On the Maxwell-Stefan Approach to Diffusion: A General Resolution in the Transient Regime for One-Dimensional Systems

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1 J. Phy. Chem. B 2010, 114, O the Maxwell-Stefa Approach to Dffuo: A Geeral Reoluto the Traet Regme for Oe-Dmeoal Sytem Erma Leoard*, ad Celeto Agel CRS4, Ceter for AdVaced Stude, Reearch ad DeVelopmet Sarda, Parco Scetfco e Tecologco, Polar, Edfco 1, I Pula, Italy, ad Dpartmeto d Chmca, UVertà d Ferrara, Va Borar 46, I Ferrara, Italy ReceVed: Jauary 26, 2009; ReVed Maucrpt ReceVed: October 13, 2009 The dffuo proce a multcompoet ytem ca be formulated a geeral form by the geeralzed Maxwell-Stefa equato. Th formulato able to decrbe the dffuo proce dfferet ytem, uch a, for tace, bulk dffuo ( the ga, lqud, ad old phae) ad dffuo mcroporou materal (membrae, zeolte, aotube, etc.). The Maxwell-Stefa equato ca be olved aalytcally (oly pecal cae) or by umercal approache. Dfferet umercal tratege have bee prevouly preeted, but the umber of dffug pece ormally retrcted, wth oly few excepto, to three bulk dffuo ad to two mcroporou ytem, ule mplfcato of the Maxwell-Stefa equato are codered. I the lterature, a large effort ha bee devoted to the dervato of the aalytc expreo of the elemet of the Fck-lke dffuo matrx ad therefore to the ymbolc vero of a quare matrx wth dmeo ( beg the umber of depedet compoet). Th tep, whch ca be ealy performed for ) 2 ad rema reaoable for ) 3, become rapdly very complex problem wth a large umber of compoet. Th paper addree the problem of the umercal reoluto of the Maxwell-Stefa equato the traet regme for a oe-dmeoal ytem wth a geerc umber of compoet, avodg the defto of the aalytc expreo of the elemet of the Fck-lke dffuo matrx. To th am, two approache have bee mplemeted a computatoal code; the frt the mple fte dfferece ecodorder accurate tme Crak-Ncolo cheme for whch the full mathematcal dervato ad the relevat fal equato are reported. The ecod baed o the more accurate backward dfferetato formula, BDF, or Gear method (Shampe, L. F.; Gear, C. W. SIAM ReV. 1979, 21, 1.), a mplemeted the Lvermore olver for ordary dfferetal equato, LSODE (Hdmarh, A. C. Seral Fortra SolVer for ODE Ital Value Problem, Techcal Report; home.html (2006).). Both method have bee appled to a ere of pecfc problem, uch a bulk dffuo of acetoe ad methaol through tagat ar, uptake of two compoet o a mcroporou materal a model ytem, ad permeato acro a mcroporou membrae model ytem, both wth the am to valdate the method ad to add ew formato to the compreheo of the pecular behavor of thee ytem. The approach valdated by comparo wth dfferet publhed reult ad wth aalytc expreo for the teady-tate cocetrato profle or fluxe partcular ytem. The poblty to treat a geerc umber of compoet (the lmtato beg eetally the computatoal power) alo teted, ad reult are reported o the permeato of a fve compoet mxture through a membrae a model ytem. It worth otcg that the algorthm here reported ca be appled alo to the Fck formulato of the dffuo problem wth cocetrato-depedet dffuo coeffcet. Itroducto The decrpto, terpretato, modelg, ad mulato of the multcompoet dffuo proce a crucal apect may reearch ad dutral actvte. For th reao, t the ubject of a large umber of book ad revew, a well a a actve reearch feld. A overvew of the dfferet theoretcal approache to dffuo beyod the cope of th paper. We retrct ourelve to remd that multcompoet dffuo ca be decrbed eetally wth three tratege. I the followg, a hort ummary of thee approache reported; the reader referred to recet revew for more detal. 3-6 Moreover, the atteto focued o oe-dmeoal ytem (z beg the * To whom correpodece hould be addreed. E-mal: ermy@cr4.t. Parco Scetfco e Tecologco. Uvertà d Ferrara. codered coordate), gve that th the ubject of the preet work. The frt approach kow a the Fck law of dffuo; the molar flux of compoet, N, wrtte a a lear combato of the cocetrato gradet, dc j /dz, of all compoet N )- j)1 Th formulato pheomeologcal; the dffuo coeffcet, D j, are obtaed from expermet ad ca how a marked depedece o the cocetrato. A example ca be foud a work of Krha ad Weelgh, 3 where a aaly of a expermetal tudy reported by Duca ad Toor o a deal terary ga mxture dcate a curou behavor for D j dc j dz (1) /jp900760c 2010 Amerca Chemcal Socety Publhed o Web 12/14/2009

2 152 J. Phy. Chem. B, Vol. 114, No. 1, 2010 Leoard ad Agel oe compoet. Here (ad the followg), cocetrato ha a geeral meag ad dcate dfferet quatte followg the problem uder tudy (for tace, t ca be a partal preure the cae of a ga mxture or a fractoal occupacy for urface dffuo mcroporou materal). The ecod approach baed o rreverble thermodyamc. I th cae, N a lear combato of the chemcal potetal gradet, dµ j /dz, whch are the drvg force for dffuo: N )- The elemet L j are called Oager pheomeologcal coeffcet ad atfy the recprocal relato, L j ) L j. Fally, the thrd approach due to Maxwell 8 ad Stefa 9 (MS) ad baed o a mcrocopc hypothe o the phycal effect cotrollg the dffuo proce. I th formulato, a clearly reported by Krha ad Waelgh, 3 two force are exerted o the molecule, ad thee two force cacel out each other. The frt force the chemcal potetal gradet, -dµ / dz, ad t the drvg force for dffuo. Th force couterbalaced by the frcto wth all of the other movg pece j; th frcto proportoal to the dfferece betwee the velocte of the two pece ad to the cocetrato of j. The proportoalty coeffcet wrtte a 1/Ð j, where the Ð j are called MS dffuvte ad repreet the vere of a drag coeffcet. After ome algebrac mapulato, the MS equato ca be cat the form - x dµ RT dz ) j)1 where x ) c /c t are the molar fracto (c t beg the total molar cocetrato). Oe ca how that the three formulato are actually fully equvalet (a reported by dfferet author; ee Wag ad LeVa 10 for a recet ad complete aaly) ad that mple equato relate the three key quatte, the Fck dffuvte, the pheomeologcal Oager coeffcet, ad the MS dffuvte. Thee three approache ca be further coected to other theore (ee, for tace, Mtrovc 11 for a dervato of the MS equato from the clacal Lagrage equato). Amog the dfferet approache, the MS formulato ha gaed the lat decade a cetral poto, partcular for the decrpto of the multcompoet dffuo proce mcroporou ytem. May techologcally relevat procee fall to th category; to report oly a few example, let u cte dffuo mcroporou membrae, 6 preure wg adorpto, 12 dffuo carbo aotube, 13 capllary dffuo, 14 remedato of cotamated groudwater, 15,16 membrae electroly proce, 17 ga traport porou fuel cell aode, 18,19 eatomer eparato chromatography, 20 ad membrae dtllato. 21 Moreover, the MS equato are ued to decrbe the dffuo, bede tadard ga ad lqud phae, more extreme tuato, uch a, for tace, hgh-temperature ga uclear reactor 22 ad plama mxture. 23,24 The MS equato ca be olved aalytcally (oly pecal cae ) or by umercal approache. The theoretcal decrpto of the phycal ytem reported above requre a geeral ad table umercal method able to olve the MS equato the traet regme for a geerc umber of compoet wthout troducg approxmato (apart, obvouly, from thoe of the L j dµ j dz (2) x j N - x N j (3) c j)1,j* t Ð j umercal procedure). Moreover, a remarkable trog pot would be the poblty to maage the varou choce for what cocer, for tace, the MS dffuvte (cotat or depedet o the cocetrato), the adorpto otherm (for mcroporou materal), ad other phycal parameter, allowg oe to treat all of thee ytem, decrbed wth dfferet phycal model, wth a uque computatoal tool. The am of th paper to preet uch a tool, from the defto of the umercal algorthm, to t mplemetato a computatoal code, ad fally, to t applcato o actual problem. The remader of the paper orgazed a follow; the ext ecto the MS equato are dcued for the dffuo dfferet phycal ytem, the umercal algorthm preeted, the method valdated by comparo wth prevouly publhed reult, ad ew applcato are reported. Fally, the Cocluo ecto preet ome cocluve remark. The Geeralzed Maxwell-Stefa Equato The geeralzed MS equato 8,9 (for a geeral revew, the reader referred to Krha ad Weelgh 3 ) decrbe the ma trafer proce a multcompoet ytem by relatg the molar fluxe, N, wth the chemcal potetal gradet, dµ /dz. Thee equato ca be cat dfferet form, followg the ature of the ytem uder coderato. Hereafter, three cae are reported detal. Dffuo wth a Bulk Flud Phae. For the dffuo wth a bulk flud phae, 3 oe ha It worth otcg that oly - 1 of thee equato are depedet due to the Gbb-Duhem equato By ug the relato - x dµ RT dz ) )1 the left-had de of eq 4 ca be wrtte a x j N - x N j (4) c j)1,j* t Ð j x T µ ) p (5) dµ -1 dz ) µ x j x j)1 j z - x dµ RT dz )- j)1 where the ( - 1) ( - 1) matrx of thermodyamc factor, Γ, ha bee troduced. It ha elemet where γ are the actvty coeffcet. Let u ote that for a deal ga mxture, Γ j ) δ j. I the cae of equmolar dffuo (o et trafer flux out or to the ytem), oe ha -1 Γ j x j z (6) (7) Γ j ) x µ l γ ) δ RT x j + x (8) j x j

3 Reoluto the Traet Regme for 1D Sytem J. Phy. Chem. B, Vol. 114, No. 1, that N t ) N ) 0 (9) -1 N )-N (10) mecham are domat, 28 the free molecular (bulk) dffuo, whe termolecular collo domate over molecule-wall collo, ad the Kude dffuo, whe molecule-wall collo domate over termolecular collo. The frt mecham more mportat for large pore ze ad hgh ytem preure, whle for mall pore ze ad low ytem preure, the ecod domate. However, both procee mut be ormally take to accout becaue may ytem, both occur. I th cae, the MS equato are Subttutg th expreo to eq 4, the rght-had de of t ca be put the matrx form c t -1 BN, where the vector N collect the frt - 1 fluxe ad the ( - 1) ( - 1) B matrx ha elemet ad B ) x Ð + k)1,k* x k Ð k (11) B j )-x ( 1 Ð j - 1 Ð ) * j (12) - x dµ RT dz ) x j N - x N j + N (18) c j)1,j* t Ð j Ð,K where Ð,K the Kude dffuvty of pece. 28 For a deal ga mxture (Γ ) 1), thee equato ca be cat the form N )-c t D ( x D ) B z) -1 (19) where N a vector collectg the fluxe of the compoet ad the ( ) B matrx ha elemet Wth thee defto, eq 4 become -c t Γ ( x ) BN (13) z) where the vector ( x/ z) collect the frt - 1 molar fracto gradet x / z. Fally, the fluxe are obtaed by the relato ad B ) 1 Ð,K + k)1,k* x k Ð k (20) B j )- x Ð j (21) wth N )-c t B -1 Γ ( x z) )-c t D ( x (14) z) Dffuo wth Mcropore. I the cae of dffuo wth mcropore, the dffug molecule alway feel the force feld of the pore urface; therefore, the proce amlated to a urface dffuo. I th tuato, the MS equato are wrtte a 29 A pecal type of bulk dffuo the cae where oe of the compoet tagat. A example of th proce the expermetal tudy of Carty ad Schrodt, 25 codered detal th work. For uch a proce, eq 9 replaced by the codto N ) 0 (for mplcty, the tagat compoet codered to be the lat oe). The oly modfcato wth repect to equmolar dffuo cocer the B matrx, whch aume the form ad B ) D ) B -1 Γ (15) k)1,k* Dffuo wth Macropore: The Duty Ga Model. I the dffuo of a mxture wth macropore, two traport x k Ð k (16) B j )- x Ð j (17) -F θ dµ RT dz ) j)1,j* q j N - q N j q at q at j Ð j + N q at Ð (22) where N are the urface fluxe, q the loadg of compoet ( molecule per ut cell or mol kg -1 ), q at the aturato loadg of compoet, θ ) q /q at are the fractoal occupace of the urface te (alo called coverage), ad F the dety ( umber of ut cell per m 3 or kg m -3 ). The fal relevat equato, th cae, are N )-FD ( θ z ) (23) D ) q at B -1 Γ (24) where q at a dagoal matrx wth the q at a dagoal elemet ad the matrx B defed by B ) 1 Ð + k)1,k* θ k Ð k (25)

4 154 J. Phy. Chem. B, Vol. 114, No. 1, 2010 Leoard ad Agel ad B j )- θ Ð j (26) The expreo of the Γ matrx elemet deped o the adorpto otherm of the mcroporou ytem. I the mplet cae, the Lagmur otherm wth equal aturato loadg, oe ha Γ j ) δ j + θ θ V (27) where θ V ) 1 - )1 θ the fracto of uoccuped te. Numercal Reoluto of the Maxwell-Stefa Equato It worth otcg that all cae dcued the lat ecto, the MS equato ca be reformulated a Fck-lke expreo, where the vector of the fluxe equal (apart from a cotat) to the product of a matrx D ad the vector collectg the cocetrato gradet. The cotuty equato for compoet, the cae of the dffuo wth a bulk flud phae oe dmeo ytem, c t x t )- N z (28) Ug eq 14 for the flux, eq 28 ca wrtte a x t ) z[ D ( x -1 ) z)] z[ D ' Thee equato are coupled ad form a ytem of olear ecod-order parabolc partal dfferetal equato (PDE). Whle the Fck formulato of dffuo the matrx D ofte codered cotat, thu mplfyg the reoluto of the ytem of PDE (eq 29), the MS approach, th a dratc approxmato. The depedece of the elemet of the D matrx (through the B -1 ad Γ matrce) o the ukow cocetrato complex; t ca be ealy derved a aalytc form for the cae of two depedet compoet, a reported varou paper, 29,30 ad t rema a affordable tak for three 31 ad perhap four depedet compoet. A a example, the cae of the membrae permeato of a two-compoet mxture, from eq 23-25, oe ca evetually obta the aalytc expreo of the fluxe 29 q at 1 Ð 1[{ Γ 11 + θ Ð 2 1 (Γ Ð 11 + Γ )} θ 1 z { + Γ 12 + θ Ð 2 1 (Γ Ð 12 + Γ )} θ 2 z ] N 1 )-F Ð 1 Ð 2 θ 2 + θ Ð Ð 12 (30) q at 2 Ð N 1 )-F 2[{ Γ 22 + θ Ð 1 2 (Γ Ð 22 + Γ )} θ 2 z { + Γ 21 + θ Ð 1 2 (Γ Ð 21 + Γ )} θ 1 z ] Ð 1 Ð 2 θ 2 + θ Ð Ð 12 (31) For a umber of depedet compoet beyod four, the dervato of the aalytc expreo of the D matrx elemet become rapdly too complex. Eve f the aalytc expreo of D kow, oe ha to deal wth the problem of the depedece of D o the ukow cocetrato (ad therefore o the poto), a relevat apect for the umercal reoluto of the ytem of PDE (29). Th pot ca be faced by ug very effcet computer package mplemetg umercal cheme of dfferet qualty; the fte dfferece cheme, the problem ca be olved wth the method of le 32 ug the em-mplct Ruge-Kutta method (uch a, for tace, the Crak-Ncolo 33,34 approach) or the more refed multtep method (a, for tace, the backward dfferetato formula, BDF, or Gear method 1 ). Oe ca alo cte other approache a, for tace, the orthogoal collocato o the fte elemet 35 algorthm coupled wth ODE olver (e.g., LSODA 36 olver). Thee computatoal program have bee largely ued the pat. 30,37,38 ')1 x ' z ] (29)

5 Reoluto the Traet Regme for 1D Sytem J. Phy. Chem. B, Vol. 114, No. 1, Whe mxture wth more tha two/three depedet compoet have bee tuded, mplfcato have bee ormally troduced the equato order to keep the problem tractable. The mot ued of thee approxmato, the gle fle dffuo model, cot of uppog Ð j ), thu makg the B 1- matrx dagoal, wth Ð a dagoal elemet. The effect of th approxmato dcued the ext ecto. I th paper, the umercal reoluto of the ytem of PDE reported eq 29 faced followg a dfferet approach; the am to have a method for the drect umercal reoluto of the ytem of PDE avodg the ymbolc vero of the B matrx, thu allowg for the poblty to treat a geerc umber of compoet wthout troducg approxmato the equato. The prce to pay for avodg the ymbolc vero of the B matrx a large umber of umercal vero of the ame matrx. Both umercal ad effcecy problem ca orgate from th choce, ad th apect dcued at the ed of th ecto. Obvouly, the ame problem met the Fck formulato f D ot uppoed cotat, ad the algorthm preeted hereafter ca be appled alo th cae. I th ecto, the umercal reoluto of the MS equato preeted for the cae where the ukow quatte are the molar fracto, but t exteo to other cae (for tace, the coverage) trval. I eq 29, D ha a olear depedece o all x j, ad th make the umercal reoluto of the equato problematc. Frt of all, D deped o z; ecod, ad mot mportat, t deped o the oluto of the et of dfferetal equato. Th problem faced th paper followg two umercal approache. The frt a fte dfferece approach already ued by our group for the tudy of the alt dffuo a olar pod (SP) ytem, where the Fck dffuo coeffcet D wa poto-depedet, gve that t wa a fucto of the alt cocetrato ad of the temperature. Here, the dervato more complex tha that the SP cae due to the depedece of the flux of oe chemcal pece o the cocetrato of all compoet. A the SP cae, the mple Crak-Ncolo 33,34 approach, wth a dcretzato of both the pace ad the tme varable, ha bee codered. The ecod approach baed o the accurate BDF alo kow a Gear method, 1 a mplemeted the ODEPACK 2 package. I both approache, the molar fracto x (z,t) are computed o a regular grd of pot z ({z 1, z 2,..., z z }, z +1 - z ) z) ad t ({t 1, t 2,..., t t }, t +1 - t ) t), t t beg the total mulato tme. Idcatg the legth of the mulated patal terval wth δ, we aume z 1 ) 0 ad z z ) δ. I order to olve eq 29, oe ha to defe the tal codto, that, all x (z j,t 1 ), the molar fracto of all compoet at all grd pot at the frt tme tep. Moreover x (z 1,t k ) ad x (z z,t k ) for all ad k are defed by the boudary codto. The cetral apect for the reoluto of eq 29 to have the umercal value of the D (z j,t k ) matrx elemet. To th am, the frt problem that they are actually ukow, depedg o the molar fracto at tme t k, that, the oluto of the equato that oe wat to olve. I order to crcumvet th dffculty, two tratege ca be followed: 42 The D (z j,t k ) are ubttuted by ther value at tme t k-1. Thee value are kow, gve that they deped o the x (z j,t k-1 ). The ue of the molar fracto of the prevou tme tep expected to be a good approxmato, gve that they ormally how modet varato from oe tme tep to the ext oe (actually, the tme tep mut be choe to atfy th requremet order to have good qualty reult). Oe ca coder the frt trategy a the frt tep a procedure whch D (z j,t k ) approached teratvely. Oce the x (z j,t k ) are computed the frt teratve tep, they are ued to compute D (z j,t k ), ad the oluto of eq 29 gve the ecod approxmato of the x (z j,t k ). The covergece reached whe x (z j,t k ) do ot how varato betwee two teratve tep. I the followg, we drop the varable t k for the D matrx elemet, wth the aumpto that oe of the two tratege reported above ued. I the actual applcato reported th paper, the frt trategy ha bee codered. However, both cae, the umercal expreo of the B ad Γ matrce ca be ealy obtaed for a gve grd pot z at the tme t k, ad the B -1 matrx computed o the ame grd pot by the umercal vero of the B matrx, thu allowg for the computato of the D matrx. The vero of the B matrx mut be performed z t tme, ad th ca be uppoed to be qute a demadg tak. The ue of the very effcet Lapack lbrary 43 ha however kept the CPU tme wth a acceptable rage for all of the practcal applcato decrbed th paper (ee the Geeral Coderato ubecto). Moreover, oe ca reaoably expect the umercal vero of the B matrx to be rather proe to umercal problem ome cae; aga, the practcal applcato reported th paper, the approach here decrbed ha bee foud to be table ad robut. Equato 29 preet a further dffculty due to the depedece of the D matrx elemet o z. Th problem faced here wth a clacal trategy ug a fte dfferece umercal approach (accurate to the ecod order pace) wth a approprate ceterg, obtag x (z j, t) ) 1-1 ' t {D ( z) 2 j+1/2 [x ' (z j+1, t) - x ' (z j, t)] + ')1 ' D j-1/2 [x ' (z j-1, t) - x ' (z j, t)]} (32) where ' D j+1/2 ' D j-1/2 ) D ' (z j+1 ) + D ' (z j ) 2 ) D ' (z j-1 ) + D ' (z j ) 2 (33) (34) By otcg that the x (z j,t) are a et of fucto of a gle varable (t), eq 32 a ytem of ordary dfferetal equato (ODE), whch ca be olved ug very effectve umercal method. The traformato of the orgal ytem of PDE, eq 29 a ytem of ODE through the dcretzato the z dmeo, kow a the method of le. 32 For the reoluto of the ytem of ODE (eq 32), two dfferet umercal approache have bee codered. The frt the Crak-Ncolo 33,34 method, whch ha bee mplemeted from cratch our code ad decrbed detal hereafter. The ecod approach, the Gear method, more complex ad effectve; th cae, ue ha bee made of the ODEPACK 2 collecto of Fortra olver by terfacg our code to the LSODE olver. The detal of th olver are ot reported here ad ca be foud the lterature. 2 The avalablty of two dfferet umercal approache ha bee exploted for a coherece tet of our code. For the ake of cocee, the followg, x j,k dcate x (z j,t k ), the molar fracto of compoet at the jth grd pot

6 156 J. Phy. Chem. B, Vol. 114, No. 1, 2010 Leoard ad Agel z ad at the tme t k. Ug th otato, the em-mplct Crak-Ncolo 33,34 cheme (accurate to the ecod order tme), eq 32 become x j,k It worth otcg that th computatoal cheme, beg a fte dfferece em-mplct model, table for ay choce of the tme tep t ad of the grd terval z. Movg all term depedg o t k to the left part of the equato ad thoe depedg o t k-1 to the rght part, oe ha x j,k where - x j,k-1 t - K -1 K ')1 ' (x j+1,k-1-1 ')1 For each tme tep, eq 36 a ytem of lear equato of the type where X a vector collectg all of the ukow x j,k (from 1 to z - 2, the x 1 j,k wth 2 e j e z - 1, from z - 1to2( z - 2), the x 2 j,k wth 2 e j e z - 1, etc.). Itroducg the uperdex ξ, whch deped o the two dce ad j accordg to the relato oe ca wrte -1 ) 1 2( z) 2 ')1 The C vector ha the form C ξ ) x j,k-1 ' ' ' D j+1/2 [(x j+1,k - x j,k-1 )] + D j-1/2 ' ' x j-1,k D j-1/2-1 K ')1 ' ' [D j+1/2 (x j+1,k K ')1 ' - x j,k ) + The A matrx ha a more complex tructure. I a compact form, oe ca wrte ' ' [(x j-1,k ' (x j-1,k-1 ' ' x j,k (D j+1/2 ' ' x j+1,k D j+1/2 ) x j,k-1 + ' - x j,k-1 K ) ' ) + D j-1/2 ' - x j,k ) + ' - x j,k-1 )] (35) ' + D j-1/2 ) - ' (x j-1,k-1 ' - x j,k-1 )] (36) t 2( z) 2 (37) AX ) C (38) ξ ) ( - 1) / ( z - 2) + j - 1 (39) -1 + K ')1 X ξ ) x j,k ' ' [D j+1/2 (x j+1,k-1 ' - x j,k-1 ) + ' ' D j-1/2 (x j-1,k-1 (40) ' - x j,k-1 )] (41) {1 ' ' + K(Dj+1/2 + D j-1/2 ) f ) ',j ) j' ' ' K(D j+1/2 + D j-1/2 ) f * ',j ) j' ' A ξ,ξ ) -KD j-1/2 f j ) j' - 1 ' -KD j+1/2 f j ) j' otherwe (42) I more detal, A ha a block tructure (( - 1) ( - 1) block) where each block a trdagoal ( z - 2) ( z - 2) quare matrx. Some elemet of the A ad C matrce are the modfed accordg to the choe boudary codto. Two poblte are codered here, that, cotat cocetrato or vahg fluxe. I the frt cae, oe ha to add to the C elemet wth ξ defed by j ) 2orj ) z - 1 the term Kx (0) -1 )1 D j-1/2 for j ) 2orKx (δ) -1 )1 D j+1/2 for j ) z - 1, where x (0) ad x (δ) are the mpoed cocetrato of the th compoet at z ) 0 ad δ, repectvely. I the cae of vahg fluxe, the dagoal elemet of the A matrx wth ξ defed by j ) 2or z - 1 are modfed by ubtractg the term K -1 )1 D j-1/2 for j ) 2or K -1 )1 D j+1/2 for j ) z - 1. The ytem of lear equato ca be olved very effectvely ug a mathematcal lbrary uch a, coderg, for tace, a FORTRAN mplemetato, the DGESV route of the Lapack lbrary. 43 Fally, oce the x j,k are obtaed wth oe of the two umercal approache (Crak-Ncolo or Gear), the flux of compoet, N j,k, ca be computed ug a fte dfferece approach N j,k 1 )-c t z ')1 ' ' [D j-1/2 (x j,k ' - x j-1,k ) + ' ' D j+1/2 (x j+1,k ' - x j,k )] (43) Summarzg, for each tme tep, the cheme to be followed a follow: For each grd pot z, compute the B matrx (defed by the pecfc problem codered), t vere B -1, ad the product B -1 Γ, thu obtag the D matrx. Compute all of the value x j,k by ug oe of the two opto, buld the A matrx (eq 42) ad the C vector (eq 41) ad olve the ytem of lear eq 38 (Crak-Ncolo) or call the LSODE olver (Gear). Compute the fluxe. Th method appled the ext ecto actual calculato o varou problem. Tetg Example ad Applcato Geeral Coderato. I the mulato of the dffuo proce govered by the MS equato, t ueful to troduce a ew et of ut, a decrbed for tace a paper by Loo et al. 44 I thee ut, the legth of the tuded ytem, δ, the legth ut ad the lower gle pece MS dffuvty, Ð m the dffuvty ut. Wth th ut ytem, tme expreed ut of δ 2 /Ð m. All calculato have bee performed o a Tohba Laptop wth a Itel Petum 2.0 GHz CPU rug the Lux operatg ytem. The program ha bee compled wth the g95 compler. The rug tme for a mulato calculato rage from a

7 Reoluto the Traet Regme for 1D Sytem J. Phy. Chem. B, Vol. 114, No. 1, TABLE 1: Acetoe ad Methaol Dffug through Stagat Ar a Stefa Tube: Expermetal ad Optmzed Iput Parameter ad Expermetal ad Computed Acetoe ad Methaol Fluxe a parameter expermetal b modfed b optmzed c, d optmzed c, e Ð f l x g x g Ñ Ñ a x 1 ad x 2 are the molar fracto of acetoe ad methaol at the vapor-lqud terface. Ð 13 ad Ð 23 are kept cotat at the expermetal value. The legth m, fluxe are mol m -2-1, ad dffuvte are m 2-1. b Carty ad Schrodt. 25 c Th work. d Frt et of optmzed parameter. e Secod et of optmzed parameter. f Calculated ug the method of Bae ad Reed 50 (ee Carty ad Schrodt 25 ). g Vapor-lqud equlbrum data. 47 few ecod to a few hour, followg the umber of compoet, grd pot, tme tep, ad the umercal approach ued. The reult are alway reported for the Crak-Ncolo method; the Gear oe gve value whch are practcally detcal (dtguhable o the cale ued the fgure). Dffuo the Bulk Flud Phae: Acetoe ad Methaol Dffug through Stagat Ar. I 1975, Carty ad Schrodt 25 (CS) preeted a expermetal tudy whch a ga mxture of acetoe ad methaol wa dffug a Stefa tube through tagat ar. Th tudy ha bee already ued for the valdato of method for the umercal reoluto of the MS equato. 45,46 Here, bede t ue a a frt tet for the umercal approache preeted the prevou ecto, ome ew coderato o th ytem are reported. I the followg, the ubcrpt 1, 2, ad 3 dcate acetoe, methaol, ad ar, repectvely. I the tudy of CS, 25 ar wa codered a a pure compoet ce the dffuvte of acetoe ad methaol oxyge ad troge are very mlar. I the expermetal etup, at the bottom of the tube, a flm of a lqud mxture of acetoe ad methaol cotuouly flowed, o that the ga-phae cocetrato cloe to the lqud-ga terface were codered cotat. From the vapor-lqud equlbrum data of Frehwater ad Pke, 47 the ga molar fracto were etmated to be x 1 ) , x 2 ) , ad x 3 ) At the top of the tube, a tream of dred ar wept away the vapor of acetoe ad methaol, o that x 1 ) 0, x 2 ) 0, ad x 3 ) 1.0. The temperature wa K ad the preure mm of Hg, ad alog the tube ( m log), ar wa tagat. At the teady tate, the meaured fluxe were mol m -2-1 for acetoe ad mol m -2-1 for methaol. The expermetal value of the dffuvte at cotat temperature (328 K) ad preure (1 atm) of acetoe ( m 2-1 ) ad of methaol ( m 2-1 ) ar were take from the lterature. 48,49 The dffuvty of acetoe methaol wa ot expermetally avalable ad wa etmated to be m 2-1. The value of thee parameter are reported Table 1. The cocetrato of the varou pece wa meaured at eve pot alog the tube ad are reported Fgure 1. For th ytem the MS equato ca be tegrated at the teady tate, obtag the cocetrato profle (ee eq 2 ad 3 of CS 25 ) a fucto of the (cotat) fluxe, of the dffuvte, ad of the expermetal cocetrato at a choe heght of the tube. Carty ad Schrodt oted that f the cocetrato of the vapor-lqud terface are choe a the kow value, the cocetrato profle how qute a large depedece o the ued value. By ug for all quatte the Fgure 1. Acetoe ad methaol dffug through tagat ar a Stefa tube: expermetal ad optmzed molar fracto profle; +, /, ad ymbol repreet expermetal pot for acetoe, methaol, ad ar, repectvely. Full le: aalytc oluto wth a et of modfed parameter. 25 Dahed le: computed value wth the ecod et of optmzed parameter (th work; ee Table 1). value reported above, a marked dagreemet of the computed cocetrato wth repect to the expermetal oe wa oberved. The agreemet betwee theory ad expermet wa mproved 25 by modfyg, the computatoal put, the cocetrato at the vapor-lqud terface (x 1 ) 0.319, x 2 ) 0.528, ad x 3 ) 0.153), the tube legth (l ) m), ad the fluxe (N 1 ) ad mol m -2-1 ), whle the dffuvte ad the cocetrato at the top of the tube were kept uchaged. The aalytc molar fracto profle wth thee parameter are alo reported Fgure 1. I both of the umercal approache reported th work, oly the dffuvte ad the cocetrato at the top ad the bottom of the tube mut be defed. The tme evoluto of the cocetrato profle computed, ad the teady tate reached whe fluxe are cotat alog the tube. Our umercal approache have bee teted ug the modfed parameter ued CS 25 (alo reported Table 1) wth z ) 101 ad t ) ad for a total mulato tme of 5000 (whch eure that the teady tate reached). The cocetrato profle, ad the fluxe (the output of our code) are practcally dtguhable from the aalytc oluto. I order to verfy f the parameter modfcato 25 gve the bet agreemet wth the expermetal data, a full optmzato of a elected umber of put parameter here preeted. The optmzato performed mmzg a fucto f(η 1, η 2,..., η m ), where η 1, η 2,..., η m are the m parameter to be optmzed. The fucto f ca be defed dfferet way; here, we ue the form -1 exp f(η 1, η 2,..., η m ) ) ( x (zj j ) - xj 2 (zj j ) + )1 j)1 xj (zj j ) ) -1 7 ( )1 Ñ - N j 2 Nj ) (44) where zj j are the poto ( umber exp ) 7) alog the tube at whch the th cocetrato, xj (zj j ), ha bee meaured ad x (zj j ) are the teady-tate computed cocetrato at the ame poto. Aalogouly, Ñ ad Nj are the teady-tate computed ad meaured fluxe, repectvely. The dfferet weght o the fluxe wth repect to the cocetrato (7 veru 1) ha bee troduced order to accout for the larger umber of meaured

8 158 J. Phy. Chem. B, Vol. 114, No. 1, 2010 Leoard ad Agel cocetrato wth repect to the umber of meaured fluxe. The mmzato of f(η 1, η 2,..., η m ) ha bee coducted ug the MINUIT 51,52 lbrary developed at CERN. Two et of optmzato parameter have bee codered; the frt cota the parameter modfed CS, 25 that, the acetoe ad methaol cocetrato at the vapor-lqud terface ad the tube legth l. The ecod et cota, bede the parameter of the frt et, the acetoe-methaol dffuvty, Ð 12, whoe value the orgal work 25 wa theoretcally etmated. The optmzed parameter for both et are reported Table 1. For the frt et, they cloely reemble the modfed parameter defed CS 25 (l beg lghtly lower, whle the acetoe ad methaol cocetrato are lghtly lower ad hgher, repectvely). The agreemet wth the expermetal cocetrato mproved at the prce of a lght woreg of the agreemet of the computed fluxe wth the expermetal oe. For the ecod et of parameter, oe ote a almot eglgble varato of the optmal parameter commo wth the frt et ad a marked varato of the ew parameter, Ð 12, whch le tha oe-half of the value etmated the orgal work. 25 Th modfcato brg about a mprovemet of the computed reult, both for the cocetrato ad for the fluxe ( partcular, for the methaol flux). The computed cocetrato wth the ecod et of parameter are how Fgure 1, thoe correpodg to the frt et of parameter beg very cloe to them (they are ot how for the ake of clarty). Wth thee parameter, the decrpto of the expermetal ytem ca be codered very atfactory. Dffuo Mcropore: Traet Uptake of Mxture Compoet Mcroporou Sorbet. The uptake proce of two-compoet ga mxture mcroporou orbet ha a practcal mportace partcular the feld of eparato cece ad for covero procee. A traet umercal tudy of uch a proce wa publhed 1992 by Loo et al., 44 wth a umercal method dfferet from the oe reported th paper. Th method wa developed for the tudy of a twocompoet ytem, ad t ot traghtforwardly geeralzable to a geerc umber of compoet. I order to valdate our umercal approach, the mulato preeted by Loo et al. 44 are reproduced th ubecto. It worth otcg that the orgal work, 44 ue ha bee made of the equato derved a prevou paper, 53 whch the B matrx the Maxwell-Stefa formulato ha a expreo dfferet from the oe reported here (compare eq 21 ad 22 of the paper of Krha 53 wth eq 25 ad 26 of th paper). I order to make the comparo meagful, the followg mulato are performed wth the defto of the B matrx reported by Krha. 53 The traet uptake of a bary mxture ha bee reported Fgure 2 of Loo et al. 44 for the cae of three dfferet dffuvty rato Ð 1 /Ð 2 (2, 10, ad 100). The tal codto vahg cocetrato the mcroporou orbet, whle the boudary codto are, at all tme tep, cotat fractoal 1 2 occupacy at oe urface (θ 1,k ) 0.10 ad θ 1,k ) 0.85) ad vahg fractoal occupacy dervatve, that, vahg fluxe, at the other urface. The uptake value a a fucto of tme are reported Fgure 2, ad the comparo wth Fgure 2 of Loo et al. 44 how a excellet agreemet of the two umercal method. Fgure 3 report a detaled aaly of the fractoal occupacy profle at varou tme (0.001, 0.01, 0.2, 0.4, 0.6, 0.8, 1.0, ad 1.2 reduced ut) for Ð 1 /Ð 2 ) 100, allowg for a drect comparo wth Fgure 5 of Loo et al.; 44 the agreemet, alo th cae, good. Fgure 2. Uptake of a bary mxture o a mcroporou materal a a fucto of tme wth the ame codto ued for Fgure 2 of Loo et al. 44 Number of grd pot z, z ) 21; umber of tme tep, t ) The umber cloe to the curve repreet the Ð 1 /Ð 2 rato ued the mulato (2, 10, 100). Fgure 3. Uptake of a bary mxture o a mcroporou materal. Occupacy profle at varou mulato tme. Same codto a thoe ued for Fgure 5 of Loo et al. 44 Number of grd pot z, z ) 21; umber of tme tep, t ) Full le, compoet 1; dahed le, compoet 2. Dffuo Mcropore: Traet Permeato of a Bary Mxture through Zeolte Membrae. A a fal tet example, the traet permeato of a bary mxture through zeolte membrae ha bee codered. All mulated cae have bee take from Martek et al. 54 (MGNF), wth whch the preet reult are compared. I MGNF, 54 a ormally doe the lterature, the umercal approach tart by the dervato of the aalytc expreo of the elemet of the B -1 matrx. A prevouly commeted, depte the effcecy of uch a approach for a bary mxture, t mplemetato dffcult for a three-compoet ytem, ad t almot mpoble for a geerc umber of compoet. I the frt mulato, the reult reported Fgure 1 of MGNF 54 have bee reproduced ad are how (o the ame cale) Fgure 4, where the feed ad permeate fluxe are reported for compoet 1 (fater-dffug, upper part) ad compoet 2 (lower-dffug, lower part) a a fucto of tme. The mulato parameter are here reported for the ake of completee, Ð 1 ) m 2-1,Ð 2 ) m 2-1, q at 1 ) q at 2 ) 2 mol kg -1, δ ) 100 µm, F)1800 kg m -3 1, θ 1,k ) θ 1,k ) 0.33, ad θ z,k ) θ z,k ) 0. The cro dffuo coeffcet Ð 12 ha bee computed from the gle-compoet MS dffuvte ug the logarthmc average Ð j ) (Ð ) θ /(θ +θ j ) (Ð j ) θ j/(θ +θ j ) (45) The parameter for the umercal mulato are 101 grd pot z ad tme tep.

9 Reoluto the Traet Regme for 1D Sytem J. Phy. Chem. B, Vol. 114, No. 1, Fgure 4. Feed (full le) ad permeate (dahed le) fluxe of compoet 1 (upper part) ad compoet 2 (lower part) a a fucto of tme for the traet mulato of a permeato of a bary mxture through a zeolte membrae. The mulato parameter are the ame a thoe ued for Fgure 1 of Martek et al. 54 (ee text for detal). Fgure 5. Feed (full le) ad permeate (dahed le) fluxe of compoet 1 a a fucto of tme (upper part) ad fractoal occupato of compoet 1 a a fucto of z at dfferet tme (lower part) for the traet mulato of a permeato of a bary mxture through a zeolte membrae. The mulato parameter are the ame a thoe ued for Fgure 4 of MGNF 54 (ee text for detal). The ecod mulato ha bee performed ug the ame parameter but coderg Ð 12 ). The reult are how Fgure 5, where the feed ad permeate fluxe for compoet 1 a a fucto of tme (upper part) ad the fractoal occupacy of compoet 1 a a fucto of z for dfferet tme (lower part) are reported, allowg for a meagful comparo wth Fgure 4 of MGNF. 54 The agreemet of the reult here obtaed wth thoe of MGNF 54 excellet. Sgle-Fle Dffuo Model: A Example of the Effect of Th Approxmato. I ome cae (a, for tace, wth more tha two depedet compoet), approxmato have bee propoed order to mplfy the MS equato ad to allow for a hady reoluto of them. A example ca be foud the paper of Krha ad va de Broeke 55 (KvB), where two approxmato are codered, (1) the gle fle dffuo model (SFDM) where Ð j ) ; the B -1 matrx dagoal, wth the Ð a dagoal elemet; ad (2) the cotat Fck model ; the Fck matrx, D, uppoed to be coveragedepedet. A pecal cae of th approxmato that whch D dagoal, wth Ð a dagoal elemet. I th cae, beyod the approxmato troduced the SFDM, the Γ matrx uppoed to be the detty matrx. Thee approxmato have bee ued KvB 55 for the mulato of the permeato of dfferet bary mxture acro a mcroporou membrae. Here, we cocetrate our atteto o two problem, both treated the frame of the SFDM, the permeato of a propee(1)/propae(2) bary mxture acro a lcalte membrae ad the permeato of methae(1)/butae(2) bary mxture acro a lcalte-1 membrae. The Fgure 6. Permeate fluxe (full le, compoet 2; dahed le, compoet 1) of a propee(1)/propae(2) bary mxture acro a lcalte membrae, ug the SFDM. The mulato parameter are thoe of Fgure 6 of KvB 55 (ee text for detal), 101 grd pot z, tme tep. permeate fluxe for both compoet a a fucto of tme are reported Fgure 6 ad 9 of KvB. 55 I order to tudy the effect of the approxmato troduced the SFDM, we frt tred to reproduce the reult of KvB, 55 ug the ame mulato parameter, here ummarzed for the ake of completee. For Fgure 6 of KvB, 55 the boudary 1 2 codto are θ 1,k ) ad θ 1,k ) at the feed 1 2 terface ad θ z,k ) 0 ad θ z,k ) 0 at the permeate terface, whle the MS dffuvty rato Ð 2 /Ð 1 ) For Fgure 9 of KvB, the boudary codto are θ 1,k ) ad θ 1,k ) at the feed terface ad θ z,k ) 0 ad θ z,k ) 0atthe permeate terface, whle the MS dffuvty rato Ð 2 /Ð 1 ) 10. The reult of our mulato are reported Fgure 6 ad 7.

10 160 J. Phy. Chem. B, Vol. 114, No. 1, 2010 Leoard ad Agel Fgure 7. Permeate fluxe (full le, compoet 2; dahed le, compoet 1) of a methae(1)/-butae(2) bary mxture acro a lcalte-1 membrae, ug the SFDM. The mulato parameter are thoe of Fgure 9 of KvB 55 (ee text for detal), 101 grd pot z, tme tep. Fgure 8. Permeate fluxe (full le, compoet 2; dahed le, compoet 1) of a propee(1)/propae(2) bary mxture acro a lcalte membrae, for the cae of fte Ð j (computed from eq 45). The mulato parameter are thoe of Fgure 6 of KvB 55 (ee text for detal), 101 grd pot z, tme tep. The comparo wth the reult of KvB 55 how that the preet approach gve lghtly dfferet reult for the frt mulated ytem (the fluxe of both compoet beg lghtly hgher our calculato), whle for the ecod ytem, the dfferece marked, wth eve a dfferet qualtatve behavor ( KvB, 55 the flux of compoet 1 lower tha the flux of compoet 2, whle the oppote happe our calculato). The reult here obtaed are cofrmed by the aalytc expreo for the teady-tate fluxe preeted by Krha ad Baur 26 (eq 37 ther paper), whch hold the weak cofemet regme (the MS dffuvte Ð are ot depedet o the factoal occupace), the oe here codered. Ug the reduced ut, th equato ca be wrtte the mplfed form N ) l(θ V z /θ V 1 ) 1/θ V V 1-1/θ B-1 π (46) z Fgure 9. Permeate fluxe (full le, compoet 2; dahed le, compoet 1) of a methae(1)/-butae(2) bary mxture acro a lcalte-1 membrae, for the cae of fte Ð j (computed from eq 45). The mulato parameter are thoe of Fgure 9 of KvB 56 (ee text for detal), 101 grd pot z, tme tep. where B -1 dagoal wth elemet (B -1 ) 11 ) 1.15 ad (B -1 ) 22 ) Moreover, π ) b (p (0) - p (δ)), where b, p (0), ad p (δ) are the parameter of the exteded Lagmur otherm, the partal preure of compoet at the feed terface, ad that at the permeate terface, repectvely (ee KvB 55 ). From eq 46, oe ha for the frt ytem (Fgure 6) N 1 ) ad N 2 ) , excellet agreemet wth the teady-tate value obtaed by our umercal mulato ( ad , repectvely). The applcato of eq 46 to the ecod ytem (Fgure 7) gve N 1 ) ad N 2 ) , alo th cae excellet agreemet wth thoe obtaed by our umercal mulato ( ad , repectvely). The dfferece betwee the preet reult ad thoe the lterature probably due to a mprt of ome of the put parameter the capto of Fgure 6 ad 9 of KvB. 55 A ad, the am of th ubecto to how a example of the effect of the SFDM approxmato ad ot to accurately decrbe a partcular ytem. The problem of the permeato of a methae/-butae mxture acro a lcalte membrae ha bee faced by Krha ad collaborator three ubequet paper, where a more refed model of the adorpto otherm ha bee codered, obtag good agreemet wth the expermetal reult of Bakker. 59 The mplemetato our code of adorpto otherm dfferet from the exteded Lagmur oe uder developmet. The effect of the approxmato troduced the SFDM clear by comparg Fgure 6 ad 7 wth the reult obtaed for the cae of fte Ð j (computed from eq 45) reported Fgure 8 ad 9. Oe ee that the frt mulated cae (propee/propae, Fgure 6 ad 8), the ue of the SFDM lead to modet varato the computed fluxe, lghtly creag the flux of the fat compoet (1) ad decreag that of the low compoet (2). O the cotrary, the ecod mulated cae (methae/-butae, Fgure 7 ad 9), the chage marked. Whle the flux of the lower compoet (2) oly lghtly reduced upo pag from the full treatmet to the SFDM, the flux of the fater compoet more tha doubled. The behavor foud the mulato baed o SFDM ca be ealy udertood; the MS formulato, each compoet exert a frcto force o the other compoet, 3 ad th force proportoal to the velocty dfferece of the compoet ad to (Ð j ) -1. I the SFDM, the frcto force are et to zero (Ð j ) ). Such a frcto force clearly reduce the velocty dfferece wth repect to the cae where t abet, lowg dow the fater pece ad peedg up the lowet pece. The modfcato troduced by the SFDM are obvouly larger f the two compoet have very dfferet MS dffuvte, Ð (that, average, larger velocty dfferece).

11 Reoluto the Traet Regme for 1D Sytem J. Phy. Chem. B, Vol. 114, No. 1, Fgure 10. Traet permeato through a mcroporou membrae of a fve-compoet mxture the weak cofemet cearo (ee text); feed (upper part) ad permeate (lower part) fluxe of the fve compoet a a fucto of tme. The MS dffuvte are defed by Ð +1 /Ð ) 2, wth Ð j computed from eq 45. Boudary codto: θ 1,k ) 0.15 ad θ z,k ) 0 at the feed ad permeate terface, repectvely. Fally, let u treat the cae whch the Fck matrx D codered dagoal ad cotat wth Ð a dagoal elemet (the Γ matrx the detty matrx). I the ytem here tuded, th approxmato correpod to coder a the cotat D matrx that computed at the permeate terface (where the Γ matrx the detty matrx) the SFDM. For the frt ytem here codered at the teady tate, the fluxe are N 1 ) 0.48 ad N 2 ) 0.57, to be compared wth N 1 ) 2.30 ad N 2 ) 2.74 computed wth the SFDM (ee Fgure 6). For the ecod ytem, the teady-tate fluxe are N 1 ) 2.26 ad N 2 ) 0.48 (N 1 ) 3.90 ad N 2 ) 0.82 the SFDM; ee Fgure 7). I both cae, oe oberve a marked reducto of the fluxe. Multcompoet Traet Permeato through a Mcroporou Membrae: Reult for a Fve-Compoet Mxture. I order to how the capablty of the umercal approache here reported to treat the oe-dmeoal dffuo problem wth the MS formulato for more tha two depedet compoet (more tha two compoet for urface dffuo; more tha three compoet bulk dffuo), ome reult are here preeted for the traet permeato of a mxture of fve compoet through a mcroporou membrae a model ytem. The model ytem ca be ee a a multcompoet geeralzato of the reult preeted MGNF 54 for a bary mxture. The mulato tart wth a empty mcroporou membrae, ad the boudary codto are cotat fractoal occupace at both membrae terface. A feed ga mxture wth fve compoet (wth cotat partal preure) cotact wth the mcroporou membrae, ad the fractoal occupace o the feed terface are equal for all compoet, Fgure 11. Traet permeato through a mcroporou membrae of a fve-compoet mxture the weak cofemet cearo (ee text); feed (upper part) ad permeate (lower part) fluxe of the fve compoet a a fucto of tme. The MS dffuvte are defed by Ð +1 /Ð ) 2 ad Ð j ) (SFDM). Boudary codto: θ 1,k ) 0.15 ad θ z,k ) 0 at the feed ad permeate terface, repectvely. θ 1,k ) O the permeate terface, the fractoal occupace are all vahg, θ z,k ) 0. The MS dffuvte are choe to atfy the rato Ð +1 /Ð ) 2. All mulato decrbed th ecto have bee performed wth 101 grd pot z ad tme tep. The fluxe a fucto of tme are reported Fgure 10 for the cae whch the MS dffuvte are depedet of loadg, whle the Ð j are defed by eq 45. The feed fluxe do ot how a partcular behavor; the hape roughly the ame for all compoet, the teady-tate fluxe beg related to the MS dffuvte. O the cotrary the permeate fluxe have a dfferet qualtatve profle for the varou compoet. The permeate flux profle for compoet 5 (the fater dffuo pece) ad 1 (the lower oe) cloely reemble thoe reported Fgure 4 for the two compoet of a bary mxture. The other three compoet how termedate profle, wth a gradual varato betwee the two lmtg cae. It teretg to ote that, a foud for bary mxture, 54 the flux for the lower compoet evolve a expected for a glecompoet dffuo, whle the flux of the fater compoet ha a qualtatvely dfferet hape, wth a marked overhoot of the teady-tate flux the tal part of the mulato ad the a mootoc decreag of the flux toward the teady-tate flux. I a ecod mulato, the SFDM (Ð j ) ) ha bee appled whle keepg the other phycal ad umercal parameter uchaged. From the flux profle reported Fgure 11, oe ote that the ue of the SFDM doe ot trogly modfy the qualtatve behavor of the ytem, whle from the quattatve pot of vew, the fater pece (compoet 4 ad 5) have creaed ther fluxe ad the lower pece (compoet 1 ad

12 162 J. Phy. Chem. B, Vol. 114, No. 1, 2010 Leoard ad Agel Fgure 12. Traet permeato through a mcroporou membrae of a fve-compoet mxture the trog cofemet cearo (ee text). Feed fluxe of the fve compoet a a fucto of tme for the cae whch Ð j are computed from eq 45 (upper part) ad that whch Ð j ) (lower part) are how. The MS dffuvte are defed by Ð +1 /Ð ) 2. Boudary codto: θ 1,k ) 0.15 ad θ z,k ) 0atthe feed ad permeate terface, repectvely. 2) have reduced ther fluxe. Compoet 3 ha roughly the ame flux a that Fgure 10. The value of the teady-tate fluxe the umercal mulato have bee cofrmed by the agreemet wth the reult of the aalytc oluto (eq 37 of Krha ad Baur 26 ) for the weak cofemet cearo (Ð depedet of loadg) for whch the formula exact (whle t oly approxmate for Ð j defed by eq 45). Prevou tude have how that the hypothe that the Ð are depedet of loadg ofte wrog, partcular, wth hgh fractoal occupacy. The depedece of Ð o the loadg ca be complex However, may cae, a lear depedece of Ð o the total fractoal occupacy, θ ) θ, ha bee oberved (ee Krha ad Baur 26 ) Ð (θ) ) Ð (0)(1 - θ) (47) ad th called the trog cofemet cearo. A mulato baed o th defto of the MS dffuvte (the other parameter beg thoe ued for Fgure 10) ha bee performed. The permeate fluxe are reported Fgure 12 for the cae whch Ð j are computed from eq 45 (upper part) ad for the SFDM (lower part). The comparo wth the permeate fluxe the weak cofemet cearo (Fgure 10 ad 11, lower part) how that all fluxe are lower the trog cofemet cearo (a expected from the reducto of the MS dffuvte the latter cae). A the precedg cae, the ue of the SFDM creae the fluxe of the fat-movg compoet 4 ad 5 ad reduce the fluxe of the tardy compoet 1 ad 2, whle the flux of Fgure 13. Traet permeato through a mcroporou membrae of a fve-compoet mxture the trog cofemet cearo (ee 3 text). Feed fluxe of the fve compoet a a fucto of tme for θ 1,k 3 ) 0.25 (upper part) ad θ 1,k ) 0.05 (lower part) a boudary codto o the feed membrae urface are how. The other boudary codto are θ 1,k ) θ 1,k ) θ 1,k ) θ 1,k ) 0.15 for the feed urface ad θ +z,k ) 0 for the permeate urface. The MS dffuvte are defed by Ð +1 /Ð ) 2 ad Ð j computed from eq 45. compoet 3 rema almot uchaged. It worth otcg that a aalytc oluto for the teady-tate fluxe ha bee preeted alo the cae of the trog cofemet cearo, ad th oluto exact the cae of the SFDM (eq 44 Krha ad Baur 26 ). The reult preeted the lower part of Fgure 12 at log tme agree wth the value computed wth the aalytc equato. Fally, the effect of the varato of oe of the fractoal occupace (that of compoet 3, whch ha termedate MS dffuvty) at the feed urface ha bee codered. All of the other parameter are thoe of the mulato reported Fgure Fgure 13 how the permeate fluxe for the cae of θ 1,k ) (upper part) ad θ 1,k ) 0.05 (lower part), for the weak cofemet cearo ad wth the logarthmc average defto of Ð j (eq 45). Oe clearly ee that, wth repect to the cae 3 3 wth θ 1,k ) 0.15, the creae (reducto) of θ 1,k lead to a creae (reducto) of the flux of compoet 3 at all tme. Th ha a coequece, eve f qute modet, alo o the other fluxe; the frt cae, compoet 3 ha a draggg effect, lghtly creag all fluxe, whle the ecod cae, t ha a lowg dow effect, wth a moderate reducto of all fluxe. Moreover, oe ote a tefcato or a reducto of the overhootg of the fater compoet flux wth repect to the teady-tate flux related to the creae or decreae of the fractoal occupace of compoet 3 o the feed urface. For all cae codered th ecto, the fluxe after 0.3 reduced ut of tme are reported Table 2, together wth the teady-tate value computed wth the aalytc equato. It worth otcg that mot cae, the teady tate ha ot bee reached the mulato (the weak cofemet cae wth the

13 Reoluto the Traet Regme for 1D Sytem J. Phy. Chem. B, Vol. 114, No. 1, TABLE 2: Permeate Fluxe (Reduced Ut) after a Smulato Tme of 0.3 Reduced Ut for the Permeato through a Mcroporou Membrae of a Fve-Compoet Mxture a ytem comp. 1 comp. 2 comp. 3 comp. 4 comp. 5 WC-LA (2.21) 1.57(1.50) 1.03(1.02) 0.68(0.69) 0.44(0.46) WC-LA (2.54) 1.82(1.77) 2.11(2.10) 0.83(0.84) 0.56(0.57) WC-LA (2.04) 1.43(1.35) 0.28(0.28) 0.58(0.59) 0.36(0.39) WC-SF (4.44) 2.23(2.22) 1.11(1.11) 0.55(0.55) 0.26(0.28) SC-LA (1.20) 1.06(0.81) 0.61(0.55) 0.34(0.37) 0.15(0.25) SC-SF (2.40) 1.23(1.20) 0.62(0.60) 0.30(0.30) 0.11(0.15) a Aalytc value are wth parethee. The teady-tate lmt oly approached ( partcular, for the mulato wth low fluxe). WC, weak cofemet; SC, trog cofemet; LA, logarthmc average of Ð j ; SF, gle-fle dffuo model. SFDM beg the cloet to the teady tate). Moreover, let u ote that the aalytc equato gve exact value oly for the SFDM, whle f eq 45 ued for Ð j, they are oly approxmate. The reult dcued th ecto, bede ther teret for the compreheo of multcompoet dffuo, clearly how that the algorthm decrbed th paper ca be uccefully appled for the tudy of ytem wth more tha two depedet compoet. Cocluo Th paper decrbe a ew algorthm for the oe-dmeoal umercal reoluto of the Maxwell-Stefa equato the traet regme. The key pot of th approach are: The MS equato are rewrtte the Fck-lke form N ) -c t D( x/ z), where D ) B -1 Γ ad B -1 ad Γ deped o the pecfc problem uder tudy. Both the patal ad the tme varable are dcretzed o regular grd of pot. The ymbolc vero of the B matrx ubttuted by the umercal vero of t o all grd pot ad at all tme tep. The dervatve are evaluated wth a ecod-order accurate tme Crak-Ncolo fte dfferece approach or ug the Gear method a mplemeted the ODEPACK package. For the Crak-Ncolo approach, the problem (cludg the defto of the boudary codto) reformulated a matrx form, leadg to a ytem of lear equato, whch olved ug very effcet mathematcal lbrare. The method ha bee appled to dfferet ytem, wth the am of verfyg t correcte (by the comparo wth prevouly publhed tude), ad for ew applcato. The MS equato have bee olved wth the ame computatoal code for the bulk dffuo, for the uptake proce a mcroporou materal, ad the for the permeato through mcroporou membrae, howg that the approach geeral, table, ad old. Oe of the ovatve apect of the preet formulato the poblty to treat a geerc umber of compoet, the lmt beg the computatoal hardware. A ecod relevat apect related to poble future developmet; deed, the formulato here decrbed allow for a eay modfcato of ome aumpto for what cocer the key quatte uch a, for tace, the adorpto otherm mcroporou urface dffuo, the depedece of the MS dffuvte o the fractoal occupace, or the ature of the Ð j coeffcet. Such a poblty, whch deped o the fact that the algorthm, the B -1 ad Γ matrce are explctly computed o the varou grd pot z, ca allow the ue of varou expreo for the relevat key quatte ad eve the ue of expermetal (kow o a grd of pot) data. The cojugato of thee two ovatve apect make th method teretg, bede t ue for the mulato of the dffuo proce ytem decrbed by kow model, alo for tetg ew hypothee o the key quatte or ew model for the decrpto of the dffuo procee at the molecular level techologcally relevat materal. Th paper report, a a example of the potetal of the method, the tudy of the effect of ome aumpto (weak/ trog cofemet, logarthmc average for the Ð j or SFDM) ad parameter (fractoal occupacy of a gve compoet at the feed urface) the cae of the traet permeato through a mcroporou membrae of a fve-compoet mxture. Two dfferet umercal cheme have bee mplemeted our code, the mple Crak-Ncolo method ad the more refed BDF or Gear method. Eve f the comparo of dfferet ODE olver ot the ubject of th paper, a hort commet ca be made. Both approache have how to be table ad robut, ad they gve pretty much the ame reult (mall dfferece are due to the dfferet tegrato cheme ad are expected). Keepg md that the Crak-Ncolo method ha bee mplemeted from cratch (plaubly wth oly a partal optmzato of the code) whle the Gear method ha bee mplemeted by ug the ODEPACK Fortra package (certaly wrtte wth a hgh level of optmzato), our calculato have how that the Gear method uperor ad fater, partcular, for the ytem wth a large umber of compoet. Neverthele, we pla to keep both method our code, gve that they allow for a cotecy check of the reult. Fally, t worth otcg that the algorthm here decrbed ca be obvouly ued alo for the reoluto of the dffuo problem the Fck formulato; actually, the MS equato are frt brought back to Fck formulato ad the olved wth the hypothe that the Fck dffuvte are potodepedet. Th poblty ca be of teret reearch feld ot dcued here, uch a, for tace, the crytal growth prote oluto 63 or old-old metallc dffuo, 64 where the Fck decrpto of the dffuo proce ormally ued. Ackowledgmet. Th work ha bee carred out wth the FISR project Stem tegrat d produzoe d drogeo e ua utlzzazoe ella geerazoe dtrbuta ad wth the facal upport of the Uverty of Ferrara through t local fudg. The author wh to thak Lorezo Parech for tmulatg dcuo o the umercal reoluto of PDE. Referece ad Note (1) Shampe, L. F.; Gear, C. W. SIAM ReV. 1979, 21, 1. (2) Hdmarh, A. C. Seral Fortra SolVer for ODE Ital Value Problem, Techcal Report; 2006; odepack/odepack_home.html. (3) Krha, R.; Weelgh, J. A. Chem. Eg. Sc. 1997, 52, 861. (4) Amudo, N. R.; Pa, T. W.; Paule, V. I. Flud Mech. Trap. Pheom. 2003, 49, 813. (5) Kerkhof, P. J. A. M.; Geboer, M. A. M. Chem. Eg. Sc. 2005, 60, (6) Gavala, G. R. Id. Eg. Chem. Re. 2008, 47, (7) Duca, J. B.; Toor, H. L. AICHE J. 1962, 8, 38. (8) Maxwell, J. C. Phlo. Tra. R. Soc. 1866, 157, 49. (9) Stefa, J. Stzugber. Akad. W. We 1871, 63, 63. (10) Wag, Y.; LeVa, M. D. J. Phy. Chem. B 2008, 112, (11) Mtrovc, J. It. J. Heat Ma Trafer 1997, 40, (12) Srcar, S.; Golde, T. C. Sep. Sc. Techol. 2000, 35, 667. (13) Krha, R.; va Bate, J. M. Id. Eg. Chem. Re. 2006, 45, (14) Do, H. D.; Do, D. D. Chem. Eg. Sc. 1998, 53, (15) Hug, H. W.; L, T. F.; Bau, C.; Sacher, F.; Brauch, H. J. EVro. Techol. 2005, 26, (16) L, S.; Tua, V. A.; Noble, R. D.; Falcoer, J. L. EVro. Sc. Techol. 2003, 37, 4007.