A comparative study of flux-limiting methods for numerical simulation of gas-solid reactions with Arrhenius type reaction kinetics

Transcription

1 A comparatve study of flux-lmtg methods for umercal smulato of gas-sold reactos wth Arrheus type reacto ketcs Hassa Hassazadeh, Jalal Abed, ad Mehra Poolad-arvsh epartmet of Chemcal ad Petroleum Egeerg, Uversty of Calgary, 500 Uversty rve NW, Calgary, AB, Caada, TN N4 Abstract Heterogeeous gas-sold reactos play a mportat role a wde varety of egeerg problems. Accurate umercal modelg s essetal order to correctly terpret expermetal measuremets, leadg to developg a better uderstadg ad desg of dustral scale processes. The exothermc ature of gas-sold reactos results large cocetrato ad temperature gradets, leadg to steep reacto frots. Such sharp reacto frots are dffcult to capture usg tradtoal umercal schemes uless by meas of very fe grd umercal smulatos. However, fe grd smulatos of gassold reactos at large scale are computatoally expesve. O the other had, usg coarse grd block smulatos leads to excessve frot dsspato/smearg ad accurate results. I ths study, we vestgate the applcato of hgher-order ad flux-lmtg methods for umercally modelg oe-dmesoal coupled heat ad mass trasfer accompaed wth a gas-sold reacto. A comparatve study of dfferet umercal schemes s preseted. Numercal smulatos of gas-sold reactos show that at low grd resoluto whch s of practcal mportace Superbee, MC, ad va Albada- flux lmters are superor as compared to other schemes. Results of ths study wll fd applcato umercal modelg of gas-sold reactos wth Arrheus type reacto ketcs volved varous dustral operatos. Keywords: gas-sold reactos; hgher-order methods; flux lmter; umercal modelg Correspodg author. Tel.: (403) ; Fax (403) ; E-mal: jabed@ucalgary.ca

2 . Itroducto Heterogeeous gas-sold reactos play a sgfcat role may dustral applcatos. These processes clude but are ot lmted to heavy ol recovery, the roastg ad reducto of ores, the pyrolyss of bomass ad coal, the combusto of solds, waste cerato, the absorpto of acd gases by lme, reactve vapor phase deposto, producg ceramc materals, extractve metallurgy, coal gasfcato, ad catalyst maufacture (Ramachadra & oraswamy, 98; Rajaah et al., 988; Hastaoglu & Berrut, 989; Patsso & Abltzer, 000; Maras et al., 00). Aalytcal solutos of the goverg partal dfferetal equatos of gas sold reactos are usually mpossble or extremely dffcult to obta. Numercal modellg of gas-sold reactos s therefore the approach oe must pursue, ad such has bee wdely used to terpret expermetal measuremets, the desg of chemcal reactors, ad large-scale reactve flow smulatos. The exothermc ature of such reactos leads to complex olear traset teracto of covecto, heat coducto, mass dffuso, ad chemcal reactos, resultg steep cocetrato ad temperature gradets. I such crcumstaces, a better uderstadg of the system behavour ca oly be accomplshed by coductg very fe grd umercal smulatos to capture the frotal behavour of the process. However, fe grd umercal smulatos of dustral scale gas-sold reactos are computatoally expesve. O the other had, coarse grd block umercal smulatos of these processes usg low-resoluto umercal schemes would yeld excessve frot dsspato/smearg that would mpar terpretato of the expermetal data ad egatvely affect the desg of dustral processes. It s for ths reaso that hghresoluto umercal schemes are desged to mprove the accuracy of the umercal solutos covecto domated flows. The oted pheomeo of a umercal soluto of covecto domated partal dfferetal equatos beg proe to umercal dffuso has gve rse to umerous studes reported the lterature whch propose umercal schemes for mtgatg ths ssue (Sweby, 984). A comparatve study of umercal methods wth respect to covecto domated problems preseted by Wag ad Hutter (00) shows that the

3 modfed TV (total varato dmshg) Lax-Fredrchs method s the most capable/comprehesve method for hadlg covecto domated problems wth a steep spatal gradet of the varables. Alhumaz et al. (003) performed a umercal aalyss of a homogeeous tubular reactor whch a cubc autocatalytc reacto s coupled to mass dffuso ad covectve trasport. Ther calculatos show that specal hgh resoluto schemes such as ENO (essetally o-oscllatory) are ecessary to track effcetly steep movg frots exhbted by strogly covectve problems. Alhumaz (004, 007) studed the accuracy of several fte dfferece schemes to solve a oedmesoal covecto-dffuso-reacto problem of a autocatalytc mutatg reacto model ad foud that the Superbee ad MUSCL (Mootoe Upstream-cetered Schemes for Coservato Laws) flux lmters are the most approprate for smulatg sharp frots for covecto-dffuso-reacto ad covecto-dffuso systems, respectvely. Atae- Ashta ad Hosse (005) ad Atae-Ashta et al. (996) developed a correcto for the trucato error assocated wth a fte dfferece soluto of covecto ad dffuso wth a frst order reacto. They compared umercal results wth aalytcal solutos ad suggested that trucato errors are ot eglgble. It was also show that the Crak Ncholso method s the most accurate scheme based o trucato error aalyss. The atteto that has bee gve the lterature to devsg sutable umercal schemes for mprovg the accuracy of covecto-domated problems has ot bee the case for problems volvg covecto-dffuso wth Arrheus type ketcs reacto; such vestgatos are rare. There are fact oly lmted studes suggestg sutable hgh resoluto umercal schemes for specfc applcatos of covecto-dffusoreacto systems. Furthermore, there s o umercal scheme yet detfed that performs a superor maer for all problems; therefore, a choce s usually made based o experece. As far as s kow to the authors, comparatve studes of hgher-order ad flux-lmtg methods for coupled heat ad mass trasfer gas-sold systems wth Arrheus type reactos have ever bee reported the lterature. The objectve of ths study s therefore to perform a comparatve study of flux-lmtg methods for umercal smulato of gas-sold reactos wth Arrheus type reacto ketcs, where the exothermc ature of the reactos coupled wth heat ad mass trasfer leads to steep 3

4 temperature ad cocetrato gradets. Prevous comparatve studes of flux lmters dd ot cosder Arrheus type reactos. Results of ths study wll fd applcato umercal modelg of gas-sold reactos wth Arrheus type reacto ketcs volved varous dustral operatos. Ths paper s orgazed as follows. Frst, umercal errors covecto-dffusoreacto systems are descrbed. Next, the goverg partal dfferetal equatos for gassold reactos are preseted. The fte dfferece dscretzato of the goverg equatos ad dfferet umercal schemes for approxmatg covectve flux are the preseted. Comparsos of dfferet methods are descrbed the Results secto, followed by Summary ad Coclusos.. Numercal errors covecto-dffuso-reacto systems. Frst order covecto-dffuso-reacto system The goverg dfferetal equato for a lear covecto-dffuso-reacto system s gve by: C t C C = u kc () where C s cocetrato of reactat, s molecular dffuso coeffcet, t s tme, x s the lear spatal coordate, u s velocty, ad k s the reacto rate costat. The forward tme ad sgle-pot upstream fte dfferece method of dscretzato lead to the followg approxmato for the goverg partal dfferetal equato: C + C t C = + C ( x) + C C u C x kc where stads for cremets tme or space, ad ad deote grd dex ad tme step dex, respectvely. Neglectg thrd ad hgher order terms, the reactat cocetrato at dfferet tmes ad locatos ca be approxmated by the followg expressos (Latz, 97; Chaudhar, 97; Moldrup et al., 99; Atae-Ashta et al., 996): + C t C C = C + t (3) t t () 4

5 C x C C + = C + x (4) C x C C = C x +... (5) The secod tme dervatve of the reactat cocetrato Eq. (3) ca be obtaed by takg the tme dervatve of the goverg equato as gve by the followg expresso: C C C C = u k t t t t Substtutg for C / t from Eq.() results : (6) t C = 4 3 C C u + u 4 3 C C C + uk k + k Neglectg the hgher order dervatves C / ad C / gves: C C C C = u k + uk + k C t Usg Eqs. (3) to (5) ad (8) ad substtutg Eq.() we obta: C C C = + u x / tu / + k t u + k t k + k t / t ( ) ( ) ( )C Comparg Eqs. (9) ad () reveals that the umercal dscretzato modfes the dffuso coeffcet, velocty, ad reacto costat. The modfed dffuso coeffcet, velocty, ad reacto costat are thus gve by ( + u x / tu / + k t) mod =, u = u( + k t) C (7) (8) (9) mod, ad kmod k( + k t / ) =, respectvely. The compoets of the dffuso coeffcet, velocty, ad reacto costat attrbutable to umercal dscretzato are the = ( u x / tu / + k t) um, u um = uk t, ad = k t /, respectvely, where the subscrpt um refers to the k um addtoal terms that are created because of the umercal dscretzato errors. We scale dstace x by L, legth of the reacto doma, ad tme by dffuso tme scale, L /. The modfed form of Eq. (9) ca therefore be represeted by: C C φ ( + Pe( Co) / + φ τ ) Pe ( + φ τ ) φ + τ C C = s τ χ χ where τ s the dmesoless tme, χ s the dmesoless dstace, Pe s = ul / s the (0) system Peclet umber, Co = u t / x ad Pe = u x / are grd block Courat ad Peclet 5

6 umbers, respectvely, ad φ kl / s the Thele modulus (Fogler, 998). The = umercal dffuso, velocty, ad reacto costat ca the be respectvely represeted by: um u um u ( )/ + φ τ = Pe Co = φ τ () () k um k φ = τ The scalg aalyss shows that, for a lear covecto-dffuso-reacto system, umercal results are sestve to both temporal ad spatal dscretzato. Therefore, the accuracy of umercal solutos s affected by grd sze. (3) 3. Goverg equatos for gas-sold reactos The mathematcal model used ths study to perform umercal expermetatos s based o a formulato preseted by Rajaah et al. (988). Based o ths model, the exothermc o-catalytc gas-sold reacto s assumed to take place a oe-dmesoal flow system. The reacto that takes place s betwee a gaseous oxdzer (oxyge or troge) ad a sold phase that geerates a gas or a sold. Ths model assumes that Fourer s law of heat coducto a sold s vald. Sterg effects are gored ad the system porosty remas costat durg the process. The physcal propertes of the materals are assumed costat. Radato effects are corporated to a effectve thermal coductvty ad the reacto s cosdered to be frst order wth respect to gas ad sold reactats. A pseudo-homogeous, oe phase model s used. The goverg dfferetal equatos for such a system are gve by: T T T E ρ p p 0 G S (4) t RT Eergy balace: ( c ) = λ uρc + k ( H ) C C exp CG CG CG E Gas mass balace: ε = ε u k0cgcs exp (5) t RT 6

7 C E ε G S (6) t RT S Sold mass balace: ( ) = k C C exp 0 where ρ = ρc ε + ρ c ( ε ) c s the effectve heat capacty, T s temperature, C s p p s ps cocetrato of reactat, ρ s desty, c p s heat capacty, λ s the average thermal coductvty, k 0 s the pre-expoetal factor, E s actvato eergy, H s heat of reacto, R s the uversal gas costat, s molecular dffuso coeffcet, u s velocty, ε s sold materal porosty, t s tme, ad x s the spatal coordate. The subscrpts S ad G deote sold ad gas, respectvely. The tal ad boudary codtos as suggested by Rajaah et al. (988) are gve by: T = 0 x, t = 0 (7.) T 0 C = 0 x, t = 0 (7.) G C G 0 C = 0 x, t = 0 (7.3) S C S 0 ( ρc ) u( T T ) T λ = p let at x = 0 (7.4) T = 0 at C ( C C ) G ε = u Glet G at = 0 x (7.5) x (7.6) C G = 0 at x (7.7) The followg dmesoless scalg groups are used to reder the equatos to dmesoless form: * ( T T ) E θ = (8.) * RT C = C / C (8.) G S G S G let C = C / C (8.3) x t S let ρc p = x (8.4) λt * Ek ( H ) C E exp t RT 0 G = * * ρc p RT (8.5) 7

8 t * ρc RT p * E = exp * Ek0 ( H ) CG C let S let RT (8.6) λ t * x * = (8.7) ρ c p Pe H uρc = λ p λt ρc * p / (8.8) Pe M = u ε λt ρc * p / (8.9) λ Le = (8.0) ρc p γ γ G S ρc p RT = ε E = * ( H ) CG let ρc p RT * ( ε ) E( H ) CG let (8.) (8.) RT * β = (8.3) E Usg the above scalg groups, the dmesoless form of the dfferetal equatos s gve by: θ θ = Pe t H θ + C G C S θ exp βθ + (9) C t G = C Le G Pe M C G γ GC G C S θ exp βθ + (0) CS θ = γ SCGCS exp () t βθ + The above fact employs a stadard scalg protocol avalable the combusto lterature to reder the equatos dmesoless (Merzhaov et al., 973; Merzhaov & Borovskaya, 975; Puszysk et al. 987; Merzhaov & Khak 988; Rajaah et al., 988; adekar et al., 990a; adekar et al., 990b). The scalg varable x * 8

9 correspods to a approxmate measure of the heatg zoe legth ad of the reacto frot velocty (adekar et al., 990b). x * / t * s a measure 4. scretzato of the goverg equatos The goverg dfferetal equatos are dscretzed usg a explct--tme fte dfferece approxmato. A block-cetered scheme s used, where the dffusve flux s calculated based o grd block ceter values, whle the covectve flux values are evaluated based o the grd block terface values. A explct dscrete fte dfferece formulato of the goverg equatos for temperature, gas cocetrato, ad sold cocetrato s gve by: θ C t + x ( ) t ( ) θ θ θ θ PeH θ+ / θ / CG CS t exp x βθ + + = θ γ C C t + G + Le x θ t exp βθ + t ( C C + C ) Pe ( C C ) + G = C G G G M G + / G / Le x G + S where G C = C S S t ad x (5) () θ γ exp SCG CS t (3) βθ + are temporal ad spatal cremets, respectvely. I the followg secto, dfferet umercal schemes approprate for approxmatg covectve flux are preseted. 4. Numercal approxmato of covectve flux There are several optos for dscretzg advecto terms the flow ad trasport equatos. The most commoly used method for approxmatg block terface propertes s sgle-pot upstream weghtg. The sgle-pot upstream weghtg scheme approxmates the value of a fucto at a grd block face wth the value the grd block o the upstream sde. The drawback of such a scheme s that artfcal dffuso s 9

10 troduced ad mght be cosderable, thereby producg erroeous results uless a large umber of grd blocks (.e., a hgh degree of grd block refemet) s employed. A alteratve method of reducg umercal dffuso s to employ a hgher order method, such as two-pot upstream weghtg (Todd et al., 97). Thrd-order methods (Leoard, 979) as well as Total Varato mshg (TV) schemes (Harte, 983) have also bee used to cotrol umercal dffuso. I the followg, a bref revew of these umercal methods s preseted. 4.. Sgle-pot upstream Ths scheme s wdely used evaluatg terface block propertes, but as oted s proe to umercal dffuso. I ths method, et covectve flux to a cell ca be approxmated as: ( uc) = ( u [( ω ) C + ωc ] u [( ω) C + ωc ]) where x + / + / x s the grd block sze ad ω s the weghtg factor ad s equal to zero or oe depedg o the flow drecto. It s oted that for the problem uder vestgato the flud velocty u s costat ad therefore the choce of terface values for velocty gve by Eq.(4) s rrelevat. (4) 4.. Two-pot upstream Ths approxmato was proposed by Todd et al. (97). I ths secod-order method, the mesh terface values are obtaed by usg the values at two adjacet blocks that are depedet o the flow drecto. Because the approxmato s a extrapolato process, t s mportat to lmt the computed values to physcally acceptable values. However, such cases, the method reverts to sgle-pot upstream weghtg whch guaratees physcally acceptable results. The block terface propertes for a arbtrary grd block ca be approxmated as follows: x f (5) ( f f ) / = f + x + x x f (6) ( f f ) + / = f + x + x 0

11 where f = uc s the covectve flux fucto. The block terface fluxes are calculated based o the sgle-pot upstream value whe the followg codtos are ot satsfed: f + / the greater of f or f + ad f / the greater of f or f 4..3 Thrd order methods For thrd order methods, troduced by Leoard (979), the grd block terface value s approxmated by usg three pots adjacet to a arbtrary grd block. I ths method the boudary grd blocks are approxmated by the sgle-pot upstream weghtg. Saad et al. (990) modfed Leoard s method to accout for varable grd block sze. For that thrd order method, the block terface propertes are calculated based o the followg approxmatos: ( f f ) + Λ ( f f ) f, (7) f / = f + Π ( f f ) + Λ ( f f ) + / = f + Π +, (8) where Π = 3 x x + x ad Λ = 3 x x + x + The boudary pots ca be approxmated by sgle-pot upstream weghtg because at least two upstream pots ad oe dow-stream pot each coordate drecto are eeded for the hgher order method. Smlar to the prevous case, the above formulato volves a extrapolato process, ad t s ecessary to costra the computed values to physcally admssble values. The other codto that must be fulflled s the mootocty costrat, whch requres that the terface values of the calculated parameter be less tha or equal to the larger cocetrato o ether sde of the grd block (Saad et al., 990). (9) 4..4 Total Varato mshg methods (TV) The TV method was frst troduced by Harte (983) ad later by Sweby (984). The TV property guaratees that the total varato of the soluto of a fucto wll ot crease as the soluto progresses tme. A umercal method s sad to be TV f: TV ( Q ) TV ( Q + ) (30)

12 where TV s the total varato gve by: Q j Q j + j TV = (3) The scheme lmts the flux betwee grd blocks ad the lmts spurous growth the grd block averages so that the above equalty s satsfed. A geeral approach s to multply the jump grd block averages by a lmtg fucto. For our problem, the terface flux ca be expressed by: ( r ) [ f f ] Φ f / = f + (3) f ( r ) [ f f ] Φ + / = f + + where f f r = (34) f f ad r f f = (35) f+ f where Φ flux lmter fucto ad r s the rato of successve gradets ad s a measure of the smoothess of the soluto. Makg Φ = 0 returs to the commoly used sglepot upstream method as descrbed prevously. These types of flux lmters are called o-lear flux lmters. There are may TV schemes foud the lterature for solvg covecto domated problems; Table gves some of the flux lmters reported the lterature. Fg. shows the acceptable flux lmter rego for secod-order TV schemes as suggested by Sweby (984). (33)

13 Table Lterature values of varous flux lmters Flux lmter Flux lmter fucto Φ ( r) va Leer (va Leer,974) ( r + r )/( r +) MC (va Leer,977) max [ 0, m(r,0.5( r + ),] va Albada - (va Albada et al., 98) r ( r + ) /( r + ) Superbee (Roe, 985,986) max[ 0, m(r,),m( r,) ] Mmod (Roe, 986) max[ 0,m( r,) ] SMART (Gaskell & Lau, 988) max [ 0, m(r,0.75r + 0.5,4) ] H-QUICK (Leoard, 987) ( r + r )/( r + 3) Kore (993) max [ 0, m(r,( r + ) / 3),] OSPRE (Waterso & ecock, 995).5r ( r + ) /( r + r + ) CHARM (Zhou, 995) r ( + 3r) /( r + ) HCUS (Waterso & ecock, 995).5( r + r ) /( r + ) va Albada - (Kerma et al., 003) r /( r + ) Φ() r Φ = r Φ( r) Φ = r Φ = r Φ = r Φ = Φ = Φ = Φ = r r Fg.. (a) TV rego ad (b) secod order TV rego (Sweby 984); Φ = ad Φ = r correspod to cetral (Lax-Wedroff) ad Beam-Warmg scheme, respectvely. 3

14 5. Results The goverg partal dfferetal equatos of gas-sold reactos are solved usg dfferet umercal schemes. Two test problems of sold combusto that gve dfferet frotal characterstcs are preseted ad results of dfferet umercal schemes are compared. To estmate the accuracy of the varous umercal solutos, we defe umercal error usg the followg expresso as a measure of umercal soluto accuracy: Ξ = ( ψ ref ) ψ dx / ψ dx (36) where ψ ca be ether temperature or cocetrato ad the subscrpt ref deotes soluto. The covergece of the umercal solutos was verfed by coductg tests for 5 0 3, 0 0 3, ad grd blocks usg the sgle-pot upstream method. The data used the umercal smulatos are gve Table. Fg. shows the umercal solutos for the two test cases. These solutos are used the aalyss that follows. Results demostrate that large umbers of grd blocks ( excess of a thousad) are eeded to fd a accurate umercal soluto usg the sgle-pot upstream method. I the followg, we compare varous flux-lmtg schemes for the purpose of seekg a approprate flux lmter. Such a approprate flux lmter wll facltate a accurate umercal soluto wth fewer grd blocks as compared to the tradtoal sgle-pot upstream method. Numercal expermets were performed usg all flux lmters gve Table ad usg 50, 00, 500, ad 000 grd blocks. Fg. 3 shows temperature ad cocetrato profles obtaed from dfferet flux-lmtg schemes ad wth 50 grd blocks. The correspodg umercal errors for N=50 are summarzed Table 3. Results reveal that, for N=50 ad for both test cases, MC (va Leer, 977), Superbee (Roe, 985,986), ad va Albada- (Kerma et al., 003) flux lmters are superor as compared to the other schemes. Results show that the sgle-pot upstream method fals to model the reacto frot accurately, whle the flux lmtg methods ted to correctly capture the sharp reacto frot. Fg. 4 compares umercal solutos obtaed by usg dfferet flux lmters ad choosg total umber of grd blocks equal to 00. Whle most of the flux 4

15 lmters could accurately capture almost all of the reacto frot characterstcs, the calculated umercal errors Table 4 reveal that MC (va Leer, 977), Superbee (Roe, 985, 986), ad va Albada- (Kerma et al., 003) are more accurate tha the other flux lmters. Smlar to the prevous case (N=50), the sgle-pot upstream method fals to resolve the reacto frot. Fgs. 5 ad 6 show the calculated profles wth 500 ad 000 total umbers of grd blocks, respectvely. Results show Fgs. 5 ad 6 ad Tables 5 ad 6 reveal that the flux lmters accurately capture the reacto frot. Aga, the sglepot upstream results are stll ot accurate at N=000. Results therefore suggest that usg a flux-lmtg approach allows coductg umercal smulatos wth sgfcatly fewer umbers of grd blocks yet wth the same accuracy of very fe grd smulato by the sgle-pot upstream scheme. Table ata used umercal smulatos Parameter γ S γ G β θ 0 θ let Pe H Pe M Le Test case Test case

16 Temperature (a) Gas cocetrato. (b) Sold cocetrato. (c) (d). (e). (f) Temperature - Gas cocetrato Sold cocetrato Fg.. mesoless temperature (a), gas cocetrato (b), ad sold cocetrato (c) for a gas-sold reacto of test case at t =. 5, ad dmesoless temperature (d), gas cocetrato (e), ad sold cocetrato (f) for a gas-sold reacto of test case at t = obtaed by coductg tests wth 5 0 3, ad grd blocks usg sgle-pot upstream method. 6

17 Temperature (a) sgle-pot upstream Temperature - (d) sgle-pot upstream Gas cocetrato (b) sgle-pot upstream Gas cocetrato (e) sgle-pot upstream Sold cocetrato (c) sgle-pot upstream Sold cocetrato (f) Fg. 3. mesoless temperature (a), gas cocetrato (b), ad sold cocetrato (c) for a gas-sold reacto of test case at t =. 5, ad dmesoless temperature (d), gas cocetrato (e), ad sold cocetrato (f) for a gas-sold reacto of test case at t = obtaed by coductg tests wth varous flux lmters gve Table ad 50 grd blocks. Table 3 Comparso of umercal error for total umber of grd blocks N=50 Flux lmter Error (fracto) Test case Test case θ C G C S θ C G C S Sgle-pot upstream va Leer (va Leer,974) MC (va Leer,977) va Albada - (va Albada et al., 98) Superbee (Roe, 985,986) Mmod (Roe, 986) SMART (Gaskell & Lau, 988) H-QUICK (Leoard, 987) Kore (993) OSPRE (Waterso & ecock, 995) CHARM (Zhou, 995) HCUS (Waterso & ecock, 995) va Albada - (Kerma et al., 003)

18 Temperature (a) sgle-pot upstream Temperature - (d) Gas cocetrato (b) sgle-pot upstream Gas cocetrato (e) sgle-pot upstream Sold cocetrato (c) sgle-pot upstream Sold cocetrato (f) Fg. 4. mesoless temperature (a), gas cocetrato (b), ad sold cocetrato (c) for a gas-sold reacto of test case at t =. 5, ad dmesoless temperature (d), gas cocetrato (e), ad sold cocetrato (f) for a gas-sold reacto of test case at t = obtaed by coductg tests wth varous flux lmters gve Table ad 00 grd blocks. Table 4 Comparso of umercal error for total umber of grd blocks N=00. Flux lmter Error (fracto) Test case Test case θ C G C S θ C G C S Sgle-pot upstream va Leer (va Leer,974) MC (va Leer,977) va Albada - (va Albada et al., 98) Superbee (Roe, 985,986) Mmod (Roe, 986) SMART (Gaskell & Lau, 988) H-QUICK (Leoard, 987) Kore (993) OSPRE (Waterso & ecock, 995) CHARM (Zhou, 995) HCUS (Waterso & ecock, 995) va Albada - (Kerma et al., 003)

19 Temperature (a) sgle-pot upstream Temperature - (d) Gas cocetrato (b) sgle-pot upstream Gas cocetrato (e) sgle-pot upstream Sold cocetrato (c) sgle-pot upstream Sold cocetrato (f) Fg. 5. mesoless temperature (a), gas cocetrato (b), ad sold cocetrato (c) for a gas-sold reacto of test case at t =. 5, ad dmesoless temperature (d), gas cocetrato (e), ad sold cocetrato (f) for a gas-sold reacto of test case at t = obtaed by coductg tests wth varous flux lmters gve Table ad 500 grd blocks. Table 5 Comparso of umercal error for total umber of grd blocks N=500 Flux lmter Error (fracto) Test case Test case θ C G C S θ C G C S Sgle-pot upstream va Leer (va Leer,974) MC (va Leer,977) va Albada - (va Albada et al., 98) Superbee (Roe, 985,986) Mmod (Roe, 986) SMART (Gaskell & Lau, 988) H-QUICK (Leoard, 987) Kore (993) OSPRE (Waterso & ecock, 995) CHARM (Zhou, 995) HCUS (Waterso & ecock, 995) va Albada - (Kerma et al., 003)

20 Temperature (a) Temperature - (d) Gas cocetrato (b) sgle-pot upstream Gas cocetrato (e) sgle-pot upstream Sold cocetrato (c) sgle-pot upstream Sold cocetrato (f) Fg. 6. mesoless temperature (a), gas cocetrato (b), ad sold cocetrato (c) for a gas-sold reacto of test case at t =. 5, ad dmesoless temperature (d), gas cocetrato (e), ad sold cocetrato (f) for a gas-sold reacto of test case at t = obtaed by coductg tests wth varous flux lmters gve Table ad 000 grd blocks. Table 6 Comparso of umercal error for total umber of grd blocks N=000 Flux lmter Error (fracto) Test case Test case θ C G C S θ C G C S Sgle-pot upstream va Leer (va Leer,974) MC (va Leer,977) va Albada - (va Albada et al., 98) Superbee (Roe, 985,986) Mmod (Roe, 986) SMART (Gaskell & Lau, 988) H-QUICK (Leoard, 987) Kore (993) OSPRE (Waterso & ecock, 995) CHARM (Zhou, 995) HCUS (Waterso & ecock, 995) va Albada - (Kerma et al., 003)

21 6. Summary ad coclusos Physcal processes volved gas-sold reactve systems clude dffusve (heat ad mass) as well as reactve processes that have dfferet trsc scalg characterstcs. I most cases, the o-learty of the processes does ot allow aalytcal soluto. Therefore, accurate umercal modelg s ecessary order to properly terpret expermetal measuremets, leadg to developg a better uderstadg ad desg of dustral scale processes. Accurate fe grd umercal smulato of sharp reacto frot propagato gas-sold reactos s a challegg task. Ideed, most practcal cases, such as stu combusto heavy ol reservors, the use of small grd blocks s ot feasble. The covetoal sgle-pot upstream scheme s also kow to smear the frots. Therefore, oe eeds to accout for the small scale gradets that caot be captured by coarse grd blocks whe usg the sgle-pot upstream method. Oe possble opto for reducg the smearg effect resultg from ths method s to use flux lmters. I ths paper, we coducted a comparatve study of varous flux lmters to fd approprate flux lmters for oe-dmesoal gas-sold reactve flow smulatos wth Arrheus type reacto. Relatvely fe grd umercal smulatos of the gas-sold reactos show that most of the methods wth the excepto of sgle-pot upstream perform well. More specfcally, for small umber of grd pots whch s of practcal mportace Superbee, MC, ad va Albada- flux lmters are superor as compared to other schemes. These results wll ad choosg proper flux lmters umercal modelg of gas-sold reactos wth Arrheus type reacto ketcs. Ackowledgmets The facal support of the Alberta Igeuty Cetre for I Stu Eergy (AICISE) s ackowledged. Nomeclature c p heat capacty, J kg -l K -l C cocetrato, kg/m 3 Co Courat umber molecular dffuso coeffcet, m /s E actvato eergy, J kmol - f covectve flux fucto H heat of reacto, J/kg

22 k pre-expoetal rate costat, ut depeds o reacto type L legth of reactg system, m Le Lews umber, dmesoless N total umber of grd blocks Pe Peclet umber Q umercal soluto TV methods r rato of successve gradets R uversal gas costat, J kmol - K -l t tme, s T temperature, K u velocty, ms - TV total varato dmshg x spatal coordate, m Greek letters β verse of dmesoless actvato eergy γ verse of dmesoless heat of reacto dfferece or cremet ε porosty θ dmesoless temperature Λ weghtg factor thrd order method λ effectve thermal coductvty, J m - s - K - Ξ umercal error Π weghtg factor thrd order method ρ desty, kg m -3 τ dmesoless tme used secto. Φ flux lmter fucto φ square root of Thele modulus χ dmesoless dstace used secto. ψ varable umercal error fucto; ca be temperature or cocetrato ω weghtg factor sgle-pot upstream method Subscrpts dmesoless G gas H heat f frot grd block dex let let codto L left M mass mod modfed um umercal ref s system S sold 0 tal value

23 * scale value Superscrpts tme dex average 3

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