NON-ADIABATIC COMBUSTION WAVES FOR GENERAL LEWIS NUMBERS: WAVE SPEED AND EXTINCTION CONDITIONS

Transcription

1 ANZIAM J. 46(2004), 1 16 NON-ADIABATIC COMBUSTION WAVES FOR GENERAL LEWIS NUMBERS: WAVE SPEED AND EXTINCTION CONDITIONS A. C. MCINTOSH 1,R.O.WEBER 2 ang.n.mercer 2 (Receive 14 August, 2002) Abstract This paper aresses the effect of general Lewis number an heat losses on the calculation of combustion wave spees using an asymptotic technique base on the ratio of activation energy to heat release being consiere large. As heat loss is increase twin flame spees emerge (as in the classical large activation energy analysis) with an extinction heat loss. Formulae for the non-aiabatic wave spee an extinction heat loss are foun which apply over a wier range of activation energies (because of the nature of the asymptotics) an these are explore for moerate an large Lewis number cases the latter representing the combustion wave progress in a soli. Some of the oscillatory instabilities are investigate numerically for the case of a reactive soli. 1. Introuction Earlier work [8, 12] has explore the classical combustion wave (premixe flame propagation) problem using a ifferent grouping of parameters to that use in the past. In particular, instea of using asymptotic analysis base on activation energy being large, the same equations have been recast in terms of the ratio of activation energy to heat release. This theory is istinct from the traitional large activation energy approach, since there is no one to one mapping between the two approaches. Nevertheless there is a goo corresponence between the two theories, an arguably the new approach has the istinct avantage of still being vali for quite low activation energies (see [12, Figure 3]). The present approach has more reaily shown the bifurcation behaviour that can occur at moerate values of the ratio (þ) of activation energy to heat release. Such instabilities have been well known in the work of Bayliss an co-authors [2, 5]. The 1 Fuel an Energy Department, University of Lees, Lees, LS2 9JT, U.K. 2 School of Mathematics an Statistics, University College of New South Wales, ADFA, Canberra, ACT 2600, Australia; row@afa.eu.au. c Australian Mathematical Society 2004, Serial-fee coe /04 1

2 2 A. C. McIntosh, R. O. Weber an G. N. Mercer [2] later work of Metcalf et al. [10] an Balmforth et al. [1] has also shown that this is a general phenomenon that can occur at general Lewis numbers an with ifferent reaction terms. For Arrhenius kinetics with infinite Lewis number (that is, soli fuels) there are inications (known experimentally [11] an confirme numerically [4, 6] that there is a possible perio-oubling route to chaos. The paper by Weber et al.[12] was able to confirm this with an accurate value etermine for the minimum wave spee. It is known from classical large activation energy analysis that heat losses have a great effect on the stability of combustion fronts, an with the new scheme for the grouping of the parameters, Mercer et al.[9] have shown numerically that oscillatory instabilities occur for non-aiabatic systems with again a perio-oubling route to chaos. This present paper concentrates on the influence of global heat loss on the propagation spees of combustion fronts for general Lewis numbers. In particular we erive moifie analytical expressions for the wave spee using large þ asymptotic analysis (rather than large activation energy asymptotics). As mentione earlier, þ is the ratio of activation energy to heat release. There is a wie range of reaction possibilities. Normally for the common hyrocarbon reactions the heat release is large but generally the activation energy is larger still, so that þ is then large. Only if the activation energy is low with high heat release, woul the analysis here not be vali, an this is not usually foun in practice. The remaining case of low heat release with moerate activation energy is feasible since then the wave propagates with a thicker reaction region (since the activation temperature is low) but with a low temperature rise. Though the kinetics can certainly not be escribe accurately by one-step chemistry as use here, qualitatively an example of low heat release with moerate activation energy is the cool flames phenomena ([7, pp ]). Thus there can be practical cases of burning where heat release is small an activation energy is not great, an consequently this theory has the avantage of being able to wien the applicability of asymptotic analysis to similar cases in gases an solis where a relatively low activation energy value pertains. Using this approach, extinction values of heat loss for these fronts are erive an we further show some numerical results concerning the soli fuel oscillatory instability. 2. Mathematical moel The erivation of the governing equations is well known from earlier papers (see, in particular, [12]). For the case of global heat loss, the non-imensional forms of the ifferential equations are 2 u ¾ 2 + c u ¾ + ye 1=u `.u u a / = 0; (2.1)

3 [3] Non-aiabatic combustion waves for general Lewis numbers 3 7 \ 7 7 D X X D \ [ f (TXLOLEULXP 7 7 I X X I \ [ +RW 7 7 D X X D \ [ f &ROG FIGURE 1. Schematic of combustion wave with heat loss. 1 L 2 y ¾ 2 + y ¾ þye 1=u = 0; (2.2) where it shoul be recognise that ¾ = x ct is a coorinate following the combustion wave travelling with spee c an time t is non-imensionalise with respect to a characteristic time t 0 c p.e=r/=qa, with istance x non-imensionalise with respect to a characteristic length 0.k=²/.E=R/=QA. The parameter groupings are as follows: Lewis number L k=.² Dc p /, heat loss ` hse=.v² QAR/, ratio of activation energy to heat release þ Ec p =RQ, temperature u RT=E, where k=²c p is thermal thermal iffusivity (m 2 s 1 )(k, ² an c p are thermal conuctivity (W m 1 K 1 ), ensity (kg m 3 ) an specific heat (J kg 1 K 1 ) respectively), h is an overall heat transfer coefficient (W m 2 K 1 ), S is surface area (m 2 ), V is volume (m 3 ), E is activation energy (J mol 1 ), Q is the heat of reaction (J kg 1 ), A is the pre-exponential reaction constant (s 1 )anr is the Universal gas constant (8:314 J mol 1 K 1 ). In these equations y represents the mass fraction of fuel an the temperature T (K) is non-imensionalise with respect to the activation temperature E=R, rather than the ambient temperature. A schematic of the propagating combustion wave is given in Figure 1. As inicate in Figure 1, the bounary conitions are ¾ =+ : u = u a ; y = 1; (2.3a) ¾ = : u = u a ; y = 0: (2.3b) A point of iscussion is relevant here concerning the approach to solving (2.1) an (2.2). If one was seeking to solve these equations without first applying some

4 4 A. C. McIntosh, R. O. Weber an G. N. Mercer [4] asymptotic limit, an with full account taken of the ambient temperature as a parameter, then the approach of using an ignition temperature (such as that use by Bayliss an Matkowsky [3]) woul be necessary, couple with a rescaling of temperature with a consequent ajustment to the form of the reaction term. We o not use an ignition temperature here since the asymptotic analysis consiere later in Section 3 naturally brings in an inner region where the reaction ominates, but is negligible outsie such a zone. The bounary conitions (2.3), giving the same bounary conition on temperature, are a result of the slow ecay to the original ambient temperature T a (non-imensionally u a )at¾ = on the equilibrium sie. Later it is assume that u a = 0 with little lost in the main results particularly concerning extinction. 3. Exact integral an asymptotic analysis, finite Lewis number 3.1. Exact integral By multiplying (2.2)by1=þ an aing it to (2.1), we have the erive result ( 2 u + y ) + c ( u + y ) `.u u ¾ 2 a / = 0: (3.1) þl ¾ þ Integrating this result from to + implies the following result for wave spee: c = þ` +.u u a / ¾: (3.2) Note that as ` 0, so the temperature tail shown in Figure 1 extens to infinity with c reaching its aiabatic value. In this work ` will always be consiere small, that is o.1/. The cooling tail is in fact characterise by a length of O.` 1 / Outer zone asymptotics Away from the reaction zone, the solution for temperature an fuel in the preheat an equilibrium zones must satisfy 2 u ¾ 2 + c u ¾ `.u u a/ = 0; 1 L 2 y ¾ 2 + c y ¾ = 0: This is on the basis of u 1 in the preheat zone, an y 0 in the equilibrium zone. For ¾>0 on the col (pre-heat) sie, the solutions fitting the bounary conitions are u = u a +.u f u a /e m¾ ; (3.3) y = 1 e L¾ ;

5 [5] Non-aiabatic combustion waves for general Lewis numbers 5 where m c c2 + 4`.> 0/; (3.4) an u f represents the flame temperature, that is, u f u ¾=0. For ¾<0 on the hot (equilibrium) sie, the solutions fitting the bounary conitions are where u = u a +.u f u a /e n¾ ; (3.5) y = 0; (3.6) n c c2 + 4`.> 0/: (3.7) Substituting these approximate solutions into the wave spee equation (3.2) implies 0 + c þ`.u u a / ¾ + þ`.u u a / ¾ 0 + [ 0 0 ] ( 1 þ`.u f u a / e n¾ ¾ + e m¾ ¾ þ`.u f u a / n + 1 ) ; m that is, c þ c 2 + 4`.u f u a /: (3.8) Note as ` 0sou f a special case. u a u a + 1=þ, so that aiabatic conitions satisfy (3.8) as 3.3. Inner zone asymptotics In the inner reaction zone we re-scale the temperature an space. Thus we efine an u = u a + þ = u a + 1 þ + þ = þ¾; (3.9a) (3.9b) where u a is the aiabatic temperature an is the rescale space variable. Consequently in this zone, the exponential in the reaction zone can be written [ e 1=u exp ( u a + ) ] 1 þ [ exp 1 ( 1 )] ( exp 1 + u a þu a u a þu 2 a ) ;

6 6 A. C. McIntosh, R. O. Weber an G. N. Mercer [6] an the two ifferential equations in the inner zone to leaing orer in þ 1 become 1 L 2 + e 2 ye 1=ua =þua = 0 an (3.10) 2 y e 2 ye 1=ua =þua = 0 so that to leaing orer, the equation eliminating the reaction rate (3.1) becomes L 2 y 2 = 0: Integrating this equation once yiels + 1 y L = c 1; (3.11) where c 1 is a constant. Matching with the equilibrium solutions at ¾ = (see (3.5) (3.7)) requires = þ u ¾ = þ.u f u a / = n.1 + /: 0 þ (3.12) Consequently uner the þ limit (3.12)gives = 0, an so c 1 is zero. Note that for ` small, then from (3.7), n = c 2 + c ` 2 c ` 2 c.` small/: (3.13) Consequently integrating (3.11) a secon time yiels where c 2 is a constant. Matching on the hot sie again gives + y=l = c 2 ; (3.14) If we efine þ = u ¾=0 u a ; with y = 0: (3.15) f þ.u f u a /; (3.16) then the conition (3.15a) becomes = f,sothatc 2 is simply f. Noting then from (3.14) that y = L. f /, we then obtain the single equation in the inner zone from (3.10): 2 2 = Le 1=ua. f /e =þu2 a ;

7 [7] Non-aiabatic combustion waves for general Lewis numbers 7 with bounary conitions on the equilibrium sie (¾ = ): = f ; = 0: This single ifferential equation can be integrate once to give 1 2 ( ) 2 = Le 1=ua. f /e =þu2 a : = f If we let p. f /=þu 2 a, p. f /=þu 2 a, then 1 2 ( ) 2 p = Lþ 2 u 4 ae f =þu2 a e 1=u a p e p p : (3.17) p =0 Upon integration, (3.17)gives ( ) 2 [( ) ] = Lþ 2 u 4 a e f =þu2 a e 1=u a f 1 e. f /=þu2a + 1 : (3.18) þu 2 a This solution for the inner graient in temperature must now be matche to the outer temperature solution (3.3) on the preheat sie where¾ =+. Thus = þ u = þ m.u f u a /; ¾ which using (3.16)gives + 1 þ m.1 + f /: (3.19) From (3.9b), þ.u u a / = þ.u u a / 1, so that as, 1. Consequently the matching of graients ((3.18) an (3.19)) as gives the result 1 m.1 + f / þ = 2Lþ 2 u 4 a e f =þu2 ae 1=ua [( 1 f þu 2 a ) ] 1 e. 1 f /=þu2 a + 1 : (3.20) This result matches a leaing orer outer zone temperature graient with an inner zone temperature graient which inclues higher orer terms. Hence it is recognise that there will therefore be extra terms which strictly coul be balance by taking the outer expansion to the next orer clearly not possible in a tractable way analytically

8 8 A. C. McIntosh, R. O. Weber an G. N. Mercer [8] here. That these terms o not affect the leaing orer calculation of wave spee is confirme by the earlier paper ([12]) where exactly the same approach was use, an it was shown that there is excellent agreement between numerical results an preictions from the large þ asymptotic analysis. This analysis is not the same as that base on large activation energy, an the large þ wave spee formula for wave spee oes not in any way actually inclue the classical large activation energy result, since (as explaine in the introuction) there is no one to one mapping between the two approaches that is, by efinition the parameter þ in the current metho is compose of the ratio of two parameters (non-imensional activation energy E=RT a an nonimensional heat release Q=c p T a ) in the classical approach. Nevertheless, as the following pages show, there are aitional terms initially carrie in this non-aiabatic analysis, an recognising that we actually only want the leaing estimates of heat loss affecte wave spee, we later in this work simplify (3.20) to(3.26) an then (3.33), so that carrie in this result are only the main terms necessary to maintain the balance of graients implie in the limit Wave spee formula for non-aiabatic conitions The result (3.20) isnow couple with the result for c ((3.8)) from the exact integral, which using (3.9b) can be written c = c 2 + 4`.1 + f /. As state after (3.13), ` is regare as o.1/, sothattoleaingorer(3.21) yiels the following connection between f, ` an c: that is, c c.1 + 4`=c 2 / 1=2.1 + f /; `=c 2 + /.1 + f /: (3.21) Thus ` c 2 + f f 2` c 2 : (3.22) Performing a similar expansion for m in (3.4)wefin an thus m c 2 + c 2 ( 1 + 2` ) c + ; that is, m c + ` 2 c (3.23) m.1 + f / c ` c : (3.24) Substituting results (3.22) (3.24)into(3.20) then yiels the following transcenental

9 [9] Non-aiabatic combustion waves for general Lewis numbers 9 equation for the wave spee c: c ` [( 1 c = 2Lþ 3 u 4 f a e f =þu2 ae 1=ua þu 2 a = 2Lþ 3 u 4 a e 1=ua 1=þu2 a ) ] 1 e. 1 f /=þu2 a + 1 [ e f =þu2 ae 1=þu2 a f þu 2 a þu 2 a [ ] 1 + 2`=c = 2Lþ 3 u 4 2 ae 1=ua 1=þu2 a 1 + e 2`=þc2 u 2 a +1=þu2 a ; (3.25) where u a = u a + 1=þ. Thus for a given ` an þ, one formally can fin the wave spee c. The expression (3.25) becomes a little easier to manipulate with little loss of generality if we make the approximation u a 0(see[12]) so that u a 1=þ an consequently the result (3.25) becomes the simpler expression c ` c = [ ( 2L þ e 2þ e þ.1 2`=c2 / 1 þ 1 2` c 2 ] )] : (3.26) For aiabatic conitions ` 0 with c c a,(3.26) collapses own to 2L c a = þ e 2þ [e þ 1 þ]; (3.27) which for L = 1 is the same result as in Weber et al. [12]. Equation (3.26) an the aiabatic solution (3.27) are plotte for L = 1 in Figure 2 for low heat loss, an clearly confirm the well-known result that for a given þ with small heatloss ` there are two main solutions with an extinction point where the two solutions merge. Theoretically there are also other low c solutions, from the full formula, but it is unlikely that these are practically relevant. Numerical solutions of the full system (2.1), (2.2) an (2.3) are also marke on this plot as the ashe lines. For large þ, yet further simplification of (3.26) an (3.27) can be mae where the terms other than the exponential in the square bracket are small in comparison to the exponential term. Hence we obtain an c.2l=þ/e þ 2þ`=c2 c a.2l=þ/e þ : (3.28a) (3.28b) Although (3.28a) is not plotte, it is close to the curves b an c illustrate in Figure 2 for ` = 0:00001 an ` = 0:00002.

10 10 A. C. McIntosh, R. O. Weber an G. N. Mercer [10] F E FIGURE 2. Non-aiabatic wave-spee formula given in (3.26) (soli lines) an numerical solutions to the full system (ashe lines) for Lewis number L = 1 an various heat losses (a) ` = 0, (b) ` = 0:00001, (c) ` = 0: Note the multiple wave spees for the non-aiabatic cases an the subsequent extinction point Extinction heat loss an wave spees From (3.28), forlargeþ wecaneuce an approximate extinction conition since þ` = c 2 ln.c a =c/; (3.29) so that.þ`/=c = 0 when c = 2c ln.c a =c/, which implies that at this turning point, where the subscript ext stans for extinction an so c ext =c a = 1= e = 0:607 ::: ; (3.30) þ`ext =c 2 a = 1=.2e/ = 0:184 ::: `ext = L=.þ 2 e 1+þ /; (3.31a,b) where the secon expression uses the approximate formula for c 2 a from (3.28b). Figure 3 plots out the simpler expression, erive from (3.29): þ`ext =c 2 a =.c=c a/ 2 ln.c a =c/; (3.32) which essentially encapsulates the major effect of heat loss which is either to preict a critical extinction heat loss, or for a given heat loss, a critical extinction value for the ratio þ of activation energy to heat release. However this approximation is not really accurate enough in etermining the extinction heat loss `ext. For the case plotte in Figure 2 for ` = 0:00002, it can be seen that the critical þ value from the full formula is about 6:3, with the corresponing

11 [11] Non-aiabatic combustion waves for general Lewis numbers 11 E λ F DG FF DG FIGURE 3. Approximate extinction heat loss an wave spee for combustion waves (finite Lewis number). critical wave spee to be near 0:01 (refer to the soli line (c) in Figure 2). From the above approximate formula, for ` = 0:00002, the critical þ is about 6, with c a 0:03, so that c 0:607 0:03 0:018, so that the error is in the region of 6%. A better approximation is to regar only `=c 1in(3.26). Near extinction conitions this is always true. One then obtains c [ ( 2L þ e 2þ e þ.1 2`=c2 / 1 þ 1 2` c 2 )] : (3.33) Differentiating an insisting that `=c is zero for the turning points leas to the following relation at extinction: c 2 þ` 2L e þ.1 2`=c2/ 1 c þ e 2þ e þ.1 2`=c 2 / 1 þ.1 2`=c 2 / : (3.34) By efining r þ.1 2`=c 2 /; (3.35) an iviing (3.34) by(3.33), it can then be euce that at extinction þ = r + er 1 r e r 1 ; (3.36) an reversing efinition (3.35), an using (3.33)forc 2, we have also at extinction that `ext = L.er 1 r/ 2 þ 2 e 2þ : (3.37) e r 1

12 12 A. C. McIntosh, R. O. Weber an G. N. Mercer [12] FIGURE 4. Extinction heat loss `ext plotte against ratio þ of activation energy to heat release for various Lewis numbers L. The soli lines use (3.36) an (3.37), whilst the otte lines are the results from the crue approximation resulting in (3.31b). Equations (3.36) an (3.37) can then be use to calculate to much greater accuracy the extinction heat loss. The results from using these are shown in Figure 4. It is evient from maxima in the plots in Figure 4 that for very small þ the other unrealistic turning points are picke up (shown in Figure 2 bottom left). The realistic section of the curves in Figure 4 is the right-han part where in the log plot the curves are almost a straight line with constant graient. The graient is consistent with (3.31b), but the shift in the values is important for accuracy, an (3.36) an (3.37) represent a significant correction to the stanar use of (3.31b) for extinction calculations. Equations (3.33), (3.36) an (3.37) represent the main finings from the analysis of this paper, namely that by using the large þ analysis (as against large activation energy asymptotics) one can obtain a more accurate formula for non-aiabatic wave spees couple with extinction conitions. 4. Analysis for soli fuels (infinite Lewis number) 4.1. Exact integral an outer zone asymptotics For infinite Lewis number (soli fuels) a similar analysis is possible with the ae benefit that an explicit formula for

13 [13] Non-aiabatic combustion waves for general Lewis numbers 13 the ratio of the fuel (y) is possible. For infinite Lewis number L, (2.2) enables the immeiate substitution of ye 1=u =.c=þ/.y=¾/ to be mae into (2.1) which now becomes 2 u ¾ + c u 2 ¾ + c y þ ¾ `.u u a/ = 0: (4.1) The exact integral analysis is similar to the last section with c governe by the integral + (see (3.2)): c = þ`.u u a/ ¾. However unlike in the previous section, the first integration of (4.1) enables an explicit formula for y to be foun: y = þ` c ¾ u a / ¾ þ.u u a /.u þ c so that a single ifferential equation in u alone can be forme, 2 u ¾ + c u [ þ` ¾ 2 ¾ + u a / ¾ þ.u u a / c.u þ u c ¾ u ¾ ; ] e 1=u `.u u a / = 0: The outer zone asymptotic analysis for large þ yiels the same leaing orer solutions for temperature either sie of the combustion wave (see (3.3) an (3.5)) u = u a +.u f u a /e m¾ an u = u a +.u f u a /e n¾ ; so that one still obtains the same form for the wave spee relationship (3.8) c þ c 2 + 4`.u f u a /: (4.2) 4.2. Inner zone asymptotics In the inner zone, the re-scaling of temperature is ientical, but matching requirements lea to a ifferent natural scaling for the new inner stretche coorinate : u = u a + =þ = u a + 1=þ + =þ; þ¾: Consequently the ifferential equation (3.35) in the inner zone becomes þ 2 + c [ ` þc f =. + 1/ þ ] e 1=ua+=.þu2 a / c `.1 + / = 0 þ an to leaing orer 2 2 = 1 c e 1=ua+=.þu2 a / : (4.3)

14 14 A. C. McIntosh, R. O. Weber an G. N. Mercer [14] In a similar way as with the general Lewis number case, the relationship for wave spee (4.2) can be written as c c 2 + 4`.1 + f /, where matching of temperature either sie of the reaction zone leas as before to = f an = 1. With ` regareas o.1/, the same result for the small rop in temperature at the flame an its connection to heat loss is foun from (4.2): f 2`=c 2 : (4.4) Integration of the inner zone ifferential equation (4.3) yiels = þu2 a c e 1=ua ( ) e f =þu2 a e =þu 2 a : Matching of this solution for the inner graient in temperature to the outer temperature solution (3.3) on the preheat sie where ¾ =+ gives + = u ¾ = m.u f u a / = m 0 + þ.1 + f /: Noting that as, 1 an using similar reasoning to the previous section (as (3.24)) that m.1 + f / c `=c, we obtain the analogue of (3.25) for the wave spee for soli fuels (L = ) with heat losses: c ` c = þ2 u 2 ) a e (e 1=ua 2`=þc2 u 2 a e 1=þu 2 a : (4.5) c 4.3. Heat loss an multiple wave spees for moerate β ratio As with the equivalent result (3.25) for non-infinite Lewis numbers, there is little loss of generality if we take u a 0, in which case the result (4.5) becomes c ` c = 1 ( ) c e þ e 2`þ=c2 e þ ; (4.6) which checks with the result in Weber et al. [12] c = e þ.1 e þ / for aiabatic conitions, that is, when ` 0. Figure 5 shows the multiple wave spees that pertain to the formula (4.6) particularly at moerate þ values. Marke on this figure are the numerical results as well, an locations where known oscillatory instabilities occur [9] (the maximum an minimum wave spees in the oscillatory region are shown). The important ifference in the case of soli fuels is that this instability is much more reaily obtaine an on the figure below correspons to the slow branch of the wave spee plot.

15 [15] Non-aiabatic combustion waves for general Lewis numbers 15 F E FIGURE 5. Soli fuel (L = ) wave spees from (4.6) shown as the soli lines (a) ` = 0, (b) ` = 0: Note the multiple wave spees for the non-aiabatic case. Also shown are the numerical results (ashe lines); the maximum an minimum wave spees have been plotte in the oscillatory regions Extinction Further simplification of result (4.4) can be obtaine by again regaring `=c 1, so that we have a very goo approximation without losing the main characteristics of the transcenental wave spee formula: ( ) c 2 = e þ e 2`þ=c2 e þ an c 2 = ( a e þ 1 e þ) ; (4.7) leaing to þ` = 1 2 ĉ2 ln [ ĉ ( 2 1 e þ) + e þ] : (4.8) c 2 a This curve is essentially the top part of the shape shown in Figure 5,so that it captures the main ual wave spee phenomenon obtaine with heat loss. If one ignore the secon an thir terms in the square bracket, then one obtains the same approximate expression (3.32) as we ha for non-infinite Lewis number combustion waves. The more accurateresult however is summarise by (4.7) an (4.8). For the infinite Lewis number case consiere in this section, it is foun in this case that for þ values over about 5, the critical extinction heat loss `ext an wave spee c ext are well approximate by (3.30) an (3.31a). Thus for soli combustion one can still use c ext =c a 0:61 an þ`ext =c 2 a 0: Conclusions Heat losses an general Lewis number have been ae to the combustion wave spee analysis of Weber et al.[12] using an asymptotic analysis base on the ratio of

16 16 A. C. McIntosh, R. O. Weber an G. N. Mercer [16] activation energy to heat release being large (as against the more traitional analysis of activation energy alone). Multiple wave spee solutions have been obtaine these are more pronounce in the soli fuel case for moerate values of the ratio þ. A turning point in the spee curves inicates the extinction conition. This was further analyse to obtain an approximate extinction conition base solely on the ratio of the activation energy to the heat release, the heat loss an the aiabatic wave spee. These formulae apply over a wier range of activation energies than the classical approximations base on large activation energy asymptotics. For both gaseous an soli fuels, extinction criteria have been establishe an goo comparison with full numerical solutions has been obtaine. Of particular note is that as reporte by Mercer et al.[9] for soli fuels, the numerical results clearly show that the oscillatory instability is much more reaily observe as the heat loss is increase. Illustrative calculations of maximum an minimum wave spees have been shown for a typical case with an without heat loss. References [1] N. J. Balmforth, R. V. Craster an S. J. A. Malham, Unsteay fronts in an autocatalytic system, Proc. Roy. Soc. Lonon Ser. A 455 (1999) [2] A. Bayliss an B. J. Matkowsky, Two routes to chaos in conense phase combustion, SIAM J. Appl. Math. 50 (1990) [3] A. Bayliss an B. J. Matkowsky, Nonlinear ynamics of cellular flames, SIAM J. Appl. Math. 52 (1992) [4] A. Bayliss an B. J. Matkowsky, From travelling waves to chaos in combustion, SIAM J. Appl. Math. 54 (1994) [5] A. Bayliss, B. J. Matkowsky an M. Minkoff, Perio oubling gaine, perio oubling lost, SIAM J. Appl. Math. 49 (1989) [6] T. Boington, A. Cottrell an P. G. Laye, A numerical moel of combustion in gasless pyrotechnic systems, Combust. Flame 76 (1989) [7] J. F. Griffiths an J. A. Barnar, Flame an combustion, 3r e. (Stanley Thomas, Lonon, 1995). [8] G. N. Mercer, R. O. Weber, B. F. Gray an S. D. Watt, Combustion pseuo-waves in a system with reactant consumption an heat loss, Math. Comput. Moelling 24 (1996) [9] G. N. Mercer, R. O. Weber an H. S. Sihu, An oscillatory route to extinction for soli fuel combustion waves ue to heat losses, Proc. Roy. Soc. Lonon Ser. A 454 (1998) [10] M. J. Metcalf, J. H. Merkin an S. K. Scott, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis, Proc. Roy. Soc. Lonon Ser. A 447 (1994) [11] K. S. Shkainskii, B. I. Khaikin an A. G. Merzhanov, Propagation of a pulsating exothermic reaction front in the conense phase, Combust. Expl. Shock 7 (1971) [12] R. O. Weber, G. N. Mercer, H. S. Sihu an B. F. Gray, Combustion waves for gases (Le = 1) an solis (Le ), Proc. Roy. Soc. Lonon Ser. A 453 (1997)