A MODIFIED ORTHOGONAL COLLOCATION METHOD FOR REACTION DIFFUSION PROBLEMS

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1 Brazilian Journal of Chmical Enginring ISSN -663 Printd in Brazil Vol. 3, No., pp , Octobr - Dcmbr,.doi.org/.59/-663.3s69 A MODIFIED ORTHOGONAL COLLOCATION METHOD FOR REACTION DIFFUSION PROBLEMS M. Soliman,, Y. Al-Zghayr A. Ajbar 3* Dpartmnt of Chmical Enginring, Collg of Enginring, King Saudi Univrsity, P.O. Box 8, Riyadh,, Saudi Arabia. yszs@ksu.du.sa Dpartmnt of Chmical Enginring, Th British Univrsity in Egypt, El-Shorouk City, Egypt. moustafa.aly@bu.du.org 3 Dpartmnt of Chmical Enginring, King Saud Univrsity, P.O. Box 8 Riyadh, Saudi Arabia. Phon: , Fax: aajbar@ksu.du.sa (Submittd: July 5, 3 ; Rvisd: January 7, ; Accptd: Fbruary, Abstract - A low-ordr mthod is oftn usful in rvaling th main faturs such as concntration tmpratur profils th ffctivnss factor for porous catalyst particls. Two modifications ar introducd in this papr to mak th mthod mor fficint. Th first modification is to add an xtra point at th cntr of th particl. It is shown that such xtra point introducs a singl variabl non-linar quation to b solvd aftr obtaining th mthod solution. In th scond modification, th polynomial solution obtaind from th application of th orthogonal mthod is transformd to a rational function form. Ths two modifications ar applid to spcific xampls it is shown that thy can improv th prformanc of mthods in gnral th on-point mthod in particular. Kywords: Orthogonal ; Jacobi polynomials; Intrpolation; Raction diffusion; Catalyst particl. INTRODUCTION In th mthod of wightd rsiduals, w sk a solution for a diffrntial quation in trms of a polynomial with unknown cofficints to b dtrmind such that crtain critria ar satisfid. Ths critria ar usually chosn such that th intgrals of th wightd rsidual, which is obtaind by substituting th assumd solution in th diffrntial quation writtn with on of its sids as zro, ar zros. Dpnding on th wight, w obtain mthods lik th Galrkin mthod, last squar mthods, th mthod of momnts, mthods. In mthods, w would lik to hav th rsidual to b zros at particular points calld points. If th points ar th zros of orthogonal polynomials, th mthod is calld orthogonal. In all mthods of wightd rsiduals, a nonlinar ordinary diffrntial quation is approximatd by a st of non-linar algbraic quations. Th orthogonal mthod was dvlopd by Villadsn Stwart (967. It was thn subsquntly studid applid to many chmical nginring problms by svral authors (Finalyson, 97; Villadsn Michlsn, 978; Finalyson, 98; Torrs t al., ; Biscaia Junior t al.,. Rcntly, rational mthods wr also dvlopd (Brrut t al., 5. Thy ar basd on approximating th solution for a diffrntial quation by a rational function, which is usually mor accurat than a straightforward polynomial. Rathr than starting with a rational function whos paramtrs ar to b *To whom corrspondnc should b adssd

2 968 M. Soliman, Y. Al-Zghayr A. Ajbar dtrmind, w us a diffrnt approach in this papr. Having obtaind a polynomial solution, w convrt this solution into a rational function. Solution of catalytic ractor modls rquirs th solution of a st of diffrntial quations rprsnting th phnomna of diffusion with chmical raction insid th pors of th catalyst particls at ach location along th ractor. It is clar that such calculations would rquir xcssiv computation tim. It is our objctiv in this papr to mak th mthod of orthogonal mor fficint in ordr to rduc th computational load. In th nxt sctions w introduc Jacobi polynomials, thn w introduc th mthod of orthogonal. This is followd by prsnting th modifications. Thn w driv formula for th ffctivnss factor calculation. Finally w apply th mthod to som problms of diffusion raction in catalyst particls. JACOBI POLYNOMIALS Usually orthogonal polynomials ar dfind on th closd intrval [-, ]. Howvr by simpl linar transformation, w can chang th dfinition to any othr closd intrval. For th purpos of this study w would lik to dfin th polynomials in th intrval [, ]. In this cas thy ar calld "shiftd" polynomials. Howvr, w will op th word "shiftd" for brvity. (, Pn α β Th Jacobi polynomials of dgr n ar dfind such that thy satisfy th orthogonality conditions (Villadsn Stwart, 967 ( αβ, ( αβ, wxp ( n ( xp m ( x ( n m ( ( αβ, ( αβ, w( x Pn ( x Pm ( x Cn > ( n m ( whr w(x is th wighting function for th orthogonality conditions Cn is a constant. For Jacobi polynomials, α β wx ( ( x x, α, β > (3 Thus α, β ar th indics of th wight function w(x. W will op th suprscripts α, β for brvity. Jacobi polynomials satisfy th diffrntial quation: d Pn( x dpn( x x( x + [( +β ( α+β + x] ( nn ( +α+β+ P( x n For th cas of α which is of intrst in this work, this quation can b writtn as, d [( x Pn( x] d[( x Pn( x] x + ( β+ [ nn ( +β+ ( +β] P( x n (5 Th first mmbr of th Jacobi polynomials, P, is of cours a constant; thus from ( for α, w hav: x β ( x Pn ( x ( n> (6 MATHEMATICAL FORMULATION AND SOLUTION BY THE STANDARD COLLOCATION METHOD Mathmatical Formulation Considr a raction with a dimnsionlss rat R(u, whr u is th dimnsionlss concntration. Th raction taks plac isothrmally insid a catalyst particl with no xtrnal rsistanc to mass transfr. W may writ th dscribing quation as: r s d s du [ r ] R( u (7 with th boundary conditions: u ( (8 du r (9 Hr u is th dimnsionlss concntration of th ky componnt in th raction, r is th dimnsionlss spac variabl in th catalyst, is th Thil modulus s is th shap factor of th catalyst (s for an infinit slab, s for an infinit cylindr s for a sphr. Th dimnsionlss raction Brazilian Journal of Chmical Enginring

3 A Modifid Orthogonal Collocation Mthod for Raction Diffusion Problms 969 rat R(u is normalizd with rspct to th raction rat at th xtrnal surfac of th catalyst, hnc, at th xtrnal surfac r, u R (. Th ffctivnss factor η is givn by: ( s + du η r s ( + rru ( ( Sinc th boundary condition (9 implis that th solution will b symmtric with rspct to (r, w introduc th transformation: x r ( so that th problm is changd to: d u du x + ( s+ R( u ( u ( (3 Th ffctivnss factor is givn by: s η ( s + rru ( s ( s ( s + x Ru ( ( Excpt for a fw cass, such as isothrmal firstordr zro-ordr ractions, th analytical solution of th boundary valu problm, Eqs. (7-(9 or Eqs. (, (3 is, in gnral, not fasibl, th problm can only b solvd numrically. In th cas whn th ffctivnss factor is calculatd rpatdly, such as in th simulation of packd bd catalytic ractors, w nd a fast fficint numrical mthod. A simpl, convnint asy approach to solv such problms is through th us of th orthogonal mthod. It is th purpos of this papr to introduc two modifications to th mthod which may yild somwhat highr accuracy for th concntration profil th ffctivnss factor than th mthod. Collocation Solution Rprsntation In th orthogonal mthod, w rprsnt th solution as a polynomial with unknown cofficints to b dtrmind such that th diffrntial quation is satisfid at crtain points. A bttr way of writing a polynomial approximation of a function in trms of its valus (ordinats at crtain points is through th Lagrang intrpolation formula. It has many advantags compard to th straightforward polynomial. First th solution is obtaind dirctly at points. Scondly, whn w hav a nonlinar diffrntial quation, w could hav a vry good initial guss for th ordinats ui's which li btwn, whras w do not know th rang of th paramtrs in th straightforward polynomial. Thirdly most important, w can asily driv formula for th drivativs using th Lagrang intrpolation formula. Fourthly, th sam Lagrang formula can thn b usd to obtain th valu of u at any point x [, ]. Th Lagrang intrpolation formula is givn by: N + i i i i ux ( ux ( l( xux ( (5 whr N + x x j li ( x Π ( j, j i x x i j (6 ar th Lagrang polynomials. From (5, th first scond ordr drivativs of any function u(x xprssd in trms of Lagrang polynomials at a point x xi can b writtn as: du d u whr N + au (7 x xi ij j j N + bu (8 x xi ij j j a l ( ( x (9 ij j i b l ( ( x ( ij j i An fficint mthod for calculating th lmnts a ij b ij is givn in th book of Villadsn Michlsn (978. W can also prsnt quaatur Radau formula (Villadsn Stwart (967 for th ffctivnss factor η in th form: Brazilian Journal of Chmical Enginring Vol. 3, No., pp , Octobr - Dcmbr,

4 97 M. Soliman, Y. Al-Zghayr A. Ajbar s i N+ ( s + η x Rux ( ( wru i ( i ( i Th wights of th quaatur ar obtaind by th procdur dscribd by Villadsn Michlsn (978. This formula uss on xtra point in x, th zros of th Lagrang polynomial ar calculatd using (α+ instad of only α (Villadsn Michlsn, 978. Th points ar th zros of th Jacobi polynomials suitabl for th gomtry of th catalyst particl. In our cas, w us th Jacobi polynomials indics α, β (s-/ Application of On-Point Stard Collocation to a First-Ordr Raction For a first-ordr polynomial with th function valu givn as u( at x u (x at x x, th Lagrang intrpolation formula is givn by: ( x x ( x ux ( u( + ux ( ( x ( x ( This formula satisfis th valus of u at x, at x x. Now if w choos x to b th zro of a Jacobi orthogonal polynomial P (x such that α, β (s-/, w obtain: s + x (3 s + 5 For a first-ordr raction, ( bcoms: d u du x + ( s+ x x ( s+ ( s+ 5 ( ( ux ux ( u(x bcoms: ( ( x ( [ s+ + / ( s+ ( s+ 5] ux ( (5 On can show that this solution is accurat up to trms containing at x x up to lswhr. In addition, th ffctivnss factor, givn by th following (6, is accurat up to. ( s s ( s + η ( s + r u x u ( s ( s+ ( x x ( x [ ( ( ] x u + u x ( x ( x (6 ( s+ ( s+ 5 [ u( + u( x ] ( s+ 3 ( s+ 3 ( + ( s+ 3( s+ 5 ( + ( s+ ( s+ 5 MODIFIED COLLOCATION METHOD Two modifications to th mthod ar now suggstd to improv its prformanc. First Modification On can show that, if w collocat th quations at an xtra point in addition to thos obtaind as th zros of th propr orthogonal polynomial, w will hav th ordinat of this xtra point apparing only in th quation of this point. This ordinat will not appar or affct th othr quations. Lt us add an xtra x point, prfrably at x, dfin u such that N+ N N+ + x xj Π i i i j, j i x i i x j i u ( x ( u( x l ( x u( x (7 W notic that th trm multiplying ux ( is givn by: ( x N x x j ux ( Π( ( x x x j j (8 which, according to (5, whn it is subjctd to th diffusion diffrntial oprator ( ( will giv a trm that contains a Jacobi polynomial whos valu is zro at th points. Thus, th quations thir solution do not chang with th addition of th xtra point. This xtra point will also not affct th ordinats at th points. Now th at this xtra point will giv a non-linar quation in a singl un- Brazilian Journal of Chmical Enginring

5 A Modifid Orthogonal Collocation Mthod for Raction Diffusion Problms 97 known variabl ux ( which can b solvd aftr solving th quations. This xtra point will also not affct th ffctivnss factor. Application of On Point Collocation First Modification (mod to a First-Ordr Raction Lt us writ th Lagrang intrpolation formula in trms of ordinats at on intrior point x bsids at th boundary x. W hav: ( x ux ( u( + [ ux ( u(] ( x (9 Lt us dfin anothr solution u such that its Lagrang intrpolation formula in trms of ordinats at an intrior point x, bsids at th boundary x any point x, is: ( x x ( x x u( x u( ( x ( x ( x ( x x + ux ( ( x ( x x ( x ( x x + u( x ( x ( x x (3 Now if x x, w not that u ( x u ( x d u du both + (s+ ( s at x x th intgral x ux ( do not contain a trm of u (x. This mans that, whatvr th valu of u (x, u(x th ffctivnss factor η will not chang. This is th major strngth of th mthod as applid to th catalyst particl problm. Th choic of x, thus u (x, will only affct th profil of u (x. Now for an arbitrary x, w could writ from (: d u du x + ( s+ x x Thus, aftr calculating u(x from (5, w calculat u (x from (3 to obtain: ( s+ 3 ( s+ 5 ( ux ( ( x u( x ( s ( x u ( x u( ( x ( s + ( + ( s+ 5( s+ ( x( x x + ( x 8( s+ ( s+ 3[ + ][ + ] ( s+ 5( s+ ( s+ 3 (3 (33 Now x can b chosn to improv th accuracy at a crtain point. x can b chosn as zro so that w obtain a highr accuracy for th valu of (u at th cntr of th catalyst particl. In Soliman (988 th choic x x was mad. In summary, th modifid on point rquirs th solution of two algbraic quations squntially it maks th whol profil xact for trms up to, whras in th original on point th solution is xact for trms up to only at th point up to lswhr. Scond Modification Lt us form a rational function ur( x from th polynomial solution ux, ( u ( x such that: ux ( + CPN ( x ur( x + CP ( x whr N j N (3 P ( x Π( x x (35 N i ( s+ 3 ( s+ 5 [ u( x] [ u( x] ( x u ( x (3 C is a constant to b stimatd such that: u( + CPN ( ur( u( + CP ( N (36 Brazilian Journal of Chmical Enginring Vol. 3, No., pp , Octobr - Dcmbr,

6 97 M. Soliman, Y. Al-Zghayr A. Ajbar Thus u(( u( C ( u(/ PN ( u ( (37 This choic maks u r (x qual to u (x at x x x x. Application of On Point Collocation Scond Modification (mod to a First-Ordr Raction Now w apply th rational function approximation to th diffusion raction problm with first ordr raction rat, whr: Ru ( u (38 W not that th has th sam valus for (u at th points as that of mod. thy diffr at u(. Th mod. mod. hav th sam valus of (u at th points at x. EFFECTIVENESS FACTOR CALCULATIONS Thr ar many ways to driv xprssions for th ffctivnss factor. ( is usful for th cass of low. If w multiply (7 by s du r intgrat both sids, w obtain: u ( x + ( s+ 5( s+ ( s+ ( s+ 3( s+ 5 ( s+ u( [( s+ 5( s+ + ] ( s+ 3 + ux ( + Cx ( x ur( x + Cx ( x Lt us choos C such that: u( Cx ur( u( Cx Thus, u(( u( C ( u(/ x ( u ( ( s + 5 (( s+ 3( s+ 5 + (39 ( ( ( (3 ( s + s du η r R( u (5 For a slab, s; η u( Rudu ( u( (6 This formula is usful for larg valu of. Going back to (7 multiplying both sids by th trm s du s s ( r + r u (7 thn carry out an intgration of both sids with rspct to (r, w rach, aftr a lngthy drivation, th following xprssion for th ffctivnss factor: η sη ( + + I I s + ( s + whr (8 [( s + ( s+ 3( s+ 5 (( s+ ( s+ 5 x] u ( r x ( [(( s+ ( s+ 5 + (( s+ 3 + ( x] W not that this rational function is xact for trms up to. In addition, as, u r (x for x [,. s du I s r u (9 s du s s I R( u( r + r u (5 Brazilian Journal of Chmical Enginring

7 A Modifid Orthogonal Collocation Mthod for Raction Diffusion Problms 973 Solving th scond ordr (7 in η, w obtain: occurs at x. Th mod. on point profil is vry clos to th analytical solution. + η + ( s ( s ( I I s (5.8 Analytical Solution On point On point modifid ( On point modifid ( Two points This simplifis for th cas of a slab to: I η (5 u. for a sphr to: 3 ( u ( I η + (53 Not that I quals zro for a slab can b intgratd analytically for a sphr. For a first-ordr raction, I I can b intgratd analytically to giv known xprssions for th ffctivnss factor. Exampl NUMERICAL RESULTS Th first xampl dals with a first-ordr raction for which th application of th mthod lads to a systm of linar quations. For a first-ordr raction occurring in a slab with, w applid diffrnt mthods discussd in this papr to plot th dimnsionlss concntration (u against dimnsionlss distanc (x. Th rsults ar shown in Figur. Th on point givs a ngativ concntration at x, whras th two point profil oscillats. For th modifid mthods, a vry small ngativ valu r Figur : Concntration profils in a slab using diffrnt mthods for a first-ordr raction,. Th ffctivnss factor is compard using diffrnt mthods for diffrnt is shown in Tabl. For on point, th ffctivnss factor is accurat using ( for small valus of, whras its accuracy improvs as incrass using (6. Th application of ( to th profil obtaind by th application of mod. to th on point givs ovrall good accuracy, but will not b abl to prdict th asymptotic valu of th ffctivnss factor as incrass. Th application of ( to th mod. mthod will giv th sam rsults as th on point bcaus th intgral dpnds on th valu of u at th point, which is th sam in th two cass. Tabl : Effctivnss factor for a first-ordr raction in a slab using diffrnt mthods ( (6 mod. ( ( mod. (& (6 ( (6 Exact Solution Brazilian Journal of Chmical Enginring Vol. 3, No., pp , Octobr - Dcmbr,

8 97 M. Soliman, Y. Al-Zghayr A. Ajbar Th application of (6 givs bttr accuracy will convrg to th asymptotic valu for larg. Th rsults will b th sam for mod. mod. bcaus th intgral in (6 dpnds on u(, which is th sam in both mthods. Th rsults for th two point using ( ar xcllnt up to, but will start to dtriorat for highr valus. Morovr, it can b notd from Tabl that th rsults for th two-point mthod using (6 ar bttr than thos using ( for >, but ar slightly wors than thos of mod. on-point for >. In summary, th rsults show th supriority of mod. on-point for th calculation of th concntration profil th ffctivnss factor using (6. Exampl W now considr a th ordr raction taking plac in diffrnt shaps; slab, infinit cylindr a sphr. Th rsults for th cas of a slab ar givn in Tabl for th ffctivnss factor for diffrnt. Hr s (6 (5 ar th sam, only trminal valus of u ar ndd. On point with mod. or (6 givs rasonabl rsults, whil xact rsults ar obtaind for two-point with mod. or. For th cas of a slab, it can b sn from Tabl that for larg Thil modulus, th rsults for th two-point mthod using (6 tak intrmdiat valus btwn on two point with mod. or using (6. Th lattr is th closst to th xact solution. For th cas of a cylindr (Tabl 3, th ffctivnss factor calculations rquir an accurat concntration profil. Th with mod. using (5 givs th bst rsults. Th rsults for two-point giv, on th othr h, a rasonabl accuracy. It can also b notd that substantial improvmnt is obtaind with two-point using ( ovr th on-point. Th sam obsrvations ar notd for th cas of a sphr in Tabl. Tabl : Effctivnss factor for a th ordr raction in a slab using diffrnt mthods. ( ( (6 (6 mod. (& (6 mod. (& (6 Exact Solution Tabl 3: Effctivnss factor for a th ordr raction in an infinit cylindr using diffrnt mthods. ( (5 ( (5 mod. ( (5 mod. ( (5 mod. ( (5 mod. ( (5 mod. ( (5 mod. ( (5 Exact Solution Brazilian Journal of Chmical Enginring

9 A Modifid Orthogonal Collocation Mthod for Raction Diffusion Problms 975 Tabl : Effctivnss factor for a th ordr raction in a sphr using diffrnt mthods. ( (5 ( (5 mod. ( (5 mod. ( (5 mod. ( (5 mod. ( (5 mod. ( (5 mod. ( (5 Exact Solution It can also b sn that, for both th cass of cylindr sphr for larg Thil modulus, th rsults for th two-point mthod using (5 giv intrmdiat rsults btwn on two point with mod. using (5. Th lattr is th closst to th xact solution. Th us of four-point (not shown gav xact rsults. Lss accurat rsults ar obtaind whn using (5 for th valuation of th ffctivnss factor. CONCLUSIONS Two modifications ar prsntd to improv th prformanc of th mthod. In th first modification, w add an xtra point to th zros of Jacobi polynomials at th cntr of th catalyst particl. Th solution is in th form of a highr ordr polynomial. In th scond modification this polynomial is transformd into a rational function with bttr accuracy of th solution. Whn this rational function is usd with th propr formula for th ffctivnss factor, xcllnt rsults ar obtaind. Th rsults ar of particular importanc to th cas of larg Thil modulus whr th concntration profil is vry stp. Such a cas usd to b tratd by th dad-zon mthod (Soliman, 989. ACKNOWLEDGEMENT Th authors xtnd thir apprciation to th Danship of Scintific Rsarch at King Saud Univrsity for funding th work through th rsarch group projct No RGP-VPP-88. REFERENCES Brrut, J. P., Baltnsprgr, R. Mittlmann, H. D., Rcnt dvlopmnt in barycntric rational intrpolation. In: Trnds Applications in Constructiv Approximation. Intrnational Sris of Numrical Mathmatics, (5. Biscaia Junior, E. C., Mansur, M. B., Salum, A. Castro, R. M. Z., A moving boundary problm orthogonal in solving a dynamic liquid surfactant mmbran modl including osmosis brakag. Braz. J. Chm. Eng., 8, 63 (. Finlayson, B. A., Th Mthod of Wightd Rsiduals Variational Principls. Acadmic Prss, Nw York (97. Finlayson, B. A., Nonlinar Analysis in Chmical Enginring. McGraw-Hill, Nw York (98. Soliman, M. A., A modifid on point mthod for diffusion raction problms. Chmical Enginring Scinc 3, 98 (988. Soliman, M. A., Collocation with low ordr polynomials for fast ractions in catalyst particls. Chmical Enginring Scinc,, 59 (989. Torrs, L. G., Martins, F. J. D. Bogl, I. D. L., Comparison of a rducd ordr modl for packd sparation procsss a rigorous nonquilibrium stag modl. Braz. J. Chm. Eng., 7, 95 (. Villadsn, J. V. Stwart, W. E., Solution of boundary-valu problms by orthogonal. Chmical Enginring Scinc,, 83 (967. Villadsn, J. V. Michlsn, M. L. Solution of Diffrntial Modls by Polynomial Approximation. Prntic-Hall, Nw Jrsy (978. Brazilian Journal of Chmical Enginring Vol. 3, No., pp , Octobr - Dcmbr,