Mass Transfer in Multicomponent Mixtures Mass Transfer in Multicomponent Mixtures J.A. Wesselingh R.Krishna DELFT UNIVERSITY PRESS CIP-data Koninklijke Bibliotheek, Den Haag Wesselingh, J.A. Mass transfer in multicomponent mixtures - Delft : Delft University Press. - TIl. ISBN 90-407-2071-1 NUGI 812,813,831 VSSD First edition 2000 Published by: Delft University Press P.O. Box 98, 2600 MGDelft, The Netherlands tel.+31152783254, telefax +31152781661, e-mail dup@library.tudelft.nl website:http://www.library.tudelft.nlldup On behalf of:: Vereniging voor Studie- en Studentenbelangen te Delft Poortlandplein 6, 2628 BM Delft, The Netherlands tel.+31152782124, telefax +31152787585, e-mail:vssd@tudelft.n1 internet: http://www.oli.tudelft.nllvssdlhlflhandleidingen.html Allrightsreserved.Nopartof thispublicationmaybereproduced,storedina retrievalsystem,ortransmitted,inanyformorbyanymeans,electronic, mechanical,photo-copying,recording,orotherwise,withoutthepriorwritten permission of the publisher. ISBN 90-407-2071-1 5 Foreword It wastwentyyearsago.Alittle before that,Ihad left theEquipment Engineering Department of Shell Research in Amsterdam for a less hectic job at Delft University. At least,so I thought at that moment.In myformersection at Shell we had worked on catalytic crackers, on polymerisation reactors, on cleaning of oil tankers and other exciting developments,but I had foundmanagingthisa bit too much.There I was, withalowersalary,teachingseparationprocessestosecondyearstudents,and running the undergraduate laboratory with one hundredth of myShell budget.I had written a little book on Separation Processes, and sent it to friends in Amsterdam. One of the pieces of equipment that wehad in Shell Research was(what was then) thelargestdistillationtestcolumnintheworld.It wastwoandahalf metresin diameter andsometwentymetreshigh.The column wassobigthat wecould only run it in the summer: the reboiler used the complete capacity of our boiler house. The operating pressure could be varied between vacuum and fifteen atmospheres. We had a beautiful time trying out allkindsof traysand packings.In time westarted to get interested in trying tounderstand not only distillation of binary mixtures, but also of mixtureswithmorecomponents.Westarted togather measurementsandtotryto understand them. However, much of what we saw was baffling, to say the least. Only gradually did we realisethat our binary masstransfer toolswerenotadequate;that weneeded totrysomethingdifferent.Thatsomethingwasayounggraduatefrom Manchester who had picked up wild ideas on mass transfer doing his PhD. His name wasKrishna. I left Shell just after he arrived. One day, Krishna came along at home to visit me.He had read my book and told me politely that myapproach tomass transfer was not all that good.I was a little vexed because I was professor, he was not,and besides, I had copied my ideas from well-knownhandbooks.Evenso,Ididtrytolisten,and threeweekslater went back to himformoreexplanation.It wasallaboutmulticomponentmasstransfer,ithad connectionswith thermodynamics and wasquite different fromanything Iknew ..I had difficulties infollowingwhat he wastelling me,and I can remember:'Hans,if you really want to understand this, we should give a course together.'That is where this book started. Wearenowtwentyyears,fifteenPhDstudents,twenty-threecoursesandsome thousandparticipantsfurther.The course hasevolved andsohasthebook.Itnow has examples frommembrane technology, reaction engineering,sorption processes, biotechnology;frommixturesofgasesorliquids,butalsoporousmediaand 6Mass Transfer inMulticomponent Mixtures polymers.The basics have not changed:theyarestill almost the same aspresented byJamesClerkMaxwellin1866and(moreclearly)byJosefStefanin1872. (Maxwell istheone of thetheoryof electricalandmagneticfields,andStefan the oneassociatedwiththename of Boltzmann.)Iam alittleashamed whenIlook in editionsoftheEncyclopaediaBritannicafromaround1900andseehowwell diffusionwasthenunderstood.It feelsasif it hastakenmemorethana century to pick upthe brilliant but simple ideasof thesetwolong-dead scientists.Krishand I hope this book will help you to do that a bit more quickly. Hans Wesselingh Groningen, May 2000 7 Contents FOREWORD5 1BEGINNING ...13 1.1Who should read this Book?13 1.2 What this Book covers14 1.3 Structure of the Book15 1.4 Guidelines to the Reader16 1.5 Guidelines to the Teacher17 1.6 Symbols18 1.7 Conventions21 2Is SOMETHING WRONG?24 2.1The Starting Point24 2.2 Three Gases26 2.3 Two Cations28 2.4 Two Gases and a Porous Plug28 2.5Summary30 2.6 Further Reading30 2.7 Exercises30 PART1.MAss TRANSFER IN GASESAND LIQUIDS33 3DRIVING FORCES34 3.1Potentials, Forces and Momentum34 3.2 Momentum (Force) Balance of a Species36 3.3 The Driving Force: a Potential Gradient38 3.4 The Maxwell-Stefan Equation39 3.5 Simplifying the Mathematics40 3.6 The Film Model41 3.7 Difference Form of the Driving Force42 3.8 Summary44 3.9 Further Reading44 3.10 Exercises45 4FRICTION46 4.1Friction Coefficients and Diffusivities46 4.2 Velocities and the Bootstrap48 4.3 Velocities and Fluxes50 4.4 The Difference Equation50 4.5 Mass Transfer Coefficients52 4.6 Summary53 4.7 Further Reading53 4.8 Exercises53 5BINARY EXAMPLES55 5.1Stripping55 5.2 Polarisation56 8Mass Transfer inMulticomponent Mixtures 5.3 Vaporisation56 5.4 Gasification of a Carbon Particle58 5.5 Binary Distillation59 5.6 Summary60 5.7 Further Reading60 5.8 Exercises61 6TERNARY EXAMPLES63 6.1From Binary to Ternary63 6.2 A Condenser64 6.3 A Ternary Distillation65 6.4 A Ternary Reaction68 6.5 Binary Approximation of a Ternary68 6.6 Summary70 6.7 Further Reading70 6.8 Exercises71 7MASS AND HEAT TRANSFER74 7.1Temperature Gradients74 7.2 Enthalpy75 7.3 Mass Transfer Relation76 7.4 Energy Transfer Relation77 7.5 Condensation in Presence of an Inert Gas79 7.6 Heterogeneous Reacting Systems80 7.7 An Ammonia Absorber80 7.8 Summary81 7.9 Further Reading81 7.10 Exercises82 8N ON-IDEALITIES84 8.1Chemical Potential and Activity84 8.2 Non-ideal Binary Distillation84 8.3 A Simple Model of Non-idealities85 8.4 Large Non-Idealities: Demixing87 8.5 Maxwell-Stefan versus Fick88 8.6 When can we neglect Non-ideality?89 8.7 Mass Transfer in Liquid-Liquid Extraction90 8.8 Summary92 8.9 Further Reading92 8.10 Exercises92 9DIFFUSION COEFFICIENTS95 9.1Diffusivities in Gases95 9.2 Diffusivities in Liquids97 9.3 How do you measure diffusivities?100 9.4 Summary102 9.5 Further Reading102 9.6 Exercises103 10TRANSFER COEFFICIENTS106 10.1Introduction106 10.2 Dimensionless Groups106 Contents9 10.3 Tubes and Packed Beds108 10.4 Packed Gas-Liquid Columns110 10.5 Single Particles, Bubbles and Drops111 10.6 Using'Single' Coefficients for Swarms117 10.7 Using Binary Coefficients in Multicomponent Calculations118 10.8 Summary118 10.9 Further Reading119 10.10 Exercises120 11ELECTRICAL FORCES AND ELECTROLYTES122 11.1Electrolytes122 11.2 The Electroneutrality Relation123 11.3 Electrical Forces124 11.4 Transport Relations124 11.5 Diffusion of Hydrochloric Acid125 11.6 Plus a Trace of Sodium Chloride126 11.7 Diffusion of Proteins126 11.8 Conduction and Friction between Ions128 11.9 Diffusivities in Electrolyte Solutions130 11.10 Summary132 11.11 Further Reading133 11.12 Exercises134 12CENTRIFUGAL AND PRESSURE FORCES137 12.1Volumetric Properties137 12.2 The Pressure Gradient138 12.3 Gravitational Force138 12.4 Centrifuges139 12.5 Gas and Protein Centrifugation140 12.6 Difference Equation for the P r e s s u r ~Force141 12.7 The Maxwell-Stefan Equations (again)142 12.8 Summary143 12.9 Further Reading143 12.10 Exercises143 13WHY WE USE THE MS-EQUATIONS146 13.1Three Ways146 13.2 A Mixture of Three Gases147 13.3 The Fick Description148 13.4 Thermodynamics of Irreversible Processes150 13.5 The Maxwell-Stefan Description150 13.6 Units152 13.7 Further Reading152 13.8 Exercises153 PART 2.MASS TRANSFER THROUGH A SOLID MATRIX155 14SOLID MATRICES156 14.1The Applications156 14.2 Membrane Processes156 14.3 Adsorption and Chromatography158 14.4 Heterogeneous Catalysis160 10Mass Transfer inMulticomponent Mixtures 14.5 Structured and Non-structured Matrices161 14.6 Effects of a Matrix on Mass Transfer162 14.7 Compositions with a Matrix163 14.8 How Further?164 14.9 Further Reading165 14.10 Exercises165 15PROPERTIES OF POLYMERS167 15.1A Few Words on Polymers167 15.2 Thermodynamics of Mixtures in a Polymer171 15.3 Summary177 15.4 Further Reading_177 15.5 Exercises177 16DIFFUSION IN POLYMERS180 16.1Behaviour of Diffusivities180 16.2 The Free Volume Theory182 16.3 Summary188 16.4 Further Reading189 16.5 Exercises189 17DIALYSIS AND GAS SEPARATION191 17.1Dialysis191 17.2 Gas Separation194 17.3 Summary197 17.4 Further Reading197 17.5 Exercises198 18PERV APORA TION AND REVERSE OSMOSIS200 18.1Pervaporation200 18.2 Reverse Osmosis204 18.3 Summary208 18.4 Further Reading208 18.5 Exercises209 19ELECTROLYSIS AND ELECTRODIALYSIS211 19.1Introduction211 19.2 Polarisation in Electrolysis212 19.3 Electrodialysis214 19.4 Summary220 19.5 Further Reading220 19.6 Exercises221 20ION EXCHANGE223 20.1Fixed-Bed Processes223 20.2 Ion Exchange EquiIibria224 20.3 Linear Driving Force Model225 20.4 Ion Exchange (Film Limited)227 20.5 Ion Exchange (Particle Limited)228 20.6 Summary230 20.7 Further Reading230 20.8 Exercises231 Contents11 21GAS PERMEATION233 21.1Transport in Cylindrical Pores233 21.2 The Diffusion Coefficients233 21.3 Looking Back235 21.4 Transport in a Bed of Spheres236 21.5 The Dusty Gas Model238 21.6 Summary240 21.7 Further Reading240 21.8 Exercises241 22IN POROUS CATALYSTS242 22.1Introduction242 22.2 Pressure gradients inside a particle243 22.3 Separate Transport Equations245 22.4 Single-variable Pressure and Rate Expressions245 22.5 Solution for a Slab247 22.6 Summary249 22.7 Further Reading249 22.8 Exercises250 23IN ADSORBENTS252 23.1Adsorption252 23.2 Equilibria - Langmuir Isotherm253 23.3 Maxwell-Stefan and Fick Diffusivities254 23.4 Macropore Diffusion257 23.5 Transport Equations258 23.6 Transient Adsorption of a Binary259 23.7 Membrane Applications260 23.8 Summary261 23.9 Further Reading261 23.10 Exercises262 24ULTRAFILTRATION264 24.1The Module264 24.2 Membrane and Permeants265 24.3 Osmotic Pressure (No Ions, no Charge)266 24.4 Size Exclusion266 24.5 Polarisation267 24.6 Transport Equations269 24.7 Inside the Membrane272 24.8 Electrical Effects273 24.9 Summary275 24.10 Further Reading276 24.11Exercises277 25......... ENDING279 25.1Looking Back279 25.2 Thermodynamic Models - the Potentials279 25.3 Driving Forces281 25.4 Friction Terms281 25.5 Friction Coefficients282 12Mass Transfer inMulticomponent Mixtures 25.6 Additional Relations (Bootstraps) 25.7 The Many Variants 25.8 Goodbye 25.9Further Reading 25.10 Last Exercise THANKS ApPENDIX1. READING MATHCAD ApPENDIX 2. UNITS A2-1Molar basis A2-2 Mass basis A2-3 Volume basis A2-4 Molar and mass diffusivities A2-5 Molar and volume diffusivities A2-6 Molar and mass driving force A2-7 Molar and volume driving forces A2-8 Difference equation for a'film' A2-9 Difference equation, molar basis A2-10 Difference equation, mass basis A2-11Difference equation, volume basis A2-12 Summary ApPENDIX 3. PoRING OVER PORES A3-1Introduction A3-2 The System A3-3 Forces and Velocities A3-4 The Two Transport equations A3-5 Comparing the Two Models A3-6 Summary A3-7 Further Reading A3-8 Exercises ANSWERS INDEX CD-ROM FOR'MASS TRANSFER IN MULTICOMPONENT MIXTURES' 283 285 286 287 287 289 293 296 296 297 297 298 299 300 300 301 301 302 302 303 304 304 305 305 308 310 313 314 314 317 326 329 13 1 Beginning ... 1.1Who should read this Book? This book is about the diffusion and mass transfer processes that are really important, but which are neglected in most textbooks: those with three or more species, the'multicomponent'mixtures, those with more than one driving force,including electrical or pressure gradients, and those with a solid matrix such as a polymer or a porous medium. If youwant toknowmoreabout these subjects,but findexisting textstoodifficult, then this is the book for you. Also, if you already understand the intricacies of multi-component mass transfer,you may find it enjoyable tosee how faryou can get with simple means. We are assuming that you are interested in processes or products. This may be in an academicor industrialsetting:in chemicals,water treatment,food,biotechnology, pharmaceuticals...younameit.Thebookassumesthatyouhaveaworking knowledge of: thermodynamicsandphaseequilibria:chemicalpotentials,enthalpies,activity coefficients, partial molar volumes and distribution coefficients, transportphenomena:simplemassbalances,binarydiffusionandmasstransfer coefficients, and fluid flow,especially around particles and in porous media. If youarenot toosure,donotdespair.Wewill repeatallimportantconceptsina leisurelymanner.However,thisisnotabookforacompletebeginnerinmass transfer;youmust have heardof theaboveconcepts.Becausetherearemanynew ideas toget used to,we have tried to avoid mathematical complexity. For the greater part of the text you do not need more than theabilitytosolve three linear equations with three unknowns.You can even do a fairbit with pencil,paper and a calculator. Of courseyouwillneedacomputer forlarger problems,butnottoobtainafirst understanding. 14Mass Transfer inMulticomponent Mixtures 1.2 What this Book covers Thisbooktakesmotioninamixturetobegovernedbyforcesontheindividual species. There are two kinds of forces: driving forces,which stem from the potential gradient of a species, and friction forces between the species, which arise from their velocity differences. MaxwellandStefanalreadyusedthismethodmorethanacenturyago.This mechanical viewpoint ismuch more general than Fick's law,which isusually taken asthebasisofdiffusiontheory.It hasnotcaughton,probablybecausethe mathematics is thought to be difficult. This is not really a problem however: There are simple approximations to the solutions of the equations. Thecomputerandnumericaltechniquesnowmake'exact'calculationsmuch easier. Using potential gradients allows the incorporation of different driving forces: composition gradients (or more pr:ecisely:activity gradients), electrical potential gradients, pressure gradients, centrifugal fields and others. The frictionapproach to interaction between the species allows a consistent handling of any number of components. Working with force balances makes it easy to link the subjectof masstransfertoother partsof science.Thermodynamicsandtransport processesbecomeneighbours:equilibriumissimplythesituationwheredriving forces have disappeared. The use of forceson the species in a mixture fitsin the way of thinkingof engineers:it isalogicalextensionof mechanicsof asinglespecies. For friction coefficients we can make use of the many relations for hydrodynamics of particlesor porousmedia.Theseshowthat flowand diffusionaretwosidesof the same coin. With our starting points we can describe almost any mass transfer process. Examples in this book cover: multicomponent distillation, absorption and extraction, mUlticomponent evaporation and condensation, sedimentation and ultracentrifugation, dialysis and gas separations, pervaporation and reverse osmosis, electrolysis and electrodialysis, iop exchange and adsorption heterogeneous catalysis and ultrafiltration. 1.Beginning ...15 The examples treat diffusion in gases,in liquids, in electrolytesolutions, in swollen polymersandinporousmedia.Thebookincludesmethodsforestimatingmulti-component diffusivities and mass transfer coefficients. Amajorlimitationof thebookisthatitmainlycoversexampleswithasingle transferresistance,notcompletepiecesof equipment.Sucharesistancewillbea buildingblockforthesimulationof separationcolumns,membranemodulesor chemical reactors.The reader must be prepared toincorporate the equations into his ownsimulations.Theapproximationsusedshould besufficientlyaccurate formost engineering applications. Withthisbook,wehopetomakeyoufeelathomeintheequationsofmulti-componentmasstransfer.However,wedonotderivetheseequations.If youare inquisitive and have some perseverance, you will be able to retrace the fundamentals in the references that we give. 1.3 Structure of the Book Thisbook hastwenty-five chapters,covering a rangeof subjects.You mayfeelthat it is a jumble of factsand problems, but there is an underlying structure. The theme is the development of the Maxwell-Stefan equations. There are two main parts:on transfer in gasesand liquids (Chapters 3 ...13),and on transferthroughasolid matrix(Chapters14 ... 24).Chapters3and4introducethe twosides of the Maxwell-Stefan equations:thedriving forcesformasstransfer and the frictional forces between moving species.Chapters 5 and 6 apply the equations to simple binary and ternaryexamples.Chapters 8,11and12 complete the description of thedrivingforcesbyincludingtheeffectsofnon-idealityinamixture(8), electricalforces(11)andcentrifugalandpressureforces(12).Chapters9and10 consider parametersin thefrictionterms:diffusivitiesor masstransfer coefficients. Chapter13discussestherelationbetweentheMS-equationsandotherwaysof describing mass transfer. YoumaywonderwhathashappenedtoChapter7.It introducestheeffectsof a temperature gradient,and doesnot quitefitinto thestructure.We could have put it almost anywhere. The second part of the book considers mass transfer through solid matrices.Chapter 14 gives a preview of the subject and discusses the two types of matrix: polymer matrices (Chapters15 ... 20) and structures with defined pores (Chapters 21 ... 24). Chapters15and16giveabrief descriptionof polymersandof thebehaviourof diffusioncoefficientsinpolymers.Thesechaptersareasideline,introducing conceptsthatweneedfurtheron.Wecontinuewithaseriesof exampleswith 16Mass Transfer inMulticomponent Mixtures differentdrivingforces:compositiongradients(Chapter17),pressuregradients (Chapters 17 and18) and electrical gradients (Chapters19 and 20). Inthechapterson porousmedia,wemainlyfocuson thefrictionsideof theMS-equations.Chapter21coverstransportof non-adsorbinggasesandintroducesthe effectsof viscousflow.Chapter22showshowtheMaxwell-Stefanequationsare applied when chemical reactionsare taking place.Chapter 23considers diffusion of specieswhichdoadsorb,suchasinmicroporousadsorbents.InChapter24we consider an example where viscous flowis very important: ultrafiltration. Wefinishbylookingbackatthemanydifferentaspectsof theMS-equationsin Chapter 25. 1.4 Guidelines to the Reader Thetext was written toaccompany overhead transparencies ina fullweek's course on multi-component mass transfer. Most transparencies have found their way into the figures:they are important, not just illustrations. The figures contain alland muchoftheotherinformation.Notallchaptersareequallyimportant.Asa minimum,werecommend that you work through Chapters 3,4,5, 6,14,17and 21. Together, these will give you a working idea of multicomponent mass transfer theory for about two day's work. Other comments: If you are convinced that you know all about mass transfer (as we used to be!) you should read Chapter 2. It may contain a fewsurprises. Chapters 7,8 and12 cover subjects which,although important, can be omitted on first reading. The Chapters 9,10,11(second half),15 and16 are on the estimation of properties and model parameters such as diffusivities and mass transfer coefficients. You can skip these on first reading. If youarenotinterestedinions,electrolytesandelectricalfieldsyoucanskip Chapters11,19,20 and partsof 24.(However,donotethat electricalfieldsare much more prevalent than thought by most chemical engineers!) If you never encounter polymers, you will not need Chapters15 ... 20. When porous media play no role in your life, you can omit Chapters 21 ... 24. It is all up to you. Chapters2 ... 6containanumberof questionsandsmallsumsinthetext.We recommend that you try these.The answersareburied in the text or figures.Behind each chapter is a series of exercises. These are to help you to go through the material morethoroughlythanyouwillwithasinglereading.Thereareshortquestions, discussionsand additionsto thematerial of themain text.The answersaregiven at the end of the book. 1.Beginning ...17 BeginningwithChapter 5,thereareassignmentsinMathcad - afairlyaccessible programming language.There aretwo kinds:short ones, which youare to program yourself andlonger files,which aredemonstrationsof more complicated problems. Our students very much favour the first type.We have marked them with* and we hope that you willtrya fair number of these.Thesecond kindof filesis for you to useforyourownproblems,tolookat,toplaywith,tomodifyor tocriticise.We leave it to you.Many of these fileswork out examples given in the text.The Math-cad assignmentsareinthefolderExercises/Questions on theCD-ROM.Completed Mathcad filesarein thefolderExercises/Answerson theCD-ROM.Youcan read these,and change their parameters, using the freeprogram Mathcad Explorer which isalso on the CD-ROM as a self extracting filein MathcadlExplorer. Appendix1 in the book contains an introduction toMathcad; enough to allow you to read the files. You can further improve your Mathcad skills with the tutorial in Mathcad Explorer. Tomakefulluseof theMathcadexercises,youwillneedMathcad7.0,Student Edition or higher. Before our courses, we give students a short self-instruction course inMathcad.YouwillfindthisintheMathcadlTutor7folder.It consistsof ten Mathcad files;theyshould help you to get a good start in Mathcad in less than half a day. Theregular textof thebookcontinuesinChapter2.However,youshouldglance through the list of symbols, and the list of conventions at the end of this chapter. You may not understand all detailson first reading, but you should know where theyare so you can look back later. 1.5 Guidelines to the Teacher Thisbookhasevolvedinaseriesof twenty-threecoursesthatwehavegivenat differentuniversitiessince1982.TheparticipantsweremainlyPhDandMasters students, but we have also had many participants from industry and a fair number of our colleagues: together about nine hundred of them. Most have been from chemical engineering, but we have also had mathematicians, chemists, physicists,mechanical engineers, and the occasional pharmacy or biology student. Becausewealwayshaveanaudiencecomingfrommanydifferentplaces,our courses have mostlybeen in five days consecutively. In such a course we have about 36 hours forlecturesand computer assignments.We divide these into(roughly)16 hours of lecturing and 20 hoursof computer assignments in changing groups of two. Except for Chapters1 and 25(which require nolecturing time)and 2 and14 (which take less than half an hour), all chapters need about an hour. This means that you will have to make a choice of about15chapters from the19 others that we provide. The 'Guidelines to the Reader'above should help you in making a choice. AsaminimumforacoursewerecommendChapters3,4,5,6,14,17and21. Together,thesewillgiveparticipantsaworkingideaofmulticomponentmass 18Mass Transfer inMulticomponent Mixtures transfer theory for about two day's work. The CD-ROM that accompanies this book contains a complete set of PowerPoint 7.0 filesof thecolour transparenciesthatweuseinthecourse.Theyareinthefolder Transparencies.You can use and edit these freely for your own teaching, but you are not allowed to use them for commercial purposes. They are our property! On assessing the knowledge of students.You can of course do that in the traditional way. We also have good experience with giving each student some article on a mass transfer problemandaskinghim or her toconstructanewMathcad example.Our students find this difficult but instructive. Success! 1.6 Symbols Symbols used only at one or two points, are defined there and not listed here. flAvogadro constant# mol-I Anon-ideality p'arameter-aactivity-cmolar concentrationmol m-3 cp molar heat capacityJ morl DFick diffusivitym2 S-I ddiameterm DMaxwell-Stefan diffusivitym2 S-I Eenergy fluxJ m-2 S-I Eactivation energyJ mol-I Fforce per moleN mol-I '.FFaraday constantC mol-I gacceleration of gravitymS-2 HenthalpyJ mol-I hheat transfer coefficientW m-2 KI Ielectrical current densityA m-2 Jflux with respect to the mixturemol m-2 S-I kmass transfer coefficientm S-I Kequilibrium constant-Mmolar masskg mol-I mmasskg Nfluxwith respect to an interfacemol m-2 S-I nnumber of molesmol ppressureN m-2 1.Beginning ... Rgas constant J mol-IK"I Rretention rradiusm rreaction rate mol m-3 S-I TtemperatureK ttimes Tg glass transition temperatureK udiffusive velocity ms-I vviscous velocity ms-I Vmolar volume m3 mol-I wwhole (overall) velocity ms-I xmole fraction-ymole fraction-zdistance (position) zcharge number-Greek symbols aviscous selectivity-Llincrease of... m Evolume (void) fraction electrical potentialV 'Yactivity coefficient-11viscosityPa s Athermal conductivityJ m-IK"I /lchemical potentialJ mol-I Vstoichiometric coefficient-1tosmotic pressureN m-2 Pdensitykg m-3 crinterfacial tensionN m-I 'tstressN m-2 'ttortuosity-'\)diffusion volumem3 mol-I COangular speedrad S-I ~friction coefficient, structuredN s mol-Im-I 'Ifpotential of a speciesJ mol-I Sfriction coefficient, unstructuredN s mol-Im-I 19 20Mass Transfer inMulticomponent Mixtures Superscripts * PI 11* o DI,2 d VI,2 pr DxI=I 1,2 pfq T,.ej vT ui X',X" Subscripts Cw dp Dt,ejf \j (I,M VC xIa' xIf3 T,.ej UI, U2 UI. Ui Uj (v average of x in a film .: pressure of pure '1' (vapour pressure) .: volume of pure' 1' Q in free space in the dispersed phase boiling pressure V1,2at x,= 1 pressure of '1' if a reaction were to go to s;guilibrium reference thermal velocity velocity of '1' at the average composition x in two different phases concentration of water diameter of particle or pore effective Fick diffusivity of '1' freevolume friction coefficient between'1' and the matrix molar volume of a chain element species'1' at different positions 2 there is a domain where the component activity decreaseswith increasing mole fraction(see alsoFigure8.6).Thisdomainisboundedbythetwospinodal points.Withinthe spinodal domain the mixture is completely unstable:a component tends to diffuse to regions where it has a higher concentration. The mixture then splits in two parts:one with a high, and one with a low mole fraction of that component. spinodal points ..,.."."" ....., /I ~~'i' moves to regions with ,Whigher concentrations ~demixing Fig.8.6Instability between the spinodalpoints Acompleteanalysisof whathappensisoutsidethescopeof thisbook.We have, however,summarised the main results in Figure 8.7.For A< 2 there is nodemixing. The critical point (Xl= 0.5, A= 2)is at the boundary of the demixing zone. For A> 2 we see the spinodal zone where demixing occurs spontaneously. It is bounded by two supersaturationzones.Fluidswithacompositioninsuchazonecanexist indefinitely,buttheyaremetastable.Anydisturbancetendssplitthemintotwo liquids with compositions on the binodal curve or phase boundary. i A, non-ideality 4 2 critical point -jone liquid f--0 0 Xl 1 unstable supersaturation spinodal curve binodal curve (phase boundary) Fig.8.7Demixing due to non-ideality 88Mass Transfer inMulticomponent Mixtures 8.5 Maxwell-Stefan versus Fick Thisbook usestheMaxwell-Stefan relationstodescribemasstransfer.Most other texts use theclassical Fick description,and most diffusivitydata you willencounter in literature areFick coefficients.So it isimportant tohaveanidea of the relations between the two kindsof coefficients.In the conventional theory the effects of non-idealityareincorporatedin theFick diffusivity.In thegeneralised Maxwell-Stefan (MS) approach they are part of the driving force (Figure 8.8). Maxwell Stefan: non-ideality in the driving force: t d(ln(Ylxl)) _uI -u2 -----'-----'-'-'----'-'-'---x2--dzDI,2 Fick: non-ideality in the diffusivity: \dx JI =-CDI2-1 ,dz FickD1,2=f=l+xl d(lnYl) MS-Dt,2/,dx1 ", thermodynamic correction factor D example:InYI DI,2 Fig.8.8Difference between the Maxwell-Stefan and the Fick descriptions of diffusion For a binary mixture therelationbetween Fick and MSdiffusivities isgiven in the figure.The relation is not difficult toderive:you need to replace the velocities in the MSequation by fluxeswith respect to the mixture,and then to arrange the equations in the Fick form.For our simple modeltheratio of the twodiffusivitiesisgivenat thebottomof thefigure.Proponentsof theFicktheoryoftencallthisratio'the thermodynamic correction factor' . Figure8.9summarises the behaviour of theFick diffusivities;itshowsthe ratioof theFick to the MS-diffusivities. The MSdiffusivity isusually a positive, monotonic and well-behaved function.For an ideal binary solution the Fick and MSdiffusivities areequal.Innon-idealsolutionstheFickcoefficientshowsaminimumat intermediateconcentrations.Thisminimumbecomesprogressivelydeeperasthe non-idealityincreases.AtthecriticalpointtheFickdiffusivitybecomeszero.A further increase in the non-ideality gives a spinodal zone where the Fick coefficients arenegativeandsupersaturationzoneswheretheyrapidlyfalltozero.Clearlythe behaviourof Fickdiffusivitiesisconsiderablymorecomplicatedthanthatof MS diffusivities. This is even more so in multicomponent mixtures.

1 '-2 \ o 8.Non-idealities D's positive, well behaved in ideal binaries D,,2 =),,2 in non-ideal binaries, D,,2 varies considerably \\............../---- ---- @ D,,2 =0 at a critical point D, 2 falls rapidly in super-saturation zones, is zero at spinodal points and negative in between. forIn y\= A(l- x} -1'-----'------' o Fig.8.9Relation between theMS and the Fick diffusivities 8.6 When can we neglect Non-ideality? 89 Even in this book, you will see that we often neglect the effect of non-ideality - there are often no activity coefficients in our equations.When is thisallowable? To put it another way:howlarge is the error that we make? Figure8.10 gives the ratioof the realdrivingforcetotheforcewhen thesolution isconsidered ideal.It isthesame formulathatweencountered when comparingthebinary Fick andMaxwell-Stefan diffusivities (Figures 8.8and 8.9). F_dln(xi) ideal- dz F'.xact= dln(rix)1+ dln(r) F;deald In( X)d In( X) Fig.8.10Theeffect of neglecting non-ideality in the driving force AlookatFigure8.9tellsusthatwecanneglectnon-idealities(forthesystem considered here): for any sufficiently dilute solution, and over the whole concentration range if the parameter Ais small (sayIAI < 0.1). It will also be clear that neglecting non-idealities leads to large errors in concentrated non-ideal solutions. It is unacceptable near a critical point. In multicomponent mixtures,wemust furtherspecifythederivative in Figure8.10. Forexample,if weconsidercomponent'1'inaternarymixture,thenwemust specifyhowtheothercompositionschangeaswetakethederivative.Wemight 90Mass Transfer inMulticomponent Mixtures specifyx2 =constant,X3= constant or something in between. What we do can make a difference to the effect ofnon-ideality. 8.7Mass Transfer inLiquid-Liquid Extraction Severalseparationprocessesmakeuseof strongnon-idealitiesinliquids.Well-knownexamplesareliquid-liquidextraction,andextractiveandazeotropic distillation.In these processesthereareatleast threecomponents.It isoutside the scope of this book totreat them properly,but we will discuss a fewaspectsof their mass transfer characteristics. The example used isa liquid-liquidsystem that extractswater fromacetone,using glycerolasasolvent.Figure8.11showsthecompositiondiagramof thesystem. Acetoneandglycerolarepoorlymiscible:theyhaveademixingzonealongthe bottomlineof thetriangle.Water ismisciblewithbothacetoneandglycerol.For water concentrations that are not too highthereare two phases:one rich inacetone andonerichinglycerol.Pointsonthephase boundarythatarein equilibriumare connected bytielines.Theseslopeupwardtotheright,showingthatwaterhasa preference for the glycerol phase. t water (3) 0.4 0.2 0.0 0.0 acetone (2) 0.40.8XlI glycerol (l) Fig.8.11Ternary diagram for liquid-liquid extraction,showing the demixing zone, tie lines,anda possible set of compositions in the two phases andat the phase interface Incomputercalculations,athermodynamicmodelprovidesinfonnationonthe activitycoefficientsof thecomponents.Thismodelcanalsocalculatethephase boundaries and tie lines in the diagram.The extraction is done in a series of counter-currentstages.Each of theseconsistsof amixer formasstransfer andasettler for separatingthephases.Wewillonlyconsiderasinglestageandneglectanymass transfer in the settler (Fig8.12).In the mixer,both phasesare considered tobe well 8.Non-idea/ities91 mixed.Thereisamasstransfer resistanceon eithersideof the interface. In sucha stronglynon-idealsystem,wecannotassumeabootstraprelationbeforehand;we have tosolve all mass balance and transport relationssimultaneously.The variables in this model are shown in the figure.There are thirteen unknowns!This complexity, bytheway,hasnothingtodowithourusingtheMSequations.Thethirteen equations can be chosen as follows: three overall mass balances for the three components, three equilibrium relations at the interface, two transport relations in the glycerol film, two transport relations in the acetone film and threepartialmassbalancesrelatingthefluxesthroughtheinterfacetotheflow changes in one of the phases. ,, X I ~XI'YNI XI XlaX I ~ ,, X 2 ~X2'YN2X2a X 2 ~ L' / L N3 X2 , / , ,,{ .//jlows X3 13 unknowns,but thetransport relations remainsimple: _Llal x2NI-xIN2 X3 N I- ~ N 3 - X- =+ -"---'---'---"-1lljk1,2Ck1,3C et cetera... Fig.8.12Evena single, wellmixed stage,hasmany unknowns Asyoucansee,the bootstrapping of thisproblem isnot atrivial task.On theother hand,thetransportrelationsof eachfilmseparatelyarenothingspecial.Wehave writtenoneof themdownfortheglycerolphaseinFigure8.12.It isasimple extension of the binary case: hardly anything to be remarked upon. Youshouldnothavetheillusionthatyoucansolveallproblemsinliquid-liquid extraction bysolving the MS-equations.Extractioncalculationsarestill unreliable. Some reasons are: poor accuracy of thermodynamic data, lack of good data on multicomponent diffusivitiesand mass transfer coefficients, and theoccurrenceof interfacialflowsandemulsionformationduetointerfacial tension gradients. 92Mass Transfer inMulticomponent Mixtures 8.8 Summary For non-idealmixturesyouneedathermodynamicmodelthatpredictsactivity coefficients.Withsuchamodel,includingthenon-idealityeffectsinour mass transfer theory is trivial. Figure 8.3 shows how that is done. Dilute systems behave as ideal systems. Non-idealitiesbecomeparticularlyimportantinthevicinityof acriticalpoint. There is,however, nothing special in the behaviour of the MSdiffusivities around suchapoint.Thesameistrue inthesupersaturation zones.Thisisbecausethe effect of non-idealities is included in the driving forces. In contrast to the MSdiffusivities the Fick diffusivities of classic mass transfer do show a complicated behaviour in non-ideal mixtures. Inliquid-liquidextraction(afairlytypicalnon-idealprocess)bootstrapping requiresthesimultaneoussolutionof manyequations.Thetransportrelations, however, are nothing special. 8.9 Further Reading Reid,RC. Prausnitz, J.M.and Poling, B.,1987, The properties of gases and liquids. Fourth edition, McGraw-Hill, New York. A good introduction to the estimation of activity coefficients in non-ideal liquid mixtures. Krishna,R, Low, C.Y., Newsham, D.M.T., Olivera-Fuentas, C.G.and Standart, G.L. (1985)Ternarymasstransfer inliquid-liquid extraction.Chem.Eng.Sci.,40,893-903. Experimental data on diffusionin the system glycerol-water-acetone to demonstrate the strong influence of thermodynamic non-idealities near the critical point. Pertler, M.,Blass, E.and Stevens, G.W.(1996)Fickian diffusion in binary mixtures that form two liquid phases.A.l.Ch.E.1., 42, 910-920. Taylor,Rand Krishna,R.(1993)MulticomponentMassTransfer.Wiley,New York. A good compilation and analysis of literature data. 8.10 Exercises 8.1Distillation of Methanol-Water(Mathcad).In thisfilewehavetaken theliquid phasetobe ideal.Youaretoincludetheeffectof non-idealityandtonotehowit affectsthefluxes.Thisproblem issimilar tothat in Figure8.3.Themodelforthe activity coefficients hasalready been programmed for you. 8.Non-idea/ities93 8.2Non-idealBinaryDistillation(Mathcad).Thiscalculationshowsthatthe differenceapproximationisquitegood,eveninstronglynon-idealmixtures.You must,however,includetheeffectofnon-idealityinthedrivingforce.Inthe calculationwecomparetheone-stepdifferenceapproximationwithaneight-step calculation(which issupposed tobe accurate).The part that youare to program is quitesmall;theassignment takessome time because you need to read a rather large file. 8.3CompositionProfilesinaNon-idealBinaryMixture.Belowyouseeasimple diffusioncell,withaporousplatebetweenthetwocompartments.Theleft-hand compartmentcontainsonlycomponent'1'ofabinarymixture:theright-hand compartment only component'2'. The twocomponents have equal molar volumes, so the volume and mole fluxesmust add upto zero.Diffusivities in the mixture are given in Figure 8.8.On the right side of the figure,you see the concentration profiles forsteadydiffusionforfourvaluesof thenon-idealityparameter A.Youareto answer the following questions: a)What happenstothetluxwithincreasingnon-ideality?Youcaneasilyseethis qualitativelywhenyourealisethat allsolutionsareidealin thedilutelimits,so you can use Fick's law there. b)For A = 2 and Xl = 0.5 the Fick diffusivity becomes zero. Does the fluxgo to zero? How can this be? c)What happens when A > 2? d)For A> 2,Figure8.9showsnegative Fick diffusivitiesin thecentral part of the compositionrange.Whatwouldthesemean?Whydotheynotshowupin the figure above? The calculation of the profiles above isgiven in a separate Mathcad file.You do not need to read this file toanswer the questions. A=OA=lA=2 increasing non-ideality ) A=3 8.4A SingleStageLiquid/LiquidExtractor(Mathcad).Thisfilecalculatesthe compositions and fluxes in a single stage such asin Figures 8.11and 8.12. It uses the 94Mass Transfer inMulticomponent Mixtures ternary Margules equation for the solution thermodynamics.The result isa large and complicated file:it took usaweektoget it running.Youwillneed timetoread it, andgetanideaofwhatitdoes.Youwillseethatthefluxesinliquid/liquid extraction can behave in ways which are counter-intuitive. 8.5MyersonandSenol (A.1.Ch.E.J., 30,1004-1006 (1984))havemeasured Fick diffusioncoefficientsnear thespinodalcurveforureain aqueoussolutions.Using their data tryto estimate theMaxwell-Stefan diffusivitiesand check the validityof the trends shown in Figure 8.9. 95 DiffusionCoefficients Our transport relations require Maxwell-Stefan diffusion coefficients or diffusivities. Thischaptersummarisesourknowledgeof theseforgasesandnon-electrolyte liquids. 9.1Diffusivities in Gases Gasdiffusivities can be estimated quite well from the kinetic theoryof gases.Inthe simplest form of this theory, molecules are hard spheres.Friction between species is caused by momentum transfer in binary collisions of unlike molecules. The theory is not extremely difficult, but too long to give here. Important results (Figure 9.1) are: friction is proportional to the velocity difference between the species, the diffusion coefficients do not depend on the gas composition, diffusivitiesincreaserapidlywithtemperatureandareinverselyproportionalto the pressure, diffusivities are lower for larger and heavier molecules. molecules are little (hard) spheres moving around with their thermal velocity and occasionally bumping into each other d= dl+d2 1,22 Fig.9.1Diffusion coefficients ingases Realmoleculesarenothardspheres.Athighertemperaturestheyundergoharder collisionsandthenshowalower effectivediameter.The empiricalmodificationof thetheoryin Figure9.2 takesthisinto account.To usetheempiricalequation you needmolar diffusionvolumes.Thesearetabulatedinhandbooks;theirvaluesare roughly two thirdsof themolar volume of the liquid at its normal boiling point.The equation is easy to use.Errors are less than10%up to pressures of 1 MPa.At higher pressures the assumptions of the kinetic theory, only binary collisions, and a'free path'much larger than the molecule diameters, are not correct. 96Mass Transfer inMulticomponent Mixtures empirical equation: .. ' diffusion volume::; liquid volume,nf mo[-l IH2N2 CO2 I NH3I H2O i I7.0717.926.914.9I 12.7I Fig.9.2Modification of the kinetic gas theory Figure9.3givesanexampleof theuseof theempiricalformula.Diffusivitiesof gases at ambient conditions are around10-5 m2 s-'. example: N2(1)CO2 (2)T=300Kp=105Pa 3001.75 DJ2= 3.16xlO-8 2 ,105[(17.9 X 10-6)113 + (26.9 x 10-6)113] Fig.9.3Calculation of the diffusivity inN2- CO2 mixtures Frictionisdeterminedbybinarycollisions.Formixturesofmorethantwo components you can therefore simply use the binary coefficients (Fig.9.4).This isa distinct advantage of the Maxwell-Stefanapproach.However,youcan only dothis with ideal gases at low to moderate pressures. binary: o D,,2 does not depend on composition muIticomponent: D. = f)bmary I,'1,1 also does not depend on composition mole 12 Fig.9.4Using binary gas diffusivities inmulticomponent calculations 9.Diffusion Coefficients97 9.2 Diffusivities in Liquids For liquids thesituation islesssatisfactory.There is nogood,accuratetheory.The 'sphere in liquid'theory that we have seen earlier gives the right orders of magnitude fordilutebinarysolutions:around10-9 m2 S-I(Figure9.5).The'constant'inthe equation dependson the ratio of the diametersof thetwospecies molecules.When thesoluteismuchlargerthanthesolvent,theconstanthasavalueof31t;for molecules of similar size it is nearer 21t. f)_RT 1')1t)l112dl\. /dl d2 this 'constant' varies with 31tthe size ratio of the species:' ,--------------..21t---,--- dl = d2 D - RT-10-92-1J 12 - 2/- msI ,21t)llh d1 d2 I1t:J' "'-- ------

o24 Fig.9.5Diffusivity of a dilute sphericalspecies ina liquid For thesphere-in-liquid model,weneed thediametersof bothmolecules.We can estimatethesefromthecubicrootof thediffusionvolumes.For thelatter,wecan also take two thirds of the molar volume of the liquid at its boiling point. These rules onlyapplytosimpleliquidssuchashydrocarbons,wheretherearenospecial interactions between the different molecules. In many mixtures,thesolute binds to solvent molecules. This is especially so,if the two can form a hydrogenbond. Molecules that can doso, usually contain positively charged hydrogen (such asin hydroxyl or ammonia) and negatively charged oxygen (alsoin hydroxylgroups).Bonds can alsooccur betweensolvent molecules.When suchbondsoccur,youshouldcalculate diametersfromthevolumeof thebonded molecule.Hydrogen bonding always occurs in mixtures involving water - this is one of the reasons why diffusivities in aqueous solutions are difficult to estimate. If thebinarydiffusivitiesatinfinitedilutionareknown,valuesatintermediate compositionscanbeestimatedbylogarithmicinterpolation(Figure9.6).The exampleshowninthefigureconcernstwosimplehydrocarbons,hexaneand hexadecane.Theirmixturesarealmostideal,andtheMaxwell-StefanandFick 98Mass Transfer inMulticomponent Mixtures diffusivities almost coincide. The two species differ in size, and the diffusivity varies by a factor of three. It is highest when the more mobile species (hexane) dominates. hexane (1)-hexadecane (2) DI,2,DI,2 1O-9m2s-1 o MS and Fick diffusivities almost equal~2""DI 2 ologarithmic interpolation: oD larger in the mobile fluid 2 0.5'-----____---'-____----l o Fig.9.6Diffusivity inanidealliquid mixture Ethanol-water(Figure9.7)isanon-idealmixture.Here,theFick coefficientgoes throughaminimum,where itsvalue isone third of the Maxwell-Stefan diffusivity. TheMS-diffusivitydoesnotvarymuchwithcomposition;againlogarithmic interpolation is not bad. 5 ethanol (1) - water (2) DI,2,DI,2 1O-9m2s-1 o MS and Fick diffusivities differ-DI2~Dt 2 o only atXI= 1 and x2 = 1 is D1,2=D1,2 ointerpolation roughly logarithmic forD 2 0.5 25C ro0DD0 LS DD DI,2 r. 0 --vD12 u ~ ' o ~ o Fig.9.7Diffusivities ina non-idealliquid mixture JL..il"", OCT 0 9.Diffusion Coefficients 99 Many non-ideal mixtures behave like ethanol-water, but there are also systems where theFickdiffusivityislargerthantheMS-diffusivity.Thiscanbecausedby dimerisation or oligomerisation of one of the species. Our last binaryexampleisastronglynon-idealmixture:methanol-hexane(Figure 9.8).Here,thetwosetsof diffusivities differ by asmuch asa factorof thirty.If the temperatureisafewdegreeslower,thissystemdemixesandtheFick coefficient goes through zero.The Maxwell-Stefan coefficients donothing special; you can still estimate them roughly using logarithmic interpolation. 5 methanol (l) -hexane (2) 2 D[,2,D[,2 1O-9m2s-[ 0,5 0,2 I:;----~ 000 0 [:gD12 0 .of 0 uu 0 40C 0 ~ 0 0 ~ 2 ~ Pr> v 0 0.1?'---___--'I_l___----'1 x Fig.9.8Diffusivities ina strongly non-idealliquid mixture Inmulticomponent liquids the situation is worse again.Experimental data exist only forasmallnumberofsystems.Alsotheyusuallyonlyencompassafew measurements.Forlackofsomethingbetteryoumightusethepseudo-binary logarithmicinterpolation of Figure 9.9.In thisternary interpolation,weneedthree diffusivitiesatinfinitedilution.We consider the diffusivity forspecies'1'and'2'. We then need: 100Mass Transfer inMulticomponent Mixtures the diffusivity of a trace of '1' in'2', the diffusivity of a trace of '2' in'1' and the diffusivity of traces of '1' and '2' in'3'. Thefirsttwoarebinarydiffusivitiesaswehavealreadydiscussed.Thethird parameter isverydifficulttomeasure.Thefewdata thatwe havesuggestavalue close to the geometrical average of the'13' and'23' diffusivities in pure'3'. Interpolation in mixtureswith three or morecomponents isstill guesswork.We can sayonethinginitsfavour.Thefriction termscountingmostheavilyarethose involving the highest mole fractions.These are the terms in which the compositions are relatively close to the corresponding binary compositions. for lack of better ... logarithmic scale mole fraction composition triangle i 32 Fig.9.9Using logarithmic interpolationinmulticomponent liquids We finishbyshowingasingleexample of data onaternarymixture(Figure9.10). Thesystemistoluene(l)-chlorobenzene(2)-bromobenzene(3).Thespeciesare numbered in order of decreasing mobility and increasing molar mass. We see that the toluene-chlorobenzene'12'diffusivitiesarehighest,andalsothattheyhavethe highestvaluesinthemobileliquid.Conversely,thelowestdiffusivityisthatof chlorobenzene-bromobenzene'23'inbromobenzene.For thissimple,almostideal mixture,logarithmic interpolation doesa good job; the measured points (not shown, but six in each diagram) differ less than10% from the values drawn. 9.3 How do you measure diffusivities? We couldspend a whole book on that;here you onlyget a brief descriptionof two commonmethods.Theyarethediffusioncellwithaporousmembraneandthe Taylor or dispersion capillary Figure 9.11). 9.Diffusion Coefficients 2 [!]3/2 mole traction composition triangle Fig.9.10Three diffusivities in the mixture toluene (1 )-chlorobenzene (2)-bromo-benzene(3)at 30C. The triangles are the best-fit logarithmic interpolations (2)Taylor capillary sharp peak in. CDdiffusion cell followconcentration changes~105 s residence time ~1000 s broad peak out peak smeared out by velocity profile smearing reduced by radial diffusion Fig.9.11Two techniques for measuringdiffusivities 101 102Mass Transfer inMulticomponent Mixtures The diffusion cell is filled with twosolutionswith different compositions.You then followthe compositions in the two compartments asa function of time. From the rate of equilibration you can deduce the binary diffusivity. Diffusion cell experiments are simple, but they take a long time and have to be done carefully. Amoderntechniqueusesa'Taylorcapillary'.Herethebulkfluidispumped continuously throughanarrow capillary.At a certain moment apulse isinjected in theinletoffluidwithadifferentcomposition.Thepulseisdispersedbythe combined action of the flow profile (which pulls the pulse apart) and radial diffusion, whichnarrowsthepulse.Fromthepulsebroadeningyouobtainthebinary diffusivity.Measurements with a capillary take much less time than with a diffusion cell. However, they do require a very stable and sensitive detector. BothtechniquesyieldtheFickdiffusivity'withrespecttothevolumeaveraged velocity of the mixture'.You need species volumesand anaccurate thermodynamic modeltotransformtheseintoMaxwell-Stefandiffusivities,usingformulafrom Chapter 8 and the Appendix 2 on Units. 9.4 Summary Foridealgases,thekinetictheoryofgasespredictsbinarydiffusivities. MulticomponentgasdiffusivitiesintheMaxwell-Stefantheoryareidenticalto these binary diffusivities. For liquids, you should try at least to get experimental data for dilute solutions. If youcannotget anything,usethemodifiedsphere-in-liquidmodelor one of the empirical relations in the literature. Fornon-dilutebinarysolutions,usedilutediffusivitiesandlogarithmic interpolation. For non-dilute multicomponent mixtures you have little choice but touse pseudo-binarylogarithmicinterpolation.The method isnot reliable,but there isnothing better. Exceptforgases,ourknowledgeof multicomponentdiffusivitiesisstillvery incomplete. Measuring multicomponent diffusivities is not easy. 9.5 Further Reading Clark,W.M.andRowley,RL.(1986)Themutualdiffusioncoefficientof methanol-n-hexane near the consolute point. A.!. Ch.E.]., 32,1125-1131. Source of datain Fig.9.7. Fuller, E.N., Schettler, P.D.,and Giddings, J.C.(1966).A new method for prediction 9.Diffusion Coefficients of binary gas-phase diffusion coefficients, Ind. Eng.Chem., 58,19-27. Theempirical modification of the kinetic gas theory for estimation of binary dif.fusivities is from these authors. 103 Reid,RC. Prausnitz, J.M.and Poling, B.(1987) The properties of gasesand liquids. Fourth edition, McGraw-Bill, New York. This text gives all standard methods to estimate dif.fusivities in binary mixtures of gases and liquids. Taylor,R.andKrishna,R(1993)MulticomponentMassTransfer.Wiley,New York. Discusses estimation of dif.fusivities in binary and ternary mixtures along with worked numerical examples.Extensive literature references. Tyn,M.T.andCalus,W.F.(1975).Temperatureandconcentrationdependenceof mutual diffusion coefficients of some binary liquid systems, J.Chem.Eng.Data, 20, 310-316 Source of data in Fig.9.6. Vignes,A.(1966)Diffusion in binarysolutions.Ind.Eng.Chem.Fundam.,5,189-199. Thelogarithmic interpolation formula for binary liquid dif.fusivities used in Fig.9.5 was first suggested by Vignes. Kooijman,B.A.andTaylor,R(1991)Estimationof DiffusionCoefficientsin Multicomponent Liquid Systems, Ind.Eng.Chem.Res., 30,1217-1222. This develops the logarithmic interpolation formula for multicomponent liquids in Fig.9.B. Rutten,Ph.W.M, Diffusion inLiquids(1992)PhDthesis,Universityof Delft.Delft University Press, Delft, ISBN 90-6275-838-X / CIP 9.6 Exercises 9.1Spheresin aGas.We consider small spheres(2)movingin agas(1).When the size of thespheres reducestomolecular dimensions,the frictioncoefficient will be given by the kinetic gas theory: 104Mass Transfer inMulticomponent Mixtures (Thisisthe fonnula in Figure 9.1,rearranged.)Larger spheres willmove according to Stokes'equation. For such spheres, we expect the friction coefficient to follow: Sl,2= 3njl1hd2 Sketch the behaviour of the frictioncoefficient asafunctionof the diameter of the sphere.Useafigurewithlogarithmicscales.Atwhichdiameteristhecrossover between the two fonnulae? Isthe friction coefficient of the smallest spheressmaller or larger than predicted by Stokes'law? Use the following parameters: jl = 6.02 X 1023 mol-1 R = 8.31Jmol-1 K-1 T = 300 K p = 105 Nm-2 Ml= 0.03kg mol-1 9.2MutualandSelf-Diffusivities(Mathcad).Thediffusivitiesandfriction coefficientsthatweuseinthisbookaremostlymutualcoefficients.Theyare characterised by two different subscripts such asin DI2 or 1;12'In literature you will oftenfindself-diffusivities.Thesearediffusivitiesofaspecieswiththesame physicalpropertiesasthesolvent.Theycanbemeasuredfairlyeasilybyusing radioactiveisotopes,orbytaggingmoleculeswithNMR-techniques.Theself-coefficients have two identical subscripts. It is often assumed that the mutual friction coefficientscanbecalculatedfromtheself-frictioncoefficientsasthegeometric average: Sl,2= ~ S I , 1. S2,2 Themutualcoefficientsinabinarygasarefoundbyrearrangingthefonnulain Figure 9.1: The self-coefficients are given by the same fonnulae: _(;3 jlp( 2dd2 (2)-0.5 SI,1- fn (RT)o.5Ml You are to investigate how well the geometric average fonnula agrees for ideal gases withdifferentspeciesdiameters.Youmayassumethatthemolarmassis proportional to the cube of the diameter. 9.3 Diffusivity of Gaseous NH3-H20(Mathcad).This fileuses the empirical fonnula in Figure 9.2 to calculate the diffusivity in a mixture of two gases. You can use it for other gases than the example by modifying the constants. 9.Diffusion Coefficients105 9.4* Diffusivity of Dilute Spheres in a Liquid (Mathcad). Calculate the diffusivities of hexanediluteinhexadecane,and of hexadecanedilutein hexane.Youaregiven viscosity and volume data; you are to use the method in Figure 9.5. 9.5*Fick and MSDiffusivitiesof Acetone-Benzene(Mathcad).Thisfileusesthe NRTL model for solution thermodynamics and logarithmic interpolation (the Vignes equation)to calculate the two diffusivities ina non-ideal binary. If you haveNRTL parametersanddiffusivitiesinthedilutesolutions,youcanalsouseitforother mixtures. 9.6*PlottingtheDiffusivitiesofAcetone-Water(Mathcad).Fromalistof experimentalFickdiffusivitiesandthermodynamiccorrectionfactorsfromthe UNIQUAC equation,we obtain MSdiffusivities.We then compare the two in a plot of diffusivity versus composition. 9.7*Diffusion near theCritical Point (Mathcad).Inthisfileyou use experimental Fick diffusivitieson thesystemmethanol-n-hexane.Thesystemis just abovethe temperaturebelowwhichdemixingoccurs,andtheFick diffusivitiesshowadeep minimum. (The result is shown in Figure 9.8.) 9.8Three-dimensional Plotting of Ternary Diffusivities(Mathcad).This fileallows you to explore in three dimensions the behaviour of the MS-diffusivities in a ternary liquid mixture. 9.9 The six infinite dilution MSdiffusivities in the system acetone (1) - benzene (2) - carbon tetrachloride (3) at 298K are: = 2.75 x 10-9 m2 Is;=4.15x1O-9 m2 /s = 1.70 x 10-9 m2 /s;D;r-71= 3.57 x 10-9 m2 /s = 1.42 X10-9 m2 /s; = 1.91 X10-9 m2 /s Estimate thevaluesof Dijata compositionof Xl= 0.7, x2 = 0.15,X3=0.15.Usethe formula shown in Figure 9.9. Also use the data given here asinputs to Example 9.8. 106 Transfer Coefficients This chapter contains relations toestimate mass transfer coefficients, and shows how they are used in multicomponent calculations. 10.1Introduction TheengineeringformoftheMaxwell-Stefanequationcontainsmasstransfer coefficients:one coefficient for each pair of species.At this moment, we can usually only estimate transfer coefficients forsolutes in binary mixtures.The MS-equations allow us to interpolate between these binary extremes. So far,we have used the film theory,where the transfer resistance is due to a stagnant layernexttotheinterface.Thistheoryisonlystrictlyapplicabletothin homogeneousmembranes.For other geometries,and forflowingmixtures,wedo havebetter models.Here,wewillalsoseehowthesecan be used in theMaxwell-Stefan equations. Eventhedilutebinarytransfercoefficientsareoftennotknownaccurately.They depend on the pattern of flowpast the interface,and the flowoften shows transitions thatarenotcompletelypredictable.Also,withdeformingbubblesanddrops,the geometry of the interface may not be known.So predictability varies. It is fairly good for tubes and packed beds, less for single drops, bubbles and particles, and poor for swarms of bubbles, drops and fluidised particles. 'Fairly good'means within thirty percent,'less' is usually within a factor of two,and 'poor'within a factor of ten. For the'poor'cases,you can sometimes find better data for specific situations. Below,wegivea collection of our favouriterelations fordilutebinarymixtures.In these systems,the Fick and theMaxwell-Stefan approachesare identical,sowe can replace theFick diffusivitiesby MSdiffusivities.Even thoughwe have spent much timeon thiscollection,wemustwarn that youuseit atyour own risk.Thispart of the subject of mass transfer is still far from reliable. 10.2 Dimensionless Groups Relationsformasstransfercoefficientsaretraditionallygivenintermsof dimensionlessgroups.Figure10.1showstheimportantgroups:theSherwood, Reynolds and Schrnidt numbers. 10.Transfer Coefficients107 Youseeonedifferencewithothertexts:inourproblemsthereareoftenmany Sherwood numbersandmanySchmidt numbers.Thereisonesetof suchnumbers foreachpair of components.Thereis,however,onlyoneReynoldsnumberfora givenproblem.BoththeSherwoodnumberandtheReynoldsnumbercontaina diameter.Which diameter depends on the problem considered.For transfer toa tube wall,itwillbethetubediameter;fortransfertoasphericalparticle,theparticle diameter.For relationsdescribingmorecomplicatedgeometries,youmayhaveto check how the diameter has been defined. Sherwood number kddiameter Sh. =-'-"-mass transfor '"fJ. filmthickness '" Reynolds number Re. =pvd Re < 1 laminar fluid flow I,'11 Re1 turbulent Schmidt number _11Sc "'"1 in gases Scij---mixture property ,pDi,j Sc "'"103 in liquids Fig.10.1Dimensionless groups usedinmass transfer TheSherwood numberscontain themasstransfer coefficients.Youmayregardthe Sherwood number asthe ratio of system diameter to film thickness.However, do not takethistooseriously.The'film'concept isnot a verygood one and wewill often find that the'film thickness' is different for every pair of components. The Reynolds number describes the character of the flow.For small values, the flow islaminar;forlargevaluesit isturbulent.Inflowthroughtubesthereisasharp transition atRe "'"2000.This,however,isanexception,not the rule.In other flows thetransitionisgradualandmorecomplicated.Manyflowsshowwhirlsand instabilitieswhenRe;::: 100.TheReynoldsnumbercontainsavelocity.Thisis usually the'superficial velocity': the volume flowdivided by the whole cross section of the equipment. TheSchmidtnumberonlycontainsfluidproperties.It tellshowquicklyvelocity fluctuationsare evened out by viscosity, compared tothe smoothing of concentration differencesby diffusion.For gasesthe Schmidt number hasavalue close to one.In ordinaryliquidsthisgoesuptoaroundonethousand.Fordiffusionofsmall molecules in viscous liquids, the Schmidt number might have a value of one million. Heat Transfer Coefficients We end thisparagraph withasideline.Aswe haveseen in Chapter 7,masstransfer and heat transfer oftenoccur simultaneously.To describe this,wealsoneed aheat transfer coefficient.Diffusion in adilutestagnant mixture followsFick's law;heat conduction in a stagnant mixturefollowsFourier's law.These have thesame form. 108Mass Transfer inMulticomponent Mixtures Alsothemechanismsof convectivetransportof matterandheataresimilar.The result is that you can use the same relations for obtaining heat transfer coefficients as for mass transfer coefficients. You only need to change a fewvariables. Inallrelations,theSherwoodnumberisreplacedbyaNusseltnumber,andthe Schmidt number by a Prandtl number (Figure 10.2). This is the same as replacing the ratioof themasstransfercoefficienttothediffusivitybythegroupshowninthe figure,and the diffusivity by the thermal diffusivity. mass transferheat transfer Sh= f(Re,Sc)same asNu= f(Re,Pr) replacekif)byhit..1/ thermal diffusivity f)bya/a = t../(pcp) Fig.10.2Heat transfer coefficients from mass transfer relations Thisanalogybetweenmassandheattransferisonlyvalidforarestrictedsetof conditions.However,theseare just the conditionswheremasstransfer coefficients areusuallydetermined:indilutesolutionswithonlyconcentrationgradientsasa driving force.So the analogy is a truly useful tool. 10.3 Tubes and Packed Beds Tubes,capillariesandhollowfibres,areusedformasstransferinmembrane modules.Figure10.3showssucha tube,with fluidflowingfromleft toright.The flowis laminar. The fluid contains a species that is being removed through the wall. At first,the concentration is high across the whole diameter.However, material near thewallisrapidlyremoved,andaconcentrationprofileforms.Afteracertain distance,the concentration in the middle of the tube begins to decline aswell.From that point onward, the profile retains its shape, but becomes lower and lower.So we havetwozones:the inlet zoneand a zonewith a fullydeveloped declining profile. The behaviour of the masstransfer coefficient between fluidand wall isdifferent in thetwozones.In the inlet region,it decreaseswith increasingdistance;in the fully developed zone it has a constant value. The relation for the inlet zone givesthe average valuesuptoa distance z.To obtain thelocaltransfercoefficientsyoushouldreplacetheconstant1.62by1.08.In liquids,where diffusivitiesarelow,the inlet zoneisoftenquitelong;in gasesit is usually negligible. In a large tube (FigurelOA)flowwill be turbulent.The masstransfer coefficient to thewallcanbe muchhigher inthiscase.Alsohere,transfer isfasterin theinlet. 10.Transfer Coefficients109 However, the inlet zone is short (of the order of ten tube diameters)and the effect of the inlet correction is usually small. laminarRe < 2000 -v +-----41 fully developed11-----J:3.66 Sh .. = 1.62 Re 1/3 ScY3. ()-1/3 I"I"dd20D1,/ Fig.10.3Transfer coefficients ina tube frombulk to wall-Iaminar flow turbulentRe> 2000Shi,; = 0.027(1+ (see text)'--v----' dinlet correction 1--J00 integration with thermodynamics 00 looks like'chemical engineering'0 Fig.13.8Advantages anddisadvantages of the three descriptions 13.6 Units Not only can we choose different systems to describe diffusion; there are also many different ways of choosing compositions.Three of these arecovered in Appendix 2 on Units. They use: 1.mole fractions (as in the greater part of this book), 2.mass fractions (when mass transfer is combined with hydrodynamics) and 3.volume fractions (which often show diffusion theory most simply). 13.7 Further Reading Kuiken,G.D.C.,1994,Thermodynamicsof irreversible processes:Applicationsto diffusion and rheology. John Wiley, Chichester. Thisbookpresentsarigoroustreatmentof irreversiblethermodynamicsand provides a theoretical foundation for the Maxwell-Stefan approach. Taylor,R.andKrishna,R.(1993)MulticomponentMassTransfer.Wiley,New York. Interrelationship between the Fick,Maxwell-Stefan and TIP approaches. 13.Why we use the MS-equations153 13.8 Exercises Thefirstthree filesare not really exercises.They just demonstratethat the different descriptionsof masstransfer,described in thischapter,givethesameresults.You can skip them if you believe us! 13.1The Effect of the Component Numberingon theFick Diffusivities(Mathcad). Showsthat just changing thesequenceof thespecieschangesthesignsof theFick cross coefficients (as in Figure 13.5). 13.2 MSand Fick are thesame (Mathcad).Shows that you get thesame fluxeswith thetwosetsof equations,independentof theinputvalues.Youwillseethatthe transformationbetweenthetwodescriptionsisquitecomplicated,evenforthis ternary mixture of ideal gases. We do not derive the transformation; we only use it. 13.3MSand TIP are the same(Mathcad).Shows that you get thesame fluxeswith thetwosetsof equations,independent of theinput values,just asintheprevious example. 13.4 Molar,Massand Volume Based Diffusivities(Mathcad).The Maxwell-Stefan equations can be written in different units,asexplained in the Appendix on'Units'. In this exercise you investigate how the choice of units influences the dependence on composition of the Maxwell-Stefan diffusivity in a binary mixture. Mass Transfer through a Solid Matrix 156 1 SolidMatrices In thesecondpartof thisbook,weconsider masstransfer throughmixtureswhich contain a solid matrix. This chapter introduces the subject and paves the way. 14.1The Applications Permeablematriceshaveagreatnumberof applicationsintechnology.Themost important ones are: as membranes in separation technology, as sorbents in adsorption and chromatography columns,and as heterogeneous catalysts for chemical reactions. This book is on mass transfer,and not on the separate applications. So we will not be discussingthesetechnologiessystematically:onlyasexamplesof masstransfer problems.However,toputtheexamplesinsomeperspective,wegiveabrief descriptionofthethreetechnologiesandtheirclassification.Webeginwith membrane processes. 14.2 Membrane Processes Amembranemoduleconsistsof(atleast)twocompartments,separatedbythe membrane (Figure 14.1). The feed enters the upstream side and splits into a retentate andapenneateflow.Theseflowsshouldhavedifferentcompositions.How different,dependsonthemembrane,thedrivingforcesusedtotransfermatter throughthemembrane,andthepropertiesof thecomponentsinthefeed.The components moving through the membrane are called penneants. feedretentate drivingforcemembranepermeate Fig.14.1A simple membrane module Membrane processes can be roughly classified according to the main driving force in theprocess(Figure14.2).Thefigurehasthreerows.Processesin thetoproware mainly driven by composition gradients, in the second by electrical gradients, and in the third by pressure gradients. 14.Solid Matrices 157 Of thecomposition-driven processes,dialysisistheoldestandmost important.In dialysisthesolvent(usuallywater)isthesameonbothsidesof themembrane. Dialysis hasfoundconsiderable application in thelaboratory,in medical techniques and in the foodindustry.In pertraction,twodifferentsolventsare used.Pertraction hasmuch incommon with theprocessof liquid-liquid extraction.It isnot(yet)of great significance. Of theelectricallydrivenprocesseswewillonlydiscusselectrolysis,membrane electrolysis and electrodialysis. Electrodialysis uses two kinds of membranes: positively charged membranes which only transfer negative ions, and negatively charged membranes which transfer positive ions. Electrodialysisisusedonafairlylargescale,bothforpurifyingandforconcen-trating electrolyte solutions. driving forces C;imple fluids,you can estimate this minimum volume from the liquid branch of their of state.It isusuallyaboutonequarter of thecriticalvolume.)At higher :emperatures(orlower pressures),thefluidexpandsandwegetfreevolume.The Teevolume isan almost linear function of temperature.Asweshall see, only a small lart of the free volume is actually accessible to diffusing molecules. )tatistical modelswithspherical particles tell usthat thefreevolumedistributesto holes withdimensionssimilar tothose of the particles.The chance of findinga lole larger than a certain volume isa simple function of the freevolume per particle. IV eareinterested inthefractionof thevolumeconsistingof holeslarger thanour nolecules.This is thevolumeavailablefordiffusionalmotion.Youshould havea lookattheformulafortheavailablevolumeinFigure16.4asitplaysan mportant role in the rest of this discussion. Within the exponent, we see a minus sign nd afraction.Themolecularvolumeisontopandthefreevolumeper molecule dow. If we plot thisavailable volume against the ratio of free volume to molecular olume, we see that it stays near zero until the free volume exceeds about one quarter f the molecular volume. It then begins to rise rapidly. closely packed minimum volume Vo (per molecule) this distributes itselfas'holes ': v o actual volume ........ free volume vf T chance of a hole> Vo -I-_--.-''---,__ V f o0.25Vo Fig.16.4Free volume andholes ina pure fluid lassTransition and Self Diffusion "ecanalsoplottheavailablevolumeagainsttemperature(Figure16.5).The mperature at which it begins to rise rapidly is the glass transition temperature. this temperature there isinsufficient available volume to allow much motion. lefluidwillbehaveasasolid,andthere willbelittlediffusion.Abovetheglass msitionpointthischangesrapidly.Simpleliquidsusuallycrystalliseon cooling 184Mass Transfer inMulticomponent Mixtures anddonotshowaglasstransitiontemperature.However,withprecautions,many fluidscan be sub-cooled toform viscous liquidsandglasses.These glasses have an amorphous structure, like that of the liquid frozen in.The free volume theory predicts the glass transition temperature fairly well. little internal motion fraction of volume 'available' T - + - ~ - f - - - -0/1Tg glass transition temperature internal motion possible Fig.16.5The glass transition temperature Our ideas are useful in predicting the self-diffusivity of our fluid(Figure16.6).This isa measure of the local rate of diffusional mixing of molecules.You can determine it by followinga radioactive tracer'1-' with properties identical tothose of the fluid '1' . Aslightlyartificial,butsimplemodelofdiffusionassumesthatmoleculesare arrangedonalattice.Aftera certain period(extremelyshort in practice),theycan moveoneposition.Thishappensrandomlyinoneof sixdirections:up,down, forward,backward,leftorright.Thishappensmany,manytimespersecondand causes diffusion.The theory tells usthat the diffusivity on the lattice is proportional tothe frequencyof thestepsand tothesquareof thelatticedistance.Wecanalso write thisasthe product of displacement distance and the velocity.For molecules in afluid,wewouldexpectthedistancetoberoughlythemoleculediameter.The velocity should roughly equal the thermal velocity. In the free volume theory, we assume that a displacement on the lattice will only take place if thereisa freeposition(a holeof sufficient size)next tothemolecule.The combination of these two ideas yields the formula in Figure16.6. This formula gives reasonableestimates of self-diffusivities.Note that the formula contains the ratioof twovolumes.Thisallowsustoreplacethemolecular volumeand freevolumeper molecule by molar volumes. InaTernary Mixture We will work out the theory fora ternary mixture;the ideas can be extended toany number of components.For mixtures,weneed twomoreassumptions.Thefirstis that allspecies in the mixture contribute tothe freevolume;it is thesum of the free 16.Diffusion in Polymers185 volumespermolemultipliedbytheirmolefractions(Figure16.7).Withthis assumption,wecancalculate thediffusivityof anytracer species'1','2' or'3' throughastagnantmixtureof'1','2'and'3'.Theseeffectivediffusivitiesarea combination of the different Maxwell-Stefan diffusivities in the system, including the self-diffusivities.Thefurthercalculationsaremoreeasilydonewithfriction coefficients. 'self diffusivity'of tracer identical to'1': 1,1'DI exp[- '1.J /'Vfl.J, // -'' '\molar volumes factor from lattice theory: T .n_VIdl-10-82-1 '-1--.-,- ms ,/6\ thermal velocity':"'olecular diameter 300 mSl3 x 10-10 m Fig.16.6Self diffusivity of a tracer ina pure fluid CDfree volumes are additive: @tracer friction coefficients (3x): RT_I'_1'0["1]- I'I'I' --- -exp- - + +D1'. effVf relation between tracer and normal coefficients (3x): Sl.2= Sl',lS2.,2= Sl.2 . 3= @solving forFig.16.7Derivationof the ternary free volume theory Thesecond assumption isthe relation between theself-friction coefficientsand the Maxwell-Stefan frictioncoefficients.Weassumethat themutualcoefficient isthe geometricalaverageof theself-frictioncoefficients.Thissoundsplausibleandhas some theoretical justification. ('Some' means weak, but that we have nothing better.) WenowhavethreeequationsforthethreeMS-frictioncoefficients.Solvingthe equationsforthethreefrictioncoefficientsyieldstheformulaeandcalculation scheme of Figure 16.8. 186Mass Transfer inMulticomponent Mixtures input D1,D2,D3 Vf =x1Vf1 + x2 Vf2 + X3Vf3 I ". ...... s=RT = exp[1 _1D 12 -1 '+ +S =RT r+ = exp[1r+ _13-2D '+ +2 S =RT = exp[ 1 _23 -3D '+ +3 RTRTRT output Dr,2=r- Dr,3=r-D23=-1,21,3 'Fig.16.8Calculationscheme for the free volume theory Some Results Thefreevolumetheoryisreallyasimpletheoryforliquidmixtures.However,it turnsout that we can use it todescribe themovement of small permeants through a polymer. We do that by regarding the polymer as a stagnant component consisting of itschainunits.Apparently,thepermeatingspeciesonly'see'localpartsof the polymer,roughlyof thesizeof achain unit.(The chain unitsdohavealower free volume than the other components, but are otherwise similar). Figure16.9 shows three examples of what the free volume theory can do. It is typical forwhatwewould expect forthediftusivitiesin asystem consisting of a polymer (1), a solvent (2) and a larger solute (3).The solute has a very low concentration. Figure(a)showsthebehaviourof astronglyswollen,rubberymembrane,suchas mightbeusedindialysis.Thediffusivitiesincreasewiththeswelling(ormole fraction of the solvent), but not strongly.With the parameters as chosen, the solvent-solute (2,3)diffusivity ishighest,followedbythe polymer-solvent (1,2)diffusivity. Figure(b)showsthesamemembrane,butnowwithmuchlessswelling.The diffusivities are much lower, especially the polymer-solute diffusivity. If we decrease the freevolume of the polymer (Figure c)we obtain a system with low diffusivities. Theseareaverystrongfunctionof swellingand of thesizeof thepermeant.This simulates the behaviour of a glassy polymer. Asfarasweknow,nobodyhasyetcompletelymeasuredsuchdiagramsforareal ternary mixture with a polymer. .. 16.Diffusion inPolymers 187 r---D1,2-~ I----)2.3 ID2,3 ----f----::----,,---r---D1.3 r---I"" )1,2 f-------i::== D1,3 I---~ D23,--------'" .-./ === )1,27 10-12/ 77' (ab Vf1 -01= ~ . --10-13 (bb Vf1 -01= ~ . --I D== .L:icb r===== 1,3'r -- -Vf1) ~ / '-==O.OJ - ~ 10-14 0.40.5x2 0.60.00.1x2 0.20.00,1x2 0.2 (a) astrongly swollen rubbery polymer (b) a slightly swollen rubbery polymer ( c) a slightly swollen glassy polymer DiO/m2s-1 v,/m3mor1 polymer (1)2xlO-8 Vc=lOxlO-5 solvent (2)2xlO-8 V2 =6xlO-5 solute (3)lxlO-8 l'3==8x 10-5 VfiIV; see above 0.4 (0.3) Fig.16.9Threeresults from the free volume theory Remarks onthe Model Coefficients You may be surprised that the loose set of arguments in the free volume theory leads anywhere.One reason for the success of the theory is the large number of adjustable constants.You will have tofit these in some way to experimental data;in its current form the theory is not accurate enough for real predictions. In the example above, we have nine constants: three pre-exponential factors, three minimum molar volumes, and three free volumes. 188Mass Transfer inMulticomponent Mixtures These parameters usually have physically realistic values. Even so,you can probably better regard them asfitting parameters and not expect them to have much meaning. Before searching forparameters,youshould haveanidea of thepropertiesof each component: the molar mass, molar volume and viscosity. For the mass and volume of the polymer, you may start with that of a chain unit. Thepre-exponentialfactorsusuallyhavevaluesintherange0.5 ... 2 x 10-8 m2 S-I. The values are roughly what you would find from lattice theory. You may expect the smallest or lightest component tohave the highest value.The diffusivitiesvary in a roughly linear fashion with these constants. You can get an idea of the minimum volume from the molar mass and molar volume of a pure species.Youwill probably have toadjust the minimum volumes to .obtain goodresults.Theformof thefreevolume theorypresented hereoverestimatesthe effect of volume.So you may have to choose compressed volume parameters nearer to each other than the pure component data would suggest. Theresultsareverysensitivetothevaluesof thefreevolumesof thedominant components. Rough starting values for the pure components are: ordinary liquids viscous liquids and rubbery polymers glasses and glassy polymers Vfi"'"0.3 V; Vfi"'"0.1 V; Vfi"'"0.05 V; Usually, there will be other constraints, simplifying the search of good constants. For instance,in the example of the previous paragraph, the mole fractionof the solute is zero,soitsfreevolumeplaysno role.Evenso,youwillfindthatyouhardlyever have enough data to fit all parameters properly. Such is life. 16.3 Summary In thefirstpart of thischapter wehavehada look at the diffusivitiesof asmall permeant through a polymer. We have seen - that thediffusivityincreases rapidly with temperature,especially near theglass transition, - that large permeants have very low diffusivities in glassy polymers and - that swelling of the polymer by the permeant (plasticising) greatly increases the diffusivity . In thesecondpart,we havederivedthe freevolumetheory.This theorygivesa fairlygooddescriptionofthebehaviourofdiffusivitiesofbothliquidsand mixturescontainingapolymer.Wehaveextendedittoternarymixturesand shownthatitgivesresultswhichagree(atleastqualitatively)withexperience. The theory does require a large number of fitting parameters. 16.Diffusion inPolvmers189 16.4 Further Reading Wesselingh,J.A.andBoIlen,A.M.,(1997)Multicomponent Diffusivitiesfromthe Free Volume TheoryTrans IChemE,75, Part A,590-601. Amoredetailedversionof thetheoryinthischapter,withmanyexamples.This theoryusessurface fractionsinstead of thevolume fractionsused here;theseare better when the species differ in size,but also a little more complicated. Curtiss,C.F.and Bird,R.B.(1996)Multicomponent diffusion in polymeric liquids. Proc.Nat.Acad. Sei., 93, 7440-7445. The Maxwell-Stefan approach applied to diffusionin polymers. Cussler,E.L.and Lightfoot,E.N.(1965)Multicomponent diffusioninvolvinghigh polymers. m. Ternary diffusion in the system polystyrene 1 - polystyrene 2 - toluene. J.Phys.Chem.,69, 2875-2879. Data showing the strong coupling effects in polymeric systems. Peppas,N.A.andMeadows,D.L.,1983,Macromolecularstructureandsolute diffusion in membranes:An overview of recent theories.J.MembraneSci., 16,361-377. A review. Vieth,W.R.DiffusionInandThroughPolymers,(1991)CarlHanserVerlag, MUnchen / Oxford University Press One of themorerecent books on thistopic.Hardlyconcerned withmulticomponent diffusion. Bitter,J.G.A.TransportMechanismsinMembraneSeparationProcesses(1991) Plenum Press, New York Thisbookcontainsmanyinterestingand novelideas,and alsovaluabledataon diffusionof binariesinpolymers.Weadviseyoutobecriticalonthe formof the Maxwell-Stefan equations used. 16.5 Exercises 16.1BinaryFreeVolumeTheory.Consideramixtureof applymer(1)anda permeant (2). a)Write out the free volume theory equation for SI.2'using Figure16.8. b)Determine the value of SI,2whenx2 ~O. c)Now consider thecase that the permeant hasa mole fractionsmall,but not zero. Rememberthatthefreevolumeof thepermeantmaybeasmuchastentimes higher than that of the polymer. Derive an approximation for SI,2in this case. 190Mass Transfer inMulticomponent Mixtures 16.2 Diffusion in a Polymer (Mathcad).This is a short exercise to let you explore the sharpeffectsofvaryingtheparametersinthefreevolumetheory.It usesthe approximation derived in the previous exercise. 16.33DPlotsof FreeVolumeDiffusivities(Mathcad).Thisfiledemonstratesthe freevolumetheorycalculationsforaternarymixture(Figure16.8).It plotsresults such asin Figure16.9 in a three dimensional fashion.Nice colours ... but you cannot do much except vary the parameters. 191 Dialysis andGas Separation Dialysisand membrane gasseparation aretwoof thesimpler membrane processes. Both use polymer membranes,which we will regard ashomogeneous. These are our first examples where we have a solid matrix; the matrix gives a few complications. 17.1Dialysis Moneywise,dialysisisthe most important membrane process;it isusedon alarge scalefortreatingof kidneypatients(Figure17 .1).Thedialysismoduleremoves substancessuch asurea from blood.The module containsa large number of hollow cellulose fibres.Blood flowsin the fibres,theaqueous rinsingsolution outside.The dimensionsshownaretypicalforthismoment,butsmallermodulesarebeing developed. t 50 mm ~ rinsing solution blood-- - - - - - + - ~ membrane module ./ rinsing solution + urea 200 JlffiJ, t~ : .Llz=lOllm ~..m / hollow cellulose fibres Fig.17.1Treating of blood with dialysis Inamoment,wewillbesettinguptheMaxwell-Stefantransportequationsfor species passing through the membrane (the permeants). Before doing so,we will first havealookattheirequilibriumsolubilitiesinthemembrane(Figure17.2).The membraneisstronglyswollen - theliquidoccupiesavolumefractionof 0.6.The polymer matrixoccupies theother 0.4.Inthisexample,wetakethe matrix tobe a 192Mass Transfer inMulticomponent Mixtures separate phase,not part of themixture.Thetwopermeants,waterandurea donot adsorb specifically,so their concentrations inside the membrane are 0.6 times those outside. at equilibrium: c' = 55 kmolm-3 (1) water (2) urea 0.6 x 0.6 x c= 33kmolm-3 Fig.17.2Compositions inand around the membrane Undertransportconditions(Fig17.3),thecompositionsonthetwosidesof the membrane differ.However,atthetwo membrane-liquid interfaces,westillassume equilibrium.(Theinterfaceisonlyafewmoleculesthick,andpotentialchanges across this zone are usually very small.) This again means that both components have concentrations just insidethemembrane,whichare0.6timestheexternalvalues. Because of this,themole fractionsdonot changeacrosstheinterface.Please note that the accents in the figure now denote a different phase. in this example: same mole fractions on both sides ofan interface bloodrinsing solution 0.03 0.01 Fig.17.3Compositions at the membrane interfaces Figure17.4 gives the transport equations. In dialysis, the only driving forcesare due to activity (composition)gradients.For simplicity we takethe solutions to be ideal. Thefrictionsidesof thetwo equationscontain permeant-permeant'12'termsand permeant-matrix 'lM' and '2M' terms. The velocity of the matrix is zero. Intheswollenmembraneusedhere,alldiffusivitiesareofthesameorderof magnitude.(Mostothermembraneprocessesusedensermembraneswherethe permeant-matrix diffusivities are small. Then friction with the matrix dominates.) 17.Dialysis and Gas Separation 193 difference equations-Ax= x2NI-xI N2 Ikl,2Ckl,MC -Ax2kl,2Ck2,MC k= DI,2k= DI,M 1,2!lzI,M!lz D k2,M!lz Fig.17.4Transport equations for a dialysis membrane For our exerciseweusetheroundedvaluesfordiffusivitiesgiveninFigure17.5. Calculatingthevelocitiesandfluxesofthetwocomponentsissimple.The interesting thing to note is the value of the water flux;it goes into the blood and has a higher value than the flux of urea. transport coefficients: MS-equations !lz =10-5m f) 510-102-Ik5010-6 -I 12 =Xms12=Xms ,-10'-6 kl,M=100xlO D2,M=2xlOk2,M = 20 X 10-6 (1)002 - 0.02N1 - 0.98N2+NI -.- (50xlO-6)(33xI03)(l00xlO-6)(33xI03) (2)+0.02 =0.98N2 -0.02NI+N2 (50 x 10-6)(33 x 103)(20 x 10-6)(33 x 103) Fig.17.5Calculation of fluxes Unless we do something (by adding other solutes to the rinsing solution or applying a pressure difference) this will have disastrous effects (Figure 17.6). Asusual,wehaveonlyneededtransportequationsof twoof thethreespecies. However,in thisexample,theforcebalanceon the matrixalsotellsussomething interesting(Figure17.7).If thereisnopressuredifferenceacrossthemembrane, therewill be nosupport force,and thetwofrictiontermsmust cancel.Weseethat the permeant-matrixtransfercoefficientsgoverntheratioof thetwofluxes.This remains so even in extremely open matrices; friction with the matrix is never totally unimportant. Here,we haveonly looked at transport through the membrane on a localscale.We haveneither considered transportthroughthe boundary layerson bothsidesof the membrane, nor the effects of the flowpattern in the module.We come back to these points in other examples and in the exercises. 194Mass Transfer inMulticomponent Mixtures the patient will swell and burst (not shown) Fig.17.6Theeffect of water entering the patient no support/orce on the membrane: kl,MC k2,MC ratio o/the twofluxes: NI=_ kl,M N2k2,M Fig.17.7Force balance on the matrix and flux ratio 17.2 Gas Separation Gas separation is one of the fastest growing parts of membrane technology. There are alreadymanyunitsinoperationforhydrogenrecoveryfromprocessgasesandfor recoveryoforganicsfromnitrogenandair.Purificationofnaturalgaslooks promising and large-scale enrichment of air is in sight. Ingasseparationprocesses,weapplyapressuredifference.Thiscausesgasesto diffuse through the membrane. The rates depend mainly on: the applied pressure difference, the solubility of the gas in the membrane, and the diffusivity of the gas in the membrane. There are two groups of processes (Figure17.8): those with high upstream pressures, andthosewith(roughly)atmosphericupstreampressures.High-pressuresystems mostlyuseglassypolymermembranes;low-pressuresystemsuserubbery membraneswhichhaveahigherpermeability,butlowerselectivitiesthanglassy membranes.In both typesof process,compression costsaresubstantial,and weuse extremelythinmembranes.Thesemaybeasthinas0.1/lm;theycanonlybe 17.Dialysis and Gas Separation195 handled if they have a thicker (porous)support layer.In the example that wediscuss here, we neglect the effect of the support layer. p""O.1MPa N 2' organicsN 2 ~ - - - - - - ~ - - ~ pS; 10 MPa1ps; 0.01 MPa ~ - - - - ' "N,organicq Fig.17.8High-pressure andlow-pressure gasseparations The modules can use hollow fibres(usually with the high pressure on the outside) or flatmembranesthatarewoundup.Thehollowfibremodulesaresimilartothose discussed under dialysis,but much larger.We discussthespiral woundmodules in Chapter18under reverseosmosis.Toobtain thedesired flowswemayneedvery large membrane areas. These may be hundreds or thousands of square metres. Our example considers theseparation of carbon dioxide from natural gas(methane). Thegasisatapressureof 10 MPa;it containstenpercentof carbondioxide.The scheme shown in Figure 17.9 is to reduce this to five percent. We are to calculate the required membrane area and the methane losses. F=10molS-1 YCH4=0.90 YC02=0.10 I p=10MPa F-P YCH4= 0.95 YC02=0.05 Fig.17.9Asimple separationscheme Tosimplifytheproblem,wetaketheupstreamcompartmentof themembrane system to be well mixed. It then has the same composition throughout. We nowhave a look at the membrane (Figure17.10). The active layer is one micrometre thick and we take it to be a homogeneous polymer. The solubility of both gases in the polymer is proportional to their partial pressures. That of carbon dioxide is higher than that of methane;we can see this from the two Henry constants. Even so,both gases have a low solubility in themembrane material;even at these high pressures it isnot more than a few mass percent. ThedrivingforcesintheMaxwell-Stefanequationsarethepartialpressure differences(as discussed in Chapter12). They are multiplied by the respective mole 196Mass Transfer inMulticomponent Mixtures fractions because we are using the flux form ofthe equations (Figure 17.11). porous support layer ,P2 C=-2RT equilibrium solubilities: CH4(l)CO/2) Cl= HelPIc2 = He2P2 Hel=lxlO-5 He2=2xlO-3 ///------------:.------'Henry constants'mol m -3Pa-1 Fig.17.10The membrane and the solubilities of CH4 andCO2 _ X/).pI= J(2Nr;-" ~ N 2+ _NIN= -kC;/1p1 Ik'kII,MI PI., , ,i,2C".,'I,MC""Pi An,_N-'-NNYN - k/1p2 "'-'f/2_XI2Xz I22- - 2MC"2--x:-- ""+-- ,P zPZ.>kl2 ~k2 MC2 /', / / / small ('no coupling ')DtM =2x1O-11 kl,M = 2 X 10-5 D2M=2X1O-l2 m2s-l ~ , M= 2 X 10-6 ms-l Fig.17.11The transport equations Thefrictionsideof theMS-equationscontainspermeant-permeantandpermeant-matrixfrictionterms.Becauseof thelowpermeantconcentrations,thepermeant-permeant termsarenot important.Thisleadstothesimpledifferenceequationsin Figure17 .11. We take the diffusi vities to be constant; here, that of methane is higher thanthatof carbondioxide.If thedownstreampressureismuchlowerthanthe upstream pressure then we can calculate the fluxes directly.(Check that you do know allparameters inthe fluxequations).Amass balance forcarbon dioxidethengives the requiredmembranearea(here,abouttwenty-fivehundredsquare metres).With thisarea,wethencalculatethemethanelosses(here,aboutfivepercentof the throughput).If thedownstream pressure isnot negligible,you willhavetosolvea number of equations simultaneously, using the results with zerodownstream pressure as a starting point. In a real design, we would do several things more accurately: Wewouldmakeabettermodelof theflowthroughthesystem.Thiswould probably be a plug flow model, or some multi-module configuration . We would usea better thermodynamicmodelforthesolubilities,one that takes the influence of the two components on each other into account. 17.Dialysis and Gas Separation197 Wemight taketheeffectsof pressureandcompositioninthemembraneon the diffusivities into account. Wemighttaketheeffectsof masstransferresistancesonthetwosidesof the membrane into account.(These effects are small in gasseparation because of the high diffusivities in gases.) Finally,fortheseveryhighpressureswemightneedabettermodelforthe potentials in the gases, using fugacities instead of partial pressures. The first point is the most important, but in a multi million euro plant also the effects of the other refinements could be worthwhile.We will explore a fewof these points in the exercises. 17.3 Summary In this chapter we have had a first look at membrane processes. Our first example was on dialysis, a process driven by activity gradients. We have seen that the two components are driven in opposite directions. Theratesaredetermined by frictionbetweenthepermeantsand(usuallymore important) by friction between the permeants and the matrix. Diffusion coefficients between permeants and matrix are very small in tight (non-swollen) membranes, but they increase rapidly when the matrix swells.They also tend to be low for large molecules. Permeant-permeantdiffusivitiesarelowerthaninfreesolution.Theyare important in strongly swollen membranes. Our second example was on the separation of two gases. This process uses a tight membrane. Here,thedrivingforcescontainthegradientsof thepartialpressuresofthe components (at least for ideal gases). The fluxesdepend on thedriving forces,but alsoon solubilitiesand diffusivities of the components. Finding the membrane area and the flowsrequiresa simultaneoussolution of the transport equations and mass balances (and several auxiliary equations). 17.4 Further Reading Wankat, P.C. Rate Controlled Separations, Elsevier 1990 Keurentjes,I.T.F., Ianssen, A.E.M.,Broek, A.P.,van der Padt,A.,Wesselingh,I.A. andvan'tRiet,K