C H A P T E R2 Diffusion in Dilute Solutions In this chapter,we considerthe basiclaw thatunderliesdiffusionandits application to severalsimpleexamples.The examplesthat will be givenarerestrictedto dilute solutions. Resultsfor concentratedsolutionsare deferreduntil Chapter3. This focuson the specialcaseof dilute solutionsmay seemstrange.Surely,it would seem more sensibleto treatthe generalcaseof all solutionsandthen seemathematicallywhat the limit is like. Most booksusethis approach.Indeed,becauseconcentrated dilute-solution solutionsare complex,thesebooks often describeheattransf'eror fluid mechanicsfirst and then teachdiffision by analogy.The complexityof concentrateddiffusion then becomesa mathematicalcancergraftedonto equationsof energyand momentum' I haverejectedthis approachfor two reasons.First,the mostcommondiffusionproblems do take place in dilute solutions.For example,diffusion in living tissuealmost alwaysinvolvesthe transportof small amountsof soluteslike salts.antibodies,enzymes,or steroids' Thus many who are interestedin diffusion neednot worry about the complexitiesof concentratedsolutions;they can work effectivelyand contentedlywith the simplerconceptsin this chapter. Secondand more important,diffusion in dilute solutionsis easierto understandin physical terms. A diffusion flux is the rate per unit areaat which massmoves. A concentration profìle is simply the variation of the concentrationversustime and position. Theseideas are much more easily graspedthan conceptslike momentumflux, which is the momentum per areaper time. This seemsparticularlytrue for thosewhosebackgroundsarenot in engineering,thosewho needto know aboutdiftision but not aboutother transpol'tphenomena. This emphasison dilute solutionsis found in the historicaldevelopmentof the basiclaws sections2.2and2.3ofthischapterfocusontwosimple rnvolved,asdescribedinSection2.L diffusion into diffusion acrossa thin film and unsteady-state casesof {iffusion: steady-state .rninfinite slab. This focus is a logical choicebecausethesetwo casesare so common. For L'xample,diffusion acrossthin films is basic to membranetransport,and diffusion in slabs rr importantin the strengthof welds and in the decayof teeth. Thesetwo casesare the two r.xtretn€sin nature,and they bracketthe behaviorobservedexperimentally.In Sections2'4 .tnd2.5. theseideas are extendedto other examplesthat demonstratemathematicalideas Lrsefulfbr other situations. 2.1 Pioneersin Diffusion 2.1.1 ThomasGraham Our modern ideason diffusion are largely due to two men, ThomasGrahamand \dolf Fick. Grahamwas the elder. Born on December20, 1805,Grahamwas the son of i successfulmanufacturer.At 13 yearsof age he enteredthe University of Glasgowwith :hc intention of becoming a minister, and there his interestin sciencewas stimulatedby ThomasThomson. l-l 2 / Di/Jusionin Dilute Solutions l4 / Pioneers irt Dir t-r \c.--=/ il- D i f f u s i n gg o s ti ( o) F i g . 2 . l - 1C . : fiee diflìsri: F i g . 2 . l - 1 . G r a h a m ' s d i f f u s i o n t u b e f o r g a s eTs h . isapparatuswasusedinthebestearlystudyof diftìsion. As a gas like hydrogen difTusesout through the plug, the tube is lowered to ensure that there will be no Dressuredil'ference. Graham'sresearchon the ditTusionof gases,largelyconductedduring the years 1828to I 833, dependedstronglyon the apparatusshownin Fig. 2. 1-I (Graham,1829, | 833). This apparatus,a "diffusion tube," consistsof a straightglasstube, one end of which is closed with a densestuccoplug. The tubeis filled with hydrogen,and the end is sealedwith water, as shown. Hydrogendiflises throughthe plug and out of the tube, while air diffusesback throughthe plug and into the tube. Becausethe diffusion of hydrogenis fasterthanthe diffusion of air, the waterlevelin this tube will rise during the process.Grahamsaw that this changein water level would leadto a pressuregradientthat in turn would alterthe diffusion. To avoid this pressuregradient,he continuallyloweredthetubesothatthe waterlevelstayedconstant.His experimentalresults then consistedof a volume-changecharacteristicof each gas originally held in the tube. Becausethis volume changewas characteristicof diffusion, "the diffusion or spontaneous intermixtureof two gasesin contactis effèctedby an interchangeof position of infinitely minute volumes,being,in the caseof eachgas,inverselyproportionalto the squareroot of the densityof the gas" (Graham,1833,p. 222). Gtaham'soriginal experimentwas unusual becausethe diffusiontook placeat constantpressure,not at constantvolume(Mason, 1970). Grahamalso perfbrmedimportantexperimentson liquid diffusion using the equipment he workedwith dilutesolutions. shownin Fig.2.l-2 (Graham,I 850);in theseexperiments, In one seriesof experiments,he connectedtwo bottlesthat containedsolutionsat diffèrent concentrations;he waiteil severaldays and then separatedthe bottles and analyzedtheir contents.In anotherseriesofexperiments,he placeda small bottle containinga solutionof kno'uvnconcentrationin a largerjar containingonly water. After waiting severaldays,he removedthe bottleand analyzedits contents. Graham'sresultswere sirnpleand definitive. He showedthat diffusion in liquids was at leasrseveralthousandtimes slowerthandiflision in gases.He recognizedthat the diffusion that "diffusion mustnecessarilyfollow processgot still slowerastheexperimentprogressed, Most important,he concludedfrom the resultsin Table2.1-1 a diminishingprogression." that "rhe quanritiesdiftised appearto be closely in proportion . . . to the quantity of salt in the ditTusionsolution" (Graham,1850,p. 6). In other words, the flux causedby difTusion is proportionalto the concentrationdiffèrenceof the salt. 2.t.2.1d, lE r|lllr' p F l ' f r" r . $sr iW i!ì-.]j .--' :,,lLttiotts l5 2.1 / Pioneersin Dffision t+ \/ ' /h fi Gloss -- otot' H ( o) Fig.2.l-2. Graham's diffusion apparatusfbr liquids. The equipment in (a) is the ancestorof fiee diffusion experiments; that in (b) is a forerunner of the capillary method. :---'.tudr of , Table 2.1-l. Graham's resultsfor liquid difrusion ::1\ UI.e Weight percentof sodiumchloride 1 2 3 : , , : ' 'l S l E t o \ :.ì I. ThiS -:r r. Jlo\ed r .,. :h \\ater, ...j. back - . :: rnthiS : .cld to j ' ' . - t 3 n t .l ì e : . , : .f - . U l t S ' ' . : ì at u b e . -:- ....Lllel)US :.:'.lr'ritely - ---:l:' ltrL)t Of .r.:' ,ìllllsUOl I. . . . . I9 " 0 ) . -:-.,.ìPlllÈnt '- .LltrrnS. .--r-::l'ierent .- . ,:J their . , . : t o nO f - -., . : . r r: . h g .. .t. riiìS ot J irl'iusion ..: ' itrÌlow . , , ' 1 . l.. l - 1 : . , , r s a l ti n . irtTusion ,1 Relativeflux r.00 1.99 3.01 4.00 Source:Data from Graham(1850). 2.1.2 Adolf Fick The next major advancein the theory of diffusion came from the work of Adolf Eugen Fick. Fick was born on September3, 1829, the youngestof fìve children. His ofbuildings. During his secondaryschooling, father,a civil engineer,was a superintendent Fick was delightedby mathematics,especiallythe work of Poisson.He intendedto make mathematicshis career.However,an older brother,a professorof anatomyat the University of Marlburg,persuadedhim to switch to medicine. In the spring of 1847, Fick went to Marlburg, where he was occasionallytutored by Carl Ludwig. Ludwig strongly believedthat medicine, and indeedlife itself, must have a basis in mathematics,physics,and chemistry. This attitudemust have been especially appealingto Fick, who saw the chanceto combine his real love, mathematics,with his medicine. ehosenprofession. In the fall of 1849.Fick's educationcontinuedin Berlin. where he did a considerable irmountof clinical work. In 185t he returnedto Marlburg,wherehe receivedhis degree.His rhesisdealtwith the visualerrorscausedby astigmatism,againillustratinghis determination t o c o m b i n e s c i e n c e a n d m e d i c i n e ( F1i c8k5,2 ) .I n t h e f a l l o f 1 8 5l , C a r l L u d w i g b e c a m e professorof anatomy in Zurich, and in the spring of 1852 he brought Fick along as a prosector.Ludwig movedto Menna in 1855,but Fick remainedin Zurich until 1868. do not dependon diffuParadoxically,the majority of Fick's scientifìcaccomplishments (Fick, 1903).He did physiology .ion studiesat all, but on his more generalinvestigationsof outstandingwork in mechanics(particularlyas appliedto the functioning of muscles),in hydrodynamicsand hemorheology,and in the visual and thermalfunctioningof the human l6 2 / Diffusion in Dilute Solutions body. He was an intriguing man. However,in this discussionwe are interestedonly in his developmentof the fundamentallaws of difTusion. In his first diffusion paper,Fick (1855a) codified Graham's experimentsthrough an impressivecombinationof qualitativetheories,casualanalogies,and quantitativeexperiments. His paper,which is refreshinglystraightforward,deservesreadingtoday. Fick's introductionof his basic idea is almost casual: "[T]he diffusion of the dissolvedmaterial . . . is left completelyto the influenceof the molecularforcesbasic to the samelaw . . . for the spreadingof warmth in a conductorand which hasalreadybeenappliedwith suchgreat s u c c e s s t o t h e s p r e a d i n geol fe c t r i c i t y " ( F i c k1, 8 5 5 a , p . 6 5 ) .I n o t h e r w o r d s , d i f f u s i o n c a n be describedon the samemathematicalbasisas Fourier'slaw for heatconductionor Ohm's law for electricalconduction.This analogyremainsa usefulpedagogicaltool. Fick seemedinitially nervousabout his hypothesis. He buttressedit with a variety of argumentsbasedon kinetic theory. Although theseargumentsare now dated,they show physicalinsightsthatwould be exceptionalin medicinetoday. For example,Fick recognized that diffusion is a dynamic molecular process. He understoodthe differencebetweena true equilibrium and a steadystate,possibly as a result of his studieswith muscles(Fick, 1856).Later,Fick becamemore confidentashe realizedhis hypothesiswas consistentwith Graham'sresulrs(Fick, 1855b). Using this basic hypothesis,Fick quickly developedthe laws of diffusion by meansof analogieswith Fourier's work (Fourier, 1822). He defined a total one-dimensionalflux ./1 as Jt: Ajr: -AD? ! r li ù a ; ffiT: rfr:l][email protected] ffiù @: I F,út lhr (2.1-1) d7. whereA is the areaacrosswhich diffusionoccurs,/1 is the flux per unit a;tal., c1is concentration, and z is distance.This is the f,rst suggestionof what is now known as Fick's law. The quantity D, which Fick called"the constantdependingof the natureof the substances," is, of course,the diffusion coefficient.Fick alsoparalleledFourier'sdevelopmentto determine the moregeneralconservation equation 3r'r / d2c, I AA At' \ a , : D [ * * A , ' à r ) (2.1-2) When the area A is a constant,this becomesthe basic equation for one-dimensional unsteady-state diffusion, sometimescalled Fick's secondlaw. Fick next had to prove his hypothesisthat diffusion and thermal conductioncan be describedby the sameequations.He was by no meansimmediatelysuccessful.First, he tried to integrateEq.2.l-2 for constantarea,but he becamediscouragedby the numerical effort required. Second,he tried to measurethe secondderivativeexperimentally.Like manyothers,he foundthatsecondderivativesaredifficult to measure:"the seconddifference increasesexceptionallythe efTectof [experimental]errors." His third effort was more successful.He used a glass cylinder containingcrystalline sodium chloride in the bottom and a large volume of water in the top, shown as the lower apparatusin Fig. 2.1-3. By periodicallychangingthe water in the top volume,he was able to establisha steady-state concentrationgradientin the cylindrical cell. He found that this gradientwas linear,as shownin Fig. 2. 1-3. Becausethis resultcan be predictedeitherfrom Eq. 2.1-1or from Eq.2.l-2, this was a triumph. GItlN Jlil5 hIr nafl, IJJ Nlffi &iltuf E Àr, [lm' î,& pmrrr I| )11.t ) I / Pioneersin Diffusion l t nls '-- ,--:l .in . -,.1 . '. .- ., .. ,À\ r .-.::lfl i-tl1 - , *::'.ìt -- E (! (J o '6 o CL a , ,ill Distance,z . ìt .-ù . - I : l . t -\ . Fig. 2 I -3. Fick's experimental results. The crystals in the bottom of each apparatussaturatethe adjacent solution, so that a fixed concentration gradient is establishedaìong the narrow, lower partoftheapparatus. Fick'scalculationofthecurveforthefunnelwashisbestproofofFick's law. -h Table 2.1-2. Fick's law for diffu,sionwithout conNection For one-dimensionaldiffusion in Cartesiancoordinates - j t : D ,dc' az For radial diffusion in cylindrical coordinates - j r : D ,d c , ar For radial diffusion in spherical coordinates - j t : D , dc, ar Nole..Moregeneralequations aregivenin Table3.2_l. :le Ir JI - , - : :b e : -. . he But this successwas by no meanscomplete. After all, Graham'sdata for liquids anJlpatedFq.2.l-1. To try to strengthenthe analogywith thermal conduction,Fick used - lower apparatusshown in Fig. 2.1-3. In this apparatus,he establishedthe steady-state - 'ncentrationprofìle in the samemanneras before. He measuredthis profile and then tried Dredicttheseresultsusing Eq. 2.1-2,in which the funnel areaA availablefor diffusion ,ned with the distance:. When Fick comparedhis calculationswith his experimental -r'.ults,he found the good agreementshownin Fig.2.l-3. Theseresultswere the initial . r'rilìcationof Fick's law. ::ìJill . Lrke - J:3ilCe - - - ,l t n e ....,ble ': , - .Ì h l s - i 1llÌl 2.1.3 Forms of Fick's Inw useful forms of Fick's law in dilute solutionsare shown in Table 2.1-2. Each r.lurìtioncloselyparallelsthat suggestedby Fick, that is, Eq. 2.1-I. Each involvesthe 'úrlle phenomenologicaldiffision coefficient.Each will be combinedwith mass balances , analyzethe problemscentralto the rest of this chapter. One must rememberthat theseflux equationsimply no convectionin the samedirection ',' the one-dimensionaldiffusion. They are thus specialcasesof the generalequations lr\en in Table3.2-1. This lack of convectionoften indicatesa dilute solution. In fàct. t8 2 / Diffusíon in Dílute Solutions Fig.2.2-1. Diîfusion across a thin fìlm. This is the simplest diffusion problem, basic to perhaps 80% of what follows. Note that the concentration profìle is independentof the diffusion coelhcient. the assumptionof a dilute solution is more restrictivethan necessary,for there are many concentratedsolutionsfor which thesesimple equationscan be used without inaccuracy. Nonetheless,for the novice, I suggestthinking of diffusion in a dilute solution. 2.2 SteadyDiffusion Acrossa Thin Film In the previoussectionwe detailedthedevelopmentof Fick's law, thebasicrelation for diffusion. Armed with this law,we cannow attackthe simplestexample:steadydiffusion acrossa thin film. In this attack.we want to find both the diffusionflux andthe concentration profile. In other words, we want to determinehow much solutemovesacrossthe film and how the soluteconcentrationchangeswithin the film. This problemis very important. It is one extremeof diffusion behavior,a counterpointto diffusion in an infinite slab. Every reader,whethercasualor diligent, shouldtry to master this problem now. Many will fail becausefilm diffusion is too simple mathematically. Pleasedo not dismiss this important problem; it is mathematicallystraightforwardbut physically subtle.Think aboutit carefully. b,["s 2.2.1 The Physical Situation Steadydiffusion acrossa thin film is illustratedschematicallyin Fig. 2.2-1. On eachside of the film is a well-mixed solutionof one solute,speciesI . Both thesesolutions are dilute. The solutediffusesfrom the fixed higher concentration,locatedat z < 0 on the lefrhand side of the film, into the fixed, lessconcentratedsolution,locatedat z > / on the right-handside. We want to find the soluteconcentrationprofile and the flux acrossthis film. To do this, we first write a massbalanceon a thin layer Az, locatedat somearbitrarypositionz within the thin film. The massbalancein this layer is / I solute \ l: / r a t eo î d l f l u s t o n \ | | / rate ol diffusion \ . . 1 . î . , _ . , _ _ . . . .ì || O U t o l t n e l a v e rI \ a c c u m u l a t i o n /\ i n t o t h e l a y e r a t , / \ " " ; ; ; ' , ' i ? ' ' / Becausethe processis in steadystate,the accumulationis zero. The diffusion rate is the {N :llu - \r tlutictttr - ) / SteadyDiffusion Across a Thin Film 19 .t'iusionflux times the film's areaA. Thus 0 : A ( j t l . - - / rr . + r . 7 (2.2-1) ):i iding this equationby the film's volume,AAz, andreaffanging, -/rl') o:-(rr '-r' \(:+Ar)-zl hen Az becomesvery small,this equationbecomesthe definitionof the derivative " d 0:_;jt (2.2-3) 47. ' .erhaps ,nrbiningthis equationwith Fick's law, - j t : D :r'I11&Iì! r))-)t dct ' 47. () )-4\ .' find, for a constantdiffusion coefficientD, , Juracy. '1) " 0 : D';',' - r? )-51 :r. differentialequationis subjectto two boundaryconditions: z :0, r-..1I1On cr: clo (2.2-6) cl : ct! (2.2-1) . : ' r l\ i O n z: ::rtion .:il and - ,u]t to t:Jster . : : . ,i t l l y . I, :.:.rrn.becausethis systemis in steadystate,the concentrations c16and c17are independent ' lime. Physically, this meansthat the volumes of the adjacentsolutionsmust be much :'rter than the volume of the fìlm. _::J but 2.2.2 Muthematical Results The desiredconcentrationprofile and flux arenow easilyfound. First, we integrate - ).2-5 twice to find ct:albz --r On ..ltlons . , nt h e , ì nt h e itr this. . \\ithin -- ;' constantsa andb canbe found from Eqs.2.2-6 and2.2-l, so the concentrationprofile is 'Z c l : c l o + ( c l /_ c1o) t () ) _a\ :r' li1s41variationwas,of course,anticipated by the sketchtnFig.2.2-1. The flux is found by differentiatingthis profile: jt:-D dc, D , :;(cro-crr) 07. : I: the (2.2-8) (2.2-10) I r3.'rìusethe systemis in steadystate,the flux is a constant. .\s mentionedearlier,this caseis easymathematically.Although it is very important,it . ,itten underemphasized becauseit seemstrivial. Before you concludethis, try someof r' e\amplesthat fbllow to make sureyou understandwhat is happening. 20 2 / Diffusion in Dilute Solutions 2 / SteadyDffisit .:cidentsthatresult 'rembrane. (a) (b) (c) Fig.2.2-2. Concentration profiles across thin membranes. In (a). the solute is more soluble in the membrane than in the adjacent solutions; in (b), it is less so. Both casesconespond to a chemical potential gradient like that in (c). This type of diff . ,,lctly,in termsof : -rossthe membra ,1.which drops .: :ce responsiblefì. npletely is Secti The flux scroS:; ' . r ì l ew i t h F i c k ' : i. . rI - IDH r . is parallelto E. Exampfe 2.2-l: Membrane diffusion Derive the concentrationprofile and the flux for a single solutediffising acrossa thin membrane. As in the precedingcaseof a fìlm, the membraneseparates two well-stirredsolutions.Unlike thefilm, themembraneis chemically differentfrom thesesolutions. Solution As before,we first write a massbalanceon a thin layer Az: -'rerhilitrr on'l .' :r'nneabilitrPs the diffusion; - : 1 t è r e n c ei ns . , 0 : A ( j t l . - , / rl : + r : ) This leadsto a differentialequationidenticalwith Eq. 2.2-5: r) O-Cr 0: Daz' However,this new massbalanceis subjectto somewhatdifferentboundaryconditions: z:0, z,: I, ct:HCrc ' , : m P l € 2 . 2 - 2 :P o r , : l e a r ec h a n s : Solution , n s e ro n È - . 1 "' -:.:ne. Rath:: r ctreficre: - - . r l È t h e n; * ct: HCtt where 11 is a partitioncoeffìcient,the concentrationin the membranedivided by that in the adjacentsolution. This partitioncoeffìcientis a equilibriumproperty,so its useimplies that equilibrium existsacrossthe membranesurface. The concentrationprofìle that resultsfiom theserelationsis l . l - . 1 :\ 1 . : --' ^ . . : .' ct:H('tl+HlC11 -C,r)ì which is analogous to Eq. 2.2-9.This resultlooksharmlessenough.However,it suggests concentrationprofìleslikes thoseinFig.2.2-2, which containsuddendiscontinuitiesat the interface.If the soluteis more solublein the membranethan in the surroundingsolutions, then the concentrationincreases.If the solute is less soluble in the membrane.then its concentrationdrops. Either caseproducesenigmas.For example,at the lefi-hand side of the membranein Fig. 2.2-2(a),solutediffusesfiom the solution at r:y6into the membrane at hi pher concentration. This apparentquandaryis resolvedwhen we think carefully aboutthe solute'sdiffusion. Diffusion often can occur fiom a region of low concentrationinto a region of high concentration;indeed,this is the basisof many liquid-liquid extractions.Thus the jumps in concentrationinFig.2.2-2 arenot asbizarreasthey might appear;rather,they aregraphical ,!L-"' ,|[- \olution.s 2.2 / Steudt'DiJJusionAcrcssa Thin Film 21 accidentsthatresultfrom usingthe samescaleto representconcentrations insideandoutside membrane. This type of diffusion can also be describedin terms of the solute'senergy or, more exactly,in termsof its chemicalpotential.The solute'schemicalpotentiaÌdoesnot change acrossthe membrane'sinterface,becauseequilibrium existsthere. Moreover,this potential, which drops smoothly with concentrarion,as shown in Fig. 2.2-2(c), is the driving fbrce responsiblefor the diffusion. The exact role of this driving force is discussedmore completelyis Sections6.4 and1.2. The flux acrossa thin membranecanbe found by combiningthe foregoingconcentration profilewith Fick's law: IDHl -C11.) ,/t :--(C11y \ fu)r . .r h e .-lllr This is parallelto Eq. 2.2-10. The quantity in squarebrackersin this equationis calledthe permeability,and it is ofien reportedexperimentally.Sometimesthis sameterm is called the permeabilityper unit length. The partition coefficient11 is found to vary more widely than the diffision coefficientD, so differencesin diffusion tend to be less importantthan the differencesin solubilitv. Example 2.2-2:Porous-membranediffusion Determinehow the resultsof the previous exampleare changedif the homogeneousmembraneis replacedby a microporouslayer. Solution The differencebetweenthis caseand the previousone is that diffusion is no longer one-dimensional;it now wiggles along the tortuouspores that make up the rnembrane. Rather than try to treat this problem exactly, you can assumean effective diffusion coefîcient that encompasses all ignoranceof the pore'sgeometry.All the earlier answersare then adopted;for example,the flux is lr: :t the . rhat . - !ú\L\ '-.,ttne - .i l o n s . :l;'tl ltS - r J eo f îrane ,lrion. : ùon- I D-oHf l-,LltCro-Crr) t t l rvhereD.s i, u n"*, "effective" diffusion coefficient.Sucha quantityis a flnction not only of soluteand solventbut also ofthe local seometrv. Example 2.2-3: Membrane diffusion with fast reaction Imagine that while a solute :s diffusing steadily across a thin membrane,it can rapidly and reversibly react with Ither immobile solutesfìxed within the membrane. Find how this fast reaction affects :hesolute'sflux. Solution The answeris surprising:The reactionhasno efTect.This is an excellent :rample becauseit requirescareful thinking. Again, we begin by writing a massbalance .rn a layer Az locatedwithin the membrane: solute \ / / s o l u t ed i î f u s i o ni n \ / a m o u n rp r o d u c e d \ * \ u v c h e m i c a l r e a c r i /o n \ a c c u m u l a t i o " i : \ m i n u s t h a r o u ri Becausethe systemis in steadystate,this leadsto rrt-ìs in 'rh ì,..1 0 : A( jrl: - ,/rl.:+r;)- rrAL.z 22 2 / Diftusion in Dilute Solutions nùúuil! ||ùnu"tr I Porous Diaphragm ,UllL ), ft": fE rems '!,!r fi"-: tilHflraul riil*r Fig. 2.2-3. A diaphragm cell for measuring diffusion coefîcients. Becausethe diaphragm has a much smaller volume than the adjacent solutions. the concentration profile wìthin the diaphragm has essentially the linear, steady-statevalue. ,Jili = ír ftrH,*,r' ff ) u 0:-..À--rr dZ where rl is the rate of disappearance of the mobile speciesI in the membrane.A similar massbalancefbr the immobile product2 gives , ' 't-*# d U - - . . / 1 f / ' 1 47. But becausethe productis immobile, j2 is zerc,and hence11is zero. As a result,the mass balancefor speciesI is identicalwith Eq. 2.2-3,leavingthe flux and concentrationprofìle unchanged. This result is easierto appreciatein physicalterms. After the diffusion reachesa steady state,the local concentrationis everywherein equilibrium with the appropriateamountof do not changewith time, the the fast reaction'sproduct. Becausetheselocal concentrations amountsof the productdo not changeeither.Diffusion continuesunaltered. This casein which a chemicalreactiondoesnot affect diffusion is unusual.For almost any other situation,the reactioncan engenderdramaticallydifferent masstransfer. If the reactionis irreversible,the flux can be increasedmany ordersof magnitude,as shown in Section I 6.1. If the difTusionis not steady,the apparentdiffusion coefficientcan be much greaterthan expected,as discussedin Example 2.3-3. However,in the casedescribedin this example,the chemicalreactiondoesnot affect diffusion. Example 2,2-4: Diaphragm-cell diffusion One easy way to measurediffusion coefîcients is the diaphragmcell, shown in Fig. 2.2-3. Thesecells consistof two well-stirred volumesseparated by a thin porousbarrieror diaphragm.In the more accurateexperiments, the diaphragmis ofien a sinteredglassfrit; in many successfulexperiments.it is just a piece of filter paper (seeSection5.5). To measurea diffusion coefficientwith this cell, we fill the lower compartmentwith a solutionof known concentrationand the uppercompartment with solvent. After a known time, we sampleboth upper and lower compartmentsand measuretheir concentrations. Find an equationthat usesthe known time and the measuredconcentrationsto calculate the diffusion coefficient. ,un &l I ,F \ttlurions 2.2 / SteadyDiJfusionAcross a Thin Film L-') Sucha Solution An exactsolutionto this problemis elaborateand unnecessary. solutionis known but neverused(Barnes,1934). The usefulapproximatesolutiondepends value on the assumptionthat the flux acrossthe diaphragmquickly reachesits steady-state flux is approachedeventhoughthe concen(Robinsonand Stokes,1960).This steady-state trationsin the upperand lower compartmentsare changingwith time. The approximations introducedby this assumptionwill be consideredagainlater. In this pseudosteadystate,the flux acrossthe diaphragmis that given for membrane diffusion: rrl,grn has the I DHI , - l t C r . r o * . r - C lr p p . r ) /r :| t t l Here, the quantity H includes the fraction of the diaphragm'sarea that is availablefor Jiffusion. We next write an overall massbalanceon the adjacentcompartments: dCt.lu*", Vtower# dt -Ajt dCr.upp., Vuoocr---.::lAjt " d Í .' -\ similar .vhereA is the diaphragm'sarea. If thesemassbalancesare divided bY Vru*",and yuppcr, -espectively,and the equationsare subtracted,one can combine the result with the flux tquationto obtain te mass d - nrnlìle ,: .tead) -.nLÌnt of : n t e .t h e :: : rlntost , :': If the .'-:tr\\flilì -3 nluch ,: -,:lbedin . , r coeffi:. l-stirred ,: rriments, , . . ra p l e c e -: .. rie fiÌl ì.lrrlment .-:r-nts and : Jrlculate ^ ;;{Cr : Df(Ct,opp.,- Cl.lo*.,) tu*.,- Ct.upp",) r which 1 \ A H l I YR- - - l - r - l / \vt"*t''v',,") . a geometricalconstantcharacteristicof the particulardiaphragmcell being used. This -.l-ferential equationis subjectto the obviousinitial condition : CÎ.to*..- Cf.uno.t Cl.ror..- Ct.upp., / : 0, - rheuppercomparrmentis initially filled with solvent,then its initial soluteconcentration :ll be zero. Integratingthe differentialequationsubjectto this condition givesthe desiredresult: Cl.lo*". - Cl.upp., : ( ^-ftDr Cî.,n*.. - C?.uoo"' I D:-ln| pt 'tt \ / c P ''. ,' ,c. r - L l u P P c r I \ C'.'u*",- Cr.upper / ,-an measure the time r and the various concentrations directly. We can also determine coeffìcient Seometric factor B by calibration of the cell with a species whose diffusion ro.uvn.Then we can determine the diffusion coefficients of unknown solutes. ) 1 2 / Diffusion in Dilute Solutions Stead,tDiffusion At There are two major ways in which this analysiscan be questioned.First, the diffusion coefficientusedhereis an effèctivevalue alteredby the tortuosityin the diaphragm.Theoreticiansoccasionallyassertthat differentsoluteswill havedifferenttortuosities,so that the diffusion coefficientsmeasuredwill apply only to that particulardiaphragmcell andwill not be generallyusable.Experimentalistshavecheerfullyignoredtheseassertionsby writing D- | /c9. -cP clo \ - r n 1 ' i h ' u c r ' l u P PI e r - Ct.,pr,, B't \C'.'"*", / wherep' is a new calibrationconstantthat includesany torluosityefTects.So far, the experimentalistshavegottenaway with this: Diffusion coefficientsmeasuredwith the diaphragm cell do agreewith thosemeasuredby other methods. The secondmajor questionaboutthis analysiscomesfrom the combinationof the steadystateflux equationwith an unsteady-state massbalance.You may find this combinationto be one ofthose areaswheresuperfìcialinspectionis reassuring,but wherecarefulreflection is disquieting.I havebeentemptedto skip overthis point, but havedecidedthat I had better not. Heregoes: The adjacentcompartmentsare much larger than the diaphragm itself becausethey containmuch more material. Their concentrationschangeslowly, ponderously,as a result of the transfer of a lot of solute. In contrast, the diaphragm itself contains relatively little material. Changesin its concentrationprofile occur quickly. Thus, even if this profile is initially very diffèrent from steadystate,it will approacha steadysratebefore the concentrationsin the adjacentcompartmentscan changemuch. As a result,the profìle acrossthe diaphragmwill alwaysbe closeto its steadyvalue,eventhoughthe compartment concentrationsare time dependent. Theseideascan be placedon a more quantitativebasisby comparingthe relaxationtime of the diaphragm,t21o, with that of the comparrmenrs,1l(Dp) The analysisusedhere will be accuratewhen (Mills, Woolf, and Watts, 1968) trr,t.l,/!:" | /( p Dr1, ( 1 :vai"p,.ogn' ".r + \ Vt,,*., I S m ol l ---diffusion coefficient trio ra-J C the diffu'ior. - :r film. [ì rtc:ir (. i - t l _ ci -- 'j..ìlt.thei.-.. - -D. ) Vuppr, / This type of "pseudosteady-state approximation"is common and will be found to underlie mostmasstransfercoefficients. Example 2.2-5: concentration-dependent diffusion In all the examplesthus far, we haveassumedthat the diffusion coefficientis constant.However.in somecasesthis is not true; the diffusion coeffìcientcan suddenlydrop from a high value to a much lower one. Suchchangescan occur for water difTusionacrossfìlms and in detergentsolutions. Find the flux acrossa thin film in which diffision variessharply. To keep the problem simple, assumethat below somecritical concentrationc1., diffusion is tast,but abovethis concentrationit is suddenlymuch slower. Solution This problem is best idealized as two films that are stuck together (Fi$.2.2-4). The interfacebetweenthesefìlms occurswhen the concentrationequalsc1.. ll = t ) D [)í,iùsiorrAcrossa Thin Film . Solutions :.: :ic diffision -:::,rgllì. . TheO- J.. \(l that the L0rge d if f u s i o n coefficient : ' - : . . L n dw i l l n o t - ' h1 rvriting \ :.ii.theexpef: : : .: : - . d: i a p h r a g m : :: I thesteady: , , : : b r n a t i o nt o , . : r : r . i rl e f l e c t i o n - : .::I hadbetter : : 'iailusethey : ,-. ., J\ a result - ' . : : : ì -r e l a t i v e l y . : 'e. n i f t h i s : -., -:-,tr'betbre - - . . : : t ep r o t ì l e '_ - 'ìrt.Lrtment .ri - Lrll tlllle . z=Q z=zc z=l ::g. 1.2-.1.Concentration-dependentdiffusion acrossa thin fiim. Above the concentration t 1,, ::e drffusion coeffìcient is small; below this critical value, it is larger. '- j: tÌm. a steady-state massbalanceleadsto the sameequation: , d j t dz - -;.ult. the flux j1 is a constanteverywherein the film. However,in the leffhand film - -': .oncentrationproducesa small diffusion coefficient: -.iJ here dc, / t: - D , a?. . ::.ult is easilyintegrated: t.: .rnderlie f, I ita, -D I .ttt dct J,,u - : t h er e s u l t - -.. îdr. wO . . : . : : - . ìi.s n o t . . \e r o n e . : '.:J rroblem - ,.: -:'!r\e this :.-\ tlrgether " 3 . l u a l cs t c . - - (Dc r o - c r . ) ,/r z(. --' risht-handfilm, the concentrationis small, and the diffusion coefficientis large: t.. l r : - D ?az. D :, L _ - aa (clr_cu) 2 / Difrusionin Dilute Solutions /(r The unknownposition2,.can be found by recognizingthat the flux is the sameacrossboth films: I t' : o!!r-r-t' l(c19 - c1,.) The flux becomes D ( c ' r o- c r , . )* D ( c r , - c t t ) .Il - If the critical concentrationequalsthe averageof c1sand c11,then the apparentdifTusion coefficientwill be the arithmeticaverageof the two diffusion coefficients. In passing,we shouldrecognizethat the concentrationprofile shown in Fig' 2.2-4 im' - . efluxacrossthefilmisconstantand p l i c i t l y g i v e s t h e r a t i o otfh e d i f f i s i o n c o e f f l c i e n tTs h is proportionalto the concentrationgradient.Becausethe gradientis largeron the left, the difTusioncoeffìcientis smaller. Becausethe gradientis smalleron the right, the diffusion coefficientis larger. To test your understandingof this point, you should considerwhat the concentrationprofìle will look like if the diffusion coeffìcientsuddenlydecreasesas will help you understandthe next and final the concentrationdrops. Such consi<lerations + :rt(, examplein this section. ui Example 2.2-6: Skin diffusion The diffusion of inert gasesthrough the skin can cause itching,burning rashes,which in turn can lead to vertigo and nausea.Thesesymptomsare believedto occur becausegaspermeabilityand diffision in skin are variable.Indeed,skin behavesas if it consistsof two layers,eachof which has a different permeability(Idicula et at., 1976).Explain how thesetwo layerscan lead to the rashesobservedclinically. Solution This problemis similarto Examples2.2-l and2.2-5,butthe solution is very complex in terms of concentration.We can reducethis complexity by defining a new variable: the gas pressurethat would be in equilibrium w'ith the locctlconcentration. The "concentrationprofìles" acrossskin are much simpler in terms of this pressure'even though it may not exist physically. To make theseideasmore specific,we label the two layersof skin A and B. For layer A, !!u e]l-:u MÍ i ,lliirr .,uÍil. lttfir! [r r, ]|]:,ù, !Îlr]l]rrt[|r i'ì trtrlnr{u gî:L'r |lnùtlmF[]L, fi ilÍlÍ0tclúl'h{r lr"*l !l|r,[. le. u u-s J úuftrv:r nr îîìr !Ín{rXT P t : P t . g a+s - Pt go') i(ttti 3, 5!:i: @lg rùifîrn\E Pil muu:1 and for layer B, f pt : Pti + z f{rt.ti..ue - Pl;) -Ìc The interfacialpressure I ' l t - r ulf iilrhlt(xl.l.;. / D,qHa\ / DnHa\ ,^ r, \ f )Ptc"*\ h - /Pt ilfullilÌ$hì ri"ue k can be found fiom the fàct that the flux throughlayer A equalsthat throughlayer B. li:t:t1 M'urii:'-r' {m:! 2l 2.2/ Stead,-DiJJusionAc'rossa Thin FiLm tn I P1iîP?i p1îp? Loyer P1,qos P2, tissue . , n :ll- -,:'lJ . G o so u t s i d e the body :|lr' 't] ::.1t ':\ 'L.11 z =O . ifc' z= lA z=lA + lA Fig. 2.2-5. Gas difiusion across skin. The gas pressuresshown are those in equilibriurn with the actual concentrations. In the specifìc case considered here, gas 2 is more permeable in layer B, and gas I is more permeable in layer A. The resulting total pressurecan have major physiologic effècts. - rìll -,|a . . , .J I ì - i I ' , t l, J.-lì :',\o Theseprofiles,which are shownin Fig. 2.2-5,imply why rashesform in the skin. In particular,thesegraphsillustratethe transportofgas I from the suffoundingsinto the tissue and the simultaneousdiffusion of gas 2 acrossthe skin in the oppositedirection. Gas I is more permeablein layer A than in layer B; as a result, its pressureand concentration gradientsfall lesssharplyin layer A thanin layer B. The reverseis true fbr gas2; it is more permeable in layer B thanin A. Thesedifferent permeabilitieslead to a total pressurethat will have a maximum at the rnterfacebetweenthe two skin layers. This total pressure,shown by the dotted line in Fig. 2.2-5, may exceedthe surroundingpressureoutsidethe skin and within the body. If it doesso, gasbubbleswill form aroundthe interfacebetweenthe two skin layers. These bubblesproducethe medicallyobservedsymptoms.Thusthis conditionis a consequence of unequaldifTusion(or, more exactly,unequalpermeabilities)acrossdiffèrentlayersof skin. The examplesin this section show that diffusion acrossthin films can be diffìcult to understand.The difficulty doesnot derivefrom mathematicalcomplexity; the calculation rs easy and essentiallyunchanged.The simplicity of the mathematicsis the reasonwhy Jrffusion acrossthin films tends to be discussedsuperfìciallyin mathematicallyoriented books. The difîculty in thin-film diffusion comes from adaptingthe same mathematics ttr widely varying situationswith different chemical and physical effects. This is what is Jifîcult to understandaboutthin film diffusion. It is an understandingthat you must gain befbreyou can do creativework on hardermasstransferproblems. 2 / Diffusion in Dilute Solutions 28 2.3 UnsteadyDiffusionin a SemiinfiniteSlab We now turn to a discussionof diffusion in a semiinfìniteslab. We considera volume of solutionthat startsat an interfaceand extendsa very long way' Such a solution can be a gas,liquid, or solid. We want to find how the concentrationvariesin this solution as a result of a concentrationchangeat its interface. ln mathematicalterms, we want to fìnclthe concentrationand flux as functionsof position and time' This type of masstransferis often calledfiee diffusion (Gosting, 1956)simply because this is briefer than "unsteadydiffusion in a semiinfiniteslab."At first glance,this situation may seemrare becauseno solutioncan extendan infinite distance.The previousthin-film examplemademore sensebecausewe can think of many more thin films than semiinfinite slabs.Thus we might concludethat this semiinfinitecaseis not common. That conclusion would be a seriouserrol. The important caseof an infinite slab is common becauseany diffusion problem will behaveas if the slabis infinitely thick at shorlenoughtimes. For example,imaginethat one of the thin membranesdiscussedin the previoussectionseparatestwo identical solutions, so that it initially containsa solute at constantconcentration.Everything is quiescent,at equilibrium. Suddenlythe concentrationon the leffhand interfaceof the membraneis raìsed,as shown in Fig. 2.3-1. Just after this suddenincrease,the concentrationnear this left interf'acerisesrapidly on its way [o a new steadystate. In thesefirst few seconds,the concentrationat the right interfaceremainsunaltered,ignorantof the turmoil on the left' The left might as well be infinitely far away; the membrane,for thesefirst few seconds, might as *"ll b. infinitely thick. Of course,at largertimes,the systemwill slitherinto the steidy-statelimit in Fig. 2.3-l(c). But in those first seconds,the membranedoesbehave like a semiinfìniteslab. This example points to an important corollary, which statesthat casesinvolving an infinite slab an<la thin membrane will bracket the observed behavior. At short times, dif1ision will proceedas if the slab is infìnite; at long times, it will occur as if the slab is thin. By focusing on these limits, we can bracket the possiblephysical responsesto different diflision Problems. 2.3.1 The PhYsical Situation The diffusion in a semiinfiniteslab is schematicallysketchedin Fig. 2'3-2' The as slab initially contains a uniform concentrationof solute clÉ. At Sometime, chosen although increased, abruptly and is suddenly interface the time zero, the concentrationat the solute is always presentat high dilution. The increaseproducesthe time-dependent concentrationprofile that developsas solutepenetratesinto the slab. We want to fìnd the concentrationprofile and the flux in this situation,and so again we needa massbalancewritten on the thin layer of volume A Az: / soluteaccumulation\ -in volumeAAz / \ rate of diffusion / \ \into thelaYeratz ) f rateof diffusion\ / [*:::'l'i1"' (2.3-r) ln mathematicalterms,this is !o +u rt1 : A (.i tl-. ,/rl.+r .) 12.3-2) 2.3 / UnsteadyDffision in a SemiinfiniteSlab C o n c e n î r o t i o np r o f i l e i n o m e m b r o n eo l e q u i l i b r i u m !l -4 . 29 .i-Lrll .i1'rll '': Ir) ..,1Ìl : .L n l C o n c e n t r oi fo n p r o fi l e s l i g h îl y o f î e r î h e c o n c e n t r o f i o no n îhe lefî is roised .ilg ..'rll I ncreose lll 'ne Il \. rl ..'ìi is ,.: :his Limiîing concenlrolion p r o fi l e o f l o r g e f i m e J -ait. , - I - .,uì. "j an Fig. 2.3- I . Unsteady- versus steady-statediffusion. At small times, difTusion will occur only near the lefrhand side of the membrane. As a result. at these small times. the diffusion will be the same as if the membrane was infinitely thick. At large times, the results become those in the thin fìim. ..:ì1es. ' j .ìiìb ' .1. tO - The .ifl AS ::10ugh : sndent :rln we 1 . 3I-) ').3-2) Fig.2.3-2. Free diffusion. ln this case,the concentration at the left is suddenly increasedto a higher constant value. Diflìsion occurs in the region to the right This case and that in Fig.2.2-1 are basic to most diffusion problems. 30 2 / Dilt'usionin Dilute Solutions tr'î 'î : "". We divide by AA: to find Dcr at : - l - ( j t l , + t , - , r rl . \ l (2.3-3) \(:+a:)-:/ We then let A: go to zero and usethe definitionof the derivative àcr _ à/r _ 3t 3: (2.3-4) Combining this equation with Fick's law, and assumingthat the diffusion coefficient is independeo n lt 'c o n c e n t r a l i o w n .e g e t ^ d C r' - D ^1 d'Ct (2.3-s) ' ò ::2 At This equationis sometimescalled Fick's secondlaw, and it is often referred to as one exampleof a "diffusion equation."In this case,it is subjectto the following conditions: /:0, a l l: , cl:cre />0. z:0. cl:cto (2.3-6) (2.3-1) i:@, cl:clx (2.3-8) Notethatbothcl6ÀDdct0aretakenasconsta Tnhtesc. o n c e n t r a t i o n c l - i s c o n s t a n t b e c a u s e it is so fàr fiom the interfàceas to be unaffèctedby eventsthere; the concentrationc1eis kept constantby adding materialat the interface. 2.3.2 Mathematical Solution The solutionof this problem is easiestusing the methodof "combinationof variables."This method is easy to fbllow, but it must have been difficult to invent. Fourier, Graham,and Fick failed in the attempt;it requiredBoltzman'storturedimagination(Boltzm a n ,1 8 9 4 ) . The trick to solving this problem is to definea new variable 'rrom r*' :IIlr :". q Lr U (2.-r-9) '/4Dt The differential equation can then be w ntten as dt, /à(\ dl,, /;t. 1: . I ti1 l : D d, (-- ; t . ,I \àt / \azl (2.3r0) nu or ,, d ' c r' Lt<' , | 1r d r:t - n d< ( 2 . 3r -l ) In otherwords,the partialdiff'erentialequationhasbeenalmostmagicallytransformedinto an ordinarydifferentialequation.The magic also works for the boundaryconditions;from F,q.2.3-7, (:0' cl:cro (2.3-12) 'ffius** "* ) -l / Unstead) DrJJusionin a Semiinfinite Slab a t J 1 nd fiom Eqs.2.3-6and 2.3-8, (:oo, ' l_11 (2.3-13) cl:ctn '\ ith the methodof combinationof variables,the transformationof the initial andboundary ,nditionsis often more critical than the transformationof the differentialequation. The solutionis now straightforward.One integrationof Eq. 2.3-1I gives 'l-:ll dc:1 - :ae \' () 7-14\ d< . rJnt ls :erea is an integrationconstant.A secondintegrationand useofthe boundarycondition .- \ 't-5r cr - clo clc - (2.3l s) - È r t r clu ,t. Olì9 :'ft . , \ l l\ : 'ì-6r ì ì-Rì 1i i a u s e ... ! :,- 1 r )r ò e r f1 : l : ) r q I e "ds t/Í (2.3-t6) .lo - ir is the errorfunctionof {. This is the desiredconcentrationprofile giving the variation - rìcentrationwith position and time. nranypracticalproblems,the flux in the slabis ofgreaterinterestthantheconcentration ' .: itself.This flux can againbe foundby combiningFick's law with Eq. 2.3-15: j, : Ar, -O'i) t11 : 1/OJ"t, :'l4Dt(crc- cr-) (2.3-t7) :.rrticularly useful limit is the flux acrossthe interfaceat z - 0: 'I Van- F',urier, Boltz- I l-g) t -r-10) r 3 - l1 ) :nlrd into , ' n \ :f r o m (2.3-18) ./rl::o: f Dlrt(c11'-cr!) - ' .' lr is the valueat the particulartime / andnot that averagedovertime. This distinction , -,' ulÌportantin Chapter13. - . ::rispoint, I havethe samepedagogicalproblem I had in the previoussection:I must '.e vou that the appzrentlysimple resultsin Eqs. 2.3-15and 2.3-18 are valuable. -'" . . , :r'\ultsare exceededin importanceonly by Eqs.2.2-9 and2.2-10. Fortunately, the î.- - :ìr.rticsmay be difficult enoughto sparkthought and reflection;if not, the examples r,.,rrl shclulddo so. [,r,--nple 2.3-1: Diffusion across an interface The picture of the processin Fig. 2.3-2 nr-- -. ihat the concentrationat z : 0 is continuous.This would be true, for example,if * " . - ' ( ) t h e r e w aas s w o l l e ng e l ,a n d w h e n z< 0 t h e r ew a sa h i g h l y d i l u t es o l u t i o n . -- : r\ er. a much more common caseoccurswhen thereis a gas-liquid interfàceat - = t lrtlinarily,the gas at ; < 0 will be well mixed. but the liquid will not. How will î i::.rcerffect the resultsgivenearlier? Solution Basically,it will haveno effect. The only changewill be a newboundary Pro '1 ì-l?\ tl r 2 / Dffision ín Dilute Solutions -)z wherec I is the concentrationof solutein the liquid, x I is its mole fraction,p ro is its partial pressurein the gas phase,H is the solute'sHenry's law constant,and c is the total molar concentrationin the liquid. The difficulties causedby a gas-liquid interfaceare anotherresult of the plethoraof units in which concentrationcan be expressed.Thesediffìculties require concernabout units, but they do not demandnew mathematicalweapons. The changesrequired for a liquid-liquid interfacecan be similarly subtle. Example 2.3-2: Free diffusion into a porous slab How would the foregoingresultsbe changedif the semiinfiniteslab was a poroussolid? The diffusion in the gas-filledporesis much fasterthan in the solid. Solution This problem involves diffusion in all three directionsas the solute moves through the tortuouspores. The common method of handling this is to define an effectivediffision coefficientD.6 and treatthe problem as one-dimensional.The concentration profile is then ('t - cto - : e ('r\-cru z. " îJ 4 D , r l and the interfacial flux is --*:,lrlr:o: 1/D.xlnt(crc - c1r) This type of approximationoften works well if the distancesover which diffusion occurs are largecomparedwith the size of the pores. Example 2.3-3: Free diffusion with fast chemical reaction In many problems, the diffusing solutesreactrapidly and reversiblywith sunoundingmaterial. The surrounding materialis stationaryand cannotdiffuse. For example,in the dyeing of wool, the dye can react quickly with the wool as it diffusesinto the fiber. How does such a rapid chemical reactionchangethe resultsobtainedearlier? Solution In this case,the chemical reaction can radically changethe process by reducingthe apparentdiffusion coeflìcientand increasingthe interfacialflux of solute. Theseradical changesstandin stark contrastto the steady-state result,wherethe chemical reactionproducesno elTect. To solve this example,we first recognizethat the solute is effectively presentin two forms: (l) free solutethat can diffuse and (2) reactedsolutefixed at the point of reaction. If this reactionis reversibleand fasterthan diffusion. c2: Kct where c2 is the concentrationof the solutethat has alreadyreacted,c1 is the concentration of the unreactedsolutethat can diffuse,and K is the equilibriumconstantof the reaction.If the reactionis minor, K will be small; as the reactionbecomesirreversible,K will become verv larse. l{lllnri;n,.'* ltb nu"^, nÌ urt- - .1/ UnsteadyDffision in a Semiinfinite Slah : r - . :.rfiial - :.nrllar , -:'ì rbout \ IL)f a _ . : : 3 \ 1 5 , .,,lute - / a c c u m u l a t i o\ n I - _ - - . . - . 1b, e ' With thesedefinitions,we now write a massbalancefbr eachsoluteform. Thesemass --,lancesshouldhavethe form - : l.,rfî Of -:- . i : : 1 3a n : ì .e n - J-') - inAA: l - / / | d i f f u s i o ni n \ | \ m i n u st h a to u r) / a m o u n to r o d u c e d bv\ J' | \ r e a c t i oi n A A :, " l / ,r thediffusingsolute,thisis À -d t [ A A z c r l: A ( j t l . - , t rl . + r . )+ r l A L z . :irre r; is therateof productionper volumeof speciesI, the diffusingsolute.By arguments - r.rÌogous to Eqs.2.3-2to 2.3-5,this becomes d(r rl'('t : D * dI u d7' t i ' t :r' term on the left-handsideis the accumulation;the first term on the right is the diffusion " nrinusthe diffusion out; the term 11is the effect of chemicalreaction. \\'hen we write a similar massbalanceon the secondspecies,we find À -[AAec2] : ot -rrALz ocz At -. the ' . .r' ,.i"ì .n,o. è ,,3 aan ::.nical '1-' do not get a diffusion term becausethe reactedsolutecannotdiffuse. We get a reaction :::l that has a different sign but the samemagnitude,becauseany solutethat disappears -. 'peciesI reappears as species2. To solvethesequestions,we first add them to eliminatethe reactionterm: ò à2c, I t ' - l -r ' r ) : D - dt dz' ,r i now usethe fàct that the chemicalreactionis at equilibrium: I lLrieSS . ,r l u t e . Jn l l C a l :l îwo :-::tiOn. .:,ttlon t , , n .I f :J()mg 3 i Jl c , -(triKcll:Ddt dz.' dct D ò 2 ,t Ar lfK 0:2 .:.ri\ result is subjectto the sameinitial and boundaryconditionsas before in Eqs. 2.3-6, - -ì-7. and 2.3-8. As a result, the only differencebetweenthis example and the earlier :oblemis that D/(1 f K) replacesD. This is intriguing. The chemicalreactionhas left the mathematicalform of the answer -:r;hanged,but it has alteredthe diffusion coefficient.The concentrationprofile now is ct -cto (r\-(ro _ - .-,'-J q ù i + z- R Í l l __2 / Dilfusion in Dilute Solutions 34 and the interfacialflux is r K l l " t ( ( r o- ( r à ) ,/rl::o: V6i The flux hasbeenincreasedby the chemicalreaction. These effects of chemical reaction can easily be severalorders of magnitude. As will be detailedin Chaprer5, diffusion coeffìcientstend to fall in fairly narrow ranges. Thosecoefficientsfor gasesare around0.3 cm2/sec;thosein ordinaryliquids clusterabout l0-5cm2/sec.Deviationsfromthesevaluesofmorethananorderofmagnitudeareunusual. However,differencesin the equilibrium constantK of a million or more occur frequently. Thus a fast chemicalreactioncan tremendouslyinfluencethe unsteadydiffusion process. Example 2.3-4: Determining diffusion coefficients from free diffusion experiments Diffusion into a semiinfiniteslab is the geometryusedfor the most accuratemeasurement determinethe concentration of diffusion coefficients.Thesemost accuratemeasurements interferometer,uses Rayleigh the profile by interferometry.One relativelysimple method, index (Dunlop in refractive function a rectangularcell in which there is an initial step shining collinated by profile is followed et al. 1912). The decay of this refiactive index refractive index record the These fringes light throughthe cell to give interferencefringes. v e r s u sc a m e r ap o s i t i o na n dt i m e . Find equationsthat allow this informationto be usedto calculatediffusion coefficients. Solution The concentrationprofìlesestablishedin the diffision cell closely approachtheprofilescalculatedearlierfor a semiinfiniteslab. The cell now effectivelycontains two semiinfiniteslabsjoined togetherat z : 0. The concentrationprofile is unalteredfrom Eq.2.3-15 ct-clg ctx - 7, ,- - :efl Threr t cto "/4Dt wherecle[: (cr-*cr--)/2] istheaverageconcentrationbetweenthetwoendsofthecell. How accuratethis equationis dependson how exactly the initial changein concentration can be realízed in practicethis changecan routinely be within 10 secondsof a true step function. We must convert the concentrationand cell position into the experimentalmeasured refractiveindex and cameraposition. The refractiveindex n is linearly proportionalto the concentratron: p:4.o1u"n,f/rr.1 is the refractiveindex of the solvent. Each position in the camerais proporwhere nro1u.n1 tional to a positionin the diffusion cell: Z:az, wherea is the magnificationof the apparatus.It is experimentallyconvenientnot to measure the positionof one fiinge but ratherto measurethe intensityminima of many fringes. These minima occur when n-no l ttx-tto J12 iffi* \ , 1 i l I 1 (r fl , t 35 ).4 / ThreeOther Examoles \\'herer?{ and r?0are the refiactiveindicesat z - oo and z - 0, respectively;./ is the total numberof interferencefringes,and j is an integercalledthe fringe number. This number r\ most convenientlydefìnedas zero at í - 0, the center of the cell. Combining these iquations, :-- -..Ì3. As :- i . - . .. : 3 r l b o u t l: -:l: -,llUrUOl. , - :'J,lì.lentlY. . j l:\rCeSS. ',"" érl iilfl9e S. J1 2 zi - uJ4a .thereZlis the intensityminimum associated with the 7th tiinge. Becausea and t are -'\perimentallyaccessible,measurementsof 210. "/) can be used to find the diffusion - Lrefficient D. While the accuracyof interferometricexperimentslike this remainsunrivaled, ic useofthese methodshas declinedbecausethey are tedious. r\prriments -- ::'-ifÈl]l€lÌt -.'nîration ' i : 3 r .u s e s :- '- - llinated : - - . . - :. ' i n d e x - - ::lìcients. : , .eÌr'aP' , : .. ,'.r1113i15 ,. :::r'J frOm ,- ::ht'cell. '' - r':'rtration - - ::!re slep . - :-tJit\Ufed :r.11 to the 2.4 ThreeOther Examples The two previoussectionsdescribedifTusionacrossthin films and in semiinfinite ..lbs. In this section,we turn to discussingmathematicalvariationsof diffusion problems. . his mathematicalemphasischangesboth the pace and the tone of this book. Up to now, . e haveconsistentlystressedthe physicalorigins of the problems,constantlyharping on -ituraleffectslike changingliquid to gas or replacinga homogeneousfluid wìth a porous . 'lid. Now we shift to the more common textbook composition,a sequenceof equations . ,metimesasjarring as a twelve-toneconcerto. In theseexamples,we havethreeprincipal goals: ( I ) We want to show how the diff-erentialequationsdescribingdiffusion are derived. (2) We want to examinethe effectsof sphericaland cylindrical geometries. (3) We want to supply a mathematicalprimer for solving thesedifferent diffusion equations. :: all threeexamples,we continueto assumedilute solutions.The threeproblemsexamined '.j\t are physicallyimportantand will be referredto againin this book. However,they are -rroducedlargely to achievethesemathematicalgoals. \ 2.4.1 Decay of a Pulse (Inplace Transforms) propor- :.r nlcasure _r3\.These As a first example,we considerthe diffusion away from a sharppulse of solute rc that shownin Fig.2.4-l. The initially sharpconcentrationgradientrelaxesby diffusion :r the z direction into the smoothcurvesshown (Crank, 1975). We want to calculatethe . rape of thesecurves.This calculationillustratesthe developmentof a differentialequation .-:riiits solutionusing Laplacetransforms. As usual, our first step is to make a massbalanceon the differential volume AAz as - 1rlwn: / solute \ solute \ solute \ / / airiusion outof I aitfusion inro I | | volume volume this this / \ / \ l"::r'f:"',) : Q.4-tl / 2 / Diffusion in Dilute Solutions 36 'itr'r'e Oîlt, rble cond I .-.l:L'gfJII' . . ' ,e t h r - P o s i î i o nz : 0 diffuses as Fig. 2.4- l. Diffusion of a pulse. The concentratedsolute originally located at z t h e t h r e e m o s t i m p o r t a n t c a s e s , alongwiththose T h i s i s t h e t h i r d o f theGaussianprofileshown. in Figs.2.2-l and2.3-2. In mathematicalterms,this is a ;dlt A A : r ' 1l : A i r l .- A . llr,rr . t) L-)\ Dividing by the volume and taking the limit as Au goesto zero gives 3r'r }jt àt ò2. r) 4-7t Combining this relationwith Fick's law of diffusion, 0 c'' - D At à2c, .' àz.' (2.4-4) This is the samedifferent equationbasic to the free diffusion consideredin the previous section.The boundarycon<litionson this equationare as follows' First, far from the pulse, the soluteconcentrationis zero: / > 0. : : oQ, cr :0 Q'4-5) Second.becausediffusion occurs at the samespeedin both directions,the pulse in symmetric: r>0, ':0, dct -o (2.4-6) oz This is equivalentto sayingthat at z : 0, the flux has the samemagnitudein the positive and negativedirections. The initial condition for the pulse is more interestingin that all the solute is initially locatedatz:0: / : 0, M cr : -ó(:) lfilL : - tLa t '' (2.4-1) [email protected] where A is still the cross-sectionalareaover which diffusion is occurring, M is the total amountof solutein the system,and 6(z) is the Dirac function. This can be shownto be a *ttrr ile*- : 'Tltrer Otlter Erttrttltle.s ), '..lil( )/1.\ 3l ,rronabl€ conditionb1,a massbalance: /./ . t,Att::./ f'M , iu,.tAdz: M (2.4-8) thisintegration, we shouldremember tható(z)hasdimensions of (length)-r. rotur this problem,we fìrsrtakerhet-uptu." rransfbrm of Eq.z.+r+withrespectro ,,ll .."0 ' t t '-t. ! . t ( t ' _0 , - o ? a7' ':ere c-1is the transformed concentration. The boundary conditions are d|t z -('l dz .li tìuses as : u ith those a : oc, _ _M/A 2D Q'4-g\ (2.4_10) f't :0 (2.4_r1) - ;' first of thesereflectsthe properties of the Dirac function, bul the secondis routine. ' -.ration2.4-9 canthen easily by integratedto give r14)t cr : aey'slDz r be-/4 nz -:re (2.4-3) cl - | 2.4_4) previous hr'pulse, rl.4-5) in sym- t2.4-6) ìi positive . initially \2.4-7) . rhe total ',ntobea (2.4-t2) a and ó areintegrationconstants.Clearly, a is zeroby Et1.2.4_11. UsingEq.2.4-10, : fìndà andhencelr: : M/E ';; T, yro-1tr- 1/ s/ Dz. (2.4-13) .: inverscLaplacetransformof this function gives _ M/A,-z2141nt lq; Dt- (2.4-11) l:ch is a Gaussiancurve. You may wish to integratethe concentrationover the entire .rcmto checkthat the total solutepresentisM. This solutioncan be usedto solve many unsteadydiffusion problemsthat haveunusual ' ral conditions(crank, l9?5). More important, it is often ur.à to corelate the dispersion ' rollutants, especiallyin the air, as discussed in Chapter4. 2'4'2 steady Dissorution of a sphere (sphericat coordinates) our secondexample,which is easiermathematicary, is the steadydissorutionof ':herical particle,as shown inFig.2.4-2. The sphereis of a sparingly sorubremateriar, hat the sphere'ssizedoesnot changemuch. However,this materiarquickly dissorvesin ': rulroul.ìding solvent'so that solute'sconcentrationat the sphere'ssurfàceis saturated. r"ausethe sphereis immersedin a very largefluid volume,,l..on."nirltion far from -:r-rels the zero. The goal is to find both the dissolution rate and the concentrationprofile around -rere' the Again, the fìrst stepis a massbarance. In contrastwith the pi.urou, exampres, ' nlassbalanceis most conveniently made in sphericalcoordinateso'rigrnating from the 2 / Dift'usionin Dilute Solutions 38 2.4/ ThreeOther This basicdiffere r:Ro. S o l u t ef l u x owoy the sphere werr rhanged, but tl E q . 2 . 4 - l 8f i e dct dr ' -':e a is an inte c 1: f i . D i s t o n c ef r o m s D h e r es c e n i e r \ t diffusion asphere.Thisproblemrepresentsanextensionof Fig.2.4-2.steadydissolutionof this dissolutioncanbe thJoryto a sphericallysymmetricsiìuation.In actualphysicalsituations, '12) by fiee convectioncausedby diffision (seeChapter complicated shell of thickness ar center of the sphere. Then we can make a mass balance on a spherical shell is like the rubber located at some arbitrary clistancer from the sphere. This spherical of a balloon of surface area4T rz and thickness Ar' earlier: A mass balance on this shell has the same general fbrm as those used ai[1sio1 d i f r u s i o\ n_ l / soluteaccumulation \ _ .,) - ( s h e l l / t h e i n t o within the shell ) \ \ o u t o l t h es h e l l/ \ \2.4-t5) In mathematicalterms,this is a " i),*6, tru) : o : (4trr2 i;, - (4trr2 fi6rrt \2.4-r6) is The accumulationon the left-handside of this massbalanceis zero' becausediffusion point by this steady,not varying with time. Novices frequently make a seriousefror at term r2;1 is The wrong. is This side. righlhand on the terms both of ;";""li"g the ,i oit at (r + Ar) in evaluateclat r in the first term; that is,-it is r21;r l,). The term is evaluated the secondtem; so it equals(r * Lr)'(irl'+a'). the Imit If we divide both sidesof this equationby the sphericalshell'svolume and take as Ar '+ 0. we find t d 0:-,;(r'j1) t2.4-17) rar is constant' Combinins this with Fick's law and assumingthat the diffusion coefficient o" - D d .dct --y'dr 12dr2 :thet$oL' (2.4-18) - r \oltrÍions 2.4 / Three Other Examples 39 This basicdifferentialequationis subjectto two boundaryconditions: r : Ro, ct : ct (sat) r:@, ct:0 (2.4-19) () 4-)O\ If the spherewere dissolvingin a partiallysaturatedsolution,this secondcondition would be changed,but the basicmathematicalstructurewould remain unaltered.One integration of Eq. 2.4-18 yields dct a dr 12 () 4-)1t where a is an integration constant. A secondintegrationgives () 4i)\ r Use of the two boundaryconditionsgivesthe concentrationprofile cl : -l \ 1On ,ncanbe . .R tl(sat) - dc, ar D R,, :_jr.1(sat) () 4-)4\ which, at the sphere'ssurface,is lr : r.-1-l5) () 4-)11 The dissolutionflux can then be found from Fick's law: jr:_D, , \ n e S SA r .:rerubber 0 r D -cr(sal) R6 t) 4-)5\ ifthe sphereis twice as large,the dissolutionrate per unit areais only halfas large,though the total dissolutionrate over the entire surfaceis doubled. This examplesforms the basisfor suchvariedphenomenaas the growth of fog droplets .rndthe dissolutionof drugs. It is included here to illustrate the derivationand solution rf differential equationsdescribingdiffusion in sphericalcoordinatesystems. Different .oordinatesystemsare also basicto the final examplein this section. l.:1-I 6) , : : i u s i o ni s . point by r:lll rlJrt is - -1,r)rn . . r t h el i m i t t) . 1 - 1 7 ) li \tant, rl.;1-18) 2.4.3 Unsteady Diffusinn Into Cylinders (Cylindrical Coordinates and Separatio n of Variable s) The final example,probably the hardestof the three,concernsthe diffusion of a .oluteinto the cylindershownin Fig.2.4-3. The cylinderinitially containsno solute.At :mezero,it is suddenlyimmersedin a well-stirredsolutionthatis of suchenormousvolume iet its soluteconcentrationis constant.The solutediffusesinto the cylindersymmetrically. ri-oblemslike this are importantin the chemicaltreatmentof wood. We want to find the solute'sconcentrationin this cylinder as a function of time and Jation. As in the previousexamples,the first stepis a massbalance;in contrast,this mass .,ìanceis madeon a cylindrical shell locatedat r, of area2r Lr, andof volume 2n Lr Lr. -re basicbalance . o l u t ea c c u m u l a t i o n _ solutediffusion - / solutediffusion \ \ / \ n t h i sc y l i n d r i c asl h e l l/ \ into the shell / \ out of the shell / (2.4-26) 2 / Dffision in Dilute Solutions 40 ( c) z tr É. F z trj z o P O S I ft o N Fig.2.4-3.Waterproofìngafencepost.Thisproblemismodeledasdiffusioninaninfinite situation. In reality, the .yiin.l.r, and so representsan extenslon to a cylindrically symmetrìc the grain is faster than with diffusion because ends ofthe post must be considered,especially acrossthe grain. becomesin mathematicalterms LPrrLLrc,l dt : (2trLj), - (2rrL11),',6, Q'4-27) small: we can now divide by the shell'svolume and take the limit as ^r becomes l a a 'òt": -i u'l' t) 4-)?\ equation combining this expressionwith Fick's law givesthe requireddifferential (2.4-2e) à c t_ D _ r , . d ! 'òt r òr òr which is subjectto the fbllowing conditions: 1<0, allr, / > 0, r : Ro, r:0, ct:0 ct : ct (surface) Ac =o lr : o r) 4-30ì (2.4-3r) t) l-7)\ - : / Three Other Examples 4l -. lheseequations, c1(surface)is the concentrationat the cylinder's surfaceand Rs is the -. :nder'sradius. The fìrst of the boundaryconditionsresultsfrom the large volume of -l\)unding solution,and the secondreflectsthe symmetryof the concentrationprofiles. Problemslike this are often algebraicallysimplified if they are written in terms of di- :rrionlesq variables. This is standardpracticein many advancedtextbooks.I ofien find - ' procedure confusing,becausefor me it producesonly a small gain in algebraat the - r 3 n s e o f a l a r g e l o s s i n p h y s i c a l i n s i gNh ot .n e t h e l e s s , w e s h a l l f o l l o w t h i s p r o c e d u r e h e r e lustratethe simplificationpossible.we first definethreenew variables: ('l dimensionlessconcentratio : ln ' . -0 (2.4-33) c I (surface) r dimensionless position: f : & Dt Ré d i m e n s i o n l e s s t i m e : z :.-..- (2.4-34) (2.4-3s) liflerential equationand boundaryconditionsnow become a 0_ l a , l ) e ò'-4afta6 (2.4-36) r:0, all{. r>0, f :1, :\. tlìe ,1.-+-28) (1.4-30) (2.4-40) Si:t)J(€) Eqs.2.4-36 and2.4-40are combined,the resultingtangleof termscan be separated . r o nw i t h S G ) f ' ( € ' ) : K- {. t. tq d . d f ( E- t î , - . d R ( r ): J ' 5 t - , . d r d € € d q t ds(r) e(r) dr t).4-29) (2.4-39) oE r'novice, this manipulationcan be more troublesomethan it looks. 'olve theseequations,we first assumethat the solutionis the productof two functions, : time and one of radius: p(r,€): ).1-21) (2.4-37) (2.4-38) 0:0 'Àe :" - g È:0. '-:la :': ihan 0:l 1 (t) d "df d€' d€ €.fG) (2.4-4t) : rrnefìxesf and changes,, ./(6) remainsconstantbut g(z) varies.As a result. I dg(t) ((r) dt ) t) 4-4) \ \2.4-31) i\ a constant.Similarly, if we hold z constantand let f change,we realize t2.4-32) -, -'r (€) riq' r/6 1, n u , l f tr ts \ F u . , / 2 (2.4-43) 2 / Diffusion in Dilute Solutions A ) Thusthepartialdifferentialequation2.4-36hasbeenconvertedinto two ordinarydifferential equations2.4-42 and2.4-43. The solutionof the time-dependentpart of this result is easy: (2.4-44) wherea'is an integrationconstant.The solutionfor /(6) is more complicated,but straightforward: -r [email protected] J G) : [email protected]) (2'4-45) where -/sand Is areBesselfunctionsand a andb are two more constants.From F,q.2.4-39 we seethafb :0. From Eq. 2.4-38.we seethat (2.4-46) 0: alo@) Becaused cannotbe zero,we recognizethat theremust be an entirefamily of solutionsfor which Je(a,) : (2.4-41) Q ú.. .llllll",, The most generalsolutionmust be the sum of all solutionsof this form found for different integralvaluesof n: ll,llf."! f ù : É t r . È ) : f1 - " ( . 1 , r ' . )J,s 1 a , , € , \ e ' l ' ' (2.4-48) n:l We now usethe initial conditionF,q.2.4-37to find the remainingintegrationconstant(aa'),,'. !; t-f (2.4-4e) We multiply both sidesof this equationby f Je(cv,,f ) and integratefrom f :0 flnd (aa'\. The total resultis then o : itI- t to t : ill r Jll 1 to - I ["- lrn.,-.É)e-o,,' (2.4-50) L a , , J 1 ( a ], , ) nirrrl Aú,i or, in terms of our original variables, I ùu-u -'- @h' ( t(surface) e-DoÎtlRih@,llno) _ | _ )-{ ,t-, (2.4-5r) u , , J 1 ( u , , t ' fR 1 1 l i[],--. -- mtul- This is the desiredresult,though the cy,must still be founclfrom 8q.2.4-41 . This problem clearly involvesa lot of work. The seriousreadershouldcertainly work one more problemof this type to get a feel for the idea of separationof variablesand for the practiceof evaluatingintegrationconstants.Even the seriousreaderprobablywill embrace the ways of avoidingthis work describedin the next chapter. 1lîr' rem-i: i' , .\olutk)ns ,rtvection and Dilute Dffision +) . Jrîferential t2.1-44) Stolic eleclrode of which soluîe c o n c e n l r o î i o ni s r10 D i r e c l i o no f diffusion :, rut straighti I J-215'l Moving electrode ol which soluîe c o n c e n l r o l i o ni s e 1 7 - : Eq.2.4-39 \).4-46) ' . rLrtions for r).1-47) ', "o"'nrn,,rl'ou'o Fig. 2.5- l. Steady diflision in a moving lìlm. This case is mathematically the same as diffusion acrossa stagnant 1ìlm, shown in Fig. 2.2-1. It is basic to the film theory of mass tlanstèr ciescribedin Section I l. L ,r Jifterent 2.5 Convection and Dilute Diffusion (1..1-48) ritl (aa'),r'. t).4-49) ri :l t o In many practicaìproblems,both diffusion and convectiveflow occur. In some --...speciallyinf-astmasstransferinconcentratedsolutions,thediffusionitselfcausesthe :,tion. This type of masstransfer,the subjectof Chapter3, requiresmore complicated r'- -.rl and mathematical analyses. :'reis anothergroup of importantproblemsin which diffusion and convectioncan be : '. :rsily handled.Theseproblemsarisewhen diffusion and convectionoccur normal to : ". - ther. In other words,diffusion occursin one direction,and convectiveflow occursin . -':-:c'ndiculardirection. Two of theseproblemsare examinedin this section. The first, - ..-:f)nacross a thin fowing f1m, parallelsSection2.2;thesecond,diffusioninto a liquid ' .\ a less obvious analogueto Section2.3. Thesetwo examplestend to bracketthe - . :,.ed experimentalbehavior,and they are basic to theoriesrelatingdiffusion and mass (seeChapterI 3). :- .::r coefficients (1.4-50) 2.5.1 Steady Diffusion Across a Falling Film (2.4-51) ,:r.unly work ^ :. .rndfor the .,.ilì embrace The first of the problemsof concernhere,sketchedin Fig. 2.5-1, involvesdiffusion - .- a thin, moving liquid film. The concentrations on both sidesof this film are fixed by , ::.rchemicalreactions,but the fìlm itself is moving steadily.I havechosenthis example ' îr'ccuSeit occursofien but becauseit is simple. I hopethat readersorientedtowardthe -:Lial will wait for later examplesfor resultsof greaterapplicability. . , solvethis problem,we make threekey assumptions: . r The liquid solutionis dilute. This assumptionis the axiom for this entire chapter. I r The liquid is the only resistanceto masstransfer. This implies that the electrode reactionsare fast. 2 / Dffision in Dilute Solutions A A ++ (3) Masstransportis by diffusionin thez directionandby convectionin thex direction' Transportby the other mechanismsis negligible. It is the lastof theseassumptionsthat is most critical. It impliesthat convectionis negligible in the z direction. ln fact, diffusion in the z direction automaticallygeneratesconvection in this direction.but this convectionis small in a dilute solution. The last assumptionalso suggeststhat thereis no difTusionin the x direction. Theremay be suchdiffusion,but it is assumedmuch slower and hencemuch lessimportantin the x directionfhan convectton. This problem can be solved by writing a mass balance on the differential volume W Lx Lz, where W is the width of the liquid film, normal to the plane of the paper: / s o l u t ea c c u m u l a t i o\n -in WA,r'A.: \ i / s o l u t ed i f f u s i n gi n a t z m i n u s\ \ s o l u t ed i f f u s i n go u t a t . r L z .) , / soluteflowing in at x minus \ r \ sofute flowing out at r + Lr ) ( 2 . s1- ) or, in mathematicalterms, (l ^ (crl{ArA:): (tt [(lrWLx). - (71WAr).*o.l * [ ( c r u .W , A z ) - .- ( c ru . ,W A z ) , . + l ' l (2.s-2) The term on the left-handsideis zerobecauseof the steadystate.The secondterm in square bracketson the right-handside is also zero,becauseneithercl nor ur changeswith x. The concentrationc1 <loesnot changewith x becausethe fìlm is long, and thereis nothing that will causethe concentrationto changein the x direction. The velocity u., certainlyvaries with how far we are acrossthe film (i.e., with z), but it doesnot vary with how far we are alongthe film (i.e.,with -r). After dividing by lVA,rA: and taking the limit as this volume goes to zero, the mass balancein Eq. 2.5-2 becomes .I: ^ uJ1 r? 5-31 47. This can be combinedwith Fick's law to give 0 - D r) O-Cl , 1 (2.s-4) a 7.- This equationis subjectto the boundaryconditions z:0, ('r:clo r?5-5ì z: I, cr: clt (2.s-6) When theseresultsarecombinedwith Fick's law, we haveexactlythe sameproblemas that in Section2.2. The answersare cl :t'lnttCti-{lgl7 (2.s-7) D jr:-(.cn-ctt) (2.s-8) I The flow has no effèct. Indeed,the answeris the sameas if the fluid was not flowing. lb uaunl,: ; ' - 1 . t ! ( ,S o l u t i o n s ,JlreCIlOn. Convectionand Dilute Diffusion 45 L i qu i d s o lv e nt :r:uli_eible - :l\ ection ::1\lnalSO :r.but it is ' r. S a t l o n . S o l uî e g o s ,,. r rllume Li. C o nv e c li o n r 5_?\ :l \quare :r .i..The :nc that .r Varies r \\e are hemass Liquid with dissolved solulegos F i g . 2 . 5 - 2u. n s t e a d y - s t a t e d i f f u s i o n i n t o a f a l r i n T gh f ìirsma.n a l y s i s t u r n s o u t t o b e mathematically equivalentto fiee diflusionlseeFig. 2.3-2). rtis basicto the penetratron theory of masstransfèrdescribedin SectionI 1.2. .fhisansweristypical ofmanyproblemsinvolvingdiffusionandflow. Whenthesolutions lilute' the diffusion and convection often are perpendicular to each other and the solution (1.5_4) I ' \ - \ ì (2.s-6) :- -.ent asthat (2.5-7) (2.5-8) i ng. ":aightfbrward. You may armostfeergyppediyou girdedyourserftbr a diffìcurtprobrem - iound an easyone. Restassuredthat more diflìcutt problemsfollow. 2.5.2 Diffusion Into a Falling Fitm The secondproblem of interestis iilustrated schematicailyin Fig. 2.5-2 (Bird, ' 'irr' and Lightfoot, r 960). A thin liquid film flows slowly and wirhour ripptes down a 'urlàce. one side of this firm wets the surfàce;the other side is in contactwith a gas, 'h is sparinglysolubrein the liquid. we want to find out how muchgascrissorves in the J ' ' solvethis problem,we again go throughthe increasingryfamiliar ritany; we write a " balanceas a differentialequation,combine this with Fick's law, and then integrate 'r fìnd the desiredresurt.we do this subject to four key assumptions: . r The solutionare alwaysdilute. I Mass transportis by 3 diffusion anc.lr convection. 2 / Dilfusion in Dilute Solutions ( 3 ) T h e g a si s P u r e . (4) The contactbetweengas and liquid is short' example' The third The fìrst two assumptìonsare identicalwith thosegiven in the earlier liquid. The final in the only phase, gas in the meansthat therels no resistanceto diffusion assumptionsimplifiesthe analysis. shownin the inset we now make a massbalanceon the differentialvolume IV in width, in Fis.2.5-2: / \ massaccumulation : \ w i t h i nW A x A : I / mass<liffusingin at z minus \ diffusing out at ^z+ Lz.) \ r-t-tuts massflowingin at x minus _ - / -l Lx )/ \ massflowingout at x 6/AFinalPerspecti rs again reflectsthe .ult, the solutecan di the exact location o Ì be infinitely far au lhis problemis des ..rsionin a semiinfì , t. Becausethe ml p leis t t c flux at the inte: tr., ,l : Y D This result is parallelto thosefound in earliersections: l à | - (IVA/r).+r.] | - t c 1 a , l l : l v ) I : [(WAjr/r): tdt ì * [ ( l 4 z A z c r u ' ) '- ( W L z c t u . ) , . + 1 . , ] ( 2 . sl -0 ) the left-handside when the systemrs at steadystate,the accumulationis zero. Therefore, vary with both z andx' of the equaiionis zero. No otherterms are zero,because71and c1 goesto zero,we find tf we divide by the volume lv ax az and takethe limit as this volume ol1 0:-;. dz. ,l - (2.5-r r) o ^ (lui dx law and set u' equal We now maketwo further manipulations;we combinethis with Fick's of short assumption the reflects change second This to its maximum vaìue, a consùnt. and ìt interface' the cross to a chance has barely contacttimes. At such times, the solute the reaches velocity fluid the region' interfacial diffusesonly slightly into the fluid. In this a serious not probably is value constant of a maximum suggestedin Fig. 2.5-2, so the use assumption.Thus the massbalanceis ,òct .tetnative vier' balance.ln t . , h i c hl i q u i d :. -statediffer ,ngwith the . equationli . .rer is that t - nethod use , : . c r i b e da : ': of watch - : i d g e .u e 3. \\'erea \ Fina the soluteflow out minusthatin; theright-hand The left-handsideof this equationrepresents side is the diffusion in minus the diffusion out' This massbalanceis subjectto the following conditions: .r :0 . r: 0 . all:, r>0. z:0, cr:ct(sat) --t ( ' 'I - - ( t \ ' , (2.5-l3r (2.5-l4r (2'5-15 with the gasitself, and / where c1(sat)is the concentrationof dissolvedgasin equilibrium threeboundaryconditions is the thicknessof the falling film in Fig. 2.5-2. The last of these is replacedwith z:oo, -" ' ,rrethe answef\ -r answersapP l . T h o s es t u d r J think about : i masstransfe now conside (2.5-12l : DA:r-: dz' 0 (.r/u-"*) x>0. : l - c1(sat) cl:0 (2.5-t6 . .rnd:l ' ,,ndtll :3 lllS: -' Th -.::er : ì--Ji . - '::ì , Solutions The third The final t h ei n s e t ).6 / A Final Perspective This again reflectsthe assumptionthat the film is exposed only a very short time. As a :esult,the solutecan diffuse only a short way into the film. Its diffusion is then unaffected îy the exact location of the other wall, which, from the standpoint of diffusion, might as n ell be infinitely far away. This problem is describedby the samedifferentialequationand boundary conditionsas Jiffusionin a semiinfiniteslab.The soledifferenceis thatthe quantity x /u^u"replacesthe lime /' Becausethe mathematicsis the same,the solutionis the same. The concentration profileis ct c 1( s a )t rI 5-9t ' : - r r i n ds i d e . : r. a n dx . ::'. $e find il.-5-ll) -,'lr'Ì equal n of short " . . ! i e .a n d i t :iitchesthe I I SenOUS (2.5-12) : rishrhand ( 2 . 51- 3 ) (.2.5-14) ( 2 . 5l-s ) ." itself,andI .1r\conditions (2.s-t6) : l-erf-è J4Dx f u^r^ (2.s-11) . r n dt h e f l u x a t t h e i n t e r l a c e is . t rl . - o : r. 5r-0 ) 41 V'Dr^^J,"-cr(sat) ( 2 . 5l 8 -) Theseare the answersto this problem. Theseanswersappearabruptly becausewe can adopt the mathematical resultsof Secrion 2'2. Those studyingthis materialfor the first time often find this abruptnessjarring. Stop and think about this problem. It is an important problem, basic to the penetration theoryof masstransferdiscussedin Section 13.2.To supplya forum for furtherdiscussion, rveshall now considerthis problem from anotherviewpoint. The alternativeviewpoint involveschangingthe differentialvolume on which we make the massbalance.In the foregoingproblem, we chosea volume fixed in space,a volume rhrough which liquid was flowing. This volume accumulated no solute, so its use led to a steady-state differentialequation. Alternatively,we can choosea differentialvolume f l o a t i n g a l o n g w i t h t h e f l u i d a t a s p e e d uT- "h*e. u s e o f t h i s v o l u m e l e a d s t o a n u n s t e a d y - s t a t e differentialequarionlike Eq. 2.3-5. Which viewpoint is conect? The answeris that both are correct;both eventuallylead to the same answer.The fixedcoordinatemethodusedearlieris often dignified as "Eulerian," and the movrng-coordinate plcture is describedas "Langrangian."The difference betweenthem can be illustrateclby the situationof watchingfish swimming upstreamin a fast-flowing river. If we watch the tìshfrom a bridge,we may seeonly slow movement,but if we watch the fish from a freelv floatingcanoe,we realizethat the fìsh are moving rapidly. 2.6 A Final Perspective This chapteris very important,a keystoneof this book. It introduces Fick's law for dilute solutionsand showshow this law can be combinedwith massbalancesto calculate concentratlonsand fluxes. The massbalancesare madeon thin shells. When theseshells are very thin, the massbalancesbecomethe ditferentialequations necessaryto solve the Variousproblems.Thus the bricksfrom which this chapteris built arelargelymathematical: shellbalances,diffèrentialequations,and integrationsin different coord-inate systems. However, we must also see a different and broader blueprint based on physics, not mathematics.This blueprint includesthe two limiting casesof'diffusion acrossa thin fìlm anddiffision in a semiinfiniteslab. Most diffusion problemsfall between thesetwo limits. The first, the thin fìlm, is a steady-stateproblem, mathematically easy and sometimes physicallysubtle.The second,the unsteady-state problemof the thick slab,is a little harder to calculatemathematically,and it is the limit at short times. 48 2 / Dffision in Dilute Solutions In many cases,we can use a simple criterionto decidewhich of the two centrallimits is more closely approached.This criterion hingeson the magnitudeof the Fourier number (length)2 / diffusion \ I ;. .- . lltime) \ coelnclent/ This variableis the argumentof the error function of the semiinfiniteslab, it determines the standarddeviation of the decayingpulse, and it is central to the time dependenceof diffusion into the cylinder. In other words, it is a key to all the foregoing unsteady-state problems.Indeed,it can be easily isolatedby dimensionalanalysis. This variablecan be usedto estimatewhere limiting caseis more relevant.If it is much larger than unity, we can assumea semiinfinite slab. If it is much less than unity, we should expect a steadystateor an equilibrium. If it is approximatelyunity, we may be forced to make a fancier analysis. For example,imagine that we are testinga membrane for an industrial separation. The membraneis 0.01 centimetersthick, and the diffusion coefficient in it is l0 7cm2/sec. If our experimentstake only 10 seconds,we have an unsteady-state problem like the semiinfiniteslab; it they take three hours we approacha steady-state situation. In unsteady-state problems,this samevariablemay also be usedto estimatehow far or how long masstransferhasoccurred.Basically,the processis significantlyadvancedwhen this variableequalsunity. For example,imagine that we want to guesshow far gasoline has evaporatedinto the stagnantair in a glass-fiberfilter. The evaporationhas beengoing on about 10 minutes,and the diffusion coefficientis about0. lcm2/sec. Thus ( l e n g t h) 2 (0.I cm2/sec)(600sec) - l; length: 8cm Alternatively,supposewe find thathydrogenhaspenetratedabout0. 1 centimeterinto nickel Becausethe diffusion coefficientin this caseis about l0-8 cm2lsec.we can estimatehou long this processhasbeengoing on: (10-1cm2) : (10-8 cmzlsec)(time) l: t i m e: lOdays This sort ofheuristic argumentis often successful. A secondimportantperspectivebetweenthesetwo limiting casesresultsfrom compering their interfacialfluxesgivenin Eqs.2.2-10and 2.3-18: jr : D 7 tcr (rhin fitm) i, : f D 1nt Lc1 (rhickslab) Although the quantitiesDll and (Dlrt;l/2 vary differently with diffusion coefficients. they both have dimensionsof velocity; in f'act,in the lifè sciences,they sometimesare called "the velocity of diffusion." In later chapters,we shall discoverthat thesequantities are equivalentto the masstransfercoefficientsusedat the beginningofthis book. Final PersJ Further s , C . ( 1 9 3 4 ) .f R. B., Stewan nann,L. ( 189 . l. (1975) T; r . P J . ,S t e e l e d s .G . \ \ ; r . E . ( 1 8 5 1 r' :. E. ( 185-5r : E.( 185-ir , E . ( 1 8 5 ó ,' - E.( l90i , rB . .( l s : : - L. J.(l!:- T .r l E l " T rlS-r-: r ,l\:, trr,l:; - IJ \ ..tnons ll]ltS 1S :ber l:1: lllllfl9S - -1.'nceOf : .,-:r - statg i . llluCh - 'i\' we r.j '-'lùv be a:-:lbrane r i.:iusion : i.r\ e iìfl ::::'rach a - .\ tar or . - iJ when :: i,t\()line -:r-n -9ollì$ nickel. rtehow nparlng r:llClents, : Illes afe -,r.tntities 1A Final Perspective 49 Further Reading - .:nes.C. (1934).Pà,i'.sics,5.4. - -r. R. 8., Stewart,W. E., and Lightfbot, E. N. (1960). TransportPhenomena.New york: Wiley. - irmann,L. ( 1894).Annalender Physikund Chemie,53, 959. -.,nk.J. (1975). TheMathematics of Dffision, 2nd ed. Oxford: ClarendonPress. . :rìop,P J., Steele,B. J.. and Lane, J. E. (1972). ln.. PhysicalMethodso.fChemistry,-, eds.G.Weissberger and B. W. Rossiter.New York: Wiley. ,.. A. E. (1852). Zeítschríftfùr RationelleMedicin,2,83. -.. A. E. (1855a). Poggendorff'sAnnelender Physik,94, 59. -.. A. E. (1855b).PhilisophicalMagazine,l0,30. ,. A. E. (1856). MedizinischePlrysik Burnswick. -.. A. E. (1903). GesammelteAbhandlungen.W'jrzburg. .-rier,J. B. (1822). Théorieanal,-îiquede la chaleur. Pa,ris. .ting, L. J. (1956). Advancesin Proîein Chemístry,ll, 429. -.ham, T. (1829). Quarterly Journal of Science,Literature and Art,27,74. ',ram, T. (1833).Phiktsophical Magazine,2,175,222,351. -'ram, T. (1850). P/zrlosophicalTransaction.s of the Rov-alSociett:of London, l40,1. , - ula. J., Graves.D. J., Quinn, J. A., and Lambersten,C. J. ('|976). In. IJndem-aterPhysiology, Vol.5, ed.C. J. Lambersten,p. 355. New York: Academic. '-.or. E. A. (1970). Philosophit:al Journal,7,99. 1 i.. R., Woolf, L. A., and Watts,R. O. ( I 968). American Instituteof ChemicatEngineersJoumal, t 4 . 6 7t . .rnson, R. .A.,and Stokes,R. H. (1960). ElectrolyteSolutions.London: Butterworth.

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