Few asic Facts but Isthermal Mass Transfer in a inary Miture David Keffer Department f Chemical Engineering University f Tennessee first begun: pril 22, 2004 last updated: January 13, 2006 dkeffer@utk.edu crrectins and additins: Carrie Ga Department f Chemical Engineering University f Tennessee ugust, 2004 Table f Cntents Intrductin 1 I. Prblem Frmulatin in Terms f Mass 2 II. Prblem Frmulatin in Terms f Mles 7 III. Prblem Frmulatin in Terms f Hybrid Mass/Mles Reference 12 IV. Steady State nalysis (cnstant center-f-mass velcity) 13 V. Steady State nalysis (variable center-f-mass velcity) 16
Intrductin The purpse f these ntes is t prvide a cmplete analysis f a very simple system, namely isthermal mass transfer in a binary miture. s engineers, we ften think f a diffusin prcess as a membrane separating a reservir with a high-cncentratin f slute frm that with a lw-cncentratin f slute. We recall that the steady-state slutin is a linear prfile frm ne bundary cncentratin t the net. This picture is ver-simplified and cntains numerus assuptins. It is nt the general case. crrect frmulatin f this prblem is essential in btaining a rigrus, generalized slutin. Part f the cnfusin assciated with mass transfer is that there are many degrees f freedm in frmulating the prblem, resulting in ptentially an infinity f prblem frmulatins. Each f these frmulatins, if slved in a manner cnsistent with the arbitrary chices that went int the prblem frmulatin will yield results cnsistent with any ther frmulatin. In sectins I and II f this hand-ut, we present tw cmmn frmulatins f isthermal mass transfer in a binary miture. In sectin III, we present a third, uncmmn frmulatin t illustrate the arbitrary nature f the chices as well as the ability t rectify the results frm different frmulatins. In sectin IV, we eamine the steady-state slutin f the evlutin equatins fr a simple system (an inviscid, ideal gas) under the assumptin that the center-f-mass velcity is cnstant. We wuld like t make this assumptin because it allws us t decuple the mmentum balance and the mass balances. If this assumptin is nt true then we have t slve the mmentum and mass balances simultaneusly. In the slutin f the evlutin equatins, we cme upn a cntradictin that indicates that the assumptin that the center-f-mass velcity is cnstant is an unphysical assumptin. In sectin V, we eamine the steady-state slutin f the evlutin equatins fr a simple system (an inviscid, ideal gas) withut the assumptin that the center-f-mass velcity is cnstant. We find that the cupled mass and mmentum balances cannt be slved analytically but can be slved numerically. We prvide plts f the numerical slutin fr different cnditins. In this hand-ut, we attempt t fllw the ntatin used in Transprt Phenmena by ird, Stewart & Lightft, Secnd Editin, in terms f nmenclature, capitalizatin, subscripts, and asterisks. 2
I. Prblem Frmulatin in Terms f Mass In this sectin f the dcument, we frmulate the prblem f isthermal mass transfer in a binary system cmpletely in terms f mass and mass related variables, such as the mass density, mass fractin, mass-averaged velcity, and mass flues. We prvide () variable definitins, () evlutin equatins, (C) cnstitutive relatins, (D) assumptins, and (E) cnclusins. Parts f the tet refers back t Mlecular Dynamics (MD) simulatins because this wrk was initially prmpted by a desire t have a macrscpic descriptin that wuld serve as a carse-grained thery upn which mlecular simulatins culd be mapped. I.. variable definitins mass density ρ In an MD simulatin ρ has an unambiguus definitin: m + m ρ (I.1) V where and are the number f mlecules f and in the simulatin, and m and m are the masses f a mlecule f and respectively. mass fractins w and w In an MD simulatin w and w have unambiguus definitins: w m m + m w m m + m (I.2) cmpnent mass densities ρ and ρ In an MD simulatin ρ and ρ have unambiguus definitins: ρ wρ ρ w ρ (I.3) cmpnent velcities relative t statinary labratry frame f reference v and v In an MD simulatin v and v have unambiguus definitins: 1 v v, i i 1 1 v v (I.4) i 1 where v,i is the velcity f the i th mlecule f cmpnent relative t the labratry frame f reference., i 3
ttal mass flu f each cmpnent relative t statinary labratry frame f reference n and n The ttal flu f each cmpnent can be unambiguusly determined as n ρwv n ρwv (I.5) mass-averaged (r center-f-mass) velcity relative t labratry frame f reference v The mass-averaged velcity is defined as v w v + w v (I.6) cnvective mass flu f each cmpnent relative t statinary labratry frame f reference χ and χ The cnvective flu f each cmpnent (carried by the mass-averaged velcity) can be unambiguusly determined as χ ρw v χ ρw v (I.7) diffusive flu f each cmpnent relative t mass-averaged velcity and The diffusive flu f each cmpnent can be unambiguusly determined because the ttal flu is defined as the sum f the cnvective and diffusive flu. n χ n χ (I.8) Substituting equatins (5) and (7) int equatin (8) yields ρ ( v v) ( v v) w Substituting equatins (6) int equatin (9) yields Cmments: w ρ w w ( v v ) w w ( v v ) ρ (I.9) ρ (I.10) 1. Yu can see that the diffusive mass flu is defined withut ever intrducing any cnstitutive relatin such as Fick s law. 2. Yu can als see frm equatin (10) that there is n ttal center f mass mtin due t the diffusive flues. 0 (I.11) + 4
I.. evlutin equatins We assume here that there is n reactin. evlutin equatin n ttal mass (cntinuity equatin) ρ ρ ( n + n ) ( χ + + χ + ) ( χ + χ ) ( ρw v + ρw v) ( ρv) (I.12) (I.13) ecause the diffusive flu has n net mass transprt, it drps ut f the verall mass balance. evlutin equatin n mass f cmpnent ρw n (I.14) ρw ( + ) ( ρwv) χ (I.15) We can use the prduct rule n the accumulatin term and the cnvectin term in equatin (15) and we can substitute in equatin (13) int (15) t btain w ρ w + ρ ρv w w ( ρv) (I.16) w ρ ρv w (I.17) Equatin (17) is the epressin typically used as the evlutin equatin fr the mass f cmpnent. It is nly true in the absence f reactin. n analgus epressin can be written fr cmpnent. I.C. cnstitutive relatins One can write Fick s law in a virtual infinity f frms. Hwever, given ur decisins t (i) define material transprt in terms f mass flues and (ii) define diffusin relative t the massaveraged velcity, there is nly ne intelligent chice fr Fick s law, namely ne where (iii) the driving frce is given in terms f the gradient f the mass fractin. ρd w ρd w (I.18) 5
Given this chice D D D (I.19) Cmments: 1. Fr any ther chice f driving frce, these tw diffusivities will nt be the same. 2. Since there is nly ne independent variable in an isthermal binary diffusin prcess, regardless f the chice f gradient, there will always be sme relatin between D and D. With this cnstitutive relatin, the evlutin equatin fr the mass fractin f, equatin (17) becmes w ( ρd w ) ρ ρv w + (I.20) We can use the triple prduct rule n the diffusive term. w ρ 2 ρv w + ρd w + ρ D w + D ρ w (I.21) If the diffusivity is nt a functin f ρ and w, then equatin (21) drps ne term and becmes w ρ 2 ρv w + ρd w + D ρ w fr cnstant D (I.22) If in additin t a cnstant diffusivity, we als have an incmpressible fluid, equatin (22) becmes w w + D 2 v w fr cnstant D and incmpressible fluid (I.23) 6
I.D. assumptins side frm the definitin f the system being binary, isthermal, and free f chemical reactin, there are three imprtant assumptins in this derivatin. 1. We assumed that we wuld wrk the prblem with mass flues. 2. We assumed that we wuld measure the diffusive flu relative t the mass-averaged velcity. This assumptin was ttally arbitrary, but is a gd chice because equatin (11) is nt satisfied withut ther chices f reference velcity. 3. We assumed that the driving frce in Fick s was epressed as a gradient in mass fractin. These three assumptins define ur diffusivity. If we change any f these assumptins, then we will have t cmpute a different numerical value f the diffusivity. I.E. cnclusins There is n ambiguity in a well-defined descriptin f isthermal binary mass transprt. It is imprtant t realize what assumptins are made. There is nly ne independent diffusivity in this system. The diffusive flu is defined befre the intrductin f the cnstitutive relatin, Fick s law. ll Fick s law des is define the diffusivity. 7
II. Prblem Frmulatin in Terms f Mles In this sectin f the dcument, we frmulate the prblem f isthermal mass transfer in a binary system cmpletely in terms f mles and mle-related variables, such as the mlar density, mle fractin, mlar-averaged velcity, and mlar flues. In a precisely analgus prcedure t what was dne in sectin I., we prvide () variable definitins, () evlutin equatins, (C) cnstitutive relatins, (D) assumptins, and (E) cnclusins. dditinally, we relate the diffusivity frm sectin I t the diffusivity frm sectin II. II.. variable definitins mlar density c In an MD simulatin c has an unambiguus definitin: c + (II.1) V where and are the number f mlecules f and in the simulatin. mle fractins and In an MD simulatin and have unambiguus definitins: + + (II.2) cmpnent mlar densities c and c In an MD simulatin c and c have unambiguus definitins: c c c c (II.3) cmpnent velcities relative t statinary labratry frame f reference v and v In an MD simulatin v and v have unambiguus definitins: 1 v v, i i 1 1 v v (II.4) i 1 where v,i is the velcity f the i th mlecule f cmpnent. relative t the labratry frame f reference. These are the same definitins as were used in the mass-referenced case., i 8
ttal mlar flu f each cmpnent relative t statinary labratry frame f reference and The ttal flu f each cmpnent can be unambiguusly determined as c v cv (II.5) mlar-averaged (r center-f-mles) velcity relative t labratry frame f reference v* The mlar-averaged velcity is defined as v* v + v (II.6) cnvective mlar flu f each cmpnent relative t statinary labratry frame f reference Χ and Χ The cnvective flu f each cmpnent (carried by the mlar-averaged velcity) can be unambiguusly determined as Χ c v * Χ c v * (II.7) diffusive mlar flu f each cmpnent relative t mlar-averaged velcity J* and J* The diffusive flu f each cmpnent can be unambiguusly determined because the ttal flu is defined as the sum f the cnvective and diffusive flu. J * Χ * Χ J (II.8) Substituting equatins (5) and (7) int equatin (8) yields J * c ( v v *) J * ( v v *) Substituting equatins (6) int equatin (9) yields c (II.9) J * c ( v v ) J * c ( v v ) (II.10) Cmments: 1. Yu can see that the diffusive mlar flu is defined withut ever intrducing any cnstitutive relatin such as Fick s law. 2. Yu can als see frm equatin (10) that there is n ttal mlar-averaged mtin due t the diffusive flues. J * J 0 (II.11) + * 9
II.. evlutin equatins We assume here that there is n reactin. evlutin equatin n ttal mles (cntinuity equatin) c ( + ) (II.12) c ( Χ + J * + Χ + J * ) ( Χ + Χ ) ( c v * + c v *) ( cv *) (II.13) ecause the diffusive flu has n net mass transprt, it drps ut f the verall mass balance. evlutin equatin n mles f cmpnent c (II.14) c ( + J * ) ( c v *) J * Χ (II.15) We can use the prduct rule n the accumulatin term and the cnvectin term in equatin (15) and we can substitute in equatin (13) int (15) t btain c + ( cv *) J c cv * * c cv * J * (II.16) (II.17) Equatin (17) is the epressin typically used as the evlutin equatin fr the mles f cmpnent. It is nly true in the absence f reactin. n analgus epressin can be written fr cmpnent. II.C. cnstitutive relatins One can write Fick s law in a virtual infinity f frms. Hwever, given ur decisins t (i) define material transprt in terms f mlar flues and (ii) define diffusin relative t the mlar-averaged velcity, there is nly ne intelligent chice fr Fick s law, namely ne where (iii) the driving frce is given in terms f the gradient f the mle fractin. J * cd * * cd * J (II.18) 10
Given this chice D * D * D * (II.19) Cmments: 1. Fr any ther chice f driving frce, these tw diffusivities will nt be the same. 2. Since there is nly ne independent variable in an isthermal binary diffusin prcess, regardless f the chice f gradient, there will always be sme relatin between D* and D*. With this cnstitutive relatin, the evlutin equatin fr the mle fractin f, equatin (17) becmes c ( cd * ) cv * + (II.20) We can use the triple prduct rule n the diffusive term. c cv * + cd * 2 + c D * + D * c (II.21) If the diffusivity is nt a functin f c and, then equatin (21) drps ne term and becmes c cv * + cd * 2 + D * c fr cnstant D * (II.22) If in additin t a cnstant diffusivity, we als have a fluid with cnstant mlar density, equatin (22) becmes * + D * 2 II.D. assumptins v fr cnstant D * and c (II.23) side frm the definitin f the system being binary, isthermal, and free f chemical reactin, there are three imprtant assumptins in this derivatin. 1. We assumed that we wuld wrk the prblem with mlar flues. 2. We assumed that we wuld measure the diffusive flu relative t the mlar-averaged velcity. This assumptin was ttally arbitrary, but is a gd chice because equatin (11) is nt satisfied withut ther chices f reference velcity. 3. We assumed that the driving frce in Fick s was epressed as a gradient in mle fractin. These three assumptins define ur diffusivity. If we change any f these assumptins, then we will have t cmpute a different numerical value f the diffusivity. 11
II.E. cnclusins We have the same cnclusins as thse fr the mass-referenced system. II.F. relatinship between diffusivities In Case I, we had a single diffusivity, D, and in Case II, we had a single diffusivity, D *. These diffusivities are related. The easiest way t see the relatin is t recall that the definitins f v and v are the same in bth cases. s a result, if we can epress, each diffusivity in terms f these velcities, then we can btain a relatin between the tw diffusvities. First cmbine equatins (10), (18), and (19) fr bth cases: ρww ( v v ) ρd w ρww ( v v ) ρd w (II.24) J * c ( v v ) cd * * c ( v v ) cd * J (II.25) In each case (fr either species) slve fr the difference in velcities and equate. ( v ) D w D * v (II.26) ww Slving fr the rati f diffusivities we have: D D * ww w ww w (II.27) We can analytically evalute the partial derivative w w w (II.28) s a result, D D * (II.29) The diffusivity frm the mass-referenced case and the diffusivity frm the mlar-referenced case turn ut t be the same diffusivity. 12
III. Prblem Frmulatin in terms f hybrid mass/mles reference The purpse f this demnstratin is t indicate that ne can chse any arbitrary frmulatin f a diffusive prcess. It als demnstrates that sme chices are mre easy t live with than thers. In this case we again limit urselves t a binary, isthermal system free f reactin. dditinally, we assume that ur descriptin f mass transprt will use mass flues. These mass flues will be references relative t the mass-average velcity. s a result ur derivatin, will be the same up t equatin (I.17). w we chse a different frm f Fick s law, which emplys the gradient f the mle fractin. ρ D D ρ (III.1) The flues are still equal and ppsite, s we can slve fr the relatinship between the tw diffusivities, D and D. D D D (III.2) We can equate the flues in equatin (I.18) and (III.1) t btain ρ ρ ρ ρ (III.3) D w D D w D Slving either species fr the rati f diffusivities, we have: D D w ww (III.4) Therefre, if we knw either f the diffusivities, we can btain the ther. 13
IV. Prblem Frmulatin in terms f a density driving frce One cmmnly sees Fick s law written where the driving frce is nt the gradient f the mass fractin but rather the gradient f the mass density, D ρ D ρ (IV.1) The flues are still equal and ppsite, s we can slve fr the relatinship between the tw diffusivities, D and D. D ρ D ρ (IV.2) ρ D D ρ (IV.3) The partial derivative that appears in equatin (IV.3) des nt vanish as it did in case I, which is because w w but, in general, ρ ρ. We will further discuss this factr mmentarily. Regardless, we can can equate the flues in equatin (I.18) and (IV.1) t btain ρd w D ρ ρd w D ρ (IV.4) Slving fr the rati f diffusivities, we have: ρ ρ D w D ρ ρ D w D (IV.5) Therefre, if we knew the diffusivity, D, and had a thermdynamic descriptin f the fluid, we culd btain the diffusivities, D and D. In ther wrds, they are nt independent. 14