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Chapter24 Fundamentals of Mass Transfer The previous chapters dealing with the transport phenomena of momentum and heat transfer have dealt with one-component phases that possessed a natural tendency to reach equilibrium conditions. When a system contains two or more components whose concentrations vary from point to point, there is a natural tendency for mass to be transferred, minimizing the concentration differences within the system. The transport of one constituent from a region of higher concentration to that of a lower concentration is called mass transfer. Many of our day-to-day experiences involve mass transfer. A lump of sugar added to a cup of black coffee eventually dissolves and then diffuses uniformly throughout the coffee. Water evaporates from ponds to increase the humidity of the passing air stream. Perfume presents a pleasant fragrance that is imparted throughout the surrounding atmosphere. Mass transfer is the basis for many biological and chemical processes. Biological processes include the oxygenation of blood and the transport of ions across membranes within the kidney. Chemical processes include the chemical vapor deposition (CVD) of silane (SiH ) onto a silicon wafer, the doping of a silicon wafer to form a 4 semiconducting thin film, the aeration of wastewater, and the purification of ores and isotopes. Mass transfer underlies the various chemical separation processes where one or more components migrate from one phase to the interface between the two phases in contact. For example, in adsorption or crystallization processes, the components remain at the interface, whereas in gas absorption and liquid–liquid extraction processes, the components penetrate the interface and then transfer into the bulk of the second phase. If we consider the lump of sugar added to the cup of black coffee, experience teaches us that the length of time required to distribute the sugar will depend upon whether the liquid is quiescent or whether it is mechanically agitated by a spoon. The mechanism of mass transfer, as we have also observed in heat transfer, depends upon the dynamics of the system in which it occurs. Mass can be transferred by random molecular motion in quiescent fluids, or it can be transferred from a surface into a moving fluid, aided by the dynamic characteristics of the flow. These two distinct modes of transport, molecular mass transfer and convective mass transfer, are analogous to conduction heat transfer and convective heat transfer. Each of these modes of mass transfer will be described and analyzed. As in the case of heat transfer, we should immediately realize that the two mechanisms often act simultaneously. However, in the confluence of the two modes of mass transfer, one mechanism can dominate quantitatively so that approximate solutions involving only the dominant mode need be used. 39824.1 Molecular Mass Transfer 399 24.1 MOLECULAR MASS TRANSFER Asearlyas1815,Parrotobservedqualitativelythatwheneveragasmixturecontainstwoor more molecular species, whose relative concentrations vary from point to point, an apparentlynaturalprocessresults,whichtendstodiminishanyinequalitiesofcomposition. This macroscopic transport of mass, independent of any convection within the system, is defined as molecular diffusion. In the specific case of gaseous mixtures, a logical explanation of this transport phenomenon can be deduced from the kinetic theory of gases. At temperatures above absolute zero, individual molecules are in a state of continual yet random motion. Within dilute gas mixtures, each solute molecule behaves independently of the other solute molecules, as it seldom encounters them. Collisions between the solute and the solvent moleculesarecontinuallyoccurring.Asaresultofthecollisions,thesolutemoleculesmove alongazigzagpath,sometimestowardaregionofhigherconcentration,sometimestowarda lower concentration. Let us consider a hypothetical section passing normal to the concentration gradient within an isothermal, isobaric gaseous mixture containing solute and solvent molecules. Thetwothin,equalelementsofvolumeaboveandbelowthesectionwillcontainthesame numberof molecules, as stipulated byAvogadro’slaw.Althoughitis notpossible tostate whichwayanyparticularmoleculewilltravelinagivenintervaloftime,adefinitenumber ofthemoleculesinthelowerelementofthevolumewillcrossthehypotheticalsectionfrom below,andthesamenumberofmoleculeswillleavetheupperelementandcrossthesection fromabove.Withtheexistenceofaconcentrationgradient,therearemoresolutemolecules inoneoftheelementsofvolumethanintheother;accordingly,anoverallnettransferfroma regionofhigherconcentrationtooneoflowerconcentrationwillresult.Thenetflowofeach molecular species occurs in the direction of a negative concentration gradient. As pointed out in Chapters 7 and 15, the molecular transport of momentum and the transportofenergybyconductionarealsoduetorandommolecularmotion.Accordingly, oneshouldexpectthatthethreetransportphenomenawilldependuponmanyofthesame characteristicproperties,suchasmeanfreepath,andthatthetheoreticalanalysesofallthree phenomena will have much in common. The Fick Rate Equation Thelawsofmasstransfershowtherelationbetweenthefluxofthediffusingsubstanceand theconcentrationgradientresponsibleforthismasstransfer.Unfortunately,thequantitative description of molecular diffusion is considerably more complex than the analogous descriptions for the molecular transfer of momentum and energy that occur in a one- componentphase.Asmasstransfer,ordiffusion,asitisalsocalled,occursonlyinmixtures, itsevaluationmustinvolveanexaminationoftheeffectofeachcomponent.Forexample,we willoftendesiretoknowthediffusionrateofaspecificcomponentrelativetothevelocityof Molecule of themixtureinwhichitismoving.Aseachcomponentmaypossessadifferentmobility,the species A mixture velocity must be evaluated by averaging the velocities of all of the components present. Inordertoestablishacommonbasisforfuturediscussions,letusfirstconsiderdefinitions and relations that are often used to explain the role of components within a mixture. Figure 24.1 Elemental Concentrations. Inamulticomponentmixture,theconcentrationofamolecularspecies volume containing a multicomponent mixture. canbeexpressedinmanyways.Figure24.1showsanelementalvolumedVthatcontainsa400 Chapter 24 Fundamentals of Mass Transfer mixtureofcomponents,includingspeciesA.Aseachmoleculeofeachspecieshasamass,a massconcentrationforeachspecies,aswellasforthemixture,canbedefined.Forspecies A,massconcentration,r ,isdefinedasthemassofAperunitvolumeofthemixture.The A totalmassconcentrationordensity,r,isthetotalmassofthemixturecontainedintheunit volume; that is, n r¼ r (24-1) i i¼1 where n is the number of species in the mixture. The mass fraction, v , is the mass A concentration of species A divided by the total mass density r r A A v ¼ ¼ (24-2) A n r r i i The sum of the mass fractions, by definition, must be 1: n v ¼ 1 (24-3) i i¼1 The molecularconcentration ofspecies A, c ,isdefinedas thenumberofmoles of A A present per unit volume of the mixture. By definition, one mole of any species contains a mass equivalent to its molecular weight; the mass concentration and molar concentration terms are related by the following relation: r A c ¼ (24-4) A M A where M is the molecular weight of species A. When dealing with a gas phase, A concentrationsareoftenexpressedintermsofpartialpressures.Underconditionsinwhich the ideal gas law, p V ¼ n RT, applies, the molar concentration is A A n p A A c ¼ ¼ (24-5) A V RT wherep isthepartialpressureofthespeciesAinthemixture,n isthenumberofmolesof A A speciesA,Visthegasvolume,Tistheabsolutetemperature,andRisthegasconstant.The totalmolarconcentration,c,isthetotalmolesofthemixturecontainedintheunitvolume; that is, n c¼ c (24-6) i i¼1 or fora gaseous mixture that obeystheideal gas law, c¼ n /V ¼ P/RT,where Pisthe total totalpressure.Themolefractionforliquidorsolidmixtures,x ,andforgaseousmixtures, A y , are the molar concentrations of species A divided by the total molar density A c A x ¼ (liquidsandsolids) A c (24-7) c A y ¼ (gases) A c Foragaseousmixturethatobeystheidealgaslaw,themolefraction, y ,canbewrittenin A terms of pressures c p /RT p A A A y ¼ ¼ ¼ (24-8) A c P/RT P24.1 Molecular Mass Transfer 401 Equation (24-8) is an algebraic representation of Dalton’s law for gas mixtures. The sum of the mole fractions, by definition, must be 1: n x ¼ 1 i i¼1 (24-9) n y ¼ 1 i i¼1 A summary of the various concentration terms and of the interrelations for a binary system containing species A and B is given in Table 24.1. Table 24.1 Concentrations in a binary mixture of A and B Mass concentrations r ¼ total mass density of the mixture r ¼ mass density of species A A r ¼ mass density of species B B v ¼ mass fraction of species A ¼ r /r A A v ¼ mass fraction of species B ¼ r /r B B r¼ r þr A B 1¼ v þv A B Molar concentrations Liquid or solid mixture Gas mixture c ¼ molar density of mixture ¼ n=Vc¼ n/V ¼ P/RT c ¼ molar density of species A¼ n /Vc ¼ n /V ¼ p /RT A A A A A c ¼ molar density of species B¼ n /Vc ¼ n /V ¼ p /RT B B B B B x ¼ mole fraction of species A¼ c /c¼ n /ny ¼ c /c¼ n /n¼ p /p A A A A A A A x ¼ mole fraction of species B ¼ c /c¼ n /ny ¼ c /c¼ n /n¼ p /p B B B B B B B p p P c¼ c þc A B A B c¼ c þc ¼ þ ¼ A B RT RT RT 1¼ x þx 1¼ y þy A B A B Interrelations r ¼ c M A A A v /M A A (24-10) x or y ¼ A A v /M þv /M A A B B x M y M A A A A v ¼ or (24-11) A x M þx M y M þy M A A A A A A B B EXAMPLE 1 The composition of air is often given in terms of only the two principal species in thegas mixture oxygen, O , y ¼ 0:21 2 o 2 nitrogen, N , y ¼ 0:79 2 N 2 Determinethemassfractionofbothoxygenandnitrogenandthemeanmolecularweightoftheair 5 whenitismaintainedat258C(298K)and1atm(1.01310 Pa).Themolecularweightofoxygenis 0.032 kg/mol and of nitrogen is 0.028 kg/mol.402 Chapter 24 Fundamentals of Mass Transfer As a basis for our calculations, consider 1 mol of the gas mixture oxygenpresent¼ (1mol)(0:21)¼ 0:21mol (0:032kg) ¼ (0:21mol) ¼ 0:00672kg mol nitrogenpresent¼ (1mol)(0:79)¼ 0:79mol (0:028kg) ¼ (0:79mol) ¼ 0:0221kg mol totalmasspresent¼ 0:00672þ0:0221¼ 0:0288kg 0:00672kg v ¼ ¼ 0:23 O 2 0:0288kg 0:0221kg v ¼ ¼ 0:77 N 2 0:0288kg As1molofthegasmixturehasamassof0.0288kg,themeanmolecularweightoftheairmustbe 0.0288. When one takes into account the other constituents that are present in air, the mean molecular weight of air is often rounded off to 0.029 kg/mol.  Thisproblemcouldalsobesolvedusingtheidealgaslaw,PV ¼ nRT.Atidealconditions,0 C 5 or 273 K and 1 atm of 1:01310 Pa pressure, the gas constant is evaluated to be 5 3 3 PV (1:01310 Pa)(22:4m ) Pam R¼ ¼ ¼ 8:314 (24-12) nT (1kgmol)(273K) molK The volume of the gas mixture, at 298 K, is  3 Pam (1mol) 8:314 (298K) nRT molK V ¼ ¼ 5 P 1:01310 Pa 3 ¼ 0:0245m The concentrations are 0:21mol molO 2 c ¼ ¼ 8:57 O 2 3 3 0:0245m m 0:79mol molN 2 c ¼ ¼ 32:3 N 2 3 3 0:0245m m n 3 c¼ c ¼ 8:57þ32:3¼ 40:9mol/m i i¼1 The total density, r,is 0:0288kg 3 r¼ ¼ 1:180kg/m 3 0:0245m and the mean molecular weight of the mixture is 3 r 1:180kg/m M¼ ¼ ¼ 0:0288kg/mol 3 c 40:9mol/m Velocities. In a multicomponent system the various species will normally move at different velocities; accordingly, an evaluation of a velocity for the gas mixture requires the averaging of the velocities of each species present.24.1 Molecular Mass Transfer 403 The mass-average velocity for a multicomponent mixture is defined in terms of the mass densities and velocities of all components by n n rv rv i i i i i¼1 i¼1 v¼ ¼ (24-13) n r r i i¼1 wherev denotes theabsolutevelocity of species irelative to stationary coordinate axes. This i is thevelocity that would be measured by a pitot tube and is thevelocity that was previously encountered in the equations of momentum transfer. The molar-average velocity for a multicomponentmixtureisdefinedintermsofthemolarconcentrationsofallcomponentsby n n cv cv i i i i i¼1 i¼1 V¼ ¼ (24-14) n c c i i¼1 The velocity of a particular species relative to the mass-average or molar-average velocity is termed a diffusion velocity. We can define two different diffusion velocities v v; thediffusionvelocityofspeciesirelativetothemass-averagevelocity i and v V, thediffusionvelocityofspeciesirelativetothemolar-velocity average i AccordingtoFick’slaw,aspeciescanhaveavelocityrelativetothemass-ormolar-average velocity only if gradients in the concentration exist. Fluxes. The mass (or molar) flux of a given species is a vector quantity denoting the amount of the particular species, in either mass or molar units, that passes per given increment of time through a unit area normal to the vector. The flux may be defined with referencetocoordinatesthatarefixedinspace,coordinatesthataremovingwiththemass- average velocity, or coordinates that are moving with the molar-average velocity. Thebasicrelationformoleculardiffusiondefinesthemolarfluxrelativetothemolar- 1 averagevelocity,J .Anempiricalrelationforthismolarflux,firstpostulatedbyFick and, A accordingly,oftenreferredtoasFick’sfirstlaw,definesthediffusionofcomponentAinan isothermal, isobaric system: J ¼D rc A AB A For diffusion in only the z direction, the Fick rate equation is dc A J ¼D (24-15) A,z AB dz whereJ isthemolarfluxinthezdirectionrelativetothemolar-averagevelocity,dc /dz A,z A is the concentration gradient in the z direction, and D , the proportionality factor, is the AB mass diffusivity or diffusion coefficient for component A diffusing through component B. A moregeneral flux relation that is not restricted to isothermal, isobaric systems was 2 proposed by de Groot who chose to write  overall diffusion concentration flux¼ density coefficient gradient 1 A. Fick, Ann. Physik., 94, 59 (1855). 2 S. R. de Groot, Thermodynamics of Irreversible Processes, North-Holland, Amsterdam, 1951.404 Chapter 24 Fundamentals of Mass Transfer or dy A J ¼cD (24-16) A,z AB dz As the total concentration c is constant under isothermal, isobaric conditions, equation (24-15)isaspecialformofthemoregeneralrelation(24-16).Anequivalentexpressionfor j ,themassfluxinthe z direction relative to the mass-average velocity, is A,z dv A j ¼rD (24-17) A,z AB dz where dv /dz is the concentration gradient in terms of the mass fraction. When the A density is constant, this relation simplifies to dr A j ¼D A,z AB dz InitialexperimentalinvestigationsofmoleculardiffusionwereunabletoverifyFick’s law of diffusion. This was apparently due to the fact that mass is often transferred simultaneously by two possible means: (1) as a result of the concentration differences aspostulatedbyFickand(2)byconvectiondifferencesinducedbythedensitydifferences thatresultedfromtheconcentrationvariation.Steffan(1872)andMaxwell(1877),usingthe kinetictheoryofgases,provedthatthemassfluxrelativetoafixedcoordinatewasaresultof twocontributions:theconcentrationgradientcontributionandthebulkmotioncontribution. Forabinarysystemwithaconstantaveragevelocityinthezdirection,themolarfluxin the z direction relative to the molar-average velocity may also be expressed by J ¼ c (v V ) (24-18) A,z A A,z z Equating expressions (24-16) and (24-18), we obtain dy A J ¼ c (v V )¼cD A,z A A,z z AB dz which, upon rearrangement, yields dy A c v ¼cD þc V z A A,z AB A dz For this binary system, V can be evaluated by equation (24-14) as z 1 V ¼ (c v þc v ) z A A,z B A,z c or c V ¼ y (c v þc v ) A z A A A,z B B,z Substituting this expression into our relation, we obtain dy B c v ¼cD þy (c v þc v ) (24-19) A A,z AB A A A,z B B,z dz As the component velocities, v and v , are velocities relative to the fixed z axis, the A,z B,z quantitiesc v andc v arefluxesofcomponentsAandBrelativetoafixedzcoordinate; A A,z B B,z accordingly,wesymbolizethisnewtypeoffluxthatisrelativetoasetofstationaryaxesby N ¼ c v A A A24.1 Molecular Mass Transfer 405 and N ¼ c v B B B Substituting these symbols into equation (24-19), we obtain a relation for the flux of component A relative to the z axis dy A N ¼cD þy (N þN ) (24-20) A,z AB A A,z B,z dz This relation may be generalized and written in vector form as N ¼cD =y þy (N þN ) (24-21) A AB A A A B It is important to note that the molar flux, N , is a resultant of the two vector quantities: A cD =y themolarflux,J ,resultingfromtheconcentrationgradient:This AB A A termisreferredtoastheconcentrationgradientcontribution; and y (N þN )¼ c V themolarfluxresultingascomponentAiscarriedinthebulkflowof A A B A thefluid:Thisfluxtermisdesignatedthebulkmotioncontribution: Either or both quantities can be a significant part of the total molar flux, N . Whenever A equation(24-21)isappliedtodescribemolardiffusion,thevectornatureoftheindividual fluxes,N andN ,mustbeconsideredandthen,inturn,thedirectionofeachoftwovector A B quantities must be evaluated. IfspeciesAwerediffusinginamulticomponentmixture,theexpressionequivalentto equation (24-21) would be n N ¼cD =y þy N A AM A A i i¼1 where D is the diffusion coefficient of A in the mixture. AM Themassflux,n ,relativetoafixedspatialcoordinatesystem,isdefinedforabinary A system in terms of mass density and mass fraction by n ¼rD =v þv (n þn ) (24-22) A AB A A A B where n ¼ r v A A A and n ¼ r v B B B Under isothermal, isobaric conditions, this relation simplifies to n ¼D =r þv (n þn ) A AB A A A B As previously noted, the flux is a resultant of two vector quantities: D =r , themassflux,j ,resultingfromaconcentrationgradient;the AB A A concentrationgradient contribution: v (n þn )¼ r v; themassfluxresultingascomponentAiscarriedinthebulk A A B A flowofthefluid;thebulkmotioncontribution:406 Chapter 24 Fundamentals of Mass Transfer Table 24.2 Equivalent forms of the mass flux equation for binary system A and B Flux Gradient Fick rate equation Restrictions n =v n ¼rD =v þv (n þn ) A A A AB A A A B =r n ¼D =r þv (n þn ) Constant r B A A AB A A A N =y N ¼cD =y þy (N þN ) B A A A AB A A A =c N ¼D =c þy (N þN ) Constant c A A AB A A A B j =v j ¼rD =v A AB A A A =r j ¼D =r Constant r AB A A A j =y J ¼cD =y A A AB A A =c J ¼D =c Constant c A A AB A Ifaballoon,filledwithacolordye,isdroppedintoalargelake,thedyewilldiffuseradially asaconcentrationgradient contribution.Whenastickisdropped intoamovingstream,it will float downstream by the bulk motion contribution. If the dye-filled balloon were dropped into the moving stream, the dye would diffuse radially while being carried downstream; thus both contributions participate simultaneously in the mass transfer. Thefourequationsdefiningthefluxes,J ,j ,N ,andn areequivalentstatementsof A A A A theFickrateequation.Thediffusioncoefficient,D ,isidenticalinallfourequations.Any AB oneoftheseequationsisadequatetodescribemoleculardiffusion;however,certainfluxes areeasiertouseforspecificcases.Themassfluxes,n andj ,areusedwhentheNavier– A A Stokes equations are also required to describe the process. Since chemical reactions are describedintermsofmolesoftheparticipatingreactants,themolarfluxes,J andN ,are A A used to describe mass-transfer operations in which chemical reactions are involved. The fluxes relative to coordinates fixes in space, n and N , are often used to describe A A engineeringoperationswithinprocessequipment.ThefluxesJ andj areusedtodescribe A A themasstransferindiffusioncellsusedformeasuringthediffusioncoefficient.Table24.2 summarizes the equivalent forms of the Fick rate equation. Related Types of Molecular Mass Transfer According to the second law of thermodynamics, systems not in equilibrium will tend to move toward equilibrium with time. A generalized driving force in chemical thermo- dynamictermsisdm /dzwherem isthechemicalpotential.Themolardiffusionvelocity c c of component A is defined in terms of the chemical potential by dm D dm AB c c v V ¼ u ¼ (24-23) A,z z A dz RT dz where u is the ‘‘mobility’’ of component A, or the resultant velocity of the molecule A while under the influence of a unit driving force. Equation (24-23) is known as the Nernst–Einstein relation. The molar flux of A becomes D dm AB c J ¼ c (v V )¼c (24-24) A,z A A,z z A RT dz Equation(24-24)maybeusedtodefineallmolecularmass-transferphenomena.Asan example,considertheconditionsspecifiedforequation(24-15);thechemicalpotentialofa componentinahomogeneousidealsolutionatconstanttemperatureandpressureisdefinedby 0 m ¼ m þ RTlnc (24-25) A c24.2 The Diffusion Coefficient 407 0 wherem isaconstant,thechemicalpotentialofthestandardstate.Whenwesubstitutethis relation into equation (24-24), the Fick rate equation for a homogeneous phase is obtained dc A J ¼D (24-15) A,z AB dz There are a number of other physical conditions, in addition to differences in con- centration, which will produce a chemical potential gradient: temperature differences, pressuredifferences,anddifferencesintheforcescreatedbyexternalfields,suchasgravity, magnetic, and electrical fields. We can, for example, obtain mass transfer by applying a temperature gradient to a multicomponent system. This transport phenomenon, the Soret effectorthermaldiffusion,althoughnormallysmallrelativetootherdiffusioneffects,isused successfullyintheseparationofisotopes.Componentsinaliquidmixturecanbeseparated withacentrifugebypressurediffusion.Therearemanywell-knownexamplesofmassfluxes beinginducedinamixturesubjectedtoanexternalforcefield:separationbysedimentation undertheinfluenceofgravity,electrolyticprecipitationduetoanelectrostaticforcefield,and magnetic separation of mineral mixtures through the action of a magnetic force field. Although these mass-transfer phenomena are important, they are very specific processes. Themolecularmasstransfer,resultingfromconcentrationdifferencesanddescribedby Fick’s law, results from the random molecular motion over small mean free paths, independent of any containment walls. The diffusion of fast neutrons and molecules in extremely small pores or at very low gas density cannot be described by this relationship. Neutrons,producedina nuclearfissionprocess,initiallypossesshighkinetic energies andaretermedfastneutronsbecauseoftheirhighvelocities;thatis,upto15millionmeters per second. At these high velocities, neutrons pass through the electronic shells of other atomsormoleculeswithlittlehindrance.Tobedeflected,thefastneutronsmustcollidewitha nucleus,whichisaverysmalltargetcomparedtothevolumeofmostatomsandmolecules. Themeanfreepathoffastneutronsisapproximatelyonemilliontimesgreaterthanthefree pathsofgasesatordinarypressures.Afterthefastneutronsaresloweddownthroughelastic- scattering collisions between the neutrons and the nuclei of the reactor’s moderator, these slowermovingneutrons,thermalneutrons,migratefrompositionsofhigherconcentrationto positionsoflowerconcentration,andtheirmigrationisdescribedbyFick’slawofdiffusion. 24.2 THE DIFFUSION COEFFICIENT Fick’s law of proportionality, D , is known as the diffusion coefficient. Its fundamental AB dimensions, which may be obtained from equation (24-15)  2 J M 1 L A,z D ¼ ¼ ¼ AB 2 3 dc /dz L t M/L 1/L t A are identical to the fundamental dimensions of the other transport properties: kinematic viscosity,n,andthermaldiffusivity,a,oritsequivalentratio,k/rc .Themassdiffusivityhas p 2 2 4 beenreportedincm /s;theSIunitsarem /s,whichisafactor10 smaller.IntheEnglish 2 system ft /h is commonly used. Conversion between these systems involves the simple relations 2 D (cm /s) AB 4 ¼ 10 2 D (m /s) AB (24-26) 2 D (ft /h) AB ¼ 3:87 2 D (cm /s) AB408 Chapter 24 Fundamentals of Mass Transfer The diffusion coefficient depends upon the pressure, temperature, and composition of the system. Experimental values for the diffusivities of gases, liquids, and solids are tabulated in AppendixTablesJ.1,J.2,andJ.3,respectively.Asonemightexpectfromtheconsiderationof the mobility of the molecules, the diffusion coefficients aregenerally higher forgases (in the 6 5 2 10 9 2 range of 510 to 110 m /s), than for liquids (in the range of 10 to 10 m /s), 14 10 2 which are higher than the values reported for solids (in the range of 10 to 10 m /s). Intheabsenceofexperimentaldata,semitheoreticalexpressionshavebeendeveloped whichgiveapproximations,sometimesasvalidasexperimentalvaluesduetothedifficulties encountered in their measurement. Gas Mass Diffusivity Theoretical expressions for the diffusion coefficient in low-density gaseous mixtures as a 3 4 function of the system’s molecular properties were derived by Sutherland, Jeans, and 5 ChapmanandCowling, baseduponthekinetictheoryofgases.Inthesimplestmodelofgas dynamics,themoleculesareregardedasrigidspheresthatexertnointermolecularforces. Collisionsbetweentheserigidmoleculesareconsideredtobecompletelyelastic.Withthese assumptions,asimplifiedmodelforanidealgasmixtureofspeciesAdiffusingthroughits isotope A yields an equation for the self-diffusion coefficient, defined as 1 D ¼ lu (24-27) AA 3 and l is the mean free path of length of species A, given by kT l¼pffiffiffi (24-28) 2 2ps P A where u is the mean speed of species Awith respect to the molar-average velocity rffiffiffiffiffiffiffiffiffiffiffi 8kNT u¼ (24-29) pM A Insertion of equations (24-28) and (24-29) into equation (24-27) results in  1=2 3/2 3 2T k N D ¼ (24-30) AA 2 3/2 M 3p s P A A where M is the molecular weight of the diffusing species A, (g/mol), N is Avogadro’s A 23 number (6:02210 molecules/mol), P is the system pressure, T is the absolute 16 temperature (K), K is the Boltzmann constant (1:3810 ergs/K), and s is the AB Lennard–Jones diameter of the spherical molecules. UsingasimilarkinetictheoryofgasesapproachforabinarymixtureofspeciesAandB composed of rigid spheres of unequal diameters, the gas-phase diffusion coefficient is shown to be  1/2 1 1  þ 3/2 2 K 2M 2M A B 1/2 3/2 D ¼ N T (24-31) AB  2 3 p s þs B A P 2 3 W. Sutherland, Phil. Mag., 36, 507; 38, 1 (1894). 4 J. Jeans, Dynamical Theory of Gases, Cambridge University Press, London, 1921. 5 S. Chapman and T. G. Cowling, Mathematical Theory of Non-Uniform Gases, Cambridge University Press, London, 1959.24.2 The Diffusion Coefficient 409 Unlike the other two molecular transport coefficients for gases, theviscosity and thermal conductivity, the gas-phase diffusion coefficient is dependent on the pressure and the temperature. Specifically, the gas-phase diffusion coefficient is an inverse function of total system pressure 1 D / AB P a 3/2 power-law function of the absolute temperature 3/2 D /T AB Asequation(24-31)reveals,andasoneoftheproblemsattheendofthischapterpointsout, the diffusion coefficients for gases D ¼ D . This is not the case for liquid diffusion AB BA coefficients. Modern versions of the kinetic theory have been attempted to account for forces of 6 attraction and repulsion between the molecules. Hirschfelder et al. (1949), using the Lennard–Jones potential to evaluate the influence of the molecular forces, presented an equation for the diffusion coefficient for gas pairs of nonpolar, nonreacting molecules:  1/2 1 1 3=2 0:001858T þ M M A B D ¼ (24-33) AB 2 Ps V D AB 2 where D is the mass diffusivity of A through B,incm /s; Tis the absolute temperature, AB in K; M , M are the molecular weights of A and B, respectively; P is the absolute A B pressure, in atmospheres; s is the ‘‘collision diameter,’’ a Lennard–Jones parameter, in AB ˚ A;andV isthe‘‘collisionintegral’’formoleculardiffusion,adimensionlessfunctionof D the temperature and of the intermolecular potential field for one molecule of A and one molecule of B. Appendix Table K.1 listsV as a function of kT/e , k is the Boltzmann D AB 16 constant,whichis1:3810 ergs/K,and e istheenergyofmolecularinteractionfor AB the binary system A and B, a Lennard–Jones parameter, in ergs, see equation (24-31). Unlike the other two molecular transport coefficients, viscosity and thermal conductivity, the diffusion coefficient is dependent on pressure as well as on a higher order of the absolute temperature. When the transport process in a single component phase was examined, we did not find any composition dependency in equation (24-30) or in the similarequationsforviscosityandthermalconductivity.Figure24.2presentsthegraphical dependency of the ‘‘collision integral,’’ V , on the dimensionless temperature, kT/e . D AB The Lennard–Jones parameters, r and e , are usually obtained from viscosity data. D Unfortunately,thisinformationisavailableforonlyaveryfewpuregases.AppendixTable K.2 tabulates these values. In the absence of experimental data, the values for pure components may be estimated from the following empirical relations: 1/3 s¼ 1:18V (24-34) b 1/3 s¼ 0:841V (24-35) c  1/3 T c s¼ 2:44 (24-36) P c e /k¼ 0:77T (24-37) A c 6 J. O. Hirschfelder, R. B. Bird, and E. L. Spotz, Chem. Rev.,44, 205 (1949).410 Chapter 24 Fundamentals of Mass Transfer 3.00 2.50 2.00 1.50 1.00 0.50 0.00 Figure 24.2 Binary gas- 0.00 2.00 4.00 6.00 8.00 10.00 phase Lennard–Jones kT/e Dimensionless temperature, ΑΒ ‘‘collision integral.’’ and e /k¼ 1:15T (24-38) A b 3 where V is the molecular volume at the normal boiling point, in (cm) /g mol (this is b 3 evaluated by using Table 24.3); V is the critical molecular volume, in (cm) /g mol; T is c c the critical temperature, in K; T is the normal boiling temperature, in K; and P is the b c critical pressure, in atmospheres. Table 24.3 Atomic diffusion volumes for use in estimating D by method of AB Fuller, Schettler, and Giddings Atomic and structure diffusion-volume increments, v C 16.5 Cl 19.5 H 1.98 S 17.0 O 5.48 Aromatic ring 20.2 N 5.69 Heterocyclic ring 20.2 Diffusion volumes for simple molecules, v H 7.07 Ar 16.1 H O 12.7 2 2 D 6.70 Kr 22.8 CCIF 114.8 2 2 He 2.88 CO 18.9 SF 69.7 6 N 17.9 CO 26.9 Cl 37.7 2 2 2 O 16.6 N O 35.9 Br 67.2 2 2 2 Air 20.1 NH 14.9 SO 41.1 3 2 For a binary system composed of nonpolar molecular pairs, the Lennard–Jones parametersofthepurecomponentmaybecombinedempiricallybythefollowingrelations: s þs A B s ¼ (24-39) AB 2 and pffiffiffiffiffiffiffiffiffi e ¼ e e (24-40) AB A B Collision integral for diffusion, W D24.2 The Diffusion Coefficient 411 Theserelationsmustbemodifiedforpolar–polarandpolar–nonpolarmolecularpairs;the 7 proposed modifications are discussed by Hirschfelder, Curtiss, and Bird. TheHirschfelderequation(24-33)isoftenusedtoextrapolateexperimentaldata. For moderaterangesofpressure,upto25atm,thediffusioncoefficientvariesinverselywiththe pressure. Higher pressures apparently require dense gas corrections; unfortunately, no satisfactorycorrelationisavailableforhighpressures.Equation(24-33)alsostatesthatthe 3/2 diffusion coefficient varies with the temperature as T /V varies. Simplifying equation D (24-33), we can predict the diffusion coefficient at any temperature and at any pressure below 25 atm from a known experimental value by  3/2 V j P T D 1 2 T 1 D ¼ D (24-41) AB AB T ,P T ,P 2 1 1 1 P T V j 2 1 D T 2 In Appendix Table J.1, experimental values of the product D Pare listed for several gas AB pairs at a particular temperature. Using equation (24-41), we may extend these values to other temperatures.  EXAMPLE 2 Evaluate the diffusion coefficient of carbon dioxide in air at 20 C and atmospheric pressure. Compare this value with the experimental value reported in appendix table J.1. From Appendix Table K.2, the values of s and e/k are obtained 8 s; inA e /k; inK A Carbondioxide 3:996 190 Air 3:617 97 The various parameters for equation (24-33) may be evaluated as follows: s þs 3:996þ3:617 A B 8 s ¼ ¼ ¼ 3:806A AB 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e /k¼ (e /k)(e /k)¼ (190)(97)¼ 136 AB A B T ¼ 20þ273¼ 293K P¼ 1atm e 136 AB ¼ ¼ 0:463 kT 293 kT ¼ 2:16 e AB V (TableK:1) ¼ 1:047 D M ¼ 44 CO 2 and M ¼ 29 Air Substituting these values into equation (24-33), we obtain 3/2 1/2 0:001858T (1/M þ1/M ) A B D ¼ AB 2 Ps V D AB 3/2 1/2 (0:001858)(293) (1/44þ1/29) 2 ¼ ¼ 0:147cm /s 2 (1)(3:806) (1:047) 7 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc., New York, 1954.412 Chapter 24 Fundamentals of Mass Transfer From Appendix Table J.1 for CO in air at 273 K, 1 atm, we have 2 2 D ¼ 0:136 cm /s AB Equation (24-41) will be used to correct for the differences in temperature  3/2 D T V jT AB,T 1 D 2 1 ¼ D T V jT AB,T 2 D 1 2 Values forV may be evaluated as follows: D 136 atT ¼ 273 e /kT ¼ ¼ 0:498 V j ¼ 1:074 2 AB D T 2 273 atT ¼ 293 V j ¼ 1:074 (previouscalculations) 1 D T 1 The corrected value for the diffusion coefficient at 208Cis  3/2 293 1:074 2 5 2 D ¼ (0:136)¼ 0:155cm /s (1:5510 m /s) AB,T 1 273 1:047 We readily see that the temperature dependency of the ‘‘collision integral’’ is very small. Accordingly, most scaling of diffusivities relative to temperature include only the ratio 3/2 ðT /T Þ . 1 2 Equation (24-33) was developed for dilute gases consisting of nonpolar, spherical monatomic molecules. However, this equation gives good results for most nonpolar, 8 binarygassystems overawiderangeoftemperatures. Otherempiricalequations have 9 beenproposed forestimatingthediffusioncoefficientfornonpolar,binarygassystems at low pressures. The empirical correlation recommended by Fuller, Schettler, and Giddings permits the evaluation of the diffusivity when reliable Lennard–Jones para- meters, s and e, are unavailable. The Fuller correlation is i i  1/2 1 1 3 1:75 10 T þ M M A B D ¼ (24-42) AB   2 1/3 1/3 P (Sv) þ(Sv) A B 2 where D is in cm /s, T is in K, and P is in atmospheres. To determine the v terms, the AB authorsrecommendtheadditionoftheatomicandstructuraldiffusion-volumeincrements v reported in Table 24.3. 10 Danner and Daubert have recommended the atomic and structure diffusion-volume incrementsforCtobecorrectedto15.9andforHto2.31andthediffusionvolumesforH to 2 be corrected to 6.12 and for air to 19.7. 8 R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids, Third Edition, McGraw-Hill Book Company, New York, 1977, Chapter 11. 9 J. H. Arnold, J. Am. Chem. Soc., 52, 3937 (1930). E. R. Gilliland, Ind. Eng. Chem., 26, 681 (1934). J. C. Slattery and R. B. Bird, A.I.Ch.E. J., 4, 137 (1958). D. F. Othmer and H. T. Chen, Ind. Eng. Chem. Process Des. Dev., 1, 249 (1962). R. G. Bailey, Chem. Engr., 82(6), 86, (1975). E. N. Fuller, P. D. Schettler, and J. C. Giddings, Ind. Eng. Chem., 58(5), 18 (1966). 10 R. P. Danner, and T. E. Daubert, Manual for Predicting Chemical Process Design Data, A.I.Ch.E. (1983).24.2 The Diffusion Coefficient 413 EXAMPLE 3 Reevaluatethediffusioncoefficientofcarbondioxideinairat208Candatmosphericpressureusing the Fuller, Schettler, and Giddings equation and compare the new value with the one reported in example 2.  1/2 1 1 3 1:75 10 T þ M M A B D ¼ hi AB 2 1=3 1/3 P (v) þ(v) A B  1/2 1 1 3 1:75 10 (293) þ 44 29 ¼ 1/3 1/3 2 (1)(26:9) þ(20:1) 2 ¼ 0:152cm /s 2 ThisvaluecomparesveryfavorablytothevalueevaluatedwithHirschfelderequation,0.155cm /s, and its determination was easily accomplished. 11 Brokaw has suggested a method for estimating diffusion coefficient for binary gas mixturescontainingpolarcompounds.TheHirschfelderequation()isstillused;however, the collision integral is evaluated by 2 0:196d AB V ¼V þ (24-43) D D 0 T where 1/2 d ¼ (d d ) AB A B 3 2 (24-44) 1:9410 m p d¼ V T D D m ¼ dipolemoment,Debye p 3 V ¼ liquidmolarvolumeofthespecificcompoundatitsboilingpoint,cm /gmol b T ¼ normalboilingpoint,K b and T ¼ kT/e AB where  1/2 e e e AB A B ¼ k k k (24-45) 2 e/k¼ 1:18(1þ1:3d )T b d is evaluated with (24–44). And A C E G V ¼ þ þ þ (24-46) D 0 B exp(DT ) exp(FT ) exp(HT ) (T ) 11 R. S. Brokaw, Ind. Engr. Chem. Process Des. Dev., 8, 240 (1969).414 Chapter 24 Fundamentals of Mass Transfer with A¼ 1:060,36 E ¼ 1:035,87 B¼ 0:156,10 F ¼ 1:529,96 C¼ 0:193,00 G ¼ 1:764,74 D¼ 0:476,35 H¼ 3:894,11 The collision diameter, s , is evaluated with AB 1/2 s ¼ (s s ) (24-47) AB A B with each component’s characteristic length evaluated by  1/3 1:585V D s¼ (24-48) 2 1þ1:3d 12 Reid, Prausnitz, and Sherwood noted that the Brokaw equation is fairly reliable, permittingtheevaluationofthediffusioncoefficientsforgasesinvolvingpolarcompounds with errors less than 15%. Mass transfer in gas mixtures of several components can be described by theoretical equations involving the diffusion coefficients for the various binary pairs involved in the 13 mixture. Hirschfelder, Curtiss, and Bird present an expression in its most general form. 14 Wilke hassimplifiedthe theory and has shownthat a close approximation to thecorrect form is given by the relation 1 D ¼ (24-49) 1mixture 0 0 0 y /D þy /D þþy /D 12 13 1n n 2 3 where D is the mass diffusivity for component 1 in the gas mixture; D is the 1mixture 1n 0 mass diffusivity for the binary pair, component 1 diffusing through component n; and y n is the mole fraction of component n in the gas mixture evaluated on a component-1-free basis, that is y y 2 2 0 y ¼ ¼ 2 y þy þy 1y 2 3 n 1 In Problem 24.7 at the end of this chapter, equation (24-49) is developed by using Wilke’s approach for extending the Stefan and Maxwell theory in order to explain the diffusion of species A through a gas mixture of several components. EXAMPLE 4 Inthechemicalvapordepositionofsilane(SiH )onasiliconwafer,aprocessgasstreamrichinan 4 inert nitrogen (N ) carrier gas has the following composition: 2 y ¼ 0:0075, y ¼ 0:015, y ¼ 0:9775 SIH H N 4 2 2 Thegasmixtureismaintainedat900Kand100Patotalsystempressure.Determinethediffusivityof silane through the gas mixture. The Lennard–Jones constants for silane are e /k¼ 207:6Kand A ˚ s ¼ 4:08A. A 12 R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids, Third Edition, McGraw-Hill Book Company, New York, 1977, Chapter 11. 13 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, p. 718. 14 C. R. Wilke, Chem. Engr. Prog., 46, 95–104 (1950).24.2 The Diffusion Coefficient 415 The binary diffusion coefficients at 900 K and 100 Pa total system pressure estimated by the Hirschfelder equation (24-33) are 3 2 3 2 D ¼ 1:0910 cm /s and D ¼ 4:0610 cm /s SiH N SiH H 4 2 4 2 The binary diffusion coefficients are relatively high because the temperature is high and the total system pressure is low. The composition of nitrogen and hydrogen on a silane-free basis are 0:9775 0:015 0 0 y ¼ ¼ 0:9849 and y ¼ ¼ 0:0151 N H 2 2 10:0075 10:0075 Upon substituting these values into the Wilke equation (24-49), we obtain 2 1 1 cm 3 D ¼ ¼ ¼ 1:1010 SiH mixture 4 0 0 0:9849 0:0151 y y s N H 2 2 þ þ 3 3 1:0910 4:0610 D D SiH N SiH H 4 2 4 2 Thisexampleverifiesthatforadilutemulticomponentgasmixture,thediffusioncoefficientofthe diffusing species in the gas mixture is approximated by the binary diffusion coefficient of the diffusing species in the carrier gas. Liquid-Mass Diffusivity In contrast to the case for gases, where we have available an advanced kinetic theory for explaining molecular motion, theories of the structure of liquids and their transport characteristics are still inadequate to permit a rigorous treatment. Inspection of published experimental values for liquid diffusion coefficients in Appendix J.2 reveals that they are severalordersofmagnitudesmallerthangasdiffusioncoefficientsandthattheydependon concentrationduetothechangesinviscositywithconcentrationandchangesinthedegree of ideality of the solution. Certainmoleculesdiffuseasmolecules,whileothersthataredesignatedaselectrolytes ionize in solutions and diffuse as ions. For example, sodium chloride, NaCl, diffuses in þ  water as the ions Na and Cl . Though each ion has a different mobility, the electrical neutralityofthesolutionindicatesthattheionsmustdiffuseatthesamerate;accordingly,it is possible to speak of a diffusion coefficient for molecular electrolytes such as NaCl. However,ifseveralionsarepresent,thediffusionratesoftheindividualcationsandanions mustbeconsidered,andmoleculardiffusioncoefficientshavenomeaning.Needlesstosay, separatecorrelationsforpredictingtherelationbetweentheliquidmassdiffusivitiesandthe properties of the liquid solution will be required for electrolytes and nonelectrolytes. Two theories, the Eyring ‘‘hole’’theory and the hydrodynamicaltheory, have been postulated as possible explanations for diffusion of nonelectrolyte solutes in low- concentration solutions. In the Eyring concept, the ideal liquid is treated as a quasi- crystalline lattice model interspersed with holes. The transport phenomenon is then describedbyaunimolecularrateprocessinvolvingthejumpingofsolutemoleculesinto theholeswithinthelatticemodel.ThesejumpsareempiricallyrelatedtoEyring’stheory 15 ofreactionrate. Thehydrodynamicaltheorystatesthattheliquiddiffusioncoefficient is related to the solute molecule’s mobility; that is, to the net velocity of the molecule while under the influence of a unit driving force. The laws of hydrodynamics provide 15 S. Glasstone, K. J. Laidler, and H. Eyring, Theory of Rate Processes, McGraw-Hill Book Company, New York, 1941, Chap. IX.416 Chapter 24 Fundamentals of Mass Transfer relations betweenthe force andthevelocity. Anequation thathasbeendeveloped from the hydrodynamical theory is the Stokes–Einstein equation kT D ¼ (24-50) AB 6prm B where D is the diffusivity of A in dilute solution in D,k is the Boltzmann constant, Tis AB the absolute temperature, r is the solute particle radius, and m is the solvent viscosity. B This equation has been fairly successful in describing the diffusion of colloidal particles or large round molecules through a solvent that behaves as a continuum relative to the diffusing species. The results of the two theories can be rearranged into the general form D m AB B ¼ f(V) (24-51) kT in which f(V) is a function of the molecular volume of the diffusing solute. Empirical correlations, using the general form of equation (24-51), have been developed, which attempt to predict the liquid diffusion coefficient in terms of the solute and solvent 16 properties. Wilke and Chang have proposed the following correlation for none- lectrolytes in an infinitely dilute solution: 8 1/2 D m 7:410 (F M ) AB B B B ¼ (24-52) 0:6 T V A 2 where D is the mass diffusivity of A diffusing through liquid solvent B,incm /s;m is AB B the viscosity of the solution, in centipoises; T is absolute temperature, in K; M is the B molecularweightofthesolvent;V isthemolalvolumeofsoluteatnormalboilingpoint, A 3 in cm /gmol; and F is the ‘‘association’’ parameter for solvent B. B Molecular volumes at normal boiling points, V , for some commonly encountered A compounds,aretabulatedinTable24.4.Forothercompounds,theatomicvolumesofeach element present are added together as per the molecular formulas. Table 24.5 lists the contributionsforeachoftheconstituentatoms.Whencertainringstructuresareinvolved, corrections must be made to account for the specific ring configuration; the following Table 24.4 Molecular volumes at normal boiling point for some commonly encountered compounds Molecular volume, Molecular volume, 3 3 Compound Compound in cm /gmol in cm /gmol Hydrogen, H 14.3 Nitric oxide, NO 23.6 2 Oxygen, O 25.6 Nitrous oxide, N O 36.4 2 2 Nitrogen, N 31.2 Ammonia, NH 25.8 2 3 Air 29.9 Water, H O 18.9 2 Carbon monoxide, CO 30.7 Hydrogen sulfide, H S 32.9 2 Carbon dioxide, CO 34.0 Bromine, Br 53.2 2 2 Carbonyl sulfide, COS 51.5 Chlorine, Cl 48.4 2 Sulfur dioxide, SO 44.8 Iodine, I 71.5 2 2 16 C. R. Wilke and P. Chang, A.I.Ch.E.J., 1, 264 (1955).24.2 The Diffusion Coefficient 417 y Table 24.5 Atomic volumes for complex molecular volumes for simple substances Atomic volume, Atomic volume, 3 3 Element Element in cm /gmol in cm /gmol Bromine 27.0 Oxygen, except as noted below 7.4 Carbon 14.8 Oxygen, in methyl esters 9.1 Chlorine 21.6 Oxygen, in methyl ethers 9.9 Hydrogen 3.7 Oxygen, in higher ethers Iodine 37.0 and other esters 11.0 Nitrogen, double bond 15.6 Oxygen, in acids 12.0 Nitrogen, in primary amines 10.5 Sulfur 25.6 Nitrogen, in secondary amines 12.0 y G. Le Bas, The Molecular Volumes of Liquid Chemical Compounds, Longmans, Green & Company, Ltd., London, 1915. corrections are recommended: for three-membered ring, as ethylene oxide deduct 6 for four-membered ring, as cyclobutane deduct 8.5 for five-membered ring, as furan deduct 11.5 for pyridine deduct 15 for benzene ring deduct 15 for naphthalene ring deduct 30 for anthracene ring deduct 47.5 Recommended values of the association parameter, F , are given below for a few B common solvents. Solvent F B 17 Water 2.26 Methanol 1.9 Ethanol 1.5 Benzene, ether, heptane, and other unassociated solvents 1.0 Ifdataforcomputingthemolarvolumeofsoluteatitsnormalboilingpoint,V ,arenot A 18 available, Tyn and Calus recommend the correlation 1:048 V ¼ 0:285V A c 3 where V is the critical volume of species A in cm /g. mol. Values of V are tabulated in c c 19 Reid, Prausnitz, and Sherwood. 17 The correction ofF is recommended by R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties B of Gases and Liquids, Third Edition, McGraw-Hill Book Company, New York, 1977, p. 578. 18 Tyn, M.T. and W.F. Calus, Processing, 21, (4): 16 (1975). 19 R.C. Reid, J.M. Prausnitz and, T.K. Sherwood, The Properties of Gases and Liquids, Third Edition, McGraw-Hill Book Company, New York, 1977, Appendix A.