Simulation of multi-component mass diffusion pellet model using least squares spectral element method for steam methane reforming process

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1 Availabl onlin a Procdia Enginring 42 (2012 ) h Inrnaional Congrss of Chmical and Procss Enginring CHISA Augus 2012, Pragu, Czch Rpublic Simulaion of muli-componn mass diffusion pll modl using las suars spcral lmn mhod for sam mhan rforming procss K. R. Rou a, H. A. Jaobsn a a* a Dparmn of Chmical Enginring, orwgian Univrsiy of Scinc and Tchnology, TU Trondhim, orway Absrac Mass basd pll modl has bn solvd using las-suars formulaion o dscrib h voluion of spcis composiion, prssur, vlociy, oal concnraion, mass diffusion flux in porous plls for h sam mhan rforming (SMR) procss. Th diffusion-racion problms ar compuaionally innsiv, ruiring fficin numrical mhods for daling wih hm. This papr prsns formulaion and algorihm of las-suars spcral lmn mhod (LS-SEM) for solving mulicomponn mass diffusion pll modls. Th mass diffusion flux is dscribd according o h rigorous Maxwll Sfan modl. Th ffcivnss facors hav bn calculad for h SMR procss and compard wih h liraur daa. Th modl valuaions rvald ha h las-suars mhod is wll suid for solving h mulicomponn mass diffusion pll modls for h SMR procss, achiving xponnial convrgnc Publishd by Elsvir Ld. Slcion undr rsponsibiliy of h Congrss Scinific Commi (Pr Kluson) Opn accss undr CC BY-C-D licns. Kyword: SMR; Maxwll-Sfan modl; mass basd pll modl; LS-SEM 1. Inroducion Th susainabl nrgy supply sysm of h fuur faurs lcriciy and hydrogn as h dominan nrgy carrirs. Today, almos all hydrogn is producd via sam mhan rforming of naural gas. * Corrsponding auhor addrss: rou@n.nnu.no Publishd by Elsvir Ld. doi: /.prong Opn accss undr CC BY-C-D licns.

2 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) Hnc, h sam mhan rforming is on of h imporan procsss by h orwgian gas indusry for uilizaion of naural gas. Sam mhan rforming (SMR) is a hrognous caalyzd procss whr mhan and sam rac ovr a nicl-basd caalys a high mpraurs o produc synhsis gas which is basically a gas mixur of H 2, CO and CO 2. Th wo rformr racions (I), and (II) and h war-gas shif racion (III) ar h mos imporan racions whn mhan is convrd in h prsnc of sam o yild synhsis gas. Th hr main racions in a SMR ar rprsnd by following uaions [1]: CH ( g) H O( g) CO( g) 3 H ( g) H 206.2J/mol (I) CH ( g) 2 H O( g) CO ( g) 4 H ( g) H 164.7J/mol (II) CO( g) H O( g) CO ( g) H ( g) H 41.5J/mol (III) In his sudy h inic xprssions wr an from liraur. Th rforming and shif racion inics for h SMR procss wr obaind using Langmuir-Hinshlwood mhodology by Xu and Fromn [1]. Mahmaical modling of inraparicl mass- and ha ransfr in porous pll has bn sudid by many rsarchrs. Elnashai and Abashar [2] dvlopd a mahmaical modl o sudy h phnomna of diffusion and chmical racions in porous caalys plls for sam mhan rforming procss. Thy hav compard h rigorous dusy gas modl o h simplr Wil-Bosanu modl. Howvr, in hir modl hy hav an h assumpions of sady sa, ngligibl viscous flow and isohrmal condiion. W hav focusd a mass basd pll modl wih Maxwll-Sfan inics. Following sablishd pracic for hrognous caalyic racion sysms, h inrnal - and ovrall ffcivnss facors wr dfind in our sudy [3]. In rcn yars, h wor by Bochv [4] has shown h applicabiliy of h las-suars spcral mhod (LSM) o solv nginring problm uaions. Sporldr al. [5] hav solvd fixd bd racor modl using h LS-SEM. Dorao and Jaobsn [6] applid las-suars mhod o solv populaion balanc problms. This papr prsns h formulaion and h algorihm of h LS-SEM o solv h diffusion-racion pll modl. Th SMR is a vry imporan procss, nown o b srongly diffusion limid. I has bn sudid xnsivly by many rsarchrs and a variy of numrical mhods applid o dal wih h srong concnraion gradins. Orhogonal collocaion [7], in paricular has bn found o b an adua mhod ha is now rouinly usd in diffusion-racion problms. Bu in his papr w hav applid h LS-SEM mhod o solv h diffusion-racion problm in a porous pll o chc h suiabiliy of h mhod on h diffusion racion problm. Th main goal of his papr is o chc h suiabiliy of LS-SEM for mulicomponn mass diffusion pll modl. 2. Mahmaical Modl Formulaion In his sudy w hav usd a gnral modl for h pll in which h racions a plac on aciv sis wihin h porous body which rprsns an assmbly of individual grains or channls. A man por diamr is assumd and h raio bwn h porosiy and oruosiy is usd o characriz h fixd srucur of h pll. Possibl pll srucural changs ar no considrd. Tim dpndn mass basd pll modl, conaining h mulicomponn Maxwll-Sfan diffusion flux and including convcion ims hav bn givn by Rou. al. [8]. Th iniial- and boundary condiions givn by Rou. al. [8]

3 1664 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) hav bn applid in h modl uaions. A sysm of non-dimnsionalizd modl uaions is usd in h simulaions. 3. Th las-suars spcral mhod (LSM) Th basic ida in h LSM is o minimiz h ingral of h suar of h rsidual ovr h compuaional domain. Considr h gnralizd formulaion of an arbirary s of parial diffrnial uaions and boundary condiions: L f g in (1) B f f on (2) In which dnos h domain, indicas h boundaris of h domain and f is h unnown funcion. Hr, w assum ha L is a linar opraor which corrsponds o h sysm of uaions and B is h boundary condiion opraor drmining h problm domain., h norm- Sinc h rsidual form of h uaions (1) and (2) ar R uivaln funcional may b wrin as Lf g and R B f f J ( f ) d d. 2 L f g 2 B f f (3) Furhrmor, h funcion f( x) can b approximad by a runcad sris xpansion adoping nodal basis li, 0 f( x) f ( x) f h ( x) (4) whr f is h basis cofficin associad wih h basis funcion. In nodal bas, h basis funcions consis of Lagrangian polynomials hrough Gauss-Labao-Lgndr (GLL) collocaion poins. Du o nodal basis propry, h basis cofficin f is ual o h valu of h discr soluion f ( x ) a h GLL poins (nods) x, i.., f f ( x ) h x Inroducing soluion funcion xpansion and minimizing h norm-uivaln funcional, w found: h d h d0, f 2 f L g 0 f 2 f B f (5) 0 whr [ x, x ] [ y, y ] [ z, z ] in hr dimnsional spac. Afr diffrniaion, min max min max min max h uaion (5) can b wrin as; flh glhd fbh f B hd0 (6) 0 0 As our aim is o find h valu of f, h uaion (6) can b rprsnd on h form, f LhLhd f BhBhd glhd f B hd (7) 0 0 This samn can b xprssd in h innr produc form as,

4 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) f Lh, Lh f Bh, Bh g, Lh f, B h (8) 0 0 In h marix form, h uaion 8 can b wrin on h form: [ A] f [ B] f F F (9) [ A] Lh, L h, [ B] Bh, Bh, F gl h and F f, Bh. Morovr, h marix A from h uaion 9 can hn b xprssd in h poin wis form by h Gaussian uadraur rul: P P T i i i i ii i i 0 i 0 [ A] w Lh ( x ) Lh ( x ) [ L] [ ] [ L] [ L] [ L ] (10) whr h lmns in h marix L is dfind as, [ L] i ( ). L h xi Hr, i is nod ha w i and x ar h wighs and poins of uadraur [9], and is h diagonal marix conaining wighs of h i uadraur. Hnc, h marix A can b wrin in h compac marix form as: A T L L (11) Thn, h sourc vcor F from uaion 9 can b wrin in h poin wis form: P T wi ( x ) ( ) [ ] i h xi i 0 F g L L g (12) So, h vcor F can b wrin in h compac marix form as: T F L g (13) 3.2 Implmnaion of LSM o h mass basd pll modl Figur 1 illusras h algorihm for solving h dynamic pll modl by using h LS-SEM. I sars wih iniial guss f a simulaion im, 0. Thn, w hav dividd h whol compuaional domain ino svral sub-domains. Furhr, h non-linar flux xprssion dmands a suiabl linarizaion procdur. Hnc, h Picard mhod is usd o linarizaion procdur [10]. Th Picard mhod, also 1 nown as succssiv approximaion iraion, calculas h currn valu f from h prvious valu f 1. f G( f ), whr G is a problm dfiniion funcion which is h xprssion of h prvious valu of f. Thrfor, h iniial gussd valu of unnown f has bn dividd for diffrn lmns wih linarizaion. So, h iniial gussd valu for ach lmn is f. Toghr wih iniial gussd val of ach lmn f, h linarizd opraor L ( f ) for ach lmn wih hir corrsponding sourc vcor g( f ) ar calculad. Compuaion of sysm marics A ( f ) and F ( f ) SEM. W hav assmbld h marics A ( f ) and ( f ) marics for h whol compuaional domain, i.., A( f ) and F ( f ) ar carrid ou for ach lmn by h LS- F for ach lmn ino global sysm. In h nx sp, w hav 1 solvd h ovrall sysm marics o g valus of f. W did upda f 1 wih f unil spcifid convrgnc criria is rachd. W hav dfind wo convrgnc criria li: Rsidual ( ) g( ), ( ) g( ) 10 5 L f f f L f f f (14)

5 1666 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) iraion f f, f f 10 (15) Th rsidual dnos a masur for h ovrall rror obaind for h sysm of uaions discrizd by h las suars mhod. Th iraion ovrall rror dnos a masur of h diffrnc in variabl valus bwn h scond las- and h las prcding iraions. Afr w hav rcivd our spcifid convrgnc criria, w wn o nx im sp. W chcd h currn simulaion im wih h spcifid final simulaion im, final. If boh wr no sam, hn, w did upda f wih f. Fig. 1. Ingraion algorihm for solving h dynamic pll modl by using LS-SEM. 4. Rsuls and Discussion Th mass basd dynamic modl dscribs h voluion of spcis mass fracion, prssur, dnsiy, mpraur, gas vlociy, mass diffusion flux, ha flux and convcion for h SMR procss.

6 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) Th prsn papr focuss h algorihm of diffusion modl using h LS-SEM and o chc h suiabiliy of h LS-SEM for solving diffusion-racion problm. Figur 2(a) shows h dpndnc of h rror wih xpansion ordr,. I has bn shown ha h rror is rducd wih an xponnial convrgnc ra. Th norm of h rsidual, shown in h figur 2(a) dcrass unil raching a poin of limiing accuracy, clos o numrical prcision. Th convrgnc ra is srongly affcd by h capabiliy of h solvr, which allows for h aainmn of mor accura rsuls. (a) orm of h rsidual (b) Shap of h problm marix (c) Mass fracions (d) Tmpraur () Prssur (f) Vlociy

7 1668 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) (g) Concnraion (h) Flux Fig. 2. Th rsuls of mass basd Maxwll-Sfan modl Th shap of h problm marix has bn shown in h figur 2(b). Whn h uaions involv an idniy opraor, h choic of h sarch affcs apprciably h fill-in of h problm marix, h Lagrang polynomials valuad a h GLL uadraur poins yilds zros and ons. Figur 2(c) shows h sady-sa mol fracion profils of diffrn componns across h pll for h SMR procss. In h cas of h SMR simulaions, h mpraur variaion bwn h surfac and h cnr of h pll is lss han 1K which has bn shown in h figur 2(d), which ar in agrmn wih h liraur [11]. Hnc, hr is uniform mpraur across h pll. Figur 2() shows ha hr is no variaion of prssur from h surfac o h cnr of h caalys. This validas h assumpion of h mass basd modl as hr is no mass formaion nar h pll surfac. Sinc hr is no prssur gradin, hr is no viscous flow and as a consunc no convciv flux. As an illusraion h convciv flux of H 2 has bn shown in figur 2(h). Figur 2(h) shows ha clos o xrnal surfac, h diffusion fluxs clarly domina ovr h convciv fluxs, hnc nglcing h convciv flux rms in h govrning uaions is a rasonabl modl approximaion. Figur 2(g) shows ha h dnsiy incrass from h surfac o h cnr as hr is a n producion of gass in h pll. Tabl 1. Inrnal and ovrall ffcivnss facors for h SMR procss Racion I II III Th inrnal ffcivnss facors and h ovrall ffcivnss facors for h racions I, II and III hav bn givn in abl 1. for h shif racion is snsiiv o h gas composiion and i may chang h sign a h caalys surfac [3]. Th ffcivnss facor for h shif racion may nd owards infiniy [3]. For h ohr wo racions h inrnal ffcivnss facors ar in h rang of 0.01 o 0.001, which ar in agrmn wih h liraur valus [12].

8 K. R. Rou al. / Procdia Enginring 42 ( 2012 ) Conclusion In his wor, mass basd mahmaical modl has bn formulad for h SMR procss and validad wih liraur daa. Th modl rsuls gnrally suppor h convnional modl approximaions li uniform mpraur across h pll as h mpraur variaion bwn h surfac and h cnr of h pll is lss han 1K and consan prssur wihin h plls. Morovr, h magniud of h diffusion fluxs gnrally dominas ovr h convciv fluxs. Th LSM is wll suid for h soluion of h pll modl uaions, achiving xponnial convrgnc in h mhod ordr. This mhod has bn dscribd and can b applid for h soluion of h ranspor and racions problms consiuing a pll modl. Acnowldgmn Th PhD fllowship (Rou, K. R.) financd by h Rsarch Council of orway hrough h GASSMAKS program is grafully apprciad. Rfrncs [1] Xu J, Fromn GF. Mhan sam rforming, mhanaion and war-gas shif:i. Inrinsic inics. Am Ins Chm Eng J 1989;30: [2] Fromn GF, Bischoff KB. Chmical Racor Analysis and Dsign. John wily and sons, Inc, Rivr Sr, Hobon, w Jrsy. 2nd dn (1990). [3] Fromn GF, Bischoff KB. Chmical Racor Analysis and Dsign. John wily and sons, Inc, Rivr Sr, Hobon, w Jrsy. 2nd dn (1990). [4] Bochv P. (2004) Fini Elmn Mhods basd on las-suars and modifid variaional principls. Tchnical Rpor, Univrsiy of Txas a Arlingon, Dparmn of Mahmaics [5] Fdrico S, Dorao CA, Jaobsn HA. Simulaion of chmical racors using h las suars spcral lmn mhods. Chm Eng Sci 2010;65: d 1 [6] Dorao CA, Jaobsn HA. Applicaion of h las-suars mhod for solving populaion balanc problms in R. Chm Eng Sci 2010;61: [7] Finlayson BA. Th mhod of wighd rsiduals and variaional principls. Mahmaics in scinc and nginring. Vol. 87. Acadmic Prss, Inc. 111 Fifh Avnu. w Yor, (1972). [8] Rou KR, Solsvi J, aya AK, Jaobsn HA. A numrical sudy of mulicomponn mass diffusion and convcion in porous plls for h SE-SMR and dsorpion procsss. Chm Eng Sci 2011;66: [9] Karniadais G, Shrwin SJ. Spcral/hp lmn mhods for CFD. Oxford Univrsiy Prss, w Yor. (1999). [10] D Marschalc B, Grrisma M. Las suars spcral lmn mhod for non-linar hyprbolic diffrnial uaions. J Compu Appl Mah 2008;215: [11] Olivira ELG, Grand CA, Rodrigus AE. Sam Mhan Rforming in a i/ Al 2O 3 Caalys: Kinics and Diffusional Limiaions in Exrudas. Can J Chm Eng 2009;87: [12] Elnasha SSH, Adris AM, Soliman MA, AL-Ubaid AS. Digial Simulaion of Indusrial Sam Rformrs for aural Gas Using Hrognous Modls. Can J Chm Eng 1992;70: