MASS DIFFUSION Mass Transfer
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1 What it is and what it isn't
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1 What it is for. Applications
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2 How to study it. Similarities and differences between Mass
Transfer and Heat Transfer .................... 2 Forces and fluxes
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3 Specifying composition. Nomenclature
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4 Specifying boundary conditions for composition
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6 Species balance equation
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8 Diffusion rate: Fick's law
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9 The diffusion equation for mass transfer
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12 Some analytical solutions to mass diffusion
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13 Instantaneous point-source
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13 Semi-infinite planar diffusion
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14 Diffusion through a wall
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16 Summary table of analytical solutions to diffusion
problems.........................................................
17 Evaporation rate
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20 References
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21 MASS TRANSFER WHAT IT IS AND WHAT IT ISN'T
ThesubjectofMassTransferstudiestherelativemotionofsomechemicalspecieswithrespectto
others(i.e.separationandmixingprocesses),drivenbyconcentrationgradients(really,animbalance
inchemicalpotential,asexplainedinEntropy).Fluidflowwithoutmasstransferisnotpartofthe
Mass Transfer field but of Fluid Mechanics.
Heattransferandmasstransferarekineticprocessesthatmayoccurandbestudiedseparatelyor
jointly. Studying them apart is simpler, but it is most convenient
(to optimise the effort) to realise that both processes are
modelled by similar mathematical equations in the case of diffusion
and convection
(thereisnomass-transfersimilaritytoheatradiation),anditisthusmoreefficienttoconsiderthem
jointly.Ontheotherhand,thesubjectofMassTransferisdirectlylinkedtoFluidMechanics,where
thesingle-componentfluid-flowisstudied,buttheapproachusuallyfollowedismoresimilartothat
used in Heat Transfer, where fluid flow is mainly a boundary
condition empirically modelled; thus, the
teachingofMassTransfertraditionallyfollowsandbuildsuponthatofHeatTransfer(andnotupon
FluidMechanics).Infact,developmentinmass-transfertheorycloselyfollowsthatinheattransfer,
withthepioneeringworksofLewisandWhitmanin1924(alreadyproposingamass-transfer
coefficienthmsimilartothethermalconvectioncoefficienth),andSherwood'sbookof1937on
"Absorption and extraction". Even more, since the milestone book on
"Transport phenomena" by Bird
etal.(1960),heattransfer,masstransfer,andmomentumtransfer,areoftenjointlyconsideredasa
new discipline.
Asusual,thebasicstudyfirstfocusesonhomogeneousnon-reactingsystemswithwell-defined
boundaries(notonlyinMassTransfer,butinHeatTransferandinFluidMechanics),touchingupon
moving-boundaryproblemsandreactingprocessesonlyafterwards.Asfortheothersubjects,itis
basedonthecontinuummediatheory,i.e.withoutaccountingforthemicroscopicmotionofthe
molecules(sothatfieldtheoryandthefluid-particleconceptareappliedheretoo).Diffusiontheory
onlyappliestomolecularmixtures(d10-5 m) Newtonian mechanics
applies. Notice that we only consider here mass diffusion due to a
concentration gradient, what might be called
concentration-phoresisinanalogytoothermechanismsofmassdiffusionlikethermo-phoresis(Soret
effect),piezo-phoresis(diffusionduetoapressuregradient),orelectrophoresis(diffusionduetoa
gradient of electrical potential applied to ionic media).
Traditionally,thefieldofMassTransferhasbeenstudiedonlywithintheChemicalEngineering
curriculum, except for humid-air applications (evaporation) and
thermal desalination processes, which has been always studied in
Mechanical Engineering. But mass transfer problems are
proliferating in so
manycircumstances,especiallyathightemperatures(drying,combustion,materialstreatment,
pyrolysis,ablation...),thatthesubjectshouldbecoveredondifferentgroundstoencourageeffective
interdisciplinary team-workWHAT IT IS FOR. APPLICATIONS
ApplicationsofMassTransferincludethedispersionofcontaminants,dryingandhumidifying,
segregation and doping in materials, vaporisation and condensation
in a mixture, evaporation (boiling
ofapuresubstanceisnotmasstransfer),combustionandmostotherchemicalprocesses,cooling
towers, sorption at an interface (adsorption) or in a bulk
(absorption), and most living-matter processes as respiration (in
the lungs and at cell level), nutrition, secretion, sweating, etc.
Acommonprocesstoseparateagasfromagaseousmixtureistoselectivelydissolveitinan
appropriateliquid(thisway,carbondioxidefromexhaustgasescanbetrappedinaqueouslime
solutions, and hydrogen sulfide is absorbed from natural-gas
sources; when water vapour is removed,
theabsorptionprocessiscalleddrying.Strippingisthereverseofabsorption,i.e.theremovalof
dissolved components in a liquid mixture. Distillation is the most
important separation technique.
HOWTOSTUDYIT.SIMILARITIESANDDIFFERENCESBETWEENMASSTRANSFERAND HEAT
TRANSFER
MassTransfereducationtraditionallyfollowsandbuildsuponthatofHeatTransferbecause,onthe
one hand, mass diffusion due to a concentration gradient is
analogous to thermal-energy diffusion due
toatemperaturegradient,andthusthemathematicalmodellingpracticallycoincides,andthereare
many cases where mass diffusion is coupled to heat transfer (as in
evaporative cooling and fractional
distillation);onanotherhand,HeatTransferismathematicallysimplerandofwiderengineering
interest than Mass Transfer, what dictates the precedence. But
there are important differences between both subjects.
-Radiation.Firstofall,fromthethreeheattransfermodes(conduction,convection,and
radiation),onlythetwofirstareconsideredinmasstransfer(diffusionandconvection),
radiationofmaterialparticles(asneutronsandelectrons)beingstudiedapart(inNuclear
Physics). Notice, by the way, that the word diffusion can be
applied to the spreading of energy (heat diffusion), or species
(mass diffusion), or even momentum in a fluid or electric charges
in conductors, but the word conduction is more commonly used than
heat diffusion (whereas mass conduction is rarely used).
-Solidsversusfluids.HeatTransferstartswith,andfocuseson,heatdiffusioninsolids,which
havehigherthermalconductivitiesthanfluids,thelatterbeingconsideredgloballythrough
empirical convective coefficients, whereas Mass Transfer focuses
ongases and liquids, which
havehighermassdiffusivitiesthansolids.Theexplanationforsuchadifferenceisthatheat
conduction propagates by particle contact (for the same type of
particles, the shortest separation
thebetter),whereasmassdiffusionpropagatesbyparticlesmovingthroughthematerial
medium (for the same type of particles, the largest voids the
better). Moreover, Heat Transfer
problemsinsolidsaresimpleandrelevanttomanyapplications,whereasMassTransfer
problemsinsolidsareofmuchlesserrelevance,andMassTransferproblemsinfluidsare
muchmorecomplicatedbecausethesimplestmass-diffusionproblemsareoflittlepractical
interest, convection within fluids being the rule (fluids tend to
flow). When diffusion in solids is wanted, as in doping silicon
substrates in microelectronics, or in surface diffusion of carbon
ornitrogeninsteelhardening,hightemperatureoperationistherule(diffusioncoefficients
show an Arrhenius' type dependence with
temperature).-Slowness.Thermaldiffusivitiesdecreasefromsolidstofluids,withtypicalvaluesofa~10-4
m2/s for metals and a~10-5 m2/s for non-metals, down to a~10-5 m2/s
for gases and a~10-7 m2/s
forliquids.Onthecontrary,massdiffusivitiesdecreasefromfluidstosolids,withtypical
values of Di~10-5 m2/s for gases and Di~10-9 m2/s for liquids, down
to Di~10-12 m2/s for solids. -Bulk flow. There is no bulk flow in
heat diffusion (either within solids or fluids), whereas there is
always some bulk flow associated to diffusion of species (except in
the rare event of counter-diffusion of similar species); i.e. mass
diffusion generates mass convection, in general.
-Numberoffieldvariables.Onemayconsiderjustoneheat-transferfunction,thetemperature
field T (the heat flux is basically the gradient field), but
several mass-transfer functions must be
considered,onemassfraction,yi,foreachspeciesi=1..C(Cbeingthenumberofdistinct
chemical species), although most problems are modelled as a binary
system of just one species of interest diffusing in a background
mixture of averaged
properties.-Continuityatinterfaces.Mass-transferboundaryconditionsatinterfacesaremorecomplex
thanthermalboundaryconditions,becausetherearealwaysconcentrationdiscontinuities,
contrarytothecontinuoustemperaturedictatedbylocalequilibrium(chemicalpotentialsare
continuous at an interface, not
concentrations).-Diffusion'uphill'.Besidestheeffectofcoupledfluxes,itisimportanttorealisethatmass
diffusioncanbefromalowconcentrationwithinacondensedmediumtowardsahigh
concentration within a more disperse medium, because, as said, it
is not concentration-gradient but chemical-potential-gradient, what
drives mass diffusion (e.g. see Diffusion through a wall, below).
Forces and fluxes
Mixing,i.e.decreasingdifferencesincomposition(really,inchemicalpotential)ortemperature,isa
natural process (i.e. it does not require an energy expenditure),
driven by the gradients of temperature,
relativespeedandchemicalcomposition(withthenaturalstratificationinthepresenceofgravityor
another force field).
Itisinterestingtorealisethatthethermalandmechanicalforcestowardsequilibriumhavebeen
harnessed toyield useful power (heat engines, wind and water
turbines),butthe chemical forces that
drivemasstransferhavenotyetbeenrenderedusefulasenergysource,nodoubtbecauseofitslow
specificenergy(therehasbeenproposalstobuiltpowerplantsdrivenbythedifferenceinsalt
concentration at river mouths).
Thegradientoftemperature,momentumandconcentrations,giverisetocorrespondingfluxesin
thermalenergy,momentumandamountofspecies.Therelationbetweenforcesandfluxesarethe
transportconstitutiveequations:FourierlawforHeatTransfer,Newton(orStokes)lawforFluid
Mechanics, and Fick law for Mass Transfer (to be presented below),
and the purpose of the subject is
tosolvegenericfieldbalanceequations(energybalance,momentumbalance,andspeciesbalance),
with the help of constitutive equations, and the particular
boundary conditions and initial conditions. But before developing
the theory, it must be understood that mixing is a slow physical
process, if not
forcedbyconvectionandturbulence,andevenso.Manypracticalprocessesarelimitedbythe
difficulty to increase the mass transfer rate. An order of
magnitude analysis shows that the relaxation
timefordiffusion-controlledphenomena(thermal,momentum,species)acrossadistanceLis
trelax=L2/a,whereaisthediffusivitythat,asexplainedbelow,isoforder10-5m2/singases,what
teaches that diffusion across a 1 m distance takes some 105 s, i.e.
one whole day. Of course, everybody knows that heating onemetre of
air doesn't takeone day, neither it takesso long for odours to
travel one metre, or for putting in motion or arresting a gas; the
explanation is that fluids are very difficult to
keepatrestwhenperturbed,andtheconvectionthatdevelopsgreatlyincreasesthemixingrateand
lowers the required time.
Thermodynamicsteachesthat,withinanisolatedsysteminabsenceofexternalforces,temperature,
relativemotionandchemicalpotentialtendtogetuniform,byestablishingathermal-energyflux,a
momentumfluxandamass-diffusionflux,proportional(toafirstapproximation)tothegradientsof
temperature,velocityandconcentration,thattendtoequilibratethesystem.Noticehoweverthat,
besides those direct fluxes, other smaller cross-coupling fluxes
may appear, as mass-diffusion due to a temperature gradient in a
uniform concentration, or heat transfer due to a concentration
gradient in an
isothermalfield,which,inthelinearapproximation,arerelatedamongthembyOnsager'sreciprocal
relations. Specifying composition. Nomenclature Mass transfer may
take place within gases, liquids, solids or through their
interfaces, always involving a mixture, but mass diffusion in a gas
is of main interest for two reasons: first, it is the best
understood,
andsecond,itisthebestdiffusingmedium(diffusioninliquidsandsolidsismuchslower).Forthat
reason, and for simplicity, we start here with a gaseous (single
phase) multi-component mixture.
Amixtureisanymulti-componentsystem,i.e.onewithseveralchemicalspecies.The
thermodynamicsofmixturesingeneral(gaseous,liquidorsolid)hasbeenconsideredunderthe
headingMixtures,mainlydevotedtoidealmixtures.Weassumetruesolutions,i.e.homogeneous
solutions, and do not consider colloids and suspensions, treated
under the heading Mixture settling. Although, from the theoretical
point-of-view, molar fractions and concentrations should be
preferred,
themostcommoncompositiondeterminantinasingle-phasemixtureisthemassfraction,yi,orthe
mass density i. Only one of those parameters is needed, but all of
them are made use of in practice, so a common (an tedious) task in
mass-transfer calculations is to pass from one variable to another,
based on their definitions: mass fractions:ymmx Mx Miiii ii i = (1)
mass densities (or mass concentrations): iiimVy = (2) molar
densities (or molar concentrations):cnV Mxx Mii iiii i = = (3)
molar fractions:xnny My Miiii ii i = //(4) partial pressure of a
species i in a gas mixture:IGMui i i u iiRp x p c R T TM = = .(5)
ThemolarmassofthemixtureisdefinedasMmm/n=/c=xiMi,althoughitisonlyusedforgas
mixtures.Therearestillotherspecialvariablesinusetodefineamixturecomposition,asair-to-fuel
ratio and richness (equivalence ratio) in combustion problems.
Exercise1.Dryaircanbeapproximatedasamixtureof79%N2and21%O2byvolume(meaning
that,byletting79volumesofpurenitrogentomixwith21volumesofpureoxygen,
without changes in pressure and temperature, i.e. by just removing
the partition, we obtain
100volumesofamixturecloselyresemblingdryair).Determineotherpossible
specifications of dry air composition, from (1-5).
Solution.Assumingidealgasbehaviour,i.e.pV=nRT,atconstantpandT,volumesVare
proportional to amounts of substance, n, and thus volume percentage
coincides with molar fractions (4); i.e., we can consideras
dataxN=0.79 and xO=0.21 (mind that subindices are just labels, not
meaning atoms but
molecules).From(1),withMN=0.028kg/molandMO=0.032kg/mol,onegets
xiMi=0.79 0.028+0.21 0.032=0.029kg/mol,yN=0.79
0.028/0.029=0.77andyN=0.23,
indicatingthatthemolarmassforthemixtureisaweightedaverageofthoseofthe
components,Mmm/n=/c=xiMi=0.029kg/mol,andthattheheavierspeciesshowsa
larger concentration-value in terms of masses than in terms of
amounts of substance.
From(2)wegetmassconcentrations(massdensities)intermsofthemixturedensity,
whichdependsontemperatureandpressure;forT=288Kandp=100kPa,wegetforthe
densityofair=1.21kg/m3,andforthespeciesN=0.77
1.22=0.93kg/m3,andO=0.23
1.22=0.28kg/m3.Noticethatsomeauthorsusemiorwiinsteadofyiformass
fractions. From (3) we can get molar concentrations, again
depending on actual p-T values; with the
previouschoice,cN=N/MN=0.93/0.028=33mol/m3,andcO=O/MO=0.28/0.032=9mol/m3
(in total, c=cN+cO=p/(RT)=105/(8.3 288)=42 mol/m3).
Fro(5)wegetthepartialpressures,pN=xNp=0.79 105=79kPa,andpO=xOp=0.21
105=21 kPa. Finally notice that we can equally say that air has
0.79/0.21=3.76 times more nitrogen than
oxygen,byvolume(oramountofsubstance),or0.770.23=3.29timesmorenitrogenthan
oxygen, by mass. Thefindingof qualitative or
quantitativecomposition in a mixture is known as chemical analysis,
or
simply'theanalysis'.Wefocushereonquantitativeanalysis,assumingthesubstancesarealready
known. Most methods of concentration analysis are based on
measuring mixture density (provided the
densitydependenceonspeciesconcentration,m=m(T,p,xi),isknownbeforehandbycalibration),by
one of the different techniques: -Absorption radiometry. By light
transmittance (in the visible, infrared, or
monochromatic).-Refractometry. By ray tracing. Refractive index
varies almost linearly with density.
-Gravimetry.Weightingaknownvolumeofliquid.Thisisperhapstheeasiestandquickest
method to measure solution concentration, but requires sampling.
-Resonant vibration. The natural frequency of anencapsulated liquid
sample precisely metered
dependsonitsmass.Maybeappliedtoaliquidflowingalongabendconnectedbysoft
bellows to the
pipes.-Sonicvelocimetry.Densityisobtainedfrom=E/c2,whereEisthebulkmodulusofthe
solution and c the sound speed through it.
-Electricconductivity.Thisisthebestmethodforverylowconcentrationofelectrolytic
solutions.Themeasuringelectrodesmaybegeneric,orselectiveforsomespecificion(e.g.
Ca2+, NH4+, Cl-, NO3-). Specifying boundary conditions for
composition Composition at boundaries or internal interfaces in a
mixture usually shows a discontinuity, contrary to
temperatureinheattransferproblems,whencontinuityistherule(exceptforthespecialtopicof
thermal joint conductance). The typical boundary conditions for a
species concentration are, as for heat transfer, a known value of
thefunction(imposedconcentrationortemperature,respectively),oraknownvalueofitsgradient
(imposedspeciesfluxorheatflux,respectively),thespecialcaseinthelatterbeingtheimpermeable
interface or adiabatic wall, respectively; here: impermeable
interface: interface0, or in 1D 0iix xxx nx=cV = =c(6) n
beingtheunitnormalvectortotheinterface.Imposinganon-zeromassflux,oragiven
concentration value, is done as in Heat Transfer, i.e. by providing
large sources of the chemical species (a solid chunk, a liquid
pool, a gas reservoir), similarly to large metal blocks to specify
the temperature
atawall.Localthermodynamicequilibriumthenteachesthatthetemperatureofthesystemnearthe
wallisequaltothatofthewall,butthesameisnottrueforconcentrations,wherelocalequilibrium
implies equality of its chemical potential, not of its
concentration.
Theboundaryconditioninagasmixturemaybeanothergasphase,aswhenmixingalongatube
connectedtoalargereservoirofagivengas;ifoneassumesthatthelargereservoiriswell-stirred,
thence,theboundaryconditionforthegasmixtureinthetubemaybeapproximatedbytheknown
concentration at the reservoir.
Foragasmixtureincontactwithacondensedphase,thetypicalboundaryconditionsforaspecies
concentration, assuming ideal mixtures, is Raoult's law (deduced in
Mixtures): * *,gas pure condensed phase,vap,condensed( ) ( )exp/ii
i uii uxp T p T p Bx Ax p p p C T T| |= = = |+\ . (7)
whereAntoine'sfittingcoefficientsforthevapourpressurecurvehavebeenexplicitlyshown(see
Phase Change for an explanation). Notice that sublimation
vapour-pressure data should be used when
thesourceissolid,e.g.wheniceisthesourceofwatervapour,insteadofliquidwater.Forinstance,
theboundaryvalueforwater-vapourdiffusioninambientairclosetoawaterpoolat15Cis
xi,vap=0.017,correspondingtothetwo-phaseequilibriumpressureofpurewaterat15C:1.7kPa.
When gases are sparingly soluble, Henry's law must be used instead
of Raoult's law (see Solutions). Exercise 2.Find the concentration
of carbon dioxide at a water surface at 25 C, when exposed to a gas
stream with a partial pressure of CO2 of 300
kPa.Solution.Henry'slawdatacanbefoundinabewilderingvarietyofmanners,andwithdifferent
units,usuallyunderthecommonnameof'Henryconstant',KH.ForsolubilityofCO2in
waterat25C,wemayfind,fromthesolubilitydata(Table3)inSolutions,
KH=ci,liq/ci,gas=0.80, meaning that, for CO2 to be at equilibrium
between the aqueous phase and the gas phase, there must be 0.80
mol/m3 of CO2 dissolved in water per each 1 mol/m3 of CO2 dissolved
in the gas phase (or pure). We might find the same number but
referring to mass concentrations, since they are just proportional
with the factor MCO2=0.044 kg/mol,
(3),KH=i,liq/i,gas=0.80,meaningthat,forCO2tobeatequilibriumbetweentheaqueous
phaseandthegasphase,theremustbe0.80kg/m3ofCO2dissolvedinwaterpereach1
kg/m3ofCO2dissolvedinthegasphase(orpure).Thosearetheonlynon-dimensional
'constants' (constant in Henry's law, and other equilibrium laws in
Chemistry, means that it only depends on temperature, not on
pressure).We might findKH=ci,liq/pi,gas=32 (mol/m3)/bar, meaning
that, for CO2 to be at equilibrium between the aqueous phase and
the gas phase, there must be 32 mol/m3 of CO2 dissolved in
waterpereach1bar(100kPa)ofpartialpressureofCO2dissolvedinthegasphase(or
pure);ofcourse,wecancheckforconsistency:ci,liq/ci,gas=RTci,liq/pi,gas,butitisproneto
trivialerrorsonunitconversion(e.g.the105in
ci,liq/ci,gas=RTci,liq/pi,gas=0.80=8.3 298 32/105.We might find
KH=xi,liq/pi,gas=580 ppm_mol/bar, meaning that, for CO2 to be at
equilibrium between the aqueous phase and the gas phase, there must
be 580 parts-per-million in molar
baseofCO2dissolvedinwaterpereach1bar(100kPa)ofpartialpressureofCO2
dissolvedinthegasphase(orpure);wecancheckforconsistency:
ci,liq/ci,gas=(mRT/Mm)xi,liq/pi,gas,wheresubindexmreferringtothesolution,whichcanbe
approximatedaspurewater,andthence
ci,liq/ci,gas=(mRT/Mm)xi,liq/pi,gas=0.80=(1000 8.3 298/0.018) 580
10-6/105.
WemightfindKH=ci,liq/ci,gas,STP=0.73m3(STP)/bar,meaningthat,forCO2tobeat
equilibriumbetweentheaqueousphaseandthegasphaseat25C,theamountofCO2
dissolved in 1 m3 of solution, per each 1 bar (100 kPa) of partial
pressure of CO2 dissolved in the gas phase (or pure), would occupy
0.73 m3 at STP-conditions of 0 C and 100 kPa;
wecancheckforconsistency:ci,liq/ci,gas,STP=(ci,liq/ci,gas)/(TSTP/T)=0.80
273/298=0.73, where subindex m referring to the solution, which can
be approximated as pure water, and thence
ci,liq/ci,gas=(mRT/Mm)xi,liq/pi,gas=0.80=(1000 8.3 298/0.018) 580
10-6/105. In summary, if we assume that pure carbon dioxide at 300
kPa (or a gas mixture with that partial pressure of CO2) is at
equilibrium with water at 25 C, the CO2 concentration in the
gasphaseisci,gas=xip/(RT)=300 105/(8.3
298)=121mol/m3,andtheCO2concentrationin
solutionisci,liq=KHci,gas=0.80
121=97mol/m3,i.e.0.17%ofthemoleculesintheliquid phase are CO2, and
99.8% are H2O molecules (assuming no other solute is present); it
can also be concluded that, if all the CO2 dissolved in 1 m3 of
water at equilibrium at 25 C and 100 kPa, were extracted and put at
STP-conditions (0 C and 100 kPa), it would occupy a volume of 2.2
m3.
Foraliquidmixtureincontactwithanothercondensedphase(asolidoranimmiscibleliquid),the
boundary condition for a species concentration, i, called a solute,
cannot be modelled in a simple form
asRaoult'slaw;atmost,intheidealcase,fromtheequalityofthesolutechemicalpotentialinboth
phases one gets: ,liq ,sol ,liq ,sol-liq ,sol-liq ,sol-liq,sol( ) (
) ( ) ( ) ( )lni i i i i ii u u u ux T T g T h T s Tx R T R T R T R
A A A= = = + (8) where the other phase has been labelled 'sol' both
for the case of a solid or an immiscible liquid. In the
caseofapuresolidasasourceofsolute,theboundarycondition(8)yields
xi,liq=exp((i,soli,liq)/(RuT)),anditisknownasthesolubilityofthesolidsoluteintheliquidsolvent
specified (i.e. the maximum molar fraction of solute the liquid can
hold). Solubility data for solid and liquid solutes in a liquid
solvent can be found aside. Diffusion of species within a solid is
much more intricate, particularly when the solid is porous or is in
agranularstate,wherehydrodynamicflowappears(seepage).Diffusionthroughone-piecesolidsis
nearly negligible in most cases at room temperature, but can be
studied with Henry's law (some values aregiven inSolutions). Gas
solubility in solids increases with temperature, contrary to what
happens in liquids, and subsequent degassing on cooling may be a
nuisance (may even ruin a casting process by
creatingporosityandvoids).Besides,chemicalreactionsmayoccuratroomtemperature(e.g.
oxidation) but particularly when the temperature is increased to
enhance mass transfer. SPECIES BALANCE EQUATION
Foragivenspeciesiinamixture(solid,
liquidorgaseous),itsmassbalanceforacontrolvolumeis (accumulation =
flux + production): ,surfacesii i gen i iA Vdmm m j n dA wdVdt= + =
+} }(9) where mi is the mass of species i in the volume V, ijis the
local mass-flux of species i at the surface
areaA,andwiapossiblelocalspeciesgenerationdensityduetochemicalreactions.Foracontrol-volume
system of differential volume dxdydz, with the continuum model: ( )
( )d d d d d d d d d d d d d ... d d di iyi ix ii ix i ix i iy i iy
ivvxyz v yz v x yz v zx v y zx wxyzt x y ( | |cc | | c (| = + + + +
+ | |c c c (\ .\ . ( )ii i iv wtc +V =c(10) where i is its mass
density and ivthe local velocity of the i-component fluid in a fix
reference frame.
Foraone-componentfluid,themassbalance(10)reducesto( ) / 0 t v c c+V
= ,thewell-known continuity equation of Fluid Mechanics, that can
be recovered by summation in (10) for all the species i in the
mixture; i.e.: ( )( )( )with/0i iv vii i iv w vt t EcE c+V E = E +V
=c c(11)
Noticethatasimilarargumentmighthavebeenfollowedwithmolardensitiesinsteadofmass
densities,andamolar-averagedvelocitydefinedthatwouldnotcoincideingeneralwiththemass-averaged
velocityv , that is traditionally used.
Besidesthespeciesbalanceinagenericdifferentialvolume(10),thespeciesbalanceinageneric
interface must be established in many problems: , , ,0i out i in i
surface genm m m= + (12)
wherethelastterm,speciesgenerationattheinterface,onlyappearsinthecaseofheterogeneous
reactions at the interface. Diffusion rate: Fick's law
Actualmixingofchemicalspeciesisgovernedbymass-transferlawsverysimilartoheat-transfer
laws, establishing a linear proportion between forces and fluxes:in
Heat Transfer, a linear proportion
betweenthetemperature-gradient,andtheenergyflowasheat;inMassTransfer,alinearproportion
betweenthespeciesdensity-gradient,andtherelativevelocityofthespecies-fluidtothemean-fluid.
The basic kinetic-law for mass diffusion was proposed in 1855 by
the German physiologist A. Fick for a homogeneous media without
phase changes or chemical reactions, namely: ( ) , withdii i i di
di i i i i dimn v v v j D j y j jA = = V = +(13)
thatreads:themass-flow-rateofspeciesidiffusingperunitareainthenormaldirectionn
(mass-diffusionfluxofspeciesi), dij
,whichisitsdensitytimestherelativevelocityofthespecies-fluidto
themean-fluid(thelatterdifferencesimplycalleddiffusionspeed di iv v
v ),isproportionaland
opposestothespeciesdensity-gradient,Vi,withtheproportionalityconstantDinamedmass-diffusivityforspeciesiinthegivenmixture,andi=yithemass-densityforspeciesiinthegiven
mixture.NoticethatFick'slaw, di i ij D = V
,onlyaccountsformass-flow-ratesandfluxesdueto
diffusion(byagradientinconcentration);ifthereisaconvectivefluxj v
(notassociatedto gradientsinconcentrationbuttobulktransportatspeedv
),thenthenetfluxofspeciesiis i i dij y j j = + , or i i i i div v v
= + , which was used in (13).
TheoriginalFick'slaw(13),whichheproposedjustemulatingFourier'slaw(of1822),perfectly
matchesexperimentswithdilutesolutions,i.e.whenthepropertiesofthemediumcanbeassumed
independentofthespeciesiconcentration,and(13)canalsobewrittenas i i
ij D y = V ,themost
generalFick'slawstatement,extending(13)tocoverdiffusionathighconcentrations.Eveninthe
originalcasehetried,saltdiffusionalongatesttubefromasaturatedbrinebelowtoafresh-water-swept
zero-concentration at the mouth, with a density jump from 1200
kg/m3 at the salt-brine interface and 1000 kg/m3 at the top
surface, deviations from the linear density profile corresponding
to the
one-dimensionalsteady-stateproblemwithconstantDiarelessthana1%atmost;hefound
Dsalt,water=0.12 10-9 m2/s.
Fick'slawissimilartoFourierslawforheattransferq k T = V (or ( )pq a
c T = V foraconstant-property medium), and applies to gases,
liquids and solid mixtures, with Di depending on the diffusing
speciesi,themediumanditsthermodynamicstate.Fick'slawisalsosimilartoDarcy'slawofmean
fluidvelocitythroughporous-media( ) /( ) v h p g k = V +
,andtoNewton'slawofmomentum transport by viscosity( ) v t v = Vfor
a constant-density fluid of kinematic viscosity v, wheretis the
stress tensor. In fact, for gases, a simplified analysis dictates
that Di=o=v.
NoticethatonlythefluxassociatedtothemaindrivingforceisconsideredinEq.(13),i.e.mass-diffusion
due to a species-concentration gradient (as for heat-diffusion due
to a temperature gradient).
Therearealsosecondaryfluxesassociatedtootherpossiblegradients(e.g.mass-diffusionduetoa
temperaturegradient,knownasSoreteffect,andmass-diffusionduetoapressuregradient;
alternatively,theremaybeheat-diffusionduetoaspecies-concentrationgradient,knownasDufour
effect, and heat-diffusion due to a pressure gradient), but most of
the times those cross-coupling fluxes are negligible. Besides,
selective force fields may yield diffusion (e.g. ions in an
electric field). Typical
valuesforDi(andthethermaldiffusivitya=k/(cp))aregiveninTable1,withSchmidtnumbers,
Sc=v/Di,toevaluatenon-idealityingases(kinetictheoryofidealgasespredictsSc=1);forair
solutions,thedynamicviscosityispracticallythatofair,v=15.9
10-6m2/sat300K.foraqueous solutions, the dynamic viscosity is
practically that of water, v=0.86 10-6 m2/s at 300 K. Table 1.
Typical values for mass and thermal diffusivities,Di and a, and
Schmith number, Sc, all at 300 K (extracted from Mass diffusivity
data). SubstanceDiffusivityTypical valuesExampleSc=v/Di Gasesa)a105
m2/saair=22106 m2/s aCH4=24106 m2/s Di105 m2/sDwaterapour,air=24106
m2/sDCO2,air=14106 m2/s (390106 m2/s at 2000 K) DCH4,air=16106 m2/s
0.66 1.14 0.99 Liquidsb)a107 m2/sawater=0.16106 m2/s Di109
m2/sDN2,water=3.6109 m2/s DO2,water=2.5109 m2/s 240 340
Solidsc)a106 m2/sasteel=13106 m2/s aice=1.3106 m2/s afresh
food=0.13106 m2/s Di1012 m2/sDN2,rubber=1501012 m2/s
DH2,polyethylene=870001012 m2/s DH2,steel=0.31012 m2/s a) For gas
diffusion, both for a and Di, a general dependence with temperature
and pressure of the form Tn/p can be used, with 1.50.Tends to a
point-source (Gauss bell) for tContinuos planar-source,
one-dimensional deposition 2exp4( , )2xQtatTx tc a t| | |\ .=(0, )2
QtT tc a t= Continuos line-source deposition 2Ei4( , )4rQatTr tca
t| | |\ .= Singular at r=0. For4 r at R. Moving planar-source
one-dimensional deposition ( )0expxVT T Ta | |= |\ . ( )0expi i
iixVy y yD| | = |\ . t0: T(0)=T0, yi(0)=yi0. Moving point-source
tri-dimensional deposition ( )exp24U r xqaTar t | | |\ .=Only valid
for steady state (t ). r2=x2+y2+z2. U is the constant relative
speed. Planar contact with forced jump at the surface ( )0erfc2 xT
T Tat| |= |\ . ( )0erfc2i i iixy y yDt| |= | |\ . t0: T(0)=T0,
yi(0)=yi0. Planar contact erf2 xT A Bat| |= + |\ . erf2iixy A BDt|
|= +| |\ . 1 ,1 1 1 2 ,2 2 201 ,1 1 2 ,2 2p pp pc k T c k TTc k c k
+=+ 1 1 1 2 2 201 1 1 2 2 2i i i iii iDy DyyD k Dk +=+ Contact
value is T0 for heat transfer, or yi0 for mass transfer. t=0,
Heaviside(x). If equal properties: T0=(T1+T2)/2 and
yi0=(yi1+yi2)/2. Continuos one-dimensional planar plate immersion (
)0 i i iy y y= ( )22202 1exp412 1inn DtLntt= | |+ | |\ . + ( ) 2
1sinnxLt + | |`| \ .) t0: T(0)=T0, yi(0)=yi0, T(L)=T0, yi(L)=yi0.
One-dimensional planar contact, steady in a moving frame (e.g.
mixing layer of two equal-speed streams) erf2 /xy A Baz v| |= + |\
. It is the same as the latter changing t=z/v, where v is the
common speed One-dimensional planar steady diffusion through a wall
or gap (e.g. evaporation from a test tube) 0 1 01 0/10 00 1 01 0*0
, 1 11 1, ( )interface equilibrium: izLV D Li ii ii i i iii iv viaT
T T T zq kT T L Ly yy yy y y y zj D Vy y L LMp TyMp
