (which should be the same for the mixture). Further, the gas phase was assumed to behave ideally. If the adsorbate phase also behaves ideally, γiσ ¼ 1. Under this condition,

150

Physicochemical basis for separation P 0i ðπÞ x iσ ; P

ð3:3:111bÞ

i.e., the equilibrium behavior is similar to that of Raoult’s law in vapor–liquid equilibrium (equation (3.3.64)) if P 0i ðπÞ can be thought of as the pure adsorbate vapor pressure for component i at temperature T and spreading pressure π of the mixture. The procedures for obtaining the gas–solid adsorption isotherms briefly outlined above are based on a solution thermodynamic Gibbs approach. There are two other approaches: the potential theory and the Langmuir approach. The latter approach is based on a dynamic equilibrium between the rates of adsorption and desorption of any species from adsorption sites on the solid surface. Since most data correlation in separation processes employs the Langmuir approach (Yang, 1987), with the adsorbate amount expressed as a function of species partial pressure, such isotherm types will be briefly identified here. For a comprehensive introduction to adsorption isotherms and phenomena, consult Ruthven (1984) and Yang (1987). Consider a few types of isotherms observed in practice (Figure 3.3.8B); the first one is the common Langmuir type. For such an isotherm, initially the amount adsorbed is linearly proportional to the gas pressure P. After further increases in gas pressure, the amount adsorbed approaches a limit, which does not change with further increases in gas pressure. Under these conditions, all adsorption sites become covered with gas molecules, the adsorbate, in a monolayer fashion. The fraction of sites occupied, or the fraction of the maximum gas that can be adsorbed under other conditions, is given, for a pure gas i, by θ¼

qiσ bil P ; ¼ ðqiσ Þmax 1 þ bil P

ð3:3:112aÞ

where ð1=bil Þ is the Langmuir constant for species i. For a gas mixture of n species, θt is the total fraction of sites n X θi ; where θi is covered by all species and is equal to i¼1

the fraction of sites covered by species i: θi ¼

bil pig : n X 1þ bil pig

ð3:3:112bÞ

i¼1

An additional isotherm type is the Freundlich isotherm. For a pure gas (species i), the Freundlich isotherm has a power law relation θ ¼ bif P 1=βi ;

ð3:3:112cÞ

where βi < 1, ¼ 1 or > 1. Figure 3.3.8B illustrates the case corresponding to βi < 1. A log-log plot of the data satisfying this type of isotherm will yield from the slope ð1=βi Þ and bif from the intercept.

Langmuir type

Amount adsorbed

x ig ¼

BET type Freundlich type

Species pressure Saturation pressure

Figure 3.3.8B. Typical gas–solid adsorption isotherms.

For a mixture of n species, θi ¼

bif pig 1=βi : n X 1þ bif pig 1=βi

ð3:3:112dÞ

i¼1

The similarities between the Langmuir gas–solid isotherm for a pure gas or a mixture and the Langmuir component of gas–membrane equilibrium in equation (3.3.81) or (3.3.82a) should be obvious. The BET (Brunauer–Emmett–Teller)-type isotherm in Figure 3.3.8B reflects multilayer adsorption of the adsorbate. After a monolayer adsorbate coverage is achieved in the adsorbent pores, additional molecular layers are formed on top of the adsorbed monolayer by condensation of vapors. In adsorbents having small-diameter pores, multilayer condensation of the adsorbate vapor can fill the pore completely with the liquid adsorbate. This phenomenon is called capillary condensation (Figure 3.3.8C). Consider the curved interface between the vapor phase and the condensed phase of species i in the micropore in this figure. The vapor pressure of the condensed liquid above the concave curved liquid surface in the capillary P sat i;curved is less than that over a plane condensed liquid surface P sat i;Pl :   12 2γ V ij cos θ sat ðP sat ð3:3:112eÞ i;curved =P i; Pl Þ ¼ exp  R Tr (see also relations (3.3.50)–(3.3.53)). This equation is also identified as the Kelvin equation (compare (3.3.54)). As a result, capillary condensation or pore condensation in a fine capillary or pore can take place at a lower value of equilibrium vapor pressure than the saturation vapor pressure P sat i;Pl at that temperature (so far we have used this quantity as P sat everywhere). The magnitude of this i effect may be illustrated for pore condensation of benzene (Ruthven, 1984) (where γ12 for benzene ¼ 29 dyne/cm; V ij ¼ V benzene ¼ 89 cm3 =gmol; T ¼ 293 K; θ 0) for two sat pore sizes, r ¼ 5 nm and 60 nm: the (P sat i;curved =P i; Pl ) value is 0.67 and 0.96, respectively, for 5 and 60 nm pore size. The

3.3

Equilibrium separation criteria closed vessel

151

q r p

A

Monolayer adsorbate

Adsorbate in two layers

Surface tension

Capillary condensation

Figure 3.3.8C. Capillary condensation of adsorbate in the adsorbent micropore.

Kelvin effect becomes important for fine capillaries/ small pores. Another important case of a fluid–solid adsorption system is liquid–solid adsorption. The general principles of adsorption equilibrium between a vapor and an adsorbent are also applicable to liquid–solid adsorption, except the adsorbate concentrations are often quite high for a liquid phase unlike that in gas–solid adsorption (unless trace concentrations of solute are being adsorbed). Adsorption equilibrium may be viewed here in two different ways. In the more conventional approach, both solute in a solution as well as the solvent molecules get adsorbed on the solid adsorbent. Thus the exchange equilibrium is between a solute molecule and a solvent molecule in both phases. An ith species solute molecule in a solution ( j ¼ 2) displaces a solvent molecule (s) adsorbed on any site in the adsorbent ( j ¼ 1) and occupies the adsorbent site: ið j ¼ 2Þ þ sð j ¼ 1Þ , ið j ¼ 1Þ þ sð j ¼ 2Þ:

ð3:3:113aÞ

If the equilibrium constant for this exchange is K, and if the activities of solute i and solvent s molecules in the adsorbed phase are assumed ideal (i.e. ail ffi xil and as1 ffi xs1), then K¼

ai1 as2 x i1 as2 ¼ : ai2 as1 ai2 x s1

ð3:3:113bÞ

Further, in the adsorbed phase, x il þ x s1 ¼ 1. Therefore x i1 ¼



   K K ai2 x s1 ¼ ai2 ð1  x i1 Þ ) x i1 ¼ as2 as2

the sites occupied by the solute molecules. This isotherm relating xi1 to ai2 (or xi2) is the equivalent of the Langmuir adsorption isotherm (3.3.112b) for liquid–solid adsorption. Often this isotherm is also expressed in terms of the molar solute concentration in the liquid phase, Ci2, and the moles of solute i adsorbed per unit mass of adsorbent, qil : qi1 ¼ qi1 jmax

ei C i2 ; 1 þ ei C i2

ð3:3:113dÞ

where qi1 jmax is the maximum value of qi1 for complete coverage of all sites by the solute species i and ei is a constant for solute i. Sometimes, the isotherm is similar to the Freundlich isotherm (3.3.112d) and is represented as θ ¼ ei ðC i2 Þ1=βi :

ð3:3:113eÞ

The graphical method for determining parameters of this isotherm has been described in the text following (3.3.112c). In an alternative approach, the adsorbent surface is assumed to have a distinct layer of solvent (in a monolayer); the solute i merely partitions between the solvent in solution and the solvent in the monolayer on the adsorbent: ið j ¼ 2Þ , ið j ¼ 1Þ:

ð3:3:114aÞ

If the solute i is such that an infinite dilution standard state  K is necessary, then, from relation (3.3.79), we get ai2 as2   ¼ θ: K C i1 ai2 1þ ¼ κi1 : as2 C i2 

ð3:3:113cÞ

Under ideal conditions of almost equal sized solute and solvent molecules and fixed surface area per adsorption site on the adsorbent, xi1is equivalent to the fraction θ of

This relation, where Ci1 and Ci2 have the units of gmol/ cm3 of adsorbent and gmol/cm3 of bulk solution, may be expressed in terms of qi1, gmol/mass of adsorbent, and Ci2 as

152

Physicochemical basis for separation qi1 ^ a; ¼ κil V C i2

ð3:3:114bÞ

^ a is the volume of the adsorbed monolayer phase where V per unit weight of adsorbent (cm3/mass of adsorbent). When a hydrophobic porous adsorbent is in contact with an aqueous solution which does not wet the adsorbent pores, a hybrid system is formed. Inside the pores containing air, there are solutes in vapor form which are adsorbed on the surfaces in the pore of the adsorbent (Rixey, 1987). Outside the pores, the aqueous solution exists. Only solutes that are volatile are adsorbed on the adsorbent. Solutes that would not be separated by conventional adsorption in the wetted state may be fractionated now. A liquid–solid adsorption system becomes a de facto vapor–solid adsorption system. 3.3.7.7

Ion exchange systems

Ion exchangers are porous insoluble solid materials having fixed charges.27 When immersed in an electrolytic solution, ions having charges opposite to the fixed charges, counterions, will enter the porous structure; the ion exchanger as a whole is electrically neutral. If such an ion exchange material in particulate form is immersed in another solution containing a different set or type of counterions, there will be an exchange of the counterions already present in the solution in the ion exchanger’s porous structure with those present in the external solution; hence, the characterization of such systems as ion exchange systems. The process is reversible. Porous natural and synthetic insoluble solid materials can act as ion exchangers. These include natural ion exchange minerals (primarily of the alumino-silicate type), synthetic inorganic ion exchangers (e.g. zeolites, zirconium phosphates, etc.), ion exchange coals having weak carboxylic acids and, most importantly, ion exchange resins prepared from organic polymers. Synthetic ion exchange resin particles consist of a three-dimensional crosslinked porous polymeric structure having a high porosity and pore dimensions exceeding 10 nm. The basic polymeric material in such resins is hydrophobic. Hydrophilic ionic groups (see Figure 3.3.9 for an illustration of such groups) are present at a high density throughout this polymeric network. Although the þ introduction of such ionic groups, e.g. SO 3 H ; makes the polymer soluble in water, crosslinking of different polymer chains in the structure makes them insoluble in water. A common polymer is linear polystyrene; divinyl benzene is employed to prepare a crosslinked polymeric matrix.  Resins having fixed negative group, e.g. SO 3 , COO , etc., can exchange cations from an external solution. These

27

Liquid ion exchangers are not considered here; see Section 5.2.2.4.

Cation exchangers : - SO -3, -COO -, -PO 32-, -AsO 32Anion exchangers : - NH +3 ,

NH +2 ,

N+,

S+

Figure 3.3.9. Ionic groups in the polymeric matrix of ion exchange resins.

are called cation exchange resins; they are acidic. Resins þ having fixed positive charges, e.g. NHþ 3 ;  N ðCH3 Þ3 , etc., exchange anions from an external solution. They are called anion exchange resins; they are basic in nature. Consult Helfferich (1995) for a detailed account. The governing criterion for equilibrium distribution of a solute between an ion exchange resin and an external solution is relation (3.3.8) when the solute is a weak electrolyte or a nonelectrolyte. The nature of the equilibrium is similar to that with nonionic adsorbents. When the solute is a strong electrolyte, criterion (3.3.29) has to be used due to the presence of fixed ionic charges and counterions in the resin. The distribution equilibrium of a weak electrolyte or a nonelectrolytic solute i between an external solution (w) (generally an aqueous solution) and the ion exchange resin particles ( j ¼ R) is usually described by a distribution coefficient: κiR ¼

C iR : C iw

ð3:3:115Þ

Often, the concentration C iR is described in terms of moles of species i per unit weight of ion exchange resin particle or per unit weight of solvent in the resin particle. In order to make κiR dimensionless, C iw should have similar units. Sometimes κiR is constant; for example, acetic acid has a linear isotherm in styrene-type cation exchangers (Helfferich, 1962, p. 126). In general, there are a variety of interactions between the solute and the resin system, leading to a complex behavior. Distribution equilibria of strong electrolytes between a solution and ion exchange resin will now be considered. First, however, certain unusual aspects of ion exchange systems have to be illustrated. Specifically, the phenomenon of swelling pressure in a resin particle is important. When water or a polar solvent enters ion exchange resin particle pores, it has a tendency to dissolve the resin material. The resin swells due to the incorporation of solvent molecules. The resin particle is, however, prevented from dissolution in the solvent due to crosslinks between the neighboring polymer chains. This essentially implies the presence of elastic forces of resin matrix – the whole resin phase including the solvent is now at a higher pressure – higher than that of the surrounding solution. The pore liquid pressure PR, the pressure in the resin

3.3

Equilibrium separation criteria closed vessel

153

Counterions

Matrix with fixed charges

Co-ions

Figure 3.3.10. Structure of an ion exchange resin (schematic). (After Helfferich (1962, 1995).)

phase, can be related to the external solution (say aqueous) pressure Pw using criterion (3.3.16) for equilibrium. Suppose the external solution is just pure solvent (i ¼ s). Then μsR ¼ μsw ;

ð3:3:116aÞ

where j ¼ R is the resin phase and j ¼ w is the external phase. From expression (3.3.21) for the chemical potential μij ðP j ;T;x ij Þ, where j ¼ R, i ¼ s, we have μsR ðP R ;T ;x sR Þ ¼ μ0s þ V sR ðP R  P 0 Þ þ RT ℓn asR : ð3:3:116bÞ Similarly, μsw ðP w ;T;x sw Þ ¼ μ0s þ V sw ðP w  P 0 Þ þ RT ℓn asw :

ð3:3:116cÞ

0

Assume standard state pressure P and external solution pressure Pw to be the same. Since there is pure solvent in the external solution, asw ¼ 1. Therefore μsR ðP R ;T;x sR Þ ¼ μ0s þ V sR ðP R  P 0 Þ þ RT ℓnasR ¼ μsw ðP 0 ;T;1Þ ¼ μ0s þ V sw  0 þ 0 ¼ μ0s ; so that

ϕR  ϕw ¼

   1 aiw  V i ðP R  P w Þ ; ð3:3:118bÞ RT ℓn ZiF aiR

using the assumption that the two solute partial molal volumes V iw and V iR are equal to V i . This equilibrium relation is valid for any ionic species i; note that (PR – Pw) is the swelling pressure of the resin and (ϕR – ϕw) is the Donnan potential, independent of the species i in a given system. An important feature of relation (3.3.118b) needs to be reemphasized: it is valid for any ionic species i. The electrolyte or solute i in general is not just one ionic species. Thus the distribution of a strong electrolytic solute AY (which produces Aþ and Y ions) between the solution and the resin needs to be known. Let the fixed charge in the resin be negative (Figure 3.3.10). Then Aþ is a counterion. Suppose one mole of electrolyte AY dissociates to give νA moles of ions Aþ and νy moles of ions Y  ; we can follow Helfferich’s (1962) treatment: Z A υA ¼ Z Y υY :

ð3:3:119aÞ

The partial molar volume of the electrolyte AY is 0

V sR ðP R  P Þ ¼ RT ℓn asR ;

ð3:3:117aÞ

0

where (PR  P ) is the swelling pressure in the resin phase and is related to the solvent activity asR in the resin phase. If asw 6¼ 1 and Pw 6¼ P0, then, for V sw ¼ V sR , V sw ðP R  P w Þ ¼ RT ℓn ðasR =asw Þ:

ð3:3:117bÞ

It is now possible to develop the distribution equilibrium relation for a strongly electrolytic solute i in an ion exchange system using equilibrium criterion (3.3.29): μiw  μiR ¼ Z i F ðϕR  ϕw Þ: We know from expression (3.3.21) for μij , μiw ¼ μ0i þ V iw ðP w  P 0 Þ þ RT ℓn aiw and μiR ¼ μ0i þ V iR ðP R  P 0 Þ þ RT ℓn aiR : Substitute these into relation (3.3.118a) to get

ð3:3:118aÞ

V AY ¼ υA V A þ υY V Y :

ð3:3:119bÞ

Use relation (3.3.118b) separately for i ¼ A and i ¼ Y and simplify to get     aAw υA aYw υY ¼ ðPR  PW ÞV AY : ð3:3:119cÞ RT ℓn aAR aYR Replace the single cation and single anion activities by the mean electrolyte activity a as follows: aυAA aυAY ¼ ða Þυ ; where υ ¼ υA þ υY . Now use result (3.3.117b) to obtain  υ  V AY =V sw aR asR ¼ ; aw asw

ð3:3:119dÞ

which relates the electrolyte activity in the external solution to that in the resin. Such a relation for the electrolyte activity in the two phases of an ion exchange system can

154

Physicochemical basis for separation

be reduced to an explicit relation between the molalities of the co-ion Y between the two phases. Consider a cation exchange resin with fixed negative charges – it prefers to exchange only cations from the solution. To demonstrate this preference, rewrite relation (3.3.119d) for the electrolyte AY as A Y A Y aυAR ðasR =asw Þ V AY =V sw : ¼ aυAw aυYw aυYR

ð3:3:120aÞ

Assume υA ¼ 1; υY ¼ 1 (for example, NaCl). Rewriting the activities in terms of molalities and activity coefficients, we obtain mA;R mY;R ¼ mA;w mY;w ðγAw γYw = γAR γYR ÞðasR =asw Þ

V AY =V sw

:

ð3:3:120bÞ

Now employ electroneutrality relation (3.1.108a) in terms of molalities for the resin phase and the solution phase: solution phase : mA;w ¼ mY;w ;

resin phase : mA;R ¼ mY;R þ mF;R ;

ð3:3:120cÞ

28

where mF;R is the molality of the fixed negative charges in the resin phase. Relation (3.3.120b) can now be rearranged via the electroneutrality relations to yield mY;R ðmY;R þ mF;R Þ ¼ m2Y;w ðγAw γYw =γAR γYR ÞðasR =asw ÞV AY =V sw ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} b

ð3:3:120dÞ

mY;R ¼

bm2Y;w m2Y;w ffi b; mY;R þ mF;R mF;R

ð3:3:120eÞ

where we have assumed that the fixed molal charge density in the resin, mF;R , is much larger than mY ;R ; it can be easily in the range of 5 molal. Relation (3.3.120e) suggests that, as long as the external solution is dilute ðmY;w Tc ¼ 9.4 ºC, of ethylene). The variation of the density of ethylene near its critical point may be seen in Figure 3.3.15(b), a plot of reduced density, ρR ¼ ðρ=ρc Þ against reduced pressure PR ¼ (P/Pc) and reduced temperature

TR ¼ (T/Tc). The 12 ºC (TR ffi 1.01) isotherm of ρR vs: P R shows a rapid rise in ρ by more than an order of magnitude around PR ¼ 1.0. The shape of this curve is quite similar to that of the naphthalene solubility curve, suggesting that the solvent power of ethylene for naphthalene is directly related to the SCF solvent density near Tc, Pc. To develop an estimate of the solute mole fraction x il in the SCF solvent in equilibrium with a solid mixture, the equilibrium relation is ^f ¼ ^f ; il is where ^f ¼ x il Φ ^ il P il

^ is P: and ^f is ¼ x is Φ

ð3:3:126Þ

The solid mixture is considered to be an agglomeration of pure species, with each component having its pure component solid fugacity f is ; i:e: ^f is ¼ f is . At low pressures, this fugacity is almost equal to the sublimation pressure P sub i ðT Þ. However, at the high pressures characteristic of SCF extraction, "ð   # P Vis sub sub fis ¼ P sub dP ; ðTÞ Φ ðT; P Þ exp i i i RT P sub i ð3:3:127Þ Φsub i

where is the fugacity coefficient at T and P sub i , and the exponential term is the Poynting pressure correction for the fugacity of pure solid having a molar volume Vis. Therefore, the mole fraction of solute i in the SCF solvent is

3.4

Interphase transport: flux expressions

x il ¼

sub P sub ðT;P sub i ðTÞ Φi i Þ exp

^ il P Φ

"ð  P P sub i

159

 # V is dP RT

Liquid g LG

:

Liquid

ð3:3:128Þ A review of the literature and thermodynamic calculation procedures for such systems are available in McHugh and Krukonis (1986). The same reference may be studied for thermodynamic equilibrium calculations for the solute distribution between a liquid and a supercritical fluid solvent.

Gas Bubble q g SL g SG

Solid

Figure 3.3.16. Interfacial tensions and contact angle for a gas bubble–liquid–solid system with a flat solid surface.

3.3.8 Particle distribution between two immiscible phases So far, we have been concerned with the distribution of molecules, ions or macromolecules between two immiscible phases. The molecules may have been solutes present in small quantities or major constituents of either or both phases. Classical principles of thermodynamics were used to develop estimates of such solute distributions. When it comes to large particles, such as ore fines, cells or other particulate matter, classical thermodynamics may not appear to be of any use. However, using the phenomenon of wetting based on interfacial thermodynamics, particle separation from one phase is achieved by introducing a second immiscible phase. There are two types of systems here depending on whether the two immiscible phases are gas–liquid or liquid–liquid. Particle separation in a gas–liquid system is much more common and is called flotation. The following three paragraphs will focus on the basic thermodynamic principles in such a system. Consider two types of mineral particles in an aqueous suspension. If air bubbles can be attached to one type of particle only, the latter will float to the surface of the suspension, due to reduced density, and can be separated from the other type of particles. Normally, mineral particles are wetted completely by water so that air bubbles cannot attach to them. However, if the particle surface can be made sufficiently hydrophobic to prevent wetting, air bubble attachment is possible. The criterion for air bubble attachment is that the free energy change should be negative: ΔG ¼ γSG  ðγSL þ γLG Þ

ð3:3:129Þ

where γSG , γSL and γLG are the interfacial tensions for the systems solid–gas, solid–liquid and liquid–gas, respectively. The general relation between these interfacial tensions and a contact angle θ for a gas–liquid–solid system with a flat solid surface (shown in Figure 3.3.16) is given by Young-Dupre’ equation: γSG ¼ γSL þ γLG cos θ Substituting this into expression (3.3.129), we get

ð3:3:130Þ

ΔG ¼ γLG ðcos θ  1Þ:

ð3:3:131Þ

If the particle is wetted by the liquid, the contact angle θ should be zero since the liquid spreads over the solid surface completely. Thus, air or a gas bubble cannot become attached to the particle surface. If, however, θ is finite, ΔG is negative; there is now a gas–solid interface, and bubble attachment to the particle is possible. To create a nonzero contact angle in a gas–liquid–solid particle system where the liquid normally wets the particle (e.g. water and most mineral matter), surface active species called collectors are dispersed in the liquid for adsorption on particle surfaces. In aqueous systems, addition of surfactants such as long-chain fatty acids, and their adsorption on particle surfaces, make the surface hydrophobic enough, leading to a nonzero θ. Equation (3.3.40b) may be utilized here to study how the various interfacial tensions are changed by changes in the bulk concentrations of the added surfactants (Fuerstenau and Healy, 1972). Instead of a gas or air, it is possible to use a second immiscible liquid, usually water or an aqueous solution, to separate particles or cells from the first immiscible liquid. The second immiscible liquid may serve as the collector phase collecting the particles into it (see Raghavan and Fuerstenau (1975)). Alternatively, the solid particles may be pushed to the liquid–liquid interface. Whether the particles are going to be located at the liquid–liquid interface or in the bulk of one of the liquid phases can be determined by relations between the three interfacial tensions γSL ;γSL2 ;γL1 L2 (see Henry (1984)), where L1 and L2 are the two immiscible liquid phases.

3.4

Interphase transport: flux expressions

In Section 3.1.4, we studied how integrated flux expressions can be used to describe mass transport in singlephase systems. In separation processes involving two or more phases, solute and/or solvent is transferred from one phase to another. In processes using membranes, a species is transferred from the feed to the membrane phase and

160

Physicochemical basis for separation

then from the membrane phase to the permeate (or product or receiving) phase; thus three phases are involved in the direction of species transport. It is necessary to have flux expressions, especially integrated ones, for such interphase transport. In general, the approach is to use integrated flux expressions for each phase and couple them together to develop an overall integrated flux expression using interphase equilibrium relations at the phase boundaries. Another objective is to develop a relation between an overall mass-transfer coefficient for the overall interphase transfer process and the mass-transfer coefficient for transfer in each phase. The following is a partial list of two-phase systems encountered in separation processes involving interphase transport: gas–liquid (alternatively vapor–liquid), liquid– liquid, solid–liquid, liquid–ion exchange resin, solid– supercritical fluid, liquid–supercritical fluid, etc. The first four systems are used much more frequently. Note that the two phases in each system are immiscible. It is possible to have three-phase combinations of the above-mentioned two-phase systems: gas–liquid (1)–liquid (2) (or vapor–liquid (1)–liquid (2)), liquid–ion exchange resin (1)–ion exchange resin (2), etc. We are not going to consider here such three-phase systems. The three-phase (or region) systems of interest here involve a membrane, the feed phase and the permeate phase. Some of the common three-phase systems of this type are: liquid– membrane–liquid, gas–membrane–gas, liquid–membrane– gas, gas–membrane–liquid. The two fluid phases on two sides of the membrane are, in general, miscible (exceptions include the liquid–membrane–gas system, etc.). An elementary introduction to species transport through membranes that are either nonporous or porous will be provided here to facilitate development of flux expressions in a few commonly used membrane processes. 3.4.1 3.4.1.1

Interphase transport in two-phase systems Gas–liquid systems

Figure 3.4.1 shows the gas-phase and liquid-phase concentration profiles of species A being transferred from a gas phase into a liquid phase. In the gas phase, the bulk composition is indicated either by the partial pressure pAb or the gas-phase mole fraction xAgb. This value is reduced to pAi or xAgi at the gas–liquid interface. In the liquid phase, the corresponding values are CAli or xAli at the gas–liquid interface. These values are reduced to the bulk liquidphase values CAlb or xAlb at some distance away from the gas–liquid interface. In general,29 the interfacial compositions in the two phases are related to each other by equilibrium criteria 29

At high mass-transfer rates and with a contaminated interface, interfacial resistance is possible (see Sherwood et al. (1975)).

Gas–liquid interface Gas phase pAb , xAgb

Liquid phase

Increasing xAl , CAl

xAgi, pAi

Increasing xAg , pA

CAli, xAli

xAlb CAlb Direction of transport of species A Figure 3.4.1. Concentration profiles of species A being absorbed from a gas into a liquid.

developed in Section 3.3. Thus pAi or xAgi is related to CAli or xAli for an ideal gas–liquid system by Henry’s law (relation (3.3.60b)): pAi ¼ H A x Ali ;

x Agi ¼ ðH A =PÞx Ali ¼ H PA x Ali

ð3:4:1aÞ

H CA ¼ H A =C tl :

ð3:4:1bÞ

or pAi ¼ H CA C Ali ;

For nonideal systems, the equilibrium relations pAi ¼ f eq ðx Ali Þ;

x Agi ¼ f eq ðx Ali Þ

ð3:4:2Þ

are to be determined from the equilibrium criterion of fugacities ^f Ag ¼ ^f Al for species A (similarly for all other species). The flux of species A in both gas and liquid phases can be expressed by either N Az ¼ k xl ðx Ali  x Alb Þ ¼ k xg ðx Agb  x Agi Þ;

ð3:4:3Þ

N Az ¼ k g ðpAb  pAi Þ ¼ k c ðC Ali  C Alb Þ;

ð3:4:4Þ

where kxl, kxg, kg and kc are mass-transfer coefficients defined for a particular phase, either gas or liquid. We recall from our earlier discussion in Section 3.1.4 that integrated flux expressions are useful since only concentrations or mole fractions or partial pressures at the two ends of the diffusion/transport path are required instead of their gradients. Here, however, we find that there are quantities like pAi, xAgi, xAli corresponding to the gas–liquid interface in single-phase integrated flux expressions; these quantities are very difficult to obtain. On the other hand, the bulk concentrations in each phase, xAgb, CAlb, xAlb and pAb are much easier to obtain. The answer is provided by an overall mass-transfer coefficient K at any location.

3.4

Interphase transport: flux expressions

161

The strategy is to define the overall mass-transfer coefficient K with respect to a single phase, either gas or liquid, and known bulk concentrations: N Az ¼ K xg ðx Agb  x Ag Þ ¼ K xl ðx Al  x Alb Þ

ð3:4:5Þ

N Az ¼ K g ðpAb  p A Þ ¼ K c ðC Al  C Alb Þ:

ð3:4:6Þ

or

Here x Ag is a hypothetical gas-phase mole fraction which is in equilibrium with x Alb ; thus x Ag is known. Similarly, x Al is a hypothetical liquid-phase mole fraction which is in equilibrium with x Agb ; similarly, p A is in equilibrium with CAlb and C Al is in equilibrium with pAb. Now express x Agb  x Ag as x Agb  x Ag ¼ ðx Agb  x Agi Þ þ ðx Agi  x Ag Þ

ð3:4:7Þ

and utilize equilibrium relation (3.4.1a) as well as the definitions of Kxg, kxg, kxl: N Az K xg

¼

N Az H A N Az H A N Az þ þ ; ðx Ali  x Alb Þ ¼ k xg P k xg P k xl

1 K xg

¼

1 HP þ A; k xg k xl

p

HA ¼

HA : P ð3:4:8Þ

Therefore, the overall mass-transfer coefficient Kxg is known in terms of the individual phase mass-transfer coefficients kxg and kxl and Henry’s law constant H PA . Further, if the bulk compositions of the two phases are known, the flux N Az of species A from gas to the liquid can be predicted. Note that relation (3.4.8) is also valid if the direction of transport of species A was reversed. Equation (3.4.8) relates the overall mass-transfer coefficient Kxg to the individual mass-transfer coefficients kxg and kxl. Relations between the overall transfer coefficient and the individual phase transfer coefficients for other types of concentration driving gradients are also of considerable use. Consider N Az ¼ K xl ðx Al  x Alb Þ ¼ k xg ðx Agb  x Agi Þ ¼ k xl ðx Ali  x Alb Þ: But x Al  x Alb ¼ ðx Al  x Ali Þ þ ðx Ali  x Alb Þ ¼ )

x Agb P x Agi P N Az  þ k xl HA HA

1 P 1 1 1 1 ¼ þ ¼ þ : K xl H A k xg k xl H PA k xg k xl ð3:4:9Þ

Similarly, 1 1 HC 1 HC ¼ þ A¼ þ A K g kg kg kc kl

ð3:4:10Þ

and 1 1 1 1 1 ¼ þ ¼ þ : K c k c H CA k g k l H CA k g Note that kl and kc will be used interchangeably.

ð3:4:11Þ

An important characteristic of the four relations (3.4.8)–(3.4.11) is that the overall mass-transfer resistance indicated by the 1/K term is the sum of the individual gasand liquid-phase resistances, regardless of the form of the concentration driving gradients. Such linear additive behavior of resistance we will find, again and again, is the rule in interphase transport, although there are a few exceptions. The individual phase transfer coefficients kxl and kxg are often of the same order of magnitude. However, the value of H PA can vary over a wide range depending on the nature of species A and the liquid phase. If a gas species is highly soluble in the liquid (Figure 3.3.3 illustrates such a system), H PA has a small value, which makes ðH PA =k xl Þ > 1 (the solute prefers the organic phase strongly over the aqueous phase), then the aqueous phase resistance is in control: 1 1 ffi : K Aw k Aw

ð3:4:20Þ

3.4

Interphase transport: flux expressions

163

On the other hand, if κAo > 1 (Yang, 1987). Therefore the external film resistance may be neglected. The internal diffusion process controls the rate of species adsorption/desorption. Since intraparticle diffusion/transport controls the species transport rate, a linear transport rate expression is sought to describe it to facilitate solution of the overall adsorption process taking place in a packed bed. The transient rate of adsorption/desorption of a species i may be described by the time rate of change of a particlep averaged species concentration C is defined for a spherical particle of radius rp by 

rðp  4 3 p p πr p C is ¼ C is 4πr 2 dr: 3 0

ð3:4:36aÞ

p

Here, C is is obtained from the mass balance equation in a spherical particle (see (7.1.19a)): # # " "  p p p p ∂C is 1 ∂ ∂C ∂2 C is 2 ∂C is ¼ Dip : ¼ Dip 2 þ r 2 is ∂t ∂r ∂r 2 r ∂r r ∂r p C is

ð3:4:36bÞ

The time rate of change of is described by a lineardriving force based expression:   p 4 3 ∂C is 15Dip p p ¼ 2 ðC is  C is Þ; ð3:4:36cÞ πr p rp ∂t 3 p

where C is is a hypothetical averaged pore-phase concentration in equilibrium with the instantaneous bulk gasphase concentration at the particle–fluid boundary ðC ii ;C ib Þ (both being essentially equal to each other since kig is, on a relative basis, quite high). This approximation, identified as the linear driving force (LDF) approximation, was developed first by Glueckauf and Coates (1947), and is widely used in particle–fluid adsorption/desorption processes. For a basis of this approximation, see the development in Yang (1987, p.127) based on a solution of the governing diffusion equation (equation (3.4.36b)) for diffusion of gases within a spherical particle. Sircar and Hufton (2000) have shown for gas–solid systems that any adsorbable species concentration profile describable by p

C is ðr;tÞ ¼ aðtÞ þ bðtÞ FðrÞ;

ð3:4:37aÞ

where F(r) is any monotonic and continuous function of r in the range 0 r r p satisfying the boundary condition  dFðrÞ  ð3:4:37bÞ r¼0 ¼ 0; dr 

3.4

Interphase transport: flux expressions

167

will satisfy the LDF approximation of (3.4.36c), where ð15Dip =r 2p Þ may be replaced, in general, by a mass-transfer coefficient k. The restriction for the LDF approximation of Glueckauf and Coates (1947) is ðDip t=r 2p Þ > 0:1, which often invalidates its use during initial gas sorption periods in activated carbon and zeolites. The linear driving force approximation (3.4.36c) is also used commonly for separation/transport between a liquid and porous adsorbent particles. In order to account for any deviation between the actual rate of adsorption and the rate obtained by the LDF approximation, a correction factor ψ p has also been introduced:   p 15Dip p 4 3 ∂C is p ¼ ψ p 2 ðC is  C is Þ: πr p ∂t rp 3

ð3:4:38Þ

On a comparison of the observed value of the rate of uptake of species i with time (as given by the left-hand side of (3.4.36c)) and the rate predicted by the right-hand side, ψ p may be determined. Some estimates of ψ p are available in Vermuelen et al. (1973). An alternative approach for estimating the rate of adsorption of species i on the adsorbent employs the method used in chemical kinetics to determine the rate of reaction as an algebraic sum of the forward and backward reaction rates for the second-order reaction (whose reaction equilibrium constant is K ¼ ð¼ k f =k b Þ) (Thomas, 1944): adsorbing species ðiÞ þ adsorbent site ðsÞ

kf

⇄ adsorbed complex ðisÞ: ð3:4:39aÞ kb

If Cif is the concentration of adsorbing species i in the pore fluid phase, Cis is the concentration of adsorbed species i on the adsorbent phase (s) occupying some of the sites of the adsorbent phase, where C m ss is the maximum value of the site concentration (s) (first subscript) in the solid phase, then the forward adsorption reaction rate for this secondorder reaction is kf C if ðC m ss  C is Þ and the backward reaction rate is k b C is . The net rate of species i adsorption is given by   ∂C is C is m : ¼ k f C if ðC m  C Þ  k C ¼ k C ðC  C Þ  is b is f if is ss ss ∂t K ð3:4:39bÞ

The quantity within the brackets on the right-hand side is called the kinetic driving force; kf is the corresponding kinetic coefficient in this reaction–kinetic treatment (Vermeulen et al., 1973). When the rate is zero, the equilibrium concentration, C is , of the adsorbent for species i is obtained for Cif as follows: C if ðC m ss  C is Þ 

C m K C if C is ¼ 0 ) C is ¼ ss : 1 þ K C if K

ð3:4:39cÞ

This is a form of Langmuir adsorption isotherm illustrated in Figure 3.3.8B and equations (3.3.112a) and (3.3.113c, d),

since ðC is =C m ss Þ is essentially equal to θ, the fraction of the sites occupied. Often, instead of a molar concentration for the adsorbed species, Cis (mole/volume), moles of species i per unit mass of adsorbent, qis, is used. In such a notation, qm ss is the maximum value of the site concentration. The corresponding form of the rate of adsorption equation (3.4.39b) and the Langmuir adsorption isotherm (3.4.39c) are changed, respectively, to  ∂qis qis  ¼ k f C if ðqm ; ð3:4:39dÞ ss  qis Þ  ∂t K q is ¼

qm ss K C if ; 1 þ K C if

ð3:4:39eÞ

where q is is the equilibrium concentration or saturation capacity of the adsorbent for species i for the fluid phase concentration Cif. A nondimensional form of equation (3.4.39d) is frequently used when qis is nondimensionalized with respect to qm ss :   1 0 q ð1þK C Þ K C if is K C if if  qis  qis k ∂ðqis =qm Þ f ss A ¼ @ qm K ∂t ss   kf qis qis  m : ð3:4:39fÞ ¼ ð1 þ K C if Þ m K qss qss An alternative form of this equation for the rate of adsorption is often used. First, rewrite it as ) ( kf ∂ðqis =qm q is qis ss Þ ¼ ð1 þ K C if Þ m  m ð1 þ K C if Þ : ∂t K qss qss ð3:4:40aÞ C ref if x if

and corresponding to C ref Now define C if ¼ if in the adsorption system. Then the Langmuir adsorption isotherm (3.4.39e) for C ref if and Cif may be written, respectively, as m qref is ¼ qss

K C ref if 1þ

K C ref if

and

qref is

q is ¼

ref qm ss K C if x if

1 þ K C ref if x if

; ð3:4:40bÞ

leading to q is =qref is ¼

1 þ K C ref if

1 þ K C ref if x if

x if :

ð3:4:40cÞ

ref Temporarily assuming that qm ss equals qis , we obtain, for (3.4.40a), ( ) kf ∂ðqis =qref qis ref ref is Þ ð1 þ K C if Þx if  ref ð1 þ K C if x if Þ : ¼ K ∂t qis

ð3:4:40dÞ Alternative forms of this equation are used in analyzing the rate of adsorption in adsorbent beds using separation factor relations.

168

Physicochemical basis for separation

3.4.1.5

Table 3.4.1. A few diffusion coefficients in ion exchange systemsa

Ion exchange resin–solution systems

Mass transfer in the ion exchange resin–solution system is briefly considered here for the case where counterion AZ A in the ion exchanger is exchanged with counterion BZ B in the external solution. An exchange of this kind may be described by five steps: (1) the diffusion of ion BZ B from the bulk of the external solution to the outside surface of the ion exchange resin particle; (2) the diffusion of ion BZ B through the solvent-filled pores of the ion exchange matrix to a site containing ion AZ A ; (3) the instantaneous exchange of places between ions B and A; (4) the diffusion of ion A through the solvent-filled pores of the ion exchange matrix to the outside surface of the resin particle; (5) the diffusion of ion A from the particle surface to the bulk of the external solution. Such a scheme was proposed by Boyd et al. (1947). Since step (3) is instantaneous, we need only consider steps (1), (2), (4) and (5). However, there are essentially two types of resistances here: steps (1) and (5) account for diffusion through the liquid film on the outside of the resin particle, whereas steps (2) and (4) describe the diffusion of counterions through the pores of the ion exchange resin particle. When steps (1) and (5) control the exchange rate, the ion exchange is said to be film diffusion controlled. If steps (2) and (4) are the ratedetermining steps, the mechanism is particle diffusion controlled (Helfferich, 1962). The diffusional flux expression for particle diffusion control in the exchange of counterion AZ A in the ion exchanger with the counterion BZ B in the external solution will now be derived. The Nernst–Plank flux expression for species A in the resin ( j ¼ R) in the absence of any pressure gradient is obtained from (3.1.106) as J A ¼ DAR rC AR  DAR

C AR Z A F RT

rΦ:

ð3:4:41Þ

Since there is no electric current in the system, obtain from the current density expression (3.1.108c) Z A N A þ Z B N B ¼ 0 ¼ Z A J A þ Z B J B ) J A ¼ 

ZB J ; ZA B ð3:4:42Þ

where N i ¼ J i due to the absence of any convection in the pores of the resin particle. Further, electroneutrality must be maintained in the resin with a fixed molar charge density of X (see relation (3.3.30b)): Z A C AR þ Z B C BR ¼ ωX : Taking the gradient of this expression yields Z A rC AR þ Z B rC BR ¼ 0:

ð3:4:43Þ

Ions

Naþ

Csþ

Baþþ Srþþ

Mnþþ

Infinitely dilute external solution, Dil  107 cm2/s Dowex 50 W-X8 resin,b DiR 107 cm2/s

133

205

84

70.8

a b

20.5

30.0

77.8

1.16

1.95

2.22

Bajpai et al. (1974). Hþ of sulfonic acid group is the exchangeable cation.

Use these three equations to eliminate rΦ from J A . First, substitute for J A and J B in (3.4.42) using the Nernst–Plank expression (3.4.41) for both A and B and rearrange: ðrΦÞ ¼

RT Z A DAR rC AR þ Z B DBR rC BR : F Z 2A DAR C AR þ Z 2B DBR C BR

Substitute this into expression (3.4.41) for J A : J A ¼ DAR rC AR þ

Z 2A C AR D2AR rC AR þ Z A Z B C AR DAR DBR rC BR : Z 2A DAR C AR þ Z 2B DBR C BR

Use here Z B rC BR ¼ Z A rC AR from equation (3.4.43) and simplify: J A

¼ ¼

Z 2A C AR D2AR  Z 2A C AR DBR DAR rC AR Z 2A DAR C AR þ Z 2B DBR C BR " # DAR DBR ðZ 2A C AR þ Z 2B C BR Þ  rC AR ¼ DABR rC AR : Z 2A DAR C AR þ Z 2B DBR C BR DAR rC AR þ

ð3:4:44Þ An unstated assumption in this derivation was that the co-ion was excluded completely from the ion exchanger by Donnan exclusion. Expression (3.4.44) for the flux of counterion A in the particle diffusion controlled exchange of counterions A and B is the equivalent of Fick’s first law; however, the diffusion coefficient, DABR for the exchange between A and B in the resin particle is, in general, not constant and depends on concentrations of counterions A and B. For example, if CAR 0

(b)

Feed : gaseous

Feed : vapor with/without gas

Pf , xif

Pf , xif

Transport direction

z

z = dm

Pp , xip

Pp , xip

Permeate : gaseous 1

Permeate : vapor with/without gas 2

Gas permeation

Vapor permeation Pf xif > Pp xip

Pf xif > Pp xip (c)

Feed : liquid

Feed : liquid

Pf , C 0if

Pf , Cif , C 0il z =0

Transport direction

z

z Pp , Cip , C dil

Pp , C dil Pf > Pp Permeate : liquid 1

z =0

z

Permeate : liquid Cip < Cif : Pf ≅ Pp 2

Ultrafiltration/microfiltration

(d)

z = dm

Dialysis/diffusion

Feed : gaseous Pf Pp ≤ Pf

z Feed : gaseous Pp 1

Gaseous diffusion/convection

Figure 3.4.5. Schematics of four broad feed phase–membrane type categories including subcategories based on permeate phases and driving forces (electrical potential gradient excluded); membrane based two-phase contacting is not included here. (a) Liquid feed– nonporous membrane; (b) gaseous feed–nonporous membrane; (c) liquid feed–porous membrane; (d) gaseous feed–porous membrane.

solution; the phenomenon is called reverse osmosis (RO) (see also Figure 1.5.1). How much salt will also go to the low-pressure product solution depends, amongst other things, on how permeable the membrane is to the salt. In water desalination using a reverse osmosis process,

membranes are highly permeable to water and not so much to salt. In this example, one species in the feed (namely salt) is nonvolatile. However, the following treatment is also going to be valid for volatile solutes provided the permeated phase is liquid.

172

Physicochemical basis for separation used. Inside the membrane phase, rμ0i ¼ 0. Membrane concentrations are indicated by Csm and Cim for the solvent and solute, respectively. For the solvent (subscript, s)     C sm V s dP RTC sm d ℓn asm  d : ð3:4:47Þ J sz ¼  d dz dz f sm f sm

Membrane

a dsm

Feed solution Cif as0m

Pp ( < Pf )

Pf Ci0m Increasing

Cip

For solute species i,     C im V s dP RTC im d ℓn aim  d J iz ¼  d : dz dz f im f im

In terms of diffusion coefficients, the frictional coefficients may be replaced by 1 Dsm ¼ RT f dsm

Ci

Cidm

Increasing

asm

z=0

z=d z

Direction of solvent and solute transport through membrane in reverse osmosis

Figure 3.4.6. Solvent and solute transport through a membrane in reverse osmosis with well-mixed feed and permeate solutions.

We ignore the problems, if any, of the transport of solute i and solvent s in the feed solution and the product solution for the time being by assuming them to be wellmixed. In the three-component system of solute–solvent– membrane, the membrane is assumed stationary. Both solvent and solute diffuse through the membrane under the driving force of a chemical potential gradient across the membrane. Such a condition is best studied at first by using the solution-diffusion theory (Lonsdale et al., 1965). Both the solvent and the solute are assumed to be dissolved in the membrane at the high-pressure feed solution–membrane interface where the species concentrations in the two phases are related by equilibrium. They then diffuse through the membrane, each under its own chemical potential gradient, and are finally desorbed into the product solution at the low-pressure solution– membrane interface. The system should then be analyzed as if we have two separate binary systems: solvent– membrane and solute–membrane. The membrane thickness is δm in the z-direction, the direction of transport. This is a case of diffusion in the membrane. Therefore, we use J i ¼ N i  C i v t and equation (3.1.84a) for N i for both i and s. Note that no external forces are present. But, since ΔP is nonzero, expression (3.1.84d) for U i has to be

ð3:4:48Þ

and

RT ¼ Dim : f dim

ð3:4:49Þ

We could have also obtained the two flux expressions given above by simply considering the uncoupled flux of any solute in the solvent from the irreversible thermodynamic formulation based expression (3.1.208). But the solvent in (3.1.208) is the membrane here. Further, both species being transported, the solute i and the solvent s, are simply solutes: J i ¼ Lii F i ¼ Lii ðrμi Þ;

ð3:4:50Þ

Lii ¼ ðDim C im =RTÞ:

ð3:4:51Þ

where

Integrate now the flux expression (3.4.47) from z ¼ 0 to z ¼ δm , assuming CsmDsm and V s to be constants, to obtain at steady state   asm jδm C sm Dsm V s C sm Dsm : J sz ¼ þ ðP f  P p Þ  ℓn RTδm δm asm j0

ð3:4:52Þ

Now asm jδm ¼ asp and asm j0 ¼ asf , corresponding to the product and feed solutions. However, relation (3.3.86a) for osmotic equilibrium between two solutions of different solvent activities and osmotic pressures provides   asp V s ðπf  πp Þ ¼ RTℓn : ð3:4:53Þ asf Therefore J sz ¼ ¼

C sm Dsm V s ½ðP f  P p Þ  ðπf  πp Þ RTδm C sm Dsm V s Q fΔP  Δπg ¼ sm ðΔP  ΔπÞ ¼ AðΔP  ΔπÞ; RTδm δm ð3:4:54Þ

where ΔP ¼ P f  P p and Δπ ¼ πf  πp :

This is an integrated flux expression for the solvent permeating through the reverse osmosis membrane in

3.4

Interphase transport: flux expressions

173

terms of bulk quantities like ΔP and Δπ obtainable outside the membrane with a mass-transfer coefficient-like proportionality constant, ðQsm =δm Þ. Here Qsm is the permeability coefficient of the solvent through the membrane of thickness δm . In reverse osmosis literature, the quantity ðQsm =δm Þ is often called the pure water permeability constant A. It is essentially a mass-transfer coefficient. Note: For water to permeate through the membrane from feed to the permeate, ΔP > Δπ. In expression (3.4.48) for the solute flux through the reverse osmosis membrane, the first term is usually unimportant (Merten, 1966). Therefore, the vectorial expression for the solute flux is given by   ∂ ℓn aim J i ¼ C im Dim rℓn x im : ð3:4:55Þ ∂ ℓn x im T;P Assume ideal solution in the membrane and a constant total molar density Ctm across the membrane. Then J i ¼ 

C im Dim rx im ¼ Dim rC im : x im

ð3:4:56Þ

For one-dimensional steady state transport in the z-direction, on integrating J iz one obtains Dim 0 ðC im  C δim Þ; J iz ¼ δm

ð3:4:57Þ

where C 0im and C δim are the solute concentrations in the membrane at the two solution–membrane interfaces. The solute distribution coefficient κi may now be used to relate the membrane concentration to the external solution concentration at the two interfaces because of equilibrium: C 0im Cδ ; κip ¼ im : C if C ip

ð3:4:58Þ

Dim κim ðC if  C ip Þ: δm

ð3:4:59Þ

κif ¼ If κif ¼ κip ¼ κim , then J iz ¼

The product, Dim κim , is a permeability coefficient of solute i through the membrane of thickness δm . The product ðDim κim =δm Þ has been called the solute transport parameter in reverse osmosis literature (Sourirajan, 1970). Expression (3.4.59) is an integrated flux expression for solute transport through a reverse osmosis membrane in terms of the differences in concentrations of the external solutions and a mass-transfer coefficient-like quantity ðDim κim =δm Þ. A few distinguishing features of transport of any species through a membrane with respect to interphase transport in two-phase systems are as follows. (1) There are two immiscible phase interfaces: the feed– membrane interface and the membrane–product interface.

(2) There is equilibrium partitioning of species between the solution and the membrane at both interfaces (the two partition coefficients were assumed here to be equal; in general, they are different). (3) The feed and the product phases here are miscible, unlike conventional interphase transport in two-phase systems (liquid–liquid, gas–liquid, etc.), which are immiscible. (4) Species transport in both feed and product phases have to be considered in general. Both J sz and J iz are diffusive fluxes through the membrane. If there are large pores in the membrane, there will be convection through such pores or defects. The total flux of any species will no longer be completely diffusive and therefore should be expressed in terms of N i (Soltanieh and Gill, 1981). A simple model for the fluxes through a reverse osmosis membrane having large pores (or defects or imperfections) has been provided by Sherwood et al. (1967); it is called the solution-diffusion-imperfection model: N sz ¼ J sz þ Ap ΔPC sf ¼ AðΔP  ΔπÞ þ Ap ΔPC sf ; ð3:4:60aÞ where Csf is the solvent concentration on the high-pressure feed–membrane interface for a well-mixed feed. Thus there is simply Darcy or Poiseuille flow through the defects (see Section 3.4.2.3 for these types of solvent flow), and the feed solution at the feed–membrane interface moves unchanged through the defects. Similarly, for the solute, N iz ¼ J iz þ Ap ΔPC if ¼

Dim κim ðC if  C ip Þ þ Ap ΔPC if ; δm ð3:4:60bÞ

where Cif is the solute concentration on the high-pressure feed–membrane interface for a well-mixed feed. There are three transport coefficients in this model, A, ðDim κim =δm Þ and Ap; the solution-diffusion model had only two transport coefficients, A and ðDim κim =δm Þ. In practical reverse osmosis, molar fluxes are not usually measured; volume fluxes are. The volume flux (in units of cm3/cm2-s), often denoted by Jvz, is defined by J vz ¼ J sz V s þ J iz V i

ð3:4:60cÞ 30

for conditions when diffusive models hold. When there is some convection through the membrane defects as well, the volume flux measured is to be obtained from the sum of Nsz and Niz. In most cases, Niz pip Þ, C 0im ¼ Sim pif ; C δm ¼ Sim pip :

ð3:4:71Þ

Then the flux J iz may be expressed as J iz ¼

Dim Sim Q Q ðpif  pip Þ ¼ im ðpif  pip Þ ¼ im Δpi : ð3:4:72Þ δm δm δm

Here, Qim is the permeability coefficient of species i through the membrane and is a product of Dim, the diffusion coefficient, and Sim, the solubility coefficient, of species i in the membrane. For permanent gases and rubbery nonporous polymeric membranes, Qim for any species is constant at any temperature. The unit commonly used for Qim is cm3(STP)-cm/cm2-s-cm Hg. The unit “barrer,” often used for Qim, is 1010 cm3(STP)-cm/cm2-s-cm Hg; cm3(STP) is sometimes written as scc. The units for δm ; pi and J iz are cm, cm Hg and cm3(STP)/cm2-s, respectively. In general, the gas concentration in the membrane at any interface (z ¼ 0 or δm ) is related to the gas partial pressure pi by C im ¼ Sim ðC im Þpi ;

ð3:4:73Þ

where the solubility coefficient Sim(Cim) is a function of the concentration Cim. For low levels of Cim, with a number of permanent gases like H2, He, Ar, O2, N2, CH4, at T > Tci (critical temperature of gas i), Henry’s law is observed (Stern and Frisch, 1981) and lim Sim ðC im Þ ¼ Sim ; a constant:

C im !0

ð3:4:74Þ

(For deviations from this behavior, see equations (3.3.81) and (3.3.82a, b). Similarly, the diffusion coefficient Dim of species i in the membrane is also generally dependent on Cim, but lim Di ðC im Þ ¼ Dim ; a constant:

C im !0

ð3:4:75aÞ

When such a limiting behavior does not hold, Qim is defined by Qim ¼

δðm

1 ðÞ Dim dC im : ðpif  pip Þ 0

ð3:4:75bÞ

This process of gas transport through a membrane is called permeation, and the mechanism has been identified as solution-diffusion. Gas species i dissolves at the feed– membrane interface (z ¼ 0); by molecular diffusion, the dissolved gas molecules move through the membrane and are finally desorbed into the product gas phase at the product– membrane interface ðz ¼ δm Þ. Under the simplest of conditions, each species in a mixture diffuses independently of the others according to flux expression (3.4.72). The nature of the dependence of Dim on the effective diameter of the gas molecules, the temperature and the polymer for an activated diffusion is illustrated in Section 4.3.3 for an amorphous polymer.

Ideally, it is necessary to use mass-transfer equations in the feed-gas phase and the product-gas phase along with the permeation equation (3.4.72). However, in general, the gas permeation rate through a nonporous membrane is so slow that mass-transfer equations in the feedgas and product-gas phases are not needed. Note that, for species i transport through the membrane from the feed to the product gas, pif > pip, but Pf need not be greater than Pp, although in practice it generally is. In such a case, flux expression (3.4.72) may also be expressed as J iz ¼

Qim Q ðp  pip Þ ¼ im ðP f x if  P p x ip Þ; δm if δm

ð3:4:76Þ

where xif is the mole fraction in the feed mixture and xip is the mole fraction in the product mixture. If the gas is condensable and/or we have a vapor in the gas mixture, the solubility of the permeating species in the polymeric membrane is higher. The condensable organic/ inorganic vapor/gas may plasticize the membrane; the polymeric chains have higher segmental mobility. As a result, the diffusion coefficient is increased drastically. The diffusion coefficient Dim can vary with the concentration Cim of species in the membrane linearly, or even higher, exponentially: Dim ¼ Dim0 expðAi C im Þ:

ð3:4:77aÞ

Assuming the solubility coefficient of species i to be constant, Sim0, corresponding to Henry’s law, the permeability coefficient Qim may be expressed as Qim ¼ Sim0 Dim0 expðAi C im Þ ¼ Qim0 expðAi C im Þ; ð3:4:77bÞ where Qim0 ¼ Sim0 Dim0 . This exponential variation of the permeability coefficient is observed at high partial pressures of condensable gas/vapor species i in rubbery amorphous membranes. Introducing such a variation of Dim with Cim into Fick’s first law, we get J iz ¼ Dim

dC im dC im ¼ Dim0 expðAi C im Þ : dz dz

Integrate it from z ¼ 0 to z ¼ δm to obtain i Dim0 h expðAi C 0im Þ  expðAi C δim Þ J iz ¼ A i δm i Dim0 h expðAi Sim0 pif Þ  expðAi Sim0 pip Þ : ¼ Ai δm

ð3:4:77cÞ

ð3:4:77dÞ

One could rewrite this as i Dim0 h expðAi Sim0 Δpi Þ  1 expðAi Sim0 pip Þ: ð3:4:77eÞ J iz ¼ Ai δm

Taking logarithms of both sides and plotting log ðJ iz Þ against pip, " # Dim0 fexpðAi Sim0 Δpi Þ  1g þ Ai Sim0 pip ; log J iz ¼ log Ai δm ð3:4:77fÞ

3.4

Interphase transport: flux expressions

179

a straight line results; the slope yields Ai Sim0 and the intercept can be used to determine Dim0. The permeability coefficient Qim will therefore be a strong function of Δpi . Knowing Sim0, the slope yields Ai. For gas transport through nonporous polymeric membranes of the glassy type (i.e. the temperature of permeation, T < Tg, the glass transition temperature of the polymer), it has been postulated that the dissolved gas species exists in the membrane in two forms: the Henry’s law species and the Langmuir species. The Henry’s law species dissolves according to the Henry’s law and diffuses in the manner described earlier. The Langmuir species dissolves according to the Langmuir isotherm (see equation (3.3.81)); further, it has a limited mobility at a fraction of that of the Henry’s law species (Paul and Koros, 1976). One-dimensional transport of gas molecules i through such a nonporous glassy polymeric membrane according to this dual sorption–dual transport model is then described by J iz

∂C d ∂C H ¼ DiD im  DiH im ¼ N iz ; ∂z ∂z

ð3:4:78Þ

where DiD is the diffusion coefficient of the Henry’s law species whose concentration in the membrane is C dim ; DiH is the diffusion coefficient of the Langmuir species of molar concentration C H im and ðDiH =DiD Þ 1. Paul and Koros (1976) have solved the onedimensional unsteady state diffusion equation (use equation (3.2.2); alternatively simplify equation (6.2.3a)), ∂C im ∂N iz ¼ ; ∂t ∂z

ð3:4:79Þ

for diffusion of pure species i through the glassy membrane. They assumed that the ratio ðDiH =DiD Þ was constant; further, d H C im ¼ C dim þ C H im . Additionally, the C im and C im species are in equilibrium with each other. For the limiting case of C δim ¼ 0, i.e. pip= 0, the species profile in the membrane is linear, and the permeability of pure species i, Qim, was found to be   0 1 DiH C 0Hi bi B N iz δm N iz δm DiD Sim C C: Qim ¼ ¼ Sim DiD B ¼ A @ 1 þ 1 þ bi p p Pf if

if

ð3:4:80Þ

The quantities C0Hi , bi and Sim are available from the membrane–gas equilibrium behavior (3.3.81). Note the difference between this permeability expression and that for Qim for rubbery polymeric membranes (¼ Sim DiD with DiD ¼ Dim ). As the feed pressure increases, the permeability coefficient in the glassy membrane decreases. The permeability coefficient has the highest value in the limit of pif ð¼ P f Þ ! 0. The integrated flux expression continues to be simply given by N iz ¼

Qim ðp  pip Þ: δm if

ð3:4:81aÞ

For the permeation of a binary gas mixture through a glassy polymeric membrane, Koros et al. (1981) have developed expressions for the permeability coefficient of each species (their equilibrium behavior is described by (3.3.82a,b)):   0 D1H C H1 b1 p1f =ðp1f  p1p Þ D1D S1m Q1m ¼ S1m D1D 1 þ 1 þ b1 pf þ b2 p2f

"



 0 D1H C H1 b1 p1p =ðp1f  p1p Þ D1D S1m  1 þ b1 p1p þ b2 p2p

#

:

ð3:4:81bÞ

The expression for Q2m is  Q2m ¼ S2m D2D 

"

 0 D2H C H2 b2 p2f =ðp2f  p2p Þ D2D S2m 1þ 1 þ b1 p1f þ b2 p2f

 0 D2H C H2 b2 p2p =ðp2f  p2p Þ D2D S2m  1 þ b1 p1p þ b2 p2p

#

:

ð3:4:81cÞ

The permeation flux expressions (3.4.76) and (3.4.81a) are valid for membranes whose properties do not vary across the thickness. Most practical gas separation membranes have an asymmetric or composite structure, in which the properties vary across the thickness in particular ways. Asymmetric membranes are made from a given material; therefore the properties varying across δm are pore sizes, porosity and pore tortuosity. Composite membranes are made from at least two different materials, each present in a separate layer. Not only does the intrinsic Qim of the material vary from layer to layer, but also the pore sizes, porosity and pore tortuosity vary across δm . At least one layer (in composite membranes) or one section of the membrane (in asymmetric membranes) must be nonporous for efficient gas separation by gas permeation. The flux expressions for such structures can be developed only when the transport through porous membranes has been studied. Flux expressions (3.4.72), (3.4.76) and (3.4.81a) for a diffusing gas species i through a membrane of thickness δm describe the observed behavior at steady state achieved after an initial unsteady period. The initial unsteady behavior begins when the gas containing the permeating species i at concentration Cif (partial pressure pif) is introduced at time t ¼ 0 to the z ¼ 0 surface of the membrane, the feed side. The rate of penetration of the membrane by species i is governed by the unsteady state diffusion equation (3.4.79), where N iz ¼ J iz is governed by Fick’s first law (3.4.68). After some time, the species appears through the membrane into the permeate side, where z ¼ δm . For a

180

Physicochemical basis for separation

constant value of Dim, the solution of equation (3.4.79) may be obtained for the downstream boundary condition of pip ffi 0ðC ip ffi 0Þ; from this solution one can calculate the moles of the diffusing species, miD jt , that have crossed the membrane in time t from t ¼ 0 to be (Crank and Park, 1968)   ∞ miD jt Dim t 1 2 X ð1Þn Dim n2 π2 t : ¼ 2   2 exp  6 π 1 n2 δm C if δm δ2m ð3:4:82aÞ

As time t increases and tends to infinity, a steady state is approached where the exponential term is negligibly small; then, a plot of miD vs. t becomes a straight line, i.e.   Dim C if δ2 t m : ð3:4:82bÞ miD jt ¼ δm 6Dim The intercept of this line on the time axis is ðδ2m =6Dim Þ, called the time lag in simple diffusive permeation through a membrane for constant diffusion coefficient in linear diffusion. The value of the time lag provides a good order of magnitude estimate of the time from t ¼ 0 after which steady state permeation through the membrane may be assumed. Correspondingly, equations (4.3.24) and (4.3.25) for gas permeation in a device (see also Chapters 6, 7 and 8) are valid only after this initial unsteady state. Experimental measurement of the time lag allows one to measure the value of Dim. Knowing Qim from steady state experiments, one can determine Sim. Gas transport through nonporous inorganic membranes falls into two categories. It is known that the conventional solution-diffusion permeation mechanism is valid for nonporous membranes of silica, zeolite and inorganic salts. It is no longer so when the membrane is metallic in nature (Hwang and Kammermeyer, 1975). Diatomic gases such as O2, H2 and N2 dissolve atomically in the metallic membrane (see (3.3.67)). While a conventional flux expression is valid for atomic species i dissolved in the membrane, i.e. N iz ¼ Dim

C 0im  C δim ; δm

ð3:4:83Þ

the flux expression changes to Q 1=2 1=2 N iz ¼ im ðpif  pip Þ δm

ð3:4:84Þ

in terms of the species partial pressures in the gas phases, in view of the solubility relation (3.3.70). As long as there is no chemical reaction of the dissolved gas species with the liquid, the gas species flux expression through a thin liquid layer acting as a membrane is identical to that through a rubbery polymeric membrane, as discussed earlier. The major difference comes about in the magnitude of the permeability coefficients. The diffusion coefficient of a gas species in a liquid membrane will, in

general, be at least two to three orders of magnitude larger than that through a rubbery polymeric membrane. Furthermore, the magnitude of the solubility coefficient of the gas in the liquid will be larger. However, the liquid layer thickness will, in general, be considerably higher. 3.4.2.3

Liquid transport through porous membranes

The nature of solute transport in a porous membrane with solvent flow is complex; the regime is influenced strongly by the ratio of solute diameter to pore diameter at any given transport rate of the solvent. The solvent transport rate is, however, commonly described by Poiseuille flow through the pores. The nonporous section of the membrane is assumed impermeable. There are two categories of porous membranes: Ultrafiltration (UF) membranes, having pore diameters in the range 1–30 nm; microfiltration (MF) membranes, having pore diameters between 0.1 and 10 μm (100–10 000 nm). All pores in the membrane may have the same pore size, but generally there is a pore size distribution (Zeman and Zydney, 1996, chap. 4; Cheryan, 1998; Kulkarni et al., 2001). Figure 3.4.5(c) illustrates the basic schematic of such processes. According to IUPAC (the International Union of Pure and Applied Chemistry), pores less than 2 nm in a diameter are called micropores, pores between 2 and 50 nm are called mesopores and larger ones are known as macropores. Correspondingly, microporous membranes should have only pores less than 2 nm, etc. The membrane literature does not follow this practice uniformly. Consider a membrane where the pores of uniform diameter31 2r p occupy a fraction εm of the membrane area. Let the membrane thickness be δm and the pore tortuosity τ m . If a pressure ΔP is imposed across the membrane from the feed to the permeate (or the filtrate) side, the volume flux vz through the pores (in units of cm3/s-cm2 of pore area) of a liquid of viscosity μ, and therefore the volume flux εm vz through the membrane (in units of cm3/s-cm2 of membrane area), is given by the Poiseuille law: " # εm r 2p ΔP εm r 2p 1 dP εm vz ¼  ¼ : ð3:4:85Þ 8μ τ m δm 8μ τ m dz That Poiseuille flow exists through track-etched mica membranes with a pore radius around 5.6 nm has been verified by Anderson and Quinn (1972). The membrane may have a pore size distribution f (rp) such that the fraction of pores in the size range rp to rp þ drp is f (rp)drp, and they contribute a fraction dεm to the membrane pore area. If N sz V s is the total volume flux through the

31

Cylindrical pore in a membrane is an idealization widely practiced in membrane literature.

3.4

Interphase transport: flux expressions

181

pores of such a membrane, then the contribution of pores of radius rp to rp þ drp to the volume flux is given by ðdεm Þr 2p ΔP εm r 2p f ðr p Þdr p ΔP ¼ : ðdN sz ÞV s ¼ τ m δm 8μ 8μ τ m δm For a membrane with pores of all radii between rmax and rmin, N sz V s

¼ ¼

ð

membrane 0

ðdN sz ÞV s ¼

εm ΔP 8μτ m δm

1 ΔP @ A: 8μ τ m δm

r max ð

r min

r 2p f ðr p Þdr p

ð3:4:86Þ qffiffiffiffiffiffi The quantity r 2p is the hydraulic mean pore radius, defined by r max ð

r min

r 2p f ðr p Þdr p :

ð3:4:87Þ

The volume flux (3.4.86) may also be represented by N sz V s ¼

Qsm ΔP : μ δm

ð3:4:88Þ

This relation between volume flux, pressure drop, membrane thickness and the fluid viscosity is commonly used for a porous medium and is known as Darcy’s law. Here, Qsm is called the solvent permeability. Observe that all characteristics of the porous medium that is the membrane, namely εm , r 2p and τ m , are incorporated in Qsm. Example 3.4.4 The utility of equation (3.4.86) for the determination of the solvent flux through a porous membrane will be briefly illustrated with an example worked out by Cheryan (1987). For an XM100A ultrafiltration (UF) membrane, the mean pore diameter (ffi 2  hydraulic mean pore radius) ¼ 17.5 nm; the number of pores/cm2 of the top membrane surface area (skin) ¼ 3  109; δm ¼ membrane thickness ¼ 0.2 μm (only of the skin (to be explained later)); viscosity of water (20  C), μ ¼ 1 cp; Δ P ¼ 105 Pa (gauge pressure); V s ¼ 18:05 cm 3 =gmol of water. From equation (3.4.86), obtain the following estimate of the solvent volume flux (here, water flux):   cm3 εm ð175  108 Þ2 ¼ cm2 N sz V s 2 4 cm -s 8 g 105 Pa  10 2 - Pa 1 cm-s ;  4 2 g cm 0:2  10 1  10 cm-s

where we have assumed that τ m ¼ 1 (since no information is available). The membrane porosity εm ¼

So only 0.72% of the membrane surface is covered with pores, i.e. εm ¼ 0.0072. We have

N sz V s ¼

7:21  103  ð175  108 Þ2  106 32  0:2  106

¼ 3:45  103 cm=s; cm s liter 1 m  3600  1000 3  N sz V s ¼ 3:45  103 s hour m 100 cm ¼ 124:3

εm r 2p

ðr 2p Þ ¼

¼ 7:21  103 :

number of pores  pore area ðcm2 Þ cm2

π 3  109  π ¼ 3  109  ð2r p Þ2 ¼  1752  1016 4 4

liter : m2 -hr

The experimentally observed value of the volume flux at 20  C is 80 liter/m2-hr (lmh).

There are several sources of discrepancy: there is a lack of information about the tortuosity τ m ð> 1Þ; the deviation of the pores from circularity; the existence of dead-end pores, etc. In the calculation, the membrane thickness of 0:2 μm refers to a thin “skin” on top of the total membrane of substantial thickness (100–300 μm). The rest of the membrane has pores that are orders of magnitude larger (with very little hydraulic resistance), and its porosity is also much higher (as much as 0.3–0.6). Such membranes are called asymmetric membranes when the membrane is prepared from one material. Often they are prepared from two different materials to yield a composite membrane. The volume flux given above is for the pure solvent. If there are solute (macrosolute) molecules in the feed solvent, the velocity of the solute molecule in the pore may or may not equal the local pore velocity of the solvent. The latter is especially likely to be true for macrosolutes, macromolecules, proteins, etc. In general, the solute/ macrosolute molecule can also diffuse along the pore, down its own concentration gradient. The solute flux Niz through the membrane is related to the solute flux through p the pores, N iz , as follows: p

N iz ¼ εm N iz ;

ð3:4:89aÞ

where, from flux expression (3.1.113), we can write p

dC p p N iz ¼ C im Gi vz εm þ J iz εm ¼ C im Gi vz εm  Dip εm im : dz diffusive convective flux flux ð3:4:89bÞ p C im

Here, is the concentration of solute i averaged over a pore cross section, Gi is a convective hindrance factor, accounting for the ratio between the velocity of solute molecules and the averaged pore velocity of the solvent vz (usually less than 1), and Dip is the diffusion coefficient of solute molecules of species i in the pore (Anderson and Quinn, 1974). We have seen earlier from (3.1.113) that

182

Physicochemical basis for separation

Dip ¼

Dil GDr ðr i ;r p Þ : τm

ð3:4:89cÞ

Depending on the dimensions of the pores and solute molecules, as well as any possible partitioning between the external solution and the pore solution, the nature of the particular terms in the above equation will vary. Integration of the flux expression will be carried out here only for a particular case, p namely when the pore concentration of solute C im is related to the solution concentration external to the membrane by a partition coefficient or a distribution coefficient, p

ðC im =C il Þ ¼ κi :

ð3:4:90aÞ

Since there is a feed side and a permeate side of the membrane, in general, at z ¼ 0, p

p0

ðC im =C il Þ ¼ κif ¼ C im =C 0il ;

ð3:4:90bÞ

and, at z ¼ τ m δm , p

pδ

ðC im =C il Þ ¼ κip ¼ C im =C δil :

ð3:4:90cÞ

Here, the diffusion length in the z-direction is τ m δm , where δm is the membrane thickness and τ m is the membrane tortuosity. At steady state, N iz in expression (3.4.89b) is constant, and pδ

C im

ð

p0 C im

p dC  im

p

C im  N iz =Gi vz εm

¼

This provides

ℓn

"

N iz G i v z εm N iz p0 C im  G i v z εm " pδ

C im 

¼

0



Gi vz εm C im 1 

:

Gi vz τ m δm ; Dip

pδ

C im

dz Dip =Gi vz

(

Gi vz τ m δm Dip C im "

# Gi vz τ m δm 1  exp  Dip p0

N iz ¼

#

τ mðδm

p0 exp



ð3:4:90dÞ

ð3:4:91aÞ )#

:

ð3:4:91bÞ Membrane pore concentrations may be changed to external concentrations using relations (3.4.90b,c): ( " )# κip C δil Gi vz τ m δm 0 Gi vz εm κif C il 1  exp  Dip κif C 0il " N iz ¼ ;

# Gi vz τ m δm 1  exp  Dip ð3:4:91cÞ where C 0il and C δil are the solute concentrations in the feed and the permeate (the filtrate) solution, respectively. The quantity fðGi vz Þτ m δm =Dip g is called the pore Péclet number, Pem i , of solute i since Gi vz is the effective solute

velocity in the pore, τ m δm is the effective pore length and Dip is the effective solute diffusivity in the pore. When Pem i 1, the contribution of solute diffusion to solute flux is unimportant and N iz ffi Gi vz εm κif C 0il :

ð3:4:92aÞ

If there is no solute partitioning, κif ¼ 1 and Gi ¼ 1, then N iz ¼ εm vz C 0il ;

ð3:4:92bÞ

a result valid for a membrane with pores very large compared to the solute molecules and no solute partitioning. The integrated flux expression (3.4.91c) for Niz contains the permeate solute concentration C δil , which is often related to the solvent velocity vz in the pore by N iz ¼ εm vz C δil ;

ð3:4:93aÞ

since the effective solvent flux through the membrane is εmvz. Substitution of this into the integrated flux expression (3.4.91c) provides the following relation between C δil and C 0il (after rearrangement):   Gi κif exp Gvz τ m δm =Dip Gi κif exp ðPem C δil i Þ  ¼ : ¼ 0 C il Gi κip  1 þ exp Gvz τ m δm =Dip Gi κip  1 þ exp ðPem i Þ ð3:4:93bÞ

In the absence of any concentration polarization, C 0il and C δil are equal to Cif and Cip, respectively. The extent of concentration polarization and its effects on the solvent flux and solute transport for porous membranes and macrosolutes/proteins can be quite severe (see Section 6.3.3). This model is often termed the combined diffusion–viscous flow model (Merten, 1966), and it can be used in ultrafiltration (see Sections 6.3.3.2 and 7.2.1.3). The relations between this and other models, such as the finely porous model, are considered in Soltanieh and Gill (1981). The preceding development assumed that solute molecules were present in each pore. The pore size distribution of porous membrane may be such that the solute molecules can enter only pores larger than the solute molecule. This has two effects. First, the development of the solvent flux expression (3.4.86) assumed no effects due to any solute molecules; in reality, the solute molecules increase the solvent viscosity. Second, the solute flux expressed by (3.4.91c) may have to be corrected if all of the membrane pores are not available to solute molecules. Simplified analysis of such a case has been provided by Harriott (1973). 3.4.2.3.1 Solute diffusion through porous liquid-filled membrane We consider here solute diffusion through a porous liquid-filled membrane under the condition of no convection (Figure 3.4.5(c)). Examples of separation processes/techniques where such a situation is encountered are: isotonic dialysis, membrane based nondispersive gas absorption/stripping or solvent extraction, supported or

3.4

Interphase transport: flux expressions

183

contained liquid membrane technique. The governing equation for solute flux Niz of species i per unit membrane cross-sectional area is obtained from Fick’s first law applied to a porous membrane having a porosity of εm (see equation (3.1.112c)):

inside the membrane, the partition coefficient being given by p0

κim ¼

pδ

C im C im ¼ δ : C 0il C il

ð3:4:96aÞ

p

N iz ¼ Dim

dC im ; dz

ð3:4:94aÞ

where Dim is the diffusion coefficient of species i in the p membrane and C im is the pore fluid concentration of species i. If the pore radial dimension, rp, is much larger than the solute dimension, ri (i.e. ri 1, then a lower pressure is P to be employed as the second guess. If ni¼1 x il < 1, then

4.1

Two-phase equilibruim separation: closed vessel

215

101.3 110

20

150

10

200

0

250 300

600 n-Hexane n-Heptane

Pressure (kPa)

500

700 800 900 1000

4000

6000

Methane

5000

–50

Isobutane n-Butane

Ethylene Ethane

3000

Propylene Propane

2000

–30

–40

Isopentane n-Pentane

1500

–20

Temperature (°C)

400

n-Octane n-Nonane

–10

–60

Figure 4.1.5. Low-temperature nomograph for Ki factors in light hydrocarbon systems. Reprinted, with permission, from D. Dadyburjor, Chem. Eng. Prog., 74(4), 85 (1978). Copyright © [1978] American Institute of Chemical Engineers (AIChE).

a higher pressure should be used as the second guess. Identify the species ethane, propane and n-butane as i ¼ 1,2 and 3. Assume the pressure to be 550 kPa at 32  C. Obtain the Ki values as shown in Table 4.1.2. For the next guess, take the pressure to be 500 kPa (see Table 4.1.3). The dew-point liquid composition is 1.85% ethane, 14.5% propane and 84.8% n-butane. (b) At the bubble-point temperature of 32  C, assume 690 P kPa pressure (see Table 4.1.4). The value of xiv is too high. Increase the pressure to, say, 1050 kPa (Table 4.1.5). This guess of 1050 kPa is just about right.

The procedure for calculating vapor–liquid equilibrium for continuous chemical mixtures will be briefly illustrated here for a bubble-point calculation. Let fl(M) be the

known molecular weight density function of the liquid phase. If now the temperature is specified, one would like to know the molecular weight density function fv(M) of the first vapor bubble formed and the total pressure P of the system. It is known for discontinuous mixtures that the first vapor bubble formed has a composition xiv, which is related to xil by (relation (3.3.61)) ^ iv P ¼ x il γil f 0il : x ivΦ At low to moderate pressures, one can easily assume that sat f 0il ¼ P sat i Φi :

216

Separation in a closed vessel

Table 4.1.2.

Table 4.1.3. 550 kPa

500 kPa x il ¼ x iv =K i

i¼1 i¼2 i¼3

0.12 0.32 0.56

6 2 0.6

0.02 0.16 0.93 P x il ¼ 1:11

i¼1 i¼2 i¼3

0.12 0.32 0.56

6.5 2.2 0.66

0.0185 0.145 0.848 P x il ¼ 1:011 ffi 1

101.3 110

n-Decane

Ki

n-Hexane n-Heptane n-Octane n-Nonane

xiv

Isopentane n-Pentane

Species

Isobutane n-Butane

x il ¼ x iv =K i

Propylene Propane

Ki

Ethylene Ethane

xiv

Methane

Species

200 190 180 170 160 150

150

200

140 250

130

300

120 110 100

500

90

600

80

700 70

800 900 1000

60 50

1500

Temperature (°C)

Pressure (kPa)

400

40

n-Decane

2000 2500 3000 3500

30

20

n-Nonane

n-Heptane n-Octane

n-Hexane

Isopentane n-Pentane

10 Isobutane n-Butane

Propylene Propane

Methane

5000

Ethylene Ethane

4000

0 –5

Figure 4.1.6. High-temperature nomograph for Ki factors in light hydrocarbon systems. Reprinted, with permission, from D. Dadyburjor, Chem. Eng. Prog., 74(4), 85 (1978). Copyright © [1978] American Institute of Chemical Engineers (AIChE).

4.1

Two-phase equilibruim separation: closed vessel

Table 4.1.4.

217    M sat þ Psat , M ¼ P ðM, TÞ ¼ P exp A 1 − T

690 kPa Species

xil

Ki

x iv ¼ x il K i

i¼1 i¼2 i¼3

0.12 0.32 0.56

5 1.6 0.5

0.6 0.512 0.28 P x iv ¼ 1:392 (too high)

Table 4.1.5. 1050 kPa xil

Ki

x iv ¼ x il K i

i¼1 i¼2 i¼3

0.12 0.32 0.56

3.4 1.3 0.34

0.408 0.416 0.19 P x iv ¼ 1:014

Therefore, sat ^ iv P ¼ x il γil P sat x ivΦ i Φi :

The equivalent relation for a continuous chemical mixture is     sat ^ Mv ¼ f l ðMÞdM γMl P sat f v ðMÞdM PΦ ð4:1:33aÞ M ΦM ,

where P sat M is the vapor pressure of the species of molecular ^ Mv , weight M at temperature T; other quantities, namely Φ sat ΦM and γMl correspond to the similar quantities for the species of molecular weight M for the given phase at the system condition. For simplicity, assume an ideal gas phase and an ideal ^ Mv ~ 1, γMl ~ 1, Φsat liquid solution: Φ M ~ 1. We obtain, from (3.3.66), ð4:1:33bÞ

which can be written as f v ðMÞ ¼

P sat M f l ðMÞ , P

where Pþ and A are constants. If fl(M) is known, e.g. it is a Gaussian or Γ distribution, then P can be determined from (4.1.33d) by integration over the molecular weight range of the continuous mixture. From (4.1.33c), the molecular weight density function of the first vapor bubble can now be determined; it is clear from this Raoult’s law relation that if fl(M) is, say, a Γ distribution, then fv(M) will also be a Γ distribution, with different values of the mean and the standard deviation. The two molecular weight density functions frequently used in such types of calculations are: • Γ distribution function

Species

P f v ðMÞdM ¼ P sat M f l ðMÞdM,

ð4:1:33eÞ

ð4:1:33cÞ

the expression which represents Raoult’s law for continuous chemical mixtures. Here P is unknown and can be determined from ð ð P sat M f l ðMÞdM P¼ ð ð4:1:33dÞ ¼ P sat M f l ðMÞdM f v ðMÞdM over the whole range of molecular weights since fl(M) is known and P sat should be available. For example, M Trouton’s rule may be used to describe P sat M for a pure species of molecular weight M as follows:

f ðMÞ ¼


  ðM−γÞα−1 M−γ exp − : α β GðαÞ β

ð4:1:33fÞ

• For a value of M ¼ γ, f(M) is zero. The mean of this distribution function is αβ þ γ and the standard deviation is αβ2. • Gaussian distribution function
  1 M−Mmean : ð4:1:33gÞ f ðMÞ ¼ pffiffiffiffiffi exp − 2σ2 2πσ • The mean is Mmean and the standard deviation is σ.

4.1.3

Liquid–liquid systems

If a species i is distributed between two immiscible liquid phases j ¼ 1,2, the separation achieved between the two liquid phases may be determined by the ratio (ai1/ai2) defined by equation (4.1.1). If separation between two species i ¼ 1,2 distributed between two phases j ¼ 1,2 is desired, the separation factor is given by equation (4.1.4). The standard state of each solute species i ¼ 1,2 has to be specified before these equations can be utilized. When pure solute at the same temperature and pressure of the system under consideration is the standard state, then μ0i1 ¼ μ0i2 :

ð4:1:34aÞ

At equilibrium, the distribution of solute i from relation (4.1.1) is given by ai1 γ x i1 ¼ 1 ¼ i1 ; ai2 γi2 x i2

ð4:1:34bÞ

x i1 γ ¼ K i ¼ i2 : γi1 x i2

ð4:1:34cÞ

Nonideal behavior is therefore a prerequisite to separation of solute i between bulk phases 1 and 2. In the process of solvent extraction of solute i from phase 2 (feed solution) into the solvent, the extract (phase 1), it is important that Ki > 1. The activity coefficients γi1 and γi2 are to be estimated from commonly used standard equations (Treybal, 1963, pp. 70–71) e.g. the two-constant based

218

Separation in a closed vessel

van Laar equation or Margules equation, the three-constant based Redlich–Kister equation. Although the γij in these equations are complex functions of concentrations of species i, as well as the concentrations of the two liquids constituting the bulk of the two immiscible liquid phases j ¼ 1,2, simple results are obtained for the γij under certain limiting conditions (Treybal, 1963, pp. 104–105). The limiting conditions are low concentrations of solute i, the two liquids constituting bulk phases 1 and 2 are very insoluble in each other so that x32 and x41 (here i ¼ 3,4 refer to the species making up the two bulk phases j ¼ 1,2, respectively) are essentially zero; the corresponding result is log10 K i ¼ Ai2 − Ai1 ,

ð4:1:34dÞ

V i ðδi − δj Þ2 , RT

ð4:1:34eÞ

where the Aij are appropriate constants in the two-constant or three-constant equations for activity coefficients mentioned earlier (Treybal, 1963, pp. 104–105). Application of regular solution theory (Prausnitz,1969) leads to the following estimate of γij for a solution of i in phase j containing primarily liquid j: ln γij ¼

where the δi and δj are the solubility parameters of liquids i and j, respectively. Therefore, from (4.1.34c), V i ½ðδi − δ4 Þ2 −ðδi − δ3 Þ2  ln K i ¼ : RT

ð4:1:34fÞ

Estimates of solubility parameters for a variety of liquids are available in the literature. Table 4.1.6 illustrates the values of solubility parameters for some common liquids. If the solute prefers the solvent (phase 1, species 3), then δi is closer to δ3 than δ4, and therefore, Ki > 1. (Note: Generally, the more polar the liquid, the larger the value of the solubility parameter; water, which is highly polar, has a δi of 21, whereas n-pentane, which is extremely nonpolar, has a δi of only 7.1. Further, the higher the required energy of vaporization, the higher the solubility parameter.) Result (4.1.34d) allows rapid estimation of whether Ki is much larger than 1 for solute i being extracted into phase 1 from phase 2. This may be used to select a particular Table 4.1.6. Solubility parameters for some common liquids at 25  C

Liquid Perfluoroalkanes n-Pentane n-Hexane n-Octane Carbon tetrachloride Toluene Benzene Ethanol

δi (cal/cm3)1/2 Liquid 6.0 7.1 7.3 7.5 8.6 8.9 9.2 11.2

Acetonitrile Acetic acid Dimethylsulfoxide Methanol Propylene carbonate Ethylene glycol Formamide Water

δi (cal/cm3)1/2 11.8 12.4 12.8 12.9 13.3 14.7 17.9 21.0

solvent over another solvent. There are many other considerations in the selection of a solvent for extracting a solute from a feed solution. These include: insolubility of the solvent in the feed liquid and vice versa; the solvent should be recoverable easily for reuse; the densities of the solvent and the feed solution when in equilibrium with each other must be different; the interfacial tension of the liquid–liquid system should not be too small to facilitate coalescence of the dispersed phase, since most commercial solvent extraction devices disperse the feed or the solvent phase in the other phase as drops to achieve extractive transfer of solute (membrane solvent extractors are an exception). When extractive separation between two solutes i ¼ 1,2 present in a feed solution (phase j ¼ 2) by an extraction solvent (phase j ¼ 1) is desired, the process is called fractional extraction. The system now becomes quaternary; however, the separation between solute species i ¼ 1,2 can be determined using equation (4.1.4) as if it is a binary system of species 1 and 2. For the standard state condition (4.1.34a), the separation factor α12 is given by α12 ¼

x 11 x 22 γ21 γ12 K 1 ¼ ¼ : x 12 x 21 γ11 γ22 K 2

ð4:1:34gÞ

Treybal (1963, p. 108) has illustrated ways of estimating α12 by estimating the different activity coefficients γ21, γ12, γ11 and γ22: the system consists of alkyl benzene (i ¼ 1), paraffin (i ¼ 2), ethylene glycol or furfural (solvent) and n-heptane (feed phase). The solubility-parameterbased approach using relation (4.1.34f) may also be employed. The estimation of Ki or α12 for cases where an infinite dilution standard state for the solute(s) is employed, can be made from equation (4.1.1) as follows:
 ai1 γi1 x i1 ðμ0 − μ0 Þ ¼ ¼ exp − i1 i2 , ð4:1:34hÞ RT ai2 γi2 x i2 where μ0i1 6¼ μ0i2 . It follows therefore that
 x i1 γ Δμ0 ¼ K i ¼ i2 exp − i : x i2 γi1 RT

ð4:1:34iÞ

For two solutes i ¼ 1,2,
 x 11 x 22 γ γ ðΔμ01 − Δμ02 Þ ¼ α12 ¼ 12 21 exp − : RT γ11 γ22 x 12 x 21

ð4:1:34jÞ

If the solutions behave ideally, both γi1 and γi2 tend to 1, resulting in
 Δμ0 K i ¼ exp − i ; ð4:1:34kÞ RT
 ðΔμ01 − Δμ02 Þ α12 ¼ exp − : ð4:1:34lÞ RT If the value of the Nernst distribution coefficient κN i1 is known (see (3.3.78)), then, for dilute solutions,

4.1

Two-phase equilibruim separation: closed vessel

219

1

1 (Ethanol)

Water as solvent

Weight fraction ethanol in extracting solvent layer, solvent-free basis

0.9 Fe

P F M

3 (Benzene)

Benzene as solvent

0.7 0.6 0.5 0.4 0.3 0.2 0.1

E

R

Fr

0.8

0 A B

4 (Water)

(a)

0

0.1 0.2 0.3 0.4 0.5 0.6 Weight fraction ethanol in nonsolvent layer, solvent-free basis (b)

Figure 4.1.7. (a) Selective extraction of ethanol (1) from phase 3 (benzene) by means of a solvent phase 4 (water); (b) selectivity plot on a solvent-free basis for a Type 1 system at 25  C: ethanol (1)–benzene (3)–water (4); water is the solvent. (After Treybal, 1963, p.124.)

α12 ¼

κN 11 : κN 21

ð4:1:34mÞ

When two solutes 1 and 2 are distributed between two immiscible phases j ¼ 1,2, we have four species; for the partitioning of one solute between immiscible phases j ¼ 1,2, there are three species. To make mass balance and separation calculations, it is useful to recognize that often the two so-called “immiscible” phases may have partial miscibility. For example, consider the ternary system benzene (i ¼ 3), water (i ¼ 4) and ethanol (i ¼ 1, the solute) (see Figure 1.2.2 and replace picric acid by ethanol) whose liquid–liquid equilibrium type is shown in the triangular diagram (see Figure 1.6.1) of Figure 4.1.7(a). As shown, at the given temperature, ethanol and benzene are completely miscible; so are ethanol and water over the whole composition range. However, benzene and water have clearly only limited solubility in each other. So, of the three possible binary pairs, only one pair (benzene–water) has partial miscibility (Type 1 system) (Treybal, 1963). On the other hand, consider a ternary mixture of n-heptane (i ¼ 3), aniline (i ¼ 4) and methylcyclohexane (i ¼ 1) (the solute) shown schematically in Figure 4.1.8(a). As long as the temperature is less than 59.6  C, the two bulk liquids n-heptane and aniline will form two immiscible layers with partial solubility (see Example 1.5.2). But aniline and methylcyclohexane also form two immiscible layers having partial solubility of each in the other phase.

However, n-heptane and methylcyclohexane are completely miscible with each other over the whole composition range. Such systems having two pairs of partially miscible liquids are called Type 2 systems (Treybal, 1963, p. 15). Obviously, if there is another solute (i ¼ 2) beside methylcyclohexane (i ¼ 1) in a significant amount, the system will become complex if it is partially miscible with some of the other species in the system. To understand the separation in solvent extraction in either system (Type 1 or Type 2), focus on a solution of composition F in Figure 4.1.7(a) for a Type 1 system. If the solvent is water (i ¼ 4) and it is added to the system, the overall two-phase mixture will be located at M (the object here is to extract ethanol (i ¼ 1) from benzene (i ¼ 3) into water (i ¼ 4)) somewhere along the line joining F and 4; M can be located using the lever rule (see equation (1.6.2)). After sufficient time has been allowed for the two phases (water phase and the benzene phase) to come to equilibrium, the two phases will be located at, say, E and R along a straight line which is called the tie line. The locus of these points E and R for various twophase mixtures for different feed compositions, ARPEB, provides an envelope in the triangular diagram: the area inside the envelope represents the two-phase region (the water phase and the benzene phase), while the area outside represents a single-phase region. Points A and B represent, respectively, a benzene phase saturated with water and a water phase saturated with benzene without

220

Separation in a closed vessel

1

1 (Methylcyclohexane)

Weight fraction 1 in aniline-rich phase, aniline free-basis

0.9 Fe

F Fr R M

E

0.7 0.6 0.5 0.4 0.3 0.2 0.1

E

3 (n-heptane)

0.8

4 (Aniline) (a)

0 0

0.2 0.4 0.6 0.8 Weight fraction 1 in n-heptane rich phase, aniline-free basis

1

(b)

Figure 4.1.8. (a) Selective extraction in Type 2 systems; (b) selectivity plot on a solvent-free basis for a Type 2 system at 25  C: methylcyclohexane (1)–n-heptane (3)–aniline (4); aniline is the solvent. (After Treybal, 1963, p.125.)

any ethanol. The envelope ARPEB, called the solubility or binodal curve, represents compositions of individual solutions which are in equilibrium with another immiscible solution on the same curve. Focus now on the mixture M which splits into two immiscible phases R (raffinate) and E (extract) (Figure 4.1.7(a)); the compositions of R and E are the equilibrium compositions of the two immiscible layers. The water phase representing point E has primarily water, a considerable amount of ethanol and a little bit of benzene. The benzene phase representing point R has primarily benzene, a small amount of ethanol and a very small amount of water. In solvent extraction processes, after the extraction is carried out, the solvent is removed from the extract and the raffinate so that the solvent may be reused. If the solvent (here, water) is removed from the extract (point E), then, by the lever rule, the composition of the water-free organic phase represented by Fe is obtained by extending line 4E to intersect with the line 13. Correspondingly, the water-free raffinate composition is obtained by extending line 4R to intersect with the line 13 at Fr. The net result of the solvent extraction of the benzene–ethanol feed mixture F with water, and the subsequent removal of water from the extract phase and the raffinate phase, are two fractions having composition Fe (ethanol-rich) and Fr (benzene-rich). Two immiscible phases having different concentrations of the solute to be extracted is a prerequisite to the application of solvent extraction. In Figure 4.1.7(a), the binodal curve ARPEB has two branches: the ARP branch,

which is mostly benzene-rich, and the BEP branch, which is mostly water-rich. However, as the curves tend toward P, the concentration of the solute (ethanol) increases and the immiscibility of the two phases disappears at point P, the Plait point. It is not necessarily the point in ARPEB which has the highest amount of ethanol. The quantitative behavior of the distribution of ethanol (i ¼ 1) between the two phases (i ¼ 3, benzene; i ¼ 4, water) can be obtained from the two branches of the binodal curve. A more useful approach would be to plot the ethanol concentration at point Fe against ethanol concentration at point Fr, i.e. on a solvent-free basis after removing water, as if the original feed mixture of benzene–ethanol has been split into two fractions (not phases, since the two fractions are miscible). Figure 4.1.7 (b) illustrates the weight fraction of ethanol in the water layer on a solvent-free basis vs. the weight fraction of ethanol in the benzene layer on a solvent-free basis; here water is the solvent. This figure also has a plot for the same system, with benzene being considered as a solvent to separate ethanol from water. For a Type 2 system, as shown in Figure 4.1.8(a), the area in the triangle in between the two curves represents the two-phase region. Any feed mixture F of n-heptane and methylcyclohexane when mixed with the solvent aniline will produce a mixture of overall composition M. If we now follow the notation of Figure 4.1.7(a) and the corresponding steps, we will end up with two fractions, Fe and Fr, of n-heptane and methylcyclohexane, free of the solvent.

Two-phase equilibruim separation: closed vessel

A plot of the methylcyclohexane weight fraction in the aniline phase vs. that in the n-heptane phase (based on an aniline-free basis) illustrates the separation achieved (Figure 4.1.8(b)). The partitioning of a solute i between two immiscible liquid phases j ¼ 1 and 2 and the selectivity between two solutes in the two-phase system have so far been considered primarily in the context of liquid extraction/solvent extraction in large-scale operations. Many such aspects and some other considerations are also important for the basic equilibria in liquid–liquid chromatography (LLC) (see Section 7.1.5). In LLC, a mobile liquid phase ( j ¼ 2) flows over a porous, finely divided solid phase whose surface has been coated with a liquid, the stationary phase ( j ¼ 1). Any solute introduced via the mobile phase is partitioned between the two liquid phases. For cases where μ0i1 ¼ μ0i2 , the regular solution-theory based expression (4.1.34f ) for Ki may be written as ln K i ¼

V i ½ðδi − δ2 Þ2 −ðδi − δ1 Þ2  : RT

ð4:1:34nÞ

To achieve a high value of Ki between the two phases, δi should tend to δ1, i.e. the solute i should be similar to the stationary phase j ¼ 1. Further, K11 should be different from K21 for any chromatographic separation between two solutes 1 and 2. Note that the two liquid phases must also be immiscible, for which, generally, ðδ1 − δ2 Þ  4 (Karger et al., 1973). There are many combinations of highly polar organic liquids and water where ðδ1 − δ2 Þ  4 does not guarantee phase immiscibility. However, the criterion of liquid immiscibility is important for solvent selection. In liquid–liquid extraction, generally one of the two immiscible liquid phases is aqueous and the other phase is organic, which is nonpolar or mildly polar. There are many examples where both immiscible phases are primarily organic: aromatic species such as toluene and benzene are extracted from an essentially nonpolar hydrocarbon feedstock by an immiscible highly polar organic solvent. Large-scale applications utilize highly polar organic solvents, e.g. ethylene glycol, dimethylsulfoxide, n-methylpyrrolidone, etc., with or without a small amount of water (Lo et al., 1983). Figure 4.1.9A illustrates the behavior of many such solvents in terms of their selectivity for aromatics with respect to nonaromatics vs. their capacity for the aromatics. The solvent power or capacity is directly proportional to the equilibrium ratio Ki1 with the extracting solvent being phase 1. An ideal solvent has a high selectivity as well as high capacity. Extraction of solute species from one liquid to another immiscible liquid is also carried out when both phases are primarily aqueous. Large-scale purification processes for proteins employ aqueous two-phase systems (Albertsson, 1986) containing two water-soluble but incompatible polymers in water, e.g. polyethylene glycol (PEG) and dextran. The PEG-rich layer is at the

221

(a) 100 80 60

Thiodipropionitrile • Ethylene glycol • • Ethylene carbonate Diethylene glycol •

40

Selectivity

4.1

20

10 8

• Sulfolane

• Nitromethane • Dimethyl sulfoxide • Methylformamide

Triethylene glycol • Tetraethylene glycol •

• Oxidipropionitrile

Propylene • Carbonate

• Ethylene diamine

Methylcarbamate • Furfural • Diethylene glycol methylether • • Triethylene tetramine Propylene glycol •

• n-Methylpyrrolidone

6 4 3 0.06 0.08 0.1

• Dipropylene glycol

Capacity

0.2

0.4

0.6 0.8 1.0

2.0

Figure 4.1.9A. (a). Capacity and selectivity of polar organic solvents for aromatics (Lo et al., 1983). Reprinted, with permission, from T.C. Lo, M.H.I. Baird and C. Hanson, Handbook of Solvent Extraction, Figure 2, p. 525, Wiley-Interscience, 1983. Copyright © 1983, John Wiley & Sons.

top ( j ¼ 1); the dextran-rich layer is at the bottom ( j ¼ 2), with a clear boundary in between. Proteins, cells, etc. are partitioned between the two layers. The partition coefficient, κp1 , decreases from about 2 to 0.2 as the molecular weight increases (Figure 4.1.9B). The partition coefficient is strongly influenced by the salt (e.g. KCl, K3PO4, K2SO4, etc.) usually present in such systems. The presence of the salt creates an electrostatic potential gradient across the interface between the twophase system, which leads to preferential partitioning of the protein into one of the phases, depending on the sign and magnitude of the protein surface charge. Haynes et al. (1991) have related the interfacial electrostatic potential difference between the two aqueous phases j ¼ w1 and j ¼ w2, namely Δϕ ¼ ðϕw2 −ϕw1 Þ,to the salt distribution coefficient κm salt in terms of the salt molar concentrations for a 1:1 electrolyte:   RT RT ms, w1 Δϕ ¼ ϕw2 −ϕw1 ¼ Þ ¼ , ð4:1:34oÞ lnðκm ln salt ms, w2 F F where ms,j is the salt molality in phase j (mol/kg). For low concentrations of electrolytes and proteins, the protein partition coefficient, κp1 , with lighter phase 1 at the top, has been related to that in the absence of any Δϕ, κ0p1 , by a model based on virial expansion (King et al., 1988): lnðC p1 =C p2 Þ ¼ ln κp1 ¼lnκ0p1 þ ðZ p F =RTÞ Δϕ ¼ B1p ðC 12 −C 11 ÞþB2p ðC 22 −C 21 ÞþðZ p F =RTÞΔϕ: ð4:1:34pÞ

Here B1p and B2p are the second virial coefficients for interaction between polymers 1 and 2, respectively, Cij is

222

Separation in a closed vessel

(b) 3.0

Liquid solution of i and j

Temperature

b-Galactosidase (from E.coli K12 and WL)

Bovine serum albumin Transferrin (human)

Papain Trypsin a -Chymotrypsin

kp1

Insulin RNase Lysozyme (hen and turkey)

1.0

a -Amylase (bacterial)

Ovalbumin

2.0

Pure solid i and liquid

Pure solid j and liquid

TE

E Immiscible mixture of pure solid i and pure solid j 0 xj = 1

xi

xi = 1

0.1

Figure 4.1.10. Temperature vs. composition diagram for two species i and j in solid–liquid systems where the two solid phases are immiscible.

0.05 1

2

3

4

5

6

Protein molecular weight (x 10

7 −4

8

50

Da)

Figure 4.1.9B. Protein molecular weight vs. protein partition coefficient in a PEG 6000-dextran 500 system with pH at pI for all proteins (Sasakawa and Walter, 1972). Reprinted, with permission, from Biochemistry, 11(15), 2760 (1972), Figure 2. Copyright (1972) American Chemical Society.

the molar concentration of polymer i in region j and Zp is the algebraic charge number (valence) of the protein. Haynes et al. (1993) have developed a model to predict κp1 for a protein in biphasic aqueous systems containing strong electrolytes amongst other things.

x il γis f 0is ¼ : x is γil f 0il Therefore, for two species i and j, αij ¼

0 x il x js γis γjl f 0is f jl ¼ : x is x jl γjs γil f 0js f 0il

For ideal solutions in both phases, γil ffi 1 ffi γjl and γis ffi 1 ffi γjs , leading to αij ¼

4.1.4

Liquid–solid systems

In Section 3.3.7.5, the equilibrium partitioning of a species between a liquid phase and a solid phase was briefly considered for three types of liquid–solid equilibria. The separation between two species i and j for such liquid– solid two-phase systems is briefly considered here. There are systems where three phases can be present; for example, two immiscible solid phases and a saturated solution, as in the case of solid salt, ice and a saturated salt solution. Figure 4.1.10 shows a temperature vs. composition phase diagram where solid phase 1 coexists with solid phase 2 and a saturated liquid solution at the eutectic point E. Below the eutectic temperature TE, immiscible pure solid phase 1 and 2 are present together. For these and more complex systems, the reader should refer to appropriate texts (Darken and Gurry, 1953; DeHoff, 1993). Separation between species i and j in simpler two-phase systems described in Figures 3.3.6A, where the solid phase is a homogeneous solution, will be determined now. For a solid solution in equilibrium with a liquid, relation (3.3.93) for the equilibrium ratio of species i is

ð4:1:35Þ

0 f 0is f jl : f 0js f 0il

ð4:1:36Þ

At temperature T and pressure P for the system in equilibrium, for any species i, f 0is ðT, PÞ f 0 ðT, PÞ f 0is ðT mi , PÞ f 0il ðT mi , PÞ ¼ 0 is   : 0 f il ðT, PÞ f is ðT mi , PÞ f 0il ðT mi , PÞ f 0il ðT, PÞ

ð4:1:37Þ

Here Tmi is the melting point of species i. For a pure species i at T mi , f 0is ðT mi , PÞ ¼ f 0il ðT mi , PÞ. Exact expressions have been developed for the product of the two remaining ratios. An expression practically useful and based on particular approximations (Smith et al., 2001, pp. 526–531) is  

f 0is ðT, PÞ ΔH i T − T mi : ð4:1:38Þ ffi exp RT mi T f 0il ðT, PÞ Therefore  9 8 1, namely the adsorbate phase is enriched in species 1 compared to species 2, we have P 02 ðπÞ > P 01 ðπÞ. Obviously,

!

ð4:1:45Þ

P ¼ P 01 ðπÞx 1σ þ P 02 ðπÞx 2σ :

þ

qiσ x iσ :

i¼1

These results follow directly from ideal adsorbed solution theory (Myers and Prausnitz, 1965), whose governing equations are (4.1.46) to (4.1.48). The key quantities are P 01 ðπÞ and P 02 ðπÞ. One needs pure component adsorption data for each species to determine P 01 ðπÞ and P 02 ðπÞ and to calculate the separation factor α12 for a binary system. The procedure suggested for determining the compositions of the gas phase (xig) and the adsorbate phase (xiσ) at any given total pressure P is as follows.

Here, qiσ is the number of moles of i per unit mass of adsorbent when adsorbed from a pure gas i at the same surface pressure π and temperature T as the mixture. The determination of the activity coefficients γiσ for the adsorbed phase requires accurate experimental data at constant temperature and pressure for the entire range of gas-phase compositions (Myers and Prausnitz, 1965). However, if the adsorbed solution phase is considered ideal, i.e. γiσ ¼ 1, then Px ig ¼ P 0i ðπÞx iσ ; 1 X2

q i¼1 iσ

qiσ ¼

2 X

2 X x iσ

ð4:1:47Þ

qiσ x iσ :

ð4:1:48Þ

i¼1

qiσ

!

i¼1

p

(1) Plot graphically qiσ =P 0i against P 0i at any given T. Calculate the area for different integration limits of P 0i (equation (4.1.49)). This will yield a plot of ðπSσ =RTÞ against pressure P, correspondingly P 0i p for each i. If qiσ can be described analytically as a function of P 0i , the same integration can also be carried out analytically to develop the same plot. See Figure 4.1.11. (2) To determine relevant quantities for the mixture, select a total pressure P and a value of π for the mixture. In Figure 4.1.11, draw a line parallel to the abscissa for the corresponding π and a vertical line at the selected P. The point of intersection of these two lines, M, defines the mixture location at a certain total pressure P, mixture spreading pressure π and temperature T. Correspondingly, the value of P 0i is obtained from the abscissa of the intersection of the line parallel to P-axis (for the corresponding π) and the pure component equilibrium curve. The mixture compositions are obtained via the lever rule (equation (1.6.2)) using equations (4.1.50) and (4.1.51). For example, equation (4.1.51) may be written as

ð4:1:46Þ

p;

¼

Note that, from the definition of P 0i ðπÞ (Section 3.3.7.6), πSσ ¼ RT

ð P01 0

p

q1σ dP ¼ P

ð P02 0

p

q2σ dP, P

ð4:1:49Þ

where each integral is to be used for pure gas adsorption data. For a binary gas mixture, this is the equivalent of a Raoult’s law type of situation in vapor–liquid equilibria (see (3.3.64) and (4.1.21b)). The separation factor α12 for species 1 and 2 between the adsorbed phase (σ) and the gas phase (g) is α12 ¼

x 1σ x 2g P 02 ðπÞ ¼ : x 1g x 2σ P 01 ðπÞ

ð4:1:50Þ

F

B

A

ð4:1:51Þ

Pure Component (1)

M C

Pure Component (2)

pSσ E

RT

0

P10

P

P20

Figure 4.1.11. Graphical calculation of adsorption equilibrium of a gas mixture from pure component spreading pressure information. (After Myers and Prausnitz (1965).)

4.1

Two-phase equilibruim separation: closed vessel

225 Table 4.1.8.

Px 1σ þ Px 2σ ¼ P 01 ðπÞx 1σ þ P 02 ðπÞx 2σ , which may be rearranged to

O2 at 0  C

ðP − P 01 ðπÞÞx 1σ ¼ ðP 02 ðπÞ − PÞx 2σ , i.e. x 1σ P 02 ðπÞ − P length of line CM ¼ ¼ : x 2σ P − P 01 ðπÞ length of line MB Further x 1σ ¼

length of line CM length of line CB

and

x 2σ ¼

length of line MB , length of line CB ð4:1:52aÞ

since x1σ þ x2σ ¼1 from (4.1.48). Correspondingly, x 1σ ¼

P 02 ðπÞ − P 0 P 2 ðπÞ − P 01 ðπÞ

x 2σ ¼

and

x 2g ¼

P 02 ðπÞx 2σ : P

¼

P 01 ðπÞ x 1σ P 01 ðπÞ ðP 02 ðπÞ − PÞ ¼ P 02 ðπÞ x 2σ P 02 ðπÞ ðP − P 01 ðπÞÞ

πSσ πSσ P − RT RT P 02 ðπÞ   ¼ ¼ P − P 01 ðπÞ πSσ P − P 01 ðπÞ 0 0 RT P 1 ðπÞ P 1 ðπÞ 1−

¼

P P 02 ðπÞ

Pressure (mm Hg)

Volume adsorbed, observed (cm3)

83.0 142.4 224.3 329.6 405.1 544.1 602.5 667.5 760.0

3.32 5.57 8.73 12.68 15.48 20.42 22.48 24.86 28.03

95.5 127.4 199.7 272.4 367.4 463.7 549.7 647.9 760.0

7.09 9.48 14.37 19.41 25.54 31.70 37.01 42.65 48.89

Volume of oxygen adsorbed (cm3) pO2

pCO

230.2 529.8 8.98 391.1 368.9 15.05 585.1 174.9 21.93

and x 2g ¼

Isotherm

Obs.

8.31 14.1 21.73

35.90 25.82 12.8

35.02 24.69 11.63

0

ð4:1:53Þ

Data from Table 4.1.8 for pure oxygen (observed) at 0  C were used to develop the following relation: p

qO2 σ ¼ −4:228  10−6 P 0O2 þ 0:0339 P 0O2

So ME EF

Obs.

the data provided in Table 4.1.8 for the adsorption of each pure species; take the pressure as P 0i and the volume adsorbed (in cm3) to number of moles of i per gram of p p adsorbent as qiσ . Now relate qiσ =P0i to P0i .

PM − PE ME ¼ : MF MF

x 1g ¼

Isotherm

Volume of carbon monoxide adsorbed (cm3)

generated. In order to generate these plots, the integral ð P0 i p ðqiσ =PÞdP has to be developed for each species. Employ

Therefore, x 1g x 2g

Volume adsorbed, observed (cm3)

P − P 01 ðπÞ : 0 P 2 ðπÞ − P 01 ðπÞ

ð4:1:52bÞ

P 01 ðπÞx 1σ , P

Pressure (mm Hg)

CO–O2 Mixtures at 0  C

To determine x1g and x2g, we employ (4.1.46) or (4.1.50) along with the above expressions for x1σ and x2σ in Figure 4.1.11. From (4.1.56), x 1g ¼

CO at 0  C

MF : EF

ð4:1:54Þ

Example 4.1.2 Markham and Benton (1931) studied experimentally the adsorption of pure O2, pure CO as well as CO– O2 mixtures on 19.6 g of silica at 0  C. The data obtained are provided in Table 4.1.8. The column for the mixtures under “isotherm” contains values obtained from pure component adsorption isotherms via interpolation. Develop a plot of x O2 g vs. x O2 σ for the O2–CO mixture adsorption on silica at 0  C at a total pressure of 1 atmosphere using the ideal adsorbed solution theory. Plot the three experimental points for the mixture provided in Tabel 4.1.8 in the same diagram. Solution To develop a plot of xig vs. xiσ, where i ¼ O2, first two plots of (πSσ/RT) vs. P 0O2 and (πSσ/RT) vs. P0CO have to be

ð4:1:55Þ

and r2 ¼ 0.967 (Figure 4.1.12). Then we have πSσ ¼ RT

ð P0

O2

0

p

qO2δ 1 dP ¼− 4:22810−6 ðP 0O2 Þ2 þ0:0339P 0O2 : P 2 ð4:1:56Þ

Data from Table 4.1.8 for pure CO (observed) at 0  C were used to obtain the following relation: p

qCOσ ¼ −1:508  10−5 P 0CO þ 0:0755 P 0CO

ð4:1:57Þ

and r2 ¼ 0.987 (Figure 4.1.13). Then we have πSσ ¼ RT

ð P0

CO

0

p

qCO 1 dP ¼ 1:50810−5 ðP 0CO Þ2 þ0:0755P 0CO : P 2 ð4:1:58Þ

226

Separation in a closed vessel

0.076

0.0405 Relationship between Pi0 and qi pσ /Pi0 at 0°C

0 Relationship between PCO and q pi σ /Pi0 at 0°C

i = O2

0.0400

−6

i = CO

0.074

0

mipσ /PO20 = −4.228 x 10 PO + 0.0339

miσp /PCO0 = −1.508 x 10−5P CO + 0.0755 0

0.0395

qi σ /PCO (cm3/mm Hg)

0.0390

0

0.0385 0.0380

0.072 0.070 0.068

p

q piσ /PO02 (cm3/mm Hg)

2

0.0375

0.066 0.064

0.0370

0.062

0.0365 0

100

200

300

400

500

600

700

0

800

100

Pressure,PO20 (mm Hg)

200

300 400 500 600 Pressure, PCO0 (mm Hg)

700

800

Figure 4.1.13. Plot for carbon monoxide.

Figure 4.1.12. Plot for oxygen.

200

1.0 CO

180 160

0.8

140

100

O2,g

0.6 O2

x

πSσ /RT

120

80

0.4

60 40

0.2

20 0

0

500

1000 1500 2000 2500 3000 3500 4000 Pressure, P (mm Hg)

0.0 0.0

0.2

0.4 x O ,σ

0.6

0.8

1.0

2

Figure 4.1.14. Calculation for mixture of O2–CO adsorption equilibia from pure component spreading pressure at 0  C.

Equations (4.1.56) and (4.1.58) have been plotted in Figure 4.1.14 which is ready to use. Employ equations (4.1.52b) and (4.1.54) to develop estimates of x O2 , σ ,x CO, σ ,x O2 , g and xCO,g. The predicted behavior of x O2 , g vs. x O2 , σ is shown in Figure 4.1.15. The experimental data have also been plotted to indicate how well the ideal adsorbed solution theory predicts the observed behavior. Another interfacial adsorption system, where one of the bulk phases is a solid, involves adsorption of solutes from a solvent onto the surface of solid adsorbents. Such liquid–solid adsorption systems are frequently used in

Figure 4.1.15. Prediction of adsorption of O2–CO mixture on silica at 0  C, total pressure ¼ 1 atm.

large-as well as small-scale applications. A particular example is the adsorption of organic pollutants from industrial and municipal waste waters onto activated carbon adsorbents. The equilibrium adsorption behavior of a waste water containing multiple solutes may be predicted by the ideal adsorbed phase model developed by Radke and Prausnitz (1972) for dilute solutions, provided the single-solute equilibrium adsorption behavior of each solute species is available. The approach is very similar to that of the ideal adsorbed solution theory of Myers and Prausnitz (1965) for gas–solid adsorption.

4.1

Two-phase equilibruim separation: closed vessel

Radke and Prausnitz (1972) suggested that when the solute species adsorb simultaneously from a dilute solution onto the adsorbent surface at constant temperature and spreading pressure π, the adsorbed phase forms an ideal solution; activity coefficients for all species are unity in the adsorbed phase. The following relations were proposed for a system containing n solutes: n X 1 x iσ ¼ p ðconstant T, πÞ; n X q i¼1 iσ qiσ

ð4:1:59Þ

i¼1

C tl x il ¼ C 0i ðπÞx iσ ðconstant TÞ,

ð4:1:60Þ

where qiσ ¼ x iσ

n X

qiσ :

i¼1

ð4:1:61Þ

Here, Ctl is the total concentration of all solutes in the liquid phase, xil is the solvent-free mole fraction of species i in the liquid phase, xiσ is the surface phase mole fraction of species i without the solvent, C 0i ðπÞ is the unknown liquid-phase concentration of the pure solute i at spreading pressure π and temperature T, which provides the same surface phase concentration of i as the mixture, and p qiσ is the number of moles of i per unit mass of adsorbent when adsorbed from a pure solution of i at the same π and T as the mixture. The first relation suggests that the total number of moles of adsorbed solutes is determined by the sum total of the adsorptions of single solutes at the same values of T and π. For the second result, relating the equilibrium mole fractions of a given solute in the two phases, C 0i ðπÞ is unknown. One has to evaluate the spreading pressure π in the manner of equation (4.1.49) from each pure comp ponent adsorption data of qiσ vs. Ci, πSσ ¼ RT

ð C 01 0

p

qiσ ðC 0i Þ dC 0i : C 0i

ð4:1:62Þ

Then, for a given π, C 0i can be determined for each species i (just as P 0i was determined earlier for gas adsorption), which will yield xiσ from relation (4.1.60) for known Ctl and xil. If experimental liquid–solid adsorption equilibrium data are available for each solute species i at a given temperature, then, to obtain the equilibria in a multisolute system, employ relation (4.1.62) for each solute species to obtain a relation between πi and fi ðC 0i Þ. For example, for a system of two solutes being present (i ¼ 1,2), obtain π 1 ¼ f1 ðC 01 Þ;

ð4:1:63aÞ

π 2 ¼ f2 ðC 02 Þ:

ð4:1:63bÞ

C 0i

is to be defined at the

For any given π for the mixture, same π; therefore,

227

π1 ¼ π ¼ π2 :

ð4:1:63cÞ

For a system of two solutes, relation (4.1.60) provides C tl x 1l ¼ C 01 ðπÞx 1σ ;

ð4:1:63dÞ

C tl x 2l ¼ C 02 ðπÞx 2σ ,

ð4:1:63eÞ

where x 1σ þ x 2σ ¼ 1. Choose any Ctl and x1l (equivalent to choosing the concentration of two solutes, C tl x 1l ¼ C 1l and C tl ð1−x 1l Þ ¼ C tl x 2l ¼ C 2l ). Then unknowns C 01 , C 02 , π1, π2 and x1σ (or x2σ since x1σ þ x2σ ¼ 1) may be determined from equations (4.1.63a–e). One can adopt a graphical procedure of the type suggested by Myers and Prausnitz (1965) and illustrated earlier for gas–solid adsorption. If the single-solute isotherms illustrated by equations (4.1.63a) and (4.1.63b) can be obtained in an integrable analytic form, the set of equations may be solved by computer using the Newton–Raphson iteration scheme (Radke and Prausnitz, 1972). This is especially relevant for a system of solutes numbering more than two. In some liquid–solid adsorption processes, the number of different components to be adsorbed is very large. One can adopt the formalism of continuous chemical mixtures and represent the class of similar adsorbable components by one variable M; a useful choice may be a parameter of the adsorption isotherm, e.g. the slope at infinite dilution (Annesini et al., 1988). In Section 3.3.7.6, the distribution of a solute between a bulk phase and an interfacially adsorbed phase was considered for a gas–liquid system containing surfactants, gas–solid adsorption systems and liquid–solid adsorption systems. So far, in this section, the separation factor for two solutes has been determined for an air–water system containing surfactants and for gas–solid adsorption. We could do the same also for the liquid–solid adsorption considered in the preceding paragraphs. In liquid–solid adsorption, if microporous adsorbents are involved, then partitioning of the solute between the external solution and the pore solution has to be considered to obtain accurately the surface adsorption equilibria. If C 0il is the initial bulk liquid concentration of solute i in the initial bulk solution volume V 0l , and Cil is the bulk liquid concentration of solute i in the final bulk liquid volume Vl, after equilibration of the microporous adsorbp ent having a specific pore volume (cm3/g of support) of V l , 2 mass ws (g) and specific surface area Sσ (m /g), then the following solute balance holds: p

p

C 0il V 0l ¼ C il Vl þ C il V l ws þ Γ iσ ws Sσ ,

p C il

ð4:1:64aÞ

is the pore liquid concentration of species i and where Γiσ is the concentration of adsorbed species i per unit surface area of the microporous adsorbent. The different liquid volumes are related by p

V 0l ¼ Vl þ V l ws :

ð4:1:64bÞ

228

Separation in a closed vessel

We can now rewrite (4.1.64a) as C 0il V 0l

¼

C il V 0l

p − C il V l ws

þ

p p C il V l ws

þ Γ iσ ws Sσ ,

and rearrange to obtain  p 

C il ðC 0il − C il ÞV 0l Γ iσ Sσ −1 þ ¼ w : s p C il C il V pl C il V l

ð4:1:65Þ

p

The quantity ðC il =C il Þ is the geometrical partitioning factor κim defined earlier (see (3.3.88a) and (3.3.89a)) for the partitioning of a solute between a solution and a porous membrane. Here the porous membrane is replaced by the microporous adsorbent. The quantity κim is less than 1 unless there are specific or nonspecific interactions (electrostatic or van der Waals interactions) between the solute and the pores; it can be quite small if the radius ri of the solute molecules is of the order of the pore radius rp. For cylindrical pores (see (3.3.88a)),   ri 2 : κim ¼ 1− rp In any equilibrium partitioning experiment using a microporous adsorbent and an external solution, one varies C 0il and ws for a given adsorbent. From equation (4.1.65), a plot of C 0il vs. ws for a given adsorbent will be linear, the slope being equal to the quantity in brackets on the right-hand side of equation (4.1.65). Unless rp >>> ri, it is clear that p κim ¼ C il =C il will be less than 1, and the determination of the pore surface adsorption equilibrium relation (between Γiσ and Cil) will be influenced by κim. This was demonstrated clearly in the adsorption of aromatic compounds (for example, napthalene) on microporous silica gel adsorbents by Alishusky and Fournier (1990).

4.1.6

Liquid–ion exchanger systems

Ion exchange processes are employed in a variety of situations in process-scale as well as preparative separations. A situation often encountered involves the selectivity of the ion exchanger for counterion A over counterion B. To develop an expression for the selectivity, we employ relation (3.3.118b) describing the equilibrium distribution of ionic species i between an aqueous solution and the ion exchange resin for ionic species A and B: 2 3 0 1 1 4 a Aw A − V A ðP R − P w Þ5 ¼ ϕR − ϕw RT ln @ aAR ZA F 2 3 0 1 1 4 aBw A @ − V B ðP R − P w Þ5: RT ln ¼ aBR ZB F This leads to " RT ln

aAw aAR

Z1  A

aBR aBw

Z1 # B

¼



 VA VB ðP R −P w Þ: − ZA ZB ð4:1:66Þ

If the swelling pressure, PR − Pw, is negligible, we obtain " 1 1 # aAw Z A aBR Z B ¼ 0, ð4:1:67Þ ln aAR aBw which leads, after rearrangement, to     aAw Z B aBw Z A ¼ : aAR aBR Suppose ZA ¼ 2, ZB ¼ 1 and aAw ¼ aBw. Then
 aAR aAw 1 aAR aBR ) ¼ : ¼ 2 ¼ a2BR aBw aBw aBR aBw

ð4:1:68Þ

ð4:1:69Þ

This result indicates that, if, for counterion B, the resin phase is selective over the aqueous phase, i.e. aBR > aBw, then aAR > aBR: the resin phase is selective toward counterion A having a higher valence or charge number over counterion B with a lower valence. This result reflects the strong influence of counterion valence on the selectivity: from a mixture of two counterions, A and B, the counterion having a higher valence, A, is always preferred by the ion exchange resin. Correspondingly, the separation factor αaAB defined using activities, namely αaAB ¼

aAR aBw , aAw aBR

ð4:1:70Þ

will be greater than 1, for example, when aAw ¼ aBw . For a cation exchange resin, for example, Caþþ will be preferred over Naþ. Helfferich (1962) has provided a detailed analysis of the influence of other factors on ion exchanger selectivity: the ion exchanger tends to prefer (there are exceptions) (1) the counterion of higher valence, (2) the counterion having the smaller (solvated) equivalent volume, (3) the counterion having greater polarizability, (4) the counterion which interacts more strongly with the fixed ionic groups or the matrix, (5) the counterion which has a lower tendency of complex formation with the co-ion. Generally, the order of selectivity among univalent cations is as follows (Helfferich, 1962): Agþ > Csþ > Rbþ > Kþ > Naþ > Hþ > Liþ : For weak-acid resins, the position of Hþ will depend on the acid strength of the fixed anionic group. The corresponding sequence for bivalent cations is: Ba2þ > Pb2þ > Sr2þ > Ca2þ > Ni2þ > Cd2þ > Cu2þ > Co2þ > Zn2þ > Mg2þ > UO2 2þ : For anions, the selectivity sequence appears to be citrate > SO4 2− > oxalate > I− > NO3 − > CrO4 2− > Br− > SCN− > Cl− > formate > acetate > OH− > F− :

4.1

Two-phase equilibruim separation: closed vessel

For weak-base resins, the position of OH− is variable, but is generally further to the left. Equation (4.1.70) expresses the selectivity between two particular counterions A and B by a resin. One would also like to know the nature of the corresponding relations that exist between a counterion and a co-ion or two co-ions. Consider the case where there are two counterions, A and B, and a co-ion Y. Using relation (3.3.118b), we obtain 2 3 0 1 1 4 a Aw A− V A ðP R − P w Þ5 RT ln @ aAR ZAF 0 1 2 3 1 4 a Bw A− V B ðP R − P w Þ5 ¼ ϕR −ϕw ¼ RT ln @ aBR ZB F 0 1 2 3 1 4 a Yw A− V Y ðP R − P w Þ5: RT ln @ ¼ ZY F aYR A rearrangement leads to " 1 1#   aAw Z A aYR ZY VA VY − ðP R − P w Þ: ¼ RT ln aAR aYw ZA ZY

ð4:1:71Þ Relation (4.1.66) is another one of these relations. Ignoring the swelling pressure in (4.1.71) leads to " 1 1# aAw Z A aYR Z Y ln ¼ 0: aAR aYw

aAw aAR

Z1

A

¼



aYw aYR

Z1

Y

¼



aBw aBR

Z1

B

,

ð4:1:72Þ

where we have employed the result (4.1.67). In fact, this result is valid for all other ions in the solution (Helfferich, 1962). Consider another co-ion in the system, say X. It follows from the above that 

aYw aYR

Z1

Y

¼



aXw aXR

Z1

X

:

ð4:1:73Þ

Let ZY ¼ −1 and ZX ¼ −2. For the cation exchange resinbased system, 

aYw aYR

−1

(correspondingly NaCl over Na2SO4). See Problem 4.1.14 for a related exercise. So far, we have activities of species to arrive at a number of conclusions. In practice, species concentrations are measured. Therefore, let us consider relation (4.1.68) for the selectivity of the ion exchanger for counterion A over counterion B: 1Z B 0 1Z A 0 1 0 1Z A =Z B a a a a Aw Bw Aw Bw @ A ¼@ A )@ A¼@ A aAR aBR aAR aBR 0 1 0 1Z A =Z B x AR γAR @ x BR γBR A x AR x Bw A @ ) ¼ ) x Aw γAw x Bw γBw x Aw x BR  0 1Z A =Z B 0 1Z A =Z B − 1 γ γ @ x BR A ; ¼ αAB ¼ Aw @ BR A γAR γBw x Bw  0 1 Z A =Z B − 1 ZA =Z B γAw ðγBR Þ @ x BR A αAB ¼ : ZA =Z B γAR x Bw ðγBw Þ 0

ð4:1:75aÞ In a dilute solution of counterions A and B in water, one may assume γAw ffi 1, γBw ffi 1. Therefore, ZA

αAB ¼

ðγBR Þ =Z B γAR

¼

 1     aXw −2 aYw 2 aXw ) ¼ : aXR aYR aXR

ð4:1:74aÞ

If aYw ¼ aXw and aYR < aYw for the cation exchange resin, then the separation factor αaYX ¼

aYR aXw aYw ¼ > 1: aYw aXR aYR

ð4:1:74bÞ

Therefore, a co-ion of lower valence (Y) is more strongly preferred by a cation exchange resin; for example Cl− will be preferred over SO42− by a cation exchanger



 ZA = x BR ð Z B Þ − 1 : x Bw

If ZA ¼ 2, ZB ¼ 1, then αAB ¼

Therefore, 

229

γ2BR γAR



 x BR : x Bw



ð4:1:75bÞ

For very low xBw, αAB will have a large value. For larger xBw, αAB will be much smaller. Consider the case of A ¼ Ca2þ and B ¼ Naþ, as encountered in the removal of hardness (Ca2þ) from water. For low values of salt concentration in water (i.e. low xBw), the selectivity of the ion exchanger for Ca2þ over Naþ will have some value. If, however, xBw is high, αAB will be much lower, leading to removal of the Ca2þ ions from the ion exchanger. At low values of xBw, the ion exchanger will have many more Ca2þ ions, thus removing the cause of hardness from water. It is useful to consider now another kind of counterion exchange process between the ion exchanger and the counterions in the external solution. The counterions to be considered now are, macroions, specifically proteins of large molecular weight. Any such protein molecule in solution has positive as well as negative charges distributed on its surface. The net charge of the macromolecular protein due to pH-based interactions of various constituent groups is positive if the pH is less than the isoelectric point (I.E.P ¼ pI); the net charge is negative if the solution pH > pI. Thus, as long as the solution pH is different from the pI, the protein surface has some net positive or net negative

230

Separation in a closed vessel

Ion exchange polymer chain

Protein Protein

Ion exchange polymer chain

Before ion exchange

After ion exchange

Figure 4.1.16. “Ion exchange” occurring when a negatively charged protein adsorbs to an anion exchanger. Seven positively charged ions (e.g. HTrisþ) associated with the protein molecule are displaced, together with seven negative ions (Cl −) from the exchanger. (After Scopes (1987).)

charges. Interactions due to pH based chemical reactions in general are considered in Chapter 5. Due to the principle of electroneutrality, small counterions present in the solution will be distributed around the protein surface to ensure local electroneutrality. Consider now a negatively charged protein molecule (with positively charged microions distributed around it) to be adsorbed on an anion exchanger resin bead (Figure 4.1.16). The fixed positive charges on the anion exchange resin have counterions (e.g. Cl−) present to start with. The protein molecule has to displace these counterions (Cl−) near the resin surface in order for it to be ion exchanged. As shown in the figure, after the protein molecule is adsorbed, seven counterions (Cl−) near the ion exchange resin surface and seven counterions present around the protein in solution (e.g. HTrisþ)3 are displaced together into the solution. This addition of Tris–Cl into the solution increases the ionic strength of the solution. The electrostatic attraction between the negative charges on the protein surface and the positive fixed charges on the anion exchange resin can be considerably weakened by two methods. If the pH of the buffered solution is decreased to a lower value, then the electrostatic binding force is considerably reduced. Alternatively, if the ionic strength of the solution is increased, the interaction between the protein and the ion exchanger is reduced and that between the microions (e.g. Cl−) and the ion exchanger is considerably enhanced. This method is generally preferred if desorption of the adsorbed protein is desired.

3

Tris stands for tris (hydroxylmethyl) aminomethane.

Another form of ion exchange of interest is that between two different proteins, i ¼1 and 2, and an appropriate active charged site on the ion exchange resin surface. For any protein i, one can represent this interaction by ið j ¼ 2Þ þ Sð j ¼ 1Þ ↔ iSð j ¼ 1Þ,

ð4:1:76Þ

where the resin phase is j ¼1 and S represents the resin site. The equilibrium constant for this binding is usually identified as Kd and represents the processes of adsorption (rate constant is ka) and the dissociation (rate constant is kd). At equilibrium, the rates are equal: k a C i2 C S1 ¼ k d C iS, 1 :

ð4:1:77aÞ

Now CS1 representing the concentration of active charged sites, can be expressed as the difference between the maximum number of charged sites, C m S1 , and the number of sites already occupied by the protein, CiS,1: C S1 ¼ C m S1 − C iS, 1 :

ð4:1:77bÞ

k d C i2 ðC m S1 − C iS, 1 Þ ¼ : C iS, 1 ka

ð4:1:77cÞ

At equilibrium, Kd ¼

Correspondingly, at equilibrium, the protein concentration in the resin phase, CiS,1 will be related to the protein concentration in the surrounding fluid phase, Ci2, by C iS, 1 ¼

C i2 C m S1 : K d þ C i2

ð4:1:77dÞ

At any temperature, this relation behaves as a Langmuir adsorption isotherm (see Section 3.3.7.6). The equilibrium constant, Kd, has a low value for proteins which bind

4.1

Two-phase equilibruim separation: closed vessel

231

strongly to the resin. For any resin, the comparative binding strengths of different proteins will be indicated by the respective values of Kd. Patel and Luo (1998) have indicated that Kd values for many different proteins and a variety of ion exchange resins vary between 10−5 and 10−7 M. Experimental protein adsorption data plotted as (1/Cis,1) vs. (1/Ci2) will yield (1=C m S1 ) as intercept and (K d =C m ) as the slope: S1

4.1.7

ð4:1:77eÞ

Supercritical fluid–bulk solid/liquid phase

Section 3.3.7.9 describes how a supercritical fluid (SCF), e.g. supercritical CO2, can extract effectively a solute from a liquid or a solid; the solute is recovered later by reducing the pressure: the SCF becomes a gas having very little solubility of the solute, which precipitates and is recovered. This section briefly considers the systems where two or more solutes are extracted simultaneously. There are two goals here. First, one would like to increase the solubility of a particular species. Second, it would be desirable to enhance the selectivity of the SCF for a particular solute with respect to the second solute. Generally, it has been found that for a pure SCF (e.g. CO2), the extent of enhancement of the solubility for a particular solute is very similar for other solutes being extracted over the entire range of the density of the SCF. Thus pure SCFs are not helpful in developing selectivity unless other entrainers (e.g. methanol) are added in small amounts. (Note: One of the lures of extraction by SCF is the opportunity to avoid organic extractants, e.g. methanol.) However, Chimowitz and Pennisi (1986) have suggested the use of the crossover region for developing selective SCF separation. Consider Figure 4.1.17 for the crossover regime of a SCF extracting two solid solutes, components 1 and 2, from a solid mixture, where the mole fraction of each solute, xil, in the SCF phase has been plotted against the SCF pressure for two temperatures TH and TL ( pI. A third category of molecular aggregates are called vesicles, liposomes, etc. Like reversed micelles, they enclose water inside a spherical shell. However, unlike reversed micelles, the shell is made of a lipid bilayer structure present in biological cell membranes. Specifically, the shell has a bilayer structure with the polar hydrophilic head of two layers of lipids forming the two surfaces of a sandwich of the two layers of lipids, with the long two-tailed nonpolar hydrophobic chains of each lipid in the interior of the sandwich (Figure 4.1.22). The specific lipids have a polar phosphate head and are therefore usually called phospholipids; an example is phosphatidylcholine. The hydrocarbon chains in any such lipid are generally 14 to 18 carbon atoms long, resulting in a typical lipid bilayer thickness of around 3.7 nm. The spherical vesicle dimensions can vary over a wide range, for example 0.03–0.1 μm. Sometimes

Carriers

Phosphatidylcholine lipid

Figure 4.1.22. Vesicle structure.

they can be much bigger, incorporate many layers of the lipid bilayer and are called multilamellar vesicles (MLVs), as opposed to those having a single lipid bilayer (Lasic, 1992). Both structures are also called liposomes. The 14–18 carbon atoms long chain of the lipid in the lipid bilayer presents a rigid hydrocarbon gel-like structure which is essentially impermeable to charged/polar molecules larger than water. The transfer of polar/charged species from outside to inside of the vesicle requires a carrier; for transferring metal ions, one uses an ionophore as a carrier of the charged/polar species (see Sections 4.1.9.1.3 and 5.4.4) incorporated into the lipid bilayer (Walsh and Monobouquette, 1993). The lipid bilayer is a new material phase that acts as a selective membrane between an external aqueous phase and an aqueous phase enclosed by the lipid bilayer, the vesicle. However, these noncovalent assemblies of lipids are damaged in the presence of detergents, water-soluble alcohols, etc. Covalently closed vesicles can be formed by using polymerizable vesicle-forming surfactants (Shamsai and Monobouquette, 1997). 4.1.9 Partitioning between a bulk fluid phase and an individual molecule/macromolecule or a collection of molecules for noncovalent solute binding A variety of chemical interactions are possible between two molecules of two different species. Those that result in chemical reactions are considered in Chapter 5 for the purpose of separation of mixtures. There are a number of other, weaker, noncovalent interactions where the bond energy is less than 50 kJ/mol. These include chelation, clathration, hydrogen bonding, hydrophobic interactions, ionic binding, pi bonding, van der Waals interactions, etc. These result in weak chemical complexes, which are

4.1

Two-phase equilibruim separation: closed vessel

reversible in nature. Further reversing the complexes without destroying the complex-forming molecule is possible due to weaker interactions. King (1980) has enumerated the variety of interactions as originally identified by George K. Keller with respect to the bond energy vs. bond type. These interactions lead to reversible binding of a solute molecule from a bulk fluid phase with a complex-forming individual molecule or collection of individual molecules introduced from outside. The complexing agent may also be a macromolecule of biological (e.g. proteins, etc.) or nonbiological (e.g. conventional polymers) origin. The complex so formed may remain in the original solution phase or may form a second phase, usually a solid. This solid phase may be withdrawn and treated to reverse the complexation. To avoid the need for separation of the solid phase thus formed, sometimes these molecules/macromolecules may be individually bonded to or deposited on a porous or nonporous bead/particle or the pore surfaces of a membrane at one end of the molecule (for example), while the other ends or parts of the molecule are available to bind the solute species from the solution. Species present in the gas phase may also bind to such molecules on a bead or particle. When the solute molecule is a protein or a large entity, a large number of binding sites are necessary to hold the solute; thus, a number of individual complexing molecules bonded to the porous particle at one end will have their other ends bind at different locations of the solute, which is a protein/large entity. A natural extension of this concept has led to the bonding/coupling of protein A to agarose beads for extremely selective binding of immunoglobulins from a solution containing immunoglobulins and other materials in antibody purification. Separation generally also requires recovery of the solute species after it has been selectively bound to specific molecules added to the solution or bound to a porous or nonporous bead/particle or membrane pores. When the process takes place without any particles, beads or membranes, the complex produced has to be removed from the solution and decomplexation carried out. The steps usually taken to achieve decomplexation (with or without the use of beads, particles or membranes), i.e. regeneration of the molecules/macromolecules for reuse in binding solute molecules, include: changing the ionic strength of the solution by adding electrolytes; changing the pH; raising the temperature; decreasing the pressure, etc. Although the binding of the solute molecule with the complexing agent is reversible and noncovalent in nature, the mathematical description is generally identical to situations where chemical reactions take place. Thus, a number of such reversible low binding energy based separations are described mathematically in Chapter 5. Here we illustrate the basic nature of the phenomenon of solute partitioning in such processes in a closed system under the following categories:

235 (1) (2) (3) (4)

inclusion compounds; ionic binding of metals with charged species; pi bonding/complexation; hydrophobic interaction.

The phrase “solute partitioning” mentioned above should not suggest that the solute must necessarily be partitioned into a separate phase. There are many examples (to be discussed in the following) where the solute molecule is bound to a different type of molecule in the same phase; such binding facilitates subsequent separation. 4.1.9.1 Inclusion compounds: adducts, clathrates, crown ethers, cyclodextrins, liquid clathrates Different types of molecules acting as mass-separating agents may be added to a liquid mixture in gaseous, liquid or solid form to form preferentially a solid phase of the added external agent molecules and the solvent/one of the solute species in the liquid mixture when cooled. The solid phase of the solvent/solute and the external added agent is generally called an inclusion compound (Atwood et al., 1984). We now consider a variety of inclusion compounds: adducts; clathrates; crown ethers; cyclodextrins; liquid clathrates. 4.1.9.1.1 Adducts When the external agent added to the solution forms the bulk of the new crystalline solid phase, the process is called adductive crystallization and the compounds so formed are called adducts. The solute molecules from the liquid feed will fit into the crystal lattice of this crystalline solid phase, which acts as the “host;” the solute molecules included in the host crystal are the “guest” (an inverse scenario happens in the case of clathrates, as we will soon see). The hosts could be either organic or inorganic. The slurry that is produced has to be removed from the raffinate liquid and may be heated to recover the “guest.” An example of this type of process of selective solute partitioning is urea based adductive crystallization. A saturated solution of urea in water at 70  C may be mixed with a mixture of aromatic and paraffinic hydrocarbons present in a solvent at 40  C (the Edeleanu process). Under conditions of appropriate refrigeration, lumps of urea–n-paraffin adducts appear as crystals (Findlay, 1962; Fuller, 1972). As shown in Figure 4.1.23(a), the host compound, urea, has crystallized into a form having a central tunnel open at both ends; the tunnel accommodates the “guest” paraffinic hydrocarbon molecule and holds it by van der Waals forces with m urea molecules forming the tunnel: mðureaÞ þ paraffin ↔ adduct:

ð4:1:79aÞ

The guest paraffin has significant freedom in the channel, akin to adsorption rather than a chemical reaction. The

236

Separation in a closed vessel

(a)

(b) Ni NH3

2.1Å

N C

8.3 Å

5.0 Å

5.1 Å

Cross section of n-paraffin molecule within urea unit cell

Figure 4.1.23. (a) Representation of a cross section of a urea–n-paraffin adduct showing the urea molecules forming the walls of a tunnel in which the n-paraffin is held by attractive forces. (b) Structure of clathrate formed between monoaminenickel cyanide and benzene, showing the benzene molecule trapped inside a cage formed by the crystal lattice of the former (Findlay, 1962). Reprinted, with permission, from R.A. Findlay, “Adductive crystallization,” Chapter 4, Figure 7, p. 283 and Figure 5, p. 275 in Schoen, H.A. (ed.), New Chemical Engineering Separation Techniques, Interscience Publishers 1962. Copyright © 1962, John Wiley & Sons.

4.1.9.1.2 Clathrates Another class of inclusion compounds are clathrates, in which, instead of a channel as in urea adducts, a clathrate or a cage is formed by the “host” molecules, and the “guest” molecule is trapped in this cage (Figure 4.1.23(b)). Generally, the guest molecule is brought into the liquid phase of the host molecules and the clathrate phase crystallizes out as the temperature is lowered. When an aqueous solution is under consideration and the guest is a low molecular weight gas/vapor, e.g. CH4, SO2, C2H4, H2S, C2H5Cl, Cl2, Br2, CO2, C2H4O, etc., the clathrate is called a gas hydrate. The clathrate compound does not have a fixed chemical formula in terms of the number of guest molecules (GM) which can be accommodated in the cage created by the host molecules; however, there is a maximum number of host molecules. Findlay (1962) has listed the following among others for water as host: for CH4 guest, 46H2O8GM; for propene guest, 136H2O8GM; for SO2 guest, 46H2O8GM. In the gas hydrate, water molecules are linked through hydrogen bonding with a cavity diameter inside the cage

80 60 Hydrate + SO2 (l)

40

A Pressure (psia)

moles of urea per mole of the pseudoreactant, paraffin, varies, for example, between 6.1 and 23.3: 6.1 for n-heptane and 23.3 for n-detriacontane (C32H66). For a given paraffin, m is not fixed; the compounds are therefore nonstoichiometric. Apparently, urea forms adducts with straight-chain paraffinic and olefinic compounds as long as the carbon number is 6 or higher. The heat of this pseudoreaction is approximately 1.6 kcal per carbon atom, which is less than that for heats of adsorption (see Ruthven (1984) for heats of adsorption). Thiourea has also been employed as an adduct-former (Findlay, 1962).

B

20

10

Hydrate + SO2 (g)

C

8 6

Liquid water + SO2 (g)

4

2

30

35

40

45

50

55

Temperature (°F) Figure 4.1.24. Phase diagram for the sulfur dioxide–water system. Reprinted, with permission, from G.N. Werezak, in Unusual Methods of Separation, Chem. Eng. Prog. Symp. Ser., No. 91, Vol. 65, AIChE, New York, p. 6 (1969). Copyright © [1969] American Institute of Chemical Engineers (AIChE).

varying between 780 and 920 pm; the guest molecules, which do not interfere with hydrogen bonding and have diameters in the range of 410–580 pm, stabilize the structure under appropriate temperature and pressure (see

4.1

Two-phase equilibruim separation: closed vessel

237

Table 4.1.10. Inclusion compounds: types and examples

Multiple molecules create host structure: solid state compounds

organic molecules

Host

Guest (G)

Examples

urea

n-alkanes, n-alkenes, straight-chain acids, straightchain esters cyclopentane, isoparaffins HCl, HBr, SO2, CS2

adductive crystallization; e.g. 6.1 moles urea per mole of G

thiourea phenol hydroquinone inorganic molecules

Single molecule is host

organic molecules

H2O monoaminenickel cyanide Ni(4-methyl pyridine)4 (SCN)2 crown ethers

cyclodextrins (CD)

Multiple molecules create liquid host structure

Organic molecules

benzene

Englezos (1993) for a review of clathrate hydrates). Such molecules are often called hydrate-formers. The clathrate hydrates are formed only in given ranges of temperature and pressure. Figure 4.1.24 illustrates the pressure–temperature diagram for an SO2–water system (Werezak, 1969). Such a diagram will allow determination of the conditions under which the hydrates are formed, or the hydrate crystals decompose, releasing the trapped molecule SO2 for recycle and reuse. Such clathrate hydrate formation provides a way of recovering almost pure water or concentrating aqueous solutions using an appropriate hydrate-former. Clathrate compounds are also formed with organic nonaqueous host species, e.g. hydroquinone, phenol, monoaminenickel cyanide, methylnaphthalene, etc. (Findlay, 1962). Clathrate compounds formed with monoaminenickel cyanide as host and benzene/thiophene as guest (Figure 4.1.23 (b)) are of interest in petroleum refining; the selectivity of the host for benzene is much higher. The nature and types of inclusion compounds are more numerous than the two types, namely adducts and clathrates, considered so far. A broader list is provided in Table 4.1.10. The list is not comprehensive but illustrative of each category. In both adducts and clathrates, the

SO2, HCl, HBr, HCN, H2S, CO, etc. CH4, C3H8, SO2, Cl2, H2S, etc. benzene, thiophene, aniline, etc. p-xylene

alkali metal ions, Liþ, Kþ, Csþ, Sr2þ, etc.

small nonpolar organic molecules; organic isomers; p-xylene organometallic salt K[Al2(CH3)6N3]

clathrate compounds; e.g., 12C6H5OH  5G 3C6H4(OH)2  G 46H2O  8G

Werner-complex based clathration

extraction of such metal salts into an organic phase from aqueous phase/bonded phase in chromatography extraction of such species into aqueous phase containing CD or bonded CD phase in chromatography liquid clathrate containing 6 benzene to 1 salt molecules

number of molecules needed to develop the host structure is generally larger than 1 and can be as high as 136 (in clathrate hydrates). On the other hand, organic compounds, such as, crown ethers, cryptands, cyclodextrins, etc., incorporate a metallic ion or an organic molecule as guest inside the existing cavity present in the single host molecule. 4.1.9.1.3 Crown ethers, cyclodextrins Crown ethers are cyclic ethers having repeating –OCH2CH2– units; therefore, they are often called macrocyclic polyethers. Figure 4.1.25 identifies a variety of crown ethers, namely 14-crown-4, benzo-15-crown-5, dicyclohexyl-18-crown-6, dibenzo-21crown-7. The number 14, 15, 18, 21, etc., in this nomenclature identifies the number of atoms in the polyether ring, whereas 4, 5, 6, 7, etc., specify the number of oxygen atoms or repeating units in the ring (Pedersen, 1967). Table 4.1.11 provides an estimate of the diameter of the cavity in the polyether ring in a few crown ethers. This table also provides the diameters of a variety of cations, primarily alkali metal cations (Pedersen, 1988; Steed and Atwood, 2000). As long as the cavity diameter is close to that of an alkali metal cation (Mþ), the metal ion is held in the cavity as a guest with several oxygen atoms in the ring

238

Separation in a closed vessel

Table 4.1.11. Crown ethers, their properties and cationsa Cavity diameter (nm)

Crown ether All 14-crown-4 All 15-crown-5 All 18-crown-6 All 21-crown 7

b

0.12–0.15 0.17–0.22 0.26–0.32 0.34–0.43

Table 4.1.12. Extraction of alkali metal picrates by different crown ethers into methylene chloridea

Cation diameter (nm) Picrate extracted (%) Group I þ

Li Naþ Kþ Rbþ Csþ NH4þ

Group II 2þ

0.136 0.194 0.266 0.294 0.266 0.286c

Ca Zn2þ Sr2þ Cd2þ Ba2þ Ra2þ

0.198 0.148 0.226 0.196 0.268 0.280

Polyether

Li

þ

Naþ

Kþ

Csþ

Dicyclohexyl-14-crown-4 Cyclohexyl-15-crown-5 Dibenzo-18-crown-6 Dicyclohexyl-18-crown-6 Dicyclohexyl-21-crown-7 Dicyclohexyl-24-crown-8

1.1 1.6 0 3.3 3.1 2.9

0 19.7 1.6 25.6 22.6 8.9

0 8.7 25.2 77.8 51.3 20.1

0 4.0 5.8 44.2 49.7 18.1

a

Adapted from Pedersen (1988). “All” refers to various substitutent groups, e.g. dicyclohexyl, dibenzo, etc. c Nonmetallic cations are also amenable to complexation.

a From Table 4, p. 538, of “The discovery of crown ethers,” Charles J. Pedersen, Science, Vol. 241, July 1988, pp. 536–540. Reprinted with permission from AAAS.

b

O 14 O

O

O O

O

O 15

O O

Benzo-15-crown-5

O

O 18

O O

14-crown-4

O

O

O O

Dicyclohexyl-18-crown-6

O 21

O O

O O

Dibenzo-21-crown-7

Figure 4.1.25. A few crown ethers.

acting as donors to complex the strongly electropositive alkali metal cations. For example, 18-crown-6 prefers Kþ strongly over Liþ and Naþ. If the cation is too small or too large, the resulting complexes are not very stable. However, even in such cases, two molecules of crown ethers may form a sandwich to hold one cation. Crown ethers have low solubility in water but have considerable solubility in organic solvents. Crown ether complexes of alkali metal cations are therefore quite stable in organic phases. The anions of the alkali metal cations present in the aqueous phase, Y−, are simultaneously extracted into the organic phase; the complex in the organic phase may be represented as (crownM)þY−. Due to the likely absence of solvation of such anions in the organic phase, they are likely to be highly reactive. Nevertheless, crown ethers and similar host compounds have been successfully incorporated into organic solvents to extract alkali metal ions/salts from aqueous solutions. When the organic solvent containing crown ether is used in the form of a liquid membrane between two aqueous solutions (see Section 5.4.4), an alkali metal salt can be selectively transferred through the organic liquid membrane from one aqueous feed solution to an aqueous strip solution. Illustrative treatment of the kinetics and mechanisms of formation and dissociation of the metal complexes with crown ethers is available in Burgess

(1988). Crown ethers have been incorporated into numerous polymers; the resulting polymers show the expected order of alkali metal selectivity. Crown ethers have also been bonded to silica. The bonded crown ethers do show selectivity for various metallic cations (Alexandratos and Crick, 1996). In the case of clathrates and adducts described earlier, the so-called “compounds” formed are separated by crystallization. Although handling of slurry/solid in an industrial context is sometimes not desired, the solid phase conclusively demonstrates the nature of the nonstoichiometric dissociable host–guest compound formed. That macrocyclic ethers form reasonably stable complexes has also been demonstrated by the isolation of the complexes as crystals (Pedersen, 1988). That a particular crown ether having a certain cavity diameter will prefer a certain alkali metal cation having a certain diameter is illustrated in Table 4.1.12 for the extraction of a particular alkali metal picrate salt from water into methylene chloride containing the crown ether. From the table it appears that potassium picrate is most efficiently extracted by 18-crown-6 compared to other crown ethers, since the cavity size of 18-crown-6 is quite close to that of Kþ (see Table 4.1.11). Cyclodextrins (CDs) are cyclic oligosaccharides having the shape of an asymmetrical doughnut (see Figure 4.1.26 (a)). The three common types of cyclodextrins, α-CD, β-CD

4.1

Two-phase equilibruim separation: closed vessel

(a)

(b) HO O

O OO O OH H H OH HO O

239

O

CH3

OH

OO HO

OH HO OH O H H HO O OO O O HO O OH

O OH

CH3

Figure 4.1.26. (a) Schematic of α-cyclodextrin; (b) schematic of p-xylene as a guest in an α-CD host.

and γ-CD, each have a cavity whose diameters are, respectively, 0.57 nm, 0.78 nm and 0.95 nm. The corresponding molecular weights and aqueous solubilities are: 972, 1135, 1297 (molecular weights); 14.5, 1.85, 23.2 (solubilities in g/100 cm3). (See Bender and Komiyama (1978) for an introduction.) Due to the presence of hydroxyl groups on the outside surface, CDs are soluble in water. However, the interior cavity lined with hydrogen and oxygen atoms is relatively hydrophobic and provides sites for inclusion complex formation with smaller “guest” organic solutes having limited polarity (Figure 4.1.26(b)). A wide variety of smaller organic guests have been found to form inclusion complexes with a cyclodextrin molecule as host. The stability of the inclusion complex depends on a variety of factors, specifically the geometric fit with the cavity opening and the nature of the interactions. Such differences are sufficient to create different complexation tendencies of different isomers with cyclodextrins. There is a considerable literature on the use of CDs for chiral separations in chromatography; in fact, they are used in a variety of chromatographic columns in analytical chemistry. To this end, the CD is bonded to polysiloxane (Figure 4.1.27), which can be coated onto appropriate silica columns for use in chromatography (see Section 7.1.5) (Armstrong et al., 1993; Jung et al., 1994). The bonding of CD to polysiloxane phase eliminates the need for any crystallization based phase separation. Instead, one has to go through the process of complexation and decomplexation in sequence. Selective liquid–liquid extraction of xylene isomers (m-, o- and p-) has been demonstrated between organic

CH3

O

CH3

O

Si

CH3

Si

O

(CH2)8 n

O

(CH3O)6

β-CD

(CH3O)7

(OCH3)7

Figure 4.1.27. Structure of polysiloxane-bonded permethylated β-cyclodextrin (Chirasil-Dex) (n  60). (After Jung et al. (1994).)

and aqueous phases using branched α-CDs; the branched CDs have considerably higher aqueous solubility, and p-xylene is strongly preferred (Figure 4.1.26(b)) (Uemasu, 1992). An aqueous solution of β-CD containing urea and NaOH was employed by Armstrong and Jin (1987) and

240

Separation in a closed vessel

Mandal et al. (1998) as a liquid membrane (see Section 5.4.4) to separate different types of isomeric mixtures effectively. 4.1.9.1.4 Liquid clathrates Unless crown ethers and cyclodextrins are used either in a liquid membrane format, in a bonded state or in solvent extraction–back extraction methodology, the inclusion compound has to be crystallized out. Crystallization, however, has to be practiced in the case of “adducts” and “clathrates.” The handling of large-scale solids is difficult in such processes; furthermore, in the case of “adducts,” the solute-bonding capacity is limited due to the large load of the host per unit guest molecule. Liquid clathrates have therefore been developed so that solvent extraction methodology is available and solids handling can be avoided (Atwood et al., 1984). Liquid clathrates essentially consist of a liquid made out of guest molecules entrapped in a host species. They are primarily based on low-melting organometallic salts, which have a high solubility in aromatic liquids, as well as a high selectivity for liquid aromatic hydrocarbons. For example, K½Al2 ðCH3 Þ6 N3 ðlÞ þ 6C6 H6 ðlÞ ↔ K½Al2 ðCH3 Þ6 N3   6C6 H6 ðlÞ, ð4:1:79bÞ

indicating that a liquid of composition 1(salt): 6(aromatic) has been formed; this liquid is immiscible with and heavier than pure benzene phase. The low melting salt K [Al2(CH3)6N3] has weak interionic interactions, and the benzene molecules surround the anion with the cation outside the cage. There is some order, such that the cations attached to different adjacent cages cannot interact with one another. The stripping/back extraction can be easily implemented, only with particular liquid clathrates, by changing the temperature by only 10  C. 4.1.9.2

Ionic binding of metals with charged species

Metal ions are often present in a variety of aqueous solutions. In hydrometallurgy, the metal ion is to be recovered; in environmental separations, the heavy metal ions5 have to be removed from water and the water purified; in process streams, it could be either. Weak ionic binding of metal ions to a variety of charged species may be utilized to achieve such goals. The charged species can be polyelectrolytes, ionic surfactants, chelating agents, polyamino acids, etc. The charged species may be an individual

5

There are a variety of definitions for heavy metals. A few useful definitions are given here: any metal having a specific gravity greater than 5; any metal generally toxic to biological systems; any metal readily precipitated from solution as a sulfide; any metal located in the lower half of the periodic table, etc.

molecule in solution or bound to the surface of a bead or a pore in a porous membrane; alternatively, the individual charged molecules may be part of a collection of such molecules, e.g. a surfactant micelle formed from ionic surfactants. Metals are most often present as cations Mnþ in a solution: Cu2þ, Ca2þ, Zn2þ, Naþ, Pb2þ, Cd2þ, etc. Some metals exist in anionic forms as well: CrO42−, HCrO4−, etc. Consider now a polyelectrolyte, polystyrene sodium sulfonate (PSS), the sodium salt of the polystyrene sulfonic acid (Figure 4.1.28(a)). If such polyelectrolyte macromolecules are added to water containing hardness-causing metal ions, e.g. Ca2þ, Mg2þ, sodium ions will be replaced by Ca2þ or Mg2þ, which will remain bound to the polymer via the charged sulfonic acid group. Since a polymer molecule is easily removed by filtration, the undesirable ions remain bound to the anionic polyelectrolyte filtered and concentrated. Each polyelectrolyte molecule (polymer) will have many metallic ions bound to it at every location where there is an oppositely charged ion (Tabatabai et al., 1995). Such polyelectrolyte macromolecules may also be bound at one end on the surface of a bead or on the surface of a pore in a membrane. The whole polyelectrolyte molecule is essentially available for binding the metal ions in solution. However, since the polyelectrolyte is bound at one end, filtration is no longer necessary to recover it and the bound metal ions. An example of a polyelectrolyte bound at one end on the pore surface of a microfiltration membrane and having various interactions with metal ions is schematically shown in Figure 4.1.28(b). Here, poly (amino acid) molecules (e.g. poly(L-glutamic acid)) are bound to the pore surfaces of porous cellulosic membranes (Hestekin et al., 2001). The COO− groups created at higher pHs bind directly with the metal ion (shown as Me2þ) after deprotonation of COOH groups in the poly (amino acid). In addition, the electrostatic potential field created by the neighboring COO− groups leads to loose retention of metallic counterions (counterion condensation). The reverse of the metal ion binding process is important in the process of recovering a concentrated solution of metal ions and the polyelectrolytes. The bound heavy metal ions in a homogeneous solution of polyelectrolytes may be released by contacting with a concentrated brine. In the case of PSS in solution used to bind Ca2þand Mg2þ, Naþ ions from brine will replace Ca2þ and Mg2þ, etc. Such a solution, when filtered, will yield a filtrate concentrated in Ca2þ, Mg2þ, etc. The concentrate or retentate solution will contain the polyelectrolyte in the Naþ form and can be recycled for re-use (Tabatabai et al., 1995). An alternative strategy of using high acid/low pH will lead to release of the bound metal from the poly(amino acid)s bound to the membrane pore surface (Figure 4.1.28(c)) (Hestekin et al., 2001).

4.1

Two-phase equilibruim separation: closed vessel

241

(a)

SO3H

(b)

SO3H

(c) COO

Me2+

-

COO

Me2+

COOElectrostatically bound (counterion Me2+ condensation)

COO-

Me2+

Me2+

COOElectrostatically bound (counterion Me2+ condensation)

-

COO

Me2+

+

-

COO

Me2+

COO-

Polyamino acid chains

-

Me2+

COO-

Me2+

COO-

COO-

Me2+

Me2+

Me2+

COO-

COOMe

COO-

2+

Figure 4.1.28. (a) Polystyrene supported sulfonic acid. (b) Metal binding by two poly(amino acid) chains bound at one end to the pore surface of a membrane (Hestekin et al., 2001). Reprinted, with permission, from I&EC Research, 40, 2668 (2001), Figure 1(a). Copyright (2001) American Chemical Society. (c) Poly(amino acid) chains bound on the surface of a pore in a membrane.

Ionic surfactants will form micelles in a solution if their concentrations are above their critical micelle concentration (CMC). The ionic surfactant, for example sodium dodecyl sulfate (SDS), is prepared as the salt in which Naþ is the counterion and DS−, representing dodecyl sulfate (C12SO4−), is the surfactant ion (here an anion). A micelle of such a surfactant will have numerous ionic headgroups, DS−, around the periphery of individual micelles. If there are other metallic ions present in the system, e.g. Ca2þ, Cu2þ, Zn2þ, Cd2þ, etc., there will be an exchange between these ions and Naþ as the counterions for the DS− ions. Thus heavy metals, and other metals, present in solution will be bound to the headgroups of the micelle formed from the ionic surfactant. By using an appropriate membrane/filter, one can concentrate the ionic micelles and their bound heavy metallic counterions. The surfactants may be recovered from the concentrated solution by precipitating the surfactant using high concentrations of a monovalent counterion (e.g. Naþ(aq)). For example, neutral SDS salt may be precipitated via

C12 SO−4 ðaqÞ þ Naþ ! NaC12 SO4 ðsÞ

ð4:1:80aÞ

(see Brant et al. (1989)). A multivalent counterion, such as Ca2þ, may also be used to precipitate the surfactant: 2C12 SO−4 ðaqÞ þ Ca2þ ! CaðC12 SO4 Þ2 ðsÞ:

ð4:1:80bÞ

Thus, regeneration of metal ions, surfactants, polyelectrolytes, poly(amino acid)s, etc., participating in the ionic binding of metal ions in solution will require chemical reactions. Chemical reactions are considered in Chapter 5 in general for their effects on separations. Whether the metal-binding agent is a polyelectrolyte or an ionic surfactant headgroup in a micelle, it is of interest to know the extent of binding of a metal ion to such a species/agent. Based on the work of Oosawa, Scamehorn et al. (1989) have suggested equations that relate the fraction of a metal ion that is bound to the micelle to the free (unbound) metal ions. The estimation of these fractions requires additional relations, such as the electroneutrality condition, concentrations of surfactants present as micelles

242

Separation in a closed vessel

(a)

Na+ NaCl

Cl-

Cl-

Cl-

Na+

Cl-

Na+

Na+

Na+

(b)

ClNaCl

ClNa+

ClNa+

Figure 4.1.29. (a) The polyelectrolyte effect; (b) the anti-polyelectrolyte effect (Lowe and McCormick, 2001). Reprinted, with permission, from Stimuli-Responsive Water Soluble and Amphiphillic Polymers, ACS Symposium Series 780, 2000, p. 352, C.L. McCormick (ed.), A.B. Lowe and C.L. McCormik (Chapter 1 authors), Figures 1 and 2 on pp. 2 and 3 of Chapter 1. Copyright (2000) American Chemical Society.

and other parameters involving the CMC value, the charges of various ions, etc. The binding or presence of metallic ions or their counterions in aqueous solutions of polyelectrolytes considered so far can influence the conformation of free polyelectrolytes in solution. At high concentrations of salt (NaCl), there can be precipitation of polyelectrolytes (equation (4.1.80a)). A range of behavior is possible, having significant effects on separation. Consider a polyelectrolyte, poly (methacrylic acid) (PMMA), which will carry negative charges from carboxylate groups at high pH. These will repulse each other, leading to an extended conformation; there will be counterions present near each charged group in the backbone. The end-to-end distance will be large. When the concentration of the polyelectrolyte (in this case, polysalt) is increased, or the concentration of the salt (NaCl) is increased, the radius of the ionic atmosphere at every charge location is reduced; the polymer chains undergo shrinkage, resulting in a smaller configuration (Figure 4.1.29(a)) often called the polyelectrolyte effect (Lowe and McCormick, 2001). On the other hand, if we have a polyzwitterion (i.e. the polymer chain has both anionic and cationic groups) such that the ratio of the anionic to cationic groups is around 1, there will be attractive electrostatic interactions between such groups, leading to a collapsed globular structure. If, however, salt (NaCl) is

added, the individual ionic groups in the polyelectrolyte will be shielded by Naþ and Cl− ions; the attractive interactions will be reduced, and the polymer will take up a more expanded configuration (the anti-polyelectrolyte effect) (Lowe and McCormick, 2001) (Figure 4.1.29(b)). Such phenomena are the basis of a number of separation techniques. For example, if the polyelectrolyte has a collapsed globular structure, then the diffusional resistance of molecules through such a polyelectrolyte based medium will be drastically increased; larger molecules, e.g. proteins, may not be able to diffuse effectively. On the other hand, a collapsed structure may open up large gaps between the polyelectrolytes and allow easy transport through the overall medium. If a gel-like structure can be created using polyelectrolytes, the swelling or contraction of the gel can open up or shrink transport corridors based on the pH of the solvent system. Sometimes, this is carried out in the presence of an uncharged polymer or inside the pores of a rigid membrane. As an extreme case, a separate gel phase can be formed. This gel phase (e.g. of crosslinked, partially hydrolyzed polyacrylamide) acts as a sizeselective extractive solvent removing water and smaller solutes from an aqueous solution. At low pH, the gel shrinks and rejects extra water (soaked in earlier) in a separate vessel (Cussler et al., 1984), just as in solvent extraction and back extraction processes.

4.1

Two-phase equilibruim separation: closed vessel

243

Antibonding (p*) orbital

-

Outer d orbital

+ + C

Bonding (p) orbital +

-

+

Metal

C

+

-

-

Outer s orbital

s component of electron donor/acceptor interaction p component of electron donor/acceptor interaction Figure 4.1.30. Dewar–Chatt model of π-complexation (Safarik and Eldridge, 1998). Reprinted, with permission, from Ind. Eng. Chem. Res., 37(7), 2572–2581(1998), Figure 1. Copyright (1998) American Chemical Society.

4.1.9.3

Pi bonding/complexation

Mixtures of olefins and paraffins having similar or the same carbon numbers can be successfully separated by what is called pi bonding or π-complexation. In such a process, the olefin–paraffin mixture in a gaseous or liquid state is contacted with transition metals like Cu, Ag, etc., in a solution or dispersed on a solid substrate having a high surface area. The liquid solution/solid substrate will contain Agþ ions; in the case of Cu, cuprous salts are used to make Cuþ ions available. The π bonds of an olefin molecule interact with the d orbitals of such a metal/ion; also, the metal ion forms σ-bonds with the carbon in the olefin. Collectively this type of metal ion–olefin coordination is identified as π-complexation (Figure 4.1.30). Such complexations are highly selective for olefins; however, they are weak enough to be broken by low pressure or high temperature to release the olefin molecule bound to the transition metal ion. Olefins have been separated with high selectivity from a mixture with paraffins in gas phase by permeation through a liquid membrane/polymeric membrane containing AgNO3 (aqueous solution, Hughes et al. (1986);

glycerol solution, Kovvali et al. (2002); polyvinyl-alcoholbased polymeric membrane, Ho and Dalrymple (1994); polymeric perfluorosulfonate membrane, Koval et al. (1989)). By chemical absorption in a solvent, olefins have been separated successfully from paraffins: aqueous AgNO3 solution for C2H4 absorption at 240 psia and 30– 40  C (Keller et al., 1992); cuprous aluminum tetrachloride in an aromatic solvent (commercially used, Gutierrez et al. (1978)); AgNO3 solution in water in a microporous membrane based absorber (Davis et al., 1993); cuprous diketonate in α-methylstyrene solvent (Ho et al., 1988). Solid SiO2 sorbent on which Agþ was spread showed considerable selectivity for butene over n-butane at 1 atm and 70  C (Padin et al., 1999). A number of different silver salts have been used for complexation with olefins, e.g. AgNO3, AgBF4, etc. The system that has been studied in great detail is AgNO3 in water. The following complexation reactions have been suggested (Herberhold, 1974): Agþ þ olefin ↔ ½Ag − olefinþ ; þ

þ

ð4:1:81aÞ 2þ

½Ag ðolefinÞ þ Ag ↔ ½Ag2 ðolefinÞ ;

ð4:1:81bÞ

244

Separation in a closed vessel ½Ag ðolefinÞþ þ olefin ↔ ½AgðolefinÞ2 þ :

ð4:1:81cÞ

For a review of π-complexation for olefin–paraffin separations, see Safarik and Eldridge (1998). 4.1.9.4

Hydrophobic interaction

Hydrophobic interaction is utilized generally in the separation of protein mixtures via hydrophobic interaction chromatography. Protein molecules have a distribution of charged hydrophilic regions, as well as hydrophobic regions, on their surface. The hydrophobic regions arise from the side chains of a variety of amino acids making up the protein. The extent of such regions will be dependent on the protein. Such regions will, in aqueous solutions, prefer to interact with other such regions in other protein molecules; alternatively, if there are adsorbent particles having hydrophobic chains, short or long, sticking out, the hydrophobic patches on the protein molecule prefer to interact with such chains. The chains commonly used have a free benzyl group at the end or consist of a (–CH2–)n chain, where n varies from 3 to 10; such chains are bonded to the adsorbent particle via various chemistries. Interactions between such chains on the adsorbent surface and the hydrophobic patches on the protein surface lead to binding of the protein to the hydrophobic adsorbent surface. If water molecules were to solvate the hydrophobic regions of the protein surface, they have to be highly ordered, requiring a strong decrease in entropy for the process represented as follows (Tanford, 1980): protein þ sH2 O ↔ protein  sH2 O; ΔS0 ¼ −ve:

ð4:1:82Þ

This process does happen (so ΔG0 is negative, correspondingly ΔH0 is also negative; neglect here any interaction with the charged/hydrophilic regions of the protein). If hydrophobic chains are available on an adsorbent, the same arguments are valid. If the hydrophobic patches on a protein molecule react with the hydrophobic chains of the adsorbent, protein  rH2 O þ adsorbent  sH2 O ↔ protein  adsorbent 0

þ ðr þ sÞH2 O; ΔS ¼ þve;

ð4:1:83Þ

the entropy increases and water molecules are released. If salt ions are available (AþY−) in solution, released water molecules are drawn away to hydrate the salt ions: Aþ Y − þ ðr þ sÞH2 O ↔ Aþ  r 1 H2 O þ Y −  s1 H2 O; ð4:1:84Þ where (r1 þ s1) ¼ (r þ s). This will facilitate the forward reaction (4.1.83), enhancing the binding of the protein with the hydrophobic adsorbent. Thus, in hydrophobic interaction chromatography, high concentrations of salt are deliberately maintained to promote reaction (4.1.83). The effect of a salt on the adsorption of hydrophobic regions of a protein molecule on the hydrophobic patches

of an adsorbent has been analyzed by Perkins et al. (1997), who have performed a model-independent thermodynamic analysis employing what is known as the preferential interaction analysis. A general result of this analysis relates the change in the capacity factor k0 il of the protein species i ¼ 2 ( j ¼ 1, adsorbent), k 0 21 , to the variation in the molal salt concentration, m3, w , in the solution phase ( j ¼ 2): ∂ ln ðk 0 21 Þ ∂ ln ðm3, w Þ






T , P, eq:

¼

ðΔυþ þ Δυ− Þ n Δυ1 − m3, w , g m1, w g ð4:1:85Þ

where subscripts i ¼ 1, 2, 3 correspond to the solvent (e.g. water), the protein and the salt, respectively; further, for the salts which are electrolytes, n is the total number of anions and cations per formula unit, m1,w is the molal concentration of water (i ¼ 1) (55.5), Δυ1 , Δυþ and Δυ

are, respectively, the number of water molecules, the number of cations and the number of anions released during the binding process. The quantity g is defined as g ¼ ð∂lnðm3, w Þ=∂ln ða ÞÞT , P . Here a is the mean ionic activity for electrolytes (see (3.3.119d)) and is given as nþ n− 1=n n− Þ , a ¼ m3, w γ ðnþ

ð4:1:86Þ

where γ is the mean ionic activity coefficient and nþ and n− are the number of cations and anions per unit of salt. For a few common electrolytes, e.g. NaCl, (NH4)2SO4, NaSCN, in hydrophobic interaction based processes, n is, respectively, 2, 3 and 2. The constant g can be calculated for each system and has a range of values (around 1.1–1.9 in Perkins et al. (1997)) for a variety of systems. The values of Δυ1 can vary over a wide range (5–500) depending on the system. Perkins et al. (1997) have integrated equation (4.1.85) for the system of ovalbumin and ammonium sulfate to obtain ln ðk 0 21 Þ ¼ a þ

ðΔυþ þ Δυ− Þ n Δυ1 lnðm3, w Þ − m3, w : g m1 , w g ð4:1:87Þ 0

From the experimental data for k 21 obtained against m3, w , one can obtain values of a, ðΔυþ þ Δυ− Þ=g and ðn Δυ1 =m1, w gÞ. An alternative scenario is possible in the absence of these hydrophobic chains on the adsorbent/any adsorbent, namely hydrophobic patches in two neighboring protein molecules are bound together; this leads to protein precipitation. This is the salting-out phenomenon for protein solutions, wherein a high concentration of a salt, typically ammonium sulfate (1.5–3M), added to the protein solution, precipitates the protein molecules from the solution. Generally, the nature of the cation in the salt is not important. However, the anions are all important. The precipitation ability of the anions follow in decreasing order:

4.2

Equilibrium separation in an external force field PO34 > SO24 > acetate− > Cl− > NO−3 > SCN− : −

−

How does one reverse the binding between the hydrophobic patches in a protein with the hydrophobic chains on hydrophobic adsorbents? Detergents, organic solvents (ethylene glycol 50% v/v, i-propanol 30% v/v, acetonitrile), lower temperature, low salt concentration, increased pH, chaotropic salts, etc., have been used successfully to desorb the proteins from hydrophobic adsorbents. If hydrophobic adsorption was facilitated by high concentration of a salt, lower salt concentration may be employed. Chaotropic salts,6 such as iodide, lithium bromide, thiocyanate, at high concentrations, are often successful in facilitating desorption since they are less polar and bind water loosely. So water molecules hydrate protein surfaces. On the other hand, a kosmotropic salt such as (NH4)2SO4, has ions having high polarity, which bind water strongly; therefore they will induce the exclusion of water molecules from the protein surfaces, which will lead to protein precipitation or adsorption on hydrophobic adsorbents. Scopes (1994) has provided a more detailed account of such techniques for eluting proteins in hydrophobic interaction chromatography. A recent analysis (Marmur, 2000) of the solubility of nonpolar solutes in water suggests that such solutes prefer to aggregate rather than to be dispersed as single molecules. Such aggregation leads to a sufficient loss in entropy, as opposed to a corresponding loss of entropy via ordering of water molecules around hydrophobic patches/chains. This analysis is also valid for the dissolution of water in nonpolar solvents. Therefore, molecular aggregation is due to the more general “solvophobic” effect rather than a “hydrophobic” effect. 4.1.10 Gas–solid particle–liquid system in mineral flotation In Section 3.3.8, the principle of particle separation via flotation in a gas–liquid system was briefly identified. Those particles whose surfaces are hydrophobic, and to which gas/air bubbles can be attached, will float to the surface of an aqueous suspension due to reduced density. Those particles whose surfaces are hydrophilic and wetted by the liquid (commonly, it is water) cannot become attached to gas/air bubbles and will not therefore float to the surface. Here, following Fuerstenau and HerreraUrbina (1989), we will briefly illustrate systems where such separations are achieved by converting particular mineral surfaces to a hydrophobic type via the adsorption of

6

Chaotropic anions create an increase in entropy resulting from the disruption of water structure around these ions in solution. This facilitates solubilization of hydrophobic proteins by water since many more water molecules are available.

245 surfactants. The system of interest contains two minerals, hematite (Fe2O3) and quartz, which can be separated by froth flotation; these two minerals are usual constituents of iron ores. The primary step involves the use of surface-active collectors, which displace water molecules from the surface of the mineral; if the mineral surface is charged, appropriate long-chain ionic surfactants, which can act as a counterion to the charged surface, are used. The long hydrophobic chains of such surfactants present a hydrophobic surface. Surface-active collectors, which are chemisorbed at the mineral–water interface, can also create a hydrophobic surface. An additional approach employs substances called depressants, which inhibit the attachment of gas bubbles to the minerals that should not undergo flotation. At pH 6–7, quartz particles have a negative charge on their surface, whereas hematite particles essentially have no charge. The addition of an alkylamine salt, which is adsorbed on the quartz particle surface, leads to the flotation of quartz particles. Commercially employed alternative separation strategies include the use of starch as a depressant for hematite; simultaneously, calcium ions are used to activate surfaces of quartz particles, which are then floated using sodium oleate at pH 11–12 (Fuerstenau and Herrera-Urbina, 1989). Another industrial approach employs flotation of hematite. At pH 2–4, the charges on hematite surfaces are positive; an anionic alkyl sulfonate collector is employed to float hematite. Since the quartz surface is negatively charged at these pHs, the anionic collector does not become adsorbed on the quartz surface. The key to any approach is knowing the electrical charge and potential on the surface of the mineral particle in an aqueous suspension. The following four phenomena contribute to the development of the surface charge: specific adsorption of surface-active ions; preferential dissolution of lattice ions; dissociative adsorption of water molecules; isomorphous substitution of ions comprising the mineral lattice (Fuerstenau and Herrera-Urbina, 1989).

4.2 Equilibrium separation in a single phase in an external force field Consider a liquid solution or a gaseous mixture in a closed vessel subjected to an external force field. We would like to know the change in composition in the solution or in the gaseous mixture in the direction of the force field and calculate the value of the separation achieved between any two locations in the vessel. The external force fields illustrated are centrifugal, electrical and gravitational. This section also explores the separation achieved in the closed vessel subjected to an external force field when a property gradient, e.g. density gradient, pH gradient, etc., exists in the single-phase system. Density gradient is important

246

Separation in a closed vessel dμi jT ¼ −dϕext i :

ð4:2:1Þ

For the centrifugal field, it is known that

F ext i

r r2

mole of i

r ¼ M i ω2 rr ¼ −

dϕext i r: dr

Further, using relation (3.3.9),

dμi ¼ RT dlnf^ij ¼ M i ω2 r dr, T

r1

ð4:2:2Þ

where subscript j refers to the radial location r under consideration. Integrating between two locations, r ¼ r1 (j ¼ 1) and r ¼ r2 (j ¼ 2), where r2 > r1, leads to lnð f^i2 =f^i1 Þ ¼ ðM i =RTÞ

ω2 2 2 ðr − r Þ: 2 2 1

ð4:2:3Þ

If the gas mixture behaves as an ideal gas, then w Figure 4.2.1. Closed hollow cylinder rotating at ω radian/s: closed gas centrifuge.

when a centrifugal or gravitational force field is present; a pH gradient is useful in the presence of an electrical force field. Finally, the role of a thermal gradient in separation is briefly identified in the absence of any external force field. 4.2.1

Centrifugal force field

Three types of mixtures/solutions are considered here: a gaseous mixture, liquid solutions of low molecular weight species and high molecular weight species. We begin with a gas mixture in a centrifuge; gas centrifuges have been used at large scale for the separation of uranium isotopes present in a gaseous mixture of U235F6 and U238F6. Macromolecules and biological particles in solutions are routinely separated and studied in the laboratory using ultracentrifuges. Centrifuges are also extensively used in the chemical process industry to remove particulate matter from a liquid/gas (see Section 7.3.2). 4.2.1.1

Gas separation

A simple closed gas centrifuge (Figure 4.2.1) is a rotating hollow cylinder containing a gas or a gas mixture introduced into the stationary hollow cylinder at time t ¼ 0. Within a short time,7 the gas mixture as well as the cylinder are assumed to start rotating at a constant angular velocity ω radian/s. Assume isothermal conditions. To determine the conditions existing at equilibrium, employ the criterion (3.3.25) for equilibrium in the centrifugal field for species i:

f^i1 ¼ x i1 P 1

and f^i2 ¼ x i2 P 2 ,

ð4:2:4Þ

where P1 and P2 are the total pressures at radial locations r ¼ r1 and r ¼ r2. Result (4.2.3) can now be expressed as   x i2 P 2 ω2 ln ð4:2:5Þ ¼ ðM i =RTÞ ðr 22 − r 21 Þ: x i1 P 1 2 A general relation between the gas mixture at any location j (r ¼ r) and that at r ¼ r1 is

! x ij P j r¼r ω2 ð4:2:6Þ ¼ ðM i =RTÞ ðr 2 − r 21 Þ: ln 2 x i1 P 1 The equilibrium separation factor α12 between species 1 and 2 in a binary gas mixture between two locations j ¼ 1 (r ¼ r1) and j ¼ 2 (r ¼ r2) is easily obtained for ideal gas behavior:
 x 11 x 22 ðM 2 − M 1 Þ 2 2 2 ¼ α12 ¼ exp ω ðr 2 − r 1 Þ : x 12 x 21 2RT

ð4:2:7Þ

Two more results are needed to provide a reasonably complete picture of separation in a closed gas centrifuge. The first provides the mole fraction profile xij(r) of species i at radius r (location j) along the centrifuge radius; the second describes the variation in total pressure. Using relation (4.2.7) for two locations (r ¼ r and r1 ¼ 0), and species i ¼ 1,2, one obtains
 x 1j ðrÞ x 1j ð0Þ ðM 2 − M 1 Þ 2 2 ¼ exp − ð4:2:8Þ ω r : x 2j ðrÞ x 2j ð0Þ 2RT Algebraic manipulation leads to x 1j ðrÞ ¼

x 1j ð0ÞexpðAr 2 =2Þ , x 1j ð0Þ½expðAr 2 =2Þ −1 þ 1

ð4:2:9Þ

where 7

Auvil and Wilkinson (1976) provide estimates of this time.

  A ¼ ðM 1 −M 2 Þω2 =RT :

ð4:2:10Þ

4.2

Equilibrium separation in an external force field

Mole fraction, xij x102

0.235 0.23 0.22 0.21

A. Charge composition, xif = 0.002 B. SO2-H2, xij = Mole fraction SO2 C. SO2-N2, xij = Mole fraction SO2 D. UF6 (235, 238 isotopes), xij = Mole fraction U235

r2 r1

Cylindrical cell

D

0.20

A

0.19

247

w

90⬚ Cylindrical cell

90⬚

C B

0.18

0.17 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless radius, r / rwall Figure 4.2.2. Mole fraction profiles developed in a simple centrifuge with a peripheral speed of 150 m/s for the gas pairs: sulfur dioxide–hydrogen, sulfur dioxide–nitrogen and UF6 (235, 238 isotopes) at 20  C. Reprinted, with permission, from Auvil and Wilkinson, AIChE J., 22, 564 (1976). Copyright © [1976] American Institute of Chemical Engineers (AIChE).

From relation (4.2.7), we get, for r1 ¼ 0 and r2 ¼ r,
 Ar 2 α12 ¼ exp − : ð4:2:11Þ 2 Substituting this into the expression (4.2.9) for x1j(r), we obtain x 1j ðrÞ ¼

x 1j ð0Þ : x 1j ð0Þ þ α12 ð1−x 1j ð0ÞÞ

ð4:2:12Þ

By our convention, species 1 is lighter than species 2, making (M2 − M1) positive. Therefore α12 is greater than 1 as long as r2 > r1; it also follows that x1j(r) < x1j(0). Lighter species concentrate near the center of the centrifuge and heavier species concentrate near the wall (r ¼ rwall). Further, the extent of this composition change is related to the molecular weight difference (M 2 − M1) since α12 increases exponentially with (M2 − M1). The extent of separation, ξ, for such a centrifuge is calculated in Problem 1.4.3, and the result has been provided there. The behavior of the mole fraction profile in a gas centrifuge along the centrifuge radius is illustrated in Figure 4.2.2 for a number of binary gas mixtures (Auvil and Wilkinson, 1976). The simple gas centrifuge rotates with a peripheral speed (¼ ωrwall) of 150 m/s at 20  C; the gas initially fed into the centrifuge contained a mole fraction of 0.002 of the species shown. Consider systems B and C, both containing SO2. The mole fraction of SO2 (¼1 − x1j(r)) changes much more rapidly with r in the case of system B since (M2 − M1) is much larger for a SO2–H2 than for a SO2–N2 system (system C). Similarly, the mole fraction variation with radius for U235F6 and U238F6 is much less since (M2 − M1) is only 3. It is also of interest to know the magnitudes of α12 achievable in such gas centrifuges. Consider a gas centrifuge of radius 4 cm rotating at an extremely high angular

w

Figure 4.2.3. Swing-bucket rotor with centrifugal tubes which swing 90 upward during centrifugation.

velocity (ω) of 4000 radian/s at 20  C. The value of α12 for a mixture of uranium isotopes (U235F6 and U238F6) will be only 1.0157 in such a centrifuge, whereas that for a SO2–N2 mixture is 1.207. However, uranium isotopes are often separated by high-speed centrifuges since other isotope separation methods yield usually a lower α12. A gas centrifuge is never used for SO2–N2 separation, however, since other methods provide a far larger separation factor. The mechanical problems encountered at these high speeds are also not inconsiderable. We will now provide the second result, namely the profile of the total gas pressure along the centrifuge radius. The partial pressure of any species is indicated by relation (4.2.6); the total pressure is obtained by summing over all species at any location r (region j):   n

X M i ω2 r 2

P j ¼ PðrÞ ¼ Pð0Þx i1 ð0Þexp − , ð4:2:13Þ r¼r 2RT i¼1

where region j ¼ 1 corresponds to r ¼ 0, there are n number of species in the gas mixture and n X i¼1

4.2.1.2

x i1 P 1 ¼ P 1 ¼ Pðr ¼ 0Þ ¼ Pð0Þ:

ð4:2:14Þ

Liquid separation

Liquid solutions subjected to a centrifugal field also undergo separation. Such separations, primarily employed in laboratories, are achieved by placing the solution in a small cylindrical cell mounted at the tip of an extremely high-speed rotor (Figure 4.2.3) in an ultracentrifuge. When equilibrium is achieved in the centrifugal force field, criterion (3.3.25) can be applied to species i:

2 dμij T ¼ −dϕext ij ¼ M i ω r dr:

For the solution, dμij T is expressed as

dμij T ¼ R T d lnaij þ V ij dP ¼ M i ω2 r dr:

248

Separation in a closed vessel

Using the assumption that the pressure gradient may be determined from the rotation of the solvent only, we have obtained the following result (see relation (3.1.52)): "   # V ij Ms : −V ij dP þ M i ω2 r dr ¼ M i ω2 r 1 − Mi Vs Therefore, "

RT dlnaij ¼ M i 1 −



V ij Mi



Ms Vs

#

ω2 r dr:

Integrating between two radial locations r1 and r2 (> r1), we obtain ) (
   V ij ai2 Mi Ms 1− ¼ exp ω2 ðr 22 − r 21 Þ : ð4:2:15aÞ Mi ai1 2RT Vs For an ideal solution, aij ¼ xij; therefore ðai2 =ai1 Þ ¼ ðx i2 =x i1 Þ. Further, for dilute solutions, ðx i2 =x i1 Þ ffi ðci2 =ci1 Þ. This last result allows us to relate the concentration of a species at two radial locations at equilibrium: ( )
    V ij C i2 Mi Ms 2 2 2 1− ¼ exp ω ðr 2 − r 1 Þ : C i1 2RT Mi Vs ð4:2:15bÞ Consider a dilute aqueous solution of ovalbumin (Mi ~ 45 000) at 24  C; the density of ovalbumin is 1.34 g/cm3 ðffi M i =V ij Þ. For a dilute solution, ðM s =V s Þ ffi 1g=cm3 . From the expressions given above for (ai2/ai1) and (Ci2/Ci1), it is clear that ai2 > ai1, Ci2 > Ci1 and xi2 > xi1. At equilibrium, there develops a continuous solute concentration profile along the cell length. Solute concentrates at the largest radial location in the cell at the cell wall, and the solution nearer the center becomes depleted in ovalbumin. Such a profile does not change with time at equilibrium. At any radial location, the concentration gradient and the pressure gradient (pressure increases also as the radius increases in the centrifuge) force ovalbumin molecules toward the center (r ¼ 0) to balance the centrifugal force driving the molecule away from the center. There is no net radial force on ovalbumin molecules at any radial location once equilibrium separation profile is established. Ultracentrifuges used in laboratories for solutions employ very high speeds (as much as 60 000 rpm). Measurement of the concentration profile (e.g. (4.2.15b)) is achieved optically; the refractive index gradient is measured (it being proportional to the concentration gradient). For an introduction, see Hsu (1981). The difference in concentration between r ¼ 0 and r ¼ rwall for species like ovalbumin in an ultracentrifuge is substantial. If, however, we had subjected a solution of two low molecular weight liquids, e.g. water and ethanol, to the centrifugal field in an ultracentrifuge, the difference (Ci2 (r ¼ rwall) − Ci1(r ¼ 0)) would be very small. For the same values of ω, T, r2, r1, the change in

concentration from one radial location to another is much smaller due to the small value of Mi, since the magnitude of f1 − ðV ij =M i ÞðM s =V sj Þg remains comparable. In fact, the higher the molecular weight, the larger the separation. Some of the more common applications of ultracentrifuges, therefore, involve large macromolecules, viruses, cell fragments for preparative scale separations, etc. In centrifuges used for preparative applications, the liquid solution is mounted in the rotor. After centrifugation, a pellet is obtained near the radius r ¼ rwall, containing the particles, large macromolecules, cells, etc., separated from the clear supernatant solvent. We will now focus on the velocity of such types of cells/ particles/macromolecules during centrifugal separation, the time required to form the pellet and the relative ease with which different biologically relevant macromolecules/ cells, etc., may be pelleted. The analysis of the behavior of larger centrifuges that are open with flow in and out are considered in Section 7.3.2 as well as in Section 6.3.1.3. For a spherical particle of radius rp, density ρp and mass mp in a liquid medium of density ρt, the net external force acting on the particle (from equation (3.1.59)) in the radial direction (without any gravitational force in that direction) is mp rω2 ð1−ðρt =ρp ÞÞr if the angular velocity of the particles is ω radian/s. The drag force experienced by this particle from the liquid resisting this radial particle motion according to Stokes’ law is 6πμrpUpr where Upr, is the radially outward particle velocity. Reformulating the basic equation of motion (3.1.60) for the particle in the r-direction, we have mp

dU pr d2 r r ¼ mp r ω2 ð1 − ðρt =ρp ÞÞr − 6πμr p U pr r: r ¼ mp dt dt 2 ð4:2:16aÞ

At steady state the particle radial acceleration is reduced to zero and the terminal radial particle velocity achieved, Uprt, is

2 3 2 dr

4 πr p ρp rω ð1− ðρt =ρp ÞÞ 2 r p ðρp − ρt Þ 2 ¼ U prt ¼ ¼ ω r;

dt 3 9 6πμr p μ terminal velocity

U prt ¼ sp ω2 r,

ð4:2:16bÞ

where sp is called the sedimentation coefficient of the particle of size rp (correspondingly sp1 and sp2 for particles of sizes rp1 and rp2). If ω2 r may be considered as the centrifugal force per unit effective mass8 (i.e. mp ð1−ðρt =ρp ÞÞ ¼ 1Þ of the particle under a given centrifuge condition, then sp is an effective indicator of the radial sedimentation particle terminal velocity per unit of centrifugal force. Its units are svedbergs, 1 svedberg having the value 10−13 second (Svedberg and Pederson, 1940). The quantity ω2 r is often expressed as so many times (e.g. n times) gravity, ng, where g is the acceleration due to gravity.

8

Magnitude of the centrifugal field strength.

4.2

Equilibrium separation in an external force field

Ivory et al. (1995) have collated in svedberg units the sedimentation coefficients of a variety of biologically relevant particles in aqueous systems: yeast and red blood cells ð106 > sp > 105 Þ; mitochondria and bacteria ð105 > sp > 104 Þ; viruses and phages ð103 > sp > 102 Þ; proteins (e.g. serum albumin) ðsp < 10Þ. The higher the value of sp, the easier it is for the particles to settle and the shorter the particle settling time in a given centrifuge for a given angular velocity, ω. Consider now the time required for such a particle to settle in the centrifuge, i.e. reach the centrifuge wall, rwall. Employ equation (4.2.16b); integrate between the starting radius rstart anywhere in the centrifuge and rwall:

t wall r wall

ð ð dr

dr r wall 2 2 ω r ) ¼ s ¼ sp ω ; dt ¼ sp ω2 t wall ¼ ln p

dt r r start settling velocity r start

t wall

o

1 r wall r r wall ln ¼ : ln ¼ sp ω2 r start sp ω2 r r start

ð4:2:16cÞ

Sometimes, this time is replaced by tpellet, the time needed to form the pellet in an ultracentrifuge. Example 4.2.1 Harvesting time for a virus in an ultracentrifuge. Lotfian et al. (2003) have investigated the centrifugal recovery of a disabled herpes simplex virus type-1 (HSV-1) vector, potentially useful for gene therapy, from a culture medium containing recombinant cells of a suitable kind. The extracellular virus is obtained first in the supernatant of the broth centrifuged at a low value of ω2r ¼ 1600 g. This supernatant is then subjected to ultracentrifugation using a swing-bucket rotor and 15 ml volume centrifuge tubes. Consider operation at ω2 r ¼ 26000g. The radius of the base of the centrifuge tube from the axis of rotation is 13.7 cm. (a) Calculate the settling velocity of the virus at the following two locations in the ultracentrifuge tube: 15 mm and 50 mm away from the base of the tube. (b) Estimate the time needed for the virus to deposit if the starting locations are identified in part (a). You are given the sp for virus ¼ 102 svedberg. Solution (a) The settling velocity of a particle having a sedimentation coefficient sp at a radial location r is given by

U prt ¼ sp ω2 r cm=s; sp ¼ 102 svedberg ¼ 102  10−13 ¼ 10−11 s; cm 2 ω r ¼ 26 000 g ¼ 26 000  980 2 ; s Therefore,

U prt ¼ 10−11 s  26 000  980 ¼

2:6  0:98  10−4 m s 102

ffi 2:55  10−6 m=s:

cm cm ¼ 2:6  0:98  10−4 s2 s

249 If the value of ω is known, then U prt may be calculated for each individual viral location. (b) To estimate the time needed for the virus to settle, we employ equation (4.2.16c):

r r wall ln : r start sp ω 2 r

t wall ¼

Since ω is not explicitly given here, we will employ ω2r ¼ 26 000 g and r ¼ (rwall þ rstart)/2. For particles starting at 15 mm from the base, rstart ¼ 137 − 15 ¼ 122 mm. Therefore r ¼ 129.5 mm and

t wall

¼

¼

12:95 cm

2

10  10

−13

137 cm ln 122 s  26 000  980 2 s

12:95  104 12:95  1145  0:1145 s ¼ s 2:6  0:98 2:6  0:98

¼ 5734 s ¼ 95:5 minutes: For particles starting at 50 mm away from the base of the tube, rstart ¼ 137 − 50 ¼ 87 mm. Therefore r ¼ 112 mm (averaged) and

t wall

¼

¼

11:2 cm

10

−11

137 cm ln 87 s  26 000  980 2 s

11:2  104  2:303  0:197 s 2:6  0:98

¼ 19941 s ¼ 332 minutes: Since the sedimentation coefficient sp for proteins is around 1–10 svedberg, the settling time needed for proteins in Example 4.2.1 would be an order of magnitude larger. Ultracentrifuges are therefore not used for protein separation. So far we have focused on liquid separation or particle separation where the solvent density was essentially assumed constant throughout the centrifuge. This condition did not pose any problem for separating a given cell or macromolecular species from the solvent; the cells or macromolecular species are obtained as a pellet at the centrifuge tube base. A different strategy has to be pursued if it is necessary to separate different cells or macromolecular species from one another. We focus on that in Section 4.2.1.3. The mathematical expressions used so far for liquid systems subjected to a centrifugal field have assumed that the solvent density ðM s =V sj Þ was essentially constant across the centrifuge. In general, it will vary in the radial direction due to the variation of pressure which influences V sj . It is particularly true when there is at least one solute species present in the system at a substantial concentration. Under such a condition, we use expression (3.1.58) for centrifugal equilibrium and expression (3.3.17) for dμij jT , Pj at any location. This leads to

  

n−1  X ∂μij V ij

ω2 r dr ¼ ¼ M i 1 − ρtj dμij dx kj

T , Pj Mi ∂x kj T , Pj , x l , l6¼k k¼1

ð4:2:17Þ

when there are a number of solute species present at substantial concentrations. Consider a system of solvent

250

Separation in a closed vessel

ð4:2:18Þ

The use of such equations will be illustrated in the following for a special case.

4.2.1.3

(a) CsCl-based solution density

(i ¼ s) and two solutes i ¼ 1 and i ¼ 2. The particular forms of the above relations are as follows.

∂μ1j ∂μ1j

2

i¼1: ðM 1 −ρtj V 1j Þω r dr¼ dx 1j þ dx 2j ;

∂x 1j

T ,Pj ,x 2 ,x s ∂x 2j T ,Pj ,x 1 ,x s

∂μ2j ∂μ1j

2

i¼2: ðM 2 −ρtj V 2j Þω r dr¼ dx þ dx 2j :

1j ∂x 1j

T ,Pj ,x 2 ,xs ∂x 2j T ,Pj ,x 1 ,x s

r0|species 2 r0|species 3

Isopycnic sedimentation

Just as a particle denser than a liquid falls through the liquid in the direction of the gravitational force, similarly a particle or species of density ρi ð¼ M i =V ij Þ will be thrown radially outward in the direction of the centrifugal force if ρi > ρt , the solution density. This phenomenon is identified in analytical/preparative chemistry as sedimentation. When ρt < ρi , the centrifugal force is radially outward. However, if ρt > ρi , the centrifugal force on species i will be radially inward. When ρt ¼ ρi , species i or particle i does not experience any radial centrifugal force (outward or inward); therefore its radial location does not change in the centrifuge with time; it no longer undergoes sedimentation. These phenomena provide the basis for separating a multicomponent mixture in a liquid centrifuge provided a radial density gradient is created in the centrifuge to start with. Consider Figure 4.2.4A illustrating a density gradient established using a low molecular weight salt such as CsCl in water in the centrifuge. Into such a solution, where ρt is a function of r, is introduced a small amount of a mixture of different macromolecular species. Assume that the small amount of sample will not affect the density gradient established in the CsCl solution. Each macromolecular species i will establish its own centrifugal equilibrium and will move to the radial location r, where ρtj ðrÞ ¼

Mi ; V ij

ð4:2:19Þ

i.e. the solution density is equal to the species density (which is also the inverse of the partial specific volume of species i). As long as the value of (M i =V ij ) is different for each ith species, each ith species will be concentrated at different radial locations – provided the range of ρtj(r) in the centrifuge accommodates the range of the effective macromolecular density (M i =V ij ). The separation of different species in a centrifugal force field (the imposed primary force field) having a radial density gradient (secondary physical property based field) is identified as isopycnic sedimentation or density-gradient sedimentation (Giddings and Dahlgren, 1971). Each species

Distance from rotor center

Figure 4.2.4A. Equilibrium density gradient in a centrifuge due to CsCl solution.

forms a small band around a center point where relation (4.2.19) is satisfied. Determinations of the width of such a band and the separation between two neighboring bands are needed to determine the usefulness of isopycnic sedimentation. Following Meselson et al. (1957), we consider the separation of different macromolecules in an aqueous solution of CsCl. For the purpose of analysis, one macromolecular species, i ¼ 2, only needs to be considered, where i ¼ 1 represents CsCl and i ¼ s is water. To simplify matters, the macromolecular species may be represented as a macromolecular electrolyte P (Cs)n, where P is the macromolecular backbone (e.g. DNA (deoxyribonucleic acid)). We will soon find out the utility of such an assumption. The centrifugal equilibrium of the macromolecular species i ¼ 2 in an aqueous solution of CsCl (i ¼ 1) is governed by equation (4.2.18) for i ¼ 2:

∂μ2j

∂μ2j

2 ðM 2 − ρtj V 2j Þω r dr ¼ dx 1j þ dx 2j :

∂x 1j T , Pj , x 2 , x s ∂x 2j T , Pj , x 1 , x s The radial dependence of the CsCl solution density around the location r ¼ r0, where ρtj ðr 0 Þ ¼ M 2 =V 2j , may be described by a Taylor series expression:

dρtj

ρtj ðrÞ ¼ ρtj ðr 0 Þ þ

ðr − r 0 Þ: dr r0

ð4:2:20Þ

ð4:2:21Þ

There is an implicit assumption here that the macromolecular band has a narrow width around the location r ¼ r0. For ideal solutions of species 1 and 2, we have

4.2

Equilibrium separation in an external force field

∂μ2j

dx 2j ffi RT d ln C 2j :

∂x 2j T , Pj , x 1 , xs

Further, we may define a dimensionless factor b, the binding coefficient (Hsu, 1981), to be   ∂x 1 b¼ : ð4:2:22Þ ∂x 2 P, T , μ1 Additionally, by Maxwell’s relation for any extensive quantity, e.g. Gibbs free energy Gtj (Pitzer and Brewer, 1961),

∂μ2j

∂μ1j

∂Gtj

¼ ¼ , ð4:2:23Þ

∂x 1j T , Pj , x 2 , x s ∂x 2j T , Pj , x 1 , xs ∂x 1j ∂x 2j T , Pj , x s which leads to

∂μ2j

∂μ1j

dx ¼ dx 1j ¼ bRT d ln C 1j

1j ∂x 1j T , Pj , x 2 , x s ∂x 2j T , Pj , x 1 , xs

ð4:2:24Þ

for an ideal solution. Here, b is the effective number of cesium counterions which are to be moved along with the charged polymer molecule P(Cs)n−b to maintain electoneutrality (so far we have gone about this as if we were dealing with nonelectrolytes). The governing equation now becomes, for ideal solutions,

) ( dρtj

M 2 − ρtj ðr 0 ÞV 2j − V 2j

ðr − r 0 Þ ω2 r dr dr r0 ¼ bRT d ln C 1j þ RT d ln C 2j :

ð4:2:25Þ

Just as the radial dependence of the CsCl solution density around r ¼ r0 was described by a Taylor series expansion (4.2.21), similarly the C1j profile around r ¼ r0 may be described by

dC 1j

ð4:2:26Þ C 1j ðrÞ ¼ C 1j ðr 0 Þ þ

ðr − r 0 Þ: dr r0 Introducing this into equation (4.2.25), we get

i h dρtj

2 M 2 − ρtj ðr 0 ÞV 2j ω2 r dr − V 2j

ω rðr − r 0 Þ dr dr r0

# " dC 1j

¼ bRTd ln C 1j ðr 0 Þ þ

ðr − r 0 Þ þ RTd ln C 2j : dr r 0

We integrate from r0 to r to obtain 2 3 ðdC 1j =drÞr0 C 2j ðrÞ ðr − r 0 Þ5 ¼ −b ln41 þ ln C 1j ðr 0 Þ C 2j ðr 0 Þ i ðr 2 − r 2 Þ h 0 þ M 2 − ρtj ðr 0 ÞV 2j ω2 2RT

−V 2j

0 12 0 1

dρtj 2 r − r 0 r − r r 0 0

ω @ A @ þ A: ð4:2:27Þ dr

r0 RT 3 2

251 Since the variation of CsCl concentration over the bandwidth of polymeric species 2 is small,
 ðdC 1j =drÞr 0 ðr − r 0 Þ ln 1 þ : ðr − r 0 Þ ffi ðdC 1j =drÞr0 C 1j ðr 0 Þ C 1j ðr 0 Þ Further, the bandwidth is small: jr−r 0 j T2) such that the two bulbs are maintained at two different temperatures. If the thermal diffusion ratio kT for a given species 1 is negative, then, from definition (3.1.45) and force relation (3.1.44), we can conclude that species 1 moves due to the temperature gradient from the colder bulb to the hotter bulb. This builds up its concentration in the hotter bulb, setting up its back diffusion into the colder bulb. At equilibrium, the temperature-gradientdriven flux of species 1 is balanced by that due to the concentration gradient in the opposite direction, since the two forces in the opposite direction must balance each other. To exploit this dynamic equilibrium, we need to calculate the flux due to the temperature gradient first. Employ expression (3.1.44) for the force on a gmol of species 1 (say) due to the temperature gradient in a mixture of species 1 and 2: F T1 C 1 ¼ temperature-gradient-driven molar flux of 1 ¼ J T1 f d1 0 1 RT DT1 C1 DT ¼ −@ A ðr ln TÞ d ¼ − 1 ðr ln TÞ; M1 D12 C 1 M 1 f1

ð4:2:62Þ

where f d1 is the frictional coefficient of species 1 (¼ RT/D12) and there is no bulk velocity. This flux is opposed by the diffusive flux J1 of species 1 from a concentration gradient: J T1 ¼ −

DT1 C 2 M 1 M 2 D12 r ln T ¼ −J 1 ¼ t rx 1 : M1 ρt M 1

ð4:2:63Þ

Use the one-dimensional form of this relation since the species movement is only in one coordinate direction: −

DT1 ρt dT dT ¼ dx 1 ¼ −k T : T D12 C 2t M 1 M 2 T

Often, kT is expressed as −k 0T x 1 ð1−x 1 Þ, where k 0T is a thermal diffusion constant; this leads to

262

Separation in a closed vessel

k 0T

dT dx 1 ¼ : T x 1 ð1−x 1 Þ

(a)

ð4:2:64Þ

Feed

Integrate between region 1 at T1 and region 2 at T2. The extent of separation generated by thermal diffusion is rather low; therefore the product x1 (1 − x1) is considered to be a constant at an averaged value x 1 ð1−x 1 Þ. Integrating equation (4.2.64) now yields k 0T x 1 ð1 − x 1 Þ ln

T1 ¼ x 1 jT 1 − x 1 jT 2 ¼ x 11 − x 12 : T2

Membrane Dialysate

Feed

x 11 − x 12 ðx 11 Þð1 − x 12 Þ T1 ¼ − 1 ¼ α12 − 1 ¼ ε12 ¼ k 0T ln : ðx 12 Þð1 − x 11 Þ ðx 12 Þð1 − x 11 Þ T2 ð4:2:65Þ

Strip

4.3 Equilibrium separation between two regions in a closed vessel separated by a membrane In Sections 4.1 and 4.2, either a two-phase system or a single-phase system was introduced into a vessel which was then kept closed with no mass additions or withdrawals. After allowing the system to reach equilibrium, the separation achieved between the two regions in the closed vessel was calculated. There was always some separation achieved between the two regions. In this section, either we introduce a feed mixture into one region of a membrane device, or we introduce two different mixtures into two regions of the membrane-containing device. We then allow a long time to elapse and calculate the separation achieved at equilibrium between the two regions. This has been studied here for the membrane processes of simple dialysis, Donnan dialysis, gas permeation, reverse osmosis and ultrafiltration. 4.3.1

Separation by dialysis using neutral membranes

Dialysis is a membrane process in which a microsolute present in a feed solution is removed from the feed to a receiving solution of the same phase called the dialysate; the membrane is such that the macrosolutes

C1d, Vd

(b)

Since x11 and x12 are quite close, we may substitute fx 1 ð1 − x 1 Þg by (x12)(1 − x11) (i.e. as a product of two boundary values). Therefore,

This relation provides an expression for the enrichment factor ε12 for two species 1 and 2 between the two regions in terms of k 0T and the two temperatures. Consider now an equimolar mixture of H2 and N2 subjected to a temperature difference of 260  C in the hot bulb and 15  C in the cold bulb. In the above result, (x11 − x12) may be estimated if we can estimate k 0T ðx 12 Þð1 − x 11 Þ, which is equal to −k T ; a rough estimate of this value for illustration is 0.066 (Ibbs et al.,1939). The value of (x11 − x12) we obtain is 0.0406, indicating about a 4% difference in mole fraction for the lighter species H2 between the hot region (T1) and the cold region (T2).

C1f, Vf

BN

Cation exchange membrane AN Figure 4.3.1. (a) Dialyzer with closed inlets and outlets. (b) Closed vessel having two aqueous solutions separated by a cation exchange membrane: Donnan dialysis.

present in the feed solution are not allowed to pass through it to the dialysate stream. This process is the basis for blood purification using an artificial kidney (or hemodialyzer). The membrane does not have substantial convection of the solvent, namely water. Solutes diffuse through fine pores to the dialysate; metabolic wastes, e.g. urea, uric acid, creatinine, etc., diffuse through the water-filled pores or hydrogels from the blood to the dialysate, but macrosolutes, e.g. blood proteins (albumin, etc.), and larger species are essentially excluded by the membrane. In actual operation, the blood (or feed solution) flows on one side of the membrane and the dialysate flows on the other side into the device and out. If, in such a device, blood is introduced into the feed side and the dialysate stream is introduced into the receiving side, and the device inlets and exits are closed (Figure 4.3.1(a)), one would like to know what will happen as t ! ∞ when equilibrium will be achieved. Would the two regions show separation with respect to the microsolute as t ! ∞? Let the microsolute (e.g. urea) to be removed from feed be species 1, the feed solution region j ¼ f and the dialysate solution region j ¼ d. Let the feed and the dialysate concentrations of the solute 1 at time t ¼ 0, be C 01f and C 01d ð¼ 0Þ. We are interested in knowing the values of C1f and C1d as t ! ∞. Assume the two solutions to be well stirred; therefore the two surfaces are exposed to the bulk solution concentrations C1f and C1d at any time. Let the volumes of the feed solution and the dialysate solution be Vf and Vd , respectively. The solute flux expression through the membrane of thickness δm and area Am at any time may be written, following relation (3.4.99), as

4.3

Equilibrium separation: membrane, closed vessel Q1m ðC 1f − C 1d Þ: δm

N 1z ¼ k 1m ðC 1f − C 1d Þ ¼

ð4:3:1Þ

263

C 1d

¼

A total solute balance on the feed side and the permeate side leads to dC 1f Q ¼ − 1m Am ðC 1f − C 1d Þ; Vf δm dt dC 1d Q1m Vd ¼ Am ðC 1f − C 1d Þ: δm dt

ð4:3:2Þ ð4:3:3Þ

Laplace transforms of these two coupled ordinary differential equations will convert the t-domain to the s-domain and yield the following two equations: sC 1f − C 1f ðt ¼ 0Þ ¼ − sC 1d − C 1d ðt ¼ 0Þ ¼

Q1m Am ðC 1f − C 1d Þ; δm V f

Q1m Am ðC 1f − C 1d Þ, δm V d

where C 1f and C 1d are the transformed variables in the s-domain. Introducing the values of C1f and C1d at t ¼ 0, we get sC 1f − C 01f ¼ sC 1d ¼

Q1m Am ðC 1d − C 1f Þ; δm V f

Q1m Am ðC 1f − C 1d Þ: δm V d

ð4:3:4Þ ð4:3:5Þ

After eliminating C 1f from these two equations, the following equation is obtained for C 1d :      Q1m Am Q Am Q1m Am Q1m Am C 1d − C 1d s þ 1m δm V f δm V d V d δm V f δm   Q1m Am 0 C 1f ; ð4:3:6Þ ¼ δm V d 3 2   7 6 Q1m Am 6 1 7:   C 1d ¼ C 01f Q1m Am 1 1 5 δm V d 4 2 þ s þs Vf Vd δm




sþ

ð4:3:7Þ

If we define 

Q1m Am δm



1 1 þ Vf Vd



¼ a,

a constant, then C 1d ¼ C 01f



Q1m Am δm V d

 1 : s2 þ as



ð4:3:8Þ

Taking an inverse transform of this, we get C 1d ¼ C 01f



Q1m Am δm V d

 1 ð1 − e − at Þ : a




As t ! ∞, this is reduced to

ð4:3:9aÞ

¼

0

1

Q Am C 01f @ 1m A 0 δm V d C 01f

1 10

1 Q A 1 1 m @ 1m A@ þ A δm Vf Vd

Vf ) ðV f þ V d ÞC 1d ¼ V f C 01f : Vf þ Vd V f C 01f

ð4:3:9bÞ

Thus, the total amount of solute present initially in the feed is distributed between the feed and the dialysate. As t ! ∞, C1d has become a constant, ðdC 1d =dtÞ in equation (4.3.3) must be zero and C 1f ¼ C 1d . The process of dialysis has reduced the solute concentration in the feed solution in the closed vessel; at equilibrium, the value of C1f has, however, become equal to that of C1d, the concentration in the dialysate. Thus, if separation of solute 1 between the two regions is the goal, at equilibrium, there is no separation. Further, if separation between two solutes 1 and 2 in the feed is considered and both solutes are permeable through the membrane, then, at equilibrium, each solute will be present in each region of the closed vessel at the same concentration, resulting in no separation. In practice, separation is achieved in dialysis since the device is operated as an open system: fresh dialysate is introduced and then taken out continuously. Thus C1d is always maintained lower than C1f; ultimately, the feed solution concentration of species 1 can be reduced to a very low level. The continuous renewal/removal of the permeate side fluid, in this case the dialysate, is essential to most membrane based processes, which are based on different intrinsic rates of transport of species 1 and 2 through the membrane. Thus, membrane devices have to be open, in general, to achieve separation. A closed device employing a dialysis membrane as described above can achieve high purification of the feed solution if special conditions exist. For example, if in the dialysate side there is another species, which reacts chemically with species 1 and produces a product 3 which cannot diffuse through the membrane to the feed side, then it may be possible to purify the feed solution. Obviously, the added reactant in the dialysate side should not be able to diffuse through the membrane to the feed solution if the feed solution has to be purified. One such example has been provided in Section 5.4.3 based on the work carried out by Klein et al. (1972, 1973). The intrinsic separation capability of a dialysis membrane located in a closed vessel can be determined by considering the transport of two solutes i ¼ 1 and 2 through the membrane during the initial time period when t ! 0. The separation factor between two solutes, α12, between the two regions ( j ¼ 1, dialysate ! j ¼ d; j ¼ 2, feed ! j ¼ f ) for a two-species (i ¼ 1,2) system is

264

α12

Separation in a closed vessel C 2f C 1d C 1d þ C 2d C 1f þ C 2f x 11 x 22 x 1d x 2f ¼ ¼ ¼ : C 1f C 2d x 12 x 21 x 1f x 2d C 1f þ C 2f C 1d þ C 2d

Heparin (12 000) PEG (4000) Myoglobin (17 000) Ovalbumin (45 000) Albumin (66 000)

ð4:3:10Þ 0.1

Therefore, α12 ¼ C 1d C 2f =C 1f C 2d . To determine its value as t ! 0, we assume that, as t ! 0, C 1f ffi C 01f ,

C 2f ffi C 02f :

Avisco wet gel Cuprophan PT-150

ð4:3:11Þ

0.01

Further, C 1d jt!0 ¼ C 1d jt¼0 þ

dC 1d

Δt þ    dt t!0

ð4:3:12Þ

0.001

(Taylor series expansion around t ¼ 0). But C 1d jt¼0 ¼ 0. From relation (4.3.7), define    Q1m Am 1 1 a1 ¼ þ and δm Vf Vd    Q2m Am 1 1 : a2 ¼ þ Vf Vd δm From relation (4.3.9a),   dC 1d Q1m Am − a1 t ¼ C 01f ) C 1d jt¼Δt e dt δm V d   Q1m Am Δt : ¼ C 01f δm V d ð1 þ a1 Δt þ   Þ

In the limit of Δt ! 0,

Q1m : Q2m

ð4:3:13Þ

This relation identifies the intrinsic separation capability of the dialysis membrane for two solutes 1 and 2. The Δt value must be larger than the time required for the solutes to diffuse through the membrane and appear in the dialysate. For porous membranes, one can employ the flux expression (3.4.97) to estimate Qim: Qim ¼

Dil GDr ðr i , r p Þεm κim : τm

ð4:3:14Þ

Therefore, α12 jt!0 ¼

D1l GDr ðr 1 , r p Þ κ1m : D2l GDr ðr 2 , r p Þ κ2m

0.0001

Dextran (16 000) Vitamin B-12 (1355) Inulin (5200) Sucrose (342) Uric acid (5mg %)(168) Creatinine (113)

Urea (60) Sodium chloride (0.15M)(58)

0.00001 0

5

10

15 r (Å)

20

25

30

35

Figure 4.3.2. Permeability reduction as a function of characteristic solute radius (molecular weight in parenthesis) (Colton et al., 1973). Reprinted, with permission, from C.K. Colton, K.A. Smith, E.W. Merrill, P.C. Farrell, J. Biomed. Mater. Res., 5, 459, (1971), Figure 5, p. 483, © 1971, John Wiley & Sons.

Similarly for C 2d jt¼Δt . Therefore . Q A   1m m ð1 þ a2 ΔtÞΔtC 02f C 01f δ V α12 ffi .  m d  :  Q2m Am 0 ð1 þ a1 ΔtÞΔtC 02f C 1f δm V d

α12 ffi

Qim Dil

ð4:3:15Þ

An estimate of GDr ðr i , r p Þ may be obtained from the Faxen relation (3.1.112e); the partition coefficient κim will be determined by the geometric partitioning effect. Such an

approach is likely to be quite fruitful, provided the membrane pore size distribution is quite narrow around an average rp. For many commercially available porous membranes, used, for example, in hemodialyzers, the structure is that of a water-swollen gel, where the transport corridors and channels have varying dimensions. Therefore, models based on cylindrical capillaries of radius rp may not be accurate enough. However, Klein et al. (1979) suggest that their data for Cuprophan 150PM membrane for a variety of solutes may be reasonably modeled by an average pore diameter of 3.5 nm; on the other hand, such an average pore model could not describe the behavior of a number of other membranes with a wider pore size distribution. Estimates of Qim for a variety of solutes and Cuprophan membranes may be obtained from the data of Colton et al. (1971) (see Figure 4.3.2) and Farrell and Babb (1973). 4.3.2 Separation between two counterions in two solutions separated by an ion exchange membrane: Donnan dialysis Consider two electrolytes AN and BN in two different aqueous solutions separated by a cation exchange membrane (as in Figure 4.3.1(b)). Assume that the anion N is impermeable through the membrane. Suppose, at time

4.3

Equilibrium separation: membrane, closed vessel

t ¼ 0, the solution on one side of the membrane (the corresponding quantities are identified by the prime) contains only electrolyte AN, whereas the solution on the other side of the membrane (the corresponding quantities are identified by the double prime) contains only electrolyte BN. After a long time, equilibrium is expected to be established. Therefore the expression for Donnan potential (relation (3.3.118b)),     1 aiw ϕ R − ϕw ¼ − V i ðP R − P w Þ RTln Zi F aiR for ionic species i, should be valid for both solutions on both sides of the membrane at equilibrium. For cation A and the side of the membrane corresponding to AN,  0    1 a RT ln Aw ϕ0R − ϕ0w ¼ − V A ðP 0R − P w Þ , ð4:3:16aÞ 0 aAR ZAF where ϕ0w is the feed solution potential, if any. For cation A and the side of the membrane corresponding to BN,    00  1 a Aw 00 00 00 RT ln 00 ϕ R − ϕw ¼ − V A ðP R − P w Þ : ð4:3:16bÞ a AR ZAF The solution pressure, Pw, is the same on both sides. Assume swelling pressure effects to be such that 00 P 0R − P R ffi 0. Then (

00 Z1 ) 1 RT a0 Z A aAw A 00 00 : − ln ϕ0R − ϕ0w − ϕR þ ϕw ¼ ln Aw 00 a0AR F aAR ð4:3:16cÞ Now there is no potential difference across the membrane, 00 so φ0R ¼ φR . Therefore (
1) RT a0Aw Z A 00 0 ln 00 ð4:3:17aÞ ϕw − ϕ w ¼ F aAw 00

since a0AR ¼ aAR , both being merely the activity of the 00 counterion A in the membrane. But φw − φ0w is independent of any ionic species, which suggests that the relation (
1) RT a0Bw Z B 00 0 ln 00 ϕw − ϕw ¼ ð4:3:17bÞ F aBw is also valid. Consequently (Helfferich, 1962),


a0Aw 00 aAw

Z1

A

¼


0 Z1 aBw B ¼ a constant: 00 aBw

ð4:3:18Þ

The implications of this result may be understood if we know the concentrations of anion N in the two chambers. Suppose the chamber for species AN (0 ) has a high concentration of AN to start with, whereas the chamber for BN (00 ) has a low concentration of BN to start with. Due to the requirement for electroneutrality and the impermeability of the membrane to anion N, the concentrations of cations (A and B) in the chamber for BN will always be low. 00 Therefore the ratio ða0Aw =aAw Þ will always be large.

265

Consider now the case where Z A ¼ Z B ¼ 1. This 00 implies that, at equilibrium, ða0Bw =aBw Þ has a large value. Since the initial concentration of BN in chamber 00 was low to start with, distribution of B between the two chambers will drastically reduce the concentration of B in chamber 00 at equilibrium from its initial concentration. Therefore, if cation B is undesirable in the chamber 00 liquid, the concentration of B in chamber 00 can be drastically reduced via this phenomenon, called Donnan dialysis, using an acceptable cation A introduced via chamber 0 . Further, by reducing the volume of chamber 0 to a small value, undesirable cation B may now be concentrated to a high level in chamber 0 ; the solution in chamber 0 is called the strip solution, whereas the original solution in chamber 00 containing BN only is called the feed solution. After equilibrium is reached, the solution in chamber 00 is called the raffinate (in analogy to liquid–liquid extraction). Wallace (1967) illustrated this technique by concentrating uranyl ions (UO2þþ) from a 0.01 M uranyl nitrate  UO2 ðNO3 Þ2 feed with a 2 M nitric acid strip using a cation exchange membrane (see Section 3.4.2.5). The concentration of UO2þþ in the raffinate was reduced to 0.67% of the feed solution ( 0.01 M), whereas that in the strip was raised to 0.148 M. Wallace (1967) used an open system with feed and the strip solutions flowing on two sides of the membrane. A closed system will also achieve similar partitioning and separation. Another application is water softening. Using a strip solution containing the cation Naþ at large concentrations, divalent cation Caþþ present in dilute concentrations in the feed solution (hard water) on the other side of a cation exchange membrane can be removed substantially and replaced by Naþ. The difference between the basic result achieved at equilibrium under Donnan dialysis with that achieved in conventional dialysis using a neutral membrane (Section 4.3.1) is due to the assumption of perfect rejection of anion N by the cation exchange membrane in Donnan dialysis. Under such a condition, the requirements of the electroneutrality condition ensure that the two solutions will have radically different cation concentrations at equilibrium for any cation since the anion concentrations of the two chambers are so different. In practical Donnan dialysis in open systems, with less than perfect rejection of co-ions, the calculation of ion transport rates through the membrane requires a knowledge of the ionic concentrations at the two membrane– solution interfaces on the two sides of the ion exchange membrane. These concentrations are to be determined for the electrolyte AN based on the Donnan potential based equilibrium relation (4.1.73) developed for ion exchange resins: 

a0 Aw a0 Am

Z1

A

¼



a0 Nw a0 Nm

Z1

N

:

ð4:3:19Þ

266

Separation in a closed vessel

Here, subscript m refers to the membrane, w refers to the solution and the 0 side of the membrane is under consideration. To determine a0 Am or C 0 Am as a function of known quantities, we need also the electroneutrality relations for the solution and the membrane: Z A C 0Aw þ Z N C 0Nw ¼ 0;

solution

ð4:3:20Þ

Z A C 0Am þ Z N C 0Nm − C m ¼ 0,

membrane

ð4:3:21Þ

where Cm is the molar concentration of the negatively charged fixed ionic groups in the cation exchange membrane. Consider a specific electrolyte CaSO4, so ZA ¼ 2 and ZN ¼ −2. It follows from (4.3.19) that ða0Aw a0Nw Þ ¼ ða0Am a0Nm Þ:

ð4:3:22aÞ

Using molar concentrations and appropriate activity coefficients, 0

0

0

0

M M M M γNw Þ ¼ ðC 0Am C 0Nm Þ ðγAm γNm Þ: ðC 0Aw C 0Nw Þ ðγAw 0

0

0

ð4:3:22bÞ

ðC 0Aw C 0Nw Þ ffi ðC 0Am C 0Nm Þ:

ð4:3:22cÞ

From (4.3.20), C 0Aw ¼ C 0 Nw , and, from (4.3.21), C 0Nm ¼ ð − Z A C 0Am þ C m Þ=Z N :

ð4:3:22dÞ

Therefore,

C 02 Am −

Z A 02 C 0Am C m C 0Am C m ; C Am þ ¼ þC 02 Am − ZN ZN 2

C 0Am C m 2

− C 02 Aw ¼ 0: ð4:3:22eÞ

The acceptable solution of this quadratic is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cm Cm2 0 þ þ C 02 C Am ¼ Aw : 4 16

ð4:3:22fÞ

The molar concentration of the anion in the membrane at the interface C 0Nm can now be obtained from (4.3.21) since C 0Am is known. A similar analysis may be carried out for the 00 side of the membrane. Note that if the AN electrolyte concentration in the solution at the membrane–solution interface is C 0sw , then, for CaSO4, C 0Aw ¼ C 0sw .

4.3.3

pAf ¼

0

M M M M Assume that γAw γNm . Then γNw ffi γAm

C 02 Aw ¼ −

enriched in the preferentially permeating species. The residual feed gas, as well as the permeated gas mixture, are removed continuously from the gas permeation membrane device. Consider such a device. At time t ¼ 0, a mixture containing m0Af moles of species A and m0Bf moles of species B is introduced into the feed side, the permeate side does not have any gas molecules, and both feed and permeate side are kept closed. One would like to know the condition at equilibrium as t ! ∞. The volumes of the feed and permeate sides are Vf and Vp, respectively; the temperature is T, the membrane area is Am, the membrane thickness is δm and ideal gas behavior may be assumed. Due to ideal gas assumption, the partial pressures of species A in the feed chamber, pAf, and the permeate chamber, pAp, are related to the corresponding moles of species A, mAf and mAp, respectively, at any time t by

Separation of a gas mixture by gas permeation

Gas transport through a nonporous polymeric membrane, called gas permeation, has been considered in Section 3.4.2.2. In practical gas separation processes, the feed gas mixture is allowed to flow on one side of the membrane; one of the species preferentially permeates through the membrane to the other side. The permeated gases are

mAf RT Vf

and pAp ¼

mAp RT : Vf

ð4:3:23Þ

The rates of change in the number of moles of species A in the two chambers are   dmAf Q Am mAf RT mAp RT ; ð4:3:24Þ ¼ − A − Vf Vp δm dt   dmAp QA Am mAf RT mAp RT ¼ − ð4:3:25Þ dt δm Vf Vp Laplace transforms of these two domain to the s-domain yield  Q RTAm − mAf s mAf − m0Af ¼ A V f δm  Q RTAm s mAp ¼ A mAf − V f δm

equations from the t Vf mAp ; Vp  Vf mAp : Vp þ

ð4:3:26Þ ð4:3:27Þ

From these two equations, one can easily obtain the following expression for mAp :   QA RTAm 1 ð4:3:28Þ mAp ¼ m0Af V f δm s2 þ a 1 s where a1 ¼



 QA RTAm QA RTAm : þ V p δm Vf δ

On inverting relation (4.3.28), we get    QA RTAm 1 mAp ¼ m0Af ð1 − e − a1 t Þ : a1 V f δm

ð4:3:29Þ

ð4:3:30Þ

At time t ! ∞, mAp ¼

m0Af

  Vp QA RTAm : ¼ m0Af a1 Vf þ Vp V f δm

ð4:3:31Þ

4.3

Equilibrium separation: membrane, closed vessel

Correspondingly, as t ! ∞, mAf ¼ m0Af − mAp jt!∞ ¼ m0Af

Vf : Vf þ Vp

ð4:3:32Þ

pAf ¼

Vf þ Vp

and

pAp ¼

m0Af RT Vf þ Vp

:

pBf ¼

RT

Vf þ Vp

and

pBp ¼

m0Bf

RT

Vf þ Vp

ð4:3:34Þ

at time t ! ∞. Therefore, at equilibrium, ðpAf =pBf Þ ¼ ðm0Af =m0Bf Þ ¼ ðp0Af =p0Bf Þ,

ð4:3:35Þ

where p0Af and p0Bf are the partial pressures of species A and B in the feed chamber in the feed gas mixture at time t ¼ 0. Therefore, no separation has taken place in the gas mixture left on the feed side at equilibrium. Similarly, at equilibrium, ðpAp =pBp Þ ¼ ðm0Af =m0Bf Þ ¼ ðp0Af =p0Bf Þ:

a2 ¼

ð4:3:33Þ

Thus, the partial pressures of species A on the two sides of the membrane are equal at equilibrium. The same result will be obtained for species B present originally in the feed reservoir, i.e. m0Bf

Since the general expression for mBp may be written as    QB RTAm 1 ð4:3:38Þ ð1 − e − a2 t Þ , mBp ¼ m0Bf V f δm a2 where

Therefore, as t ! ∞, m0Af RT

267

ð4:3:36Þ

Therefore, the permeate side gas mixture at equilibrium has a composition equal to that of the original feed gas mixture at t ¼ 0; no separation has taken place in the closed system. This is why gas permeation devices in practice are operated as an open system: any gas mixture produced by permeation is immediately withdrawn from the device. Since the intrinsic membrane transport rates of the two species for unit partial pressure difference between the feed and the permeate are different, the gas mixture produced as a permeate has a composition different from that of the feed. The permeate mixture withdrawn from the device must have a composition different from that of the feed mixture for separation. See Sections 6.3.3.5, 6.4.2.2, 7.2.1.1 and 8.1.9 for open systems. Continuous energy input will yield separation in a closed system – see Figures 8.1.4(b) for a system operated at total reflux. One can demonstrate such separation capabilities, even in the closed vessel considered so far, if we focus, for example, on the initial time period in the vessel when t ! 0. We can calculate the moles of species A and B in the permeate chamber when t ! 0. For species A, differentiate the expression (4.3.30) for mAp with time t to obtain, by Taylor series expansion around t ¼ 0,

dmAp

mAp ¼ mAp þ

ðΔtÞ t¼0 t!0 dt t!0   QA RTAm − a1 Δt ¼ 0 þ m0Af e Δt: ð4:3:37Þ V f δm



 QB RTAm QB RTAm , þ V p δm V f δm

we get, similarly, mBp jt!0 ¼ m0Bf



 QB RTAm − a2 Δt Δt: e V f δm

ð4:3:39Þ

The separation factor αAB as t ! 0 may be expressed as    mAp mBf

mAp þ mBp m þ mBf m m  Af  ¼ Ap Bf

: αAB jt!0 ¼  mAf mBp mBp mAf t!0 mAf þ mBf mAp þ mBp

ð4:3:40Þ

For a system where very little of the feed permeates in the short time interval Δt near t ! 0, we can assume easily that ðmBf =mAf Þ ffi ðm0Bf =m0Af Þ. Therefore    Q ea2 Δt QA 1 þ a2 Δt þ    : ð4:3:41Þ αAB jt!0 ¼ A a1 Δt ¼ QB e QB 1 þ a1 Δt þ    This result identifies (QA/QB) as the initial time separation factor (only after the time lag period is over); as time increases, this factor decreases from this high value. As t ! ∞, there is no separation. In practical gas separation processes in open separators using gas permeation through nonporous membranes, the selectivity (QA/QB) of gas species A over B is of considerable importance. From the flux expression (3.4.72), we may write    Q DAM SAM : ð4:3:42Þ αAB jt!0 ¼ A ¼ DBM SBM QB The selectivity, αAB, is often broken up into a product of two factors, the diffusivity selectivity (also known as the mobility selectivity), (DAM/DBM), and the solubility selectivity, (SAM/SBM). Table 4.3.1 illustrates the contributions of Table 4.3.1. Transport, solubility and selectivity properties for the gas pair helium (A)–methane (B) at 25  C for a variety of polymersa Polymer Silicone rubber Natural rubber Hydropol (hydrogenated polybutadiene) Low-density polyethylene, ρ ¼ 0.9137 g/cm3 High-density polyethylene, ρ ¼ 0.964 g/cm3 Poly (vinyl acetate), glassy a

After Paul (1971).

DA/DB

SA/SB

0.38 1.08 1.21

5 24 28

0.075 0.044 0.043

1.70

35

0.048

2.94

55

0.054

355.00

5000

0.071

αAB

268

Separation in a closed vessel

Table 4.3.2. Kinetica sieving diameter of a gas/vapor based on the smallest zeolite window where it can fit Molecule

di, kinetic sieving diameter (nm)

He H2 NO CO2 Ar O2 N2 CO CH4 C2H4 C3H8 n-C4 CF2Cl2 C3H6 CF4 i-C4

0.26 0.289 0.317 0.33 0.34 0.346 0.364 0.376 0.38 0.39 0.396 0.43 0.43 0.44 0.45 0.47

a

Breck (1974).

mobility and solubility selectivity to the overall selectivity αAB for the gas mixture of helium (A)–methane (B) through a variety of polymeric membranes (Paul, 1971). Due to the smaller size of helium compared to methane (Table 4.3.2), the diffusivity of helium is higher than that of methane through all membranes identified in Table 4.3.1. Therefore, the diffusivity selectivity (DAM/DBM) is greater than 1 in all cases. On the other hand, we know from Section 3.3.7.3, that the more condensable the gas (higher Tc), the higher its solubility coefficient. Since methane is more easily condensable, (SAM/SBM) is less than 1; further, the value of this ratio is independent of the polymer material, whereas (DAM/DBM) depends very much on the structure of the polymer. Generally, the larger the size or molar volume of the gas/vapor molecule, the more it is impeded in its motion by the polymer chains in a polymer membrane. Thermal and other motions of the chains, creating openings in the membrane through which gas/vapor species can diffuse strongly, influence the value of Dim of any species i. For polymers that are glassy at a given temperature T ( Tg of the polymer) are very flexible and offer much less hindrance to the permeation of gas/vapor molecules. Therefore, Dim values are much larger for any given species in a rubbery polymeric membrane; further, the effect of a change in the size of the gas/vapor molecule is much less. These types of behaviors are illustrated in Figure 4.3.3 for a wide variety of gases and vapors for a rubbery polymer such as natural rubber and a glassy polymer such as polyvinyl chloride.

The basis for the wide variation in the mobility selectivity between a rubbery and a glassy polymer in Table 4.3.1 should now be clearer. Figure 4.3.4 illustrates the solubility coefficient Sim (also called the sorption coefficient) of a number of vapors and a few gases in a natural rubber membrane against the molar volume of the vapor/gas species. Generally, the larger the molecule, the more condensible it is. Correspondingly, its solubility in the membrane material is also higher. Therefore, a larger species which has higher condensibility, and therefore higher solubility in the membrane, will have higher solubility selectivity with respect to a smaller species. The general characteristics of the solubility selectivity and the mobility selectivity, and therefore the membrane selectivity, of gas species A over B through a polymer membrane as described above can be given in a more quantitative basis as follows (Michaels and Bixler, 1968). First, consider the solubility coefficient Sim of a gas species i in a polymer membrane. Jolley and Hildebrand (1958) have shown that the Henry’s law constant for a lot of gases in simple organic liquids at a reference temperature T0 can be correlated with the Lennard–Jones force constant εi for the gas i by an equation of the form ln Sim, T 0 ¼ aðεi =kB Þ þ ln bm ; Sim, T 0 ¼ bm expðaεi =kB Þ,

ð4:3:43aÞ

where the quantity a depends on the temperature, k B is the Boltzmann constant and bm is a constant dependent on the solvent. This equation has been found to describe the gas solubility coefficients Sim, T 0 in amorphous polymers also, where polymer–gas interactions may be neglected. Further, for the solubility of such gases in amorphous polymers, the van’t Hoff equation, dlnSim ΔH s ¼ − , dð1=TÞ R

ð4:3:43bÞ

has been found to be valid, just as in the case of gas solubilities in organic liquids (see equation (4.1.10)). Here, ΔHs, the enthalpy of the solution of gas i in polymer m, is related to the Lennard–Jones constant εi via ΔH s ¼ b0m − a0m ðεi =k B Þ,

ð4:3:43cÞ

0

where a0 m and b m depend on the polymer. One can now combine (4.3.43a,b,c) to obtain   b0 − a0 ðεi =kB Þ T0 : ð4:3:44aÞ lnðSim =Sim, T 0 Þ ¼ m m 1− T R T0 The solubility selectivity of a polymer for two gases A and B at T0 may be obtained from (4.3.43a) as follows:  

SAm, T 0 ðεA − εB Þ ¼ exp a : ð4:3:44bÞ SBm, T 0 kB Thus, the solubility selectivity as a first approximation depends only on the gas pair, and does not depend on

4.3

Equilibrium separation: membrane, closed vessel

10−4

269

He H2 He

Infinite dilution diffusion coefficient, D0 (cm2/s)

10−6

H2

O2 N2 CO2 A

C2H4 CH4 C2 H6 nC3

O2 N2 H2O

10−8

nC4

nC5

SF6 Bz

Natural rubber

CO2 A

CH4

Kr

10−10

MeOH

C2H6

VCM 10−12

Polyvinyl chloride EtOH

Acetone nC4

nC3OH 10−14

10−16

Bz

0

20

40

60

80

100

120

Van der Waals volume

nC5

nC4OH

nC6

140

160

180

(cm3/gmol)

Figure 4.3.3. Dependence of diffusivity on gas/vapor molecular size (Chern et al., 1985). Reprinted, with permission, from Material Science of Synthetic Membranes, ACS Symposium Series 269, 1985, p. 492, D. Lloyd (ed.), R.T. Chern, W.J. Koros, H.B. Hopfenberg and V.T. Stannett, (Chapter 2 authors), Figure 1 on p. 28 of Chapter 2. Copyright (1985) American Chemical Society.

the polymer membrane. However, if we use relation (4.3.44a), we obtain for solubility selectivity, at any temperature T, 0  

    SAT 0 SAm a m ðεB − εA Þ=kB T0 : 1− exp ¼ SBT 0 RT 0 SBm T ð4:3:45Þ

There is a weak dependence on the membrane. Consider now the diffusion coefficient of a gas species i through a polymer membrane for the purpose of determining the mobility selectivity. The diffusion coefficient of a gas i at a reference temperature T0 has been found to vary with the effective diameter di of the gas species as (Michaels and Bixler, 1968) DiT 0 ¼ g m exp ð − f m dni Þ,

ð4:3:46aÞ

where gm and fm are characteristic of the polymer forming the membrane; n has the value of 2 for rotationally hindered stiff-chain macromolecules such as polypropylene, poly(ethylene) terephthalate, cellulose, polyimide, etc.; n is 1 for flexible-chain polymers such as natural rubber, polyethylene, etc. The estimate of di based on the kinetic sieving diameter, which is the smallest zeolite

window through which a gas molecule can fit, is considered to be quite appropriate (Table 4.3.2). As in conventional theories of activated-statebased diffusion in liquids, diffusion of a gas molecule through an amorphous polymer membrane is assumed to be an activated process; it involves the cooperative movements of the gas molecule and the local polymer chain segments around it. Correspondingly, the temperature dependence of Dim depends on an Arrhenius relation, dlnDim ED ¼ − i, dð1=T Þ R

ð4:3:46bÞ

which implies 

 ED 1 1 Dim ¼ DiT 0 exp − i , − T T0 R

ð4:3:46cÞ

where the activation energy, E Di , has been related to the effective diameter di of the gas species i via E Di ¼ E Dm þ f 0m d ni :

ð4:3:46dÞ

Here, EDm and f 0m are constants, which depend on the polymer. Thus, Dim may be expressed as follows:

270

Separation in a closed vessel

0.4

Qim ¼ Dim Sim ¼ g m bm exp C5H12

0.3

Sorption coefficient (cm3(STP)/ 0.2 cm3-cmHg)

Natural rubber

CH4

0

0 He H2

 f 0 ðd n − dn Þ − a0 ðεA − εB Þ ) m B m A kB : RT 0

ð4:3:50Þ

A few comments can now be made about the effect of some important quantities on the ideal selectivity between species A and B, where dA is, say, smaller than dB and ðεA =k B Þ is larger than, say, ðεB =kB Þ (Michaels and Bixler, 1968):

Iso-C4H10 C3 H8

N2

For a binary gas pair i ¼ A and B, the ideal selectivity αAB j t!0 ¼ α AB displayed by the membrane is therefore ( QAm εA − ε B ¼ exp a − f m ðdnA − d nB Þ αAB ¼ αAB jt!0 ¼ QBm kB T0 þ 1− T

0.1 C2H2

)   T 0 E Dm þ f 0m d ni þ b0m − a0m ðεi =k B Þ þ 1− : ð4:3:49Þ RT 0 T



C4H10

(
   εi a B − f m d ni k

C2H6 40

80 120 160 200 Van der Waals molar volume O2 (cm3/mol)

Figure 4.3.4. Henry’s law sorption coefficient of vapors and gases as a function of molar volume for natural rubber membranes. From Figure 3, p. 361, in “Membrane Separation of Organic Vapors from Gas Streams’”, R.W. Baker and J.G. Wijmans, Chapter 8 in Polymeric Gas Separation Membranes, D.R. Paul and Y.P. Yampolskii (eds.) 1994, CRC Press; reprinted by permission of the publisher (Taylor & Francis Group, http://www. informaworld.com.)

(1) For a smaller gas A (dA < dB), other items remaining favorable, α AB > 1; i.e. the smaller molecules of species A are more permeable than the molecules of species B. The membrane is A-selective. (2) For a more condensable gas, which therefore has a higher critical temperature (say, TcjA > TcjB), the selectivity α AB > 1; i.e. the membrane is more selective to species A. This is so since ðεi =k B Þ is roughly proportional to the critical temperature (e.g. εi =kB ¼ 0:77T c j i Þ. (3) At the reference temperature T0, the ideal selectivity α AB is simply expressed by



Q εA − ε B α AB jT 0 ¼ Am

¼ exp a − f m ðdnA dnB Þ : B QBm k T 0

Dim ¼ g m exp

ð − f m dni Þ

exp



 

E Dm þ f 0m dni T0 : 1− RT 0 T

ð4:3:47Þ

Consequently, the mobility selectivity of species A over B through an amorphous polymer membrane is given by 0  

DAm f ðd n − d nB Þ T0 , 1− ¼ expf − f m ðdnA − dnB Þg exp m A RT 0 DBm T ð4:3:48Þ

which shows that the mobility selectivity depends almost solely on ðd nA − d nB Þ, i.e. on the relative difference between the nth power of the effective diameters of two types of gas molecules, dA and dB, for a particular membrane. Further, it is much larger for stiff-chain polymers (n ¼ 2) compared to flexible-chain polymers (n ¼ 1). One can now develop an illustrative expression for the permeability Qim of species i through a membrane by combining expression (4.3.47) for Dim and relation (4.3.44a) for Sim (Michaels and Bixler, 1968):

ð4:3:51Þ

At higher temperatures, the ideal selectivity α AB j T will be lower than α AB j T 0 . (4) Membranes fabricated from stiff-chain polymers, where n ffi 2, have much higher selectivity α AB than membranes prepared from flexible-chain polymers, where n ffi 1. Since different polymers have been observed to possess, for a gas, permeability coefficients that vary over many orders of magnitude, a question of general interest is: what happens to the corresponding selectivities for a particular gas pair? We adopt here an approach suggested by Freeman (1999) to illustrate how αAB is likely to vary with, say, QAm where A is the smaller gas molecule. Freeman (1999) may also be consulted for an overview of the subject by the current author (following Michaels and Bixler (1968)), albeit with somewhat different notation and the literature, and also for numerical estimates of various quantities. First, the ideal selectivity α AB may be described via equations (4.3.42) and (4.3.48) as

4.3

Equilibrium separation: membrane, closed vessel

ln α AB ¼ ln ðSAm =SBm Þ þ ln ðDAm =DBm Þ 2

¼ ln ðSAm =SBm Þ þ ðd nA − d nB Þ4 − f m þ 20

1n

3

2

0

Table 4.3.3. Permeabilitiesa of carbon dioxide and nitrogen through various polymers at 30  C

13

f 0m @ T0 1 − A5 RT 0 T 0

271

QCO2  1011 QN2  1011

Film 13

dB f0 T0 ¼ ln ðSAm =SBm Þ − 4@ A − 15d nA 4− f m þ m @1 − A5: dA RT 0 T

ð4:3:52Þ

From expression (4.3.49) for the permeability coefficient Qim and expression (4.3.47) for Dim , we get ln Qim ¼ ln Sim þ ln Dim 8 0 19 = < n E þ f d T Dm 0 m i @ ¼ ln ðSim g m Þ þ − f m d ni þ 1− A ; : RT 0 T ; ln Qim − ln ðSim g m Þ 8 0 1 0 19 = < E Dm @ T 0A f T 0 − ¼ d ni − f m þ m @1 − A : 1− : RT 0 T T ; RT 0

ð4:3:53Þ

Saran 0.29 Mylar 1.53 Nylon 1.6 Pliofilm NO 1.7 Hycar OR 15 746 Butyl rubber 518 Methyl rubber 75 Vulcaprene 186 Hycar OR 25 186 Pliofilm P4 182 Perbunan 309 Neoprene 250 Polyethylene 352 Buna S 1240 Polybutadiene 1380 Natural rubber 1310

0.0094 0.05 0.10 0.08 2.35 3.12 4.8 4.9 6.04 6.2 10.6 11.8 19 63.5 64.5 80.8

αCO2 −N2 ¼ QCO2 =QN2 30.9 30.6 16.0 21.2 31.7 17.4 15.7 37.9 30.9 29.4 29.1 21.1 18.5 19.6 21.4 16.3

a From Stannett (1968). Units of Qi are scc-cm/cm2-s-cm Hg.

Introduce i ¼ A in this relation and substitute it into relation (4.3.52) to obtain 1000

Rearrange it to get
 n  dB − 1 ln QAm ln α AB ¼ − dA (
 n  dB þ ln ðSAm =SBm Þ − −1 dA  )   E Dm T0 : ð4:3:54Þ 1− − ln SAm  − ln g m − RT 0 T

100




dB dA

n

 

E Dm T0 1− RT 0 T

ln α AB ¼ −

−1



ln QAm − ln ðSAm g m Þ −

þ ln ðSAm =SBm Þ:

To understand the implications of this expression for α AB , recognize that the solubility selectivity ðSAm =SBm Þ for the gas pair A–B changes very little with a change in the polymer (see, e.g., Table 4.3.1); further, the variations of the solubility coefficient of gases with different polymers are very limited (thus, gm, EDm, etc., may not vary much). On the other hand, it is well known that Qim, as well as α AB , varies widely with the polymer. The relation (4.3.54) developed above suggests that a plot of lnα AB against lnQAm  will decrease linearly with a slope of − ½ðd B =d A Þn − 1 . In fact, the so-called “upper bound” line of Robeson (1991) observed for a variety of gas-pair systems over a wide variety of polymers is well described by this type of suggested behavior (Freeman, 1999). The gas pairs useful in this analysis are: He–H2; He–CO2; He–N2; He–CH4; H2–O2; H2–N2; H2–CO2; H2–CH4; O2–N2; CO2–CH4. Figure 4.3.5 illustrates the suggested behavior

Upper bound

αH2–N2 10

1 10−2 10−1 100 101 102 103 H2 Permeability x 1010

104

[cm3(STP)cm/(cm2 s cmHg)] Figure 4.3.5. Relationship between hydrogen permeability and H2/N2 selectivity for rubbery (○) and glassy (●) polymers and the empirical upper bound relation of Robeson (1991). (From Freeman (1999).) Reprinted, with permission, from Macromolecules, 32, (1999) 375, Figure 1, Copyright (1999) American Chemical Society.

(shown by the solid line) for a H2–N2 system for a wide range of polymers. Over a smaller range of permeability variation of a gas with a series of polymers, the data in Figure 4.3.5 appear to indicate that the separation factor may be almost constant. If we consider flexible polymers (n ¼ 1) and equation (4.3.50) at, say, T ¼ T0 (for simplicity), then

272

Separation in a closed vessel

εA − ε B α AB ¼ exp a − f m ðdA − dB Þ : B k

ð4:3:55Þ

Since, for flexible polymers, the mobility based selectivity plays less of an important role, it is likely that α AB may depend primarily on the gas-pair A–B. Table 4.3.3 illustrates that, for a number of polymers, the selectivity for the CO2–N2 system varies by less than a factor of 2, whereas the permeability coefficient Qim varies by 5000 times (Stannett, 1968). A simplistic interpretation is provided by the following approach, where the permeability coefficient of species i through a polymeric membrane m, Qim, is assumed to be described as the product of three quantities: Qim ¼ F m ðpolymerÞ Gi ðgasÞ γði, mÞ,

ð4:3:56aÞ

where γði, mÞ accounts for the specific interaction between the gas and the polymeric membrane; however, Fm depends only on the polymer and Gi depends only on the gas species i. If γði, mÞ ffi 1, then α AB

Q ¼ Am ¼ GA =GB : QBm

ð4:3:56bÞ

The limited set of data in Table 4.3.3 lends support to this simplistic explanation since the selectivity for the CO2–N2 system does not appear to depend much on the nature of the polymers listed. Example 4.3.1 (a) Determine the solubility selectivity of helium (A) and methane (B) at T0, corresponding to 25  C for a polydimethylsiloxane membrane. Compare with the value quoted in Table 4.3.1. The value of a in equation (4.3.43a) is 0.023 K−1 (Freeman, 1999); (ε/kB) for He and CH4 are 10.2 and 149 (K), respectively (Freeman, 1999). (b) Determine the mobility selectivity of a helium (A)– methane (B) pair at 25  C for a polydimethylsiloxane membrane. Compare with the value quoted in Table 4.3.1. Assume T ¼ T0, corresponding to 25  C; the value of fm in equation (4.3.46a) may be obtained from Freeman (1999) as

fm ¼ c

ð1 − aÞ cal ð1 − 0:64Þ ¼ 250 : 2 RT RT mol-Å

Solution (a) From equation (4.3.44b), the solubility selectivity for the gas pair helium (He)–methane (CH4) is     SHe, T 0 aðεHe − εCH4 Þ ¼ exp ¼ exp 0:023 K − 1 ð10:2 − 149ÞK B SCH4 , T 0 k 1 1 ¼ expð − 0:023  138:8Þ ¼ ¼ ¼ 0:041: expð3:192Þ 24:23 Therefore, the solubility selectivity of a He–CH4 pair at T0 ¼ T ¼ 25  C is 0.041. Table 4.3.1 quotes a value of 0.075. For other polymers, such as like natural rubber, the values are close to 0.041.

(b) From equation (4.3.48), at T ¼ T0, the mobility selectivity for the helium–methane pair for any membrane is

ðDHem =DCH4 m Þ ¼ exp ð − f m ðdnHe − dnCH4 ÞÞ:

ð4:3:57Þ

From Table 4.3.2, dHe ¼ 2.69 Å (¼ 0.269 nm) and d CH4 ¼ 3:87 Åð¼ 0:387 nmÞ; further, f m ¼ 250

cal

2

0:36 250  0:36 1 ¼ ; cal 1:987  298 Å2 298 K mol K 0 1

mol Å 1:987

2C B 250  0:36 1 ðDHem =DCH4 m Þ ¼ exp @ − ð7:23 − 14:9Þ Å A 1:987  298 Å2

¼ exp ð1:165Þ ¼ 3:205:

The result quoted in Table 4.3.1 is 5.

4.3.4 Separation of a pressurized liquid solution through a membrane Consider a liquid solution of a microsolute or a macrosolute under pressure (e.g. by a piston) on one side (side 1) of a membrane; let the other side of the membrane (side 2) be empty at time t ¼ 0. As the solvent passes through the membrane (porous/nonporous) under pressure, the other side will fill up, if it has a fixed volume. At this time, and afterwards, what will happen to the solute with respect to the two chambers? If the membrane is semipermeable, i.e. it is impermeable to the solute but permeable to the solvent, then we have seen in Section 3.3.7.4, that, due to osmotic equilibrium, the pressures on the two sides will be related to the osmotic pressure of the feed solution (side 1) on the side of the piston by V s ðP 1 − P 2 Þ ¼ V s π 1 ¼ RT ln

as2 : as1

ð4:3:58Þ

One side will have pure solvent at activity as2; the other side will have solvent at activity as1 as well as the solute, and separation will be achieved at equilibrium. Note: P1 > P2 and as2 > as1. Consider now a porous membrane for the process of ultrafiltration separation of a protein solution (Section 3.4.2.3). Suppose the regions on the two sides of the membrane have become filled with liquid (from the feed region 1 to product region 2). If the membrane does not reject the protein (a macrosolute) completely, the solute protein will slowly diffuse from region 1 to region 2. Ultimately, both sides will have the same protein concentration. The separation achieved initially will be lost. Therefore, in practice, it is necessary to have an open system so that any permeate appearing from the feed side into the permeate side is immediately withdrawn. The same argument is equally valid if the membrane in reverse osmosis (Section 3.4.2.1) has some permeability of the microsolute present in the feed side.

Problems

273

Problems 4.1.1

Dilute waste streams from a chemical plant contain low concentrations of a variety of volatile organic compounds. We would like to know the effectiveness of air stripping of such waste streams at ambient temperature and pressure. Contaminated groundwater at many sites may also have very low concentrations of a variety of organic compounds. The groundwater is pumped to an air stripper and then pumped back into the ground by what is known as “pump-and-treat” processes. We would like to know the effectiveness of such processes for removing volatile organic compounds having similar vapor pressures. Consider two sets of such compounds: (1) n-hexane and chloroform; (2) 1-octene and tetrachloroethene. It is known that, in each set, the first compound is more hydrophobic and will have much higher activity coefficients. The vapor pressures of n-hexane, 1-octene and chloroform at 25  C are 0.2, 0.029 and 0.258 atm, respectively. The 25  C activity coefficients for air-stripping conditions are: n-hexane, 5  105; 1-octene, 2.3  106; chloroform, 7.98  102. The following property values are also available: K ∞i for chloroform and tetrachloroethene are 2.1  102 and 1.5  103, respectively. (Data from Hwang et al. (1992b).) Obtain the air-stripping separation factors for the two sets of compounds. Comment on the strippability of individual compounds in each set.

4.1.2

The wastewater stream from a pharmaceutical plant on two different days had two different sets of pollutants: day (1) benezene and n-hexane; day (2) chloroform and 2-propenal (acrolein). They are removed by air stripping at 25  C. For the wastewater of each day: (a) Which compound on a given day is more easily stripped? (b) What is the separation factor between the two volatile organic compounds on a given day? The vapor pressures at 25  C are: benzene ¼ 0.125 atm; n-hexane ¼ 0.2 atm; chloroform ¼ 0.258 atm; 2-propenal ¼ 0.361 atm. Also K ∞i at 25  C are, for chloroform, 2.1  102, and, for benzene, 3  102. The activity coefficients at 25  C under air stripping conditions are, for n-hexane, 5  105, and, for 2-propenal, 2.16  10.

4.1.3

One step in the preparation of ultrapure water needed for various applications involves the removal of dissolved gaseous species 1 and volatile organic compound 2. This removal is to be implemented using N2 and/or vacuum to strip species 1 and 2. Develop an expression for the separation factor between species 1 and 2 in terms of the thermodynamic constants and temperature-dependent physical properties of the two compounds 1 and 2. Assume nonideal behavior in the liquid phase, an infinitely dilute solution of each species and ideal gas behavior.

4.1.4

For the determination of steam/air stripping based separation of sparingly soluble hydrophobic volatile organic compounds (VOCs) i ¼ 1, 2 from water, K ∞i , an infinite dilution vapor–liquid equilibrium ratio for any species i is quite useful. Consider two such species having nearly the same vapor pressure at the stripping temperature. x 2w=solubility Show that, in such a case, if the two species are liquids at the stripping temperature, α12 ffi x 1w=solubility :

(Hint: To determine γ∞il , consider an equilibrium between the pure organic phase of species i and water.) 4.1.5

Consider the simple vapor–liquid equilibrium in the following binary systems: (1) N2–O2 at 1 atmosphere and 90 K; (2) benzene–toluene at 1 atmosphere and 100  C; (3) ethylbenzene–styrene at 0.133 atmosphere and 80  C. Calculate the mole fraction of each species in the vapor phase and liquid phase, respectively, as well as α12, assuming ideal liquid solution and ideal gas behavior. The following information is available: (1) the vapor pressures of N2 and O2 at 90 K are 3.53 atm and 750 mm Hg, respectively (Ruhemann, 1949); (2) the vapor pressures of benzene and toluene at 100  C are 1.78 atm and 0.73 atm, respectively; (3) the vapor pressures of ethylbenzene and styrene are 0.166 atm and 0.119 atm, respectively. State your assumptions.

4.1.6

A vapor mixture of 7% methane, 8% ethane, 20% propane and 65% n-butane is present at 32  C in a closed vessel. (a) Determine the dew-point liquid composition and pressure using the Ki-factor charts. (b) Determine the bubble-point vapor composition and pressure using the Ki-factor charts.

274

Separation in a closed vessel

4.1.7

(a) In a closed vessel, mtf moles of a mixture of n volatile species are present in vapor–liquid equilibrium. If mtv and mtl are the total number of moles present in the vapor and liquid phases, respectively, and if Ki is defined as K i ¼ ðx iv =x iℓ Þ, obtain the following relations: x iv ¼

x if K i ; mtv ðK i − 1Þ 1þ mtf

n X i¼1

1þ

x if K i ¼ 1: mtv ðK i − 1Þ mtf

The temperature is given as T  C. (b) Consider the feed mixture having the following composition in mole %: methane 7%, ethane 8%, propane 20% and n-butane 65%. It is present in a closed vessel at 32  C and 1000 kPa. What fraction of this mixture is present in the vapor phase? Obtain the vapor phase composition. 4.1.8

An organic feed mixture present at 80  C and 1 atm consists of 40 mol% acetone, 30 mol % acetonitrile and 30 mol % toluene. Assume that the liquid phase is an ideal solution and that the vapor phase behaves as an ideal gas. Further, the system has both liquid and vapor phases. Caculate the fraction of the feed present in the vapor phase and the compositions of the vapor phase and the liquid phase. The P sat values at 80  C for acetone, i acetonitrile and toluene are 196 kPa, 98 kPa and 39 kPa, respectively.

4.1.9

Consider a continuous chemical mixture present in the vapor phase. Its molecular weight density function is known to be fv(M). Determine the expression of fl(M) for the first liquid drop formed as the vapor is cooled to temperature T. Make appropriate simplifications for low-pressure gas and ideal liquid solution.

4.1.10

We have observed in Chapter 3 (equation (3.1.25a)) that the reversible work needed to transfer 1 mole of species i from state 1 to state 2 is equal to the difference between the values of the partial molar free energy at the two states: ðGi j2 − Gi j1 Þ ¼ ðμi j2 − μi j1 Þ. Consider the solution of an impurity B in a solvent A. Comment about the possibility of obtaining a pure solvent with respect to the energy required.

4.1.11

Markham and Benton (1931) studied experimentally the adsorption of pure O2 and pure CO as well as CO–O2 mixtures on 19.6 g of silica at 100  C. The data obtained are provided in Table 4.P.1. Develop a plot of x O2 g vs. x O2 σ for the O2–CO mixture adsorption on silica at 100  C at a total pressure of 1 atmosphere using the ideal adsorbed solution theory. Plot the three experimental points for the mixture in the same figure.

4.1.12

Consider the following cation exchange process: 2NaR þ CaCl2 ðaqÞ , CaR2 þ NaClðaqÞ, where R represents the resin phase. Develop an estimate of the activity-based separation factor between the sodium ion and the calcium ion in terms of the activity ratio of the resin phase to that of the water phase for one of the two cations. Neglect swelling effects.

Table 4.P.1. Pure oxygen at 100  C

Pure CO at 100  C

Volume adsorbed

Volume adsorbed

Press. (mm)

Obs. (cc)

Press. (mm)

Obs. (cc)

17.9 88.6 133.9 221.9 329.0 417.1 471.4 570.5 653.2 760.0

0.08 0.54 0.90 1.52 2.19 2.87 3.18 3.92 4.38 5.01

26.3 127.9 224.4 321.0 434.5 537.2 639.4 760.0

0.26 1.37 2.36 3.37 4.42 5.30 6.33 7.40

a

CO–O2 mixtures at 100  C Volume of oxygen adsorbed

Volume of carbon monoxide adsorbed

pO2

pCO

Isotherma

Observed

Isotherma

Observed

210.3 286.4 335.7

549.7 473.6 424.3

1.45 1.95 2.30

1.56 1.96 2.33

5.50 4.80 4.34

5.27 4.40 4.05

Contains values obtained from pure component adsorption isotherms via interpolation.

Problems

275

4.1.13

Consider a cation exchange resin in an aqueous solution containing Na2 SO4 . Obtain the selectivity of the resin for the sodium cation over the sulfate anion if the activity based distribution coefficient of the anion between the resin and the external solution is 0.001. (Ans. 31, 630.)

4.1.14

Consider two anions Cl− and SO4−− and an anion exchange resin. If the activities of both anions in the aqueous solution are equal, determine which anion will be preferred by the resin phase.

4.1.15

The activity based separation factor αaAB between two counterions A and B in an ion exchange resin system has been defined by (4.1.70). Assume an ideal system. (a) Show that, for an ideal solution in both phases, the separation factor αAB based on mole fractions is equal to the separation factor αm AB based on molalities. (b) Redefine the expression (3.3.121e) for the law of mass action based equilibrium constant for an “ion exchange reaction” between counterions A and B for ideal solutions as Km AB ¼

ðmA, R ÞjZ B j ðmB, w ÞjZ A j

ðmA, w ÞjZ B j ðmB, R ÞjZ A j

:

m jZ A j Show that K m when ZA ¼ ZB. AB ¼ ðαAB Þ m (c) Show that in ideal solutions, where V A ¼ V B , the value of K m AB ¼ 1 when ZA ¼ ZB. The quantity K AB has been defined as the electroselectivity or selectivity coefficient (Helfferich, 1995). Any nonideality and specific preference for an ion by the ion exchanger leads to ln K m AB > 0 when ZA ¼ ZB.

4.1.16

Define the equivalent ionic fraction x ij for i ¼ A, B, a system of two counterions A and B in phase j ¼ w, R, by Z A mA, j : x ij ¼ Z A mA, j þ Z B mB, j Develop the following general relation between x ij and K m ij defined in part (b) of Problem 4.1.15:    1 x AR ðjZ B j=jZ A jÞ x Aw jZ B j=jZ A j Z A mA, R þ Z B mB, R jZ A j , ¼ ðK m AB Þ Z A mA, w þ Z B mB, w 1 − x AR 1 − x Aw where K m counterions A and B. AB is the selectivity coefficient between P Rewrite this result in terms of mF , R and i Z i mi, w , where mF , R is the molality of fixed charges in the resin phase.

4.1.17

Calculate the equivalent ionic fraction x iR of sodium and magnesium ions in a cation exchanger whose selectivity coefficient K m AB is unity due to ideal behavior. (See problem 4.1.16.) Assume mF , R ¼ 9  103 mequiv:=1000 g H2 O of fixed ionic groups. The total molality of ions in solution is 11 mequiv./1000 g H2O; the salts NaCl and MgCl2 are present in equal equivalent amounts (assume x Mgw ¼ x Naw ¼ 0:5 ). (Hint: Employ the results obtained in Problem 4.1.16.) (Ans. x MgR ¼ 0:9756.)

4.1.18

Chimowitz and Pennisi (1986) have measured the supercritical phase ( j ¼ ℓ ) mole fractions of 1,10-decanediol (species i ¼ d) and benzoic acid (species i ¼ b) in supercritical CO2 at two temperatures, 318 K and 308 K for a number of pressures. The results from their extensive experiments are summarized in Table 4.P.2.

Table 4.P.2. Pressure (bar)

Temperature (K)

x dℓ  104 (mole fraction)

x bℓ  103 (mole fraction)

306.8

318 308 318 308 318 308 318 308

5.335 3.064 4.107 2.542 3.411 1.814 2.03 1.53

4.84 3.874 3.843 3.246 2.755 2.338 1.72 1.79

228.5 163.8 132.2

276

Separation in a closed vessel

TH TH

C

TH TL

TL TH

Mole fraction

TL

TL

TH TH

B

TL

TL TL

TH

TL TH

TH TL

TL

TL

A

TH TL TH TH

P2

P1

P3

Pressure

Figure 4.P.1. Crossover behavior for a ternary system.

Determine the solid phase yield (defined as the ratio of moles of solid deposited/mole of CO2 in feed) of each of the two species, as well as the purity enhancement factor10 of 1,10-decanediol in the deposited solid phase when the temperature is reduced from 318 K to 308 K at each one of the four pressures identified above. 4.1.19

The separation of a pure solid from a binary solid mixture extracted by supercritical CO2 via cooling of the mixture in the crossover region has been experimentally demonstrated by Chimowitz and Pennisi (1986). Suppose there is a ternany solid mixture of three species A, B and C. If their solubility behavior in the crossover region is as given in Figure 4.P.1, describe the process sequence you will have to follow to get deposits of pure A, pure B and pure C in different vessels under appropriate conditions.

4.2.1

Calculate the separation factor for each of the following gas mixtures: (1) SO2–N2, (2) U235F6–U238F6, present in a gas centrifuge of radius 4 cm rotating at a high angular velocity, ω ¼ 4000 rad/s at 20  C. The two regions are r2 ¼ 4 cm and r1 ¼ 0 cm. (Ans. αN2 − SO2 ¼ 1:207; αU235 – U238 ¼ 1.0157.)

4.2.2

The mole fraction of species 1 in a binary gas mixture of species 1 and 2 introduced into a gas centrifuge is x1f. After the gas centrifuge is rotated at an angular velocity of ω rad/s, a concentration profile is developed for each species in the radial direction. If the centrifuge radius is r0, indicate a procedure to determine the mole fraction profile, x1j(r).

4.2.3

Calculate the maximum number of macromolecular species whose peaks may be clearly resolved in isopycnic sedimentation if the cell length lz is 1.5 cm. For the purpose of calculations, use the data provided in the example given in the text to calculate the resolution (Rs) for the isotopically labeled and unlabeled DNA molecules in a CsCl gradient; the density gradient is 0.08 g/cm4. You may use the values of Rs or σ developed/used in the text calculations for an estimate of the peak width. How does this number change if the Rs value is 1.5? (Ans. nmax ¼ 18:25; nmax ffi 13.)

10

Purity enhancement factor ¼ ratio of mole fraction of decanediol/benzoic acid in the deposited solid phase to that in the feed phase.

Problems

277

4.2.4

A sedimentation cell has a length of 1 cm; it is located 6.0 cm from the rotation axis (r1 ¼ 6.0, r2 ¼ 7.0 cm). A macromolecular solute of molecular weight 105 is rotated at 25  C at an angular speed of ω rad/s. When the rotational speed is 5000 rpm, the concentration of the macromolecule in the solution varies from a maximum to 1% of the maximum over the cell length. (1) Determine the rotational speed (rpm) which will concentrate the macromolecule to the same extent, over a distance of only 0.01 cm from the tip of the cell (r ¼ 7.0 cm). (2) Determine the ratio of the macromolecular concentration between the two locations r2 ¼ 7 cm and r1 ¼ 6.5 cm for the macromolecular solute when the rotational speed is 5000 rpm. Assume R ¼ 8.315  107 erg/ gmol-K. (Ans. (1) 48 180 rpm; (2) 2.54.)

4.2.5

The Gaussian concentration profile of protein species i around location z0 in a cell used for isoelectric focusing, where the pH is equal to the pI of the protein, can also be determined by a balance of the diffusive flux of the protein, leading to band broadening, and the flux, due to the uniform electrical force field E forcing the protein to concentrate at z0. Obtain the result (4.2.38a), employing appropriate assumptions.

4.2.6

Develop an expression for the maximum number of proteins whose peaks may be clearly resolved via isoelectric focusing with a value of resolution Rs ¼ 1 in a cell of length ℓz . The values of pH at locations ℓz and 0 are pH ℓ and pH0, respectively. The variation of Zi with pH will be incorporated as (dZi/dpH).   ! F E ð− dZ i =dðpHÞÞ ℓz ðpH ℓ − pH 0 Þ 1=2 : Ans: nmax ¼ 16 RT

4.2.7

In a natural gas well assumed to consist of methane and butane only, the mole fraction of butane at 3055 m below the surface is 0.16. Calculate the value of the butane weight fraction in the gas well at the surface level, assuming ideal gas behavior and isothermal conditions. (Ans. 0.69.)

4.2.8

Measurements of isotopic ratios or ratios of noble gases (84Kr and 36Ar) in Greenland ice show that (Craig and Wiens, 1996) gravitational separation in the unconsolidated firn layer above the ice is responsible for the observed enrichments relative to atmosphere ratios. Define R to be the ratio of the mole fraction of noble gas 1 over the noble gas 2 at a height z1, and let R0 be the corresponding ratio in free atmosphere at the surface z2 (>z1). Obtain, as a first order of approximation, the following expression: " , ! #
   R x 11 x 12 ðM 1 − M 2 Þ ðz2 − z1 Þ 3 − 1 10 ¼  1:18, − 1  103 ¼ Δ¼ R0 T x 21 x 22 where region 1 corresponds to z1 (meters) and region 2 corresponds to z2; species 1 is heavier than species 2. If (z2−z1) ~ 70 m, and species 1 and 2 are 84Kr and 36Ar, respectively, obtain an estimate of Δ when the ice temperature is −20  C. (Ans. Δð84 Kr=36 ArÞ ¼ 15:6.)

4.3.1

The extent of solubilization of a hydrophobic solute present in water into a micelle is determined by semiequilibrium dialysis (SED) techniques. This technique employs a dialysis cell (see Figure 4.3.1(a)), where the pore size of the dialysis membrane completely rejects the spherical micelle. However, the solute must be able to pass freely through the membrane. The surfactant monomer may also pass freely through the membrane. On addition of surfactant at a concentration of Csur mol/liter to the chamber on one side of the membrane to which C0i mol/liter of solute species i has been added, binding of the solute to the micelles will occur if the surfactant concentration exceeds the CMC for the surfactant. (a) Identify what will happen after a sufficient time has been allowed to lapse. (b) After equilibrium has been established, the free solute concentration in the solution chamber on the other side of the membrane has been determined to be Cip mol/liter The volumes of the two chambers are Vp and Vf liter, respectively, where subscripts p and f refer to the permeate side and the feed side (where solute was introduced at concentration C0i ), respectively. Determine the number of moles of solute i which have been solubilized into the micellar phase. (c) Assume now that Csur mol/liter >> C0i mol/liter. Further, assume that Csur mol/liter is >> [CMC] and that the moles of surfactant in micelle >> free surfactant moles. Determine the micellar equilibrium ratio, Kim, defined as K im ¼ x im =x if , where xim and xif are the mole fractions of i in the micellar phase and the feed chamber, respectively. Assume that Vp ¼ Vf ¼ 0.05 liter.

278

Separation in a closed vessel

4.3.2

Cuprophan membranes made from regenerated cellulose are frequently used in hemodialysis. Model this membrane as one consisting of cylindrical capillaries of radius 18 Å. Determine the separation factors of the dialysis membrane for two solutes, urea and vitamin B12. The characteristic radii of urea and vitamin B12 are 2.8 Å and 8.5 Å, respectively. The diffusion coefficients of urea and vitamin B12 at infinite dilution in isotonic saline at 37  C are 1.81  10−5 and 0.38  10−5 cm2/s, respectively. Compare the result for the given pore size estimate with that obtained from the data of Colton et al. (1971), namely 16 (based on effective diffusion coefficients without any consideration of equilibrium partition coefficients in their Figure 6). (Ans. 23.5.)

4.3.3

Through a microporous denitrated cellulose membrane, the separation factor for two solutes, sodium sulfate and sucrose, has to be determined for the case of simple diffusion with no convection. The properties of the membrane, the operating conditions and the solute properties are available in Example 3.4.5.

4.3.4

In Donnan dialysis based removal and concentration of Cu2þ ions from a wastewater into an acidic strip solution through a cation exchange membrane, the equilibrium pH of the strip solution is 1 and the equilibrium pH of the feed solution is 4. Determine the concentration ratio of the Cu2þ ions in the two chambers. The membrane-impermeable anion in both chambers is SO24 − . (Ans. Strip solution concentration is 106 larger than that in the feed.)

4.3.5

In seawater desalination processes by thermal evaporation or a membrane based technique, the hardness of the seawater in terms of calcium carbonate or sulfate (also MgCO3, MgSO4) is a problem, which leads to scaling of evaporator surface or membrane fouling. It has been suggested that one could employ Donnan dialysis using the concentrated seawater from the process as the strip solution to reduce substantially the Caþþ/Mgþþ content of the seawater, which is to undergo desalination by a thermal or a membrane process. If the strip solution Naþ concentration is twice that of the seawater to be subjected to desalination, identify the type of membrane and the maximum possible extent of reduction of CaSO4 from that present in the feed seawater.

4.3.6

Develop an expression (in terms of known quantities) for the molar concentration of ion Naþ in a cation exchange membrane at the membrane–solution interface, where the salt Na2SO4 is present at a molar concentration of C 0sw in the feed solution. The molar fixed charge density in the membrane is Cm. Make appropriate assumptions.

4.3.7

A mixture of CO2 and CH4 containing 20 mol% CO2 has been introduced into one side of a synthetic membrane in a closed vessel; the other side of the membrane is empty to start with. We would like to know the composition of the gas mixture that appears first on the other side of the membrane. The permeability coefficients for CO2 and CH4, QCO2 and QCH4 , are provided in units of Barrers as QCO2 ¼ 15, QCH4 ¼ 0:5. (Ans. x CO2 , p ¼ 0:882:)

4.3.8

In molecular distillation, a liquid mixture of A and B is distilled at a very low pressure of 0.001 mm to prevent heat-sensitive materials from spoiling. At such low pressures, if we are dealing with a pure liquid, the flux of the species escaping the liquid surface is given by equation (3.1.119a): P sat A ffi, N A ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πRTM A

where P sat A is the vapor pressure of pure A of molecular weight MA. The escaping vapor molecules condense on another surface close by and do not come back to the liquid. For a mixture, the vapor pressure is to be replaced by the corresponding partial pressure. Obtain an expression for the separation factor αAB for species A and B in a liquid mixture subjected to molecular distillation with the product liquid mixture obtained from the condensing surface, assuming ideal behavior. 4.3.9

A feed gas mixture of species A and B at atmospheric pressure is exposed to a microporous membrane of thickness δm and pore radius rp. The permeate side is under vacuum at time t ¼ 0. Determine the value of the membrane selectivity αAB as t ! 0 in a closed vessel, with the feed gas mixture at total pressure P at time t ¼ 0. Assume Knudsen diffusion exists. The membrane porosity is εm .

4.3.10

Consider two gases A and B being separated through a microporous carbon membrane. The gas transport may be described as that by Knudsen diffusion through the microporous membrane as well as by surface diffusion of the adsorbed gas layer on the pore surface (3.4.112). (1) Determine the value of αAB as t ! 0 in a closed vessel in the manner of Section 4.3.3.

Problems

279

(2) Speculate what will happen if one of the species (between A and B, for example H2 and NH3) is condensable and forms a multilayer condensate on the pore wall, blocking gas-phase transport. 4.3.11

The solubility Sim of a gas species i in a polymer has been empirically expressed as ln Sim ¼ M þ 0:016 T ci , where M depends on the polymer; the value for M for the polydimethylsiloxane polymer at 30  C is –8.8. The diffusivity of a gas species i has been correlated as Dim ¼ ðτ=V ηci Þ, where the value of η for the polydimethylsiloxane (PDMS) polymer is 1.5 and τ has the value of 2.6  10−2. Here, Tci and Vci are the critical temperature and the critical molar volume of gas species i. Calculate the solubility selectivity, mobility selectivity and the ideal selectivity for a PDMS polymer membrane for the two gas pairs O2–N2 and CO2–N2. Assume Tci (K) ¼ 154.6 (O2), 126.2 (N2), 304.2 (CO2). ðAns: ðSO2 m =SN2 m Þ ¼ 1:575; ðSO 2 m =SN 2 m Þ ¼ 17:24; ðDO 2 m =DN 2 m Þ ¼ 1:35;ðDCO 2 m =DN 2 m Þ ¼ 0:929:Þ

4.3.12

You are in a boat on a lake of brackish water. To recover water from the brackish water (salt concentration ~1000−10 000 ppm) of salt concentration Csalt into a concentrated sugar solution C 0sugar , you have a forward osmosis device separating the two solutions by a perfect semipermeable membrane. In forward osmosis, water from the salt solution with an osmotic pressure lower than that of the concentrated sugar solution will permeate to the sugar solution. The initial volume of the concentrated sugar solution is V 0sugar . We know that the osmotic pressure of each solution is defined as π sugar π salt

ffi ffi

bsugar C sugar ; bsalt C salt ,

where bsalt > bsugar. Determine the final volume of the sugar solution, Vsugar, that you can achieve and the corresponding sugar concentration, Csugar. Assume that the membrane is perfectly semipermeable.

5

Effect of chemical reactions on separation

Chemical reactions occur in many commonly practiced separation processes. By chemical reactions, we mean those molecular interactions in which a new species results (Prausnitz et al., 1986). In a few processes, there will be hardly any separation without a chemical reaction (e.g. isotope exchange processes). In some other processes, chemical reactions enhance the extent of separation considerably (e.g. scrubbing of acid gases with alkaline absorbent solutions, solvent extraction with complexing agents). In still others, chemical reactions happen whether intended or unintended; estimation of the extent of separation requires consideration of the reaction. For example, in solvent extraction of organic acids, the extent of acid dissociation in the aqueous phase at a given pH should be taken into account (Treybal, 1963, pp. 38–41). Chemical equilibrium has a secondary role here, yet sometimes it is crucial to separation. Familiarity with the role of chemical reactions in separation processes will be helpful in many ways. This is especially relevant since a few particular types of chemical reactions occur repeatedly in different separation processes/techniques. These include ionization reactions, acid–base reactions and various types of complexation reactions. The complexation reactions also include the weaker noncovalent low binding energy based bonding/interactions identified in Section 4.1.9. A better understanding and quantitative prediction of separation in a given process is possible, leading to better process and equipment design. In processes where a chemical agent is used, different agents can be evaluated systematically. On occasions, it may facilitate the introduction of reactions to processes for enhancing separation. The relation between the extent of separation and the extent of reaction is briefly considered in Section 5.1. How chemical reactions alter the separation equilibria in gas–liquid, vapor–liquid, liquid–liquid, solid–liquid, surface adsorption equilibria, etc., is described in Section 5.2. The role of chemical reactions in altering the separation in

rate-controlled equilibrium processes is treated in Section 5.3. Section 5.4 illustrates how chemical reactions affect the separation in rate-governed processes using membranes and external field based processes; for example, electrochemical gas separation using membranes in an external electric field.

5.1 Extent of separation in a closed vessel with a chemical reaction Consider a closed vessel with two regions j ¼ 1, 2 and two species i ¼ 1, 2. The extent of separation at any time t (or at equilibrium, as the case may be) can be written as follows:


m11 ðt Þ m21 ðt Þ

: ð5:1:1Þ ξ ðt Þ ¼ jY 11 ðt Þ − Y 21 ðt Þj ¼

− m01 m02


Suppose now that region 1 in the separation system has another species 3 with which species 1 reacts to produce a fourth species 4, i.e. 1 þ 3 , 4:

ð5:1:2Þ

Then species 1 is present in region 1 not only as species 1, but also as species 4. Assume that species 3 and 4 cannot go to region 2 and that species 2 does not react with either species 3 or 4. If, at time t, the total number of moles of species 1 present as free species 1 and species 4 in region 1 as a result of reaction is mr11 (t), the extent of separation with the reaction is


mr ðt Þ m21 ðt Þ


ξ r ðt Þ ¼ jY r11 ðt Þ − Y r21 ðt Þj ¼
11 0 −
; ð5:1:3Þ
m1 m02


where we assume that Y r21 ðt Þ ¼ Y21(t) since the distribution of species 2 is unaffected by the reaction. The difference between the two extents of separation is given by


mr ðt Þ m ðt Þ


mr ðt Þ − m ðt Þ

11

11 11 ξ r ðt Þ − ξ ðt Þ ¼
11 0 − : ð5:1:4Þ
¼
m1 m01 m01


5.2

Chemical reactions change separation equilibria

For the reaction under consideration, the stoichiometric coefficient υi for i ¼ 1 is −1. The molar extent of reaction is traditionally defined for any ith species participating in a reaction as the number of moles consumed or produced by the reaction : jυi j

281 5.2.1.1.1 Solute ionization in aqueous solution The solubility of a gas in a liquid is often described by Henry’s law (see equation (3.3.60b)). Thus the gas-phase partial pressure of SO2, pSO2 , is related to its aqueous phase mole fraction x SO2 by pSO2 ¼ H SO2 x SO2

ð5:2:1Þ

We note that the difference in (5.1.4) of the two extents of separation is merely the molar extent of reaction of the ith species normalized with respect to the total number of ith species originally present in the system. Thus, the traditional formalism of reaction engineering or reaction kinetics may be used to estimate directly the change in the extent of separation due to a chemical reaction. Such a straightforward result may not be valid if some of the assumptions made earlier break down.

for ideal gas-phase behavior, where H SO2 is the Henry’s law constant for SO2 in water. However, it is valid only for molecular SO2 in water. Dissolved SO2 in water ionizes to some extent (Danckwerts, 1970):

5.2 Change in separation equilibria due to chemical reactions

The ionization equilibrium of (5.2.2) is described by1

The chemical equilibrium in a multiphase system and the corresponding species concentrations in chemical separations are altered when chemical reactions take place. We will illustrate this now for gas–liquid equilibrium (as in gas absorption), vapor–liquid equilibrium (as in distillation), liquid–liquid equilibrium (as in solvent extraction), stationary phase–liquid equilibrium (as in ion exchange, chromatography and crystallization), surface adsorption equilibrium (as in foam fractionation) and Donnan equilibrium. 5.2.1

Gas–liquid and vapor–liquid equilibria

First we consider gas–liquid equilibrium as in gas absorption/stripping. 5.2.1.1

SO2 þ H2 O , HSO−3 þ Hþ :

Thus, SO2 is present in water also as HSO−3 ; the total SO2 concentration in water is given by   ð5:2:3Þ C tSO2 ¼ C SO2 þ C HSO−3 gmol=liter: Kd ¼

C Hþ C HSO−3 : C SO2

ð5:2:4Þ

The aqueous solution has no net charge anywhere (see the electroneutrality condition (3.1.108a)); therefore, C HSO−3 ¼ C Hþ ¼ ðK d C SO2 Þ1=2 :

ð5:2:5Þ

It is convenient to write Henry’s law (5.2.1) for free SO2 as pSO2 ¼ H CSO2 C SO2 :

ð5:2:6Þ

But the relation between pSO2 and C tSO2 is of interest here, where h i 00 C tSO2 ¼ C SO2 þ ðK d C SO2 Þ1=2 ; pSO2 ¼ H CSO2 C tSO2 : ð5:2:7Þ

We find h i 00 ¼ H CSO2

Gas–liquid equilibrium

Gas mixtures are commonly separated in industry by absorption in a solution where chemical reactions may take place (the solution is regenerated subsequently in a separate vessel by stripping). The amount of gas absorbed by simple physical absorption is often insufficient; it can be increased considerably by incorporating in the solution chemicals with which the dissolved gas undergoes a chemical reaction. Removal of acid gases, e.g. CO2, H2S, SO2, COS, by alkaline solutions are well-known examples (Danckwerts, 1970; Astarita et al., 1983). Aqueous absorbent solutions conventionally employed to increase absorption of a species, e.g. CO2 from a gas, may have a variety of reagents, e.g. K2CO3, Na2CO3, NaHCO3, NaOH, monoethanolamine, diethanolamine, etc. Even without a reagent, the absorption equilibrium of a species may be affected by reaction. Absorption of SO2 into water where SO2 is ionized illustrates such a case.

ð5:2:2Þ

where

h

H CSO2

i00

1

H CSO

2

þ

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; K d

H CSO P SO2

ð5:2:8Þ

2

is the pseudo-Henry’s law constant to

determine the total SO2 concentration in the liquid phase h i 00 for a given pSO2 . As expected, when Kd ¼ 0, H CSO2 ¼ H CSO2 .

Alternatively, the total molar concentration of SO2 in water is given by C tSO2

1

p 2 þ ¼ SO H CSO2

K d pSO2 H CSO2

!1=2

:

ð5:2:9Þ

We have generally avoided describing equilibrium constants for any reaction using activities; therefore activity coefficients are absent in (5.2.4) as well as in most relations in this chapter.

282

Effect of chemical reactions on separation

Now H CSO2 ¼ (1/1.63) atm · liter/gmol and Kd ¼ 1.7  10−2 gmol/liter at ordinary temperatures (~25  C) (Danckwerts, 1970). Suppose pSO2 ¼ 0.01 atm, then C SO2 ¼ 0.0163 gmol/ liter but C tSO2 ¼ 0.033 gmol/liter, showing the considerable effect of SO2 ionization in water on its total aqueous solubility (Prausnitz et al., 1986). The gas CO2 also ionizes in water in the manner of (5.2.2), producing HCO−3 and Hþ. However, the equilibrium constant Kd for such a reaction is a couple of orders of magnitude smaller than that of SO2. Hence, unless alkaline conditions are maintained to shift the reaction to the right, the effect of CO2 ionization in water has a negligible influence on CO2 solubility in water. The solubility of an acid gas such as CO2 or H2S in water is, however, strongly affected by ionization if a basic gas like NH3 is present. Ammonia ionizes in water as − NH3 þ H2 O , NHþ 4 þ OH ,

ð5:2:10Þ

whereas CO2 ionizes as CO2 þ H2 O , HCO−3 þ Hþ :

primary amine CO2 þ 2RNH2 , RNH COO− þ RNHþ 3;

ð5:2:13Þ

secondary amine CO2 þ 2R2 NH , R2 N COO− þ R2 NHþ 2:

ð5:2:14Þ

Consider an aqueous solution of a primary amine in contact with a gas containing CO2. The following equilibria are valid: CO2 ðgÞ ↑ ↓ K CO2 ðw Þ þ 2RNH2 ðw Þ , RNH COO− ðwÞ þ RNHþ 3 ðw Þ:

ð5:2:15Þ

The molar concentration based equilibrium constant K for this aqueous-phase reaction is given by K¼

ð5:2:11Þ

The Hþ ions and OH− ions are participants in the ionization H2 O , Hþ þ OH− ,

may be represented by RNH2 and R2NH, respectively. The respective overall reactions with CO2 in an aqueous amine solution are as follows:

ð5:2:12Þ

where the equilibrium lies far to the left, since water has very little tendency to ionize. Thus both reactions (5.2.10) and (5.2.11) are pushed to the right, resulting in considerable − amounts of NHþ 4 and HCO3 . This means that the partial pressure of CO2 (or NH3) needed for a certain total liquidphase concentration of CO2 (or NH3) is much reduced from that when only either CO2 or NH3 is present (Prausnitz et al., 1986). A detailed analysis of the equilibria in a NH3–CO2– H2O system is available in Edwards et al. (1978). Conversely, if one tries to strip NH3 and CO2 from the water with steam or air, one finds that the efficiency is quite low compared to the cases when either CO2 or NH3 only is present. In fact, sour waste streams in refineries, steel mills and coal conversion wastewaters are typical examples of this difficulty. It has been suggested therefore that one should extract NH3 by solvent extraction and remove CO2 (or H2S) by steam stripping (Cahn et al., 1978; MacKenzie and King, 1985). Such a process has been called extripping (simultaneous extraction by a solvent and stripping by steam or air). 5.2.1.1.2 Reaction with an absorbent in solution We now consider enhancement of gas absorption into a liquid by liquid-phase reactions with an absorbent species. One of the common industrial absorbents for removal of CO2 from gas streams is an aqueous solution of a primary or a secondary amine. A primary amine and a secondary amine

C RNHCOO−  C RNHþ3 C CO2  ðC RNH2 Þ2

:

ð5:2:16Þ

At 20  C, its value is 1.1  105 gmol/liter (Danckwerts, 1970). The concentration of free CO2 in water, C CO2 , is very small when there is a substantial amount of free amine RNH2 present in water. Further, since the concentration of OH− is very much less than that of RNHCOO−, by the electroneutrality condition, C RNHCOO− ffi C RNHþ3 . The distribution coefficient of CO2 (species i) between the aqueous (j ¼ w) phase and the gas phase (j ¼ g) in the absence of the amine is κiw ¼

C iw : C ig

ð5:2:17aÞ

In the presence of a large amount of amine and the chemical reaction in water, the effective distribution coefficient 00 κiw based on the total CO2 concentration in water is 00 ¼ κiw

C iw þ C RNH COO− C tiw ¼ , C ig C ig

ð5:2:17bÞ

where Ciw is the molar free CO2 concentration in water. Using the electroneutrality condition and the equilibrium 00 constant relation in the above definition of κiw , we find that 00 κiw ¼

C iw ðK C CO2 Þ1=2 C RNH2 þ , C ig C ig

where C CO2 ¼ Ciw. Therefore, "

00 ¼ κiw 1 þ κiw

K 1=2 C RNH2 ðC iw Þ1=2

#

:

ð5:2:17cÞ

From Danckwerts (1970, p.198), for example, some typical estimates for monoethanolamine at 20 ºC are C RNH2 ¼ 1.75 gmol/liter, Ciw ¼ 4.1  10−7 gmol/liter. These values lead to

5.2

Chemical reactions change separation equilibria

CH3

CH3 H H2N

C

C

OH

C

H3C

H

CH3

NH2

CH3

2-amino-2-methyl-1-propanol

t-butylamine

Figure 5.2.1. Examples of hindered amines.

00 κiw

¼ κiw

"

103 1:75  ð107 Þ  1þ 3 ð4:1Þ1=2

1=2

#

:

ð5:2:17dÞ

This illustrates the extraordinary increase in the effective distribution coefficient of CO2 due to the chemical reaction with a primary amine added to water. An alternative way to illustrate the same effect would be to use the partial pressure pig of species i in the gas phase instead of Cig. An even higher absorption of CO2 is possible if a hindered amine is used. Both reactions (5.2.13) and (5.2.14) indicate that two moles of amine are utilized per mole of CO2 absorbed and one mole of an amine carbamate RNHCOO− is produced. This carbamate can hydrolyze as follows: RNHCOO þ H2 O ¼ RNH2 þ HCO−3 :

ð5:2:18aÞ

An overall reaction in the case of carbamate hydrolysis for CO2 absorption may be written, using (5.2.13), as follows: − CO2 þ RNH2 þ H2 O ¼ RNHþ 3 þ HCO3 :

ð5:2:18bÞ

Thus, for unstable carbamates, only one mole of amine is utilized per mole of CO2 absorbed; more amine is available for CO2 absorption. For conventional amine absorption processes, the amines used (monoethanolamine, diethanolamine, dimethylamine, etc.) are unhindered; a hindered amine has a bulky alkyl group attached to the amino group as in Figure 5.2.1 (Sartori et al., 1987). The carbamates of such hindered amines are much less stable than the carbamates of the corresponding unhindered amines. For example, the equilibrium constant for the hydrolysis reaction (5.2.18a), K ¼ C RNH2 C HCO−3 =C RNHCOO− ,

ð5:2:18cÞ

has the value of 0.143 for t-butylamine and 0.06 for the unhindered n-butylamine HO–CH2–CH2–NH2 (see Sartori et al., 1987). Thus, reaction (5.2.18b) will occur if the alkyl group in the amine is bulky and the carbamate is unstable. The net result with a highly hindered amine is that we can approach the theoretical absorption capacity of one mole of CO2 per mole of amine.

283 There is an added advantage to having a hindered amine. For gas purification by absorption in an aqueous amine solution in a continuous process with a fixed absorbent, the amine solution has to be regenerated by heating and CO2 is stripped from the solution. Reactions (5.2.13) and (5.2.18b) now go from right to left. In reaction (5.2.13), all of the absorbed CO2 cannot be stripped because RNHCOO− for an unhindered amine is stable. On the other hand, there is no such problem in reaction (5.2.18b), leading to much better stripping. It is useful to consider reactive absorbents in the aqueous solution other than amines, e.g. K2CO3. In the hot potassium carbonate process widely practiced for CO2 and H2S removal, the absorbent is simply K2CO3, which ionizes to give Kþ and CO¼ 3 . When CO2 is present in the gas phase, we have dissolved CO2. Of course, HCO−3 , OH− and Hþ ions are also present: − CO2 þ H2 O þ CO¼ 3 ¼ 2HCO3 ,

ð5:2:19aÞ

which follows from the set   − − CO¼ 3 þ H2 O , HCO3 þ OH : − þ CO2 þ H2 O , HCO3 þ H Two moles of HCO−3 are obtained per mole of CO2 absorbed. Depending upon the conditions, all or some of the CO¼ 3 introduced into the system may have reacted. In the amine absorption example, we observed how a chemical reaction increased CO2 absorption. Here we study how much of the chemical absorbent is being utilized in the reactive absorption; the fractional consumption f of the chemical absorbent has been defined as the degree of saturation (Astarita et al., 1983). The equilibrium constant K for reaction (5.2.19a) is K¼

C 2HCO−3 C CO2 C CO¼3

:

ð5:2:19bÞ

Since 1 mole of CO¼ 3 reacts with 1 mole of CO2, the total molar concentration of CO¼ 3 introduced as absorbent, C tCO¼3 , includes the CO¼ concentration present at equilib3 rium, C CO¼3 , and the CO¼ 3 concentration which has reacted to produce HCO−3 , 0:5C HCO−3 . Therefore, C tCO¼3 ¼ C CO¼3 þ 0:5 C HCO−3 :

ð5:2:19cÞ

Further, from the definition of f (f ¼ 1 means all of the absorbent has been consumed; f ¼ 0 means no consumption at all),   f C tCO¼3 ¼ 0:5 C HCO−3 : ð5:2:19dÞ We substitute these relations into (5.2.19b) to get C CO2 ¼

4 t f2 : C CO¼3 ð1−f Þ K

ð5:2:19eÞ

284

Effect of chemical reactions on separation

102

Equilibrium partial pressure of CO2, psia

40 wt% K2 CO3 140°C 130 120 110 90 70

101

100

102

10−1

coal gasification streams have H2S and CO2; only H2S needs to be removed to avoid pollution. Yet there will be absorption of both H2S and CO2 in the absorbents traditionally employed. Selective removal of H2S over CO2 from gases with a high CO2/H2S ratio is highly desirable for several reasons: increased CO2 absorption increases cost for absorbent regeneration and recirculation; reduction of CO2 in the acid gas obtained from the stripper-regenerator is useful for sulfur recovery by the Claus process (Astarita et al., 1983; Savage et al., 1986). Under conditions of absorption equilibrium, the thermodynamic selectivity of an absorbent for H2S over CO2 has been defined as (Astarita et al., 1983)

101

20 wt% K2CO3

100

130 °C 110 90 70

10−1 10−2

10−1 −1

0

10

10

1

10

f2 1−f

Using Henry’s law for free CO2, pCO2 ¼ H CCO2 C CO2 , we obtain 4 H CCO2 t f2 , C CO¼3 ð1 − f Þ K

!

! pCO2 , C tCO2

ð5:2:19fÞ

which relates the partial pressure of CO2 in the gas at equilibrium with the total molar absorbent concentration and its fractional consumption (Figure 5.2.2). For a given C tCO¼3 , a higher equilibrium CO2 partial pressure implies a larger value of the fractional absorbent consumption. The temperature dependence of K is also crucial to the partial pressure of CO2 at equilibrium. Further, the amount of CO2 absorbed is directly proportional to C tCO¼3 ; therefore, K2CO3 is preferred to Na2CO3, which has a lower solubility. Table 5.2.1 lists common alkaline reagents used for acid gas scrubbing in aqueous or organic solutions.

ð5:2:20Þ

where C ti is the total molar concentration of species i in the liquid phase whose equilibrium vapor pressure is pi. The liquid-phase concentration C ti includes the free ith species and the chemically combined form of the ith species. Most chemical absorbents are aqueous alkaline solutions containing a base, B. The reactions of H2S and CO2 are as follows: K

Figure 5.2.2. CO2 equilibrium vapor pressures over potassium carbonate solutions (Astarita et al., 1983.) Reprinted, with permission, from G. Astarita, D.W. Savage, A. Bisio, Gas Treating with Chemical Solvents, Figure 2.4.1, p. 70, John Wiley & Son, 1983. Copyright © 1983, John Wiley & Sons.

pCO2 ¼

αH2 S−CO2 ¼

C tH2 S pH2 S

H2 S þ B ¼S BHþ þ HS− ; K −1 CO

1

CO2 þ B þ H2 O ¼2 BHþ þ HCO−3 :

ð5:2:21Þ ð5:2:22Þ

There is an additional reaction for H2S, a simple dissociation of HS−, given by HS− ¼ Hþ þ S¼ :

ð5:2:23Þ

The equilibrium constant for this reaction is very small (in the range of 10−14 to 10−12 gion/liter between 20 and 100  C) compared to that for reaction (5.2.21), which is in the range 0.1 to 10 (Astarita et al., 1983). Therefore we can neglect dissociation of HS−. We can conveniently consider reactions (5.2.21) and (5.2.22) in terms of K

HCO−3 þ H2 S ¼CS H2 O þ CO2 þ HS− :

ð5:2:24Þ

The equilibrium constant for this reaction, KCS, in terms of the molar concentration based equilibrium constants, KS and K CO2 1 , is K CS ¼

KS C CO2 C HS− , ¼ C K −1 HCO−3 C H2 S CO2 1

ð5:2:25Þ

where 5.2.1.1.3 Selective absorption in reactive solution Some industrial gas streams contain two volatile species, only one of which needs to be removed. For example, some

KS ¼

C BHþ C HS− C CO2 C B ; K CO2 1 ¼ : C H2 S C B C BHþ C HCO−3

ð5:2:26Þ

5.2

Chemical reactions change separation equilibria

285

Table 5.2.1. Common alkaline reagents used in chemical solventsa

C Monoethanolamine (MEA)

N

C

OH

C

OH

C

OH

H H C

Diethanolamine (DEA)

N

H C

OH

C Diisopropanolamine (DIPA)

N

C

OH

C

C

H C

OH C β, β0 Hydroxyaminoethylether (DGA)

N

Potassium carbonate (with promoters)

K2CO3

C

O

C

C

OH

H H O C

Potassium glycinate

N

Caustic soda

NaOH

C H

OK

H a

Reprinted with permission from G. Astarita, D.W. Savage, A. Bisio, Gas Treating with Chemical Solvents, John Wiley & Sons (1983), Table 1.2.3, p. 9. Copyright © 1983, John Wiley & Sons.

Now, C tH2 S ¼ C H2 S þ C HS− . Assume that C H2 S x2. Further, since x1 is small, y2 can be much larger

5.2

Chemical reactions change separation equilibria

equilibrium enrichment achieved in one stage, a closed vessel. When two isotopic compounds, say XAn (i ¼ 1) and XAn−1B (i ¼ 2), are present in a mixture containing two isotopes A and B, the separation factor α12 may be easily sat obtained from (4.1.21b) as α12 ¼ P sat 1 =P 2 as long as the gas phase behaves ideally and the liquid phase is an ideal solution. However, in the case of heavy water, there is a third isotopic compound, HDO, present; there is the following reaction in the liquid phase:

1.0 Kx = 10 Kx = 5

Kx = 1

0.8

0.6 Kx = 0 and x3 = 0

y1 0.4

287

0.2

Kx

H2 O þ D2 O , 2HDO: 0 0

0.2

0.4

0.6

0.8

1.0

x1 Figure 5.2.3. Effect of reaction equilibrium constant on phase equilibrium (x3 ¼ 0.1 unless stated otherwise) (Terrill et al., 1985). Reprinted, with permission, from I & E Chem. Proc. Des. Dev., 24, 1062 (1985). Figure 2. Copyright (1985) American Chemical Society.

than y1. For example, suppose Kx ¼ 10, x3 ¼ 0.1 and x1 ¼ 0.1. From the above relations, we get y1 ¼ 0.125, x2 ¼ 0.7 and y2 ¼ 0.875. Thus the vapor is highly enriched in species 2 over species 1. Suppose x2 ¼ 0.7 (x3 ¼ 0 ¼ x4) and there is no chemical reaction. Then x1 ¼ 0.3, y1 ffi 0.3 and y2 ffi 0.7. Obviously, the chemical reaction has led to a much purer vapor in terms of species 2, and therefore to a better separation. The converse approach is sometimes practiced to improve the conversion from a reaction. By withdrawing selectively one or more products from the reaction mixture, a reversible reaction can be driven to the right, e.g. CH3 COOH þ C2 H5 OH , CH3 COOC2 H5 þ H2 O: ð5:2:39Þ Removal of water would push the reaction to the right in this esterification process, an example of separation facilitating the main objective, namely a chemical reaction (Suzuki et al., 1971). 5.2.1.2.1 Separation factor in distillation of isotopic mixtures Isotopic mixtures are often separated by distillation. The production of heavy water, D2O, is a prime example. In the distillation of natural water, which contains variable amounts of deuterium depending on the source (see Table 13.1 in Benedict et al. (1981)), deuterium is invariably concentrated in the liquid phase. Water in the tropical oceans contains about 156 ppm deuterium. Multistage distillation (see Section 8.1.3) has to be carried out to obtain a liquid fraction substantially enriched in deuterium. Here we will consider only the

ð5:2:40Þ

We will now obtain a separation factor α12 between H2O (i ¼ 1) and D2O (i ¼ 2) in the presence of HDO (i ¼ 3), all three species being distributed between the liquid phase and the vapor phase. Assume (Benedict et al., 1981) that (1) we have ideal gas behavior and an ideal solution in liquid phase; (2) the vapor pressure of HDO is a geometric mean of the vapor pressures of H2O and D2O, i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sat ð5:2:41Þ P sat P sat 3 ¼ 1 P2 ;

(3) reaction (5.2.40) is at equilibrium in the liquid phase; (4) the equilibrium constant K x of reaction (5.2.40) based on mole fractions is 4.

The separation factor α12 based on H2O (i ¼ 1) concentrating in the vapor phase and D2O (i ¼ 2) concentrating in the liquid phase is calculated using the atom fraction of hydrogen and the atom fraction of deuterium, which in turn are calculated from the mole fractions of individual compounds (see (1.6.1d)). For example, the atom fraction of hydrogen in vapor is given by aH2 v ¼ 2x H2 O, v þ x HDO, V

ð5:2:42aÞ

and the atom fraction of deuterium in vapor is given by aD2 v ¼ x HDO, v þ 2x D2 O, v ,

ð5:2:42bÞ

yielding   aH2 v aD2 l aD2 v aH2 l    2x H2 O, v þ x HDO, v x HDO, l þ 2x D2 O, l : ¼ x HDO, v þ 2 x D2 O, v 2 x H2 O, l þ x HDO, l

α12 ¼



ð5:2:43Þ

From assumption (1), x H2 O, v ¼

P sat H2 O x H2 O, l ; P

x D 2 O, v ¼

From assumptions (2), (3) and (4),

P sat D2 O x D2 O, l : ð5:2:44aÞ P

288

Effect of chemical reactions on separation

P sat HDO ¼ Kx ¼ 4 ¼

x HDO, v ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sat P sat H2 O P D2 O ;

ð5:2:44bÞ

x 2HDO, l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) x HDO, l ¼ 2 x H2 O, l x D2 O, l ; x H2 O, l x D2 O, l

P sat HDO x HDO, l P

ð5:2:44cÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sat P sat H2 O P D2 O x H2 O, l x D2 O, l ¼2 : P ð5:2:44dÞ

Substituting these results into the expression for α12, we get 2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sat 2 P sat Psat H2 O P D2 O x H2 O, l x D2 O, l H2 O x H2 O, l þ 62 7 P P α12 ¼ 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 sat Psat 2 Psat x D O , l H2 O P D2 O x H2 O, l x D2 O, l D2 O 2 þ 2 P P  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 x H2 O, l x D2 O, l þ 2x D2 O, l ð5:2:45Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2x H2 O, l þ 2 x H2 O, l x D2 O, l 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 sffiffiffiffiffiffiffiffiffiffi sat

pffiffiffiffiffiffiffiffiffiffiffiffi P sat x D2 O, l 6 P H2 O x H2 O, l 7 H2 O : ð5:2:46Þ α12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 ¼ x H2 O, l P sat D2 O P sat x D2 O

D2 O , l

This value varies between 1.12 at 0  C to 1.026 at 100  C (see Table 13.4 of Benedict et al. (1981)). Apparently, using similar assumptions, this type of result has been generalized for a mixture of isotopic compounds. Benedict et al. (1981) have suggested that, for a mixture of isotopic compounds, XAn, XAn−1B, XAn−2B2, . . .., XBn, the separation factor αAB for isotopes A and B may be approximately obtained as sffiffiffiffiffiffiffiffiffiffi sat n P XA n : ð5:2:47Þ αAB ¼ P sat XBn For example, in the distillation of ammonia, the separation factor between NH3 and ND3 is quite close to the value sffiffiffiffiffiffiffiffiffiffi sat 3 P NH3 : ð5:2:48Þ αNH3 , ND3 ¼ P sat ND3

(see Chapters 8 and 9), it is possible to produce a highly enriched C13 fraction containing as much as 60–70% C13 as HCN vapor. One could react this HCN vapor with an aqueous NaCN to obtain a NaCN solution in which 60–70 atom % of C is C13. Any separation factor describing such an isotope enrichment process has to take into account the vapor– liquid equilibrium for the vapor species HCN. Consider now the isotope exchange reaction (5.2.49) equilibrium along with the vapor–liquid equilibrium of HC12N and HC13N. There are a total of six mole fractions, xij, to deal with: HC12N (v), i ¼ 2, j ¼1: x21; HC13N (v), i ¼1, j¼1: x11; HC12N (l), i ¼ 2, j ¼ 2: x22; HC13N (l), i ¼ 1, j ¼ 2: x12; NaC13N (l), i ¼ 3, j ¼ 2: x32; NaC12N (l), i ¼ 4, j ¼ 2: x42. First, the mole fraction based separation factor in vapor–liquid equilibrium between HC13N and HC12 N is given by x 11 x 22 αvl : ð5:2:50Þ 12 ¼ x 22 x 12 Second, the overall isotope reaction (5.2.49) may be written for the liquid phase as follows: Kl

HC12 NðlÞ þ Naþ C13 N− ðlÞ , HC13 NðlÞ þ Naþ C12 N− ðlÞ:

ð5:2:51Þ

One can now develop the following relation between the various concentrations and the reaction equilibrium constant Kl: Kl ¼

HC12 NðvÞ þ Naþ C13 N− ðlÞ , HC13 Nðv Þ þ Naþ C12 N− ðlÞ: ð5:2:49Þ As a result of this exchange between C12 and C13 isotopes of carbon, the HCN vapor becomes preferentially enriched in the C13 isotope. Natural carbon contains 1.11% C13 (Benedict et al., 1981). However, the reaction above will lead to a higher percentage of C13 in the HCN vapor. If such a behavior could be multiplied in a cascade of many stages

ð5:2:52aÞ

However, this is also equivalent to Kl ¼

This separation factor was measured to be around 1.042. 5.2.1.2.2 Isotope exchange reactions in a vapor–liquid system Consider the following overall isotope exchange reaction between HC12N vapor and an aqueous NaC13N solution:

C 12 C 42 : C 22 C 32

x 12 x 42 : x 22 x 32

ð5:2:52bÞ

Define the overall separation factor αov 12 between isotopes 1 and 2 and the two phases j ¼ 1, 2 as αov 12 ¼

x 11 ðx 22 þ x 42 Þ : x 21 ðx 12 þ x 32 Þ

ð5:2:53Þ

Pratt (1967) has defined a quantity w as the mole fraction of both isotopic forms of the nonvolatile reactant in the total dissolved reactants as w¼

x 32 þ x 42 : x 22 þ x 12 þ x 32 þ x 42

ð5:2:54Þ

vl He has then claimed that αov 12 is related to Kl, w and α12 by vl αov 12 ¼ α12 ½ð1 − wÞ þ wK l 

ð5:2:55Þ

5.2

Chemical reactions change separation equilibria

as long as x11 þ x21 ffi 1.0 by neglecting the humidity in the vapor phase. For the reaction under consideration, αvl 12 ¼ 0.995 and Kl ¼ 1.031 at 25  C. It becomes clear that αov 12 > 1 as long as w is considerable. For example, if w is 0.8, αov 12 ¼ 1.0188. The experimentally obtained value of αov 12 is 1.013 (Pratt, 1967, chap. 6). Many other isotopes have been enriched by isotope exchange reactions. However, they have primarily employed gas–liquid systems. A few of them are listed below (for a more detailed introduction, see Pratt (1967, chap. 6) and Benedict et al. (1981)): (1)

CO16 2 ðgÞ

18

þ H 2 O ðl Þ ,

CO16 O18 ðgÞ þ H2 O16 ðlÞ, T ¼ 25 C;

− (2) N15 H3 ðgÞ þ N14 Hþ 4 NO3 ðlÞ ,

−  N14 H3 ðgÞ þ N15 Hþ 4 NO3 ðlÞ, T ¼ 25 C;

34 þ 32 − (3) S O2 ðgÞ þ Na HS O3 ðlÞ ,

289 5.2.2.1

Organic solutes to be extracted from an aqueous phase into an organic solvent phase, or vice versa, are often weak acids or bases. Only unionized solutes are extracted into the organic solvent. But the unionized weak acid solute, HA, will partially dissociate in an aqueous solution2 as follows: HA , Hþ þ A− :

B þ Hþ , BHþ :

ð5:2:58Þ

Thus, the concentration of free HA or free B in water is different from the concentration without ionization. Consider the distribution equilibrium of the weak acid HA (i ¼ 1) between water and a nonionizing organic solvent: , HA , Hþ þ A− : HA |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} solvent phase aqueous phase ð j ¼ 1 ¼ oÞ ð j ¼ 2 ¼ wÞ

Liquid–liquid equilibrium

The effective distribution of a solute i between two immiscible liquid phases 1 and 2 is often changed if any one, or some or all of the following happen: species i dissociates in one phase; species i associates in another phase; species i or its dissociated form reacts in one of the phases with another species deliberately added to this phase or deliberately added to the other phase in another form. This may be brought about by using acid–base equilibrium, simple complexation, chelation, ion pairing, solvation, etc. In some cases (for example, in metal extraction), partitioning of the solute species into another phase is generally not possible without some kind of chemical reaction. In the following, we first consider extraction of nonmetallic species in the presence of an acid–base equilibrium, association, dissociation and simple complexation. Metal extraction is treated later. In many extraction systems in practice, species i can exist not only as i, but also in other forms. The effective distribution (or partition) coefficient κil00 will then be based on the total concentration of i in all forms in the two phases. The intrinsic distribution coefficient κil of only the free species i between the two phases in the absence of chemical reactions will, in general, be different from κil00 . The intrinsic distribution coefficient κil0 of only the free species i between the two phases in the presence of chemical reactions may be different from κil . If the presence of other species does not influence the distribution coefficient, then κil0 ¼ κil . Assume the mutual solubility of water and the organic solvent to be unaffected by any of these reactions. We now study how to determine κi100 for several different cases.

ð5:2:57Þ

Similarly, an unionized weak base B will partially ionize in the aqueous phase as follows:

S32 O2 ðgÞ þ Naþ HS34 O−3 ðlÞ:T ¼ 25 C: ð5:2:56Þ

5.2.2

Dissociation of an organic acid or base in water

ð5:2:59Þ

Now, 00 00 κ11 ¼ κ1o ¼

C 1o C 1o ¼ , C 1w þ C 2w ðconc:ofHA þ conc:ofA− Þwater

ð5:2:60Þ

−

where ionic species A is i ¼ 2. Note that the hydrogen ion concentration, C Hþ w , here is also equal to C2w. For simplicity, we write C Hþ w ¼ C Hþ . The dissociation or ionization constant Kd1 of solute 1 is given by K d1 ¼

C Hþ C 2w C 2w K d1 ) ¼ : C 1w C 1w C Hþ

ð5:2:61aÞ

Further, by definition of κi10 , 0 κ1o ¼ ðC 1o =C 1w Þ:

Using these relations for Kd1 and 00 κ1o , we get 00 κ1o ¼

0 κ1o

ð5:2:61bÞ in the expression for

0 0 0 κ1o C 1w κ1o κ1o ¼ : K d1 ¼ K d1 C 1w 1 þ C þ 1 þ CC 2w C 1w þ C þ 1w H

H

ð5:2:62Þ

00 0 < κ 1o in general. Note here that Thus κ1o

κ1o ¼ ðC 1o =C 1w Þ

ð5:2:63Þ

0 ; i.e. in the absence of any ionization; κ1o may equal κ 1o the distribution relation of the free acid HA without any

2

The reaction in water is HA þ H2O , H3Oþ þ A−. For dilute solutions, we will avoid writing down water, whose concentration may be assumed to be essentially constant at 55 gmol/liter.

290

Effect of chemical reactions on separation

ionization (i.e. no other solute species) may equal the distribution relation of the free acid HA when ionization 0 00 is present. But κ1o unless the ionization constant Kd1 6¼ κ1o is vanishingly small. On the other hand, in an alkaline solution of a weak acid HA (Treybal, 1963) C þ H  K d1 , and 00 0 ¼ κ1o C Hþ =K d1 , κ1o

ð5:2:64Þ

leading to 00 0 ¼ log κ1o − pH − log K d1 , log κ1o

00 0 κ1o ¼ κ1o

00 κ1o ¼

log

¼

0 logκ1o −pH

ð5:2:65aÞ

þ pK 1 :

acid pH − pK 1 ¼ log

pH−pK 1 ¼ logðC 2w =C 1w Þ ) ðC 2w =C 1w Þ ¼ 10pH−pK 1 : ð5:2:65cÞ Correspondingly, expression (5.2.62) becomes 00 ¼ κ1o

0 κ1o : 1 þ 10pH−pK 1

ð5:2:65dÞ

We now consider the case of a weak organic base B (i ¼ 1) and the ionization reaction (5.2.58) with species BHþ being i ¼ 2: aqueous phase ðj ¼ 2 ¼ wÞ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B þ Hþ , BHþ m B |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} solvent phase ðj ¼ 1 ¼ oÞ

ð5:2:66Þ

ð5:2:70bÞ

0  κ1o − 1 ; 00 κ1o

base −pK 1 − pH ¼ log

ð5:2:65bÞ

If we now use an alternative form of relation (5.2.61a) using pH and pK1, we get

0 κ1o −pK 1 −pH : 1 þ 10

ð5:2:70aÞ

The general relations (5.2.62) for a weak organic acid and (5.2.70a) for a weak organic base can also be expressed in the following forms (Robinson and Cha, 1985):

where pH ¼ −log10C þ H . Also, since log Kd1 ¼ −pK1, an alternative result is 00 κ1o

0 C 1w κ1o ¼ ; C 1w þ C Hþ C 1w K d1 1 þ C Hþ K d1

0  κ1o − 1 : 00 κ1o

ð5:2:71Þ ð5:2:72Þ

From these relations, we find that, for a weak organic acid, 00 decreases, since most as the pH increases, the value of κ1o of the acid is ionized and cannot be extracted. This leads to a poor extraction of the unionized acid into the organic phase when compared with the total amount of acid available in the aqueous phase. Conversely, for an organic base, 00 as the pH decreases, κ1o decreases, reducing the extent of extraction of the unionized base. Figure 5.2.4(a) illustrates these two relations for a weak organic acid (e.g. phenols, formic acid, benzoic acid, etc.) and a weak organic base (Robinson and Cha, 1985). For carboxylic acid extraction, consult Kertes and King (1986). The extraction of pencillins, which are weak acids, is illustrated in Figure 5.2.4(b) (Souders et al., 1970). For the application of solvent extraction to pharmaceutical manufacturing processes for antibiotics and nonantibiotics, see Ridgway and Thrope (1983). The values of pK1 of a number of biological solutes, e.g. carboxylic acids, amino acids, etc. are provided in Tables 5.2.2 and 5.2.5.

Here, 00 κ1o ¼

C 1o C 1o ¼ : C 1w þ C 2w ðconc:ofB þ conc:ofBHþ Þwater

ð5:2:67Þ

The dissociation constant3K d1 of base 1 is 1 C þ C 1w C 2w ) C Hþ K d1 ¼ : ¼ H C 2w C 1w K d1

ð5:2:68aÞ

An alternative form is C 2w ¼ 10−pK 1 −pH : C 1w

ð5:2:68bÞ

Now, 0 κ1o

¼ C 1o =C 1w :

Substitute these definitions of Kd1 and 00 sion for κ1o to obtain

ð5:2:69Þ 0 κ1o

into the expres-

Table 5.2.2. pK values of selected solutesa Solute

pK1

Formic acid Acetic acid Propionic acid n-Butyric acid Lactic acid Tartaric acid Oxalic acid Citric acid Ammonium (NHþ 4) Trimethylamine Phosphate H3PO4 H2 PO−4 HPO2− 4

3.74 4.76 4.87 4.82 3.73 3.03 1.25 3.13 9.25 9.87 2.15

a

3

Note that this definition of Kd1 is inverse of that used by Robinson and Cha (1985).

pK2

pK3

4.37 3.67 4.76

6.40

7.22 11.50

Values obtained from Belter et al. (1988), Garcia et al. (1999) and Table 6.3.1 at 25 C. See Table 5.2.5 for amino acids.

5.2

Chemical reactions change separation equilibria

291

(b) 100

(a) Scale for equation (5.2.71)

Penicillin F

pH − pK1 −3 −2 −1

0

Penicillin K

1

2

3

Phenylacetic acid Distribution coefficient, solvent / aqueous

1.0

0.8

k ⬙1o k⬘1o

0.6

0.4

0.2

0

−3

−2

−1

0

1

2

10

CX − 1 CX − 2 TX − 2

1

Penicillin G TX − 1 Acetic acid

3

−pK1 − pH

0.1 1

2

3

4

5 pH

6

7

8

Figure 5.2.4 (a) Solutions of equations (5.2.71) and (5.2.72), respectively. (After Robinson and Cha (1985).) (b) Distribution coefficient data for penicillin broth contents for a solvent. Reprinted, with permission, from Souders et al., Chem. Eng. Prog. Symp. Ser., 66(100), 41 (1970), Figure 2. Copyright © [1970] American Institute of Chemical Engineers (AIChE).

Some carboxylic acids have two ionizable protons. They are called dicarboxylic acids, e.g. oxalic acid, HOOC– COOH. We may represent the unionized acid, species i ¼ 1, by H2A. The first ionization of this acid, represented by H2 A ) HA− þ Hþ , ði ¼ 1Þ ði ¼ 2Þ

ð5:2:73aÞ

has a dissociation constant Kd1 corresponding to pK1 via C Hþ C HA− K d1 C HA− ¼ ) , C Hþ C H2 A C H2 A

ð5:2:73bÞ

−pK 1 þ pH ¼ log10 ½C HA− =C H2 A :

ð5:2:73cÞ

K d1 ¼ which leads to

ð5:2:74aÞ

has a dissociation constant Kd2, corresponding to pK2 via K d2

C þ C 2− ¼ H A : C HA−

−pK 2 þ pH ¼ log10 ½C A2− =C HA− :

ð5:2:74bÞ

ð5:2:74cÞ

We may rewrite this as −pK 2 þ pH ¼ log10 ½C A2− =C H2 A þ log10 ½C H2 A =C HA− , −pK 2 − pK 1 þ 2pH ¼ log½C A2− =C H2 A ,

ð5:2:74dÞ

by using the result (5.2.73c). We can now define an effective partition coefficient for H2A between water and an organic solvent in the absence of any complexing extractant for the dicarboyxlic acid H2A as 00 ¼ κ1o

The second ionization, represented by HA− ) A2− þ Hþ , ði ¼ 2Þ ði ¼ 3Þ

Rearrangement leads to

00 ¼ κ1o

0 C 1o κ1o ; ¼ C 2w C 1w þ C 2w þ C 3w 1 þ C þ CC 3w 1w 1w

ð5:2:74eÞ

0 κ1o : þ 10ð2pH−pK 1 −pK 2 Þ

ð5:2:74fÞ

1 þ 10

ð−pK 1 þpH Þ

Note: Here the total concentration of the acidic species in water is contributed by three species: H2A (species 1), HA−(species 2) and A2− (species 3). In the case of

292

Effect of chemical reactions on separation

tricarboxylic acids, which may be represented by H3A, the number of such species will be four: H3A, H2A−, HA2−, A3−. There are substances that are amphiprotic: at low pH values they behave as a base, and at high pH values they act like organic acids. At intermediate pH, the neutral species predominates. An example is 8-quinolinol: 5

3

log

N

OH

If we represent it as HQ, then, at lower pH, H2Qþ, resulting from the protonation of the nitrogen, will be present in substantial amounts, whereas at high pH, Q− will be preponderant. The neutral species HQ is the major species at intermediate pH. Problem 5.2.10 considers the development of a distribution coefficient for such a substance over the whole pH range (Karger et al., 1973). A most important example of similar behavior is obtained with amino acids. If we represent an amino acid NH2CHRCOOH as HA, then, at low pH, H2Aþ dominates; at high pH, A− is preponderant. At intermediate pH, the zwitterionic form HA± is the major species. See Section 5.2-3.1-2 for an analysis of their ionization behavior. 5.2.2.2 Two weak organic acids or bases: dissociation extraction Many compounds used in practice, especially in the pharmaceutical industry, are weak acids or bases. If two compounds are similar, their distribution coefficients between the aqueous and organic phases would be similar. Since one of the compounds is preferentially sought for its particular properties over the other, conditions for preferential extraction of that compound are required. Select two compounds A and B, both being bases (acids are treated later). Assume for the present that κ 0 Ao > κ 0 Bo , i.e. species B is more soluble in the aqueous phase than species A. Remember, however, that κ 0 Ao is quite close to κ 0 Bo . Consider pH conditions in the aqueous solution (acidic) so that ionization of A and B has taken place (obviously the bases will hardly ionize at very large pH, say pH > 12). Then from (5.2.70a), 0 κAo ; 1 þ ðC Hþ K dA Þ

0 κBo : 1 þ ðC Hþ K dB Þ

00 κAo 0 1 þ ðC Hþ K dB Þ 00 ¼ αAB 1 þ ðC þ K Þ : κBo dA H

ð5:2:77Þ

Rewrite this relation as

2

7

00 κBo ¼

00 ¼ αAB

4

6

00 κAo ¼

00 00 00 0 0 0 (κAo =κBo ) ¼ αAB and κAo =κBo ¼ αAB . The separation 00 factor with aqueous dissociation, αAB , and the separation 0 factor αAB unaffected by the dissociation are then related by

ð5:2:75Þ ð5:2:76Þ

00 00 Now, the ratio ðκ Ao =κ Bo Þ ¼ ½C Ao ðC B þ C BHþ Þw = ðC A þ C AHþ Þw C Bo . For a dilute solution, we consider the mixture of these two bases on the following basis: organic phase on a solvent-free basis; aqueous solution on a water-free and acid-free basis. Then it is obvious that

00 

 00 αAB K dB αAB − − 1 ¼ −pK − pH þ log : ð5:2:78Þ A 0 0 αAB K dA αAB

Only cases where KdA 6¼ KdB are useful for our purpose 0 00 since αAB if KdA ¼ KdB. Therefore, we consider cases ¼ αAB where the dissociation constants of species A and B are different. For given pH, KdA and KdB, one can now solve  00  00 0 0 for the ratio αAB is assumed known, αAB =αAB ; since αAB  00  0 =αAB can be determined. Figure 5.2.5 illustrates how αAB varies with −pKA − pH for different positive values of pKA − pKB (Robinson and Cha, 1985). Here B is a stronger base than A (KdB > KdA); it is also more water soluble. Thus, when the solution becomes more acidic, species B ionizes more and its water solubility increases further. 00 Therefore, κBo will decrease much more rapidly, leading to a high separation factor; species A will be extracted much more into the solvent compared to species B, leading to a higher purity of the extract in species A. Of course, the 00 value of κAo will be decreased due to an increase in pH. Consequently, an optimum between the purity of A and its extent of recovery in the solvent has to be struck. The case where pKB > pKA (KdA > KdB) has not been considered here. A relationship similar to (5.2.77) can be developed (Treybal, 1963, p. 48) when both species A and B are weak acids: 00 αAB ¼

00 κAo 00 1 þ ðK dB =C Hþ Þ 00 ¼ αAB 1 þ ðK =C þ Þ : κBo dA H

ð5:2:79Þ

This can be expressed for KdB >> C Hþ and KdA >>C Hþ (i.e. pH >> pKA and pKB) as  00  0 ffi ðK dB =K dA Þ, αAB =αAB

ð5:2:80Þ

reflecting the effect of different dissociation constants. Thus, if acid B ionizes more, acid A will be extracted more into the organic solvent at higher pH values. For lower pH 00 0 values, if KdB KdA

9Ao

>

pKA − pKB

9Bo

2

100

α⬙AB α⬘AB 1

10

0.5

0.1

1 −4

−3

−2

−1

0

1

2

3

−pKA − pH Figure 5.2.5. Solution of equation (5.2.78) for two bases. (After Robinson and Cha (1985).)

293 alternative technique of fractional neutralization may be adopted. Add a neutralizing alkali corresponding to the weaker base A, which will then revert to A from AHþ (but B is present as BHþ). This undissociated A can now be extracted by an organic solvent. For additional information on, and analysis of, dissocation extraction, the work by Wise and Williams (1963), the description by Pratt (1967, pp. 327–332) and the papers by Anwar et al. (1971) and Wadekar and Sharma (1981) may be consulted. 5.2.2.3 Association and/or complexation of a solute in organic solvent Solutes to be extracted sometimes associate in the organic solvent. This happens especially when the solute is polar, e.g. organic acids, while the solvent is quite nonpolar, e.g. hexane, benzene, toluene, etc. For example, acetic acid (i ¼ 1) is known to dimerize (through hydrogen bonding) significantly in benzene. (Similarly, benzoic acid dimerizes in benzene.) Thus the equilibrium distribution of free acetic acid between the aqueous and organic phase will be different from that of the total amount of acetic acid in the two phases. With a polar solvent, dimerization of solute is likely to disappear. The equilibrium distribution will also be influenced if an agent, present in the organic phase, complexes or reacts with acetic acid. We consider now the distribution of acetic acid, for example, under such conditions. For the time being, we neglect the dissociation of the acid in the aqueous phase. The analysis is useful in general for carboxylic acids. Conventionally, oil-phase (j ¼ o) concentration of acetic acid is not large due to a low value of κ1o. To increase it, organic amines, for example, are added to the oil phase. The basic amine (i ¼ 3) complexes with acetic acid to form a complex i ¼ 4. In the oil phase, acetic acid is present in three forms, whose concentrations are: free acetic acid C1o, acetic acid dimer C2o and the complex C4o, where i ¼ 2 refers to the dimer. The equilibrium constants for the various reactions (shown in Figure 5.2.6) are as follows (j ¼ w for the aqueous phase):

4

Oil phase

K3 3

+

K2 CH3

CH3COOH + CH3COOH Kd 1 CH3 COOH

O C OH

OH O

C

CH3

H+ + CH3COO−

Aqueous phase

Figure 5.2.6. Distribution of acetic acid between an oil phase and an aqueous phase with attendant reactions.

294

Effect of chemical reactions on separation K2

C 1o þ C 1o , C 2o ; K3

C 1o þ C 3o , C 4o :

3

ð5:2:81Þ

Equation (5.2.88) 1o =

ð5:2:82Þ

K2 = 0

Assume no acetic acid dissociation in the aqueous 0 is unaffected by these reactions: phase and that κ1o C 1o 0 ¼ κ1o : C 1w

ð5:2:83Þ

00 , however, should be based on The effective value of κ1o , κ1o t C 1o , the total acetic acid concentration in the oil phase: 00 ¼ κ1o

C t1o C 1o þ C 4o þ 2C 2o ¼ : C 1w C 1w

2

K3 = 260 Experimental data

κ 1o ⬙

κ1o ¼

0.024

1

ð5:2:84Þ

From (5.2.82), K 3 ¼ C 4o =C 1o C 3o :

ð5:2:85Þ

0 0

0.1

K 2 ¼ C 2o =C 21o :

ð5:2:86Þ

C 1o C 1o C2 þ K3 C 3o þ 2K 2 1o C 1w C 1w C 1w

¼ κ1o þ κ1o K 3 C 3o þ 2κ1o K 2 C 1o : Using the definition of κ1o , we get a simpler result: 00 ¼ κ1o þ κ1o K 3 C 3o þ 2κ21o K 2 C 1w : κ1o

ð5:2:87Þ

The concentration of free amine C3o in the organic phase is the only unknown here (assume that κ1o , K3 and K2 are available from the literature and that C1w is known). If C t3o , the total amine present initially, is known, then C t3o ¼ C 3o þ C 4o ¼ C 3o þ K 3 C 1o C 3o ¼ C 3o ½1 þ K 3 κ1o C 1w :

Substitute into (5.2.87) to get 00 ¼ κ1o þ 2κ21o K 2 C 1w þ fκ1o K 3 C t3o =ð1 þ K 3 κ1o C 1w Þg: κ1o

ð5:2:88Þ

00 , κ1o

0.4

0.5

Ct3o

Substituting these relations into (5.2.84), we get 00 κ1o ¼

0.3

0.2

Similarly,

the net distribution coefficient of acetic acid, is Thus, considerably increased from κ1o . Kuo and Gregor (1983) have studied the applicability of such a relation for the solvent decahydronaphthalene (Cl0H18, decalin) and the complexing agent trioctylphosphine oxide (TOPO) under conditions where the extent of dimerization was negligible (K2 ffi 0). Figure 00 5.2.7 shows the distribution coefficient, κ1o , of acetic acid with a fixed initial concentration (0.0893 M) as the TOPO concentration was varied from 0 to 0.52 M in decalin. They had earlier observed that a value of K3 ¼ 260 described the 00 value of κ1o well when κ1o ¼ 0.024 for a C t3o ¼ 0.1295 M.

Figure 5.2.7. Effective distribution coefficient of acetic acid as a function of C t3o . (After Kuo and Gregor (1983).)

This figure shows that, beyond C t3o ffi 0.2 M, relation (5.2.88) no longer holds. This is apparently due to the association or micellization of TOPO at higher concentrations. However, the addition of TOPO has increased the extraction of acetic acid into the solvent phase by an extraordinary amount. We will briefly identify now the results for extraction of acetic acid into an organic solvent in the absence of any complexing agent in the organic phase; however, the acid can dissociate in the aqueous phase, as shown in Figure 5.2.6. If the dissociation or the ionization constant of acetic acid (i ¼ 1) is Kd1 (see relation (5.2.61a) and Table 5.2.2) and there is dimerization of the acid in the oil phase, then it can be shown that

 0 þ 00 0 1 þ 2K 2 κ1o C 1w C H κ1o ¼ κ1o ð5:2:89Þ C Hþ þ K d1 in general. If the acid is the only solute, then C1w can be expressed in terms of C Hþ using the electroneutrality condition. 5.2.2.4

Extraction of metals

Metallic compounds generally ionize in aqueous solution; the metal ion, in addition, is hydrated, i.e. surrounded by basic molecules of water. To extract the ionic metal species into an immiscible organic solvent, an uncharged metalcontaining species which is soluble in a water-immiscible organic solvent has to be produced before it is extracted

5.2

Chemical reactions change separation equilibria

295

Table 5.2.3. Structure and properties of various metal extractantsa

Type

Name

Formula

1a Acidic extractants

di-2-ethylhexyl phosphoric acid

(C4H9CH(C2H5)CH2O)2POOH

Mol. wt. of Active species active species (wt%)

Specific gravity

322

100

0.98

175

99.6

0.91

339

39.2

0.88

311

74–80

0.99

~392 351–393 ~442

95 100 >88

0.81 0.83 0.88

R2 Versatic 10

R1

C

COOH

R3 1b Chelating extractants

LIX 65N aromatic β-hydroxyoximes

R3

R1

R1 = φ R2 = H OH R3 = C9H19

C N

R2

Kelex 100 oxime derivative

R R=Dodecyl

N OH

2 Anion exchangers

3 Solvating extractants

a

tertiary amine, alamine 336 secondary amine, LA-2 quaternary ammonium compounds, aliquat 336 phosphorus-oxygen-bonded donors: tri-n-butyl phosphate trioctyl phosphine oxide methyl isobutyl ketone

R3N (R¼ C8–C10) R2NH (R ¼ C12–C13) (R3N(CH3)þCl−) (R¼ C8-C10) R1R2R3PO

0.97

R1 ¼ R2 ¼ R3 ¼ C4H9O R1 ¼ R2 ¼ R3 ¼ C8H17 R1COR2, R1 ¼ CH3, R2 ¼ (CH3)2CHCH2

226 386 100

— — —

— 0.804

Prepared from Tables 2, 3, 4 and 5 in Flett et al. (1991).

into the organic solvent. This involves metal charge neutralization as well as replacing some or all waters of hydration by solvent compatible agents. Conventional approaches to achieve this goal are the following (Ritcey and Ashbrook, 1984a, p. 6). (1) Compound formation between the metallic ion and an extractant is achieved by using acidic extractants or chelating extractants. Acidic extractants have a group like –COOH, –SO3H, whose hydrogen is exchanged for the metal. A chelating extractant has two or more sites to complex with a metal ion into a cyclic compound. A chelating agent dissolved in the organic solvent forms an uncharged metal chelate in the aqueous phase for extraction of the chelate into the organic solvent. The acidic extractant acts similarly. (2) Ion-pair formation is initiated in the aqueous phase between the metallic ion and a large counterion containing a bulky organic group. This large counterion is obtained from a complexing agent added to the organic phase; the agent dissociates in the aqueous phase to provide the counterion. The neutral ion pair is extracted into the organic phase. (3) The formation of a loose nonchelating uncharged compound between the metal ion and the organic solvent

itself is achieved with many oxygen-containing solvents like tributyl phosphate. This is identified as solvation of the metal ion. Table 5.2.3 identifies a few extracting agents/extractants/ solvents from each of the above categories. In the second category of ion-pair formation, two types of extractants have been identified as anion exchangers: long-chain amines (tertiary, secondary or primary) and quaternary ammonium compounds. Some properties of each extractant have been identified in the table. We study the first of these three approaches now. Consider the chelation4 of metal ions Mnþ with an organic ion K− obtained from the chelating agent HK. This agent, dissolved in the organic phase, partitions into the aqueous phase and then dissociates into Hþ and K−: , HK , Hþ þ K− HK |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

organic phase

ðj ¼ 1 ¼ oÞ 4

aqueous phase

ðj ¼ 2 ¼ wÞ

ð5:2:90Þ

Quantitative representations for acidic or chelating extractants are quite similar. Macrocyclic compounds can complex metal ions in a more stable fashion (Burgess, 1988).

296

Effect of chemical reactions on separation

The metal ion Mnþ in the aqueous phase diffuses to the interface, reacts with the ion K− and forms MKn, a neutral species which is extractable into the organic phase: Mnþ þ nK− , MKn , |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} aqueous phase

ðj ¼ 2 ¼ wÞ

MKn |ffl{zffl}

organic phase

ðj ¼ 1 ¼ oÞ

ð5:2:91Þ

Generally, the reaction is thought to occur in the aqueous side of the interfacial region; MKn then partitions into the organic phase. The overall chemical reaction may be represented as Mnþ ðaqÞ þ nHKðoÞ , MKn ðoÞ þ nHþ ðaqÞ:

ð5:2:92Þ

To determine the net distribution coefficient of the metal 00 between the two phases, κM1 , defined by 00 κM1 ¼

C MKn , o C MKn , w þ C Mnþ , w

ð5:2:93Þ

we proceed as follows. Define the following two distribution coefficients of species MKn and HK between the organic and the water phase: C MKn , o C HK, o 0 0 ¼ : ð5:2:94Þ ; κHK κMK ,1 ¼ n, 1 C MKn , w C HK, w The equilibrium constants for the two aqueous-phase reactions (5.2.90) and (5.2.91) are K d, HK ¼

C Hþ w C K− w ; C HK, w

K d, MKn ¼

C MKn , w : C Mnþ , w ðC K− w Þn

ð5:2:95Þ

00 as Using these relations, we can express κM1 00 κM1 ¼

κ 0 n , 1 ðK d, HK Þn K d, MKn ðC HK, o Þn nMK  0 n : C Hþ , w κHK þ K d, MKn ðK d, HK Þn ðC HK, o Þn ,1

ð5:2:96Þ

Thus, as the concentration of the chelating agent HK in the 00 increases. Similarly as organic phase, CHK,o, increases, κM1 00 the pH increases (C Hþ , w decreases), κM1 increases. Define fM as C Mnþ , w , ð5:2:97Þ fM ¼ C Mnþ , w þ C MKn , w

i.e. the fraction of the total metal ion concentration in the aqueous phase present as Mnþ. Then, using (5.2.94) and 00 (5.2.95), we can rewrite κM1 as follows: 00 ¼ κM1

whereby

0 κMK ðK d, HK Þn K d, MKn ðC HK, o Þn f M n, 1  n  0  n , C Hþ , w κHK ,1

00 logκM1 ¼ nðpH Þ þ nlogC HK, o   0 ðK d, HK Þn K d, MKn f M κMK n, 1 : þ log ðκ 0 HK, 1 Þn

ð5:2:98Þ

ð5:2:99Þ

00 Therefore, log κM1 varies linearly with pH, provided the other quantities are unaffected by the pH variation. It is obvious that by choosing the right pH range, metal extraction into the organic phase can be drastically increased. If we have to separate two metals M and N having the same valence n, at a given pH and a chelate concentration CHK,o, then

" # 0  00  f M K d, MKn κMK 00 n, 1 log κM1 =κN1 ¼ log , 0 f N K d, NKn κNK n, 1

ð5:2:100Þ

which is likely to be unaffected by pH, at least over a certain range. To a first approximation, the curves of log 00 00 κM1 vs. log pH and log κN1 vs. log pH are therefore parallel, with slope n. Figure 5.2.8(a) shows such a plot for metals 00 M and N. Recognize that if there is a pH at which log κM1 00 is −2, the two metals are almost completely is 2 and log κN1 separated. Here the difference in the two metal dissociation or hydrolysis constants K d, MKn , K d, NKn , are crucial. For the extraction of a given metal, depending on the purity and the recovery desired, the pH can be chosen. Figure 5.2.8(b) illustrates the extraction of different metals by a particular extractant as a function of pH (Cox and Flett, 1991). Note, however, that, as pH increases, both 00 00 log κM1 and log κN1 flatten out due to the pH variation of fM and fN. If the pH of the aqueous solution is increased to a high value, at which the metal hydrolyses, the solvent extractability will drop drastically. The flat behavior we observed in Figures 5.2.8(a) and (b) will change into one 00 where κM1 will decrease rapidly with pH. At very low 00 pH, κM1 also has a very low value since high Hþ concentration prevents ionization of the chelating agent HK (see equation (5.2.90)) and thus hinders formation of the complex MKn. Some examples of acidic and chelating agents (Karger et al. (1973), Ritcey and Ashbrook (1984a, p. 24) and Lo et al. (1983) provide more detailed lists) are provided in Table 5.2.3. Additional examples are given in Figure 5.2.9. Note than an acidic extractant, such as di-2-ethylhexyl phosphoric acid (DEHPA), may have a more complex chemistry than a chelating agent, HK, due to its dimerization or self-association in the organic phase (Ritcey and Ashbrook, 1984a, p. 28). We have observed such behavior earlier with carboxylic acids in nonpolar organic solvents. In the extraction of metals by chelation, the neutral product MKn is a chelated compound. Sometimes, the neutral species is simply a pair of oppositely charged ions which is soluble in the organic phase. Basic extractants, like primary, secondary or tertiary organic amines, are often used in solvent extraction. A tertiary amine R3N (where R is usually an aliphatic long-chain group) in the presence of an acid HA in the aqueous phase will undergo the following reactions and produce an ion pair R3NHþA−:

5.2

Chemical reactions change separation equilibria

297

100

(b)

M

N

Fe(III)

Zn

Cd Cu

Mg

Co Ni

0

n=2

n=2

−1

1

2

3

4

5 6 7 8 Equilibrium pH

9

10 11 12

0

−3

20

−2

60

1

40

log k ⬙M1 or log k ⬙N1

2

80

3

Extraction (%)

(a)

0

1

2 3 Equilibrium pH

4

5

Figure 5.2.8. (a) Equilibrium distribution of a metal species M or N as a function of pH for Mnþ or Nnþ. (b) Extraction of some metals by DEHPA from a sulfate solution (Cox and Flett, 1991). Reprinted, with permission, from M. Cox and D.S. Flett, “Metal extraction chemistry,” in T.C. Lo, M.H.I. Baird and C. Hanson, Handbook of Solvent Extraction, Wiley-Interscience (1983), Figure 4, p. 59. Copyright © 1983, John Wiley & Sons.

8-Quinolinol (HOx)

producesan anion forcom plexation

N

N O

OH

–

1,3-diketonethenoyltrifluoracetone

HC

CH

HC

C

C

CH

C

CF3

S O

OH

Figure 5.2.9. Examples of acidic and chelating extractants, e.g. both HOx and M(Ox)2 with 8-quinolinol.

In addition, we have the distribution equilibrium of R3NHþ· A− between the two phases:

R3 NðorgÞ þ HAðaqÞ , R3 NHþ A− ðorgÞ;

ði ¼ 1 Þ

þ

ði ¼ 2Þ

R3 NH A ðaqÞ , R3 NHþ ðaqÞ þ A− ðaqÞ: ði ¼ 3 Þ

−

ði ¼ 4 Þ

ði ¼ 5 Þ

ð5:2:101Þ

00 κ31 ¼

C 3o : C 3w

ð5:2:102Þ

298

Effect of chemical reactions on separation

1 −n)− {(R3NH+)m −nMA(m }2 m 2 1 [R3NH+A−]2 2

−n)− {(R3NH+)m −nMA(m } m

R3NH+A−

R3N

Organic

R3NH+ + A−

R3N + H+ + A−

R3NH+A−

HA −n)− (m – n)R3NH+ + MA(m m

−n)− {(R3NH+)m −nMA(m } m

Aqueous Figure 5.2.10. Ion-pair extraction of a metal ion.

In the aqueous solution, the metal ion Mnþ in the presence an acid HA exists as various complexes (charged or ðm−nÞ− uncharged) MAm . Typical examples for HA will be HCl, HI, etc. Now, the bulky counterion R3NHþ in the aqueous phase will form an ion pair with a charged metalðm−nÞ− lic species in aqueous solution, say, MAm according to ðm − nÞR3 NHþ ðaqÞ þ MAðmm−nÞ− ðaqÞ , ðR3 NHþ Þm−n MAðmm−nÞ− ðaqÞ: ði ¼ 4Þ

ði ¼ 7Þ

ði ¼ 6Þ

ð5:2:103Þ

ðm−nÞ− The ion pair (R3NHþ)m−n MAm with no charge will partition into the organic phase −

−

ðm−nÞ ðm−nÞ ðaqÞ , ðR NHþ Þ ðR3 NHþ Þm−n MAm ðorgÞ 3 m−n MAm ↑↓ ði ¼ 6Þ ði ¼ 6Þ − 1 ðorgÞ, ½ðR3 NHþ Þm−n MAðm−nÞ m 2

ði ¼ 8Þ

ð5:2:104Þ

and may even dimerize or polymerize. The various chemical reactions and partition equilibria are schematically illustrated in Figure 5.2.10, where we have included the dimerization of R3NHþA− in the organic phase. Note that the anionic metal complex MAðmm−nÞ− is formed with the anion A− from the acid according to Mnþ ðaqÞ þ mA− ðaqÞ , MAðmm−nÞ− ðaqÞ:

ð5:2:105Þ

Further, there can be a number of such complexes at any time, depending on the value of m.

One can also argue that, prior to the ion-pair formation of (R3NHþ)m−n with MAðmm−nÞ− to enable metal extraction, there was an ion pair R3NHþA− whose anion A− was exchanged with another anion MAðmm−nÞ− to effect metal extraction. The tertiary amine compound R3N had to be protonated before anion exchange is possible. Secondary or primary amines also require protonation before anion exchange. On the other hand, quaternary ammonium compounds, used as an extraction agent and representable as R4NþA− (e.g. R3N(CH3)þCl−, where R may be C8, C9 or C10), can readily undergo anion exchange with other anions in the system: R4 Nþ A− ðorgÞ ↑↓ R4 Nþ A− ðaqÞ , R4 Nþ ðaqÞ þ A− ðaqÞ

ð5:2:106Þ

ðm−nÞ R4 Nþ ðaqÞ þ MAðmm−nÞ− ðaqÞ , ðR4 Nþ Þm−n MAðmm−nÞ− ðaqÞ ↓↑ ðR4 Nþ Þm−n MAðmm−nÞ− ðorgÞ:

ð5:2:107Þ

Micellization of R4NþA− is also a problem in the organic phase. Such anion exchange is also a powerful technique used in the solvent extraction of amino acids under appropriate pH conditions so that the amino acid (Am) is in a form (Am−) (see Section 5.2.3.1.2) ready for anion exchange, for example using a quaternary ammonium compound: R4 Nþ A− ðorgÞ þ Am− ðaqÞ , R4 Nþ Am− ðorgÞ þ A− ðaqÞ:

ð5:2:108Þ

5.2

Chemical reactions change separation equilibria

The third approach to metal extraction uses oxygencontaining solvents like TBP (tributyl phosphate, (BuO)3PO). Widely adopted in the extraction of many radioactive metals in the nuclear industry, this technique may be illustrated for uranium extraction in the presence of nitrate ions in the aqueous phase by − UO2þ 2 ðaqÞþ2NO3 ðaqÞþ 2TBPðorgÞ , UO2 ðNO3 Þ2 2TBPðorgÞ:

ð5:2:109Þ

To shift the equilibrium to the right, the NO−3 ion concentration is increased in the aqueous phase by adding, say, HNO3 and the level of free TBP in the organic phase is raised. Under such conditions, nitric acid is also extracted into the organic phase by Hþ ðaqÞ þ NO−3 ðaqÞ þ TBPðorgÞ , HNO3 TBPðorgÞ:

299 5.2.3.1

Ion exchange resin–liquid solution equilibrium

There is an extensive literature in this area. Helfferich (1962, 1995) has provided a comprehensive description of the effect of chemical reactions on ion exchange equilibria. Three types of chemical reactions of interest are: complex formation in solution; complex formation with the ion exchanger; ionization/dissociation in the solution. 5.2.3.1.1 Complex formation The complex or complexes formed in solution or in the ion exchanger can strongly affect ion exchange equilibria. Consider first the equilibria with cation exchangers in the presence of a complexing anion (say, Cl−). Let the two competing cations (counterions) A and B be Zn2þ and Hþ. In the presence of a free anion (Y), Cl−, obtained from, say, ZnCl2, and HCl, the cation Zn2þ will form a variety of complexes (Helfferich, 1962):

ð5:2:110Þ

Detailed considerations of equilibria in such systems for the extraction of radioactive metals are available in Benedict et al. (1981). Normally water molecules form a shell around the metal atom, creating the hydration sphere. TBP has a highly polar group P ¼ O, which can replace water around the metal. Other extractants, like ethers, alcohols and ketones, have also been used to the same end, except water molecules are not completely excluded. Table 5.2.4 illustrates the variety of solvent extraction agents used commercially for extracting metals in hydrometallurgy. Extracting agents, such as TPB, are rarely used by themselves. The extracting agent is normally used along with a diluent solvent, e.g. kerosene. The purposes behind the use of a diluent are several. First, the viscosity of the extractant is considerably reduced by the diluent. Second, the inventory of costly extractant is decreased. Diluents, in addition, control the surface-active tendencies of most extractants, leading to improved dispersion-coalescence behavior. A detailed discussion on diluents is available in Ritcey and Ashbrook (1984a, chap. 4). In addition, sometimes modifiers like TBP are added to the organic phase to avoid what is called the third-phase separation (Flett et al., 1991), namely to prevent the splitting of the metal complex-rich organic phase into a metal complex-rich phase at the aqueous interface and a diluent-rich phase above. 5.2.3

Stationary–mobile phase equilibria

Here we consider briefly the effect of chemical reactions on two types of phase equilibrium: ion exchange resin–liquid solution equilibrium; stationary medium–mobile liquid solution equilibrium, as in chromatography (Section 7.1.5). For the first case, the resin phase is generally stationary in a fixed bed; however, the resin phase may also be mobile.

K1

Zn2þ þ Cl− ! ZnClþ : K 1 ¼

C ZnClþ ; C Zn2þ C Cl−

K2

Zn2þ þ 2Cl− ! ZnCl2 : K 2 ¼ K3

Zn2þ þ 3Cl− ! ZnCl−3 : K 3 ¼ K4

C ZnCl2 ; C Zn2þ C 2Cl− ð5:2:111Þ

C ZnCl−3 ; C Zn2þ C 3Cl−

Zn2þ þ 4Cl− ! ZnCl2− 4 : K4 ¼

C ZnCl2− 4

C Zn2þ C 4Cl−

,

where K1, K2, K3 and K4 are the stability constants for the complexes formed. Although the overall concentration of zinc in the solution is now given by C tZn ¼ C Zn2þ þ C ZnClþ þ C ZnCl2 þ C ZnCl−3 þ C ZnCl2− ; 4

 C tZn ¼ C Zn2þ 1 þ K 1 C Cl− þ K 2 C 2Cl− þ K 3 C 3Cl− þ K 4 C 4Cl− ,

ð5:2:112Þ

the concentration of the free cation Zn2þ is now substantially lower. The extent of reduction depends on how strong the complexes are. Further, some of the divalent cation Zn2þ is now either a monovalent cationic species or even an anionic species; therefore the preference of the cation exchanger for Zn is now substantially reduced. On the other hand, the selectivity of the competing cation B (Hþ) is increased substantially in the ion exchange reactions with the cation exchanger (R−) having fixed negative charges since ZnCl−3 , ZnCl2− 4 do not participate in the cation exchange process: Zn2þ ðaqÞ þ 2HR , ZnR2 þ 2Hþ ðaqÞ, ZnClþ þ HR , ZnClR þ Hþ ðaqÞ:

ð5:2:113Þ

The expressions for the selectivity of A over B are available in Helfferich (1962). The net result in this case is that

300

Effect of chemical reactions on separation

Table 5.2.4. Solvent extraction reagents for hydrometallurgya Class

Type

Acidic carboxylic acids extractants alkyl phosphoric acids

Acid chelating extractants

Examples

Manufacturer

naphthenic acids

Shell Chemical Co. copper–nickel separation

di-2-ethylhexyl phosphoric acid (D2EHPA) octylphenylphosphoric acid (OPPA) aryl sulfonic acids SYNEX 1051 hydroxyoximes LIX63, LIX64N, LIX65N, LIX 70

oxime derivatives

SME 529 P5000 series Kelex 100

β-diketones

Hostarex DK16 LIX54 X151

alkarylsulfonamide LIX34 polyols Anion primary amines Primene JMT exchangers secondary amines LA-2 Adogen 283 tertiary amines

various alamines; in particular, Alamine 336 various adogens; in particular, Adogen 364, Adogen 381, Adogen 382 quaternary amines Aliquat 336

Union Carbide Mobile Oil Co.

Commercial use

yttrium recovery, europium extraction, nickel– cobalt separation uranium extraction

King Industries, Inc. magnesium extraction Henkel Corporation copper and nickel extraction

Shell Chemical Co. Acorga Ltd. Sherex Chemical Cob. Farbwerke Hoechst AG Henkel Corporation Henkel Corporation

copper extraction copper extraction proposed for copper extraction

proposed for copper extraction from ammoniacal solution copper extraction from ammoniacal solution proposed for cobalt extraction from ammoniacal solution Henkel Corporation proposed for copper extraction from acidic leach Dow Chemical Co. liquors boron extraction Rohm and Haas no known commercial use

Rohm and Haas zinc and uranium extraction Sherex Chemical zinc and tungsten extraction b Co. Henkel Corporation widley used; cobalt, tungsten, vanadium, uranium extractions, etc. Sherex Chemical Co.b

cobalt, vanadium and uranium extractions

Henkel Corporation vanadium extraction; other possible uses are chromium, tungsten and uranium extraction Adogen 464 Sherex Chemical similar to Aliquat 336 b Co. Solvating phosphoric, tributylphosphate Union Carbide, nuclear fuel reprocessing, U3O8 refining, iron extractants phosphonic and (TBP) Albright and extraction, zirconium–hafnium separation, phosphinic acid Wilson niobium–tantalum separation, rare earth esters separations, acid extraction phosphonic acid esters Farbwerke Hoechst no known commercial use trioctylphophine oxide AG, Henkel recovery of uranium from wet process (TOPO) Corporation, phosphoric acid liquors (with D2EHPA) Cyanamid various alcohols, butanol–pentanol various phosphoric acid extraction ethers, ketones diisopropyl ether various phosphoric acid extraction methyl isobutyl ketone various niobium–tantalum separation, zirconium– (MIBK) hafnium separation alkyl sulfides Di-n-hexyl sulfide palladium extraction a

Reprinted with permission from D.S. Flett, J. Melling and M. Cox, “Commercial solvent systems for inorganic processes,” in T.C. Lo, M.H.I. Baird and C. Hanson, Handbook of Solvent Extraction, Table 1, p. 631, Wiley-Interscience (1983). Copyright © 1983, John Wiley & Sons. b Previously Ashland Chemical Co.

5.2

Chemical reactions change separation equilibria

301

Table 5.2.5. A few amino acids: their structures and dissociation constantsa Name

R–

pK1 COOH

pI

pK2NHþ 3

pK3 R-group

Glycine Alanine Valine Leucine Phenylalanine Glutamic acid Lysine

H– CH3– (CH3)2–CH– (CH3)2CHCH2– C6H5CH2– HOOCCH2CH2– H2NCH2CH2CH2CH2–

2.34 2.31 2.33 2.27 2.17 2.18 2.19

5.97 6.02 5.97 5.97 5.48 3.22 9.74

9.6 9.70 9.76 9.57 9.11 9.59 9.12

– – – – – 10.68 12.48

a

General formula:

NH3+

COO–

CH R

See Bailey (1990); Martell and Smith (1974, 1982); Lehninger (1982); Saunders et al. (1989).

uptake of the metal zinc from the solution is reduced by complexation of the metal with a complexing anion (Cl−) in the solution. Additional examples of excellent complexing agents are anions of weak acids, such as citric acid, ethylenediaminetetraacetic acid (EDTA), etc. (Helfferich, 1962). The situation regarding cation A uptake by an anion exchange resin is different if there is a complexing anion (e.g. Cl−) present. For the same cation Zn2þ, which would not be preferred by an anion exchanger, we now have anionic complexes ZnCl−3 , ZnCl2− 4 . These can now successfully participate in the ion exchange process and lead to substantially increased metal uptake when very little would have been possible in the absence of the complexation with the anion Cl−. The above two examples considered the role of complexation reactions taking place in the solution on the ion exchange equilibria. The effect of complex-forming cations in the ion exchangers on the partitioning of anionic and nonionic ligands from the solution is also of interest. Examples of complexes in the cation exchange resin (fixed charge group, R−) formed by cations like Cu2þ (and Ni2þ,

 2þ − Agþ, etc.) are (R−)2 CuðNH3 Þ2þ 4 , (R )2CuðH2 OÞ4 . Potential ligands from the solution are ammonia, aliphatic amine, polyhydric alcohols, etc. Consider the following ligand exchange between ammonia present in the resin as a ligand with ethylene diamine (NH2C2H4NH2) (EDA):

 ðR− Þ2 CuðNH3 Þ2þ þ 2NH2 C2 H4 NH2 ðaqÞ , 4

 − ðR Þ2 CuðNH2 C2 H4 NH2 Þ2þ ð5:2:114Þ þ 4NH3 ðaqÞ: 2

Ammonia present in the solution had previously formed a complex with the metal ion in the ion exchanger, which acts as a solid carrier for the complexing metal ion. Addition of EDA in the solution allows for an exchange between EDA and ammonia, the two ligands in this case. Since complex formation of a ligand with a metal ion is a

highly specific interaction, high selectivities can be obtained. It depends on the coordination valence of the ligands and the solution concentrations (Helfferich, 1962). 5.2.3.1.2 Ionization/dissociation of amino acids Largescale production of amino acids frequently use ion exchange processes. Amino acids are amphoteric molecules that can exist as anions and cations depending upon the pH of the solution. There are three types of amino acids depending on the nature of the side chain R if we represent the amino acid as NH2CHRCOOH: the R-group is not ionizable; the R-group is negatively charged at pH 7; the R-group is positively charged at pH 7. Consider now the first case with a nonionizable R-group (examples are alanine, leucine, glycine, valine, etc.) (see Table 5.2.5). The following forms of dissociation equilibria exist in solution: K1

þ − þ NHþ 3 −CH−COOH , NH3 −CH−COO þ H j j R  R  ðAmþ Þ Am

ð5:2:115aÞ

K2

− þ − NHþ 3 −CH−COO , NH2 −CH−COO þ H j j R  R  Am ðAm− Þ

ð5:2:115bÞ

Here Am± is the zwitterion form of the amino acid, whereas Amþ is the cationic form and Am− is the anionic form of the amino acid.5 From equation (5.2.115a), the dissociation equilibrium constant K1 is

5

The forms should be more appropriately written as HAm± for the zwitterion form, H2Amþ for the cationic form and Am− for the anionic form, with HAm representing the amino acid.

302

Effect of chemical reactions on separation

K1 ¼

C Am w C Hþ w : C Amþ w

þ þ − ½R− ½Hþ  þ NHþ 3 −CH−COOHðaqÞ , ½R ½NH3 −CH−COOH þ H ðaqÞ j j R R

ð5:2:116aÞ

Correspondingly,

ð5:2:119Þ

C Am− w C Hþ w K2 ¼ : C Am w

ð5:2:116bÞ

If we have a cation exchange resin, we should be interested in the concentration of C Amþ w , especially as a fraction of the total amino acid concentration, C tAmw : C tAmw ¼ C Amþ w þ C Am− w þ C Am w :

ð5:2:117Þ

Employing the two dissociation equilibrium relations (5.2.116a,b), we obtain C tAmw

K 2 C Am w K 1 C Amþ w þ : ¼ C Amþ w þ C Hþ w C Hþ w

A repeat application of (5.2.116a) leads to C Amþ w ¼ 

C tAmw 1 þ CKþ1 þ CK21 K 2 H w

Hþ w

C tAmw 1þ

C Hþ w K2

C2

þ KH1 þKw2

ð5:2:118bÞ

i:

To get an idea of which charged form of the amino acid dominates under what pH condition, consider the amino acid valine, where R ¼ (CH3)2–CH, K1 ¼ 10−2.33 and K2 ¼ 10−9.76. Further, let the pH be 1, i.e. the solution is highly acidic. Then C Amþ w ¼ h

C tAmw 1 þ 1010−1 þ 10 −2:33

whereas

C Am− w ¼ h

i¼h −ð2:33þ9:76Þ 10−2

C tAmw 1 þ 1010−9:76 þ 1010 −ð12:09Þ −1

−2

i¼

5.2.3.2 Stationary–mobile phase equilibria in chromatography6 In chromatography, the equilibrium distribution of a species between the mobile phase and a stationary phase is crucial. To separate two species, it is essential that the 0 distribution ratio k i1 for two species i ¼ A and B is significantly different. Note that (from equation (1.4.2))

ð5:2:118aÞ

:

Correspondingly, the concentration of C Am− is obtained as a fraction of C tAmw as C Am− w ¼ h

Further, if there are other cations in the system, there will be additional ion exchange reactions. Illustrations of the detailed behaviors of the equilibrium sorption of a number of amino acids by a cation exchange resin are provided in Saunders et al. (1989) and Dye et al. (1990).

C tAmw 1 1 þ 1011:33 þ 1010:09

½1 þ 10

C tAmw 8:76

i,

þ 1010:09 

0 k i1 ¼

where V1 is the volume of the stationary phase (j ¼ 1) and V2 is the volume of the mobile phase (j ¼ 2). For a mobile liquid phase, the stationary phase may be a solid with bonded liquid ion exchange resin or a liquid phase. The chemical reactions may be acid–base equilibrium, complexation, ion pairing, etc. First, we consider acid–base equilibrium for two weak acids HA and HB distributed between the mobile liquid phase and a stationary solid phase, with or without any bonded liquid phase. Assume 0 0 that their distribution coefficients κA1 and κB1 are almost identical but their dissociation constants KDA, KDB are quite different. Assume further that only the form HA or HB partitions; A− and B− remain in the mobile phase. Using relation (5.2.62), write the effective distribution coefficients for A and B as 00 0 κA1 ¼ κA1

:

Thus, the cationic form Amþ dominates at low pH and is very close to C tAmw . At very high pH, the anionic form Am− dominates. In addition, one can conclude from relation (5.2.116a) that when pH ¼ pK1, C Amþ w ¼ C Am w . Similarly, from (5.2.116b) when pH ¼ pK2, C Am− w ¼ C Am w . Table 5.2.5 provides the isoelectric point of the amino acid, pI, where the concentrations of the positively charged species balance those of the negatively charged amino acids, leading to no charge: C Amþ w ¼ C Am− w . Although the above conclusions are generally useful, there are other considerations. Consider a low pH where the Amþ form dominates and is expected to have successful exchange with a cation exchanger; however, there is a competing ion exchange reaction with the Hþ ion:

mi1 C i1 V 1 V1 ¼ ¼ κi1 , mi2 C i2 V 2 V2

C Hþ ; C Hþ þ K dA

00 0 κB1 ¼ κB1

C Hþ : C Hþ þ K dB ð5:2:120Þ

For species A, there are two limits of the distribution ratio 0 0 00 k A1 for , i.e. κA1 corresponding to the two values of κA1 0 K dA  C Hþ and ðκA1 C Hþ =K dA Þ for K dA  C Hþ since (V1/V2) is fixed for a system. Similarly for species B. It is obvious that if KdA and KdB (therefore pKA and pKB) are substantially different, then choosing a pH at an intermediate value anywhere between pKA and pKB will lead 0 0 to substantially different k A1 and k B1 , even though 0 0 κA1 ffi κB1 . This is vitally important for an effective separation of the peaks in chromatography. (Note: The detector records both the undissociated and the dissociated species of, say, HA as one species and therefore one peak.)

6

To be read along with Section 7.1.5.

5.2

Chemical reactions change separation equilibria

Next, we briefly examine the role of weak acid (or weak base) dissociation imposed on ion exchange equilibrium between the mobile phase and the stationary ion exchange resin (having, say, fixed positive charge, Rþ). The solute of interest is a weak acid HA with a dissociation constant KdA. The mobile phase has a monovalent anion C−, the counterion in the ion exchange resin, which exchanges with A− obtained from the dissociation of HA: þ

−

−

þ

−

−

ðR þ C Þresin þ A , ðR þ A Þresin þ C :

ð5:2:121Þ

We assume that the undissociated acid HA does not partition into the resin. The equilibrium constant for this reaction is K AC ¼

C A − R C C− , C C− R C A−

ð5:2:122Þ

where C A− R and C C− R refer to the ion exchange phase and C C− and C A− are for the aqueous eluent phase. The value of the distribution coefficient for species A between the aqueous phase and the ion exchange resin phase (j ¼ 1) is  A  C A− R K C C C− R C A− 00 κA1 ¼ : ð5:2:123Þ ¼ C C− C A− þ C HA C A− þ C HA 00 Now, KdA ¼ C Hþ C A− =C HA ; use it to rearrange κA1 to obtain  A  K C C C− R K dA 00 κA1 : ð5:2:124Þ ¼ C C− K dA þ C Hþ 00 By choosing the pH properly, κA1 can be varied over a wide 00 limit. Since KdA > KdA, κA1 is 0 small; therefore k A1 is also small. When, however, KdA >> C Hþ , the acid is fully dissociated, leading to a much higher 0 0 00 κA1 and k A1 . Note that the value of k A1 for a stationary liquid or solid (uncharged) phase under this condition is the lowest since A− cannot partition into it. Complexing agents can be added to the mobile phase to alter the equilibria between the stationary and mobile phases. This is especially useful for separating metal ions using ion exchange resins. For example, suppose the mobile phase has ligands L− and that a metal ion can complex with it as Mnþ þ nL− , MLn or as

Mnþ þ mL− , MLðmn−mÞ−

ð5:2:125Þ

in general. Now different metals will have different extents of binding with the ligand L. Therefore the partitioning between the mobile phase and the stationary phase will be different for different metals (Karger et al., 1973). On the other hand, when various ligands have to be separated, one can introduce metal ions in the eluent for complexation, as in (5.2.125). Chelating agents like 8-hydroxyquinoline are examples of ligands used for these purposes. Many ionic solutes are poorly retained by nonpolar stationary phases. However, if a counterion is introduced into an eluent such that it can form a neutral ion pair with the solute, then the ion pair can partition into the nonpolar stationary

303 phase and its retention will be altered. For example, tetramethylammonium chloride ((CH3)4NþCl−) can be used for pairing with simple anions; sodium dodecyl sulfate (CH3(CH2)11OSO−3 Naþ) can be utilized for simple cations. Still another method is to use a solute–micelle interaction: S þ M , SM,

ð5:2:126Þ

where S is a solute and M represents a micelle (see Figure 4.1.18), which is a spontaneous aggregate of surfactants. The solute distributes itself between the eluent and the micelle; the latter has a very limited affinity for the stationary phase. Thus, effectively, the retention of the solute is decreased. Different solute–micelle distribution equilibria will then help in separating different solutes. The solutes of interest are ionic or ionizable. The following aspect of some common mobile phases used in liquid chromatography is important. Polar mobile phases are rarely aqueous; generally they are partially aqueous, e.g. methanol–water or acetonitrile–water, etc. For water, neutral pH is defined as C Hþ ¼ C OH− ¼ 10−7 gmol/liter at 25  C, but what is the neutral pH in a water– methanol system? For water, the ion product or the autopyrolysis constant Kwater is (using concentrations instead of activities) Kw ¼ (C Hþ ) (C OH− ) ¼ 10−14. Correspondingly, for methanol, which produces Hþ and CH3O−, the autopyrolysis constant is Kmethanol ¼ ðC Hþ ÞðC CH3 O− Þ ¼ 10−16.7 (see Bates (1964)). Thus in methanol–water mixtures, with the autopyrolysis constant varying between 14 and 16.7, neutral pH may be said to vary between 7 and 8.35, depending on the mixture composition. For water– acetonitrile mixtures, the range will be even broader since Kacetonitrile is 26.5. The subject is quite complex; refer to Bates (1964) to develop a more correct picture of the state of the protons and anions in such mixed polar solvents. 5.2.4

Crystallization and precipitation equilibrium

We will deliberate first on the role of pH, dissociation, etc., on various crystallization/precipitation equilibria. Consider aqueous solutions of amino acids. We have seen in Section 5.2.3.1.2 that amphoteric molecules of amino acids can exist in three forms in solution: a cationic form Amþ at low pH; an anionic form Am− at very high pH; and at intermediate pH, the zwitterionic form, Am±, is important. The relative ratio of Amþ or Am− to Am± depends on the pH with respect to pK1 or pK2. Only the Am± form exists at the pI of the amino acid. On the other hand, it is the neutral form Am± which forms the crystal (j ¼ s): 

κiw  − þ − NHþ 3 RCHCOO Þs , NH3 RCHCOO Þw ,

ð5:2:127Þ

where κiw is the equilibrium solubility parameter (¼ − − þ (NHþ 3 RCHCOO )w/(NH3 RCHCOO )s) which depends on the solute, temperature and solute concentration.

304

Effect of chemical reactions on separation

(a)

(b)

0.4

25

0.2

Solubility, M

g Ile

0.6

total Ile in solution

30

20

Solubility

(

100g solution

)

1.0

35

15

0.06 0.04

10

0.02

5 0 0

0.1

0.01

2

4

6

pH

0

5

10

15

HCI concentration

(

20

25 g HCI

30

100g solution

)

0.05

2

3

4

pH

5

6

7

8

Figure 5.2.11. (a) The solubility of L-isoleucine as a function of pH and HCl concentration at 25 ºC (Zumstein and Rousseau, 1989). Reprinted, with permission, from Ind. Eng. Chem. Res., 28, (1989), 1226, Figure 4. Copyright (1989) American Chemical Society. (b) Solubility–pH profiles of amino penicillin at 37  C. The points are experimental values. The solid curves were generated from equation (5.2.130) and other parameters. Key: Δ, cyclacillin anhydrate; ◯, ampicillin anhydrate; ●, ampicillin trihydrate; □, amoxicillin trihydrate; and , epicillin anhydrate (Tsuji et al., 1978). Reprinted, with permission, from A. Tsuji, E. Nakashima, S. Hamano, T. Yamana, J Pharmaceut. Sci., 67(8), 1059 (1978), © 1978, John Wiley & Sons.

For amino acids there is a band of about 2–3 pH units around the pI where the net charge is zero. This is identified as the isoelectric band; a pH value beyond this band on either side increases the amino acid solubility. For example, from equation (5.2.118a), we find that as C Hþ w increases, C Amþ w increases; therefore the amino acid solubility increases since C Amþ w increases. Zumstein and Rousseau (1989) have illustrated that, for the amino acid L-isoleucine, the C Am w species dominates above pH ¼ 2 leading to the lowest solubility. As the pH decreases, the C Amþ w species forms more and more and the amino acid solubility increases sharply. At pH ¼ 1, it reaches a maximum, then it decreases sharply at lower pH (Figure 5.2.11a). As the pH is decreased below 2 by the addition of HCl acid, more and more of the acid form of the amino acid is present in solution. However, there is a solubility limit of this form of amino acid which is subject to an equilibrium with the hydrochloric acid salt (Cl− · NþH3RCOOH · H2O)s in the solid phase via ðCl− Nþ H3 RCOOH H2 OÞs , Nþ H3 RCOOHðwÞ þ Cl− ðwÞ þ H2 O:

ð5:2:128Þ

There is a solubility product Ksp of the two ions in the aqueous solution: K sp ¼ C Nþ H3 RCOOH, w  C Cl− , w :

ð5:2:129Þ

As HCl is added, initially C Amþ w will increase, but so does C Cl− , w . Correspondingly, at some point of acid addition, the

increase in C Cl− , w will reduce C Amþ w when their product has already reached Ksp. This decreases C Amþ w and therefore the total amino acid solubility (¼ C Amþ w þ C Am w ) from the ionization equilibrium (5.2.116a). One therefore observes a maximum and then a signifcant drop off in the total solubility of the amino acid L-isoleucine on increasing the amount of added HCl and a corresponding decrease in pH (Figure 5.2.11(a)). If the complication of the salt solubility limit being exceeded is not present, then the total concentration of the amino acid in solution in terms of the concentration of the zwitterionic species C Amþ , w may be obtained from equations (5.2.116a, b) as C tAm w ¼ C Am w þ C Amþ w þ C Am− w ¼ C Am w þ C Am w 2

C Hþ K2 þ C Am w C Hþ K1

3 þ C K 2 5: ¼ C Am w 41 þ H þ C Hþ K1

ð5:2:130Þ

While the crystals are formed from the uncharged Am± species, the solubility of the amino acid in the aqueous solution is obtained from (5.2.130). This solubility decreases at first at a low pH as the pH is increased, then it has a minimum around the pI band since both C Amþ w and C Am− w are zero at pI. Then, as the pH is increased further beyond pK2, the total amino acid solubility increases again.

5.2

Chemical reactions change separation equilibria

This type of behavior has been observed also for proteins. The solubility of a protein built out of amino acids is minimum when pH ¼ pI, the isoelectric point. The minimum is a point not a band. As the pH moves away from the isoelectric point, the protein solubility increases. The presence of a net charge on the protein molecule increases its solubilization due to its interaction with dipolar water molecules (Hþ as well as OH−). Kirwan and Orella (1993) have illustrated broadly a similar solubility behavior for other biological molecules of smaller molecular weight which produce zwitterions. Observe the solubility variation with pH of amphiprotic solutes such as the β-lactam antibiotics, ampicillin, amoxicillin, etc., as determined by Tsuji et al. (1978) (Figure 5.2.11(b)). If the species is susceptible to hydrolysis at high pH, then pH effects are observed only at lower pH in the range of pH < pK1  pI. Optical isomers in solution can be separated by a number of techniques, e.g. enzymatic methods, mechanical resolution, etc. In mechanical resolution of racemic mixtures of optical isomers, a supersaturated solution of a racemic mixture is seeded with the pure crystal of one of the isomers. This crystal grows and one of the isomers is separated from the solution. However, the solution remains supersaturated in the other isomer, which tends to precipitate, resulting in poor separation of the isomers. If the racemic mixture consists of ionizable species, the addition of acid or base to the solution has been found to stabilize the solution (Asai and Ikegami, 1982). This has been found to be true for ionizable amino acids, e.g. the L and D forms of glutamic acid, the L and D forms of DOPA (3,4-dihydroxy-β-phenylalanine), etc. (Asai, 1985). Here both the free and the salt forms of the acid are racemic mixtures. But the salt forms are much more soluble. The reaction that takes place, for example, in the presence of NaOH is as follows: L − A þ D − S , L − S þ D − A,

ð5:2:131Þ

where A refers to the free form of glutamic acid, S refers to the sodium salt of glutamic acid and L and D refer to the two different isomers. When the D–A is seeded, the L–A form of the acid (not seeded) will be pushed to the L–S form since D–A is disappearing by crystallization on the growing crystal. Remember, the L–S form, being more soluble, does not precipitate. An alternative strategy relies on differing crystal sizes of the two reaction products, which therefore can be successfully separated. For example, in the resolution of DL-DOPA and its hemihydrochloride DL-DOPA · 0.5 HCl, the following reaction occurs:

of the amino acid are much greater than those of the free form. Thus, the mixture of the crystals of L-DOPA and DDOPA · 0.5 HCL can be easily separated by sieving (Asai, 1985). A comprehensive account of the effect of chemical reactions on crystallization in enantiomeric systems has been provided by Jacques et al. (1981). In the resolution of a racemic mixture of acid AH (particular forms are identified as D-AH and L-AH) by an optically active base B, the following dissociations/ionizations, acid–base reactions (leading to the salt, e.g. AHB (undissociated form)) and partitioning equilibria are relevant: Kd

AH ! A− þ Hþ ;

ð5:2:132Þ

Not only are the solubilities of the salt form much greater than the free form, but also the crystal sizes of the salt form

Kd¼

C A− C Hþ : C AH

ð5:2:133Þ

For both D- and L- forms, K dAH ¼

C DA− C Hþ ; C DAH

K dAH ¼

K dB

B þ Hþ ! BHþ ;

K dB ¼

K

K¼

AH þ B ! AHB;

C LA− C Hþ ; C LAH

ð5:2:134Þ

C BHþ ; C B C Hþ

ð5:2:135Þ

C AHB , C AH C B

ð5:2:136Þ

for both D-AHB (KD) and L-AHB (KL). Also, κiw

ðAHBÞsolid ! ðAHBÞsolution :

ð5:2:137Þ

Here, Kd represents the dissociation constant of the enantiomeric acid, L- or D-form; KdB represents the dissociation constant of the optically active base B, and K represents the acid–base reaction equilibrium constant, KD for the D-form and KL for the L-form. Further, κiw represents the equilibrium solubility parameter of the salt i. However, the process is useful because κiw varies with the enantiomeric salt species i, i.e. whether it is D-AHB or L-AHB. We identify this via κDw and κLw. To start with, we have a certain total concentration of the racemic mixture C tAH : it consists of C tDAH (¼ 0.5 C tAH ) and C tLAH (¼ 0.5 C tAH ). However, this total initial concentration after the process is over will consist of, for example, C tDAH ¼ C DAH þ C DA− þ C DAHB þ C Dp ,

ð5:2:138aÞ

where C Dp is the concentration of the precipitate or crystals of the D-form of AHB (CLp for L-form). Correspondingly, C tLAH ¼ C LAH þ C LA− þ C LAHB þ C Lp : From (5.2.134), C LAH þ C LA−

L  DOPA 0:5 HCl þ D  DOPA , L  DOPA þ D  DOPA 0:5 HCl:

305

0

1

K dAH A ; ¼ C LAH @1 þ C Hþ

C LAHB ffi κLw ,

C tLAH − κLw − C Lp

ð5:2:138bÞ

C DAHB ffi κDw ; 0 1 K dAH A: ¼ C LAH @1 þ C Hþ

ð5:2:139Þ

306

Effect of chemical reactions on separation

Correspondingly,

C tDAH − κDw − C Dp ¼ C DAH 1 þ



K dAH : C Hþ

ð5:2:140Þ

If the total initial concentration of the base is C tB , then C tB − κLw − κDw − C Dp − C Lp ¼ C B ð1 þ K dB C Hþ Þ: ð5:2:141Þ Now, from (5.2.136), (5.2.140) and (5.2.141), we get, for the L-form,  t   C LAH − κLw − C Lp C tB − κLw − κDw − C Dp − C Lp ¼

 κLw K dAH 1þ þ K dB C þ ð5:2:142Þ H þ K dAH K dB : KL C Hþ Correspondingly, for the D-form, we obtain   t  C DAH − κDw − C Dp C tB − κLw − κDw − C Dp − C Lp

 κDw K dAH ð5:2:143Þ ¼ 1þ þ K dB C þ H þ K dAH K dB : KD C Hþ Knowing C tDAH ¼ C tLAH ¼ 0.5 C tAH , we divide the expression for the L-form by that for the D-form to obtain  t 

 C − κLw − C Lp κLw K D  tLAH ¼ , κDw K L C DAH − κDw − C Dp

which may be rearranged to give the following:  

  κLw K D κLw K D − 1 − C tDAH −1 κDw K L κDw K L

 KD −1 : ð5:2:144Þ þκLw KL

C Lp − C Dp ¼ C Dp



Here we have arbitrarily assumed that the D-form of the salt is more soluble so that CLp − CDp > 0. The result above is pH independent, even though the individual value of CLp or CDp depends on pH. Consult Jacques et al. (1981, sect. 5.1) for quantitative estimates of various constants for an enantiomeric system. Knowing κLw , κDw , K D and K L for a system, and the value of C tDAH , one can find out the difference in the amount of crystals of the two enantiomers.

5.2.5

Surface adsorption equilibrium

We have considered the equilibrium distribution of a nonelectrolytic surfactant solute between the bulk liquid phase and the interfacial phase in a gas–liquid system via relation (3.3.106). We have thereby illustrated the application of the Gibbs adsorption isotherm (3.3.40a) to a single nonionic surface-active solute. Chemical reactions can influence such adsorption isotherms in a number of ways. If the surface-active solutes are ionic, the adsorption equilibria are affected. In other cases, the solute to be removed (the colligend) is not surface active but it reacts with or is

adsorbed on a solute (collector) which is surface active. We treat first the ionic surface-active solute distribution between a bulk liquid phase j ¼ l and the interfacial phase j ¼ σ for a gas–liquid system. 5.2.5.1

Ionic surface-active solute R± X− in solution

An ionic surface-active solute RþX− in a dilute solution may or may not have an excess of another electrolyte present simultaneously in the bulk solution. Further, this electrolyte may or may not have a common ion (RþY−). Select first an ionic surface-active solute RþX− without any other electrolyte. From the Gibbs isotherm (3.3.40a), dγ12 jT ¼ −Γ ERþ σ RTdlnC Rþ 1 jT − Γ EX− σ RTdlnC X− 1 jT ð5:2:145aÞ by considering the Rþ and X− ions as separate species. If the solute RþX− is completely dissociated in the bulk solution (j ¼ 1), then C Rþ 1 ¼ C X− 1 ¼ C RX, 1 ,

ð5:2:145bÞ

where CRX,1 is the bulk concentration of the surfactant solute to start with. The equilibrium relation is now simplified to (since Γ ERþ σ ¼ Γ EX− σ ¼ Γ RX;σ by (3.3.42b)) dγ12 jT ¼ −2Γ ERX, σ RTdlnC RX, 1 ;   1 dγ12 jT : Γ ERX, σ ¼ − 2RT dlnC RX, 1

ð5:2:146Þ

If γ12 ¼ a1 − b1C RX, 1 , then Γ ERX, σ ¼

b1 C RX, 1 : 2RT

ð5:2:147Þ

Thus, ionization of the solute has reduced its surface excess by a factor of two (compare relation (3.3.106)) from that for a nonionic solute. When there is also a large excess of an electrolyte with a common ion (RþY−) present in the liquid, the Gibbs isotherm becomes dγ12 jT ¼ −Γ ERþ σ dμRþ 1 − Γ EX− σ dμX− 1 − Γ EY− σ dμY− 1 : ð5:2:148aÞ Suppose the concentration of the surface active solute RþX− is changed in the bulk solution; the concentration of the anion Y− is essentially unchanged, i.e. dμY− l ¼ 0. For a dilute solution, we can then write

 dγ12 jT ¼ −Γ ERþ σ dlnC Rþ 1 − Γ EX− σ dlnC X− 1 RT: ð5:2:148bÞ But, due to any addition of RþX−, dC Rþ 1 ¼ dC X− 1 . Further, C Rþ 1 >> C X− 1 due to a large excess of RþY−. Then, assuming Γ ERþ σ and Γ EX− σ to be of the same order of magnitude, Γ ERþ σ which leads to

dC Rþ 1 dC X− 1 , > KB, which is true for, say, copper (A) against silver

Now, if CC2 is 0, KB (pK1 þ2), m μm HA;eff ffi μA since essentially almost all acid molecules are ionized at a high pH. One-half of the acid molecules are ionized when pH ¼ pK1. A not-as-illustrative expression equivalent to (6.3.30a) is m μm HA;eff ¼ μA

ðK d1 =C Hþ Þ : 1 þ ðK d1 =C Hþ Þ

ð6:3:30bÞ

The effective mobility of a weak base is considered in Problem 6.3.4. The ionic mobility of a few smaller species and their pK values (primarily pK1) are provided in Table 6.3.1. For a protein, the value of the solution pH in relation to its pI is important. Since the net charge (Zi) on a protein is positive at pH < pI (see Figure 4.2.5(c)), the direction of a protein’s ionic mobility will correspond to that of a cation. When pH > pI, its mobility direction will be reversed. For compounds which are quite similar (e.g. enantiomers), inclusion complexes formed with cyclodextrin (CD) molecules can lead to selective separation. Cyclodextrin molecules are uncharged and are therefore moved by the electroosmotic flow at a velocity hvEOF, z i. Any racemic mixture which exists as ions in the solution may be

6.3

Bulk flow parallel to force direction

383

Table 6.3.1. Ionic mobilities of selected ionsa

Ions, i Ammoniumb Hydrogenb Lithiumb Octadecyl tributylammoniumb Potassiumb Sodiumb Triethylammoniumb TRIS+c Acetate Chloride Hydroxyl Phosphate

ACESd BESe HEPESf MESg

mA−

5 μm i  10 cm2/V-s

pK

76.2 362 40 17.2 76 53 35.2 29.5 –42.4 –79.1 –206 –34.1 –58.3 –71.5 –31.3 –24 –21.8 –26.8

– – – – – – – 8.3 4.74 – – 2.15 7.22 11.50 6.84 7.16 7.51 6.13

a

Data from Janini and Issaq (1993), Karger and Foret (1993) and Fu and Lucy (1998). b Cations; μm i chosen positive. c tris (hydroxylmethyl) aminomethane. d N-2-acetamide-2-aminoethanesulfonic acid. e N,N-bis (2-hydroxylethyl)-2-aminoethanesulfonic acid. f N-2-hydroxyethylpiperazine-N-2-ethanesulfonic acid. g 2-(N-morpholino) ethanesulfonic acid.

separated if one of the compounds preferentially forms an inclusion complex with the CD (see Figure 4.1.26) and has its charge sticking out of the hydrophobic CD cavity. The compound forming the inclusion complex will now move at a velocity different from that of the compound that does not. See Fanali (1993) for details of CD based facilitation of separation in CE. It has been already pointed out in expression (6.3.8g) that the diffuse double layer around an ion/charged particle will reduce the effective charge on an ion to Zi,eff from the true charge Zi. The two are approximately related as follows (Newman, 1973; Wieme, 1975): ðZ i,eff =Z i Þ ¼

ð1=hÞ , ð1=hÞ þ r i

ð6:3:31aÞ

where (1/h) is a characteristic thickness of the double layer (equivalent to the Debye length λ in equation (6.1.22)) around the ion/charged particle of radius ri. From Debye–Hückel theory, h is proportional to the square root of the solution ionic strength. Thus, for larger charged particles (larger ri), at high ionic strengths, Zi,eff will be related to Zi by ðZ i,eff =Z i Þ ffi ð1=hÞ=r i :

ð6:3:31bÞ

Now, Zi, for a large ion/charged particle, is proportional to the surface area of the particle/ion, 4πr 2i , if we assume it to

m HA,eff mA− 2

0

pH =pK1

pH

Figure 6.3.6. Dependence of the effective mobility μ of a weak acid on pH. (After Karger and Foret (1993).)

be spherical. Therefore its effective mobility, μm i,eff , (Giddings, 1991), μm i,eff ¼ /

Qi,eff Z i,eff F Z i F ð1=hÞ ¼ ¼ ~ 6πμr 2i ~ 6πμr i ~ N N f di =N 4πr 2i F ð1=hÞ 6¼ f ðr i Þ, ~ 6πμr 2i N

ð6:3:32Þ

is no longer a function of the size of the particle. For colloidal particles, ionic moblility is therefore not influenced much by the size of the particle in free solution. Consequently, capillary electrophoresis is no longer useful. The capillary has to have a porous gel/matrix which will allow smaller colloids to pass through with less resistance in the applied electrical field. The profile development analysis of an injected sample carried out earlier led to the concentration profile (6.3.15) based on a number of assumptions, which included a constant Uiz and therefore a constant μm i . Constancy of μm along the capillary length ensures that the peak disperi sion will not have any contribution from variations in μm i . Further, constancy of vEOF, z requires constancy of the zeta potential of the silica surface, ζ, which, in turn, requires constancy of pH. Therefore, the pH has to be constant and uniform along the capillary, especially for acidic/basic species, suggesting that the solution has to be buffered. The buffer solution usually employed in CE/CZE is 0.01– 0.05 M phosphate (0.2 M solution of monobasic sodium phosphate and 0.2 M solution of dibasic sodium phosphate in different ratios). For solutes that are bases, the same buffer is recommended at a low pH. For acidic solutes, a borate buffer is recommended at a high pH. Sometimes a mixed phosphate–borate buffer may be used. For the analysis of DNA samples, the capillary is filled with a gel, generally of crosslinked polyacrylamide. Proteins and oligonucleotides have also been separated in such capillaries. The gel provides a sieving matrix, having

384

Open separators: bulk flow parallel to force and CSTSs

pores through which smaller biomacromolecules pass much faster. The gel also eliminates electroosmotic flow and substantially decreases the solute diffusion coefficient; thus band broadening is reduced. In fact, for DNA sample analysis, CE employing a gel has become highly successful. DNA sequencing is an area where gel-filled CE has been critically important to develop the speed and selectivity needed to sequence the entire human genome (Zubritsky, 2002). To solve the problem of gel stability, fresh polymer solution was introduced for each sample analysis (Karger and Foret, 1993) and polymerization carried out. After each run, the polymer solution may be blown out and the capillary reloaded with fresh solution. A run may last for a maximum of 80 minutes. The sample injection volume in conventional CE/CZE is around 1–10 nanoliter. The sample amount obtained at the exit in one run is often insufficient for spectral measurement/structure determination. If a semipreparative scale of operation is employed, then the sample amount at the exit would be significant. Since the sample-loading capacity is likely to be proportional to the capillary/column crosssectional area, use of larger capillaries up to 180 μm diameter (including those of rectangular cross section (Tsuda, 1993)), sometimes packed with octadecyl silica particles (Chen et al. (2001) employed capillaries with diameters

550 μm containing 1.5 μm octadecyl silica particles), will allow larger sample injection volumes up to 1 microliter. Alternatively, a bundle of multiple capillaries (up to five) may be introduced into one column to increase the sample-loading capacity (Tsuda, 1993). This last concept has been extended to what is known as capillary array electrophoresis (CAE) (Huang et al., 1992; Kheterpal and Mathies, 1999), wherein there may be as many as 96 capillaries in parallel in instruments currently available (Zubritsky, 2002). This increases the throughput drastically. Such devices were the workhorse of the human genome analysis. Scaling up to 1000 capillaries has also been reported (Kheterpal and Mathies, 1999). Capillary array electrophoresis enables easy and parallel loading of multiple samples as well as rapid and parallel separation and detection using appropriate detection techniques. An uncharged compound, if present in the injected sample, will move in CE with the bulk liquid at the electroosmotic velocity vEOF, z toward the cathode. Different uncharged compounds would have the same velocity, namely vEOF, z . On the other hand, if a charged phase could be created into which different uncharged compounds would partition in a varying fashion, the situation is changed. This is achieved in CE by having an appropriate solution of an ionic surfactant, for example sodium dodecyl sulfate (SDS), as the medium in CE. If the SDS concentration is higher than the critical micelle concentration (CMC), then the excess surfactants will form spherical micelles (see Section 4.1.8) with negatively charged headgroups ðSO2 4 Þ. These micelles will tend

to migrate toward the positive electrode in the electric field. However, it is observed that, at pH > 5, the value of vEOF,z toward the cathode is higher than the electrophoretic migration velocity of the negatively charged micelles toward the anode, vEPMC, z ; the net effective direction of movement of the micelles is therefore toward the cathode: vMC, z ¼ vEOF, z  vEPMC, z ,

ð6:3:33Þ

where vMC, z is the net velocity of the negatively charged micelles toward the cathode. Uncharged compounds in such an environment will partition into the hydrophobic core of the charged micelles. Some will partition much more, the more polar compounds much less, as we have seen in the results illustrated in Table 4.1.9. Therefore, the net velocity of each such compound toward the cathode will be quite different. This is the basis of micellar electrokinetic chromatography (MEKC), first developed by Terabe et al. (1984). However, this technique will be covered in Chapter 8, since the partitioning of an uncharged compound from a mobile phase into a micelle, both of which have a bulk motion perpendicular to the force direction, causing the compound partitioning into the micelle, is more appropriate for Section 8.2.2.1. In CE using silica capillaries having a charged surface, oppositely charged solutes may undergo electrostatic adsorption on the capillary surface. Biopolymers are especially susceptible to adsorption. The capillary surface is therefore often coated with a neutral hydrophilic polymer (e.g. methylcellulose) in a thin layer. A thick neutral coating can eliminate the electroosmotic flow. For these and related issues, the reader should consult a number of relevant chapters in Guzman (1993). Additional details on the experimentally observed effects of wall adsorption of polycations in particular (Towns and Regnier, 1992), and the theoretically predicted consequences of adsorption in general (Schure and Lenhoff, 1993), provide a useful perspective. 6.3.1.3

Centrifugal elutriation

In Section 6.3.1.1, we studied elutriation based separation of a mixture of particles in the gravitational field in the presence of a bulk flow of the liquid in the vertical direction. To separate a mixture of different particles (different sizes and/ or densities), the liquid flow in the vertical direction occurred in a vessel with changing flow cross-sectional area. Different sized particles (or particles of different densities) found equilibrium positions at different heights where the liquid velocity changing with height equaled the critical terminal velocity of the particle. Gravitational force is not strong enough to fractionate microscopic particles, such as bacterial cells, etc., in the size range of 1–50 μm. However, as we have seen earlier, in Section 4.2.1.2, centrifugal force in the radially outward direction can. In centrifugal elutriation, a mixture of particles is introduced into a rotating chamber through which the liquid

6.3

Bulk flow parallel to force direction

(a) Bulk liquid flow direction

385

(b)

ω

ω Centrifugal force

Centrifugal force direction Liquid flow

2RL

2R

q

Liquid flow

r rL

L

Axis of centrifuge rotation

Bulk flow based drag force direction

Figure 6.3.7. (a) Schematic of a truncated cone as an elutriation chamber, with bulk flow toward the axis of rotation of the centrifuge. (b) Mixture of cells separated according to their size in centrifugal elutriation. (After Figdor et al. (1998).)

flows in a direction opposite to that of the centrifugal force. The rotating chamber is essentially a conical tube with its base near the center of rotation and its apex far away (Figure 6.3.7(a)). If the liquid in which the cells/particles are suspended is made to flow in this conical tube in a direction from the apex to the base, then the liquid velocity increases as the distance r from the center of rotation increases. Correspondingly, at steady state, the drag force on a cell/particle (assumed Stokesian) of radius rp moving with the liquid velocity vr at radial location r is 6πμr p vr ; the magnitude of this force increases as r increases. This force is, however, opposed by the centrifugal
radially outward  force on the particle, mp rω2 1  ðρt =ρp Þ , which increases linearly with r. Under appropriate conditions, there will be a radial location r where the two forces will balance each other and the cell/particle of specific size/density will concentrate at the radial location: 
6πμr p vr ¼ mp rω2 1  ðρt =ρp Þ : ð6:3:34aÞ The inwardly radial medium velocity vr is related to the medium volume flow rate Qf and the local cross-sectional area A(r), which depends on the radial location r. Replacing the cell mass mp by ð4=3Þ πr 3p ρp , we get r 2p ¼

Qf 9 μ : 2 ω2 ðρp  ρt Þ rAðrÞ

ð6:3:34bÞ

This equation provides a relation between the radial location and the dimension rp of the cell population (assuming they are of the same density) which accumulates at r. (Generally, ðρp  ρt Þ in equation (6.3.34b) is very small; therefore this technique is not used to separate particles of the same rp but differing only slightly in ρp .) The exact relation will depend on how A(r) varies with r. Consider

the conical tube shown in Figure 6.3.7(a), where RL is the largest radius of the conical tube, L is the distance of this radius from the apex of the conical tube, rL is the radial distance of the apex of the conical tube from the center of rotation and r is the radial distance from the center of rotation of the cross-sectional area of radius R under consideration. The expression for A(r) is then
2 πR2 ¼ π tan θ  ðr L  rÞ ¼ πðRL =LÞ2 ðr L  rÞ2 : ð6:3:34cÞ

Correspondingly, equation (6.3.34b) may be reduced to the following expression for rp: 3 rp ¼ ωðr L  rÞðRL =LÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μQf : 2πðρp  ρt Þr

ð6:3:34dÞ

From this equation, we can infer that the smaller the particle/ cell size rp, the closer to the center of rotation the equilibrium location of the particles, as illustrated schematically in Figure 6.3.7(b). The withdrawal/separation of each size group of cell population may be accomplished by either a stepwise increasing of the flow velocity enough or a stepwise decreasing of the centrifugal rotational speed, so that the smallest particles in the conical rotating tube come out with the bulk liquid flow. As a practical design, there is a counter-taper at this exit section (closest to the center of rotation), as shown by the dashed-dotted line in Figure 6.3.7(b). This concept of centrifugal elutriation was developed first by Lindahl (1948, 1956). A recent review of the extensive literature in the biological field on various types of cell separation is available in Figdor et al. (1998). Conventionally, the rotating separation chamber volume useful for separation is around 3–4 milliliter. Larger separation chambers with volumes up to 40 milliliter have been

386

Open separators: bulk flow parallel to force and CSTSs

Feed Feed

Particles

Particles to be filtered Vessel wall

Filter cloth

Membrane Vessel wall

Packed bed of particles/ fibers

Filtrate

Filtrate Principle of depth filtration using a packed bed of particles / fibers

Principle of surface filtration using a membrane or filter cloth

Figure 6.3.8. Principles of surface filtration and depth filtration for removing particles from a fluid.

Small particles in air or water are conventionally removed by passing the air/gas/water through a filter. Particles in the size range of 0.02 to 10 μm are removed by microfiltration as well as granular filtration. The latter is also called depth filtration. Microfiltration is a general term which describes filtration processes used to remove micron and submicron particles from air/gas/water/solvent. There are two basic mechanisms: surface filtration and depth filtration (Figure 6.3.8). In surface filtration, the fluid passes through the pores of a relatively thin membrane/filter/cloth, whereas the particles are retained on top of the membrane/filter/cloth whose pores/openings are smaller than the particles. This filtration technique is covered in Section 6.3.3.1. In depth filtration, taking place in a granular/porous/ fibrous medium having a considerably larger thickness in the flow direction, the interstitial openings are usually larger than the particles to be removed. The particles are carried into the interior of the filter by the flowing fluid; the particles are deposited on the surface or collectors of the filter medium via a number of different mechanisms. For aerosols20 in a gas, the mechanisms are: inertial impaction, interception, gravitational settling, electrostatic

deposition and Brownian diffusion. Of these, only inertial impaction involves bulk flow of the gas parallel to the direction of the force on the particle and will be considered here. The rest of the mechanisms are mentioned in Section 7.2.2 since bulk gas flow will generally be perpendicular to the forces involved in the other mechanisms. The same is true for the capture of hydrosols21 from an aqueous solution in a depth filter/granular filter, except additional forces are involved. It is useful to illustrate briefly how the mechanism of inertial impaction leads to the capture of the particles from a gas stream onto filter elements. Figure 6.3.9A illustrates the streamlines of gas flow in the depth filter around a cylindrical fiber. Far away from the fiber, the gas velocity is uniform: v∞. For small particles, the velocity is equal to that of the fluid, and the particle flow path follows the streamline. However, near the fiber, the fluid streamline is changed as the fluid goes around the fiber. Due to the inertia of the particle, especially those of diameter larger than 1μm, the particle trajectory does not coincide with the fluid streamline near the fiber. The particle continues on a path (dashed line in Figure 6.3.9A) which takes it straight to the fiber, and a collision takes place. If the adhesion forces between the particle and the fiber are strong enough for the particle to remain stuck to the fiber surface, the particle is captured. This situation describes the fate of those particles present in an envelope bounded by distant streamlines called the limiting streamlines. The dimension of this envelope is of the order of the fiber diameter and has a width of 2b; the gas flow streamline at a distance b from the centerline is the limiting streamline, and the particle trajectory that

20

21

developed. The highest rotational speed used is 5000 rpm. Each sample fraction that is obtained has a relatively large volume. The volume fraction sizes may vary from 25 to 150 milliliter (Figdor et al., 1998). Pumps of various types are used to generate the flow, whose rate may be around 15–18 milliliter/min. 6.3.1.4 Inertial deposition of particles on a filter/collector in depth filtration

A dispersion of tiny particles of solid or liquid in a gas.

A dispersion of tiny solid particles or liquid droplets in water.

6.3

Bulk flow parallel to force direction

Particle rp

387 For spherical particles of size r p , density ρp and mass mp , if we nondimensionalize equation (6.3.36) using the following nondimensional variables:

y rp

Gas streamline vz = v ∞

2b

rf

z

Gas streamline Solid fiber Limiting particle trajectory Figure 6.3.9A. Gas flow streamlines around a solid fiber of radius rf in a filter bed and the trajectory of a particle of radius rp due to inertial force near the fiber.

coincides with this streamline far away from the fiber is called the limiting trajectory. Far away from the fiber, particles present in the streamlines at distance greater than b from the centerline are not captured. As the streamline changes near the cylindrical fiber in the filter, the fluid accelerates since both axial and normal fluid velocities change rapidly. Heavier aerosol particles in air, larger than 1 μm in size, are unable to follow the accelerating gas; the effect increases as the free stream velocity v∞ and the particle mass increases. In the absence of any external forces, equation (6.2.45) may be written as ¼ mp F iner p

dU p ¼ F drag p : dt

ð6:3:35Þ

Depending on the Reynolds number, Re ¼ 2r p v∞ =ν (based on a fixed particle), the drag force may or may not be described by Stokes’ law. When Stokes’ law (3.1.63) is valid (Re r i ¼ C ∘il

dN sz V s ,

ri

ð6:3:146aÞ where ðdN sz ÞV s is the contribution of the pores of radius rp to rp þ drp to the solvent volume flux:   εm r 2p f ðr p Þdr p ΔP ðdN sz ÞV s ¼ : ð6:3:146bÞ 8μ τ m δm

macrosolute through a microporous ultrafilter depends solely on the characteristics of the pore size distribution of the membrane and the macrosolute dimension. If the membrane has a broad pore size distribution, macromolecules having a wide range of molecular weights and sizes will be able to pass through the membrane. Such a membrane is said to have a diffuse cut off. On the other hand, if the membrane has a narrow pore size distribution, the membrane is assumed to have a sharp cut off; the membrane is such that the size difference between the macrosolute which is completely retained and the macrosolute which passes through with very little retention is quite small. The macrosolute retention behaviors of these two types of membranes are illustrated in the semilog plot of Figure 6.3.27(b). It is now useful to explore the factors which control the selectivity of the UF membrane for one macrosolute (i ¼ 1) of solvated radius r1 over another macrosolute (i ¼ 2) of solvated radius r2 (>r1). If we define the selectivity/separation factor of the membrane for species 1 over 2 via α12 ¼

The total solvent flux (volume flux) from (3.4.86) is εm r 2p

N sz V s ¼

8μ



 ΔP : τ m δm

Since the macrosolute concentration, Cip, in the filtrate (permeate) is given by N iz ¼ C ip N sz V s ,

ð6:3:146cÞ

From equation (6.3.146d) we obtain, in the absence of concentration polarization,

C ip N iz ¼ ¼ ∘ C 0il C il N sz V s

¼

r max ð ri

6 C 0il 4

r 2p f ðr p Þdr p ¼

r 2p

13 εm r 2p f ðr p Þ ΔP A7 dr p @ 5 τ m δm 8μ 0 1 ε r2 0 m p @ ΔP A C il 8μ τ m δm 0

r max ð ri

r max ð ri

r max ð

r min

Rtrue ¼

r min

ð6:3:146dÞ

r1

r 2p f ðr p Þdr p r 2p

:

ð6:3:147cÞ

:

ð6:3:147dÞ

Similarly,

C 2p C 2p ¼ ¼ C 2f C 02l

α12 ffi

r max ð r1

r max ð r2

r 2p f ðr p Þdr p r 2p

r max ð

r max ð r2

r 2p f ðr p Þdr p r 2p

Therefore

r 2p f ðr p Þdr p

Therefore ðri

C 1p C 1p ¼ ¼ C 1f C 01l

r 2p f ðr p Þdr p

¼ Strue ¼ 1  Rtrue:

ð6:3:147aÞ

then, for dilute solutions, we may rewrite this as  ,  C 1p C 2f C 1p C 2p : ð6:3:147bÞ ¼ α12 ffi C 1f C 2p C 1f C 2f

we obtain 2

x 1p x 2f , x 1f x 2p

r2

r 2p f ðr p Þdr p r 2p f ðr p Þdr p

¼

ð

r1

r 2p f ðr p Þdr p þ r max ð r2

r max ð r2

r 2p f ðr p Þdr p

r 2p f ðr p Þdr p

r2

:

ð6:3:146eÞ

This interesting, but highly simplified, result suggests that the membrane rejection/retention/transmission of a

ð

r

r 2p f ðr p Þdr p

1 ¼ 1 þ r max ð

r2

: r 2p f ðr p Þdr p

ð6:3:147eÞ

426

Open separators: bulk flow parallel to force and CSTSs

Suppose now that r2 is sufficiently small compared to rmax, i.e. the membrane allows a substantial amount of species 2 to pass through. Therefore, unless r1 and r2 are far apart, the integral in the numerator on the right-hand side of (6.3.147e) is small compared to the denominator. The selectivity of a “diffuse cut off” microporous ultrafilter is not very high unless the macrosolutes differ considerably in the values of their solvated radii. Traditionally, therefore, fractionation of macrosolutes like proteins by ultrafiltration is limited to systems where the two macrosolutes differ in size by about seven to ten times (Cherkasov and Polotsky, 1996). Enhancement of the membrane selectivity for a given macrosolute has been achieved by considering a number of other factors, including increasing/decreasing the effective radius of a charged protein molecule. If the UF membrane has some charge on the surface, then macrosolutes having the same charge will be effectively rejected/repulsed by the membrane. For proteins, this can be achieved by changing the solution pH. If the solution pH is greater than the pI of the protein, the protein will have a net negative charge (see Figure 4.2.5(c)); if the membrane has a negative charge, this protein will be excluded from the membrane pores. If the ionic strength of the solution is, however, increased substantially, then the extent of electrostatic shielding of the charged protein molecule will be substantially increased. The negatively charged membrane, for example, will not be able to reject the protein as much, resulting in decreased selectivity in relation to another protein which may be uncharged (for example, bovine serum albumin, pI ¼ 4.7) at the solution pH (say 4.7). (See the following references: Saksena and Zydney (1994); van Eijndhoven et al. (1995); Nyström et al., (1998).) By stacking three such membranes one over the other, Feins and Sirkar (2004, 2005) were able to obtain one pure protein in the permeate for a binary mixture of proteins whose molecular weight ratio was as low as 1.03 to 2.05. This internal-staging concept has not been treated further. There are additional factors that influence membrane selectivity or retention. If the macrosolute has tendencies of adsorption on the membrane surface, the adsorbed macromolecules may block the pore entrance or form a layer on top of the membrane changing the membrane selectivity (Figure 6.3.27(c)). If the pore diameter changes along the membrane thickness, a macrosolute which could enter the pore at the feed side may plug the pore, render it useless and change the pore size distribution. Due to lack of convection, the top of this pore surface becomes a stagnant zone. A gel layer formed on top of the membrane from one macrosolute may substantially influence the transport behavior of a smaller macrosolute. A globular macrosolute/protein is likely to have a higher solute rejection than a linear macromolecule of the same molecular weight.

A variety of UF membranes are commercially available. Table 6.3.8 provides an illustration of the properties and performance characteristics of a series of polymeric flat membranes that are used in a batch cell. The data on solute rejection were acquired in a stirred batch UF cell of the type to be considered in Section 6.4. Example 6.3.8 Saksena and Zydney (1994) have studied the protein transmission characteristics of a 100 000 MWCO ultrafiltration membrane (OMEGA 100K) of polyethersulfone using bovine serum albumin (BSA) as a model protein (mol. wt. 66 430) in a solution of pH ¼ 7.0 and ionic strength of 0.15 M NaCl solution. The batch cell operation characteristics and the protein transport properties are as follows:

k iℓ ¼ 5:2  106 m=s; Pem i ¼ ðS ∞ J v δm =Dieff Þ; S∞ ¼ 0:016; Dieff ¼ 1  1013 m2 =s; δm ¼ 0:5 μm; J v ¼ jvz j ¼ 106 m=s: Determine the values of the observed and true transmission coefficients, Sobs and Strue , respectively, of BSA under these conditions. (See Table 6.3.8 for a definition of MWCO.) Solution We will first employ relation (6.3.145c) between S∞ expðPem Þ Strue and S∞ : Strue ¼ S þexp Pemi 1. ð iÞ ∞ Here

S∞ ¼ 0:016;

Pem i ¼

S ∞ J v δm Dieff 0:016  106

¼ Pem i ¼ 0:08 ) S true ¼ ¼

m  0:5  106 m s

1  1013 m2 =s

;

0:016expð0:08Þ 0:016 þ expð0:08Þ  1 0:016  1:0833 ) Strue ¼ 0:1745: 0:0993

Next we utilize relation (6.3.144a) between Sobs and Strue :

Strue ¼

0:1745 ¼

¼

Sobs   ; jvz j þ Sobs ð1  Sobs Þexp k iℓ Sobs  106 þ Sobs ð1  Sobs Þexp 5:2  106 

Sobs ð1  Sobs Þexpð0:192Þ þ Sobs

) 1:212  0:1745ð1  Sobs Þ þ 0:1745Sobs ¼ Sobs Sobs ð1 þ 0:211  0:1745Þ ¼ 0:211 ) Sobs ¼ For BSA, Sobs ¼ 0:204.

0:211 : 1:0355

Table 6.3.8 Properties and performance characteristics of flat Amicon UF membranesa

Nominal MWCOb Average pore diameter (nm) Water fluxc (ml/min/cm2) Material

UM05

UM2

UM10

YM10

YM30

PM10

PM30

XM50

XM100A

XM300

500 2.1

1000 2.4

10 000 3.0

10 000 –

30 000 4.0

10 000 3.8

30 000 4.7

50 000 6.6

100 000 11

300 000 48

–

–

–

0.10–0.20

0.7–1.1

1.5–3.0

2.0–6.0

1.0–2.5

-

0.5–1.0

polyelectrolyte complex

Polyelectrolyte complex

Polyelectrolyte complex

Regenerated cellulose

Regenerated cellulose

Polysulfone

Polysulfone

PAN-coPVCd

PAN-coPVC

PAN-coPVC

Solute rejectione

Macrosolute (MW)

D-Alanine (89) DL-Phenylalanine (165) Tryptophan (204) Sucrose (342) Raffinose (594) Inulin (5,000) Dextran T10 (10 000) Myoglobin (18 000) α-Chymotrypsinogen (24 500) Albumin (67 000) Aldolase (142 000) IgG (160 000) Apoferitin (480 000) IgM (960 000) a

pH5

pH10

15 20

80 90

0 0

0 0

– –

– –

0 0

0 0

0 0

0 0

0 0

20 70 90 – – >95 >95

80 80

0 50 – 80 90 >95 >98

0 25 50 60 90 95 >95

– – 10 45 – 80 –

– – – – – – >80

0 0 0 – 5 80 >95

0 0 0 0 – 35 75

0 0 0 0 – 20 85

0 0 0 0 – – 25

0 0 0 0 – – 0

>98 >98 >98 >98 >98

>98 >98 >98 >98 >98

>90 – – – –

>98 – >98 – >98

>98 >98 >98 >98 >98

>90 >98 >98 >98 >98

>90 >95 >98 >98 >98

45 – 90 >95 >98

10 50 65 85 >98

>98 >98 >98 >98 >98

From Amicon catalogs. MWCO ¼ molecular weight cutoff; it means Ri ¼ 0.9 for a solute of the specified molecular weight. c Measured in a stirred cell after 5 minutes of pressure. All membranes at 55 psi, except XM300 at 10 psi. d Copolymer of acrylonitrile and vinyl chloride. e At 55 psi, except 10 psi for XM100A and XM300 b

428 6.3.3.3

Open separators: bulk flow parallel to force and CSTSs Reverse osmosis

In Section 3.4.2.1, the phenomenon of reverse osmosis (RO) through a nonporous membrane was introduced. If the hydraulic pressure of a solution containing a microsolute, e.g. common salt, on one side of a nonporous membrane exceeds that of another solution on the other side of the same membrane by an amount more than the difference of the osmotic pressures of the same two solutions, then, according to the solution-diffusion model, the solvent will flow from the solution at higher pressure to the one at a lower pressure (equation (3.4.54)) at the following rate: J sz ¼

C sm Dsm V s vz ððP high  P low Þ  ðπ high  π low ÞÞ ffi : RT δm Vs ð6:3:148aÞ

The solution at a higher pressure Phigh has a higher solute concentration, therefore a higher osmotic pressure πhigh, and is the feed solution; the corresponding quantities on the other side of the membrane are Plow and πlow. For reverse osmosis to take place, ΔPð¼ P high  P low Þ > Δπð¼ π high  π low Þ:

ð6:3:148bÞ

The equation describing the solute transport according to the solution-diffusion model is (see equation (3.4.59)) J iz ¼

Dim κim ðC if  C ip Þ: δm

ð6:3:149Þ

Consider Figure 6.3.28(b) for a batch cell containing an RO membrane and a feed solution; as the permeation is going on, we can picture it as if the piston is driving the solution toward the membrane, i.e. the bulk flow of the feed solution is parallel to the direction of the force driving the permeation velocity vz through the membrane. As the solvent permeates through the membrane, if the RO membrane is effective, the solute (e.g. salt) is rejected and the salt concentration builds up on the feed side. Whether we

have a batch cell or not, the salt concentration on the feed side of the membrane will change with time (Nakano et al., 1967). Assume, however, a pseudo steady state for the sake of the following analysis. At any instant of time, t, we can assume (Nakano et al., 1967) that we have a bulk solute concentration, C iℓb , which increases to C 0iℓ at the RO membrane surface (Figure 6.3.28(a)). Therefore we have back diffusion of the solute (i.e. salt) from the membrane surface to the bulk solution. From the pseudo steady state analysis of concentration polarization carried out for ultrafiltration, resulting in equation (6.3.142b), and the corresponding equation for RO, namely (3.4.65c), we have     ðC 0iℓ  C ip Þ jvz j jvz j : ð6:3:150Þ ¼ exp ¼ exp ðC iℓb  C ip Þ k iℓ ðDiℓ =δℓ Þ If the value of kiℓ is low and that of jvz j is high, one can have a situation where the value of C 0iℓ can become large enough so that, for a given ΔP, ΔP can become equal to (πwall,feed  πpermeate); at this time, there will not be any solvent flux, due to a zero driving force. The membrane is considered polarized. In practice, the k iℓ values are sufficiently high so that the extent of the concentration polar



ization modulus C 0iℓ  C ip = C iℓb  C ip is low and not too far from 1. However, in the mode of operation shown in Figures 6.3.28(a) and (b), the value of k iℓ is quite low; in practice, therefore, different configurations of flow vs. force are adopted. However, we will use this configuration and pseudo steady state assumption to illustrate here the extent and nature of separation achieved through the membrane in RO; specifically we will derive expressions for the solute rejection, Ri, and the separation factor, αsi , between the solvent s and the solute i in RO using the solution-diffusion model. We will also point out the inadequacy of the solution-diffusion model, especially, at high ΔP values. Consider a dilute solution of a solute (specifically, a microsolute such as NaCl) i in a solvent s (say, water). By definition,

(a) Batch cell wall

Cil Pf > Pp

Pf

| vz |

Pf - Pp = ∆P

Cilb z z=0

(b) | vz |

Piston

CV

C 0il

Piston driven operation

dl

RO membrane

RO membrane | vz |

Cip Pp Figure 6.3.28. (a) Reverse osmosis (RO): RO in a batch cell: solute concentration profile in feed side. (b) Piston-driven RO in a batch cell: bulk flow parallel to the force.

6.3

Bulk flow parallel to force direction

x sp 1  x sf x x

¼ sp if , αsi ¼ x ip x sf 1  x sp x sf

429

ð6:3:151aÞ

where x sp and x sf are the mole fractions of the solvent in the permeate and the feed stream, respectively. Corres



pondingly, 1  x sp and 1  x sf are the mole fractions of the salt in the permeate and the feed, respectively, in this binary system. The definition for separation factor may be reexpressed as follows: αsi ¼

x sp x if ¼ x ip x sf

C sp C tp C ip C tp

C if C tf C sf C tf

¼

C sp C if , C ip C sf

ð6:3:151bÞ

where Ctj is the total concentration in region j. For a dilute solution, C sp ffi C sf ; therefore, αsi ffi

C if : C ip

ð6:3:151cÞ

On the other hand, by definition, the solute rejection Ri is (definition (2.2.1b) Ri ¼ 1 

C ip ; C if

ð6:3:152aÞ

consequently, Ri ffi 1  ð1=αsi Þ:

ð6:3:152bÞ

For the flow vs. force configuration of Figures 6.3.28(a) and (b) J iz C ip ffi V s C ip ¼ J sz C sp

ð6:3:153aÞ

for a dilute salt solution. Substitute here the expressions for J iz and J sz from expressions (6.3.148a) and (6.3.149) resulting from the solution-diffusion model: J iz ¼ V s C ip ¼ J sz

Dim κim RT Dim κim ðC if  C ip Þ δm ðC if  C ip Þ ¼ : C sm Dsm V s C sm Dsm V s ðΔP  ΔπÞ RT δm ðΔP  ΔπÞ

ð6:3:153bÞ

Now, αsi ffi

C if V s C if C sm Dsm V s ðΔP  ΔπÞ ¼ Dim κim RTðC if  C ip Þ C ip

ð6:3:154aÞ

Ri ffi 1 

Dim κim RT

:

ð6:3:154dÞ

2

αsi ffi

C sm Dsm V s ðΔP  ΔπÞ Dim κim RT

ð6:3:154eÞ

for C ip t1

t3>t2

t4>t3

C 0i 2

Purified liquid

Purified liquid

(b)

Force direction D

z+∆z L

Stationary adsorbent phase

z

z

z=0

z=0

Feed

A

C

B

z+∆z Mobile liquid phase

z

z Feed, C 0i 2

Packed-bed schematic

Packed-bed idealization in pseudo-continuum approach

Figure 7.1.1. (a) Movement of the color front in the packed adsorbent bed with time and the corresponding concentration of the coloring species at the bed outlet; (b) packed-bed schematic and its idealization in pseudo-continuum approach for adsorption from a liquid. (After Lightfoot et al., 1962.)

solution C i2 (z, t). This may be achieved by averaging over the packed-bed cross section in the x- and y-directions. However, to allow for species transport from the mobile phase to the stationary (fixed-bed particles) phase and vice versa, it is necessary to consider each phase separately without any x and y dependence, as shown in Figure 7.1.1(b) (Lightfoot et al., 1962). This figure shows the cross section of only a small length, ∆z, of the packed bed. Note: C i1 ðz;tÞ represents the species i concentration per unit volume of the solid phase. A mass balance on species i over the shell ABCD which spans the whole cross section of the packed bed of length ∆z may now be carried out. The total bed cross-sectional area is Sc; the cross-sectional area for flow of fluid is εSc .

Here, ε is the void volume fraction of the packed bed2 and the interstitial fluid velocity3 in z-direction is vz. The effective diffusion coefficient of solute i in the liquid in the z-direction is Di, eff, z. The particle surface area per unit particle volume is av. The particles may be porous, with a porosity εp. We will consider it later; for now, εp ¼ 0. Referring to Figure 7.1.1(b), a species i mass balance may be written in the absence of any chemical reaction as

2

In a capillary bundle model of the packed bed. From now on, we will use vz instead of vtz for the average fluid velocity. 3

490

Bulk flow perpendicular to the direction of force

0

1 rate of accumulation @ of species i A¼ in volume ABCD 1 0 1 0 rate of outflow rate of inflow @ of species i into A − @ of species i out of A; ð7:1:1Þ volume ABCD volume ABCD 2 3 ∂C i2 ∂C i1 5 Sc Δz 4ε þ ð1 − εÞ ¼ ∂t ∂t 2 3 2 3 ∂C i2 5 ∂C i2 5 4 4 εSc vz C i2 − Di;eff;z − εSc vz C i2 − Di;eff;z : ∂z ∂z z

zþΔz

ð7:1:2Þ

The mobile phase has been identified here by j ¼ 2 and the stationary phase by j ¼ 1. The assumptions are: (1) species i enters the volume ABCD by convection and diffusion at location z; (2) species i leaves by similar mechanisms at location z þ ∆z; (3) the particle phase (j ¼ 1) occupies a volume fraction ( 1 − ε) of the packed-bed volume Sc∆z. We now implement a limiting process where, as Δz ! 0, the differential equation   ∂C i2 ð1 − εÞ ∂C i1 ∂ðvz C i2 Þ ∂ ∂C i2 ð7:1:3Þ þ þ ¼ Di;eff;z ∂t ∂t ∂z ε ∂z ∂z is obtained for C i2 . Note: C i1 is the molar concentration of species i based on particle volume only. Conventionally, for liquid-phase systems, vz is independent of z and so is Di, eff, z. Therefore ∂C i2 ð1 − εÞ ∂C i1 ∂C i2 ∂2 C i2 þ þ vz ¼ Di;eff;z : ∂t ∂t ∂z ∂z2 ε

ð7:1:4Þ

This is the starting differential equation for almost all liquid-phase based fixed-bed processes for a single adsorbable solute in an inert liquid under isothermal conditions with particles, where εp ¼ 0. (Note: We could have obtained this equation directly from equation (6.2.35).) However, this equation does not take into account the transfer of species i from liquid phase 2 to adsorbent particle phase 1; such a relation between C i1 and C i2 is needed before a solution of C i2 ðz;tÞ may be obtained. In terms of an overall mass-transfer coefficient K il based on the liquid-phase concentration and the surface area of the particle (where a is the particle surface area per unit bed volume and av is the specific surface area of the particle, i.e. particle surface area per unit particle volume so that a ¼ av ð1 − εÞ), the mass-transfer rate per unit bed volume is ð1 − εÞ 

∂C i1  ¼ K il aðC i2 − C i2 Þ; ∂t

ð7:1:5aÞ

where C i2 is a hypothetical mobile-phase concentration in equilibrium with C i1 . (We could have obtained this equation also from equation (6.2.32) for the stationary phase.) If  the equilibrium relation is known, C i2 is easily expressed in terms of C i1 . Thus, the solution for C i2 ðz;tÞ can be obtained

in principle for appropriate initial and boundary conditions imposed on the fixed bed of adsorbents. Extensive heat effects from adsorption will introduce additional complexity. Species i is transferred from the mobile liquid phase to an adsorbed state on the particle surface via a number of steps: diffusion in the fluid phase around the particle, diffusion across the fluid–particle interface, diffusion in liquid in the pores of the particle, adsorption on available sites on the particle pore surface and surface diffusion, if any (see Sections 3.1.3.2.3/4 and 3.4.2.3/4). A fundamental approach would be to develop a species balance for such a particle phase and couple it with equation (7.1.4) via an additional mass-transfer relation for diffusion in the fluid phase around the particle. Such an approach, with some simplifications by Rosen (1952, 1954), has been illustrated at the very end of this section. There are a number of other solutions primarily based on various assumed mechanisms of mass transfer. One such approach replaces the mass balance equation for diffusion within the particle by simplifying assumptions. In what is known as the linear driving force assumption, Glueckauf (1955b) suggested that, for ðDip t=r 2p Þ > 0:1,



 ∂C i1 15Dip   ¼ 2 C i1 − C i1 ; rp ∂t

ð7:1:5bÞ

ρb ¼ ρs ð1 − εp Þ ð1 − εÞ:

ð7:1:5cÞ

where C i1 is in equilibrium with the bulk liquid concentration C i2 , and Dip is the effective diffusion coefficient of species i in the pores of particle of radius rp. A nonlinear driving force approximation has been suggested by Vermuelen (1953). In both models the liquid-phase resistance around the particle is neglected. Before focusing on some of the simpler solutions of such a system to develop an understanding of the single solute separation capabilities of a fixed-bed adsorption process, we will provide a few more definitions for a packed adsorbent bed in general. The density of the solid material of the adsorbent particles is ρs . The bulk density of the adsorbent bed ρb is related to the void volume ε of the packed bed, solid material density ρs and the porosity of the particle εp via

This definition assumes the pores inside the particles to be empty. If the bulk density is defined with respect to particles containing a fluid phase j ¼ 2 inside the pores, then ρb ¼ ρs ð1 − εp Þ ð1 − εÞ þ εp ρ2 ;

ð7:1:5dÞ

where ρ2 is the fluid-phase density. Unless specified, we will assume (7.1.5c) to be the valid relation for ρb . We can now write down the differential equation for solute i in the manner of equation (7.1.3) for porous adsorbent particles whose pores contain the external solution (therefore C i2 ):

7.1

ε

Force −rμi in phase equilibrium: fixed-bed processes

∂C i2 ∂C i2 ∂C i1 ∂ðvz C i2 Þ þ εp ð1 − εÞ þ ð1 − εÞ þε ∂t ∂t ∂t ∂z   ∂ ∂C i2 Di;eff;z : ð7:1:5eÞ ¼ ε ∂z ∂z

This equation is based on a number of assumptions, one being that partitioning of solute i between the external solution and that in the pores of the adsorbent is such that the partitioning coefficient is 1. This is true when the solute size is small in relation to the adsorbent pore size (by about two orders of magnitude). An alternative form of this equation is as follows:   ð1 − εÞ ∂C i2 ð1 − εÞ ∂C i1 ∂ðvz C i2 Þ 1 þ εp þ þ ε ε ∂z ∂t ∂t   ∂ ∂C i2 ¼ Di;eff;z : ð7:1:5fÞ ∂z ∂z When the solute size is larger (e.g. proteins) or the pore sizes are smaller, the partitioning of the solute between the mobile phase and the pore phase liquid (i.e. p κim ¼ ðC im =C i2 Þ as in (3.3.89a)) has to be taken into account:   ð1 − εÞ ∂C i2 ð1 − εÞ ∂C i1 ∂ðvz C i2 Þ 1þ εp κim þ þ ε ε ∂z ∂t ∂t   ∂ ∂C i2 : ð7:1:5gÞ ¼ Di;eff;z ∂z ∂z The simplest solution of a fixed-bed adsorption problem is provided by isothermal equilibrium nondispersive operation of the fixed bed. For single solute adsorption from the liquid phase to the adsorbent particles (where εp ¼ 0) under isothermal conditions, (1) assume that the liquid-phase concentration Ci2 (z, t) in the column everywhere is locally in equilibrium with the solid-phase concentration Ci1 (z, t), and (2) neglect the contribution of the axial diffusion and dispersion term, Di;eff;z ð∂2 C i2 =∂z 2 Þ, in equation (7.1.4) (the plug flow assumption). Define qi1 to be moles of species i in solid phase 1 per unit mass of solid phase. Then C i1 may be replaced by qi1 using the following relation: C i1 ¼ qi1 ρb =ð1 − εÞ;

ð1 − εÞ

∂C i1 ∂q ¼ ρb i1 ; ∂t ∂t

ð7:1:6Þ

where ρb is the bulk density of the packed bed. Describe the equilibrium relation of the first assumption now by   qi1 ¼ qi C i2 : ð7:1:7Þ

Using this and the second assumption in equation (7.1.4), we get   ρ q 0 ðC i2 Þ ∂C i2 ∂C i2 þ vz ¼ 0; ð7:1:8Þ 1þ b i ∂t ∂z ε

491

where q 0i ðC i2 Þ ¼ ∂qi ðC i2 Þ=∂C i2 . Note that relation (7.1.7) serves the purpose of relating C i2 to C i1 , i.e. qi1 , and we now have only one equation, (7.1.8), to solve to obtain C i2 ðz;tÞ. This equation is sometimes called the De Vault equation (De Vault, 1943). An alternative form of equation (7.1.8) is more convenient: ∂C i2 þ ∂t

vz ∂C i2 ¼ 0:  ρb q 0i ðC i2 Þ ∂z 1 þ ε

ð7:1:9Þ

A key question in fixed-bed adsorption separation is: what is the time needed for the fixed bed to become saturated if it is fed continuously and steadily with a feed of constant concentration C 0i2 ? The feed initially introduced into the column (Figure 7.1.1(a)) displaces the liquid already present in the column. If the column particles did not have any species i to start with, the liquid displaced from the column and appearing at the outlet will be free of species i. Meanwhile, species i from the feed liquid will be adsorbed near the feed entry and, soon after, feed liquid free of species i will appear at the column outlet. As this process continues, the particle surfaces will become saturated with species i. Ultimately, all particles will lose their capacity of adsorbing species i; the feed solution of concentration C 0i2 will appear at the column outlet. At this time, the adsorber is taken off the feed line and subjected to regeneration treatment so it may be used again for adsorption. Obviously, the velocity with which the concentration C 0i2 moves down the column (the concentration wave velocity) is less than that of the interstitial liquid. If one could ride with the wave having a concentration C 0i2 , one will only witness C 0i2 around oneself; therefore, for any wave having a fixed value of specific concentration C i2 , dC i2 ¼ 0. Now, C i2 is a function of the independent variables z and t; therefore, for any small change in z and t, the change in C i2 is given by     ∂C i2 ∂C i2 dt þ dz: ð7:1:10Þ dC i2 ¼ ∂t z ∂z t For any particular concentration C i2 , the concentration wave velocity must satisfy dC i2 ¼ 0. Compare a reformulated (7.1.10) under such a condition,     ∂C i2 dz ∂C i2 þ ¼ 0; ð7:1:11Þ ∂t z dt ∂z t with equation (7.1.9) to obtain (De Vault, 1943) vCi ¼

dz vz ¼ : dt ρb q 0i ðC i2 Þ 1þ ε

ð7:1:12aÞ

Here, vCi is the velocity with which a wave of concentration C i2 travels along the column (the z-direction). For a given C i2 , the above expression provides a unique relation between the location z along the column and time t since

492

Bulk flow perpendicular to the direction of force

q 0i ðC i2 Þ is fixed. The velocity vCi is variously called the concentration wave velocity of species i, the migration rate of species i or the concentration front propagation velocity of species i (Sherwood et al., 1975). A more formal method of arriving at expression (7.1.12a) is given below. This formal procedure is based on the method of characteristics (Aris and Amundson, 1973). If a solution for C i2 were available, we may write any change in C i2 as     ∂C i2 ∂C i2 dC i2 ¼ dt þ dz ¼ dC i2 : ð7:1:12bÞ ∂t z ∂z t Now one can solve for the two derivatives of C i2 ðz;tÞ using equation (7.1.9):     ∂C i2 ∂C i2 vz þ   ¼ 0: ∂t z ∂z t ρb q 0i ðC i2 Þ 1þ ε The two expressions that result are   vz   2 3  0   0   ρ q ðC Þ 41 þ b i i2 5     ε         dC i2 dz ∂C i2 ¼   ; v z 1 2 ∂t z 3     0 ρ q ðC Þ  i2 41 þ b i 5     ε     dt dz     1  0      dC i2 dt  ∂C i2 ¼ : vz ∂z t  1 2 3     0  41 þ ρb q i ðC i2 Þ5     ε     dt dz

ð7:1:12cÞ

The denominator (i.e. the coefficient determinant) in both expressions is zero under certain conditions. One such condition is dz ¼ 

vz dz vz ¼  dt ) : dt ρb q 0i ðC i2 Þ ρb q 0i ðC i2 Þ 1þ 1þ ε ε

ð7:1:12eÞ

If the denominator is zero, for the derivatives to be finite the numerator also has to be zero. From relation (7.1.12c), this implies dC i2 ¼ 0:

ð7:1:12fÞ

Result (7.1.12e) is valid if C i2 is constant along the characteristic line traced by (7.1.12e), which is the same as (7.1.12a). It is clear from relation (7.1.12a) that the value of vCi for a given C i2 will depend on the value of q 0i ðC i2 Þ, i.e. on the nature of the equilibrium relation qi1 ¼ qi ðC i2 Þ. Figure 7.1.2(a) shows two linear adsorption equilibrium isotherms for two different species i ¼ 1 and i ¼ 2. As shown, species i ¼ 1 is more strongly adsorbed than species i ¼ 2 and for C 12 ¼ C 22 : q0 1 ðC 12 Þ > q0 2 ðC 22 Þ:

ð7:1:13aÞ

Therefore, by relation (7.1.12a) vC1 < vC2

ð7:1:12dÞ

(a)

ð7:1:13bÞ

in equilibrium nondispersive operation; the species which is more strongly adsorbed moves through the column more slowly and will take longer to appear at the column exit. There is another way to visualize the migration velocity of a solute species down the column. When some extra solute is added to the packed-bed section of length ∆z

(b)

o Fav

q i1

r

ea

n Li

Un

fa

vo

ra

bl

i =2

e

q i1

le

rab

i =1

Ci 2

Ci 2

Figure 7.1.2. Adsorption isotherms: (a) two linear isotherms for two species i ¼1, 2; (b) three isotherms, favorable, linear and unfavorable.

7.1

Force −rμi in phase equilibrium: fixed-bed processes

which may already contain some solute (Figure 7.1.1(b)), the following will happen. Due to a higher solute concentration in the mobile phase, more solute will be adsorbed now on the adsorbent. Therefore the fraction of the extra solute added that remains in the mobile phase will be less than 1, reducing the mobile-phase concentration. Suppose the small increase in mobile-phase concentration and stationary-phase concentration of species i are, respectively, ΔC i2 and ΔC i1 , then 0 1 fraction of solute εSc ΔC i2 @ added that remains A ¼ þ ð1 − εÞSc ΔC i1 εS ΔC i2 c in the mobile phase ¼

1 :  1 − ε ΔC i1 1þ ε ΔC i2 

From relation (7.1.6), this ratio is given by ΔC i ! 0

¼

lim

1þ

1 : ρb 0 q i ðC i2 Þ ε

ð7:1:13cÞ

We now focus on beds with no dispersion and with a constant interstitial fluid velocity vz (used to derive equation (7.1.12a)). Due to adsorption, the probability of any solute molecule being in the mobile phase after introduction into the control volume is reduced from 1 by the ratio given above. Thus the velocity with which these molecules migrate down the column is reduced from the fluid velocity vz by the same fraction: vCi ¼

vz : ρb q 0i ðC i2 Þ 1þ ε

ð7:1:13dÞ

The mechanistic basis of the result (7.1.13b) is clearer from such an approach. The result (7.1.13d) was based on ε being the void volume fraction occupied by the mobile phase in the packed bed. If the adsorbent particles are also porous with a void volume fraction of εp , and ρs is the actual density of the solid adsorbent particle material, then, using arguments as before, one can show that vCi ¼

1þ

ð1 − εÞ ε εp

þ

vz : ð1 − εÞ 0 ε ð1 − εp Þρs q i1 ðC i2 Þ

ð7:1:13eÞ

The denominator in (7.1.13e) has (unlike that in expression (7.1.13c)) three sources that contribute: the fraction of total solutes that remain in the mobile phase, the fraction of total solutes present in the liquid in the pores of the adsorbent and the fraction of total solutes adsorbed on the surfaces of the pores of the adsorbent and other outside surfaces. Note that ρs ð1 − εp Þð1 − εÞ is the bulk density, ρb , of the bed used in earlier expressions. The above expression is valid when the solute size is small with respect to the adsorbent pore size so that the solute concentration in the adsorbent pore liquid phase is the same

493

as that in the mobile phase. For larger solute sizes, the partitioning of the solute between the mobile-phase and the pore-phase liquid has to be taken into account. The corresponding result will be (see equation (7.1.5g)) vCi ¼

vz : ð7:1:13fÞ ð1 − εÞ ð1− εÞ 1þ εp κim þ ð1−εp Þρs q0 i1 ðC i2 Þ ε ε

This result indicates that larger solutes (larger with respect to the pore size) will have larger values of vCi . The solute molecules of a particular species staying in the mobile phase move down the column with the fluid velocity vz , whereas those that are adsorbed cannot move down the column. Thus, a given species concentration moves down the column at an effective velocity vCi lower than vz . For a given species concentration, the higher the fraction of the species in the mobile phase, the higher its speed through the column. Figure 7.1.2(b) illustrates a few more adsorption equilibrium isotherms. For the isotherm identified as favorable, it is obvious that if two concentrations C i2 j1 and C i2 j2 are significantly apart and C i2 j2 > C i2 j1 , then q0 i2 ðC i2 j2 Þ < q0 i2 ðC i2 j1 Þ:

ð7:1:14aÞ

From (7.1.12a) and (7.1.13f), vC j2 > vC j1 :

ð7:1:14bÞ

Therefore, a higher solute concentration exits the column faster for a favorable isotherm in equilibrium nondispersive operation. For the isotherm identified as unfavorable, it is easily shown that an exactly reverse behavior holds. These features help one to visualize the movement of trajectories of constant C i2 in the (z, t)-plane shown in Figure 7.1.3(a). These lines (called characteristics) which start at z ¼ 0 represent different concentrations entering the column (z ¼ 0) at different times. If a feed of constant concentration C 0i2 enters the column, then all lines starting at z ¼ 0 and t  0 will be parallel; the slope of each line is equal to vCi for C i2 ¼ C 0i2 . The interstitial liquid at the column inlet is represented by the trajectory AB, whose slope is vz , the liquid velocity. To the left of it lie the trajectories of liquid originally present in the column at different locations in the column (different z, t ¼ 0). On the right side of line AB are characteristics of the feed liquid which have C i2 ¼ 0. The wave velocity of this zero concentration vCi (for C i2 ¼ 0) should provide the slope of these characteristics. If the adsorption isotherm is favorable (Figure 7.1.2(b)), vCi jC i2 ¼0 < vCi jC i2 6¼0 . These intersect the characteristics for C 0i2 along the line AS. At any point on line AS, characteristics having two different values C i2 ¼ 0 and C i2 ¼ C 0i2 simultaneously exist. Line AS is like a shock wave having two different concentrations on two sides of a line, a discontinuity (Sherwood et al., 1975). The point of intersection of this line of discontinuity AS with z ¼ L, the column length, is crucial. The value of time t

494

Bulk flow perpendicular to the direction of force

corresponding to the coordinates (L, t) of this point defines the time when the bed is completely saturated. As shown in Figure 7.1.3(b), the exit concentration in the liquid suddenly jumps from C i2 ¼ 0 to C i2 ¼ C 0i2 at this time; the feed solution breaks through the bed at this time, and the adsorption operation must stop. The sudden jump part of the square concentration wave and its subsequent constant value at C 0i2 is identified as the column breakthrough curve. If the isotherm is linear instead of favorable, the characteristics for C i2 ¼ 0 in between AB and AS in Figure 7.1.3(a) will be parallel to the characteristics for C 0i2 . Even then, when the characteristics for C 0i2 first hit z ¼ L, there will be a concentration discontinuity from C i2 ¼ 0 to C i2 ¼ C 0i2 at the column outlet. For more details, see Wankat (1986, vol. I, pp 16–22). To determine the value of time t ð¼ tÞ when the feed solution breaks through a column of length L, a solute mass balance is carried out over the column from time t ¼ 0 to the time t of breakthrough. For the sake of generality, the mass balance is carried out in a column which may have some solute present in the column at t ¼ 0, i.e. C i2 ðz; 0Þ and C i1 ðz; 0Þ are nonzero. Further, εp 6¼ 0. Before the feed concentration breaks through the column end in the liquid effluent, the liquid phase everywhere in the column would have C 0i2 ; thus the total number of moles of solute in the column at time t of breakthrough is given by

LSc εC 0i2 þ ð1 − εÞ εp C 0i2 κim þ ð1 − εÞC 0i1

¼ LSc εC 0i2 þ ð1 − εÞ εp C 0i2 κim þ ρb qi1 ðC 0i2 Þ : This must equal the number of moles introduced into the column by the steady inflow of feed from time t ¼ 0 to time t for breakthrough, i.e. εvz Sc tC 0i2 plus the moles of solute originally present in the column, Sc

ðL 0



εC i2 ðz;0Þ þ ð1 − εÞ εp C i2 ðz;0Þ κim þ ð1 − εÞ C i1 ðz;0Þ dz:

(a) B

S

A t=0

t

z L

z=0

t

(b)

C 0i 2 C i2

exit

t 0

t

Figure 7.1.3. (a) Characteristics for feed concentration C 0i2 for adsorption in the (z, t)-plane for the equilibrium nondispersive model for a favorable isotherm. (b) Breakthrough curve for equilibrium nondispersive model with constant feed concentration in an initially empty column.

L ¼ t

vz  ð1 − εÞ ρ q ðC 0 Þ εp κim þ b i 0 i2 1þ ε ε C i2

vz L ¼  ¼ ¼ vCi jshock : t ð1 − εÞ ð1 − εÞ C 0i1 1þ εp κim þ ε ε C 0i2

ð7:1:15cÞ

This also happens to characterize the square wave front The solute balance relation is therefore of concentration C 0i2 . For time t < t (breakthrough) ¼ t, the L ð  εvz Sc tC 0i2 þ Sc εC i2 ðz;0Þ þ ð1 − εÞ εp C i2 ðz;0Þ κim þ ð1 − εÞC i1 ðz;0Þ dz ð7:1:15aÞ 0

¼ LSc εC 0i2 þ ð1 − εÞ εp C 0i2 ðz;0Þ κim þ ρb qi ðC 0i2 Þ :

For the special case studied so far, C i2 ðz; 0Þ ¼ 0 ¼ C i1 ðz; 0Þ; in such a case,   L ð1 − εÞ ρ q ðC 0 Þ ð7:1:15bÞ εp κim þ b i 0 i2 : 1þ t¼ vz ε εC i2 Alternatively, the direct relation between the column length and the time for breakthrough for a liquid feed of constant concentration C 0i2 fed at a constant interstitial velocity vz is

location of the front will be inside the column where z < L; for such times, relation (7.1.15c) should be expressed as ðz=tÞ instead of ðL=tÞ. A more general expression for t or vCi jshock may be derived for the case where C 0i2 ðz;0Þ ¼ C ii2 and C 0i1 ðz;0Þ ¼ C ii1 (in equilibrium with C ii2 ). The relation equivalent to (7.1.15a) is

ε vz Sc tC 0i2 þ L Sc ε C ii2 þ ð1 − εÞ εp C ii2 κim þ ð1 − εÞ C ii1

¼ L Sc ε C 0i2 þ ð1 − εÞ εp C 0i2 κim þ ρb qi ðC 0i2 Þ : ð7:1:15dÞ

7.1

Force −rμi in phase equilibrium: fixed-bed processes

On rearranging, one can obtain L L ¼ ¼ vCi jshock t t ¼ 

Ci 1− i2 C 0i2

vz   : ð1−εÞ ð1−εÞ C 0i1 −C ii1 1þ εp κim þ ε ε C 0i2

ð7:1:15eÞ

Is this operation of fixed-bed adsorption with bulk liquid phase flow perpendicular to the direction of the chemical potential gradient force from the liquid to the solid beneficial for separation when compared with just plain equilibration between the feed liquid and the adsorbent without any bulk flow in a given direction? There are a number of ways of looking at this issue. First, the bulk liquid-phase flow perpendicular to force produces a very substantial volume of absolutely solute-free feed liquid at the column exit. Simple equilibration in a batch process in a vessel will never yield solute-free liquid. For example, feed solution of concentration C 0i2 will be reduced in plain batch equilibration to C i2 , determined by (for εp ¼ 0) εSc vz tC 0i2 ¼ εSc vz tC i2 þ ð1 − εÞSc vz tC i1 ;

ð7:1:16aÞ

which leads to C i2 ¼

C 0i2 : ð1 − εÞ κi1 1þ ε

ð7:1:16bÞ

Here, κi1 is the distribution coefficient of solute i between phases 1 and 2 at equilibrium, i.e. κi1 ¼ C i1 =C i2 . The fraction of the total solute i present in the liquid phase is obtained from the ratio R0 i , εSc vz tC i2 εC i2 ¼ εSc vz tC i2 þ ð1 − εÞSc vz tC i1 εC i2 þ ð1 − εÞC i1 1 1 1 ¼ ¼ : ð7:1:16cÞ ¼ ð1 − εÞ C i1 1 þ ðmi1 =mi2 Þ 1 þ k0 i1 1þ ε C i2

R0 i ¼

Thus, the purification capability of this pattern of bulk flow vs. force is excellent for plain batch equilibration. Second, one can compare the total number of moles transferred to the adsorbent phase in rival modes of operation. If C 0i1 is the solid-phase concentration in equilibrium with the influent concentration C 0i2 , then the moles of species i transferred to the solid adsorbent phase per unit volume of adsorbent is ðC 0i1 − 0Þ for an initially solute-free adsorbent bed when the bulk flow is perpendicular to the force. A batch operation without bulk flow will transfer according to relation (7.1.16a) only ðC i1 − 0Þ moles per unit volume of adsorbent, where C i1 < C 0i1 . Thus, bulk flow of feed along the bed length perpendicular to the force direction achieves a better utilization of the intrinsic adsorption capacity of the adsorbents. Third, the longer the bed, the larger the volume of feed liquid that can be purified almost completely. A batch adsorption process lacks any such feature.

495

The volume V of feed solution needed to be passed through the column of length L so as to saturate it with C0i2 under equilibrium nondispersive mode of operation is:

V C 0i2 ¼ Sc L εC 0i2 þ ð1 − εÞεp C 0i2 κim þ ð1 − εÞC 0i1

¼ Sc L εC 0i2 þ ð1 − εÞεp C 0i2 κim þ ρb qi ðC 0i2 Þ ;   ρ q ðC 0 Þ V ¼ Sc L ε þ ð1 − εÞεp κim þ b i 0 i2 : C i2

ð7:1:16dÞ

It will be found later that V is a useful quantity in the study of the actual separation behavior of real columns which may not satisfy equilibrium nondispersive conditions. The value of t corresponding to V is t, and it is obtained from the expression (7.1.15c) for t. The concentration wave front moving out of a column in equilibrium nondispersive operation need not be a square wave front as shown in Figure 7.1.3(b). The square wave front resulted from an initially solute-free column and a constant influent concentration C 0i2 . Suppose the column has in its entry region a linear distribution of solute concentration from C 0i1 to 0, from column inlet to some distance into the column. The liquid in immediate contact is in equilibrium, and its concentration then varies from C 0i2 to 0, as shown in Figure 7.1.4. As the feed liquid enters the column, concentration waves having values from C 0i2 to 0 will be moving down the column due to this initial column loading. The concentration wave velocity of each concentration will depend on the value of q 0i ðC i2 Þ for the particular C i2 , which in turn will depend on the nature of the adsorption isotherm. Consider first the “favorable isotherm” of

Figure 7.1.2(b).  2 This type of isotherm is characterized by d2 qi =dC i2 < 0 for all values of C i2 . For such an isotherm, if the values of C i2 at two locations z1 and z2 ð> z1 Þ are such that C i2 jz1 > C i2 jz2 , then q 0i ðC i2 Þjz1 < q 0i ðC i2 Þjz2 and vCi jz2 < vCi jz1 . Therefore, the velocity vCi of the higher concentration nearer the column inlet is higher than that of the lower concentration further from the column inlet. After some time t (Figure 7.1.4), an entirely different concentration profile will be observed along the column. The concentration profile has become quite sharp, almost like a square wave, since the higher concentrations have moved much faster and have caught up with the much slower moving lower concentrations further down. This is identified as the self-sharpening wave front. When C 0i2 is zero, that is pure solvent (called sometimes eluent) comes in, the desorption behavior of the solute is often called elution. We would now like to focus on calculating such elution behavior from a preloaded column (Figure 7.1.4). The nondispersive isothermal equilibrium operation of the packed adsorption bed described by equation (7.1.9) may be written using the following new independent variable s (instead of time, t),

496

Bulk flow perpendicular to the direction of force

Favorable isotherm based Linear isotherm based

C 0i 2

C 0i 2

Ci 2

Ci 2

0

z

0

Unfavorable isotherm based

Figure 7.1.4. Movement of concentration profile along the column for three different types of isotherms when the initial column section is loaded linearly with solute from C i2 ¼ C 0i2 at z ¼ 0 to C i2 ¼ 0 some distance down.

C 0i 2 Ci 2

0

z

z

ðds=dtÞ ¼ εvz

ð7:1:17aÞ

(where s is the volume of solution fed to the column per unit empty cross section of the bed (Lightfoot et al., 1962)) as (for εp ¼ 0)

∂C i2 ∂C i2 þ ε þ ρb q 0i ðC i2 Þ ¼ 0; ð7:1:17bÞ ∂z ∂s Ðt since s ¼ 0 ε vz dt ¼ ε vz t (for a constant feed flow rate). In general, we may write dC i2 ¼ ð∂ C i2 =∂zÞdz þ ð∂ C i2 =∂sÞds:

ð7:1:17cÞ

Following the procedure used in equations (7.1.12b) and (7.1.12e), we get 

∂s  ð7:1:17dÞ  ¼ ε þ ρb q 0i ðC i2 Þ : ∂z C i2

(Note: Equation (7.1.12e) for vCi continues to be valid for any C i2 .) For a constant feed concentration, C i2 (0, t), we can integrate this equation to obtain

fs= ε þ ρb q 0i ðC i2 Þ g ¼ z − z0 ðC i2 Þ; ð7:1:17eÞ   where z0 C i2 is the distance corresponding to any C i2 at the start of the operation. We may rewrite this relation as  

s   ¼ ε þ ρb q 0i C i2 : z − z 0 C i2

ð7:1:17fÞ

This relation may be interpreted as follows. Consider a column of cross-sectional area Sc. The volume of solution being passed, V, is given by sSc. Let the volume of solution passed before elution is started be Vo. Then the elution   solution volume passed is V – Vo. If z0 C i2 ¼ 0, we obtain from relation (7.1.17f)



V − V 0 ¼ Sc ðz − z0 ðC i2 ÞÞ ε þ ρb q 0i ðC i2 Þ  

dC i1 : ¼ Sc z ε þ ρb q 0i ðC i2 Þ ¼ Sc z ε þ ð1 − εÞ dC i2 ð7:1:17gÞ This expression allows us to calculate the eluent volume V that has to be passed corresponding to a given C i2 at the column outlet. If the isotherm is asimple linear one in Figure 7.1.2(b)

2 (therefore d2 qi =dC i2 ¼ 0 for all C i2 ), then all concentrations at all locations in the initial part of the column (Figure 7.1.4), have the same wave velocity. As time progresses, the same linear concentration profile is transported down the column without any change in shape. On the

other hand, if the isotherm is unfavorable 2 d2 qi =dC i2 > 0 in Figure 7.1.2(b)), lower concentrations further down from the inlet of the column will have higher vCi than higher concentrations near the beginning of the column inlet. This will lead to a more spread out profile along the column as time increases and concentrations move down the column. Such a condition is identified as a dispersive (or diffusive) wave front. So far, the adsorption based purification of a liquid feed flowing down a bed of adsorbent particles has been analyzed using the equilibrium nondispersive approximation. For a liquid feed of constant concentration C 0i2 and a bed initially free of any solute i, such an approximation suggests a square concentration wave exiting the column end at time t defined by result (7.1.15c). Alternatively, a volume V (given by relation (7.1.16d)) of feed liquid has to pass through the column before the feed solution concentration C 0i2 suddenly breaks through. In reality, the column effluent concentration has more of an S-shape (Figure 7.1.5(a)) than a square wave. This requires

7.1

Force −rμi in phase equilibrium: fixed-bed processes

497

(c)

(a)

Species 1, 2 and 3 Total solvent concentration in effluent

C 0i 2 C i2

br

Ci 2 0

t V

Species 3

tbr V t

t3

C 0i 2 tbr t1 < t 2

t2 < tbr

C i2

z

LMTZ

t

t2

t1

Figure 7.1.5. (a) Effluent concentration profile from a fixed bed (breakthrough curve) for one solute i in feed. (b) Column concentration profile at three different times t1, t2 (>t1) and tbr for one solute i in feed. (c) Breakthrough curve from a fixed bed for a feed containing three solutes i ¼ 1, 2, 3: species 3 is least strongly adsorbed; species 1 is most strongly adsorbed.

(b)

0

Species 2 and 3

L

termination of the adsorption operation at time tbr less than that ðtÞ defined by (7.1.15c); the volume V br of purified liquid obtained would be less than V . The breakthrough concentration of the solute C br i2 at t br is determined by the nature of the operation. This value is usually around 0.05–0.1 of C 0i2 ; when the liquid is to be purified, its value is much lower. Such a diffuse breakthrough of solute instead of a sharp front may be caused by a lack of equilibrium between the two phases or by the presence of axial diffusion and dispersion, or both. The lack of equilibrium between the two phases may come about due to a finite mass-transfer resistance in the liquid film surrounding the particle or due to diffusional resistances in the pores of the particle, or both. If one were to determine the solute concentration profile in the liquid phase in the packed bed at any time prior to breakthrough, one will find a shape inverse to that shown in Figure 7.1.5(a). Figure 7.1.5(b) illustrates two bed solute profiles at two instants of time t 1 and t 2 ð> t 1 Þ that are far apart. In both profiles, the initial section of the profile has a constant concentration C 0i2 corresponding to the feed liquid coming in; the bed is saturated here and no longer has the capacity to adsorb any more solute. The final sections of the profiles in each case have a value of C i2 ¼ 0; they are without any solute and retain their intrinsic capacity for solute adsorption. In between, each profile changes from C i2 ¼ C 0i2 to 0; this section of the bed has some solute adsorption capacity left. It is obvious that in the first section, where C i2 ¼ C 0i2 , solute transfer is no

longer going on, although some occurred at an earlier time; in the final section, with C i2 ¼ 0, no mass transfer has started yet. Thus, mass transfer occurs only in the intermediate region: this region of the bed is called the adsorption zone or the mass-transfer zone (MTZ). As time increases, this adsorption zone travels through the bed; finally, it shows up in the bed effluent as concentration increasing with time (Figure 7.1.5(a)). The length of this zone is identified as LMTZ. A number of analyses have been developed to account for such features in real fixed-bed adsorption processes. Two such analyses and their major results will be briefly identified below. A linear equilibrium model with dispersion is considered first. Lapidus and Amundson (1952) solved equation (7.1.4) which included the axial dispersion term under the following assumptions: (1) local equilibrium exists everywhere between the liquid and solid phase and it is linear, e.g. C i1 ¼ κi1 C i2 ;

ð7:1:18aÞ

where κi1 is a constant; (2) the column is infinitely long. The conditions used for the solution are: a steady liquid feed of constant inlet concentration C 0i2 , uniform solute concentration of C ii2 and C ii1 (at equilibrium) throughout the bed for time t  0. Define a new dependent variable

498

Bulk flow perpendicular to the direction of force θi ðz;tÞ ¼ ðC i2 ðz;tÞ − C ii2 Þ=ðC 0i2 − C ii2 Þ:

ð7:1:18bÞ

Under the assumption of local equilibrium, we can rewrite equation (7.1.4) as   ð1 − εÞ C 0i1 ∂θi ðz;tÞ ∂θi ðz;tÞ ∂2 θi ðz;tÞ þ vz ¼ Di;eff;z 1þ : 0 ε C i2 ∂t ∂z ∂z2

ð7:1:18cÞ

The boundary and initial conditions are as follows: z ¼ 0; θi ¼ 1 for t > 0; t ¼ 0; θi ¼ 0 for z>0:

z ¼ ∞; θi ¼ 0 for t > 0; ð7:1:18dÞ

The solution for equation (7.1.18c) under the boundary and initial conditions is 8 9 2 9 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi> 8 >

2 4t εDi;eff;z > Di;eff;z ; : 4Di;eff;z γ ; : 9# 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi> =

4t εDi;eff;z > ; : 4Di;eff;z γ where

γ ¼ ð1 þ ðð1 − εÞ=εÞ C 0i1 =C 0i2 Þ:

ð7:1:18fÞ

The erfc quantity is a small number, and under conditions where ðvz z=Di;eff;z Þ is not too large it may be neglected. Then we get " ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)# 1 vz t γ ; ð7:1:18gÞ θi ¼ 1 þ erf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − z 2 4t Di;eff;z 4Di;eff;z tγ which may be rearranged to yield ( " 1=2 #) Pez;eff ðV − V Þ C i2 ðz;tÞ − C ii2 1 ¼ ; 1 þ erf θi ¼  1=2 2 C 0i2 − C ii2 2 VV

ð7:1:18hÞ

where V ¼ εvz Sc t is the volume of feed solution which has passed through the column of length z from t ¼ 0 to time t,   C0 ¼ εvz Sc t V ¼ Sc z½ε þ ð1 − εÞκi1  ¼ Sc z ε þ ð1 − εÞ i1 0 C i2 ð7:1:18iÞ

is the volume of solution required to saturate a column of length z in the absence of dispersion, and   Pez;eff ¼ zvz =Di;eff;z is a Péclet number based on the effective axial diffusion coefficient. Equation (7.1.18e) provides the liquid-phase solute concentration in an infinitely long column at location z from the inlet. Although the solution was developed for an infinitely long column, the value of the concentration profile at any z will also provide the output

concentration profile for a column of length z. The truncated form4 (7.1.18h) of the solution (7.1.18e) represents the complete solution accurately, provided Pez;eff is not too large (Lapidus and Amundson, 1952). Sometimes it is desirable to have the solution (7.1.18h) expressed in terms of t and t (remember erfð − xÞ ¼ − erfðxÞ): ( " #) C i2 ðz;t Þ − C ii2 1 1 − ðt=t Þ ¼ 1 − erf  1=2 : ð7:1:18jÞ 2 C 0i2 − C ii2 2 Di;eff;z t=vz zt

Lapidus and Amundson (1952) have shown that, as the value of vz increases or Di;eff;z decreases (increasing Pez;eff ), the S-shaped breakthrough curve approaches the square wave form of nondispersive operation. On the other hand, if we consider the extent of dispersion around the center of the profile t and V , it varies as t 1=2 or V 1=2 : the extent of dispersion increases with time or volume. We will employ equation (7.1.18h) sometimes to describe the breakthrough. There are a number of models which do not assume the existence of equilibrium between the liquid and the solid phases. However, they do not incorporate the effect of axial dispersion. In such models of nonequilibrium nondispersive operation of the column, the mass-transfer rate between the liquid and the surface of the adsorbent (primarily in the pores) is not infinitely fast; rather it is finite. Further diffusion in the porous adsorbent particle is quite important. One of the earlier models of this type is by Rosen (1952, 1954). Description of the mass-transfer rate between the liquid and the adsorbent particle is facilitated by a specification of the particle geometry. Relatively large particles are often used in industrial adsorbents to reduce pressure drops. Rosen considered spherical particles; the mass balp ance equation for the concentration, C i2 , of species i in the pore liquid of such an adsorbent particle (where the effective diffusion coefficient of species i is Dip ) is   p p  ∂C i2 1 ∂ ∂C r i2 ¼ Dip 2 ð7:1:19aÞ r ∂r ∂t ∂r if spherical symmetry is assumed. (See Table 6.2.1; v ¼ 0; J iθ ¼ J iϕ ¼ 0.) In adsorption, the rate of transfer of species i from the liquid to this particle can be described by means of a mass-transfer coefficient k c and a particle-average p concentration C i2 of species i for particles of radius r p : p C i2



4 3 πr 3 p



¼

rðp 0

p C i2 4πr 2

dr

)

p C i2

3 ¼ 3 rp

rðp

p

C i2 r 2 dr;

0

ð7:1:19bÞ

4

This solution is often the basis for what is called linear chromatography based on linear adsorption equilibrium (7.1.18a).

7.1 

Force −rμi in phase equilibrium: fixed-bed processes

 p

 4 3 ∂C i2 p πr p ¼ k c 4πr 2p C i2 − C i2 jr¼rp 3 ∂t p  ∂C 3k c p C i2 − C i2 jr¼r p : ) i2 ¼ ∂t rp

ð7:1:19cÞ

Rosen had assumed linear adsorption equilibrium between p the liquid-phase concentration in the pore, C i2 , and the p solid-phase concentration in the pore, C i1 : p

p

C i1 ¼ κi1 C i2 : A particle-average concentration C i1 based on the particle volume is then obtained simply from definitions (7.1.19b) as C i1 ¼

3 r 3p

rðp

p

κi1 C i2 r 2 dr:

0

ð7:1:19dÞ

These four equations, along with the form of equation (7.1.4) without any axial dispersion, namely ∂C i2 ð1 − εÞ ∂C i1 ∂C i2 þ þ vz ¼ 0; ε ∂t ∂t ∂z

ð7:1:19eÞ

have been solved by Rosen for the following conditions: p

at t ¼ 0; C i2 ¼ 0 for z > 0 and 0  r  r p ; 0

for t  0; at z ¼ 0; C i2 ¼ C i2 :

ð7:1:19fÞ

The solution is obtained as an integral: ð∞  dλ C i2 1 2 ¼ expf − ηH 1 ðλ;υÞg sin τλ2 − ηH 2 ðλ;υÞ þ ; λ C 0i2 2 π 0

ð7:1:20aÞ

where 0 1 3Dip κi1 z @1 − εA η¼ ; vz r 2p ε

υ¼

Dip κi1 ; kC rp

τ¼

2Dip ðt − fz=vz gÞ ; r 2p

499

Due to the restriction of υ  0:01, this solution also represents negligible external mass-transfer film resistance if the bed is long. Note that this solution is symmetrical around   C i2 =C 0i2 ¼ 1=2. Since these solutions are expressed in dimensionless quantities, results from one particular column could be used to predict the breakthrough curve for other columns. The value of k c needed to make calculations in such systems may be obtained from (Dwivedi and Upadhyay, 1977) !   k c ðSc Þ2=3 0:458 2r p G − 0:407 ; ð7:1:20dÞ ¼ jD ¼ μ ε vz where G is the superficial mass velocity based on empty column cross-sectional area and the Reynolds number   2r p G=μ > 10. There are a number of other analytical solutions for C i2 ðz;tÞ available in the literature for linear isotherms. These take into account axial dispersion, model particles using macropores and micropores, etc.; they have been summarized by Ruthven (1984) in his Table 8.1. Widespread use of powerful computers and sophisticated numerical methods have reduced the importance of such analytical solutions for breakthrough curves. For a constant liquid feed of concentration C 0i2 fed into the bed, an immediate result of the diffuse breakthrough of the feed solute through the column end is that the total solute adsorption capacity of the masstransfer zone of length LMTZ cannot be fully utilized. At the breakthrough time tbr for the breakthrough solute concentration C br i2 at the column end z ¼ L, the solute concentration profile in the solution in the mass-transfer zone is shown in Figure 7.1.5(b). This profile, assumed to be symmetrical, shows that the hatched area representing the integral

H D1 þ υðH 2D1 þ H 2D2 Þ ; H 1 ðλ;υÞ ¼ ð1 þ υH D1 Þ2 þ ðυH D2 Þ2

H 2 ðλ;υÞ ¼ H D1 ¼

S c ρb

0

H D2 ; ð1 þ υH D1 Þ2 þ ðυH D2 Þ2 λ½sinh2λ þ sin2λ − 1; ½cosh2λ −cos2λ

LMTZ ð

H D2 ¼

λ½sinh2λ−sin2λ ; ½cosh2λ−cos2λ

 0  

qi C i2 − qi C i2 dL

represents the additional moles of solute i which could have been adsorbed if there were no diffuse breakthrough. Due to the symmetric concentration profile, one can argue that essentially the total number of solute moles adsorbed in this region,

ð7:1:20bÞ and λ is the variable of integration. This exact solution of the solute breakthrough curve is numerically determined due to the nature of the integral in solution (7.1.20a). If the adsorbent bed length L is very large (specifically, η  50 and υ  0:01), an asymptotic solution of (7.1.20a) is "

C i2 1 ð3=2Þτ − η ¼ 1 þ erf C 0i2 2 2ðη=5Þ1=2

!#

:

ð7:1:20cÞ

ð7:1:21aÞ

S c ρb

LMTZ ð 0

  qi C i2 dL;

ð7:1:21bÞ

is approximately equal to the number of moles not adsorbed, S c ρb

LMTZ ð 0

  qi C i2 dL ¼ Sc ρb

LMTZ ð 0

 0  

qi C i2 − qi C i2 dL;

500

Bulk flow perpendicular to the direction of force

unless the sorption process is highly nonlinear. Therefore, 2 3 LMTZ LMTZ ð ð    0 

14 qi C i2 dL ¼ qi C i2 dL5 S c ρb S c ρb 2 0

0

¼ Sc ρb

LMTZ ð 0

 0  

qi C i2 − qi C i2 dL;

ð7:1:21cÞ

where the second expression in brackets represents the total solute adsorption capacity of LMTZ at tbr for the feed solution concentration C 0i2 . The total adsorption capacity of the bed of length L for the same feed solution concentration C 0i2 under the nondispersive equilibrium condition is ðL     Sc ρb qi C 0i2 dL ¼ Sc ρb qi C 0i2 L:

ð7:1:21dÞ

0

Correspondingly, the lost adsorption capacity of the bed from (7.1.12c) is 2 3 LMTZ LMTZ ð ð  0  0  

14 S c ρb qi C i2 dL5 qi C i2 − qi C i2 dL ¼ Sc ρb 2 0

0

  1 ¼ Sc ρb qi C 0i2 LMTZ : 2 ð7:1:21eÞ

An estimate of the fractional loss of the bed adsorption capacity due to the diffuse breakthrough is therefore given by  0 1 LMTZ 1 LMTZ 2 Sc ρb qi C  i2  ffi : ð7:1:21fÞ 2 L Sc ρb qi C 0i2 L

The quantity the length of the unused bed (LUB), is often used to describe this loss:   1 LMTZ LUB ¼ L: ð7:1:21gÞ 2 L The smaller the length of the MTZ, the lower the loss of bed adsorption capacity due to the diffuse breakthrough. Methods for calculation of LUB are available in Treybal (1980) and Wankat (1990, pp. 366–375). It is useful to recall the basic liquid adsorption system considered so far. There is one solute i to be adsorbed from an inert solvent flowing through a column of adsorbent particles under isothermal conditions. There can be many systems, however, with more than one adsorbable solute species in a solvent which may or may not be inert. For an overview, the reader is referred to Ruthven (1984); original studies of considerable importance in this area are by Glueckauf (1949), Helfferich and Klein (1970) and Rhee et al. (1970a). The adsorption behavior and the breakthrough curve for systems containing a number of adsorbable solutes in a solvent, as illustrated in the above-mentioned references,

are generally quite complex. It is useful, however, to consider an elementary (but highly inexact) analysis of breakthrough in such a system containing more than one solute. Assume that there are, say, three solutes i ¼ 1, 2, 3 present in the feed solution at concentrations of C 012 , C 022 and C 032 , respectively. Assume further that each solute adsorbs independently of the other two and that the adsorption isotherm for each is linear in the following fashion (Figure 7.1.2(a)):       q11 C 12 q21 C 22 q31 C 32 > > : ð7:1:22aÞ C 12 C 22 C 32 Therefore species 3 is least adsorbed, whereas species 1 is most strongly adsorbed, with species 2 being in between. In the context of a nondispersive equilibrium adsorption model, the breakthrough times t i (7.1.15c) for the three species are related by       L ρ q C0 L ρ q C0 t1 ¼ 1 þ b 1 0 12 > t 2 ¼ 1 þ b 2 0 22 > ε C 12 ε C 22 vz vz  0   L ρ q C ð7:1:22bÞ 1 þ b 3 0 32 : t3 ¼ vz ε C 32 The least strongly adsorbed species 3 will come out of the adsorber first (Figure 7.1.5(c)). Correspondingly there will be a pure solution of species 3 at the adsorber outlet during the time period t 2 − t 3 . At time t 2 , species 2 will break through and will be present along with solute 3; Figure 7.1.5(c) plots the total concentration of all solutes at the column outlet as a function of time. We observe a staircase behavior, with all three solutes present at the outlet after t 1 . Figure 7.1.5(c) displays this behavior in the context of a dispersive model. Such an outlet breakthrough profile development is called a frontal development. Frontal development is employed in large-scale purification applications, for example in decolorization of sugar, corn syrup, for removal of oxidation products from waxes, used oils, etc. Adsorbents, such as activated charcoal, are used to adsorb the strongly adsorbing color-causing impurity, etc. One pure species, the least adsorbing one, is obtained. However, the feed concentration of the least adsorbed species can be substantial, making this technique industrially useful. Example 7.1.1 Breakthrough calculations for equilibrium nondispersive operation of a fixed bed with a liquid feed. (a) Lightfoot et al. (1962) have illustrated the breakthrough behavior of a solution of lauric acid in petroleum ether flowing through a packed bed of activated carbon adsorbent, 3 cm long, having ε ¼ 0.4 and a cross-sectional area of 1.32 cm2. The concentration of lauric acid is 0.035M. The equilibrium adsorption isotherm (Freundlich type, see . Determine the break(3.3.112c)) is given as C i1 ¼ 2:26 C 0:324 i2 through volume of liquid when the solute feed concentration will appear as a shock wave at the column end for a similar column 6 cm long.

7.1

Force −rμi in phase equilibrium: fixed-bed processes

(b) Baker and Pigford (1971) have studied the adsorption of acetic acid on activated carbon from a dilute aqueous solution at 60  C. The sorption behavior of the solute was found to be

qi

C i1 ðmg=cm3 Þ ¼ 40 C i2 ðmg=cm3 Þ: Determine the breakthrough solution volume. Solutions (a) The breakthrough volume V of the solution to be passed through the column for the feed concentration to break through in equilibrium nondispersive operation is given by relation (7.1.16d):    ρ q C0 V ¼ Sc L ε þ b i 0 i2 : C i2 Employing definition (7.1.6), we can rewrite this as "  0:324 #   2:26 C 0i2 C0 ; V ¼ Sc L ε þ ð1−εÞ i1 ¼ 1:32  6 0:4 þ 0:6 C 0i2 C 0i2 " # 1 0 C i2 ¼ 0:035M ) V ¼ 7:92 0:4 þ 1:356 ð0:035Þ0:676

) V ¼ 7:92 ½0:4 þ 13:1 ¼ 7:92  13:5 ¼ 107cm3 :

(b) Since the adsorbent particle porosity has been provided, we will employ equation (7.1.15c) to determine the time t for breakthrough. For acetic acid, assume κim ¼ 1. Here the interstitial fluid velocity

vz ¼ Now

volumetric flow rate 15ðcm3 =minÞ ¼ ¼ 11:6cm=min: cross-sectional areaε 3  0:43cm2

L ¼ t

2 3 100 4 1:037  3:025 t¼ 1:755 þ 11:6 ð0:1Þ0:59 2 3 3:13 5 4 ¼ 8:62  13:925: ¼ 8:62 1:755 þ 0:257

gmol ¼ 3:02 C 0:41 i2 ; kg dry carbon

for C i2 in gmol/liter. The activated carbon bed characteristics are as follows: ε ¼ 0.43; εp ¼ 0.57; ρs ¼ 1:82 g=cm3 ; L ¼ 100 cm; Sc ¼ 3 cm2. The feed solution at a concentration of 0.1 gmol/liter is introduced at 15 cm3/min. Determine the time for solute breakthrough and the volume of solution that will have passed through the bed at the time of breakthrough. (c) An enzyme present in a dilute solution is to be adsorbed on cellulosic adsorbent particles in a 100 cm long packed bed 4 cm in diameter, having a bed porosity of 0.4. The enzyme feed concentration is 1 mg/liter; the linear adsorption isotherm is as follows:

vz   ; ρ q C0 ð1 − εÞ 1þ εp κim þ b i 0 i2 ε ε C i2

501

Now, t ¼ 120 min. From equation (7.1.16d), the breakthrough volume   ð1 − εÞ ρ q C0 εp κim þ b i 0 i2 V ¼ Sc Lε 1 þ ε ε C i2

¼ 3  100  0:43  13:925 ¼ 1796 cm3 :

(c)

  C0 V ¼ Sc L ε þ ð1 − εÞ 0i1 C i2 π 2 ¼ ð4Þ  100½0:4 þ 0:6  40 cm3 4 ¼ π  400½24:4 ¼ 30:66 liter: Note: Use of equations (7.1.13d, e) is not recommended for determining t ¼ ðL=vCi Þ for adsorption examples with nonlinear equilibrium behavior since the characteristic lines for different concentrations can overlap, a physically impossible situation (Sherwood et al., 1975; Wankat, 1986). Example 7.1.2 Elution calculation for equilibrium nondispersive operation of a fixed bed with a liquid feed. In Example 7.1.1(a), 150 cm3 of the solution of lauric acid was passed. The bed was completely saturated and the exiting solution had feed concentration 0.035 M. Now pure petroleum ether flow is initiated to elute the adsorbed lauric acid. Calculate the volume of pure petroleum ether passed corresponding to the following concentrations of lauric acid at the bed outlet: 0.03 M, 0.02 M, 0.01 M and 0.005 M. Solution From equation (7.1.17g), we can write

V − V 0 ¼ Sc ðz − z0 ðC i2 ÞÞ ε þ ð1 − εÞ ðdC i1 =dC i2 Þ : Here

z0 ðC i2 Þ ¼ 0; 0

ð − 0:676Þ

ðdC i1 =dC i2 Þ ¼ 2:26  0:324  C i2

:

3

Now, V ¼ 150 cm . Therefore h

i 0:676 V ¼ 150 þ 1:32 z 0:4 þ 0:6  2:26  0:324  1=C i2 h i − 0:676 ) V ¼ 150 þ 3:168 þ 3:47=C i2

(since z ¼ 6 cm for the outlet concentration). The volume of pure petroleum ether passed is h i − 0:676 V − 150 ¼ 3:168 þ 3:47=C i2 for the C i2 values of interest (see Table 7.1.2).

So

100 ¼ t

¼

11:6 ρ ð1 − εp Þð1 − εÞ qi ðC 0i2 Þ 0:57 1þ  0:57  1 þ s ε 0:43 C 0i2 1:755 þ

11:6 ; 1:82  0:43  0:57 3:02 0:59 0 0:43 ðC i2 Þ

Table 7.1.2. Outlet concentration, C i2 (M)

(V − 150) cm3

0.03 0.02 0.01 0.005

~34 ~52 ~82 ~120

502

Bulk flow perpendicular to the direction of force

(a)

(b)

r0 rh

r r

dr

dr

L

Figure 7.1.6 Radial flow fixed bed (adsorbents not shown): (a) flow direction radially outward; (b) flow direction radially inward. Arrows show flow direction.

7.1.1.1.1 Fixed-bed adsorption – radial flow Fixedbed adsorption from a solution is sometimes carried out in a packed bed where the liquid flows radially instead of axially along the column length. Typically there is a central duct or hole in the packed column. The feed solution may be added to the central hole and it flows radially outward (Lapidus and Amundson, 1950); alternatively the feed solution may be forced to enter the packed bed from the periphery, move through the bed radially into the central duct of radius r h (Huang et al., 1988a, b). These two configurations are shown in Figures 7.1.6(a) and 7.1.6(b), respectively. Note that in each case the radial bulk flow is perpendicular to the force direction, which is vertically upward or downward depending on the location of the flow and the particle. A differential equation for the concentration C i2 of species i in the mobile phase (averaged in the z- and θ-directions) may be developed in the manner of equation (7.1.2). Consider a cylindrical layer of adsorbent, of thickness Δr and bed length L, at radial location r for the flow configuration of Figure 7.1.6(a) with a radially outward flow. A species i balance, in the absence of any chemical reaction, is ðrate of accumulationÞ ¼ ðrate of inflow at r Þ

•

where V is the radial volume rate of flow of solution from the inside hole to the periphery of the packed bed. In the limit Δr ! 0, one obtains   • ∂C i2 ∂C i2 ∂C i1 þ ð1 − εÞ þV 2πrL ε ∂t ∂t r ∂r   ∂ ∂C i2 r ; ð7:1:23bÞ ¼ 2πLεDi;eff;r ∂r ∂r •

where V and Di;eff;r are assumed constants. Further, r < r < r þ Δr, and r varies from r h to r 0 . An alternative form is given by (this could also be obtained from (6.2.32) in the manner of (6.2.35), with the z-coordinate replaced by r)   • ∂C i2 ∂C i1 V ∂C i2 Di;eff;r ∂ ∂C i2 ε þ ð 1 − εÞ þ ¼ε : r ∂t ∂t r ∂r ∂r r 2πrL ∂r

ð7:1:23cÞ

If diffusion and dispersion in the radial direction are neglected, this equation is reduced to •

ε

∂C i2 ∂C i1 V ∂C i2 þ ð 1 − εÞ þ ¼ 0: ∂t ∂t 2πrL ∂r

A solution of this nondispersive packed-bed adsorption with radial flow is available (Lapidus and Amundson, 1950) under the condition of elution chromatography (we

− ðrate of outflow at r þ Δr Þ; 9 8 0 1 0 1 = < ∂C

•   ∂C ∂C i2 A • ∂C i2 A i2 i1 @ @ 2πrΔrL ε þ ð1 − ε Þ ¼ V C i2 − 2πL rDi;eff;r ε − V C i2 þ 2πL rDi;eff;r ε : ∂t ∂t ; r rþΔr ∂r ∂r r

ð7:1:23dÞ

r

rþΔr

;

ð7:1:23aÞ

7.1

Force −rμi in phase equilibrium: fixed-bed processes

will soon learn about this) for two cases: (1) equilibrium theory with linear adsorption isotherm; (2) nonequilibrium theory based on a suitable rate equation. The governing equation analogous to (7.1.23b) for the radial flow packedbed adsorption where the effluent flows into the central duct from the bed periphery is   • ∂C i2 ∂C i2 ∂C i1 − 2πrL ε þ ð1 − εÞ þV ∂t ∂t r ∂r   ∂ ∂C i2 : ð7:1:24aÞ r ¼ 2πLεDi;eff;r ∂r ∂r If the diffusion term is neglected, this equation is simplified as follows: •

ε

∂C i2 ∂C i1 V ∂C i2 þ ð 1 − εÞ − ¼ 0: 2πrL ∂r ∂t ∂t

ð7:1:24bÞ

The recovery of bioactive products from their dilute solutions is generally carried out using a fixed bed of adsorbents. Since such solutions are dilute, often large solution volumes have to be processed; also, the solutions may be viscous. In a conventional fixed bed (Figure 7.1.1) of considerable length, this will lead to large pressure drops. A radial flow fixed bed will, however, have a much lower pressure drop. Results of the purification of an enzyme and the removal of proteases from human plasma using radial flow cartridges are available in Huang et al. (1988b). 7.1.1.2

Fixed-bed adsorption: mobile feed gas

Preferential adsorption of a gas species or a vapor species from a flowing gas mixture by solid adsorbent particles in a packed bed is frequently practiced in industry (see Table 7.1.1). A large number of gas purifications (solvents or odors from air, sulfur compounds from natural gas, vent streams, etc., CO2 from natural gas, H2O from a variety of gas streams) and bulk separations (O2–N2, nparaffins from iso-paraffins, mixtures of aromatics, etc.) are achieved by adsorption in a packed bed of adsorbent such as activated carbon, zeolites, carbon molecular sieves, silica gel, etc. (Yang, 1987). When the adsorbent particles are saturated with adsorbates, the bed is regenerated, by heating, by a purge gas or by lowering the pressure. The latter process, known as pressure swing adsorption (PSA), is quite important and is treated in Section 7.1.2. An analysis of fixed-bed adsorption with mobile feed gas (instead of feed liquid) should start with the mass balance equation (7.1.3) for species i. Unlike liquid feeds, the gas velocity vz is a function of z unless some trace compounds are being removed. (For trace compound removal, the volume V of feed gas needed to saturate the column is given by (7.1.16d); the shock velocity is obtained from (7.1.15c)). Therefore, instead of (7.1.4), the general

503

mass balance equation for species i (phase j ¼ 1, solid; phase j ¼ 2, gas) is ∂C i2 ð1 − εÞ ∂C i1 ∂C i2 ∂vz ∂2 C i2 ; þ þ vz þ C i2 ¼ Di;eff;z ε ∂t ∂t ∂z ∂z ∂z2 ð7:1:25Þ where C i2 is the gas-phase molar concentration and C i1 is the solid-phase molar concentration. Analyses of adsorber performance for a gaseous feed often assume nondispersive operation. Models of nondispersive operation either postulate equilibrium between the two phases or describe the mass-transfer rate of species in particular ways. Analysis of the adsorber using an equilibrium nondispersive mode of operation is considered next. It is customary to use the mole fraction x i2 of the gas phase instead of C i2 , the two being related by C i2 ¼ x i2 C t , where C t is the total molar concentration. Here, x i2 is the value of x i2 space-averaged over the x- and y-coordinates in the same manner as C i2 is developed from C i2 . If the gas pressure drop along the packed bed is very small, C t may be considered constant along the bed since C t is proportional to the total gas pressure. The mass balance equation (7.1.25) may now be written in terms of x i2 as (Ruthven, 1984) ∂x i2 ð1 − εÞ ∂C i1 ∂ x i2 ∂vz þ þ vz þ x i2 ¼0 ∂t ∂z ∂z C t ε ∂t

ð7:1:26Þ

under nondispersive condition. The variation of the gas velocity vz with the axial coordinate z is due to adsorption of species onto the adsorbent particles. For a binary gas mixture, either both species may be adsorbed or only one will. Assuming only one species is adsorbed, a mass balance for species i being adsorbed is ð 1 − εÞ

∂C i1 ∂vz ¼ − ε Ct ; ∂t ∂z

ð7:1:27aÞ

where, following relations (7.1.6) and (7.1.7), we can write   ∂C i2 ∂C i1 ρb ∂qi1 ρb ¼ q0 C i2 ¼ ∂t ð1 − εÞ ∂t ð1 − εÞ i1 ∂t   ρ q0 C i2 C t ∂x i2 ¼ b i1 : ð7:1:27bÞ ∂t ð1 − ε Þ Note that if the species adsorbed is in trace amounts, the change in vz along z due to adsorption would be negligible. Thus, for nontrace systems, using these two relations, relation (7.1.26) can be simplified to     ρ q0 C i2 ∂C i2 ∂C i2 1 þ b i1 þ vz ¼ 0: ð7:1:28Þ ð1 − x i2 Þ ∂t ∂z ε Following (7.1.9) and (7.1.11), we can obtain for the wave velocity vCi of concentration C i2 for species i in a gaseous feed of one adsorbed species i (inert carrier)

504

Bulk flow perpendicular to the direction of force

vCi ¼

dz vz : ¼ dt ρb q0 i1 ðC i2 Þ 1 − ð x Þ 1þ i2 ε

ð7:1:29Þ

v 0z

Note that vz changes with bed distance z and mole fraction x i2 ; it would therefore be convenient to replace it by terms containing x i2 . Assume the isotherm to be linear so that   q0 i1 C i2 ¼ constant. Then, combining relations (7.1.27a) and (7.1.27l), the following expression is obtained: ∂vz ρ q0 ∂x i2 ¼ − b i1 : ε ∂t ∂z

C 0i 2

ε

ð7:1:30Þ

T0

Zone I

ð7:1:31Þ

ε

vCi

Ci 1

Zone II

This can be integrated between the limits of v0z , x 0i2 (at the adsorber inlet) and vz , x i2 anywhere else in the adsorber to yield  ρ q0  vz 1 þ b ε i1 1 − x 0i2 ; ð7:1:32Þ ¼ 0 v0z 1 þ ρb qi1 ð1 − x i2 Þ h i ρb q0i1  0 0 dz vz 1 þ ε 1 − x i2 ¼ h ¼ i2 : 0 dt 1 þ ρb εqi1 ð1 − x i2 Þ

Ci2

C 0i 1

Using now relation (7.1.29) in the form of dz=dt in the above equation, we get ∂vz ρ q0 vz i: ¼ − b i1 h ∂x i2 ε 1 þ ρb q0i1 ð1 − x i2 Þ

vz

ð7:1:33Þ

This result implies that, as x i2 increases, vCi increases. In liquid systems with a linear isotherm, equation (7.1.12a), however, indicated no such concentration dependence of vCi . The effect of concentration on vCi in a nontrace gaseous system is, then, similar to that observed with a favorable isotherm (see Figure 7.1.4). A spread out profile along the column is compressed to a front sharper than would have been possible otherwise. This type of behavior aids in creating what is known as constant-pattern behavior. Consider the mass-transfer zone (MTZ) in Figure 7.1.5(b). Near the entrance to the column, such a profile is created spontaneously by the effects of axial dispersion and mass-transfer effects. Further down the column, as the gas velocity decreases to vz from the inlet value v0z due to substantial adsorption, the concentration front tends to become compressed: this compression acts counter to the tendency of axial dispersion and mass transfer to expand the front (as in an unfavorable isotherm of Figure 7.1.4). Often these two opposing tendencies balance each other, so that the MTZ (Figure 7.1.5 (b)) travels through a long column without any change. Such constant-pattern behavior may be achieved also in systems where the adsorbed species is present in trace systems, provided the species displays a favorable isotherm. The adsorption of moisture from air onto silica gel adsorbent is a case in point. Although dispersion and mass-transfer effects, if any, tend to broaden the front,

Zone III

L Figure 7.1.7. Constant-pattern behavior in isothermal singlecomponent adsorption for a gas mixture. (After Sircar and Kumar (1983).)

the favorable isotherm tends to sharpen the front, leading often to a constant-pattern behavior. The nature of the constant-pattern behavior for isothermal adsorption of a single gas species i in the presence of an inert gas is shown in Figure 7.1.7, where the profiles of gas velocity vz , mobile-phase concentration C i2 of species i, adsorbent-phase concentration C i1 and temperature T are shown along the column. There are three zones in the column: Zone I is saturated with species i, therefore the values of vz , C i2 and C i1 are constant; Zone II is the MTZ, where the values of these quantities vary from saturation to values characteristic of the bed where no adsorption has taken place yet, the latter being Zone III. Using the constant-pattern model, Sircar and Kumar (1983) have provided analytical expressions relating the time difference (t2 − t1) in the breakthrough curve corresponding to two arbitrary composition levels C i2 j1 and C i2 j2 for a number of cases. They have also provided analytical expressions for the length of the MTZ. The cases studied include bulk single-component adsorption satisfying Langmuir adsorption isotherm under either gas film control or solid diffusion control. 7.1.1.3

Fixed adsorbent bed regeneration

After the passage of a feed gas mixture or liquid solution through a fixed adsorbent bed for some time, the adsorbent particles are saturated with the solute. For reuse, the bed has to be regenerated; the particles in the bed acting as mass-separating agents need to be restored to their original state so that they may be useful again as massseparating agents. The process of desorption needed for the regeneration of adsorbents may be understood from

Force −rμi in phase equilibrium: fixed-bed processes

505

(c)

(b) TAdsorption

A

qi 1A qi 1

T1

A

qi1A

qi 1

B TDesorption p12

qi1|B

B

Adsorption

qi 1

qi1|B

A

Adsorption

(a)

Desorption

7.1

B T1 p12½Desorption

Figure 7.1.8. Adsorption isotherms for a gas mixture and conditions for adsorption–desorption: (a) thermal swing; (b) purge gas or pressure swing; (c) combined thermal swing and purge gas.

the adsorption isotherms shown in Figure 7.1.8 for gas separation, where pi2 is the partial pressure of species i in the mobile gas phase; pi2 is related to C i2 for ideal gas behavior by pi2 ¼ C i2 RT. If, at the end of adsorption, the adsorbate concentration on adsorbent particles is represented by point A (Figure 7.1.8(a)) and the temperature is, say, TAdsorption, for desorption the temperature is increased to TDesorption, which is higher. The adsorption isotherm now is considerably lower; qi1 is reduced to a lower value, even if pi2 remains the same. The temperature of the bed is raised by supplying either a hot gas or steam. The hot gas may be obtained using the feed gas itself or some other gas, e.g. air. When the partial pressure of species i in the hot gas is lower than that at point A, there is additional desorption since qi1 jB < qi1 jB0 (Figure 7.1.8(c)). After desorption, the bed is ready for adsorption again at a lower temperature, in a cyclic fashion, in processes characterized as thermalswing adsorption (TSA). An alternative procedure to desorbing by a hot gas or vapor is to reduce the pressure of the bed. As shown in Figure 7.1.8(b), although the temperatures of A and B are the same (they are on the same isotherm), the partial pressure of species i in gas phase at B is much lower; therefore, the adsorbate concentration is much lower. The pressure reduction in the bed is achieved either by lowering the pressure of the bed from the higher pressure, in a process usually called blowdown, or by pulling a vacuum. Both strategies lower the value of pi2 , which can also be lowered by passing a purge gas. The process whereby the absolute pressure level of the adsorber is reduced essentially at constant temperature, with or without a purge gas, is called pressure-swing adsorption (PSA)

(see Section 7.1.2). Such processes are cyclic in nature; after desorption, the bed is ready for adsorption again. An elementary basis for quantifying the movement of concentration of the adsorbed species in the column during desorption will now be developed. If we consider the adsorption/desorption of a trace amount of gaseous species from/into a carrier gas vis-à-vis an adsorbent, then equation (7.1.25) is reduced to equation (7.1.4), which is normally used for liquid-phase feeds. If now nondispersive operation is assumed, the governing species balance equation in the column is (7.1.8). Thus, the relevant equation continues to be equation (7.1.9), ∂C i2 vz ∂C  i2 ¼ 0; þ ∂t ∂z ρb q 0i ðC i2 Þ 1þ ε whereby the concentration wave velocity vCi is obtained as (equation (7.1.12a)) vCi ¼

dz vz : ¼ dt ρb q 0i ðC i2 Þ 1þ ε

Let the fluid-phase concentration of a uniformly saturated bed be C 0i2 . This fluid, of concentration C 0i2 , will be pushed out by the desorbing mobile phase entering the column at a velocity vz . The fluid at z ¼ 0 and t ¼ 0 (having a concentration C 0i2 ) will exit from the column of length z ¼ L at time t ¼ ðL=vz Þ. Fluid elements at z > 0 and t ¼ 0 having the initial saturation concentration C 0i2 will exit earlier (Figure 7.1.9(a)). Let the desorbing mobile-phase concentration be C i2 ¼ 0 (it should be < C 0i2 ). The value of vCi corresponding to any C i2 is obtained from the expression (7.1.12a)

506

Bulk flow perpendicular to the direction of force

(a) B

z

the column outlet concentration, C i2 (z ¼ L), changes with t (Figure 7.1.9(b)). To obtain such concentrations, i.e. C i2 (z ¼ L) as a function of time, use the definition for vCi , namely

C 0i 2 C

S

L

dz vz ; ¼ dt ρb q0i ðC i2 Þ 1þ ε

Ci 2 = 0

ð7:1:35Þ

for any given favorable isotherm, e.g. Langmuir isotherm (equation (3.3.112b)), A

t

θi ¼ (b)

bi1 C i2 q ¼ i1 ; 1 þ bi1 C i2 qSi1

ð7:1:36aÞ

where qSi1 is the moles of species i in solid phase 1 at saturation per unit solid-phase mass. Therefore

C 0i 2

q 0i ðC i2 Þ ¼ ðdqi1 =dC i2 Þ ¼

Ci2 exit

bi1 qSi1 : ð1 þ bi1 C i2 Þ2

ð7:1:36bÞ

Substitute this into (7.1.35) using C i2 instead of C i2 and integrate between z ¼ 0 and t ¼ 0 to z and t for any given C i2 :

0 t Figure 7.1.9. (a) Characteristics for desorption of a bed saturated with C 0i2 with an inert purge: equilibrium nondispersive model, favorable isotherm. (b) Column exit concentration for desorption of column in (a).

ðz z ¼ dz ¼  0

vz 1þ

ρb bi1 qSi1 εð1þbi1 C i2 Þ

2

ðt  dt ¼  0

vz t 1þ

ρb bi1 qSi1 εð1þbi1 C i2 Þ

2

:

ð7:1:37aÞ

Rearrange this as given earlier. Consider any favorable adsorption isotherm (Figure 7.1.2(b)). Obviously,     ð7:1:34aÞ q0i1 C i2 jC i2 ¼0 > q 0i1 C i2 jC 0i2 : Therefore

vCi jC i2 ¼0 < vCi jC 0i2 < vz :

ð7:1:34bÞ

The characteristics lines for C i2 ¼ 0 have been shown in Figure 7.1.9(a). When such a line, AS, starting at t ¼ 0 intersects z ¼ L, the desorbing fluid breaks through the column, i.e. the column desorption is complete. Between this line AS and the line AB (whose slope is that of the fluid velocity, vz) lie two regions. The first region is between lines AB and AC, where the desorbing fluid has arrived but its concentration is still C 0i2 ; such lines have origins on AB at different column locations and they exit the column with velocity vCi jC 0i2 . Such a line originating at z ¼ 0 and t ¼ 0 is shown as AC. To the right of such a line, the mobile-phase column concentrations are changing. Further, the fluid exiting the column now has concentrations between C 0i2 and C i2 ¼ 0. The column exit concentrations as a function of time are shown in Figure 7.1.9(b). For a favorable isotherm, this implies that the slopes of lines originally at t ¼ 0, z ¼ 0 now decrease. Ultimately, the line AS is reached when the bed is completely desorbed. Note that

C i2 1 ¼ C 0i2 bi C 0i2

(

zρb bi1 qSi1 εðvz t − z Þ

1=2

)

−1 :

ð7:1:37bÞ

For z ¼ L, the bed length, this expression (Walter, 1945) provides the breakthrough curve for desorption at the column outlet, a relation between C i2 jexit and t, as shown in Figure 7.1.9(b). Further, the whole bed is completely desorbed when C i2 ¼ 0 at z ¼ L; this happens when L vz i: ¼h ρ b qS t 1 þ b i1 i1 ε

ð7:1:37cÞ

Recall that these two results are valid for nondispersive equilibrium desorption of a bed, initially saturated throughout at a level of C 0i2 by a mobile phase without any solute species, when Langmuir adsorption isotherm characterizes the adsorption equilibrium between the solute in the mobile phase and the adsorbent. The breakthrough curve developed above for desorption was based on a pure mobile phase, i.e. C i2 ¼ 0, sent in to desorb. Often such a phase is at a higher temperature, e.g. a hot purge gas. In such a case, the species balance equation (e.g. (7.1.9)) is coupled with an overall heat balance equation. A solution of such a system is schematically illustrated in Figure 7.1.10(a) for nonequilibrium, nonisothermal desorption of a single adsorbed species

7.1

Force −rμi in phase equilibrium: fixed-bed processes

507

Figure 7.1.10. (a) CO2 concentration profile ( C i2 ) and temperature profile along the bed length at any time in desorption by a hot purge. (After Kumar and Dissinger (1986).) Schematics of the (b) purity (% normal paraffins) and (c) flow rate of the hydrocarbon stream leaving the sieve bed during steady state cyclic operation of the Ensorb linear paraffins unit at Baytown Refinery (arbitrary units). The lines represent the system behavior based on theoretical curves and plant data. (After Ruthven (1984).)

(a)

T Ci2

L z

z=0 (b)

(c) Adsorption

3

10

50

Cyclic operation One cycle: 27 min

2

Inlet Flow rate

100 Purity (%)

Purity (%)

Desorption

Hydrocarbon flow rate

15

HC NH3

1

5

0

14

Adsorption (13.5 min)

Desorption (13.5 min) 0

0

5

10 15 20 Time (min)

25

30

0 0

(e.g. CO2) from an adsorbent (0.5 nm molecular sieve) by a hot purge of N2 (Kumar and Dissinger, 1986). This figure shows the temperature and CO2 concentration profile along the bed length at any instant of time. The concentration of CO2 in the saturated bed is shown near the column end (near z ¼ L), where no desorption process has been initiated yet. In the initial sections of the column near z ¼ 0, where desorption is complete, the very low value of C i2 reflects the purge concentration of CO2, if any, just as the high purge gas temperature is present. Then there is a MTZ, where CO2 concentration in the bed rises and the temperature falls. Finally, there is an additional MTZ further down the column which pulls down the higher temperature and gas-phase concentration to the values existing in the saturated column prior to purge introduction. There have been a number of investigations into the behavior of a column subjected to desorption. Although these include studies on isothermal desorption (Zwiebel et al., 1972; Garg and Ruthven, 1973), nonisothermal desorption based studies are often more realistic. The reader interested in detailed understanding should refer to papers by Rhee at al. (1970b, 1972), Basmadjian et al. (1975a,b), Kumar and Sircar (1984) and Kumar and

5

10

15

20

25

30

Time (min)

Dissinger (1986). A comprehensive introduction to bed regeneration is available in the treatment of cyclic batch processes by Ruthven (1984). There are additional methods for regenerating an adsorbent bed beside TSA and PSA. In the case of gas/ vapor mixtures, the introduction of a displacement purge gas (as opposed to an inert purge gas) into the bed serves first the same function as in PSA, namely the reduction of the partial pressure, pi2 , which facilitates desorption of the adsorbed gas species i. Second, there is a separate and more important function: the displacement purge gas adsorbs strongly and competes with the adsorbed feed species to be separated. Further, if the temperature of the displacement purge gas/vapor is higher, as in the case of steam stripping of activated carbon to strip organic solvents or volatile organic compounds, then we have the combined effects of TSA, the reduction of pi2 and desorption via displacement. A major requirement for a successful displacement purge gas/vapor, the displacer, however, is that it should not be too strongly adsorbed compared to the feed species to be desorbed. If it is, then it would be difficult to desorb it during the adsorption part of the next cycle. In fact, ideally, it should have an almost identical affinity for adsorption on

508

Bulk flow perpendicular to the direction of force

the adsorbent as the feed species to be desorbed. Then, during the beginning of the next cycle, the species in the feed mixture which is preferentially adsorbed will displace the displacer adsorbed during the regeneration part of the cycle. Even under optimal conditions, this displacer will contaminate the preferentially nonadsorbed (raffinate) species/stream, leaving the adsorber during the adsorption part of the cycle. It is desirable therefore to use a displacer which is easily separated from the raffinate stream. An example of such a displacer is ammonia in the large-scale separation of linear paraffins of medium molecular weight (C10–C16) from branched chain and cyclic isomers. The adsorbent is a 0.5 nm molecular sieve, which, at 550–600  F and slightly above atmospheric pressure, adsorbs the linear paraffins strongly from the feed mixture (Figures 7.1.10(b) and (c)). Such a high feed temperature precludes a TSA process for desorption lest it should lead to cracking/coking. Ammonia is, therefore, used to desorb these paraffins. A higher ammonia flow rate is required since it is adsorbed somewhat less strongly than the paraffins (Ruthven, 1984), but it is easily separated from the paraffins due to its high volatility after cooling the mixture and flashing it out via distillation. Another displacer for the system is n-hexane, which also possesses a high volatility with respect to the other species in the system. A brief introduction to these and related processes has been provided by Ruthven (1984). An introduction to the problems of modeling in such multicomponent systems is provided in Wankat (1990, pp. 394–400). Displacer based desorption is practiced more often in liquid-phase processes, particularly in ion-exchange resin based processes. Conventional adsorption based liquidphase processes are not particularly suitable for displacer based processes. Further, displacer based processes are invariably used in multicomponent separations (Rhee and Amundson, 1982) and are properly considered under chromatographic processes (see Section 7.1.5) (Frenz and Horvath, 1985). 7.1.1.4

Ion exchange beds

To remove a particular ion from an aqueous solution, a column containing ion exchange resin beads is often used, especially if the ion is present at low concentrations. The ion exchange bed/column may also be used to replace one particular ion in solution by another ion. The resin beads are typically introduced in a vertical cylindrical column and are supported from the bottom; the top section is free so that, as the resins expand, there is room for expansion; this is operationally important since resin beads swell in water. Figure 7.1.11(a) illustrates a column filled with vertical resin beads. Consider the following process scenario. The ion exchange resin beads are to participate in an ion exchange process between two counterions A and B, with Y being the

co-ion; the ion exchange resin beads in the column to start with are in B form and the solution being continuously fed from the column top contains essentially AY. As the solution keeps coming down the column, the beads at the top of the column are converted to the A form. Just as we observed in Figure 7.1.5(b) how the column concentration profile developed in an adsorbent column fed continuously with a solution of concentration C 0i2 , similarly here the top of the column will be converted to the A form. There will also be a mass-transfer zone (MTZ, as in Figure 7.1.5(b)) where, at one end, the column is in A form and, at the other end, it is in B form, with a continuous transition going on in between (Figure 7.1.11(b)). What would be of interest here is to find out when the feed ionic species A is going to break through from the column end, as shown in Figure 7.1.11(c). As in Figure 7.1.5(a), the actual breakthrough curve is going to be diffuse, shown by the dashed line in Figure 7.1.11(c). However, we will calculate the time corresponding to a step jump in the column outlet concentration of ionic species A (solid line in Figure 7.1.11(c)) in the manner of Figure 7.1.3(b). The governing equation for the concentrations of ionic species A in the two different regions, mobile phase ( j ¼ 2) and the ion exchange resin phase ( j ¼ 1), namely C A2 and C A1 , respectively, will continue to be equation (7.1.4)): ∂C A2 ð1 − εÞ ∂C A1 ∂C A2 ∂2 C A2 þ þ vz ¼ DA;eff;z : ∂t ∂t ∂z ∂z 2 ε An item to be noted here is that, even though the resin is swollen and highly porous (εp 6¼ 0), the ions inside the ion exchange resin beads are essentially next to the fixed charges in the ion exchange resin beads. Therefore, the concept employed to develop equation (7.1.13e) is not valid here; we therefore employ, de facto, εp ¼ 0. The analysis carried out earlier to arrive at the concentration wave velocity expression (7.1.12a) for vCi is also valid here. Therefore, for equilibrium nondispersive operation of an ion exchange column, we can write, for ionic species A, vCA ¼ h

vz 1þ

ρb q0 A ðC A2 Þ ε

i:

What we will do now is to deliberate on the ion exchange equilibrium behavior of species, say A. However, note what ρb ð∂ qA ðC A2 Þ=∂C A2 Þ ¼ ρb q0 A ðC A2 Þ used for solute A stands for here (see notation (3.3.121f–n)): ρb

ð7:1:38Þ the nature of a given ionic the quantity adsorption of in equations

∂ qA ðC A2 Þ ∂qðC A2 Þ ∂ fρR qðC A2 Þg ¼ ρR ð1 − εÞ ¼ ð1 − εÞ ∂ C A2 ∂ C A2 ∂ C A2 ¼ ð1 − εÞ

∂ C AR : ∂ C A2

ð7:1:39aÞ

7.1

Force −rμi in phase equilibrium: fixed-bed processes

(a)

(b)

509

CA2

Feed Feed A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

B

A

A

B

B

A

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

Treated solution

Column

Exhausted part of the column

Mass-transfer zone

Unconverted part of the column

Treated solution

Resin beads

Diffuse

(c) 0

CA 2

Sharp (shock wave) 0 t V

t br

Figure 7.1.11. Mass-transfer/concentration profiles in an ion exchange column: (a) sections of the column where it is in A form, B form or in mixed AB form; (b) concentration profile of the A form of the resin in the column at any time; (c) breakthrough behavior at the column bottom, sharp breakthrough and diffuse breakthrough.

However in view of relations (3.3.121g) for the resin phase and (3.3.121h) for the external solution phase, we can express the resin phase and the external solution phase concentrations in terms of mole fractions x AR and x Aw in the resin and water phase, respectively:5 x AR ¼

5

C AR C AR ¼ ; C AR þ C BR C FC

x Aw ¼

C Aw C Aw ¼ : C Aw þ C Bw C tw ð7:1:39bÞ

The bar “–” on top of any quantity refers to a quantity averaged over the column cross section.

Here we define the system in terms of the binary mixture of counterions A and B and ignore the presence of any co-ion Y on other ions. Therefore, the concentration wave velocity vCA of ionic species A through the column is given by (for a given concentration, say C Aw or mole fraction x Aw ) vz vCA ¼  : ð1 − εÞ C FC ∂ x AR 1þ ε C tw ∂ x Aw

ð7:1:40aÞ

The corresponding expression for vCA jshock may be obtained from a mass balance (as in equation (7.1.15c)):

510

Bulk flow perpendicular to the direction of force

vz vCA ¼   ð1 − εÞ C FC Δ x AR 1þ ε C tw Δ x Aw vz ¼  : ð1 − εÞ C FC x AR jshock − x AR jinitial 1þ ε C tw x Aw jshock − x Aw jinitial

ð7:1:40bÞ

For an initially empty column, x AR jinitial ¼ x Aw jinitial ¼ 0. (Note: In determining ðdx AR =dx Aw Þ, the overbar has no relevance since the equilibrium relation is independent of column cross section. However, there may be variations along the cross section of x Aw , therefore x Aw is relevant.) In such a context, the Langmuir ion exchange isotherm (3.3.121n) for counterion A may be written as qAR C AR K AB C Aw =C tw : ¼ ¼ x AR ¼ qmax R C FC ð1 þ ðC Aw =C tw ÞðK AB − 1ÞÞ

ð7:1:41Þ

Note that here the counterionic species under consideration along with counterion A is counterion B, so that (compare (3.3.121g,h)) C AR þ C BR ¼ C FC ; x AR þ x BR ¼ 1 ;

C Aw þ C Bw ¼ C tw ; x Aw þ x Bw ¼ 1;

ð7:1:42aÞ ð7:1:42bÞ

and the equilibrium constant K AB for the ion exchange process between counterions A and B is K AB ¼

C AR C Bw : C Aw C BR

ð7:1:42cÞ

From isotherm (7.1.41), one can calculate ð∂x AR =∂x Aw Þ for the Langmuir isotherm as dx AR K AB ¼ : dx Aw ð1 þ x Aw ðK AB − 1ÞÞ2

ð7:1:42dÞ

Following the column mass balance based development of (7.1.15c), one can determine the concentration wave velocity of the shock wave at time t when the exit concentration jumps from C i2 ¼ 0 to C 0i2 , the incoming concentration: L vz ¼ vCi ¼ vCi jshock ¼  0 : t ð1 − εÞ C i1 1þ ε 0

ð7:1:43aÞ

C i2

The corresponding quantities in the ion exchange process for the counterion A are L vz  ¼ vCi jshock ¼  ð7:1:43bÞ t ð1 − εÞ C AR 1þ ε C Aw

for an initially empty column. Therefore L ¼ vCi jshock ¼  t

vz ð1 − εÞ C FC x AR 1þ ε C tw x Aw

:

ð7:1:43cÞ

Example 7.1.3 Using the equations developed above, we will now illustrate an ion exchange system worked out by

Wankat (1990, pp. 465–470) for removal of Kþ from a solution using an ion exchange bed in the form of Naþ. Langmuir isotherm describes this ion exchange with KAB ¼ 1.54, where A ¼ Kþ and B ¼ Naþ. The mole fraction of potassium in the 0.2 N feed solution x Kw ¼ 0:7. Determine the concentration wave velocity of Kþ and determine when this shock wave exits the column of length 30 cm and diameter 2 cm if the volumetric flow rate of the solution is 20 cm3/min. Next, regeneration of the bed is implemented by passing a solution of 0.2N NaCl. Calculate the Kþ concentration in the effluent as a function to time (eluent volume). You are given: ε ¼ 0.4, C FC ¼ 3.96 equivalent/liter of resin. Solution To employ equation (7.1.43c) for the shock wave velocity vCA jshock , we need vz, the interstitial velocity:

20

cm3 π 2 π 20 ; ¼ d  vz  ε ¼  4cm2  vz  0:4 ) vz ¼ 4 π  0:4 min 4 15:9cm=min 3: 0 10 1 41þ @0:6A @3:96A x AR 5 0:4 0:2 x Aw

vz ¼ 15:9cm=min ) vCA jshock ¼ 2

For the Langmuir isotherm, (7.1.41),

x AR ¼

K AB x Aw 1:54  0:7 1:078 ¼ ) x AR ¼0:782: ¼ 1þ x Aw ðK AB − 1Þ 1þ0:7ð0:54Þ 1:378

Therefore

15:9 cm=min ½1 þ 1:5  18  ð0:782=0:7Þ 15:9 cm=min ¼ 0:465 cm=min ¼ vCK jshock : ¼ 34:18

vCA jshock ¼

Since the column length is 20cm, this shock wave will appear at the column end at 43 ¼ 20=vCK jshock minutes. In a process to remove Kþ from the solution, the feed solution flow will be stopped somewhat earlier so that the effluent solution from the column is Kþ-free. We will now focus on the regeneration of the bed by the 0.2 N NaCl stream at t ¼ 43 minute. If we employ equation (7.1.17g) for z 0 ðC i2 Þ ¼ 0 for C K2 (at the column inlet), then

 ð1 − εÞ V − V 0 ¼ Sc z ε 1 þ ε  ð1 − εÞ ¼ Sc z ε 1 þ ε

 dC K1 dC K2  C FC dx AR : C tw dx Aw

We know that V 0 ¼ 43 minute  20 cm3 =min ¼ 860 cm3 . Now

K AB ½1 þ x Aw ðK AB − 1Þ2 K KNa 1:54 ¼ : ¼ ½1 þ x Kw  0:542 ½1 þ x Kw ðK KNa − 1Þ2

ðdx AR =dx Aw Þ ¼

The results in Table 7.1.3 illustrate the diffuse nature of the Kþ-containing stream exiting the column. The book by Wankat (1990, pp. 463–494) should be read for greater details of such equilibrium nondispersive calculation procedures for ion exchange columns. See Sherwood et al. (1975) for models involving mass-transfer resistance.

Force −rμi in phase equilibrium: fixed-bed processes

Table 7.1.3.

x Kw 0.7 0.5 0.3 0.1 0.0

Dry air out

  ð1−εÞ C FC dx AR ¼ ðV −V 0 Þ cm3 Sc z ε 1 þ ðdx AR =dx Aw Þ ε C tw dx Aw 0.81 0.95 1.14 1.4 1.54

630 734 850 1045 1174

7.1.2 Pressure-swing adsorption process for gas separation The fixed-bed processes studied earlier use adsorbents which are mass-separating agents. These mass-separating agents have to be regenerated for reuse. An introduction to the process of fixed-bed adsorbent regeneration was provided in Section 7.1.1.3. The bed may be regenerated by a temperature increase, a pressure decrease, a purge stream, a displacer stream or various combinations of these. In an actual fixed-bed separation process, the adsorption step and the desorption step are combined in a cyclic fashion. The manner in which these cycles are operated has led to a few distinct classes of fixed-bed processes. The nature of the introduction of the feed (semicontinuous, pulse, temperature programmed, etc.) is integral to the development of such distinct processes. However, the direction of the force between the fluid and the fixed bed in the following processes always remains perpendicular to the bulk fluid flow direction: (1) (2) (3) (4) (5) (6) (7)

511

thermal-swing adsorption; pressure-swing adsorption (PSA); inert purge; potential-swing adsorption; parametric pumping; cycling zone adsorption; chromatographic processes.

We focus first on pressure-swing adsorption processes. Potential-swing adsorption is mentioned in passing. In the three sections that follow, we consider parametric pumping, cycling zone adsorption and chromatographic processes. Thermal-swing adsorption and inert purge have already been briefly considered. The basic mode of operation in pressure-swing adsorption will be illustrated first using the Skarstrom cycle (Skarstrom, 1960, 1975) employed for air drying (Figure 7.1.12). The process uses two identical beds of a granular solid adsorbent which preferentially adsorbs moisture from air. Wet air at 40 psig enters bed A through the connection shown by a solid line in a four-way solenoid valve. The air leaving the other end of bed A is essentially dry air at 40 psig. About half of the wet air entering is taken off as dry air product; the other half, also produced as dry air, is

Adsorbent bed B

Wet air out

7.1

Adsorbent bed A

Solenoid valve

Wet air in Figure 7.1.12. Continuous adsorption drying system using pressure-swing adsorption. (After Skarstrom (1960, 1975).)

throttled down to essentially atmospheric pressure and introduced into bed B from the opposite end (i.e. the exit end of bed A). This low-pressure dried air acts as a purge to desorb the moisture from bed B, which has already been saturated with H2O during a previous cycle. The resulting wet air is taken out through the front end of bed B and discharged to the atmosphere. This process is continued for 3 minutes, by the end of which bed A is saturated with moisture and bed B is essentially dry and regenerated. Then the solenoid valve connection is changed. Wet air is introduced at 40 psig into bed B by a connection in the solenoid valve, shown in Figure 7.1.12 as dashed lines. This wet air leaves bed B at the other end dry. Half of this exiting dried air is taken out as dry product at 40 psig; the other half is throttled down to near atmospheric pressure and sent to the previous product end of bed A, whose pressure has now been quickly reduced. This dry air, acting as a purge, flows through bed A in the opposite direction and exits at the other end as wet air through the connection, indicated by the dashed lines of the four-way solenoid valve. It is released to the atmosphere. This process continues for another 3 minutes, when bed B becomes saturated and bed A is regenerated. At this time, the cycle is complete, and wet air is sent to bed A ready for adsorption as before, while bed B has to be regenerated. Meanwhile, dry air at 40 psig is obtained continuously as the product. Such a process was originally known as heatless fractionation or heatless adsorption since no heat was used to regenerate the adsorbents. Currently the term

512

Bulk flow perpendicular to the direction of force

pressure-swing adsorption (PSA) is used instead. Skarstrom (1975) demonstrated that air containing 3800 volume ppm of H2O (11 mm Hg H2O partial pressure) at 40 psig was dried to 1 ppm H2O at the same pressure. However, it took some time (5 days) to achieve a steady state in the moisture concentration in the dried air output. In modeling PSA processes, therefore, steady state achievement is usually indicated by n ! ∞, where n is the number of cycles. Since the original patent of Skarstrom was issued in 1960 (Skarstrom, 1960), there has been considerable progress in the research and development of PSA processes. A large number of commercial plants based on PSA are used to purify or fractionate a variety of gas mixtures and vapor mixtures. These include: drying of air and other gases; air separation by zeolites (N2 is more strongly adsorbed than O2 at equilibrium) and by carbon molecular sieves (O2 more rapidly adsorbed, therefore the high-pressure product is a highly purified N2 stream); H2 purification to very high levels (~99.9999%) from various refinery streams; adsorption removal of n-paraffins from mixtures containing branched-chain isomers and cyclic hydrocarbons, etc. (vacuum desorption is used for n-paraffins with carbon numbers less than 10). The adsorbent particle diameter varies between 1/1600 pellets to as low as 60 mesh. The number of beds used simultaneously is often more than two, going all the way up to ten. The cycle time can go as low as a few seconds, especially if a single-bed based rapid PSA is used (Keller, 1983). For feed gases slightly above atmospheric pressure, vacuum-swing adsorption has been used to desorb via vacuum (Sircar and Zondlo, 1977). An excellent treatment of PSA based processes is available in Yang (1987), which also provides a critical account of PSA models and experiments. Before we consider PSA models, one should recognize that separation by adsorption in PSA can take place in two general ways: equilibrium separation and kinetic separation. In equilibrium separation, the adsorbent preferentially adsorbs one or more species in preference to others on an equilibrium basis. In zeolite adsorbents, as well as molecular sieves, molecules having an appropriately small dimension can enter the zeolite pores and are adsorbed, whereas others cannot enter the pores due to steric effects and are excluded. Such adsorption processes are considered under equilibrium separation processes (Yang, 2003). On the other hand, adsorption separation by kinetic effects involves adsorbents through whose pores one of the species diffuses much more readily than others (e.g. O2 diffuses 30 times faster than N2 through carbon molecular sieves having the right pore opening dimensions). Such separations come under a kinetic separation mechanism in PSA processes. 7.1.2.1

An equilibrium nondispersive PSA model

Consider the two columns or two beds used in Skarstrom’s scheme for air drying. A simple PSA cycle for this process involves four steps, shown schematically in Figure 7.1.13(a), where two beds, bed 1 and bed 2, are identified

by 1 and 2, respectively. We provide first the cycle description according to Chan et al. (1981). A high-pressure feed mixture is continuously supplied to the top of the bed 1 in step 1 and the adsorbable species are taken up by the bed. The effluent from the end of the bed is sufficiently purified and still at a high pressure. A part of this effluent becomes the desired product; the pressure of the remaining effluent is reduced without any change in mole fraction to the low-pressure level of bed 2, where it is introduced to the column top as a low-pressure purge to desorb species adsorbed by bed 2 during an earlier cycle. Figure 7.1.13(b) shows the essentially constant bed 1 pressure and steady flow rates of high-pressure feed and product and low-pressure purge during step 1 (Weaver and Hamrin, 1974). After some time, bed 1 approaches breakthrough conditions; step 2 is therefore initiated, wherein bed 1 is depressurized and there is removal of the adsorbed species in the blowdown stream from the end of the bed where the highpressure feed was introduced earlier. Simultaneously, bed 2, ready for adsorption, is pressurized by the high-pressure feed gas mixture. Step 2 is of short duration, and is followed by step 3, in which bed 2 is fed with high-pressure feed gas continuously. In step 3, bed 2 behaves as bed 1 in step 1, just as bed 1 behaves in step 3 as bed 2 did in step 1. Similarly, in step 4, bed 1 is subjected to pressurization with high-pressure feed (as bed 2 was in step 2); while bed 2 undergoes blowdown prior to complete regeneration by a low-pressure product purge (as in step 1) A model developed by Chan et al. (1981) for the purification of a binary gas mixture of species A and B by such a PSA cycle when A is present at a trace level will be presented below. Focus on bed 1’s operation. The assumptions employed in this model are: (1) isothermal operation; (2) negligible pressure drop along the bed during highpressure feed flow or low-pressure purge flow; (3) gas–solid equilibrium exists always and everywhere; (4) linear adsorption isotherms are valid, with species A being preferentially adsorbed: κA1 > κB1 : C A1 ¼ κA1 C A2

ð7:1:44aÞ

C B1 ¼ κB1 C B2 ;

ð7:1:44bÞ

(5) no axial dispersion; (6) ideal gas law is valid; (7) the interstitial gas velocity vz is constant during constant-pressure steps 1 and 3. Any model for PSA should provide ways of calculating the fractional recovery of the less adsorbed species, its purity, the purity of the strongly adsorbed species dominated fraction, the minimum bed length, etc. We follow to these ends the treatment by Chan et al. (1981) and focus on steady state operation obtained after initial transients. The governing

7.1

Force −rμi in phase equilibrium: fixed-bed processes

513

(a) Step 1 High-pressure feed

Step 2 Blowdown

Step 3 Low-pressure purge

Step 4 Pressurization with feed

1

1

1

1

Product

Product

2

2

2

2

Low pressure purge

Pressurization with feed

High pressure feed

Blowdown

(b) Bed 2 Step #

1

2

Bed 1 3

1

4

2

3

4

High Pressure Low 0

0

0

0

0

0

0

0

Feed flow rate

Product flow rate

Purge flow rate

Figure 7.1.13. (a) Four steps in a PSA cycle. (b) Diagrams of pressure and flow rate changes for beds 1 and 2 during one cycle. (After Weaver and Hamrin (1974).)

514

Bulk flow perpendicular to the direction of force

balance equations for species A and B are obtained from equation (7.1.3) under nondispersive condition as    ∂C A2 ∂ vz C A2 ∂C A1 þ ð1 − εÞ þ ¼ 0; ð7:1:45aÞ ε ∂z ∂t ∂t    ∂C B2 ∂ vz C B2 ∂C B1 þ ð 1 − εÞ þ ¼ 0: ð7:1:45bÞ ε ∂z ∂t ∂t Recall that C A2 and C B2 refer to (z, t)-dependent molar concentrations of species A and B in the gas phase, while C A1 and C B1 refer to those in the solid adsorbent phase, as before. Replace κig by yi here. Due to ideal gas behavior, C A2 ¼ pA =RT ¼ ðPyA =RT Þ;

C B2 ¼ pB =RT ¼ ðPyB =RT Þ;

ð7:1:46Þ

where P is the total pressure and y A and yB are the local (z, t)-dependent mole fractions of species A and B in the gas phase. Since species A is present in very dilute concentration, we may assume that y B ffi 1 and represent y A henceforth only by y. Substituting these into equations (7.1.45a) and (7.1.45b) (where we now use the equilibrium relations (7.1.44a) and (7.1.44b)) leads to   ∂ðPy Þ ∂ðvz PyÞ ∂ðPyÞ þ ¼ 0; ð7:1:47aÞ þ ð1 − εÞκA1 ε ∂t ∂z ∂t   ∂P ∂ðvz P Þ ∂ðP Þ þ ¼ 0: ð7:1:47bÞ þ ð1 − εÞκB1 ε ∂t ∂z ∂t Since ð∂P=∂zÞ is zero by assumption in either bed/column at any time, we get     ∂P ∂y ∂y ∂vz ∂P ∂y ε y þ P þ vz P þ Py þ ð1 − εÞκA1 y þ P ¼ 0; ∂z ∂t ∂t ∂z ∂t ∂t ð7:1:48aÞ

ε





∂P ∂vz ∂P þP þ ð1 − εÞκB1 ¼ 0: ∂t ∂t ∂z

ð7:1:48bÞ

Now multiply equation (7.1.48b) by y and subtract it from equation (7.1.48a) to obtain ½ε þ ð1 − εÞκA1 

∂y ∂y ∂ln P þ εvz ¼ − ð1 − εÞðκA1 − κB1 Þy : ∂t ∂z ∂t ð7:1:49aÞ

To use the method of characteristics (see equations (7.1.12b) to (7.1.12f)), we can write any solution of y as     ∂y ∂y dt þ dz ¼ dy ð7:1:49bÞ ∂t ∂z to obtain    dy dz     d ln P  − ð1 − εÞðκA1 − κB1 Þy εvz     dt  ∂y   ¼   ∂t dt dz      ε þ ð1 − εÞκA1 εvz 

ð7:1:50Þ

If the determinant in the denominator is zero, we get dz εvz vz i: ¼ ¼h dt ½ε þ ð1 − εÞκA1  1 þ ð1 − εÞ κA1 ε

ð7:1:51aÞ

For ð∂y=∂t Þ to be finite, the numerator has to be zero, which leads to εvz

d ln y d ln P ¼ − ð1 − εÞðκA1 − κB1 Þ : dz dt

Using (7.1.51a), this may be rearranged to yield   d ln y ð1 − εÞðκA1 − κB1 Þ d ln P d ln y β d ln P ¼− ) ¼ A −1 ; dt dt dt ½ε þ ð1 − εÞκA1  dt βB ð7:1:51bÞ

where βA ¼

ε ε ; βB ¼ ; βA < βB : ð7:1:52aÞ εþ ð1−εÞκA1 εþ ð1−εÞκB1

Note that, if the species are strongly adsorbed, i.e. ð1−εÞκA1 >> ε and ð1−εÞκB1 >> ε, then the separation factor αAB for species A and B is αAB ffi

κA1 βB 1 ¼ ¼ : κB1 βA β

ð7:1:52bÞ

Since the interstitial velocity vz is constant during steps 1 and 3 by assumption (7), the characteristic velocity of any y value, ( dz=dt), is also constant from (7.1.51a) since κA1 is constant. On the other hand, vz varies during steps 2 and 4; the pressure also changes, just as mole fraction y also changes. The interstitial velocity during steps 1 and 3 may be obtained by dividing the volumetric flow rate by εSC , where SC is the column cross-sectional area. We will now calculate the changes in the position of the characteristics for these two kinds of changes. For steps 1 and 3, integrate (7.1.51a) and obtain, respectively, ΔzH ¼ βA vzH Δt;

ð7:1:53aÞ

Δz L ¼ βA vzL Δt:

ð7:1:53bÞ

Here ΔzH and ΔzL , called the penetration distances, are the net displacements of the concentration wave front in the high-pressure feed step 1 (subscript H), having a steady interstitial gas velocity vzH , and the low-pressure purge step 3 (subscript L), having a steady interstitial gas velocity vzL , respectively. For the blowdown (step 2) and pressurization with feed (step 4), consider equation (7.1.51b) first. This allows the gas mole fraction y to be related to pressure P in the column in two cases: as the high pressure, PH, is reduced very quickly to a low pressure, PL, during blowdown step 2, y is increased from yH to yL; as the low pressure, PL, is increased to PH during the pressurization with feed in step 3, y is decreased from yL to yH. Integration of equation (7.1.51b) leads to estimates of such changes:

7.1

Force −rμi in phase equilibrium: fixed-bed processes

(a)

515

(b) Pressurize Feed

Blowdown Purge

G

A

1.0

Pressurize Feed

B

∆zH

0.8

D

0.6

z/L

yH = y f

0.4

C

A

1.0

∆zL

F

Blowdown Purge

D

0.8 0.6

z/L

E

0.4

yH = yH (z)

E

B

yH = yf yH = yH (z)

0.2

0.2

0.0

C

0.0

t0 t1

t2 t3

t4

t0 t1

t2 t3

t4

t

t

Figure 7.1.14. Characteristics movement during a PSA cycle mixture of h i for the adsorption of a binary h i A and B very dilute in A; G ¼ Gcrit, ðL=ΔzL Þ ¼ 2, ðPH =PL Þ ¼ 10. (a) β ¼ 0:05, 1 − ½PH =PL −β L − ΔzH < 0; (b) β ¼ 0:4, 1 − ½PH =PL −β L − ΔzH  0. Reprinted from Chem. Eng. Sci., 36, (1981), p. 243, Y.N.I Chan, F.B. Hill, Y.W. Wong, “Equilibrium theory of a pressure swing adsorption process,” copyright (1981), with permission from Elsevier.

ðy H =y L Þ ¼ ðP H =P L Þβ − 1 ;

ð7:1:54aÞ

Rearrange it as d ln z d ln P ¼ −β : dt dt

where β ¼ ðβA =βB Þ < 1:

ð7:1:54bÞ

It is necessary to calculate how the characteristics position will change as the transition takes place rapidly from PH to PL (blowdown) or from PL to PH (pressurization with feed). For this purpose, we use equation (7.1.48b) in a rearranged form,   ∂vz ε þ ð1 − εÞκB1 ∂ ln P ; ¼ − ∂t ∂z ε and integrate with respect to z (since the right-hand side is not a function of z, remember ð∂P=∂zÞ ¼ 0) to obtain   1 ∂ln P vz ¼ − z þ constant: ð7:1:55aÞ βB ∂t If the coordinate system of the column is such that the closed end of the column of length L (Figure 7.1.14) is identified as z ¼ 0, then, during the blowdown or repressurization steps, vz ¼ 0 at z ¼ 0. That the constant in equation (7.1.55a) is zero implies that   1 ∂ln P z: ð7:1:55bÞ vz ¼ − βB ∂t Substitute this expression for vz into the basic equation (7.1.51a) for characteristics movement along the column to get   dz ∂ln P zβA ¼ − : ð7:1:55cÞ βB dt ∂t

ð7:1:55dÞ

Integrate to obtain ln

h i . zH ¼ ln P H−β PL−β ) ðzH =z L Þ ¼ ðPH =P L Þ −β ; ð7:1:55eÞ zL

where zH and zL are the z-coordinates of the characteristics when the pressures are PH and PL, respectively. Obviously, the absolute value of z at the end of any such process will depend on the z-coordinate at the beginning of the process, which will be influenced by prior steps. It is useful now to calculate the net displacement of a concentration front during a complete cycle of four steps in PSA. Assume the column/bed length to be L and that step 1 is as shown in Figure 7.1.13(a) with the high-pressure feed continuously entering at the top end (z ¼ L). Prior to this, step 4 occurred wherein the gas (mole fraction yf, feed mole fraction) entered the column (z ¼ L) at a low pressure, PL. Gas entering the column a short time later enters at a slightly higher column pressure, and the final gas in this step enters at pressure PH. The characteristics location of the initial entering gas is obtained from (7.1.55e) as zH ¼ LðP H =P L Þ − β ;

ð7:1:56aÞ

where zL is assumed to be L. Based on this assumption, the zH location is now at B. The specific characteristic path is difficult to plot for this step. This gas concentration front (Figure 7.1.13(a)) (now at pressure PH) moves down the column by ΔzH during step 1 (high-pressure feed flow) so that the lowest z-coordinate at the end of step 1 (point C in Figure 7.1.14(a)) has the value

516

Bulk flow perpendicular to the direction of force

z H − ΔzH ¼ LðP H =P L Þ − β − βA vzH Δt:

ð7:1:56bÞ

The triangular region above AD has only a feed-gas composition. The gas composition changes between C and D, it being highest at D (¼ yf). The blowdown step that follows (step 2) pushes the characteristics back. The zH and zL for this step are still related by (7.1.55e). Since zH in this relation is valid at the start of this step, it must be equal to that of the location described by (7.1.56b): zL ðP H =P L Þ − β ¼ LðP H =P L Þ − β − βA vzH Δt

) z L ¼ L − βA vzH Δt ðP H =P L Þβ ;

ð7:1:56cÞ

corresponding to point E, just as point F corresponds to starting point D. Now comes the low-pressure purge step, which pushes the characteristics still further toward z ¼ L by Δz L (equation (7.1.53b)); thus, the final z-coordinate (location G) at the end of step 3 is L − βA vzH Δt ðP H =P L Þ − β þ βA vzL Δt:

ð7:1:56dÞ

At the start of step 4, the characteristics front was at z ¼ L (the feed gas entered the column at P ¼ PH). Therefore, in the whole cycle, the net displacement of the characteristics front from z ¼ L has been Δzjoverall ¼ − βA vzH Δt ðP H =P L Þ − β þ βA vzL Δt h i ð7:1:57aÞ ¼ βA Δt vzL − vzH ðP H =P L Þ − β :

Figure 7.1.14(a) is drawn such that Δzjoverall ¼ 0. The ratio of vzL , the velocity during the low-pressure purge step, and vzH , the velocity during the high-pressure feed step, is an important parameter, γ, in PSA: γ ¼ ðvzL =vzH Þ:

ð7:1:57bÞ

If Δzjoverall for a cycle is zero, then a critical value of ðvzL =vzH Þ is reached:    β vzL PH ¼ : ð7:1:57cÞ γcrit ¼ vzH crit PL This definition is valid only when Δt for the low-pressure purge step 3 is equal to that for the high-pressure feed step 1. Consider now step 4 and then step 1. Assuming that the characteristics start at zL ¼ L (top of the column where fresh feed enters) with P ¼ PL and y ¼ yf in step 4, zH by equation (7.1.55e) for any P is z H ðP Þ ¼ LðP L =P Þβ ;

PL  P  PH :

ð7:1:58aÞ

By equation (7.1.54a), the y value at any P and zH(P) is then related to yf by yH ðP Þ ¼ ðP L =P Þ1 − β yf ;

PL  P  PH

ð7:1:58bÞ

so that yH ðzÞ ¼ y f ðz=LÞðð1=βÞ − 1Þ :

ð7:1:58cÞ

This expression provides the value of the gas mole fraction of species A in the advancing front in the pressurization step 4 (Figure 7.1.5(b) illustrates how the composition decreases in the front from the feed composition to the exiting composition). When P ¼ PH ¼ Pf, step 1 begins, and the characteristics are pushed further into the column toward z ¼ 0 by an amount ΔzH . As long as zH ðP H Þ from (7.1.58a) is larger than ΔzH , the feed-gas composition never appears at z ¼ 0 and therefore never breaks through into the high-pressure product leaving the column at z ¼ 0. This condition is identified by z H ðP H Þ ¼ LðP L =P H Þβ  ΔzH ¼ βA vzH Δt:

ð7:1:58dÞ

If, in addition, ðvzL =vzH Þ  γcrit , then Chan et al. (1981) have indicated that, at steady state, complete removal of trace component A from the high-pressure product stream is obtained. The same result was theoretically demonstrated earlier by Shendalman and Mitchell (1972) for the same system, except that species B was inert and was not adsorbed at all. A more general index to identify the onset of complete purification is Gcrit, the value of the fraction of the feed introduced during steps 1 and 3, just required as purge in order to obtain complete purification: Gcrit ¼

vzL P L amount of species B in purge ¼ : ð7:1:58eÞ vzH P H amount of species B in feed

On the basis of (7.1.57c), Gcrit ¼ ðPL =P H Þβ − 1 :

ð7:1:58fÞ

A plot of Gcrit against ðP H =P L Þ for various values of β is provided in Figure 7.1.15. The fraction of feed required as purge decreases as (1) the ratio of the high feed pressure to the low purge pressure increases and (2) the value of β decreases, i.e. the separation factor increases. However, too high a ratio of ðPH =P L Þ does not reduce Gcrit by much. The fraction of the major component (species B) fed to the process which is recovered in pure form in the highpressure product stream at steady state (identified by subscript ∞) is f∞ ¼

mB;feed − mB;purge ; mB;feed þ mB;blowdown

ð7:1:59aÞ

where mB represents the number of moles of species B introduced to or leaving the process per half-cycle. Assuming the ideal gas law to be valid, for any column/ bed having a total volume V, cross-sectional area Sc (¼ V/ L) and a void volume fraction ε, mB;feed ¼ εðV =LÞvzH Δt ðP H =RT Þ;

ð7:1:59bÞ

mB;purge ¼ εðV =LÞvzL Δt ðP L =RT Þ;

ð7:1:59cÞ

since the feed gas occupies the void volume at pressure PH, whereas the purge occupies the void volume at pressure

7.1

Force −rμi in phase equilibrium: fixed-bed processes

1.0

b = 1.0

0.8

Gcrit

0.6

0.8

0.4

0.0 0

4

8

12

16

20

PH /PL Figure 7.1.15. Critical purge-to-feed ratio. (After Chan et al. (1981).)

PL. The number of moles removed during blowdown (when pressure is reduced from PH to PL) is mB;blowdown ¼ ½ε þ ð1 − εÞκB1 V ½P H − P L =RT ¼

εV ½P H − P L  ; βB RT ð7:1:59dÞ

which includes the gas leaving the void volume of the bed over a pressure range (PH−PL) and the adsorbed gas in the pressure range (PH−PL). Therefore an expression for f ∞ is f∞¼

based theory. The column/bed length was given. An alternative design oriented goal would be to calculate the shortest length of the column, Lmin, i.e. the least amount of adsorbent required for complete removal of species A. This is achieved when G ¼ Gcrit (equation (7.1.58f)) and L ¼ ΔzL . These conditions ensure that component A is just completely purged from a column at the end of purge step 3; full adsorption capacity of all adsorbents in the column is available in step 4 onwards. An expression for Lmin is obtained by replacing vzH in (7.1.59b) using (7.1.57c) and (7.1.53b):

0.4

0.2

0.0

517

Δt εðV =LÞ RT ðvzH P H −vzL P L Þ

Δt vzH P H εðV =LÞ RT

þ εV ½βPHRT−PL  B

¼

½P H =PL 1−β −1 h i ; Lβ ½P H =P L 1−β þ PPHL −1 Δz L ð7:1:59eÞ

where we have used the definition for γcrit from (7.1.57c) to replace ðvzH =vzL Þ. If the preferentially adsorbed species A has to be recovered in a more enriched form, the blowdown and the purge stream would be the source; species A mole fraction yBDPG in this combined stream may be obtained from  1 − β

 βL PH PH ΔzL PL − 1 þ PL y BDPG mB;feed þ mB;blowdown  : ¼  ¼ βL PH yf m B;purge þ mB;blowdown −1 þ 1 ΔzL

PL

ð7:1:59fÞ

This ratio is identified as the enrichment of species A by Chan et al. (1981). When β ! 0, this enrichment increases linearly with an increase in ðP H =P L Þ. The above expressions provide theoretical estimates of the productivity and process stream quality from a PSA process for a trace component removal by an equilibrium

Lmin ¼

βA mB;feed ðP H =P L Þβ : εSc ðP H =RT Þ

ð7:1:60Þ

This equilibrium based analytical theory of a conventional PSA process predicts that the mole fraction of species A in the high-pressure product gas, y, will continue to decrease with time (Shendalman and Mitchell, 1972; Chan et al., 1981). This is contrary to experimental observations. More exact theories based on axial diffusion and mass-transfer effects are needed to predict the observed behavior. A number of other analyses for PSA systems have been carried out. Shendalman and Mitchell (1972) studied analytically and experimentally the PSA separation of trace amounts of CO2 from helium, assuming that helium is completely inert and a linear isotherm is valid for CO2 with instantaneous equilibrium. Fernandez and Kenney (1983) studied air separation in a single-column PSA using a linear equilibrium adsorption model without any axial diffusion. Knaebel and Hill (1985) studied two-column equilibrium PSA for a general binary system using the same framework. Raghavan et al. (1985) numerically simulated the CO2–He system of Shendalman and Mitchell (1972) using axial dispersion and finite gas-to-solid mass-transfer resistance. Bulk separation of a ternary gas mixture, H2/CH4/CO2, has been experimentally and numerically studied by Doong and Yang (1986). Variations in the simple PSA cycle, especially in the pressurization and blowdown steps, have been studied. Suh and Wankat (1987) studied a combined cocurrent– countercurrent blowdown cycle; Liow and Kenney (1990) investigated the backfill part of the cycle involving pressurization of the adsorption bed with the product in a direction opposite to that of the feed mixture. The design of PSA systems has been treated by White and Barkley (1989). Example 7.1.4 A refinery gas stream at 10.13  105 Pa contains primarily H2 besides some impurities. Consider this as a binary mixture where the impurity species A adsorbs at the gas temperature onto the porous adsorbent used according to qA1 (gmol/g adsorbent) ¼ 40CA2 (gmol/cm3). This H2 stream has to be purified by a two-bed PSA process using the above adsorbent (ρs ¼ 2 g/cm3); the porosity of the adsorbent is 0.3. The adsorbent bed porosity is ε ¼ 0.4. The bed diameter in a small-scale study to be conducted is 4 cm; the high-pressure feed-gas flow rate is 0.52 liter/s. The low pressure, PL, is 1.013  105 Pa. The durations of different

518

Bulk flow perpendicular to the direction of force

steps in the four-step (Figure 7.1.13(a)) process are: pressurization, 3 s; high-pressure feed flow, 80 s; blowdown, 3 s; lowpressure purge, 80 s. Determine the length of the bed needed for essentially complete purification of H2. Determine the values of γcrit and Gcrit. Calculate the enrichment of the impurity in relation to its feed concentration in the blowdown–purge. Employ the equilibrium nondispersive PSA model of Chan et al. (1981).

But

ðz L − zH Þ ¼ 0:027zL ) 99:95 þ 0:027z L ¼ zL ) zL ¼

99:95 : 0:973

The required bed length

zL ¼ 102:7cm ¼ L: The critical value of the (purge/feed) ratio, γcrit, is obtained from (7.1.57c):

Solution The equilibrium model of Chan et al. was formuγcrit ¼ ðvzL =vzH Þcrit ¼ ðP H =P L Þβ ¼ ð10Þ0:0117 lated and solved using an equilibrium relation of the type for ) γcrit ¼ 1:027: impurity A: C A1 ¼ κA1 C A2 . We will now convert qA1 ¼ 40C A2 to this form using 0 1 0 1 3 g adsorbent cm of adsorbent A ð1 − εp Þ @ A qA1 ðgmols A=g adsorbentÞ ρs @ 3 cm of adsorbent material cm3 of porous adsorbent

¼ C A1 ðmols A=cm3 of porous adsorbentÞ ¼ 40  ρs ð1 − εp ÞC A2 ¼ 40  2  0:7C A2 ¼ 56 C A2 :

We will adopt an approximate approach. If L is assumed to be equal to ΔzL, then, by equation (7.1.60),

Lmin ¼ βA mB;feed ðP H =P L Þβ =fε Sc ðP H =RTÞg ¼ ΔzL : But from (7.1.59b),

mB;feed ¼ εðV =LÞ vzH ΔtðP H =RTÞ: Therefore

βA εðV =LÞvzH ΔtðP H =RTÞðP H =P L Þβ ε Sc ðP H =RTÞ

Lmin ¼

) Lmin ¼ βA vzH ΔtðP H =P L Þβ :

But β ¼ (βA/βB) ¼ βA since H2 is assumed not to be getting adsorbed, and

βA ¼

ε 0:4 0:4 ¼ ¼ ¼ 0:01176: ε þ ð1 − εÞκA1 0:4 þ 0:6  56 0:4 þ 33:6

From result (7.1.58e),

Gcrit ¼ ðP H =P L Þβ − 1 ¼ ð10Þ − 0:9883 ¼

To determine the enrichment of the impurity species in the blowdown–purge stream in relation to the feed stream, we use (7.1.59f): 1 0 11 − β 0 10 βL P PH H @ A@ − 1A þ @ A Δz L PL PL yBDPG 1 0 10 ¼ yf @ βL A @P H − 1A þ 1 Δz L PL

0

¼

Further, Δt ¼ 80 s, so

520 cm3 =s 520 cm ¼ vzH ¼ π 1:6π s 42  0:4 cm2 4 ) Lmin ¼ 0:01176  ) Lmin

520  80  ð10Þ0:01176 1:6π

0:01176  520  80  1:027 ¼ cm 1:6π

¼ 99:95 cm: We could have also obtained this length by employing equation (7.1.58d) for the high-pressure feed step. However, there is a pressurization step ahead of this step where, by equation (7.1.55e),

ðzH =zL Þ ¼ ðP H =P L Þ − β ) ðzH =z L Þ ¼ ð10Þ − 0:0117 ¼ 0:973:

To account for the short length of the bed (near z ¼ L) which is consumed as the pressure rises from PL to PH in 3 s, we employ the above result:

99:95 cm þðzL − z H Þ ¼ zL :

1 ¼ 0:102: 9:735

¼ ðy BDPG =y f Þ ¼

1 0:01176  102:7 @ A ð10 − 1Þ þ ð10Þ0:9883 99:95 0:01176  102:7 ð10 − 1Þ þ 1 99:95

0:012  9 þ 9:73 ; 0:012  9 þ 1 0:1087 þ 9:73 ¼ 8:87: 1:1087

This analysis ignores the second term, ((1 − ε)εp/ε), in the denominator of relation (7.1.13e). To include this term, terms in the analysis of Chan et al. (1981) have to change.

Except for the study by Fernandez and Kenney (1983), all the studies mentioned above were based on multicolumn arrangements. Turnock and Kadlec (1971) and Kowler and Kadlec (1972) initiated studies of PSA processes using only a single bed. There are basically two steps in such a process, adsorption and desorption; the adsorption step consists of rapid pressurization and then high-pressure feed flow, while desorption is achieved by depressurization and adsorbed species removal from the feed introduction end. Short cycle times (  20 seconds) are used, and product can be continuously withdrawn from the opposite end in a

7.1

Force −rμi in phase equilibrium: fixed-bed processes

variety of ways using a surge tank and pressure-reducing valve. Hill (1980) used a single-column PSA to recover a weakly adsorbed impurity. A somewhat different singlecolumn PSA process for oxygen production has been described by Keller (1983). The PSA processes mentioned so far generally produce a high-pressure product stream purified of the species strongly adsorbed. Further, the models employed assume selectivity between the different species generated by different equilibrium relationships (see (7.1.52a) and (7.1.52b)). There are PSA processes already developed or being developed that are based on differing rates of diffusion of gases through molecular-sieve carbons (MSCs). For example, oxygen diffuses very rapidly through MSCs compared to nitrogen, although their equilibrium uptakes are very similar. This property is being exploited in a few PSA processes; see Yang (1987) for an introduction. Schork et al. (1993) have provided a model for such a process. We have in this section used porous solid adsorbents to adsorb selectively specific gas species from the feed gas stream and then go through a cycle of desorption followed by adsorption in PSA processes. Absorption–desorption of specific gas species from a feed gas stream vis-à-vis an absorbent liquid has been carried out also using a porous hollow-fiber membrane based rapid pressure-swing absorption (RAPSAB) process (Bhaumik et al., 1996). In this cyclic separation process, a well-packed microporous hydrophobic hollow-fiber module was used to achieve nondispersive gas absorption (see Sections 3.4.3.1 and 8.1.2.2.1) from a high-pressure feed gas into a stationary absorbent liquid on the shell side of the module during a certain part of the cycle, followed by desorption of absorbed gases from the liquid in the rest of the cycle. The total cycle time was varied from 20 s upwards. Separation of mixtures of N2 and CO2 (around 10%), where CO2 is the impurity to be removed, was studied using an absorbent liquid such as pure water and a 19.5% aqueous solution of diethanolamine (DEA). Three RAPSAB cycles studied differ in the absorption part. Virtually pure N2 streams were obtained with DEA as absorbent, demonstrating the capability of bulk separation to very high levels of purification. 7.1.3

Potential-swing adsorption

Sometimes, the amount of species adsorbed from a solution onto an adsorbent, e.g. activated carbon, depends on the electrical potential applied to the adsorbent: the amount adsorbed at equilibrium depends on the potential applied. This equilibrium phenomenon of potential dependent adsorption could provide a basis for potentialswing adsorption. Eisinger and Keller (1990) have utilized such a phenomenon (see Zabasajja and Savinell (1989) for electrosorption of alcohols on graphite surfaces and references to earlier studies) to develop a potential-swing adsorption process. They have experimentally studied

519

DCV

Reference electrode

Outlet streams enriched and depleted in organic component

Granular activated carbon

Ion exchange membrane

Direction of current flow

Electrical terminal

Organic-containing inlet stream Figure 7.1.16. A suggested design of a single cell of two layers of carbon particles separated by an ion exchange membrane for potential-swing adsorption. Reprinted, with permission, from Figure 5 of R.S. Eisinger and G.E. Keller, “Electrosorption: a case study on removal of dilute organics from water,” Env. Progr., 9(4), Nov., 235 (1990). Copyright © [1990] American Institute of Chemical Engineers (AIChE).

adsorption of ethylenediamine (EDA) on activated carbon from an aqueous alkaline brine at negative potentials of upto −1.0 volt; the potential was then changed to 0.0 volt or positive to desorb the EDA into a suitable aqueous solution such that the solution will be highly concentrated in EDA. An engineering assessment of this concept, also called electrosorption, was carried out by Eisinger and Keller (1990) using flat cells: a single cell consisted of two layers of granular activated carbon separated by an ion exchange membrane (Figure 7.1.16). Organic-containing inlet streams flow through each carbon layer. The carbon on one side of the membrane is held at the adsorbing potential (−1.0 volt) while the other side is at the desorbing potential (0 volt). When the carbon on the adsorbing side approaches saturation with EDA, and therefore breakthrough is imminent, the potentials are switched: the adsorbing side becomes the desorbing side, and vice versa. At the same time, the outlet stream switching valves would be activated. Note that the electrical potential in the carbon particles is generated by a current flow transverse to the carbon layer thickness via electrodes on the two sides. The electrical field here merely generates the requisite

520

Bulk flow perpendicular to the direction of force

electrical potential on the adsorbents. The economic advantages and technical uncertainties of this concept have been considered. Scale-up problems have also been identified. Desalination of water by a similar technique, called capacitive deionization (CDI), is being studied (Farmer, 1995; Farmer et al., 1996). Here two carbon electrodes kept at 1.3 volts apart are used during ion collection.

feed, and the bottom reservoir has a solute concentration less than that in the initial feed. More cycles lead to a higher solute enrichment in the top reservoir and greater solute depletion in the bottom reservoir. A very high degree of separation can be achieved after multiple cycles, limited only by axial dispersion and finite mass-transfer rate between the mobile and fixed phases.

7.1.4

7.1.4.1 A nondispersive equilibrium theory: batch operation

Parametric pumping

Parametric pumping is a cyclic fixed-bed process in which the axial fluid flow direction and the driving force direction (perpendicular to the axial flow direction) are changed periodically and synchronously throughout the bed in the presence of reflux6 of the fluid at one or both ends of the bed (Sweed, 1972). The most common driving force is − rμi created by a difference in temperature between the mobile phase and the stationary phase. Other driving forces used include a pH difference or a partial pressure difference. Some of the earliest descriptions of parametric pumping appear in Wilhelm et al. (1966, 1968). Consider Figures 7.1.17(a) and (b), illustrating the batch operation of a thermal parametric pump in the direct mode. To start with, the packed bed is filled with the feed liquid containing an adsorbable species i. The bottom reservoir is also filled with the same feed fluid, whereas the top reservoir is essentially empty. The cyclic process was initiated earlier by suddenly heating the fluid and the solid in the packed bed to Thot via the jacket and simultaneously forcing the fluid to flow upward into the top reservoir in a synchronous fashion (Figure 7.1.17(c)). As this figure shows, the velocity and temperature at any point in the bed remain constant for a time equal to half the cycle if a square wave is used (Pigford et al., 1969a). Generally, the amount of species i retained by the adsorbent is reduced as the temperature is increased. Further, the bottom reservoir has the same concentration as that in the packed bed. The fluid flowing into the top reservoir, however, is enriched or concentrated in species i, compared to conditions where the bed is colder. During the next half of the square wave cycle, the fluid from the top reservoir is pushed downward into the packed bed, which is suddenly cooled to Tcold; the colder fluid is now pushed into the bottom reservoir with the same speed. Since the adsorbent particles are cold, their adsorption capacity is much higher, thus purifying the fluid of solute i. The fluid coming into the bottom reservoir is substantially purified of species i. This cycle is ended when the volume of fluid introduced into the bottom reservoir during the second half of the cycle is equal to that removed from the same reservoir during the first half of the cycle. A new cycle is then initiated. Note that the top reservoir now has a solute concentration more than that in the initial

6

Reflux is considered in general in Chapter 8.

The earliest nondispersive linear equilibrium based theory of parameteric pumping was developed by Pigford et al. (1969a). A number of others have since appeared (Aris, 1969; Gregory and Sweed, 1970; Rhee and Amundson, 1970; Chen and Hill, 1971). We illustrate here the model of parametric pumping developed by Pigford et al. (1969a) for a batch system. The model has the following assumptions. (1) There is perfect local equilibrium between the solid and the fluid phases. (2) A linear equilibrium relationship exists that depends on bed temperature. (3) No axial dispersion or diffusion occurs, and there is no variation in the radial direction. (4) There is an instantaneous change in bed temperature and velocity direction in a square-wave pattern, as shown in Figure 7.1.17(c). (5) During each half-cycle, the interstitial velocity and temperature have the same value throughout the length of the bed. Employing assumptions (3) and (5) in equation (7.1.3), the mass balance equation for species i is obtained as (j ¼ 1, solid phase, j ¼ 2, mobile phase): ε

∂C i2 ∂C i2 ∂C i1 þ εvz þ ð1 − εÞ ¼ 0: ∂t ∂z ∂t

ð7:1:61Þ

If ρs is the density of the solid particles (mass/volume) and qi1 represents the moles of species i per unit mass of solid particles, then C i1 ¼ qi1 ρs :

ð7:1:62aÞ

If y i (instead of κi2 for phase 2) represents the moles of species i per mole of fluid, and ρf m stands for the number of moles per unit fluid volume, then C i2 ¼ yi ρf m :

ð7:1:62bÞ

The linear equilibrium assumption (2) is used to relate qi1 to y i by qi1 ¼ ai ðT Þ yi ;

ð7:1:63Þ

where ai ðT Þ is the local temperature-dependent species i equilibrium constant. Introducing definitions (7.1.62a) and (7.1.62b) into (7.1.4.1) leads to

7.1

Force −rμi in phase equilibrium: fixed-bed processes

521

(a)

Driven piston

(b)

Packed bed of adsorbent particles

Heating and cooling jacket

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Heating half-cycle

Cooling half-cycle

Driving piston

(c)

Tcold

Velocity

Temperature

Tmean

Thot

v0 0 −v 0

0

0.2

0.4

0.6

0.8

1.0

Fraction of cycle Figure 7.1.17. (a) Diagram of column for parametric pumping in direct mode. (b) Heat flow and fluid flow direction during two halfcycles (Wilhelm et al., 1968). Reprinted, with permission, from I & EC Fund., 7(3), 337 (1968), Figure 3. Copyright (1968) American Chemical Society. (c) Velocity and temperature at a point in the bed as a function of time.

522

Bulk flow perpendicular to the direction of force

ερf m

∂y i ∂y ∂q þ εvz ρf m i þ ð1 − εÞ ρs i1 ¼ 0: ∂t ∂z ∂t

ð7:1:64Þ

Introduce now the equilibrium relation (7.1.63) into the above equation. This results in    ∂y ∂y dAi ðT Þ ∂T ð1 þ Ai ðT ÞÞ i þ vz i ¼ − y ; ð7:1:65Þ ∂t ∂z dT ∂t i

temperature, qi1 decreases and y i increases because of reduced adsorption capacity; hence Ai ðt Þ is reduced from A0i , the value at the mean temperature (Tmean), leading to a negative sign before ^a i sqðωt Þ. Introduction of these square-wave variations into the two governing equations (7.1.69) and (7.1.70) gives (here v0z ¼ v0 ) dz v0z sqðωt Þ v0 sqðωt Þ ¼ ¼ 0 dt 1 þ Ai − ^ a i sqðωt Þ 1 − bi sqðωt Þ

where A i ðT Þ ¼

ð1 − εÞ ρs ai ðT Þ ερf m

ð7:1:66Þ

is a modified equilibrium constant. To use the method of characteristics (equations (7.1.12b)–(7.1.12f)), write     ∂y i ∂y i dt þ dz ¼ dyi ð7:1:67Þ ∂t ∂z to obtain   dz   dy i    

    − y ∂T  dAi ðT Þ v  z i ∂t dT ∂yi  : ¼   ∂t dt dz      1 þ A i ðT Þ v z 

ð7:1:68Þ

If the determinant in the denominator is zero, for ð∂y i =∂t Þ to be finite the determinant in the numerator has to be zero. These two determinants with a zero value lead, respectively, to dz vz ¼ dt 1 þ Ai ðT Þ

ð7:1:69Þ

and vz

  d ln yi ∂T dAi ðTÞ ¼ − : dz ∂t dT

ð7:1:70Þ

Use of equation (7.1.69) in (7.1.70) yields   d ln yi ∂T 1 dAi ðT Þ d ln ð1 þ Ai ðT ÞÞ ¼ − : ¼ − ∂t 1 þ Ai ðT Þ dT dt dt

ð7:1:71Þ

Equations (7.1.69) and (7.1.71) represent, respectively, the characteristics and yi variation along the characteristics. The square-wave mode of variation of fluid velocity and bed temperature (and therefore equilibrium constant Ai ðt Þ) may be represented (via assumption (4)), respectively, by (Figure 7.1.17(c)) vz ¼ vz ðωt Þ ¼ v0z sqðωt Þ ¼ v0 sqðωt Þ

ð7:1:72aÞ

Ai ðt Þ ¼ A0i − ^a i sqðωt Þ:

ð7:1:72bÞ

and

When the bed is heated, the fluid flows upward through the bed into the top reservoir. Due to the higher

ð7:1:73Þ

and d ln y i d ln ð1 þ Ai ðT ÞÞ d ln ½1 − bi sqðωt Þ ¼− ¼− : ð7:1:74Þ dt dt dt Equation (7.1.73) provides the velocity of propagation of any yi through the bed and represents the characteristics. During the half-cycle when the bed is hot (T ¼ Thot), the concentration wave velocity is given by  0  dz v : ð7:1:75aÞ ¼ T ¼ T hot ; dt 1 − bi During the half-cycle when the bed is colder (T ¼ Tcold), the concentration wave velocity is given by  0  dz v T ¼ T cold ; : ð7:1:75bÞ ¼ − 1 þ bi dt Focus now on Figure 7.1.18, which illustrates the characteristics lines for the process initiated by upflow of the feed fluid in the column to the top reservoir while the bed temperature is instantaneously raised to Thot. The feed fluid species concentration in equilibrium with the bed adsorbents at this temperature is C 0i2 and the corresponding yi value is y 0i . Since this fluid is in equilibrium with the adsorbents, as this concentration moves along the z–t characteristics, it remains unchanged during the half-cycle 0 < ωt < π. Those characteristic lines which cross z ¼ L, the column length, discharge a hot fluid of y i equal to y0i into the top reservoir. Denoting by hy iT in the mean composition of the fluid entering the top reservoir from the column during the nth cycle, it would appear from the figure that only the characteristics in region (A) of Figure 7.1.18 enter the top reservoir during the first hot period (n ¼ 1) 0 < ωt < π. All characteristics in the column, however, have y i ¼ y0i . Therefore hy iT i1 ¼ y0i :

ð7:1:76Þ

Consider the next half-period π < ωt < 2π, when the bed temperature and the fluid temperature have been lowered to Tcold. The fluid entering the top of the column from the top reservoir will now have the composition y 0i , and their characteristics with changed slopes lie in region (C) of the z–t diagram. These characteristics do not cross z ¼ 0, the column bottom. The characteristics in region (D) do. These characteristics originated earlier from z ¼ 0 when T ¼ Thot during 0 < ωt < π and therefore had a value of

7.1

Force −rμi in phase equilibrium: fixed-bed processes

á yi Tñ2

á yi Tñ1 = yi0 e

z=L

523

a C

c

á yi Tñ3 d

a

c

d

C

A B

B

B z

b

D

D

D

D

0 Cold

Hot 0

2π

π yi0

á yi Bñ1

Cold

Hot 3π

wt

4π

á yi Bñ2

Cold

Hot 5π

6π

á yi Bñ3

Figure 7.1.18. Characteristic lines used for solution by equilibrium theory: upflow and heating in phase (Pigford et al., 1969a). Reprinted, with permission, from I & EC Fund., 8(1), 144 (1969), Figure 3. Copyright (1969) American Chemical Society.

y i ð0;t Þ ¼ y0i . This composition will be changed in region (D) of π < ωt < 2π, where the characteristic slope is changed to that of (7.1.75b) when ωt  π. Further, from the second governing equation (7.1.74), we get y i ð1 − bi sqðωt ÞÞ ¼ constant; i.e. fyi ð1 − bi sqðωtÞÞgcold ¼ fyi ð1 − bi sqðωtÞÞghot : ð7:1:77aÞ Therefore yi jcold 1 − bi : ¼ 1 þ bi yi jhot

ð7:1:77bÞ

Since, for ωt  π, the characteristics for y i jhot change to those for yi jcold ,     1 − bi 1 − bi ¼ y0i : ð7:1:78Þ y i jcold ¼ yi jhot 1 þ bi 1 þ bi The characteristics crossing z ¼ 0 will enter the bottom reservoir. For the period π  ωt  2π, only the characteristics in region (D) enter z ¼ 0. Therefore, the composition of this fluid entering the bottom reservoir, represented by hy iB i1 , is   1 − bi hyiB i1 ¼ y0i : ð7:1:79Þ 1 þ bi In the second cycle, beginning 2π  ωt  3π, this very same fluid with hy iB i1 will enter the column bottom, which is now at T ¼ Thot from the bottom reservoir, and the characteristic line slope is changed. However, we are more interested in those characteristics which cross z ¼ L and

therefore enter the top reservoir. There are two regions in 2π  ωt  3π, region (B) and region (C), through which such characteristics pass. The characteristics in region (C) have originated from those in region (C) of π  ωt  2π, with ultimate origin from the top reservoir having a composition of y 0i . This composition, remaining constant up to ωt ¼ 2π, is changed in the ωt  2π region since the temperature is changed to Thot from Tcold:     1 þ bi 1 þ bi ¼ y 0i : region ðCÞ; ωt  2π : y i jhot ¼ y i jcold 1 − bi 1 − bi ð7:1:80Þ

These characteristics thus yield a higher species i composition in the fluid entering the top reservoir. But the top reservoir also receives characteristics in region (B), 2π  ωt  3π. These characteristics originated in region (B) for π  ωt  2π, where they originated from region (B) of 0  ωt  π. The composition of these last characteristics is simply y0i ; as these characteristics cross ωt ¼ π, from Thot to Tcold, the composition is lowered to y0i ð1 − bi Þ=ð1 þ bi Þ. This last composition is increased to y0i as these characteristics cross ωt ¼ 2π and change from Tcold to Thot. We now know the compositions of the characteristics in regions (B) and (C) for 2π  ωt  3π, where the fluid in upflow enters the top reservoir at z ¼ L. To determine the mean composition hy iT i2 of these characteristics entering the top reservoir via regions (B) and (C), the weighted contribution of each region is needed since each region occupies a different time interval:

524        1 þ bi ac cd y 0i  þ y0i g; ad ad 1 − bi

ð7:1:81Þ

where ac, cd and ad are identified in Figure 7.1.18 for 2π  ωt  3π. To determine ratios ðac=ad Þ and ðcd=ad Þ, focus on the characteristics lines in both regions π  ωt  2π and 2π  ωt  3π. Specifically, in region (C) of time period π  ωt  2π, characteristic line eb has a slope of magnitude ðv0 =ð1 þ bi ÞÞ. Therefore, the distance ab is πðv0 =ð1 þ bi ÞÞ (ideally the length of time t used should be π=ω, but ω cancels out later anyway). From the slope of the characteristics in region (C) for 2π  ωt  3π,  0  ab v 1 ðabÞω ωπð1 − bi Þ ¼ ) ac ¼ 0 : ¼ ac ð1 þ bi Þ ðv =ð1 − bi ÞÞ 1 − bi ω Since ad is ωπ, we get ðac=ad Þ ¼ ð1 − bi Þ=ð1 þ bi Þ:

ð7:1:82aÞ

The ratio ðcd=ad Þ is obtained as cd ac 2bi ¼ 1− ¼ : ad ad 1 þ bi

ð7:1:82bÞ

Thus hyiT i2 ¼ y0i



    1 þ bi ð1 − bi Þ 2bi 2bi þ ¼ y 0i 1 þ : 1 − bi ð1 þ bi Þ 1 þ bi 1 þ bi

Thus the bottom reservoir solution is depleted in species i after the second cycle, while the top reservoir is enriched in species i. Pigford et al. (1969a) have provided general expressions for hyiT in and hyiB in after n cycles (n > 0) in terms of those immediately proceeding:      1 þ bi 1 − bi 2bi þ hy iB in − 3 ; hy iT in ¼ hyiT in − 1 1 − bi 1 þ bi 1 þ bi ð7:1:85aÞ

 1 − bi : 1 þ bi



ð7:1:85bÞ

ð7:1:86aÞ

Use of both relations (7.1.85a) and (7.1.85b) yields ( "   #) 2bi 1 − bi n − 2 : ð7:1:86bÞ hy iT in ¼ y 0i 1 þ þ 1− 1 þ bi 1 þ bi The composition ratio between the top reservoir and the bottom reservoir is      hy iT in 2bi 1 þ bi n 1 þ bi 2 − : ð7:1:87Þ ¼ 2þ 1 þ bi 1 − bi 1 − bi hyiB in These results indicate that, as n increases, hy iB in , the bottom product composition, tends to zero (relation (7.1.86a)). Although as n increases hyiT in increases, relation (7.1.85a) suggests that hyiT in ¼ hy iT in − 1 . In a real system, the ratio hy iT in =hy iB in does not tend to infinity, as suggested by (7.1.87) for large n; instead, mass-transfer rate limitations between the phases and axial diffusion limit the possible enrichment (Pigford et al., 1969a). Figure 7.1.19 compares the predictions from this equilibrium theory with the experimental data on an n-heptane–toluene separation 105

ð7:1:83Þ

This result demonstrates that the hot stream entering the top reservoir at the beginning of the second cycle 2π  ωt  3π is considerably enriched beyond y0i (the value for hy iT i1 ). Although fluid of composition hy iT i2 enters the column top from the top reservoir for the next half-cycle, 3π  ωt  4π, the bottom region of the column is more important for determining hy iB i2 , the stream composition entering the bottom reservoir. Characteristics in region (D) here originated in region (D) for 2π  ωt  3π. The composition of the characteristics in this later region is hy iB i1 ; this composition is changed when ωt  3π. The changed composition yi jcold is related to hyiB i1 ¼ yi jhot by (7.1.80); thus, for 3π  ωt  4π region (D),     1 − bi 1 − bi 2 : ð7:1:84Þ ¼ y 0i hy iB i2 ¼ yi jcold ¼ y i jhot 1 þ bi 1 þ bi

hyiB in ¼ hy iB in − 1

This preceding relation leads to   1 − bi n hyiB in ¼ y 0i : 1 þ bi

Toluene composition ratio áyiT ñn / áyiB ñn

hy iT i2 ¼

Bulk flow perpendicular to the direction of force

104

103

102

10

1 0

10

20

30

40

50

60

70

Number of cycles, n Figure 7.1.19. Toluene composition ratio according to equilibrium theory and the data of Wilhelm et al. (1968) for the system nheptane–toluene in a silica gel column (Pigford et al., 1969a). Reprinted, with permission, from I & EC Fund., 8(1), 144, (1969), Figure 6. Copyright (1969) American Chemical Society.

7.1

Force −rμi in phase equilibrium: fixed-bed processes

525

in a silica gel column (Wilhelm and Sweed, 1968; Wilhelm et al., 1968). That separation achieved in batch parametric pumping can be demonstrated more easily by what is known as the Tinkertoy model developed by Wilhelm et al. (1968). This model is considered next.

It is assumed that the integral on the right-hand side of (7.1.90a) exists. To calculate its value, assume specific forms for f c ðωt þ εa Þ and vz ðωt Þ:

Tinkertoy model of parametric pumping

∂C i2 A2 ¼ cos εa ; ∂z A1

7.1.4.2

Consider Figure 7.0.1(c) from the very beginning of this chapter. The hatched section represents the stationary phase or the packings in the manner of Figure 7.1.1, i.e. a pseudocontinuum approach. Focus on a section of the column of differential length dz. As before, subscript j ¼ 1 represents solid packing phase and j ¼ 2 represents the mobile fluid phase. The mass balance equation (7.1.61) for species i at any axial column location z may be rearranged to yield   ∂C i2 ∂C i2 ð1 − εÞ ∂C i1 vz ðωt Þ ¼ − þ ð7:1:88Þ ∂z ∂t ∂t ε for nondispersive column operation. In parametric pumping, the flow velocity vz ðωt Þ is alternating in direction (a particular form is illustrated by equation (7.1.72a)); the force direction, and therefore the mass flux direction, are simultaneously alternating in direction. During fluid upflow, if the bed is heated, solute is released by adsorbents into the fluid (shown by the dashed lines in Figure 7.0.1(c)); during fluid downflow, the bed is cooled and the solute is adsorbed by adsorbents from the fluid (shown by solid lines in Figure 7.0.1(c)). It can now be argued (Wilhelm et al., 1968; Sweed, 1971, pp. 175–180) that if the force is oscillating with a frequency ω (the same as the velocity), the concentrations C i2 and C i1 will also be periodic in frequency ω. Therefore,     ∂C i2 =∂t and ∂C i1 =∂t will also be periodic in ω. Equation (7.1.88) may be expressed now as vz ðωt Þ

∂C i2 ¼ fc ðωt Þ ∂z )

ð7:1:89aÞ

∂C i2 f ðωt Þ ¼ c : vz ðωt Þ ∂z

ð7:1:89bÞ

The change in mobile-phase composition C i2 over the column of differential length dz is provided by the above relation at any instant of time t. We are more interested in knowing what this change is over one cycle (t ¼ 2π=ω), and ultimately over many cycles.   Before determining a time average of ∂C i2 =∂z over one cycle, let it be recognized that, in general, f c ðωt Þ and vz ðωt Þ are displaced by a phase angle εa . Therefore, an averaged behavior over a time period 0  t  ð2π=ωÞ is obtained from ω 2π

2π ðω  0

2π

 ðω ∂C i2 ∂C i2 ω f c ðωt þ εa Þ ¼ dt: dt ¼ ∂z ∂z 2π vz ðωt Þ 0

ð7:1:90aÞ

vz ðωt Þ ¼ A1 cos ωt; f c ðωt þεa Þ ¼ A2 cos ðωt þεa Þ; ð7:1:90bÞ

where A1 and A2 are arbitrary constants. Then ð7:1:91Þ

a result indicating that the time-averaged local axial concentration gradient in C i2 can be nonzero provided εa is anything but ðπ=2Þ or ð3π=2Þ, i.e. there will be separation provided the oscillations in velocity and concentration are not out of phase by ðπ=2Þ or ð3π=2Þ: 7.1.4.3

Continuous parametric pumping

Batch parametric pumping in a closed system was studied in the context of an equilibrium nondispersive model in Section 7.1.4.1. Continuous open parametric pumps have been modeled by a number of investigators (Gregory and Sweed, 1970; Chen and Hill, 1971). A number of experimental systems have been studied. These include pH based operations (Sabadell and Sweed, 1970; Chen et al., 1979, 1980) and pressure based gas separation operations (Keller and Kuo, 1982). 7.1.4.4

Cycling zone adsorption

It is useful to recapitulate briefly the basics of the two separation techniques described earlier, namely PSA and parametric pumping. Both are cyclic processes. In PSA, one gaseous species is preferentially adsorbed from the feed gas flowing into the bed of adsorbents at a high pressure, allowing the remaining gas species to be purified (for a binary mixture). The preferentially adsorbed species is then desorbed from the adsorbent bed at a much lower pressure, usually with a low-pressure purge flowing in the opposite direction (see Figure 7.1.13(a)). For liquid-phase adsorption–desorption processes, pressure has a very limited effect; therefore, temperature swing is useful for liquid-phase adsorption. In parametric pumping, the liquid flow up or down the adsorbent bed is synchronized with heating or cooling of the liquid and the bed, leading to the reservoirs at the top and bottom being enriched or depleted in the solute (see Figures 7.1.17–7.1.19). The reversal of the direction of liquid flow through the adsorbent bed (practiced in parametric pumping) is avoided in the technique called cycling zone adsorption, which was proposed first by Pigford et al. (1969b). Focus on Figure 7.1.20(a), which shows an adsorbent column being heated or cooled in the direct mode (via a jacket). Unlike that in Figure 7.1.17(a), there are no pistons/ reservoirs at the top and bottom of the column as such. Liquid flow in the bed is unidirectional; liquid feed is

526

Bulk flow perpendicular to the direction of force

(a)

(b) Cooling half-cycle

Heating half-cycle Thot

Cih

Tcold

Heating half-cycle Thot

Cic

Cooling half-cycle Tcold

Cih

Cic

Packed adsorbent bed Qout

Qin

Packed Tf , Cif adsorbent bed

Tf , Cif

Qin

Qout Tf , Cif

(c)

Tf , Cif

(d) 100 NaCl concentration (kmol/m3)

Temperature (⬚C)

90 80 70 60 50 40

0.5 m

30 0

3

Distance from feed inlet 6

0.0 m

9

12

15

Time (min)

0.06

0.04 Feed concentration 0.02 Behavior of experimental data 0.0

0

2.5

5

7.5

10

12.5

15

Time (min)

Figure 7.1.20. Cycling zone adsorption. (a) Direct mode of operation with heat supplied/removed directly into the bed through the wall. (b) Recuperative mode of operation with heat supplied or removed directly into the feed. (c) Temperature variation with time at two distances from feed inlet. (After Knaebel and Pigford (1983).) (d) NaCl concentration of feed outlet as a function of time. (After Knaebel and Pigford (1983).)

continuously flowing up the column of adsorbents (or down the column of adsorbents!). However, the temperature of the column is oscillating, for example, as in Figure 7.1.17(c). If the liquid feed is continuously flowing up the column, then, during the first half of the cycle when the bed is heated up (the temperature is Thot), in an empty bed at the start, there will be very little adsorption. However, in the next half-cycle, as the temperature is reduced to Tcold, the liquid feed undergoes purification since solutes are adsorbed much more at the low temperature. The liquid exiting the bed at the top has much less solute and is purified. At the beginning of the next cycle, the bed is heated up; the adsorbed solutes are desorbed and the feed liquid entering the bed becomes highly enriched in the solute as it leaves the column at the top. This half of the cycle, in succeeding cycles also, will keep on producing a soluteenriched liquid stream. Therefore, by cycling the zone

temperature, we are going through a cycling of the zone adsorption behavior. (Note: Here we have assumed that the liquid feed temperature is essentially the same as that of the bed in the direct mode of operation.) One can have an alternative arrangement where the liquid feed will be heated or cooled outside in a heater/cooler and introduced into the bed in the so-called recuperative mode (useful in parametric pumping as well) of operation (Figure 7.1.20 (b)). Correspondingly, the outlet liquid in the “hot” part of the cycle is enriched in the solute, whereas in the “cold” part of the cycle the outlet liquid is purified of the solute (in both modes of operation). A variety of systems have been investigated using the cycling zone adsorption technique. See Wankat et al. (1975) and Knaebel and Pigford (1983) for a list of useful references. Figures 7.1.20(c) and (d) illustrate schematically the recuperative mode of cycling of temperature in a bed

7.1

Force −rμi in phase equilibrium: fixed-bed processes

527

Eluting solvent

Detector Liquid sample injection

Concentration

(a)

Time

Column Pump

(b)

Detector Gaseous sample injection

Concentration

Mobile-phase reservoir

Time

Column

Carrier gas cylinder Figure 7.1.21. Schematics for a basic chromatographic system for (a) liquid sample or (b) a gaseous sample.

(Figure 7.1.20(b)) and the corresponding bed outlet liquid concentration profile for purification of a brackish water (feed NaCl concentration, 0.030 kmol/m3) using thermally regenerable ion exchange resins Amberlite XD. Two basic models of the cycling zone adsorption are available in Baker and Pigford (1971) and Gupta and Sweed (1971). 7.1.5

Chromatographic processes

The fixed-bed processes studied so far are generally useful for separating one solute or one ionic species from the solvent; in some cases, more than one solute or one ionic species are also separated from the solvent. Separation of each individual species present in a multicomponent gas mixture or liquid solution can be achieved in a fixed bed of adsorbent particles if the mode of operation, e.g. the method of feed introduction, is changed from that used so far. A number of methods are commonly used to this end. They are elution chromatography, displacement chromatography and frontal chromatography (see Figure 7.1.5(c)). 7.1.5.1

Elution chromatography

Consider first the process known as elution chromatography used to separate components from one another present in a multicomponent gas mixture or liquid solution. In such a process, a small volume, V0, of the feed solution or feed gas mixture is first introduced into the column at the column top. Different species present in this feed sample are

adsorbed in a small band of stationary adsorbent particles in the region of the column top. (The nature of the adsorbent in relation to the nature of the feed fluid, gas or liquid will be discussed in detail in Section 7.1.5.1.2.) Next the solvent used in the feed solution, another solvent or an inert carrier gas is introduced into the column top at a steady rate. If one monitors7 the effluent concentration at the column bottom or end, one observes, after some time, a series of concentration pulses of different species appearing one after another (Figure 7.1.21). By collecting effluent volumes between appropriate times (see Section 2.5), it is possible to have each particular solute species in the feed essentially segregated into separate and different effluent volumes; thus, all species present in the multicomponent feed have been separated from one another. However, they are obtained as a solution in the mobile-phase solvent or carrier gas. The solvent or carrier gas is identified as the eluent, and this process of multicomponent separation using a fixed bed of adsorbents is called elution chromatography.

7

Generally, two types of detectors are used to monitor gas compositions: a thermal conductivity detector (TCD) for common permanent gases like O2, N2, CO2, He, Ar, etc., or a flame ionization detector (FID) for common combustible hydrocarbon gases/vapors such as CnH2nþ2, Cn H2n and all sorts of volatile organic compounds. Ultraviolet spectrophometric detectors, refractive index detectors and conductivity monitors are used to monitor liquid-phase compositions.

528

Bulk flow perpendicular to the direction of force

The mechanism of separation may be described as follows. As the solute-free solvent or carrier gas is introduced into the column, it desorbs the solutes adsorbed on the column-top particles from the initial feed sample. Those solutes that are adsorbed less strongly desorb more easily and are quickly carried downstream by the eluent. All solute species are desorbed and carried downstream, where they are readsorbed on fresh adsorbents immediately downstream. As fresh eluent appears and contacts these particles, the solute molecules go through cycles of repeated desorption–adsorption along the column length. This process continues till they appear in the effluent at column exit, with the least strongly adsorbed solute appearing first and the most strongly adsorbed appearing last. 7.1.5.1.1 Liquid–solid adsorption based elution chromatography First, we consider systems where the eluent is a liquid; further, the mechanism of solute partitioning onto the solid particles is simply adsorption, e.g. adsorption on silica or alumina particles. We assume further that the eluent pressure drop along the bed does not influence the adsorption process. There are a number of ways by which elution chromatography in such a system may be described mathematically. The differential equation describing the solute adsorption–desorption process for a dilute liquid-phase system in a fixed bed of adsorbent particles continues to be equation (7.1.4): ∂C i2 ð1 − εÞ ∂C i1 ∂C i2 ∂2 C i2 þ þ vz ¼ Di;eff;z : ∂t ∂t ∂z ∂z2 ε We will develop a solution for elution chromatography in two ways. In the first method, a direct solution of this equation is developed for the initial and boundary conditions appropriate for elution chromatography using linear local equilibrium assumption everywhere. In the second method, the solution (7.1.18h) developed earlier for a fixed-bed process with a continuous liquid feed stream is differentiated to approximate the conditions for elution chromatography. We consider the first method now. Rewrite the governing equation using the local equilibrium assumption and relations (7.1.6) and (7.1.7) as ∂C i2 vz ∂C Di;eff;z ∂2 C i2 i i2 ¼ h i þh :     ρ ρ ∂t ∂z ∂z2 1 þ b qi0 C i2 1 þ b qi0 C i2 ε ε ð7:1:92Þ For linear adsorption equilibrium, encountered in dilute systems,   ð1 − εÞ q0i C i2 ¼ κi1 ρb

ð7:1:93aÞ

is a reasonable assumption, with κi1 constant. For porous adsorbent particles, employing equation (7.1.13f), we get the following form of equation (7.1.92):

∂C i2 vz ∂C 3 i2 þ 2 ∂t   ∂z 41 þ ð1 − εÞ εp κim þ ð1 − εÞð1 − εp Þρs q0i C i2 5 ε ε ∂2 C i2 : ∂z 2 ð1 − εÞð1 − εp Þρs 0  5 qi C i2 þ ε Di;eff;z

¼ 2

41 þ ð1 − εÞ εp κim ε

3

ð7:1:93bÞ

Additionally, the total local molar concentration C it of species i per unit volume (which incorporates both mobile- and stationary-phase solute contributions) is given, for nonporous particles, by C i1 C it ¼ ε C i2 þ ð1 − εÞC i1 ¼ ε C i2 þ ð1 − εÞ C i2 C i2   C i1 C i2 ¼ ½ε þ ð1 − εÞκi1  C i2 ; ð7:1:94Þ ¼ ε þ ð1 − εÞ C i2

where ½ε þ ð1 − εÞκi1  is now a constant. Equation (7.1.92) may be reformulated now with C it ðz;t Þ as the dependent variable: ∂C it þ ∂t

vz ∂C Di;eff;z ∂2 C  it ¼   2it : ð1 − εÞ ð 1 − εÞ ∂z ∂z κi1 κi1 1þ 1þ ε ε

ð7:1:95aÞ

This is a linear equation with constant coefficients and is essentially equivalent to equation (3.2.10). Only the coefficients are different. For porous adsorbent particles, we need to change equation (7.1.94) and obtain instead the following: C it ¼ C i2 ðε þ ð1 − εÞεp κim þ ð1 − εÞ κi1 :

ð7:1:95bÞ

Equation (7.1.93b) may now be reformulated with C it ðz;tÞ as the dependent variable: ∂C it þ ∂t

vz ∂C  it ð1 − εÞ ð1 − εÞ ∂z 1þ εp κim þ κi1 ε ε

Di;eff;z ∂2 C  2it : ¼  ð1 − εÞ ð1 − εÞ ∂z εp κim þ κi1 1þ ε ε

ð7:1:95cÞ

Since the initial solute band on top of the column (very near z ¼ 0) is a very thin one, we will assume that all of the solute species i (mi moles) introduced via the feed sample are essentially located at z ¼ 0 at the start of solvent elution operation (t ¼ 0); therefore, at t ¼ 0, C it ¼ ðmi =Sc Þ δðzÞ:

ð7:1:96Þ

Although the length of the column is finite, we will assume that the solution for z ! ∞ is usable for our purpose (exactly as in the boundary condition (3.2.14a)). For all t, then, at z ¼ ∞, C it ¼ 0

and

∂C it ¼ 0: ∂z

ð7:1:97Þ

The solution for C it from equation (7.1.95a) may be obtained following equation (3.2.19) as the Gaussian concentration profile,

7.1

Force −rμi in phase equilibrium: fixed-bed processes

92 3 2 8 > > > > = 7 6 < vz t 7 6   z − 7 6 > > ð 1 −ε Þ > 7 6 > ; : κ 1 þ i1 7 6 mi 1 ε 6 9 7 C it ðz;t Þ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 6 − 8 7: 7 6 Sc u > > > > u 4πDi;eff;z t  7 6 = < u 7 Di;eff;z 6 ð1− εÞ t 7 6   t 4 κi1 1þ 4 > 5 > ð1 −εÞ > ε ; : 1þ κi1 > ε

ð7:1:98aÞ

Correspondingly, the solution for the profile C i2 of species i in the mobile phase is the Gaussian concentration profile, 92 3 2 8 > > > > = 7 < 6 vz t 7 6   z − 6 > > 7 ð1 − εÞ 6 > ; 7 : 1þ κi1 > −1 7 6 mi ½ε þ ð1 −εÞκi1  ε 9 7 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 6 C i2 ðz;tÞ ¼ 7: 6− 8 u 7 6 Sc u 4πDi;eff;z t > > > > 7 6 = <   u 7 Di;eff;z 6 ð1− εÞ t 7   6 t 4 κi1 1þ 5 4 > > ð1 − εÞ > > ε ; : 1þ κi1 ε

ð7:1:98bÞ

This concentration profile may be written in a compact form as follows:  ( )  ðz − R0i vz tÞ2  : ð7:1:98cÞ C iz ðz;tÞ ¼ C iz ðz;tÞ exp max 2 σ 2ti The quantities R0i and σ ti are defined below as we characterize this profile, with C iz ðz;tÞjmax being the preexponential factor in (7.1.98b) as well as the value of C i2 ðz;tÞ when z ¼ R0i vz t. The center point of the profile of the total concentration C it of species i (also of C i2 ) at any time is located at z¼

vz t  ¼ R0i vz t ¼ zi0 : ð1 − εÞ 1þ κi1 ε

ð7:1:99aÞ

Since the quantity vz

,

1þ

ð1 − εÞ κi1 ε

!

is a constant, the z-directional velocity of the center point of the C i2 ðz;tÞ, as well as the C it ðz;t Þ, profiles is   dz vz  ¼ R0i vz ; ð7:1:99bÞ ¼ ð1 − εÞ dt center point of i κi1 1þ ε where R0 i , the fraction of the total solute i present in the mobile liquid phase, was defined earlier by relation (7.1.16c). The standard deviation of this profile is given by 8 91=2 > > > > < = 2Di;eff;z t  : ð7:1:99cÞ σ ti ¼  > > ð 1 − εÞ > : 1þ ; κi1 > ε The retention time, t Ri , of any species i is obtained from (7.1.99a) when z ¼ L, the column length:

529 t Ri ¼ ðL=R0i vz Þ:

ð7:1:99dÞ

If we can assume that the movement of each species (present in the feed sample introduced in the column prior to elution with the solvent) is independent of those of all other species (generally valid for dilute systems), then it is obvious from relation (7.1.99b) that the migration velocity of each species i down the column will be different as long as each κi1 is different. If the migration velocity of each species i down the column is different from those of others, the concentration peaks monitored in the detector at the column end (Figure 7.1.21) at different times will represent different species. Generally, this is indicated by stating that each species i has a different retention time t Ri (see (2.5.3)) or retention volume V Ri after which it appears at the column exit. The retention volume, V Ri , is the volume of elution solvent that passes through the column after which species i appears at the exit. Since the mobile-phase velocity vz is constant, this volume V Ri is obtained from V Ri ¼ ðt Ri Þ  ðvolumetric flow rate of mobile phaseÞ ð7:1:99eÞ ¼ t Ri  ε Sc vz :

If the column length is z, then, for any species i, t Ri and z are related by (7.1.99a). Substituting that relation into (7.1.99e), we get   ð 1 − εÞ V Ri ¼ z ε Sc 1 þ ð7:1:99fÞ κi1 ; ε

where, however,

z ε Sc ¼ V M

ð7:1:99gÞ

is the mobile-phase volume within the column of length z. Rewriting (7.1.99f) as   ð 1 − εÞ κi1 ¼ VM =R0 i ¼ V M þ V S κi1 ; VRi ¼ VM þ VM ε ð7:1:99hÞ

where VS is the stationary adsorbent-phase volume in the column, the contributions of the quantities VM, VS and κi1 to V Ri become clear. Therefore, as long as κi1 for each species i between the mobile fluid phase and the stationary adsorbent particles is different, each V Ri will be different. Correspondingly, each t Ri will be different. Further, both V Ri and t Ri are increased by an increase in VM, VS and κi1 . Conversely, a reduction of V Ri and t Ri is achieved for any given system by a reduction in VM and VS. The resolution, RS, between peaks of two neighboring species i and j is h i 1 1 t Ri − t Rj L R0i − R0j   ¼ RS ¼ 2σ ti þ 2σ tj 2vz σ ti þ σ tj     ð1 − εÞ ð 1 − εÞ κi1 − 1 þ κj1 L 1þ ε ε 28 ¼ 91=2 8 91=2 3 : > > > > = < < 6 2Dj;eff;z t Rj = 7 2Di;eff;z t Ri 7 þ 2vz 6 5 4> > ; ; :1 þ ð1 − εÞ κ > :1 þ ð1 − εÞ κ > i1 j1 ε ε ð7:1:99iÞ

530

Bulk flow perpendicular to the direction of force

  The resolution is proportional to t Ri − t Rj and thus to     V Ri − V Rj , i.e. to V S κi1 − κj1 . Therefore, one often uses the net retention volume, V N i , for species i: VN i ¼ VRi − VM ¼ VS κi1 :

ð7:1:99jÞ

Reflect now on what is being achieved in elution chromatography. By having the bulk flow of the mobile phase (solvent) perpendicular to the direction of force (for adsorption from the mobile phase to the stationary particle phase; for desorption from the stationary particle phase to the mobile phase), a multicomponent mixture of solutes is separated, as long as RS values are reasonable. If the feed mixtures were simply equilibrated with the adsorbent particle phase without any mobile-phase flow perpendicular to the force direction, no such separation would have been achieved. Consider, in addition, the nature of the force being used in separation. Since there is adsorption and/or desorption of solutes between the mobile fluid and the stationary solid phases, the potential profile under consideration for each solute is discontinuous, a simple step function (see Figure 3.2.2); there are no external forces. According to Section 3.2, such a system in a closed vessel without any flow does not have any multicomponent separation capability. Multicomponent separation capability is, however, achieved in elution chromatography by having bulk flow perpendicular to the direction of the discontinuous chemical potential profile. The velocity vz here func0 tions exactly like the quantity ð − ϕþ 1 Þ in equation (3.2.37). To illustrate this, ignore the diffusion coefficient based term in (7.1.95a) to obtain ∂C it ¼ − ∂t

vz ∂C  it : ð 1 þ εÞ ∂z 1þ κi1 ε

ð7:1:100Þ

In the trailing edge of the profile (along the column length z),     ∂C it =∂z is positive and ∂C it =∂t is negative. Thus, solute i is being picked up from the trailing edge region and carried forward to increasing z to make room for the slower-moving species j to come and get adsorbed in the stationary adsorbent particles. Correspondingly, in the leading edge of the C it profile, C it is decreasing with z, therefore C it is increasing with time due to solute i brought over from the trailing edge region by mobile-phase convection. This is how separate profiles of multiple species are developed along the column length, and ultimately the chromatographic detector output shown in Figure 7.1.21 is obtained as a function of time. The results of elution chromatography were illustrated above using the solution of the equation for the local value of the total solute concentration C it . In reality, a series of profiles of mobile-phase concentration C i2 show up at the column end at different times in elution chromatography. Such a result may be obtained by using the second method mentioned earlier.

Consider solution (7.1.18h) for a fixed-bed process having a liquid feed of constant inlet concentration C 0i2 into an initially solute-free column (i.e. C ii2 ¼ 0). The solution may be expressed compactly as C i2 ðz;t Þ ¼ C 0i2 θi ðz;V Þ;

ð7:1:101aÞ

where θi ðz;V Þ is the solution (7.1.18h) for the linear dispersive equilibrium condition. Suppose that a volume V 0 of solution of concentration C 0i2 is passed through the column from time t ¼ 0 to t ¼ t0. Now, for time t > t0, suppose that the interstitial liquid velocity remains the same, but that the incoming solution concentration is changed to C 00 i2 . One is interested in knowing what happens at t > t0, especially at very large times. Since the system (including the governing equation, the boundary and initial conditions) is linear, the sum of two solutions may be used to develop a solution for this changed condition. For V  V 0 , C i2 ¼ C 0i2 θi ðz;V Þ

V  V 0:

ð7:1:101bÞ

For V > V 0 , one can assume that an inlet concentration of  00  C i2 − C 0i2 has been introduced into the column for a volume flow equal to ðV − V 0 Þ; however, the effect of C 0i2 must continue. Therefore for V 0 < V < ∞, the solution is additive:8     0 0 C i2 ¼ C 0i2 θi ðz;V Þ þ C 00 V 0 < V < ∞: i2 − C i2 θi z;V − V ð7:1:101cÞ

The second term reflects the contribution of the extra   0 solute concentration C 00 passed is i2 − C i2 as if the volume  00 0 0 and the feed concentration is C 00 i2 − C i2 . When C i2 ¼ 0, we find    ð7:1:101dÞ C i2 ¼ C 0i2 θi ðz;V Þ − θi z;V − V 0 : If V 0 is small, as in elution chromatography, a limiting process can be initiated:   0 C i2 0 θi ðz;V Þ − θ i ðz;V − V Þ ¼ V ; ð7:1:101eÞ V0 C 0i2 C i2 ∂θi ðz;V Þ : ¼ V0 ∂V C 0i2

ð7:1:101fÞ

The approximation leading to the derivative is acceptable only in the limit of V 0 ! 0. Using the solution (7.1.18h) for θi ðz;V Þ, 8

This is the principle of superposition. Since equation (7.1.18c) or equation (7.1.95a) are linear equations due to constant κi1, the superposition of two simpler solutions by simple addition leads to the solution of the problem under consideration. This is the advantage of linear chromatography models.

7.1

Force −rμi in phase equilibrium: fixed-bed processes

8 93 > > 1=2  < Pez;eff V −V =7 C i2 V ∂ 6 41 þ erf  1=2 >5 0 ¼ > C i2 2 ∂V : 2 VV ; 2

0

8 9 > > 8 > > 0 19 > > > 1=2 = < < V −V 2 > = Pe 1 V −V z;eff 0 @ A exp − ¼ V pffiffiffi : 1=2 :1 þ ; > 2 V 4V V > > > 2 π VV > > > > : Pez;eff ; ð7:1:101gÞ

  Since V −V =2V is, in general, small compared to 1 in the breakthrough region, we can rewrite the above as 8 9 > > > >   > 1=2 2 < V −V > = Pez;eff  0 0 C i2 ffi V C i2 pffiffiffi ; ð7:1:102aÞ 1=2 exp > − > 4V V > > 2 π VV > > : ; Pez;eff 0

C 0i2

¼ mi is the number of moles of solute i where V injected. This is a Gaussian profile around V , with a standard deviation σ V i given by σV i ¼



2V V Pez;eff

1=2

:

ð7:1:102bÞ

For such a profile, the following approximations are valid around9 V ¼ V ¼ V i : 

VV

1=2

ffiV

and

 1=2 σ V i ffi V 2=Pez;eff : ð7:1:102cÞ

Employing the definition of V and Pez,eff in (7.1.18i), σ 2V i ¼

2S2C z2 ½ε þ ð1 − εÞ κi1 2 Di;eff;z 2Di;eff;z / z; ð7:1:102dÞ zvz vz

where z is the column length. For illustrative examples of the above procedure, consult Mayer and Tompkins (1947) and Lightfoot et al. (1962). Expressions for similar profiles may be developed for other species present in the small feed sample of volume V0 introduced initially at the top of the column. To determine κi1 and Pez;eff from V ¼ V , see Problem 7.1.13, in which column information and flow rates are provided. A simpler representation of the solute i elution profile (7.1.102a) is ( ) ðV − V Þ2 C i2 ¼ C i2 jmax exp − ; ð7:1:102eÞ 2 σ 2V i where h i pffiffiffi 1=2 C i2 jmax ¼ ðV 0 C 0i2 Þ Pez;eff =f2 πðV V Þ1=2 g

ð7:1:102fÞ

9 V here refers to expression (7.1.16d) for solute i and therefore should more correctly be represented as V i , which is not to be confused with the partial molar volume of species i.

531

is the maximum value of C i2 achieved when V ¼ V (see equation (2.5.6b)). The volume of eluent collected during the elution of a solute band is ultimately the source of recovering the solute. The presence of multiple eluting solutes with overlapping bands forces us to make a decision on the eluent volume acting as the cut point tc in time (see Figure 2.5.1(b)). Correspondingly, one is interested in knowing what is the fractional recovery of a particular solute for a given amount of solute input moles ð¼ mi ¼ V 0 C 0i2 ¼ m0i Þ. For the species 1 type of profile shown in Figure 2.5.1(b), the fractional recovery of solute i from time t ¼ 0 (alternatively t ¼ −∞) to time t ¼ tc, the cut point, is given by mii m0i

fractional solute recovery ¼

¼

ðt

−∞

Qf C i2 ðtÞ dt m0i

ðtc

¼ ð ∞− ∞

−∞

ε Sc vz C i2 ðtÞ dt ε Sc vz C i2 ðtÞ dt

: ð7:1:102gÞ

However, from equations (2.5.11)–(2.5.13), we already know that this ratio for Gaussian profiles is given by     mii 1 1 t c − t Ri 1 1 V c − V Ri p ffiffi ffi p ffiffi ffi ¼ ¼ ; þ erf þ erf 2 2 2 2 m0i 2 σ ti 2 σV i ð7:1:102hÞ where the final expression has been written by analogy, since the volumes Vc (the volume at the cut point), VRi and σ V i in the elution process are linearly proportional to t c , t Ri and σ ti ðVc ¼ ε vz S t c ; etc:Þ. Similarly, for the species 2 type of profile shown in Figure 2.5.1(b), mii m0i

fractional solute recovery ¼

¼

ð∞ tc

Qf C i2 ðtÞ dt m0i

ð∞

¼ ð ∞t c

ε Sc vz C i2 ðtÞ dt

−∞

ε Sc vz C i2 ðtÞ dt

:

ð7:1:102iÞ

From equations (2.5.14)–(2.5.16), we already know that this ratio for Gaussian profiles is given by fractional solute recovery ¼ ¼

mii m0i

    1 1 tR − tc 1 1 VR − Vc ¼ : þ erf piffiffiffi þ erf piffiffiffi 2 2 2 2 2 σ ti 2σ V i ð7:1:102jÞ

Example 7.1.5 In the purification of an enzyme by an appropriate column of adsorbents, the elution process provided the information given in Table 7.1.4 about the enzyme concentration (in suitable units) in the eluent leaving the column (volume 2 liter) and the total volume of eluent passed.

532

Bulk flow perpendicular to the direction of force

Table 7.1.4. C i2

V (liter)

6.5 15.0

1.8 2.0a

a It is known that the C i2 value of 15.0 was the maximum; therefore V ¼ 2 liter.

Determine the value of σ V i (liter) and the fractional recovery of the enzyme (assuming that the concentration profile is Gaussian) when the eluent volume passed is 2.1 liters. Solution Consider the Gaussian elution profile (7.1.102e): ( ) ðV − V Þ2 C i2 ¼ C i2 jmax exp − : 2 σ 2V i Apply the profile to the two data points provided and then divide one by the other: 9 8 < ðV − V Þ2 = 1 exp − : 2 σ 2V i ; C i2 j1 0:65 9 ) 8 ¼ 1:5 C i2 j2 < ðV − V Þ2 = 2 exp − 2 : 2 σ Vi ; 0 1 2 ð1:8 − 2:0Þ A exp @ − 2 σ 2V i 0 1 ) 0:43 ¼ 2 ð2 − 2Þ A exp @ − 2 σ 2V i 0 1 0 1 2 ð0:2Þ 0:02 A ) exp @þ A ¼ 2:325 ¼ exp @ − 2 σ 2V i 2 σ 2V i

) ð0:02=σ 2V i Þ ¼ 0:844 ) σ V i ¼ 0:154 liter:

Since the eluent volume passed, 2.1 liters, is larger than V ¼ 2 liter, the cut point tc > t Ri . So the profile corresponding to species 1 in Figure 2.5.1(b), is relevant. The formula for fractional recovery is given by (7.1.102h):   1 1 Vc − VR fractional recovery ¼ þ erf pffiffiffi i 2 2 2 σV i   1 1 2:1 − 2:0 ¼ 0:5 þ 0:5 erf ð0:459Þ ¼ þ erf 1:414  0:154 2 2 ¼ 0:5 þ 0:5  0:484 ¼ 0:742:

Elution chromatography based on liquid–solid adsorption for small molecules typically employs adsorbents such as alumina, charcoal, silica, hydrophobic silica, etc. Surface area, water content and the chemical nature of the adsorbent (e.g. polar or nonpolar) distinguish one adsorbent from another. For larger molecules/macromolecules/charged species, other adsorbents needed are briefly touched upon in Sections 7.1.5.1.6–7.1.5.1.8. 7.1.5.1.2 Types of elution chromatography The type of elution chromatography described earlier was liquid– solid adsorption chromatography. The mobile phase was

liquid, the stationary phase consisted of porous particles, and the local mechanism of separation of various species between the mobile phase and the stationary phase was simply adsorption. There can be a number of other types of elution chromatography depending on the combination of mobile and stationary phases and the separation mechanism locally operative. The more well-known and frequently used of these techniques are summarized in Table 7.1.5. There are variations in a given technique identified in Table 7.1.5. For example, in liquid–liquid chromatography (LLC), normally the stationary-phase liquid is polar and the mobile phase is nonplolar, both being immiscible with each other. In reversed-phase LLC, the stationary-phase liquid is nonpolar, whereas the mobile phase is polar; the polar phase can even be water. In many cases, the stationary liquid phase may be a monomolecular layer chemically bonded to the surface groups of the porous support material. This is often achieved with a nonpolar hydrocarbon liquid phase. A very common column configuration in elution chromatography is simply a tubular column packed with porous particles, the packings, with or without a bonded liquid phase on the particle surfaces. Other column configurations include capillary columns or open tubular columns, in which a thin liquid film of adsorbents has been applied (or bonded) to the internal surface of the capillaries. A potential variation of this is the microporous hollow fiber membrane based column, wherein the stationary phase is held in the pores of fiber wall and the eluent is passed through the bore of the fiber (Ding et al., 1989). Analytical-scale column chromatography, used to identify the components in a sample or to determine the feed sample composition, uses small columns; for example, in gas–liquid chromatography, a typical column may be 6 ft long. In preparative chromatography, column diameters of 1 to 2 cm are used, and the feed sample volumes are considerably larger. While this is used to prepare larger amounts of purified materials or compounds for laboratory use, industrial-scale column chromatography, producing 100 metric ton/year, is being practiced (Chem. Eng., 1980, 1981). The same organization (Elf Aquitane) has demonstrated a variety of separations in gas chromatography using industrial-scale columns of diameter 40 cm, length 1.5 m, containing 60–80 mesh packing operating at 180  C for injection times between 5 and 50 s and sample sizes between 10 and 50 g/s (Bonmati et al. 1980). In column or capillary chromatography, the gas or liquid eluent is driven to flow by the application of a pressure gradient along the column length. In paper or planar chromatography, the adsorbent material is in a thin granular form (thin-layer chromatography) or in fibrous form (as in paper chromatography); no such pressure gradient can be independently applied. Instead, the liquid eluent is driven along the layer by capillary action (i.e. the capillary pressure, see Section 6.1.4). The rate of

7.1

Force −rμi in phase equilibrium: fixed-bed processes

533

Table 7.1.5. Phase combinations and local retention mechanisms in chromatography Mobile phase

Stationary phase

Local mechanism of separation

Name

Liquid

porous solid adsorbent

adsorption

Liquid Liquid

porous ion exchange resin particles liquid stationary phase coated on a porous solid support

Liquid

porous gels

ion exchange partitioning (as in solvent extraction between two immiscible liquid phases) molecular size exclusion via gel pore size

liquid–solid adsorption chromatography (LSC) ion exchange chromatography liquid–liquid chromatography (LLC)

Liquid

immobilized affinant on an insoluble support having strong affinity for macromolecules liquid stationary phase with low vapor pressure coated on a porous solid support porous solid adsorbent porous solid adsorbent or liquid stationary phase coated on a porous solid support

Gas

Gas Supercritical fluid

gel permeation chromatography (GPC) or size exclusion chromatography (SEC) affinity chromatography

reversible binding between solute and affinant differential solubility in the stationary phase and differing solute vapor pressure adsorption adsorption or differential solubility and differing solute vapor pressures

gas–liquid chromatography (GLC)

gas–solid chromatography supercritical fluid chromatography

spreading of the liquid cannot be precisely controlled. Further, the movement of the liquid is two-dimensional, unlike in column chromatography, in which the movement is in one dimension only, along the column length.

acting as a porous medium (see the solvent flux expressions (3.4.85), (3.4.88) and (6.1.4g)):

7.1.5.1.3 Elution chromatography with mobile gas phase Elution chromatography with a mobile gas phase (gas chromatography, GC) can be carried out with either porous solid adsorbent or a liquid absorbent coated on porous solid support particles (see Table 7.1.5). The former is identified as gas–solid chromatography (GSC), whereas the latter is called gas–liquid chromatography (GLC). Sometimes in GLC, the liquid absorbent exists as a coating on the inside surface of a capillary tube. The liquid absorbent in GLC must have low enough vapor pressure to be considered essentially nonvolatile. The analysis of elution chromatography with a liquid eluent that was presented earlier was based on a constant eluent velocity, vz. Unlike simple fixed-bed adsorption processes with low pressure drops, analytical-scale chromatographic techniques for a mobile gas phase employ long packed columns, where the gas undergoes a considerable pressure drop. As a result, the gas velocity changes with column location. But the gas velocity is proportional to the molar gas volume at every location. Therefore, by Boyle’s law, the gas pressure P and velocity vz at any location are related to those at the inlet (subscript, in) and outlet (subscript, out) by

Remember: the Darcy permeability Qg is proportional to r 2p , where r p is the pore radius, and therefore to d 2p , where d p is the packing diameter (since it is essentially proportional to the effective diameter of the interpacking opening). Substituting for vz in terms of P from (7.1.103a) into the above expression, then integrating and rearranging, we get ( )1=2 v z ðz Þ ðP in =P out Þ2 ; ð7:1:103cÞ ¼

 vz;in ðP in =P out Þ2 − ðP in =P out Þ2 −1 Lz

Pvz ¼ P in vz;in ¼ P out vz;out :

ð7:1:103aÞ

The gas velocity vz and the gas pressure gradient (dP/dz) at any location are related by Darcy’s law for the packed bed

vz ¼ −

Qg dP : εμ dz

ð7:1:103bÞ

where μ is assumed to be constant and independent of gas pressure. The retention time t Ri may now be obtained from relation (7.1.99b) if vz is considered to be a function of the z-coordinate: t Ri

ð 0

dt ¼

ðL 0

dz ¼ t Ri : R0i vz ðz Þ

ð7:1:103dÞ

Substituting into this an expression for vz(z) from (7.1.103c) and integrating, one obtains t Ri ¼

L 1 ; R0i vz;avg

ð7:1:103eÞ

where " # vz;avg 3 ðP in =P out Þ2 − 1 ¼ ; vz;out 2 ðP in =P out Þ3 − 1

ð7:1:103fÞ

534

Bulk flow perpendicular to the direction of force

as demonstrated first by James and Martin (1952). This last ratio allows the conversion of the known gas velocity at column outlet, vz,out, to an average gas velocity, vz,avg, in the column for calculating the retention time t Ri of species i. Correspondingly, the retention volume of gas needs to be corrected due to this gas compressibility. In addition, if the column temperature T is different from the exit temperature Tout at which the gas flow rate is measured, an additional correction has to be made. Given the flow conditions and column dimensions, the retention time or volume of a species is known, if κi1 is known. Since the variation in V Ri or t Ri between two neighboring peaks or species depends on values of κi1 for i ¼ 1, 2 (say), it is useful to enquire what properties of the gas–stationary phase combination influence κi1 . For gas–liquid chromatography (GLC) with liquid coating on packings, assume equilibrium to exist between the two phases for any species i. Then, from Section 3.3.7.1, we get ^f ig ¼ ^f il , ^f ig ¼ x ig ϕig P and ^f il ¼ γil x il f 0il . Since the gas phase may be assumed to behave ideally at the lowpressure characteristics of GLC, ϕig ffi 1. For condensible solute species (vapors) at low pressures, f 0il ffi P sat from i equation (3.3.62). Further, the solute concentration in the stationary liquid phase on packing is very low, so that γil ffi γ∞il , corresponding to infinite dilution. Therefore x il P ¼ x ig γ∞il P sat i

and

α12 ¼

x 1l x 2g γ ∞2l P sat 2 ¼ ; x 1g x 2l γ ∞1l P sat 1 ð7:1:104aÞ

where species 2 elutes earlier from the column than sat species 1. Generally, P sat 2 > P 1 . However, the ratio of the activity coefficients is quite important, especially in the separation of two species of essentially equal vapor pressure. The selectivity is obtained by different molecular interactions between the solute species to be separated and the stationary liquid phase. For a brief introduction to the contribution of the activity ratio to the selectivity, see Karger et al. (1973). The net retention volume, V N i (definition (7.1.99j)), for and γ∞i . species i can also be expressed in terms of P sat i Consider the distribution coefficient κi1 first and assume ideal gas behavior: C i1 C i1 C i1 RT mil RT ¼ ¼ ¼ x ig P V Sl x ig P C i2 C ig       mil RT msl x il RT msl ¼ ffi ; x ig P V Sl msl x ig P V Sl

κi1 ¼

where mil denotes the moles of solute i in a stationary liquid phase of volume V Sl and solvent moles msl . Therefore,   RT msl κil ¼ ∞ sat : ð7:1:104bÞ V Sl γil P i To get this result, we have used relation (7.1.104a) for   x il =x ig and assumed mil vzjmin , the increase in HETP is very little (Majors, 2005). This same reference provides an estimate of the number of plates in a 15 cm column as the particle size is varied: 100 μm, 200; 50 μm, 1000; 10 μm, 6000; 5 μm, 12 000; 2.5 μm, 25 000; 1.8 μm, 32 500. Additional developments have led to a different form of the Van Deemter equation, where the plate height H may be represented in the following fashion (Giddings, 1965, eq. (2.11.2), 1991; Horvath and Lin, 1978): H ¼

X B 1 : þ C vz þ vz ð1=AÞ þ ð1=C m vz Þ

ð7:1:107dÞ

The following alternative representation is also employed (Jennings, 1987; Karger et al., 1993): H ¼ H D þ H M þ H SM þ H S ;

ð7:1:107eÞ

540

Bulk flow perpendicular to the direction of force

where

Table 7.1.6.

HD is due to the longitudinal molecular diffusion, HM is due to the mobile-phase contributions in a packed bed, HSM is due to the stagnant mobile phase in particle pores, HS is due to the stationary-phase sorption–desorption.

He flow rate (cm3/min)

Retention time (cm)

Peak base width (cm)

H (cm)

7.1 34.4 78.9

14 5.3 3.28

3.3 0.95 0.66

0.529 0.306 0.386

Expressions for the individual plate height contributions are given below: 2Dil γ1 ; HD ¼ vz

ð7:1:107fÞ

where Dil is the diffusion coefficient of species i in the mobile-phase liquid and γ1 is the obstruction factor ( t Ri j2 :

ð7:1:109oÞ

In the separation of a number of proteins by elution based ion exchange chromatography (see Ladisch (2001) for much greater detail), first the sample is injected. Different proteins undergo ion exchange based binding with the resin (see Figure 4.1.16). Soon after the sample injection, the salt concentration in the mobile phase is increased with time, say, in a linear or stepwise fashion. Therefore, the proteins which are bound more strongly, and therefore have a larger value of V Ri or t Ri at a given asw, will come out much earlier, since asw has been increased. Figures 7.1.25(b) and (c) illustrate this behavior. Figure 7.1.25(b) illustrates the elution profiles for proteins A and B, where the salt concentration in the eluent is constant at a low value of asw j1 ; as a result, t RA and t RB are far apart. Figure 7.1.25(c) shows that gradient elution is started close to the time when protein A elutes by a step increase in salt

544

Bulk flow perpendicular to the direction of force

(a) D

3 B 2

D logk i l⬘

il

4 3

log

logVNi

4

2

A

B A

1

1

−1

−0.8 −0.6 −0.4 −0.2 log asw (c) Protein concentration

(b) Protein concentration

−0.8 −0.6 −0.4 −0.2 log asw

−1

3 2 1 0

0 tinj

tRA

asw

t

tRB

constant 1

Salt concentration increased

3 2 1 0

0 tinj

ts

tRA

t

asw

1 from t = 0 to t = ts

asw

2 > asw

tRB

1 from t ³ ts

Figure 7.1.25. Ion exchange chromatography. (a) Variations of net retention volume V N i and partition coefficient κi1 as well as the capacity factor ðk0i1 Þ with eluent ion activity. Gradient elution is depicted in (b) for constant eluent ion concentration and in (c) for stepwise variation of eluent ion concentration.

concentration. The result is that the protein B peak comes out much earlier. Instead of a step change in salt concentration, the salt concentration in the eluent may be varied continuously, for example in a linear fashion. The typical concentration variation of NaCl used as a result is from 0.01 M to 0.1 M (up to 0.3 M). The discussion of elution of two protein/ionic species has so far employed species A and B identified in Figures 7.1.25(a)–(c); both species had identical charges. Species D in Figure 7.1.25(a), however, has a different charge, in fact a higher charge jZD j > jZA j ¼ jZB j. Therefore, the negative slope of log V N D or log κD1 vs. log asw is larger than that for species A or B. The difference in the two slopes leads to the following: at lower salt concentrations, D will elute much later; however, at the salt concentration where the two lines intersect, two species will elute at the

same time (no separation). Beyond the intersection, at higher salt concentrations, species D will elute faster than species B (or A if there is an intersection). Such behavior leads to complications; alternatively, it may be an opportunity for improved separation. 7.1.5.1.7 Elution chromatography with a mobile liquid phase – size exclusion chromatography We will now briefly consider size exclusion chromatography (SEC), which is also called gel permeation chromatography (GPC), gel filtration or gel chromatography. In Section 3.3.7.4, we came across porous crosslinked polymers called hydrogels and the partitioning behavior of macromolecules/proteins between their solutions and the hydrogels (e.g. κim given by (3.3.90i)). Generally, the lower the molecular weight and the smaller the effective radius (e.g.

Force −rμi in phase equilibrium: fixed-bed processes

(a)

(b) 7

(c) 6

6

9 8

5 log Mi

log Mi

545

5

log [h]i Mi

7.1

4

4

7 6

3 3

0

1

1.5

im

5 VRi

VRi

Figure 7.1.26. Characteristics of size exclusion chromatography/gel permeation chromatography. (a) Partition coefficient of macromolecular species i between solution and gel particle: Mi is the molecular weight of macromolecular species. (b) Elution volume of macromolecular species i. (c) Universal calibration plot of many polymers in tetrahydrofuran (THF): log [η]i Mi vs. elution volume V Ri . (After Grubisic et al. (1967).)

the radius of gyration, rg; hydrodynamic radius, rh), the higher the value of κim (Figure 7.1.26(a)). A larger protein/macromolecule has a lower κim ; consequently, it will spend less time in the porous hydrogel phase and will be eluted faster in a chromatographic column containing hydrogel beads as the packing/sorbent phase. Let there be no other type of interaction between the macromolecular solute and the porous hydrogel except geometrical partitioning between the pore liquid and the external liquid (e.g. (3.3.88a) and (3.3.89a)); this interaction is dependent on the pore size, solute size, pore shape and solute shape. Here the value of κim ! 1 when the solute size tends to zero. In such a situation, there is no solute adsorption as such, so that q0i1 ðC i2 Þ in equation (7.1.13f) is zero and vCi ¼

vz : ð1 − εÞ 1þ εp κim ε

ð7:1:110aÞ

Therefore, a large macromolecule/protein which cannot enter the pores of the hydrogel beads will have a concentration velocity vCi ¼ vz , the interstitial fluid velocity. On the other hand, a small solute such as salt will probably have κim ¼ 1 and its vCi P p Þ: N im ðQim =δm Þ ðP f x if − P 0p x 0ip Þ : ¼ N jm ðQjm =δm Þ ðP f x jf − P 0p x 0jp Þ

ð7:2:2bÞ

If there is essentially no pressure drop in the transport of these permeated gases through the porous substrate, then P 0 p ffi P p , which is the pressure in the bulk gas on the permeate side. Further, if this gas mixture does not mix with any other gas stream emerging from other membrane locations, then only x 0 ip ¼ x ip and x 0 jp ¼ x jp , leading to the crossflow relation (7.2.2a). These conditions may be achieved in a hollow fiber permeator where the gas is fed through the hollow fiber bore; the permeate is radially withdrawn from the shell side and there is a low pressure drop in the shell side in the radial direction (Figure 7.2.1(c)). Two other commonly used configurations, a shell-side fed hollow fiber module and a spiral-wound device shown in Figures 7.2.1(d) and (e), will bring permeates emerging from different feed concentrations together at every pore mouth. The hollow fibers in these membrane processes may have internal diameters varying between 50 and 1000 μm and wall thicknesses varying between 5 and 300 μm. The analysis of such configurations is best considered in Chapter 8 since the compositions and pressure drops in both streams flowing perpendicular to the force direction are important. However, the crossflow analysis is still useful because the analytical results allow a quick estimation of the separation performance. We will now develop the governing equations for such a crossflow permeator. For the permeator configuration of Figure 7.2.1(a) and the assumptions of (1) no concentration polarization and (2) no longitudinal diffusion/dispersion in the flow direction (z-coordinate), a control volume analysis (in the manner of Figure 6.2.1 but without any x-coordinate dependence) leads to (under conditions of no chemical reaction (Ri ¼ 0) and steady state ((∂Ci/∂t) ¼ 0)) 0 ¼ ΔxΔy ðN iz jz − N iz jzþΔz Þ − ΔxΔz N im :

ð7:2:3Þ

Here, the control volume dimensions Δx and Δy are along the membrane flow channel of width W and height h, respectively. Further, the expression for Niz in view of assumption (2) is simply

7.2

Crossflow membrane separations, granular filtration

(a)

557

z

(b) L xA2

Feed xAf

Feed gas flow

Exit Wt 2L xA1L Concentrate

Wt2

Nonporous skin

xif Pf

Porous substrate

Membrane Permeate

x ⬘ip,P ⬘p

xip,Pp

Permeate xA1L, Wt1L (c)

Permeate

Reject Hollow fiber membrane (d)

Feed

Permeate

Permeate

Reject (e) Module housing Residue flow Permeate flow Residue flow

Feed flow Collection pipe Feed flow Feed flow

Spacer Membrane Permeate flow after passing through membrane

Spacer

Figure 7.2.1. (a) Crossflow membrane module for gas permeation; (b) model for crossflow gas permeation in an asymmetric or composite membrane; (c) tube-side feed crossflow hollow fiber module; (d) shell-side feed hollow fiber module; (e) spiral-wound module.

N iz ¼ C i2 v ¼ ðW t2 x i2 Þ=W h;

ð7:2:4Þ

where the flow stream on the feed side is j ¼ 2. For the permeate j ¼ 1, p. At z ¼ 0, Niz ¼ (Wtf xif)/Wh, corresponding to the feed location. We are interested in finding out what would be the value of Wt2 and xi2 at z ¼ L, the end

of the permeator on the feed side. Rewriting equation (7.2.3) as 0 ¼ W hðN iz jz − N iz jzþΔz Þ − W Δz N im ; and taking the limit of Δz ! 0, we get

558

Bulk flow perpendicular to the direction of force

dðN iz W hÞ dðW t2 x i2 Þ ¼ −W N im ) ¼ − N im dz W dz )

x A1 ð1 − x A2 Þ ; ð1 − x A1 Þ x A2

αAB ¼

dðW t2 x i2 Þ Q ¼ − N im ¼ − im ðP f x i2 − P p x i1 Þ; dAm δm

ðαAB − 1Þ ¼

αAB x A2 ; 1 þ x A2 ðαAB − 1Þ

x A1 ¼

x A1 − x A2 ; ð1 − x A1 Þ x A2

ð1 − x A1 Þ ¼

ð1 − x A2 Þ : 1 þ x A2 ðαAB − 1Þ ð7:2:11Þ

ð7:2:5Þ where dAm ¼ W dz

Atm

and

¼ W L;

ð7:2:6Þ

L being the permeator length in the z-direction for a total membrane area Atm . This is the governing equation for the feed side. The governing equation for the permeate side is primarily dictated by the crossflow relation (7.2.2a). An alternative form of this relation for a multicomponent system is N im n X

N jm

¼

j¼1

x ip n X

x jp

¼ x ip :

ð7:2:7Þ

j¼1

However, n X

n X

Integrating this relation between the limits of Wt2 and xA2 along the permeator length from z ¼ 0, Wt2 ¼ Wtf, xA2 ¼ xAf to z ¼ L, Wt2 ¼ Wt2L, xA2 ¼ xA2L, Wt1L ¼ Wtp, xA1L ¼ xAp, we get ℓn

xð xð A2L A2L   W t2L 1 dx A2 αAB dx A2 þ ¼ ðαAB − 1Þ W tf x A2 ðαAB − 1Þ ð1 − x A2 Þ x Af



x A2L ¼ ℓn x Af

x Af

ðα

1 AB − 1Þ



1 − x Af þ ℓn 1 − x A2L

ðα αAB− 1Þ AB

W tf − W t1L W t2L ¼ 1−θ ¼ W tf W tf   1   αAB x A2L ðαAB − 1Þ 1 − x Af ðαAB − 1Þ : ¼ x Af 1 − x A2L

;

ð7:2:12Þ

n X

Therefore, knowing αAB and xAf, we have a relation N jm dAm : between the stage cut, θ, and the high-pressure outlet i¼1 i¼1 j¼1 composition (concentrate/reject) xA2L. If either θ or xA2L is specified, the other quantity is obtained from the above Therefore, equation. Note that if the integration is not carried out N im dAm dðW t2 x i2 Þ to the permeator end, this relates xA2 at any location to ¼ x ¼ x ð7:2:8Þ ¼ ip i1 n X dW t2 the corresponding Wt2. It is also important to determine N jm dAm j¼1 the value of xAp, the mole fraction of species A in the permeate stream: is the required relation for the permeate side composxð A2L ition. In addition, we have to satisfy the following two x A1 dW t2 relations: dðW t2 x i2 Þ ¼ dW t2 ¼ −

n X i¼1

x i2 ¼ 1;

N im dAm ¼ −

n X i¼1

x i1 ¼ 1:

One can now follow the solution procedure of Pan and Habgood (1978a) instead of the Weller–Steiner (1950a, b) approach, which is valid for a binary system only. We will now develop an analytical solution for a binary feed gas mixture of species A and B by assuming that the local separation factor αAB is independent of the membrane separator length/location. This approach (Stern and Walawender, 1969) is an adaptation of the Naylor and Backer (1955) approach employed for gaseous diffusion with porous barriers. Rewrite equation (7.2.8) for i ¼ A as dW t2 W t2

x Ap ¼

ð7:2:9Þ

x A2 dW t2 þ W t2 dx A2 ¼ x A1 dW t2 ;   dx A2 1 þ x A2 ðαAB − 1Þ ¼ dx A2 ; ¼ ðx A1 − x A2 Þ x A2 ðαAB − 1Þ ð1 − x A2 Þ

ð7:2:10Þ

where we have used definitions and relations such as

x Af xð A2L

ð7:2:13Þ

:

dW t2

x Af

The integral in the numerator may be rearranged by using expression (7.2.11) for xA1 in terms of xA2, expression (7.2.10) for dWt2 in terms of xA2 and expression (7.2.12) for Wt2 in terms of xA2:

x Ap ¼ −

¼−

xð A2L  x Af

αAB x A2 1þx A2 ðαAB −1Þ

 1þx A2 ðαAB −1Þ W t2 dx A2 x A2 ðαAB −1Þð1−x A2 Þ



ðW tf −W t2L Þ xð A2L x Af

 αAB  ðα 1 −1Þ  1−x Af ðαAB −1Þ αAB ð1−θÞ x A2 AB W tf dx A2 x Af 1−x A2 ðαAB −1Þð1−x A2 Þ θW tf αAB

¼−

αAB ð1−θÞ ð1−x Af ÞðαAB −1Þ ðαAB −1Þθ ðx Af ÞðαAB1 −1Þ

xð A2L x Af

1

ðx A2 ÞðαAB −1Þ ð1−x A2 Þ

2αAB −1 ðαAB −1Þ

dx A2

7.2

Crossflow membrane separations, granular filtration

2

ð1−θÞ 1 6 ¼ 4ð1−x A2L Þ θ ðx A2L ÞðαAB1 −1Þ

αAB αAB −1



x Af 1−x Af

  ααAB−1 AB

αAB αAB −1

− ðx A2L Þ

3

7 5:

x Af ¼ θ x Ap þ ð1 − θÞx A2L ;

ð7:2:15Þ

we can rewrite expression (7.2.14) for xAp as 2 ( )α αAB−1 AB αAB ð1−θÞx A2L þθx Ap ð1−θÞ 1 αAB −1 4

x Ap ¼ ð1−x A2L Þ θ ðx A2L ÞαAB1 −1 1− ð1−θÞx A2L þ θx Ap αAB αAB −1

#

ð7:2:16Þ

:

This is an implicit expression for xAp in terms of xA2L, θ and αAB. The actual separation factor, αAB , is to be determined from the ideal separation factor αAB , the pressure ratio γð¼ P p =P f Þ and the exit composition for the concentrate xA2L (since αAB is assumed to be independent of the local variation of xA2 from xAf to xA2L at the exit) via relation (6.3.202b), where xA2 can be used instead of xAf. The total membrane area Atm (¼ WL) required for this separation can be obtained by combining equations (7.2.5) and (7.2.8): dAm ¼ − Atm ¼

Qim δm

ð xAf

x A2L

dðW t2 x A2 Þ x A1 dW t2  ¼−  ; Qim ðP f x A2 − P p x A1 Þ δm ðP f x A2 −P p x A1 Þ x A1 dW t2

Qim δm



ðP f x A2 − P p x A1 Þ

ð7:2:17Þ

:

Now employ equation (7.2.10) for dWt2, equation (7.2.12) for Wt2 and equation (7.2.11) expressing xA1 in terms of xA2 and αAB in the above integral for Am, and numerically evaluate it. Pan and Habgood (1978a) have provided a solution to this crossflow permeator problem without assuming a constant value of the selectivity αAB for a binary mixture characteristic of the Naylor–Backer approach described above. We use their notation here: species 1, base component (less permeable); xi, mole fraction of species i in feed side; yi, mole fraction of species i in permeate side; αi for i 6¼ 1 is the selectivity of species i with respect to species 1, where α1 ¼ 1; βi ¼ 1 − ð1=αi Þ;Yi ¼ βi yi; γ ¼ pressure ratio (¼ permeate pressure/feed pressure); a ¼ ðγα2 β2 þ 1Þ=ðα2 β2 ð1 −γÞÞ;

b ¼ ðγα2 β2 −α2 Þ=ðα2 β2 ð1 −γÞÞ: ð7:2:18Þ

Their results are as follows: 1−θ ¼



Y2 Y 2;f

a 



1þðα2 −1Þðγþx 2 Þ−

ð7:2:14Þ

This expression relates the permeate mole fraction xAp to the residue composition xA2L as a function of xAf, θ and αAB. Since xAp, xA2L, xAf and θ are related via

−ðx A2L Þ

Y 2¼

559

β2 − Y 2 β2 − Y 2;f

b 

 1−Y2 ; 1 − Y 2;f

where Y 2;f is the value of Y2 at feed location;

ð7:2:19Þ

S¼ð1−Y 2 Þθ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1þðα2 −1Þðγþx 2 Þ2 −4γα2 ðα2 −1Þx 2 2γα2

ðY 2

;

ð7:2:20Þ θdY 2 ;

ð7:2:21Þ

S¼ðQ1 =dÞðP=Lf Þð1−γÞs;

ð7:2:22Þ

Y 2;f

where S is the nondimensional membrane area, (Q1/d) is the permeance (¼ permeability coefficient/membrane thickness) of species 1, P is the feed side pressure, Lf is the molar feed gas flow rate, s is the actual membrane area (¼ WL in our notation). For the special case of α2 >>1 (i.e. β2 ffi 1, b ffi –1). These results are reduced to θ ¼1 − ðY 2 =Y 2;f Þa ; ð7:2:23Þ  a   a Y2 a S¼ 1− 1− Y 2;f − Y 2 : ð7:2:24Þ Y 2;f aþ1 aþ1 Example 7.2.1 Consider the production of nitrogen-enriched air from atmospheric air using an asymmetric cellulose acetate membrane having an ideal selectivity of ðQO2 m =QN2 m Þ ¼ 6:0. If the permeate side is maintained at a considerable vacuum, and if you can assume crossflow and no permeate side pressure drop in the module (spiral-wound or hollow fiber), then determine the stage cut and the permeate composition for the following values of oxygen mole fraction in the concentrate: xA2L ¼ 0.1, 0.05, 0.02, 0.01, where species A is oxygen. Plot the oxygen mole fraction in the permeate and concentrate as a function of the stage cut; include values for θ ¼ 0 and 1. Solution There is no specification of the permeate side pressure, except we have considerable vacuum. Since the pressure ratio γ ¼ (permeate pressure/feed pressure) is very low, we may assume that αAB ffi αAB ; therefore, αAB ¼ αO2 − N2 ¼ 6:0. Since αAB is constant throughout the module (due to negligible pressure drop along the permeate side and very low γ), we can use the result (7.2.12) based on the Naylor–Backer method to calculate θ from α AB

1

  1 − θ ¼ ðx A2L =x Af ÞðαAB − 1Þ ðð1 − x Af Þ=ð1 − x A2L ÞÞðαAB − 1Þ

given xAf ¼ 0.21, αAB ¼ 6 and xA2L (different values; see Table 7.2.1). Table 7.2.1. 1

α AB

xA2L ðx A2L =x Af ÞðαAB −1Þ ðð1 − x Af Þ=ð1−x A2L ÞÞðαAB −1Þ 

0.1 0.05 0.02 0.01 0.21

(0.1/0.21)0.2 (0.79/0.9)1.2 ¼ 0.73 (0.05/0.21)0.2 (0.79/0.95)1.2 ¼ 0.600 (0.02/0.21)0.2 (0.79/0.98)1.2 ¼ 0.481 (0.01/0/21)0.2 (0.79/0.99)1.2 ¼ 0.3297 1



θ

xAp

0.27 0.40 0.519 0.6703 0

0.507 0.45 0.386 0.308 0.61a

xAp is determined from (7.2.15): xAf ¼ θ xAp þ (1– θ) xA2L. a In this case, xAp has to be determined from the relations (6.3.199) for γ ffi 0 and (6.3.201) for the nonzero γ.

560

Bulk flow perpendicular to the direction of force

(a)

1

1

xAp xO2p

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

xN2L = xBL

0

0 0

0.2

0.4

0.6

0.8

1

q

(b) Permeator 1

Fresh feed

Permeator 2

Concentrate retentate reject

Recycle

Compressor or vacuum pump

Permeate highly enriched

(c) 1.0 0.8

Wt2L xN2L Wtf xN2f

0.6

a O2 − N 2

0.4 0.2 0 0.05

0.04

0.03

0.02

0.01

0

xO2L Figure 7.2.2. (a) Oxygen mole fraction in permeate vs. stage cut for a membrane having αO2 −N2 ¼ 6:0. (b) Two permeators in series operation with recycle of permeate from the second permeator to the feed. (c) Fractional nitrogen recovery vs. residual oxygen mole fraction in NEA for different membrane selectivities. (part(c) after Baker (2004).)

Since γ ffi 0 and αAB ffi αAB here,

x Ap ¼

αAB x Af 6  0:21 1:26 ¼ ¼ ¼ 0:61; 1 þ x Af ðαAB − 1Þ 1 þ 0:21  5 2:05

the maximum value achievable anywhere in the permeator. The plot of xAp vs. the stage cut, θ, is illustrated in Figure 7.2.2(a).

For θ ¼ 1, xAp ¼ xAf ¼ 0.21, since all of the feed gas appears as the permeate. If a value of permeate side pressure was provided, then γ would be known; αAB may then be determined from (6.3.202b) knowing αAB , γ and xAf.

What is clear from this example is that, as the value of the stage cut, θ, increases, the permeate mole fraction of O2 in the oxygen-enriched air (OEA) decreases and the mole fraction of N2 in the retentate (the nitrogen-enriched air

7.2

Crossflow membrane separations, granular filtration

(NEA)) increases. However, the loss of nitrogen in the permeate is also increased. If the goal is to obtain oxygenenriched air (OEA) as well as nitrogen-enriched air (NEA), the behavior of xAp vs. θ shown in Figure 7.2.2(a) provides guidance. One can take the permeate up to a stage cut of, say, 0.1–0.2; the corresponding permeate oxygen mole fraction will be around 0.55. On the other hand, the concentrate mole fraction of nitrogen for a stage cut of 0.6703 will be 0.99, a reasonable quality nitrogen-enriched air. These two product streams are shown in Figure 7.2.2(b). However, in order to show what to do with the rest of the permeate, Figure 7.2.2(b) shows that the gas permeator has been split into two permeators: the first permeator produces OEA as the permeate; the concentrate from this permeator goes to a second permeator having a high stage cut and producing the NEA as the concentrate. The composition of the permeate from this second permeator is of interest. Let us make a rough calculation concerning the composition of this second permeate, identified as x Apj2 (the first one being x Apj1 ) for xA2L leaving the permeators being 0.01 (99% N2). For 1 mole of feed air (21% O2) entering the system per unit time, and W t1j1 and W t1j2 being the permeate flow rates from the two permeators,

561

higher oxygen selectivity will lead to a lower loss of N2 in the permeate and therefore a higher fractional N2 recovery in the NEA. For a more detailed introduction to different membranes used in O2/N2 separation, see Zolandz and Fleming (2001) and Baker (2004). Other gas separation systems of interest are as follows. (1) Separation of H2 from a variety of gas streams: H2 from N2 in purge streams in ammonia plants; recovery of H2 from various streams in petroleum refineries and petrochemical plants; H2/CO ratio adjustment in synthesis gas for oxo-alcohol plants. The membrane materials used are polysulfone, cellulose acetate, polyimides and polyaramides. Selectivities of H2 over N2, CH4, CO in these applications vary between 30 and 200. (2) Separation of CO2 from hydrocarbons: CO2 separated from light hydrocarbons in natural gas streams at high pressures; CO2 separated from biogas (containing CO2 and methane); CO2 separated from breathing gas mixture. (3) Dehydration of air. (4) Recovery of C3þ condensable hydrocarbons from natural gas using rubbery polymeric membranes. (5) Recovery of helium from natural gas.

1  0:21 ¼ W t1 j1  x Ap j1 þ W t1 j2  x Ap j2 þ W t2 x A2L : A quantity of considerable practical utility is the membrane area required to treat a gas stream of given flow rate to Let the overall stage cut ¼ 0.67 ! Wt2 ¼ 0.33; W t1j1 þ W t1j2 achieve a certain concentrate mole fraction or permeate ¼ 0.67. Let W t1j1 ¼ 0.1, x Apj1 ¼ 0.55, xA2L ¼ 0.01. Then mole fraction under given conditions; these conditions 0:21 ¼ 0:1  0:55 þ 0:57x Apj2 þ 0:33  0:01 ! x Apj2 ¼ 0:266; involve specification of the value of (Qim/δm) for one of the species, namely species i, αAB for a binary mixture of a somewhat O2-enriched permeate. This stream, if recycled gases A and B, feed pressure and permeate pressure, alterto the fresh feed air, will increase the O2 concentration of the natively, one of the stream pressures and a pressure ratio. net feed entering the permeator 1. Therefore the oxygen In general, there is a pressure drop in the feed side and the concentration in the permeate from permeator 1 will be even permeate side. However, the analysis provided so far in higher (as discussed in Section 2.2.2). On the other hand, if this section is based on zero pressure drops on either side. we operate permeator 2 as a new permeator with a N2- Quite often, such an assumption does provide reasonable enriched feed stream coming as the reject from permeator estimates unless the pressure drop is such that the γ value 1, then we may have a different situation if we operate it at a changes considerably along the permeator length. high θ. The oxygen concentration in this feed may be around 0.15. If we operate the second permeator with a very high Example 7.2.2 We wish to estimate the membrane stage cut (say θ ¼ 0.67), it is entirely likely that x Apj2 < 0.21. area required to produce nitrogen-enriched air having x N2 2L ¼ 0:95 using a silicone rubber-coated hollow fiber Such a stream, if recycled to the fresh feed air, will increase membrane, with ðQN m =δm Þ at 25 ∘ C ¼ 4:6  10 − 10 gmol/sN2 concentration in the feed gas. This will lead to a better cm2-cm Hg and α 2 ¼ 2:1. The feed air is introduced at O 2 − N2 NEA from permeator 2, but a poorer OEA from permeator 1. 10 bar; the permeate is at 1 atm. In the microporous hollow If the production of NEA is the goal, then the variation fiber membrane with a silicone coating on the fiber internal of the production rate as a function of nitrogen concen- diameter, the feed flows through the fiber bore. The permetration is also of interest. The higher the production rate of ate is withdrawn in crossflow on the shell side all along the this NEA stream, the higher will be the fractional recovery module length (Figure 7.2.1(c)). Neglect any pressure drop  of N2 from the feed air in this product stream. However, as along the module length. The feed air flow rate at 25 C is 900 kgmol/hr. the purity of N increases in this product, the fractional N 2

2

recovery ðW t2L x N2 2L =W tf x N2 f Þ, will decrease. Figure 7.2.2 (c) illustrates this behavior qualitatively for membranes having different O2–N2 selectivities (varying between 2 and 8) (after Baker (2004)). Obviously, membranes having

Solution Since there is no pressure drop in either stream along the module length, and we have pure crossflow, we can use the Pan and Habgood (1978a) analysis for nonconstant αAB , i.e. αAB can change with composition.

562

Bulk flow perpendicular to the direction of force

Table 7.2.2. x 2 ð¼ x O2 2 Þ

Y2

θ

0.21 0.18 0.14 0.10 0.05

0.178 0.156 0.124 0.092 0.0476

0 0.219 0.47 0.66 0.858

The equations of relevance are (7.2.18) for αO2 , βO2 , a and b, (7.2.19) for θ, (7.2.20) for Y O2 ;f and Y O2 , (7.2.22) for the definition of nondimensional membrane area S and (7.2.21) for the calculation of this S to obtain the actual membrane area s. In this approach

αN2 ¼ 1;

αO2 ¼ 2:1;  1 β O2 ¼ 1 − ¼ 0:524; γ ¼0:1; αO2

a ¼ ðγ αO2 βO2 þ 1Þ=ð αO2 βO2 ð1 − γÞ 

0:1  2:1  0:524 þ 1 ¼ 1:12; 2:1  0:524  0:9

b ¼ ðγ αO2 βO2 − αO2 Þ=ð αO2 βO2 ð1 − γÞ ¼

¼

0:1  2:1  0:524 − 2:1 ¼ − 2:01: 2:1  0:524  0:9

To determine S from equation (7.2.21), we need to have the value of θ for a few values of Y2 and integrate. Determine Y2,f ¼ 0.178 (from (7.2.20)) for x2 ¼ x2f ¼ x O2 f ¼ 0.21. Now determine Y2 for different values of x 2 ð¼ x O2 2 Þ from (7.2.20) and then determine the value of the stage cut θ from (7.2.19). The results of these calculations are summarized ðY 2 in Table 7.2.2. θ dY 2 in (7.2.21) is 0.0698. The value of the integral Y 2;f Therefore

S ¼ ð1 − Y 2 Þθ þ 0:0698 ¼ ð1 − 0:0476Þ0:858 þ 0:0698 ¼ 0:887: So,

S¼ ¼ s¼

   Q N2 m P ð1 −γÞs Lf d

cm Hg 4:6  10 −10 gmol 10 atm  76 atm  3600 hrs  0:9  s; 1000gmol s-cm2 -cmHg 900 kgmol hr  kgmol

0:887  900  1000  1010 1 m2 cm2 4 3 10 cm2 4:6  10  76  3:6  10  0:9

¼ 7:05  104 m2 : 7.2.1.2

Reverse osmosis

As we have seen in Section 6.3.3.3, the dead-end mode of operation of reverse osmosis creates major problems: steady state is not possible; it is difficult to generate a high value of the solute mass-transfer coefficient k iℓ ; further, as in Section 6.3.3.5 for gas permeation, it is often easier to pack a lot of membrane surface area when the bulk liquid flow is not parallel to the direction of the force.

The configuration studied here is crossflow (see Figure 7.0.1(f)): feed solution (e.g. brine in desalination) flows along the membrane module length parallel to the membrane length direction while the permeate flows perpendicular to the feed solution flow direction, i.e. in the direction of the driving force. In many devices, the permeated liquid collected along the membrane length flows in a particular direction. This second bulk flow (of the permeate) may have a significant effect on separation; this case of two bulk flows, where both are perpendicular to the direction of the force across the membrane, is studied in general in Chapter 8. Here we study those cases where the second bulk flow is of no consequence in so far as separation is concerned. Thus crossflow as in Figure 7.0.1(f) is sufficient for analysis. The physical configurations studied are: (1) spiralwound module; (2) tubular module. Let us consider first a highly simplified, but reasonably useful, model for a spiral-wound reverse osmosis module. The basic structure of a single-leaf spiral-wound membrane module is shown in Figure 7.2.1(e). An actual module consists of a number of leaves. A single-leaf module as shown consists of a rectangular membrane packet (sandwich) spirally wound around a central collection pipe. Each membrane packet (sandwich) is separated from the membrane in the next and nearest membrane packet by a spacer screen. There are two membranes in each membrane packet, with a product water side backing material in between (Figure 7.2.3(a)). Three ends of this packet are glued and sealed, and the fourth end empties into the collection pipe through a glued joint. As brine (or any other aqueous feed) flows along the module length over the spacers in between the membrane packets, permeation occurs and permeate flows along the spirally wound permeate channel in the membrane packet through the product water side backing toward the central collection pipe (see Figure 7.2.1 (e)). By having a number of leaves in parallel, the length of the membrane permeate side is reduced; correspondingly, the permeate pressure drop is reduced. If we open a spirally wound membrane channel and lay it flat, then, as shown in Figure 7.2.3(b), brine flows at a high pressure in the mean flow direction (the residue flow direction in Figure 7.2.1(e)), designated here as the z-coordinate; the membrane length in this direction is L. The permeate flows inside the membrane packet in the x-direction, the so-called permeate channel toward the central collection tube; the width of the membrane packet is W (in the x-direction). Our basic assumptions in the following analysis are as follows. (1) The pressure difference ΔP changes very little along the z-direction from the value ΔPf at the feed entrance location (z ¼ 0). (2) The permeate channel pressure drop along the x-direction is negligible.

7.2

Crossflow membrane separations, granular filtration

(a)

563

(b)

Membrane

Membrane

Backing material bf

Feed flow direction z

Permeate tube Permeate flow direction x y

bf Mesh spacer

Permeate channel

Glue line (c)

Feed channel

Glued end of permeate channel

Membrane (d) 0.9 0.8 0.7

5

0

0.6 re 0.5

Mesh wire

γ=

γ=

Boundary layer

0.0 γ=

0.2

0.4 0.3 0.2

.3

γ=0

0.1

Membrane

0

0.2

0.4

0.8

0.6

1.0

1.2

L+ Figure 7.2.3. (a) Details of a spiral-wound module. (After Ohya and Taniguchi (1975).) (b) Channels and flow configurations in an unwrapped spiral-wound module. (c) How mesh wires in the brine side spacer screen disturb the brine flow and create new boundary layers (schematically). (d) Fractional water recovery re as a function of normalized channel length Lþ for various parametric values of γ, indicating the feed osmotic pressure level with respect to ΔPf.

(3) The membrane has a very high solute rejection (salt rejection 99%þ, i.e. Rsalt ffi 0.99 ! 1.0). (4) The solute mass-transfer coefficient k iℓ in the membrane feed channel has a high value and can be considered to be essentially constant along the z-coordinate. The species conservation equation for solute i (here, salt) in the feed flow channnel may be written using the species conservation equation (6.2.5b) for steady state, no external force based species velocities (Ui ¼ 0) and no chemical reactions as r ðC is v Þ ¼ − r ðJ i Þ ¼ r ðDis r C is Þ:

ð7:2:25Þ

The flow conditions in the thin feed brine channel with feed brine spacers in between is complex (see Schwinge et al. (2002, 2003) for the unsteady flow, vortex shedding and mass-transfer enhancement in spacer-filled channels). The complexity is captured by using an empirically

derived mass-transfer coefficient k iℓ reflective of the flow conditions (Schock and Miquel, 1987). One could represent the flow in a simplistic way as a chopped laminar flow (Solan et al., 1971) with a boundary layer beginning at each cross wire in the spacer (illustrated schematically in Figure 7.2.3(c)). However, the mass-transfer coefficient k iℓ in the very thin boundary layer is high even if the Reynolds number is quite low (essentially laminar flow at this low Reynolds number, even though there is a lot of mixing). We will represent this velocity field as twodimensional, vy (normal to the membrane) and vz (along the membrane length, z-direction). Equation (7.2.25) may be represented in this context in the (y, z)-coordinate system as ∂ðC is vy Þ ∂y

þ

∂ðC is vz Þ ∂ ¼ ∂z ∂y

    ∂C is ∂ ∂C is þ : Dis Dis ∂y ∂z ∂z ð7:2:26Þ

564

Bulk flow perpendicular to the direction of force

Since the axial concentration gradient of the solute, Cis,, is very small,12 the axial diffusion term may be neglected, leading to   ∂ðC is vy Þ ∂ðC is vz Þ ∂ ∂C is ¼ Dis þ ; ð7:2:27Þ ∂z ∂y ∂y ∂y   ∂ðC is vz Þ ∂ ∂C is ¼ − þC is vy − Dis : ð7:2:28Þ ∂z ∂y ∂y Integrating this equation along the membrane feed channel height bf (as if it is a flow between two flat plates with a gap bf in between), we get  ð bf  ð d bf ∂C is : ð7:2:29Þ C is vz dy ¼ − d C is vy − Dis ∂y dz 0 0

If we use the solution–diffusion model to describe the solvent flux, then, from equations (3.4.60c–f), Nsy ¼ Nwy ¼ J wy : jN wy j ¼ jJ wy j ¼ AðΔP − ΔπÞ ¼ AðΔP − bC 0il þ bC ip Þ;

ð7:2:35Þ

where (see relation (3.4.64a)) C 0il is the wall concentration of the solute and Cip is the solute concentration in the permeate. By assumption (3), we can neglect Cip here. Further, using the film model analysis and assumption (4) with jN wy j V w =k il > 1, and zþ ¼ (L/zcr) >> 1 with the cake resistance controlling, we can obtain

ð7:2:144aÞ

þ

vs javg ¼ 1:31 γ_ μ ðϕs Þ

!1=3

!1=3

;

r 4p Qþ p ðϕs Þjcr ϕs L

!1=3

;

ð7:2:145Þ

where we have employed relations (7.2.133), (7.2.118) and (7.2.127); γ_ is the value of the shear rate in the boundary layer. The averaged filtration flux is proportional to the shear rate in the boundary layer; it is independent of the pressure drop applied. However, in the case of membrane resistance control, vs ¼ vs0 / ΔP; ð7:2:146Þ i.e. the averaged filtration flux is proportional to the applied ΔP. We will now illustrate graphically the behavior of the length-averaged flux and the cake layer thickness with distance along the membrane length. In Figure 7.2.6(c), the filtration flux profile along the membrane length has been shown for a given set of conditions. The flux is constant till z ¼ zcr, after which it decreases steadily. In Figure 7.2.6(d), the nondimensional filtration flux profile along the nondimensional membrane length has been illustrated parametrically for β as a parameter; it shows how the filtration flux decreases as the cake resistance increases, relative to the membrane resistance. This decrease is due to increasing cake thickness along the ^ cδ , as membrane length for nonzero values of β, i.e. R shown in Figure 7.2.6(e). We observed earlier in dead-end cake filtration (equa^ cδ varies inversely with the square of tion (6.3.135k)) that R the particle radius; therefore, in effect, the filtration flux vs varies with the square of the particle radius for cake dominated filtration. The larger the particle radius, the higher the filtration flux. As shown in equation (7.2.145), in crossflow microfiltration also the averaged filtration flux increases with particle radius, here as (rp)4/3. Romero and Davis (1988) have shown via calculations based on their integral model that, for suspensions having particles larger than 4.3 μm radius, the flux of solvent will correspond to

7.2

Crossflow membrane separations, granular filtration

that with zero cake resistance, i.e. the flux will correspond to that of a clean membrane. However, the shear rate is also important. Unless the shear rate is around 660 s−1, the flux for a suspension of particles of 5 μm radius was shown by Romero and Davis (1988) to be less than that of a clean filter.17 An additional quantity to be taken into account is the deformability of the particles, especially valid for cellular suspensions, e.g. red blood cells in blood; the cake resistance will be much higher and the cake will be more compact. One would now like to determine the length-averaged flux for a given membrane undergoing microfiltration under given conditions. Although we have the nondimensional expressions (7.2.143a) for the solvent flux for the membrane dominated case and (7.2.143b) for the cake resistance controlled situation, determination of the length-averaged flux, as well as the local flux for the latter, is complicated due to two quantities, Qþ p ðϕs Þjcr and zcr. Knowledge of one will yield the other from relation (7.2.133). The quantity Qþ p ðϕs Þjcr has to be obtained from the double integral (7.2.130) for the limit of ϕw ¼ ϕmax for þ appropriate dependences of Dþ p ðϕÞ and μ ðϕÞ on ϕ. Davis (1992) and Romero and Davis (1988) have provided results (1) and (2), and (3), respectively: −5 ð1Þ lim Qþ p ðϕs Þjcr ¼ 9:79  10 ; ϕs !0

ð7:2:147aÞ

−5 ð2Þ Qþ p ðϕs Þjcr ¼ 9:79  10 ð1 − 4:38 ϕs Þ; for ϕs  0:1;

ð3Þ

"

Qþ p ðϕs Þjcr

lim ϕs !0 Qþ p ðϕs Þjcr

#

ð7:2:147bÞ

¼ 1 − 3:8 ϕs ; for ϕs  0:2: ð7:2:147cÞ

One can now estimate the required quantities, at least for ϕs  0.2, which is a high enough particle volume fraction for suspensions. Example 7.2.7 In a ceramic tubular microfilter of diameter 0.30 cm and length 40 cm, a suspension having particles of radius 0.35 μm flows; the suspension particle volume fraction is 0.005. The value of the viscosity, μ0, is 10 2 g/cm-s (1 centipoise) and the wall shear stress is 10 g/cm-s2. The observed flux for the microfilter without any particles is 3.5  10−3 cm/s for an applied pressure difference of 105 kPa. The conditions are such that one can assume a cake resistance controlled operation. Make an estimate of what the average filtration flux will be at 25  C. Estimate the same if the particle radius were 4.3 μm. Solution Since we can assume cake resistance control, we will employ equation (7.2.145) for the average filtration flux:

17 As the particle size becomes larger, the phenomenon of inertial lift (Green and Belfort, 1980) becomes important.

581

þ

vs javg ¼ 1:31 γ_ μ ðϕs Þ γ_ μþ ðϕs Þ ¼

r 4p Qþ p ðϕs Þjcr ϕs L

!1=3

;

τw 10 g=cm-s2 ¼ 103 s − 1 ; ¼ μ0 10 − 2 g=cm-s

r p ¼ 0:35  10 − 4 cm; ϕs ¼ 0:005; L ¼ 40 cm: From equation (7.2.147b), −5 Qþ ð1 − 4:38  0:005Þ p ðϕs Þjcr ¼ 9:79  10 ¼ 9:79  10 − 5  0:979;

from equation (7.2.131b),   ð0:58 − 0:13  0:005Þ 2 ffi 1:01; μþ ðϕs Þ ¼ ð0:58 − 0:005Þ 0 11=3 3:54 10−20 9:790:97910−5 A 3 @ vs javg ¼1:3110 1:01 0:00540

0 11=3 1509:5810−25 A @ ¼1310 ¼1186710−8 ¼0:00011cm=s: 0:2

We will now calculate the value of vs javg by assuming the particle radius to be 4.3 μm. We should not assume either membrane control or cake resistance control. However, that will require an extensive numerical effort. We will use the same formula as used earlier to estimate the effect of a particle size increase:

0

14=3 4:3 A cm ¼ 1:18  10 −4 ð12:28Þ4=3 vs javg ¼ 1:18  10 −4 @ 0:35 s ¼ 12:28  2:308  1:18  10 −4

) vs javg ¼ 3:344  10 −3 cm=s; which is close to the value of the clean filter flux.

The above example illustrates clearly how large a flux reduction can take place in microfiltration compared to the flux in a clean filter. In general, when crossflow microfiltration is initiated, one observes immediately a sharp flux decline with time. This immediate flux decline is due to the increase in the filtration resistance due to the buildup of the cake layer on the membrane; this buildup occurs within minutes to an hour from startup and can easily reduce the value of vs javg to one-tenth of the clean filter flux value, or an even greater reduction. A brief introduction to modeling such a transient flux behavior on its way to steady state has been provided by Davis (1992). There is an additional flux decay over a much longer period of time, e.g. days, which takes place due to slow processes of membrane fouling, compaction of the cake under the applied ΔP and membrane compaction. Such losses of membrane solvent flux is avoided now a days in practice by what is known as backflushing. The cake layer on the membrane is built up by the applied ΔP over a period of time as the liquid flows perpendicular to this

582

Bulk flow perpendicular to the direction of force

1.0

(c)

0.9 0.8

ánsñ ns 0

0.7 0.6 0.5 0.4 0.3

(b)

0.2 (a)

0.1 0 1000

2000

3000

Time (seconds) Figure 7.2.7. Filtration flux decay with time after starting and the effect of backflushing: (a) no backflushing; (b) slow backflushing introduced after some time (low frequency); (c) rapid backflushing with high frequency initiated immediately after starting. (After Bhave (1991).)

force direction. If the force direction can be reversed, the cake layer may be completely lifted off the membrane and swept away by the tangential flow. The backflushing mode is based on this strategy. Every so often, the permeate side pressure is suddenly increased to above the feed pressure. A small amount of earlier permeated filtrate is forced back into the feed side (and is therefore lost from the permeate): as it comes, it lifts off the cake and cleans the membrane pores as well if there is some membrane fouling. The tangential flow on the feed side sweeps off the dislodged debris from the destruction of the cake layer. Figure 7.2.7 illustrates this behavior. Curve (a) illustrates the rapid flux decay during the first 30 minutes to 1 hour of conventional crossflow microfiltration. Curve (b) illustrates the flux behavior for low values of backflush frequency started after some time. Curve (c) illustrates the flux for a high backflush frequency initiated after a very short interval, as shown in the figure. Experience indicates that it is necessary to introduce rapid backflushing if one wants to obtain high flux, and further that it must be introduced immediately after the flow is introduced (Bhave, 1991). Recent research into the reduction of filtration flux that occurs as soon as the microfiltration process is started (Figure 7.2.7(a)) indicates that it is very much dependent on the level of the flux and therefore the applied ΔP. Beyond a critical flux, vs,cr, deposits form on the membrane since the forces dragging the colloidal particles in suspension toward the membrane are larger than the forces causing the particles to move away from the membrane. If this critical flux value is reached at the end of the membrane channel, where the boundary layer is the thickest, then the

membrane is unlikely to experience fouling. Thus the value of applied ΔP should be reduced and operation continued at a subcritical flux level to avoid membrane fouling (Field et al., 1995; Howell, 1995). 7.2.1.4.1 crossflow microfiltration–operational configurations In Section 6.3, we observed in many examples that the magnitude of the bulk flow velocity which was parallel to the direction of the force could not be arbitrarily increased, otherwise the separation achieved could be severely damaged or the separation process halted. Further, in many processes, there was unsteadiness due to this bulk flow vs. force configuration. Clearly, in crossflow microfiltration we have overcome this unsteadiness. Further, in order to maintain a thin caker layer, we maintain a high wall shear rate in the tangential flow field. In most crossflow microfiltration devices18 this means that the bulk flow velocity in the tangential flow direction (z) is quite high; on the other hand, the filtration velocity continues to be limited by the membrane, the cake or both. Thus, the ratio (vz/vs) in crossflow microfiltration is high, unlike that in dead-end microfiltration, where its value is 1. In addition, in the latter, vs is likely to be significantly smaller than that in crossflow microfiltration due to a larger cake thickness; also, in the dead-end mode, with time, the flux decreases at constant ΔP. 18

In those devices where the tangential shear rate can be generated by means other than bulk motion, namely Dean vortices or Taylor vortices, the correlation between shear rate and the bulk velocity may be weak or zero.

7.2

Crossflow membrane separations, granular filtration

(a)

583

(b)

(c) fs

z=L Q bleedfs

Qrecy

Qfiltrate

Qf fs fs

tank

z=L

z=0

Q tank

Figure 7.2.8. Operational configurations of microfiltration: (a) batch concentration – open system; (b) batch concentration – closed system; (c) feed and bleed operation in the continuous mode.

A result of this high bulk flow velocity and high shear rate in crossflow microfiltration is that the suspension circulation rate is high when compared with the microfiltration rate through the membrane. Either one provides a very long membrane device or one keeps on circulating the solution from a reservoir/tank over a much smaller length of the membrane for an extended length of time. The latter is the most common mode of crossflow microfilter operation and is shown in Figure 7.2.8. This mode of batch concentration can be carried out in two ways: open system and closed system (Bhave, 1991). In an open system, the concentrate from the module is sent back to the reservoir for recirculation through a single pump; in a closed system, there are two pumps, and the energy loss associated with bringing the liquid suspension from the tank to full pressure is avoided. Mass balance considerations for a suspension volume of Vs(t) having a particle volume fraction ϕs(t) being microfiltered through a membrane area Am leads to V s ðtÞjt¼0 − V s ðtÞjt ¼ Am

ðt

0

vs javg dt;

ð7:2:148Þ

where vs javg for the membrane of area Am changes with time since the suspension particle volume fraction ϕs(t) is time-dependent, as the suspension volume is reduced with time. The corresponding increase in the volume fraction of particles in suspension from ϕs ðtÞjt¼0 to ϕs ðtÞ at any time t is related to the suspension volume reduction by

V s ðtÞjt¼0 ϕs ðtÞjt¼0 ¼ V s ðtÞ ϕs ðtÞ;

ð7:2:149Þ

since all solid particles are assumed to be rejected by the membrane. One feature of the closed system described above is a rapid increase in ϕs in the recirculation loop. One could avoid it by continually bleeding off a small portion of the retentate/concentrate, as shown in Figure 7.2.8(c), in a method called feed and bleed. This bleed volume is small since the increase in concentration of ϕs per pass through the filter is also small. If we indicate by ϕs jz¼0 and ϕs jz¼L the values of ϕs at the inlet and outlet of the microfilter, respectively, then, for a totally rejecting filter, Qf ϕs jz¼0 ¼ Q2 ϕs jz¼L ;

ð7:2:150Þ

where Qf and Q2 are the volumetric suspension flow rates at the filter inlet and the concentrate end, respectively. The flow rate ratio Qf/Q2 (>1) is close to 1 since the recirculation rates are high; therefore ϕs jz¼L = ϕs jz¼0 is also close to 1 (but >1). However, this flow rate Qf into the filter consists of the sum of the flow rate Qtank out of the tank and the flow being recirculated Qrecy (¼ Q2), the latter being R times the bleed flow rate Qbleed: Qf ¼ Qtank þQrecy ¼ Qtank þ R Qbleed ;

Qf ϕs jz¼0 ¼ Qtank ϕs jtank þ Qrecy ϕs jz¼L :

ð7:2:151Þ

584

Bulk flow perpendicular to the direction of force

From the second equation, one can obtain

Filter cake

ϕs jz¼0 ¼ ðQtank =Qf Þϕs jtank þ ðR Qbleed =Qf Þϕs jz¼L : ð7:2:152Þ

w

However, to obtain steady state operation, the increase in solids volume flow rate due to filtration ðQf − Q2 Þ ϕs jz¼0 must not affect steady state operation: Dry

volume flow rate balance : Qf ¼ Qrecy þ Qbleed þ Qfiltrate ;

Drum

Qf ϕs jz¼0 ¼ Qrecy ϕs jz¼0 þ Qbleed ϕs jz¼0 þ Qfiltrate ϕs jz¼0 ¼ Qrecy ϕs jz¼L þ Qbleed ϕs jz¼L :

ð7:2:154Þ

Rearrange the last equation to obtain

Vacuum

¼ ðQbleed þ Qrecy Þ ðϕs jz¼L − ϕs jz¼0 Þ; Qf − Q2 ϕj Q ¼ s z¼L − 1 ¼ filtrate ; ϕs jz¼0 Q2 Qbleed þ Qrecy ð7:2:155Þ

where the ratio on the left-hand side is also called the volume concentration factor (VCF). When the desired VCF value is higher, then the bleed stream from the first stage could be used as if it were the feed stream from the tank for the second stage in a multistage arrangement. 7.2.1.5

Vacuum Filter cake discharge

Filter cloth

q

qf

Filter cake

Filtrate

Suspension

Figure 7.2.9. Schematic of a rotary vacuum filter.

Qfiltrate ϕs jz¼0 ¼ ðQf − Q2 Þ ϕs jz¼0

Qf ϕs jz¼L Q ¼ 1 þ filtrate ¼ ; Q2 ϕs jz¼0 Q2

qw

Vacuum

ð7:2:153Þ

The particle volume fraction balance around the filter is given by

Wash nozzles

Rdrum

Rotary vacuum filtration

Consider Figure 7.2.9 which illustrates schematically a drum rotating around its axis. The drum length (not shown) is L. As the drum rotates at an angular velocity of ω radian/s, part of the outside of the drum remains submerged in the suspension to be filtered. The drum surface has a filter medium exposed to the suspension at atmospheric pressure. Inside the porous drum surface, a partial vacuum is maintained so that the filter medium has a ΔP across it from the suspension. As filtration takes place, particles are deposited on the outside of the filter medium surface. As a particular section of the drum/filter medium rotates through the suspension bath, there is continuing filtration; so there is a cake buildup on the filter medium as it goes into the suspension and then comes out of the suspension after some time. During this movement through the suspension, the partial vacuum applied brings out filtrate into the inside of the drum. As this section of the drum comes out of the suspension, nozzles on the outside spray clean liquid onto the drum filter medium surface and wash out any solution remaining in the cake built up on the filter. This removes

any valuable product left in the cake via filtration into the filtrate inside the drum operating under partial vacuum. Only a section of the drum circumference is exposed to this cake washing. The next environment encountered by the cake on the rotating drum surface (Figure 7.2.9) is essentially dry air. The vacuum inside the drum continues to remove any remaining liquid in the cake on the filter medium. The fourth and final step in this cyclic process involves removal of the filter cake formed on the filter medium via a knife blade. In conventional application, no coating is applied to the filter medium. In the filtration of fermentation broths, however, a thin but controlled layer of a filter aid, e.g. diatomaceous earth, is applied onto the filter medium before the drum enters the suspension as a “precoat.” The filtration of the fermentation broth leads to a buildup of the cake over it. When the knife blade is applied at the end of one cycle, most often only a particular fraction of this “precoat” of filter aid is removed. The porous precoat facilitates extended treatment of the broth, which tends to create a complex compressible cake. If the filtrate contains the desired and valuable material, the cake discharged by the knife blade is waste; filter aid material added as a precoat contaminates the cake and increases the volume of this waste. It also changes the nature of the waste, which will now contain, in addition to cellular debris, cells, etc., siliceous material. We have already pointed out in the introduction to Section 7.2 that the separation configuration of this rotary vacuum filtration technique corresponding to Figure 6.3.25(a) may be conceived of as a crossflow filtration technique (Figure 6.3.25(b)) if we fix the coordinate system to the filter medium/drum surface. In this configuration, the suspension/slurry will be moving with the linear velocity of

7.2

Crossflow membrane separations, granular filtration

the drum surface where the filter medium is located. However, this velocity, and the corresponding tangential shear stress magnitude, are rather low and are incapable of affecting the cake thickness buildup (unlike that in crossflow microfiltration studied in Section 7.2.1.4) (Davis and Grant, 1992). Therefore, one should consider the cake buildup with rotation (as angle θ in Figure 7.2.9 increases) as if we have dead-end filtration (Section 6.3.3.1), where (θ/ω) may replace time, t. Correspondingly, one can employ expressions developed earlier in dead-end cake filtration for the time-dependent cake thickness δc(t) (6.3.138c) and the filtration rate/volume flux for constant ΔP, vs(t), (6.3.138d) and write here (Davis and Grant, 1992) " #  ^ cδ ϕs ΔP θ 1=2 2 R ^ cδ Þ 1 þ δc ðθÞ ¼ ðRm =R −1 ; ð7:2:156Þ ðϕc − ϕs Þμ R2m ω vs ðθÞ ¼

ΔP μRm



1þ

^ cδ ϕs ΔP θ 2R ðϕc − ϕs Þμ R2m ω

 −1=2

ð7:2:157Þ

:

Note that here Rm includes the resistance due to any precoat applied to the membrane. The question of interest is: what is the volumetric rate of filtration in this rotating drum (of total surface area 2π RdrumL) due to the submerged fraction of its surface? That will yield the device filtration rate (unless the suspension properties change with time – they will for a batch operation!) since, as long as the drum rotates, part of it is always carrying out filtration (fractional surface area submerged, θf/2π; total surface area submerged at any time, LRdrum θf). This quantity may be obtained by integrating expression (7.2.157) for the filtration flux over the range of angles θ ¼ 0 to θf, the angle of contact with the suspension (over the time period the surface area of the drum is in contact with the suspension): ð θf Qfiltrate ¼ Rdrum L vs ðθÞdθ 0

¼ Rdrum L

Qfiltrate ¼

ð θf 0

  ^ cδ ϕs ΔP θ − 1=2 ΔP 2R 1þ dθ; μ Rm ðϕc − ϕs Þ μ R2m ω

Rdrum Lðϕc −ϕs ÞRm ω ^ cδ ϕs R

(

1þ

ð7:2:158Þ )

 −1=2

^ cδ ϕs ΔP θf 2R ðϕc −ϕs Þμ R2m ω

−1 :

ð7:2:159Þ If the cake resistance is much larger than the membrane ^ cδ δ >> Rm ; correspondingly, (þ precoat) resistance, then R we have already observed in Example 6.3.4 that vs ðtÞ ¼



^ cδ ϕs μ t 2R ðϕc − ϕs Þ ΔP

 − 1=2

:

We follow now the same procedure we followed earlier to obtain expression (7.2.159) for Qfiltrate from expression (7.2.157) for vs(θ):

585

vs ðθÞ ¼



^ cδ ϕs μ θ 2R ðϕc − ϕs Þ ΔP ω

Qfiltrate ¼ Rdrum L

θðf 0

 − 1=2

;

ð7:2:160Þ

vs ðθÞ dθ

91=2 8 90% PE at a recovery of >60%. Wei and Realff (2003) have discussed two-stage operations with or without recycle to improve purity and considered several design options. They have also illustrated the charging tendencies of different plastics via a triboelectric series of plastics. There is another useful geometrical configuration of an electrostatic separator for separation of plastic particles: the drum-type separator shown in Figure 7.3.10(c). There is a rotating drum charged with a high voltage (–V) and an outer plate also charged with a high voltage (þV). As shown, the outer plate is curved, but it can be straight as well. Particles coming over the belt are subjected to three forces as they approach the angular location of the outer plate: the electrostatic force due to the electrostatic potential gradient between the two electrodes, a centrifugal force and the gravitational force. If the net outwardly radial force is positive for a particle, it will become detached from the belt. The location and force of detachment will determine the particle trajectory and therefore the collection bin where it lands. The use of electrostatic separators for mineral particle separation is common. It is primarily used for separating mineral ores, ore tailing, pulverized coals containing clay particles (Inculet, 1984), coal ash, etc. For example, triboelectrification of coal ash from power plant exhaust leads to two fractions: carbon-rich material, which becomes positively charged, and the rest of the ash, which is negatively charged (Holusha, 1993). Similarly, high mineral matter containing coal will be separated via triboelectrification into a positively charged coal-rich fraction and a negatively charged mineral-rich fraction. Very-large-scale separation of raw salts by electrostatic separators is practiced at the level of 900 tons/hr (Beier and Stahl, 1997). In these processes, used for the separation of the potash ore for example, the ore is first ground to free the minerals of individual crystals, then the salt crystals undergo a conditioning treatment, where they are coated with a reagent (or a combination of the reagents), followed by warming and adjustment of the relative humidity, which allows charge exchange to take place. The chemical conditioning and humidity treatment are generally carried out in a fluidized bed using an air stream having the required

617

relative humidity. The charged salt mixture is then separated by letting the mixture fall through a vertical charged separator. Multiple stages are used to process various fractions into increasingly pure fractions. The conditioning agent’s role is quite important. Beier and Stahl (1997) have shown that aliphatic monocarboxylic acids allow one to recover almost all of KCl in a mixture of KCl, NaCl and MgSO4 · H2O in the positive electrode, with very limited recoveries of NaCl and MgSO4 · H2O. The addition of aromatic monocarboxylic acid or NH4 salts of aliphatic monocarboxylic acids leads to different patterns of recovery. In the above examples, for the successful separation of minerals at the electrodes, a different arrangement is used. A major problem is the adherence of dust/particles on the electrodes in the configurations of Figures 7.3.10(a) and (b) which reduces the field strength. Instead, a number of vertical tubes standing side by side constitute the electrode; however, these tubes slowly rotate around their axis, and any adhering dust/minerals on their surfaces are scraped away by brushes at the back (Figure 7.3.10(d)). 7.3.1.5

Separation of cells using flow cytometry

Flow cytometry is a successful technique for “cytometry” (i.e. cell measurements) when a suspension of cells in a liquid stream “flows” through the device and the cells are separated one by one. Cells of interest introduced via a sample injected into the core liquid stream surrounded by a much thicker liquid flowing stream (called the sheath flow) in laminar flow are analyzed via their optical signals, usually by a laser beam (Figure 7.3.11). Prior to the sample injection, fluorescent dyes are introduced into the mixtures of cells; particular cells are tagged with particular fluorochromes. The laser beam “interrogates” individual cells as they flow downward. The cells emitting the desired signals are identified extremely rapidly and then tagged individually in the following fashion. The liquid jet/stream beyond the laser beam sensing zone is broken down into tiny droplets (Figure 7.3.11). Knowing which droplet contains a particular cell, the system introduces a charge on such a droplet; then, by electrostatic forces generated by charged deflection plates (as in Section 7.3.1.4), the droplet containing a particular cell is directed to the right or left by a certain distance and collected in several containers. The central container is for the liquid waste stream consisting of droplets (without or with cells) having no charge on them since they are not of interest. For a comprehensive introduction to flow cytometry, see Shapiro (1995). For shorter accounts, consult Hoffman and Houck (1998) and Robinson (2004). A few more details about the technique will provide a better prespective. The cell population has different subgroups having (potentially, for example) different antigens on the cell surface. The sample to be analyzed is treated

618

Bulk flow perpendicular to the direction of force

Core liquid stream Sample introduced here Sheath flow Sheath flow

Cells

Core Sheath Laser beam

+

Collection bins

–– -

+ + +–

Waste

Charged deflection plates Collection bins

Figure 7.3.11. Schematic view of a flow cytometer based on fluorescence-activated cell sorting via droplet sorting.

(stained) with, say, a number of different antibodies (Abs) which bind with specific antigens (Ags) on the surface of the cells. The antibodies are labeled with (e.g.) specific fluorochromes, each of which provides a distinct color and therefore a distinct identity to the laser probe, of which there may be more than two in some systems. The analysis of the spectral emission from the cell surface is carried out extremely rapidly, at rates as high as 10 000–25 000 cells per second. To ensure that the fluid flow regime is correct, laminar flow is maintained. The fluid jet diameter varies between 50 and 400 μm. The frequency of droplet formation varies between 2000/s and 100 000/s. Vibration at frequencies of 10 000–300 000 Hz via a piezoelectric crystal oscillator is most often used to produce drops. Drops of interest are imparted positive or negative charges. The voltage applied between the two plates can be as high as 5000 volts to separate various drops. The technique is often identified in general as flow sorting via droplet sorters, sorting out one type of cell from another via different droplets. The overall technique is also known as fluorescence-activated cell sorting (FACS). The efficiency of cell sorting or cell purification has two aspects: (1) How pure is the cell sorted in a particular droplet? (2) What is the rate at which this sorting is taking place? For the technique to be successful, the rate of cell sorting has to be reasonably high; unduly slow techniques are not acceptable. High rates of cell sorting can lead to situations where a cell that is undesirable appears in the same drop with a cell that is desirable; this happens because of the presence of an undesirable cell with a desirable cell in the liquid in the laser interrogation zone.

The presence of the desirable cell triggers droplet formation, which can enclose an undesirable cell. Hoffman and Houck (1998) have summarized the different modes of sorting, not only for droplet sorters, but also for other enclosed sorting methods, e.g. catcher-tube sorting, fluidic-switching sorting (here droplets are not generated from the sample stream; rather when there are desired cells, a catcher tube catches it, etc.). These modes are: (a) single-cell mode: reject if there is more than one cell even though both may be desirable; (b) enrichment mode: collect as long as there is a desirable cell, even though there may also be an undesirable cell; (c) exclusion mode: reject when there is an undesirable cell; collect even if there are two desirable cells. It would be useful to illustrate quantitative measures of purity of the sorted cells with respect to the rate, ra, at which cells appear in the sensing zone, the time period, T, during which a cell can be sorted for selection or rejection and subfr, the fraction of the subpopulation of desirable cells to be sorted. The results, illustrated in Hoffman and Hauk (1998), based on analyses available in literature are as follows. The efficiency, E, of sorting cells is defined as the fraction of desired cells passing through the sensing zone that are captured/sorted since sometimes the desired cells may not be sorted due to the presence of undesired cells. (i) Single-cell mode: E ¼ expððraÞT Þ:

ð7:3:86Þ

(ii) Exclusion mode: E ¼ expðð1  sub fr ÞðraÞT Þ:

ð7:3:87Þ

Illustration of the values of the parameters is useful. The time period, T, may vary from 25 to 200 μs. As T increases, E decreases substantially since unwanted cells appear more often. As the rate of arrival, ra, of cells to the sensing zone increases (ra varying between, say, 300 cells/s and 10 000 cells/s or more), the efficiency decreases. As the value of the subpopulation fraction, subfr, decreases, the efficiency decreases, reaching a limit when subfr ! 0.01. The purity of the droplets sorted out depends on the probability that no other type of cell appears along with the sorted cell in the droplet sorter or other sorters. Expressions (7.3.86) and (7.3.87) provided above for the efficiencies are also used to estimate the purity of the cells in the sorted droplet population.

7.3.2

Centrifugal force field

Large-scale devices in industry employ centrifugal forces to separate particles from a fluid or droplets from an immiscible continuous phase. Usually, these large devices are single-entry separators: a multiphase feed stream enters

7.3

External force field based separation: bulk flow perpendicular to force

the separator and two multiphase product streams, an overflow and an underflow, leave the equipment. Centrifuges of much smaller dimensions are used for preparative work or in analytical studies. The centrifugal force is imparted to the multiphase fluid by rotating the whole device around an axis or by imparting a swirl to the feed, introduced tangentially to a stationary cylindrical device, as in a cyclone or hydrocyclone separator. For a review, see Svarovsky (1977, 1981) and Hsu (1981). In all such separators, the multiphase fluid has a tangential motion as it rotates around a central axis. However, there exists an axial movement, which is superimposed on the primary circulatory motion, which generates the centrifugal force. This axial movement is essential to continuous removal through an outlet of the entering fluid from the centrifugal separating equipment. More important, from a separation point of view, is that this bulk motion is perpendicular to the direction of the centrifugal force generated by the rotation of the multiphase fluid around the main axis. This enhances the multicomponent separation capacity considerably for larger-scale operations. However, this capability is generally unutilized since industrial processing focuses more on particle separation from a fluid or phase separation from a feed which is a multiphase dispersion. Note that, even without the axial bulk flow, the primary circulatory flow is perpendicular to the direction of the centrifugal force. In this section, particle separation from a liquid having one component of bulk motion along the axis of a tubular bowl centrifuge is studied first. Following the particle separation, the separation of two immiscible liquids of different densities is briefly considered. The performance of a disk centrifuge is treated next; here the bulk liquid motion toward the outlet is not perpendicular, but is at an angle to the centrifugal force. Separation of particles from a gas in a cyclone separator is then considered, followed by brief treatments of the separation nozzle process for gas separation and hydrocyclones. 7.3.2.1

Tubular bowl centrifuge

Consider first particle separation from a liquid suspension in a simple tubular bowl centrifuge; a schematic of this centrifuge is shown in Figure 7.3.12. It is essentially a cylindrical bowl rotating around its axis at a high rpm (revolutions per minute). Smaller centrifuges may have rpms as high as 15 000. A dilute feed suspension is introduced through a central nozzle at the bottom. The feed suspension may be assumed to attain the angular velocity of the bowl very quickly as it spreads out from the feed nozzle to the bottom of the bowl at z ¼ 0. However, the feed liquid moves axially upward from z ¼ 0 to z ¼ L, the length of the centrifuge. The radial location of the free surface of this liquid, rf, is set essentially by the liquid outlet weirs at the top of the bowl.

619

The feed liquid is assumed to be uniformly distributed at the bottom of the centrifuge (i.e. z ¼ 0) from r ¼ rf to r0. Therefore, particles of all sizes found in the feed suspension are present at all radial locations (between r ¼ rf to r ¼ r0) at the centrifuge bottom (z ¼ 0). However, as soon as the particles are at various r-values at z ¼ 0, they are immediately subjected to sedimentation. The particles are thrown toward the bowl wall by the centrifugal force as they move vertically upward (z > 0) due to the axial vertical motion of the liquid. (The circumferential liquid velocity due to the rotation is not important in itself except for the calculation of the centrifugal force on the particles.) Since each particle has a velocity in the z-direction due to the z-directional liquid motion, as well as a centrifugally generated radially outward velocity, it follows a particular path. These paths are identified in Figure 7.3.12 as effective particle trajectories. If the trajectory of a particle hits the walls of the centrifuge (i.e. r ¼ r0) before z ¼ L, then the particle is separated from the liquid and is deposited along the wall. When the deposit becomes thick, the centrifuge is stopped to remove the particle deposits, since a thick deposit reduces the cross-sectional area for liquid flow. This particle deposit may be considered as the underflow in a continuous single-entry particle fractionators, even though, on an extended-time basis, the operation is intermittent. A particle trajectory is a series of (r, z)-coordinates occupied by a particle as it moves in the centrifuge. A particle trajectory, whose r value is less than r0 but z ¼ L, will not hit the centrifuge wall at r ¼ r0; this particle will leave the centrifuge with the liquid overflow and is not captured by the device. A separation analysis should be able to predict the particle size density function, f0(rp), in the liquid overflow for a given feed liquid particle size density function ff (rp). Alternatively, the separation analysis should provide an expression for the grade efficiency function, Gr(rp) of the device. Separation analysis in a centrifuge for particles begins with a particle trajectory analysis. This is conveniently initiated by following the z-coordinate and the rcoordinate of the particle with time. Assuming that there is no slip between the liquid and the particle, the zdirection particle velocity (Upz) is equal to that of the liquid, vz(z, r). Therefore, U pz ¼ ðdz=dtÞ ¼ vz ðz;rÞ:

ð7:3:88Þ

If Qf is the volumetric flow rate of the liquid, and the liquid is assumed to have plug flow between radial locations r0 and rf, then

 U pz ¼ ðdz=dtÞ ¼ vz ðz;rÞ ¼ vz ¼ Qf =π r 20 − r 2f : ð7:3:89Þ For the r-direction motion of a particle of radius rp, the general equation of particle motion (equation (6.2.45)) is

620

Bulk flow perpendicular to the direction of force

Outlet weir Outlet weir

w

Free surface of liquid

Liquid overflow with fine particles

r0

Liquid overflow with fine particles

r0 L

Effective particle trajectories

rf

Effective particle trajectories

rf

r r

z

Deposited solid particles

Deposited solid particles z=0

Dilute feed suspension in

Figure 7.3.12. Schematic for particle separation from a liquid in a tubular bowl centrifuge.

mp

dU pr d2 r drag ¼ mp 2 ¼ F ext pr − F pr : dt dt

Here centrifugal force is the external force along with the centrifugal buoyancy term (see equation (3.1.59)) and its    magnitude is given by mp rω2 1 − ρt =ρp . If the size of the particle (assumed spherical) is small enough for Stokes’ law to be valid, the radial resistive drag force magnitude is 6πμ r p vr . If we further assume that the particles settle always with their     terminal radial velocity (i.e. dU pr =dt ¼ d2 r=dt 2 ¼ 0 is achieved very quickly), we obtain ! ρt 2 ¼ m r ω 1 − ð7:3:90bÞ ¼ 6πμ r p U pr ¼ F drag F ext p pr pr : ρp This leads to an expression for the radial particle terminal velocity: U pr ¼ U prt ¼

2 2  dr 2r p ω r ρp − ρt : ¼ 9μ dt

9Qf μ dz :

 ¼ dr 2π r 2 ω2 r r 2 − r 2 ρ − ρ p t 0 p f

ð7:3:90aÞ

ð7:3:91Þ

(Note: If gravity is the external force, Upz is obtained by replacing ω2 r by g; U pr jcentrifuge ¼ U pz jgravity ðω2 r=gÞ, where the ratio ðω2 r=gÞ is called the centrifuge effect.) Combining this equation with equation (7.3.89), one gets

ð7:3:92Þ

Rearrange it and integrate between (r, z ¼ 0) and (r ¼ r0, z ¼ L) to obtain 

 rð0

r  2π r 2p ω2 ρp − ρt r 20 − r 2f dr 0 ¼ ln L: ð7:3:93Þ ¼ 9Qf μ r r r

Since r and z are the coordinates of a particle of radius rp and density ρp as it moves along its trajectory, equation (7.3.93) defines the smallest value of the radial location r of the particle at z ¼ 0, where it must be located if it is to hit the wall (r ¼ r0) and be captured when z ¼ L. If the particle is located at a value of r greater than that defined by equation (7.3.93) at z ¼ 0, then it will surely hit the wall (r ¼ r0) before z ¼ L. However, if the particle at z ¼ 0 is located at an r smaller than that defined by equation (7.3.93), it will not hit the wall (r ¼ r0) by the time z ¼ L; then the particle escapes with the overflow. A critical radius r is thus identified with each particle size rp via equation (7.3.93).

7.3

External force field based separation: bulk flow perpendicular to force

An alternative form of equation (7.3.93) is also quite useful. Suppose the particle starts out at r ¼ rf when z ¼ 0. Then, if its r ¼ r0 when z ¼ L,

    2π r 2 ω2 ρ − ρ r 20 − r 2f p t max r0 ¼ ln L; ð7:3:94Þ 9Qf μ rf where rp is identified as rmax. This is the largest particle size with any nonzero probability of not being captured. Any particle with a smaller size ( > > > > >7 > 6 2 2 2> < =7 4π ð 5000 Þ 4π  0:09  ð100  16Þ  90 r 6 100 p 7 6 1 − exp − Gr ðr p Þ ¼ 7 1406 > > ð100 − 16Þ 6 −2 2 > > 5 4 60  9   1:5  10 > > > > : ; 60 93 8 93 8 2 2 = < 16  31  25  106  680:4 < 16π3  25  106  8:1  84 r 2 = 100 4 p 2 5 ¼ 1:19 41 − exp − 1 − exp − r p 5: ¼ ; : ; : 84 11388 540  14:06  1:5

Example 7.3.4 Determine the expression for the grade efficiency function Gr(rp) for the centrifuge under the conditions of Example 7.3.3(a). Calculate the value of the grade efficiency function for cells of two sizes, rp ¼ 0.25 μm and rp ¼ 0.5 μm.

So

h n oi Gr ðr p Þ ¼ 1:19 1 − exp − 0:7404  109  r 2p ;

where rp is in cm, and

radii (r1i, r2o) and different elevations via two weirs (see McCabe and Smith (1976, p. 966) Hsu (1981) and Perry and Green (1984, pp. 21–65)). The feed liquid mixture

624

Bulk flow perpendicular to the direction of force

enters through a bowl opening in the bottom through a stationary nozzle. Two layers of liquid are formed in the bowl: the outer one is for the heavier liquid and the inner one is for the lighter liquid. Both leave through separate outlets at the top of the device. There is a liquid–liquid interface of radius rint between the two liquid layers: rint > r1i, rint > r2o. At rint, the radial location of the interface, the pressure, Pint is the same for both liquids. Equation (3.1.51) for centrifugal equilibrium in any pure liquid integrated between two radial locations will yield for the light liquid (ρ ¼ ρ1 ) and the heavy liquid ( ρ ¼ ρ2 > ρ1 ), respectively,    2   2  Ms r − r2 r − r2 P int − P 1i ¼ ω2 int 1i ¼ ρ1 ω2 int 1i ; 2 2 Vs 1    2   2  2 Ms r −r r − r2 ω2 int 2o ¼ ρ2 ω2 int 2o : P int − P 2o ¼ 2 2 Vs 2 ð7:3:107Þ

w

Light liquid

Light liquid

Heavy liquid

Heavy liquid

r20

rli

rint

rint

Centrifuge wall

Light liquid Heavy liquid

Solids

w

which leads to r 2int ¼

Feed

Figure 7.3.13. Schematic of a tubular centrifuge for separating two immiscible liquids.

   ρ ρ r 22o − 1 r 21i 1− 1 : ρ2 ρ2

ð7:3:108Þ

When ρ1 ! ρ2 , the location of the interface becomes unstable. The above relation also shows that if r2o is increased, rint is also increased, pushing the liquid–liquid interface closer to the wall of the bowl. Meanwhile, if there are any solid particles in the fluids, they are thrown to the bowl wall and can be collected from the bottom. Such liquid–liquid centrifuges have a narrower and taller bowl than those used for particle separations (Figure 7.3.12). Note that the bulk flow of each liquid phase is perpendicular to the direction of the external force causing the separation.

7.3.2.2

r20

rli

The two liquid outlet radii (r1i, r2o) are known (via mechanical adjustments); the unknown radial interface location is obtained by equating the hydraulic pressure drops in the two liquid layers reflecting mechanical equilibrium with atmospheric pressure being exerted at both r1i and r2o. Therefore  2   2  r − r2 r − r2 ρ1 ω2 int 1i ¼ ρ2 ω2 int 2o ; 2 2

Disk centrifuge

This liquid–particle separation device provides an example in this chapter where the bulk liquid motion toward the device outlet is at an angle to the centrifugal force. The rotating bowl in this device has a flat bottom and a conical top rotating at an angular speed ω rad/s around the vertical axis (Figure 7.3.14). A stack of closely spaced conical disks is set at an angle θ to the vertical axis between the conical top and the bottom. The gap between the consecutive disks is 2b; the disks rotate with the bowl at the same angular velocity. The feed suspension introduced at the top of the device through a pipe flows through the bottom of the device between the disk stack and the device flat bottom to the periphery of the bowl. From this region, the liquid suspension goes up, enters the gap between two consecutive disks and flows toward the central device axis. Particles in the suspension having a density ρp greater than the fluid density ρt are radially thrown outward and hit the bottom surface of the top disk in every channel. The collected particles slide outside and, at the end of each top disk surface, get thrown outward to the bowl wall. The clarified liquid leaves the device top through an annulus between the end of the disk stack and the feed inlet pipe. Figure 7.3.14 illustrates the trajectory of a particle in between two consecutive disks (dashed lines). Define two coordinate axes: the positive z-axis parallel to the disks, and in the direction of main liquid flow toward the main vertical device axis, and the positive r-axis radially

7.3

External force field based separation: bulk flow perpendicular to force

625

Feed suspension

Overflow r

q

rin

dr dz y

2b Upy

Upr

Disks Upz

Particle trajectories w rex rin

Figure 7.3.14. Schematic of a disk centrifuge.

outward. A particle entering the channel between two consecutive disks moves along the positive z-axis carried by the main liquid flow at a velocity Upz; simultaneously, the centrifugal force imparts a radially outward velocity Upr (¼ dr/dt) to the particle. This radially outward particle velocity, Upr, can be broken down into two component velocities, Upy (¼ dy/ dt) normal to the disk wall, or the z-direction, and another component in the negative z-direction reducing the positive z-direction velocity, Upz. The device operation will be affected if Upz is negative or zero. The y-component of the radially outward velocity causes the particle to hit the bottom surface of the top disk in every channel as it moves in the positive z-direction. Those particles which are small enough not to hit the top disk by the time the channel end is reached escape with the liquid and are not captured by the centrifuge. We assume first that the positive z-component of the particle velocity Upz (¼ dz/dt) is essentially equal to the positive z-component of the liquid velocity vz. We further assume that this liquid is in plug flow. To calculate this velocity vz, the flow cross-sectional area has to

be determined. Since the gap 2b between the two plates is small compared to the value of r at any radial location, this flow area is 2πrð2bÞ at any radial location r. So U pz ¼ ðdz=dtÞ ffi vz ffi

Qf ; 4π r b nc

ð7:3:109Þ

where nc is the total number of channels through which the liquid slurry having a volumetric flow rate Qf flows. The total number of disks is therefore nc þ 1. Since the two coordinates of interest in the particle trajectory before the particle hits the bottom of the top plate in any channel are y and z, we need next Upy (¼ dy/dt). But U py ¼ U pr cos θ ¼ ðdy=dtÞ:

ð7:3:110Þ

To determine Upr, we assume terminal settling velocity conditions and applicability of Stokes’ law (equation (7.3.91)) (but ignore the contribution (− vz sin θ): U pr ¼

2 2 dr 2r p ω r ¼ ðρp − ρt Þ: dt 9μ

626

Bulk flow perpendicular to the direction of force leading to

Therefore dy ¼ dt

2r 2p ω2 r 9μ

ðρp − ρt Þcos θ:

ð7:3:111Þ

One can now obtain a relation between y and z for the particle trajectory, 2 2 2 dy 8 π b nc r p ω r ðρp − ρt Þcos θ; ¼ 9μQf dz

ð7:3:112Þ

and integrate between the limits of y and z from the entrance to the flow channel between any pair of disks to the exit. It is, however, more useful to express the particle location in terms of the radial coordinate r instead of the z-coordinate. Define rex and rin to be the radial coordinates, respectively, for the inner location (where liquid exits the channel) and the outer location (where liquid enters the channel). Any differential positive change dz in the z-coordinate of a particle along the channel involves a differential change in r which is negative: dr ¼ − dz sin θ:

ð7:3:113Þ

Therefore 8π b nc ðρp − ρt Þ 2 2 2 dy ¼ − r p ω r cot θ: dr 9μQf

 8π b nc ðρp − ρt Þcot θ 2 2  3 r p ω r in − r 3ex : ð7:3:115Þ 27μQf

For a particle to settle on the bottom surface of the top plate at r ¼ rex, yex should equal 2b, the channel gap. At the inlet end (r ¼ rin), the particle can enter the channel with the liquid at any value of y ¼ yin. Thus, for any particle of size rp entering at any yin to settle by the time the liquid exits the channel,  8π b nc ðρp − ρt Þcot θ 2 2  3 r p ω r in − r 3ex : 2b − y in ¼ 27μ Qf

ð7:3:116Þ

We can now assume, as we did for the tubular bowl centrifuge, that particles of size rp are present uniformly at all y-values at r ¼ rin. Then the grade efficiency Gr(rp) is given by Gr ðr p Þ ¼

2b − y in ; 2b

ð7:3:117Þ

since those particles of size rp which have y < yint at r ¼ rin do not settle. Now, for rp  rmax, Gr(rp) ¼ 1 only if yin ¼ 0, i.e. all particles entering the channel settle before r ¼ rex. Therefore   8πbnc ðρp − ρt Þcot θ 2 r max ω2 r 3in − r 3ex ; 2b ¼ 27μQf

ð7:3:118Þ

ð7:3:119Þ

Further, the largest particle size with a nonzero probability of not being captured is obtained from equation (7.3.118) as (Ambler, 1952) r 2max ¼

27μQf   : 4πnc ðρp − ρt Þcot θ r 3in − r 3ex ω2

ð7:3:120Þ

All particles of size rp > rmax will be captured in the device. Thus the grade efficiency function for a disk-stack centrifuge does not depend on the channel height 2b but instead depends on the flow rate per channel. Also, the larger the value of ω, the smaller the value of rmax. Note that the particle capture efficiency E for particles of a particular size is equal to the grade efficiency Gr(rp) for that particle size. An important assumption in the derivation of equation (7.3.120) was that the value of the particle z-velocity component (dz/dt) was equal to the liquid velocity vz. By this assumption, the component of the velocity in the negative z-direction due to the centrifugal force was neglected. If we do not neglect it, then the radial velocity of the particle is (using equation (7.3.91)) 2 2 dr 2r p ω r ðρp − ρT Þ − vz sin θ: ¼ 9μ dt

ð7:3:114Þ

Integration of this relation between the limits (rin, yin) and (rex, yex) yields y ex − yin ¼

Gr ðr p Þ ¼ ðr 2p =r 2max Þ for r p  r max :

ð7:3:121Þ

Employ now expression (7.3.111) for (dy/dt) and expression (7.3.109) for vz to obtain dy r 2 cos θ ¼ 2 2 ; dr r − b1

ð7:3:122Þ

where b1 ¼

9Qf μ sin θ 8π b nc ω2 r 2p ðρp − ρt Þ

!1=2

:

ð7:3:123Þ

Integration has to be carried out between the limits (rin, yin) and (rex, yex), where yex ¼ 2b: yðex

yin

dy ¼ cos θ

rðex

r in

r 2 dr : r 2 − b21

ð7:3:124Þ

This analysis was developed by Jury and Locke (1957); they have also provided an analytical solution to the integral in terms of the grade efficiency function:    2b − yin cos θ b1 b1 þ r in Gr ðr p Þ ¼ ¼ ln 2b 2b 2 b1 − r in    b1 − r ex  ð7:3:125Þ − ðr in − r ex Þ : b1 þ r ex Jury and Locke (1957) have provided, in addition, an expression for the thickness of the sludge film flowing down the cone wall.

7.3

External force field based separation: bulk flow perpendicular to force

Typical values and ranges for the operating conditions and dimensions of a disk centrifuge are provided now for perspective: 35∘  θ  50∘ ; nc 23; 2b 0:17 cm; bowl diameter 15 cm–100 cm; rotational speeds up to 12 000 rpm ðω ¼ 1257 rad=sÞ; Qf up to 16 m3/min. The disk centrifuges generally employed are larger than tubular bowl centrifuges and can be run continuously. 7.3.2.3

Cyclone dust separator

To separate particles from gaseous streams, cyclones are frequently used in large-scale practices. In fact, cyclone based dust collectors are one of the most widely used devices for removing larger-sized particles. Virtually all cyclones used industrially are reverse-flow cyclones. There are two other types of cyclones: rotary-flow (Ciliberti and Lancaster, 1976a, b) and uniflow (Ter Linden, 1949). Only the reverse-flow cyclone will be considered here. A reverse-flow cyclone (Figure 7.3.15(a)) is a simple hollow structure consisting of two parts: a cylindrical cyclone barrel having an annular vortex finder or exit pipe; and a truncated cone at the bottom joined at the top to the cyclone barrel. The dusty gas is introduced tangentially into the cyclone barrel through the inlet. The gas, having a swirling motion, goes around the cyclone as it moves down toward the dust exit; however, the swirling gas soon reverses its gross axial movement, rises up and exits through the top of the vortex finder. The rotational motion of the gas generates the radial centrifugal force on dust particles, which makes them hit the wall of the cyclone as shown in the idealized flow pattern in an ideal cyclone (Flagan and Seinfeld, 1988) shown in Figure 7.3.15(b). The centrifugal force on the particle may be as much as 5–2500 times larger than the gravitational force (Perry and Green, 1984, p. 20–83). If there is no reentrainment, the dust particles settle and are removed through the dust exit at the bottom. Figure 7.3.15(b) shows only the circular gas motion in an idealized flow. The gas and the particles have an axial (z-directional) motion, which is not shown. An analysis of the ideal flow cyclone for particle separation is available in Flagan and Seinfeld (1988). The flow pattern of the vortex motion of the gas in reverse-flow cyclone is quite complex. First, it is threedimensional; second, the flow is turbulent. An exact analysis is therefore difficult. Soo (1989) has summarized a fundamental analysis of velocity profiles and pressure drops in such a cyclone. He has also analyzed the governing particle diffusion equation in the presence of electrostatic, gravitational and centrifugal forces. He has then provided an analytical expression for particle collection efficiency under a number of limiting conditions. We will, however, opt here for a much simpler model of particle separation in a cyclone developed by Clift et al. (1991). This approach is based on a modification of the original model by Leith and Licht (1972). The model will be

627

presented in the framework of a three-region model of cyclone by Dietz (1981) to introduce the inherent complexities. The treatment begins with the Dietz (1981) model. The three regions of a cyclone particle separator in the model of Dietz (1981) consist of the entrance region (region 1), the downflow region (region 2) and the upflow region (region 3) (see Figure 7.3.15(a)). Regions 2 and 3 are sometimes called the annular region and the core region, respectively. In these cyclone models, the cyclone of Figure 7.3.15(a) is replaced by a right circular cylinder of radius rc and length L below the exit tube, equal to the length of the actual cyclone below the exit pipe; the exit pipe is of radius rt (Figures 7.3.15(b), (c)). Regions 1 and 2 have the swirling gas flowing downward at a volumetric flow rate of Qv0 and Qv(z), respectively, where Qv0 is the total volumetric gas flow rate entering the cyclone. In Dietz’s model, Qv(z) is related to Qv0 by Qv ðzÞ ¼ Qv0 f1 − ðz=LÞg:

ð7:3:126Þ

The linear decrease in Qv(z) with increasing z is due to the radial gas velocity, –vr(z), from the annular to the core region (the upflow region): jvr ðzÞ j ¼ vr0 ¼

Qv0 : 2πr t L

ð7:3:127Þ

Dietz (1981) assumed (i) jvr ðzÞ j to be a constant, vr0 , and (ii) that, due to turbulent mixing, the particle mass concentration profile in each region was radially uniform. However, each such profile,22 ρp1 ðzÞ, ρp2 ðzÞ and ρp3 ðzÞ for regions 1, 2 and 3, respectively, changes with the zcoordinate due to particle deposition at the cyclone wall and/or particle exchange between regions 2 and 3:  d ρp1 ðzÞ Qv0 ¼ − 2π r c Γ w ðzÞ; ð7:3:128Þ dz  d ρp2 ðzÞ Qv ðzÞ ¼ − 2π r c Γ w ðzÞ − 2π r t Γv ðzÞ; region 2 : dz ð7:3:129Þ region 1 :

region 3 : −

 d ρ ðzÞQv ðzÞ ¼ 2πr c Γ v ðzÞ: dz p3

ð7:3:130Þ

Here the quantity Γ w ðzÞ is the particle mass flux at the wall at axial coordinate z. For region 1, Γ w ðzÞ is the product of the particle mass concentration, ρp1 ðzÞ, and the radial particle velocity at the wall, Uprw: region 1 : Γ w ðzÞ ¼ ρp1 ðzÞU prw :

ð7:3:131Þ

For regions 2 and 3, ρp1 ðzÞ has to be replaced by the corresponding particle mass concentrations, ρp2 ðzÞ and ρp3 ðzÞ, respectively. Further, Uprw is a function of the z-coordinate:

22

Instead of Dietz’s number density profiles, n1(z), n2(z) and n3(z), the particle density profiles, are used here.

628

Bulk flow perpendicular to the direction of force

(a)

Inlet Vortex finder (Exit pipe)

b rc

a

Region 1 s

(b)

rt

Exit pipe surface Region 2

rt

h

rc y

q H

rt

vr

vq

r

Region 3

Particle trajectory Particle hits outer wall

x

Gas flow streamline

Truncated cone

Dust exit

(c)

Region 1

rt

s-

Figure 7.3.15. (a) Geometry of a conventional reverse-flow cyclone; (b) particle trajectory in a cyclone having an idealized flow pattern. (After Flagan and Seinfield (1988).) (c) Modified cyclone geometry for analysis. (After Dietz (1981).)

a 2

Q v0

Qv 0

z=0 Gw (z)

Region 3

vro Gw (z)

rt

Qv (z)

r Qv (z)

Gv (z)

L Qv (z)

rc Gv (z)

Region 2 Region 2 z=L

7.3

External force field based separation: bulk flow perpendicular to force Region 2 : Γ w ðzÞ ¼ ρp2 ðzÞ U prw ðzÞ:

ð7:3:132Þ

Note: In each region the radial particle flux due to the radial centrifugal force is perpendicular to the bulk flow of the fluid and the particles in the main flow (z) direction. We need to know what Uprw and Uprw (z) are; further, an expression for Γ v ðzÞ has to be found if the three equations (7.3.128) – (7.3.130) are to be solved. We assume that • Stokes’ law is valid; • the particle is hitting the wall at the terminal velocity without any radial acceleration. From equation (7.3.90b), we obtain at the wall (r ¼ rc)

 ð7:3:133Þ mp r c ω2 1 − ðρt =ρp Þ ¼ 6πμr p U prw :

  However, r c ω2 is simply U 2ptw =r c , where Uptw is the tan    gential particle velocity ¼ U pθ at the wall ¼ U pθw . Now, if one assumes no slip between the particle and the gas in the tangential direction ( θ-direction), the tangential particle velocity at the wall, Uptw, is also equal to the corresponding tangential fluid velocity, vtw, at the wall. At the wall, the fluid velocity vtw is zero. Note that, in the boundary layer close to the wall, it is nonzero. Further, the tangential gas velocity vt(r) in the vortex may be related to the radial location by an equation valid for a free vortex in an ideal fluid:

r m c : ð7:3:134Þ vt ðrÞ ¼ vtw vt ðrÞ r m ¼ constant; r There is little error in replacing vt (r) in the wall region by vtw for m ¼ 1 (ideal vortex); in general for cyclones, the vortex index m is between 0.5 and 1. We can therefore write U prw ¼

2 r 2p ρp v2tw 9μr c

ð7:3:135Þ

from equation (7.3.133) if mp ¼ 4 π r 3p ρp =3 and ρt Upztc settle to the bottom plate before reaching the overflow: ð∞ 0

Q1 ϕs1 f 1 ðU pzt ÞdU pzt ¼ Q1 ϕs1 ¼ ϕsf ) ϕs1 ¼ ϕsf

ð∞ 0

ð U pztc 0

where we have employed equations (7.3.179) and (7.3.187). Substituting for f1(Upzt) from equation (7.3.189), we get f 2 ðU pzt Þ ¼

ðQf − ðQ1 − Qc ðU pzt ÞÞÞ f f ðU pzt Þ ð U pztc : Qf − ðQ1 − Qc ðU pzt ÞÞ f f ðU pzt Þ dU pzt 0

ð7:3:194Þ

ðQ1 − Qc ðU pzt ÞÞ f f ðU pzt Þ dU pzt Q1 − Qc ðU pzt Þ f f ðU pzt ÞdU pzt þ Q1

The second integral on the right-hand side is zero from definition (7.3.186), leading to ð U pztc Q1 − Qc ðU pzt Þ f f ðU pzt Þ dU pzt : ð7:3:187Þ ϕs1 ¼ ϕsf Q1 0 The particle volume fraction in the underflow, ϕs2 , is now obtained from the total particle balance (7.3.180) as   ð U pztc fQ1 − Qc ðU pzt Þgf f ðU pzt Þ dU pzt ϕsf Qf − 0 ϕs2 ¼ : Qf − Q1 ð7:3:188Þ

One can obtain the probability density function f1(Upzt) of the overflow stream by substituting (7.3.187) in to relation (7.3.184):

ð∞

U pztc

Q1 − Qc ðU pzt Þ f f ðU pzt ÞdU pzt : Q1

Note: We have generally employed particle size density functions ff (rp), f1(rp) and f2(rp), where the probability density function depends on the random variable rp, the particle radius. In the analysis considered here for inclined settlers, we are dealing with density functions ff(Upzt), f1 (Upzt) and f2(Upzt). Since, by relation (6.3.1), the relation between Upzt and rp is (if Stokes’ law is valid) U pzt ¼ ð2=9Þr 2p ðρp − ρt Þg=μ; we have f ðU pzt Þ ¼ f ðr p Þ=jðdU pzt =d r p Þj;

ð7:3:195aÞ

f ðU pzt Þ ¼ 9 μ f ðr p Þ=ð4 ðρp − ρt Þ g r p Þ:

ð7:3:195bÞ

640

Bulk flow perpendicular to the direction of force

7.3.4 Field-flow fractionation for colloids, macromolecules and particles This technique is primarily utilized in a relatively small scale of operation. It involves a somewhat different interaction of the bulk flow with the force operating perpendicular to the bulk flow direction compared to what we have seen so far. So far, whenever we have employed the bulk flow, limited attention was paid to the fact that there is a velocity profile in the bulk flow taking place in the channel/device. Such velocity profiles can often be damaging to the separation achievable (for example, see, the comments in the paragraph preceding equation (6.3.6)) or they can introduce complications/distortions in the separation achieved (see Figures 7.3.2(b) and 7.3-3 in thinfilm continuous-flow electrophoresis). In the field-flow fractionation (FFF) technique, however, the velocity profile of bulk flow in the channel is crucial to the separation achieved. Before we discuss this feature, let us describe the effect of a force field applied perpendicular to the bulk flow direction in a channel where a suspension of colloids, macromolecules or particles is flowing. The force field can be electrical, centrifugal, gravitational, thermal diffusion or crossflow from a pressure gradient across the channel wall. Let the dimensions of such materials in suspension vary between the wide range of 1 nm and around 100 μm (Giddings, 1993). As shown in Figure 7.3.21(a) , a force field applied in the negative y-direction in the channel creates a flux of macromolecules/colloids/particles toward the wall. For macromolecular species i, we may write the following expression23 (See 3.1.88): N iy ¼ C is ðvty þ U iy Þ − Dis

dC is : dy

ð7:3:196Þ

If the flow in the channel is laminar and fully developed, then vty ¼ 0 and N iy ¼ C is U iy − Dis

dC is ; dy

ð7:3:197Þ

where Uiy is the force field induced migration velocity of species i in the y-direction. For particles, an expression for the particle number flux npy in the y-direction in terms of the total number density Nt of particles may be written from (3.1.70) as npy ¼ N t U py − Dp

dN t : dy

ð7:3:198Þ

This field induced macromolecular migration velocity Uiy normal to the channel wall and toward the wall at y ¼ 0

may be created by a variety of fields (to be discussed soon). At steady state, there should be no net flux of species i in the y-direction as the flux due to the field is counteracted by back diffusion of species i away from the wall: N iy ¼ 0 ¼ C is U iy − Dis Dis

There is also an axial flux Niz since Cis(y,z) is a function of y and z. Here we assume that Cis(y,z) may be represented as Cis(y) f(z) and focus on Cis(y).

dC is ¼ C is U iy : dy

ð7:3:199aÞ ð7:3:199bÞ

A solution of this equation is obtained from   U iy U iy dC is U iy dy ) ℓnC is ¼ y þ a ) C is ¼ C i0 exp y ; ¼ C is Dis Dis Dis

ð7:3:200Þ

where Ci0 is the concentration of species i at the wall, y ¼ 0. The direction of the force field applied is such that Uiy is a negative quantity. We can therefore write the above concentration profile as   jU iy j C is ¼ C i0 exp − y ; ð7:3:201Þ Dis where jU iy j is the magnitude of the force field induced velocity of species in the y-direction. One could define a characteristic24 thickness δi of this concentration profile by defining Dis ¼ δi : jU iy j

ð7:3:202Þ

Rewrite the concentration profile (7.3.201) now as   y : ð7:3:203Þ C is ¼ C i0 exp − δi If there are two macromolecular species a and b, the characteristic thicknesses of their profiles extending from the wall out are δa and δb , respectively. Species subjected to a lower jU iy j and possessing a higher Dis will have a larger δi. Therefore the larger species, whose diffusion coefficient at infinite dilution D0is may be described by the Stokes–Einstein equation (3.3.90c) D0is ¼

kB T ; 6π r i μ

ð7:3:204Þ

will have a smaller D0is , and therefore a smaller Dis. For two macromolecular species a and b having similar Uiy (say), if ra > rb, Day < Dby; therefore δa < δb. Thus the smaller macromolecular species will have a profile whose average extends out further from the wall (Figure 7.3.21(a)).

24 23

dC is ; dy

The characteristic thickness here corresponds to a distance where the value of Cis has been reduced to 36% of its value at the wall; therefore the bulk of the molecules are contained in the region y ¼ 0 to y ¼ δi.

7.3

External force field based separation: bulk flow perpendicular to force

641

(a) Parabolic velocity profile

Force field

Channel wall

b

Detector response

y b z Channel wall

a

a

b

da

time

db

(b) 1.0

(c)

Ri 0.8

Liquid in crossflow

6li

Porous wall Equation (7.3.214)

0.6 Ri

a

b

0.4 0.2

Crossflow out

Membrane (10 000 MWCO) Porous wall

0 0 0.1 0.2 0.3 0.4 0.5 0.6 li = (di / b)

(e)

(d) Flow inlet (sample injection)

Flow to detector

Flow

W

b

b

Flow

Flow Spin

Exploded view

a

L

Figure 7.3.21. Field-flow fractionation (FFF). (a) Basic configuration of field-flow fractionation device and the detector response to a sample. (b) Retention ratio of species i vs. retention parameter λi. (c) Schematic for flow field-flow fractionation with a crossflow. (d) Physical configuration/dimensions of a FFF channel with inlet/outlet. (e) Configuration of the channel in sedimentation FFF. (After Giddings (1993).)

Field-flow fractionation exploits this difference in the distance of the mean of the species profile from the channel wall by coupling it with the velocity profile in the channel. In laminar channel flow with a parabolic velocity profile (equation (6.1.2b)), the larger molecules are then concentrated in the slower axial velocity zones closer to the

wall, whereas the smaller molecules reside primarily in higher axial velocity zones further away from the wall. Therefore the smaller molecules will show up (via a detector) at the channel exit faster than the larger molecules if there is an injection of a sample containing different macromolecular species upstream in the channel

642

Bulk flow perpendicular to the direction of force

(f)

(g)

Hot wall

Electrode Membrane Force field

a

Cold wall

b

Membrane Electrode

(h) Force field

y

Channel wall

Detector response r3

z Parabolic r1 velocity profile

r2

r3 > r2 > r1

r2

r1

Time

(i) Particle product fraction a

Sample liquid flow Channel wall Inlet splitter

Force field

Transport laminae Outlet splitter

Carrier liquid flow

Channel wall

Particle product fraction b

Figure 7.3.21. (cont.) (f) Thermal FFF. (g) Electrical FFF. (h) Steric FFF and the detector response to a sample injection. (i) SPLITT fractionation.

(Figure 7.3.21(a)). It is as if fluid lamellae (laminae) of each species located at different distances from the wall (y ¼ 0) travel at different velocities; therefore they arrive at different times at the channel end. (For a comparison, in elution chromatography (Section 7.1.5.1) different species also arrive at different times at the end of the column; however, each species exists throughout the column cross section as the species peak travels. Similarly, in capillary electrophoresis (Section 6.3.1.2) species arriving at the end of the capillary at different times exist throughout the capillary cross section.) This technique was first proposed by Giddings (1966). Quantitative analysis of the retention time, t Ri , of a macrosolute species in the channel after sample injection may be carried out as follows (Giddings, 1991). The axial velocity profile, vz ðyÞ, of a liquid of viscosity μ in a thin rectangular channel formed between two infinite parallel plates (Figure 7.3.21(a)) spaced a distance b apart is

given by25 (Happel and Brenner, 1965; Grushka et al., 1973; Giddings, 1991)   y y2 vz ðyÞ ¼ 6vz;avg ð7:3:205Þ − 2 ; b b where vz;avg ¼

ΔPb2 ; 12 μ L

ð7:3:206Þ

L is the channel length and ΔP is the axial liquid pressure drop. If a macrosolute which is not affected by the force field is injected into the liquid upstream, then t R0 , its

25

Equation (6.1.2b) provided earlier is for a slit flow where the channel gap rb),

t Ra Rb rb ¼ ¼ ; t Rb Ra ra

where

We can safely assume that for such low λi, Ri ffi 6λi ¼ 0.423. For the T7 virus particle of Mi ¼ 240  106, and having the same density 1.57 as before,

λi ¼ 0:0705 

49 ¼ 0:0144; 240

Ri ffi 6λi ¼ 0:0863;

a substantially different retention parameter and ratio (almost inversely proportional to Mi).

7.3.5

Magnetic force field

A magnetic force field has been employed to separate particles having differing magnetic properties, primarily from a liquid, via the technique of high-gradient magnetic separation (HGMS). This technique requires the presence of ferromagnetic cylinders (steel wools, etc.), to develop strong variations in the magnetic force field on the particles in the local regions around the ferromagnetic cylinders, leading to the capture of particles on those cylinders. The technique of free-flow magnetophoresis, on the other hand, is simpler: it does not employ additional ferromagnetic cylinders or spheres, and is somewhat similar to free-flow electrophoresis (Section 7.3.1). We will describe initially the free-flow magnetophoresis based particle separation. Then we will provide a brief treatment of HGMS based particle separation from a fluid. However, a few basic features of particle motion in a magnetic field will be considered first. As pointed out in Section 3.1.2.4, there are three classes of magnetic materials: ferromagnetic materials (Fe, Co, Ni), which are strongly magnetic; paramagnetic materials (which are far less magnetized compared to ferromagnetic materials); and diamagnetic materials. Expression (3.1.24) for the magnetic force on a nonferromagnetic spherical particle is

7.3

External force field based separation: bulk flow perpendicular to force

3 F mag ¼ 2π μm p s rp

m μm p − μs m μm p þ 2μs

!

2

rH m 0 :

ð7:3:244Þ

m If the magnetic permeabilities μm p and μs of the particle and solution may be described, respectively, via m μm p ¼ μ0 ð1 þ χ p Þ;

m μm s ¼ μ0 ð1 þ χ s Þ;

ð7:3:245Þ

where μm 0 is the magnetic permeability of free space and χp and χs are, respectively, the magnetic susceptibilities of the particle and the solution, then !   χp − χs B0 2 mag m 3 F p ¼ 2π μs r p r m : ð7:3:246Þ μ0 χp þ 2 χs þ 3 Here the magnetic field strength H m is related to the  0 magnetic flux density B0 by B0 =μm 0 . This relation may be rewritten as follows: 0 1 2 χ − χ 3V p s p @ A rðB0 Þ F mag ¼ 2 μm 0 ð1 þ χ s Þ p 2 m 4 χ p þ 2χ s þ 3 ðμ0 Þ 0

¼ 3ð1 þ χ s ÞV p @

1 2 χp − χs A rðB0 Þ : 2μm χ p þ 2χ s þ 3 0

ð7:3:247Þ

When χp and χs are much smaller than unity, we obtain F mag ¼ p

V p ðχ p − χ s Þ rðB0 Þ2 ; 2μm 0

ð7:3:248aÞ

which is essentially expression (3.1.22), sometimes written as ¼ V p ðχ p − χ s ÞðH m F mag p 0 rÞB0 :

ð7:3:248bÞ

In one-dimensional form (say the z-direction), this force expression is given by   d B20 dB0 F mag : ¼ V ðχ − χ Þ ¼ V p ðχ p − χ s ÞH m p p s pz 0 dz dz μm 0

ð7:3:248cÞ

For paramagnetic particles, (χp − χs) is positive; for diamagnetic particles, it is negative. A nonferromagnetic particle moving in the direction of the nonuniform magnetic force field would encounter a drag force from the fluid and very soon achieve a terminal velocity, usually identified as the magnetic migration velmag ocity U pzt , where, if Stokes’ law is valid,  2 d B0 mag F drag ; ð7:3:249Þ r pz ¼ 6πr p μU pzt ¼ V p ðχ p − χ s Þ 2μm dz 0    2 χ p −χ s d 2 B0 mag U pzt ¼ r 2p : ð7:3:250Þ μ 9 dz 2μm 0 mag

This magnetic migration velocity U pzt (Watson, 1973), also called the magnetic velocity, may be expressed in terms of a mag magnetophoretic mobility mop and the magnetic field force strength Smag (Moore et al., 2004) by

mag

U pzt

¼ momag Smag ; p

Smag ¼ 7.3.5.1

649

momag ¼ p

2 2 ðχ p − χ s Þ ; r μ 9 p

  d B20 m : dz 2μ0

ð7:3:251Þ

Free-flow magnetophoresis

We now briefly illustrate free-flow magnetophoresis in analogy to free-flow electrophoresis (Section 7.3.1.1). Consider Figure 7.3.22, in which a rectangular flat separation chamber is shown. Here the vertical coordinate y is normal to the (x,z)-plane. There are a number of inlet and corresponding outlet channels at the two ends of the flat separation chamber. Buffer liquid comes in through all inlet channels except one at one end, where a particle mixture is introduced along with the liquid flow. A nonuniform magnetic field is applied perpendicular to this laminar liquid flow. Those particles which are nonmagnetic, e.g. polystyrene microspheres (Pamme and Manz, 2004), go straight through without any deflection in the direction of the magnetic field gradient to the corresponding outlet channels in the z-direction. Paramagnetic particles, however, follow a trajectory deflected in the direction of the magnetic field (x-coordinate), the extent of deflection depending on the magnetic susceptibility χp and the particle radius rp since mag

U pxt

/ r 2p ðχ p − χ s Þ:

ð7:3:252Þ

The particle trajectory equations are (see equations (7.3.88) and (7.3.91)):   ðχ p − χ s Þ d B20 dx 2 mag ; ð7:3:253aÞ ¼ U pxt ¼ r 2p dt 9 dx 2μm μ 0 dz ¼ U pzt ¼ vz ; dt

ð7:3:253bÞ

where vz is the fluid velocity in the mean flow direction (z-direction). Typical magnitudes of these flow velocities in microfluidic environments employed by Pamme and Manz (2004) were 0.1–0.4 mm/s. As shown in Figure 7.3.22, paramagnetic particles are deflected in the x-direction; different particles follow different trajectories, arriving at different outlet channels, and are therefore separated from one another. The larger the particle volume (or size, rp) and the higher the magnetic susceptibility χp of the particle material, the higher the deflection in the x-direction since mag U pxt is higher. The particle trajectory is obtained from equations (7.3.253a,b) as  2 dx 2 2 ðχ p − χ s Þ d B0 rp ; ð7:3:254aÞ ¼ μ dz 9vz dx 2μm 0 ðχ p − χ s Þ m d H m dx 2μm 0 0 H0 : r 2p ¼ μ dx 9vz dz

ð7:3:254bÞ

To identify the exact extent of deflection of the particle in the x-direction, one has to know the manner in

650

Bulk flow perpendicular to the direction of force

Rectangular flat laminar flow chamber

S N Paramagnetic particles of increasing size and susceptibility

Liquid (buffer) Flow channels

Particle mixture sample

x

Nonmagnetic particles

z

y

Figure 7.3.22. Free-flow magnetophoresis in a horizontal channel. (After Pamme and Manz (2004).)

Method 1

Mag, bead

(Mab) Method 2

Mag. bead with coated Mab targeting cell

Mag, bead

+ Cell

+ Cell

Mab-coated target cell

(Mab)

Mag, bead

Mag, bead

+

Bead coated with anti-IgG antibody Figure 7.3.23. Two methods of cell separation using magnetic beads and antibody–antigen binding. (After Gee (1998).)

which the nonuniform magnetic field is varying in the xdirection. Pamme and Manz (2004) have provided estimates of the magnetic force being exerted on small superparamagnetic particles.26 Particles of diameter 4.5 μm were found to traverse a distance 2 mm in the x-direction when the value of the distance tranversed in the z-direction was 6 mm; the axial fluid velocity was 0.3 mm/s. From equation (7.3.249), we obtain for a liquid consisting essentially of water (viscosity μ ¼ 1 cp) mag

F mag px ¼ 6πr p μU pxt ¼ 6π  2:25 μm  1cp  ¼ 6π  2:25  10 − 6 m  1  10 − 3 ¼ 6π  2:25  0:1  10 − 12 ¼ 4:24  10 − 12

2 mm  0:3 6 s

kg 10 − 3 m  0:1  m-s s

kg-m s2

kg-m ¼ 4:24 pN ðpiconewtonÞ: s2

The force on the smaller particles of similar χp will be correspondingly smaller by the volume ratio (equation (7.3.248c)) of the particles.

26

Dynabeads, Dynal Biotech, Oslo, Norway.

The manner in which such free-flow magnetophoresis (FFM) is applied in practice is related to the needs of life sciences vis-à-vis separation of native biological particles. These native biological particles by themselves do not lend to practical separation by FFM. However, they can be fixed to magnetic or magnetizable microspheres having specific antibodies (Abs) on their surfaces via antigen (Ag)–Ab bridges from antigens on the surfaces of the biological particles. Two examples of this strategy are illustrated in Figure 7.3.23 (Gee, 1998). A common form of the magnetic or magnetizable microspheres consists of superparamagnetic, polystyrene based particles 2.0–4.5 μm in diameter having iron oxide homogeneously distributed throughout the particle (Hausmann et al., 1998); these are called Dynabeads (as mentioned earlier). For the separation of cells and organelles, the FFM method has been identified as Continuous Immunogenic Sorting. 7.3.5.2

High-gradient magnetic separation

A high-gradient magnetic separator contains a loosely packed bed of ferromagnetic wool of stainless steel; there is one inlet at the bottom for feed fluid to come in and one outlet at the top for treated feed fluid to go out (Figure 7.3.24). A liquid feed containing paramagnetic particles to be removed enters the device from the bottom. A uniform magnetic field is applied in the same direction

7.3

External force field based separation: bulk flow perpendicular to force

Cleaned stream containing nonmagnetics Water stream for backflushing Electromagnetic coils

Iron enclosure

Stainless steel wool Magnetics obtained during backflushing Feed stream Figure 7.3.24. Schematic of a HGMS unit for cyclic operation.

as that of the fluid flow. Around each strand of the stainless steel wool, the uniform magnetic field becomes highly distorted, creating a nonuniform magnetic field whose gradient can be of the order of 105 tesla/m. Particles in the size range of microns are forced to the stainless steel strands and retained there. The nonmagnetic particles pass through with the fluid. The matrix of stainless steel wool will soon be covered with paramagnetic particulates captured from the fluid, at which time the liquid feed flow is stopped. A wash water stream is passed. Then the magnet is turned off and a water stream is passed in the direction opposite to that of the feed stream; this stream carries away the paramagnetic particles, which fall off the strands of stainless steel wool. After these particles are removed, the magnet is turned on and the feed liquid starts flowing in. The process thus operates cyclically; its operation is very similar to the operation of the packed-bed adsorber in pressureswing adsorption (Figure 7.1.13(a)). The only difference here is that, instead of the adsorption process, we have magnetic force based particle capture in a direction perpendicular to the main flow direction. Shutting off the magnetic force field leads to release, as in desorption processes. This HGMS technique is widely employed in industry for the separation of paramagnetic particles from a stream with or without nonmagnetic particles. This technique can also be employed to remove paramagnetic particles from a gas stream (Gooding and Felder, 1981). The stainless steel wool matrix is very open, with a porosity around 0.95. However, it is important that the matrix openings are not very large and that the wire dimensions are quite small since the magnetic field variation around a wire is strongest over a range of distances very close to the wire (Birss and Parker, 1981) and of the order of the particle diameter. We will now provide a very constricted glimpse into the modeling of such a separation device. The particle trajectory equation (see equation (6.2.45) in such a device is as follows:

mp

dv dt

inertial force

¼ F mag

þ F gr

651

− F drag :

magnetic gravitational drag force force force

ð7:3:255Þ

The strands in the stainless steel wool may be modeled as a thin cylinder of radius rw. The random configurations of the wires are ignored at this time; they may be considered as the sum total of interactions involving three arrangements: longitudinal (uniform magnetic field H m 0 perpendicular to the wire axis and particle velocity parallel but opposed); transverse ( H m 0 perpendicular to the wire axis and perpendicular to the particle velocity, which is perpendicular to the wire axis); axial ( H m 0 perpendicular to the wire axis as in the longitudinal arrangement, but particle velocity parallel to the wire axis). One could represent a general configuration of the particle-to-wire interaction as shown in Figure 7.3.25 (a) (Oak, 1977; Birss and Parker, 1981; Liu and Oak, 1983). (The physical configuration is similar to that of a particle flowing with air around a solid fiber of radius rf in depth filtration; See Figure 6.3.9A.) Based on such a configuration and the analyses of Birss and Parker (1981) and Oak (1977), we can provide the following expressions for the rcomponent and the θ-component of the individual force terms on a spherical particle of radius rp and volume Vp in equation (7.3.255), after neglecting the gravitational and inertial force components vis-à-vis the other forces. In the following expressions, Ms stands for the saturation magnetization of the wire, i.e. the value of Mw (for example, in equation (3.1.23) when it is saturated). For the: r-component:     1 m 4M s r 3w M s r 2w m F mag cos 2θ þ ðχ −χ ÞV ¼ − H ; μ p 0 pr 2r w r 3 2r 2 2 0 p s

ð7:3:256aÞ    dr F drag − v∞ cos ðθ−γÞð1− ðr w =rÞ2 Þ : ð7:3:256bÞ pr ¼ 6πμr p dt

As long as (χp − χs) is positive, the radial magnetic force F mag is attracted pr on the particle is negative; i.e. the particle  to the ferromagnetic wire as long as H m 0 cos 2θþ ðM s r 2w =2r 2 ÞÞ > 0; the force magnitude is largest for θ ¼ 0 (for γ ¼ 0). The configuration becomes that of flow parallel to force (Section 6.3.1.4). There are regions around the wire where the particle is repulsed by the wire (Birss and Parker, 1981). For diamagnetic particles ((χp − χs) is negative), the attractive force is largest for θ ¼ π/2. For the θ-component:   1 4M s r 3w H m mag 0 sin 2θ ; ð7:3:257aÞ F pθ ¼ − μm 0 ðχ p −χ s ÞV p 3 2r w r 2   

r  2    dθ w drag : r þ v∞ sin ðθ −γÞ 1 þ F pθ ¼ 6 π μr p r dt

ð7:3:257bÞ

652

Bulk flow perpendicular to the direction of force

(a)

y Particle accumulation H0 rp r S, steel wire

rw

Particle

q v∞

g

Free stream velocity

z Ms Uniform magnetic field

H0

Particle

(b)

Particle trajectory

2 (y/rw)

(ylim/rw) = 2.39

1

mag

(Upz /v∞) = 10

1

2

4

3

5

6

(z/rw) S, steel wire Figure 7.3.25. (a) Configuration of the paramagnetic particle, magnetized stainless steel wire and a particle; coordinate system for analysis (Birss and Parker, 1981; Liu and Oak, 1983). (b) The trajectory of a particle being captured. (After Watson (1973).)

Therefore the governing equations of particle motion in the r-direction and the θ-direction are as follows. For the r-component:     1 4 M s r 3w M s r 2w m − μm cos 2θ þ ðχ − χ Þ V H p 0 2 rw r3 2 r2 2 0 p s    dr − v∞ cos ðθ − γÞ ð1 − ðr w =rÞ2 Þ ¼ 0: − 6 π μ rp dt ð7:3:258aÞ

This leads to the following for the r-component equation from equation (7.3.255):    dr rw 2 ¼ v∞ cos ðθ − γÞ 1 − r dt    μm 4M s r 3w M s r 2w 0 ðχ p − χ s Þ V p Hm : − 0 cos 2θ þ 3 2 2r w r 2r 12π μ r p ð7:3:258bÞ

Similarly for the θ-component:   1 4M s r 3w H m 0 sin 2θ − μm 0 ðχ p − χ s Þ V p 2 2r w r 3   

r 2  dθ w ¼ 0; − 6 π μ rp r þ v∞ sin ðθ − γÞ 1 þ r dt

ð7:3:259aÞ

leading to r



r  2  dθ w ¼ −v∞ sin ðθ − γÞ 1 þ dt r −

  μm 4M s r 3w H m 0 ðχ p − χ s Þ 0 sin 2θ: ð7:3:259bÞ Vp 12π μr p 2r w r 3

Dividing equation (7.3.258b) by equation (7.3.259b), we get the particle trajectory equation:

7.3

External force field based separation: bulk flow perpendicular to force

   μm ðχ − χ Þ V   p 2 p s 4M s r 3w M s r 2w − 012π μ r v Hm cos ðθ − γÞ 1 − rrw 0 cos 2θ þ 2 r 2 2r w r 3 1 dr p ∞

 ; ¼  2  μm0 ðχ p − χ s Þ  4M s r 3w H m r dθ 0 sin 2θ − 12π μ rp v∞ V p − sin ðθ − γÞ 1 þ rrw 2r w r 3 1 dr ¼ r dθ

1−

 r w 3  r w 2  2μm ðχ p − χ s Þ r 2p M s H m 0 Ms cos 2θ þ 2H cos ðθ − γÞ − 0 9 r μ v m r r w ∞ 0 :

2 M Hm   r 2   2μm ðχ − χ Þ r s 3 p s p 0 rw sin ðθ − γÞ − 0 9 r v μ sin 2θ − 1 þ rw r

r w 2  r

653

ð7:3:260aÞ

ð7:3:260bÞ

w ∞

From definition (7.3.260) for the magnetic migration velmag ocity U pzt , we have ! 2 m 2 2 ðχ p − χ s Þ d H m mag 0 μ0 r U pzt ¼ 2 μ 9 p dz ¼ mag

U pzt ¼

2 2 μm Hm dH m 0 r 0 0 ðχ p − χ s Þ ; 9 p μ dz

ð7:3:261aÞ

m m m 2 2 2μm 2 μ0 r p ðχ p − χ s Þ H 0 M s 0 r p ðχ p − χ s Þ H 0 M s : ¼ 9 rw 9 μ rw μ ð7:3:261bÞ

angle between θ ¼ 0 and 90 . When θ ¼ 0 and γ ¼ 0, it may appear as a case of bulk flow parallel to force (as in Section 6.3.1.4). However, if θ 6¼ 0, there is always a component of force perpendicular to the bulk flow (as long as γ ¼ 0; it is true even for γ 6¼ 0 as long as θ 6¼ γ). But the overall configuration continues to be particles taken out of the bulk flow in the perpendicular direction and deposited in the woolen medium.

7.3.5.2.1 Performance of a HGMS filter We will develop now an expression for the fractional capture of the paramagnetic particles as the liquid flows through the Using this definition, we can rewrite the expression HGMS filter bed of ferromagnetic wool. We assume that (7.3.260b) as follows: the trajectory model has yielded a value of the y-coordinate

    mag     U pzt r w 3 rw 2 M s rw 2 of the limiting particle trajectory, y ℓim , equal to b (see cos 2θ þ 1− cos ðθ −γÞ− r r 2H m r v∞ 1 dr o ¼ :

   Figures 7.3.25(b) and 6.3-9A). If the void volume fraction mag   U 2 3 r dθ sin ðθ −γÞ − vpzt∞ rrw sin 2θ − 1 þ rrw of the bed is ε, then (1 − ε) is the bed volume fraction ð7:3:262aÞ occupied by strands of the ferromagnetic wool. For a length dz in the feed flow direction of the filter bed of unit þ Defining a nondimensional radial coordinate r ¼ (r/rw), cross-sectional area perpendicular to fluid flow, the length we get 

    mag 1 3 cos 2θ þ 2HMm srþ2 1 − ð1=r þ Þ2 cos ðθ − γÞ − U pzt =v∞ rþ 1 dr þ 0

  : ð7:3:262bÞ ¼   mag 1 3 r þ dθ − 1 þ ð1=r þ Þ2 sin ðθ − γÞ − U pzt =v∞ sin 2θ rþ

The solution of this trajectory equation can be developed numerically. What is of interest, however, is the ycoordinate of the limiting trajectory of a particle which will be captured: if the particle starts out at a y-value in the range ðy lim =r w Þ  ðy=r w Þ  − ðy ℓim =r w Þ, where the magnitude of the y-value of the limiting trajectory is jyℓim j ðat z ¼ ∞Þ, then its trajectory will surely hit the stainless steel wire (r ¼ rw) at differing values of θ. Luborsky and Drummond (1975), as well as Liu and Oak (1983), have provided a few simplified expressions for yℓim : pffiffiffi mag mag ðyℓim =r w Þ ¼ ð3 3=4Þ ðU pzt =v∞ Þ1=3 if ðU pzt =v∞ Þ > 1;

ð7:3:263aÞ p ffiffi ffi mag mag ðyℓim =r w Þ ¼ ðU pzt =2v∞ Þ if ðU pzt =v∞ Þ  2: ð7:3:263bÞ

Watson (1973) also provided the last result. This limiting trajectory coordinate y ℓim far away from the stainless steel wire is similar to the coordinate b in Figure 6.3.9A except the particle trajectory does not follow the streamline at all near the wire (Figure 7.3.25(b)). A few general features are to be noted here. In Figure 7.3.25(a), the particle may come in and hit the wire at any

of the ferromagnetic wire in the wool is ðð1 − εÞdz=π r 2w Þ. As in Figures 7.3.25(b), and 6.3.9A and 6.3.9B, a total width of 2y ℓim ð¼ 2bÞ in the y-direction and unit length of the wire provides a fluid flow cross-sectional area of 2y ℓim per unit wire length: all particles coming in through this capture cross section are ultimately captured by the stainless steel wires. Therefore the total capture cross section for the filter bed of thickness dz is ð2ð1 − εÞy ℓim dz=π r 2w Þ (assuming that there are no overlaps between the capture cross sections for different location of the wire). For a velocity v∞ of the fluid entering the bed and a feed fluid particle number concentration of n(rp)drp (the total number of particles of size rp to rp þ drp per unit fluid volume), the number of particles removed per unit time per unit flow cross-sectional area is given by

v∞  nðr p Þjz dr p  2ð1 − εÞ y ℓim dz ¼ v∞ ε ð − dðnðr p ÞÞdr p jz ; πr 2w ð7:3:264aÞ where the right-hand side provides the corresponding total change in particle flow rate over the distance dz (see the

654

Bulk flow perpendicular to the direction of force

development for equation (6.3.43)). We can rearrange this relation for particles of size rp as follows:

− dnðr p Þ=nðr p Þjz ¼ 2 y ℓim ð1 − εÞ=ε π r 2w dz: ð7:3:264bÞ

Integrate from z ¼ 0 to z ¼ L, the bed length, to obtain



ℓn nðr p Þjz¼0 =nðr p Þjz¼L ¼ 2y ℓim ð1 − εÞ=ε π r 2w L;

ð7:3:264cÞ

ðnðr p Þjz¼0 =nðr p Þjz¼L Þ ¼ expf 2yℓim ð1 − εÞ=ε π r 2w Lg: ð7:3:264dÞ

When all particles have the same size rp, r max ð

r min

2r

nðr p Þjz¼0 dr p ¼ 4

max

ð

r min

3

nðr p Þjz¼L dr p 5 expf 2y ℓim ð1−εÞ=επr 2w Lg

) Nðr p Þjz¼0 ¼ Nðr p Þjz¼L ) Nðr p Þjz¼L ¼ Nðr p Þjz¼0

ð7:3:264eÞ

expf 2y ℓim ð1−εÞ=επr 2w Lg

ð7:3:264fÞ

expf− 2yℓim ð1−εÞ=επr 2w Lg; ð7:3:265Þ

where N(rp) represents the total concentration of particles, all of which have the same size, rp. If there is a particle size distribution, then we will obtain, instead of (7.3.264e), Nðr min ; r max Þjz¼L ¼ ¼

r max ð

r min

r max ð

r min

pffiffiffi 100–400). For values of the parameter less than 2, we mag know that ðyℓim =r w Þ ¼ 0:5ðU pzt =v∞ Þ (equation (7.3.263b)). For higher values of the parameter, ðyℓim =r w Þ keeps on mag increasing and can be as high as 7 for ðU pzt =v∞ Þ ¼ 100. mag Let us, for the sake of illustration, take ðU pzt =v∞ Þ to be 1. Then   Nðr p Þjz¼L =Nðr p Þjz¼0 9 8 < 2  0:5  1  r  0:5  51= w ¼ exp − ; : 0:95  π  r 2w 9 8 = < 0:05  51 ¼ exp − − 4 : 0:95  π  45  10 ; 9 8 < 5  51  102 = ¼ exp f − 189g: ¼ exp − : 0:95  π  45 ; This result shows that the exiting fluid stream will have very few of the particles left; almost all of them will be captured by the bed. Performances of HGMS units have generally been found to support such conclusions. The influence of the captured particle buildup on the stainless steel wires is important; see Liu and Oak (1983) for a detailed analysis.

7.3.6 nðr p Þjz¼L dr p



nðr p Þjz¼0 expf− 2yℓim ð1 −εÞ=επ r 2w Lgdr p ; ð7:3:266Þ

since y ℓim depends on the particle size rp. Some comments are needed to qualify the results developed above. As shown in Figure 7.3.25(a), the magnetic field H0 is perpendicular to the stainless steel wire. However, about one-third of the strand length is parallel to H0. Therefore the argument of the exponential in equation (7.3.265) should be multiplied by 2/3 (Watson, 1973). Further, any change in the bed void volume fraction ε due to particle buildup on the wires is neglected. Ignoring additional assumptions inherent in the above analysis, let us make an estimate of the particle concentration reduction   Nðr p Þjz¼L ¼ Nðr p Þjz¼0 achieved in a bed of length L ¼ 51 cm (say) following Liu and Oak (1983). The bed void volume fraction ε is around 0.95; therefore (1 − ε) ~ 0.05. An average wire radius is around 45 μm (¼ rw). The value of yℓim depends very mag strongly on the ratio ðU pzt =v∞ Þ; the value of this last parameter in practice is generally greater than 0.5 (the minimum value needed for particle buildup (Liu and Oak, 1983)) and can go to quite high values (as much as

Radiation pressure – optical force

It is known (equation (3.1.47)) that radiation pressure from continuous wavelength (cw) visible light is known to apply a force on and accelerate freely suspended particles in the direction of the light. This optical force (radiation pressure force) induced migration of particles, especially colloidal particles, is called photophoresis (PP). Such forces have very little impact on larger particles. However, micron and submicron particles can encounter substantial acceleration. Different refractive indices of transparent particles vis-à-vis the surrounding liquid, as well as light absorption, are the bases of this force. Helmbrecht et al. (2007) have demonstrated that a simple crossflow setup, in which a laser beam illuminated perpendicular to the mean flow direction of the fluid containing suspended microparticles causes a lateral shift in different particles. Different particles follow different trajectories, as in other cases of bulk flow perpendicular to the direction of the force. To describe such trajectories, consider the net photophoretic force F rad on a particle of radius rp by a light beam (whose waist size wa is much larger, i.e. rp k B . Show that the two key equations corresponding to equations (7.1.51a) and (7.1.51b) are, in this case, dz β A vz ¼ dt 1 þ ðβ − 1Þy

Table 7.P.1. O2

N2 −1

ðqAσ Þmax (mmol/g) bAℓ (atm ) LiXðSi=Al ¼ 1Þ NaX

2.653 0.982

0.946 0.901

ðqBσ Þmax (mmol/g) bBℓ (atm−1) 2.544 0.276

0.086 0.624

658

Bulk flow perpendicular to the direction of force Table 7.P.2. C i2 , enzyme concentration in eluent exiting the column (in suitable units)

Volume of eluent passed (in liter), V

15 6.5

2.0 2.2

and dy ðβ − 1Þ ð1 − yÞy ¼ dP ½1 þ ðβ − 1Þy P where β ¼ ðβA =βB Þ; βA and βB are defined in equation (7.1.52a). (Suggestion: Work with the mass balance equations for species A and total moles (i.e. A plus B); then obtain the following equation: ∂y β A vz ∂y ðβ − 1Þ ð1 − yÞy 1 dP þ ¼ 1 þ ðβ − 1Þy ∂z ∂t 1 þ ðβ − 1Þy P dt and employ the method of characteristics.) 7.1.11

For liquid–solid adsorption based chromatography, two separate expressions were developed for the chromatographic output profile C i2 ðz;tÞ by two methods: See (7.1.98b) and (7.1.102a). Show that these two expressions are identical.

7.1.12

Elution based purification of an enzyme in an adsorbent column yielded significant data, some of which are listed in Table 7.P.2. It is known that the C i2 value of 15.0 was the maximum. Determine the value of σ iV (liter) and the percent recovery of enzyme when the eluent volume passed is 2.2 liter. The elution profile may be assumed to be Gaussian.

7.1.13

The elution profile of praseodymium chloride (PrCl3) by a citrate buffer at pH ¼ 3 from a Dowex 50 resin column (originally studied by Mayer and Tompkins (1947)) can be described by equation (7.1.102a) (Lightfoot et al. (1962) have described it in detail) in the presence of cerium (CeCl3); the sorption equilibria are linear and noninterfering. (1) From an elution profile of praseodymium, show how you can determine the value κi1 and Pez;eff for praseodymium for the given device and system. (2) Calculate the values of κi1 and Pez;eff for praseodymium when you have the following experimental values available: the maximum effluent concentration of C i2 ¼ 0:0505 ðC 0i2 V 0 =ε L Sc Þ is reached at an elution volume V ¼ 113ε L Sc; ε ¼ 0.4. (3) Calculate the plate number. (Ans. (2) κi1 ¼ 74.6; Pez,eff ¼ 409; (3) N ¼ 204.5.)

7.1.14

In the recovery of a product from a clarified fermentation broth, the adsorption time needed is 80 minutes. In a given cycle, the individual steps of washing, elution and regeneration (see Section 7.1.6) require the same amount of time. The data were acquired in a small laboratory column having a flow rate of x cm3/min. This process has to be scaled up for ten times higher production rate of the bioproduct using the same overall cycle time and individual step times. Consider three cases: (1) same pressure drop in both columns; (2) same σiV in both columns during elution; (3) same plate number. Determine for each case a relation between the lengths and diameters of the two columns. Assume that the adsorbent particle diameter remains unchanged. The value of Di;eff may be assumed to be the same in the two columns.

7.1.15

Dai et al. (2003) obtained the following linear isotherm behavior for the two proteins myoglobin and αlactalbumin (α-LA) with the ion exchange resin DEAE-Sepharose fast flow, at low concentrations in 20 mM Tris-HCL buffer solution (pH ¼ 8.5): qi1 ðmg=mliterÞ ¼ K i C i2 ðmg=mliterÞ; K i ðmyoglobinÞ ¼ 43:3; K i ðα-lactalbuminÞ ¼ 31 099:

Problems

659

Myoglobin elution was easily accomplished using 0.05 M NaCl, whereas α-LA elution needed 0.5 M NaCl solution; α-LA was quite tightly bound. which elution procedure, gradient or stepwise, would you recommend if it is known that small amounts of α-LA start coming out at salt concentrations less than 0.5 M? 7.1.16

Size exclusion chromatographic separation of a protein mixture from salt, as well as the separation of two proteins present, in the mixture, are being carried out in a column of diameter 3 cm and length 30 cm. The column void volume fraction is 0.39. The gel particle void volume fraction is 0.6. The values of the partition coefficient κim of the two proteins 1 and 2 are κim j1 ¼ 0:8; κim j2 ¼ 0:1, with protein 2 being much larger than protein 1. (1) Estimate the values of the elution volumes for the salt, protein 1 and protein 2. (2) What would be the corresponding value for blue dextran if it cannot access the pores in the gel particle at all? (3) If the pore diameter of the gel is estimated to be around 20 nm, estimate the diameters of protein 1 and protein 2, assuming that they are spherical and the pores are cylindrical.

7.1.17

Many organic synthesis reactions employ metal-catalyzed processes. As a result, the solution containing the reaction products or waste streams from the synthesis process will contain residual metals in solution. Such residual metals have to be removed. In one organic synthesis process, the organic solution/aqueous waste stream contained residual metals such as Cu, Al and Ni. It is proposed to remove these metals via functionalized macroporous polystyrene resin beads in a packed-bed format. The beads are crosslinked enough to undergo only limited swelling in organic solvents. Suggest appropriate functional groups in the resin to remove such metals in solutions.

7.1.18

Consider capillary electrochromatography based separation of two enantiomers identified as Q1 and Q2 respectively. A chiral complexing agent (C) is bonded to the stationary phase in the capillary. The enantiomers Q1 and Q2 in the mobile phase (subscript m) react differently with the stationary phase complexing agent (C with subscripts): Kf1

Q1m þ Cs ↔ ½Q1 Cs ; Kf2

Q2m þ Cs ↔ ½Q2 Cs ;

Kf1 ¼

½Q1 Cs ; Q1m Cs

Kf2 ¼

½Q2 Cs : Q2m Cs

The enantiomers also partition between the mobile phase and the stationary phase in the free form: K1

Q1m ↔ Q1s ;

K2

Q2m ↔ Q2s :

m Assume that K1 ¼ K2, μm 1 ¼ μ2 for the enantiomeric pair 1 and 2. Develop an expression for α12 jt R1 , the separation factor based on the retention time t R1 . It is known that one enantiomer forms the complex [QC]s much more readily than the other one, so that Kf 1 and Kf 2 are far apart. Further, the mobile-phase volume and stationary-phase volume in the column are Vm and Vs respectively. Assume that the Cs concentration is essentially very high and unaffected by the reactions. (Ans.  . Vm ðK f 1 þ K 1 Þ : α12 jt R ¼ ðV M =V s Þ ðK f 1 − K f 2 Þ 1 þ 1 Vs

7.1.19

In problem 7.1.18, there was a stationary phase containing a chiral complexing agent C which reacted differently with the different enantiomers. Suppose in a column in which the buffer solution flowing vertically has a chiral selector molecule C (sulfated β-cyclodextrin chiral selector) added as the two enantiomers flow with the buffer. The chiral selector binds to form enantiomer–ligand complexes with each enantiomer. However, these complexes have different electrophoretic mobilities. If there are two electrodes at the two ends of the column as shown in Figure 7.P.1, then, by selecting an appropriate hydrodynamic velocity in the counterflow at the column bottom, one could separate the fast enantiomer-ligand complex (mobility μm iF ) from the slow enantiomer–ligand complex (mobility μm ). iS (1) Write down the electrophoretic velocities of the two complexes as functions of their ionic mobilities. (2) Obtain the hydrodynamic velocities in the four sections of the column shown. The column cross-sectional area is Sc. The various volumetric flow rates entering or leaving the column are shown in Figure 7.P.1.

660

Bulk flow perpendicular to the direction of force

QR − Electrode 4 4 Product 1 Q1

3

2 Product 2 Q2

Feed in Qf

1 + Electrode

Counterflowing buffer Qb Figure 7.P.1.

(3) If both enentiomer–ligand complexes have negative charges, identify which of the two product streams will have the faster complex. Identify the criteria vis-à-vis the hydrodynamic velocities in a given section. Neglect competitive binding aspects. Assume uniform electrical field gradient. 7.2.1

Air is being separated into nitrogen-enriched air (NEA) and oxygen-enriched air (OEA) in a crossflow membrane separator (See Figure 7.2.1(a)). The permeate side total pressure is very low so that the pressure ratio γ ! 0. The nitrogen mole fraction in the NEA is 0.96. Calculate the highest and the lowest oxygen mole fractions achieved in the permeate at different locations in such a permeator for two membranes, one from cellulose acetate ðαO2 − N2 ¼ 6Þ and the other from an unknown glassy polymer ðαO2 − N2 ¼ 9Þ.

7.2.2

To remove the acid gas CO2 from natural gas at 500 psia, a polymeric membrane module is being used. The permeate side pressure may be assumed to be quite low. The feed gas has 10% CO2. The purified natural gas should have only 2% CO2. The permeability values (in barrer) for CO2 and the dominant consitutent of natural gas CH4, are 200 and 5 units, respectively. Calculate the highest and lowest values of the CO2 mole fraction in the permeate side if you can assume pure crossflow in the module. Identify the locations. What will be the values of CH4 in the permeate at these locations? Assume: a binary CO2–CH4 system.

7.2.3

In crossflow gas permeation, we have identified in Figure 7.2.1(c) a tube-side feed crossflow hollow fiber module. Unlike flat membrane systems, the radial cross-sectional area for permeation through the hollow fiber wall (inner radius r1, outer radius r2) increases as the radius increases. Develop the following expression for the molar permeation rate of species i through a hollow fiber of length Δz:   Qim ðpi1 − pi2 Þ; ðΔzÞπd ℓm δm where dℓm ¼

ð2r 2 − 2r 1 Þ

 ; 2 ln 2r 2r 1

Qim ¼ Dim Sim ;

δm ¼ ðr 2 − r 1 Þ:

You are given: that C im jr1 ¼ Sim jr1 pi1 ; C im jr2 ¼ Sim jr2 pi2 ; Sim ¼ Sim jr 1 ¼ Sim jr2 . Assume ideal gaseous mixture behavior. Employ N ir ¼ J ir from Table 3.1.3C; neglect U i ; the partial pressure of species i in the feed gas at radius r1 is pi1; the partial pressure of species i in the permeate gas at radius r2 in crossflow is pi2; d ℓm is the logarithmic mean diameter of the hollow fiber membrane. 7.2.4

This problem is concerned with removing small amounts of solvent vapor (volatile organic compounds, VOCs) from an air stream through a hollow fiber membrane which is highly selective for the VOC over air. The feed air stream moves through the bore of the hollow fiber, as in Figure 7.2.1(c). The permeate side in crossflow is maintained under high vacuum such that pif >>pip, where subscripts f and p refer to the feed and the permeate,

Problems

661

respectively. The change in the total feed gas flow rate with the permeator length z may be neglected. Employing the notation of Figure 7.2.1(a), develop an analytical relation between the length L of the permeator needed to reduce the VOC mole fraction in air from xif (feed) to xi2 (concentrate, reject). Assume a negligible feed gas pressure drop along the length of the hollow fiber module. Employ the result of Problem 7.2.3. You are given that (Qim/δm) for VOC, i ¼ a exp(bPf xif ), i.e. the VOC permeability varies exponentially with the feed-gas mole fraction of VOC. 7.2.5

For a crossflow gas permeator and a binary system, adopt the Naylor–Backer approach and the starting equation (7.2.10). Define Wt2 and xA2 as the molar gas flow rate and mole fraction of species A at any location in the feed side of the permeator; let the values of Wt2 and xA2 at the permeator exit be W ot2 and x oA2 . Develop a relation somewhat similar to the result (7.2.12) relating W t2 to W ot2 as a function of x A2 , x oA2 and αAB, where the more permeable species is A in the binary mixture of species A and B.

7.2.6

Consider the spiral-wound module for reverse osmosis desalination described in Example 7.2.3. Determine the fractional water recovery if the feed brine has 10 000 ppm salt. All other conditions are as in the Example 7.2.3. What will be the fractional recovery if the feed brine has 20 000 ppm salt?

7.2.7

Calculate the value of the average permeate salt concentration for the two cases of spiral-wound RO desalination given in Problem 7.2.6: feed brine has 10 000 ppm salt; feed brine has 20 000 ppm salt. Make comments about the possible errors in calculation. You are given that the salt permeation parameter (Dim κim/ δm) ¼ 2.4  10−5 cm/s.

7.2.8

Consider reverse osmosis desalination in a spiral-wound module for a feed having negligible osmotic pressure. If the fractional water recovery, re, is such that the osmotic pressure of the concentration from the reverse osmosis process still has a negligible osmotic pressure vis-à-vis the liquid pressure, derive the result (7.2.44), i.e. Lþ ¼ re, from simple mass balance considerations. Assume that the membrane has a very high rejection of salt.

7.2.9

The Colorado River has become brackish at the level of 600 ppm. This water is passed at 34 atm (gauge) through a spiral-wound module. The gap between the membranes lining the brine feed channel is 1.1 mm. The membrane length in the brine flow direction is 70 cm. The length of the unwrapped membrane in the permeate flow direction is 150 cm. The pure water permeability constant for the membrane exposed to the feed brine at 20 C is 66.15  10−6 gmol/cm2-min-atm. The osmotic pressure of the brine solution is provided via π f ¼ 0:0115 (ppm)f , where π f is in psi and the salt concentration is in ppm. The average brine velocity in the brine channel is 320 cm/min. Estimate the fractional water recovery by assuming a highly rejecting membrane and any other appropriate assumption. (Note: You do not have any method to estimate the value of k iℓ for this spiral-wound module.)

7.2.10

Milk is to be concentrated by ultrafiltration in a hollow fiber module having the following dimensions and characteristics: number of hollow fibers, 500; fiber internal diameter, 800 μm; fiber length, 60 cm. The milk will flow on the tube side at an average flow velocity of 90 cm/s. The relevant properties of milk are: viscosity, 0.8 cp; density, 1.02 g/cm3; diffusion coefficient of proteins in milk, 6.5  10−7 cm2/s; milk protein contents, 2.8% w/v; gel concentration Cigel, 25% w/v. (1) Determine the value of the water flux through this module using the length-averaged Leveque solution. (Ans. 25.3 liter/m2-hr). (2) Determine the fractional feed water removal through this module. Justify the use of the form of Leveque solution recommended for use. (Ans. 0.0235.)

7.2.11

Consider a protein solution having the characteristics of the feed solution in Examples 7.2.5 and 7.2.6. This solution has to be desalted via continuous diafiltration (Section 6.4.2.1) in the hollow fiber unit of Example 7.2.6. The diafiltration configuration is illustrated in Figure 7.2.5(e). For 2000 liters of feed protein solution to be desalted, determine the crossflow membrane area required if the following levels of salt removal are desired in 2 hours: (a) 99% of the salt removed; (b) 99.9% of the salt removed; (c) 99.99% of the salt removed. It is known that salt has zero rejection. Specify the volume of buffer solution required in each case.

7.2.12

Consider Example 7.2.5 dealing with the concentration of a dilute solution of BSA. The diffusion coefficient of BSA has been found to vary as Diℓ ¼ D0iℓ

tanhð0:159 ðρsolu  100ÞÞ : 0:159 ðρsolu  100Þ

662

Bulk flow perpendicular to the direction of force It is known further that the Leveque solution (see Example 7.2.6) is valid here. If all other conditions in the problem are similar to those of Example 7.2.6, determine the fractional reduction in solvent flux as the BSA solution is concentrated from 2 g/100 cm3 to 15 g/100 cm3.

7.2.13

For the UF based protein concentration example provided in Example 7.2.5, develop a graphical illustration of (vz(z)javg/vs0) against C þ isbL as per equation (7.2.86).

7.2.14

Consider the batch ultrafiltration configuration of Figure 7.2.5(d). We would like to determine the time required to go from an initial volume Vf0 of the solution to the final volume Vfe ( 1.0. Assume that the Sherwood number ðk iℓo di =Diℓ Þ for the mass-transfer process is such that Sh ¼ ðk iℓo di =Diℓ Þ / ðReÞb1 . (1) Show that, if gel polarization exists,

7.2.15

b1 d bi 1 − 1 ; vs0 / vz;avg

where vz;avg is the bulk average velocity and di is the tube internal diameter. (2) Define ep, a measure of power consumption, as ep ¼

ΔP loss Qf power dissipated in circulating fluid ¼ : volume rate of permeate production vs0 Am

Employ the Blasius relation (6.1.3a,b) for the pressure drop in turbulent flow through a smooth pipe and assume Am is the tubular membrane surface area. If we require equal flux performance from tubes of diameters d1 and d2, show that ðep1 =ep2 Þ ¼ ðd1 =d 2 Þð− 3b1 þ2:75Þ=b1 : 7.2.16

It has been observed that particle flux expressions developed based on shear-induced particle diffusivity describe the observed solvent flux through a microfiltration membrane much better than those based on Brownian diffusivity of a particle. For the following system properties, determine the ratio of the solvent fluxes based on shear-induced particle diffusivity and Brownian diffusivity. You should employ the particle volume fraction based solvent flux expression based on the gel polarization model used in ultrafiltration (equation (7.2.72)): vs ¼ ðDp =δp Þ ℓn ðϕw =ϕs Þ: You are given the following: ϕs ¼ ϕp ¼ 0:1; r p ¼ 10 − 4 cm; γ_ ¼ 103 s − 1 ; μ ¼ 10 − 2 g=cm-s; kB ¼ 1:380  10 − 16 g-cm2 =s2 -K; temperature ¼ 25  C.

7.2.17

In a ceramic tubular microfilter of diameter 0.2 cm and length 30 cm, a suspension having a particle volume fraction of 0.1 (¼ ϕs) and 1 μm particle radius is flowing at 25  C. The suspension viscosity μ0 is 1 cp (¼ 10−2 g/ cm-s); the wall shear stress is 10 g/cm-s2. The clean membrane resistance is Rm ¼ 109/cm. The applied pressure difference is 10 psi (7  105 g/cm-s2). The filter cake porosity εc ¼ 0:4. (1) Determine whether the cake layer resistance controls or the membrane resistance controls. (2) Obtain an estimate of zcrit and determine the ratio (L/zcrit).

Problems

663

7.2.18

In a ceramic tubular microfilter of radius 0.13 cm and length 37 cm, a suspension having particles of radius 0.32 μm flows; the value of ϕs ¼ 0:00015. The wall shear rate at inlet γ_ 0 ¼ 2400 s − 1 . The observed flux in the microfilter at the inlet, vs0 , is 0.0034 cm/s. It is known that β >> 1. Calculate the value of vs javg for this microfilter. The values given above are valid for 25  C.

7.2.19

Many aqueous solutions used as a feed for separations contain a mixture of particulate matter such as cells, cell fragments, macromolecules and low molecular weight solutes. These are found in the food industry, the biopharmaceutical industry, etc. One could use a combination of reverse osmosis (RO) membrane devices, ultrafiltration (UF) units and microfiltration (MF) membrane modules. What order of feed solution processing should you follow: RO ! UF ! MF or MF ! UF ! RO? Why?

7.2.20

Employing the correct sequence of crossflow membrane processes from Problem 7.2.19, suggest a solution for the following processing problems. (1) Consider the fractionation of a mixture of the following proteins and other solutes: aldolase (MW, 142 000), ovalbumin (MW, 45 000), cytochrome C (MW, 12 400), cyanocobalamin (MW, 1355). Draw an appropriate sequence of crossflow membrane processes. For ultrafiltration based processes, identify an appropriate membrane for the particular separation from Table 6.3.8. (2) Animal blood from slaughterhouses can be fractionated to recover constituent components: red blood cells (0.8 μm), bacteria (> 0.2 μm), virus (MW, 2–3  106), serum albumins (MW, 60 000 to 100 000). Draw an appropriate sequence of crossflow membrane processes. Identify the membrane pore size or MWCO selected in each process. Specify which species is concentrated in the reject stream and which species is primarily in the permeate.

7.2.21

Rotary vacuum filtration of a fermentation broth is to be carried out at the rate of 5000 liter/hr. The properties of ^ cδ are provided in Example 6.3.7. The vacuum the dilute suspension, e.g. ϕs, ρs, μ and the cake properties, ϕc ;R based applied ΔP is 55 cm Hg. Assume an incompressible cake whose resistance dominates filtration resistance. The filter cycle time is 80 s; the filtration time duration is 20 s. Determine the total filter area Am of the rotary filter needed for this operation. The temperature of operation is the same as that in Example 6.3.7 so that the physical properties are identical.

 Determine the value of the normalized volume concentration of particles C vp =C ivp at 3 hours from the start at bed locations at a distance z ¼ 30 and 70 cm, respectively, from the inlet of a granular filter. You have been provided the following information about the bed characteristics and operating conditions: ε0 ¼ 0.50; C ivp ¼ 100  10 − 6 cm3 =cm3 ; λ0 ¼ 0:1 cm − 1 ; vz ¼ 0.28 cm/s; a ¼ 50 (Ornatski parameter).

7.2.22

7.2.23

In deep-bed filtration, very fine particles of uniform radius rp in suspension in water are deposited on the surface of the filter bed material as the water flows down. The porous medium of filter bed may be modeled as a bundle of straight capillaries of radius rc and length L through which water flows vertically downward at a velocity vz(r), considered parabolic. Figure 7.P.2 shows the limiting trajectory of a particle which enters the capillary (z ¼ 0) at a radius r cr c such that it is deposited on the capillary wall at z ¼ L. Particles entering at a smaller r are not captured. (1) Obtain an integral expression for the value of Gr, the grade efficiency of this capillary, and ET, the overall efficiency, assuming that the particle concentration entering the capillary is uniform across the capillary radius. (2) If the rate of particle deposition per unit volume of the bed is first order with respect to the volume concentration of particles Cvp in water (¼ k1Cvp, where k1 is the rate constant), develop a simple first-order differential equation for Cvp in the capillary by mass balance in a differential control volume at location z. Use vz javg . (3) Solve this equation for Cvp and related Cvp at z ¼ L to C ivp at z ¼ 0. Using these two values of Cvp, develop an estimate of Gr in terms of L, vz javg and k1.

7.3.1

For typical proteins such as albumin being separated by continuous free-flow electrophoresis in a rectangular device (Figure 7.3.1) the following information is available: 2b ¼ 4 mm; L ¼ 30 cm; vmax ¼ 1 cm/s; E ¼ 30 volt/ −4 cm; μm cm2 =s − volt; Dis ¼ 6  10 − 7 cm2 =s. i ¼ 1:6  10 m Calculate (xiL – x0), σ i , collection port width and xiL – xjL (if μm j ¼ 1:1  μi and Djs ffi Dis ). Comment on whether it is possible to separate species i from species j. Notes: (1) Average axial velocity is equal to (2/3)vmax for parabolic velocity profile; (2) assume plug flow and neglect electroosmotic velocity.

664

Bulk flow perpendicular to the direction of force

Flow rccr

z=0

r

Particle trajectory

L

Capillary wall rc Figure 7.P.2.

7.3.2

A cylindrical electrophoresis column with electrodes at the center radius (rb) and at the outer circumference (radius ra) with lengthwise flow (z-direction) through inert packings can be used for continuous fractionation of ionic solutes on a scale approaching kilograms per hour. A feed mixture is fed in a ring of radius rF at the upstream end, and solute fractions are withdrawn from eight concentric rings at the downstream end at a distance L from the inlet. A background electrolyte (“elutant”) is fed downward continuously and uniformly across the bed cross section with a linear velocity of vz . (1) It is desired to determine the radial location of the ring at the product withdrawal end for any given ith species. (2) Obtain a relation between the radial product withdrawal locations ri and rj of two ionic species (i and j) with the closest ionic mobilities. (Ans. m  2 L E ab ðμm i − μj Þ ðr 2i − r 2j Þ ¼ : vz ℓnðr a =r b Þ 2 You are given the following: ionic mobility of solute i ¼ μm i cm /s-volt; Eab is the voltage difference between the two electrodes; the voltage gradient varies inversely with the radius (see problem 3.1.5); the solution conductivity is uniform throughout.

7.3.3

The rotationally stabilized continuous free-flow electrophoretic (CFE) separator of Figure 7.3.4, sometimes known as the Philpot–Harwell electrophoretic separator, is of interest for determining the trajectory of a protein introduced at z ¼ z0 and r ¼ ri, the radius of the inner cylinder, which is stationary and acts as the cathode. The outer rotating cylinder of radius r ¼ ro acts as the anode. Neglect solute diffusion, electroosmotic flow velocity, Joule heating and any change in physical properties. Determine the trajectory of the protein for this annular flow configuration if the outer electrode has a voltage Vo while the inner electrode has a voltage Vi ( 0, the right-hand plate temperature is raised to a high value (e.g. see Example III, 50% H2–50% N2 separation by thermal diffusion in Figure 1.1.3; the gas mixture in the hot bulb becomes enriched in the lighter species, H2, whereas the cold bulb becomes enriched in the heavier gas, N2 (see calculations after equation (4.2.65)); the other plate is maintained at the initial low temperature. If species 1 is the lighter species, e.g. H2, then the gas mixture next to the hot plate will become enriched in H2, whereas the gas mixture next to the cold plate will become enriched in the

671

heavy species 2, e.g. N2. In Figure 8.1.1(b), we illustrate the changed composition in terms of light gas (e.g. H2) by assuming a 4% change in thermal diffusion based equilibrium composition: the gas mixture next to the hot plate is 52% H2, whereas that next to the cold plate is 48% H2. The enclosed channel with the particular temperature profile will lead spontaneously to natural convective motion of the gaseous regions (see Figure 6.1.3): the gaseous mixture next to the heated plate will go up, whereas that next to the cold plate will come down. As the gaseous regions move up and down slightly, we will have somewhat changed situations near the top and bottom of the column. We illustrate this by dividing the vertical channel into eight identical sections at different vertical locations. As Grew and Ibbs (1952) first illustrated (since described also by Powers (1962, pp. 1–98, 36–37)), the top section will have both regions of the channel containing a gas mixture of 52% H2, whereas the bottom section will contain 48% H2 (Figure 8.1.1(c)). However, due to the temperature difference between the two plates, there will be thermal diffusion based separation, creating a changed situation, as shown in Figure 8.1.1(d), in these two sections. Now the bottom section will have 50% species 1 near the hot plate and 46% near the cold plate, whereas, in the top section, the region next to the hot plate will have 54% species 1 and there will be 50% species 1 near the cold plate. This separation is achieved after thermal diffusion based equilibriation is achieved following the natural convective motion. This way of describing the separation development assumes consecutive processes: Figure 8.1.1(a) – time t ¼ 0; Figure 8.1.1(b) – first thermal diffusion takes place; Figure 8.1.1(c) – first natural convection occurs; Figure 8.1.1(d) – second thermal diffusion based equilibriation takes place. In reality, these steps/processes go on simultaneously. Let us continue with this process. As the natural convection continues (the second natural convection, Figure 8.1.1(e)), we have gas compositions in the top two sections and bottom two sections that are different from those based on local thermal diffusion equilibrium. Establishment of local thermal diffusion based equilibrium will lead to the compositions shown in Figure 8.1.1(f) (third diffusion), which shows that the top section now has 55% species 1 near the hot plate and 51% species 1 near the cold plate; further, the bottom section now has 45% species 1 near the cold plate and 49% species 1 near the hot plate. What is clear from this description is that the top section near the hot plate now has 55% species 1, whereas the bottom section near the cold plate has 45% species 1. Figure 8.1.1(g) shows the further enhancement in this composition difference between the top and the bottom column locations after the third natural convection and the fourth thermal diffusion based equilibriation steps have taken place. There are two basic differences that exist here with respect to the separation achieved in any one section

Bulk flow of two phases ⊥ to direction of force

672

(a)

(b)

(c)

(d)

Cold Hot

Cold Hot

Cold Hot

Cold Hot

50 50

48 52

52 52

50 54

50 50

48 52

48 52

48 52

50 50

48 52

48 52

48 52

Flow

48 52 48 52 Flow

48 52

Hot plate

Flow

48 52

48 52

48 52

48 52

48 52

50 50

48 52

48 52

48 52

50 50

48 52

48 48

46 50

Starting condition t=0

After 1st thermal diffusion based equilibrium

After 1st convection

After 2nd thermal diffusion based equilibrium

(e)

(f)

(g)

(h)

Cold Hot

Cold Hot

Cold Hot

54 52

51 55

52 56

50 52

49 53

48 52

48 52

z

50 50

Cold plate

50 50

48 52

Flow

50 50

Hot plate

Cold plate

z = L

48 52 48 52 47.5 51.5

47 51

46.5 50.5

48 46

45 49

44 48

After 2nd convection

After 3rd thermal diffusion based equilibrium

After 3rd convection and 4th thermal diffusion based equilibrium

48 50

Flow

48 52

48 52

z

48 52

Cold region flow direction Hot region flow direction

48.5 52.5

Hot plate

48 52

48 52

49.5 53.5 Cold plate

48 52

Hot plate

Cold plate

Cold Hot

Flow

z = 0

Thermal diffusion Ordinary diffusion

Force and flow directions

Figure 8.1.1. Type (1) system. Development of concentration profile for a binary gaseous system along the height of a vertical column of parallel plates having closed ends and subjected to thermal diffusion equilibrium and countercurrent flow of two regions by natural convection. Numbers in boxes in each figure represent mole % of lighter species (e.g. H2). (After Grew and Ibbs (1952) and Powers (1962).)

of this channel (or column), the latter simulating the condition in Figure 1.1.3. First, there is countercurrent flow of two regions perpendicular to the directions of the two forces operating here: those due to thermal diffusion and ordinary diffusion (not really a force as such) (Figure 8.1.1(h)). Second, due to the top end of the column being closed, the hot fluid enriched in species 1 is turned around (recycled) in this case, or refluxed in the case of two-phase systems, to the cold part of the channel (total recycle or total reflux). This allows even further enhancement of the hot side concentration of species 1 due to the thermal

diffusion phenomenon, since the cold side has higher species 1 concentration than before. Similarly the cold fluid is turned around in the bottom part of the channel to the hot part, leading to further enrichment on the hot side. (If we did not have this recycle or reflux, and the top and bottom sections were open, we would just have 52% species 1 in the hot stream exiting the top and 48% species 1 in the cold stream exiting the bottom.) By making the column taller, one can achieve a very large change in concentration between the top right and the bottom left section; one can achieve almost pure streams at the top

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

and at the bottom for an exceedingly tall column. However, in practice, one should recognize that one may not achieve the separation corresponding to equilibration in each section of the vertical column (called the Clusius–Dickel column). Let us reflect on the net directional movement of species 1 and species 2 in this countercurrent column. It is clear that the lighter species 1, H2, moves up the column and the heavier species 2, N2, moves down the countercurrent column. One could argue that, in effect, the concentration wave velocity, vC1 , of species 1 is positive vis-à-vis the z-coordinate in Figure 8.1.1, whereas that for species 2, N2, is negative. We illustrate the quantitative criterion for this type of species movement in a countercurrent system of two immiscible phases in Section 8.1.1.3, where we illustrate also that only two species may be separated in such a countercurrent column having steady state flow. We will now consider a system of type (2). Figure 8.1.2(a) illustrates a vertical channel as in Figure 8.1.1(a). However, the top and the bottom ends of this vertical separator are open. Let a liquid absorbent stream enter at the top on the left-hand side and flow down the column. Let a gas to be cleaned up containing at least two species, 1 and 2, enter on the right-hand side at the column bottom and flow up. (Conventionally the two phases are contacted in a dispersive mode; in Section 8.1.2 on gas absorption and

673

stripping, we will discuss the actual physical configuration of the column.) Consider what happens as species 2 in the gas stream is absorbed in the liquid and species 1 is not absorbed. As we assumed in our description of the type (1) system, the gas and the liquid may be assumed to be in equilibrium everywhere in the column, even as they move countercurrently. If the liquid absorbent entering at the top does not have any species 2 at all, and the liquid, as well as the gas stream, are assumed to be in equilibrium, then the gas stream exiting the column top through section 1 will be essentially purified of species 2. Correspondingly, the liquid stream leaving at the column bottom through section n will be in equilibrium with the gas stream entering at the bottom of the column; so the liquid absorbent leaving will be saturated with species 2 corresponding to the feed-gas composition. Therefore, in this type of bulk flow vs. force schematic, the extent of purification of the gas stream leaving the column top is dependent on the purity of the liquid stream entering at the top of the column. For simplicity, we have not divided the vertical column into n sections as we have done in Figure 8.1.1 (where n ¼ 8). Note: We can argue that the unabsorbed gas species 1 moves up the column in the positive z-direction, whereas species 2, absorbed by the liquid absorbent, moves in the negative z-direction. A common example

(b)

(a) Absorbent liquid

Cleaned gas stream

Heavier liquid inlet

Lighter liquid exiting

z

Force

Gas flow

Absorbent liquid

Section 1

direction

Droplets of lighter liquid

Section n Continuous-phase heavier liquid

Spent Gas stream absorbent needs to be regenerated

Lighter liquid introduced

Heavier liquid exiting

Figure 8.1.2. (a) Type (2) systems. Countercurrent flow of a gas stream and an absorbent liquid in a vertical column for absorption of a species from the gas; countercurrent separation system without recycle or reflux. (b) Countercurrent solvent extraction column: lighter liquid introduced at column bottom by dispersing it as droplets as it rises through a continuous heavier liquid, which flows downward through the column. In aqueous–organic extraction systems, usually the aqueous phase is heavier.

Bulk flow of two phases ⊥ to direction of force

674 would be as follows: species 2 is CO2 being absorbed in the absorbent liquid, whereas species 1, say N2 or O2, etc., is absorbed very little. If, instead of absorption, we are engaged in stripping a volatile species from the absorbent liquid entering at the top of the column by means of a stripping gas stream (e.g. air) or a stripping vapor stream (e.g. steam) flowing countercurrently, similar considerations will hold, but in reverse. The extent of stripping based purification of the liquid stream leaving the column bottom will depend on how pure the stripping gas/steam is as it enters the column at the bottom. An important distinction between this mode of operation for gas absorption or stripping and the mode of operation illustrated in Figures 8.1.1(a)–(h) is that there is recycle/reflux (in two-phase systems) of the streams in the latter arrangement. Usually there is no such recyle or reflux in Figure 8.1.2(a). A comparable separation system of type (2) is routinely encountered in liquid–liquid systems for solvent extraction or back extraction processes. Figure 8.1.2 (b) illustrates a dispersive countercurrent extraction column where the heavy liquid flows down the column as the immiscible lighter liquid rises up the column as droplets. A system of type (3) will now be discussed. Consider a tall column (Figure 8.1.3(a)), in which a binary volatile liquid mixture containing a more volatile species 1 and less volatile species 2 is flowing down the column, and a binary vapor mixture of the same species moves up the column. Details of the actual flow arrangements and column internals will be considered later. At the top of the column, the

vapor stream goes out, is condensed in a condenser by an external coolant, and the condensed liquid is refluxed back to the column as the liquid stream going down the column. At the bottom of the column, the liquid is taken out to a reboiler, vaporized completely by an external heat source, and introduced into the bottom of the column as the vapor stream going up the column. Witness, however, the basic similarity in the flow pattern and the arrangements of recycle (here reflux) of both streams at the top of the column and the bottom of the column between Figures 8.1.1(a)–(h) and the present arrangement (Figure 8.1.3(a)). We may assume that, everywhere in the column, the liquid and the vapor streams are in vapor–liquid equilibrium. Just as in Figures 8.1.1(a)–(h), one can develop a very large difference in composition between the top section, containing a stream highly enriched in the lighter species 1, and the bottom section, containing a stream highly enriched in the heavier species 2. This arrangement is known as total reflux in the process of column distillation; in the case of a tall column, it can provide compositions at the top and bottom that are quite close to pure species. During the actual operation of such a distillation column for continuous separation of a feed mixture to obtain two product streams, the feed stream (shown by the dashed lines in Figure 8.1.3(a)) is introduced somewhat near the middle of the column. From the total condenser at the top, a fraction of the total condensate stream is continuously withdrawn as the top product stream; similarly, a fraction of the liquid to be introduced into the reboiler at (b)

(a)

Melting section

Water

Total condenser Vapor

xif

Liquid flow

Binary feed mixture

Vapor flow

Liquid reflux stream

High-melting product

Countercurrent conveying column

Condensed liquid

xild = xid

Distillate product

Force

Feed

Slurry of crystals and adhering liquid Force directions

directions Liquid xilb = xib Reboiler

Steam

Bottoms product Freezing section

Low-melting product

Figure 8.1.3. (a) Type (3) systems. Countercurrent flow of a vapor stream and a liquid stream in a distillation column operated at total reflux, no feed, no bottom product. (b) Countercurrent melt crystallizer (column crystallizer) for contacting of a slurry of crystals and adhering liquid being conveyed countercurrent to a liquid reflux obtained by melting crystals.

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

the bottom is continuously withdrawn as the bottom product stream (also shown by a dashed line in Figure 8.1.3(a)). The same strategy is employed in the column of Figures 8.1.1(a)–(h) for continuous feed introduction and product withdrawals, except the two streams are of identical phase. One must note here that this countercurrent column is such that the lighter species 1 has a net velocity in the positive z-direction (considered vertically upward) and the heavier species 2 has a net velocity in the negative z-direction. An analogous separation system of type (3) is illustrated in Figure 8.1.3(b) for a solid–liquid system as in melt crystallization (an alternative title might be column crystallization). The solid solution of Figure 3.3.6A would be one such system; eutectic systems are additional examples. Crystals and the liquid adhering to crystals are carried by means of a spiral conveying arrangement from one end of the column (the freezing section end) to the other end: this flow is countercurrent to the flow of the free liquid. During this process, impurities from the crystals and the liquid adhering to the crystals are transferred to the counterflowing free liquid stream. Once the crystals and the liquid adhering to the crystals reach the column end, they are taken to a melting section, and the molten liquid is sent (refluxed) back to the column for countercurrent flow (after withdrawing some amount of product which melts at a higher temperature). Conversely, the free liquid exiting the other end of the column is introduced to a freezing section to produce some low-melting product, while a slurry of crystals and adhering liquid is introduced to the conveying arrangement (Albertins et al., 1967). We will now briefly consider the membrane gas permeation process to illustrate a separation system of type (4). A binary gas mixture at a high pressure flows along the device length on one side of the membrane. The gas mixture permeating through the membrane and emerging on the other side enriched in the more permeable species flows countercurrent to the feed gas stream. No separate gas stream is introduced in the permeate side. Additional variations of this type include: (1) a sweep gas/vapor stream introduced on the permeate side in countercurrent flow (shown by a dashed line in Figure 8.1.4(a)); in this case one can have the same total pressure on both sides of the membrane; (2) a fraction of the permeated gas stream (shown by a dashed line) is mixed with the feed gas stream (recycled), compressed and introduced back into the column top. An additional countercurrent configuration appears somewhat analogous to the distillation column of Figure 8.1.3(a); the permeated low-pressure gas is compressed and introduced (recycled) back into the column top (after withdrawal of a product stream), with the feed gas introduced somewhere in the middle of the column (with a resemblance to the top section of the distillation column (Figure 8.1.4(b)). This latter configuration is called the continuous-membrane column.

675

Figure 8.1.4(c) introduces through the example of dialysis (see Section 4.3.1) a countercurrent flow configuration of a membrane device where two separate feed streams are entering the separator as in the separation system type (2). In the electrodialysis (Section 3.4.2.5) process of selective transport of ions through an ion exchange membrane, the liquid solutions on two sides of any ion exchange membrane are sometimes in countercurrent flow. The countercurrent flow system for two phases/regions with the force responsible for separation acting perpendicular to both phases/regions is very effective in developing separation, especially with high levels of purifications, as we will see later in this section. It is convenient to implement this physically for many two-phase/two-region separation systems. However, for a few systems, either (1) other modes of operation of two phases/regions are much more convenient or (2) there are inherent problems to achieving direct countercurrent flow. In both cases, separation schemes/arrangements have been developed to achieve de facto countercurrency, even though locally countercurrent flow is not present. We will now illustrate each case. Consider situation (1) identified above. In the separation processes using a vapor–liquid system (distillation) or a gas–liquid system (absorption/stripping), a most common and convenient mode of phase contacting employs crossflow on a horizontal plate (see Figure 2.1.2(b)): vapor/gas coming from the bottom and bubbling through a liquid layer flowing perependicular to this vapor/gas flow in crossflow. However, many such plates are located in a vertical column (Figure 8.1.5) such that one can achieve, on an overall basis, a countercurrent flow of the vapor and the liquid or the gas and the liquid. Each plate acts as a separator or a stage; but the vertical column integrates the vapor or gas and the liquid flow as if we have countercurrent flow of gas and liquid phases (Figure 8.1.5). The vertical column in such a case becomes a multistaged countercurrent separation device instead of a device where the two phases/regions are in continuous countercurrent flow/contact throughout the device/column length. We now illustrate examples corresponding to situation (2). In some two-phase configurations based on a fluid– solid system, with the solid phase consisting of, say, solid adsorbent particles, the flow of the solid particles in a vertical, or any other, direction is difficult to implement without encountering substantial attritional losses, eroding the device wall, etc. To avoid such losses, solid particles remain in a fixed bed, as in Sections 7.1.1 and 7.1.5; however, the arrangement of the fluid inlets and exits is manipulated as if there is countercurrency via the movement of the fluid past the particles. In Figure 8.1.6 one such arrangement (simulated moving bed (SMB)) is shown. A vertical packed-bed column may be divided into four sections, with the liquid flowing up from the bottom to the

Bulk flow of two phases ⊥ to direction of force

676

(a) Permeate

Low pressure

Sweep gas or vapor

Feed

Force

Recycle

Force

gas mixture

High pressure

Membrane (M) Reject gas mixture

gas mixture (b)

Compressor Permeate

Dialysate

Dialyzing liquid

M

M

Low pressure

High pressure

(c)

Feed solution

Reject gas mixture

Purified feed solution

M

Feed gas mixture

Figure 8.1.4. (a) Type (4) systems. Countercurrent flow of feed gas mixture and permeated gas mixture in a membrane device. (b) Continuous membrane column method of gas mixture separation. (c) Countercurrent dialyzer with the feed solution and the dialyzing liquid entering the device countercurrently on two sides of the membrane.

top section of the column. Liquid streams are introduced or withdrawn at different locations, shown as an eluent (desorbent), extract, feed, raffinate, etc., as if we had true countercurrent flow with the adsorbent particles flowing vertically downward, as shown in Figure 8.1.6. However, in a SMB, adsorbent particles do not come down. Therefore the upward relative velocity of the liquid vis-à-vis the particles has to be increased. This is achieved as follows: at selected time intervals, the two liquid inlet points and the two liquid exit points are advanced upward in the column by means of an external rotary valve in a programmed sequence (e.g. the Sorbex® process of UOP Inc.) as if we have countercurrent flow. Figure 8.1.6 has been drawn for true countercurrent flow of two phases, solid adsorbent and the liquid, for the separation of two species 1 and 2. Species 1 is preferentially

adsorbed by the adsorbent. Therefore in sections 2 and 3, it comes down with the adsorbent if the adsorbent particles were in countercurrent flow. In section 4, desorbent liquid or eluent introduced into the column bottom strips both species 1 and 2 from the adsorbent and moves them up the column in the direction of liquid flow. Part of this stream is withdrawn as the product/extract in between sections 3 and 4. This stream is primarily enriched in the strongly adsorbed species 1, since the species 2 content of the adsorbent entering section 4 is quite low. In section 3, the upflowing desorbent/eluent liquid strips the adsorbent stream of species 2. The adsorbent had picked up species 1 and 2 from the feed liquid stream introduced in between sections 2 and 3. However, the conditions are such that species 1 moves down and species 2 moves up. Species 2 is removed by the stream labeled “Raffinate” between

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

Scrubbed gas

Fresh liquid absorbent

1 2 3 4 5 6 7 8

Inlet for gas to be scrubbed

9

8.1.1.1 Introduction to equations of change of concentration in a countercurrent device Now that we have an understanding of the variety of separation systems where two phases/regions are in continuous countercurrent flow along the device length, it would be useful to specify the equations of change of concentration of species i in the two phases/regions along the direction of the mean flow coordinate, z (positive for one stream, negative for the other). For systems where Uij is zero and molecular diffusion and partitioning between two phases driven by − rμi is the primary driving force, equations (6.2.30) and (6.2.31) are needed. For systems in incompressible flow, where Uij is nonzero, the governing equation (6.2.40) is relevant. In practice, equations employing quantities averaged over the cross section of the bulk flow streams are employed. For two immiscible phase based separation systems, the corresponding equations for phases j ¼ 1 and j ¼ 2 may be obtained from equation (6.2.33) as ε1

Liquid absorbent out (spent)

ε2 Figure 8.1.5. Nine-plate (nine-stage) gas–liquid scrubber with the absorbent liquid in crossflow over each perforated plate, through which the gas bubbles move up; the overall pattern of flow of the gas and liquid phases is countercurrent.

sections 2 and 1. In Section 8.1.1.3, the concentration wave velocity of a species has been developed for countercurrent flow. We will employ this information to provide an elementary analysis of the SMB in Section 8.1.6. In the following part of this section, we provide simple mathematical descriptions of a few common features of two-phase/two-region countercurrent devices, specifically some general considerations on equations of change, operating lines and multicomponent separation capability. Sections 8.1.2, 8.1.3, 8.1.4, 8.1.5 and 8.1.6 cover two-phase systems of gas–liquid absorption, distillation, solvent extraction, melt crystallization and adsorption/SMB. Sections 8.1.7, 8.1.8 and 8.1.9 consider the countercurrent membrane processes of dialysis (and electrodialysis), liquid membrane separation and gas permeation. The subsequent sections cover very briefly the processes in gas centrifuge and thermal diffusion. In the following developments, we rely on the results of Section 6.2.1.1 and identify the equations of change of concentration of a species i in a countercurrent tworegion/two-phase system; we focus on two-phase systems. Next we consider the equations for operating lines in such devices. The multicomponent separation capability of such systems is treated next in the context of a two-phase system.

677


∂C i1 ∂C i1 ∂2 C i1  þ hv1z i ¼ ε1 Di, eff , 1 − K i1c a C i1 − C i1 ; ∂t ∂z ∂z 2 ð8:1:1aÞ
  ∂C i2 ∂C i2 ∂2 C i2 þ hv2z i ¼ ε2 Di, eff , 2 þ K i2c a C i2 − C i2 ∂t ∂z ∂z2 ð8:1:1bÞ

if species i is being transferred from phase 1 to phase 2, with a being the interfacial area between phases 1 and  2 per unit total volume. Here C i1 is the hypothetical concentration of species i in phase 1, which will be in equilibrium with C i2 , the bulk concentration of species i in phase 2 averaged over the flow cross section. Correspondingly,  C i2 is the hypothetical concentration of species i in phase 2, which will be in equilibrium with C i1 , the bulk concentration of species i in phase 1. Further, K i1c and K i2c are the corresponding overall mass-transfer coefficients for molar concentrations of species i based on phases 1 and 2, respectively. Note: One of the two velocities, hv1z i and hv2z i, is positive in the z-direction, whereas the other is negative. Note further that, in equations (8.1.1a,b),

   K i1c C i1 − C i1 ¼ K i2c C i2 − C i2 : ð8:1:1cÞ

We will briefly develop equation (8.1.1a) now by developing a mass balance over the cross section of the countercurrent column with respect to phase 1 (one can do it similarly for phase 2 in the manner of equation (7.1.4)). Figure 8.1.7 illustrates a countercurrent flow based column of length L, in which phase 1 flows upward in a positive zdirection and phase 2 flows downward; phase 1 is losing species i and phase 2 is gaining species i. Focus on the column cross-sectional area bounded by axial locations z and z þ dz: phase 1 is imagined to flow through the region ABCD and phase 2 is imagined to flow through the region EFHG, wherein the interface between phases 1 and 2 is

Bulk flow of two phases ⊥ to direction of force

678

1

2 1

Raffinate

Adsorbent circulated

2 2 Feed (1 + 2)

1

2 3

Species movement 1

Desorbent (eluent) recycle

1

Extract

2 4 Eluent (desorbent)

Make-up

Figure 8.1.6. Various sections, species movement and phase velocity directions in a true vertical countercurrent system for solid adsorbent and liquid phase for separation of two species 1 and 2.

Phase 1

Phase 2

z =L

z + Δz

B

C E G

z

H A

D F

z =0

Phase 1

Phase 2

Figure 8.1.7. Countercurrent flow of two phases in a column: schematic and control volume ABCDEFGH.

located between lines CD and EF (lines CD and EF are identical). In reality, the phases are most often dispersed in each other and therefore flow over the whole cross-section. The column cross-sectional area is Sc (see Figure 7.1.1); the cross-sectional area occupied by flowing phase 1 is ε1Sc and that by phase 2 is ε2Sc. We now adopt the procedure employed to develop equations (7.1.1) and (7.1.3), except we focus only on phase 1 (adopting the pseudocontinuum approach). (Note: v1z is the actual velocity of phase 1 based on the phase 1 cross section; therefore ε1v1z is the superficial velocity of phase 1 through the column, hv1z i.) We have 1 0 1 0 rate of rate of convective and B accumulation C C C B diffusive=dispersive B C B of species i C ¼ B A C @ inflow of species i into B A @ in volume volume ABCD through AD at z ABCD 1 0 rate of convective B and diffusive=dispersive C C B C −B B outflow of species i out C A @ of volume ABCD through BC at zþ Δz 0 1 mass-transfer ð8:1:1dÞ − @ rate out through A; interface CDEF

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces 2 ∂C i1 ¼ ε1 Sc 4 v1z C i1 − ε1 Sc Δz ∂t convective 2

3 ∂C i1 5 Di, eff , 1 diffusive=dispersive ∂z

− ε1 Sc 4 v1z C i1 − convective

− K i1c ðaSc ΔzÞ




hv1z iSc

z

Di, eff , 1 diffusive=dispersive

C i1 − C i1

 ,

3

∂C i1 5 ∂z

where a is the interfacial area between phases 1 and 2 at CDEF per unit of total volume in ABCDEFHG and ScΔz is the total volume of ABCDEFHG. Dividing all terms by ScΔz and taking the limit of Δz ! 0, we get
 ∂ v1z C i1 ∂C i1 ∂2 C i1 ε1 þ ε1 Di, eff , 1 ¼ − ε1 2 ∂z ∂t diffusive=dispersive ∂z
 − K i1c a C i1 − C i1 : ð8:1:1fÞ If v1z is constant along z, then we can write ε1


∂C i1 ∂C i1 ∂2 C i1  þ hv1z i ¼ ε1 Di, eff , 1 −K i1c a C i1 − C i1 : 2 ∂t ∂z ∂z diffusive=dispersive

ð8:1:1gÞ

To be noted here is that Di, eff , 1 is developed based on the whole cross section, but, for the phase 1 equation, ε1 is needed. Generally, countercurrent flow of two immiscible phases contacting each other in one continuous contacting device (Figures 8.1.2 and 8.1.3) is carried out under steady state conditions. Therefore equations (8.1.1a) and (8.1.1b) are reduced to hv1z i hv2z i


∂C i1 ∂2 C i1  ¼ ε1 Di, eff , 1 − K i1c a C i1 − C i1 ; ∂z ∂z2

ð8:1:2aÞ


  ∂C i2 ∂2 C i2 ¼ ε2 Di, eff , 2 þ K i2c a C i2 − C i2 : ∂z ∂z 2

ð8:1:2bÞ


∂C i1  ¼ − K i1c a C i1 − C i1 ; ∂z

ð8:1:3aÞ

For a given two-phase based countercurrent flow system, there will be appropriate boundary/initial conditions. If we neglect any axial/longitudinal dispersion/diffusion in the flowing system, these two steady state equations are reduced to hv1z i hv2z i


  ∂C i2 ¼ þK i2c a C i2 − C i2 : ∂z

ð8:1:3bÞ

Correspondingly, an overall balance of species i for both phases j ¼ 1, 2 is available from equation (6.2.34) as (or by adding equations (8.1.3a) and (8.1.3b)) hv1z i

∂C i1 ∂C i2 þ hv2z i ¼ 0: ∂z ∂z

C i1 ¼ C t1 x i1 ;

If we multiply this equation by the device flow crosssectional area Sc, we get

ð8:1:4bÞ

C i2 ¼ C t2 x i2 ,

ð8:1:4cÞ

where C t1 and C t2 are the total molar concentrations of phases 1 and 2, respectively. If the total molar flow rates of phases 1 and 2 are W t1 and W t2 , respectively, then, if W t1 and W t2 do not change along the z-coordinate, we get hv1z iSc C t1

∂x i1 ∂x i2 þ hv2z iSc C t2 ¼ 0; ∂z ∂z

ð8:1:4dÞ

∂x i1 ∂x i2 þ Wt2 ¼ 0, ∂z ∂z

ð8:1:5Þ

Wt1

provided C t1 and C t2 are essentially constant along the z-coordinate or change very little. This is valid under the following two conditions: (1) species i being transferred from one phase to another is present in a very dilute solution; (2) the species transport between the two phases is taking place such that the total molar flow rate in a given phase does not change in the direction of mean flow (z-direction). This situation is realized approximately under the condition of equimolar counterdiffusion (see Figure 3.1.4); this condition is also characterized by the assumption of constant molar overflow1 in column distillation. In the absence of axial diffusion/dispersion, equation (8.1.5) is the governing equation for transfers in dilute solutions or under conditions of constant molar overflow. There are separation systems where the molar flow rate in each phase/region (j ¼ 1, 2) changes very substantially along the z-direction (Figures 8.1.4(a), (b)). Then, instead of equation (6.2.33), we will obtain the following equation for region j from the general equation (6.2.30) if Uik ¼ 0:
   ∂C ij ∂ hvjz iC ij ∂2 C ij  þ εj ¼ εj Di, eff , j − K ijc a C ij − C ij : 2 ∂t ∂z ∂z ð8:1:6Þ The final term, describing transport of species i via a masstransfer coefficient, may be replaced by a membrane transport rate expression for a membrane process. For steady state, if we neglect the contribution of longitudinal

1

ð8:1:4aÞ

∂C i1 ∂C i2 þ hv2z iSc ¼ 0: ∂z ∂z

If the molar concentrations of the individual species may be expressed in terms of the corresponding mole fractions x i1 and x i2 , then

zþΔz

ð8:1:1eÞ

679

Requires the latent heat of the two species (e.g.) to be identical and the dependences of the enthalpy of both the vapor and the liquid mixture to vary linearly with the mole fraction based composition (see Treybal (1980, pp. 402–403) and Doherty and Malone (2001, pp. 507–512)).

Bulk flow of two phases ⊥ to direction of force

680 diffusion/dispersion, we get, for species i balance in region/phase j,
   ∂ hvjz iC ij  ¼ − K ijc a C ij − C ij : ∂z

of the species transferred along the column length. We have not considered here any specific one-dimensional z-directional balance equations for external force field based countercurrent separation systems. These will be considered at the end of Section 8.1.

ð8:1:7aÞ

Multiplying by the device flow cross-sectional area Sc, we get

8.1.1.2 device


   ∂
x W  ∂ Sc hvjz iC ij  ij tj ¼ − K ijc aSc C ij − C ij ¼ : ð8:1:7bÞ ∂z ∂z

To develop a better understanding of how the stream compositions change with respect to each other along the countercurrent device length in the absence of dispersion and axial diffusion, equations (8.1.5) or (8.1.8) are frequently used. However, instead of a differential equation, algebraic equations relating C il to C i2 or x i1 to x i2 are developed. The lines represented by such equations are usually called operating lines; sometimes the relationship is linear. Consider equation (8.1.5) and a countercurrent device for a separation system of type (2), as shown in Figure 8.1.8(a). We may rewrite this equation as

Summing such a relation for two phases/regions (j ¼ 1, 2), we get ∂ðx i1 W t1 Þ ∂ðx i2 W t2 Þ þ ¼ 0, ∂z ∂z

ð8:1:8Þ

a very useful result for a number of membrane systems where W t1 or W t2 is likely to vary considerably along the device length. This equation is also needed for those phase equilibrium based systems where the flow rate changes substantially due to substantial changes in concentration

xi 1L

Phase 1 exit

Wt 1L

Phase 2 inlet

(a)

Wt 2L

Equation for the operating line in a countercurrent

(b)

Top product

t

xi 2L

W t 1 ,xi 1L

Wt 2t , xi 2t t W t2 ,

Phase 1

Phase 2

z=L xi 1

xi 2

xi 1

xi 2

xi 2L

Feed

z xi 10 xi 2

xi 1

b W t1

xi 20 b

W t2 Wt 2b

z=0

Ci 10 Wt10

Phase 2 exit

xi 10

Phase 1 inlet

xi 2b

Bottom product

xi 20 Ci 20 Wt 20

Figure 8.1.8. (a) Schematic of the countercurrent flow device for a separation system of type (2). (b) Schematic of the countercurrent flow device for a separation system of type (3).

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces hv1z iSc C t1 d x i1 ¼ − hv2z iSc C t2 d x i2 ; d x i2 hv1z iSc C t1 jW t1 j , ¼ ¼ d x i1 − hv2z iSc C t2 jW t2 j

ð8:1:9Þ ð8:1:10Þ

since phase 2 flows in the negative z-direction and jW t1 j and jW t2 j are the magnitudes of the total molar flow rates in phases/regions 1 and 2. For dilute systems or systems satisfying the condition of constant molar overflow, the ratio jW t1 j=jW t2 j is essentially constant along the z-coordinate. Integrating from z ¼ 0 (phase 1 inlet, x i10 , C i10 ; phase 2 exit, x i20 , C i20 ) to any z (x i1 , C i1 ; x i2 , C i2 Þ, we get x i2 ¼

jW t1 j ðx i1 − x i10 Þ þ x i20 jW t2 j

ð8:1:11Þ

as the equation for the operating line for separation systems of type (2) relating the two species i mole fractions, x i2 and x i1 , in the two phases/regions at any value of z along the device for dilute systems or systems having constant molar overflow. An alternative representation of the operating line may be developed by integrating equation (8.1.10) between any location z and z ¼ L (where x i1 ¼ x i1L and x i2 ¼ x i2L ): x i2 ¼

jW t1 j ðx i1 − x i1L Þ þ x i2L : jW t2 j

ð8:1:12Þ

We now develop the equation for an operating line for a separation system of type (3), illustrated in Figure 8.1.8(b), with a reflux at the top and at the bottom. Under the condition of constant molar overflow, we use equation (8.1.10) again between any axial location z and the top of the column z ¼ L. After integration, we get  t        W  x i2L − W t  x i2 ¼ W t  x i1L − W t  x i1 , ð8:1:13Þ t2 t1 t1 t2 where the superscript t on the molar flow rate indicates the top half of the column; subscripts t and b imply top product and bottom product, respectively. However, a molar balance on species i between stream 1 leaving the column top, stream 2 entering the column top and the top product ðx i2t Þ flowing out at the molar flow rate, W t2t , yields    t  W  x i1L ¼ W t  x i2L þ Wt2t x i2t : ð8:1:14Þ t1 t2

Introducing this species i molar balance into relation (8.1.13), we get  t    W  x i2 ¼ W t  x i1 − W t2t x i2t ; t1 t2  t  W  W  x i1 −  t2t  x i2 ¼  t1 ð8:1:15Þ W t  x i2t : Wt  t2

t2

An alternative result is  t  W  W  x i2 þ  t2t  x i1 ¼  t2 W t  x i2t , Wt  t1

t1

ð8:1:16Þ

relating the phase 1/region 1 composition x i1 with that (x i2 ) for phase 2/region 2 at location z in the column and

681

the top product flow rate, W t2t , and composition x i2t . These represent the equations for the operating line of the top half of the column, the so-called enriching section, since the top product stream becomes enriched in the more volatile species in, for example, distillation. Separation systems of type (3), with the feed entering near the middle of the column and two product streams, one at the top and the other at the bottom with reflux/ recycle of the products at the top and the bottom, have two sets of operating lines. The operating line equation given by either (8.1.15) or (8.1.16) relates the two local compositions x i1 and x i2 in the top part of the column above the feed entry location (often called the enriching section) to the top product stream (W t2t , x i2t ). Now we follow a similar procedure and develop a molar balance relation between x i1 and x i2 in the bottom half of the column and the bottom product stream (W t2b, x i2b ). First, we require a relation between x i1 , x i2 and x i10 , x i20 :  b       W  x i2 ¼ W b  x i1 − W b  x i10 þ W b  x i20 , ð8:1:17Þ t2 t1 t1 t2

where the superscripts b on W t1 and W t2 reflect the bottom half of the column. However, a molar balance of species i between stream 1 entering the column bottom, stream 2 leaving the column bottom and the bottom product stream (W t2b , x i2b ) yields  b   W  x i20 ¼ Wt2b x i2b þ W b  x i10 : ð8:1:18Þ t2 t1 Introducing this relation into (8.1.17), we get  b   W  x i2 ¼ W b  x i1 þ W t2b x i2b , t2 t1

which may be rewritten as either  b W  W  x i1 þ  t2b  x i2 ¼  t1 W b  x i2b Wb  t2

t2

ð8:1:19Þ

ð8:1:20Þ

or

 b W  W   x i2 −  t2b x i1 ¼  t2 W b  x i2b : W bt1  t1

ð8:1:21Þ

These are considered to be equations for the operating line of the bottom half of the column, the so-called stripping section, since the bottom product stream gets stripped of the more volatile species in, for example, distillation. As mentioned earlier, in Section 8.1.1, in separation systems of both type (2) and type (3) (illustrated in Figures 8.1.2 and 8.1.3), the local flow conditions of the two phases/ regions vis-à-vis each other may not be countercurrent; often, it is crossflow, even though the overall arrangement of these two flow streams is in countercurrent flow. We consider here an arrangement characteristic of separation systems of type (2), where we have many local two-phase contacting stages in crossflow, with the connection between the stages in countercurrent flow (Figure 8.1.9). The stage characterizing number increases toward the inlet of phase 1 from n ¼ 1 at the top of the column to n ¼ N at

Bulk flow of two phases ⊥ to direction of force

Phase 2 inlet

Wt 1L xi 1b

Phase 1exit

682

any location of the column to those at the column exit/inlet at the top: Wt 2L

x i1ðn þ 1Þ ¼

xi 2L

z= L

Wt2n Wt1L x i1L Wt2L x i2L x i2n þ − : ð8:1:24Þ Wt1ðn þ 1Þ Wt1ðn þ 1Þ Wt1ðn þ 1Þ

In the case of dilute streams 1 and 2 or a constant molar overflow assumption, this relation will be simplified to

1

x i1ðn þ 1Þ ¼

2

¼

n Wt 2n xi 1(n+1)

xi 2n

Wt2 Wt2 x i2n þ x i1L − x i2L Wt1 Wt1 Wt2 ðx i2n − x i1L Þ þ x i1L : Wt1

ð8:1:25Þ

We can also develop an ith species balance over an envelope covering the bottom exit of the column and a location between the stages n and n þ 1:

Wt 1(n+1) n +1

W t1ðn þ 1Þ x i1ðn þ 1Þ þ W t20 x i20 ¼ W t10 x i10 þ W t2n x i2n :

ð8:1:26Þ

N –1

Correspondingly, an operating line equation relating x i1ðn þ 1Þ and x i2n with respect to the quantities at the bottom exit/inlet of the column has the following form:

N z =0 xi 10 Wt 10 Phase 1 inlet

x i1ðn þ 1Þ ¼

xi 20 Wt 20

Wt2n Wt10 x i10 Wt20 x i20 x i2n þ − : ð8:1:27Þ Wt1ðn þ 1Þ Wt1ðn þ 1Þ Wt1ðn þ 1Þ

8.1.1.3 Multicomponent separation capability in a device with a countercurrent flow system

Phase 2 exit

Figure 8.1.9. Multistage countercurrent flow schematic for a separation system of type (2) with individual stages having, say, crossflow.

the bottom. At z ¼ 0, the molar flow rates and compositions are indicated by: phase 1, W t10 , x i10 ; phase 2, W t20 , x i20 . At z ¼ L, the top of the column, the corresponding quantities are W t1L , x i1L ; W t2L , x i2L . Stream 2 entering a stage, say n þ 1, from stage n above it will be characterized as W t2n, x i2n . Stream 1 entering a stage, say n, from stage n þ 1 below it will be characterized as W t1ðnþ1Þ, x i1ðnþ1Þ . We will now consider molar balances on species i, first around plate n and then from plate n to the top of the column (z ¼ L). Consider the balance around plate n as follows: Wt1ðn þ 1Þ x i1ðn þ 1Þ þ Wt2ðn − 1Þ x i2ðn − 1Þ ¼ Wt1n x i1n þ Wt2n x i2n : ð8:1:22Þ

If we now consider a balance over the envelope covering the top exit of the column and a location between stages n and n þ 1, we get Wt1ðn þ 1Þ x i1ðn þ 1Þ þ W t2L x i2L ¼ Wt1L x i1L þ Wt2n x i2n :

ð8:1:23Þ

It is now possible to develop a relation between x i1ðn þ 1Þ and x i2n , the compositions of the two phases/streams at

We now focus on the multicomponent separation capability of a countercurrent flow configuration in a two-phase system. For two phases j ¼ 1, 2 moving with superficial velocities of hv1z i and hv2z i in the z-direction, an overall balance of species i in both phases j ¼ 1, 2 is obtained from equation (6.2.34) as ε1

∂C i1 ∂C i2 ∂C i1 ∂C i2 þ ε2 þ hv1z i þ hv2z i ∂t ∂t ∂z ∂z ¼ ε1 Di, eff , 1

∂2 C i1 ∂2 C i2 þ ε2 Di, eff , 2 : ∂z2 ∂z2

ð8:1:28Þ

Note that with the z-coordinate being vertically upward and, say, the phase j ¼ 2 moving vertically upward, hv2z i is a positive quantity; phase j ¼ 1 will be then moving vertically downward, and therefore hv1z i is a negative quantity. We will represent it as − jhv1z ij, where jhv1z ij is the magnitude of the velocity hv1z i. (To allow comparison with the fixed-bed processes of Section 7.1.1, we assume here that phase j ¼ 2 moves up along the positive z-coordinate.) We now make the following three assumptions: that the operation is isothermal and nondispersive, and that the phases are locally in equilibrium. (These are exactly the same ones made to develop the de Vault equation (7.1.8) for fixed-bed adsorption.) The assumption that the two phases j ¼ 1 and 2 are everywhere in equilibrium

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

in the vertical column with respect to species i may be illustrated by the following equilibrium relation: C i1 ¼ κi1 C i2 :

Alternatively,

∂C i1 ∂C i2 ∂C i1 ∂C i2 þ ε2 − jhv1z ij þ hv2z i ¼ 0 : ð8:1:30Þ ∂t ∂t ∂z ∂z

If the magnitudes of the actual/interstitial velocities of the two phases through the column are jv1z j and v2z for phases j ¼ 1 and 2, respectively, then the superficial velocities and the interstitial velocities are related as follows: jhv1z ij ¼ ε1 jv1z j,

hv2z i ¼ ε2 v2z :

ð8:1:31Þ

Employing these two relations, as well as the linear equilibrium relation (8.1.29), the overall balance equation (8.1.30) is simplified to   ∂C i2 κi1 jhv1z ij ∂C i2 þ hv2z i 1 − ¼ 0: ð8:1:32Þ ðε1 κi1 þ ε2 Þ ∂t ∂z hv2z i Using now the relations (8.1.31), we get 

   ε1 ∂C i2 κi1 ε1 jv1z j ∂C i2 1 þ κi1 þ v2z 1 − ¼ 0, ε2 v2z ε2 ∂t ∂z

ð8:1:33Þ

which may be rewritten as 1 0 ε1 jv1z j 1 − κ i1 B ∂C i2 ε2 v2zC C ∂C i2 ¼ 0: þ v2z B @ A ∂z ε1 ∂t 1 þ κi1 ε2

ð8:1:34Þ

This equation has an exactly similar form to equation (7.1.9) (the de Vault equation). Therefore, following a procedure similar to that used to obtain equations (7.1.12a) and (7.1.12e), we can obtain the following expression for the concentration wave velocity, vCi , of species i (Fish et al., 1989):   ε1 jv1z j 1 − κi1 ε2 v2z  : ð8:1:35Þ vCi ¼ v2z  ε1 1 þ κi1 ε2

When κi1 ðε1 jv1z jÞ=ðε2 v2z Þ < 1, vCi is positive, and species i moves up the column along the positive z-direction, the direction of movement of phase 2. When κi1 ðε1 jv1z jÞ= ðε2 v2z Þ > 1, vCi is negative, and species i moves down the column along the negative z-direction, the direction of movement of phase 1. Therefore, if there is a binary mixture of species A and B (i ¼ A, B), and we want species A to go up the column and species B to go down the column, the following relation has to be satisfied: κA1

ε1 jv1z j ε1 jv1z j < 1 < κB1 : ε2 v2z ε2 v2z

κA1

>  x ig0 − H Pi x iℓ0 = 1   ℓn , jW tg j > x igL − H Pi x iℓL > > > : 1 − H Pi ; jW tℓ j

ð8:1:53aÞ

where, to obtain the denominator of the logarithmic term, we have employed equation (8.1.43b) at location z ¼ L. Here the units of Kigx are gmol/cm2-s-mole fraction, a is in
 cm2/cm3 and jW tg j=Sc is given in gmol/cm2-s. Therefore

  the unit of jW tg j=Sc =K igx a is cm, that of length or height, since mole fraction has no units. An alternative form of the above results for the absorber length, first derived by Colburn (1939), is

 jW tg j 1   L¼ K igx a Sc P jW tg j 1 − Hi jW tℓ j

     P P H jW tg j x ig0 − H Pi x iℓL H i jW tg j ℓn 1 − i : þ jW tℓ j jW tℓ j x igL − H Pi x iℓL ð8:1:53bÞ

Often the results (8.1.53a) and (8.1.53b) for the length or height of the countercurrent column are expressed as 

L ¼ HTU og  NTU og , ð8:1:54aÞ where

HTU og ¼ height of a transfer unit based on overall gas phase
 ¼ jW tg j=Sc =K igx aÞ,

NTU og ¼ number of transfer unitsðoverall gas-phase basedÞ: ð8:1:54bÞ

The latter is defined by NTU og ¼

xð ig0

x igL



dx ig x ig − x ig

:

ð8:1:54cÞ

Bulk flow of two phases ⊥ to direction of force

692 These definitions follow from equation (8.1.50). The quantity NTUog provides an estimate of how difficult the separation is as we change the gas composition from x ig0 (inlet) to x igL (outlet) at the column top. If (x ig − x ig ) may be assumed to be a reflection of the composition difference between the two phases driving the transport of species i from the gas to the liquid phase, then the larger this value is to achieve a given change in x ig , the smaller is going to be the value of NTU, and the easier will be the separation. We see this interpretation reflected in the following result if (x ig − x ig ) is assumed to be constant over the desired gasphase concentration change from x ig0 to x igL in definitions (8.1.54b, c): x ig0 − x igL : NTU og ¼  x ig − x ig

ð8:1:54dÞ

An expression for NTUog may also be written down for the Henry’s law case from equation (8.1.53b) as

NTU og

8 < 1 ¼  : 1 − HP

9 =  jW tg j ;

i jW tℓ j

     P H P jW tg j x ig0 − H Pi x iℓL H i jW tg j ℓn 1 − i : þ P jW tℓ j jW tℓ j x igL − H i x iℓL ð8:1:54eÞ

Note the following points. (1) We have used x iℓ0 in many equations here. Usually in gas absorption x iℓL is known from the entering liquid composition; x iℓ0 is unknown. It has to be calculated by mass balance, knowing x ig0 and x igL . (2) These derivations were carried out for species i absorption from the gas phase to the liquid phase. However, they are equally valid for the stripping of species i from the liquid to the gas phase (or vapor phase if steam or some other vapor is used). (3) All derivations were carried out using gas-phase concentration change. We will now provide a few corresponding results using liquid-phase concentration changes. (4) When species i is a vapor, use Raoult’s law (3.3.64): x ig

¼




P sat i =P



x iℓ :

ð8:1:54fÞ

Therefore

ðL 0

  x ig jW tℓ j d x iℓ ¼ − K iℓx a x − iℓ ; dz Sc H Pi


 dC iℓ jW tℓ j d x iℓ ¼ ¼ − K iℓx a x iℓ − x iℓ : dz Sc dz


 x ig =H Pi ¼ x iℓ :

ð8:1:55bÞ

jW tℓ j=Sc dx iℓ   x ig K iℓx a x iℓo P − x iℓ Hi xðiℓo jW tℓ j dx iℓ  : ¼ x ig Sc K iℓx a x − x iℓL iℓ H Pi

ð8:1:55dÞ

Introducing the operating line equation (8.1.43b) for x ig in terms of x iℓ , we get xðiℓ0

d x iℓ  jW tℓ j 1 jW tℓ j x iℓ0 x ig0 − 1 − þ P x iℓL x iℓ jW tg j H Pi jW tg j H Pi Hi 8 9 > > > > > 2 3> > > > > < = ð jW j Þ 1 tℓ 5 ! ¼4 > Sc K iℓx a > > > jW tℓ j 1 > > > > > : jW tg j H P − 1 > ; i 0 1 3 2 jW tℓ j 1 jW tℓ j x iℓ0 x ig0 @ A 6 x iℓ0 −1 − þ P7 6 jW tg j H Pi jW tg j H Pi Hi 7 7 6 7; ð8:1:56aÞ 6 0 1 ℓn6 7 7 6 x jW j 1 jW j x ig0 tℓ iℓ0 4x @ tℓ A −1 − þ P5 iℓL jW tg j H Pi jW tg j H Pi Hi

L¼

jW tℓ j Sc K iℓx a

8 39 2x > > igL > > > − x iℓL > < P 7= 6 Hi jW tℓ j 1 7 ;   ℓn 6 L¼ 5 4 x ig0 jW tℓ j 1 Sc K iℓx a > > > − x iℓ0 > > > : 1 − jW j P ; H Pi tg H i

ð8:1:56bÞ

L ¼ ½HTU oℓ ½NT U oℓ

where

HTU oℓ ¼

ð8:1:57aÞ

jW tℓ j Sc K iℓx a

ð8:1:57bÞ

and

NTU oℓ ¼

ð8:1:55aÞ

If we assume Henry’s law to be valid for the dilute gas stream, then

xð iℓL

dz ¼ L ¼ −

The equation corresponding to equation (8.1.48) for a liquid-phase based calculation is hvℓz i

ð8:1:55cÞ

NTU oℓ

8 > >
> =

1 jW tℓ j 1 > > > > :1 − ; jW tg j H Pi 3 0 2 x ig0 1   x iℓL − P B 6 Hi C jW tℓ j jW tℓ j 7 C 7 B ℓn6 x ig0 A þ jW j H P 5; 4 1 − H P jW j @ tg tg i i x iℓo − P Hi

8 9 2 > > > > < = 6x iℓL − 1  ℓn6 ¼  4 > jW tℓ j 1 > > > x iℓ0 − : 1− ; jW tg j H Pi

3

ð8:1:57cÞ

x igL H Pi 7 7 x ig0 5: ð8:1:57dÞ H Pi

Countercurrent bulk flow of two phases/regions ⊥ to the forces

8.1

Here, HTUoℓ stands for the height of the transfer unit based on the overall liquid phase; correspondingly, NTUoℓ represents the number of transfer units based on the overall liquid phase. Example 8.1.2 Air at atmospheric pressure containing acetone vapor at the level of 0.01 mole fraction is to be scrubbed by pure water at 15  C. The value of H Pi for acetone and water under these conditions is 1.2 (Sherwood et al., 1975). To scrub the air entering the column bottom at 7 kgmol/hr, the pure water flow rate into the packed column is maintained at 23 kgmol/hr. The value of Kigxa is known to be 20 gmol/m3-smole fraction under these flow conditions for acetone in a column of diameter 33 cm. Determine the height of the column needed to scrub 90% of the acetone from air. Solution

species i is not specifically identified, these quantities will be represented as k ℓx , k gx , k gc , k ℓc ; see Section 3.4.1.1.) To develop expressions based on individual phase based mass-transfer coefficients, we need to define/locate the gas–liquid interface composition at any location in the
 absorber in Figure 8.1.14(b), namely x iiℓ , x iig , representing point B on the gas–liquid equilibrium curve, corres
 ponding to bulk gas–liquid compositions x iℓ , x ig , represented by point A on the operating line. The flux of species i, Ni, across the gas–liquid interface at this location may be expressed as (Figure 8.1.14(b))
 N i ¼ k igx x ig − x iig ¼ k iℓx ðx iiℓ − x iℓ Þ ð8:1:58Þ for species i transfer from the gas-phase bulk to the gas– liquid interface and then to the liquid bulk.4 Therefore

π π H Pi ¼ 1:2; Sc ¼ d2 ¼  ð0:33Þ2 m2 ¼ 0:085 m2 ; 4 4 jW tg j ¼ 7 kgmol=hr ¼ 7000 gmol=3600 s ¼ 1:944 gmol=s; kgmol jW tℓ j ¼ 23 ¼ 23 000 gmol=3600 s ¼ 6:389 gmol=s: hr We employ equation (8.1.53b) to calculate the column height. (Note: x iℓL ¼ 0 since pure water is being used.) The column height is given by

8 9 > > > > > 3> > > > > < = jW j 1 tg 5 4 ! L¼ > K igx a Sc > > > > P jW tg j > > > > : 1 − H i jW tℓ j > ; 2 ! !3 ! P P P H jW j x − H x H jW j tg ig0 iℓL tg i i i 5 þ ℓn4 1 − jW tℓ j jW tℓ j x igL − H Pi x iℓL 8 9 > > > > gmol > > > > 1:944 < = 1 s ¼ 2 > 1:944> gmol  0:085m > > > 20 3 >1 − 1:2 6:389> > ; m -s-mol fraction : 2 3    1:944 0:01 1:2  1:944 5: þ 2:303 log4 1 − 1:2 6:389 0:001 6:389 2

So

L¼




The previous results expressing the length of the absorber/ stripper in terms of HTU and NTU for the absorption of species i present in dilute concentrations in the feed were developed in terms of an overall mass-transfer coefficient
 K igx or K igc and K iℓx for species i. It is also useful to develop such results in terms of individual phase based mass-transfer coefficients, k iℓx , k igx , k igc , k iℓc . (When

 k iℓx x ig − x iig =ðx iℓ − x iiℓ Þ ¼ − , k igx

ð8:1:59Þ

representing the negative slope of the line AB connecting the two bulk-phase compositions to the gas–liquid interface composition. We can now rewrite equation (8.1.46) as follows:
 jW tg j dx ig ¼ − kigx a x ig − x iig ¼ − k iℓx a ðx iiℓ − x iℓ Þ, Sc dz ð8:1:60Þ which leads to ðL o

dz ¼ L ¼ −

xð igL

x ig0

¼−

xð iℓL

x iℓ0

jW tg j=Sc dx ig
 k igx a x ig − x iig

jW tℓ j=Sc dx iℓ , k iℓx a ðx iiℓ − x iℓ Þ

ð8:1:61Þ

where, for the last integral, we used equation (8.1.43a). We observed earlier (in Section 3.1.4, equations (3.1.127), repeated below for convenience) that there are basically two types of molecular diffusion processes in a binary gaseous mixture: equimolar counterdiffusion and nonequimolar counterdiffusion. In general, we may express the mass-transfer coefficient for a general case in terms of that for equimolar counterdiffusion via

 1:944 m 2:303 log½ð0:635Þ  10 þ 0:365 20  0:085 1 − 0:365

¼ 1:143  3:626  log½6:715 ¼ 4:144  0:827 ¼ 3:42 m:

693

k ix ¼

k 0ix ; ϕN

k ic ¼

k 0ic ; ϕN

k ig ¼

k 0ig ϕN

,

where ϕN is the bulk flow correction factor. Expression (8.1.61) for the gas absorber column height L may also be expressed in terms of the individual phase based masstransfer coefficient for equimolar counterdiffusion as


 The quantity x ig
− x iig may be considered as the cross sectional averaged x ig − x iig .

4

Bulk flow of two phases ⊥ to direction of force

694

L¼ −

¼

xð igL

x ig0

"



 jW tg j=Sc d x ig   
x ig − x iig k 0igx a=ϕN

jW tg j=Sc k 0igx a

# xðig0 x igL

For




ϕN d x ig : x ig − x iig

L ¼ HTU g  NTU g ,




HTU g ¼ jW tg j=Sc =k 0igx a;

NT U g ¼

ð8:1:62aÞ

xð ig0


 mixture), we may express the difference x ig − x iig as varying linearly with x ig :
 x ig − x iig ¼ b1 x ig þ b2 : ð8:1:64bÞ

Therefore,

L ¼




¼




L¼




ϕ dx
N ig , x ig − x iig

x igL

ð8:1:62bÞ

so L¼−

xð iℓL

x iℓ0

ðjW tℓ j=Sc Þ dx iℓ ðjW tℓ j=Sc Þ ¼ ðx iiℓ − x iℓ Þ k iℓx a k 0iℓx a

 xðiℓ0 x igL

ϕN dx iℓ ; ðx iiℓ − x iℓ Þ ð8:1:62cÞ

for L ¼ HTU ℓ  NTU ℓ , HTU ℓ ¼

ðjW tℓ j=Sc Þ=k 0iℓx a;

NTU ℓ ¼

xðiℓ0

x iℓL

ϕN dx iℓ : ðx iiℓ − x iℓ Þ ð8:1:62dÞ

In equations (8.1.62a,b), k 0igx corresponds to equimolar counterdiffusion and k igx may be assumed to represent the case of species i diffusing through a stagnant gas consisting of species other than i (see equations (3.1.127)). In such a case for a binary system, we note from (3.1.130) that, for NR ¼ 1,    

 1 − x δig − 1 − x 0ig 1 − x iig − 1 − x ig 
 h   i ¼ 
ϕN ¼ ℓn 1 − x iig = 1 − x ig ℓn 1 − x δ = 1 − x 0 ig




ig



ð8:1:63Þ ¼ 1 − x ig iℓm ,
 where 1 − x ig iℓm represents a logarithmic mean of the bulk and the interface mole fractions of stagnant gas species other than species i. For a dilute system vis-à-vis species i, ϕN is close to 1 and can often be neglected. When it cannot be neglected, and yet may be considered as very weakly dependent on composition in the range under consideration, we may simplify equation (8.1.62a) as follows: L¼




xð ig0   jW tg j=Sc
dx ig
: 1 − x ig iℓm k 0igx a x ig − x iig x igL

where


xð ig0  jW tg j=Sc dx ig ð1 − x Þ ig iℓm b1 x ig þ b2 k 0igx a x igL



2


3   jW tg j=Sc b1 x ig0 þ b2 1 4  5; ð1 − x ig Þiℓm ℓn
b1 k 0igx a b1 x igL þ b2 




ð8:1:64cÞ



 jW tg j=Sc
x ig0 − x igL  , 1 − x ig iℓm
k 0igx a x ig0 − x iig ℓm

x ig0 − x iig



ℓm

¼

ð8:1:64dÞ



 x ig0 − x iig0 − x igL − x iigL 

 : ð8:1:64eÞ ℓn x ig0 − x iig0 = x igL − x iigL

The corresponding expression in terms of k0 iℓx is
 jW tg j=Sc ðx iℓ0 − x iℓL Þ ð1 − x iℓ Þiℓm : ð8:1:64fÞ L¼ ðx iiL − x iℓ0 Þℓm k 0iℓx a So far, we have obtained the height of the gas–liquid absorber, L, expressed in terms of the HTUg and NTUg based on the individual gas film (equation (8.1.62b) or in terms of HTUℓ or NTUℓ based on the individual liquid film (equation (8.1.62d)) or in terms of the overall gas film (equations (8.1.54a–c)) or in terms of the overall liquid film (equations (8.1.57a–d)). It is very useful to have these quantities based on individual film coefficients related to those based on the overall film coefficients. To develop such relations, recall the basic relation (3.4.8) between the individual film transfer coefficients and the overall gas-phase based mass-transfer coefficient in a gas–liquid system: p

1 1 H ¼ þ A; K xg k xg k xℓ

p

x Agi ¼ H A x Aℓi :

Here we are using the notation kiℓx, where i refers to the species i, ℓ to the liquid phase (second subscript is usually for the phase) and x for mole fraction based calculations; similarly for kigx and Kigx. Therefore we rewrite the above relation here (species i for A) as p

ð8:1:64aÞ

If we can assume now that the equilibrium curve shown in Figure 8.1.14(b) is approximately a straight line (the operating line being essentially straight for the dilute gas

1 1 H ¼ þ i : K igx k igx k iℓx

ð8:1:65aÞ

We can rewrite this relation as p

1 1 H ¼ þ i K 0igx k 0igx k 0iℓx

ð8:1:65bÞ

Countercurrent bulk flow of two phases/regions ⊥ to the forces

8.1

for a dilute system, where ϕN ffi 1. It follows therefore that H Pi jW tg j

jW tg j jW tg j jW tℓ j þ ¼ Sc K 0igx a Sc k 0igx a Sc k 0iℓx a

jW tℓ j

HTU oℓ ¼ HTU ℓ þ

ð8:1:65cÞ

:

to get

ð8:1:65dÞ

HTU og ¼ HTU g þ HTU ℓ  λ:

ð8:1:65eÞ

If we now employ the basic relations between the column height L and the HTU and NTU values based on individual films and the overall transfer coefficients, equation (8.1.65e) is reduced to L L λL ¼ þ , NTU og NTU g NTU ℓ

ð8:1:66aÞ

1 1 λ ¼ þ : NTU og NTU g NTU ℓ

ð8:1:66bÞ

i.e.

The corresponding relations between an overall liquid and the individual film coefficients are based on relation (3.4.9): 1 1 1 ¼ þ : K xℓ k xℓ H pi k xg Rewrite this as 1 1 1 , ¼ þ k 0iℓx k 0iℓx k 0igx H pi

ð8:1:67aÞ

leading to (for ϕN ffi 1 in a dilute system) jW tg j jW tℓ j 1 jW tℓ j jW tℓ j þ ¼ ; Sc K 0iℓx a Sc k 0iℓx a Sc k 0igx a jW tg j H Pi

2r

dr

ð8:1:67bÞ

HTU g , λ

1 1 1 : ¼ þ
NTU oℓ NTU ℓ λ NTU g

Define
 λ ¼ H Pi jW tg j =ðjW tℓ jÞ

695

ð8:2:67cÞ ð8:2:67dÞ

8.1.2.2.1 Porous membrane contactor–stripper for deoxygenation of ultrapure water We will provide a simplified mathematical description of a gas stripper using a membrane contactor, as illustrated in Figure 8.1.13(a). The specific example is deoxygenation of ultrapure water by applying a vacuum through the bore of the fibers as an aqueous solution flows on the outside of the hollow fibers. The aqueous solution (fluid 1) enters through a central tube, having many openings on its surface, through which the fluid flows out radially in crossflow across the hollow fibers. As shown in the simplified schematic of Figure 8.1.15, the water flows radially outward in the first half of the device, goes to the outermost radius, turns around by flowing over the baffle to the second half of the device, where it flows radially inward, and finally enters the other half of the central tube and flows out. The baffle completely separates the two sides of the device including the central tube. Fluid 2 in Figure 8.1.13(a) is the gaseous stream taken out through the bore of the hollow fibers via vacuum. The overall pattern of flow between the two streams is countercurrent; however, locally in each half the flow pattern is crossflow. The analysis illustrated below is based on Sengupta et al. (1998). The basic assumptions are as follows. (1) The partial pressure of the gas species being stripped into the fiber bore is negligible compared to the equilibrium partial pressure of the species over the liquid. (2) The dissolved gas concentration in the liquid phase in any half of the device changes only radially; it has no axial variation.

Membrane cartridge

Baffle Fiber bed

Inner distribution tube

Liquid out

Liquid inlet Ql, Cil0

Ql, CilL

2Rci

Baffle L/2

L /2

2Rco Figure 8.1.15. Simplified configuration of hollow fiber membrane contactor device for stripping of a gas from a liquid. (After Sengupta et al. (1998).)

Bulk flow of two phases ⊥ to direction of force

696 Consider now the first half of the device where the liquid is flowing radially out. Take a thin cylinder of radial thickness dr spanning the length L/2 of the device. As the liquid having an averaged radial velocity hvℓ ri flows out through a radial cross-sectional area of Scr, oxygen (species i) is being transferred through a hollow fiber membrane surface area dAs via the liquid-phase based overall transfer coefficient Kiℓ (following (8.1.46)): hvℓr iScr


 d C iℓ ¼ − K iℓ dAs C iℓ − C iℓ : dr

ð8:1:68Þ

By assumption (1), C iℓ 1, subcooled liquid; q < 0, superheated vapor.

fraction of the feed; Hℓf is the molar enthalpy of the liquid fraction of the feed; Hvf is the molar enthalpy of the vapor fraction of the feed; Hf is the molar enthalpy of the feed. Figure 8.1.21(a) also illustrates the molar flow rates and enthalpies of the two streams, vapor as well as liquid, on both sides of the feed plate: enriching section, W ttℓð f − 1Þ , H tℓð f − 1Þ , W ttvf , H tvf ; stripping section, W btℓf , H bℓf , W btvð f þ 1Þ , H bvð f þ 1Þ . Now carry out a total molar balance and an enthalpy balance around the feed plate as follows. Molar flow rate balance around feed plate: W tf þ W ttℓð f − 1Þ þ W btvð f þ 1Þ ¼ W ttvf þ W btℓf :

ð8:1:144Þ

Enthalpy balance around feed plate: W tf H f þ W ttℓð f − 1Þ H tℓð f − 1Þ þ W btvð f þ 1Þ H bvð f þ1 Þ ¼ W ttvf H tvf þ W btℓf H bℓf :

ð8:1:145Þ

Simplification of these equations is carried out by recognizing that the variation in the molar enthalpy of the saturated liquid, as well as the saturated vapor phase, do not vary much over one plate. Therefore, H tℓð f − 1Þ ffi H bℓf ffi H ℓf ;

H bvð f þ1Þ ffi H tvf ffi H vf , ð8:1:146Þ

which simplifies equation (8.1.145) to W tf H f þ W ttℓð f − 1Þ H ℓf þ W btvð f þ 1Þ H vf ¼ W ttvf H vf þ W btℓf H ℓf )





W btℓf − W ttℓð f − 1Þ H ℓf

ð8:1:147Þ  ¼ W tf H f þ W btvð f þ 1Þ − W ttvf H vf : 

ð8:1:148aÞ

Substitute now from relation (8.1.144), W btvð f þ 1Þ − W ttvf ¼ W btℓf − W ttℓð f − 1Þ − W tf ,

ð8:1:148bÞ

into (8.1.148a), to obtain     W btℓf − W ttℓð f − 1Þ H ℓf ¼ W tf H f þ W btℓf − W ttℓð f − 1Þ H vf − W tf H vf ;



W btℓf − W ttℓð f − 1Þ W tf



¼

H vf − H f ¼ q: H vf − H ℓf

ð8:1:149Þ ð8:1:150Þ

The quantity q defined above identifies the moles of liquid added to the liquid stream in the column per mole to total feed introduced. Since (Hvf − Hℓf) represents the amount of heat needed to vaporize one mole of saturated liquid under feed plate conditions, q also represents the fraction of that heat required to vaporize completely the feed as it is introduced into the column. If the feed introduced into the column is a saturated liquid at feed plate conditions, i.e. Hf ¼ Hℓf, q ¼ 1. If the feed is introduced as saturated vapor at feed plate condition, then Hf ¼ Hvf, q ¼ 0. If the feed introduced is a mixture of liquid and vapor, then 0 < q < 1. If the feed is introduced as a subcooled liquid, then Hf < Hℓf, the molar enthalpy of the saturated liquid at the temperature and pressure of the feed plate; correspondingly, q >1. If the feed is introduced as a superheated vapor vis-à-vis the feed plate conditions, then Hf > Hvf and q < 0. Note at this time the overall material and component i balance for the column: W tf ¼ W tℓd þ W tℓb ;

W tf x if ¼ W tℓd x iℓd þ W tℓb x iℓb :

ð8:1:151Þ

We will now determine the locus of the intersection of the operating lines of the enriching section and the stripping section as a function of the condition of the feed introduced into the column. Recognize that at the locus

Bulk flow of two phases ⊥ to direction of force

714 of the intersection the vapor phase and liquid phase compositions of both sections are identical, i.e. x ivðnþ1Þ and x iℓn for the enriching section operating line equations (8.1.136b) and (8.1.138b) are identical to x ivðmþ1Þ and x iℓm , respectively, for the stripping section operating line equation (8.1.140). Therefore, we may rewrite the enriching section operating line at this location as W ttv x ivf ¼ W ttℓ x iℓf þ W tℓd x iℓd :

ð8:1:152Þ

Here x ivf and x iℓf refer to the composition vis-à-vis the feed plate. The corresponding form of the stripping section operating line is W btv x ivf ¼ W btℓ x iℓf − W tℓb x iℓb :

ð8:1:153Þ

Subtract equation (8.1.152) from the above equation to obtain

b  W tv − W ttv x ivf ¼ W btℓ − W ttℓ x iℓf − W tℓb x iℓb − W tℓd x iℓd : ð8:1:154Þ

From equation (8.1.151), we may rewrite this as

b  W tv − W ttv x ivf ¼ W btℓ − W ttℓ x iℓf − W tf x if : ð8:1:155Þ Recognize now that  
 W btℓf − W ttℓð f − 1Þ ¼ W btℓ − W ttℓ :

Therefore, from (8.1.150),
b  W tℓ − W ttℓ ¼ q W tf :

x ivf

Solution (1) Let i ¼ 1 stand for benzene. Then x1f ¼ 0.50; x1ℓd ¼ 0.95; x1ℓb ¼ 0.05. From equations (8.1.136e,f) involving total and species molar balances, we get




 x 1f − x 1ℓb 0:5 − 0:05 0:45 W tℓd =W tf ¼ ¼ ¼ ¼ 0:5: x 1ℓd − x 1ℓb 0:95 − 0:05 0:9

Therefore

W tℓd ¼ 0:5W tf ¼ 0:5  3 kgmol=min ¼ 1:5 kgmol=min: Further,

W tℓb ¼ W tf − W tℓd ¼ ð3 − 1:5Þkgmol=min ¼ 1:5 kgmol=min: Part (2) Given that the reflux ratio R equals 2, equation (8.1.138b) for the operating line for the enriching section is

x 1vðnþ1Þ ¼ 0:667 x 1ℓn þ 0:333 x 1ℓd :

ð8:1:156bÞ

The slope of this operating line is 0.667; it intersects the ordinate at ðx 1ℓd =ðR þ 1ÞÞ, namely (0.95/3) ¼ 0.317. We will now calculate the slope of the q-line and obtain its intersection with the equation for the operating line of the enriching section. From equation (8.1.150),

ð8:1:156cÞ

Use results (8.1.156b) and (8.1.156c) in equation (8.1.155): q 1 ¼ x iℓf − x if : ðq − 1Þ ðq − 1Þ

(1) Determine the molar flow rates of the distillate and the bottoms product. (2) Determine the equations of the operating lines and plot them. (3) Using the McCabe–Thiele method, obtain the number of ideal stages needed to achieve this separation.

ð8:1:156aÞ

Further, from (8.1.148b) and the above two simplifications, W btv − W ttv ¼ q W tf − W tf ¼ W tf ðq − 1Þ:

5% benzene–95% toluene. The feed contains both vapor and liquid. The latent heat of vaporization of the feed at the feed temperature is 32 000 J/gmol; further, the difference between the feed enthalpy and the saturated liquid enthalpy at the feed plate temperature is 18 000 J/gmol. The reflux ratio is 2.

ð8:1:157Þ

This is the locus of the intersection of the operating lines of the enriching section and the stripping section of the multiplate distillation column; it is often called the q-line. It is a straight line whose slope is q/(q − 1). The point x ivf ¼ x iℓf ¼ x if also satisfies the equation; it is located on the 45 line. Figure 8.1.20 has a q-line corresponding to a feed containing both vapor as well as liquid, i.e. 0 < q < 1. Figure 8.1.21(b) illustrates the q-line for a variety of feed conditions representing different values of q: q ¼ 1, vertical line, saturated liquid; q ¼ 0, horizontal line, saturated vapor; 0 < q < 1, mixed vapor and liquid; q > 1, subcooled liquid; q < 0, superheated vapor. Example 8.1.8 Figure 4.1.1(c) illustrates the vapor–liquid equilibrium diagram for a benzene–toluene system at 1 atm. A distillation column fed with 3 kgmol/min of a 50– 50 benzene–toluene mixture is producing a distillate having 95% benzene–5% toluene and a bottoms product containing

q¼

H vf − H f : H vf − H ℓf

However,

H vf − H ℓf ¼ latent heat of vaporization of the feed at feed temperature ¼ 32 000 J=gmol:

Further,



H vf − H f ¼ H vf − H ℓf þ H ℓf − H f ¼ H vf − H ℓf − H f − H ℓf ¼ 32 000 J=gmol 18 000 J=gmol ¼ 14 000 J=gmol:

Therefore

q¼

14 000 ¼ 0:4375: 32 000

The slope of the q-line (equation (8.1.156c)) is

ðq=ðq 1ÞÞ ¼ ð0:4375= ð0:5625ÞÞ ¼ 0:777:

It starts at x 1f ¼ 0:5 on the 45 line; as shown in Figure 8.1.22, draw it with the calculated slope of –0.777 to intersect the enriching section operating line. Draw a line between this point of intersection and x1ℓb ¼ 0.05 on the 45 line; that is the operating line of the stripping section. slope of this stripping section operating line is
The  W btℓ =W btv ; the determination of the values of Wtℓb ¼

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

715

1

0.75

q - line

x1v

0.5

0.317 0.25

1

0

0.05

0.25

0.5 x1l

0.75

0.95

Figure 8.1.22. McCabe–Thiele diagram for Example 8.1.8.

4.312 kgmol/min and Wtℓb ¼ 2.812 kgmol/min may be carried out af follows. We know from relation (8.1.156c) that

W btv − W ttv ¼ W tf ðq − 1Þ ¼

3 kgmol ð0:4375 − 1Þ: min

Since R ¼ 2,


 R ¼ W ttℓ =W tℓd ) W ttℓ ¼ 2 W tℓd :

From (8.1.138a),

W ttℓ þ W tℓd ¼ W ttv ¼ 3 W tℓd : Therefore

W btv ¼ W ttv − 3  0:5625 ) W btv ¼ 3 W tℓd − 1:6875 ¼ 4:5 − 1:6875 ¼ 2:812 kgmol=min; W btℓ ¼ W btv þ W tℓb ¼ 2:812 þ 1:5 ¼ 4:312 kgmol=min:

(3) To find out the number of equilibrium stages, start drawing steps from the distillate end, at the intersection of the 45 line and x1ℓ¼ 0.95. Keep drawing the steps, go over the intersection of the q-line with the operating lines and go down to the value x1ℓ¼0.05 on the 45 line. We find approximately 13 stages in total, 6 stages in the enriching section and the rest in the stripping section (Figure 8.1.22).

8.1.3.2 Two limits of reflux ratio: total reflux and minimum reflux The reflux ratio R, defined by equation (8.1.137) as (Wtℓt/ Wtℓd), has considerable influence on the separation achieved in a distillation column. We will focus here on

two limits of column operation: (a) the reflux ratio R achieved when Wtℓd¼0; (b) the reflux ratio for the highest possible value of Wtℓd. When Wtℓd¼0, the reflux ratio R becomes infinite, no distillate product is withdrawn and Wtℓt¼Wtvt (see equation (8.1.138a)). This is achieved by condensing the overhead vapor stream completely and returning it into the column with no product withdrawal. At the same time, the bottoms product withdrawal rate Wtℓb¼0 for total reboil, i.e. the liquid stream leaving the column bottom, is completely evaporated in the reboiler and the vapor is returned to the column. For steady state operation, the feed flow rate Wtf into the column has to be zero. These conditions define total reflux: W tℓd ¼ 0 ; W ttℓ

¼

W tℓb ¼ 0 ;

W ttv

;

W btℓ

W tf ¼ 0 ;

¼ W btv ,

ð8:1:158Þ

where the final two equalities follow from the bottom product withdrawal rate being zero. Correspondingly, the slope of the enriching section operating line, Wtℓt/Wtvt, and that for the stripping section operating line, Wtℓb/Wtvb, become equal to 1. Both operating line equations coincide with the 45 line where xiv¼xiℓ (Figure 8.1.23(a)). If now one were to draw the steps between the equilibrium curve and the operating line(s), we would have the least numbers of steps, and therefore the equilibrium stages needed to change the composition from the column bottom composition x iℓN ¼ x ivðNþ1Þ ¼ x iℓb (where Wtℓb¼0

Bulk flow of two phases ⊥ to direction of force

716

(a)

(b)

1.0

Equilibrium curve

Equilibrium curve

A

B

xiv

xild R+1 45° line

45° line

xiv

q-line

C

0

xilb

xild

xif xil

0

1.0

xilb

xild

xif xil

1.0

(d)

(c) 1.0

1.0 B

Equilibrium curve

A

0.9

A

0.8 0.7

x1v

xiv

0.6 45° line q-line

B

0.5 0.4 0.3 0.2 0.1

C 0

xilb

xif

xild

1.0

0 0.05 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

xil

xil

Figure 8.1.23. (a) McCabe–Thiele diagram for total reflux distillation and the minimum number of stages needed; (b) minimum reflux ratio and the operating lines; (c) minimum reflux ratio and the enriching section operating line being tangent to the equilibrium curve; (d) graphical determination of minimum reflux ratio and the number of ideal stages at total reflux for benzene–toluene distillation in Example 8.1.8.

for total reboil) to the column top composition x iv1 ¼ x iℓd (where Wtℓd¼0) in a column of N plates. Essentially, the composition change being achieved on any stage or plate is the largest possible value. Another way to look at “total reflux” would involve considering the composition change achievable in a distillation column having a certain number of plates or ideal stages. The reflux would achieve the highest possible change in composition in the distillation column containing a certain number of plates. Figure 8.1.23(a) illustrates the few steps (ideal stages) between the particular equilibrium curve and the total reflux operating line, which is the 45 line.

Fenske (1932) developed an equation to estimate the number of ideal stages needed under total reflux to go from the bottoms composition x iℓb to the top distillation composition x iℓd . We will start from the reboiler and then the bottommost plate N. The total reboil mode of operation without any bottoms liquid product means that the following relation is valid for the reboiler: 


 x ivðN þ 1Þ = 1 − x ivðN þ 1Þ ¼ ½x iℓb =ð1 − x iℓb Þ

ð8:1:159Þ


 since x iℓb ¼ x ivðN þ 1Þ . Due to the total reflux condition, we have already observed that x ivðN þ 1Þ ¼ x iℓN . Therefore

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces x iℓN =ð1 − x iℓN Þ ¼ ½x iℓb =ð1 − x iℓb Þ:

ð8:1:160Þ

Focus now on the bottommost plate N, where, under total reflux, ½x ivN =ð1 − x ivN Þ ¼ αijN ½x iℓN =ð1 − x iℓN Þ:

ð8:1:161Þ

Introduce relation (8.1.160) now in this result: ½x ivN =ð1 − x ivN Þ ¼ αijN ½x iℓb =ð1 − x iℓb Þ:

ð8:1:162Þ

Focus now on the next plate up, N − 1, where 
 
 x ivðN − 1Þ = 1 − x ivðN − 1Þ ¼ αijðN − 1Þ x iℓðN − 1Þ = 1 − x iℓðN − 1Þ ;

ð8:1:163Þ

but, under total reflux, x iℓðN − 1Þ ¼ x ivN . Therefore 
 x ivðN − 1Þ = 1 − x ivðN − 1Þ ¼ αijðN − 1Þ αijN ½x iℓb =ð1 − x iℓb Þ:

ð8:1:164Þ

In this fashion, we can go all the way up to the column top: ½x iv1 =ð1 − x iv1 Þ ¼ αij1 αij2    αijN ½x iℓb =ð1 − x iℓb Þ: ð8:1:165Þ However, with total reflux, x iv1 ¼ x iℓd . Therefore ½x iℓd =ð1 − x iℓd Þ ¼ αij1 αij2    αijN ½x iℓb =ð1 − x iℓb Þ: ð8:1:166Þ Using an average relative volatility αij ,
N ½x iℓd =ð1 − x iℓd Þ ¼ αij ½x iℓb =ð1 − x iℓb Þ,

ð8:1:167Þ

where


N αij ¼ αij1 αij2    αijN :

ð8:1:168Þ

This result, the Fenske equation, is commonly illustrated as follows:

 x iℓd ð1 − x iℓb Þ log ð1 − x iℓd Þ x iℓb , ð8:1:169Þ N ¼ N min ¼ log αij where Nmin is the minimum number of plates or stages required under total reflux to achieve a distillate composition of x iℓd and a reboiler bottoms composition x iℓb under total reboil. Often αij is used as a geometric mean of αij1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and αijN, i.e. αij1 αijN , provided the variation along the column is minimal. The total reflux condition is usually employed during distillation column startup; then slowly one can start withdrawing products from the column top and the column bottom. Normal column operation involves product withdrawls at both the column top and the column bottom. The column performance in terms of composition changes achieved will depend on the reflux ratio employed. If the reflux ratio (Wtℓt/Wtℓd) is decreased by increasing Wtℓd, the distillate product withdrawl rate, the slope of the enriching section operating line, will decrease. As one keeps on decreasing R, the number of plates required to achieve a certain extent of composition change increases. Ultimately,

717

location, B, where the enriching section operating line AB and the q-line intersect (the feed plate region), hits the equilibrium curve and we reach the limit of minimum reflux ratio, Rmin (Figure 8.1.23(b)). Point B on the equilibrium curve may be reached only via an infinite number of ever-decreasing smaller steps, effectively an infinite number of stages. Point B is identified as a pinch point, and leads to the other limit of the reflux ratio, the limit of minimum reflux,
 Rmin ¼ W ttℓ =W tℓd min : ð8:1:170Þ

Practical operating conditions in a column utilize R > Rmin; usually, it varies between 1.1Rmin and 1.5Rmin, depending on the separation factor α12. For higher values of α12, higher values of R are utilized; for lower α12, lower values are used. It is based on an optimization of the capital costs of the distillation unit and the operating costs of running the column. As the reflux ratio increases, the number of stages required decreases, which reduces the capital cost. Simultaneously, the operating costs increase since the amount of heat utilized in the reboiler/condenser increases (see equation (10.1.40)). Therefore an optimum cost is usually obtained at 1.1Rmin < R < 1.5Rmin. The equilibrium curve shown in Figures 8.1.23(a) and (b) is that of a fairly well-behaved system exhibiting ideal solution behavior. Often the systems behave nonideally. Extreme nonideal behaviors have been illustrated in Figures 4.1.3 and 4.1.4. We are not considering here such systems forming azeotropes. However, there are systems where, near the top of the column, the enriching section operating line may become a tangent to the equilibrium curve; Figure 8.1.23(c) shows such a system with the contact point B between the operating line and the equilibrium curve; contact point B is also considered as a pinch point. These behaviors occur when the relative volatility α12 is not constant over the whole composition range. It is possible to obtain an analytical expression for Rmin, provided the equilibrium curve is concave downward over the whole composition range, as shown in Figure 8.1.23(b) and we have straight operating lines. At location B on the enriching section operating line and equilibrium curve intersection, the q-line also intersects. Underwood (1932) utilized the fact that the intersection of the operating line for the enriching section (equation (8.1.138b)) and the q-line (equation (8.1.157)) lie on the equilibrium curve point B, a pinch point. Denoting this location via the composition coordinates (x iℓB , x ivB ), we can rewrite equations (8.1.138b) and (8.1.157) as x ivB ¼

Rmin 1 x iℓd ; x iℓB þ Rmin þ 1 Rmin þ 1

ð8:1:171Þ

x ivB ¼

q 1 x iℓB − x if : q−1 q−1

ð8:1:172Þ

Bulk flow of two phases ⊥ to direction of force

718 Simultaneous solution of these two equations yields x iℓB ¼

ðRmin þ 1Þx if þ ðq − 1Þx iℓd ; Rmin þ q

x ivB ¼

Rmin x if þ x iℓd q : Rmin þ q ð8:1:173Þ

Since this point lies on the equilibrium curve, the two coordinates are related by the separation factor or relative volatility relation (4.1.24) between two species i and j: αij ¼

x ivB ð1 − x iℓB Þ : ð1 − x ivB Þ x iℓB

ð8:1:174Þ

Substituting relations (8.1.173) for x iℓB and x ivB in definition (8.1.174), we obtain the following general relation: R x þ q x iℓd
min if Rmin 1 − x if þ qð1 − x iℓd Þ ¼

αij fx iℓd ðq − 1Þ þ x if ðRmin þ 1Þg
 : ðRmin þ 1Þ 1 − x if þ ðq − 1Þð1 − x iℓd Þ

ð8:1:175Þ

For specific cases, e.g. feed is a saturated liquid so that q ¼ 1, we can simplify the above to obtain " # 1 x ð1 − x iℓd Þ  iℓd − αij
 , Rmin ¼
ð8:1:176Þ x if αij − 1 1 − x if

a result often identified as the Underwood equation. When xiℓd !1 for high-purity distillate products, it is simplified to Rmin ¼


1 1  ; αij − 1 x if

W ttℓ ¼


W tf 1 W tℓd :  ¼
αij − 1 αij − 1 x if

ð8:1:177Þ

When the feed is introduced as a saturated vapor, i.e. q ¼ 0, the corresponding result is ! αij x iℓd ð1 − x iℓd Þ 1

 − 1: ð8:1:178Þ − Rmin ¼ x if αij − 1 1 − x if Example 8.1.9 Consider Example 8.1.8 for the separation of a 50–50 benzene–toluene mixture producing a distillate having 95% benzene and a bottoms product having 95% toluene. All conditions of operation, including that of the feed, remain the same as in Example 8.1.8. Determine the following: (a) the minimum reflux ratio given the separation factor αbenzene-toluene¼2.5 near the feed plate; (b) the total number of ideal equilibrium stages if the column is operated at total reflux (αbenzene-toluene¼ 2.5). Employ both graphical and analytical approaches. Solution (a) The minimum reflux ratio may be determined analytically as well as graphically. For an analytical solution, employ the general relation (8.1.175) for the unkown Rmin ; here, x 1f ¼ 0:5, x 1ℓd ¼ 0:95 and q ¼ 0.4375 for species i ¼ 1 ¼ benzene:

Rmin  0:5 þ 0:4375  0:95 Rmin  0:5 þ 0:4375  0:05 ¼

2:5f0:95ð − 0:5625Þ þ 0:5 ðRmin þ 1Þg ðRmin þ 1Þð0:5Þ þ ð − 0:5625Þð0:05Þ

) 0:375 R2min − 0:4596 Rmin − 0:1978 ) Rmin ¼ 1:56: For the graphical solution, draw the q-line in Figure 8.1.23(d) with a slope of (q/(q − 1)) ¼ –0.777 from x 1f ¼ 0:5 on the 45 line. It intersects the equilibrium curve at point B. Draw a line from this point to point A on the 45 line, where x 1ℓd ¼ 0:95. The slope of this line is (Rmin/(Rmin þ 1)); from the graph, we find the value to be 0.6, leading to Rmin ¼ 1.5. The difference between the two values is due to inaccuracies in graphical measurement of the slope. (b) We will first calculate the total number of ideal equilibrium stages needed under total reflux via the Fenske equation (8.1.169):

x 1ℓd ¼ 0:95; x 1ℓb ¼ 0:05, αij ¼ 2:5;



0:95 0:95 N min ¼ log  logð2:5Þ; 0:05 0:05 N min ¼ log½361=0:3979 ¼ 2:557=0:397 ¼ 6:427: Graphically in Figure 8.1.23(d), we have drawn the steps from x 1ℓd ¼ 0:95 on the 45 line to the equilibrium line and down to the 45 line in the manner of Figure 8.1.23(a). The number of stages so found is ~7 (¼ Nmin).

8.1.3.3

Additional modes of distillation column operation

There are a number of additional modes of operating a distillation column. These include: partial condenser; open steam introduced at the column bottom without a reboiler; enriching distillation column; stripping distillation column; operation with a side stream, etc. We will briefly describe each one of these in the context of equilibrium stages/ plates and constant molal overflow. 8.1.3.3.1 Partial condenser Sometimes an overhead condenser is operated such that all of the vapor stream leaving the top plate in the column is not condensed, and the noncondensed vapor is withdrawn as the product instead of the condensed liquid as the product. Figure 8.1.24(a) illustrates this mode of operation. The overall and more volatile species molar balance equations for this condition over the dashed envelope crossing the column between the nth and (n þ 1)th plate are: W ttv ¼ W ttℓ þ W tvd ;

ð8:1:179aÞ

W ttv x ivðnþ1Þ ¼ W ttℓ x iℓn þ W tvd x ivd :

ð8:1:179bÞ

The operating line equation of the enriching section may be obtained from equation (8.1.179b) as x ivðnþ1Þ ¼

W ttℓ W tvd x iℓn þ t x ivd : W ttv W tv

ð8:1:180aÞ

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

(a)

719

(b)

xil Partial condenser

Plate 1

xivd

2 W ttl

Plate 2

Operating line for partial condenser

Equilibrium line

xild

B Enriching section operating line

n n+1

45° line xiv

(xil1,xiv1)

Wtvd

1

3

A

(xild, xivd)

xiv1

xil1

xild xil = xivd

Figure 8.1.24. (a) Schematic for a partial condenser in a distillation column (feed and stripping section not shown). (b) McCabe–Thiele diagram for the partial condenser and the enriching section operating line.

Using the overall molar balance equation, this is simplified to   Wt Wt x ivðn þ 1Þ ¼ ttℓ x iℓn þ 1 − ttℓ x ivd , ð8:1:180bÞ W tv W tv
 a line with a slope of W ttℓ =W ttv . If the partial condenser is operated such that xiℓd and xivd are in equilibrium, then one can write for the partial condenser also W ttv ¼ W ttℓ þ W tvd , W ttv x iv1 ¼ W ttℓ x iℓd þ W tvd x ivd ,

ð8:1:181aÞ ð8:1:181bÞ

leading to x ivd ¼ −

W ttℓ Wt x iℓd þ tv x iv1 , W tvd W tvd

ð8:1:182Þ

which relates x iℓd and x ivd , the compositions of the two product streams from the condenser; this straight line has
 a slope of −W ttℓ =W tvd . Figure 8.1.24(b) shows a McCabe– Thiele diagram having two operating lines, one for the enriching section (slope, W ttℓ =W ttv Þ, the other for the par
 tial condenser (slope, −W ttℓ =W tvd . The latter operating line intersects the 45 line at x iℓ ¼ x iv1 ¼ x iℓd . 8.1.3.3.2 Open steam introduced at the column bottom without a reboiler If the bottoms product in a distillation column is essentially water, then one could avoid having a reboiler and introduce steam directly at the bottom of the column as if it is coming from the reboiler. The molar steam introduction flow rate is Wtvb and the corresponding mole fraction for the more volatile species, xiv(N þ 1), is equal to zero (Figure 8.1.25(a)). The enriching section remains unchanged in this mode of operation: if the reflux ratio R is specified, the operating line (8.1.138b) may be plotted knowing x iℓd . However, in this mode of operation, the stripping line equation is somewhat changed. (The q-line

equation is unchanged; see Figure 8.1.25(b).) To understand the column behavior, consider the following molar balances. For the overall column: W tf þ W btv ¼ W tℓd þ W btℓ ;

ð8:1:183Þ

for the overall column, more volatile species: W tf x if ¼ W tℓd x iℓd þ W btℓ x iℓN :

ð8:1:184Þ

However, x iℓN ¼ x iℓb ,

W btℓ ¼ W tℓb :

ð8:1:185Þ

So, for the stripping section operating line we have W btvðm þ 1Þ x ivðm þ 1Þ þ W tℓb x iℓb ¼ W btv x ivðN þ 1Þ þ W btℓ x iℓm :

ð8:1:186aÞ

(Compare this with equation (8.1.139a).) Equation (8.1.186a) may be rearranged to yield W btvðm þ 1Þ x ivðm þ 1Þ þ W tℓb x iℓb ¼ W btℓ x iℓm ,

ð8:1:186bÞ

leading to x ivðm þ 1Þ ¼

W btℓ ðx iℓm − x iℓb Þ: W btv

ð8:1:187Þ

This stripping section operating line has a slope of
b  W =W btv , and it intersects the x iv -axis at –
btℓ  W tℓ =W btv x iℓb (when x iℓm ¼ 0); further, it intersects the x iℓ -axis at x iℓb (when x ivðN þ 1Þ ¼ x ivðm þ 1Þ ¼ 0). Therefore this operating line may be drawn as a line joining (x iℓb , x ivðN þ 1Þ ¼ ð0Þ) at the x iℓ -axis with the point of intersection of the q-line with the enriching section operating line (Figure 8.1.25(b)). As a result, stage construction at the column bottom begins with the point ðx iℓ ¼ x iℓb ¼ x iℓN , x ivðN þ 1Þ ¼ 0Þ, goes up, hits the equilibrium diagram, a horizontal line is drawn, and so on.

Bulk flow of two phases ⊥ to direction of force

720

(a) Cooling water

Wtld Distillate product

1 2

(b)

1

q-line

Wtf f

Equilibrium curve Enriching section operating line

m

xiv

m+1

N

W btl xilb xilN

W btv (Steam flow in) xiv(N+1) = 0

45° line Stripping section operating line

0

xilb

xif

xild

1

xil

Figure 8.1.25. (a) Distillation column with open steam injection at the bottom instead of a reboiler; (b) the corresponding McCabe– Thiele plot.

8.1.3.3.3 Stage requirements for a distillation column having only an enriching section or a stripping section – Kremser equations Consider what a distillation column effectively achieves. The feed has a certain level of the more volatile species. In the enriching section of the column, the countercurrent exchange ends up developing a distillate product stream highly enriched in the more volatile species. The downflowing liquid from the feed plate has a considerable amount of the more volatile species; the stripping section strips this more volatile species from the downcoming liquid so that the bottoms product can become highly enriched in the less volatile species and is stripped of the more volatile species. These column separation capabilities suggest the following possibilities. From a feed that comprises only vapor that is introduced at the bottom of a column, one could expect to achieve a top distillate stream highly purified in the more volatile species if there is a condenser at the top and there is reflux of the condensate at the top. Therefore such a distillation column has only an enriching section (Figure 8.1.26(a)) which can produce a highly purified distillate. However, there will be considerable loss of the more volatile species in the liquid leaving the column bottom where the vapor feed is introduced since there is no stripping section in the column. Correspondingly if a liquid feed is

introduced at the top of a column, acting as a stripping section only with a reboiler at the bottom, one can achieve a highly purified bottoms product with very little contamination by the more volatile species (Figure 8.1.26(b)). However, the vapor leaving the top will have a composition close to that of the feed liquid. If the vapor–liquid equilibrium behavior can be described as linear in either of the two configurations discussed above and the operating lines are linear, we can use the Kremser equations (8.1.131)–(8.1.133) to determine the number of stages required to achieve the desired compostion change. The equilibrium relation is assumed to be x iv ¼ mi x iℓ :

ð8:1:188Þ

Note: For the conditions shown in Figures 8.1.26(a) and (b), the mode of operation is very similar to that of a type (2) separation system (Figure 8.1.2(a)), for example a gas absorber or a stripper. The condenser at the top of Figure 8.1.26(a) provides essentially a downflowing liquid stream which acts as the absorbent for the less volatile species. The reboiler at the bottom of Figure 8.1.26(b) provides an upflowing vapor stream which strips the downflowing liquid of the more volatile species as in a stripper.

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

(a)

721

(b) Vapor product

Cooling water

Distillate product

Liquid feed

Vapor feed Bottoms product

Liquid product

Figure 8.1.26. (a) Enriching distillation column; (b) stripping distillation column.

Therefore we will use different forms of the Kremser equation to estimate the number of ideal stages for the two different configurations. For Figure 8.1.26(a), an enricher, the result (8.1.133) is based on the more volatile species i in the gas stream going up from plate N þ 1 (feed stream) into the column of N plates. Here the feed vapor stream is introduced onto the last plate (N), which becomes the feed plate and goes up from plate N (composition xivN); therefore result (8.1.133) based on the more volatile species in the binary system is changed to 

   ð1 − x ivN Þ − ð1 − x iℓL Þmi 1 1 ℓn 1− þ A A ð1 − x iv1 Þ − ð1 − x iℓL Þmi N −1 ¼ , ℓn A ð8:1:189aÞ since it is based on the less volatile species. The vapor leaving the top plate n ¼ 1 has a composition xiv1 and the liquid composition entering the top from the condenser is xiℓL. The quantity A is defined as ðmi jW tℓ j=jW tv jÞ. Correspondingly, the number of plates required for the configuration of Figure 8.1.26(b), a stripping distillation column, may be obtained as follows (Treybal, 1980, p. 422): " #,
  x iℓL − x ivðN þ 1Þ =mi 1 1
 1− N þ 1 ¼ ℓn þ ℓn S, S S x iℓ0 − x ivðN þ 1Þ =mi where

S ¼ mi jW tv j=jW tℓ j:

ð8:1:189bÞ

The basis is that Figure 8.1.26(b) is acting as a stripper of the more volatile species present in the liquid feed;

therefore equation (8.1.134) is applicable for the more volatile species. Further, the reboiler acts as one more ideal stage, therefore N þ 1. 8.1.3.3.4 Column operation with a side stream/two feed streams In the simplest form of a distillation column operation, there is one overhead liquid product stream, one liquid reflux stream and one feed stream, one bottoms product stream and one vapor stream from the reboiler, in addition to a liquid stream going out at the bottom to the reboiler and one overhead vapor stream going out to the condenser. There are additional configurations in terms of the number of streams entering or leaving the column. When there is a need for additional product stream(s) beyond the two product streams (one from the top and the other from the bottom), a side stream or two may be withdrawn from the column. Figure 8.1.27(a) illustrates two such side streams: the first one, in between the feed and the liquid reflux at the top, is taken out as a liquid product of intermediate composition; the second one, in between the feed and the vapor input from the reboiler, is withdrawn as a vapor stream. The stage requirements for a column may be determined by considering the McCabe–Thiele plot for the case of only one liquid side stream between the feed and the top liquid reflux. As shown in Figure 8.1.27(b), the operating line CD for the stripping section remains unchanged since operation up to the feed introduction point from the bottom is unchanged. Similarly, the q-line for the feed plate region and its intersection with line CD is unchanged.

Bulk flow of two phases ⊥ to direction of force

722

(a)

(b)

Wtld 1

n

Wtf 1 Wtf

xild W ttl

s t W tv(s +1)

Equilibrium curve

W ttlsp , xilsp

xif

xiv

E m

Wtf 2

A

D

q-line

t W tls

m+1

B

F

W btvsp , xivsp 45°line

N

C xilb Wtlb

xif

xif xilsp

xild

xilb Figure 8.1.27. Column operation with additional product streams. (a) Schematic with two additional product streams, one in the enriching section, one in the stripping section; (b) McCabe–Thiele diagram for the column having one additional product stream in the enriching section.

The slope and the origin of the enriching section operating line AB is the same, except it ceases to be valid below the point of side stream withdrawal; at this location, due to the liquid side stream product withdrawal, the liquid flow rate up to the feed point changes and we have a new operating line EF, F being the point of its intersection with the q-line. If point E is fixed by xiℓsp, the liquid-phase composition of the liquid product being withdrawn and its intersection with the enriching section operating line AB, then knowing the slope of this intermediate operating line, one can draw it till it intersects the q-line and the stripping section operating line. For the envelope shown in Figure 8.1.27(a) around plate s somewhere above which the liquid product side stream is withdrawn at the rate W ttℓsp and the overhead product stream rate is Wtℓd, a total molar balance is given by W ttvðs þ 1Þ ¼ W ttℓs þ W tℓd þ W ttℓsp :

ð8:1:190Þ

Now, W ttℓn ¼ W ttℓ for the enriching section plates above the side product withdrawal plate, whereas W ttℓs is valid for plates below it. Therefore W ttℓ ¼ W ttℓs þ W ttℓsp :

ð8:1:191Þ

However, the vapor flow rates remain unaffected, i.e. W ttvðs þ 1Þ ¼ W ttvðn þ 1Þ ¼ W ttv ,

ð8:1:192Þ

where the plate number s is obviously larger than the enriching section plate number n. A balance on more volatile species i for the balance equation (8.1.190) yields W ttvðs þ 1Þ x ivðs þ 1Þ ¼ W ttℓs x iℓs þ W tℓd x iℓd þ W ttℓsp x iℓsp ;

ð8:1:193Þ

x ivðs þ 1Þ ¼

W ttℓsp x iℓsp

W tℓd x iℓd þ W ttℓs x iℓs þ W ttvðs þ 1Þ W ttvðs þ 1Þ

: ð8:1:194Þ

The slope of this intermediate operating line,
 W ttℓs =W ttvðs þ 1Þ ¼ W ttℓs =W ttv , is less than that of the top section operating line AB, namely W ttℓ =W ttv . The vertical step down for the plate where the side product is being withdrawn must intersect the two top operating lines, AB and EF, E being the point where they intersect. Figure 8.1.27(a) shows via dashed lines two feed streams having molar flow rates of Wtf1 and Wtf 2, respectively (see Problem 10.2.4 for an example). The condition of each feed stream dictates where it is going to be introduced into the column. For example, if one of the feed streams is

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

a vapor stream, it should be introduced in the bottom half of the column, where its composition is likely to match the local vapor stream composition. Similarly, if one of the feed streams is a liquid stream, it should be introduced at an appropriate location in the top half of the column. 8.1.3.4

Stage efficiency, tray efficiency, plate efficiency

Before we initiated the illustration of the McCabe–Thiele method of analyzing the distillation column (Section 8.1.3.1), we specified a basic assumption: on each stage/ plate/tray in the column, the exiting vapor and the liquid streams are in equilibrium with each other; therefore the stages/plates/trays are ideal stages. In reality, the two exiting streams from a given stage/plate are generally not in equilibrium. Consequently, the number of stages/plates N needed to achieve the required change in composition from the column top to the column bottom is larger than Nideal, the ideal number of stages predicted by the assumption of the exiting vapor and liquid streams for any stage being in equilibrium. This difference is usually described by an overall efficiency Eo: E o ¼ N ideal =N:

ð8:1:195Þ

The plate number N refers to the actual plates/trays in a distillation column. The correlation by O’Connell (1946) allows one to estimate the overall efficiency (given as a percentage)

when the relative volatility of the key components and the feed viscosity are known (Figure 8.1.28). Several empirical relations provide different approximations of O’Connell’s (1946) correlation. (1) Doherty and Malone (2001):  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eo − a ¼ exp − α12 ðμ=μ0 Þ , 1−a

ð8:1:196Þ

E o ¼ 50:3 ðα12 μÞ− 0:226 ,

ð8:1:197aÞ

where a ¼ 0.24, μo ¼ 10−3 Pa-s (1 cp), α12 is the relative volatility of the key components 1 and 2, μ is the viscosity of the liquid mixture at the feed composition (see Section 2.4.3 for definitions of the key components). (2) Seader and Henley (1998):

valid for a range of 0.1 to 10 cp for the product α12μ, where μ is in centipoise. Here Eo is in percent. (3) Wankat (2007): E o ¼ 0:52782 − 0:27511 log10 ðα12 μÞ þ0:044923 ðlog10 ðα12 μÞÞ2 ,

ð8:1:197bÞ

where μ is in centipoise. The overall efficiency is ultimately a reflection of the performance being achieved on each plate/tray/stage. The performance of any tray (say the nth) is usually judged by the Murphree vapor efficiency EMV or the Murphree liquid

Commercial hydrocarbon fractionation column Commercial chlorinated hydrocarbon fractionation column Commercial alcohol fractionation Laboratory fractionation of ethyl alcohol Miscellaneous

100 90 80 Plate efficiency (%)

723

70 60 50 40 30 20 10 0 0.1

0.2

0.6 0.8 1.0 2.0 4.0 0.4 Relative volatility of key component ⫻ viscosity of feed (at average column conditions)

6.0

8.0 10

Figure 8.1.28. Effect of relative volatility and viscosity on plate efficiency of fractionating columns. Reprinted, with permission, from O’Connell, Trans. AIChE, 42, 741 (1946). Copyright © [1946] American Institute of Chemical Engineers (AIChE).

Bulk flow of two phases ⊥ to direction of force

724 efficiency EML for the more volatile species i in a binary mixture: E MV ¼

x ivn − x ivðn þ 1Þ ; x ivn − x ivðn þ 1Þ

E ML ¼

x iℓn − x iℓðn − 1Þ : ð8:1:198Þ x iℓn − x iℓðn − 1Þ

Here x ivn and x iℓn are the actual vapor and liquid compositions, respectively, leaving the nth tray, while x ivn and x iℓn are hypothetical vapor and liquid compositions, respectively, leaving the nth tray that are in equilibrium with the actual liquid composition, x iℓn , and the actual vapor composition, x ivn , respectively, leaving the nth tray; correspondingly, x ivn ¼ f eq ðx iℓn Þ;

x iℓn ¼ g eq ðx ivn Þ,

ð8:1:199Þ

where functions feq and geq represent vapor–liquid equilib
 rium relations. Correspondingly, x ivn − x ivðn þ 1Þ is the actual vapor-phase enrichment achieved in plate n, whereas (x ivn − x ivðn þ 1Þ ) would have been the vapor-phase enrichment had the vapor phase been in equilibrium with the liquid phase leaving the nth tray. Similarly, (x iℓn − x iℓðn − 1Þ ) is the actual liquid-phase composition change achieved in plate n compared to the change (x iℓn − x iℓðn þ 1Þ ) which would have been achieved had the liquid phase leaving tray n been in equilibrium with x ivn . The notion of a Murphree stage efficiency was introduced in Section 6.4.1.2 for solvent extraction in a well-mixed

extractor. However, the overall flow pattern on a distillation plate/tray is quite different and involves the crossflow of two different immiscible phases with an overall direction of force being perpendicular to both flowing phases (Section 8.3.2). If the value of EMV is available for a given stage/plate,
 then knowing x ivn − x ivðn þ 1Þ one can determine the actual change achieved (x ivn − x ivðn þ 1Þ ). Such a procedure provides a basis for the graphical determination of the total number of actual plates in the McCabe–Thiele method. Figure 8.1.29 illustrates this approach, with xiℓb providing the start of the vertical line. Focus on the bottom of the column to the last plate and on the horizontal line intersecting the operating line at (x iℓN , x ivðN þ 1Þ ), say point G. When we go up from this point, we intersect the equilibrium curve at (x iℓN , x ivN ), point H. The distance GH, i.e. (x ivN − x ivðN þ 1Þ ), is the maximum vapor-phase composition change achievable in plate N, since x ivN is the equilibrium composition valid for an ideal stage. Multiply this vertical distance by EMV ( 0) axially dispersed flow of fluid phase j ¼ 2. The governing balance equation from equation (6.2.34) is therefore ε1

∂C i1 ∂C i2 ∂C i1 ∂2 C i2 ∂C i2 þ ε2 þ hv1z i þ hv2z i ¼ ε2 Di, eff , 2 2 ; ∂t ∂t ∂z ∂z ∂z ð8:1:364Þ ε1 ∂C i1 ∂C i2 ∂2 C i2 ∂C i2 ε1 ∂C i1 þ ¼ Di, eff , 2 þ jv1z j : − v2z ε2 ∂t ∂t ∂z 2 ∂z ε2 ∂z ð8:1:365Þ

Note that since there are only two phases, ε1 ¼ ð1 − ε2 Þ. Consider steady state: the terms on the left-hand side disappear. We now employ the linear driving force assumption (7.1.5b) along with linear equilibrium relation (7.1.18a), C i1 =C i2 ¼ κi1 and the relation jv1z j ¼ − ðz=t Þ relating the solid-phase velocity to time and axial coordinates: 
  ∂ C i1 15 Dip
 ¼ C i1 − C i1 ¼ k ip C i1 − C i1 r 2p ∂t ð8:1:366Þ

These assumptions allow us to express equation (8.1.365) as
 ∂C i2 ∂C i2 ð1 − ε2 Þ − k ip κi1 C i2 − C i1 ¼ 0: − v2z ε2 ∂z 2 ∂z ð8:1:367Þ

The overall species i balance over the adsorber column length L for a dilute mixture of species i to be adsorbed may be carried out by assuming that v2z and jv1z j remain essentially constant along the column:







The boundary conditions similar to those for a gas absorber (8.1.91a–d) are in this case as follows. For z ! 0 ðor z ¼ 0þ Þ, ε2 v2z C i20 − ε2 v2z C 0i20 ¼ − ε2 Di, eff , 2

d C i2 ; dz

ð8:1:369aÞ

d C i1 ¼ 0; dz

ð8:1:369bÞ

For z ! L ðor z ¼ L − Þ, d C i2 ¼ 0: dz

ð8:1:369cÞ

Using the following dimensionless quantities: ϕ ¼ C i2 =C i20 ; St ¼

k ip L ; v2z

z þ ¼ z = L; γ¼

Pe ¼

ð1 − ε2 Þ κi1 jv1z j , ε2 v2z

v2z L ; Di, eff , 2 ð8:1:370Þ

Ruthven (1984) has expressed the solution of the differential equation (8.1.367) and boundary conditions (8.1.369a–c) in a dimensionless form, ϕð1 − ð1=γÞÞ þ ð1=γÞ − ðC i10 =κi1 C i20 Þ 1 − ðC i10 =κi1 C i20 Þ m1 exp ðm1 þ m2 z þ Þ − m2 exp ðm2 þ m1 z þ Þ : ¼ m1 ðexp m1 Þ ð1 − ðm2 =PeÞÞ − m2 ðexp m2 Þ ð1 − ðm1 =PeÞÞ

ð8:1:371Þ

See equations (8.1.373) and (8.1.374) below for m1 and m2. For an adsorption process where γ > 1, C i1L ¼ 0, the overall mass balance relation (8.1.368) yields
 1 − C i2L =C i20 C i10 , ð8:1:372Þ ¼ κi1 C i20 γ which reduces the general solution (8.1.371) (Ruthven, 1984) to ϕðγ − 1Þ þ ϕL γ − 1 þ ϕL m1 exp ðm1 þ m2 z þ Þ − m1 exp ðm2 þ m1 z þ Þ ¼ , m1 ðexp ðm1 ÞÞ ð1 − ðm2 =PeÞÞ − m2 ðexp ðm2 ÞÞ ð1 − ðm1 =PeÞÞ ð8:1:373Þ


 ∂ C i1 : ¼ k ip κi1 C i2 − C i1 ¼ − jv1z j ∂z

Di, eff , 2

759



ε2 v2z C i20 − C i2L ¼ − jv1z j ð1 − ε2 Þ C i1L − C i10 : ð8:1:368Þ

where m1 (þve sign) and m2 (–ve sign) are the roots of the following equation:  1=2 o 1n , ðPe þ St Þ ðPe þ St Þ2 þ 4 PeSt ðγ − 1Þ m¼ 2 ð8:1:374Þ leading to

where A¼


 ϕL ¼ C i2L =C i20 ¼

γ−1 , ðγ=AÞ − 1

ð8:1:375Þ

m1 − m2 : m1 ðexp ð − m2 ÞÞ ð1 − ðm2 =PeÞÞ − m2 ðexp ð − m1 ÞÞ ð1 − ðm1 =PeÞÞ

ð8:1:376Þ

Bulk flow of two phases ⊥ to direction of force

760 8.1.6.3

Simulated moving bed (SMB) process

We have already introduced the basic notion of the SMB process in the material accompanying Figure 8.1.6. Here we elaborate on it. Consider Figure 8.1.45(a), which shows four packed adsorbent beds each of length L (they could be different lengths) located in one column (they do not have to be located in the same column; one can have four separate packed-bed columns connected to one another externally (Wankat, 1990, p. 527)). The solid lines with arrows in Figure 8.1.45(a) indicate the lines along which the mobile phase having a particular composition is moving. The dashed vertical line on the left of this figure identifies the hypothetical recirculation of the adsorbent particles from the column bottom to the top for the hypothetical countercurrent system. Consider now how SMB operates. As indicated earlier, if the adsorbent particles were actually coming down in any one of the four beds, the relative velocity between the upward mobile phase and the downward adsorbent particles will be significantly higher than the actual upward mobile-phase velocity. Under such a condition, the mobile phase would have been in contact with particles higher up in the bed than it is in actual contact within a fixed bed. Therefore, after a defined time period, t  , called the switching time, the mobile phase is introduced into a bed (a)

(b)

port at a higher up location in the packed-bed system. This arrangement is simultaneously implemented in four locations in between the four columns, as shown in Figure 8.1.45(b). Further, this switching is continued, as shown in Figures 8.1.45(c) and (d) till we are back to the original scheme in Figure 8.1.45(a); then the cycle is repeated. To understand the mechanics better, focus on the separation of species A (glucose) and B (fructose) from a feed solution using ion exchange resin in the Ca2þ form which adsorbs fructose strongly with respect to glucose. In bed 1 (Figure 8.1.45(a)), the goal is to produce purified desorbent for recycle to column 4 at the bottom; any glucose remaining after withdrawal of the raffinate product (enriched in glucose) should not be allowed to contaminate the desorbent product via breakthrough. One can calculate the time t needed for glucose breakthrough using the concentration wave velocity expression (7.1.13e) for vCA and the bed length L for an interstitial velocity vz L L ¼ vCA, 1 ) t ¼  t vCA, 1

(in vCA, 1 , the subscript 1 refers to column 1). Therefore, after a time t > t  where t  ¼ L=vport ,

vCA, 1 ¼ α1A vport ,

(c)

t* ³ t ³ 0

ð8:1:377Þ

α1A 1,

ð8:1:378Þ

(d)

2t* ³ t ³ t*

3t* ³ t ³ 2t*

4t* ³ t ³ 3t*

Desorbent

Hypothetical adsorbent recirculation

1

2

3

4

Raffinate

Feed

Extract

Make-up

product A

A+B

product B

desorbent

3

2

4 Extract

Feed

4

2

Make-up

Raffinate

product B

desorbent

product A

1 Desorbent recycle

Make-up desorbent (eluent)

Raffinate product A

product A

desorbent 1

Extract

4

Raffinate

Make-up

product B

A+B 3

1

2

Feed A+B 3

Feed

Extract

A+B

product B

Figure 8.1.45. Simulated moving bed operation with four beds and four equal time steps for separation of a binary mixture of glucose (A) and fructose (B).

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

vport being defined as the average port velocity (Wankat, 1990, p. 529, 2007, pp. 649–654), the liquid introduction at the top column 1 is now from the feed solution (Figure 8.1.45(b)) and the top column now behaves as if it were column 2 of Figure 8.1.45(a). Correspondingly, column 4 of Figure 8.1.45(a) now (in Figure 8.1.45(b)) behaves as if it were column 1 of Figure 8.1.45(a). Similarly, the other liquid introduction ports now have different liquid streams being introduced (see Figure 8.1.45(b))). In bed 2 of Figure 8.1.45(a), species A (glucose) moves up toward column 1 and species B (fructose), which is more strongly adsorbed, moves down. If the raffinate product stream from bed 2 (Figure 8.1.45(a)) is to be a glucose-enriched stream, then glucose will break through bed 2 during 0 to t  , i.e. before port switching. This suggests that the concentration wave velocity of glucose in bed 2, vCA, 2 , is vCA, 2 ¼ α2A vport

ðα2A  1Þ:

ð8:1:379Þ

Further, we do not want fructose to break through into the raffinate product stream. Therefore vCB, 2 ¼ α2B vport

ðα2B 1Þ:

ð8:1:380Þ

In bed 3 of Figure 8.1.45(a), we want similar behavior in terms of movement of glucose and fructose: glucose should move up and break through, whereas fructose should have a net downward movement and not break through into the stream exiting bed 3: vCA, 3 ¼ α3A vport

ðα3A  1Þ;

ð8:1:381Þ

vCB, 3 ¼ α3B vport

ðα3B 1Þ:

ð8:1:382Þ

On the other hand, in bed 4 of Figure 8.1.45(a), we want fructose to break through into the top exiting stream from which the extract product enriched in fructose is withdrawn before port switching occurs. Therefore vCB, 4 ¼ α4B vport

ðα4B  1Þ:

ð8:1:383Þ

An important item to be noted here is that the interstitial velocity vz is different in different beds because of the extract product withdrawal between beds 3 and 4, feed introduction between beds 3 and 2 and raffinate product withdrawal between beds 2 and 1 (Figure 8.1.45(a)). Therefore    vz bed 4 − vz bed 3 ¼ vz extract product , ð8:1:384Þ

assuming the same cross section for beds 3 and 4 and appropriate allowance for extract product mass balance. Similarly,    vz bed 2 − vz bed 1 ¼ vz raffinate product : ð8:1:385Þ Equations (8.1.378)–(8.1.383) have to be solved simultaneously with additional constraints of equations such as (8.1.384) and (8.1.385). See Wankat (1990, 2007) for

761

numerical illustrations. Earlier analyses of such a process developed at UOP are available in Ching and Ruthven (1984, 1985) and Ching et al. (1986). 8.1.7 Membrane processes of dialysis and electrodialysis We consider briefly the processes of countercurrent dialysis and electrodialysis. 8.1.7.1 Countercurrent dialysis, buffer exchange, hemodialyzer The process of dialysis, in which small solutes diffuse from a feed solution through a microporous membrane into the dialyzing solution, was illustrated in Section 4.3.1 via a membrane kept in a closed vessel. To achieve separation in a continuous fashion, we realized there that the dialyzer had to be open. Such a device is schematically shown in Figure 8.1.46(a). A small hollow fiber membrane based dialyzer used as an artificial kidney is illustrated in Figure 8.1.46(b). Both utilize countercurrent flow of feed solution (for example, blood) and the dialyzing solution on two sides of the membrane. In hemodialysis, blood flows through the bore of the hollow fiber, and the dialyzing solution flows countercurrently on the shell side. Small molecular weight metabolic waste products, like urea, uric acid, creatinine, etc., are transferred from the blood by diffusion through the water-filled membrane pores to the aqueous dialysate solution (Figure 8.1.46(c)). Traditionally, the pore radii of dialysis membranes are >1.5 nm (see the discussion following equation (4.3.15)) large enough for metabolic waste products to diffuse through rapidly, but not large enough to allow blood plasma proteins, for example serum albumin, to pass through. Many modern hemodialysis membranes have much larger pores, with pore radii in the range of 7.5 to 10 nm to facilitate removal of solutes of “middle” molecular weight; however, as soon as these membranes are exposed to blood, substantial protein adsorption occurs, eliminating the possibility of protein loss through membrane pores, whose open diameters become substantially reduced. The fiber wall thickness varies between 10 and 50 μm. The hollow fiber internal diameters are generally around 200 μm. For a comprehensive introduction to these topics, see Kessler and Klein (2001). A major nonhemodialysis application of countercurrent dialysis (CCD) is the buffer exchange of proteins in the biopharmaceutical industry. In multistep biopharmaceutical production processes, the buffer solution constituents often have to be exchanged for protein purification steps. The original buffer solution, for example, may contain, say, 0.5 M ammonium sulfate and 0.05 M sodium phosphate (plus protein and other constituents); the CCD

Bulk flow of two phases ⊥ to direction of force

762

(a)

(b)

Qf , Cif 0

Blood inlet

Qf , CifL

z

Header Hea

z=L

Tube Tub sheet Solution outlet Membrane Fibers Fibe Jacket Jack

z=0 Qd ,C id 0

Solution inlet

Qd , CidL

Blood outlet (d)

(c) Dialyzer

1.0 0.9 l = 3.0

0.8

Heparin pump

Air trap and air detector

0.7 hi

Air detector clamp

0.6

l = 1.0

0.5

l = 0.7

0.4 . 0.3

Venous line Arterial line

l=¥

l = 2.0

l = 0.4

0.2 0.1

l = 0.2

Blood pump 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (Qf /Qd )

Figure 8.1.46. Countercurrent dialyzer: (a) flow and process schematic; (b) hemodialyzer, device schematic; (c) hemodialysis, blood flow loop; (d) hemodialyzer efficiency ηi plotted against (Qf/Qd) for various values of parameter λ. (Part (d) after Michaels (1966).)

process should remove, say, 99% of both ammonium sulfate and sodium phosphate into an appropriate aqueous dialyzing solution (Kurnik et al., 1995). These authors have demonstrated that CCD is more efficient than size exclusion chromatography (see Section 7.1.5.1.7) for buffer exchange. Correspondingly, desalting of protein solutions can also be efficiently implemented using CCD. There have been many other industrial applications of dialysis, the earliest being the recovery of NaOH from cellulosesteeping liquors; the dialysis membrane blocked cellulosic polymers, but allowed NaOH molecules to diffuse from a 20% solution to an aqueous phase leading to a dilute

NaOH solution (~4%) as dialysate (Kessler and Klein, 2001). This reference also illustrates various types of dialysis membrane devices developed and used over the years. We will now employ a lumped model of a countercurrent dialyzer to illustrate its performance in the context of hemodialysis (Figures 8.1.46(a) and (b)). We will use a device-averaged overall solute-transfer coefficient K ic instead of employing a z-coordinate dependent analysis of the type illustrated in (8.1.46) for countercurrent phase equilibrium based devices. In Figure 8.1.46(a), a feed solution having a molar solute

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

concentration of C if 0 (for example, blood) enters at z ¼ 0 with a volumetric flow rate Qf , and treated solution (Qf , C if L ) exits at z ¼ L. The membrane area is Am. The dialyzing solution (Qd , C idL ) enters at z ¼ L and leaves at z ¼ 0 as (Qd , C id0 ). The overall molar balance relations for the transfer of solute i using a device-averaged K ic are:

 Qf C if 0 − C if L ¼ Qd C id0 − C idL ¼ K ic Am ðΔ C i Þℓm , ð8:1:386Þ

where ðΔ C i Þℓm ¼



 C if 0 − C id0 − C if L − C idL 

 ℓn C if 0 − C id0 = C if L − C idL

ð8:1:387Þ

is the overall logarighmic-mean concentration difference between the blood side and the dialysate driving the solute transport. Hemodialyzer performance is analyzed using a number of indices: solute transfer efficiency ηi , clearance C, and dialysance DB . The solute transfer efficiency ηi (Michaels, 1966), 

ηi ¼ C if 0 − C if L = C if 0 − C idL , ð8:1:388Þ represents the feed solution purification achieved as a fraction of the highest possible concentration change
 C if 0 − C idL in the bloodstream. Dialysance DB of the bloodstream is defined for a countercurrent dialyzer as (Sweeney and Galletti, 1964; Michaels, 1966) 
 
ð8:1:389Þ DB ¼ Qf C if 0 − C if L = C if 0 − C idL :

Therefore DB is merely Qf ηi . The clearance C of a solute i is essentially blood dialysance DB , where C idL ¼ 0:
 ð8:1:390Þ C ¼ Qf C if 0 − C if L =C if 0 :

We will now develop an expression for ηi in terms of known parameters and variables. From relation (8.1.386), we obtain
    C id0 − C idL  1 − Qf =Qd ¼ 1−
C if 0 − C if L

 C if 0 − C id0 − C if L − C idL
 : ð8:1:391Þ ¼ C if 0 − C if L

Employing this result, we can rewrite relation (8.1.386) as 
  C if 0 − C if L 1 − Qf =Qd 

 K ic Am ðΔ C i Þℓm ¼ K ic Am ℓn C if 0 − C id0 = C if L − C idL
 ¼ Qf C if 0 − C if L , ð8:1:392Þ resulting in

   # K ic Am 1 − Qf =Qd C if 0 − C id0 : ¼ ℓn Qf C if L − C idL "

ð8:1:393Þ

763

Define λ ¼ K ic Am =Qf :

ð8:1:394Þ

Therefore     C if 0 − C id0 ¼ exp λ 1 − Qf =Qd : C if L − C idL

ð8:1:395Þ

Now obtain the following two results:     C − C − C þ C if L idL if 0 id0 1 − exp λ 1 − Qf =Qd ; ¼ C if L − C idL

ð8:1:396Þ

    C − C   C if 0 − C id0 id0 idL ¼ − : Qf =Qd − exp λ 1 − Qf =Qd C if 0 − C if L C if L − C idL

ð8:1:397Þ

A little algebra leads to the following result, providing an expression for ηi :      1 − exp λ 1 − Qf =Qd C − C if L       ¼ if 0 ¼ ηi : C if 0 − C idL Qf =Qd − exp λ 1 − Qf =Qd

ð8:1:398Þ

Figure 8.1.46(d) illustrates a few curves from a more detailed plot by Michaels (1966) with λ as a parameter. As Qd is decreased for a given blood flow rate Qf , ηi decreases, since C id increases, leading to a reduced solute transfer rate. At any given Qf /Qd , ηi increases as λ increases, since the solute transfer rate is increased. There is an additional interpretation of the parameter λ. From an analysis of the packed countercurrent column design equation (8.1.54a), we know L¼

jW tg j NTU og : K ixg a Sc

However, K ixg a Sc L ¼ K ixg A , where A is the total interfacial area in the transfer device of total volume Sc L, the interfacial area/volume being a. In the dialyzer, A will be Am; therefore, NTU og ¼

K ixg a Sc L K ic Am ffi ¼λ Qf jW tg j

ð8:1:399Þ

for the dialyzer. Since NTU og is an estimate of how difficult the separation is, it provides a guide to how large the dialyzer should be (Kessler and Klein, 2001). These authors have plotted a quantity called the extraction ratioE, which  is equivalent to ηi when C idL ¼ 0, against λ, with Qf =Qd as a parameter. There are a number of other important issues in hemodialysis. So far, for solute transport through the membrane, we have assumed only diffusion; but the device design and pressure conditions employed may lead to ultrafiltration via convective motion through the pores. For an

Bulk flow of two phases ⊥ to direction of force

764 introduction to this topic, see Kessler and Klein (2001) for the considerable contribution of convection to the accelerated removal of larger molecules whose diffusive transport rates are quite low. Another important issue is the role of the boundary layer mass-transfer resistances. The local value of the overall solute-transfer coefficient K ic in such a configuration, illustrated by relation (3.4.103), may be described in terms of the local values of the feed side and dialysate side mass-transfer coefficients k if c and k idc : 1 1 1 1 1 ¼ þ þ : K ic k if c κim k im kidc

ð8:1:400Þ

Here the membrane mass-transfer coefficient k im can be described by (3.4.95c,d). It is known that each of the local boundary layer transfer coefficients, k if c and k idc , is a function of the distance from the liquid channel inlet z in the manner of (1/z)0.33. This z-dependence is influenced by the nature of the flow channel geometry. Consult Kessler and Klein (2001) to develop a better understanding of the important role of the boundary-layer transfer coefficients. Example 8.1.20 Countercurrent dialysis may be explored to eliminate substantially 0.05M sodium phosphate from a 1 mg/ml solution of bovine serum albumin (BSA) and 25 mM Tris buffer at a pH of 7 (via NaOH). The dialyzing buffer composition is 25 mM Tris and the pH is 7 (via HCl). The dialysate flow rate is 100 ml/min. The membrane module has 0.2 m2 surface area. The value of K ic for the phosphate salt for one membrane module is 0.020 cm/min and that for two modules in series is 0.015 cm/min. Deter
mine the values of C if 0 =C if L for sodium phosphate using the expression developed in Problem 8.1.22 for the following cases: (a) feed flow rate, 10 cm3/min; (b) feed flow rate, 20 cm3/min; (c) feed flow rate, 10 cm3/min and two membrane modules in series. Comment on the effects of feed flow rate and putting membrane modules in series.
 Solution Part (a) The expression for C if 0 =C if L is:     C if 0 Qd exp λ 1 − Qf =Qd   ¼ C if L Qd − Qf (here C idL ¼ 0);




C if 0 =C if L



 !, 0:020:2104 10 ð100 −10Þ 1− ¼ 100 exp 10 100 100 ¼  exp ð3:6Þ ¼ 40:67: 90

Part (b)     
 C if 0 =C if L ¼ Qd exp λ 1 − ðQf =Qd Þ = Qd − Qf 0 1 4 100 0:02  0:2  10  0:8A ¼ 6:18: exp@ ¼ 20 80 Part (c)




   100 0:015  0:4  104  0:9 ¼ 246: C if 0 =C if L ¼ exp 10 90

A lower feed flow rate leads to a much higher level of purification. Increasing the membrane area does lower the mass-transfer coefficient; however, the purification is increased drastically.

8.1.7.2

Countercurrent electrodialysis

We have been introduced to the ion transport processes taking place in an electrolyte-containing solution adjacent to a cation exchange membrane in the diluate chamber of an electrodialysis (ED) device in Section 3.4.2.5. Figure 3.4.9 illustrated there shows two anion exchange membranes (AEMs) and one cation exchange membrane (CEM) in the ED device. An ED device in practice has many AEMs and CEMs between the two electrodes; the assembly is sometimes known as a membrane stack. One configuration employed shows (Figure 8.1.47(a)) the brine feed to be desalted introduced into one chamber and the same brine feed introduced countercurrently as a waste stream into the adjacent chamber. However, as shown, the brine feed is introduced at one end of the cascade while the brine feed acting as the wash water is introduced at the other end of the cascade. The region containing one AEM, the diluate solution in between this membrane and the adjacent CEM on the right, the CEM and concentrate solution on the right of this CEM all the way upto the next AEM constitutes a cell pair. In Figure 8.1.47(a), the brine feed to be desalted is introduced on the left-hand side into chamber 1 between a CEM and an AEM denoted as C and A, respectively. It then is passed on to chambers 2, 3, 4 and 5, respectively. As it moves, Naþ ions from this solution are moved through CEMs to the right and Cl− ions move through the AEMs to the left. So it becomes progressively desalted as it moves further down. On the other hand, the brine waste feed introduced into chamber 9 becomes concentrated in NaCl as Naþ ions come into it from the CEMs on the left and Cl− ions come from the AEMs to the right; it finally leaves the cascade of flat cells as a concentrate. There are two electrode chambers into which brine/wash water will be introduced. The following reactions take place at the anode: ð1=2ÞH2 O ! Hþ þ ð1=4ÞO2 ðgÞ þ e − ; Cl − ! ð1=2ÞCl2 ðgÞ þ e − ,

ð8:1:401aÞ

leading to evolution of Cl2 ðgÞ and O2 ðgÞ. Various steps are taken to reduce the evolution of Cl2 ðgÞ and isolate this water (Shaffer and Mintz, 1980) exiting the anode chamber. At the cathode, the reaction H2 O þ e − ! OH − þ ð 1=2ÞH2 ðgÞ

ð8:1:401bÞ

occurs, leading to the formation of NaOH from the Naþ ions coming in and evolution of H2(g). Two alternative configurations introduce feed brine into each cell pair in a countercurrent fashion in the same cell pair (Figure 8.1.47(b)) or in a cocurrent fashion

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

(a)

765

(b) Brine feed Concentrate

Electrode wash stream

A

C

A

Anode 1

6

Electrode Brine waste feed wash stream

C

A

C

Na+

Cl–

Cl–

Na+

2

A

C

A

Diluate

C

C A C A C A C A C

Cathode 7

3

8

4

Cell pair

9

5

Diluate

(c)

Electrode wash stream

A C A C A C A C A C

HCl

Diluate H2, NaOH

Cathode

Anode

Brine feed

Electrode wash stream Brine concentrate

(d)

Concentrate Cl2, O2

Brine wash stream

CEM

AEM

dA

AEM

CEM

AEM Diluate

AEM

Concentrate

Brine feed Cell pair

Cell pair

Brine wash stream

Qbr Figure 8.1.47. Schematic of various electrodialysis cell configurations using CEMs and AEMs: (a) internally staged countercurrent configuration; (b) simple countercurrent configuration; (c) simple cocurrent configuration; (d) process/transport analysis in a cell pair.

(Figure 8.1.47(c)). A fourth configuration employs a number of such membrane stacks with the partly desalted brine product from one stack introduced into the next stack for additional desalting. We will now utilize the result (3.4.114) along with Ohm’s law and the dependences of various resistances on salt concentration to develop an expression for the extent of desalting in the diluate chamber. From equation (3.4.114), I t 1m I ðt 1s þ t 2s Þ ¼ ¼ I=F : F F

ð8:1:402Þ

Assume the CEM to be perfectly selective so that t 1m ¼ 1. So the rate at which Naþ ions are being transported through the membrane, and therefore the rate at which the solution is being desalted, is I=F gm-eq/s. We follow now the treatment of Shaffer and Mintz (1980). Consider a

distance dz along the brine flow direction in the diluate chamber, where the brine volumetric flow rate is Qbr liter/s and the entering brine normality at location z in the diluate is Ndil (Figure 8.1.47(d)); the magnitude of the decrease in brine normality due to ion transport is –dNdil, leading to the following expression for the total rate of salt transport over the length dz, −Qbr dNdil gm-equiv/s. Let the membrane area (of each of the CEMs and the AEMs) in the diluate chamber over the brine flow path length dz be dAm. The current density i amp/cm2 in this section of the cell pair (Figure 8.1.47(d)) for a total current dI is i¼

ϕp dI ) dI ¼ dAm , Rp dAm

ð8:1:403Þ

where ϕp is the voltage drop over the cell pair and Rp is the resistance of 1 cm2 cross section of the cell pair at

Bulk flow of two phases ⊥ to direction of force

766 location z. In most cases, resistance Rp (in units of ohmcm−2) may be described as Rp ¼

a1 ohm-cm−2 : N dil

δ ohm; A

ð8:1:405Þ

the unit of resistivity is ohm-cm: a 0.5 gm-equiv/liter solution of NaCl at 25  C has a resistivity value of 25 ohm-cm, whereas a 0.001 gm-equiv/liter solution’s resistivity is 8000 ohm-cm. In general, a small fraction of the current is not utilized in salt transport (it is lost, for example, in the brine connection channels at the end of the membranes); the fraction ηIF (pretty close to 1) utilized is called the current utilization factor. Introduce this factor in equation (8.1.403) and equate it to the gram-equivalents of salt transferred per unit time over 1 cm2 area at location z (Figure 8.1.47(d)): ϕp dI η dAm IF ¼ − Qbr dN diℓ : η ¼ F IF Rp F

ð8:1:406Þ

Introduce here expression (8.1.404) for the cell-pair resistance: − Qbr dN diℓ ¼

ϕp ηIF N dil dAm : a1 F

ð8:1:407Þ

Integrate over the membrane area 0 to Am,cp of the cell pair along which the diluate solution concentration decreases from N dil, 0 to N dil, L : −

Nð dil, L

N dil, o

ℓn



ϕp ηIF dN dil ¼ Qbr F a1 N dil

N dil, 0 N dil, L



¼

Amð, cp

dAm ;

0

ϕp ηIF Am, cp : Qbr F a1

ϕp ϕp i ¼ ¼ , N dil Rp N dil a1

ð8:1:404Þ

The cell pair resistance is the sum of the resistances of the AEM, the diluate chamber solution, the CEM and the concentrate chamber solution. Usually the resistance of the two membranes is quite low (around 3–20 ohm-cm−2) because of the high density of ions and ionic fixed charges in the membrane. However, the diluate chamber will have considerably fewer electrolytes than the concentrate chamber, indicating that the normality of the diluate chamber N dil will most likely control the solution resistance. For example, the solution resistance R for a NaCl solution may be represented in terms of the solution resistivity (ohm-cm), solution thickness δ (cm) and cross-sectional area A (cm2) via R ¼ ðresistivityÞ

Utilizing (8.1.403) and (8.1.404), one can express the current density i as

ð8:1:408Þ

ð8:1:409Þ

If Am, cp ¼ W  L where L is the brine path length and W is the membrane width, we get   ϕp ηIF W L: ð8:1:410Þ ℓn ðN dil, 0 =N diℓ, L Þ ¼ F a1 Qbr

ð8:1:411Þ

leading to ℓn ðN dil, 0 =N diℓ, L Þ ¼

ηIF F



i N dil



 W L ¼ ℓn ðDr Þ: Qbr

ð8:1:412Þ

The quantity ði=N dil Þ is an important one. An estimate of the limiting value of ði=N dil Þ is around 480 (milliamp-liter/ cm2-gm-equiv) when concentration polarization sets in (Shaffer and Mintz, 1980). Note: The result (8.1.412) includes the desalination ratio Dr defined by (2.2.1a) achieved in the diluate chamber. The analysis carried out and the result (8.1.412) achieved do not indicate any influence due to the overall cocurrent or countercurrent flow pattern; there is no apparent influence of the concentrate chamber concentration. Actually there is; see equation (3.4.115a). But the value of ðQim =δm Þ is so small that there is hardly any transfer of cations from the concentrate chamber to the diluate chamber through the CEM, and similarly for anions through the AEM. This is an example where an external force trumps the generally assumed influence of flow patterns in two fluid-phase systems. There is another important issue, however, which decides whether countercurrent or cocurrent flow pattern is to be used: pressure drop. The channel dimension across the flow between an AEM and an adjacent CEM is very small, ~1 mm; the membranes are thin to keep membrane resistances low. Yet the diluate stream has to flow over long distances (~150–300 cm) at a reasonable velocity to reduce polarization and increase the limiting current density ilim (see equation (3.4.117)). This leads to a considerable pressure drop along the length of the diluate channel as well as the concentrate channel. If the cocurrent flow is utilized, then the pressure difference across any membrane is reduced to a low level. In countercurrent flow, the pressure difference across the membrane is increased considerably. This is the major reason for adopting the cocurrent flow pattern (Figure 8.1.47(c)) in general in ED. However, Figure 8.1.47(a) with countercurrent flow and internal staging has one advantage – one can reduce the electrolyte concentration substantially. Configurations shown in Figures 8.1.47(b) and (c) require the diluate from one stack of membranes to be fed to additional stacks of membranes to achieve considerable demineralization. A few more details about ED devices will be useful. There may be as many as 150–300 cell pairs in an ED stack between the anode and the cathode. The value of ϕp for a cell pair varies between 0.5 and 3 volts, whereas the stack voltage may vary between 100 and

8.1

Countercurrent bulk flow of two phases/regions ⊥ to the forces

300 volts. Liquid velocities in a compartment in a cell pair may vary around 6 cm/s. Example 8.1.21 Determine the extent of desalination achieved in terms of the desalination ratio Dr in an ED stack in which the diluate chamber liquid velocity is 10 cm/s, the diluate chamber cell thickness is 0.1 cm, ηIF ¼ 0:9, the membrane length along the diluate flow path is 250 cm and ði=N dil Þ has the value of 1000 milliamp-liter/cm2-gm-equiv. The feed brine concentration is 3000 ppm. Solution We will employ equation (8.1.412):    η i W ℓn ðDr Þ ¼ IF L, F N dil Qbr where L ¼ 250 cm; ηIF ¼ 0:9; F ¼ 96 500 C/gm-equiv. So   i milliamp-liter ¼1000 2 N dil cm -gm-equiv cm3 amp liter 1000 1000 10 − 3  milliamp liter milliamp ; ¼ cm2 -gm-equiv

1 1 ¼ ; vbr  δ 10 cm  0:1 cm s amp 0:9 -cm 3 ℓnðDr Þ ¼ 2 -gm-equiv 1000 cm 96 500 C==gm-equiv 1 s=   250 cm 10  0:1 cm 2 Qbr ¼ vbr  W δ ) ðW =Qbr Þ ¼

¼

=  250 0:9  100=0 ¼ 2:3315 ) log10 ðDr Þ ¼ 1:011 =1 =0 96 50

) Dr ¼ desalination ratio ¼ 10:3:

A broader introduction to stage analysis in an ED cell pair has been provided by Mason and Kirkham (1959). Examples of applications of ion exchange membranes in ED from earlier literature is available in Mason and Juda (1959). Strathmann (2001) has provided a comprehensive overview of ED, including considerable details on ion exchange membranes. 8.1.8

Countercurrent liquid membrane separation

In Section 5.4.4, we studied a variety of chemical reaction facilitated separation where the reaction was taking place in a thin liquid layer acting as the liquid membrane; Figure 5.4.4 illustrated a variety of liquid membrane permeation mechnisms. Here we will identify first the structural configuration of the liquid membranes as they are used in separators with countercurrent flow pattern (as well as for the cocurrent flow pattern). There are three general classes of liquid membrane structures: emulsion liquid membrane (ELM); supported liquid membrane (SLM) or immobilized liquid membrane (ILM); hollow fiber contained liquid membrane (HFCLM). Each will be described very briefly. A most comprehensive description of the emulsion liquid membrane is available in Ho and Sirkar (2001,

767

chaps. 36–40); our treatment relies on these chapters. Figure 8.1.48(a) describes an emulsion liquid membrane via a large emulsion globule. The large globule diameter may vary between 100 and 2000 μm. Inside each large globule, there are many tiny droplets of diameter 1–3 μm. Although this figure does not show it, the tiny droplets are tightly packed together. The large globule is dispersed in an immiscible continuous liquid phase, in this case a feed aqueous phase. Specifically, the tiny internal droplets consist of an aqueous phase, in this case a solution of NaOH in water. These tiny droplets are inside an organic solvent phase making up the rest of the large emulsion globule. This organic solvent phase contains surfactants and additives in a solvent and acts like a membrane between the external aqueous solution and the internal tiny aqueous phase droplets. From the external solution, phenol present in the continous aqueous phase partitions into the surface of the emulsion globule, the organic membrane phase. It then diffuses through the organic liquid membrane and hits one of the tiny aqueous phase droplets, the receiving phase, the permeate phase here. This particular separation with reaction example was briefly considered at the beginning of Section 5.4.4. This liquid membrane structure has also been used with an aqueous liquid membrane phase surrounding tiny organic phase droplets in an emulsion globule which is dispersed in an external organic phase, a continuous phase. Figure 8.1.48(b) illustrates how the ELM based separation works. First, the internal droplet phase is dispersed into the membrane – the process identified as emulsification. Next this emulsion is dispersed into the external continuous feed phase, creating the large emulsion globules. For the separation system of Figure 8.1.48(a), the emulsion is oil/water (O/W) since the external continuous phase is oil; however, for the system shown in Figure 8.1.48 (b) the external continuous phase is water, therefore it is considered a W/O system. The overall system is a double emulsion W/O/W. An alternative double emulsion configuration would be O/W/O, where the feed phase and the permeate phase would be oils with the liquid membrane being water. Figure 8.1.48(b) illustrates a well-mixed stage where the permeation process goes on. After the permeation is over, the original O/W phase settles out of the feed aqueous phase in a settler, followed by withdrawal of the spent aqueous phase, the raffinate; simultaneously, the O/W emulsion is separated and the oil phase (membrane phase) is recycled while the aqueous phase becomes the extract. In actual use, a number of such stages are cobbled together in a countercurrent configuration, as shown in Figure 8.1.48(c). The countercurrent permeation column shown has the feed aqueous solution (a waste water) containing zinc being introduced at the column top. The permeate phase contains a strong solution of H2SO4; it is dispersed as tiny droplets in the organic liquid membrane

Bulk flow of two phases ⊥ to direction of force

768

(a)

Surfactant

Liquid membrane Solvent Surfactants Additives Phenol reaction inside droplets phenol + NaOH

H2O + NaOH H2O + NaOH

H2O + NaOH

Sodium phenolate (nonpermeable)

Feed

Feed waste water containing Zn++

Emulsification

Dispersion / extraction

Raffinate Settling

Aqueous feed outside

M

Organic liquid membrane phase

Emulsion breaking Extract

(c)

H2SO4-containing internal aqueous phase

External continuous phase, feed Membrane phase Internal droplet phase

Water + reagent (NaOH) H2O + NaOH

H2O + NaOH

(b)

Homogenization nozzle emulsifier

Countercurrent liquid membrane permeation column

Emulsion splitter

Concentrated Zn++ solution product

Oil separator

Treated effluent water

Recycled liquid membrane phase

Figure 8.1.48. Emulsion liquid membrane: (a) a large emulsion globule containing many tiny droplets; (b) continuous ELM process schematic; (c) countercurrent ELM column for removing Znþþ from wastewater. (After Ho and Sirkar (2001), chaps. 36, 39.)

phase containing the extractant in a diluent (see the description following equation (5.2.110)) along with surfactants needed to form a stable emulsion. This emulsion is introduced at the column bottom, and it rises up the column, allowing Znþþ to be extracted into the internal H2SO4-containing aqueous phase via reactions of the type shown in (5.4.77) and (5.4.83a) in a countertransport reaction scheme. As in Figure 8.1.48(b), the various exiting streams are separated out into the effluent treated aqueous stream (the raffinate), the aqueous H2SO4 solution contaning the extract and concentrated Znþþ product and the organic liquid membrane, which is recycled for membrane making. In this particular example, the extractant used was a sulfur-containing derivative of di-2-ethylhexylphosphoric

acid (see Table 5.2.3) at the level of 5% in an aliphatic diluent with a high flash point such as Escaid 120 containing 3% of a long-chain polyamine surfactant. The Zn concentration in the feed waste water varied between 150 and 500 mg/liter, and the raffinate Zn concentration was reduced to around 1 mg/liter. The permeate phase H2SO4 concentration was as much as 250 mg/liter. The liquid membrane was achieving first solvent extraction of Znþþ into the liquid membrane phase. Then Znþþ was back extracted and concentrated into the aqueous H2SO4containing permeate phase, from which Hþ ions were transported to the feed waste water being treated during the countertransport process. Due to the presence of a strong concentration of H2SO4 in the internal droplet phase, its osmotic pressure is far

Countercurrent bulk flow of two phases/regions ⊥ to the forces

(b)

(a) Feed gas

. . ... .. ... .. .......... ... ..... ... . .. ...... .. .. .......... ..... ..

.. .. .

........ .. ... ............... ........ .. .. . ... ... . ..... ... .. ..... . .... .. . . ............... ....... . .

Permeate gas

Liquid–liquid interface Strip liquid

(d) Feed liquid Porous polymeric film

Feed liquid .... ......... . ........ ........ .. .... . . ............

.. .... . ... . .

.. .... ... . .. .

Vacuum

. . ... .. .... .......... ... . .. .......... .. ..... .... . . ...... . .... .......... .

.. .... ... . .. .

.. . . . ..... ................ ...... ............. . ......... .. .. . .. .. . ............... ....... . . . ..........

.. .... . .. ..... . .. ..... . . .. .......... . .. ... . . ..... .. .. ..

. . ... ..... ....... . . .. .. ..... .... . . .. .. .... .... . . .. ........... .. .........

Substrate

Superhydrophobic porous coating Gas-filled pore in coating ... .... .. .. ............... ....... . . .. . ...... ......... ...... ..... .......... ...... ... ... . .. ........... .. .... . ... . ..

(c)

Pore filled with liquid membrane

.... ......... . ........ ..... .... .. ... . .

.... ......... . ........ ........ .. .... . . ............

.. .... . .. .... . ... ..... . . .. .... .... . ... . .... . ..... .. .. ..

... .. . .. .. ..... ..... . ............ . ... .. ..... ...... ..........

Liquid membrane

. .. ... ............. . . ...... . ... ..... ............ .. . . .

. .. . ........ ...... . ....... ....... .. .. .. ..... ..... . .. .. ......

Feed liquid

Porous polymeric film

Pore filled with liquid membrane ..... ... ........... .. .. .......... .. .. ..... ...... ..........

769

.... ..... ... . .. .......... . ..... .... . ... . .. ..

8.1

Vacuum

Figure 8.1.49. (a) ILM for gas separation; (b) SLM for liquid separation; (c) SLM for pervaporation with a porous membrane; (d) SLM for pervaporation with a composite membrane having a porous superhydrophobic coating on the feed liquid side.

higher than that of the feed aqueous solution. Consequently, water molecules are transported through the liquid membrane into the internal droplet phase, leading to its swelling (entrainment of the external aqueous phase also contributes). Swelling of the internal droplet phase reduces the driving force in countertransport, reduces the permeate solute concentration level and makes the liquid membrane thinner, potentially introducing instability. Another demanding aspect of the ELM process is the demulsification step (Figure 8.1.48(b), emulsion breaking) needed to recycle the liquid membrane. Demulsification is successfully implemented by using an electrostatic field, which leads to the coalescence of dispersed aqueous droplets from nonconducting oil (Ho and Sirkar, 2001, chap. 38). The countercurrent emulsion liquid membrane extractor shown in Figure 8.1.48(c) has been modeled by Ho and Li (1984b). Two types of modeling were implemented: a multistage mixer–settler and a mechanically agitated column. A multistage countercurrent mixer–settler cascade of n stages operating continuously was shown to be more effective than a cocurrent cascade of similar mixer–settlers. It was found to be true for mechanically agitated columns as well.

The two phases on two sides of the liquid membrane are liquid phases that are immiscible with the liquid membrane phase in the ELM technique. There is very little information on an ELM being employed for separation from a gas phase into another gas phase. The next liquid membrane technique to be considered, namely the SLM or ILM technique, however, allows operation with both situations: (1) feed-gas phase, permeate-gas phase; (2) feedliquid phase, sweep-liquid phase (both immiscible with the liquid membrane phase). The membrane structure in the SLM/ILM technique is as follows. Consider a porous hydrophilic film. If it is contacted with an aqueous solution, more than likely it will be spontaneously wetted. The pores will be filled with the aqueous solution, which will be held by capillary force (see equation (6.1.12)). Similarly, if we have a porous hydrophobic film and it is contacted with an organic solution, more than likely it will be wetted by the solution and the pores will be filled with the organic liquid. Consider now a feed-gas phase on one side of this film (Figure 8.1.49 (a)). If the solid polymeric region of the film is considered relatively impervious, then species in the feed-gas mixture will be absorbed in the liquid in the pores, diffuse through the liquid in the pores and be desorbed at the other side of

770 the liquid film in the pore into the gas phase on the other side of the porous membranes, provided there is a positive partial pressure difference for this gas species across the film. The liquid film stabilized by capillary forces in the pore can act as a membrane. However, if the liquid is volatile, it will be evaporated into the gas phase on either side, ultimately destroying the liquid membrane. Nonvolatile immobilized liquid membranes have been used successfully for CO2 separation (Kovvali et al., 2000; Kovvali and Sirkar, 2001; Kouketsu et al., 2007). One could use the liquid immobilized in the pores for separation of liquid mixtures as well, provided the feed liquid and the permeate (strip) liquid are immiscible with the liquid membrane in the pores (Figure 8.1.49(b)). Such membranes are called supported liquid membranes (SLMs). However, at the pore mouths, the two immiscible phases contact each other. The liquid membrane phase has some solubility, however small, in the feed/strip liquid phase. Therefore the life of the liquid membrane is limited (see Kovvali and Sirkar (2003) for a brief review), and periodically it needs to be regenerated (Yang and Kocherginsky, 2006). The liquid membrane is most likely to have extractants used with the emulsion liquid membranes or solvent extraction in the case of chemically complexing extractants. An intermediate configuration of a liquid membrane employs it for pervaporation: feed liquid contacting the liquid membrane supported in a porous membrane with vacuum or sweep gas on the other side (Qin et al., 2003). Such a configuration still cannot eliminate loss of the liquid membranes to the feed liquid (Figure 8.1.49(c)). One technique solves this problem: introduce a thin porous hydrophobic coating on the porous substrate membrane on the feed side such that neither the feed solution nor the liquid membrane can spontaneously wet the pores of the porous hydrophobic coating (Figure 8.1.49(d)). Volatile species are evaporated from the feed solution into the gas-filled pores of the coating and are then absorbed into the liquid membrane in the pores of the porous substrate as they are moved to the permeate side where the species are continuously removed by the permeate side vacuum, condensed and recovered (Thongsukmak and Sirkar, 2007, 2009). There are a few points to be kept in mind. The liquid membrane must be nonvolatile since it is subjected to vacuum on the permeate side and can evaporate on the feed side and be lost to the feed solution (Figure 8.1.49(d)). Further, wetting of the porous membrane structure spontaneously by the membrane liquid present outside requires that the surface tension of the membrane liquid should be equal to or lower than the critical surface tension γcr of the polymeric or ceramic substrate being employed. For example, γcr for polypropylene (PP) is ~33 dyne/cm. Many organic liquids will spontaneously wet it. However, γcr values for various fluoropolymers (FPs) are usually T1) (Figure 10.2.3). The difference (T2
T1) should be larger than the temperature difference needed for heat transfer to the brine (in the

To vent

To vent n -1

2

To vent Feed brine heater

n

Tb1

Cold feed brine,Tbf

Steam Vapors

Vapors

Vapors

Vapors

Distillate

Condensate Flash

Feed brine heater

Flash

Flash

Hot brine

Figure 10.2.2. Multistage-flash evaporation for desalination: n stages. (After Spiegler (1977).)

Flash

Concentrated brine

10.2

Reducing energy required for separation

839

Water vapor

Tubular heater-condenser

Evaporator

P1,T1

Compressor for water vapor

P2 T2

Compressed steam

Fresh water

Starting heater

Feed brine Concentrated brine

Brine concentrate

Heat exchanger Figure 10.2.3. Vapor-compression evaporation for desalination.

evaporator) from the condenser by the boiling-point elevation, as well as any frictional losses in the vapor conduits (El-Sayed and Silver, 1980). If the process of water vapor compression is described like that for an ideal gas undergoing an adiabatic process, then the reversible work required for the compression is given by that for an isentropic process:
W s jrev

γ P1 V 1 ¼ ðγ
1 Þ

"

P2 P1

γ
1 γ

#


1 :

ð10:2:10Þ

  Here, γ ¼ c p =c v and V1 is the volume of the vapor at temperature T1. The value of γ for water vapor is 1.32. For processes that are not completely adiabatic, γ will be replaced by another quantity δ, where PVδ ¼ constant instead of PVγ ¼ constant for an ideal gas following an adiabatic compression. In vapor-compression evaporation processes, where the ratio of the pressures (P2/P1) (¼ compression ratio ¼ r) is close to 1, one can employ the following simplification:  γ
1  γ
1   P2 γ P2
P1 γ γ
1 P2
P1 ¼ 1þ ffi 1þ , P1 P1 P1 γ


W s jgmol ¼ 2:3  10
6

¼ 430:33  10
6

ðr 2
1Þ kW-hr : r gmol

On a volumetric basis, this energy requirement turns out to be
W s jliter ¼ 430:33  10
6 ¼ 23 907  10
6

ðr 2
1Þ kW-hr 1 gmol r gmol 0:018 liter

ðr 2
1Þ kW-hr ; r liter

  kW-hr :
W s jm3 ¼ 23:9 r 2
1 =r m3

ð10:2:13Þ

Let r have the following values: 1.1, 1.2. Then the corresponding values for
W s jm3 are 4.562, 8.763 kW-hr/m3. We can compare these values with the minimum energy required per equation (10.1.18) for, say, sea water concentrated twice at 101.5  C:
W s jliter ¼

ð10:2:11Þ

ART s1 0:000537  2:31  10
6  374:2  70 kW-hr ¼ 0:018 liter Vs

¼ 18:03  10
4 kW-hr=liter ¼ 1:8 kW-hr=m3 :

yielding
W s ¼

kW-hr ðr 2
1 Þ  374:2 K  gmol-K 2r

2

0

13

γ P 1 V 1 4 γ
1 @P 2
P 1 A5 ¼ ðP 2
P 1 Þ V 1 ðγ
1Þ P1 γ

¼ ðP 1 V 1 Þ ðr
1Þ ¼

ð P 1 V 1 Þ ðr 2
1 Þ P 1 V 1 ðr 2
1Þ ffi , ðr þ 1Þ 2r

The compressed water vapor condensate is the product of this process. Its sensible heat, as well as that of the brine concentrate, are used to heat the feed brine in a heatexchanger arrangement (Figure 10.2.3).

ð10:2:12Þ for r ¼ (P2/P1) close to 1 (Spiegler, 1977). We can now obtain an estimate of the work needed for this compression work by equating P1V1 to RT1. Consider a somewhat concentrated sea water boiling at 101.05  C; now R ¼ 2.31  10
6 kW-hr/gmol-K. Therefore

10.2.2

Distillation

The three common strategies that may be employed to reduce the energy requirement in the distillation method for separation of liquid solutions involve the principles of multieffect operation, multipressure operation and heat

840

Energy required for separation

In conventional distillation processes, the main source of external energy input is in the reboiler at the highest column temperature. If, therefore, the amount of this external energy input per mole of feed is reduced, the energy efficiency ought to improve. One way to achieve this involves using the heat from the condensation of the overhead vapor in the reboiler of another column; the two columns become thermally linked. For a binary feed mixture, one method of implementing this is to have two columns connected in the fashion of Figure 10.2.4(a), with the lower column (column 1) operating at a pressure PH high enough to let the vapors from the top come out and condense in the reboiler of the upper column (column 2)

operating a lower pressure PL. However, since the feed input rate is the same for both columns, the steam input to the bottom column reboiler is only for half the total feed flow rate being processed. This energy expenditure reduction requires an increased capital investment due to two columns. Note: The column diameters are reduced due to the lower vapor flow rate. However, the total separation equipment volume is similar to that for a single-column operation. Additional details may be found in Robinson and Gilliland (1950) (as the earliest source for such a scheme), King (1980, pp. 697–698), Henley and Seader (1981) and Wankat (1988). Wankat (1993) has considered 23 different multieffect distillation schemes and has suggested heuristics to reduce the number of options. Consider the overhead vapor from column 1 in Figure 10.2.4(a). The dew-point temperature of this vapor must be greater than the bubble-point temperature of the bottom liquid of column 2. In addition, there has to be two other considerations for successful column coupling. The total heat available from condensation of the vapor from the top of column 1 should be what is needed to vaporize the required fraction of the bottoms from column 2. Correspondingly, there has to be a sufficient temperature

(a)

(b)

pump. However, unlike that for evaporation in desalination, there exists a considerable variety of approaches under each one of these broad principles. We will briefly introduce the reader to the basics of such principles one by one. All such methods are of interest since distillation continues to be the separation process which uses the largest amount of energy among all separation processes. 10.2.2.1 Multieffect distillation

N2 product vapor

Accumulator Wtf2 xif

V3

Distillate product

Column pressure

2

Liquid N2 reflux

Wtd

PL

2

V2

Top column ~1 atm

xid Bottoms Wtb 2 product xib

Wtv

Binary feed

1

Wtl

1 Wtf

1

Column pressure

1

Distillate product Wtd 1 xid

O2 product Vapor Three-stream heat exchanger

PH

xif Bottoms product

Compressed air cooled

Condenser -cumreboiler Bottom column ~5 atm

V1 Liquefied feed air Reboiler

Wtb

1

xib

O2-enriched liquefied air

Figure 10.2.4. (a) Multiple-effect distillation with two columns and feed to both columns. (b) Cryogenic air separation in a Linde doublecolumn device: forward feed of liquefied feed air to the bottom column.

10.2

Reducing energy required for separation

difference between these two streams so that the heatexchanger surface area required is not too large. The balancing of the thermal load is achieved in some multieffect distillation processes by appropriate adjustment of the feed flow rates. Regardless of the details of such balancing procedures, it is clear that, if the condenser of one distillation column is going to be the reboiler of another distillation column, there has to be a difference in pressure between the columns. Therefore multieffect distillation requires operation at multiple pressures: two columns require operation at two different pressures. Further, often fresh feed is introduced to only one column, unlike that shown in Figure 10.2.4(a). Two modes are quite common when the feed is being introduced to only one column: forward feed, where the fresh feed is introduced to the bottom column operating at a higher pressure PH; backward feed, where the fresh feed is introduced to the top column operating at a lower pressure PL. Cryogenic distillation of air provides an example of a forward feed and a Linde double-column arrangement where the lowpressure column is simply physically located above the high-pressure column (Figure 10.2.4(b)). In such a double-column device, compressed and cooled air3 is introduced at the bottom of the bottom column into a coil, where it provides a heat source (reboiler) for the evaporation of the liquid on the shell side. The compressed and cooled air then flows through a valve V1, undergoes Joule–Thompson expansion (and liquefaction) to a pressure of 5 atm. This liquefied air is introduced as feed to the bottom column. Since N2 is more volatile than O2, the liquid collecting at the bottom of this column is about 45% O2: this liquid is introduced now as feed to the middle of the top column operating at 1 atm through a pressure-reducing valve V2. Rectification in the top column produces almost pure N2 in the vapor phase at the top and almost pure O2 as liquid at the bottom of the top column. Correspondingly, the top of the bottom column produces almost pure N2 at this location. The N2 condenser–O2 evaporator at the top of this bottom column is the O2 evaporator–N2 condenser of the top column. The almost pure N2 vapor liquefied here is introduced as reflux at the top of the top column after pressure reduction through valve V3. Cold purified N2 vapor at the top of the top column and cold O2 vapor from the bottom of the top column are taken out as products through a three-channel heat exchanger, which warms up the gaseous O2 and N2 streams and cools down the compressed air feed stream going to the reboiler of the bottom column.

3

The air we are referring to here has been supplied, via a compressor, from the ambient atmosphere. The air is then cooled to ~10  C and purified of water vapor, oil, particles through a series of separators.

841 There is one potential disadvantage to the multieffect distillation schemes from energy considerations. Although the total heat input into the multieffect distillation is reduced, for example, in the binary distillation of Figure 10.2.4(a), the heat at the bottom of column 1 has to be supplied at a higher temperature due to the higher pressure. Therefore the net work consumption (equation (10.1.22)) goes up due to a higher value of Th. This has been characterized also as trading “first law heat” for “second law ΔTs” (Westerberg, 1985); the utilities cost goes up as Th goes up or Tℓ goes down. An alternative strategy employs the concept of a prefractionator before the main distillation column such that the bottoms utility temperature remains the same in both columns but the total energy consumed is reduced. Although there is a change in the capital costs, the overall costs were reduced (Agrawal et al., 1996). Figure 10.2.5 illustrates such a concept where an intermediate reboiler is used in the main lower-pressure column below the feed location to condense the overhead vapors from the prefractinator operating at a higher pressure. All of the feed is introduced to the prefractionator (this does not have to be the case). The reboilers of this prefractionator and the main column are running on the same high-temperature utility, and are therefore at the same temperature, but they have different compositions. The prefractionator in this scheme is primarily an enricher, and it produces part of the overall distillate, whereas the bottoms from the prefractionator is introduced as the feed to the main column (Agrawal et al., 1996) after a pressure reduction. The above discussion was focused on a binary mixture separation. These and other configurations require additional considerations when a ternary mixture has to be separated, as has been pointed out already in Chapter 9. 10.2.2.2 Heat pump We have been introduced to the notion of vaporcompression evaporation for desalination in Section 10.2.1.3. Its application to distillation based separation of mixtures is generally identified as a heat pump method of distillation. Figure 10.2.6 illustrates the open-loop method, which is one of the two methods of carrying it out for a binary mixture using one distillation column. The overhead vapor from the column top is compressed to a pressure high enough for it to provide boil-up at the bottom of the column as it condenses. Part of the condensate becomes the distillate product, while the pressure of the rest is reduced through a valve V before it is introduced as liquid reflux at the top of the column. The second heat pump method is called the close-loop heat pump: here, an external working fluid is used to carry out the boil-up at the column bottom at a high enough pressure. After condensation, the pressure of this liquid is reduced through a Joule–Thompson valve, cooling it substantially, and then it is used to condense the overhead vapor from the column

842

Energy required for separation

Condenser

Distillate product

Feed for column 2

2

Intermediate reboiler condenser Feed

1

Distillate product

PL

PH Reboiler Bottoms product

Reboiler

Figure 10.2.5. Prefractionator column at a higher pressure PH has been heat integrated with the main column at a lower pressure PL via an intermediate reboiler (Agrawal et al., 1996).

Feed

C O L U M N

Trim cooling

V

Distillate product

Bottoms product Reboiler Figure 10.2.6. Open-loop heat-pump arrangement for binary distillation in a single column.

as it is vaporized. The vaporized external working fluid is then prepared to act as a high-temperature utility at the column bottom by compressing the vaporized external working fluid. An introduction to heat-pump schemes for binary distillation using one column has been provided by King (1980, pp. 695–697). For ternary separation heat pump schemes using two thermally linked distillation columns, see Agrawal and Yee (1994). We will now provide the fundamental expression for the work required to pump the heat from a lower temperature T2 to a higher temperature T1 by an ideal reversible

Carnot engine. If the amount of heat Q2 extracted at the low temperature T2 is pumped by supplying an amount of work (
Ws) to a higher temperature T1 where the amount of heat rejected is Q1, then4

4

See the analysis in basic texts on applied thermodynamics, e.g., Smith et al. (2001, p. 310). Remember Ws is positive when it is done by the system on the surroundings; it is negative when the surroundings works on the system.

10.2

and

Reducing energy required for separation Q1 ¼ Q2 þ ð
W s Þ,   T1
T2 ð
W s Þ=Q1 ¼ T1 

ð
W s Þ=Q2



¼

T1
T2 : T2

ð10:2:14Þ

W max ¼

W max ¼

n  X i¼1

ð10:2:16Þ

Free energy of mixing

In Chapter 9, we came across the “no mixing condition” during cascade analysis. This condition suggests that the energy spent in developing separation should not be lost by mixing two streams having different compositions since they have different chemical potentials. This criterion is not enough since streams having the same composition but having different phases (e.g. vapor and liquid) will have different chemical potentials; if these two streams are mixed, their chemical potential difference would be lost. Therefore a more general principle guiding the mixing of two streams in cascades is: there should be no mixing of two streams having different chemical potentials. On the other hand, in real-life conditions, one may have to undertake such mixing. The strategy to be pursued then is: can one recover some energy during the process of mixing? For example, consider reverse osmosis desalination of sea water. It produces a concentrate whose salt concentration is higher than that of the sea water, the concentration level depending on the fractional water recovery. Normally, this concentrate would be dumped into the sea. However, if there was a device which could generate power resulting from the mixing of this concentrate with the sea water, then the power so generated would reduce the power required for desalination. If we consider now a relation of the type represented in (10.1.1) and (10.1.2), we may write the following expression for the maximum amount of work recoverable from mixing under ideal conditions when a mole of each feed stream ( f1 for concentrate, f2 for sea water) is mixed with the other to produce a product stream ( j ¼ p) dumped into the sea:

n  X i¼1

ð10:2:15Þ

Result (10.2.16) follows from the other two relations (10.2.14) and 10.2.15). Therefore the amount of work needed to pump one unit of heat (Q2 ¼ 1 calorie) is ((T1
T2)/T2). If this amount of work were to be generated from a thermal source at T1 rejecting the heat at T0, then the amount of heat Qs needed will have to be estimated from the Carnot principle discussed in (10.1.21), namely 0 1 T 1
T 0A T1
T2 @ ; ¼ ð
W s Þ ¼ Qs T1 T2 0 1 ð10:2:17Þ T 1 @T 1
T 2 A Qs ¼ : T2 T1
T0 10.2.3

843

 μif 1 þ μif 2
2μip ;

ð10:2:18Þ

V if P f 1 þ R T ℓn γif 1 x if 1 þ V if 2 Pf 2  þ RT ℓn γif 2 x if 2
2V ip Pp
2RT ℓn γip x ip :

ð10:2:19Þ

If P f 1 ¼ P f 2 ¼ P p and V if 1 ¼ V if 2 ¼ V ip then W max ¼ RT

n  X i¼1

 ℓn γif 1 x if 1 þ ℓn γif 2 x if 2
2ℓn γip x ip :

ð10:2:20Þ

Generally the reverse osmosis concentrate is at a high pressure (less than that of the entering high-pressure feed brine); this pressure energy is recovered to a large extent in energy recovery devices. The above expression assumes that the concentrate pressure has been reduced to atmospheric. We have seen in Section 3.4.2.5 that application of an electrical voltage in a stack of ion exchange membranes when appropriately configured can lead to desalination via the process of electrodialysis. Weinstein and Leitz (1976) and Jagur-Grodzinski and Kramer (1986) have shown how an electrodialytic stack of membranes may be operated in reverse, in what is called a dialytic battery. Concentrated brine enters one chamber and dilute brine enters the contiguous chamber: the mixing of these streams through the ion exchange membranes generates a voltage at the electrodes, and therefore power. Estimate of this salinity power, which could be potentially generated from the mixing of river water with sea water worldwide, is huge (Wick, 1978). The cost of recovering such an energy source is, however, a strong deterrent at this time. We will briefly provide an expression for the voltage generated between the electrodes in a dialytic battery. First, when a membrane is placed between two solutions of an electrolyte having mean ionic activities of a0  and a00  ð< a0  Þ (see (3.3.119d), where the mean ionic activity a± has been related to the individual ionic activity of the electrolyte AY), it develops a potential difference between the two sides called the membrane potential; it is also called concentration potential. In an electrodialysis cell, there are many cation exchange membranes and generally an equal number of anion exchange membranes. One cation exchange membrane (CEM), the two solutions on two sides of it and an anion exchange membrane (AEM) constitute a cell pair; an electrodialysis cell may have N cell pairs. If each ion exchange membrane behaves ideally, with a cation exchange membrane allowing only cations to go through and an anion exchange membrane allowing only anions to go through, then the total open-circuit voltage (no external load) for N cell pairs containing N

844

Energy required for separation

CEMs and N AEMs developed is (Weinstein and Leitz, 1976) V ostack ¼ 2N

RT ℓn ða0  =a00  Þ F

ð10:2:21Þ

as the individual ionic species moves through the corresponding ion exchange membrane from a higher activity to lower activity. For nonideal ion exchange membranes, the open-circuit voltage developed will be somewhat lower. Another way to carry out mixing of two saline solutions would be what is practiced in the concept known as osmotic power plant. Suppose you have a saline water (1) at a pressure (P1) less than its osmotic pressure (π1). Let the aqueous solution (2) on the other side of the reverse osmosis membrane (having a lower osmotic pressure π2 < π1) be present at atmospheric pressure, P0. If the osmotic pressure of this solution 2 is less than the pressure of the saline water 1 (i.e. π2 < P1), then water from solution 2 will go to solution 1 and will dilute it. However, this water will increase the volume of the saline solution present at pressure P1. Therefore one could use this extra energy through an appropriate hydraulic device/arrangement (Loeb, 1976). 10.2.4

impurities as well as a large amount of water; similarly for many other products, with water sometimes replaced by an organic solvent. In the case of uranium isotope separation, a product enriched to the level of 0.03 atom fraction of the desired isotope U235 F6 is enough. However, all such problems have a common feature:5 separation requires handling a large volume of an essentially inert or undesirable substance, e.g. water, in a number of the problems described above. There is significant literature available on the cost of a particular desired product, Costi, as a function of other quantities, such as mass of inert material in the feed per unit mass of the desired product, (win,f / wi,f) (Lightfoot and Cockrem, 1987). One such expression is   ð10:2:22Þ Costi ¼ a win, f =wi, f ,

where a is a universal constant in the so-called Sherwood plot, independent of the species. Obviously, the energy cost of treating a large volume of a feed containing a small amount of the desired product is significant. It will vary, however, with the separation method adopted. Consider a distillation column under minimum reflux for a relatively pure distillate stream. From relation (10.1.47), we can write

Dilute solutions

Dilute solutions of solutes/macrosolutes are frequently encountered in practice. Examples are: a dilute fermentation broth containing small volatile molecules such as alcohol, small nonvolatile molecules such as amino acids, large protein molecules that are secreted externally or those that are obtained in a cell lysate from a cell culture. In these cases, recovery of these molecules in a substantially pure condition is the separation goal. Fractionation of isotopes where the desired isotope is present in a very dilute solution is another problem of the same type (separation of D2O from H2O; separation of U235/U238 isotopes present in a gaseous mixture of UF6, where the U235 F6 atom fraction is 0.00711, a very low concentration). Recovery and purification of a synthetic pharmaceutically active organic compound present in a very dilute aqueous/ organic solution is essentially another related problem of a similar nature. Dilute solutions of radioactive impurities obtained as byproducts of fission confront us with a similar challenge in that water has to be purified and the radioactive impurity has to be substantially concentrated for safe disposal in a secure environment. Dilute solutions of organic pollutants in water pose a somewhat similar separation problem, even though the organic pollutant may not have much value, unlike the fermentation products/ byproducts mentioned earlier. There are considerable differences in the separation problems discussed above. In the case of purification of proteins produced by fermentation or cell culture, one needs to concentrate and then purify to obtain as pure a protein as possible that is now substantially free from

W tℓ jmin ¼ )

1 1 W td W tf ¼ ðα12
1Þ ðα12
1Þ x 1f

W tℓ jmin 1 1 : ¼ W td ðα12
1Þ x 1f

ð10:2:23Þ

Since Wtℓ is related to Wtv by the slope of the operating lines, they are not too far apart on either side of 1 for a low α12 system. The molar vapor flow rate from the reboiler provides a reasonable estimate of the energy requirement for distillation. Relation (10.2.23) suggests that as x1f decreases, i.e. the feed becomes more and more dilute in the species 1 being recovered in the overhead as an almost pure product, the energy requirement increases enormously since a very large volume flow rate of inerts has to be handled (Lightfoot and Cockrem, 1987) for this dilute feed. For example, in the separation of D2O from H2O (and just a bit of HDO), the natural abundance of deuterium in water in atom ppm varies between 119 and 163 (Benedict et al., 1981, p. 709). These authors have indicated that, in a distillation plant, 200 000 gmols of steam are required per gmol of heavy water produced (mole fraction of D2O in product is 0.998); the very low separation factor of 1.05 used here (see relation (5.2.46) for an expression for α12) contributes to this enormous energy requirement. As a result, alternative pathways to obtain D2O are followed (Benedict et al., 1981, chap. 13). Lightfoot and Cockrem (1987) have pointed out the applicability of equation (10.2.22) for a variety of separation

5

The differences between enrichment, concentration and purification have been illustrated in Section 1.7.

Problems techniques used to recover products from a dilute fermentation broth. Generally the cost of processing for separation, and therefore to some extent the energy needed for separation, is proportional to the volume of the liquid to be processed in processes such as filtration, membrane processes, centrifugation, solvent extraction, precipitation, crystallization, etc. One can argue that in processes such as solvent extraction and adsorption, if one can have highly selective solvents or adsorbents, then one can achieve a rapid reduction in processing volume via the solute transfer between the phases. It has been pointed out, however, that, in conventional dispersive extraction, due to the limitations of processing flow rates, the phase volume reduction can be only 5–10 fold, regardless of how selective the extracting solvent is (Lightfoot and Cockrem, 1987). New nondispersive membrane based solvent extraction techniques (see Section 3.4.3.2) bypass this limitation and can operate with phase volume reductions as much as 100 times or more in porous hollow fiber membrane based devices. A general problem in separation from dilute solutions may be posed as follows via an example. Suppose you have a nonvolatile solute, say a protein, a small-molecule drug or a crystalline solute, etc., present in a very dilute aqueous solution. The separation goal is to get this in as pure a solid form as possible. Lightfoot and Cockrem (1987) have identified two alternative strategies as follows. (a) One can first, in step 1, precipitate the solute from the dilute solution; the precipitate phase will have brought

845 the solute to unit activity since we will have pure solid (see Section 3.3.7.5). Then one can, in step 2, concentrate this precipitate from this solution where it is suspended. In both steps, the whole solution volume has to be treated, and consequently the energy expenditure is likely to be significant. (b) One can first concentrate the solution of the solute (step 1). Then one can bring the activity of the solute to pure solute activity (unit activity) via precipitation in step 2 from a much smaller solution volume. In both cases, there will be additional steps taken to achieve final purification. In approach (b), however, the volume of solution to be processed in step 2 is much lower; this is due to the concentration step implemented in step 1, which was not present in step 1 of approach (a). The goal in both approaches is to achieve a product whose concentration is as close as possible to 1 in terms of mole fraction (xij) and where the species activity is also as close as possible to 1 (aij). Therefore in an xij–aij plot, approaches (a) and (b) follow very different pathways. In approach (a), the first step increases aij then increases xij. In approach (b), the first step increases xij, then aij is increased. One should remember that any such discussion must be qualified by the results of the actual separation step undertaken. For example, in approach (a), the first step may be concentration of the solution via ion exchange adsorption. The desorption step should lead to a much more concentrated solution.

Problems 10.1.1

Using the expression for the minimum energy required to desalinate sea water (salinity, s ¼ 35) at 25  C, calculate the minimum energy required per m3 of fresh water produced if the final volume of the concentrated sea water (¼ V2) is 90%, 75%, 50% and 25% of the original sea-water volume V1.

10.1.2

Calculate the minimum amount of work you have to do to recover 1 cm3 and 1 liter of fresh water from a brackish water source using a salt-impermeable membrane. Show that the theoretical expression for this work is equal to RTC salt j1 , where C salt j1 , is the molar concentration of salt. The osmotic pressure of the salt solution is given by Van’t Hoff’s law. The temperature is 25  C. The brackish water contains 0.5 g of NaCl per 100 g of water.

10.2.1

To reduce the energy required for a distillation process with almost pure water as overhead vapor, a heat-pump arrangement is proposed. The column is operating at atmospheric pressure. The reboiler needs a vapor condensing at 150  C. The overhead water vapor should be compressed by a reversible adiabatic compression process such that it can be the condensing vapor in the reboiler. Determine the heat delivered by the condensing vapor per pound of the vapor; calculate the ratio of the heat delivered in the reboiler to the work required in the reversible compressor. Use the steam tables from Smith and Van Ness (1975) or any other suitable source. (Ans. 1020 Btu/ℓb (567 cal/g); 7.254.)

10.2.2

Consider the possibility of combining the vapor-compression evaporation principle for desalination with the multiple-effect evaporation principle for a two-effect system. Draw a schematic of such a plant and identify the possible advantages of such a combination with respect to Figures 10.2.3 and 10.2.1 (for a two-effect system).

10.2.3

This problem is concerned with balancing the two columns, 1 and 2, in the multieffect distillation process of Figure 10.2.4(a) by adjusting the two feed flow rates, Wtf1 and Wtf2. Assume that the feed to both the columns is

846

Energy required for separation saturated liquid. Further, the two columns are being operated with (Wtℓ/Wtd) ratios related to their minimum values in the following way: ðW tℓ =W td Þ1 ¼ r 1 ðW tℓ =W td Þ1, min ;

ðW tℓ =W td Þ2 ¼ r 2 ðW tℓ =W td Þ2, min :

Develop the following expression for the ratio (Wtf1/Wtf2) in terms of r1, r2, the minimum values of (Wtℓ/Wtd) in both columns and the latent heats of vaporization/condensation in the two columns where they are thermally coupled at the condenser–reboiler between columns 1 and 2: h i r 2 ðW tℓ =W td Þ2, min þ 1 W tf 1 n o: ¼ W tf 2 ðΔH v1 =ΔH v2 Þ r 1 ðW tℓ =W td Þ1, min þ 1

Assume that the bottoms and the distillate compositions are identical for each column. 10.2.4

A waste heat source is available at a temperature lower than the steam temperature used in the reboiler of a distillation column for a binary mixture. There are three possible ways of using this waste heat: use an additional reboiler somewhere in the stripping section; heat the feed stream to the column so that part of it is vapor; the feed is split into two streams of flow rates Wtf1 and Wtf2, where the feed stream f2 is completely vaporized by the waste heat at an intermediate reboiler and introduced at an appropriate location in the stripping section (feed stream f1 is introduced higher up). Focus on this third method of split feed, which divides the column into three sections. Assume constant molar overflow with appropriate allowances for two feed input locations. Use minimum reflux methodology and two potential pinch locations (at the pinch location or pinch point the operating line intersects the equilibrium curve). Determine the minimum vapor flow rate from the bottom reboiler for each pinch location and comment on which arrangement will lead to the reduction of energy requirement in the bottom reboiler.

10.2.5

Consider a saturated liquid feed flow rate of Wtf mol/s having a composition x1f for a binary mixture of species   sat i ¼ 1, 2 for which α12 ¼ P sat ¼ 5 over the range of temperatures and pressures of interest. We are 1 =P 2 interested in the separation achieved by distillation for the following two cases. (1) We heat this feed liquid sufficiently for it to reach vapor–liquid equilibrium in the flash drum (see Section 6.3.2.1); determine graphically the vapor and liquid compositions (x1v, x1ℓ) if x1f ¼ 0.5, q ¼ 0.5 and α12¼5. (2) We split the saturated liquid feed into two fractions having flow rates of Wtf1 and Wtf2; introduce one fraction at the top of a distillation column, vaporize the second fraction completely and introduce it at the bottom the column. Plot the q-lines for the two streams entering the column if x1f ¼ 0.5. Develop, under conditions of constant molal overflow, the operating line for this distillation column. Plot the line for the situation where Wtf1 ¼ Wtf2. Obtain graphically the compositions of the overhead vapor product and the bottoms liquid product for two values of the number of ideal equilibrium plates in column, N ¼ 1 and N ¼ 5. (3) Discuss how N ¼ 5 leads to higher separation for the same energy input for all other cases.

10.2.6

In Section 2.2.3, the separative power δU was introduced for a single-entry separator in an isotope separation plant via equation (2.2.32a). Expression (2.2.42) was developed for the value function in the case of close separation. The developments in Chapter 2 were carried out in mole fraction units and molar flow rates. In the market for nuclear fuels, an estimate of the energy requirement for isotope separation is sometimes expressed in units of separative work, SWU, which is essentially identical to the δU expression (2.2.32a) containing expression (2.2.42) for the value function, except mass units are used for the flow rates wt1, wt2, wtf :         x 1f x 11 x 12 δU ¼ wt1 ð2x 11
1Þ ℓn : þ wt2 ð2 x 12
1Þ ℓn
wtf 2x 1f
1 ℓn 1
x 1f 1
x 11 1
x 12 If the wij values are expressed in kilograms (ignore the time dependence for the time being since the wij terms are mass flow rates) and the xij values are estimated based on atom fractions, then calculate the value of δU in kilograms for the following: (1) a feed of 2 kg of natural uranium containing 0.711% U235; (2) an enriched uranium product of 0.33 kg containing 2.8% U235. Show that the value of δU is essentially 1 kg; this is identified as 1 SWU.

11

Common separation sequences

We have studied a variety of separation processes and techniques. Our focus was on developing an elementary understanding of an individual separation process/technique. In practice, more often than not, a combination of more than one separation process is employed, regardless of the scale of operation involved. Here we introduce very briefly the separation sequences employed in a few specific industries. The separation sequences of interest are considered under the following headings: bioseparations (Section 11.1); water treatment (Section 11.2); chemical and petrochemical industries (Section 11.3); hydrometallurgical processes (Section 11.4). It is to be noted here that often the separation sequences are reinforced by chemical reactions within such a sequence or before/after the separation steps. The intent here is to provide an elementary view of the complexity and demands of practical systems where certain types of separation sequences are crucial/ primary/dominant components. More often than not, we will find that certain types of separation techniques and processes are much more prevalent in certain industries. For example, solvent extraction and back extraction processes are dominant in the recovery and purification of metals and metallic compounds via hydrometallurgical processes. On the other hand, distillation and, to a much lesser extent, absorption/stripping followed by solvent extraction are the primary separation processes in the chemical/petrochemical industries. Water treatment industries/plants are however, focused much more on deactivation/removal of biological contaminants, suspended materials and dissolved impurities from water via oxidation processes, filtration, membrane techniques and ion exchange processes. Biological separations share some of these characteristics of water treatment processes in terms of the separation techniques; however, since the focus is on recovering/purifying the biologically relevant compound, processes such as chromatography are in great demand.

11.1 Bioseparations Separation processes utilized to recover, concentrate and purify biologically produced molecules, macromolecules or cellular entities in the biotechnology industry (Ladisch, 2001; Shuler and Kargi, 2002) are commonly studied under the general title of “bioseparations.” Often these and other related activities are lumped under the title downstream processing (Belter et al., 1988). Generally a sequence of separation processes is implemented to obtain the final product. This sequence is governed by the source of the compound as well as the desired final product purity and form. The variety of products from the biotechnology industry is enormous. Table 11.1.1 provides a very brief list of products in various categories, such as smaller molecules, somewhat larger molecules, proteins, enzymes, viruses and whole cells. Compounds identified in the list are/ can be/were produced via biotechnology. In general, such products are present in low concentrations or as dilute solutions, often as a result of a batch process. The process may be one of cell culture, fermentation or may involve biochemical reactions/transformations based on enzymes as such or enzymes present inside whole cells. In a few cases, the starting mixture for separation may be a natural product, e.g. corn, or a biological fluid, e.g. blood plasma. Correspondingly, one could classify the latter as biomedical separations, all others being biochemical separations. For products that are not whole cells, the products may have been secreted by whole cells and are therefore present externally in the solution: these are called extracellular products. Those products which are not secreted by whole cells, and therefore have to be recovered from the cell by destroying the cell (the so-called “lysing” of the cell), are identified as intracellular. The cost of recovery and purification of biologically produced products can be more than half of the total production cost. The more dilute the initial solution of the product, the higher will be the ultimate cost of production.

848

Common separation sequences

Table 11.1.1. Illustrative list of bioproducts in a few categories Category

Bioproduct example

Molecular weight

Smaller molecules

Ethanol, acetone, n-butanol, acetic acid, citric acid, lactic acid, butanediol, amino acids, glucose, fructose Antibiotics, steroids, disaccharides, larger fatty acids Proteins (enzymes), polysaccharides, nucleic acids Virus Bacteria, afungi (Baker's yeast), animal cells, plant cells

Generally less than 200

Somewhat larger molecules Macromolecules Particles – not free-living Particles – free-living a b

Physical dimension

Generally less than 600–700 but greater than 200 103 to 1010 100 nm 1–10 μmb

See Table 2.3, Shuler and Kargi (2002) for more details about various components of bacteria. See Tables 7.3.1 for additional details.

Product isolation

Concentration / Purification

Solid–liquid separation Aqueous back extraction agent 1

Extraction solvent 1 Broth Fermentor

Extraction 1

Filtration Cake

Aqueous waste

Solvent extract 1

Back extraction 1

Polishing Aqueous back extraction agent 2 Crystallizer 2

Crystallizer 1

Activated carbon adsorption

Aqueous extract

Back extraction 2 Solvent for recycle to 2

Vacuum drying

Solvent extract 2

Solvent for recycle to 1

Aqueous extract

Extraction 2

Extraction solvent 2

Aqueous waste

Purification / Concentration Purified product

Figure 11.1.1. Block diagram for the production process for a typical antibiotic, penicillin, an extracellular product.

Consider a typical antibiotic production process, as illustrated in Figure 11.1.1. The antibiotic is present in a fermentation broth as a very dilute solution in the presence of considerable soluble impurities, as well as a significant concentration of whole cells and cellular debris. The particulate materials, such as whole cells, cell debris, etc., are removed first by filtration or centrifugation. The filtration method is described under rotary vacuum filtration (Section 7.2.1.5) or microfiltration (Section 7.2.1.4) of the tangential-flow type (TFF). Centrifugation is illustrated in Sections 7.3.2.1 and 7.3.2.2 via different types of centrifuges. This step has been characterized as removal of insolubles (Belter et al., 1988) – it is essentially separation of cellular particles from a liquid solution of the product.

In the next step, the filtered fermentation broth is contacted with an extracting solvent in a mixer–settler type of device (Section 6.4.1.2); the solvent extracts the antibiotic from the broth, along with many related and nonrelated compounds. Countercurrent extracting cascades in the form of centrifugal extractors are employed (Section 8.1.4, Figure 8.1.35) to reduce the contact/residence time. This process is often called product isolation in that the product has been isolated from the broth; however, the solvent extraction process extracts other compounds as well from the broth. Often, adsorption (Section 7.1.1) as well as membrane processes such as ultrafiltration and reverse osmosis may be used (Sections 7.2.1.3 and 7.2-1.2). Sometimes, such a step results in a significant increase in product concentration.

11.1

Bioseparations

849

The next stage in the antibiotic processing in Figure 11.1.1 involves purification and concentration. In the case of antibiotics, first there occurs a process of back extraction into an aqueous solution, which selectively back extracts the product more than the other impurities. Then additional solvent extraction and aqueous back extraction steps are carried out under appropriate pH conditions to achieve further purification and concentration. This process is usually followed by activated carbon adsorption to remove impurities, then crystallization (Section 6.4.1), which provides a very high level of purification. For many soluble products, the purification step is usually implemented via chromatography (Section 7.1.5), in the case of proteins by the highly selective affinity adsorption/chromatography (Section 7.1.5.1.8). Traditionally, the last step in purifying the product in the desired form is often termed the polishing step (Belter et al., 1988). It involves purification via crystallization/chromatography. In the case of antibiotics, crystallization is followed by vacuum drying (Section 6.3.2.4). It is useful now to identify the concentration of the antibiotic product (i.e. the titer) during the different steps identified earlier (Belter et al., 1988). The product concentration level in the fermentation broth may vary between 0.1 and 5 g/liter. Removal of insolubles via microfiltration, rotary vacuum filter and centrifuging increases the product concentration marginally, in the range of 1–5 g/liter. Product isolation via solvent extraction enhances the product concentration to 5–50 g/liter. Purification by chromatography or crystallization leads to a product concentration level between 50 and 200 g/liter. The polishing step may not enhance the concentration in general much, but it certainly improves the purity. If a membrane process such as reverse osmosis is used to concentrate a very dilute solution of the antibiotic, 10–30 times the initial concentration may be achieved. The concentration of penicillin in the fermentation broth has recently increased to as much as 10–50 g/liter. On the other hand, the concentration of a therapeutic protein in the fermentation broth may be as low as

10−3–10−5 g/liter. At the other extreme, small molecule bioproducts, such as citric acid or alcohol, are present in the bioreactor at levels of around 100 g/liter and 70–120 g/liter, respectively. Antibiotics are not the only extracellular products obtained in this way; alcohol, citric acid, lactic acid, l-lysine are typical small-molecule examples of extracellular products. For biological macromolecules, examples of extracellular products include proteins, such as tissue-type plasminogen activator, monoclonal antibodies (mAbs), such as immunoglobulins, proteases (enzymes which degrade proteins such as subtilisins), used in a variety of applications of food processing, industrial and household applications, and polysaccharides, such as xanthan gum. We will now briefly illustrate an abstracted stepwise scheme that may be followed for recovery and purification of an intracellular product (Figure 11.1.2). Typical examples of intracellular products are: human insulin (a hormone, MW 5734), bovine growth hormone, human growth hormone (HGH), human leukocyte interferon, glucose isomerase from Streptomyces species, hemoglobin from red blood cells. Amongst many differences in the processing schematics between an extracellular product and an intracellular product are the additional steps to be pursued to get the product into the usual downstream processing steps (Figure 11.1.2). Usually, with intracellular products, the cells have to be harvested first via centrifugation (into pellets) or tangential-flow microfiltration (TFF) (retentate). These cells are then lysed (ruptured, broken open) via mechanical disruption (as in a homogenizer, bead mills, etc.), chemical disruption (via osmotic shock – which happens when cells are immersed in pure water which enters the cell and swells it, leading to rupture – or solubilization by a concentrated solution of detergents, which destroy the cell membrane, releasing the materials from the cell interior; a dilute solution (e.g. 10%) of toluene will essentially achieve the same goal except organic solvents may not be desirable as a contaminant).

Pellets Centrifuge Cell culture fermentation

Cell harvesting

via

Cell lysis

or Retentate TFF

Cell lysis

Lysate

Downstream processing

Lyophilization

Chromatography

Affinity chromatography

Dialysis or ultrafiltration

Dissolution

Precipitation

Filtration

Cake Figure 11.1.2. Block diagram containing additional steps in the production process for a typical intracellular product prior to downstream processing (TFF ¼ tangential-flow filtration).

850 Once the lysate is obtained, it is subjected to filtration to remove cellular debris; the product is in the clarified solution. Usually, it (e.g. a protein) is precipitated or extracted by aqueous two-phase extraction (Sections 4.1.3 and 8.1.4). The precipitate may be dissolved and then subjected to the usual concentration and purification steps. Such a scheme has to be preceded by a number of complex steps if the protein (for example) product is present inside the cell as an inclusion body. Inclusion bodies are specific regions in cells containing a large amount of protein, often misfolded; once they are released, after cell disruption, refolding of the protein has to be carried out. An example of such a situation is encountered in the production of human insulin, a polypeptide. An illustrative account of human insulin production is provided in Harrison et al. (2003, chap. 11). The same reference provides details of the production of extracellular products, such as citric acid as well as a monoclonal antibody. One of the earliest textbooks on bioseparations is by Belter et al. (1988). Among more recent textbooks, that by Ladisch (2001) has a greater emphasis on chromatography since it supposedly consumes about 50% of the bioseparation costs. Additional textbooks on bioseparations are those by Garcia et al. (1999) and Harrison et al. (2003). Sections on bioseparations in Bailey and Ollis (1986) and Shuler and Kargi (2002) are also useful. These books are to be consulted for additional information. Two other types of products should be mentioned. When the product is a whole cell (e.g. baker's yeast, single-cell proteins), many of the downstream processing steps considered so far are not needed. Filtration, followed by any other treatment necessary for the cell to be useful as a final product, is sufficient. For baker's yeast, the filtered cake is extruded into particles and dried at ~45–50  C. The second type of product is of much smaller molecular weight and volatile in nature, e.g. ethanol, acetone, butanol, acetic acid, etc. In all such cases, whether a concentrated or a diluted product (e.g. low-alcohol beer) is needed, often the first step is to go to a beer still, where ethanol and water, for example, are evaporated and taken out through the top whereas the suspended solids are taken out from the bottom of the still as “stillage.” The stillage is passed through screens and then through various driers to obtain dried grains, etc., as additional products. The evaporated products are purified. In these cases, evaporation and distillation are used, unlike processes for almost any other bioproduct development. Distillation is implemented almost universally for the recovery of higher concentrations of volatile bioproducts such as alcohol, acetone and butanol. However, this step consumes almost 50% of the energy cost of the alcohol production plant (Shuler and Kargi, 2002). As a result, lower energy consuming techniques, such as pervaporation (Section 6.3.3.4) (Thongsukmak and Sirkar, 2007), reverse osmosis (Section 7.2.1.2), etc., are being explored. On the

Common separation sequences other hand, to produce a low-alcohol beer, processes such as dialysis and reverse osmosis are practiced along with distillation to produce simultaneously two products, e.g. low-alcohol beer (~0.5–2.5% alcohol) and a high-alcohol content product (Tilgner and Schmitz, 1981; Moonen and Niefind, 1982; Attenborough, 1988; Ladisch, 2001). What we have observed so far in the production of bioproducts is that a large number of separation/purification steps have to be implemented consecutively, one step at a time, till the final product is obtained in the desired form and at the required purity level. These steps include: (1) cell harvesting and lysis for intracellular products; (2) separation of cells and cellular debris from the product obtained from fermentation or cell culture in solution/ suspension via centrifugation, microfiltration, filtration; (3) isolation of the product; (4) concentration and purification of the products; (5) polishing of the product. An item of considerable concern is the extent of loss of the bioproduct along the train of bioseparation steps. Each step encounters some loss. For example, suppose there are five steps and each step achieves a fractional recovery YiR ð¼ 0:95Þ of species i. The overall fractional recovery ðYiR Þov will be (Ladisch, 2001). ðYiR Þov ¼ ðYiR Þ5 ¼ ð0:95Þ5 ¼ 0:774:

ð11:1:1Þ

Therefore the overall process recovery of the product is significantly reduced. Processes have been developed that combine at least two of the steps in one device to attempt higher recovery and additional device cost reduction. Section 7.1.6 (Figure 7.1.28) illustrates the expanded-bed adsorption process for protein recovery and partial purification directly from a fermentation broth/lysate. Figure 11.1.3 illustrates a combined microfiltration/adsorption chromatography process (Xu et al., 2005). A hollow fiber microfiltration membrane module has its shell side packed with chromatographic resin beads. A cyclic process is carried out. A broth/lysate containing protein products in solution is passed through the bore of the microfiltration hollow fibers for a certain period. The permeate goes through the resin beads on the shell side and the proteins are adsorbed. After this loading period, the feed broth/ lysate flow is stopped. Then an eluent solution is passed through the fiber bore into the shell side to elute the proteins in sequence with the fiber bore end closed. Next the resin bed is regenerated by passing the starting buffer to initiate the loading–elution–regeneration cycle again. The advantages of this combined process are: recovery and separation of proteins directly from an unclarified feed; self-cleaning of the membrane fouling via the washing and elution steps; recovery of the proteins adsorbed on the membrane; prevention of product degradation in the holding tank prior to chromatography; elimination of the holding tank.

11.2

Separation sequences for water treatment

851

Fermentation broth/ cell culture harvest

Impervious coating layer

Large entities like cell or cell debris are rejected

Eluent

Permeate

Retentate

Eluted protein peaks

Target proteins permeating through the membrane are captured by the beads

Figure 11.1.3. Integrated membrane filtration and chromatography device. (After Xu et al. (2005).)

In the purification of therapeutic monoclonal antibodies (mAbs), produced as an extracellular product from bacteria, yeast and mammalian cells, the isolation and purification steps involve a number of demanding processes. For example, when a mammalian cell line such as Chinese hamster ovary (CHO) is employed, the impurities in the harvested cell culture liquid contain Chinese hamster ovary cell proteins (CHOPs), DNAs, antibody variants, viral particles, endotoxins and small molecules (Follman and Fahrner, 2004). Conventionally, affinity chromatography using protein A (Figure 7.1.27(a)) is employed to bind antibodies from such a solution, leading to >99.5% removal of the impurities. However, the protein A based affinity chromatography step is very costly; it may cost as much as 35% of the total raw material costs for downstream purification. A common process sequence used includes a three-column sequence of a protein A affinity column, a cation exchange column and an anion exchange column. Additional steps include a virus filtration step and an ultrafiltration/diafiltration step. Mehta et al. (2008) have described in detail how a twocolumn nonaffinity process, when integrated with a highperformance tangential-flow filtration (HPTFF) technique, can replace the protein A affinity column based purification sequence for the purification of mAbs. This reduces the purification cost for mAbs substantially; the plant footprint is also much smaller.

11.2

Separation sequences for water treatment

Water is essential not only to human existence, but also for a variety of industrial and agricultural activities. Purification of water is essential for potable water; ultrapurification of water is indispensable to the semiconductor industry, where the

purified water used for cleaning and rinsing of wafers is known as ultrapure water (UPW). The separation sequence followed to obtain a particular water product depends on the nature of the ultimate product water needed and the quality of the feed water. In the multistep scenario followed, processes involving chemical reactions invariably appear along with a host of different separation processes. Examples of different process sequences are related either to the water source to be treated or to the ultimate product water desired. Sea-water and brackish-water desalination, surface-water, ground-water and waste-water treatment are examples of the first category; potable water (municipal water), pharmaceutical grade water, ultrapure water, etc., are examples of the second category. We will illustrate very briefly the water treatment processes for sea-water and brackish-water desalination, ultrapure water production and municipal water production. 11.2.1

Sea-water and brackish-water desalination

Sea water usually contains total dissolved solids (TDS) at the level of 35 000 ppm; around the Middle East, this can go up to 40 000–50 000 ppm. Brackish waters can have TDS as low as 700–1000 ppm, going all the way up to 10 000 ppm plus. Desalination of sea water and brackish water involves a significant number of pretreatment steps, which are essential to the successful long-term performance of the reverse osmosis (RO) membranes increasingly and invariably employed in new desalination plants. There are also a few post-treatment steps after desalination. The exact nature and details of the pretreatment steps are influenced by the source water for the plant. As shown in Figure 11.2.1, typical pretreatment steps consist of a variety of separation

852

Common separation sequences

Cl2

Coagulant

Acid

Microfilter

Filter

Lime

Antiscalant

Cl2

Water to storage/ distribution

Degassing

Reverse osmosis

Figure 11.2.1. Schematic for a RO desalination plant.

processes, with some reaction processes taking place before, after or in between. Feed water to be desalinated undergoes the following treatment steps, in various combinations, prior to desalination by a reverse osmosis (RO) membrane: chlorination via sodium hypochlorite addition to control growth of bacteria or algae; addition of coagulants such as ferric chloride and polyelectrolytes to remove colloidal materials and suspended solids in granular filter beds (Section 7.2.2); addition of acid to convert scaling salts, e.g. CaCO3, to nonscaling salts; addition of antiscalants to prevent scale formation from salts such as CaSO4 and CaCO3; microfiltration to remove micron-size particles/precipitates, etc. (Section 7.2.1.4); degassing to remove dissolved gases, e.g. H2S, O2 (Sections 8.1.2.2.1 and 8.1.2.4). Post-treatment of the desalted permeate from the reverse osmosis unit (Section 7.2.1.2) consists of the addition of Cl2 and lime for disinfection and corrosion protection. If H2S is present, it is eliminated by air stripping. For brackish-water feeds containing hydrocarbons, an activated carbon adsorber is used to remove dissolved hydrocarbons prior to microfiltration, which is followed by steps needed during the pretreatment and post-treatment processes: dechlorination is required as a pretreatment if the RO membrane for desalination cannot tolerate residual chlorine; dissolved oxygen is often removed to avoid damaging the RO membrane via vacuum based deaeration or addition of sodium bisulfite. An introduction to the pretreatment and post-treatment processes for membrane based sea-water and brackish-water desalination has been provided by Williams et al. (2001). The scale of such desalination plants is quite large, as much as 87 million gallons per day at Ashkelon, Israel, for example. A very significant fraction of desalination is achieved via thermal desalination processes such as multistage flash, multieffect distillation, etc. (Section 10.2.1) (primarily in older plants). Pretreatment processes include filtration, chlorination, deaeration and scale control via acid addition, antiscalant treatment, etc. Post-treatment processes include building up the dissolved salt concentration to the level of 50 ppm via blending with brackish ground water or addition of lime, etc. (Howe, 1974), since distilled water is highly corrosive.

11.2.2

Ultrapure water production

Ultrapure water (UPW) is routinely used in the microelectronics industry for repeated cleaning and rinsing of wafers; a wafer may contact as much as 100–1000 liter of UPW. It is also used to clean process equipment during such manufacturing. A typical microelectronics plant may use as much as 2 million gallons per day. Such ultrapure water should not have particulate, bacterial, ionic, organic or dissolved gaseous impurities, which would all damage integrated circuits. Some of the standards to be satisfied are as follows: total organic carbon (TOC) ~ 1 ppb; dissolved O2, 1 ppb; resistivity at 25  C, 18.20 megohm-cm; metals at ppt (parts per trillion) level; cations, 1–5 ppt; anions, 20–30 ppt; reactive silica, 50 ppt; number of 0.05–0.1 μm particles per liter, 100; endotoxin units/ml, 0.03; bacteria, 1 coliform unit per 1000 ml (Baird and Williams, 2005). The plant producing UPW obtains the potable feed water from the municipality. Multimedia filters made up of sand, coal, etc., are employed first to eliminate large particles and suspended solids in the feed water (Figure 11.2.2). Microfilters used to remove 1–5 μmþ particles are included next in the loop, before this water is introduced to a reverse osmosis (RO) unit in two stages. The first RO unit, along with the preceding units, are part of the makeup system. The second RO unit is part of the main treatment system – the primary system. Microbes such as bacteria, virus and yeast present in the water (especially introduced by filter beds of carbon or otherwise) and in connecting lines are destroyed/disrupted by ultraviolet disinfection at 254 nm. The permeate from RO units still contains a significant amount of electrolytes, which are almost completely removed by continuous electrodeionization (CEDI)1 units. Any remaining trace ions are removed by a mixed ion exchange bed to produce water possessing a resistivity of >18 megohm-cm. This water is passed through a 0.2 μm filter. Part of this water is recycled

1

If the brine feed channels of an electrodialysis (ED) unit (Section 8.1.7.2) are filled with mixed ion exchange resin beads, the deionization of the water can be carried out continuously to limits much greater than that in any ED unit.

11.3

Chemical and petrochemical industries

Makeup system

853

Feed water from municipality

M RO-I RO prefilter 1–5 µm

Concentrate to drain

Primary treatment system

N2

Storage tank

M RO-II

UV sterilizer

Concentrate to drain

Multimedia filter

CEDI

Mixed ion exchange bed

to drain

Recycle water from wafer fabrication

Water to wafer fabrication plant Polishing system

UV UV Ultrafiltration

Membrane Mixed ion exchange bed

Reject

0.2 µm filter

M

O3

Advanced chemical oxidation processes

Storage tank

degasifier

Figure 11.2.2. Process flowsheet for UPW production. (After Baird and Williams (2005).)

to a storage tank (in between the RO-I and RO-II units), where it is mixed with the product of the make-up system. The rest of the water out of the 0.2 μm filter is treated as the output of the primary treatment system. This primary treatment system output undergoes an additional polishing treatment, in particular because water from the wafer fabrication is recycled to the tank receiving the primary treatment system output. Specifically, advanced chemical oxidation processes using O3 and UV are employed to destroy microbes and organics, although the latter are reduced to compounds removed in the subsequent treatment steps. These steps include: membrane degassing (Section 8.1.2.2.1) to remove dissolved O2 and CO2 (the O2 level 1000 dalton) as the last polishing filter.

WPU is subjected to the additional step of distillation (evaporation) in order to quality as WFI. However, the requirements for WPU are less stringent than those for UPW. For example, the TOC level may be as much as 500 ppb; there are no requirements for dissolved oxygen as such (in UPW it is 1 ppb); water resistivity can be as low as 0.77 megohm-cm; the bacteria level in WFI can be 10 coliform units per 100 ml, etc. (Baird and Williams, 2005). Plants producing WPU are much smaller than UPW units and typically have a capacity of 10 000 to 50 000 gallons per day. The process sequence employed to produce WPU grade water in the pharmaceutical industry typically comprises the following steps: multimedia filtration, water softening, activated carbon filtration, prefiltration, UV disinfection, reverse osmosis, continuous electrodeionization (CEDI). For WFI, as mentioned earlier, the WPU water is subjected to a distillation process, and may be supplied hot if needed.

11.3 Chemical and petrochemical industries 11.2.3

Pharmaceutical grade water

Water required for pharmaceutical purposes has to satisfy two different requirements: water for injection (WFI) and purified water (WPU). Water satisfying the requirements for

In terms of the physical dimensions of the separation equipment employed, the footprint and the volume of product produced, chemical and petrochemical operations are huge, much larger than the previous two process categories

854

Common separation sequences The process of aromatization, e.g. cyclohexane to benzene, produces a gaseous stream containing some H2 (so does cyclization, i.e. linear alkanes converted to cyclic compounds). The H2 is taken away as a bleed and the other cyclic compounds are recycled back to the feed. The liquid product from this gas–liquid separator goes to a debutanizer column, which concentrates light hydrocarbons at the top. The heavier hydrocarbons go next to a solvent extraction column, where a polar organic solvent such as ethylene glycol or propylene glycol is introduced as a solvent to extract aromatics preferentially (Figure 4.1.9(a)). The extract is taken to a solvent regenerator, which provides volatile aromatics as the top product and the much less volatile polar solvent as the bottom product recycled back to the aromatics extraction column. The volatile aromatics fraction is then taken to a distillation column producing benzene and toluene as the top product; the bottom product stream contains xylenes and heavier aromatics, which are subjected to further separation as well as isomerization.

discussed. Further, the number of products and the types of processes used are much larger. It is fruitless to attempt to describe here such variety vis-à-vis the separation processes/techniques employed. We will merely illustrate here a few flowsheets to point out the major differences from those considered in Sections 11.1 and 11.2. More often than not, the sequence of separations in chemical and petrochemical operations comprises part of a chemical production process where chemical reactions play a crucial part. Separation processes are often used to purify the feed stream entering the reactor. The products of reaction need to be separated from each other and from the residual feed, with the separated unreacted feed components recycled back to the reactor inlet where the feed stream is introduced. Figure 11.3.1 illustrates schematically this basic mode of operation, without reference to any particular process. Figure 11.3.2 provides an example. Naptha fraction (Shreve and Hatch, 1984) from crude oil distillation is taken to a reformer, where the octane number is increased by producing more olefins, compounds having lower molecular weight and achieving more cyclization and aromatization.

Feed

Separator 1

Reactor

Purifier

Residue

Product 1

Separator 2

Product 2

Reactant recycle

Figure 11.3.1. Hypothetical process schematic for production of chemicals – role of separators.

Compressor

Bleed H2 Benzene, toluene

Aromatics separation column

Aromatics

Regenerator for solvent

Polar solvent

Aromatics extraction

Separator

Debutanizer

Reformer

Light hydrocarbons Nonaromatics

Naptha

Heavier aromatics Figure 11.3.2. Separation train after reforming of naptha.

11.4

Hydrometallurgical processes

855

Tailings

Acid Solvent extraction and back extraction cascade

Leaching

Thickener

Clarifier

Residue treatment

Concentrated metal solution

Tailings Electrowinning Metal Figure 11.4.1. General schematic of a hydrometallurgical process employing electrowinning to recover the metal.

11.4

Hydrometallurgical processes

Hydrometallurgical processes involve many steps to achieve recovery of metal in the form and purity desired, starting from ores or other sources such as tailings or waste solutions/leachates. In the first step, an ore, which probably contains the metal at a low concentration (e.g. 1%), is ground in a grinding circuit and then sent to a leaching tank, where an acid, often sulphuric acid, is employed to leach the metal into solution (Figure 11.4.1). The slurry is then sent to a thickener, where flocculation2 is employed

2

Certain types of particles, for example precipitates, have a tendency to come together while in suspension. Flocculants, especially large molecular weight polymers with or without positive or negative charges, when added to such a suspension, bring the particles together much more readily and allow the larger assembly of particles, “flocs,” to settle rapidly by gravity.

with flocculating agents and the floc containing the slurry particles are settled. The liquid overflow from thickeners is filtered/clarified, and the filtered clear liquid is sent to a solvent extraction cascade, where the metal in the liquid containing the metal salt in solution is extracted into a solvent. The solvent extract is then subjected to back extraction into an aqueous solution. The back extract is generally a more concentrated solution of the metal in salt form. From such a concentrated aqueous solution of the metal, the recovery of the metal is achieved in a variety of the ways. The metal may be precipitated as a hydroxide, an oxide, or calcined as an oxide or recovered as a metal via electrowinning (e.g., Cu, Co, Ni), etc. Illustrations of process flowsheets for the recovery of a variety of metals, e.g. copper, zinc, nickel, uranium, chromium, beryllium, etc., are available in Ritcey and Ashbrook (1984b). Benedict et al. (1981) also provide process flowsheets and descriptions of metals relevant to the nuclear power industry.

Postface The area of separations is extremely broad. Most separation processes/techniques have an extensive literature. Each process has its own universe. Yet there are a few common principles and characteristics that are shared by many separation techniques/processes. This book has provided an elementary introduction to such principles and characteristics, even as it provides some details of many of the most commonly used separation processes. It is hoped that this book will help the reader to be prepared to undertake more extensive studies of individual processes/ techniques or of a certain class of techniques or certain sequences of separation processes/techniques. An important goal of this book is to develop an appreciation that separation processes/techniques may be understood on the basis of a few key features/concepts: the nature of the force causing the selective movement of molecules, macromolecules and particles; the nature of the various regions/phases in the separation system; the four types of bulk flow pattern of the regions/phases vis-à-vis the direction(s) of the forces causing selective movement of species/particles to be separated; the method of feed introduction. Further, for relatively simple systems, one could develop the governing equations that describe the separation achievable from the set of basic governing equations provided in Section 6.2 for most systems

described in the book. In addition, a useful introduction to the role played by chemical reactions of certain types for a variety of separations has been provided. Many topics could not be covered in this book. A much abbreviated list includes: the molecular basis of equilibrium partitioning of molecules between different phases; enthalpic and entropic contributions to partitioning/selectivity; the molecular basis of affinity binding in bioseparations; nonisothermal analysis of absorption columns, adsorption beds, distillation columns, etc.; multicomponent multistage separations in distillation columns; numerical methods for multicomponent multistage countercurrent separation processes; experimental methods in separation studies; hybrid separation processes; selection of separation processes for solving a separation problem; reaction–separation/ separation–reaction/reaction–separation–reaction processes and devices. Quite a few separation processes were barely touched upon. A number of separation processes could not be covered: azeotropic distillation; Calutron; cell disruption; coagulation; continuous deionization; expression; extractive distillation; forward osmosis; gaseous diffusion; ion retardation; leaching; membrane distillation; nanofiltration; pressure-swing distillation, etc. Many analytical techniques were also not covered.

Appendix A

Units, various constants and equivalent values of various quantities in different units The basic units of various quantities and fundamental dimensions are identified below for SI, CGS and FPS systems. Additional common abbreviations are also listed.

SI (Système International ď Unités) system Current, amp (ampere); length, m (meter); mass, kg (kilogram); mole, mol; kmol (kilogram mole); time, s (second); temperature, K (kelvin); force, N (newton); pressure, N/m2 (pascal, Pa); energy, N-m (joule, J); power, N-m/s (watt, W).

CGS system Length, cm (centimeter); mass, g (gram); mole, gmol (gram mole); time, s (second); temperature
C (centigrade); force, dyne; energy, erg (dyne-cm).

FPS system Length, ft; mass, lbmass;1 mole, lbmol (pound mole); time, s (second); temperature
F (Fahrenheit); force, lbforce;1 pressure, psia (pounds per square inch).

Additional common units and abbreviations atm (atmosphere), C (coulomb), cal (calorie), farad (F), hr (hour), Hz (hertz), km (kilometer), kV (kilovolt), kW (kilowatt), l (liter), ml (milliliter), rad (radian), S (Siemens), tesla (T), townsend (Td), W (watt)

1 Although conventional practice is to indicate lbmass and lbforce by lbm and lbf, respectively, we have used the nonsubscripted versions in this book.

Appendix B

Constants Fundamental constants Mathematical constants: π ¼ 3.14159…;

pffiffiffi 2 ¼ 1.41421….

Logarithmic constants: e ¼ 2.71828…; loge10 ¼ ℓn 10 ¼ 2.30258…; log10e ¼ 0.43429.

Physical constants Gas law constant: R ¼ 8.31451 J/gmol-K ¼ 1.987 cal/(gmol-K) ¼ 1.987 Btu/(lbmol ºR) ¼ 8314.3 J/kgmol-K ¼ 82.057  10−3 m3-atm/kgmol-K ¼ 8314.34 m3-Pa/kgmol-K ¼ 0.0820578 L-atm/gmol-K ¼ 82.0578 atm-cm3/gmol-K ¼ 10.73 psi-ft3(lb mol ºR) ¼ 1.314 ft3-atm/lbmol-K ¼ 0.7302 ft3-atm/lbmol-ºR. ~ ¼ 6.02214  1023 molecules/gmol. Avogadro's number: N ~ ¼ 1:381  10−23 J=K ¼ 1:381  10−16 erg=K: Boltzmann constant: k B ð¼ R=NÞ Faraday constant: F ¼ 96 485 coulomb (C)/g-equivalent. Standard acceleration of gravity: g ¼ 980 cm/s2 ¼ 9.806 m/s2 ¼ 32.174 ft/s2. (Gravitational conversion factor gc ¼ 32.174 lbmass-ft/lbforce-s2 ¼ 980.66 gmass-cm/gforce-s2.) Speed of light in vacuum: c ¼ 2.99792  1010 cm/s. Speed of sound in dry air at 0  C ¼ 331.3 m/s.

Charge of an electron: e ¼ 1.6021  10−19 coulomb (C).

Electrical permittivity of vacuum: ε0 ¼ 8.8542  10−14 coulomb/volt-cm (or farad/cm). Planck's constant: h ¼ 6.62608  10−34 J-s. Joule's constant (mechanical equivalent of heat) ¼ 4.184 J/cal ¼ 4.184  107 erg/cal ¼ 778.16 ft-lbforce/Btu. Atmospheric pressure (at sea level) ¼ 760 mm Hg at 0  C ¼ 760 torr ¼ 29.92 in Hg at 0  C ¼ 33.9 ft H2O at 4  C ¼ 1.1013 bar ¼ 14.696 psia ¼ 1.0132  105 N/m2 ¼ 1.0132  105 Pa ¼ 1 atm ¼ 76 cm Hg at 0  C.

Appendix C

Various quantities expressed in different units Acceleration: 1 m/s2 ¼ 100 cm/s2 ¼ 3.28 ft/s2 ¼ 0.425  108 ft/h2.

Area: 1 m2 ¼ 104 cm2 ¼ 10.76 ft2 ¼ 1550 in2.

Density: 1 kg/m3 ¼ 1000 g/cm3 ¼ 1 g/liter ¼ 1 g/ℓ ¼ 0.0624 lbmass/ft3 ¼ 8.34  10−3 lbmass/US gal ¼ 1.002  10−2 lb/UK gal. Diffusivity: 1 m2/s ¼ 104 cm2/s ¼ 38 750 ft2/hr ¼ 10.76 ft2/s. Energy: 1 joule ¼ 1 J ¼ 1 N-m ¼ 107 dyne-cm ¼ 107 erg ¼ 107 g-cm2/s2 ¼ 1 kg-m2/s2 ¼ 0.239 cal ¼ 9.47  10−4 Btu ¼ 1 W-s ¼ 2.7778  10−7 kW-hr ¼ 3.725  10−7 hp-hr ¼ 0.73756 ft-lbforce. These conversions are based on International Steam Table Convention of the definition of calorie. Energy flux: 1 J/m2-s ¼ 1 W/m2 ¼ 0.317 Btu/ft2-hr ¼ 2.39  10−5 cal/cm2-s. Enthalpy: 1 J/kg ¼ 2.39  10−4 cal/g ¼ 4.299  10−4 Btu/lb. Force: 1 N ¼ 1 kg-m/s2 ¼ 105 dyne ¼ 105 g-cm/s2 ¼ 7.233 lbmass-ft/s2 ¼ 7.233 poundals1 ¼ 2.2488  10−1 lbforce. Heat capacity (specific heat): 1 J/kg-K ¼ 10−3 kJ/kg-K ¼ 2.388  10−4 Btu/lb- F ¼ 2.388  10−4 cal/g- C.

Heat transfer coefficient: 1 N-m/m2-s-K ¼ 1 W/m2-K ¼ 1 kg/s3-K ¼ 10−4 W/cm2-K ¼ 103 g/s3-K ¼ 2.39  10−5 cal/cm2-s-K ¼ 1.761  10−1 Btu/ft2-hr- F.

Interfacial tension (surface tension): 1 N/m ¼ 103 dyne/cm ¼ 103 erg/cm2. Latent heat: 1 kJ/kg ¼ 0.43 Btu/lbmass ¼ 1 J/g ¼ 0.239 cal/g.

Length: 1 m ¼ 100 cm ¼ 106 μm ¼ 109 nm ¼ 1010 Å ¼ 3.28 ft ¼ 39.37 inch ¼ 1.0933 yard.

Mass: 1 kg ¼ 1000g ¼ 2.20462 lbmass ¼ 35.2739 oz ¼ 1.1022  10−3 ton ¼ 10−3 metric ton. Mass-flow rate: 1 kg/s ¼ 2.2045 lbmass/s ¼ 1000g/s.

Mass flux (mass velocity): 1 kg/m2-s ¼ 737.46 lbmass/ft2-hr ¼ 0.1 g/cm2-s. Mass transfer coefficient: Molar concentration based: 1 kgmol/m2-s-(kgmol/m3) ¼ 1 m/s ¼ 100 cm/s ¼ 11808 lb mol/ft2-hr-(lb mol/ft3) ¼ 11808 ft/hr ¼ 1 gmol/m2-s-(gmol/m3) ¼ 100 gmol/cm2-s-(gmol/cm3). Mole fraction based: 1 kgmol/m2-s-mole fraction ¼ 1000 gmol/m2-s–mole fraction ¼ 737.46 lb mol/ft2-hr-mole fraction. Partial pressure based: 1 kgmol/m2-s-(N/m2) ¼ 747  105 lb mol/ft2-hr-atm. Power: 1 W ¼ 1 J/s ¼ 1N-m/s ¼ 0.73756 ft-lbforce/s ¼ 0.239 cal/s ¼ 9.47  10−4 Btu/s ¼ 3.41 Btu/hr ¼ 1.34  10−3 hp. Power/volume: 1 W/m3 ¼ 1 N-m/s-m3 ¼ 0.0209 ft-lbforce/s-ft3. Pressure: 1 Pa ¼ 1 N/m2 ¼ 1.45  10−4 lbforce/inch2 (psi) ¼ 208.8 lbforce/ft2 ¼ 10 dyne/cm2 ¼7.5006  10−3 mm Hg ¼ 9.8692  10−6 atm. 1

1 poundal ¼ 1(lbmass-ft)/s2.

860

Appendix C

Specific heat: see “Heat capacity (specific heat).” Surface area/volume (specific area): 1 m2/m3 ¼ 0.01 cm2/cm3 ¼ 0.304 ft2/ft3.

Temperature:  C ¼ (5/9)( F – 32); 0  C ¼ 32  F ¼ 273.15 K ¼ 491.67R;  R ¼  F þ 459.67; K ¼  C þ 273.15 ¼ (1/1.8) R; 100  C ¼ 373.15K ¼ 671.67  R ¼ 212  F.

Thermal conductivity: 1 W/m-K ¼ 1 kg-m/s3-K ¼ 105 erg/s-cm-K ¼ 2.3901  10−3 cal/s-cm-K ¼ 1 J/s-m-K ¼ 5.778  10−1 Btu/hr-ft- F. Velocity: 1 m/s ¼ 3.28 ft/s ¼ 100 cm/s.

Viscosity: 1 Pa-s ¼ 1 kg/m-s ¼ 10 g/cm-s (poise) ¼ 103 cp (centipoise) ¼ 2.0886  10−2 lb force-s/ft2 ¼ 0.6719 lbmass/ft-s.

Volume: 1 m3 ¼ 106 cm3 (ml) ¼ 1000 l ¼ 35.287 ft3 ¼ 264.17 US gallon ¼ 220.83 UK gallon. Volume flow rate: 1 m3/s ¼ 35.287 ft3/s ¼ 106 cm3/s. Volume flux: 1 m3/m2-s ¼ 100 cm3/cm2-s ¼ 3.28 ft3/ft2-s.

References Abegg, C.F., J.D. Stevens and M.A. Larson, AIChE J., 14, 118 (1968). Abou-Nemeh, I., A. Das, A. Saraf and K.K. Sirkar, J. Membr. Sci., 158, 187 (1998). Adamson, A.W., Physical Chemistry of Surfaces, 2nd edn., Wiley Interscience, London (1967). Agrawal, R. and T.F. Yee, Ind. Eng. Chem. Res., 33, 2717 (1994). Agrawal R., Z.T. Fidkowski and J. Xu, AIChE J., 42(8), 2118 (1996). AIChE, Bubble-Tray Design Manual, American Institute of Chemical Engineers, New York (1958). AIChE, Equipment Testing Procedures Committee, AIChE Equipment Testing Procedure – Tray Distillation Columns, 2nd edn., American Institute of Chemical Engineers, New York (1987). Aiken, R.C., Chem. Eng. Sci., 37, 1031 (1982). Albertins, R., W.C. Gates and J.E. Powers, in Fractional Solidification, M. Zief and W.R. Wilcox (eds.), Vol. I, Marcel Dekker, New York (1967), chap. 11. Albertsson, P.-Å., Partition of Cell Particles and Macromolecules, 3rd edn., Wiley Interscience, New York (1986). Alexandratos, S.D and D.W. Crick, Ind. Eng. Chem. Res., 35, 635 (1996). Alishusky, J.J. and R.L. Fournier, AIChE J., 36(10), 1605 (1990). Allcock, H.R., F.W. Lampe and J.E. Mark, Contemporary Polymer Chemistry, 3rd edn., Pearson Education, Upper Saddle River, NJ (2003), chap 15. Alpers, A., B. Keil, O. Lüdtke and K. Ohlrogge, Ind. Eng. Chem. Res., 38, 3754 (1999). Ambler, C.M., Chem. Eng. Prog., 48, 150 (1952). Amini, A., U. Paulsen-Sorman and D. Westerlund, Chromatographia, 50, 497 (1999). Anderson, J.B., AIChE Symp. Series, 76(192), 89 (1980). Anderson, J.L. and J.A. Quinn, J. Chem. Soc. Faraday Trans., 1, 68(4), 744 (1972). Anderson, J.L., M.E. Lowell and D.C. Prieve, J. Fluid Mech., 117, 107 (1982). Anderson, J.L. and J.A. Quinn, Biophys. J., 14, 130 (1974). Andrews, A.T., Electrophoresis: Theory, Techniques, and Biochemical and Clinical Applications, Clarendon Press, Oxford (1981), pp. 39, 87. Annesini, M.C., F. Gironi and L. Marrelli, Ind. Eng. Chem. Res., 27(7), 1212 (1988). Anwar, M.M., C. Hanson and M.W.T. Pratt, Trans. Inst. Chem. Eng., London, 49, 95 (1971). Applegate, L.E. and C.R. Antonson, in Reverse Osmosis and Membrane Research, H.K. Lonsdale and H.E. Podall (eds.), Plenum Press, New York (1972).

Aris, R., Proc. Roy. Soc. London, Ser. A., 235, 67 (1956). Aris, R., Ind. Eng. Chem. Fund., 8, 603 (1969). Aris, R. and N.R. Amundson, Mathematical Methods in Chemical Engineering, vol. 2, Prentice Hall, Englewood Cliffs, NJ (1973). Armstrong, D. and L. He, Anal. Chem., 73, 4551 (2001). Armstrong, D.W. and H.L. Jin, Anal. Chem., 59, 2237 (1987). Armstrong, D.W., Y. Tang, T. Ward and M. Nichols, Anal. Chem., 65, 1114 (1993). Arsalani, V., K. Rostami and A. Kheirolomoom, Ind. Eng. Chem. Res., 44, 7469 (2005). Asai, S., Ind. Eng. Chem. Proc. Des. Dev., 24, 1105 (1985). Asai, S. and S. Ikegami, Ind. Chem. Fund., 21, 181 (1982). Ash, R., R.M. Baker and R. M. Barrer, Proc. Roy Soc. A, 299, 434 (1967). Ash, R., R.M. Barrer and J.H. Clint, R.J. Dolphin and C.L. Murray, Phil. Trans. Roy. Soc. London, 275(1249), 255 (1973a). Ash, R., R. M. Barrer, and R.J. Lawson, J. Chem. Soc. Faraday Trans. 1, 69, 2166 (1973b). Ashkin, A., Phys. Rev. Lett., 24(4), 156 (1970). Astarita, G., Mass Transfer with Chemical Reaction, Elsevier, Amsterdam (1967). Astarita, G., D.W. Savage and A. Bisio, Gas Treating with Chemical Solvents, John Wiley, New York (1983). Atkinson, G., in American Institute of Physics Handbook, D.E. Gray (ed.), 3rd edn., McGraw-Hill, New York (1972). Attenborough, W.M., Ferment, 1(2), 40 (1988). Atwood, J.L., J.E.D. Davies and D.D. MacNicol (eds.), Inclusion Compounds, Academic Press, London (1984). Auvil, S.R. and B.W. Wilkinson, AIChE J., 22, 564 (1976). Babcock, W.C., R.W. Baker, E.D. Lachapelle and K.L. Smith, J. Membr. Sci., 7, 89 (1980). Bailey, J.E. and D.F. Ollis, Biochemical Engineering Fundamentals, 2nd edn., McGraw-Hill, New York (1986). Bailey, P.D., An Introduction to Peptide Chemistry, John Wiley, New York (1990). Baird, A. and R. Williams, Chem. Eng., May, 36–43 (2005). Baird, M.H.I., “Extraction by chemical reaction,” 88th AIChE National Meeting, Philadelphia, PA, June 8–12 (1980), paper 36b. Bajpai, R.K., A.K. Gupta and M.G. Rao, AIChE J., 20, 989 (1974). Baker, B. and R.L. Pigford, Ind. Eng. Chem. Fund., 10, 283 (1971). Baker, R.W., Membrane Technology and Applications, 2nd edn., John Wiley, Hoboken, NJ (2004). Baker, R.W. and J.G. Wijmans, “Membrane fractionation process,” US Patent 5,032,148, July 16 (1991). Baker, R.W. and J.G. Wijmans, in Polymeric Gas Separation Membranes, D.R. Paul and Y.P. Yampolskii (eds.), CRC Press, Boca Raton, FL (1994), p. 361.

862 Bard, A.J. and M.V. Mirkin (eds.), Scanning Electrochemical Microscopy, Marcel Dekker, New York (2001). Barker, P.E., Progr. Separ. Purif., 4, 325 (1971). Barrer, R.M., Appl. Mater. Res., 2, 129 (1963). Basak, S. and M.R. Ladisch, Anal. Biochem., 226, 51 (1995). Basmadjian, D., D.K. Ha and C.Y. Pan, Ind. Eng. Chem. Proc. Des. Dev., 14(3), 328 (1975a). Basmadjian, D., D.K. Ha and C.Y. Pan, Ind. Eng. Chem. Proc. Des. Dev., 14(3), 340 (1975b). Basu, R., R. Prasad and K.K. Sirkar, AIChE J., 36(3), 450 (1990). Bates, R.G., Determination of pH: Theory and Practice, John Wiley, New York (1964), p. 183. Batt, B.C., V.M. Yabannavar and V. Singh, Bioseparations 5, 41 (1995). Beckwith, J.B. and C.F. Ivory, Chem. Eng. Commun., 54(1–6) 301 (1987). Begovich, J.M. and W.G. Sisson, AIChE J., 30(5), 705 (1984). Beier, P.M. and I. Stahl, “The electrostatic processing of raw salts at Kali and Salz GmbH,” Proc. XXIMPC, Aachen, Sept. 21–26 (1997), p. 657. Belfort, G., Synthetic Membrane Processes, Academic Press, New York (1984). Belter, P.A., E.L. Cussler and W-S. Hu, Bioseparations: Downstream Processing in Biotechnology, John Wiley, New York (1988). Bender, M.L. and M. Komiyama, Cyclodextrin Chemistry, Springer-Verlag, New York (1978). Benedict M. and T. H. Pigford, Nuclear Chemical Engineering, McGraw-Hill, New York, (1957), p. 500. Benedict, M., T.H. Pigford and H.W. Levi, Nuclear Chemical Engineering, 2nd edn., McGraw-Hill, New York (1981). Benitez, J., Process Engineering and Design for Air Pollution Control, Prentice Hall, Englewood Cliffs, NJ (1993). Benitez, J., Principles and Modern Applications of Mass Transfer Operations, 2nd edn., John Wiley, Hoboken, NJ (2009). Berglund, K.A., in Handbook of Industrial Crystallization, A.S. Myerson (ed.), Butterworths-Heinemann, Boston (1993), p. 89. Berglund, K.A. and E.J. de Jong, Separ. Technol., 1, 1 (1990). Bhattacharyya, D., T. Barranger, M. Jevtitch and S. Greenleaf, Separation of Dilute Hazardous Organics by Low Pressure Composite Membranes, NTIS Report no.: PB 87-214870, July (1987). Bhattacharyya, D., M.E. Williams, R.J. Ray and S.B. McCray, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Van Nostrand Reinhold, New York (1992), chap. 23. Bhattacharyya, D., M.E. Williams, R.J. Ray and S.B. McCray, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Kluwer Academic Publishers, Boston, MA (2001). Bhaumik, S., S. Majumdar and K.K. Sirkar, AIChE J., 42(2), 409 (1996). Bhaumik, D., S. Majumdar and K.K. Sirkar, J. Membr. Sci., 138, 77 (1998). Bhave, R.R., Inorganic Membranes, Synthesis, Characteristics and Applications, Van Nostrand Reinhold, New York (1991), chap. 5. Bhave, R.R. and M.M. Sharma, Chem. Eng. Sci., 38, 141 (1983). Bhave, R.R. and K.K. Sirkar, in Liquid Membranes: Theory and Applications, R.D. Noble and J.D. Way (eds.), ACS Symposium Series. 347, ACS, Washington, D.C. (1987), chap. 10. Biesenberger, J.A., Devolatilization of Polymers, Hanser Publishers, Munich (1983). Bikerman, J.J., Physical Surfaces, Academic Press, New York (1970), p. 315.

References Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley, New York (1960). Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, 2nd edn., John Wiley, New York (2002). Birss, R.R. and M.R. Parker, in Progress in Filtration and Separation, R.J. Wakeman (ed.), vol. 2, Elsevier Scientific Publishing Co., Amsterdam, (1981), chap. 4. Blakebrough, N., Biochemical and Biological Engineering Science, vol. 1, Academic Press, New York (1967). Block, B., Chem. Eng., 68(3), 87 (1961). Böddeker, K.W., G. Bengston and E. Bode, “Pervaporation of low volatile aromatics from water,” IMTEC ’88, Sydney, Australia, Nov. 15–17 (1988). Bogart, M.J.P., Trans. AIChE., 33, 139 (1937). Bonmati, R., R. Margulis and G.C. Letourneux, “Gas chromatography, a new industrial process,” 88th AIChE National Meeting, Philadelphia, PA, June 8–12 (1980). Boyd, G.E., A.W. Adamson and L.S. Myers Jr., J. Am. Chem. Soc., 69, 2836 (1947). Boyde, T.R.C. Separ. Sci., 6(6), 771 (1971). Boyle, M.D.P., E.L. Faulmann and D.W. Metzger, in Molecular Interactions in Bioseparations, T.T. Ngo (ed.), Plenum Press, New York (1993), p. 91. Bradley, D., The Hydrocyclone, Pergamon Press, Oxford (1965). Bramley, A., Science, 92, 427 (1940). Brandt, S., R.K. Goeffe, S.B. Kessler, J.L. O’Connor and S.E. Zale, Biotechnol., 6, 779 (1988). Brant, L.R., K.L. Stellner and J.F. Scamehorn, in Surfactant-based Separation Processes, J.F. Scamehorn and J.H. Harwell (eds.), Marcel Dekker, New York (1989), chap. 12. Breck, D.W., Zeolite Molecular Sieves, Wiley-Interscience, New York (1974). Brian, P.L.T., Ind. Eng. Chem. Fund., 4, 439 (1965). Brooks, C.A. and S. M. Cramer, AIChE J., 38(12), 1969 (1992). Burgess, J., Ions in Solution: Basic Principles of Chemical Interactions, Ellis Horwood Ltd., Chichester; John Wiley, New York (1988), p. 81. Burton, J.A., R.C. Prim and W.P. Slichter, J. Chem. Phys. 21, 1987 (1953). Cahn, R.P. and N.N. Li, Separ. Sci. Technol., 9, 505 (1974). Cahn, R.P. and N.N. Li, in Membrane Separation Processes, P. Meares (ed.), Elsevier, Amsterdam (1976), p. 327. Cahn, R.P., N.N. Li and R.C. Minday, Environ. Sci. Technol., 12, 1051 (1978). Calderbank, P.H. and M.B. Moo-Young, Chem. Eng. Sci., 16, 39 (1961). Caldwell, K.D. and Y-S. Gao, Anal. Chem., 65, 1764 (1993). Callahan, R.W., AIChE Symp. Series, 84(261), 54 (1988). Calvert, S., in Handbook of Air Pollution Technology, S. Calvert and H.M. Englund (eds.), Wiley, New York (1984), p. 215. Carlson, A. and R. Nagarajan, Biotechnol. Prog., 8, 85 (1992). Carslaw, H.S. and J.C. Jaeger, Conduction of Heat in Solids, 2nd edn., Clarendon Press, Oxford (1959). Catsimpoolas, N. (ed.), Methods of Cell Separation, Vol. I, Plenum Press, New York (1977). Chan, Y.N.I., F.B. Hill and Y.W. Wong, Chem. Eng. Sci., 36, 243 (1981). Chang, Y.-K. and H.A. Chase, Biotechnol. Bioeng., 49, 204 (1996). Chao, K.C. and J.D. Seader, AIChE J., 7, 598 (1961). Chapman, T.W., in Lectures in Transport Phenomena, R.B. Bird, W.E. Stewart, E.N. Lightfoot and T.W. Chapman (eds.), AIChE Continuing Education Series 4, AIChE, New York (1969).

References Chem. Eng., “Gas chromatography analysis to production,” Chem. Eng., March 24 (1980), p. 70. Chem. Eng., “Separating paraffin isomers using chromatography,” Chem. Eng., May 18 (1981), p. 92. Chen, C-J., Separ. Sci. Technol, 36(3), 499 (2001). Chen, H., A.S. Kovvali, S. Majumdar and K.K. Sirkar, Ind. Eng. Chem. Res., 38(9), 3489 (1999a). Chen, H., A.S. Kovvali and K.K. Sirkar, Ind. Eng. Chem. Res., 39(7), 2447 (2000). Chen, H-P., Z-J. Lin, D-C. Liu, X.S. Wang and M.J. Rhodes, Ind. Eng. Chem. Res., 38(4), 1605 (1999b). Chen, H.T. and F.B. Hill, Separ. Sci., 6, 411 (1971). Chen, H.T., Y.W. Wong and S. Wu, AIChE J., 25, 320 (1979). Chen, H.T., W.T. Yang, U. Pancharoen and R. Parisi, AIChE J., 26, 839 (1980). Chen, R-R., R.N. Zare, E.C. Peters, F. Svec and J.J. Frechét, Anal. Chem., 73, 1987 (2001). Chen, S., Y.K. Kao and S.T. Hwang, J. Membr. Sci., 26, 143 (1986). Cherkasov, A.N. and A.E. Polotsky, J. Membr. Sci., 110, 79 (1996). Chern, R.T. and N.F. Brown, Macromolecules, 23(8), 2370 (1990). Chern, R.T., W.J. Koros, H.B. Hopfenberg and V.T. Stannett, in Material Science of Synthetic Membranes, D. Lloyd (ed.), ACS Symposium Series 269, ACS, Washington, D.C. (1985), p. 25. Cheryan, M., Ultrafiltration Handbook, Technomic Publishing, Lancaster, PA (1986), p. 60. Cheryan, M., Ultrafiltration and Microfiltration Handbook, Technomic Publishing, Lancaster, PA (1998). Chimowitz, E.H. and K.J. Pennisi, AIChE J., 32, 1665 (1986). Ching, C.B. and D.M. Ruthven, Can. J. Chem. Eng., 62, 398 (1984). Ching, C.B. and D.M. Ruthven, AIChE Symp. Series, 81(242), (1985). Ching, C.B., C. Ho and D.M. Ruthven, AIChE J., 32(11), 1876 (1986). Choudhury, A.P.R. and D.A. Dahlstrom, AIChE J., 3, 433 (1957). Choy, E.M., D.F. Evans and E.L. Cussler, J. Am. Chem. Soc., 96, 7085 (1974). Christian, S.D. and J.F. Scamehorn, in Surfactant-based Separation Processes, J.F. Scamehorn and J.H. Harwell (eds.), Marcel Dekker, New York (1989), p. 3. Christian, S.D., S.N. Bhat, E.E. Tucker, J.F. Scamehorn and D.A. El-Sayed, AIChE J., 34, 189 (1988). Chung, J., Pharmaceut. Technol., June, 39 (1996). Ciliberti, D.F. and B.W. Lancaster, AIChE J., 22(2), 394 (1976a). Ciliberti, D.F. and B.W. Lancaster, Chem. Eng. Sci., 31, 499 (1976b). Cleary, W. and M.F. Doherty, Ind. Eng. Chem. Proc. Des. Dev., 24, 1071 (1985). Clift, R., M. Ghadiri and A.C. Hoffman, AIChE J., 37(2), 285 (1991). Cohen, K., The Theory of Isotope Separation, McGraw-Hill, New York (1951). Cohn, E.J. and J.D. Ferry, in Proteins, Amino Acids and Peptides as Ions and Dipolar Ions, E.J. Cohn and J.T. Edsall (eds.), ACS Monograph Series no. 90, Reinhold, New York (1943), p. 602. Colburn, A.P., Trans. AIChE, 35, 211 (1939). Collins, J.P. and J.D. Way, Ind. Eng. Chem. Res., 32, 3006 (1993). Colman, D.A. and M.T. Thew, “Cyclone separator,” US Patent 4,764,287, August 16 (1988). Colton, C.K., K.A. Smith, E.W. Merrill and P.C. Farrell, J. Biomed. Mater. Res., 5, 459 (1971). Colton, C.K., C.N. Sattersfield and C.J. Lai, AIChE J., 21, 289 (1975).

863 Cornish, A.R.H., Trans. Inst. Chem. Eng., London, 43, T332 (1965). Cotterman, R.L. and J.M. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev., 24, 434 (1985). Cotterman, R.L., R. Bender and J.M. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev., 24, 194 (1985). Cox, M. and D.S. Flett, in Handbook of Solvent Extraction, T.C. Lo, M.H.I. Baird and C. Hanson (eds.), Wiley, New York (1983), chap. 2.2. Reprinted, Krieger Publishing Co., Marlboro, FL (1991). Craig, H. and R. C. Wiens, Science, 271, 1708 (1996). Craig, H., Y. Horibe and T. Sowers, Science, 242, 1675 (1988). Crank, J. and G.S. Park, in Diffusion in Polymers, J. Crank and G.S. Park (eds.), Academic Press, London (1968), chap. 1. Curtiss, C.F and R.B. Bird, Ind. Eng. Chem. Res., 38, 2515 (1999). Curtiss, C.F. and J.O. Hirschfelder, J. Chem. Phys., 17, 550 (1949). Cussler, E.L., AIChE J., 17, 405 (1971). Cussler, E.L., Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, New York, (1984), p. 407. Cussler, E.L., Diffusion: Mass Transfer in Fluid Systems, 2nd edn., Cambridge University Press, Cambridge (1997), pp. 251, 304. Cussler, E.L., M.R. Stokar and J.E. Varberg, AIChE J., 30(4), 578 (1984). Dadyburjor, D.B., Chem. Eng. Progr., 74(4), 85 (1978). Dai, X.P., S. Majumdar, R.G. Luo and K.K. Sirkar, Biotechnol. Bioeng., 83(2), 125 (2003). Danckwerts, P.V., Chem. Eng. Sci., 2, 1 (1953). Danckwerts, P.V., Gas-Liquid Reactions, McGraw-Hill, New York (1970). Danckwerts, P.V. and M.M. Sharma, The Chem. Engr., no. 202, CE 244 (1966). Danesi, P.R., Separ. Sci. Technol., 19(11&12), 857 (1984–85). Danesi, P.R., E.P. Horwitz and P.G. Rickert, J. Phys. Chem., 87, 4708 (1983). Danesi, P.R., L. Reichley-Yinger, C. Cianetti and P.G. Rickert, Solv. Extract. Ion Exchange, 2, 781 (1984). Darken, L.S. and R.W. Gurry, Physical Chemistry of Metals, McGraw-Hill, New York (1953). Datta, R., Biotechnol. Prog., 6, 485 (1990). Davies, J.T. and E.K. Rideal, Interfacial Phenomena, Academic Press, New York (1963). Davis, J.C., R.J. Valus, R. Eshraghi and A.E. Velikoff, Separ. Sci. Technol., 28, 463 (1993). Davis, R.H., in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), chaps. 31, 33. Davis, R.H. and D.C. Grant, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Van Nostrand Reinhold, New York, (1992), chap. 32. Davis, R.H. and D.C. Grant, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Kluwer Academic Publishers, Boston, MA (2001). Davis, R.H. and D.T. Leighton, Chem. Eng. Sci., 42(2), 275 (1987). Davis, R.H. and J.D. Sherwood, Chem. Eng. Sci., 45(11), 3203 (1990). Davis, R.H., X. Zhang and J.P. Agarwala, Ind. Eng. Chem. Res., 28, 785 (1989). de Clerk, K. and C.E. Cloete, Separ. Sci., 6(5), 627 (1971). de Groot, S.R. and P. Mazur, Nonequilibrium Thermodynamics, North Holland, Amsterdam (1962). Debenedetti, P.G. and R.C. Reid, AIChE J., 32, 2034 (1986). DeHoff, R.T., Thermodynamics in Material Science, McGraw-Hill, New York (1993).

864 Dekker, M., K. Van’t Reit, J.J. Van Der Pol, J.W.A. Baltussen, R. Hilhorst and B.J. Bijsterbosch, Chem. Eng. J., 46, B69 (1991). Denbigh, K., The Principles of Chemical Equilibrium, Cambridge University Press, London (1971), pp. 87, 103. Deng, Y., J. Zhang, T. Tsuda, P.H. Yu, A.A. Boulton and R.M. Cassidy, Anal. Chem., 70, 4586 (1998). DePriester, C.L., Chem. Eng. Prog. Symp. Ser., 49(7), 1 (1953). Deutsch, W., Ann. Phys. (Leipzig), 68, 335 (1922). DeVault, D., J. Am. Chem. Soc., 65, 532 (1943). Dewitt, T.S., “Liquid-Liquid Extraction of COS into Caustic Soda,” 88th AIChE National Meeting, Philadelphia, PA June 8–12 (1980), paper 61b. Dietz, P.W., AIChE J., 27(6), 888 (1981). Ding, H.B., M.C. Yang, D. Schisla and E.L. Cussler, AIChE J., 35(5), 814 (1989). Diwekar, U.M., Batch Distillation: Simulation, Optimum Design and Control, Taylor and Francis, Washington, D.C. (1996). Dobby, G. and J.A. Finch, Powder Technol., 17, 73 (1977). Dodge, B.F. and A.K. Dunbar, J. Am. Chem. Soc., 49, 591 (1927). Doherty, M.F. and M.F. Malone, Conceptual Design of Distillation Systems, McGraw-Hill, New York (2001), pp. 186–199, 259– 265, 507–513. Doherty, M.F. and J.D. Perkins, Chem. Eng. Sci., 33, 569 (1978). Donovan, R.P., Fabric Filtration for Combustion Sources, Marcel Dekker, New York (1985). Doong, S.J. and R.T. Yang, AIChE J., 32(3), 397 (1986). Dose, E.V. and G. Guiochon, Anal. Chem., 62, 174 (1990). Dresner, L., Desalination, 10, 27 (1972). Dunn, R.O., J.F. Scamehorn and S.D. Christian, Separ. Sci. Technol., 20, 257 (1985). Dunn, R.O., J.F. Scamehorn and S.D. Christian, Separ. Sci. Technol., 22 (2&3), 763 (1987). Dutta, P. and A. Beskok, Anal. Chem., 73, 1979 (2001). Dwivedi, P.N. and S.N. Upadhyay, Ind. Eng. Chem. Proc. Des. Dev., 16, 157 (1977). Dye, S.R., J.P. DeCasli, II, and G. Carta, Ind. Eng. Chem. Res., 29, 849 (1990). Dyer, P.N., R.E. Richards, S.L. Russek and D.M. Taylor, Solid State Ionics, 134, 21 (2000). Eckert, J.S., Chem. Eng. Progr., 66(3), 39 (1970). Eckstein, E.C., P.G. Bailey and A.H. Shapiro, J. Fluid Mech., 79, 191 (1977). Eduljee, H.E., Hydrocarbon Proc., 54, 120 (1975). Edwards, T.J., G. Maurer, J. Newman and J.M. Prausnitz, AIChE J., 24(6), 966 (1978). Eggerstedt, P.M., J.F. Zievers and E.C. Zievers, Chem. Eng. Progr., Jan., 62 (1993). Eiceman, G.A. and Z. Karpas, Ion Mobility Spectrometry, CRC Press, Boca Raton, FL (1994). Eiceman, G.A., E.V. Krylov, N.S. Krylova, E.G. Nazarov and R.A. Miller, Anal. Chem., 76(17), 4397 (2004). Eisinger, R.S. and G.E. Keller, Environ. Progr., 9(4), Nov., 235 (1990). Ellerbe, R.W., Chem. Eng., 80, 110 (1973). El-Sayed, Y.M. and R.S. Silver, in Principles of Desalination, Part A, K.S. Spiegler and A.D.K. Laird (eds.), 2nd edn., Academic Press, New York (1980), chap. 2. Englezos, P., Ind. Eng. Chem. Res., 32, 1251 (1993). Erbar, J.H. and R.N Maddox, Petroleum Refiner, 40(5), 183 (1961). Estrin, J., in Handbook of Industrial Crystallization, A.S. Myerson (ed.), Butterworth-Heinemann, Boston, MA (1993), chap. 6.

References Evangelista, F. and G. Jonsson, Chem. Eng. Commun, 72, 69 (1988). Everaerts, F.M., F.E.P. Mikkers and Th.P.E. Verheggen, Separ. Purif. Meth., 6(2), 287 (1977). Fair, J.R., Petro/Chem. Engr., 33, 45 (1961). Fair, J.R., H.R. Null and W.L. Bolles, Ind. Eng. Chem. Proc. Des. Dev., 22, 53 (1983). Fair, J.R., D. R. Steinmeyer, W.R. Penney and B.B. Crocker, in Perry’s Chemical Engineers’ Handbook, 6th edn., R. H. Perry and D. Green (eds.), McGraw-Hill, New York (1984), sect. 18. Fanali, S., in Capillary Electrophoresis Technology, N.A. Guzman (ed.), Marcel Dekker, New York (1993), p. 731. Farmer, J.C., “Method and apparatus for capacitive deionization, electrochemical purification and regeneration of electrodes,” US Patent 5,425,859 (1995). Farmer, J.C., D.V. Fix, G.V. Mack, R.W. Pekola and J.F. Poco, J. Electrochem. Soc., 143, 159 (1996). Farrell, P.C. and A.L. Babb, J. Biomed. Mater. Res., 7, 275 (1973). Feins, M., “Novel internally-staged ultrafiltration for protein,” Ph. D. Thesis, Department of Chemical Engineering, New Jersey Institute of Technology, Newark, NJ (2004). Feins, M. and K.K. Sirkar, Biotechnol. Bioeng. 86(6), 603 (2004). Feins, M. and K.K. Sirkar, J. Membr. Sci., 248(1–2), 137 (2005). Fenske, M.R., Ind. Eng. Chem., 24, 482 (1932). Fernandez, G.F. and C.N. Kenney, Chem. Eng. Sci., 38, 827 (1983). Ferry, J.D., J. Gen. Physiol., 20, 95 (1936). Field, R.W., D. Wu, J.A. Howell and B.B. Gupta, J. Membr. Sci., 100, 259 (1995). Figdor, C.G., F. Preijers, R. Huijbens, P. Ruijs, T.J.M. de Witte and W.S. Bont, in Cell Separation Methods and Applications, D. Recktenwald and A. Radbruch (eds.), Marcel Dekker, New York (1998), chap. 3. Findlay, R.A., in New Chemical Engineering Separation Techniques, H.A. Schoen (ed.), Interscience Publishers, New York (1962), chap. 4. Fish, B.B., R.W. Carr and R. Aris, AIChE J., 35(5), 737 (1989). Fitz Jr., C.W., J.G. Kunesh and A. Shariat, Ind. Eng. Chem. Res., 38, 512 (1999). Flagan, R.C. and J.H. Seinfeld, Fundamentals of Air Pollution Engineering, Prentice Hall, Englewood Cliffs, NJ (1988). Flett, D.S., J. Melling and M. Cox, in Handbook of Solvent Extraction, T.C. Lo, M.H.I. Baird and C. Hansen (eds.), Krieger Publishing Co., Malabar, FL, (1991), chap. 24. Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY (1953). Follman, D.K. and R.L. Fahrner, J. Chromatog. A, 1024, 79 (2004). Foust, A.S., L.A. Wenzel, C.W. Clump, L. Maus and L.B. Andersen, Principles of Unit Operations, John Wiley, New York (1960), p. 532. Freeman, B.D., Macromolecules, 32, 375 (1999). Frej, A.-K. Barnfield, H.J. Johannson, S. Johansson and P. Leijon, Bioprocess Eng., 16, 57 (1997). Frenz, J. and C. Horvath, AIChE J., 31(3), 400 (1985). Friedlander, S.K., Smoke, Dust and Haze: Fundamentals of Aerosol Behavior, John Wiley, New York (1977). Froment, G.F. and K.B. Bischoff, Chemical Reactor Analysis and Design, John Wiley, New York (1979), chap. 14. Fu, S. and C.A. Lucy, Anal. Chem., 70(1), 173 (1998). Fuchs, N.A., The Mechanics of Aerosols, McMillan, New York (1964). Fuerstenau, D.W. and T.W. Healy, in Adsorptive Bubble Separation Techniques, R. Lemlich (ed.), Academic Press, New York (1972).

References Fuerstenau, D.W. and R. Herrera-Urbina, in Surfactant-based Separation Processes, J.F. Scamehorn and J.A. Harwell (eds.), Marcel Dekker, New York (1989), p. 259. Fuh, C.B., M.N. Myers and J.C. Giddings, Anal. Chem., 64, 3125 (1992). Fujita, Y., H. Nakamura and T. Muto, J. Appl. Electrochem., 16, 935 (1986). Fuller, E.J., in Separation and Purification Methods, E.S. Perry and C.J. Van Oss (eds.), vol. 1, Marcel Dekker, New York (1972), p. 253. Gandhi, S.K., VLSI Fabrication Principles, John Wiley, New York (1983). Garcia, A.A., M.R. Bonen, J. Ramirez-Vick, M. Sadaka and A. Vuppu, Bioseparation Process Science, Blackwell Science, Malden, MA (1999). Garg, D.R. and D.M. Ruthven, Adv. Chem. Ser., 121, 345 (1973). Geankoplis, C.J., Mass Transport Phenomena, Ohio State University Bookstores, Columbus, OH (1972). Geankoplis, C.J., Transport Processes and Separation Process Principles, 4th edn., Prentice Hall, Upper Saddle River, NJ (2003). pp. 487, 659, 681. Gebauer, K.H., J. Thommes and M.R. Kula, Chem., Eng. Sci., 57(3), 405 (1997). Gee, A.P., in Cell Separation Methods and Applications, D. Rectenwald and A. Radbruch (eds.), Marcel Dekker, New York (1998), chap. 9. Geller, D.A. and G.W. Swift, J. Acoust. Soc. Am., 111(4), 1675 (2002a). Geller, D.A. and G.W. Swift, J. Acoust. Soc. Am., 112(2), 504 (2002b). Gellings, P.J. and H.J.M. Bouwmeester, Catalysis Today, 12(1), Feb. (1992). Gentilcore, M.J., Chem. Eng. Progr., Jan., 56 (2002). Gerster, J.A., A.B. Hill, N.N. Hochgraf and D.G. Robinson, “Tray efficiencies in distillation columns,” Final Report, University of Delaware, Research Committee, American Institute of Chemical Engineers, New York (1958). Ghowsi, K., J.P. Foley and R.J. Gale, Anal. Chem., 62, 2714 (1990). Gidaspow, D., Y-T. Shih, J. Bouillard and D. Wasan, AIChE J., 35(5), 714 (1989). Giddings, J.C., Dynamics of Chromatography, Part I. Principles and Theory, Marcel Dekker, New York (1965). Giddings, J.C., Separ. Sci., 1, 123 (1966). Giddings, J.C., Separ. Sci., 4, 181 (1969). Giddings, J.C., Separ. Sci. Technol., 13, 3 (1978). Giddings, J.C., in Treatise on Analytical Chemistry, Part I. Theory and Practice, P.J. Elving, E. Grushka and I.M. Kolthoff (eds.), vol. 5, Wiley-Interscience, New York, (1982), chap. 3. Giddings, J.C., Separ. Sci Technol., 20 (9&10), 249 (1985). Giddings, J.C., Unified Separation Science, John Wiley, New York (1991). Giddings, J.C., Science, 260, 1456 (1993). Giddings, J.C. and K. Dahlgren, Separ. Sci., 6, 343 (1971). Giddings, J.C., E. Kucera, C.P. Russel and M.N. Myers, J. Phys. Chem., 72, 4397 (1968). Giddings, J.C., F.J. Yang and M.N. Myers, Science, 193, 1244 (1976). Gilbert, S.W., AIChE J., 37(8), 1205 (1991). Gilliland, E.R., Ind. Eng. Chem., 32, 1220 (1940). Glatz, C.E., M. Hoare and J. Landa-Vertiz, AIChE J., 32, 1196 (1967).

865 Glueckauf, E., Discuss. Faraday Soc., 7, 12 (1949). Glueckauf, E., Trans. Faraday Soc., 51, 34 (1955a). Glueckauf, E., Trans. Faraday Soc., 51, 1540 (1955b). Glueckauf, E. and J.I. Coates, J. Chem. Soc., 1315 (1947). Gnanasundaram, S., T.E. Degaleesan and G.S. Laddha, Can. J. Chem. Eng., 57(2), 141 (1979). Gobie, W.A., J.B. Beckwith and C.F. Ivory, Biotechnol. Prog., 1(1), 60 (1985). Göklen, K.E. and T.A. Hatton, Biotechnol. Prog., 1(1), 69 (1985). Göklen, K.E. and T.A. Hatton, Separ. Sci. Techol., 22, 831 (1987). Goel, V., M.A. Accomazzo, A.J. Dileo, P. Meier, A. Pitt and M. Pluskal, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Van Nostrand Reinhold, New York (1992), chap. 34. Goel, V., M.A. Accomazzo, A.J. Dileo, P. Meier, A. Pitt and M. Pluskai, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Kluwer Academic Publishers, Boston, MA (2001), chap 34. Goettler, L.A. and R.L. Pigford, AIChE J., 17, 793 (1971). Gooding, C.H. and R.M. Felder, AIChE J., 27(2), 193 (1981). Gray, W.G., Chem. Eng. Sci., 30, 229 (1975). Green, G. and G. Belfort, Desalination, 35, 129 (1980). Greenlaw. F.W., R.A. Sheldon and E.V. Thompson, J. Membr. Sci., 2, 333 (1977). Gregory, R.A. and N.H. Sweed, Chem. Eng. J., 1, 207 (1970). Grew, K.E. and T.L. Ibbs., Thermal Diffusion in Gases, Cambridge University Press, Cambridge (1952). Grieves, R.B., in Treatise on Analytical Chemistry, Part I. Theory and Practice, P.J. Elving, E. Grushka and I.M. Kolthoff (eds.), vol. 5, Wiley-Interscience, New York (1982), p. 371. Grosjean, P.R.L. and H. Sawistowski, Trans. Inst. Chem. Eng., London, 58, 59 (1980). Grosser, J.H., M.F. Doherty and M.F. Malone, Ind. Eng. Chem. Res., 26, 983 (1987). Grossman, M.W. and T.W. Shepp, IEEE Trans. Plasma Sci., 19(6), 1114 (1991). Grubisic, Z., P. Rempp and H. Benoit, Polymer Lett., 5, 753 (1967). Grushka, E., K.D. Caldwell, M.N. Myers and J.C. Giddings, Separ. Purif. Meth. 2(1), 127 (1973). Guggenheim, E.A., Thermodynamics, 5th edn., North-Holland Publishing Co., Amsterdam (1967). Guha, A.K., S. Majumdar and K.K. Sirkar, Ind. Eng. Chem. Res., 31, 593 (1992). Gupta, A.S. and G. Thodos, AIChE J., 8, 608 (1962). Gupta, R. and N.H Sweed, Ind. Eng. Chem. Fund., 10, 283 (1971). Gutierrez, A.P., D. Haase, D.A. Keyworth, D.G. Walker, D.L. Klein, “ESEP, A Process for the recovery of ethylene,” 175th ACS National Meeting, Anaheim, CA, March 12–17 (1978). Guzman, N.A. (ed.), Capillary Electrophoresis Technology, Marcel Dekker, New York (1993). Hagel, L., in Protein Purification, Principles, High Resolution Methods and Applications, J.C. Jansen and L. Ryder (eds.), VCH, New York (1989), p. 63. Haimour, N., A. Bidarian and O.C. Sandall, Separ. Sci. Technol., 22 (2&3), 921 (1987). Halliday, D. and R. Resnick, Physics, Part II, John Wiley, New York (1962), p. 921. Han, S., F.C. Ferreira and A. Livingston, J. Membr. Sci., 188, 219 (2001). Hannig, K., Z. Anal. Chem., 181, 244 (1961). Hansen, C.M., Ind. Eng. Chem. Prod. Res. Dev., 8, 2 (1969).

866 Hansen, C.M. and A. Beerbower, in Encyclopedia of Chemical Technology, 2nd edn., A. Standen (ed.), suppl. vol., Interscience, New York (1971), p. 889. Happel, J., AIChE J., 4, 197 (1958). Happel, J., AIChE J., 5, 174 (1959). Happel J. and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ (1965). Happel J. and H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague (1983). Harriott, P., Separ. Sci., 8, 291 (1973). Harrison, R.G., P. Todd, S.R. Rudge and D.P. Petrides, Bioseparations Science and Engineering, Oxford University Press, New York (2003), chap. 8, pp. 349–362. Hartland, S. and J.C. Mecklenburgh, Chem. Eng. Sci., 21, 1209 (1966). Hatta, S., Tohoku Imperial Univ. Tech. Rep. 8, 1 (1928). Hatta, S., Tohoku Imperial Univ. Tech. Rep. 10, 119 (1932). Hatton, T.A., in Surfactant-based Separation Processes, J.F. Scamehorn and J.A. Harwell (eds.), Marcel Dekker, New York (1989), p. 55. Hausmann, M., C. Cremer, R. Hartig, H.G. Liebich, G.H. Luers, A. Saalmüler and R. Teichmann, in Cell Separation Methods and Applications, D. Recktenwald and A. Radbruch (eds.), Marcel Dekker, New York (1998), chap. 10. Haynes, C.A., J. Carson, H.W. Blanch and J.M. Prausnitz, AIChE J., 37(9), 1401 (1991). Haynes, C.A., F.J. Benitez, H.W. Blanch and J.M. Prausnitz, AIChE J., 39(9), 1539 (1993). Helfferich, F., Ion Exchange, McGraw-Hill, New York (1962). Helfferich, F., Ion Exchange, Dover Publications, New York (1995). Helfferich, F. and G. Klein, Multicomponent Chromatography, Marcel Dekker, New York (1970). Helmbrecht, C., R. Niessner and C. Haisch, Anal. Chem., 79, 7097 (2007). Hengstebeck, R.J., Distillation: Principles and Design Procedures, Reinhold, New York (1961), chap. 7. Henis, J.M.S. and M.K. Tripodi, J. Membr. Sci., 8, 233 (1981). Henley, E.J. and J.D. Seader, Equilibrium-Stage Separation Operations in Chemical Engineering, John Wiley, New York (1981), pp. 690–692, 714. Henry, J.D. Jr., in Perry’s Chemical Engineers’ Handbook, R.H. Perry and D.W. Green (eds.), 6th edn., McGraw-Hill, New York (1984), sect. 17. Henry, J.D. Jr. and C.G Moyers, Jr., in Perry’s Chemical Engineers’ Handbook, R.H. Perry and D.W. Green (eds.), 6th edn., McGraw-Hill, New York (1984), pp. 17–6 to 17–12. Herberhold, M., Metal π-complexes, Part II: Specific Aspects, vol. II, Elsevier, New York (1974). Herzig, J.P., D.M. Leclerc and P. Le Goff, Ind. Eng. Chem., 62(5), 8 (1970). Hestekin, J.A., L.G. Bachas and D. Bhattacharyya, Ind. Eng. Chem. Res., 40, 2668 (2001). Heyne, L., in Solid Electrolytes, S. Geller (ed.), Springer-Verlag, New York (1977), chap. 7. Higuchi, A., Y. Ishida and T. Nakagawa, Desalination, 90, 127 (1993). Hill, F.B., Chem. Eng. Commun., 7, 37 (1980). Hill, T.L., J. Chem. Phys., 17, 520 (1949). Hills, G.J., P.W.M. Jacobs and N. Lakshminarayaniah, Proc. Roy. Soc. London, Ser. A, 262, 246 (1961). Hines, A.L. and R.M. Maddox, Mass Transfer: Fundamentals and Applications, Prentice-Hall PTR, Upper Saddle River, NJ (1985).

References Hirschfelder, J.O., C.F. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids, John Wiley, New York (1954). Ho, W.S.W and D.C. Dalrymple, J. Membr. Sci., 91, 13 (1994). Ho, W.S.W and N.N. Li, in Perry’s Chemical Engineers’ Handbook, R.H. Perry and D.W. Green (eds.), 6th edn., McGrawHill, New York (1984a), pp. 17–14 to 17–35. Ho, W.S.W and N.N. Li, in Hydrometallurgical Process Fundamentals, R.G. Bautista (ed.), Plenum Press, New York (1984b), p. 555. Ho, W.S.W. and N.N Li, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Kluwer Academic Publishers, Boston, MA (2001), chaps. 36, 37. Ho, W.S.W and K.K. Sirkar (eds.), Membrane Handbook, Kluwer Academic Publishers, Boston, MA (2001). Ho, W.S.W., G. Doyle, D.W. Savage and R.L. Pruett, Ind. Eng. Chem., Res., 27, 334 (1988). Hochhauser, A.M. and E.L. Cussler, AIChE Symp. Ser. 71(152), 136 (1975). Hoffman, R.A. and D.W. Houck, in Cell Separation Methods and Applications, D. Recktenwald and A. Radbruch (eds.), Marcel Dekker, New York (1998), chap. 11. Holusha, J., “From the ashes, a plus for a utility,” New York Times, Nov. 24 (1993). Horvath, C.S. and H.J. Lin, J. Chromatog., 149, 43 (1978). Hougen, O. A., K. M. Watson and R.A. Ragatz, Chemical Process Principles, Part I, Material and Energy Balances, 2nd edn., John Wiley, New York (1954), p. 163. Hougen, O.A., K.M. Watson and R.A. Ragatz, Chemical Process Principles, Part II, Thermodynamics, 2nd edn., John Wiley, New York (1959), pp. 968–969. Howard, A.J., G. Carta and C.H. Byers, “Novel applications of continuous annular chromatography: separation of sugars,” ACS National Meeting, Denver, CO, April, 5–10 (1987). Howe, E.D., Fundamentals of Water Desalination, Marcel Dekker, New York (1974), chap. 5 II. Howell, J.A., J. Membr. Sci., 107, 165 (1995). Hovingh, M.E., G.H. Thompson and J.C. Giddings, Anal. Chem., 42, 195 (1970). Hsu, H-W., Separation by Centrifugal Phenomena, Techniques of Chemistry, E.S. Perry (ed.), vol. XVI, John Wiley, New York (1981). Huang, S.H., W.C. Lee and G.T. Tsao, Chem. Eng. J., 38, 179 (1988a). Huang, S.H., S. Roy, K.C Hou and G.T. Tsao, Biotechnol. Prog., 4(3), 159 (1988b). Huang, X.C., M.A. Quesada and R.A. Mathies, Anal. Chem., 64(8), 967 (1992). Huckins, H.E. and K. Kammermeyer, Chem. Eng. Prog., 49, 180 (1953a). Huckins, H.E. and K. Kammermeyer, Chem. Eng. Prog., 49, 295 (1953b). Hughes, R.D., J.A. Maloney and E.F. Steiglemann, in Recent Developments in Separation Science, N.N. Li and J.M. Calo (eds.), vol. 9, CRC Press, Boca Raton, FL (1986), p. 174. Hulbert, H.M. and S. Katz, Chem. Eng. Sci., 19, 555 (1964). Humphrey, J.L. and G.E. Keller, II, Separation Process Technology, McGraw-Hill, New York (1997), p. 297. Hunter, J.B., Separ. Sci. Technol., 23(8&9), 913 (1988). Hwang, S.T. and K. Kammermeyer, “Membranes in separations,” in Techniques of Chemistry, A. Weissberger (ed.), vol. 7, WileyInterscience (1975). Reprinted, Kreiger Publishing, Melborne, FL (1984).

References Hwang, Y-L., G.E. Keller, II, and J.D. Olson, Ind. Eng. Chem. Res., 31, 1753 (1992a). Hwang, Y-L., J.D. Olson and G.E. Keller, II, Ind. Eng. Chem. Res., 31, 1759 (1992b). Ibbs, T.L., K.E. Grew and A.A. Hirst, Proc. Roy. Soc. (London), A173, 543 (1939). Inculet, I.I., Electrostatic Mineral Separation, Research Studies Press Ltd., John Wiley, New York (1984). Ivory, C.F., J. Chromatog., 195, 165 (1980). Ivory, C.F. and W.A. Gobie, Biotechnol. Prog., 6, 21 (1990). Ivory, C.F., M. Gilmartin, W.A. Gobie, C.A. McDonald and R.L. Zollars, Biotechnol. Prog., 11, 21 (1995). Jaasund, S.A., Chem. Eng., Nov. 23, 159 (1987). Jacobs, D.B. and J. Zimmerman, in Polymerization Process, C.E. Schildknecht (ed.), Wiley, New York (1977), chap. 12. Jacques, J., A. Collet and S.H. Wilen, Enantiomers, Racemates and Resolutions, Wiley-Interscience, John Wiley, New York (1981), chap. 5.1. Jafvert, C.T., W. Chu and P.L. Van Hoof, in Surfactant-enhanced Subsurface Remediation: Emerging Technologies, D.A. Sabatini, R.C. Knox and J.H. Harwell (eds.), ACS Symposium Series 594, Amercian Chemical Society, Washington, D.C. (1995), p. 24. Jagur-Grodzinski, J. and R. Kramer, Ind. Eng. Chem. Proc. Des. Dev., 25(2), 443 (1986). James, A.T. and A.J.P. Martin, Biochem. J., 50, 679 (1952). Janini, G.M. and H.J. Issaq, in Capillary Electrophoresis Technology, N.A. Guzman (ed.), Marcel Dekker, New York (1993), p. 119. Jassim, M.S., G. Rochelle, D. Eimer and C. Ramshaw, Ind. Eng. Chem. Res., 46, 2823 (2007). Jennings, W., Analytical Gas Chromatography, Academic Press, New York (1987), p. 79. Jillavenkatesa, A., S.J. Dapkunas and L.S.H. Lum, Particle Size Characterization, NIST Recommended Practice Guide, Special Publication 960–1, January, NIST (2001). Johnson, J.S., L. Dresner and K.A. Kraus, in Principles of Desalination, K.S. Spiegler (ed.), Academic Press, New York (1966), chap. 8. Johnson, J.S., L. Dresner and K.A. Kraus, in Principles of Desalination, K.S. Spiegler and A.D.K. Laird, 2nd edn., Part B, Academic Press, New York (1980), chap. 8. Johnston, K.P., S.E. Barry, N.K. Read and T.R. Holcomb, Ind. Eng. Chem. Res., 26, 2372 (1987). Jolley, J.E. and J.H. Hildebrand, J. Am. Chem. Soc., 450, 80, 1050 (1958). Jones, J.B., in Techniques of Chemistry, Vol. X, Applications of Biochemical Systems in Organic Chemistry, J.B. Jones, C.J. Sih and D. Perlman, (eds.), Part 1, John Wiley, New York (1976), p. 18. Jones, J.B. and J.F. Beck, in Techniques of Chemistry, Vol. X, Applications of Biochemical Systems in Organic Chemistry, J.B. Jones, C.J. Sih and D. Perlman (eds.), Part 1, John Wiley, New York (1976), p. 139. Jones, T.B., Electromechanics of Particles, Cambridge University Press, New York (1995). Jorgenson, J.W. and K.D. Lukacs, Anal. Chem., 53 1298 (1981). Jung, M., S. Mayer and V. Schuring, LCGC North Am., 12(6), 458 (1994). Jury, S.H. and W.L. Locke, AIChE J., 3(4), 480 (1957). Kammermeyer, K., in Progress in Separation and Purification, E.S. Perry (ed.), vol. 1, Interscience, New York (1968), p. 335. Karger, B.L. and F. Foret, in Capillary Electrophoresis Technology, N.A. Guzman (ed.), Marcel Dekker, New York (1993), p. 3.

867 Karger, B.L., L.R. Snyder and C. Horvath, An Introduction to Separation Science, Wiley, New York (1973), chap. 8, p. 269. Karlson, E.L. and S.P. Edkins, AIChE Symp. Ser., 71(151), 286 (1975). Kay, R.L., Pure Appl. Chem., 63, 1393 (1991). Keey, R.B., Drying: Principles and Practice, Pergamon Press, New York (1972). Kehlen, H., M.T. Ratzsch and J. Bergmann, AIChEJ., 31(7), 1136 (1985). Keller, G.E., A.E. Marcinkowsky, S.K. Verma and K.D. Williamson, in Separation and Purification Technology, N.N. Li and J.M. Calo (eds.), Marcel Dekker, New York (1992), chap. 3. Keller, G.E., II, in Industrial Gas Separations, ACS Symposium Series 223, American Chemical Society, Washington, D.C. (1983). Keller, G.E. II and C.H.A. Kuo, “Enhanced gas separation by selective adsorption,” US Patent 4,354,859, Oct. 19 (1982). Keller, K.H. and T.R. Stein, Math. Biosci., 1, 421 (1967). Kelsall, D.F., in Solid-Liquid Separation, J.B. Pool and D. Doyle (eds.), HMSO, London (1966). Kennard, E.H., Kinetic Theory of Gases, McGraw Hill, New York (1938). Kenyon, N.S., C. Ricordi, J.G. Gribben, L.M. Nadler, R.K. Zwerner and T.R. Russell, in Cell Separation Methods and Applications, D. Reckenwald and A. Radbruch (eds.) Marcel Dekker, New York (1998), p. 103. Kertes, A.S. and C.J. King, Biotechnol. Bioeng., 28, 269 (1986). Kessler, D.P. and P.C. Wankat, Chem. Eng., Sept. 26, 71 (1988). Kessler, S.B and E. Klein, in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.) Kluwer Academic Publishers, Boston, MA (2001), chaps. 11–15. Kheterpal, I. and R.A. Mathies, Anal. Chem., 71(1), 31A (1999). Kiani, A., R.R. Bhave and K.K. Sirkar, J. Membr. Sci., 20, 125 (1984). King, C.J., Separation Processes, 2nd edn., McGraw-Hill, New York (1980). King, C.J., Freeze-Drying of Foods, CRC Press, Cleveland, OH (1971). King, R.S., H.W. Blanch and J.M Prausnitz, AIChE J., 34, 1585 (1988). Kirchner, J.J. and E.F. Hasselbrink, Jr., Anal. Chem., 77, 1140 (2005). Kirkland, J.J. and C.H. Dilks, Jr., Anal. Chem., 64, 2836 (1992). Kirwan, D.J. and C.J. Orella, in Handbook of Industrial Crystallization, A.S. Myerson (ed.), Butterworths-Heinemann, Boston, MA (1993), chap. 11. Kister, H.Z., Distillation Operation, McGraw-Hill, New York (1990). Kister, H.Z., Distillation Design, McGraw-Hill, New York (1992). Kister, H.Z., Chem. Eng. Prog., June, 39 (2008). Kister, H.Z. and D.R. Gill, Chem. Eng. Progr., 87(2), 32, Feb. (1991). Kister, H.Z., J. Scherffius, K. Afshar and E. Akbar, Chem. Eng. Prog., 28, July (2007). Kitiyanan, B., J.H. O’Haver, J.H. Harwell and D. Sabatini in Surfactant-based Separations, J.F. Scamehorn and J.H. Harwell (eds.), ACS Symposium. Series, 740, American Chemical Society, Washington, D.C. (1999), p. 76. Klein, E., J.K. Smith, R.P. Wendt and S.V. Desai, Separ. Sci., 7, 285 (1972). Klein, E., J.K. Smith, R.E.C. Weaver, R. P. Wendt and S.V. Desai, Separ. Sci., 8, 585 (1973).

868 Klein, E., F.F. Holland and K. Eberle, J. Membr. Sci., 5, 173 (1979). Knaebel, K.S. and F.B. Hill, “Analysis of gas purification by heatless adsorption”, AIChE National Meeting, Los Angeles, CA, Nov. 14–18 (1982), paper 91d. Knaebel, K.S. and F.B. Hill, Chem. Eng. Sci., 40(12), 2351 (1985). Knaebel, K.S. and R.L. Pigford, Ind. Eng. Chem. Fund., 22, 336 (1983). Kohl, A.L. and F.C. Riesenfeld, Gas Purification, 3rd edn., Gulf Publishing Co., Houston, TX (1979), p.783. Komasawa, I and T. Otake, Ind. Eng. Chem. Fund., 22, 122 (1983) Koros, W.J., J. Polym. Sci., Polym. Phys. Ed., 18, 981 (1980). Koros, W.J., R.T. Chern, V.T. Stannett and H.B. Hopfenberg, J. Polym. Sci., Polym. Phys. Ed., 19, 1513 (1981). Kouketsu, T., S. Duan, T. Kai, S. Kazama and K. Yamada, J. Membr. Sci., 287, 51 (2007). Koval, C.A., T. Spontarelli and R.D. Noble, Ind. Eng. Chem. Res., 28, 1020 (1989). Kovvali, A.S. and K.K. Sirkar, Ind. Eng. Chem. Res., 40, 2502 (2001). Kovvali, A.S. and K.K. Sirkar, Ann. NY Acad. Sci., 984, 279 (2003). Kovvali, A.S., H. Chen and K.K. Sirkar, J. Am. Chem. Soc., 122(31), 7594 (2000). Kovvali, A.S., H. Chen and K.K. Sirkar, Ind. Eng. Chem. Res., 41(3), 347 (2002). Kowler, D.E. and R.H. Kadlec, AIChE J., 18, 1207 (1972). Kozinski, A.A. and E.N. Lightfoot, AIChE J., 18(5), 1030 (1972). Kralj, J.G., M.T.W. Lis, M.A. Schmidt and K.F. Jensen, Anal. Chem., 78, 5019 (2006). Kremser, A., Nat. Petrol. News., 22(21), May 30, 43 (1930). Krishna, R. and G.L. Standart, Chem. Eng. Commun., 3, 201 (1979). Krishna, R., H.F. Martinez, R. Sreedhar and G.L. Standart, Trans. Inst. Chem. Eng., 55, 178 (1977). Krishnamurthy, R. and R. Taylor, AIChE J., 31, 449 (1985). Kulkarni, S.S., E.W. Funk and N.N Li, in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), chaps. 26–30. Kulov, N.N., Chem. Eng. Commun., 21, 259 (1983). Kumar, R. and G.R. Dissinger, Ind. Eng. Chem. Proc. Des. Dev., 25, 456 (1986). Kumar, R. and S. Sircar, Chem. Eng. Commun., 26, 339 (1984). Kuo, Y. and H.P. Gregor, Separ. Sci. Technol., 18, 421 (1983). Kurnik, R.T., A.W. Yu, G.S. Blank, A.R. Burton, D. Smith, A.M. Athalye and R. van Reis, Biotechnol. Bioeng., 45, 149 (1995). Kurrelmeyer, B. and W.H. Mais, Electricity and Magnetism, Van Nostrand Co., Princeton, NJ (1967). Kuwabara, S., J. Phys. Soc. Jpn., 14, 527 (1959). Kynch, G.J., Trans. Faraday Soc., 48, 166 (1952). La Mer, V.K., Ind. Eng. Chem., 44, 1270 (1952). Ladisch, M.R., Bioseparations Engineering: Principles, Practice and Economics, John Wiley, New York (2001). Lahiere, R.J., J.L. Humphrey and J.R. Fair, Separ. Sci. Technol., 22(2&3), 379 (1987). Landau, L.D. and E.M. Lifshitz, Fluid Mechanics, Addison Wesley, Reading, MA (1959), p. 97. Lane, J.A., in Chemical Engineers Handbook, 3rd edn., J. H. Perry (ed.), McGraw-Hill, New York (1950), p. 753. Lane, J.A. and J.W. Riggle, Chem. Eng. Progr. Symp. Ser., 55(24), 135 (1959). Langer, S. H. and R.G. Haldeman, J. Phys. Chem., 68(4), 962 (1964).

References Langhaar, H.L., Dimensional Analysis and Theory of Models, Wiley, New York (1951). Langmuir, I., Phys. Rev., 8, 149 (1916). Lapidus, L. and N.R. Amundson, J. Phys. Colloid Chem., 54, 821 (1950). Lapidus, L. and N.R. Amundson, J. Phys. Chem., 56, 984 (1952). Lapple C., Air Pollution Engineering Manual, USEPA, AP-40, 94 (1951). Larsen, M.U. and N.C. Shapley, Anal. Chem., 79, 1947 (2007). Lasic, D., Am. Scientist, 80, 20 (1992). Lee, C-K., Ind. Eng. Chem. Res., 34, 2104 (1995). Lee, H.H., Fundamentals of Microelectronic Processing, McGrawHill, New York (1990), chap. 3. Lee, H.L., J.F.G Reis, J. Dohner and E.N. Lightfoot, AIChE J., 20, 776 (1974). Lee, H.L., E.N. Lightfoot, J.F.G., Reis and M.D. Waissbluth, in Recent Developments in Separation Science, N.N. Li (ed.), vol. III, Part A, CRC Press, Cleveland OH (1977a), p. 1. Lee, K-H., D.F. Evans and E.L. Cussler, AIChE J., 24, 860 (1978). Lee, K.L., H.B. Hopfenberg and V.T. Stannett, J. Appl. Polym. Sci., 21, 1795 (1977b). Lehninger, A.L., Principles of Biochemistry, 3rd edn., Worth Publishers, New York (1982). Leighton, D.T. and A. Acrivos, J. Fluid Mech., 177, 109 (1987). Leith, D. and W. Licht, AIChE Symp. Ser., 68(126), 196 (1972). Lemlich, R., in Progress in Separation and Purification, E.S. Perry (ed.), vol. 1, Wiley (Interscience), New York (1968). Leonard, E.F. and L.W. Bleumle, Jr., Trans. NY Acad. Sci. Ser., 2(21), 585 (1959). Leung, P.S., in Ultrafiltration Membranes and Applications, A.R. Cooper (ed.), Plenum, New York (1979), p. 415. LéVêque, J., Ann. Mines Ser., 12, 13, 201, 305–362, 381–415 (1928). Levich, V.G., Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ (1962), sect, 94. Lewis, G.N. and M. Randall, Thermodynamics, 2nd edn., revised by K.S. Pitzer and L. Brewer, McGraw-Hill, New York (1961). Lewis, W.K., Ind. Eng. Chem., 28, 399 (1936). Liddle, C.J., Chem. Eng., 75(23), Oct. 21, 137 (1968). Liepmann, H.W., J. Fluid Mech., 10, 65 (1961). Lightfoot, E.N., Transport Phenomena and Living Systems, John Wiley, New York (1974). Lightfoot, E.N., Separ. Sci. Technol., 40, 739 (2005). Lightfoot, E.N. and M.C.M. Cockrem, Separ. Sci. Technol., 22(2&3), 165 (1987). Lightfoot, E.N., R.J. Sanchez-Palma and D.O. Edwards, in New Chemical Engineering Separation Techniques, H.A. Schoen (ed.), Interscience Publishers, New York (1962), chap. 2, pp. 99–181. Lightfoot, E.N., P.T. Noble, A.S. Chiang and T.A. Ugulini, Separ. Sci. Technol., 16, 619 (1981). Lin, I.J. and L. Benguigui, Powder Technol., 17, 95 (1977). Lin, Y-S., W. Wang and J. Han, AIChE J., 40(5), 786 (1994). Lindahl, P.E., Nature, 161(4095), 648 (1948). Lindahl, P.E., Biochem. Biophys. Acta, 21, 411 (1956). Linton, W.H. and T. K. Sherwood, Chem. Eng. Progr., 46, 258 (1950). Liow, J.L. and C.N. Kenney, AIChE J., 36(1), 53 (1990). Liu, Y.A. and M.J. Oak, AIChE J., 29(5), 771 (1983). Lo, T.-C., M.H.I. Baird and C. Hanson, Handbook of Solvent Extraction, Wiley-Interscience, New York (1983), chaps. 2.1, 2.2, 3.

References Lo, T.C., M.H.I. Baird and C. Hanson, Handbook of Solvent Extraction, Kreiger Publishing Co., Malabar, FL (1991). Lockett, M.J., Distillation Tray Fundamentals, Cambridge University Press, Cambridge (1986). Lockhart, F.J. and R.J. McHenry, Petroleum Refiner, 37, 209 (1958). Loeb, S., J. Membr. Sci., 1, 49 (1976). Long, R.B., Ind. Eng. Chem. Fund., 4(4), 445 (1965). Lonsdale, H.K., in Desalination by Reverse Osmosis, U. Merten (ed.), The MIT Press, Cambridge, MA (1966), p. 117. Lonsdale, H.K., U. Merten and R.L. Riley, J. Appl. Polym. Sci., 9, 1341 (1965). Lopez-Leiva, M., in Ultrafiltration Membranes and Applications, A.R. Cooper (ed.), Polymer Science and Technology Series, vol. 13, Plenum Press, New York (1980). Lotfian, P., M.S. Levy, R.S. Coffin, T. Fearn and P. AyaziShamlou, Biotechnol. Prog., 19, 209 (2003). Lowe, A.B. and C.L. McCormick, in Stimuli-Responsive Water Soluble and Amphiphilic Polymers, C.L. McCormick (ed.), ACS Symposium. Series 780, American Chemical Society, Washington, D.C. (2001), p. 1. Lowenheim F. A. and M. K. Moran, Faith, Keyes and Clark’s Industrial Chemicals, 4th edn., John Wiley, New York (1975), p. 654. Luborsky F.E and B.J. Drummond, IEEE Trans. Magnetics, 11(6), 1696 (1975). Lye, G.J., J.A. Asenjo and D.L. Pyle, AIChE J., 42(3), 713 (1996). McCabe, W.L. and J.C. Smith, Unit Operations of Chemical Engineering, 3rd edn., McGraw-Hill, New York (1976). McCabe, W.L., J.C. Smith and P. Harriott, Unit Operations of Chemical Engineering, 5th edn., McGraw-Hill, New York (1993). Macias-Salinas, R. and J.R. Fair, Ind. Eng. Chem. Res., 41, 3429 (2002). McCoy, B.J., AIChE J., 32(9), 1570 (1986). McHugh, M.A. and V.J. Krukonis, Supercritical Fluid Extraction, Butterworths, Boston, MA (1986). MacKenzie, P.D. and C.J. King, Ind. Eng. Chem. Proc. Des. Dev., 24, 1192 (1985). McNab, G.S. and A. Meisen, J. Colloid Interface Sci., 44, 339 (1973). Maget, H.J.R. “Process for gas purification,” US Patent 3,489,670, January 13 (1970). Majors, R.E., LCGC North Am., 23(12), 1248 (2005). Majumdar, S., A.K. Guha and K.K. Sirkar, AIChE J., 34, 1135 (1988). Majumdar, S., K.K. Sirkar and A. Sengupta, in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.), Kluwer Academic Publishers, Boston, MA (2001), chap. 42. Mallinson, R.H. and C. Ramshaw, “Mass transfer apparatus and process,” European Patent 0053881 (1982). Mandal, D.K., A.K. Guha and K.K. Sirkar, J. Membr. Sci., 144, 13 (1998). Marin G.B. (ed.), Multiscale Analysis, Advances in Chemical Engineering 30, Elsevier Academic Press, Amsterdam (2005). Markham, E.C. and A.F. Benton, J. Am. Chem. Soc., 53, 497 (1931). Marmur, A., J. Am. Chem. Soc., 122, 2120 (2000). Marr, R. and A. Kopp, Int. Chem. Eng., 22(1), 44 (1982). Martell, A.E. and R.M. Smith, Critical Stability Constants, Vol. 1, Plenum Press, New York (1974). Martell, A.E. and R.M. Smith, Critical Stability Constants, Vol. 5, Plenum Press, New York (1982).

869 Martin, A.J.P. and R.L.M. Synge, Biochem. J., 35, 1358 (1941). Martin, H. and W. Kuhn, Z. Physik. Chem., 189A, 219 (1941). Martin, J.J., Chemie Ingenieur Technik, 44, 249 (1972). Martin, J.J., Chem. Eng. Edu., Summer, 119 (1983). Mason, E.A. and W. Juda, Chem. Eng. Progr. Symp. Ser., 24(55), 155 (1959). Mason, E.A. and T.A. Kirkham, Chem. Eng. Progr. Symp. Ser., 24 (55), 173 (1959). Mason, E.A. and E.W. McDaniel, Transport Properties of Ions in Gases, John Wiley, New York (1988). Mass, J.H., in Handbook of Separation Techniques for Chemical Engineers, P.A. Schweitzer (ed.), McGraw-Hill, New York (1979), sect. 6.1. Matson, S.L. and J.A. Quinn, Ann. NY Acad. Sci., 469, 152 (1986). Matson, S.L, C.S. Herrick and W.J. Ward, III, Ind. Eng. Chem. Proc. Des. Dev., 16, 370 (1977). Mattiasson, B. and T.G.I. Ling, in Membrane Separations in Biotechnology, W.C. McGregor (ed.), Marcel Dekker, New York (1986). Mattock, P., G.F. Aitchison and A.R. Thomson, Separ. Purif. Meth., 9(1), 1 (1980). Mayer, S.W. and E.R. Tompkins, J. Am. Chem. Soc., 69, 2792 (1947). Meares, P., in Membrane Separation Processes, P. Meares (ed.) Elsevier, New York (1976), chap. 1. Medina, J.C., N. Wu and M.L. Lee, Anal. Chem., 73(6), 1301 (2001). Mehta, A., M.L. Tse, J. Fogle et al., Chem. Eng. Progr., special issue on Purifying Therapeutic Monoclonal Antibodies, Bioprocessing (2008), pp. S14–S19. Meister, B.J. and A.E. Platt, Ind. Eng. Chem. Res., 28, 1659 (1989). Merten, U., Desalination by Reverse Osmosis, MIT Press, Cambridge, MA (1966). Meselson, M. and F.W. Stahl, Proc. Natl. Acad. Sci. USA, 44, 671 (1958). Meselson, M., F.W. Stahl and J. Vinograd, Proc. Natl. Acad. Sci. USA, 43, 581 (1957). Michaels, A.S., Trans. Am. Soc. Artif. Intern. Organs, 12, 387 (1966). Michaels, A., Chem. Eng. Progr., 64(12), 31 (1968a). Michaels, A.S., in Progress in Separation and Purification, E.S. Perry (ed.), vol. 1, Wiley-Interscience, New York (1968b), pp. 297–334. Michaels, A.S. and H.J. Bixler, J. Polym. Sci., 50, 393 (1961). Michaels, A.S. and H.J. Bixler, in Progress in Separation and Purification, E.S. Perry (ed.), vol. 1, Wiley-Interscience, New York (1968), p. 143. Middleman, S., An Introduction to Mass and Heat Transfer, Principles of Analysis and Design, John Wiley, New York (1998). Miers, H.A., J. Inst. Metals, 37, 331 (1927). Miers, H.A. and F. Isaac, J. Chem. Soc., London, 89, 413 (1906). Migita, H., K. Soga and Y.H. Mori, AIChE J., 51(8), 2190 (2005). Mikkers, F.E.P., F.M. Evaeraerts and Th.P.E.M. Verheggen, J. Chromatog., 169, 11 (1979). Milnes, A.G., Deep Impurities in Semiconductors, John Wiley, New York (1973). Mittlefehldt, E., Separ. Times, 8(1), 10 (2002). Miyauchi, T. and T. Vermeulen, Ind. Eng. Chem. Fund., 2, 113 (1963). Moates, G.H. and J.K. Kennedy, in Fractional Solidification, M. Zief and W.R. Wilcox (eds.), vol. I, Marcel Dekker, New York (1967), chap. 10.

870 Molinari, J.G.D., in Fractional Solidification, M. Zief and W.R. Wilcox (eds.), vol. I, Marcel Dekker, New York (1967), chap. 13. Moody, HW., J. Chem. Ed., 59(4), Apr., 290 (1982). Moonen, H. and N.J. Niefind, Desalination, 41, 327 (1982). Moore, L.R., S. Milliron, P.S. Williams, J.J. Chalmers, S. Margel and M. Zborowski, Anal. Chem., 76, 3899 (2004). Mothes, H. and F. Löffler, Chem. Eng. Process, 18, 323 (1984). Mulder, M., Basic Principles of Membrane Technology, Kluwer Academic Publishers, Dordrecht (1991). Mullin, J.W., Crystallization, Butterworths, London (1961). Mullin, J.W., Crystallization, 2nd edn., Butterworths, London (1972). Myers, A.L. and J.M. Prausnitz, AIChE J., 11, 121 (1965). Myers, A.L., C. Minka and D.Y. Ou, AIChE J., 28, 97 (1982). Myerson, A.S., Handbook of Industrial Crystallization, Butterworth-Heinemann, Boston, MA (1993), chaps. 1–2. Nakano, Y., C. Tien and W.N. Gill, AIChE J., 13(6), 1092 (1967). National Research Council, International Critical Tables, Vol. III, McGraw-Hill, New York (1929). Naylor, R.W. and P.O. Backer, AIChE J., 1, 95 (1955). Neel, J., in Membrane Separations Technology: Principles and Applications, R.D. Noble and S.A. Stern (eds.) Elsevier, New York (1995), chap. 5. Nerenberg, S.T. and G. Pogojeff, Am. J. Clin. Path., 51(6), 728 (1969). Neumann, L., E.T. White and T. Howes, “What does a mean size mean?”, AIChE Annual Meeting, San Francisco, CA, Nov. 16– 21 (2003), paper 39a. Newman, J., Electrochemical Systems, Prentice Hall, Englewood Cliffs, NJ (1973), p. 193. Newman, J. Electrochemical Systems, 2nd edn., Prentice Hall, Englewood Cliffs, NJ (1991), p. 215. Ng, P.K., J. Lundblad and G. Mitra, Separ. Sci., 11(5), 499 (1976). Noble, P.T., Biotechnol. Prog., 1(4), 237 (1985). Noble, R.D. and S.A. Stern (eds.), Membrane Separations Technology: Principles and Applications, Elsevier, Amsterdam (1995). Noble, R.D. and P.A. Terry, Principles of Chemical Separations with Environmental Applications, Cambridge University Press, Cambridge (2004). Noble, R.D. and J.D. Way (eds.), Liquid Membranes: Theory and Applications, ACS Symposium Series 347, American Chemical Society. Washington, D.C. (1987). Noble, R.D., J.D. Way and L.A. Powers, Ind. Eng. Chem. Fund., 25, 450 (1986). Norman, M.A., C.B. Evans, A.R. Fuoco, R.D. Noble and C.A. Koval, Anal. Chem., 77(19), 6374 (2005). Null, H.R., Chem. Eng. Progr., Aug., 42 (1980). Nyborg, W.L., in Ultrasound: Its Application in Medicine and Biology, F.J. Fry (ed.), vol. 1, Elsevier, New York (1978), p. 1. Nyström, M., P. Aimar, S. Luque, M. Kulovaara and S. Metsämuuronen, Colloids Surf. A: Physicochem. Eng. Aspects, 138, 185 (1998). Nývlt, J., Design of Crystallizers, CRC Press, Boca Raton, FL (1992). Nývlt, J., O. Sohnel, M. Matuchova and M. Brout, The Kinetics of Industrial Crystallization, Elsevier, Amsterdam (1985). Oak, M.J., “Modeling and experimental study of high gradient magnetic separation with application to coal beneficiation,” unpublished M.S. thesis, Auburn University, AL, Dec. 8 (1977). O’Connell, H.E., Trans. AIChE., 42, 741 (1946).

References O’Farrell, P.H., Science, 227, 1586 (1985). Ohno, M., O. Ozaki, H. Sato, S. Kimura and T. Miyauchi, J. Nucl. Sci. Technol., 14(8), 589 (1977). Ohno, M., T. Morisue, O. Ozaki and T. Miyauchi, J. Nucl. Sci. Technol., 15(5), 376 (1978). Ohya, H. and Y. Taniguchi, Desalination, 16, 359 (1975). Olander, D.R., Adv. Nuclear Sci. Technol., 6, 105 (1972). Orcutt, J.C., in Fractional Solidification, M. Zief and W.R. Wilcox (eds.), vol. 1, Marcel Dekker, New York (1967), chap. 17. Ornatski, N.V., E.V. Sergeev and Y.M. Shekhtman, Investigations of the Process of Clogging of Sands, University of Moscow, Moscow (1955). Overdevest, P.E.M., J.T.F. Keurentjes, A. Van der Padt and K. van’t Reit, in Surfactant-based Separation: Science and Technology, J.F. Scamehorn and J.H. Harwell (eds.), ACS Symposium Series 740, Amercian Chemical Society, Washington, D. C. (2000), chap. 9. Padin, J., R.T. Yang and C.L. Munson, Ind. Eng. Chem. Res., 38 (10), 3614 (1999). Pamme, N. and A. Manz, Anal. Chem., 76, 7250 (2004). Pan, C.Y., AIChE J., 29, 545 (1983). Pan, C.Y., AIChE J., 32, 2020 (1986). Pan, C.Y. and H.W. Habgood, Ind. Eng. Chem. Fund., 13, 323 (1974). Pan, C.Y. and H.W. Habgood, Can. J. Chem. Eng., 56, 197 (1978a). Pan, C.Y. and H.W. Habgood, Can. J. Chem. Eng., 56, 210 (1978b). Patel, D.C. and R.G. Luo, in Adsorption and its Applications in Industry and Environmental Protection, A. Dabrowski (ed.), Studies in Surface Science and Catalysis 120, Elsevier Science, New York (1998). Paul, D.R., Ind. Eng. Chem. Proc. Des. Dev., 10(3), 375 (1971). Paul, D.R. and W.J. Koros, J. Polym. Sci., Polym. Phys. Ed., 14, 675 (1976). Pearce, C.W., in VLSI Technology, S.M. Sze (ed.), McGraw-Hill, New York (1983). Pedersen, C.J., J. Am. Chem. Soc, 89(26), 7017 (1967). Pedersen, C.J., Science, 241, 536 (1988). Perkins, T.W., D.S. Mak, T.W. Root and E.N. Lightfoot, J. Chromatog. A, 766, 1 (1997). Perry, R.H. and D. Green, Perry’s Chemical Engineers’ Handbook, 6th edn., McGraw-Hill, New York (1984). Petenate, A. and C.E. Glatz, Biotechnol. Bioeng., 25, 3059 (1983). Peters, E.A.J.F., B. Breure, P. van den Henvel and P.J.A.M. Kerkhof, Ind. Eng. Chem. Res., 47, 3937 (2008). Petersson, F., A. Nilsson, H. Jönsson and T. Laurell, Anal. Chem., 77, 1216 (2005). Pfann, W.G. Zone Melting, 2nd edn., John Wiley, New York (1966), p. 31. Pfeffer, R., Ind. Eng. Chem. Fund., 3, 380 (1964). Phelps, D.S.C. and D.M. Ruthven, Ind. Eng. Chem. Res., 40, 2168 (2001). Philpot, J. St. L., Trans. Faraday Soc., 36, 38 (1940). Pigford, R.L., B. Baker, III and D.E. Blum, Ind. Eng. Chem. Fund., 8, 144 (1969a). Pigford, R.L., B. Baker, III and D.E. Blum, Ind. Eng. Chem. Fund., 8, 848 (1969b). Pitzer, K.S. and L. Brewer, Thermodynamics by G.N. Lewis and M. Randall, 2nd edn. McGraw-Hill, New York (1961), p. 666.

References Pohl, H.A., in Methods of Cell Separation, N. Castimpoolas (ed.), vol. 1, Plenum Press, New York (1977), chap. 3. Pohl, H.A., Dielectrophoresis: The Behavior of Matter in Nonuniform Electric Fields, Cambridge University Press, New York (1978). Pohl, H.A. and K. Kaler, Cell Biophys. 1, 15 (1979). Porath, J., Protein Expression Purif. 3, 263 (1992). Porath, J. and P. Flodin, Nature, 83, 1657 (1959). Porath, J., J. Carlsson, I. Olsson and G. Belfrage, Nature, 258, 598 (1975). Porter, H.F., G.A. Schurr, D.F. Wells and K.T. Semrau, in Perry’s Chemical Engineers’ Handbook, R.H. Perry and D. Green (eds.), 6th edn., McGraw-Hill, New York (1984), sect. 20. Porter, K.E., M.J. Lockett and C.T. Lim, Trans. Inst. Chem. Eng., 50, 91 (1972). Powers, J.E., in New Chemical Engineering Separation Techniques, H.M. Schoen (ed.), Interscience, New York (1962). Pozrikidis, C., J. Fluid Mech., 351, 139 (1997). Prasad, R. and K.K. Sirkar, Ind. Eng. Chem. Proc. Des. Dev., 24, 350 (1985). Prasad, R. and K.K. Sirkar, AIChE J., 33, 1057 (1987). Prasad, R. and K.K. Sirkar, AIChE J., 34, 177 (1988). Prasad, R. and K.K. Sirkar, in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), chap. 41. Pratt, H.R.C., Countercurrent Separation Processes, Elsevier, Amsterdam (1967). Pratt, H.R.C., Solv. Extr. Ion Exchange, 1(4), 669 (1983). Pratt, H.R.C. and G.W. Stevens, Ind. Eng. Chem. Res., 30, 733 (1991). Prausnitz, J.M., Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, Englewood Cliffs, NJ (1969), p. 273. Prausnitz, J.M. and P.L. Chueh, Computer Calculations for HighPressure Vapor-Liquid Equilibria, Prentice-Hall, Englewood Cliffs, NJ (1968). Prausnitz, J.M., R.N. Lichtenthaler and E.G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd edn., Prentice Hall, Englewood Cliffs, NJ (1986). Prausnitz, J.M., R.N. Lichtenthaler and E.G. Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, 3rd edn., Prentice Hall, Englewood Cliffs, NJ (1999). Present, R.D., Kinetic Theory of Gases, McGraw-Hill, New York (1958), p. 118. Probstein, R.F., Physicochemical Hydrodynamics: An Introduction, Butterworths, Boston, MA (1989), pp. 129–141. Pujar, N.S. and A.L. Zydney, Ind. Eng. Chem. Res., 33, 2473 (1994). Pusch, W., H.G. Burghoff and E. Staude, in Proc. 5th Int. Symp. Fresh Water From the Sea, vol. 4, Alghero, Italy May 16–20 (1976), p. 143. Qin, Y.J., J.P. Sheth and K.K. Sirkar, Ind. Eng. Chem. Res., 42, 582 (2003). Rachford, H.H. Jr. and J.D. Rice, J. Petrol. Technol. 4(10), sect. 1, Oct., p. 19 (1952a). Rachford, H.H. Jr. and J.D. Rice, J. Petrol. Technol., 4(10), sect. 2, Oct., p. 3 (1952b). Radke, C.J. and J.M. Prausnitz, AIChE J., 18(4), 761 (1972). Raghavan, S. and D.W. Fuerstenau, AIChE Symp. Ser., 71(150), 59 (1975). Raghavan, N.S., M.M. Hassan and D.M. Ruthven, AIChE J., 31(3), 385 (1985).

871 Ramkrishna, D., Population Balances: Theory and Applications to Particle Systems in Engineering, Academic Press, New York (2000). Ramshaw C. and R.H. Mallinson, “Mass transfer process,” US Patent 4,283,255, Aug. 11 (1981). Randolph, A.D. and M.A. Larson, Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization, 2nd edn., Academic Press, New York (1988). Rao, G.H. and K.K. Sirkar, Desalination, 27, 99 (1978). Ratanathanawongs, S.K. and J.C. Giddings, Anal. Chem., 64, 6 (1992). Rathore, A.S. and C.S. Horvath, Anal. Chem., 70, 3069 (1998). Rautenbach, R. and R. Albrecht, Membrane Processes, John Wiley, New York (1989). Recktenwald, D. and A. Radbruch (eds.), Cell Separation Methods and Applications, Marcel Dekker, New York (1998), p. 293. Reed, B.W., M.L. Semmens and E.L. Cussler, in Membrane Separations: Principles and Applications, R.D. Noble and S.A. Stern (eds.), Elsevier, Amsterdam (1995), p. 467. Rege, S.U. and R.T. Yang, Separ. Sci. Technol., 36(15), 3355 (2001). Reid, R.C., J.M. Prausnitz and T.K. Sherwood, The Properties of Gases and Liquids, 3rd edn., McGraw-Hill, New York (1977). Reis, J.F.G., E.N. Lightfoot and H-L. Lee, AIChE J., 20(2), 362 (1974). Reiss, L.P., Ind. Eng. Chem. Proc. Des. Dev., 6, 486 (1967). Renkin, E.M., J. Gen. Physiol., 38, 225 (1954). Reschke, M. and K. Schügerl, Chem. Eng. J., 29, 825 (1984). Rhee, H.K. and N.R. Amundson, Ind. Eng. Chem. Fund., 9, 303 (1970). Rhee, H.K. and N.R. Amundson, AIChE J., 28(3), 423 (1982). Rhee, H.K., R. Aris and N.R. Amundson, Phil. Trans. Roy. Soc. London., 267A, 419 (1970a). Rhee, H.K., E.D. Heerdt and N.R. Amundson, Chem. Eng. J., 1, 279 (1970b). Rhee, H.K., E.D. Heerdt and N.R. Amundson, Chem. Eng. J., 3, 22 (1972). Richman, D., E.A. Wynne and F.D. Rosi, in Fractional Solidification, M. Zief and W.R. Wilcox (eds.), vol. I, Marcel Dekker, New York (1967), chap. 9. Ridgway, K. and E.E. Thrope, in Handbook of Solvent Extraction, T.C. Lo, M.H.I. Baird and C. Hanson (eds.), Wiley, New York, Ch. 19 (1983). Reprinted, Krieger Publishing Co., Malabar, FL (1991). Rietema, K., Chem. Eng. Sci., 7, 89 (1957). Rietema, K., Chem. Eng. Sci., 15, 298 (1969). Ritcey, G.M. and A.W. Ashbrook, Solvent Extraction: Principles and Applications to Process Metallurgy, Part II, Elsevier Scientific Publishing Company, Amsterdam (1979). Ritcey, G.M. and A.W. Ashbrook, Solvent Extraction, Part I, Elsevier, New York (1984a). Ritcey, G.M. and A.W. Ashbrook, Solvent Extraction, Part II, Elsevier, New York (1984b), pp. 607–608. Ritter, J.A. and R.T. Yang, Ind. Eng. Chem. Res., 28, 599 (1989). Rixey, W.G., “Nonwet adsorbents for the selective recovery of polar organic solutes from dilute aqueous solution,” unpublished Ph.D. dissertation, Dept. Chem. Eng., University of California, Berkeley, CA (1987). Roberts, D.L. and S.K. Friedlander, AIChE J. 26, 593 (1980a). Roberts, D.L. and S.K. Friedlander, AIChE J. 26, 602 (1980b).

872 Roberts, D.L., J.F. Scamehorn and S.D. Christian, in Surfactantbased Separations: Science and Technology, J.F. Scamehorn and J.H. Harwell (eds.), ACS Symposium Series 740, American Chemical Society, Washington, D.C. (2000), chap. 11. Robeson, L., J. Membr. Sci., 62, 165 (1991). Robinson, C.S. and E.R. Gilliland, Elements of Fractional Distillation, 4th edn., McGraw-Hill, New York (1950), pp. 162–175. Robinson, J.P., in Encyclopedia of Biomaterials and Biomedical Engineering, G.E. Wnek and G.L. Bowlin (eds.), Marcel Dekker, New York (2004), p. 630. Robinson, J.S., D. Scott and J. Winnick, AIChE J., 44(10), 2168 (1998). Robinson, R.G. and D.Y. Cha, Biotechnol. Prog., 1, 18 (1985). Rogers, T.H., F.V. Grim and N.E. Lemmon, Ind. Eng. Chem., 18, 164 (1926). Rolchigo, P.M. and D.J. Graves, AIChE J., 34 (3), 483 (1988). Romero, C.A. and R.H. Davis, J. Membr. Sci., 39, 157 (1988). Romero, J. and A.L. Zydney, Separ. Sci. Technol., 36(7), 1575 (2001). Rony, P.R., Separ. Sci., 3(3), 239 (1968a). Rony, P.R., Separ. Sci., 3(4), 357 (1968b). Rony, P.R., Separ. Sci., 4, 413 (1969a). Rony, P.R., Separ. Sci., 4(6), 493 (1969b). Rony, P.R., Separ. Sci., 5(1), 1 (1970). Rony, P.R., in Recent Advances in Separation Techniques, N. N. Li (ed.), AIChE Symposium Series 120, no. 68 (1972), pp. 1–58. Rosen, J.B., J. Chem. Phys., 20, 387 (1952). Rosen, J.B., Ind. Eng. Chem., 46, 1590 (1954). Rothfeld, L.B., AIChE J., 9, 19 (1963). Rousseau, R.W. and J.S. Staton, Chem. Eng., July, 91 (1988). Rubin, E. and E.L. Gaden, in New Chemical Engineering Separation Techniques, H.M. Schoen (ed.), Interscience, New York (1962). Rubin, E. and J. Jorne, Ind. Eng. Chem. Fund., 8, 474 (1969). Rubow, K.L., “Submicron aerosol filtration characteristics of membrane filters,” unpublished Ph.D. Thesis, Mechanical Engineering Department, University of Minnesota, MN (1981). Rudge, S.R. and M.R. Ladisch, Biotechnol. Prog., 4(13), 123 (1988). Ruhemann, M., The Separation of Gases, 2nd edn., Oxford University Press, London (1949). Rumeau, J., F. Persin, V. Sciers, M. Persin and J. Sarrazin, J. Membr. Sci., 73, 313 (1992). Russel, W.B., D.A. Saville and W.K. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge (1989). Ruthven, D.M., Principles of Adsorption and Adsorption Processes, John Wiley, New York (1984), pp. 381–383. Sabadell, J.E. and N.H Sweed, Separ. Sci., 5, 171 (1970). Safarik, D.J. and R.B. Eldridge, Ind. Eng. Chem. Res., 37, 2571 (1998). Saffman, P.G., J. Fluid Mech., 22, 385 (1965). Sage, B.H., Thermodynamics of Multicomponent Systems, Reinhold Publishing Corp., New York (1965), p. 256. Said, A.S. (1956), AIChE J., 2(4), 477 (1956). Said, A.S., Separ. Sci. Technol., 13(8), 647 (1978). Saksena, S. and A.L. Zydney, Biotechnol. Bioeng., 43, 960 (1994). Salcudean, M., I. Gartshore and E.C. Statie, Chem. Eng., Apr., 66 (2003). Sandell, E.B., Anal. Chem., 40, 834 (1968).

References Sanders, E.S., W.J. Koros, H.B. Hopfenberg and V.T. Stannett, J. Membr. Sci., 18, 53 (1984). Sanyal, D., N. Vasishtha and D.N. Saraf, Ind. Eng. Chem. Res., 27, 2149 (1988). Sarkar, S., C.J. Mumford and C.R. Phillips, Ind. Eng. Chem. Proc. Des. Dev., 19, 665 (1980). Sartori, G., W.S. Ho, D.W. Savage, G.R. Chludzinski and S. Wiechert, Separ. Purif. Meth., 16(2), 171 (1987). Saskawa, S. and H. Walter, Biochemistry, 11, 2760 (1972). Satterfield, C.M., C.K. Colton and W.K. Pitcher Jr., AIChE J., 19, 628 (1973). Saunders, M.S., J.B. Vierow and G. Carta, AIChE J., 35(1), 53 (1989). Savage, D.W., E.W. Funk, W.C. Yu and G. Astarita, Ind. Eng. Chem. Fund., 25, 326 (1986). Scamehorn, J.F., S.D. Christian and R.T. Ellington, in Surfactantbased Separation Processes, J.F. Scamehorn and J.H. Harwell (eds.), Marcel Dekker, New York (1989), chap. 2. Schachman, H.K., J. Phys. Colloid Chem., 52, 1034 (1948). Schaetzel, P., C. Vauclair, G. Luo and Q.T. Nguyen, J. Membr. Sci., 191, 103 (2001). Schaetzel, P., C. Vanclair, Q.T. Nguyen and R. Bouzerar, J. Membr. Sci., 244, 117 (2004). Schasfoort, R.B.M., S. Schlautmann, J. Hendrikse and A. van den Berg, Science, 286, 942 (1999). Schock, G. and A. Miquel, Desalination, 64, 339 (1987). Schork, J.M., R. Srinivasan and S.R. Auvil, Ind. Eng. Chem. Res., 32, 2226 (1993). Schottky, W., Phys. Z., 25, 635 (1924). Schultz, J.S., J.S. Goddard and S.R. Suchdeo, AIChE J., 20, 417 (1974). Schure, M.R. and A.M. Lenhoff, Anal. Chem., 65, 3024 (1993). Schweitz, L., P. Spégel and S. Nilsson, Analyst, 125, 1899 (2000). Schweitzer, P.A., Handbook of Separation Techniques for Chemical Engineers, 3rd edn., McGraw-Hill, New York (1997). Schwinge, J., D.E. Wiley and D.F. Fletcher, Ind. Eng. Chem. Res., 41, 4879 (2002). Schwinge, J., D.E. Wiley and D.F. Fletcher, Ind. Eng. Chem. Res., 42, 4962 (2003). Scopes, R.K., Protein Purification: Principles and Practice, 2nd edn., Springer Verlag, New York (1987), p. 101. Scopes, R.K., Protein Purification: Principles and Practice, 3rd edn., Springer, New York (1994), chaps. 6, 7. Scott, C.D., R.D. Spence and W.G. Sisson, J. Chromatogr., 126, 381 (1976). Scott, D.S. and F.A.L. Dullien, Chem. Eng. Sci., 17, 771 (1962). Seader, J.D. and E.J. Henley, Separation Process Principles, John Wiley & Sons, New York (1998). Seader, J.D. and E.J. Henley, Separation Process Principles, 2nd edn., John Wiley, New York (2006). Sengupta, A. and K.K. Sirkar, in Progress in Filtration and Separation, R.J. Wakeman (ed.), vol. 4, Elsevier, Amsterdam (1986), p. 289. Sengupta, A. and K.K. Sirkar, in Membrane Separations Technology: Principles and Applications, R.D. Noble and S.A. Stern (eds.), Elsevier Science, Amsterdam (1995), chap. 11. Sengupta, A., R. Basu and K.K. Sirkar, AIChE J., 34, 1698 (1988). Sengupta, A., B. Raghuraman, and K.K. Sirkar, J. Membr. Sci., 51, 105 (1990). Sengupta, A., P.A. Petersen, B.D. Miller, J. Schneider and C.W. Fulk Jr., Separ. Purif. Technol. 14, 189 (1998).

References Setchenow, M., Ann. Chim. Phys., 25, 226 (1892). Shaffer, L.H. and M.S. Mintz, in Principles of Desalination, K.S. Spiegler (ed.), Academic Press, New York, Ch. 6 (1966); 2nd edn., Part A, Ch. 6, K.S. Spiegler and A.D.K. Laird (eds.) (1980). Shaffer, L.H. and M.S. Mintz, in Principles of Desalination, K.S. Spiegler (ed.), Academic Press, New York (1980). Shamsai, B.M. and H.G. Monobouquette, J. Membr. Sci., 130, 173 (1997). Shapiro, H.P., Practical Flow Cytometry, 3rd edn., Wiley-Liss, New York (1995). Sharnez, R. and D. Sammons, Strategies for Enhancing Performance in Continuous-Flow Electrophoresis, I: Selective Manipulation of Particle Trajectories, Preprints of First Separations Division Topical Conference on Separation Technologies: New Developments and Opportunities, Miami Beach, FL, Nov. 2–6 (1992), pp. 264– 270. Shelden, R.A. and E.V. Thompson, J. Membr. Sci., 4, 115 (1978). Shendalman, L.H. and J.E. Mitchell, Chem. Eng. Sci., 27, 1449 (1972). Sheng, H.P., Separ. Purif. Methods, 6(1), 89 (1977). Sherwood, T.K. and F.A.L. Holloway, Trans. AIChE J., 36, 39 (1940). Sherwood, T.K., P.L.T. Brian and R.E. Fisher, Ind. Eng. Chem. Fund., 6, 2 (1967). Sherwood, T.K., R.L. Pigford and C.R. Wilke, Mass Transfer, McGraw-Hill, New York (1975), chap. 11. Shiras, R.N., D.N. Hanson and C.H. Gibson, Ind. Eng. Chem., 42, 871 (1950). Shreve, R.N. and G.T. Hatch, Chemical Process Industries, 5th edn, McGraw-Hill, New York (1984). Shuler, M.L. and F. Kargi, Bioprocess Engineering: Basic Concepts, 2nd edn., Prentice Hall PTR, Upper Saddle River, NJ (2002). Sidhoum, M., A. Sengupta and K.K. Sirkar, AIChE J., 34(3), 417 (1988). Siegell, J.H., G.D. Dupre and J.C. Pirkle Jr., Chem. Eng. Progr., Nov., 57 (1986). Siirola, J.J. and S.D. Barnicki, in Perry’s Chemical Engineers’ Handbook, R.H. Perry and D.W. Green (eds.), 7th edn., McGraw-Hill, New York (1997), pp. 13–54. Silver, R., in Principles of Desalination, K.S. Spiegler (ed.), Academic Press, New York (1966), p. 77. Sircar, S and J.R. Hufton, AIChE J., 46 (3), 659 (2000). Sircar, S. and R. Kumar, I&E. Chem. Proc. Des. Dev., 22(2), 271 (1983). Sircar, S. and J.W. Zondlo, US Patent 4, 013, 429 (1977). Sirkar, K.K., Separ. Sci., 12(3), 211 (1977). Sirkar, K.K., Separ. Sci. Technol., 15 (4), 1091 (1980). Sirkar, K.K., “Immobilized interface solute-transfer process,” US Patent 4997569 (1991). Sirkar, K.K., in Membrane Handbook, W.S. Winston Ho and K. K. Sirkar (eds.), Van Nostrand Reinhold, New York (1992), chap. 46. Sirkar, K.K., “Immobilized interface solute transfer apparatus,” US Patent 4,789,468, Dec. 6 (1988); US Patent Re, 34, 828, Jan. 17 (1995). Sirkar, K.K., in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), chap. 46. Sirkar, K.K., Ind. Eng. Chem. Res., 47(15), 5250 (2008). Sirkar, K.K., P.T. Dang and G.H. Rao, Ind. Eng. Proc. Des. Dev., 21, 517 (1982).

873 Skarstrom, C.W., “Method and apparatus for fractionating gas mixtures by adsorption,” US Patent 2,944,627, July 12 (1960). Skarstrom, C.W., in Recent Developments in Separation Science, N. Li (ed.), vol 2, CRC Press, Cleveland, OH (1975), p. 95. Skelland, A.H.P., Diffusional Mass Transfer, Wiley, New York (1974). Slattery, J.C., Momentum, Energy and Mass Transfer in Continua, McGraw-Hill, New York (1972). Sleicher, C.A. Jr., AIChE J., 5(2), 145 (1959). Sleicher, C.A. Jr., AIChE J., 6(2), 529 (1960). Smith, D.R. and J.A. Quinn, AIChE J., 26, 112 (1980). Smith, D.R., R.J. Lander and J.A. Quinn, in Recent Developments in Separation Science, vol. 3B, N.N. Li (ed.), CRC Press, Cleveland, OH (1977), p. 225. Smith, J.M. and H.C. Van Ness, Introduction to Chemical Engineering Thermodynamics, 3rd edn., McGraw-Hill, New York (1975). Smith, J.M., H.C. Van Ness and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 6th edn., McGrawHill, New York (2001). Smith, K.A., J.K. Meldon and C.K. Colton, AIChE J., 19, 102 (1973). Snyder, L.R. and J.J. Kirkland, Introduction to Modern Liquid Chromatography, 2nd edn., Wiley-Interscience, New York (1979), pp. 260–289. Solan, A., Y. Winograd and U. Katz, Desalination, 9, 89 (1971). Soltanieh, M. and W.N. Gill, Chem. Eng. Commun, 12, 279 (1981). Song, L., Z. Ma, X. Liao, P.B. Kosaraju, J.R. Irish and K.K. Sirkar, J. Membr. Sci., 323, 257 (2008). Soo, S.L., Particulates and Continuum: Multiphase Fluid Dynamics, Hemisphere Publishing, New York (1989), chap. 2 . Soo, S.L. and L.W. Rodgers, Powder Technol., 5, 43 (1971). Souders, M., G.J. Pierotti and C.L. Dunn, Chem. Eng. Progr. Symp. Ser., 66(100), 41 (1970). Sourirajan, S., Reverse Osmosis, Logos Press, London (1970). Spiegler, K.S., Salt-Water Purification, 2nd edn., Plenum Press, New York (1977). Spiegler, K.S. and O. Kedem, Desalination, 1, 311 (1966). Spielman, L.A. and P.M. Cukor, J. Colloid Interface Sci., 43, 51 (1973). Spielman, L.A. and J.A. Fitzpatrick, J. Colloid Interface Sci., 42, 607 (1972). Spoor, P.S. and G.W. Swift, Phys. Rev. Lett., 85(8), 1646 (2000). Spriggs, H.D. and J.L. Gainer, Ind. Eng. Chem. Fund., 12, 291 (1973). Spriggs, H.D. and N.N. Li, in Membrane Separation Processes, P. Meares (ed.), Elsevier, Amsterdam (1976), chap. 2 . Stannett, V.T., in Diffusion in Polymers, J. Crank and G.S. Park (eds.), Academic Press, London (1968), p. 41. Stannett, V. T., W.J. Koros, D.L. Paul, H. Lonsdale and R. Baker, Adv. Polym. Sci., 32, 69 (1979). Steed, J.W. and J.L. Atwood, Supramolecular Chemistry, John Wiley, New York (2000). Stella, A., H.R.C. Pratt, K.H. Mensforth, G.W. Stevens and T. Bowser, Ind. Eng. Chem. Res., 45, 6555 (2006). Stern, S.A. and H.L. Frisch, Ann. Rev. Mater. Sci., 11, 523 (1981). Stern, S.A. and W.P. Walawender Jr., Separ. Sci., 4(2), 129 (1969). Sternling, C.V. and L.E. Scriven, AIChE J., 5, 514 (1959). Stewart, G.H., Separ. Sci. Technol., 13(3), 201 (1978).

874 Strathmann, H., Separ. Sci. Technol., 15, 1135 (1980). Strathmann, H., in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), chaps. 16–20. Strickland, A.D., R.G. De Krester and P.J. Scales, AIChE J., 51(9) 2481 (2005). Strigle, R.F. Jr., Packed Tower Design and Applications, 2nd edn., Gulf Publishing, Houston, TX (1994). Suen, S.-Y. and M. Etzel, Chem. Eng. Sci., 47(6), 1355 (1992). Suh, S.S. and P.C. Wankat, AIChE J., 35(3), 523 (1987). Sussman, M.V. and C.C. Huang, Science, 156, 974 (1967). Sussman, M.V., K.N. Astill, R. Rombach, A. Cerullo and S.S. Chen, Ind. Eng. Chem. Fund., 11(2), 181 (1972). Sussman, M.V., K.N. Astill and R.N.S. Rathore, J. Chromatograph. Sci., 12, 91 (1974). Sutija, D.P. and J.M. Prausnitz, Chem. Eng. Ed., Winter, 20 (1990). Suzuki, I., H. Yagi, H. Komatsu and M. Hirata, J. Chem. Eng. Jpn., 4, 26 (1971). Svarovsky, L., in Solid-Liquid Separation, L. Svarovsky (ed.), Butterworths, London (1977). Svarovsky, L., in Progress in Separation and Filtration, R.J. Wakeman (ed.), vol. 1, Elsevier, Amsterdam (1979), p. 251. Svarovsky, L., Solid-Gas Separation, Elsevier Scientific Publishing, Amsterdam (1981). Svarovsky, L., Solid-liquid Separation, Butterworths, London (1982), chap. 6. Svedberg, T. and K.O Pedersen, The Ultracentrifuge, Clarendon Press, Oxford (1940). Svrcek, W.Y. and W.D. Monnery, Chem. Eng. Progr., Oct., 53 (1993). Sweed, N.H., in Progress in Separation and Purification, E.S Perry and C.J. Van Oss (eds.), vol. 4, John Wiley, New York, (1971). Sweed, N.H., in Recent Developments in Separation Science, N.N. Li (ed.), vol. I, CRC Press, Boca Raton, FL (1972). Sweeney, M.J. and P.M. Galletti, Trans. Am. Soc. Artif. Intern. Organs, 10, 3 (1964). Tabatabai, A.J., F. Scamehorn and S.D. Christian, J. Membr. Sci., 100, 193 (1995). Talbot, J.B., Separ. Sci. Technol., 15(3), 277 (1980). Talbot, L., R.K. Chang, R.W. Schefer and D.R. Willis, J. Fluid Mech., 101, 737 (1980). Tamada, J.A., A.S. Kertes and C.J. King, Ind. Eng. Chem. Res., 29, 1319 (1990). Tamon, H., H. Mizota, N. Sano, S. Schulze and M. Oakazaki, AIChE J., 41(7), 1701 (1995). Tan, H.K.S., AIChE J., 40(2), 369 (1994). Tanford, C., Physical Chemistry of Macromolecules, John Wiley, New York (1961). Tanford, C., The Hydrophobic Effect: Formation of Micelles and Biological Membranes, 2nd edn., Wiley Interscience, New York (1980). Tanford, C., Y. Nozaki, J.A. Reynolds and S. Mikano, Biochemistry, 13, 2369 (1974). Tang, K.E.S. and V. Bloomfield, Biophys. J., 82, 2876 (2002). Taylor, G.I., Proc. Roy. Soc. London Ser. A., 219, 186 (1953). Taylor, G.I., Proc. Roy. Soc. London, Ser. A., 225, 473 (1954). Taylor, R. and R. Krishna, Multicomponent Mass Transfer, John Wiley, New York (1993). Taylor, R., H.A. Kooijman and J.-S Huang, Comput. Chem. Eng., 18, 205 (1994).

References Taylor, R., R. Krishna and H. Kooijman, Chem. Eng. Progr., July, 28 (2003). Ter Haar, G. and S.J. Wyard, Ultrasound Med. Biol., 4, 111 (1978). Ter Linden, A.J., Proc. Inst. Mech. Eng., 160, 233 (1949). Terabe, S., in Capillary Electrophoresis Technology, N.A. Guzman (ed.), Marcel Dekker, New York (1993), p. 65. Terabe, S., K. Otsuka, K. Ichikawa, A. Tsuchiya and T. Ando, Anal. Chem., 56, 111 (1984). Teraoka, Y., H-M. Zhang, S. Furukawa and N. Yamazoe, Chem. Lett., p. 1743 (1985). Terrill, D.L., L.F. Sylvestre and M.F. Doherty, Ind. Eng. Chem, Proc. Des. Dev., 24, 1062 (1985). Theodore, L., Chem. Eng. Progr., Sept., 16 (2005). Thijssen, H.A., Chem. Eng. Progr., 75(7), 21 (1979). Thomas, H., J. Am. Chem. Soc., 66, 1664 (1944). Thomas, H., Ann. NY Acad. Sci., 49, 161 (1948). Thongsukmak, A. and K.K. Sirkar, J. Membr. Sci., 302, 45 (2007). Thongsukmak, A., and K.K. Sirkar, J. Membr. Sci., 329, 119 (2009). Thorman, J.M., H. Rhim and S.T. Hwang, Chem. Eng. Sci., 30(7), 751 (1975). Thorman, J.M., S.T. Hwang and K.H. Yuen, Separ. Sci. Technol., 15(4), 1069 (1980). Tien, C., Granular Filtration of Aerosols and Hydrosols, Butterworths, London (1989), pp. 108–109. Tilgner, H.G. and F.J. Schmitz, European Patent 36, 175; assigned to Akzo GmbH (1981). Tomida, T., K. Hamaguchi, S. Tashima, M. Katoh and S. Masuda, Ind. Eng. Chem. Res., 40, 3557 (2001). Tompkins, C.J., A.S. Michaels and S.W. Peretti, J. Membr. Sci., 75, 277 (1992). Tondre, C., in Surfactant-based Separations: Science and Technology, J.F. Scamehorn and J.H. Harwell (eds.), ACS Symposium Series 740, American Chemical Society, Washington, D.C. (2000), chap. 10. Toor, H.L., AIChE J., 3, 198 (1957). Towns, J.K. and F.E. Regnier, Anal. Chem., 64, 243 (1992). Treybal, R.E., Liquid Extraction, 2nd edn., McGraw-Hill, New York, (1963). Treybal, R.E., Mass Transfer Operations, 3rd edn., McGraw-Hill, New York (1980). Tsuda, T., in Capillary Electrophoresis Technology, N.A. Guzman (ed.), Marcel Dekker, New York (1993), chap. 14. Tsuda, T. (ed.), Electric Field Applications in Chromatography, Industrial and Chemical Processes, VCH, Weinheim (1995). Tsuji, A., E. Nakashima, S. Hamano and T. Yamana, J. Pharm. Sci., 67, 1059 (1978). Turco, R.P., O.B. Toon, T.P. Ackerman, J.B Pollack and C. Sagan, Science, 247, 166 (1990). Turner, D.B., Workbook of Atmospheric Dispersion Estimates, HEW, Washington, D.C. (1969). Turnock, P.H. and R.H. Kadlec, AIChE J., 17, 335 (1971). Uemasu, I., J. Inclusion Phenom. Mol. Recog. Chem., 13, 1 (1992). Ulrich, J., in Handbook of Industrial Crystallization, A.S. Myerson (ed.), Butterworth-Heinemann, Boston, MA (1993), chap. 7. Underwood, A.J.V., Trans. Inst. Chem. Engrs., 16, 112 (1932). Underwood, A.J.V., Chem. Eng. Progr., 44, 603 (1948). Van Deemter, J.K., F.J. Zuiderweg and A. Klinkenberg, Chem. Eng. Sci., 5, 271 (1956). Van der Kolk, H., in Cyclones in Industry, K. Rietema and C.G. Verver (eds.), Elsevier, Amsterdam (1961), p. 76. Van Dongen, D.B. and M.F. Doherty, Chem. Eng. Sci., 39, 883 (1984).

References Van Ebbenhorst Tengbergen H.J. and K. Rietema, in Cyclones in Industry, K. Rietema and C.G. Verver (eds.), Elsevier, Amsterdam (1961), p. 23. Van Eijndhoven, R.H., S. Saksena and A.L. Zydney, Biotechnol. Bioeng., 48, 406 (1995). Van Ness, H.C., Ind. Eng. Chem. Fund., 8, 464 (1969). Van Ness, H.C. and M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions, McGraw-Hill, New York (1982). Van Reis, R. and S. Saksena, J. Membr. Sci., 129, 19 (1997). Van Vlasselaer, P., V.C. Palathumpat, G. Strang and M.J. Shapero, in Cell Separation Methods and Applications, D. Recktenwald and A. Radbruch (eds.), Marcel Dekker, New York (1998), p. 14. Vermeulen, T., Ind. Eng. Chem., 45, 1664 (1953). Vermeulen, T., L. Nady, J.M. Korchta, E. Ravoo and D. Howery, Ind. Eng. Chem. Proc. Des. Dev., 10(1), 91 (1971). Vermuelen, T., G. Klein and N.K. Hiester, in Chemical Engineers’ Handbook, J.H. Perry (ed.), McGraw-Hill, New York (1973), sect. 16. Vieth, W.R., J.M. Howell and J.H. Hsieh, J. Membr. Sci., 1, 177 (1976). Vilker, V.L., C.K. Colton and K.A. Smith, AIChE J., 27(4), 637 (1981). Vink, H., J. Chromatogr., 69, 237 (1972). Viovy, J-L., Rev. Mod. Phys., 72(3), 813 (2000). Von Hippel, A., Dielectrics and Waves, John Wiley, New York (1954). Von Stockar, U. and X-P. Lu, Ind. Eng. Chem. Res., 30, 1248 (1991). Vrentas, J.S., J.L. Duda and S.T. Hsieh, Ind. Eng. Chem. Prod. Res. Dev., 22, 326 (1983). Wadekar, V.V. and M.M. Sharma, J. Separ. Process Technol., 1, 1 (1981). Wagener, K., H.D. Freyer and B.A. Billal, Separ. Sci., 6(4), 483 (1971). Wakao, N. and T. Funazkri, Chem. Eng. Sci., 33, 1375 (1978). Walawender, W.P. and S.A. Stern, Separ. Sci., 7(5), 553 (1972). Wallace, R.M., Ind. Eng. Chem. Proc. Des. Dev., 6, 423 (1967). Walsh, A.J. and H.G. Monobouquette, J. Membr. Sci., 84, 107 (1993). Walter, J.E., J. Chem. Phys., 13(6), 229 (1945). Wang, M., S. Hou and J.F. Parcher, Anal. Chem., 78(4), 1242 (2006). Wankat, P.C., AIChE J., 23(6), 859 (1977). Wankat, P.C., Large Scale Adsorption and Chromatography, vols. I & II, CRC Press, Boca Raton, FL (1986). Wankat, P.C., Ind. Eng. Chem. Res., 26, 1579 (1987). Wankat, P.C., Equilibrium-Staged Separations, Elsevier, New York (1988), chap. 10. Wankat, P.C., Rate-Controlled Separations, Elsevier Applied Science, New York (1990). Wankat, P.C., Ind. Eng. Chem. Res., 32, 894 (1993). Wankat, P.C., Separation Process Engineering, 2nd edn., Prentice Hall, Upper Saddle River, NJ (2007), pp. 334–338, 649, 653. Wankat, P.C., J.C. Dore and W.C. Nelson, Separ. Purif. Meth., 4(2), 215 (1975). Ward, W.J., AIChE J., 16, 405 (1970). Ward, W.J. and W.L. Robb, Science, 156, 1481 (1967). Wark, K. and C.F. Warner, Air Pollution: Its Origin and Control, Harper and Row Publishers, New York (1976). Watson, J.H.P., J. Appl. Phys., 44(9), 4209 (1973). Wauters, C.N. and J. Winnick, AIChE J., 44(10), 2144 (1998).

875 Way, J.D. and R.D. Noble, in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.) Kluwer Academic, Boston, MA (2001), chap. 44. Way, J.D., R.D. Noble, T.M. Flynn and F.D. Sloan, J. Membr. Sci., 12, 239 (1982). Weaver, K. and C.E. Hamrin, Chem. Eng. Sci., 29, 1873 (1974). Wei, J. and M.J. Realff, AIChE J., 49(12), 3138 (2003). Weinstein, J.N. and F.B. Leitz, Science, 191, 557 (1976). Weiser, M.A.H., R.E. Apfel and E.A. Neppiras, Acoustica, 56, 114 (1984). Weller, S. and W.A. Steiner, J. Appl. Phys., 21, 279 (1950a). Weller, S. and W.A. Steiner, Chem. Eng. Progr., 46, 585 (1950b). Weaver, K. and C.E. Hamrin, Chem. Eng. Sci., 29, 1873 (1974). Werezak, G.N., Chem. Eng. Progr. Symp. Series 65(91), 6 (1969). Westerberg, A.W., Comput. Chem. Eng., 9(5), 421 (1985). Whitaker, S., Chem. Eng. Sci., 28, 139 (1973). White, D.H. and P.G. Barkley, Chem. Eng. Progr., Jan., 25 (1989). White, H.J., Industrial Electrostatic Precipitation, AddisonWesley, Reading, MA (1963). Whitney, R.P. and J.E. Vivian, Chem. Eng. Progr., 45, 323 (1949). Wick, G.L., Energy, 3, 95 (1978). Wickramasinghe, S.R., M.J. Semmens and E.L. Cussler, J. Membr. Sci., 84, 1 (1993). Wieme, R.J., in Chromatography: A Laboratory Handbook of Chromatographic and Electrophoretic Methods, E. Heftmann (ed.), Van Nostrand Reinhold, New York (1975), pp. 228–281. Wijmans, J.G. and R.W. Baker, J. Membr. Sci., 79, 101 (1993). Wijmans, J.G., A.L. Athayde, R. Daniels, J.H. Ly, H. D. Kamaruddin and I. Pinnau, J. Membr. Sci., 109, 135 (1996). Wilcox, W.R., in Fractional Soldification, M. Zief and W.R. Wilcox (eds.), vol. I, Marcel Dekker, New York (1967), chap. 3. Wilcox, W.R. and C.R. Wilke, AIChE J., 10, 160 (1964). Wiley, D.E and D.F. Fletcher, J. Membr. Sci., 211, 127 (2003). Wilhelm, R.H. and N.H. Sweed, Science, 159, 522 (1968). Wilhelm, R.H., A.W. Rice and A.R. Bendelius, Ind. Eng. Chem. Fund., 5, 141 (1966). Wilhelm, R.H., A.W. Rice, R.W. Rolke and N.H. Sweed, Ind. Eng. Chem. Fund., 7, 337 (1968). Wilke, C.R. and P. Chang, AIChE J., 1, 264 (1975). Williams, M.E., D. Bhattacharyya, R.J. Ray and S.B. McCray, in Membrane Handbook, W.S. Winston Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), chap. 24. Williams, P.S., S. Levin, T. Lenczycki and J.C. Giddings, Ind. Eng. Chem. Res., 31, 2172 (1992). Wilson, E.J. and C.J. Geankpolis, Ind. Eng. Chem. Fund., 5, 9 (1966). Wise, W.S. and D.F. Williams, “The principles of dissociation extraction,” Proc. Symp. Less Common Means of Separation, Birmingham, UK, Apr. 24–26, 1963, The Institution of Chemical Engineers, London (1963), pp. 112–118. Wolbert, D., B.-F. Ma, Y. Aurelle and J. Seureau, AIChE J., 41 (6), 1395 (1995). Wolf, D., J.L. Borowitz, A. Gabor and Y. Shraga, Ind. Eng. Chem. Fund., 15, 15 (1976). Wyatt, G.R. and S.S. Cohen, Nature, 170, 1072 (1952). Wynn, N.P., Chem. Eng. Progr., Mar., 52 (1992). Xu, Y., K.K. Sirkar, X-P. Dai and R.G. Luo, Biotechnol. Prog., 21, 590 (2005). Yanar, D.K. and B.A. Kwetkus, J. Electrostat., 35, 257 (1995). Yang, M.C. and E.L. Cussler, AIChE J., 32, 1910 (1986). Yang, Q. and N.M. Kocherginsky, J. Membr. Sci., 286, 301 (2006). Yang, R.T., Gas Separation by Adsorption Processes, Butterworths, Boston, MA (1987).

876 Yang, R.T., Adsorbents: Fundamentals and Applications, Wiley-Interscience, Hoboken, NJ (2003). Yang, Z.F., A.K. Guha and K.K. Sirkar, Ind. Eng. Chem. Res., 35(4), 1383 (1996a). Yang, Z.F., A.K. Guha and K.K. Sirkar, Ind. Eng. Chem. Res., 35(11), 4214 (1996b). Yasuda, H., A. Peterlin, C.K. Colton, K.A. Smith and E.W. Merrill, Makromol. Chem., 126, 177 (1969). Yih, C.S., Phys. Fluids, 11, 477 (1968). Yun, C.H., R. Prasad, A.K. Guha and K.K. Sirkar, Ind. Eng. Chem. Res., 32(6), 1186 (1993). Zabasajja, J. and R.F. Savinell, AIChE J., 35(5), 755 (1989). Zeman, L.J. and A.L. Zydney, Microfiltration and Ultrafiltration: Principles and Applications, Marcel Dekker, New York (1996).

References Zief M. and W.R. Wilcox, in Fractional Solidification, M. Zief and W.R. Wilcox (eds.), vol. I, Marcel Dekker, New York (1967), chap. 1. Zolandz, R.R. and G.K. Fleming, in Membrane Handbook, W.S.W. Ho and K.K. Sirkar (eds.), Kluwer Academic, Boston, MA (2001), p. 44. Zubritsky, E., Anal. Chem., 74, 23A–26A (2002). Zuiderweg, F.J. and A. Harmens, Chem. Eng. Sci., 9, 89 (1958). Zumstein, R.C. and R.W. Rousseau, Ind. Eng. Chem. Res., 28, 1226 (1989). Zwiebel, I., R.L. Gariepy and J.L. Schnitzer, AIChE J., 18(6), 1139 (1972).

Index absorbent 138 absorption 206, 281, 683 of SO2 281 absorption factor 70, 691 abundance ratio 24 acetic acid 293, 326, 342 acetone–chloroform system 211 acid–base equilibrium 302 acid–base reactions 280 acidic extractants 295 acidic extracting agent 331 acoustic contrast factor 260 acoustic force field 260 acoustic radiation force 86, 260 activated carbon 488 activated charcoal 149 activation energy 269 activity 84 activity coefficient 84, 206, 217, 433 adductive crystallization 235 adducts 235 adsorbate 135, 148 adsorbent 148 adsorbent bed regeneration 504 adsorption 148 adsorption isotherm 225 adsorption zone 497 affinity adsorption 547 Cloeth–Streat contactor 754 countercurrent adsorption 754 countercurrent fluid–solid adsorber 754 expanded bed adsorption 550 gas-fluidized multistage countercurrent adsorber 754 heatless adsorption 511 height of countercurrent adsorber 756 ideal adsorbed solution theory 224 operating line for countercurrent adsorption 756 pressure-swing adsorption 505, 511 scale up or scale down 541 silica gel adsorber 758 simulated moving bed (SMB) 754, 760 switching time in SMB 760 thermal-swing adsorption 505 adsorption isotherm 225 aerosol 90, 386 aerosol filtration 77 affinity adsorption 547 affinity binding 321 affinity chromatography 547, 851

air stripping 208 alanine 301 albumin 427, 471 Aliquat 85, 317 α-chymotrypsinogen 198, 427 α-lactalbumin 93 alumina particles 260 ambipolar diffusion 118, 194 amino acids 292, 301 ammonia 340 ammonium sulfate 244 amorphous polymer 140 amoxicillin 305 amphiprotic species 292 ampholyte 253 amphoteric ion exchanger 156 amphoteric species 301 ampicillin 305 analytical cut size 62 angular velocity 78, 249 aniline 324 anion exchange 155 anion exchange membrane 764 anion exchange resins 152, 230 anion exchangers 295 anthracene 197 antibiotics 290, 736, 742 production of 848 antibodies 259, 547, 618 antigens 617 anti-polyelectrolyte effect 242 antisolvent 147 Antoine equation 197, 209 apoferritin 427 apparent cut 48 apparent heads separation factor 48 aqueous phase control 191 aqueous two-phase systems 221 Arrhenius relation 269 artificial kidney 262 asphaltenes 260 asymmetric glassy membrane 444 rubbery coating 444 asymmetric membrane 181, 420, 556 asymmetric separator 44 atom fraction 24 atomic vapor laser isotope separation 605 laser excitation 605 autopyrolysis constant 303 axial dispersion 363 axial dispersion coefficient 111, 356, 701

back extraction 316, 742 bacteria 249 bacteriophage 251 band broadening 123, 363 barrer 178 batch crystallization supersaturation balance 456, 458 cooling curve 457 batch dialysis 262, 323 batch distillation 398–9 batch extraction 463 batch stirred tank separator 372 batch ultrafiltration 472 benzene 21, 23 benzene-cyclohexane 149 benzene/thiophene 237 benzene-toluene system 210, 714 benzoic acid 23 BET (Brunauer–Emmett–Teller)-type isotherm 150 binary diffusion coefficient 92 binary system 25–6 binodal curve 220 bioseparations 1, 9, 847 baker’s yeast 850 bovine growth hormone 849 Chinese hamster ovary cell proteins 851 combined microfiltration/adsorption chromatography 850 downstream processing 847 extracellular products 847 fractional recovery 850 human insulin 850 intracellular products 847 low-alcohol beer 850 monoclonal antibodies (mAbs) 851 polishing step 849 product isolation 848 removal of insolubles 848 volatile bioproducts 850 Biot number 166 birth and death processes 371 Blake–Kozeny equation 349, 415 blowdown 514 blue dextran 546 Boltzmann’s constant 90 Bond number 352, 355 boundary layer thickness 408, 575 bound water 412 bovine serum albumin 93, 253, 426, 574, 764 brackish water 199

878 breakthrough calculation 500 breakthrough concentration 497 breakthrough curve 508 breakthrough pressure 190 brine 21 Brownian motion 89 Brownian motion force 85 bubble point 420 bubble point calculation 212, 214 bubble point line 210 bubble point pressure 420 Buckingham’s pi theorem 109 buffer exchange 545, 761 bulk concentration 107 bulk density of adsorbent bed 490 bulk displacement 76–7 bulk domain 135 bulk flow due to capillarity 352 due to centrifugal force 354 due to drag flow 355 due to electroosmotic flow 353 gravity induced 350 hydrostatic pressure induced 348 parallel to the force 372 due to surface tension gradient 354 of two regions perpendicular to the force(s) 670 bulk flow correction factor 104, 693 bulk motion perpendicular to the force 485 bulk phases 133, 231 buoyancy force 78 cake 413 porosity 415 thickness 415 capacitive deionization 520 capacity factor 244 capillary array electrophoresis 384 capillary condensation 150 capillary electrochromatography 554, 788 capillary electrophoresis 378 capillary pressure 352 capillary zone electrophoresis 378 carbamate hydrolysis 283 carbon molecular sieves 488, 512 carbon number 24, 139 carboxylic acid 290 carrier species 326 cascades 3, 812 countercurrent 812 crosscurrent 819 flow rates between stages 817 heads separation factor 815 heavy fraction (tails) 812 ideal 812, 814 light fraction (heads) 812 multicomponent mixture separation via distillation 822 no-mixing condition 814 number of stages 816 enriching section 816 stripping section 816 recrystallization 821 series 822 single-entry separator stage 812

Index squared off 812 stage separation factor 815 tapered 812, 818 crossflow membrane modules 818 ternary nonazeotropic mixture 822 two-dimensional 821 uranium isotope separation 818 CaSO4 266 categories of separation processes 1 cation exchange 154 cation exchange membrane 264 cation exchange resins 152 cation exchangers 299 cell model 389 cellulose acetate membrane 174, 441, 559 cellulose nitrate membrane 420 centrifugal elutriation 384 centrifugal equilibrium 250 centrifugal filters 354 centrifugal force 3, 88, 250 centrifugal force field 77–8, 246, 596, 618 centrifugal force field potential 78 centrifuge 246 centrifuge effect 620 cyclone separator 619, 627 disk centrifuge 619, 624 gas centrifuge 246, 775 tubular bowl centrifuge 619 chaotropic salts 245 charge equality in an ionic gas 118 charge on a particle 608 chelating extractant 295, 318 chelation 234, 295 chemical equilibrium 83, 128, 281 chemical potential 83 chemical potential force 84 chemical potential gradient 77, 83–4 chemical potential gradient based force 3, 85, 390 chemical reactions 1, 280 chemical separations 1, 8 chemisorption 149 Chinese hamster ovary cell proteins 851 chirally selective complexing agent 323 chromatograms 67 chromatographic processes 66, 527, 546, 787 affinity chromatography 547 capacity factor 543 capillary electrochromatography 788 continuous annular chromatograph 794 continuous chromatographic separation 794 continuous gas chromatography 797 crossflow magnetically stabilized bed chromatograph 797 displacement chromatography 549 elution chromatography 527 gas chromatography 533 gas–liquid chromatography 533 gas–solid chromatography 533 Gaussian profile 531, 537 gel chromatography 544 gel permeation chromatography 544

gradient elution 543 height of a theoretical plate 537 HETP 538 hydrophobic interaction chromatography 244 immobilized metal affinity chromatography 549 ion exchange chromatography 542 isocratic mode of operation 543 liquid–liquid chromatography 549 liquid–solid adsorption based 528 membrane chromatography 548 net retention volume 530, 534 number of plates 538 paper or planar chromatography 532 plate height 538–9 Poisson distribution function 537 pseudostationary particle phase 788 resolution 529 retention time 529 retention volume 529, 545 reversed-phase LLC 532 rotating annular chromatograph 796 size exclusion chromatography 544 stationary phases (GLC) 534 theoretical plate 535 Van Deemter equation 539 chromatographic separations 66, 527 chromium 332 chymotrypsin 308 Cibacorn blue 321 clarification 586 classification of separation processes 6 clathrate 236 clathrate compound 236 clathration 234 Clausius–Clapeyron equation 195, 834 Clausius–Mossoti function 81, 83, 602 closed system 19, 29 close separation 27, 45, 50 c.m.c. 307 CO–O2 225, 274 CO2 282, 313 CO2 ionization in water 282 cocurrent bulk motion of two phases or regions 782 chromatographic separations 785 chromatography 787 cocurrent flow stages in a countercurrent cascade 782 cocurrent gas permeation 791 cocurrent hemodialyzer 789 equations of change 783 micellar electrokinetic chromatography 785 micellar phase capacity factor 786 microfluidic devices 783 migration velocity 786 multicomponent separation capability 783 operating line 783 particle separation in cocurrent gas–liquid flow 788 Venturi scrubber 788 cocurrent flow 3, 50 coefficient of volume expansion 351 coion 132, 154, 508

Index collector–colligend equilibrium 307 collectors 159 colligend 306 colloidal solution 19 column breakthrough curve 494 combined diffusion–viscous flow model 182, 423 common separation sequences 847 bioseparations 847 chemical and petrochemical industries 853 hydrometallurgical processes 855 separation sequences for water treatment 851 complexation 320 complexation reactions 280 complexing agent 235 component cut 44, 51 composite membranes 179, 181, 420, 443, 556 ideal separation factor 443 PRISM membrane 444 composition 19, 24 compressible flow of gas 350 conalbumin 253 concentrate 188 concentrate chamber 187 concentration front propagation velocity 492 concentration polarization 175, 188, 428, 570 concentration potential 843 concentration processes 1, 35 concentration pulse 119 concentration wave velocity 492, 508, 673, 683 conduction drying 411 constant filtration flux 416 constant-level batch distillation 398 constant-pattern behavior 504 constant total molar overflow 53, 711 contact angle 159, 352 continuous chemical mixture 62, 130, 138, 195, 217, 394 continuous chromatographic separations 794 continuous diafiltration 471, 572 continuous electrodeionization 852–3 continuous membrane column 52, 676 continuous stirred tank crystallizer 446 continuous stirred tank separator (CSTS) 3, 346, 348, 367, 371, 445 continuous well-stirred extractor 445 continuous well-stirred membrane gas separator 446 continuum hypothesis 358 continuum level description 347 controlled progressive freezing 409 convective diffusion 371 convective dispersion 363 convective hindrance factor 100, 181 cooling crystallization 457, 821 coordination number 321 corona-discharge reactor 605 corona discharge 605

879 deposition-type reactor 605 electron attachment 605 ion mobility spectrometry 606 sweep-out-type reactor 605 ultrahigh gas purification 606 COS 316 cosolute 135 cosolvent 135 cotransport 332 Coulomb’s law 82 counteracting chromatographic electrophoresis 551 countercurrent absorber 65 countercurrent bulk flow of two phases or regions 3, 670 concentration wave velocity 683 countercurrent two-region/twophase system 677 enriching section 681 equation for the operating line 680 equations of change of concentration 677 multicomponent separation capability 682 stripping section 681 system type (1) 671 system type (2) 671, 673, 675, 680 system type (3) 671, 674–5, 681 system type (4) 671, 675 countercurrent electrophoresis 377 countercurrent flow 50 counterion 79, 132, 152, 168, 508 countertransport 329 coupling 116 creatinine 262 creeping flow 361 criteria for equilibrium 128–30 criterion for liquid immiscibility 221 critical micelle concentration 307, 384 critical pressure 158 critical settling velocity 374 critical size 146 critical temperature 140, 158 cross coefficient 116 crossflow assumption 331, 556 crossflow membrane separations 3, 555 crossflow gas permeation 555 crossflow microfiltration 555, 575 crossflow reverse osmosis 555, 562 crossflow ultrafiltration 555, 568 crossflow of two bulk phases 3, 794 continuous annular chromatograph 794 continuous chromatographic separation 794 continuous gas chromatography 797 continuous surface chromatograph 797 crossflow magnetically stabilized bed chromatography 797 rotating annular chromatograph 796 crown ethers 237 cryogenic distillation of air 841 crystallization 145 column crystallization 675 crystal growth rate 451

crystal number density function 446 crystal shape factor 449 crystal size distribution function 448 melt crystallization 675 CsCl 250, 252 cumene 396 Cunningham correction factor 387, 594 cup-mixing concentration 107 cuprophan 264 cuprous chloride 326 current density 97, 188 curved interface 136 cut 44, 47 cut point 531 cut size 62 cycling zone adsorption 525 cyclodextrin 238, 382 cyclone dust separator 627 cyclone mist eliminator 667 electrocyclone 630 particle collection efficiency 630–1 reverse-flow cyclone 627 vortex finder 627 cylindrical pore 415 cytochrome c 198 Czochralski technique 410 Darcy’s law 181, 349, 414, 421 dead-end cake filtration 413 dead-end filtration 413 Debye length 79, 353 decahydronaphthalene 294 decalin 294, 326 decantation 77 decolorization 197 decontamination factor 28, 42 deep bed filtration 77 ΔL law of crystal growth 165 dendrimer polyamidoamine 328 denitrated cellulose membrane 184 density gradient 245 density-gradient sedimentation 250 deoxyribonucleic acid (DNA) 250–1 depth filter 386 depth filtration 386 desalination 851–2 brackish waters 851 seawater 851 desalination ratio 28, 42 desalting protein solution 545, 762 descriptions of separation 19 desorption 208 desorption processes 165 deuterium 287 DeVault equation 491 dew-point calculation 212 dew-point line 211 dextran 221 dialysate 262, 323 dialysis 262, 323, 675 buffer exchange 761 clearance 763 cocurrent hemodialyzer 789 countercurrent dialysis (CCD) 761 countercurrent dialyzer 762 dialysance 763 dialysate 262, 323

880 dialysis (cont.) extraction ratio 763 hemodialysis 761 solute transfer efficiency 763, 789 dialytic battery 843 diatomic gases 139, 180 dicarboxylic acids 291 dielectric constant 604 relative 79 dielectric spherical particle 88 dielectrophoresis 81, 596 Clausius–Mossoti function 602 continuous 602 dielectrophoretic force 602 negative 602 particle sorting 603 particle velocity 603 positive 602 separation of polystyrene particles 603 diethylamine 163 Di-2-ethylhexyl phosphoric acid 296, 300, 340 differential distillation 397 differential extraction 400 diffuse cut off membrane 425 diffusion coefficients 85, 178, 269 in binary gaseous mixtures 97 of gases through a polymeric film 204 proteins and other macromolecules in pure water 203 selected ions in water 203 solute i in a dilute liquid solution 202–3 diffusion equation 121, 179 diffusion of A through stagnant B 105, 107–8 diffusion path 104 diffusion potential 97, 99 diffusional film resistances 333 diffusiophoresis 85 diffusive ultrafilter 421 diffusive ultrafiltration 421 diffusivity selectivity 267–8, 270 diluate 188 diluate chamber 187 diluent solvent 299 dilute solutions 844 separation from 845 dilution factor (DF) 472, 572 dimensionless groups 109 dimerization 293 dimethylsulfoxide 221 dipole 81 dipole moment 81 direct mechanical conveying 77, 401 discontinuous diafiltration 472 discontinuous potential profile 127 disk centrifuge 74, 624 dispersion 123, 190, 363, 464 dispersive system 318 dispersive wave front 496 displacement 76 displacement purge gas 507 displacer 507 displacer-based desorption 508 dissociation extraction 292 distillation 4, 25, 709

Index AIChE tray-efficiency prediction method 803 batch distillation 735 benzene–toluene system 714, 718, 734 boilup ratio 712 column diameter 727 constant molar overflow 679, 711 continuous vapor–liquid contact 731 crossflow plate 799 distillation column 39, 52, 65, 709 enriching section 711, 732, 734 entrainment 727 external reflux ratio 711 feed plate 713 Fenske equation 717, 729–30 Fenske–Underwood–Gilliland method 731 flash calculation 212 flash drum 391 flash vaporization 45, 390 flooding 727 flow parameter 727 Gilliland correlation 729, 731 heavy key 729 heavy water distillation 287, 814 height of a transfer unit 733 Kremser equation 720 Lewis relation for E0 725 light key 729 McCabe–Thiele method 711, 714 minimum reflux 715, 717, 730 minimum reflux ratio 730 multicomponent mixture 729 multicomponent mixture separation via distillation 822 Murphree point efficiency 801 Murphree vapor efficiency 723, 799 number of transfer units 733 O’Connell’s correlation 723 Oldershaw column 804 operating lines 680, 711–12, 735 overall efficiency 723 packed tower 731 partial condenser 718 partial reboiler 712 plate efficiency 723 point efficiency 800 Ponchon–Savarit method 711 q-line 714 rate-based approach for modeling distillation 728 reactive entrainer 286 reboil ratio 712 reflux ratio 53, 711 relative volatility 718 side stream 721 stage efficiency 723 stripping section 712, 732, 734 ternary nonazeotropic mixture 822 total reboiler 712 total reflux 674, 715 tray spacing 727 two feed streams 721

Type (3) system 711 Underwood equation 718, 729–30 vapor–liquid equilibrium (VLE) 138, 209, 286 VLE diagram 734 weeping 727 xiv–xil diagram 709 y–x diagram 709 distribution coefficient 25, 29, 140, 152, 182, 282, 334, 739 distribution ratio 25, 45, 51, 302 D,L-DOPA 305 DNA 251 dominant crystal size 449 Donnan dialysis 189, 262, 264, 308 Donnan equilibrium 308 Donnan potential 132–3, 153 double-entry separator 39, 50, 53 downstream processing 847 antibiotic production 848 baker’s yeast 850 extracellular products 847 intracellular products 847 monoclonal antibodies (mAbs) 851 polishing step 849 product isolation 848 removal of insolubles 848 D-phenylalanine 323 drag coefficient 89, 375 intermediate range 375 Newton’s law range 375 drag flow 355 drag force 89, 117, 368 drift tube 195 drift velocity 117 drop size number density function 467 drying 411 D2EHPA 296, 300, 340 dual sorption–dual transport 179 dual sorption mode 140 dust precipitation 609 E. Coli cells 94, 622 Edeleanu process 235 EDTA 301, 308 effective binary diffusivity of i through the mixture 114 effective diffusion coefficient 100 effective distribution coefficient 282, 289 effective ionic mobility 382 effective partition coefficient 291, 408 effective solute diffusivity in the pore 182 8-quinolinol 297, 341 electrical conductivity 98 electrical double layer 80 electrical field 79, 88, 253, 260 electrical field strength 255 electrical force 88 electrical force field 3, 79, 596 electrical permittivity of vacuum 82 electrical potential gradient 97 electrochemical cell 335 electrochemical gas separator 335 electrochemical potential 132, 337

Index electrochemical valence 79, 132 electrochromatography 554, 788 electrode reactions 336 electrodialysis 188, 764 anion exchange membrane 764 anode chamber 764 cation exchange membrane 264, 764 cell pair 764 continuous electrodeionization (CEDI) 852 countercurrent electrodialysis 764 current utilization factor 766 desalination ratio 766 diluate chamber 765 membrane stack 764 solution resistivity 766 electrogravitation 352 electrokinetic force in the double layer 82 electrokinetic potential (zeta potential) 80, 82, 192 electron holes 338 electron transport 338 electroneutrality condition 132, 135, 168, 187, 266 electroosmotic flow 353, 378 electroosmotic mobility 380 electrophoresis 81, 596 continuous free-flow electrophoresis (CFE) 596 electrophoretic mobility 598 electrophoretic Péclet number 599 electrophoretic velocity 599 Gaussian concentration profile 598 Joule heating 597, 601 Philpot model 598 thin-film CFE 596 electrophoretic motion 6 electrosorption 519 electrostatic precipitation 596 electrostatic precipitator 607 charge on a particle 608 Cottrell precipitator 608 Deutsch equation 610 diffusion charging 608 dry 607 dust precipitation 609 field charging 608 grade efficiency function 609 laminar gas flow 610 migration velocity 608 percent collection efficiency 611 total efficiency 611 turbulent flow 610 wet 607 electrostatic scalar potential 79 electrostatic separation 389 electrostatic separator 612 drum-type 613, 617 free-fall 612 mineral separation 612 particle charge to mass ratio 616 particle recoveries 612 particle trajectories 612, 615 particle-charging device 611 plastics separation 612 polyethylene 611

881 polystyrene 611 polyvinylchloride 611 triboelectric charging 611 elution 495 elutriation 375 embryos 146 emulsion liquid membrane 767 enantiomeric excess 36 enantiomers 36, 308 enantioselective micelle 322 energy required for separation 827 dialytic battery 843 energy efficiency 833 evaporation of water from saline water 828 multiple-effect evaporation 836 multistage flash evaporation 838 vapor-compression evaporation 838 free energy of mixing 843 minimum energy required for separation 827 by adsorption 835 by distillation 832 by evaporation 834 by membrane gas permeation 831 by solvent extraction 834 minimum heat supplied at the reboiler 833 net work consumption 831 in distillation 833 recovery of water by reverse osmosis 828 reducing energy required for separation 836 heat pump 841 Linde double-column 841 cryogenic distillation of air 841 multieffect distillation 840 backward feed 841 forward feed 841 energy separating agent 206 energy-separating-agent (ESA) processes 35 enhancement factor 311, 314 enriched O2 441 enrichment factor 26, 34, 43 enrichment processes 35 enrichment ratio 26 entropy 129 enzymatic reactions 308 enzyme 322, 531 equal-settling particles 375 equation of motion 361 equation of particle motion 368 equation of change 358 for mass concentration 358, 361 for molar concentration 359–60 for a particle population 368 equation of continuity for a mixture 360 equilibrium constant 232, 282, 296 equilibrium limit 347 equilibrium nondispersive operation 492, 508

equilibrium nondispersive PSA model 512 equilibrium ratio 43, 50, 138, 161 equilibrium separation 512 equilibrium separation processes 205, 347 equilibrium solubility 303 equimolar counterdiffusion 103, 106–8, 693 equiprobable size 62, 622, 630 equivalent conductance 98 equivalent ionic conductances of selected ions 204 escaping tendency 83 esterification process 287 ethanol-isopropanol 399 ethylbenzene 395 ethyl cellulose 324 ethylene 158, 326 ethylene glycol 221 Eulerian approach 347 eutectic point 222 exchanging buffer solutions 545 expanded bed adsorption 550 extended Nernst–Planck equation 97 extent of purification 38 extent of reaction 280–1 extent of recovery 469 of species i in region j 25, 44 extent of separation 27, 32, 35, 44, 47, 51, 53, 65, 68, 280, 402, 573 with reaction 280 external force based separations 346, 373 external force field 2, 128, 205, 245 external potential field 128 extracellular products 847 extract 23, 220, 739 phase 463 extraction enhancement factor 317 extraction factor 466 extraction of metals 294 extripping 282 facilitated transport 326 facilitation ratio 327 Fanning friction factor 349 Faraday’s constant 79 fast reaction 311, 313 favorable isotherm 493 Faxen’s expression 100, 183–4 feed introduction mode 356 ferromagnetic material 82–3 fibrous filtration 586 Fick’s first law of diffusion 92, 118, 176–7 Fick’s law diffusivity 177 field-flow fractionation (FFF) 640 characteristic thickness of profile 640 concentration profile 640 electrical FFF 644, 646 electrophoretic mobility 646 electropolarization chromatography 647 flow FFF 644

882 field-flow fractionation (FFF) (cont.) force-field induced migration velocity 640 lift hyperlayer FFF 646 macromolecules/colloids/particles 640 normal elution order 646 normal mode FFF 646 retention parameter for species i 643 retention ratio for species i 643 retention time of species i 643 sedimentation FFF 644 steric elution order 646 steric FFF 646 thermal FFF 644–5 film pressure 135 film theory 108, 310 film thickness 313 film transfer coefficient 317 filter aid 414 filtration 412 first ionization 291 fixed-bed processes 487 adsorption/desorption 487 scale up or scale down 541 fixed carrier membranes 337 fixed charges 132 flash calculation 212 flash devolatilizer 395 flash drum 391 flash expansion desalination 45 flash vaporization 45, 390 Flory–Huggins theory 177, 395 Flory interaction parameter 177, 395 flotation 159 flow cytometry 596, 617 antibodies 618 antigens 617 cell sorting 596 charged deflection plates 617 fluorescence-activated cell sorting 618 separation of cells 617 sheath flow 617 fluid density 78 fluid–solid systems 148 Flurbiprofen 341 flux of cations 188 flux ratio 104 foam fractionation 70 focusing 205 force potential profile 119 fraction 20–1 fractional consumption 283 fractional crystallization 403 fractional extraction 218 fractional penetration 73 fractional solidification 403 fractional water recovery 432 free convection 350 free energy of mixing 843 free water 412 free-flow electrophoresis 596 free-flow magnetophoresis 649 freeze-concentration 28 freeze-drying 411 Freundlich isotherm 150

Index FricDiff 782 frictional coefficient 89, 172, 256 of solute 123 frictional force 89 frontal development 500 froth flotation 245, 538 fruit juice concentrate 28 fugacity 85 of species i 137 fugacity coefficient 137–8 gallium arsenide 223 Γ distribution function 217, 394 gas absorption/stripping 206, 673, 683 to absorb SO2 from air 698 absorbent 138 absorption factor 70, 691 capacity factor 685 channeling 684 chemical absorber 702 operating lines and equilibrium curves 705 column with axial diffusion/ dispersion 699 concentrated gas stream 706 crossflow plate 799 Danckwerts boundary conditions 700 deoxygenation of ultrapure water 695 devices used in countercurrent gas–liquid separation 684 dilute species in a gas stream 688 dispersion based devices 684 equimolar counterdiffusion 693 fast reaction 311, 313 flooding 684 flooding correlations for dispersive devices 684 flow parameter 685, 727 height of a transfer unit (HTU) 691 Henry’s law 691 HIGEE 687 hindered amine 283 hot potassium carbonate process 705 instantaneous reaction 703 Kremser equation 707–8 loading 684 membrane contactors 688 multistage devices 706 nondispersive gas–liquid absorption/ stripping devices 687 nonequimolar counterdiffusion 693 number of ideal stages required 707 number of transfer units (NTU) 691 operating line for an absorber 688 operating line for a stripper 690 packed towers 684 performance of countercurrent devices 684 plate towers 684 porous hydrophobic hollow fiber membrane 697 porous membrane contactor–stripper 695 pressure drop at the flooding point 686 process intensification 687

random packings 684 Raoult’s law 692 Raschig rings 686 rotating packed bed 687 scheme for absorption stripping 706 Sherwood–Eckert generalized ΔP correlation chart 684 stripping factor 708 structured packings 684 gas adsorption 135 gas centrifuge 246 CGC column 778 countercurrent gas centrifuge (CGC) 775 enriching section 779 flow number 780 flow pattern efficiency 780 Groth design 777 separation of uranium isotope U235F6 from U238F6 781 stripping section 779 total reflux 780 Zippe’s machine 777 gas diffusion electrode 335 gas film coefficient 108 gas hydrate 236 gas permeation 266, 675 asymmetric membrane 556 cellulose acetate membrane 559 cocurrent gas permeation 791 composite membrane 556 continuous membrane column 675 countercurrent gas permeation 771 crossflow 555 crossflow configuration 556 crossflow relation 556 dehydration of air 561 design problem 772, 793 Hagen–Poiseuille equation 775 hollow fiber permeator 556 membrane area 561, 772, 792 Naylor–Backer approach 558 nitrogen-enriched air 559 oxygen-enriched air 560 pressure ratio 559, 775 rating problem 772, 793 separation of CO2 561 separation of H2 561, 772, 792 stage cut 774 symmetrical hollow fiber membrane 772, 791 gas separation 632 separation nozzle 632 gas transport through porous materials/membranes 100 gas–liquid chromatography 207 gas–liquid equilibrium 137, 281 gas–liquid interface 189 gas–liquid system 159 gas–membrane equilibrium 140 gas-phase film coefficient 311 gas-phase resistance 190 gas–solid adsorption 148, 223 gas–solid adsorption isotherm 150 gas–solid equilibrium 149 gas–solid interface 135 gaseous diffusion 100

Index Gaussian distribution 54, 67, 122, 217 Gaussian profile 251 gel permeation chromatography 546 gel phase 242 gel polarization 423, 571 gels 64, 143 general dynamic equation 371 gene therapy 249 geometrical partitioning 183, 197 geometrical partitioning factor 141–2 Gibbs adsorption isotherm 135, 149, 306 Gibbs equation 135 Gibbs free energy 129 Gibbs phase rule 209 Gibbs–Duhem equation 87, 134 glassy membrane 140 glutamic acid 301, 305 glycine 301 grade efficiency 59, 61, 609 granular activated carbon 488, 590 granular filtration 386, 586 average grain diameter 593 backwash 586 Brownian motion 594 capillaric model 586 clarification 586 constricted tube model 591 electrostatic forces 595 filter coefficient 589 granular activated carbon 590 gravitational force 594 Happel’s cell model 593 interception 595 logarithmic law 589 mass-transfer coefficient 594 overall filter efficiency 591 particle collection efficiency 591, 594 particle trajectory analysis 593 sand filters 586 spherical collector model 593 unit bed elements 586 volume concentration of the particles 588 graphitized carbon membrane 186 Grashof number 110 gravitational field 77, 192, 246 gravitational force 77, 87, 256, 596, 634 dust collection efficiency 635 grade efficiency function 635 gravity based dust settling chamber 634, 668 inclined settlers 638 knockout drums 637 separation of droplets from a gas/ vapor 637 gravitational potential 77 growth rate dispersion 455 growth rate of crystal 164–5, 450 guest 235 Hagen–Poiseuille equation 348 heads separation factor 44, 50, 815 heatless adsorption 511 heatless fractionation 511 heavy fraction 21, 50 heavy key component 34, 64

883 heavy water 287 distillation of 287, 814 height of a transfer unit (HTU) 465, 691, 740 helium–methane pair 272 hemodialysis 201, 761, 789 hemodialyzer 262, 761 Henry’s law 138, 140, 178, 190, 691 Henry’s law constant 131, 137, 160, 206–7, 268, 281, 334 Henry’s law species 179 herpes simplex virus 249 high-density particles 259 high-performance capillary electrophoresis 378 hindered amine 283 hindered diffusion 183 hindered settling 258 hindrance factor 100 hollow fiber membrane 557, 574, 689, 697, 761, 770 hollow fiber permeator 556 homogeneous membrane 420 homogeneous nucleation 146 cluster 146 critical size 146 embryos 146 kinetic unit 146 nuclei 146 nucleus 146 host 235 hot potassium carbonate process 283, 705 hydrate formers 237 hydration layer 147 hydraulic jig 376 hydraulic permeability 349, 431 hydraulic radius 113, 349, 570 hydraulic-mean pore radius 181 hydrocyclone 46, 54, 631 oil–water emulsions 631 solid–liquid separation 631 hydrodynamic viscosity based radius 143 hydrogel particles 546 hydrogels 143, 544 hydrogen 201, 442, 444, 512, 517, 534 H2 purification 512, 517 H2S 284, 313, 328, 336 hydrolysis constants 296 hydrophobic effect 147 hydrophobic interaction 235, 244 hydrophobic interaction chromatography 244 hydrophobic membrane 189–90 hydrophobic porous adsorbent 152 hydrosols 386, 586 ice crystals 21 ideal adsorbed phase model 226 ideal adsorbed solution theory 224 ideal fluid 361 ideal gas law 31 ideal gas mixture 95 ideal selectivity 270 ideal separation factor 439 ideal solution 84

ideal stage 205, 398 IgG 427 immiscible phases 21 immobilized metal affinity chromatography 549 immunoglobulins 547 imperfect separation 20 impurity 20 impurity concentration profile 406 impurity mass fraction profile 407 impurity ratios 26, 34, 69 inclined electrosettler 260 inclined settler 259–60, 638 lamella settler 638 inclusion compounds 235, 237 incompressible flow 360, 366 indices of separation 24, 29–30 individual phase transfer coefficient 161 inertial force 77, 86, 368 inertial impaction 386 inert purge 511 infinite dilution activity coefficient 208 infinite dilution equilibrium ratio 208 infinite dilution standard state 131, 139 infinite separation factor 26 instantaneous concentration profile 122 instantaneous reactions 311, 313, 317, 703 intensive variables 209 interception 386, 389 interfacial adsorption 147, 223 interfacial concentrations 162 interfacial equilibrium relation 147 interfacial phase 135 interfacial reaction 331 interfacial region 133, 135 interfacial tension 133, 147, 223, 354 internal chemical potential 83 internal force 77 internal particle velocity 369 internal staging 426 interphase transport 2, 366 interstitial fluid velocity 367, 489 intraparticle transport resistance 165 intrinsic distribution coefficient 289 intrinsic phase average 365 inulin 427 inviscid fluid 361 ion conduction 338 ion exchange bed/column 508 ion exchange capacity 154 ion exchange equilibrium constant 155, 230 ion exchange equilibrium for proteins 156, 229 steric mass action (SMA) 156 ion exchange membrane 186, 264, 336 ion exchange processes 154, 228 ion exchange resin particles 132 ion exchanger 152, 228 ion exchanger selectivity 228 ion mobility spectrometry 118, 606, 665 ion pair 296, 332 formation of 295, 318 ion product 303 ion transport membrane 338 ionic binding 235 ionic equivalent conductance 98

884 ionic mobility 98, 377, 383, 598 in the gas phase 118 ionic strength 79, 147, 207 ionic surface active solute 306 ionic surfactants 241 ionization constant 289 ionization product 324 ionization reactions 280 irreversible second-order reaction 317 irreversible thermodynamics 116, 172 isoamyl acetate 342 isoelectric band 304 isoelectric focusing 253 isoelectric pH 253 isoelectric point 229, 305 isopropyl ether–isopropyl alcohol 400 isopycnic sedimentation 250 isotachophoresis 255 isothermal flash calculations 392 isotope 24, 28 isotope enrichment process 288 isotope exchange reaction 288 isotope separation methods 247 isotope separation plants 49 isotopic mixture 33, 287 isotopic ratios 277 JD factor 110 Kelvin effect 137 Kelvin equation 137, 150 kinetic order 452 kinetic separation 512 kinetic sieving diameter 268, 269 knockout drums 637, 668 Knudsen diffusion 117 Knudsen diffusion flux 101, 185 Knudsen diffusivity 101, 185 Knudsen flow 100 Knudsen number 117 kosmotropic salt 245 Kremser equation 707–8, 742 Lagrangian approach 347, 368 laminar flow 106, 348, 596, 610 Langmuir adsorption isotherm 148, 230 Langmuir ion exchange isotherm 510 Langmuir isotherm 140, 150 Langmuir species 179 laser isotope separation 596, 605 law of conservation of mass 25 leaching 145, 223 leading edge of a concentration profile 127 length of a transfer unit (LTU) 465 length of the unused bed (LUB) 500 Lennard–Jones force constant 268 leucine 301 Leveque solution 110, 568, 741 lever rule 33 l-hexene 326 lift force 86 light fraction 21, 46, 50 light key component 64 limiting current density 188 limiting streamline 386 limiting trajectory 387–8, 592, 663

Index linear driving force approximation 166, 490 linked footprints of a separation process 3 lipid bilayer 234 lipid particles 261 liposome 232, 234 liquid clathrates 240 liquid film coefficient 108, 160, 162, 190, 311 liquid fraction 50 liquid membrane 324 countercurrent ELM extractor 769 countercurrent liquid membrane separation 767 emulsion liquid membrane (ELM) 767 hollow fiber contained liquid membrane (HFCLM) 767 immobilized liquid membrane (ILM) 767 pervaporation 770 supported liquid membrane (SLM) 767 liquid metal 139 liquid permeation 170, 176 liquid solution 19 liquid–liquid equilibrium 139, 217, 289 liquid–liquid extraction 217, 316, 736 liquid–liquid interface 190, 315 liquid–liquid systems 400 liquid–membrane equilibrium 141 liquid–porous sorbent equilibrium 141 liquid–solid adsorption 151, 226 liquid–solid equilibrium 144, 222, 403 liquidus 144 L-isoleucine 304 local domain 135 local equilibrium 491, 497 log reduction value (LRV) 73, 419 logarithmic concentration difference 740, 763, 790 long-chain alkyl amine 295, 300, 318 L-tryptophan 323 lysine 301 lysozyme 197, 253 McCabe–Thiele method 711, 714 McCabe’s ΔL law of crystal growth 447 macrocyclic polyethers 237 macroion 229 macromolecular solution 19, 64 macromolecules 63, 77, 143, 248, 250, 546, 547 macropores 180 macroscopic particles 77 macrosolute 420 macrosolute flux 421 magma density 452 magnetic field 82, 648 magnetic field strength 82, 649 magnetic flux density 83 magnetic force 193, 648 magnetic force field 3, 596, 648 continuous immunogenic sorting 650 free-flow magnetophoresis 648–9 HGMS filter 653

high gradient magnetic separation (HGMS) 648, 650 limiting trajectory 653 magnetic field force strength 649 magnetic migration velocity 649, 653 magnetic velocity 649 magnetophoretic mobility 649 paramagnetic particles 649 particle trajectory 649, 651 magnetic induction 82 magnetic permeability 193, 649 of vacuum 82 magnetic potential 82 magnetic susceptibility 82, 649 magnetophoresis 83 Marangoni effect 355 Mark–Houwink equation 546 mass average velocity 39, 111, 685 mass balance 25, 42 mass concentration 24, 739 mass diffusion 115, 781 FricDiff 782 mass flux 41, 96, 99 mass fraction 25, 41, 739 mass transfer between a fluid and a solid sphere 111 between a fluid and a solid wall 110 between a fluid and small particles, bubbles and drops 112 between a liquid and a membrane surface 113 mass-transfer coefficient 102–3, 160 for equimolar counterdiffusion 107, 693 mass-transfer resistance 103 mass-transfer zone 497 mass-separating agent 206 mass-separating-agent process 35 maximum boiling point azeotrope 211 Maxwell relations 251 Maxwell–Stefan equations 112 mean free path 101 mean of the distribution 54 mean residence time 447 mechanical equilibrium 128 mechanical mixture 19 mechanical separations 1 process of 2 mechanical resolution 305 melt crystallization 675, 751 center-fed column 751 column crystallization 754 countercurrent crystal-melt system 751 devices 754 end-feed column 753 eutectic systems 751 reflux 753 solid solutions 751 spiral-type conveyor 751 Type (3) system 751 xylene isomers 751 membrane 21 membrane based separations 3, 346, 412, 555, 761 membrane chromatography 548

Index membrane contactors 688, 695 membrane oxygenators 189 nondispersive countercurrent solvent extraction 736, 738, 740 porous hydrophobic hollow fiber membrane 697 porous membrane contactorstripper 695 membrane gas permeation 438, 555, 771, 791 complete mixing 474 stage cut 474 membrane mass-transfer coefficient 183, 190, 741 membrane oxygenators 189, 697 membrane porosity 181, 415 membrane potential 843 membrane processes 1, 5, 12 membrane resistance 190, 416, 418, 770 membrane separations 8, 12 membrane tortuosity 182 mesophases 128, 231 mesopores 180 metal complex 320 metal extraction 289, 746 metal-binding agent 241 metallic membrane 180 metastable region 145 method of characteristics 492 methyl isobutyl ketone 318, 342 methylcellulose 384 methyldiethanolamine 312 micellar electrokinetic chromatography 384, 785 micellar phase capacity factor 786 micellar systems 231, 322 micellar-enhanced ultrafiltration 232, 322 micelles 232, 307, 322, 785 microfiltration 180, 386, 412 backflushing 581 boundary layer of particle suspension 575 cake resistance control 580 ceramic tubular microfilter 581 closed system 583 convective diffusion equation 577 critical flux 582 crossflow microfiltration 575 excess particle flux 578 external cake fouling 575 feed and bleed 583 integral model 577 internal membrane fouling 575 length averaged filtration flux 580 membrane resistance control 580 open system 583 population balance equation 575 rotary vacuum filtration 555, 584 shear stress 577 shear-induced particle diffusivity 578 solids volume fraction 575 specific cake resistance 575 tangential-pass microfiltration 575 microfiltration membrane 419 microfluidic devices 353, 603, 783 micropores 180 microporous adsorbent 227

885 microporous ultrafilter 421 microsolute 262 migration rate of species i 492, 683 migration velocity 91, 94, 97, 122, 367, 608, 786, 794 minimum boiling azeotrope 209, 400 minimum energy required for separation 827 by adsorption 835 by distillation 832 by evaporation 834 by membrane gas permeation 831 by solvent extraction 834 mixed suspension, mixed product removal (MSMPR) crystallizer 447 mixed-conducting solid oxide membrane 337 class II system 452 class I system 452 mixer–settler 462, 736 mixer 462 settler 463 mobility selectivity 268, 270 mode of feed introduction 6, 356, 399, 794 modifiers 299 molality 154 molar balances 42 molar concentration 24 molar density function 130 molar excess surface concentration 102 molar flux 91, 96 mole fraction 24, 41 mole ratio 24 molecular diffusion 91–2, 95–6, 102, 178 in liquids 106 molecular distillation 102 molecular weight cut off 321, 426, 427 molecular weight density function 138, 215, 394 molecular weight distribution 64 moment of concentration profile 123 monoaminenickel cyanide 237 monoclonal antibodies (mAbs) 259, 457, 851 monoethanolamine 281 monomers 390 moving boundary electrophoresis 255 multicomponent chemical mixture 64 multicomponent diffusion coefficient of the gas pair (i,k) 113 multicomponent mixture 64, 112 multicomponent separation 22, 33 multicomponent separation capability 118, 126, 682, 784 multicomponent system 212, 392 multiphase systems 358 multistage devices 706 Murphree extract stage efficiency 464 Murphree point efficiency 801 Murphree raffinate stage efficiency 464 Murphree vapor efficiency 723, 799 myoglobin 93, 427 Nafion® 336 nano filtration 413, 420 naphthalene 158

natural convection 109, 350 Navier–Stokes equation 361 Nernst distribution coefficient 218 Nernst distribution law 140 Nernst–Einstein relation 98 Nernst–Planck relation 97, 188, 337 net distribution coefficient 289, 294, 296 Newton efficiency 60 NH3–CO2–H2O system 282 nitrogen-enriched air (NEA) 441, 560 nonporous glassy polymeric membrane 179, 266 nonporous inorganic membrane 180 nonporous organic polymeric membrane 140, 177 nonrecycle separator 42 normal distribution 54, 122, 254, 380, 394 normal freezing 403 nth moment of the particle size density function 55 nuclear winter 193 nucleation heterogeneous 146 homogeneous 146 primary 146 secondary 146 nucleation population density parameter 448 nucleation rate 163, 450, 452 number of diavolumes 572 number of transfer units 465, 691 based on the raffinate phase 465, 740 number-based mean crystal size 448 number-based mean particle radius 57 oblate ellipsoid 93 observed sieving coefficient 421 observed solute transmission coefficient 421 Ohm’s law 192, 415 oil and water mixture 46 oily liquid membrane 324 Oldershaw column 804 olefin–paraffin mixture 243 open separator 41, 346 operating line 464, 680, 689, 712 optical force 3, 86, 654 optical isomers 305 optimum extent of separation 403 organic extract 400 organic phase control 191 organic raffinate 400 osmotic equilibrium 130, 141 osmotic power plant 844 osmotic pressure 141, 174, 196, 253 Ostwald ripening 147 ovalbumin 88, 93, 124–5, 192, 244, 248, 253, 255 overall crystal growth coefficient 163 overall mass-transfer coefficient 160, 162–3, 464 overall solute-transfer coefficient 185 overflow 46, 54 oxalic acid 291 oxygen vacancy 338 oxygen-enriched air 441, 560 O2–N2 system 211, 441, 559

886 packed bed 349 packed-bed adsorption 488 palladium 200 palladium membrane 201 parabolic velocity profile 348, 350, 642 paramagnetic particle 82–3, 193, 649 parametric pumping batch operation 520 continuous 525 nondispersive linear equilibrium based theory 520 Tinkertoy model 525 partial liquid mixing 408 partial molar Gibbs free energy 83, 129 partial molar volume 84 particle 24, 54 particle capture efficiency 387, 631 particle classification 54 particle diffusivity 89 particle number density function 55, 369, 446 particle population 54 particle population balance 369 particle Reynolds number 89 particle separation 9 in cocurrent gas–liquid flow 788 particle size density function 54 particle size distribution function 54, 61 particle trajectories 386, 612, 619 particle velocity 88 partition coefficient 100, 144, 221, 264, 545, 739 peak capacity 124, 381 Péclet number 110, 182, 381, 424, 801 penetration distances 514 penicillin G 342 penicillins 290, 742, 848 pentane-dichloromethane 400 pepsin 253 percent relative humidity 412 Percoll 253 perfect separation 19–20, 22, 30 permeability 179 permeability coefficient 173, 178–9, 183, 271 of solute 173 permeance 439 permeate 469 permeation 47, 178 permeation velocity 174 permeator 52 perovskite ceramic materials 338 Perrin factor 93 persistence of velocity 77 pervaporation 170, 176, 344, 433 concentration polarization 436 separation factor 434 pH gradient 245, 253, 255 pharmaceutical grade water 853 purified water 853 water for injection 853 phase 21 phase averages 365 phase barrier membranes 189 phase equilibrium based separations 2, 346, 390

Index phenol 324 phenomenological coefficients 116 phenylalanine 301 photophoresis 654 migration velocity 655 net photophoretic force 654 refractive index 654 phthalic anhydride 21 physical adsorption 149 physicochemical basis for separation 2, 76 pI 253 pi bonding 235, 243 π-complexation 243 π-complexation sorbents 488 picric acid 22, 35 pinch point 846 Plait point 220 planar interface 133 plasticization constant 176–7 plate height 381 plate number 381 plate-and-frame filter press 417 plug flow 77, 619 p-nitrophenol 340 Poiseuille flow 180, 185, 348, 350 polyacrylic acid 320 polycarbonate 420 polydimethylsiloxane 272 polydisperse polystyrene 63 polyelectrolyte 240 polyelectrolyte effect 242 polyelectrolyte macromolecules 240 polyethersulfone UF membrane 323 polyethylene film 196 polyethylene glycol 221 polyethylenimine 320 polymer glass transition temperature 140 polymeric membrane 140 polypropylene 420 polystyrene 395 polysulfone 321, 420 polytetrafluoroethylene 420 polyvinylidene fluoride 420 polyvinylpyrrolidone 253 population balance equation 347, 370 population density function 55, 58, 369, 446 pore concentration of solute 100, 182 pore condensation 150 pore Péclet number 182, 423 pore size density function 424 pore size distribution 180, 424 porosity 141 porous membrane 180 as a phase barrier 318 positive dielectrophoresis 81 potential-swing adsorption 519 powdered activated carbon 488 power number 468 Poynting pressure correction 158 precipitation 147, 446, 461 preferential hydration 136 preferential interaction analysis 244 pressure ratio 439, 559

pressure-swing adsorption 505, 511 primary amine 282 principle of electroneutrality 97, 230 Proabd Refiner 411 process intensification 687 projected area 89 prolate ellipsoid 93 property density function 62 property distribution function 62 property gradient 245 propionic acid 316 protein 229, 305 protein A 547, 851 protein solubility 147 proton exchange membrane 336 pseudo-component approach 62 pseudo-continuum approach 364, 488 pseudo-continuum level 347 pseudobinary systems 116 Pseudomonas diminuta 420 pump-and-treat 273 pure water permeability constant 173 purification of H2 518 purification processes 1, 35 purity enhancement factor 276 purity indices 26, 34–5 quaternary ammonium compounds 298 radial flow packed-bed adsorption 503 radiation pressure 86, 95, 654 radiation pressure force 86 radius of gyration 143 radius of solute molecule 143 raffinate 220, 265 raffinate phase 463 Ranz–Marshall type correlation 166 Raoult’s law 138, 207, 209, 217, 692 rapid pressure swing absorption 519 Raschig ring 51 rate of crystal growth 163 rate-controlled equilibrium separation 309 rate-governed processes 205 Rayleigh distillation 397 reaction equilibrium constant 282, 286, 288 reaction equilibrium limit 327 reaction plane 311 reactive entrainer 286 rectangular channel 348 recycle ratio 48 recycle separator 48 red blood cells 261 reduced efficiency 60 reflux 53 reflux ratio 53, 711, 715 region 20–2 reject 47 relative dielectric constant 79 relative displacement 76–7 relative humidity 412 residue curve maps 399 resolution 36, 68, 123, 126, 195, 252, 255, 380, 529, 797 retentate 469 retention time 67

Index reverse osmosis 5, 22, 32, 170, 199, 272, 318, 413, 428, 562, 810, 828, 851 brackish water 566 crossflow 562 desalination 174, 199, 852 feed brine spacers 563 fractional water recovery 565 membrane length 564 membrane packet 562 post-treatment steps 852 pretreatment steps 851 retentate 567 solution–diffusion model 564 solution–diffusion–imperfection model 173, 429 Spiegler–Kedem model 430 spiral-wound module 562 spirally wound membrane channel 562 reversed micelle 35, 233, 750 reversible complex 326 Reynolds number 109, 193, 348 rigid body rotation 86 rotary vacuum filtration 555, 584 dead-end filtration 585 filter aid 584 precoat 584 washing efficiency 586 nski–Hadamard formula 375 Rybczy saline water 21 salting out 244 sand filters 586 saturation capacity of protein sorption 157 saturation humidity 412 Sauter mean diameter 57, 467 Schmidt number 109 second ionization 291 second-order reaction 313 secondary amine 282, 342 secondary nucleation 453 sedimentation 250 sedimentation coefficient 248 segregation fraction 24, 44, 59 segregation matrix 24 selective absorption 284 selective removal of H2S over CO2 284 selective SCF separation 231 crossover regime 231 1, 10-decanediol 231 selectivity 267, 314 self-sharpening wave front 495 semicontinuous mixture 64 semicrystalline polymer 140 semiequilibrium dialysis 277 semipermeable membrane 21, 130, 141 separation of compressed air 21, 441, 559 descriptions of 19 of H2 441, 561 of two immiscible liquids 623 using membrane 12, 205 separation factor 26–7, 47, 205, 246, 256, 263, 307, 382, 425, 434, 439, 742 separation index 2, 25

887 separation nozzle 632 separation factor 633 separation of uranium isotope U235F6 from U238F6 632 separation process 19, 21, 35 separation quotient 26 separative power 49, 72 Sephadex™ 546 settling velocity 249, 258 shape factor 56 sharp cut off membrane 425 shear-induced diffusivity 90 Sherwood number 109 shock wave 493 sieve opening 58 Sievert’s law 139, 141, 200 sieving 413, 450 sieving coefficient 421, 423, 424, 426, 572 silicone rubber capillary 47 silver nitrate 326 simulated moving bed (SMB) 675, 754 single-entry separator 41, 44, 48, 812 single-phase system 132, 205 size-dependent crystal growth 453 slip correction factor 90, 117, 608 slip flow 100–1, 185 SO2–N2 mixture 247 SO2–water system 237 sodium dodecyl sulfate 241, 322, 384, 785 sodium glycinate 327 sodium phenolate 324 sodium sulfate 184 sol 64 solid solution 19, 144, 403, 753 solid–vapor systems 411 solidus 144, 404 solubility coefficient 140–1, 178, 196, 268 solubility curve 145 solubility parameter 218, 319 solubility product 304 solubility selectivity 267 solubilization coefficient 232 soluble macroligands 320 solute diffusion through porous liquidfilled membrane 182 solute dispersion 366 solute distribution coefficient 173 solute flux 173, 181 through porous liquid-filled material 99 solute ionization 281 solute mass-transfer coefficient 422, 563 solute permeability coefficient 173, 430, 566 solute reflection coefficient 430–1 solute rejection 42, 421, 431 solute retention 421 solute transport parameter 173 solute–micelle equilibrium 232, 786 solute–micelle interaction 173, 303, 566 solution crystallization 446 solution diffusion 172, 337 solution–diffusion theory 172, 178, 428 solution–diffusion–imperfection model 173, 429 solvating extractants 295 solvation of metal ion 295

solvent acting as complexing agent 328 solvent exchange 398 solvent extraction 23, 162, 217, 289, 400, 462, 736, 834, 848 antibiotics extraction 736, 742, 848 aqueous two-phase extraction 221, 750 backmixing 750 centrifugal devices 738 continuous countercurrent multistage device/cascade 741 countercurrent flow 462, 736 dilute solution of a solute 139, 738 dispersive 462, 736 dispersive countercurrent extraction column 674 distribution coefficient 139, 218, 289, 739 extract 739 extraction enhancement factor 317 extraction factor 740 extraction of metals 294 extraction section 745 extraction–scrubbing cascade 744 fractional extraction 744, 746 height of overall transfer unit 740 Karr column 736 Kremser equation 742 Leveque solution 741 logarithmic concentration difference 740 membrane mass-transfer coefficient 191, 741 mixer–settler 462, 736 nondispersive countercurrent solvent extraction 736 number of overall transfer units 740 packed tower 736 partial solubility of the bulk phases 747 penicillin 742 perforated-plate towers 736 Podbielniak 738 porous membrane based 738 purifying zirconium from hafnium 746 raffinate 739 reflux 750 reverse micellar extraction 750 Scheibel extractor 736 separation factor 742 spray towers 736 stage efficiency 751 stripping section 745 tributylphosphate 299, 300, 746 solvent flux 420 solvent permeability 181 solvent velocity 414 sorption 140 sorption coefficient 140 sorting classifiers 259 sound wave 261 species conservation equation 119 species-specific regions 2, 118 specific cake resistance 415, 575 specific surface area of a particle 490 spherical micelle 232 sphericity 57 spiral-wound membrane 176, 557, 563 splitter 42

888 spray scrubber 39 spray tower 376 spreading pressure 135, 149, 223 stability constant 299 stage cut 44 stage efficiency 464, 723, 724, 801 stage separation factor 43, 815 stagnant film 108 stagnant film model 568 standard deviation of a concentration profile 67, 122, 126, 252, 255 standard state chemical potential 85, 131, 206 standard state fugacity 85, 131, 206 Stanton number 110 stationary phase 367, 529 steam stripping 208 steric mass action (SMA) 156 stoichiometric coefficients for a reaction 311, 702 Stokes–Einstein relation 90, 143 Stokes flow 361 Stokes’ law 89–90, 117, 256, 374 Stokes number 387 strip solution 265 stripping 206, 208 volatile species 674 stripping factor 70, 708 styrene 395 sucrose 184 sulfonated polystyrene 154 supercooling, 145 fractional 145 supercritical fluid (SCF) 157, 231 superficial velocity 349 supersaturated region 145 supersaturation extent of 451 fractional 145, 452 molar 145 ratio 145 relative 145 surface filtration 413 surface adsorbed phase 102 surface area 135 per mole of gas 135 surface concentration 134 surface diffusion 101, 186 surface diffusion coefficient 102, 186 surface excess 134, 185 surface excess concentration of a solute 148 surface filtration 386, 413 surface flow 101 surface phase 133 surface region 133 surface shape factor 57 surface-active solute 223 surfactant 159 susceptibility 82, 193, 649 suspension 24 suspension density 56, 448–9 svedberg 248 sweep diffusion 115 swelling pressure 152–3 symmetric separator 44, 48

Index tails separation factor 44, 50 tangential stress 354 Taylor dispersion 363 t-butylamine 283 temperature gradient 85, 261 terminal drop velocity 391 terminal particle velocity 128, 256 terminal settling velocity 94 terminal velocity 374 ternary system 23 tertiary amine 296, 329, 332 thermal desalination 852 thermal diffusion 22, 28, 31, 36, 85, 262, 671, 781 Clusius–Dickel column 673 concentration wave velocity 673 thermal diffusion coefficient 85 thermal diffusion ratio 85 thermal equilibrium 128 thermal-swing adsorption 505 thermocapillary flow 354 thermodynamic equilibrium 128 thermodynamic selectivity between H2S and CO2 286 thermophoresis 85 thermostatics 116 thickeners 259 three-phase systems 160 tie line 211, 219 time lag 180 time of breakthrough 494 toluene 21 tonnage oxygen 441 tortuosity factor 100 total driving force 84, 86 total efficiency 61, 611 total external force 87 total Gibbs free energy 129–30 total molar density 24 total potential 120 trailing edge of concentration profile 127 trajectory equations 347, 368 transference number 97 transitional flow 101 transport coefficients 117, 173 transport number 97, 187 triangular diagram 33 triboelectrification 389, 596 tributylphosphate 746 tricarboxylic acids 292 tridodecylamine 332 trioctylphosphine oxide (TOPO) 294, 326 Trouton’s rule 217 true solute rejection 421 true solution 19 tubular bowl centrifuge 619 centrifuge effect 620 equiprobable size 622 grade efficiency function 619, 621 liquid–liquid centrifuges 624 particle trajectories 619 separation of two immiscible liquids 623 sigma 622 turbulent flow 106, 349, 610 two regions 19

2-amino-2-methyl-1-propanol 283 two-bulb cell 22, 28 two-dimensional pressure 135 two-phase systems 128, 160 ultimate distribution 409 ultracentrifuge 247 ultrafiltration 180, 232, 412, 420, 469, 568 average solvent flux 570 bovine serum albumin 574 concentration factor 572 concentration polarization relation 421, 570 continuous diafiltration 471, 572 crossflow ultrafiltration 568 diafiltration mode 470, 572 dilution factor 472, 572 extent of separation 573 gel concentration of BSA 574 gel polarization 571 hollow fiber UF unit 574 macrosolute 570 number of diavolumes 572 processing time 473 product purification factor 573 selectivity 572 Sherwood number 574 solute rejection 469, 570 solvent flux 570 yield 572 ultrapure water 851–2 uncoupled conditions 86, 172 underflow 46, 54 uni-univalent electrolyte 79, 187 uniform mixture 20 uniformly continuous potential profile 120 unsaturated region 145 uranium isotopes 247, 605, 775 uranyl nitrate 265, 299 uranyl sulfate 332 urea 262, 761 urease 324 uric acid 262, 761 valine 301 value function 49, 72 vancomycin 321 van’t Hoff equation 141, 196, 268, 430 vapor absorption/stripping, see gas absorbtion/stripping vapor fraction 50 vapor pressure 207 vapor–liquid equilibrium (VLE) 138, 209, 286, 710, 734 vapor–liquid system 83, 208, 390 variance 54 velocity of zone travel 406 Venturi scrubber 788 vesicle 234, 322 virus 249, 848 viscous flow 100 volatile organic compounds 435 volume concentration ratio (VCR) 469 volume flux 173, 180–1, 414

Index volume-averaged equations 358 volume-averaging procedure 364 wastewater treatment 586 water desalination 851 brackish waters 851 seawater 851 thermal desalination 852 water softening 265 water treatment 586, 851 pharmaceutical grade water 853

889 ultrapure water 851–2 water desalination 851 water–ethanol 177 weak acid 289 weak organic acid 290 weak organic base 290 Weber number 467 Wilke–Chang correlation 92 yeast cell 192, 249, 622, 850 yield 469

Young–Dupre equation 159 Young–Laplace equation 136, 420 zeroth moment 55 zeta potential 80, 260 zone electrophoresis 255 zone melting 403 zone refining 29, 403 zwitterion 301