APTEFF, 38, 1-19 (7) UC: 66.71.6:66.11 OI:1.98/APT73875M BIBLI: 145-7188 (7) 38, 75-84 Orgnal scntfc papr APPLICABILITY OF LINEARIZE USTY GAS MOEL FOR MULTICOMPONENT IFFUSION OF GAS MIXTURES IN POROUS SOLIS Jlna Markov and Radovan Omoran Th transport of gasous componnts through porous mda could b dscrbd accordng to th wll-known Fck modl and ts modfcatons. It s also known that Fck s law s not sutabl for prdctng th fluxs n multcomponnt gas mxturs, xcludng bnary mxturs. Ths modl s stll frquntly usd n chmcal ngnrng bcaus of ts smplcty. Unfortunatly, bsds th Fck s modl thr s no gnrally accptd modl for mass transport through porous mda (mmbrans, catalysts tc.). Numrous studs on transport through porous mda rval that usty Gas Modl (GM) s supror n ts ablty to prdct fluxs n multcomponnt mxturs. Its wdr applcaton s lmtd by mor complcatd calculaton procdurs comparng to Fck s modl. It should b notd that thr wr fforts to smplfy GM n ordr to obtan satsfactory accurat rsults. In ths papr lnarzd GM, as th smplst form of GM, s tstd undr condtons of zro systm prssur drop, small prssur drop, and dffrnt tmpraturs. Publshd xprmntal data ar usd n tstng th accuracy of th lnarzd procdur. It s shown that ths smplfd procdur s accurat nough compard to th standard mor complcatd calculatons. KEYWORS: usty Gas Modl, Multcomponnt dffuson, Porous solds INTROUCTION Th ndustral applcaton of porous solds s qut wdsprad. Thy ar usd as htrognous catalysts, adsorbnts, mmbrans tc. Addtonally, th porous lctrods ar usd n ful clls tchnology. Bsds xtnsv xprmntal work thr ar numrous modls concrnng porous sold proprts and multcomponnt transport (1). It should b notd that thr s no gnrally accptd modl for multcomponnt dffuson through porous solds (). Som of thm blong to th group calld contnuum modls. r Radovan Omoran, Assoc. Prof, Jlna Markovc, BSc. Assst, Faculty of Tchnology, Unvrsty of Nov Sad, Bul. Cara Lazara 1, 1 Nov Sad, Srba 75
Thy ar rlatvly asy to us and suffcntly accurat f a porous mdum dos not chang ts txtural proprts. Howvr, thy ar not wll sutd f notcabl changs n th por connctvty, por pluggng and fragmntaton occur (). Crtan progrss has bn mad n th last yars mployng dscrt modls basd on th ntwork rprsntaton of porous mdum (). Unfortunatly, th applcaton of th dscrt modls, (por ntwork modls) rqurs trmndous computaton tm f transport should b tratd ralstcally. Ths s thr maor shortcomng. Concrnng contnuum modls, th modfd Fck s law s th smplst dffuson modl and t s oftn usd for dlut or bnary systms. It s also wll-known about ts drawbacks for multcomponnt systms (3). In ordr to dscrb multcomponnt dffuson mor accuratly, modls basd on Maxwll-Stfan quatons ar prfrrd and rcommndd (3, 4, 5). Man obstacls of ths modls applcaton ar mor complcatd calculaton procdurs comparng to th Fck's approach. Thrfor, any ustfd smplfcatons of th modls or calculaton procdurs would b accptabl. Today, thr ar thr most frquntly usd modls avalabl for dscrpton of combnd transport of multcomponnt gasous mxturs through porous solds: th usty Gas Modl (GM), th Man Transport Por Modl (MTPM), and th Bnary Frcton Modl (BFM) (6). Ths modls ar basd on Maxwll-Stfan dscrpton of multcomponnt dffuson n pors and on th d'arcy quaton for prmaton. Th am of ths study was to analyz th applcaton of th smplfd procdur for transport paramtrs dtrmnaton n porous solds, basd on th lnarzaton of th usty Gas modl (7). Ths procdur was tstd usng th publshd xprmntal data (3,7). EXPERIMENTAL Flux modls through porous solds Modfd Fck's modl rprsnts th combnaton of thr transport mchansms: bulk dffuson, Knudsn dffuson and vscous flow. Fck's modl dfns componnt flux as a product of dffuson coffcnt and partal prssur (concntraton) gradnt of th partcular componnt: N ( xp ), 1,...,n [1] In small pors, th molcul-wall ntractons ar dtrmnng th procss (Knudsn dffuson), and n fr spac, molcul-molcul ntractons (bulk dffuson). In ach of ths two rgms dffuson coffcnts hav dffrnt valus. In th transton rgon t s common to us th Bosanqut formula n ordr to valuat th dffuson coffcnt, whl n bulk dffuson rgon Wlk quaton s most commonly usd (3). Whn a convctv transport contrbuts to th total transport, d'arcy quaton of vscous flow can b addd, rsultng n what s known as xtndd Fck's modl for porous mda: V 1 B xp N N N, N ( xp ) P, 1,...,n [] 76
Ths quaton s frquntly usd prmarly bcaus th componnt flux s xprssd n trms of concntraton and prssur gradnts. Mthods of solvng ths quatons wll not b dscussd hr. Th smplst cas s on dmnsonal problm whr gradnts ar rplacd by fnt dffrncs (lnar form): P B x N x P, 1,...,n [3] Contrary to th Fck's law for multcomponnt dffuson concrnng molcul to molcul ntractons, th mor corrct and thortcally basd modl s gvn by Maxwll-Stfan dffuson quatons.(3). Th dffusv flux of componnt (n ths concpt) s gvn by th xtndd Maxwll-Stfan quaton, whch ncluds both, th bulk and th Knudsn dffuson mchansms: 1 n Px 1 x N x N Accordng to th usty Gas modl (GM) of dffusv transport total flux s obtand by addng convctv (vscous) contrbutons, smlar as n quaton []. Th followng formulaton of GM s oftn usd as workng quaton n xprmntal nvstgaton and n modllng of multcomponnt gas transport procsss n gnral. P, N x B P n xn x N x 1 P K 1, K N K, 1,...,n Thr ar n ndpndnt quatons, n ndpndnt fluxs, N and, n ndpndnt gradnts x,..., x, 1 n 1 P. In ordr to solv or us GM quatons an xtnsv ffort s ndd, xcludng som spcal cass.(4,5). GM quatons fluxs ar not gvn xplctly comparng to xtndd Fck's modl, thrfor, smplfcaton of th calculaton procdur would b vry usful. Abov drvd quatons of multcomponnt transport rfr to macro and mso-porous solds. Th common approxmaton s to consdr on dmnsonal problm: P dx x B P dp n x N x N 1 dz K dz 1, N K, 1,...,n wth th followng boundary condtons : z, x () x, P() P ; z, x ( ) x, P( ) P [7] and th addtonal constrants : n dn x 1;, 1,...,n [8] 1 dz From th composton constrant follows that only n-1 molar fluxs (gradnts) ar ndpndnt and by summng th quatons [6] th followng quaton for total prssur gradnt s obtand: n N B P n x dp 1 1 [9] K 1 K dz [4] [5] [6] 77
1 Th constrant of zro componnt flux gradnts s vald whn thr ar no chmcal ractons. Th quatons [6] wth th constrants [8] rprsnt a systm of n+1 quatons wth n unknowns (n compostons, prssur, n fluxs). Solvng ths quatons n ordr to obtan concntraton and prssur profls rqurs ntal stmats of all th fluxs N ( =1,..n). Thus, th procdur ncluds fndng th approprat componnt fluxs whch satsfy th rsultng systm of ordnary dffrntal quatons. It s carrd out numrcally n th gnral cas, though analytcal solutons hav bn gvn for som vry spcal cass. (3, 4, 5). Summng ovr th n spcs wth th constrant of absnc of total prssur gradnt GM quatons rsults as n 1 / N M, whch rprsnts th gnralzd Graham's law. On th contrary, modfd Fck's low would dfn th qumolar dffuson as N. n 1 A convnnt smplfd procdur for th soluton of GM can b dvlopd as n rfrnc (7). Th fluxs can b calculatd xplctly wth an assumpton of lnar profls of composton and total prssur along th dffusonal path. Wth ths assumptons drvng forc s as follows: whr : x B P P d x z 1 [1] K Pz x x P P x x x,x, 1,...,n, P P P, P, z z z [11] Ths lnar form of GM quaton [17] ylds th valus of th n fluxs xplctly. 78 RESULTS AN ISCUSSION Tstng of lnar GM (lnar, nonlnar, xact) Gvn lnar rprsntaton, [11], could b consdrd as th smplst form of GM (4, 5, 7). Lnar GM s tstd aganst publshd xprmntal data for bnary and trnary gas mxturs ncludng dal and non-dal bhavour (7). Authors gnrally usd th GM n th form of th systm of frst ordr lnar quaton OE, [5]. Ths systm s solvd numrcally as a boundary valu problm (nonlnar problm n th followng txt). Intal stmats of componnt fluxs wr obtand by usng th lnar GM. Analytcal soluton n som cass could b found ( xact soluton n th followng txt) whn GM s rprsntd wth a lnar OE (3). In ordr to tst all of ths forms (lnar, non-lnar, xact ) of GM numrcal smulaton was prformd (3,7). Th usd gas mxtur consstd of Ar, H, N and H n ordr to avod adsorpton ffcts on pors surfac and possbl ffct of surfac dffuson (whch s not ncludd n GM). Bnary Fck's dffuson coffcnts ar calculatd by Fullr-Shttlr-Gddngs corrlaton (9). Th usual xprmntal condtons ar assumd P 11.35 kpa, T=98K. Charactrstcs of porous sold ar gvn lswhr (7).
Numrcal smulaton s prformd for dffrnt trnary combnaton of th abov mntond gass. Typcal numrcal rsults obtand for nt flux through porous solds ar gvn n th Tabl 1. Bnary mxtur on on sd (Ar + N ) wth gvn mol fracton of Ar ( x Ar ) and pur componnt (H ) on th othr sd of th porous sold ( x H 1). Th rsults ar for zro prssur drop across th porous sold ( P ). Tabl 1. Nt fluxs ( N N ) for dffrnt forms of GM modls Mol fracton- x Ar Nt fluxs - mol N 1 m s Lnar GM Non-lnar GM Exact GM.9 -.3 -.4 -.3.8 -. -. -..7-1.6-1.6-1.8.6-1. -1. -.5 -.8 -.8 -.4 -.4 -.4 -.3.1.1 -..6.6 -.1 1.1 1.1 - Rsults from all othr combnatons of gass and compostons n trnary mxturs ar not gvn bcaus th sam conclusons could b mad; also nglgbl dffrnc btwn rsults of lnar and non-lnar GM was notcd. Although th analytcal soluton (f xsts) s mor prfrrd than th numrcal soluton, n ths cas, to solv th govrnng systm of lnar frst ordr OE analytcally, som of numrcal mthods for matrx algbra must b mployd. Ths numrcal procdurs oftn fal to convrg whch was th man rason why xact soluton could not b found (3). Thrfor, focus wll b only on lnar and non-lnar solutons of GM. In Fg.1 typcal composton profls across th porous sold for all th analyzd modls ar gvn, Ar-H bnary mxtur on on sd, and th pur componnt H on th 3 othr sd of th porous sold, for lnar ( N 9.5 1,N.14, N. 4 ), 3 Ar H non-lnar ( N Ar 9.6 1,NH.14, NH. 4 ) and xact form of GM ( N Ar.11,N H.19,NH. 5 ). Although, only th lnar GM uss th assumpton of lnar composton profls, lnar composton profls ar also always obtand for both, non-lnar and xact (whn soluton s found) GM. H 79
Fg 1. Composton profls n trnary mxtur (Ar-H -H) Influnc of prssur drop Th nflunc of prssur drop, P, has bn studd also. Th analyss s basd on th rfrnc and data for trnary mxtur from (7, Tabl 1, Tabl ). In rproducng rsults from th ltratur dnotd xact solutons concd wth rsults obtand by th non-lnar procdur gvn hr. Although n th rfrnc (7) xprmntal and smulatd data hav bn analyzd only for zro prssur drop, analyss has bn xtndd by ncludng th total prssur dffrnc. As th xprmntal rsults wr not avalabl n that cas, ths analyss has bn rstrctd to th comparson th lnar and non-lnar modls. Prssur dffrnc has bn appld to th mntond data and t was obsrvd as th dvaton from man prssur. Th man prssur was kpt constant and th rato of prssur dffrnc and man prssur was calculatd as P c P, whr th constant c could b arbtrarly chosn. Th comparson btwn lnar and nonlnar modls was prformd by th calculatng th accompand componnt fluxs. Th rato of componnt ln nln N N fluxs, RN was chosn as an ndcaton of modl dffrncs. nln N In Fg.. valus of th flux ratos for dffrnt c valus and th cas of trnary mxtur (H, N, Ar) ar prsntd. At P th fluxs wr N H ~-.5, N N ~.1, N Ar ~.7. Whn P (c = ) th dffrnc btwn modls s nglgbl. Ths s n accordanc wth prvous dscusson. 8
Fg. Th comparson of lnar and nonlnar GM for dffrnt prssur dffrnc (B =r /8, /=1, =.61-6 Pa s, =9.6mm) Th dffrnc btwn modls wll ncras wth ncras of P. Postv P wll ad th flux of H and countr th fluxs of N, Ar and ngatv P wll hav th oppost nflunc. Whn th flux of componnt s hgh and addtonally contrbutd by prssur dffrnc th rlatv dffrnc btwn modls wll b lowr (H - ngatv flux, postv P). In th cas of lowr fluxs spcally whn thy ar supprssd by prssur dffrnc (N, Ar - postv flux, postv P) th rlatv dffrnc btwn modls s mor notcabl. Addtonally, f th P s hgh nough th sgn of th flux could chang. In ths cas, th fluxs of N and Ar changd from postv to ngatv. Ths could b notd as th vrtcal asymptot (brak for N, Ar n Fg.) for c valus btwn.5 and.1 bcaus n ths rgm thr s a c valu whr th fluxs calculatd by nonlnar modl ar qual to zro. In th cas of ngatv P and comparng to postv P, th modl dffrncs n fluxs for H (H - ngatv flux, ngatv P) ar gratr and for N, Ar (N,Ar postv flux, ngatv P) ar sgnfcantly lowr. Nonlnar procdur could fal for larg prssur dffrnc (gratr absolut c valus).. th systm could bcom stff and thrfor much dffcult to solv. In th rang of c valus, prsntd n Fg.., usd numrcal procdur convrgd, and outsd of ths rgon t fald. Influnc of tmpratur It s also mportant to nvstgat th tmpratur nflunc, spcally for th lvatd tmpraturs (porous catalysts, ful cll (SOFC) lctrods tc.). Th usd xprmnts (7) ar prformd on 7 o C. Fg. 3 rprsnts th tmpratur nflunc and th smulaton for th data from th sam xprmnt. It could b sn that th dffrnc n prdctng th componnt flows btwn two modls s lowr at hghr tmpraturs. GM s also usd n analyss of th transport through th porous lctrods n a ful cll systm, SOFC (1). Th opratng tmpraturs of SOFC ar vry hgh (~57-77 o C). 81
Although th authors appld nonlnar two-dmnsonal GM thy assumd th lnar concntraton and prssur gradnts. Th authors also ustfd ths assumpton ovr a larg rang of lctrod structurs and ful cll opratng condtons. Ths also mpls th us of lnar nstad of nonlnar modls. Fg 3. Th comparson of lnar and nonlnar GM for dffrnt tmpraturs CONCLUSION In gnral, th dffrnc btwn th compard modls can b consdrd accptabl for zro or smallr prssur dffrnc. On th othr hand whn th sgnfcant prssur dffrnc s appld, lnar modl s qut accptabl for smallr man por radus. Consdrng th tmpratur, th analyss ndcats that th lnar modl s mor usful whn th tmpraturs ar hghr. Thrfor, t could b concludd that for sothrmal multcomponnt dffuson th lnar GM modl wth ts advantag of smplcty can b usd wthout sgnfcant loss of accuracy for zro P (or small dvaton from zro), smallr man por radus and hghr tmpraturs. ACKNKOWLEGMENT Ths papr s part of th work don by th authors on th proct Synthss of nanopowdrs and procssng of cramcs and nanocomposts for applcaton n novl tchnologs (1459) whch s fnancally supportd by Srban Mnstry of Scnc. 8 REFERENCES 1. Kakaç S, A. Pramuanaronk and Xang Yang Zhou: A rvw of numrcal modlng of sold oxd ful clls, Intrnatonal Journal of Hydrogn Enrgy, n prss.. apk P. and A Sdl-Morgnstrn.: Multcomponnt mass transport n porous solds and stmaton of transport paramtrs, Appld Catalyss A: Gnral, 11, (1) 7-37.
3. Taylor R. and R. Krshna : Multcomponnt Mass Transfr John Wly and Sons, Nw York, (1993), p.p. 5-3, 96-99 14-19. 4. Mason E.A. and A.P. Malnauskas: Gas Transport n Porous Mda: Th usty-gas Modl (Chmcal Engnrng Monographs), Elsvr Scnc Ltd, Nw York (1983), p.p. 11-19, 3-4. 5. Jackson R.: Transport n porous catalysts, (Chmcal Engnrng Monographs), Elsvr Scnc Ltd, Nw York, (1977) p.p. 18-5, 63. 6. Krshna R. and J. A.Wsslngh : Th Maxwll-Stfan approach to mass transfr, Chmcal Engnrng Scnc, 5, 6 (1997) 861-911. 7. Krshna R.: A smplfd procdur for th soluton of th dusty gas modl quatons for stady-stat transport n non-ractng systms, Th Chmcal Engnrng Journal, 35, (1987) 75-81. 8. Schndr P.: Multcomponnt sothrmal dffuson and forcd flow of gass n capllars, Chmcal Engnrng Scnc, 33, 1 (1978) 1311-1319. 9. Novák M, K Ehrhardt., K Klusák. and P. Schndr : ynamcs of non-sobarc dffuson n porous catalysts, Chmcal Engnrng Scnc, 43, (1988) 185-193. 1. Krkhof P. J. A. M. : A modfd Maxwll-Stfan modl for transport through nrt mmbrans: Th bnary frcton modl, Th Chmcal Engnrng Journal and th Bochmcal Engnrng Journal, 64, 3 (1996) 319-343. 11. Prry, R.H.,.W. Grn: Prry's Chmcal Engnrs' Handbook (7th Edton), McGraw-Hll, (1997) p. -37. 1. Huayang Zhu and R.J. K: A gnral mathmatcal modl for analyzng th prformanc of ful-cll mmbran-lctrod assmbls, Journal of Powr Sourcs, 117, 1- (3) 61-74., -., -.., - (-,.). - (). -.., -,,,. 83
. - -. Rcvd 3 Aprl 7 Accptd August 7 84