Revista Mexicana de Ingeniería Química

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1 Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 ERIVTIO PPLITIO OF THE STEF-XWELL EQUTIOS ESRROLLO Y PLIIÓ E LS EUIOES E STEF-XWELL bstrat Stehen Whitaker * eartment of hemial Engineering & aterials Siene University of alifornia at avis Reeived 5 of June 009; eted 9 of ovember 009 The Stefan-axwell equations reresent a seial form of the seies momentum equations that are used to determine seies veloities. These seies veloities aear in the seies ontinuity equations that are used to redit seies onentrations. These onentrations are required in onjuntion with onets from thermodynamis and hemial kinetis to alulate rates of adsortion/desortion rates of interfaial mass transfer and rates of hemial reation. These roesses are entral issues in the disiline of hemial engineering. In this aer we first outline a derivation of the seies momentum equations and indiate how they simlify to the Stefan-axwell equations. We then examine three imortant forms of the seies ontinuity equation in terms of three different diffusive fluxes that are obtained from the Stefan-axwell equations. ext we examine the struture of the seies ontinuity equations for binary systems and then we examine some seial forms assoiated with -omonent systems. Finally the general -omonent system is analyzed using the mixed-mode diffusive flux and matrix methods. Keywords: ontinuum mehanis kineti theory multiomonent diffusion. Resumen Las euaiones de Stefan-axwell reresentan una forma eseial de las euaiones de antidad de movimiento de eseies que son usadas ara determinar las veloidades de eseies. Estas veloidades de eseies aareen en las euaiones de ontinuidad de eseies que son usadas ara redeir las onentraiones de eseies. Estas onentraiones son requeridas en onjunión on los onetos de termodinámia y inétia químia ara alular las veloidades de adsorión/desorión las veloidades de transferenia de masa interfaial y las veloidades de reaión químia. Estos roesos son elementos entrales en la disilina de la ingeniería químia. En este artíulo resentamos rimeramente un desarrollo de las euaiones de antidad de movimiento de eseies e indiamos omo se simlifian a las euaiones de Stefan-axwell. Posteriormente examinamos tres formas imortantes de la euaión de ontinuidad de eseies en términos de tres diferentes fluxes difusivos que se obtienen de las euaiones de Stefan-axwell. ás adelante examinamos la estrutura de las euaiones de ontinuidad de eseies ara sistema binarios y examinamos algunas formas eseiales asoiados on sistemas de -omonentes. Finalmente se analiza el sistema general de -omonentes usando métodos matriiales y de flux difusivo de modo mixto. Palabras lave: meánia del ontinuo teoría inétia difusión multiomonente. * orresonding author. whitaker@mn.org 3 Publiado or la ademia exiana de Investigaión y oenia en Ingeniería Químia..

2 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 ontents. Introdution. onservation of mass 3. Laws of mehanis 5. ass ontinuity equation 5 3. olar ontinuity equation 6 4. ixed-mode ontinuity equation 9 5. inary systems 0 5. ass diffusive flux 0 5. olar diffusive flux ixed-mode diffusive flux 5 6. Seial forms for -omonent systems 6 6. ilute solution diffusion equation 6 6. ilute solution onvetive-diffusion equation using * J ilute solution onvetive-diffusion equation using J iffusion through stagnant seies 3 7. General form for -omonent systems: onstant total molar onentration General form for -omonent systems: onstant total mass density onlusions 4 omenlature 4 knowledgment 4 Referenes 44 endix : hemial reation and linear momentum 46 endix : Thermodynami ressure 50 endix : lgebrai relations 55 endix : ssumtions Restritions and onstraints 58 endix E: Restritions for onstant total onentration and density 6 4

3 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) Introdution Our derivation of multi-omonent transort equations is based on the onet of a seies body. In Part I of Fig. we have illustrated a system ontaining both seies and seies and these are illustrated as disrete artiles. We have also illustrated a region from whih we have ut out both a seies body and a seies body. In Part II of Fig. we have indiated that the seies body will be treated as a ontinuum while the disrete harater of seies is retained for ontrast. s time evolves the two seies bodies searate beause their veloities are different. This searation is illustrated in Part III of Fig. where we have also indiated that the seies body will be treated as a ontinuum. The ontinuum veloities of seies and seies are designated as v and v. In general the ontinuum hyothesis should be satisfatory when the distane between moleules is very small omared to a harateristi length for the system.. onservation of mass In terms of the seies body illustrated in Fig. we state the two axioms for the mass of multiomonent systems as xiom I: d dt dv = r dv =... V() t V() t xiom II: = = () r = 0 () Here reresents the mass density of seies and r reresents the net mass rate of rodution er unit volume of seies owing to hemial reation. In Eqs. () and () we have used a mixed-mode nomenlature making use of both letters and numbers to identify individual seies. For examle xiom II ould be exressed in terms of alhabeti subsrits as xiom II: r + r + r + r r = 0 (3) or we ould use numerial subsrits to reresent this axiom as xiom II: r + r + r 3 + r r = 0 (4) This latter result an obviously be omated to rodue Eq. ; however the use of alhabeti subsrits to reresent moleular seies is revalent in the hemial engineering literature. eause of this we will use alhabeti subsrits to identify distint moleular seies and we will use the nomenlature ontained in Eq. to reresent the various sums that aear in this aer. Fig.. otion of seies and seies bodies In order to extrat a governing differential equation from Eq. we make use of the general transort equation (Whitaker 98 Se. 3.4 with w = v ) d dt dv = V() t V() t + v nd =... () t dv and the divergene theorem (Stein and arellos 99 Se. 7.) ( ) v nd = v dv () t V () t (6) =... in order to exress Eq. in the form + ( v ) r dv = 0 V () t (7) =... Sine V () t illustrated in Fig. is arbitrary and sine it is lausible to assume that the integrand in Eq. (7) is ontinuous the integrand in Eq. (7) must be zero and the governing differential equation assoiated with Eq. is given by (5) + ( v ) = r =... (8) 5

4 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 If we sum Eq. (8) over all seies and imose the xiom II we obtain + ( v )=0 (9) in whih the total density and the total mass flux are determined by = = = = = v = v (0) The mass average veloity v an be exressed in terms of the mass fration ω and the seies veloity v aording to = v = ω v ω = =... () = The tyial treatment of Eq. (8) involves the solution of seies ontinuity equations along with a solution of Eq. (9). This suggests a deomosition of the seies veloity into the mass average veloity v and the mass diffusion veloity u v = v + u =... () so that the seies ontinuity equations take the form aumulation + ( v) = ( u) + r onvetive transort diffusive transort hemial reation =... (3) Here we note that only of the diffusive transort terms are indeendent sine Eqs. (0) and () require the onstraint = u = 0 (4) = In order to solve Eqs. (9) and (3) we need governing differential equations for the mass diffusion veloity u and the mass average veloity v. These are determined by the axioms for the mehanis of multi-omonent systems.. Laws of mehanis Our aroah to the laws of mehanis for multiomonent systems follows the work of Euler and auhy (Truesdell 968) the seminal works of haman & owling (939) and Hirshfelder urtiss & ird (954) along with the reent work of urtiss & ird ( ). In terms of the seies body illustrated in Fig. the linear momentum rinile for seies is given by xiom I: d dt v dv = b dv V() t V() t + t( n) d + () () t V t = v V () t dv + r dv =... P (5) With an aroriate interretation of the nomenlature one finds that this result is idential to the seond of Eqs. 5.0 of Truesdell (969 age 85) rovided that one interrets Truesdell s growth of linear momentum as the last two terms in Eq. (5). In terms of the fores ating on seies we note that b reresents the body fore t ( n) reresents the surfae fore and P reresents the diffusive fore exerted by seies on seies. This diffusive fore is onstrained by P = 0 = 3... (6) The last term in Eq. (5) reresents the inrease or derease of seies momentum resulting from the inrease or derease of seies aused by hemial reation and this term is disussed in endix. The angular momentum rinile for the seies body is given by d dv = dv dt r v r b V() t V() t ( n) () () t V t = r v V () t xiom II: + r t d + r P + r dv =... dv (7) in whih r reresents the osition vetor relative to a fixed oint in an inertial frame. Truesdell (969 age 84) resents a more general version of xiom II in whih a growth of rotational momentum is inluded and ris (96 Se. 5.3) onsiders an analogous effet for olar fluids. The analysis of Eq. (7) is rather long; however the final result is simly the symmetry of the seies stress tensor as indiated by T T (8) T = =... The onstraint on P is given by Truesdell (96 Eq. ) as xiom III: = P = 0 (9) = and a little thought will indiate that this is satisfied by P = P (0) 6

5 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 One an think of this as the ontinuum version of ewton s third law of ation and reation (Whitaker 009a). Hirshfelder et al. (954 age 497) oint out that even in a ollision whih rodues a hemial reation mass momentum and energy are onserved and the ontinuum version of this idea for linear momentum gives rise to the onstraint: xiom IV: = r v = 0 () = This result along with Eq. (9) is ontained in the seond of Eqs. 5. of Truesdell (969). Returning to the linear momentum rinile we note that the analysis assoiated with auhy s fundamental theorem (Truesdell 968) an be alied to Eq. (5) in order to exress the seies stress vetor in terms of the seies stress tensor aording to t = n T () ( n) This reresentation an be used in Eq. (5) along with the divergene theorem and the general transort theorem to extrat the governing differential equation for the linear momentum of seies given by ( v) + ( vv) = b + T loal aeleration onvetive aeleration + P + r v =... diffusive fore soure of momentum owing to reation body surfae fore fore (3) Equation (3) is idential to Eq. of urtiss and ird (996) for the ase in whih r = 0 rovided that one takes into aount the different nomenlature indiated by b = G T = σ urtiss & ird: P = F One an make use of the identity (4) v v = v v + vv vv + u u (5) in order to exress Eq. (3) in the from ( v) + ( vv + vv vv) = b + ( uu) + P + v = T (6) r... This result is idential to Eq. 4.0 of earman and Kirkwood (958) for the ase in whih r = 0 rovided that one takes into aount the different nomenlature indiated by (with the subsrit α = ) earman and Kirkwood: ( T ) P = b = X u u = σ F () (7) earman and Kirkwood refer to σ as the artial stress tensor and note that it onsists of a moleular fore ontribution reresented by T and a kineti ontribution reresented by uu. Equation (3) an be reresented in more omat form using the seies ontinuity equation. We begin by multilying Eq. (8) by the seies veloity to obtain v + ( v) = r v =... (8) Subtration of this equation from Eq. (3) leads to v + v v = b + T + P + r v v =... (9) ird (995) has ointed out that haman and owling (939) first obtained this result for dilute gases by means of kineti theory rovided that r = 0. From the ontinuum oint of view Eq. (9) is given by Truesdell and Touin (960 Eq. 5.) Truesdell (96 Eq. ) and urtiss and ird (996 Eqs. 7b and 7) all with r = 0. The orresondene with Truesdell (96) is based on the nomenlature Truesdell: b = f T = div t P ˆ = (30) In its resent form Eq. (9) reresents a governing equation for the seies veloity v and we want to use this result to derive a governing equation for the mass diffusion veloity u. To arry out this derivation we need the total momentum equation that is develoed in the following aragrahs. See seies momentum equation following Eq. 6 on age

6 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) Total momentum equation The traditional analysis of momentum transort in multi-omonent systems makes use of the sum of Eqs. (3) over all seies to obtain the total momentum equation that is used to determine the mass average veloity v. The remaining indeendent seies momentum equations an then used to determine the individual seies veloities v v v. We begin by taking into aount xioms III and IV so that the sum of Eq. (3) leads to = = v + v v = = = = = b + T = = (3) The first and third terms in this result an be simlified by the definitions = = = = v = v b = b (3) and Eqs. (0) and () an be used to obtain = = = + = = v v vv v u (33) liation of Eq. (4) allows us to simlify the onvetive momentum transort to the form = = = + = = v v vv u u (34) and substitution of Eqs. (3) and (34) in Eq. (3) rovides = ( v) + ( vv) = b + ( ) t u u = T (35) onerning the last term in this result we note that Truesdell and Touin (960 Se. 5) refer to uu as the aarent stresses arising from diffusion and we note that this term also aears in the analysis of urtiss and ird (996 Eq. 7). In that ase one needs to make use of the seond of Eqs. (4) along with = = = = ( ) π = π = T uu (36) to omlete the orresondene. t this oint we an use Eq. (9) to obtain v + ( v ) =0 (37) and this allows us to exress Eq. (35) in the form = v + v v = b + ( ) T u u (38) = In order to use this result to redit the mass average veloity we need a onstitutive equation for the sum of the seies stress tensors. This roblem is onsidered in the following aragrahs... Governing equation for the mass diffusion veloity Our objetive here is to develo the governing differential equation for the mass diffusion veloity u. We begin by multilying Eq. (38) by the mass fration ω v + v v = b = + ω ( T uu ) = (39) and subtrating this result from Eq. (9) to obtain the desired governing differential equation given by u + v u + u v = ( b b) = + T ω ( T uu ) (40) = P ( v v ) + + r =... Here it is imortant to note that this result is based only on the two axioms for mass given by Eqs. and and the four axioms for the mehanis of multiomonent systems given by Eqs. (5) (7) (9) and (). In addition we have made use of lassial ontinuum mehanis to obtain the result given by Eq. (). t this oint we need to be seifi about the seies stress tensor T and to guide our thinking and onstrain the subsequent develoment we roose that: The analysis is restrited to mixtures that behave as ewtonian fluids (Serrin 959 Se. 59; ris 96 Se. 5.). Given this restrition for the mixture we follow Slattery (999 Se. 5.3) and write = T = T = I+ τ (4a) = in whih is the thermodynami ressure and τ is the extra stress tensor given by (Serrin 959 Eq. 6.; Slattery 999 Eq ; ird et al. 00 age 843) 8

7 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 T λ τ = μ v+ v + v I (4b) Given these results Eq. (38) rovides the avier- Stokes equations ontaining an additional term assoiated with the sum of the diffusive stresses. Here we need to oint out that Eqs. (4) an be obtained by following a lassi ontinuum mehanis analysis; or this result an be obtained from kineti theory (Hirshfelder urtiss & ird 954 Eqs and 7.6-9). The advantage of this latter aroah is that a method of alulating the oeffiients μ and λ is reated within the framework of the theory. The disadvantage is that the alulations assoiated with the determination of μ for a dense gas or a liquid may be muh more diffiult than the assoiated exeriment. Given that the behavior of the mixtures under onsideration is desribed by Eqs. (4) we roose that the seies stress tensor an be reresented by Proosal: T = I+ τ =... (4) in whih is the artial ressure defined by (Truesdell 969 age 97) = ψ (43) T... Here ψ is the Helmholtz free energy of seies er unit mass of seies. In general it is more onvenient to work with the internal energy and define the artial ressure by (Whitaker 989 hater 0) ( ) = e (44) s... in whih e is the internal energy of seies er unit mass of seies. detailed disussion of the artial ressure and the total ressure is given in endix. t this oint we define the total ressure and the total visous stress tensor by = = = = = τ = τ (45) and we use these definitions along with Eq. (4) in order to exress Eq. (40) as u + v u + u v = ω uu = ω + ω + ( b b) ( τ τ ) = P r v v = + + (46) v In endix we show that differene between and v should be on the order of the diffusion veloity ( ) = ( ) v v O u (47) rguments are given elsewhere (Whitaker b) indiating that several of the terms in Eq. (46) are generally negligible. This leads to the simlifiations given by u << (48a) v u + u v << (48b) = ω uu << (48) = ω τ τ << (48d) r ( ) v v << (48e) The first of these indiates that the governing equation for u is quasi-steady; the seond indiates that diffusive inertial effets are negligible the third indiates that the diffusive stresses are negligible the fourth indiates that visous effets are negligible and the final inequality indiates that the effets of homogeneous hemial reations are negligible. When the restritions given by Eqs. (48) are imosed the governing equation for the mass diffusion veloity takes the form ω ( b b) = P =... (49) Truesdell (96 Eq. 7) reresents the left hand side of this result by d and ites Hirshfelder urtiss & ird (954) as the soure. urtiss & ird (999 Eq. 7.6) reresent the left hand side of Eq. (49) by RT d and refer to it as the generalized driving fore for diffusion. t this oint we make use of the identity ( ) = (50) in order to exress Eq. (49) in the form + ω (5) ( ) = =... b b P 9

8 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 In order to see how this result is related to the work of Hirshfelder urtiss & ird (954) we make use of their Eqs and (in terms of the nomenlature used in this work) to obtain Hirshfelder et al. [ ω ] x + x x x ( b b) v v T T x x = + ln T =... (5) The right hand side of this result is aroximate in that () it is based on dilute gas kineti theory and () the binary diffusivities have been used in lae of the oeffiient of diffusion (see Hirshfelder urtiss & ird 954 age 485). The left hand side of Eq. (5) is idential to the left hand side of Eq. (5) rovided that is relaed by x and this is onsistent with the idea that Eq. (5) was develoed for ideal gases. In terms of the work of haman and owling (970) we note that their Eqs. 8.6 and 8.33 lead to Eq. (5) with the last term in Eq. (5) exressed as T T x x kt lnt = lnt (53) When dealing with ideal gases one an roeed with Eq. (5); however for more general ases that are onsistent with Eq. (4) one should make use of Eq. (5) and this means dealing with the fore P...3 on-ideal mixtures The simlest aroah for non-ideal mixtures is to use the form assoiated with dilute gas kineti theory in order to reresent the right hand side of Eq. (5) as Proosal: T T x x x x P = ( v v ) + lnt (54) Here the diffusion oeffiients are to be determined exerimentally with the idea that this form for P is an aetable aroximation and that Eq. (0) would be utilized as a solution to xiom III. Truesdell (96 Se. 6) refers to this aroximation as the seial ase of binary drags. However multiomonent diffusion in liquids is more omlex than suggested by Eq. (54) and Rutten (99) among many others has doumented these omlexities for ternary systems. Putting aside the seminal roblem assoiated with P we make use of Eq. (54) in Eq. (5) to obtain + ω ressure diffusion x x ( b b) = v v fored diffusion T T x x thermal diffusion + ln T =... (55) Here we have exliitly identified the terms assoiated with ressure diffusion fored diffusion and thermal diffusion. This form of the seies momentum equation is restrited by the following: I. The basi assumtions assoiated with ontinuum mehanis. II. The onstitutive equation given by Eq. (4) III. The simlifiations indiated by Eqs. (48). IV. The form of the terms that aear on the right hand side of Eq. (55). One should remember that Eq. (55) is the governing equation for the diffusion veloity and this beomes more aarent if we relae v v with u u. In general thermal diffusion reates very small fluxes that are diffiult to measure (Whitaker and Pigford 958) and in this study we will neglet this term to obtain + ω ressure diffusion x x b b = v v fored diffusion =... (56) haman & owling (970 age 57) disuss the imat of ressure diffusion on the distribution of hemial seies in the atmoshere and both een (998 age 45) and ird et al. (00 age 77) rovide an examle of this effet in terms of a searation roess using an ultraentrifuge. The roess of fored diffusion of eletrially harged artiles is analyzed by haman & owling (970 ha. 9) among others. Estimates (Whitaker 009b Se. 5.6) of the terms on the left hand side of Eq. (56) indiate that these terms are generally quite small leading to the relatively simle relation given by x x 0 = ( ) + ( v v) (57) =... Here one should remember that the first term in this result is based on the use of Eq. (4) and that the 0

9 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 seond term reresents a less than robust model for non-ideal mixtures in whih the binary diffusivities must be determined exerimentally...4 Ideal mixtures t this oint we are ready to make the final simlifiation given by = x ideal mixture (58) in order to obtain the lassi Stefan-axwell equations that will be examined in the remainder of this aer Seies omentum: xx ( v v) 0 = x + =... (59) To omlete our formulation of the mehanial roblem we reall Eq. (38) in the form of the avier-stokes equations Total omentum: v + = + μ v v b v (60) in whih the diffusive stresses have obviously been negleted. The determination of v v v using Eqs. (59) and (60) is a very omlex roblem and the hemial engineering literature ontains many simlified treatments of this roblem. However the domain of validity of these simlified treatments is not always lear and in the following setions we attemt to larify the basis for some of the seial forms of the Stefan-axwell equations.. ass ontinuity equation We begin this study with the total mass ontinuity equation [see Eq. (9)] Total ass: + ( v ) = 0 (6) along with seies mass ontinuity equations [see Eqs. (3)] Seies ass: + ( v) = ( u) + r (6) =... These equations an (in rinile) be used to determine all the seies mass densities in the same way that the momentum equations reresented by Eqs. (59) and (60) an be used to determine all the seies veloities v v v. The mass diffusive flux u is often reresented as (ird et al 00 age 537) j = u (63) so that Eq. (6) takes the form Seies ass: + ( v) = j + r =... (64) Here we note that the mass diffusive fluxes are onstrained by = j = 0 (65) = and we need to determine of these diffusive fluxes in order to develo a solution for Eq. (64). In many liquid-hase diffusion roesses the governing equation for the total density given by Eq. (6) is relaed by the assumtion ssumtion: = onstant (66) and we need only solve the seies ontinuity equations given by Eqs. (64). 3. olar ontinuity equation hemial engineers are rimarily interested in hemial reations interfaial mass transfer and adsortion/desortion henomena thus molar onentrations and mole frations are more useful than mass densities and mass frations. eause of this the molar form of the seies ontinuity equation is often referred. This form is obtained from Eqs. (8) by the use of the relations = r = R =... (67) This leads to the seies molar ontinuity equation given by + ( v ) = R =... (68) while the onstraint on the mass rate of reation given by Eq. rovides = R = 0 (69) = The total molar ontinuity equation is analogous to Eq. (6) and it is develoed by onstruting the sum of Eqs. (68) over all seies to obtain Total olar: + ( v ) = R (70)

10 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 Here the total molar onentration and the molar average veloity are defined by = = = = = v = v (7) The develoment in Se. indiates that Eq. (70) should be solved along with the seies ontinuity equations given by Seies olar: + ( v ) = R =... (7) This allows for the determination of all the seies molar onentrations. The form of Eqs. (70) through (7) suggests (but does not require) a deomosition of the seies veloity given by v * = v + u =... (73) * in whih u is the molar diffusion veloity. little thought will indiate that the molar diffusion veloities are onstrained by = * u = 0 (74) = When Eq. (73) is used in Eq. (7) the transort of seies an be reresented in terms of a onvetive * art v and a diffusive art u leading to + v = u + R * ( ) ( ) =... (75) The molar diffusive flux u is often identified as (ird et al 00 age 537) J = u (76) so that Eq. (75) takes the form + ( v ) = J + R =... (77) This result is similar in form to Eq. (64) for the seies mass density; however there is no governing equation for the molar average veloity v whereas the mass average veloity in Eq. (64) an be determined by the aliation of Eq. (60). In order to eliminate the molar average veloity from Eq. (77) we return to Eq. (73) multily by ω and sum over all seies to obtain ω = ω + ω v v u (78) On the basis of the seond of Eqs. (0) this takes the form v v ω u (79) = + and we are now onfronted with the mixed-mode term ω u that involves a mass fration and a molar diffusion veloity. We would like to exress ωu in terms of molar diffusive fluxes and to do so we maniulate this term as follows ω u u = u = J = x x x ( ) If we define the mean moleular mass as (80) = x + x x (8) we an exress Eq. (80) in omat form aording to ω J u = (8) t this oint we return to Eq. (79) to develo the following relation between the molar average veloity and the mass average veloity: v = v J (83) = Substitution of this exression for the molar average veloity into Eq. (77) allows us to exress that form of the seies ontinuity equation as Seies olar: + ( v) = J x J + R (84) =... in whih the molar diffusive fluxes are onstrained by = * J = 0 (85) = Here we an see that this onvetion-diffusion roblem is inherently nonlinear in terms of the diffusive flux; however if the mole fration of

11 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 seies is suffiiently small it is ossible that the term involving the sum of the diffusive fluxes in Eq. (84) an be negleted. y suffiiently small we mean that the following inequality x J J (86) << is satisfied and Eq. (84) beomes linear in the molar diffusive flux J. To omlete our formulation of the molar forms of the seies ontinuity equation we make use of Eq. (83) in Eq. (70) to obtain Total olar: + ( v) = J + R (87) t This total molar transort equation should be omared with Eq. (6) in order to areiate the omlexity assoiated with the molar form of the seies transort equations. In many gas hase mass transfer roesses Eq. (87) an be relaed by the assumtion ssumtion: = onstant (88) and we need only solve the seies ontinuity equations given by Eqs. (84). 4. ixed-mode ontinuity equation The motivation for a mixed-mode or hybrid seies ontinuity equation is based on the aliations that are dominant in the area of hemial engineering and on the mehanial roblem under onsideration. To be exliit we note two fats: () hemial reations and interfaial mass transfer are usually reresented in terms of the molar onentration or the mole fration x thus we are motivated to use the molar form of the ontinuity equation given by Eq. (68) as oosed to the mass form given by Eq. (6). () The seies ontinuity equation involves veloities that must be determined by the laws of mehanis thus we are motivated to use the mass deomosition of the seies veloity given by Eq. () as oosed to the molar deomosition given by Eq. (73). In order to obtain a mixed-mode or hybrid ontinuity equation we begin with the seies mass ontinuity equation given by Eq. (6) and divide by the moleular mass of seies to obtain + ( v) = ( u ) + R =... (89) Here the diffusive flux is reresented in terms of a molar onentration and a mass diffusion veloity. This mixed-mode diffusive flux is often referred to as a hybrid flux and identified as (ird et al 00 age 537) J = u (90) Use of this reresentation in Eq. (89) leads to Seies olar: + ( v) = J + R =... (9) The onstraint on this diffusive flux is more omlex than that for either the mass diffusive flux or the molar diffusive flux and is given by = J = 0 (9) = This hybrid diffusive flux J laks oularity; however the transort equation given by Eq. (9) has the advantage that it is linear in the diffusive flux. In terms of the mixed-mode diffusive flux the total molar ontinuity equation takes the form Total olar: + ( v) = J + R (93) and we are still onfronted with a omlex form of the total molar transort equation. This omlexity often serves to generate the assumtion that the total molar onentration is onstant as indiated by ssumtion: = onstant (94) Often gas hase diffusion roblems lead to the use of a molar form of the seies ontinuity equation beause Eq. (94) rovides a reasonable simlifiation. On the other hand liquid hase diffusion roblems suggest the use of the mass form of the seies ontinuity equation beause Eq. (66) rovides a reasonable simlifiation. The author is unaware of any solution to a diffusion roblem that does not make use of either Eq. (66) or Eq. (94) and removing these assumtions remains as a signifiant hallenge. 5. inary systems inary systems are often used to introdue the henomena of diffusion and we will follow that aroah in order to exlore the nature of the mass 3

12 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 molar and mixed-mode forms of the seies ontinuity equation. 5. ass diffusive flux For a binary system Eq. (59) redues to xx 0 = x + ( v v ) (95) and we think of this as the governing differential equation for v. The value of v is available from a solution for v and v whih an be used in the seond of Eqs. (0) to obtain v = ( v ωv ) (96) ω For a binary system the two mass ontinuity equations are given by + ( v) = ( u ) + r (97) + ( v ) = 0 (98) and we need to determine u and v in order to solve these equations. Given the form of Eq. (97) it will be onvenient to exress Eq. (95) in terms of mass diffusion veloities and the use of Eq. () leads to xx 0 = x + ( u u ) For a binary system Eq. (4) rovides (99) ω u + ω u = 0 (00) and this an be used in Eq. (99) to obtain xx 0 = x ( ωu ) ωω (0) ultilying and dividing the seond term by the total density allows us to exress this result as 0 = xx x ( ) ω ω u (0) Here we have a mixed-mode reresentation in whih the mass diffusive flux u is exressed in terms of the gradient of the mole fration x along with the mixed-mode term xx ωω. efore attaking the binary result given by Eq. (0) it is onvenient to list some results for - omonent systems. We begin with the definitions for the mass fration ω the mole fration x and the mean moleular mass. These are given by ω = x = = x + x x (03) in whih reresents the moleular mass of seies. In addition to these results we make use of = = =... (04) to obtain the following relations between the mole frations and the mass frations x ( ) = = ω = = (05) =... t this oint we diret our attention to binary systems and make use of the following relations x = x ω = ω ω ω (06) = + along with several algebrai stes (see endix ) to arrive at x = ω (07) Substitution of this exression for the gradient of the mole fration of seies into Eq. (0) leads to 0 = xx ( ) ω ω ω u (08) From Eqs. (05) we see that xx ωω = (09) and Eq. (08) simlifies to the lassi form of Fik s Law given by Fik s Law: j = u = ω (0) Returning to Eq. (97) we make use of this form of Fik s Law to obtain the following governing equation for the seies density + ( v) = r + () For liquid systems this result an often be simlified on the basis of the assumtion ssumtion: whih leads to = onstant () 4

13 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) ( v) = onstant = ( ) + r binary system (3) and emloy the form of Eq. (76) for both seies to obtain x 0 = x + J * * x J (9) Here we have an attrative linear transort equation for the seies density. When onfronted with hemial reations and interfaial transort we generally refer to work with the molar form of the seies ontinuity equation. This form an be extrated from Eq. (3) by the use of whih leads to + ( v) = r = R (4) = onstant = ( ) + R binary system (5) This is an attrative form to use with liquids where the assumtion of a onstant density is likely to be a valid aroximation. When the total density is not onstant one must solve Eq. () simultaneously with Eq. (98). 5. olar diffusive flux eause of the revalene of molar onentrations and mole frations in hemial engineering analysis the seies molar ontinuity equation is generally referred. This form an be extrated from Eq. (84) aording to Seies olar: v J J (6) + ( ) = x R t + while the total molar ontinuity equation given earlier by Eq. (87) takes the form Total olar: v J (7) + ( ) = + R + R Ignoring for the moment the diffiulties assoiated with the total molar ontinuity equation we diret our attention to the molar diffusive flux reresented * by J. We begin by using Eq. (73) to exress the single Stefan-axwell equation as * * xx ( u u) 0 = x + (8) liation of the binary version of Eq. (85) J + J = 0 (0) * * allows us to exress Eq. (9) in the lassi form of Fik s Law given by * Fik s Law: J = u = x () This is the molar analogy of Eq. (0) and substitution of this result into Eq. (6) leads to the molar analogy of Eq. (). + ( v) = + R() If we ignore variations in the total molar onentration on the basis of the assumtion ssumtion: we see that Eq. () takes the form = onstant (3) + ( v) = = onstant + R binary system (4) in whih the resene of leads to the nonlinearity assoiated with = x + (5) In order to obtain the so-alled dilute solution form of Eq. (4) we imose Restrition: x << (6) and Eq. (4) simlifies to the lassi onvetivediffusion equation given by ( ) + ( v) = = onstant + R x( ) << binary system (7) Here it is very imortant to note that this result is idential to Eq. (5). However Eq. (7) is not based on the onstraint that the density is onstant. Instead Eq. (7) is based on the assumtion of onstant total molar onentration indiated by Eq. (3) and the assumtion of a dilute solution 5

14 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 indiated by Eq. (6). For binary systems we have (see Eqs. (03) and (04)) ( x ) = x + (8) whih an be arranged in the form { } = x + (9) When the two restritions assoiated with Eq. (7) are imosed the total density is essentially onstant and Eq. (7) is onsistent with Eq. (5). Returning to Eq. (7) we note that the maximum value of the mole fration for seies will usually be known a riori and this allows us to exress the onstraint assoiated with Eq. (7) as onstraint: ( x ) << (30) max One should remember that there is a restrition assoiated with every assumtion and when one imoses the restrition one always assumes that small auses give rise to small effets (irkhoff 960). In addition one should remember that behind every restrition there is a onstraint (see endix ); however onstraints an often be very diffiult to develo. 5.3 ixed-mode diffusive flux In this ase we return to the mixed-mode seies ontinuity equation [see Eq. (89)] + ( v) = ( u ) + R (3) and diret our attention to the single Stefan-axwell equation given by Eq. (0) and reeated here as xx 0 = x ( ωu ) ωω It is onvenient to rearrange this result in the form x 0 = x ( u ) ω (3) (33) in order to obtain the mixed-mode diffusive flux given by ω u = x (34) x t this oint we an use Eq. (05) to obtain the mixed-mode form of Fik s Law given by Fik s Law: J = u = x (35) Substitution of this result into Eq. (3) rovides the following governing equation for the seies molar onentration + ( v) = ( ) ( ) + R (36) This result is idential to Eq. () indiating that both the molar reresentation given by Eq. (6) and the mixed-mode reresentation given by Eq. (3) lead to the same result for a binary system. It is of some interest to note that the mixedmode diffusive flux an be exressed as u = u (37) and on the basis of Eq. (0) this takes the form u = ω (38) Use of this result in Eq. (3) yields what aears to be an unattrative form given by + ( v) = ω + R (39) However if we imose the ondition ssumtion: = onstant (40) and make use of the first of Eqs. (4) we find ( ) + ( v) = whih was given earlier by Eq. (5). (4) = onstant + R binary system 6. Seial forms for -omonent systems Given the omlexity of the binary forms desribed in the revious setions we should exet additional omlexities for -omonent systems. This naturally leads to the searh for simlifiations and we will examine some of these simlifiations in this setion. 6. ilute solution diffusion There are mass transfer roesses in whih all the molar fluxes are the same order of magnitude and the dominant diffusing seies is dilute. In this seial ase it is onvenient to reresent the Stefan- axwell equations in terms of the molar flux defined by = v =... (4) 6

15 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 whih allows us to exress Eqs. (59) as = x x 0 = x + =... (43) = t this oint we searate the seond term to obtain = = x 0 = x + x = = (44) =... and we define the mixture diffusivity by m = x = =... = (45) so that the Stefan-axwell equations an be exressed as = m 0 = m x + x = (46) =... For some roesses suh as diffusion in orous media (Whitaker 999) in whih the flux of all the seies is driven by heterogeneous reation or by adsortion/desortion we an imose the simlifiation x (47) =... G G < = m << = when the following two onditions are satisfied: onstraint: ( x ) max << =... G < (48a) Restrition: = O( ) =... (48b) The first of Eqs. (48) is identified as a onstraint sine the maximum values of the mole frations are generally known a riori while the seond inequality is identified as a restrition sine it is not exressed in terms of quantities that are known a riori. Equation 48b should be interreted to mean that is not signifiantly larger than and if seies is stagnant would be zero. In Eqs. (47) and (48a) we have indiated that our -omonent system ontains G omonents that are dilute. For examle if we may have a five-omonent mixture in whih three omonents have mole frations that are small omared to one we have G = 3 and Eqs. (47) and (48a) alies to these three omonents. Use of Eq. (47) allows us to exress the dilute forms of Eqs. (46) as = m x =... G < (49) t this oint we reognize that Eqs. (7) an be exressed in terms of to obtain + = R =... (50) and that Eq. (49) an be used to obtain a dilute solution diffusion equation given by ( m )... = x + R = G < (5) We are still onfronted with the omlexity of the transort equation for the total molar onentration given by Eq. (87) and this diffiulty is lassially avoided by assuming that the total molar onentration is a onstant in order to obtain ( m ) R = + G < =... G other onditions (5) in whih the other onditions assoiated with this result are given by ssumtion: = onstant (53a) Restrition: = =... O (53b) onstraint: ( x ) << =... G < (53) max The onstraint identified by Eq. (53) is generally available in terms of the roblem statement and when this onstraint is satisfied it is robable that the assumtion given by Eq. (53a) and the restrition given by Eq. (53b) are also valid. 6. ilute solution onvetive-diffusion equation using J * In order to develo the onvetive-diffusion version of Eqs. (5) we begin with the generally valid form given by Eqs. (84) and reeated here as + ( v) = x t J J + R =... The Stefan-axwell equations an be exressed as (54) x x 0 = x + J J =... (55) and the summation an be searated leading to 7

16 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 J x 0 = x + x J (56) =... The definition of the mixture diffusivity given by Eq. (45) an be used to exress this result in the form m 0 = m x + x J J (57) =... In making judgments about this result we need to remember that the diffusive fluxes are onstrained by * J = 0 (58) indiating that the diffusive fluxes tend to be the same order of magnitude. This means that the following inequality Restrition: m x J << J =... G < (59) has onsiderable aeal when the mole fration of seies is small omared to one as indiated by Restrition: x << =... G < (60) Use of the inequality given by Eq. (59) in the Stefan-axwell equations given by Eqs. (57) leads to the multi-omonent form of Fik s Law Fik s Law J G < = m x =... G x << (6) whih is analogous to the result for binary systems given by Eq. (). We now turn our attention to the seies ontinuity equation given by Eq. (54). Use of the dilute solution ondition indiated by Eq. (60) and the onstraint on the diffusive fluxes given by Eq. (58) leads to the restrition Restrition: x J J (6) << Use of this inequality along with the multiomonent form of Fik s Law given by Eq. (6) in Eq. (54) leads to the following form of the onvetive-diffusion equation ( ) + ( v) = m x + R (63) =... G < In addition to the inequalities given by Eqs. (60) and (6) we assume that the total molar onentration is onstant in order to obtain the lassi linear onvetive-diffusion equation for seies. + ( v) = ( m ) G < + R =... G = onstant x << (64) This seial form of the seies ontinuity equation is ubiquitous in the hemial engineering literature; however the simlifiations assoiated with this result are generally not made lear. In addition to the dominant restritions listed in Eq. (64) one should kee in mind the restrition given by Eq. (6) that would aear to be automatially satisfied by Eqs. (58) and (60) unless there is a serious disarity in the moleular masses. 6.3 ilute solution onvetive-diffusion equation using J In this ase we begin with Eq. (9) + ( v) = J + R =... (65) and note that the Stefan-axwell equations an be exressed as = x x 0 = x + J J =... = Searating the terms in the sum leads to (66) = = J x 0 = x + x J = = (67) =... and use of the definition of the mixture diffusivity given by Eq. (45) rovides = m 0 = m x + x J J = (68) =... In making judgments about this result we need to remember that the diffusive fluxes are onstrained by 8

17 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 = J = 0 (69) = thus if the mole fration of seies is small omared to one we an make use of the restrition given by Restrition: = m x J << J =... G < = (70) Under these irumstanes the Stefan-axwell equation for seies takes the form Fik s Law : J = x =... G < (7) m Use of this result in Eq. (65) leads to the following form of the onvetive-diffusion equation + ( v) = x + R (7) m This result based on the single restrition given by Eq. (70) is idential to that given earlier by Eq. (63). To omlete the analysis of the mixed-mode diffusive flux we assume that the total molar onentration is onstant so that Eq. (7) takes the form ( ) + ( v) = + R m G < =... G = onstant x << (73) ertainly the route to Eq. (73) is simler than that followed in the develoment of Eq. (64); however the referred aroah might still be onsidered to be a matter of hoie. 6.4 iffusion through stagnant seies The ase of binary transort of seies through a stagnant seies has been treated in terms of the lassi Stefan diffusion tube (Whitaker 009b Se..7). oving beyond the binary system we onsider the ase in whih seies is diffusing and all other seies are stagnant. Under these irumstanes the Stefan-axwell equation for seies redues to 0 = x = = xxv and this an be arranged in the form = = x 0 = x (74) (75) Use of the definition of the mixture diffusivity given by Eq. (45) immediately leads to = x (76) m ote that this result is not restrited to a dilute solution; however we have imosed the ondition on the veloities given by ssumtion: v = v =... = v = 0 (77) This assumtion ould be relaed with the restrition Restrition: v << v v << v... v << v (78) in whih the use of the absolute values of the veloities is understood. Here one should remember that we are reeatedly relying on irkhoff s (960) lausible intuitive hyothesis that small auses give rise to small effets. Use of Eq. (76) in Eq. (50) leads to = x + R (79) m and we an assume that the total molar onentration is onstant to obtain = onstant = ( m ) + R (80) other onditions where the other onditions are those indiated by Eqs. (78). This result is idential to Eq. (5) exet for the fat that there is only a single omonent that ould satisfy this equation. s a reminder of the differene between Eq. (80) and Eq. (5) we summarize the onditions uon whih it is based Restrition: v << v v << v... v << v (8) Restrition: x << (8) omaring these two restritions with Eqs. (53) indiates that Eqs. (5) and (80) desribe rather different hysial henomena even though the two equations are idential. In reality it seems unlikely that a roess restrited by Eq. (8) ould involve signifiant homogeneous reation thus a more realisti version of Eq. (80) would require that we set R equal to zero. evertheless the fat that Eq. (5) and Eq. (80) are idential in form suggests that we must be very areful to understand the reise meaning of the seial forms of Eq. (68). 7. General solution for -omonent systems: onstant total molar onentration From the analysis in revious setions it seems lear that the most effiient route to the determination of the molar onentration is via the mixed-mode 9

18 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 ontinuity equation desribed in Se. 4. This is eseially true for the ase in whih we develo an exat solution of the Stefan-axwell equations. In this setion we onsider the ase of onstant total molar onentration and in the next setion we examine the ase of onstant total mass density. The omletely general ase for whih neither nor is onstant remains as a hallenge. In this treatment we make use of Eq. (9) reeated here as + ( v) = J + R =... (83) along with the onstraint on the mixed-mode diffusive flux given by = J = 0 (84) = For -omonent systems it is onvenient to work in terms of matries thus we define the following olumn matries that will be used in subsequent aragrahs. = = x J x J x J = = x J R R R [ R] = R (85) [ ] [ ] [ x] [ J] Use of the first fourth and fifth of these matries allows us to exress Eq. (83) as [] + ([] v) = [ J ] + [ R] (86) using v and J with =... is given by ird et al. (00 Se..9). In addition Quintard et al. (006) have studied the formulation and the numerial solution for this roblem using both the molar forms j. v and J and the mass forms v and We begin our analysis of the diffusive flux with the Stefan-axwell equations given by Eq. (66) and we make use of the mixture diffusivity defined by Eq. (45) to obtain m J = m x + x J (87) =... We want to use Eq. (84) to eliminate J and it will be onvenient to exress that onstraint on the mixed-mode diffusive fluxes in the alternate form given by = J ( ) = 0 (88) = t this oint we extrat J from the sum in Eq. (87) in order to obtain m m J = m x + x J + x J (89) =... and from Eq. (88) we have the following reresentation for J = ( ) J J (90) In order to use this result with Eq. (89) we need to ondition the sum with the onstraint indiated by and this leads to = ( ) ( ) J J J (9) Use of this result in Eq. (89) rovides the following form of the Stefan-axell equations m m m J + x + x J (9) = x =... m This an be exressed in omat form aording to and our single objetive at this oint is to develo a useful reresentation for [ J ]. similar aroah 30

19 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 in whih [ ] [ H] J m x m x J J m x = J m x H is an ( ) ( ) square matrix H H... H H H... H H..... [ H ] = H.... H having the elements defined by H (93) (94) m = + x (95a) =... m m H = x (95b) =.. We assume that the inverse of [ H ] exists in order to exress the olumn matrix of the mixed-mode diffusive flux vetors in the form J m x m x J J [ ] m x = H J m x (96) The olumn matrix on the right hand side of this result an be exressed as so that the matrix reresentation for the mixed-mode diffusive flux beomes... J m x J m x J = [ H] 0 0 m... 0 x m x J (98) The diffusivity matrix is now defined by [ ] = [ ] H... m m m and this allows us to exress Eq. (98) as J x x J J x = [ ] J x with the omat form given by [ ] = [ ][ x] m (99) (00) J (0) This reresents the -omonent analog of Fik s Law given by Eq. (35) that we reall here as Fik s Law: J = x (0) Use of Eq. (0) in Eq. (86) leads to [] + ([] v ) = ( [ ] [ x] ) + [ R] (03) m x m x m x... m x m x 0 m x = 0 0 m... 0 x x... m (97) One again we may be faed with the diffiult task of determining the total molar onentration on the basis of Eq. (93) and to avoid this roblem we restrit Eq. (03) to the ase of onstant total molar onentration. This leads to [] + ([] v) = ( [ ] [ ] ) + [ R] = onstant omonent system (04) Here it is imortant to remember that [ ] deends exliitly on the mole frations as indiated by the 3

20 S. Whitaker / Revista exiana de Ingeniería Químia Vol. 8 o. 3 (009) 3-43 definitions given in Eq. 95 and imliitly as indiated by the definition of the mixture diffusivity given by Eq. (45). This means that a trial-and-error numerial solution will be neessary in whih the assumed values used for the mole frations are ugraded after eah iteration. The solution for [ ] will rovide values of... and the onentration an be determined by the first of Eqs. (7). Similarly the solution for [ R ] will rovide values of R R... R and the reation rate R an be determined by Eq. (69). In the ase of omlex kinetis the olumn matrix of reation rates will need to be exressed as [ R] [ (... )] = F (05) and the trial-and-error roedure will be more omlex. 8. General solution for -omonent systems: onstant total mass density In addition to the -omonent form of the seies ontinuity equation based on the assumtion of a onstant total molar onentration it would be useful to develo the analogous form for onstant total density. Our starting oint for this analysis is Eq. (03) and the analysis requires that we exress [ x] in terms of the gradient of the mass frations ω ω et. We begin the analysis with Eq. (05) reeated here as x =... ω = (06) in whih the mean moleular mass an be exressed as in terms of the mass frations in order to obtain (see Eq. in the endix ) ω ω ω ω = (07) We an use Eq. (06) to exress the gradient of the mole fration as x = ω + ω =... (08) while the gradient of the mean moleular mass is given by ω = (09) x = ω ω ω (0) =... t this oint we an make use of the fat that the sum of the mass frations is equal to one so that the gradients are related by (... ) ω = ω + ω + ω + + ω () This allows us to eliminate ω from Eq. (0) and exress that result in the form = x = ω ω ω () =... Here we need to ondition the sum with the onstraint indiated by and this leads to x = + ω ω (3) + ω ω =... whih an be exressed as a matrix equation given by.. x W W W W ω x W W... W ω x W..... ω = x W W... W ω (4) Here the elements of this ( ) ( ) square matrix are defined as W = + ω (5a) =... W = ω =... (5b) t this oint we reall Eq. (00) and make use of Eq. (4) to obtain Use of Eq. (09) in Eq. (08) leads to 3