Analysis of non-adiabatic heat-recirculating combustors

Transcription

1 Analysis of non-adiabatic hat-rcirculating combustors Addrss corrspondnc to: Paul D. Ronny Dpartmnt of Arospac and Mchanical Enginring Univrsity of Southrn California Los Angls, CA USA Prof. Paul D. Ronny Dpartmnt of Arospac and Mchanical Enginring Univrsity of Southrn California Los Angls, CA (213) (213) (fax) Full citation: Ronny, P. D., "Analysis of non-adiabatic hat-rcirculating combustors," Combustion and Flam, Vol. 135, pp (2003).

2 Analysis of non-adiabatic hat-rcirculating combustors Paul D. Ronny Dpartmnt of Arospac and Mchanical Enginring Univrsity of Southrn California Los Angls, CA USA ABSTRACT A simpl first-principls modl of countr-currnt hat-rcirculating combustors is dvlopd, including th ffcts of hat transfr from th product gas stram to th ractant stram, hat loss to ambint and hat conduction in th stramwis dirction through th dividing wall (and hat transfr surfac) btwn th ractant and product strams. It is shown that stramwis conduction through wall has a major ffct on th oprating limits of th combustor, spcially at small dimnsionlss mass fluxs (M) or Rynolds numbrs that would b charactristic of microscal dvics. In particular, if this conduction is nglctd, thr is no small-m xtinction limit bcaus smallr M lads to largr hat rcirculation and longr rsidnc tims that ovrcom hat loss if M is sufficintly small. In contrast, vn a small ffct of conduction along this surfac lads to significantly highr minimum M. Comparison is mad with an altrnativ configuration of a flam stabilizd at th xit of a tub, whr hat rcirculation occurs via conduction through tub wall; it is found that th countr-currnt xchangr configuration provids suprior prformanc undr similar oprating conditions. Implications for microscal combustion ar discussd.

3 Introduction Rcntly intrst in hat-rcirculating xcss nthalpy burnrs, first studid ovr 30 yars ago [1, 2], has bn rnwd du to fforts in microscal combustion and powr gnration [3, 4, 5, 6]. Such work is motivatd by th fact that hydrocarbon fuls contain 100 tims mor nrgy pr unit mass than lithium-ion battris, thus dvics convrting of ful to lctricity at bttr than 1% fficincy rprsnt improvmnts in portabl lctronic dvics and othr battry-powrd quipmnt. At small scals, howvr, hat and friction losss bcom mor significant, thus dvics basd on xisting macroscal dsigns such as intrnal combustion ngins may b impractical. Consquntly, many groups hav considrd hat rcirculation using a countr-currnt hat xchangr for thrmal managmnt. By transfrring thrmal nrgy from th combustion products to th ractants without mass transfr (thus dilution of ractants), th total ractant nthalpy (sum of thrmal and chmical nthalpy) is highr than in th incoming cold ractants and thrfor can sustain combustion undr conditions (lan mixturs, small hating valu fuls, larg hat losss, tc.) that would lad to xtinguishmnt of th flam without rcirculation. Apparntly thr has bn only on modling study of xtinction limits and limit mchanisms in hat-rcirculating burnrs. In 1978 Jons t al. [7] prformd a global nrgy balanc on th ractant and product strams in hat-rcirculating combustors using mpirically spcifid minimum raction tmpraturs and prscribd hat losss. Dspit its simplicity, two xtinction limits wr prdictd, a blow-off typ limit at larg dimnsionlss mass flux (M) or Rynolds numbr (R) and anothr limit at small M du to hat losss. For high-m limit, th minimum ful concntration supporting combustion incrass with incrasing M bcaus as M incrass, th rsidnc tim dcrass, thus a highr raction rat and consquntly highr raction tmpratur and thrfor a highr ful concntration is rquird to sustain combustion. For th low-m limit th minimum ful concntration incrass with dcrasing M bcaus th hat loss rat was assumd fixd; th xtinction limit critrion of a minimum raction tmpratur corrsponds ssntially to a fixd ratio of hat loss rat to hat gnration rat, thus as th mass flow rat dcrass a highr mass fraction of ful is ndd to produc th minimum rquird hat gnration rat to avoid xtinction. Ths prdictions ar qualitativly similar to thos sn in hat rcirculating combustor xprimnts [1-3]. Morovr, dual-limit bhavior (i.. 1

4 a high-vlocity and a low-vlocity limit for fixd ful concntration) is charactristic of many combustion systms [8]. Whil instructiv, th Jons t al. modl [7] is not ntirly prdictiv bcaus an mpirical quantity is rquird, spcifically th minimum ractor tmpratur supporting combustion, plus th rat of hat loss (not just a hat loss cofficint) must b prscribd. Prhaps mor importantly, w will show that small-m xtinction limits (which ar most rlvant to microscal applications) rquirs a procss bsids hat losss, such as hat conduction along th dividing wall btwn ractant and product strams. Low-M limits wr prdictd in [7] bcaus hat losss to ambint wr prscribd indpndnt of M, whras for ralistic hat transfr modls, ths losss will dcras with dcrasing M as discussd latr. With ralistic hat rcirculation and hat loss modls, at low M th rcirculation and loss track ach othr, thus no low-m xtinction limit is prdictd unlss an additional is prsnt. Consquntly, this work aims to dvlop th simplst possibl first-principls modl (not rquiring spcification of an mpirical minimum raction tmpratur or a prscribd hat loss) of hat transfr, finit-rat xothrmic chmical raction, stramwis wall thrmal conduction and hat loss in a countr-currnt hat xchangr/combustor and dscrib th rsulting xtinction mchanisms. Comparison is mad with an altrnativ configuration also rportd in th litratur [9, 10], namly that of a flam stabilizd at th xit of a tub, whr hat rcirculation occurs via conduction through tub wall rathr than in a countr-currnt hat xchangr. Countr-currnt hat xchangr/combustor Approach W considr a linar countr-currnt hat xchangr (Fig. 1) with intak (for prmixd ractants) and xhaust (for combustion products) ports at on nd (x = 0). At th opposit nd (x = L) a wll-stirrd ractor (WSR) is stationd btwn th ractant (low-tmpratur) and product (high-tmpratur) sids of th xchangr. A thrmally conductiv dividing wall sparats th two sids of th xchangr. Th currnt analysis could b xtndd to a concntric tub countr-currnt hat xchangr, which has also bn considrd for both macroscal [1] and microscal [10] hat-rcirculating combustors, if sparat valus of h 1 ar chosn for th outr 2

5 annual ractant stram and th innr tubular product stram and (bcaus of th axis of symmtry) hat loss from th product stram is st to zro. Th us of a WSR combustion modl is not arbitrary; initial xprimntal and numrical rsults [3] show that nar xtinction limits, raction zon structurs in such burnrs ar vry diffrnt from propagating prmixd flams; instad, thr ar much largr (compard to propagating flams) raction zons, smallr tmpratur gradints, lowr pak tmpraturs, longr rsidnc tims at high tmpratur, and no visibl flam mission. Ths charactristics ar typical of flamlss or mild" combustion obsrvd in burnrs mploying highly prhatd air [11, 12, 13]. Th WSR modl rprsnts a limit of mild combustion whr hat and mass diffusion ffcts ar ngligibl (in th sns that thy ar so rapid that tmpratur and composition gradints within th ractor cannot b sustaind) and transport is limitd ntirly by convction. With th WSR modl, blow-off xtinction limits occur at larg M whr rsidnc tims in th WSR ar insufficint compard to chmical raction tims. W will show that limits also xist at small M in non-adiabatic hat-rcirculating burnrs with wall conduction in th stramwis dirction. This is bcaus as M dcrass, th fraction of hat rlas transfrrd away from th WSR du to conduction along th wall (and subsquntly transfrrd to th gas strams, thn lost to ambint) incrass. This causs th WSR tmpratur to dcras, and bcaus raction tim incrass xponntially with dcrasing tmpratur, at sufficintly small M th raction tim bcoms largr than th rsidnc tim vn though rsidnc tims and hat rcirculation gains in nthalpy both incras as M dcrass. Without wall conduction, this low-m xtinction limit mchanism is absnt. Th ffct of wall conduction on hat xchangr prformanc has bn studid prviously in th contxt of cryognic systms [14, 15] and microscal hat xchangrs [16, 17] but apparntly not in th contxt of non-adiabatic combustors. Exprimntal and numrical rsults for hat-rcirculating burnrs [3] show that nar xtinction limits, th maximum possibl hat rcirculation (thus maximum xcss nthalpy) is ndd to sustain raction, thus raction occurs insid th WSR volum as spcifid hr, whras farthr from limits lss rcirculation is rquird, thus raction may occur upstram or downstram of th WSR. Consquntly, th modl mployd hr is appropriat nar xtinction limits, which is th main intrst, but may b inappropriat away from limits. Th oprating rgim away from xtinction limits, whr chmical raction may occur outsid th WSR, could 3

6 b tratd as a convctiv-diffusiv systm of with raction in a prhat zon of finit, prscribd lngth, in a mannr similar to that analyzd by Zl'dovich [18]. In xtrm cass of sufficintly high ful concntrations and small M, this raction outsid th WSR lads to a flashback limit bcaus a flam can propagat upstram from th raction zon to th ractant inlt without any nd for hat rcirculation. Such a limit cannot b prdictd by th currnt modl sinc th raction is prsumd to stay within th WSR. Whil this rprsnts anothr limitation on th oprating conditions of a hat-rcirculating burnr, it dos not affct th xtinction limits of primary intrst in this work. Th configuration mployd hr is inhrntly two-dimnsional bcaus gas-phas hat conduction is orthogonal to stramwis convction. A on-dimnsional modl is obtaind by (1) using constant ovrall hat transfr cofficints (h 1 ) for hat transfr from ractant and product strams to th dividing wall, (2) using constant ovrall hat loss cofficints (h 2 ) for hat losss from ths strams to ambint and (3) modling th wall as "thrmally-thin." Constant hat transfr cofficints ar not only analytically convnint but ralistic for laminar flows in plan channls, whr th Nusslt numbr basd on hydraulic diamtr h 1 d H /k g 7.5 [19], and thus for channls much widr in th spanwis dirction than thir hight d, d H 2d and consquntly h k g /d. For strongly turbulnt flows roughly h ~ m 0.8 but for laboratory-scal apparatuss with d on th ordr of a fw mm and gas vlocitis of a fw hundrd cm/s at most, Rynolds numbrs will b at most a fw hundrd and thus turbulnt flow is not xpctd. Of cours, for microscal dvics laminar flow will b ubiquitous. Th thrmally-thin assumption, which is common in modling of flam sprad ovr thin solid ful bds [20], rquirs that th wall thrmal rsistanc t/k w, whr t is th wall thicknss and k w its thrmal conductivity, is small compard to th channl thrmal rsistanc 1/h 1 so that tmpratur drops across th wall ar ngligibl. Sinc Nu 3.7, h 1 3.7k g /d, th thrmally-thin assumption rquirs 0.52k w d/k g t >> 1. For ralistic matrials k w >> k g and for most practical burnrs and proposd microscal dvics d/t 1, thus this assumption is justifid. With th thrmally-thin modl T w, -T w,i << T -T i,, whr T i (x) is th man ractant-sid gas tmpratur, T w,i (x) th corrsponding tmpratur on th wall surfac, and T (x) and T w, (x) th corrsponding product-sid tmpraturs (Fig. 1). Morovr, only stramwis wall conduction nds to b calculatd, using th man wall tmpratur T w (T w, +T w,i )/2. 4

7 For simplicity flow channl ntranc ffcts on h 1 ar nglctd. For laminar flow in straight channls th ratio of ntranc lngth L to d H is approximatly 0.04 Ud H /n. This can b rarrangd to rad L /L 0.60 M/Pr, whr L is th lngth of th xchangr and M is th dimnsionlss mass flux dfind latr, and thus for Pr = 0.7, L /L 0.85 M. For th rprsntativ conditions analyzd in this work, th maximum valu of M of intrst is about 0.2 thus at most only th first 17% of th xchangr lngth is influncd by ntranc ffcts (this of cours applis to both th low-tmpratur and high-tmpratur arms of th xchangr). In th ntranc rgions h 1 will b highr than its valu for fully-dvlopd flow and thus th ovrally prformanc of th combustor will b slightly highr than that stimatd basd on fully dvlopd flow. For th low-m cass of most intrst in this study, ntranc ffcts ar ngligibl. Many hat-rcirculating combustors mploy multi-turn spiral xchangrs (rathr than linar xchangrs as analyzd hr) for which hat losss in th dirction shown in Fig. 1 ar minimal, but hat losss in th third dimnsion still xist. Ths losss ar stimatd (for laminar flows) by H h 2 /h 1 = d/w, whr w is th channl dpth in th third dimnsion. Insulation would not chang this stimat substantially bcaus insulating matrials hav thrmal conductivitis no lowr than air and th insulation thicknss in most dvics is much lss than w. Furthrmor it is assumd, as in [7], that th WSR volum is small compard to th hat xchangr, thus hat loss from th WSR is nglctd. Analysis Hat transfr is dividd into thr zons: th dividing wall, th ractants sid of th hat xchangr and th products sid of th hat xchangr. Enrgy balancs on ths thr zons radily yild (s Nomnclatur): k w t d 2 T w dx 2-2h 1T w + h 1 (T w,i + T w, ) = 0 (1a) dt m C i P dx - h 1(T w,i - T i ) + h 2 (T i - T ) = 0 (2a) 5

8 dt m C P dx - h (T - T ) - h (T - T ) = 0 1 w, 2 Using th man wall tmpratur T w (T w, + T w,i )/2, invoking th thrmally-thin assumption T w, - T i,w << T - T i, and taking th sum and diffrnc of (2a) and (3a) yild (3a) k w t d 2 T w h 1 dx - 2T 2 w = -(T i + T ) (1b) m C P m C P d ( dx T - T i ) + (h 1 + h 2 )( T i + T ) - 2h 2 T - 2h 1 T w = 0 (2b) d ( dx T + T i ) + (h 1 + h 2 )( T i - T ) = 0 (3b) Th prvious modl by Jons t al. [7] is similar to Eqs. (2b) and (3b) applid globally rathr than on an lmnt of th hat xchangr of infinitsimal lngth. Combining (1b), (2b) and (3b) yilds M 2 BH(1+ H) d 4 T w È M 2 - d x 4 H(1+ H) + 1+ H Í Î BH d 2 d x 2 T w + T w =1 (4) whr M m C P h 1 L ;B 2h 1L 2 k w t ;H h 2 ; x x h 1 L ; T T T Thus, th hat transfr problm has thr dimnsionlss paramtrs, namly th Biot numbr (B) (scald by L/t sinc convction occurs along th lngth of th dividing wall (L) whras conduction along th wall occurs through a cross-sction of hight t), a hat loss cofficint (H) and th mass flux (M). Sinc th hat transfr cofficint h 1 is assumd to b th sam on both sids of th dividing wall and th dividing wall is assumd to b thrmally thin (ngligibl thrmal rsistanc), th ovrall hat transfr cofficint for th hat xchangr is h 1 /2, and sinc m C P is th sam on both sids of th xchangr, 2M = m C P /(h 1 /2)L is quivalnt to th Numbr 6

9 of Transfr Units (NTU) in th hat xchangr litratur. Additionally, 2/MB is quivalnt to th stramwis wall conduction paramtr mployd by Krogr [14] and othrs. For laminar flow whr h 1 3.7k g /d, M = r g UdC P /(3.7k g /d)l = 0.26(d/L)RPr, whr R r g Ud/m g is th Rynolds numbr and Pr is th Prandtl numbr. For gomtrically similar burnrs (d/l = constant), small M corrsponds to small R (charactristic of most microscal dvics), thus dscribing th small-m xtinction mchanism is th primary focus of this work. Th solution to (4) is T w ( x ) =1+ c 1 a x + c 2 -a x + c 3 b x + c 4 -b x (5) whr c 1 c 4 ar unknown constants, a = l 1 1/2, b = l 2 1/2 and l 1 and l 2 ar th roots of M 2 È M 2 BH(1+ H) l2 - H(1+ H) + 1+ H Í l +1= 0 (6). Î BH Two boundary conditions ar obtaind by assuming th dividing wall is adiabatic (thus has zro tmpratur gradint) at both nds. This is rasonabl sinc th wall cross-sctional aras ar gnrally small compard to channl cross-sctional aras, plus k w >> k g, thus thr would b littl hat transfr out th bar wall nds compard to convctiv transfr to/from th wall surfacs. With this assumption wall conduction is not a hat loss mchanism, instad it only rdistributs thrmal nrgy within th dvic; it will b shown this still rsults in a major impact on burnr prformanc. Th ffct of th wall nd boundary conditions is discussd in a latr sction. From (5) this yilds ac 1 - ac 2 + bc 3 - bc 4 = 0 ac 1 a - ac 2 - a + bc 3 b - bc 4 -b = 0 (7a) (7b) Two boundary conditions ar obtaind by substituting (5) into (1b) and applying T i (0) = 1 (ambint inlt tmpratur) yilding 7

10 Ê Á a 2 Ë B -2 ˆ Ê c1 + Á a2 Ë B -2 ˆ Ê c2 + Á b 2 Ë B - 2 ˆ Ê c3 + Á b 2 Ë B - 2 ˆ c4 = -T (0) +1 (7c) Ê Á a 2 Ë B -2 ˆ Ê a c 1 + Á a2 Ë B -2 ˆ Ê -a c 2 + Á b 2 Ë B - 2 ˆ Ê b c 3 + Á b2 Ë B -2 ˆ - b c 4 = 2 - T (1) -T i (1) (7d) whr th xhaust tmpratur T (0) and WSR inlt and outlt tmpraturs T i (1) and T (1) ar all unknowns. Two additional quations ar obtaind by manipulating (2b) and (3b) to isolat T i ( x ) and T ( x ) as functions of T w ( x ) only and applying th boundary conditions at x = 0 and x = 1. In non-dimnsional form th quations for T i ( x ) and T ( x ) as functions of T w ( x ) only ar M d T i + (1+ H) T d x i - H = T w (8), M d T - (1+ H) T d x + H = -T w (9). Substituting in T w ( x ) from Eq. (5) and applying th boundary condition T i (0) = 1 to (8), th solutions to (8) and (9) ar whr T -(1+ H ) i ( x ) = -(d 1 + d 2 + d 3 + d 4 ) x / M + d 1 a x + d 2 -ax + d 3 b x + d 4 -b x +1 (10) T ( x ) = ( T (1+ H ) (0) - (1+ f 1 + f 2 + f 3 + f 4 )) x / M +1+ f 1 a x + f 2 -a x + f 3 bx + f 4 -b x (11) d 1 = c 1 am + (1+ H) ;d 2 = c 2 -am + (1+ H) ;d 3 = c 3 bm + (1+ H) ;d 4 = c 4 -bm + (1+ H) f 1 = c 1 -am + (1+ H) ; f 2 = c 2 am + (1+ H) ; f 3 = c 3 -bm + (1+ H) ; f 4 = c 4 bm + (1+ H). Equations (10) and (11) can thn b applid at x = 1 to obtain 8

11 T i (1) = -(d 1 + d 2 + d 3 + d 4 ) -(1+ H )/ M + d 1 a + d 2 - a + d 3 b + d 4 -b +1 (12) T (1) = ( T (0) - (1+ f 1 + f 2 + f 3 + f 4 )) -(1+H )/ M + f 1 a + f 2 - a + f 3 b + f 4 -b +1 (13) whr again T (0), T i (1) and T (1) ar all unknowns. Th final rlationship ndd to clos th systm of quations is obtaind from th WSR modl [21]. For th simplst cas of first-ordr singl-stp Arrhnius chmical raction, th rlationship btwn WSR inlt tmpratur T i (1), outlt tmpratur T (1) and mass flux is m = (T (1) + DT) - T (1) Ê i xpá - E ˆ (14), r g ZA R T (1) -T i (1) Ë RT (1) whr th trm T i (1)+DT is th adiabatic flam tmpratur basd on an inlt tmpratur (to th WSR) of T i (1). In dimnsionlss form Eq. (14) bcoms Ê DT M = DaÁ Ë T (1) - T i (1) -1 ˆ Ê xpá - b ˆ Ë T ;Da r C A Z g P R,b E (15). (1) Lh 1 RT whr Da is th Damköhlr numbr and b th Zldovich numbr. For fixd T i (1), th rspons of T (1) to M producs wll-known Z-shapd curvs, howvr, for th currnt problm T (1) is strongly affctd by M du to combind ffcts of hat rcirculation, hat loss and wall conduction, thus, th rlationships btwn M and T (1) tak many forms. Equations (7a 7d, 12, 13 and 15) rprsnt 6 linar and on non-linar quation for svn unknowns c 1 -c 4, T (0), T i (1) and T (1). Ths ar radily solvd for various valus of th dimnsionlss paramtrs that compltly dfin th problm, namly th hat transfr paramtrs M, B and H and th combustion paramtrs D T, Da and b. 9

12 Choic of baslin numrical paramtrs Baslin dimnsionlss paramtrs that smi-quantitativly rprsnt macroscal hatrcirculating burnr xprimnts should b chosn. H = 0.05 is mployd, corrsponding to d = 3.9 mm and w = 78 mm as in our initial xprimnts [3, 22]. A Zl dovich numbr (b) of 70 is chosn, corrsponding to E = 42 kcal/mol (typical of hydrocarbon oxidation [21]). For xprimnts [3] using 3-turn Inconl burnrs having d 3.9 mm, t 0.51 mm, L 580 mm, k w 11.4 W/mK, thus B 2900, an xtinction limit occurs at D T 1.5 for M = 0.2; Da 10 7 lads to xtinction at ths conditions and is mployd in th calculations blow. Rsults - infinit raction rats First, hat transfr charactristics of countr-currnt hat xchangrs ar xamind for infinit raction rat (Da Æ ), thus T (1) - T i (1) = D T. Figur 2 (top) shows tmpratur profils for adiabatic (H = 0) and wall-conduction-fr (BÆ ) conditions, for which profils ar linar. For all M, th xhaust tmpratur T (0) is th adiabatic flam tmpratur 1+D T. Th WSR tmpratur T (1) incrass monotonically as M dcrass (Fig. 3) bcaus as M dcrass th hat transfr rat h 1 LD T is constant but m dcrass, thus th hat transfr pr unit mass incrass. It can b shown that T (1) =1+ DT Ê 1+ 1 ˆ Ë 2M (Da Æ,H = 0,B Æ ) (16) which is quivalnt to th wll-known rsult from th hat xchangr litratur = 1/(1+(NTU) -1 ), whr is th ffctivnss, for xchangrs with no stramwis wall hat conduction and qual m C P for th two strams. Not that for M Æ, thr is no hat rcirculation, thus th ractor tmpratur T (1) approachs th adiabatic flam tmpratur for cold ractants = 1+D T. With substantial hat loss (Fig. 2, middl) th profils ar quit diffrnt; tmpraturs ar ambint xcpt nar th WSR ( x = 1). Th WSR tmpratur T (1) is lowr than for H = 0 10

13 but bcaus Da Æ was assumd, th combustion-inducd tmpratur ris T (1) - T i (1) is still D T. This cas should still b considrd an xcss nthalpy burnr bcaus raction tmpraturs ar highr than adiabatic flam tmpraturs without rcirculation, i.., T (1) > 1+D T. Figur 3 shows that T (1) asymptots to a fixd valu as M Æ 0 and dos not dcras to ambint bcaus as M Æ 0, hat rcirculation is balancd by hat loss; blow a crtain valu of M, dcrasing M furthr (.g. in an xprimnt by dcrasing th mass flow rat or incrasing th lngth of th hat xchangr) has no ffct othr than to incras th fraction of th lngth of th xchangr whr both th ractant and product strams rmain at nar-ambint tmpraturs. It can b shown that th ractor tmpratur in th limit M Æ 0 is T (1) = (1+ D T )a -1 ;a a H + 4H(1+ H) 1+ 2H - 4H(1+ H) (Da Æ,B Æ,M Æ 0) (17) This obsrvation is xtrmly important to undrstanding th low-m xtinction limits bcaus it indicats that, vn in th prsnc of hat losss, without wall conduction thr is no mans to rduc th ractor tmpratur as M is dcrasd. This is quit diffrnt from combustors without hat rcirculation, whr sufficint rduction in mass flow rat will narly always lad to xtinction du to hat losss. With substantial wall hat transfr (B 0) but adiabatic conditions (H = 0) (Fig. 2, bottom), th tmpratur profils clarly show that wall conduction rmovs thrmal nrgy from th high-tmpratur gas nar x = 1 and rturns nrgy to th gas at lowr tmpraturs (smallr x ). Figur 3 shows that at small M wall conduction dominats and th WSR tmpratur T (1) is far blow that for BÆ (vn though th systm is still adiabatic) whras for larg M wall hat transport is insignificant bcaus gas-phas convction dominats wall conduction. Th xit tmpratur T (0) is still 1+D T bcaus th systm is adiabatic. In th limit M Æ 0, T ( x ), T i ( x ) and T w ( x ) (not shown) all tak on th valu of 1 + DT xcpt vry nar x = 0, whr T i ( x ) Æ 1, and nar x = 1, whr T (1) Æ 1 + 2DT (= 4 for th numrical valus mployd in Fig. 3) du to hat rcirculation causd by thrmal conduction along th wall. 11

14 Rsults - finit-rat chmistry Figur 4 shows th ffct of mass flux on WSR tmpratur for Da = Without hat rcirculation th H = 0 curv would hav th Z-shap typical of WSRs; with hat rcirculation, at small M (thus larg T (1) and fast raction) th uppr branch follows th DaÆ limit (Eq. (16)). Th lowr branch is probably unstabl as in convntional WSRs. (For plotting clarity th lowst, xtinguishd branch whr T (1) 1 is not shown.) For no wall conduction (BÆ ), thr ar no solutions for sufficintly larg M and two solutions for all smallr M, vn for non-adiabatic conditions. Thus, without wall conduction combustion is possibl at arbitrarily small M (or R). In fact, th highst T (1) (thus highst raction rats and longst rsidnc tims, farthst from xtinction) occur at th smallst M. This could b xpctd basd on small-m bhavior for DaÆ (Fig. 3). Exprimnts [1-3] show that small-m limits do indd xist, indicating that an additional factor is rquird to modl small-m xtinction limits. In contrast, with wall conduction (finit B), th rspons curvs bcom isolas with both lowr and uppr limits on M bcaus conduction of thrmal nrgy away from th WSR vicinity through th wall bcoms significant at small M. Onc conductd away from th WSR vicinity, som thrmal nrgy is transfrrd back to th gas via convction and a portion of this nrgy is thn lost to ambint. It is mphasizd that this mchanism is important only at small M, whr wall conduction is comptitiv with gas-phas convction. Figur 4 also shows that th larg-m xtinction limit is xtndd slightly by wall conduction, sinc hat rcirculation (thus WSR tmpratur) is low at larg M (Eq. 16), thus th incras in hat rcirculation providd by wall conduction incrass th WSR tmpratur slightly. This bhavior is discussd in mor dtail in th sction on th conductiv-tub burnr analysis. For non-adiabatic small-m cass, tmpratur profils ar similar to Fig. 2 (middl) in which all hat gnratd is lost to ambint bfor th products rach th burnr xit. Thus, simply quating rats of hat gnration and loss dos not yild xtinction critria bcaus all hat gnratd can b lost to ambint rathr than xhaustd at th xchangr xit without xtinction occurring. Jons t al. [7] prdictd low-m xtinction limits bcaus fixd hat loss was assumd; at low M hat loss would xcd hat gnration and xtinction occurs. In contrast, hat loss is calculatd systmatically in this work. 12

15 Figur 5 shows th ffct of B and thus wall conduction on th ractor tmpratur T (1) for svral fixd valus of M. Th gnral C-shap of ths curvs is th sam for all valus of M but th valu of B at th turning point is a non-monotonic function of M. Th most robust valu of M, corrsponding to th lowst valu of B at th turning point, is about 0.1 as would b xpctd from th isolas sn in Fig. 4. As was sn in Fig. 3, th maximum (and minimum, on th lowr branch of solutions) valus of T (1) occur at th lowst M. As M incrass, th curv flattns until it disappars ntirly at M 0.3. Figur 6 shows th ffct of M on th minimum D T supporting combustion (corrsponding to th minimum ful concntration, thus xtinction limit). As in Fig. 4, without wall conduction (BÆ ) no small-m xtinction limit xists. For finit B, both small-m and larg- M limits xist. Consistnt with Fig. 4, th larg-m limit is slightly xtndd by dcrasing B whras th small-m limit is drastically narrowd by dcrasing B. It should b strssd that th valu of B ndd to affct xtinction is much smallr than that xpctd basd on simplistic stimats. Th ovrall ratio of stramwis convction to wall conduction is m C P /(k w t/l) = MB/2. For th B = 10,000 curv in Fig. 6, xtinction limits ar affctd for all M 0.01, thus MB/2 50. Basd on simplistic stimats, no ffct wall conduction ffcts would b xpctd unlss MB/2 1. Th powrful wall conduction ffcts rsult from th larg wall tmpratur gradints nar th WSR whn hat losss ar prsnt (Fig. 2, middl), which ar much largr than th man gradint undr ths conditions. In Fig. 6, M is plottd on a logarithmic scal for clarity. Whn plottd linarly th curv is vry stp on th small-m xtinction branch but is much shallowr on th larg-m branch. This trnd is vry similar to that sn in xprimnts [1-3]. Figur 7 shows th ffct of M on th minimum valu of B supporting combustion for D T = 1.5. Th sam trnds as dmonstratd abov can b sn: both small-m and larg-m xtinction limits xist for non-adiabatic conditions, th small-m limits ar much narrowr at small B (and disappar as BÆ ) and larg-m limits ar xtndd slightly by wall conduction. Also, for th spcial cas H = 0, B 9, as M is dcrasd conditions chang from xtinguishd to burning to xtinguishd and finally back to burning again. This odd bhavior rsults from intractions of incrasing conduction-fr T (1) (Eq. 16), incrasing wall conduction ffcts and incrasing rsidnc tim as M dcrass. In particular, as M dcrass, th drop in tmpratur lads to a small-m xtinction limit, vn for adiabatic cass, if th dcras in M dcrass th 13

16 right sid of Eq. (15) (i.., th raction rat drops du to a drop in th raction tmpratur T (1) ) mor than th dcras in M incrass th ractor tmpratur. Of cours, sinc th minimum T (1) for adiabatic conditions is 1 + 2D T, at sufficintly small M, th xtinction limit will always disappar for adiabatic conditions, maning that thr could b multipl xtinction limits changing only M. Figur 7 (and additional calculations not shown) rval that this bhavior was found only for narrow rangs of B and probably would not b xprimntally obsrvabl. Effct of boundary conditions To tst th ffcts of th assumption (Eqs. 7a and 7b) that th nds th dividing wall ar adiabatic, th opposit cas of convction boundary conditions on th nds was xamind. At th inlt nd of th xchangr, th most natural choic would probably b to assum a hat loss pr unit ara h 2 (T w (0)-T ) across th ntir thicknss t of th wall that balancs th conductiv flux k w (dt w /dx) x=0. This can b writtn as (a - HB * )c 1 + (-a - HB * )c 2 + (b - HB * )c 3 + (-b - HB * )c 4 = 0; B * B t 2L = h 1L k w (7a ) whr th additional gomtrical paramtr t/2l that did not appar as a sparat paramtr in th analysis for adiabatic wall nds must now b spcifid. Substitution of Eq. (7a ) for (7a) was found to hav ngligibl ffct on th rsults xcpt for th unralistic cas of t/2l nar or largr than unity. This is bcaus at low M, in th rgion nar x = 0, T w (0) is nar ambint tmpratur bcaus of hat loss at largr x (S Fig. 2b) and thus no additional hat loss rsults from th chang in boundary condition, whras at high M, hat losss ar inconsquntial vn ovr th much largr surfac ara of th xchangr itslf. For th WSR nd of th xchangr, th most natural choic would probably b to assum a hat transfr pr unit ara h 1 ( T (1)-T w (1)), whr T (1) = (T i (1)+T (1))/2 is th man gas tmpratur sn by th wall nd (s Fig. 1), across th ntir thicknss t of th wall that balancs th conductiv flux k w (dt w /dx) x =1. This boundary condition can b writtn as 14

17 Ê T (a + B * ) a c 1 + (-a + B * ) -a c 2 + (b + B * ) b c 3 + (-b + B * ) -b c 4 = B * i (1) + T (1) ˆ Á -1 (7b ). Ë 2 Figur 8 shows a comparison of prdictions obtaind using Eqs. (7a) and (7b) to thos obtaind with (7a ) and (7b ). It would b mislading to show rsults as a function of th nw gomtrical paramtr t/2l for fixd B and M sinc th only way to vary t/2l for fixd B and M would b by varying th wall thrmal conductivity k w according to k w ~ t -1, and varying k w alon would hav a drastic ffct burnr prformanc. Consquntly, rsults ar shown for varying t with all othr dimnsional paramtrs fixd, maning that as t/2l is incrasd, B dcrass proportionally so that B * is fixd. It can b sn in Fig. 8a that for B * = 1, which is rprsntativ of our initial xprimnts [3, 22], thr is practically no diffrnc btwn th adiabatic and convction boundary conditions, although anothr branch of solutions appars at vry low B. This branch is considrd non-physical bcaus, as Fig. 8b shows, it corrsponds to larg valus of t/2l (i.., a hat xchangr that is narly as larg in th spanwis dirction as it is in th stramwis dirction). Th currnt modl dos not includ hat losss from th WSR nd of th burnr that would bcom critical for this aspct ratio. Th high-b branch of solutions shown in Fig. 8a was found to b almost indpndnt of B* xcpt for larg valus of t/2l. Ths rsults indicat that th choic of adiabatic wall nds is rasonabl for th conditions whr th currnt modl is assrtd to b valid. Combustors with conductiv tub walls Analysis Exprimnts [9] hav shown that by holding combustion on th nd of a thrmallyconductiv tub through which th ractants pass and ar prhatd by th tub wall, combustion can b sustaind undr conditions (spcifically tubs whos diamtrs ar smallr than th qunching distanc) that could not sustain combustion without hat rcirculation. This conductiv-tub configuration provids an altrnativ to th countr-currnt or Swiss-roll hat xchangr for rcycling a portion of th hat rlas back to th ractants. Thus, it is of intrst 15

18 to compar th prformanc of countr-currnt and conducting-tub hat-rcirculating combustors. Th configuration of Fig. 1 (lowr) is mployd in ordr to mak th comparison of th two typs of hat rcirculating combustors as valid as possibl. Hat transfr is dividd into two zons: th tub or channl wall and th gas. Th thrmally-thin wall transports hat along its lngth, transfrs hat to/from th ractant stram with hat transfr cofficint h 1, and loss hat to ambint with cofficint h 2. In addition to stramwis convction, hat transport in th gas is via convction to th wall. A no-flux condition is imposd at th lin of symmtry in th gas. Th configuration shown could b ithr a cylindrical tub or plan channl, th only diffrnc bing th choic of Nusslt numbr for computing h 1 (3.7 for th plan channl vs. 4.4 for th tub [19]). As with th countr-currnt configuration, ambint gas tmpratur and adiabatic wall conditions ar assumd at x = 0. Th adiabatic wall boundary condition cannot b applid at th WSR nd of th tub, othrwis thr would b no hat transfr from th combustion zon to th tub and thus no gas prhating would occur. In gnral th boundary condition will dpnd on th dtails of way in which th raction zon is anchord to th nd of th tub and would involv additional lngth scals and/or hat transfr cofficints. To avoid nding to spcify additional paramtrs, it is assumd that th WSR combustion is still dscribd by Eq. (15) and is followd by a short zon (short nough compard to th tub lngth L that hat losss can b nglctd) downstram of th WSR whr hat is transfrrd to th wall (assumd isothrmal through th WSR and this downstram zon) until th gas and wall tmpraturs ar qual. Th thrmal nrgy transfrrd from th gas to th wall is thn conductd along th wall in th upstram dirction. Clarly this boundary condition rprsnts an uppr bound on th prformanc of th conducting-tub combustor sinc (1) th hat transfr from th burnt gas to th tub wall dos not diminish th WSR tmpratur, which would rduc th raction rat and (2) th maximum possibl nrgy is transfrrd from th burnt gas to th tub wall and subsquntly rcycld back to th gas upstram of th WSR. In practic this boundary condition would rquir a hat transfr cofficint in this downstram thrmal quilibration zon much highr than h 1 sinc this zon is much shortr than L and thus has much lss ara availabl for hat transfr to th gas. (Altrnativly this zon could hav a much lowr loss cofficint h 2.) 16

19 Enrgy balancs on th wall and th gas radily yild, using th sam nondimnsionalization as for th countr-currnt hat xchangr analysis, 2 B d 2 d x 2 T w - (1+ H) T w = -( T i + H) (18) M d T i d x + T i = T w (19) which can b combind to obtain d 3 d x 3 T w + 1 M d 2 T w B(1+ H) dt - w 2 d x 2 d x - BH T w = - BH 2M 2M (20) Th solution to (20) is T w ( x ) =1+ c 1 a x + c 2 b x + c 3 d x (21) whr c 1 c 3 ar unknown constants and a, b and g ar th roots of l M l2 - B(1+ H) l - BH 2 2M = 0 (22). On boundary condition is obtaind by assuming th tub wall is adiabatic (thus has zro tmpratur gradint) at th inlt nd, thus ac 1 + bc 2 + gc 3 = 0 (23) In dimnsional trms, th nrgy balanc corrsponding to th aformntiond wall boundary condition at x = L is -kt dt w = m C P (T w (L) - T (L)) dx x= L (24a). 17

20 whr for consistncy with th countr-currnt xchangr analysis th notation T (L) is usd to dnot th dimnsional WSR tmpratur. By applying Eq. (21) at x = 1, Eq. (24a) can b writtn as Ê Á Ë M + 2a B ˆ a c + Ê M + 2b ˆ Ê Á 1 Ë B b c 2 + M + 2g ˆ Á g c Ë B 3 = M( T (1) -1) (24b) whr th dimnsionlss WSR tmpratur T (1) is unknown. Two additional rlations ar obtaind by substituting (23) into (18) to obtain an xprssion for T i ( x ) and applying this xprssion at x = 0 and x = 1 to yild, rspctivly, Ê ˆ Ê ˆ Ê ˆ Á 1+ H - 2a2 c 1 + Á 1+ H - 2b2 c 2 + Á 1+ H - 2g2 c 3 = 0 (25) Ë B Ë B Ë B Ê ˆ Ê ˆ Ê ˆ Á 1+ H - 2a2 a c 1 + Á 1+ H - 2b2 b c 2 + Á 1+ H - 2g2 g c 3 - T i (1) = -1 (26) Ë B Ë B Ë B whr th condition T i (0) = 1 has bn usd in (25) and T i (1) is unknown. Rsults Equations (15), (23), (24b), (25) and (26) rprsnt 1 nonlinar quation and 4 linar quations for th unknowns c 1 c 3, T i (1) and T (1). In ordr to mak th comparison btwn th countr-currnt and conducting-tub hat-rcirculating combustors as valid as possibl, th sam rprsntativ raction rat paramtrs Da = 10 7 and b = 70 and hat loss paramtr H = 0.05 ar mployd in this sction. Nvrthlss, th prvious baslin hat rlas paramtr D T = 1.5 could not b mployd bcaus with ths raction rat, hat loss and hat rlas paramtrs, thr wr no solutions to th govrning quations for any valu of Biot numbr B xcpt at xtrmly low valus of mass flux M (<10-3 ). Consquntly, th baslin hat rlas 18

21 paramtr was incrasd 50% to D T = 2.25; at this valu of D T th maximum valus of M and T (1) wr similar for th two typs of combustors. Figur 9 shows th ffct of mass flux M on WSR tmpratur for th conductiv-tub configuration (analogous to Fig. 4 for th countr-currnt configuration). As with th countrcurrnt configuration, with hat loss ths plots ar isolas indicating maximum and minimum valus of M supporting combustion. Unlik th countr-currnt combustor, howvr, dcrasing B substantially incrass th maximum M bcaus in th conductiv-tub cas hat conduction along th wall (whos ffct is proportional to 1/B) is th only mans to accomplish hat rcirculation. Significantly, howvr, is that th isolas for diffrnt valus of B ar narly concntric in th countr-currnt cas but ar displacd to lowr M as B incrass for th conductiv-tub cas. This indicats that for both countr-currnt and conductiv-tub cass, incrass wall hat conduction (i.. lowr B) raiss th minimum valu of M supporting combustion. Figur 10 illustrats why wall hat conduction, which is ssntial for hat rcirculation in th conductiv-tub configuration, still rducs low-m prformanc. At highr valus of B, thus lowr k w (Fig. 10, uppr), th wall tmpratur dcrass to ambint on a scal smallr than th hat xchangr lngth whras at lowr B, thus highr k w (Fig. 10, lowr), th ntir wall is ssntially isothrmal at a tmpratur abov ambint. In th lattr cas th systm loss mor hat to ambint than at highr B without a corrsponding bnfit of incrasd hat rcirculation, thus making th systm mor suscptibl to xtinguishmnt by hat losss. Of cours if B is too larg, th tmpratur diffrnc btwn th gas and th wall incrass (bcaus of th incrasd difficulty in transfrring hat along th wall) which in turn dcrass th amount of hat rcirculation and lads to an uppr limit on B. Figur 11 shows th ffct of M on th minimum D T supporting combustion for th conductiv-tub configuration (analogous to Fig. 6 for th countr-currnt configuration). As with th countr-currnt configuration, without wall conduction (BÆ ) no small-m xtinction limit xists and for finit B, both small-m and larg-m limits xist. In contrast to th countrcurrnt combustor, howvr, B has a substantial ffct on th high-m xtinction limit for th conductiv-tub configuration for th rason discussd in th prvious paragraph. Th curvs in Fig. 11 show a chang in slop to a smallr valu for small M corrsponding to th condition whr hat losss ar so svr that th ntir hat xchangr is ssntially at ambint tmpratur and no hat rcirculation occurs. In this cas th WSR oprats with ambint inlt 19

22 tmpratur ( T i (1) = 1) and th prformanc of th systm can b dscribd by Eq. (15) only. This bhavior also occurs in a similar mannr for th countr-currnt combustor but is not sn in Fig. 6 bcaus it occurs at th sam valus of D T as for th conductiv-tub combustor, which is wll off th scal of Fig. 6. A comparison of Figs. 6 and 11 shows that th inhrnt prformanc of countr-currnt burnrs is suprior to that of conductiv-tub burnrs having th sam raction rat and hat loss paramtrs. In particular, whn B is considrd to b a dsign paramtr that can b optimizd indpndntly for th two configurations, for all M th minimum D T rquird to sustain combustion is lowr for countr-currnt burnrs. For xampl, at M = 0.1 th minimum D T is about 1.25 for th countr-currnt burnr and 2.15 for th conductiv-tub burnr and at M = 0.01 th rspctiv valus ar about 1.0 and 1.9. This diffrnc is particularly notworthy sinc th wall boundary condition at th WSR is prhaps th most favorabl possibl for combustor prformanc. Th bttr prformanc of th countr-currnt burnr is fundamntally du to th fact that thr is lss hat loss pnalty for hat rcirculation in this cas. Spcifically, as discussd in th countr-currnt analysis, wall thrmal conduction lads to lowr WSR tmpraturs whn hat losss ar prsnt. In th conductiv-tub cas this wall conduction is th only mchanism for hat rcirculation and thus must b prsnt. In contrast, for th countrcurrnt cas hat rcirculation can b accomplishd without wall thrmal conduction, thrby mor ffctivly dcoupling hat loss from hat rcirculation. Application to microscal dvics Wll-instrumntd macroscal xprimnts ar valuabl tools for prdicting microscal prformanc by invoking similitud (constant dimnsionlss paramtrs) sinc microscal dvics ar notoriously difficult to instrumnt. Th analysis prsntd hr may b usful for this purpos. For gomtrically similar dvics (d ~ L ~ w ~ t w ) with laminar flow (h ~ 1/d) it is asily shown that M ~ Ud/a g, B ~ k g /k w,, Da ~ d 2 Z/a g and H = constant, whr a g is th gas thrmal diffusivity. Th biggst challng is to maintain constant M and Da simultanously as d dcrass. This would rquir (sinc a g ~ P -1, whr P is th prssur) P ~ d -2 and U ~ d. Changing prssur is problmatic, howvr, sinc th ovrall raction rat paramtrs Z and E ar gnrally prssur-dpndnt. If prssur is fixd thn gomtrical similarity cannot b 20

23 maintaind. Similitud could b maintaind with U = constant, L ~ d 3, w ~ d and t w ~ d 5, but bcaus of manufacturing limitations this is not practical for larg rangs of d. Prhaps th most viabl scaling approach, is to mploy gomtrical similarity, constant prssur, U ~ d -1 (thus constant M and R) and to maintain raction rat similarity incras th ful concntration D T such that th right-hand sid of Eq. (15) is constant vn though Da dcrass with dcrasing d. For xampl, for th countr-currnt combustor with fixd M = 0.01, B = 10 4, H = 0.05, b = 70 and initial valus D T = 1.1 and Da = 10 7, as d is dcrasd from its nominal valu (d o ) th rquird D T ar wll fit by th rlation D T = (d/d o ) -2. Th tmpratur profils ar narly idntical for ths valus of D T. Not that according to this xprssion D T riss rapidly as th scal (d) dcrass. This point is of particular intrst for microscal combustion applications, whr it may not b of spcial valu to burn vry lan mixturs or fuls with vry low hating valu but hat losss will b xtrmly problmatic vn for nar-stoichiomtric mixturs du to th normous surfac ara to volum ratios associatd with microscal dvics. Discussion and conclusions A simpl modl of hat-rcirculating burnrs was dvlopd, including hat transfr from product to ractant strams in a countr-currnt hat xchangr, hat loss from both strams to ambint, thrmal conduction along th dividing wall btwn th two strams and chmical raction in a wll-stirrd ractor. Th prdictd ffcts of non-dimnsional hat loss (H) and ful concntration (D T ) ar found to b straightforward. (Th ffcts of Damköhlr numbr (Da) and activation nrgy (b), not shown in this papr, also follow th xpctd trnds.) In contrast, th ffcts of mass flux (M) and Biot numbr (B) ar nithr straightforward nor vn monotonic. In particular, xtinction limits ar prdictd at both larg M (du to blow-off typ limits wll known for WSRs [21]) and small M (du to hat losss which ar not important for th larg-m limit). Most significantly, th small-m xtinction limit occurs only with wall conduction (finit B) bcaus without this damping factor, as M dcrass th amount of hat rcirculatd (rlativ to hat gnration) incrass without bound. Th importanc of wall conduction cannot b ovrstrssd sinc without wall conduction (infinit B), vn with low ractivity fuls (low Da or D T ) or larg hat losss no small-m limit xists. This is bcaus as M dcrass, hat rcirculation incrass (causing highr WSR tmpraturs and raction rats) and WSR 21

24 rsidnc tims incras; at sufficintly low M this combination is always capabl of ovrcoming losss. With wall conduction, som thrmal nrgy is transfrrd away from th WSR rgion, r-dpositd into th gas, thn lost to ambint, which lads to small-m limits. It is mphasizd that wall conduction is not a hat loss mchanism, instad it r-distributs thrmal nrgy within th dvic. In th currnt modl for th countr-currnt xchangr only gas-phas thrmal nrgy can b lost to ambint. It should also b notd that in a spiral hat xchangr whr (unlik th linar xchangr modld hr) th viw factor of th dividing wall with itslf is non-zro, th radiativ hat transfr btwn walls would hav a similar ffct to stramwis wall conduction sinc this typ of radiativ transfr would also incras hat transfr within th solid phas without a corrsponding incras in hat xchang with th gas. W hav obsrvd prliminary vidnc of this in dtaild numrical computations of Swiss Roll combustor prformanc [23] th pak tmpratur in th combustor dcrass whn radiativ transfr is includd. Of cours, if th gass wr sufficintly absorbing, radiation could incras hat xchang with th gas and thus incras combustor tmpraturs, but th typical Planck man absorption lngth of combustion products (1 2 m) far xcds channl dimnsions of laboratory-scal or microscal apparatuss and thus significant participation from gas-phas radiation would not b xpctd. Whil stramwis wall conduction dominats burnr prformanc, with th thrmallythin wall modl mployd hr spanwis conduction (across th wall) rsults in no tmpratur gradints and thus dos not affct prformanc. This is considrd ralistic sinc for practical burnr matrials and dimnsions, wall thrmal rsistanc is much lowr than gas-phas thrmal rsistanc (s Countr-currnt combustor Approach). An altrnativ configuration of combustion stabilizd at th xit of a tub, whr hat rcirculation occurs via conduction through tub wall was also analyzd. Although this configuration crtainly qualifis as a hat-rcirculating combustor, its prformanc was found to b infrior to that of th countr-currnt configuration in that much highr dimnsionlss ful concntrations ar rquird to sustain combustion at th sam M. It is blivd that th conclusions of this study, whil basd on highly simplifid transport and chmistry sub-modls, ar applicabl to ral dvics also, particularly rgarding ffcts of wall conduction. Similar trnds would b xpctd with complx chmistry if th ovrall activation nrgy is larg. Using constant ovrall hat transfr cofficints is rasonabl for 22

25 laminar flow, though for turbulnt flow at high R, roughly h 1 ~ m, thus M constant (though th Damköhlr numbr Da is proportional to h 1-1 and would dcras with incrasing m, thus lading to xtinction limits at larg M vn for turbulnt flow.) Also, linar and spiral hat xchangrs will hav diffrnt H but probably similar rspons to hat loss. Th WSR combustion modl is considrd rasonabl although approximat sinc all diffusiv transport within th WSR is nglctd. Morovr, catalytic combustion is advantagous for small-m hat rcirculating burnrs [3, 5]. If catalyst is prsnt only in th WSR volum, th WSR modl is likly rasonabl for catalytic combustion at small M whr rsidnc tims ar long and raction is kintically limitd (rathr than potntially transport limitd as at larg M). Acknowldgmnts This work was supportd by th DARPA Microsystms Tchnology Offic, undr contracts DABT63-99-C-0042 and N Nomnclatur A R WSR ara (rplacs WSR volum in thr-dimnsional problms) a S Eqs. (5), (22) B Scald Biot numbr = 2h 1 L 2 /k w t b S Eqs. (5), (22) c Intgration constants (Eq. 5) C p Hat capacity at constant prssur Da Damköhlr numbr = r g C P A R Z/Lh 1 d Channl hight d H Hydraulic diamtr E Activation nrgy H Dimnsionlss hat loss cofficint = h 2 /h 1 g S Eq. (22) h 1 Hat transfr cofficint to dividr wall h 2 Hat loss cofficint to ambint k Thrmal conductivity L Hat xchangr lngth M Dimnsionlss mass flux = m C P /h 1 L m Mass flow rat pr unit dpth NTU Numbr of Transfr Units Nu Nusslt numbr = h 1 d/k g Pr Prandtl numbr m g C P /k g R Gas constant R Rynolds numbr = m d/µ T Tmpratur 23

26 T U WSR w x x Z Dimnsionlss tmpratur = T/T bulk flow vlocity Wll-stirrd ractor Channl dpth Stramwis coordinat Dimnsionlss stramwis coordinat = x/l Pr-xponntial factor in raction rat xprssion b Non-dimnsional activation nrgy (Zldovich numbr) = E/RT DT Tmpratur ris for adiabatic complt combustion Hat xchangr ffctivnss µ Dynamic viscosity r dnsity t Dividing wall thicknss Subscripts g i w product sid of hat xchangr gas ractant sid of hat xchangr dividing wall ambint conditions Rfrncs 1. S. A. Lloyd, F. J. Winbrg, Natur 251 (1974) S. A. Lloyd, F. J. Winbrg, Natur 257 (1975) L. Sitzki, K. Borr, S. Wussow, E. Schustr, P. D. Ronny, A. Cohn, AIAA Papr (2001). 4. F. J. Winbrg, D. M. Row, G. Min, P. D. Ronny, Proc. Combust. Inst. 29 (2002) J. Vican, B. F. Gajdczko, F. L. Dryr, D. L. Milius, I. A. Aksay, Proc. Combust. Inst. 29 (2002) A. C. Frnandz-Pllo, Proc. Combust. Inst. 29 (2002) A. R. Jons, S. A. Lloyd, F. J. Winbrg, Proc. Roy. Soc. Lond. A. 360 (1978) P. D. Ronny, Proc. Combust. Inst. 27 (1998) B. Cooly, D. Walthr, A. C. Frnandz-Pllo, Exploring th Limits of Microscal Combustion, 1999 Fall Tchnical Mting, Wstrn Stats Sction/Combustion Institut, Irvin, CA, Octobr 25-26,

27 10. R. R. Ptrson, J. M. Hatfild, A Catalytically Sustaind Microcombustor Burning Propan, Proc Intrnational Mchanical Enginring Congrss and Exposition (IMECE), Nw York, Novmbr 11-16, J. A. Wünning, J. G. Wünning, Prog. Enrgy Combust. Sci. 23 (1997) M. Katsui, T. Hasgawa, Proc. Combust. Inst. 27 (1998) K. Maruta, K. Muso, K. Takda, T. Niioka, Proc. Combust. Inst. 28 (2000) P. G. Krogr, in: Advancs in Cryognic Enginring, K. D. Timmrhaus (Ed.), Plnum Prss, Nw York, 1967, pp G. Vnkatarathnam, S. P. Narayanan, Cryognics 39 (1999) X. Yin, H. H. Bau, J. Hat Trans. 188 (1996) R. B. Ptrson, Microscal Thrmophysical Enginring 3 (1999) Y. B. Zl'dovich, Combust. Flam 39 (1980) R. K. Shah, A. L. London, Laminar Flow: Forcd Convction in Ducts, Acadmic Prss, J. N. dris, Twlfth Symposium (Intrnational) on Combustion, Combustion Institut, 1969, p I. Glassman, Combustion (3 rd Ed.), Acadmic Prss, K. Maruta, K. Takda, J. Ahn, K. Borr, L. Sitzki, P. D. Ronny, O. Dutchman, Proc. Combust. Inst. 29 (2002) J. Kuo, C. Eastwood, L. Sitzki, K. Borr, P. D. Ronny, Numrical modling of hat rcirculating burnrs, 3 rd Joint US Sctions Mting, Combustion Institut, Chicago, IL, March

28 Figur Captions Figur 1. Schmatic diagram of hat xchangr / combustor configurations analyzd. Uppr: countr-currnt systm; lowr; conductiv tub. Figur 2. Tmpratur profils in th countr-currnt hat xchangr for Da =, M = 0.2, D T = 1.5. Top: H = 0, B = ; middl: H = 1, B = ; bottom: H = 0, B = 10. Figur 3. Effct of mass flux on WSR tmpratur in th countr-currnt combustor for infinit raction rats (DaÆ ) with D T = 1.5. Figur 4. Effct of mass flux on WSR tmpratur in th countr-currnt combustor for finit raction rats (Da = 10 7 ) with D T = 1.5. For rfrnc, th adiabatic, conduction-fr infinitrat curv from Fig. 3 is also shown. Figur 5. Effct of Biot numbr (B) on WSR tmpratur in th countr-currnt combustor for finit raction rats (Da = 10 7 ) with H = 1.5 and D T = 1.5, for svral fixd valus of th mass flux M. Figur 6. Effct of mass flux on ful concntration (xprssd as D T ) at th xtinction limit in th countr-currnt combustor for varying valus of th Biot numbr (B). Da = 10 7, H = Figur 7. Effct of mass flux on Biot numbr (B) at th xtinction limit in th countr-currnt combustor for varying valus of th hat loss cofficint (H). Da = 10 7, D T = 1.5. Figur 8. Effct of th boundary condition on th hat xchangr dividing wall on th prdictd xtinction limits of th countr-currnt combustor. (a): Comparison of convction boundary conditions (for B * = 1) to adiabatic walls in Biot numbr (B) mass flux (M) spac. (b): sam rsults plottd in trms of limit mass flux (M) as a function of wall thicknss (t/2l). Figur 9. Effct of mass flux on WSR tmpratur in th conductiv-tub combustor for finit raction rats (Da = 10 7, b = 70) with D T = Also shown for rfrnc is th cas for 26