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Steady state Molecular diffusion in Fluids

steady state molecular diffusion in liquids and steady state molecular diffusion in fluids at rest in laminar flow steady state convection diffusion equation
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Published Date:25-10-2017
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Chapter26 Steady-State Molecular Diffusion In this chapter, we will direct our attention to describing the steady-state transfer of mass from a differential point of view. To accomplish this task, the differential equation and the boundary conditions that describe the physical situation must be established. The approach will parallel those previously used in Chapter 8 for the analysis of a differential fluid element in laminar flow and in Chapter 17 for the analysis of a differential volume element of a quiescent material for steady-state heat conduction. During our discussion of steady-state diffusion, two approaches will be used to simplify the differential equations of mass transfer as recommended in Section 24.4. First, Fick’s equation and the general differential equation for mass transfer can be simplified by eliminating the terms that do not apply to the physical situation. Second, a material balance can be performed on a differential volume element of the control volume for mass transfer. In using both approaches, the student will become more familiar with the various terms in the general differential equation for mass transfer c A : = N þ R ¼ 0 (25-11) A A t To gain confidence in treating mass-transfer processes, we will initially treat the sim- plest case, steady-state diffusion in only one direction, which is free of any chemical production occurring uniformly throughout the process (i.e., R ¼ 0). We will then A obtain solutions for increasingly more complex mass-transfer operations. 26.1 ONE-DIMENSIONAL MASS TRANSFER INDEPENDENT OF CHEMICAL REACTION In this section, steady-state molecular mass transfer through simple systems in which the concentration and the mass flux are functions of a single space coordinate will be consi- dered. Although all four fluxes,N ,n ,J , andj , may be used to describe mass-transfer A A A A operations,onlythemolarfluxrelativetoasetofaxesfixedinspace,N ,willbeusedinthe A following discussions. In a binary system, the z component of this flux is expressed by equation (24-20) dy A N ¼cD þy (N þN ) (24-20) A;z AB A A;z B;z dz Unimolecular Diffusion Thediffusioncoefficientormassdiffusivityforagasmaybeexperimentallymeasuredinan Arnolddiffusioncell.ThiscellisillustratedschematicallyinFigure26.1.Thenarrowtube, 45226.1 One-Dimensional Mass Transfer Independent of Chemical Reaction 453 Flow of gas B which is partially filled with pure liquid A, is maintained at a constant tem- z = z peratureandpressure.GasB,whichflowsacrosstheopenendofthetube,hasa 2 negligiblesolubilityinliquidAandisalsochemicallyinerttoA.ComponentA vaporizes and diffuses into the gas phase; the rate of vaporization may be N Az z+∆ z physicallymeasuredandmayalsobemathematicallyexpressedintermsofthe molar mass flux. ∆ z Recall that the general differential equation for mass transfer is given by N Az z c z = z A 1 : = N þ R ¼ 0 (25-11) A A t In rectilinear coordinates, this equation is Pure liquid A N N N c Ay Ax Az A þ þ þ R ¼ 0 (25-27) Figure 26.1 Arnold diffusion cell. A x y z t Assumethat(1)themass-transferprocessisatsteadystatewithc /t¼ 0;(2)thereisno A chemical productionof A in the diffusionpath so that R ¼ 0; and (3)the diffusionis only A in the z direction, so that we are only concerned with the z component of the mass flux vector, N . For this physical situation, equation (25-11) reduces to A dN Az ¼ 0 (26-1) dz We can also generate this governing differential equation by considering the mass transfer occurring in the differential control volume of SDz,where S is the uniform cross-sectional area of the control volume andDz is the depth of the control volume. A mass balance over thiscontrolvolumeforasteady-stateoperation,freeofanychemicalproductionofA,yields SN j SN j ¼ 0 Az Az zþDz z Dividingthroughbythecontrolvolume,SDz,andtakinglimitasDzapproacheszero,we once again obtain equation (26-1). A similar differential equation could also be generated for component B d N ¼ 0 (26-2) B;z dz and,accordingly,themolarfluxofBisalsoconstantovertheentirediffusionpathfromz 1 to z . Considering only the plane at z and the restriction that gas B is insoluble in liquid 2 1 A, we realize that N at plane z is zero and conclude that N , the net flux of B, is zero Bz 1 Bz throughout the diffusion path; accordingly, component B is a stagnant gas. The constant molar flux of Awas described in Chapter 24 by the equation dy A N ¼cD þy (N þN ) (24-20) A;z AB A A;z B;z dz this equation reduces, when N ¼ 0, to B;z cD dy AB A N ¼ (26-3) A;z 1y dz A This equation may be integrated between the two boundary conditions: atz¼ z y ¼ y 1 A A 1 and atz¼ z y ¼ y 2 A A 2454 Chapter 26 Steady-State Molecular Diffusion Assumingthediffusioncoefficienttobeindependentofconcentration,andrealizingfrom equation (26-1) that N is constant along the diffusion path, we obtain, by integrating, A;z Z Z z y 2 A 2 dy A N dz¼ cD  (26-4) A;z AB 1y z y A 1 A 1 Solving for N , we obtain A;z cD (1y ) AB A 2 N ¼ ln (26-5) A;z (z z ) (1y ) 2 1 A 1 The log-mean average concentration of component B is defined as y y B B 2 1 y ¼ B;lm ln(y /y ) B B 2 1 or, inthe case of a binary mixture, this equationmaybe expressed interms of component A as follows: (1y )(1y ) y y A A A A 2 1 1 2 y ¼ ¼ (26-6) B;lm ln½(1y )/(1y ) ln½(1y )/(1y ) A A A A 2 1 2 1 Inserting equation (26-6) into equation (26-5), we obtain cD (y y ) AB A A 1 2 N ¼ (26-7) A;z z z y 2 1 B;lm Equation (26-7) may also be written in terms of pressures. For an ideal gas n P c¼ ¼ V RT and p A y ¼ A P The equation equivalent to equation (26-7) is D P (p  p ) AB A A 1 2 N ¼ (26-8) A;z RT(z z ) p 2 1 B;lm Equations (26-7) and (26-8) are commonly referred to as equations for steady-state diffusion of one gas through a second stagnant gas. Many mass-transfer operations involve the diffusion of one gas component through another nondiffusing component; absorption and humidification are typical operations defined by these two equations. Equation(26-8)hasalsobeenusedtodescribetheconvectivemass-transfercoefficients bythe‘‘filmconcept’’ orfilmtheory.InFigure26.2,theflowofgasoveraliquidsurfaceis Main gas stream z = d Flow of Slowly moving N z = 0 Az gas film gas B Liquid A Liquid A Figure 26.2 Film model for mass transfer of component A into a moving gas stream.26.1 One-Dimensional Mass Transfer Independent of Chemical Reaction 455 illustrated. The ‘‘film concept’’ is based upon a model in which the entire resistance to diffusionfromtheliquidsurfacetothemaingasstreamisassumedtooccurinastagnantor laminar film of constant thicknessd.In other words,for this model,d is a fictitious length which represents the thickness of a fluid layer offering the same resistance to molecular diffusionasisencounteredinthecombinedprocessofmoleculardiffusionanddiffusiondue to the mixing by the moving fluid. If this model is accurate, the convective mass-transfer coefficientmaybeexpressedintermsofthegasdiffusioncoefficient.Ifz z issetequal 2 1 to d, equation (26-8) becomes D P AB N ¼ (p  p ) A,z A A 1 2 RTp d B,lm and from equation (25-30), we have N ¼ k (c c ) A;z c A A 1 2 and k c N ¼ (p  p ) A;z A A 1 2 RT Comparison reveals that the film coefficient is expressed as D P AB k ¼ (26-9) c p d B;lm when the diffusing component is transported through a nondiffusing gas. Although this model is physically unrealistic, the ‘‘film concept’’ has had educational value in supply- ing a simple picture of a complicated process. The film concept has proved frequently misleading in suggesting that the convective mass-transfer coefficient is always directly proportional to the mass diffusivity. Other models for the convective coefficient will be discussed in this chapter and in Chapter 28. At that time wewill find that k is a function c of the diffusion coefficient raised to an exponent varying from 0.5 to 1.0. Frequently, in order to complete the description of the physical operation in which mass is being transported, it is necessary to express the concentration profile. Recalling equation (26-1) d N ¼ 0 (26-1) A,z dz and equation (26-3) cD dy AB A N ¼ (26-3) A,z 1y dz A we can obtain the differential equation that describes thevariation in concentration along the diffusing path. This equation is  d cD dy AB A  ¼ 0 (26-10) dz 1y dz A As c and D are constant under isothermal and isobaric conditions, the equation AB reduces to  d 1 dy A ¼ 0 (26-11) dz 1y dz A456 Chapter 26 Steady-State Molecular Diffusion This second-order equation may be integrated twice with respect to z to yield ln(1y )¼ c zþc (26-12) A 1 2 The two constants of integration are evaluated, using the boundary conditions at z¼ z y ¼ y 1 A A 1 and at z¼ z y ¼ y 2 A A 2 Substitutingtheresultingconstantsintoequation(26-12),weobtainthefollowingexpres- sion for the concentration profile of component A:  (zz )/(z z ) 1 2 1 1y 1y A A 2 ¼ (26-13) 1y 1y A A 1 1 or, as y þy ¼ 1 A B   (zz )/(z z ) 1 2 1 y y B B 2 ¼ (26-14) y y B B 1 1 Equations (26-13) and (26-14) describe logarithmic concentration profiles for both species. The average concentration of one of the species along the diffusion path may be evaluated, as an example for species B,by Z z 2 y dz B z 1 Z y ¼ (26-15) B z 2 dz z 1 Upon substitution of equation (26-14) into equation (26-15), we obtain Z  (zz )/(z z ) z 2 1 2 1 y B 2 dz y B z 1 1 y ¼ y B B 1 z z 2 1 (y y )(z z ) y y B B 2 1 B B 2 1 2 1 ¼ ¼ ln(y /y )(z z ) ln(y /y ) B B 2 1 B B 2 1 2 1 ¼ y ð26-6Þ B,lm Thefollowingexampleproblemillustratestheapplicationoftheforegoinganalysistoa mass-transfer situation. EXAMPLE 1 VapordegreasersliketheoneshowninFigure26.3arewidelyusedforcleaningmetalparts.Liquid solventrestsatthebottomofthedegreasertank.Aheatingcoilimmersedinthesolventvaporizesa smallportionofthesolventandmaintainsaconstanttemperature,sothatthesolventexertsaconstant vapor pressure. The cold parts to be cleaned are suspended in the solvent vapor zone where the concentrationofsolventvaporsishighest.Thesolventcondensesonthepart,dissolvesthegrease, andthendripsbackdownintothetank,therebycleaningthepart.Vapordegreasersareoftenleftopen totheatmosphereforeaseofdippingandremovingpartsandbecausecoveringthemmightreleasean explosivemixture.Whenthedegreaserisnotinuse,moleculardiffusionofthesolventvaporthrough the stagnant air inside the headspace can result in significant solvent emissions, because the surrounding atmosphere serves as an infinite sink for the mass-transfer process. As the amount26.1 One-Dimensional Mass Transfer Independent of Chemical Reaction 457 Atmosphere ofsolventinthedegreasertankislargerelativetotheamountofvapor z = z = 5 m 2 y = 0 emitted,asteady-statediffusionprocesswithaconstantdiffusionpath A2 Still air length takes place. At present, a cylindrical degreaser tank with a diameter of 2 m TCE Still and total height of 5 m is in operation, and the solvent level height is vapors TCE vapors air keptconstantat0.2m.Thetemperaturesofthesolventandheadspace ofthedegreaserarebothconstantat358C.Thesolventusedforvapor degreasingistrichloroethylene(TCE).Currentregulationsrequirethat the degreaser cannot emit more than 1.0 kg TCE per day. Does the Metal estimated emission rate of the degreaser exceed this limit? TCE has a part molecularweightof131.4g/molandavaporpressureof115.5mmHg 2 at 358C. The binary-diffusion coefficient TCE in air is 0.088 cm /s at Vapor zone 358C, as determined by the Fuller–Schettler–Giddings correlation. The source of TCE mass transfer is the liquid solvent at the z = z = 0.2 m 1 Heater bottom of the tank,andthesinkfor mass transfer isthesurrounding TCE liq. 35°C atmosphereoutsidethetank.Thesteady-statemoleculardiffusionflux y = P /P = 0.152 A1 A of TCE vapor through the stagnant gas headspace of the degreaser in Vapor degreaser tank the z direction is described by 2 m diameter  cD 1y AB A Figure 26.3 TCE emissions from a vapor degreaser. 2 N ¼ ln Az z z 1y 2 1 A 1 withspeciesArepresentingTCEvaporandspeciesBrepresentingair.Thetotalmolarconcentration of the gas, c, is determined from the ideal gas law. P 1:0atm kgmol c¼ ¼ ¼ 0:0396 3 3 RT 0:08206m atm m (273þ35)K : kgmol K ThemolefractionofTCEvaporatthesolventsurface(y )isdeterminedfromthevaporpressureof A 1 the solvent at 358C. P 115:1mmHg 1:0atm A y ¼ ¼ ¼ 0:152 A 1 P 1:0atm 760mmHg The mole fraction of TCE vapor at the exit of the degreaser tank is taken as zero (y ¼ 0), as A 2 the surrounding atmosphere serves as an infinite sink for mass transfer. The path length for diffu- sion is simply the difference between the solvent level height and the top of the degreaser tank z z ¼ 5:0m0:2m¼ 4:8m 2 1 From these input values, the flux of TCE vapor from the degreaser is  2 2 kgmol cm 1m 0:0396 0:088  3 2 m s ð100cmÞ 10 N ¼ ln Az 4:8m 10:152 kgmolTCE 8 ¼ 1:19710 2 : m s TheTCEemissionsrate(W )istheproductofthefluxandthecross-sectionalareaofthedegreaser A tank of diameter D 2 2 pD kgmolTCE p(2:0m) 8 W ¼ N ¼ 1:19710 A Az 2 : 4 m s 4  131:4kgTCE 3600s 24h kgTCE ¼ 0:423 kgmolTCE 1h day day458 Chapter 26 Steady-State Molecular Diffusion TheestimatedTCEvaporemissionrateisbelowthecurrentregulatorylimitof1.0kgTCEperday. In a real degreaser, it may be difficult to ensure a completely still gas space, as local air currents inducedfromavarietyofsourcesmayoccur.Theaircurrentswouldincreasethemass-transferflux byconvection.Consequently,thisanalysisconsidersonlythelimitingcasefortheminimumvapor emissions from a diffusion-limited process. Pseudo-Steady-State Diffusion In many mass-transfer operations, one of the Flow of gas B boundaries may move with time. If the length z = z of the diffusion path changes a small amount 2 over a long period of time, a pseudo-steady- state diffusion model may be used. When this N Az z + ∆ z condition exists, equation (26-7) describes themassfluxinthestagnantgasfilm.Reconsider ∆ z Figure 26.1, with a moving liquid surface as N Az z illustrated in Figure 26.4. Two surface levels z = z at t 1 0 are shown, one at time t and the other at time 0 z = z at t 1 1 t . If the difference in the level of liquid A over 1 Pure liquid A the time interval considered is only a small fraction of the total diffusion path, and t t 1 0 Figure 26.4 Arnold diffusion cell with isarelativelylongperiodoftime,atanyinstant moving liquid surface. inthatperiodthemolarfluxinthegasphasemay be evaluated by cD (y y ) AB A A 1 2 N ¼ (26-7) A,z zy B,lm where z is the length of the diffusion path at time t. The molar flux N is related to the amount of A leaving the liquid by A,Z r dz A,L N ¼ (26-16) A,z M dt A where r /M is the molar density of A in the liquid phase. Under pseudo-steady-state A A,L conditions, equations (26-7) and (26-16) may be combined to give r dz cD (y y ) A,L AB A A 1 2 ¼ (26-17) M dt zy A B,lm Equation (26-17) may be integrated from t¼0to t¼ t from z¼ z to z¼ z as follows: t t o Z Z t z t r y /M B,lm A A,L dt¼ zdz cD (y y ) AB A A t¼0 z 1 2 t 0 This yields 2 2 z z r y /M B,lm A A,L t t 0 t¼ (26-18) cD (y y ) 2 A,B A A 1 226.1 One-Dimensional Mass Transfer Independent of Chemical Reaction 459 2 2 Therefore,forthepseudo-steady-statediffusionprocess,aplotofz z vs.timetshould t t 0 be linear. Rearranging this expression, we obtain the equation commonly used to evaluate the gas-diffusion coefficient from Arnold cell experimental data. This equation is 2 2 r y /M z z A,L B,lm A t t 0 D ¼ (26-19) AB c(y y )t 2 A A 1 2 As illustrated by the Arnold diffusion cell above, pseudo-steady-state diffusion processes usually involve the slow depletion of the source or sink for the mass-transfer processwithtime. Below,weconsider another process thatis modeledbypseudo-steady- state diffusion, the thermal oxidation of a silicon wafer. EXAMPLE 2 Theformationofasiliconoxide(SiO )thinfilmonasilicon(Si)wafersurfaceisanimportantstep 2 in the fabrication of solid-state microelectronic devices. A thin film of SiO serves as a barrier to 2 dopant diffusion or as a dielectric insulator to isolate various devices being formed on the wafer. In one common process, silicon is oxidized by exposure to oxygen (O ) gas at temperatures 2 above 7008C Si(s)þO (g)SiO (s) 2 2 Molecular O dissolvesinto the SiO solid,diffusesthrough theSiO film, and thenreacts withSi 2 2 2 at the Si/SiO interface, as shown in Figure 26.5. Assuming that the diffusion of O through the 2 2 SiO filmlimitstheoxidationprocess,developamodeltopredictthethicknessoftheSiO layer(d) 2 2 3 as a function of time at 10008C. The density of solid SiO (r ) is 2.27 g/cm , and the molecular 2 B weightof SiO (M )is60g/mol. The moleculardiffusioncoefficient of O in SiO (D )is2:7 2 B 2 2 AB 9 2 8 3 10 cm /s at 10008C, and the maximum solubility of O in SiO (c )is9:610 molO /cm 2 2 As 2 1 solid at 10008C and 1 atm O partial pressure, using data provided by Norton. 2 O (g) 2 z = 0 c = c A As O 2 SiO (nonporous) 2 z = δ c = 0 Ad Si (crystalline) z = L Figure 26.5 Thermal oxidation Si(s) + O SiO (s) 2 2 of a silicon wafer. The physical system is represented in the rectilinear coordinate system. The model develop- mentfollowstheapproachoutlinedearlierinSection25.4.Theassumptionsformodeldevelopment arelistedhere.(1)TheoxidationofSitoSiO occursonlyattheSi/SiO interface.TheunreactedSiat 2 2 theinterfaceservesasthesinkformolecularmasstransferofO throughthefilm.(2)TheO inthe 2 2 gasphaseabovethewaferrepresentsaninfinitesourceforO transfer.TheO molecules‘‘dissolve’’ 2 2 intothenonporousSiO solidatthegas/solidinterface.(3)TherateofSiO formationiscontrolled 2 2 by the rate of molecular diffusion of O (species A) through the solid SiO layer (species B) 2 2 to the unreacted Si layer. The reaction is very rapid, so that the concentration of molecular O at the interface is zero, that is, c ¼ 0. Furthermore, there are no mass-transfer resistances 2 A,d 1 F. J. Norton, Nature, 191, 701 (1961).460 Chapter 26 Steady-State Molecular Diffusion inthegasfilmabovethewafersurface,asO isapurecomponentinthegasphase.(4)ThefluxofO 2 2 throughtheSiO layerisonedimensionalalongcoordinatez.(5)TherateofSiO filmformationis 2 2 slow enough so that at a given film thickness d, there is no accumulation of reactants or products withintheSiO film.However,thethicknessofthefilmwillstillincreasewithtime.Consequently, 2 thisisa‘‘pseudo-steady-state’’process.(6)Theoverallthicknessofthewaferdoesnotchangeasthe result of the formation of the SiO layer. (7) The process is isothermal. 2 Based on the previous assumptions, the general differential equation for mass transfer reduces to dN Az ¼ 0 dz andFick’sequationforone-dimensional diffusionofO (species A)throughcrystallinesolidSiO 2 2 (species B)is dc c dc c A A A A N ¼D þ (N þN )¼D þ N Az AB Az Bz AB Az dz c dz c Usually,theconcentrationofmolecularO intheSiO layerisdiluteenoughsothatc /ctermisvery 2 2 A small in magnitude relative to the other terms. Therefore, Fick’s equation reduces to dc A N ¼D Az AB dz It is interesting to note here that unimolecular diffusion (UMD) flux mathematically simplifies to the equimolar counter diffusion (EMCD) flux at dilute concentration of the diffusing species. As N isconstantalongz,thedifferentialfluxequationcanbeintegrateddirectlyfollowingseparationof A dependent variable c from independent variable z A d 0 Z Z N dz¼D dc Az AB A 0 c As or simply D c AB As N ¼ Az d whichdescribesthefluxofO throughtheSiO layerofthicknessd.Thesurfaceconcentrationc 2 2 As 3 refers to the concentration of O dissolved in solid phase SiO (mol O /cm solid). 2 2 2 Weknowthatdincreasesslowlywithtime,eventhoughthereisnoaccumulationtermforO 2 in the SiO layer. In other words, the process operates under the pseudo-steady-state assumption. 2 Inordertodiscoverhowdincreaseswithtime,consideranunsteady-statematerialbalanceforSiO 2 within the wafer (molarrateofSiO formation)¼ (molarrateofaccumulationofSiO ) 2 2 or  r Sd B d M B (molarrateofaccumulationofSiO )¼ 2 dt 3 where r is the density of solid SiO (2.27 g/cm ), M is the molecular weight of the SiO layer B 2 B 2 (60g/mol),andSisthesurfaceareaofthewafer.Giventhestoichiometryofthereaction,onemoleof SiO is formed for every mole of O consumed. Therefore, 2 2 D c AB As ðrateofSiO formationÞ¼ N S¼ S 2 Az d26.1 One-Dimensional Mass Transfer Independent of Chemical Reaction 461 or r dd D c B AB As ¼ M dt d B Separationofdependentvariabledfromtheindependentvariablet,followedbyintegrationatt¼ 0, d¼0to t¼ t, d¼ d gives d t Z Z M D c B AB As ddd¼ dt r B 0 0 or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2M D c B AB As d¼ t r B TheaboveequationpredictsthatthethicknessoftheSiO thinfilmisproportionaltothesquareroot 2 9 2 of time. Recall that the molecular diffusion coefficient of O in SiO (D )is2:710 cm /s at 2 2 AB 8 3 10008C,andthesolubilityofO inSiO (c )is9:610 molO /cm solidat10008C.Figure26.6 2 2 As 2 2 compares the predicted film thickness d vs. time to process data provided by Hess for 1 atm O 2 at10008C.Asonecansee,themodeladequatelypredictsthedatatrend.Thefilmisverythin,less than 0.5 mm, in part because the value for D c is so small. AB As 0.4 Data Model 0.3 0.2 0.1 0.0 Figure 26.6 SiO film 2 02 4 6 810 12 14 16 thickness vs. time at Time, t (h) 10008C. Thisexampleillustrateshowachemicalreactionataboundarysurfacecanserveasthedriving force for molecular diffusion. This concept is formally presented in Section 26.2. 2 D. W. Hess, Chem. Eng. Educ., 24, 34 (1990). SiO film thickness, δ (µm) 2462 Chapter 26 Steady-State Molecular Diffusion Equimolar Counterdiffusion Aphysicalsituationthatisencounteredinthedistillationoftwoconstituentswhosemolar latent heats of vaporization are essentially equal stipulates that the flux of one gaseous component is equal to butacting in the opposite direction from the othergaseous compo- nent; that is, N ¼N . Equation (25-11) A;z B;z c A : = N þ R ¼ 0 (25-11) A A t for the case of steady-state mass transfer without chemical reaction may be reduced to : = N ¼ 0 A For the transfer in the z direction, this equation reduces to d N ¼ 0 A,z dz ThisrelationstipulatesthatN isconstantalongthepathoftransfer.Themolarflux,N , A,z A,z for a binary system at constant temperature and pressure is described by dc A N ¼D þy (N þN ) (24-20) A,z AB A A,z B,z dz The substitution of the restriction, N ¼N , into the above equation gives an equa- A,z B,z tion describing the flux of Awhen equimolar-counterdiffusion conditions exist dc A N ¼D (26-20) A,z AB dz Equation (26-20) may be integrated, using the boundary conditions at z¼ z c ¼ c 1 A A 1 and at z¼ z c ¼ c 2 A A 2 giving Z Z z c 2 A 2 N dz¼D dc A,z AB A z c 1 A 1 from which we obtain D AB N ¼ (c c ) (26-21) A,z A A 1 2 (z z ) 2 1 When the ideal gas law is obeyed, the molar concentration of A is related to the partial pressure of A by n p A A c ¼ ¼ A V RT Substituting this expression for c into equation (26-21), we obtain A D AB N ¼ (p  p ) (26-22) A,z A A 1 2 RT(z z ) 2 1 Equations(26-21) and (26-22) are commonly referred to as the equations for steady-state equimolar counterdiffusion.26.2 One-Dimensional Systems Associated with Chemical Reaction 463 The concentration profile for equimolar-counterdiffusion processes may be obtained bysubstitutingequation(26-20)intothedifferentialequationwhichdescribestransferinthe z direction d N ¼ 0 A,z dz or 2 d c A ¼ 0 2 dz This second-order equation may be integrated twice with respect to z to yield c ¼ C zþC A 1 2 The two constants of integration are evaluated, using the boundary conditions at z¼ z c ¼ c 1 A A 1 at z¼ z c ¼ c 2 A A 2 to obtain the linear concentration profile c c zz A A 1 1 ¼ (26-23) c c z z A A 1 2 1 2 Equations (26-21) and (26-23) may be used to describe any process where the bulk- contribution term is zero. Besides the equimolar-counterdiffusion phenomenon, a neglig- iblebulk-contributiontermisalsoencounteredwhenasolutediffusesintoorthroughasolid as both the mole fraction, x , and the flux of the diffusing species, N , are very small. A Az Accordingly, their resulting product is therefore negligible. Itisinterestingtonotethatwhenweconsiderthe‘‘filmconcept’’formasstransferwith equimolarcounterdiffusion,thedefinitionoftheconvectivemass-transfercoefficientisdiffer- ent from that for diffusion in a stagnant gas film. In the case of equimolar counterdiffusion D AB 0 k ¼ (26-24) d The superscript on the mass-transfer coefficient is used to designate that there is no net molar transfer into the film due to the equimolar counterdiffusion. Comparing equation (26-24) with equation (26–9), we realize that these two defining equations yield the same results only when the concentration of A is very small and p is essentially equal to P. B;lm 26.2 ONE-DIMENSIONAL SYSTEMS ASSOCIATED WITH CHEMICAL REACTION Manydiffusionaloperationsinvolvethesimultaneousdiffusionofamolecularspeciesand thedisappearanceorappearanceofthespeciesthroughachemicalreactioneitherwithinor attheboundaryofthephaseofinterest.Wedistinguishbetweenthetwotypesofchemical reactions, defining the reaction that occurs uniformly throughout a given phase as a homogeneous reaction and the reaction that takes place in a restricted region within or at a boundary of the phase as a heterogeneous reaction. TherateofappearanceofspeciesAbyahomogeneousreactionappearsinthegeneral differential equation of mass transfer as the source term, R A c A : = N þ R ¼ 0 (25-11) A A t464 Chapter 26 Steady-State Molecular Diffusion Examplesofthesourceterm,R ,includethefirst-orderconversionofreactantAtoproduct A P, thus R ¼k c , where k is the first-order rate constant in 1/s, and the second-order A 1 A 1 reaction of reactants A and B to form the product P with R ¼ k c c , where k is the A 2 A B 2 2 second-order rate constant in units cm /mols. The rate of disappearance of A by a heterogeneous reaction on a surface or at an interfacedoesnotappearinthegeneraldifferentialequationasR involvesonlyreactions A within the control volume. A heterogeneous reaction enters the analysis as a boundary conditionandprovidesinformationonthefluxesofthespeciesinvolvedinthereaction;for example, if the surface reaction is O (g)þC(s)CO (g), the flux of CO (g) will be the 2 2 2 same as the flux of O (g), leaving in the opposite direction. 2 In this section, we shall consider two simple cases involving both types of chemical reactions.Foratreatmentofmorecomplicatedproblems,thestudentisreferredtothetwo 3 4 excellent treatises by Crank and Jost. Simultaneous Diffusion and Heterogeneous, First-Order Chemical Reaction: Diffusion with Varying Area Manyindustrialprocessesinvolvethediffusionofareactanttoasurfacewhereachemical reactionoccurs.Asbothdiffusionandreactionstepsareinvolvedintheoverallprocess,the relativeratesofeachstepareimportant.Whenthereactionrateisinstantaneousrelativeto therateofdiffusion,thentheprocessisdiffusioncontrolled.Incontrast,whenthereaction rateofthetransferringspeciesatthesurfacelimitsthemass-transferrate,thentheprocessis reaction controlled. In many power plants, pulverized coal particles are fluidized within a hot combustion chamber,whereoxygenintheairreactswithcoaltoproducecarbonmonoxideand/orcarbon dioxidegas.Thisprocess,whichproducesenergybytheheatofcombustion,isanexampleofa simultaneous diffusion and a heterogeneous reaction process that is diffusion controlled. Letusconsiderthesteady-state,one-dimensionaldiffusionofoxygentothesurfaceofa sphericalparticleofcoalalongthercoordinate.Atthesurfaceoftheparticle,oxygengas (O )reactswithsolidcarbon(C)inthe 2 coal to form carbon monoxide gas Surrounding still air (CO) and carbon dioxide (CO ) gas 2 according to the heterogeneous reac- N CO ,r 2 tion equation 3C(s)þ2:5O (g)2CO (g)þCO(g) 2 2 R asillustratedinFigure26.7.Nohomo- geneous chemical reaction occurs r N ∆ r O ,r 2 along the diffusion path so that R ¼ o 2 0. As the coal particle is oxidized, the particleshrinkswithtimeasthecarbon Carbon sphere is converted to carbon monoxide and carbon dioxide. It is desired to predict N CO, r the size of the particle with time. Based on the above physical Figure 26.7 Diffusion through a spherical film. situation, the general differential 3 J. Crank, The Mathematics of Diffusion, Oxford University Press, London, 1957. 4 W. Jost, Diffusion in Solids, Liquids and Gases, Academic Press, New York, 1952.26.2 One-Dimensional Systems Associated with Chemical Reaction 465 equation for mass transfer (25-29) reduces from 2 c 1 (r N ) 1 (N sinu) 1 N A Ar Au Af þ þ þ R ¼ 0 (25-29) A 2 t r r rsinu u rsinu f to 2 1 d(r N ) Ar ¼ 0 2 r dr or 2 d(r N ) Ar ¼ 0 (26-25) dr where A represents O , the transferring species. 2 As this is the first time we have encountered diffusion through a varying area, let us also derive equation (26-25) by making a mass balance in terms of moles of oxygen per time over the control volume bounded by the spherical surfaces at r and rþDr 2 2 N 4prr j N 4pr j ¼ 0 (26-26) O r O r 2 r 2 rþDr Dividing equation (26-26) by 4pDr and evaluating the limit as Dr approaches zero, we obtain 2 d(r N ) O r 2 ¼ 0 (26-25) dr 2 This equation specifies that r N is constant over the diffusion path in the r direction, O r 2 so that 2 2 r N j ¼ R N j (26-27) r O r O 2 r 2 R When we compare equation (26-25) with equation (26-1), we observe that for spherical 2 coordinates, r N is constant along the r direction, whereas for rectilinear coordinates, Ar N is constant along the z direction. Az Fick’s equation (24-21) can be simplified once we recognize the relationship among the fluxes of the involved species. From the stoichiometry of the surface reaction, we recognize that 2.5 moles of oxygen are transferred to the surface, whereas 2 moles of carbon dioxide per 1 mole of carbon monoxide are transferred away from the surface. Therefore, N ¼2:5N and N ¼1:25N O r COr O r CO r 2 2 2 There is no net transfer of the nitrogen in the air, as it is an inert. Therefore, N ¼ 0. N 2 Fick’s equation can now be written in terms of only oxygen dy O 2 N ¼cD þy (N þN þN þN ) N O r O -mix O O r COr CO r 2 2 2 2 2 2 dr or  dy 1 1 O 2 N ¼cD þy N  N  N þ0 O r O -mix O O r O r O r 2 2 2 2 2 2 dr 2:5 1:25 Consequently, Fick’s equation reduces to dy O 2 N ¼cD 0:2y N O r O -mix O O r 2 2 2 2 dr466 Chapter 26 Steady-State Molecular Diffusion or cD dy O -mix O 2 2 N ¼ (26-28) O r 2 1þ0:2y dr O 2 2 2 Both sides of equation (26-28) must be multiplied by r to obtain the constant r N O r 2 term. The diffusivity, D , and the total molar concentration, c, can be treated as con- O -mix 2 stants if evaluated at an average temperature and an average composition. Two boundary conditions are also needed. At the surface of the sphere, the reaction is instantaneous so that the oxygen concentration is zero r¼ R, y ¼ 0 O 2 However, a long r-distance away from the sphere r¼1, y ¼ 0:21 O 2 Finally, equation (26-28) becomes Z Z 1 0:21 dr cD 0:2dy O -mix O 2 2 2 (r N ) ¼ O r 2 2 r 0:2 1þ0:2y O R 0 2 which upon integration yields   1 cD 1 O -mix 2 2 (r N ) ¼ ln O r 2 R 0:2 1:042 The moles of oxygen transferred per time is the product of the oxygen flux and the cross- 2 sectional area 4pr cD O -mix 2 2 W ¼ 4pr N ¼4pR ln(1:042) (26-29) O O r 2 2 0:2 It is important to recognize that this equation predicts a negative value for the rate of oxygen being transferred. The reason is that the direction of oxygen flux from the bulk gas to the particle surface is opposite to the increasing r direction from r¼ R to r¼1. ThepurecarbonparticleisthesourcefortheCO fluxandthesinkforO flux.Asthe 2 2 coalparticleisoxidized,therewillbeanoutputofcarbonasstipulatedbythestoichiometry ofthereaction.Althoughthesphericalparticlewilldecreaseinsizewithtime,thediffusion path from r¼ R to r¼1 will be essentially constant with time. This allows us to use a pseudo-steady-state approach for describing the material balance on the carbon particle. The material balance for carbon stipulates that (inputcarbonrate)(outputcarbonrate)¼ (rateofcarbonaccumulation) TheoutputrateofcarbonfromthesphericalparticleinmolC/timeisrelatedtooutputrateof CO that is in turn related to the input mass transfer rate of O 2 2 3 3 3 cD O -mix 2 W ¼ W ¼ W ¼þ 4pR ln(1:042) (26-30) C CO O 2 2 2 2:5 2:5 0:2 The carbon accumulation rate in the spherical particle in mol C/time is r dV r dR C C 2 ¼ 4pR M dt M dt C C26.2 One-Dimensional Systems Associated with Chemical Reaction 467 wherer isthedensityofsolidcarbon,M isthemolecularweightofcarbon,andVisthe C C totalvolumeofthesolidcarbonsphere.Bysubstitutingthesetermsintothecarbonbalance, we obtain 3 cD r dR O -mix C 2 2 0 4pR ln(1:042)¼ 4pR 2:5 0:2 M dt C Rearrangement of this equation followed by separation of variables R and t gives R u f Z Z r C 6cD ln(1:042) dt¼ RdR O -mix 2 M C 0 R i whereR isthefinalradiusofthesphericalparticle,R istheinitialradiusoftheparticle,and f i uisthetimerequiredfortheparticletoshrinkfromtheinitialtothefinalradius.Following integration, we obtain the final expression for estimating u r C 2 2 (R R ) i f M C u¼ (26-31) 12cD ln(1:042) O -mix 2 Alternative reaction equations may be proposed for the combustion process. For example, if only carbon dioxide were produced by the reaction C(s)þO (g)CO (g) (26-32) 2 2 then bulk contribution term to Fick’s equation is zero as N ¼N . Therefore, O r CO r 2 2 Fick’s equation reduces to dy O 2 N ¼cD O r O -mix 2 2 dr wherethegasmixtureconsistsofCO ,O ,andN .Itisnotdifficulttoshowthatthemoles 2 2 2 of oxygen transferred per time is W ¼4pRcD y (26-33) O O -mix O 1 2 2 2 Forheterogeneousreactions,informationontherateofthechemicalreactioncanalso provide an important boundary condition N j ¼k c A s As r¼R wherek isthefirst-orderreaction-rateconstantforasurfacereaction,inunitsofm/s.The s negativesignindicatesthatspeciesAisdisappearingatthesurface.Ifthechemicalreaction isinstantaneousrelativetothediffusionstep,thentheconcentrationofthereactingspecies Aatthesurfaceisessentiallyzero;thatis,c ¼ 0,asweassumedabove.However,ifthe As reaction is not instantaneous at the surface, then c will be finite. As Considernowthatthereactiondescribedbyequation(26-32)isnotinstantaneous,and its heterogeneous rate equation can be expressed as a first-order surface reaction with respecttoO .Consequently,theO concentrationatthesurfaceisnotequaltozero.Inthis 2 2 context, it is easy to show that equation (26-33) becomes W ¼4pRcD (y y ) (26-34) O O -mix O 1 O s 2 2 2 2 where y is the mole fraction of O in the bulk gas and y is the mole fraction of O O 1 2 O s 2 2 2 at the surface (r¼ R). For a first-order surface reaction N j ¼k c , the mole As s As R468 Chapter 26 Steady-State Molecular Diffusion fraction of O at the surface can be expressed as 2 c N O s O R 2 2 y ¼ ¼ (26-35) O s 2 c k c s The minus sign indicates that the direction of O flux is opposite to increasing r. Substi- 2 tution of equation (26-35) into equation (26-34) yields  N O R 2 W ¼4pRcD y þ (26-36) O O -mix O 1 2 2 2 k c s Now recall 2 2 W ¼ 4pR N ¼ 4pR N (26-27) O O R O r 2 2 2 Combination of equations (26-36) and (26-27) to eliminate N results in O R 2  D O -mix 2 2 r N 1þ ¼RcD y O r O -mix O 1 2 2 2 k R s Finally, the oxygen-transfer rate for the combined diffusion and reaction process is 4pRcD y O -mix O 1 2 2 W ¼ (26-37) O 2 D O -mix 2 1þ k R s Note that as k gets very large, equation (26-37) is approximated by equation (26-33). s EXAMPLE 3 Afluidizedcoalreactorhasbeenproposedforanewpowerplant.Ifoperatedat1145K,theprocess will be limited by the diffusion of oxygen countercurrent to the carbon dioxide, CO , formed at 2 3 3 theparticlesurface.Assumethatthecoalispuresolidcarbonwithadensityof1:2810 kg/m and 4 that the particle is spherical with an initial diameter of 1:510 m(150mm). Air (21% O and 2 2 79% N ) exists several diameters away from the sphere. Under the conditions of the combustion 4 2 process,thediffusivityofoxygeninthegasmixtureat1145Kis1:310 m /s.Ifasteady-state process is assumed, calculate the time necessary to reduce the diameter of the carbon particle to 5 510 m(50mm). ThesurroundingairservesasaninfinitesourceforO transfer,whereastheoxidationofthecarbonat 2 the surface of the particle is the sink for O mass transfer. The reaction at the surface is 2 C(s)þO (g)CO (g) 2 2 Note that the reaction establishes a EMCD process where the flux of O to the particle is equal to 2 but opposite in the direction of the CO flux away from the particle, that is, 2 N ¼N O r CO r 2 2 At the surface of the coal particle, the reaction is so rapid that the concentration of oxygen is zero.Underthisassumption,theinstantaneousmasstransferofoxygentothesurfaceofthecoal particle is W ¼4pRcD y (26-33) O O -mix O 1 2 2 2 The stoichiometry of the surface reaction stipulates that 1 atom of carbon will disappear per each mole of oxygen reacting at the surface. Therefore, W ¼ W ¼W ¼þ4pRcD y C CO O O -mix O 1 2 2 2 226.2 One-Dimensional Systems Associated with Chemical Reaction 469 The total carbon balance can be written as r dR C 2 04pRcD y ¼ 4pR O -mix O 1 2 2 M dt C which simplifies to r RdR C dt¼ M cD y C O -mix O 1 2 2 This equation can be integrated between the following limits 5 t¼ 0, R¼ R ¼ 7:510 m(75mm) i 5 t¼ u, R¼ R ¼ 2:510 m(25mm) f to give 2 2 r (R R ) C i f u¼ 2M cD y C O -mix O 1 2 2 The total gas molar concentration, c, is obtained by the ideal gas law P 1:0atm kgmol c¼ ¼ ¼ 0:0106 3 3 RT m 0:08206m atm 1145K kgmolK Finally,  kg 3 5 2 5 2 1:2810 ((7:510 m) (2:510 m) ) 3 m u¼ ¼ 0:92 s  2 12kg kgmol m 4 2 0:0106 1:310 (0:21) 3 kgmol m s Diffusion with a Homogeneous, First-Order Chemical Reaction Intheunitoperationofabsorption,oneoftheconstituentsofagasmixtureispreferentially dissolved in a contacting liquid. Depending upon the chemical nature of the involved molecules, the absorption may or may not involve chemical reactions. When there is a production or disappearance of the diffusingcomponent,equation(25-11)maybeusedtoanalyzethe z Gas mixture Liquid mass transfer within the liquid phase. The following analysis ( A and inert gas) surface illustrates mass transfer that is accompanied by a homogeneous z = 0 chemical reaction. N Consider a layer of the absorbing medium as illustrated in Az z ∆ z Figure26.8.Attheliquidsurface,thecompositionofAisc .The A 0 N thickness of the film, d, is defined so that beyond this film the Az z +∆ z concentrationofAisalwayszero;thatis,c ¼ 0.Ifthereisvery A d Liquid B littlefluidmotionwithinthefilm,andiftheconcentrationofAin z = δ the film is assumed small, the molar flux within the film is des- cribed by dc A Figure 26.8 Absorption with homogeneous N ¼D (26-38) A;z AB dz chemical reaction.470 Chapter 26 Steady-State Molecular Diffusion For one-directional steady-state mass transfer, the general differential equation of mass transfer reduces to d N R ¼ 0 (26-39) A;z A dz The disappearance of component A by a first-order reaction is defined by R ¼ k c (26-40) A 1 A wherek isthechemicalreactionrateconstant.Substitutionofequations(26-38)and(26- 1 40) into equation (26-39) gives a second-order differential equation that describes simul- taneous mass transfer accompanied by a first-order chemical reaction  d dc A  D þk c ¼ 0 (26-41) AB 1 A dz dz or with a constant diffusion coefficient, this reduces to 2 d c A D þk c ¼ 0 (26-42) AB 1 A 2 dz The general solution to equation (26-42) is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ c cosh k /D zþc sinh k /D z (26-43) A 1 1 AB 2 1 AB The boundary conditions at z¼ 0 c ¼ c A A 0 and at z¼ d c ¼ 0 A permittheevaluationofthetwoconstantsofintegration.Theconstantc isequaltoc ,and 1 A 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c is equal to (c )/ðtanh k /D d), where d is the thickness of the liquid film. 2 A 1 AB 0 Substituting these constants into equation (26-43), we obtain an equation for the concen- tration profile pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c sinh k /D z A 1 AB 0 c ¼ c cosh k /D z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (26-44) A A 1 AB 0 tanh k /D d 1 AB The molar mass flux at the liquid surface can be determined by differentiating equa- tion (26-44) and evaluating the derivative, (dc /dz)j . The derivative of c with res- A A z¼0 pect to z is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dc c k /D cosh k /D z A A 1 AB 1 AB 0 ¼þc k /D sinh k /D z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 AB 1 AB 0 dz tanh k /D d 1 AB which, when z equals zero, becomes  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  dc c k /D c k /D A A 1 AB A 1 AB 0 0  ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (26-45)  dz tanh k /D d tanh k /D d 1 AB 1 AB z¼0 Substituting equation (26-45) into equation (26-38) and multiplying by d/d, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  D c k /D d AB A 1 AB 0 N j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (26-46) A,z z¼0 d tanh k /D d 1 AB26.2 One-Dimensional Systems Associated with Chemical Reaction 471 It is interesting to consider the simpler mass-transfer operation involving the absorp- tionofAintoliquidBwithoutanaccompanyingchemicalreaction.ThemolarfluxofAis easily determined by integrating equation (26-38) between the two boundary conditions, giving D c AB A 0 N ¼ (26-47) A,z d pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi It is apparent by comparing the two equations that the term½( k /D d)/(tanh k /D d) 1 AB 1 AB showsthe influence of the chemical reaction.This term is a dimensionless quantity, often 5 called the Hatta number. Astherateofthechemicalreactionincreases,thereactionrateconstant,k ,increases 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi andthehyperbolictangentterm,tanh k /D d,approachesthevalueof1.0.Accordingly, 1 AB equation (26-46) reduces to pffiffiffiffiffiffiffiffiffiffiffiffiffi N j ¼ D k (c 0) A,z AB 1 A z¼0 0 A comparison of this equation with equation (25-30) N ¼ k (c c ) (25-30) A;z c A A 1 2 reveals that the film coefficient, k , is proportional to the diffusion coefficient raised to c 1 the power. With a relatively rapid chemical reaction, component Awill disappear after 2 penetrating only a short distance into the absorbing medium; thus, a second model for convective mass transfer has been proposed, the penetration theory model, in which k is c 1 considered a function of D raised to the power. In our earlier discussion of another AB 2 model for convective mass transfer, the film theory model, the mass-transfer coefficient was a function of the diffusion coefficient raised to the first power. We shall reconsider thepenetrationmodelinSection26.4andalsoinChapter28,whenwediscussconvective mass-transfer coefficients. The following example considers diffusion with a homogeneous first-order chemical reaction under a different set of boundary conditions. EXAMPLE 4 Diluteconcentrationsoftoxicorganicsolutescanoftenbedegradedbya‘‘biofilm’’attachedtoan inert,nonporoussolidsurface.Abiofilmconsistsoflivingcellsimmobilizedinagelatinousmatrix. Biofilms are not very thick, usually less thana few millimeters. A toxic organic solute (species A) diffusesintothebiofilmandisdegradedtoharmlessproducts,hopefullyCO andwater,bythecells 2 withinthebiofilm.Forengineeringapplications,thebiofilmcanbeapproximatedasahomogeneous substance(i.e.,speciesB).Therateofdegradationofthetoxicsoluteperunitvolumeofthebiofilmis described by a kinetic rate equation of the form R c A,max A R ¼ A K þc A A 3 whereR isthemaximumpossibledegradationrateofspeciesAinthebiofilmandK (mol/cm )is A,max A the half-saturation constant for the degradation of species Awithin the biofilm at hand. Consider the simple ‘‘rotating disk’’ process unit shown in Figure 26.9 for the treatment of phenol (species A) in wastewater. The biofilm contains a microorganism rich in the enzyme peroxidasethatoxidativelydegradesphenol.TheconcentrationofspeciesAinthebulk-fluidphase overthebiofilmisconstantifthefluidphaseiswellmixed.However,theconcentrationofAwithin the biofilm will decrease along the depth of the biofilm z as species A is degraded. There are no 5 S. Hatta, Technol. Rep. Tohoku Imp. Univ., 10, 199 (1932).