EFFECT OF VOLUMETRIC HEAT LOSS ON TRIPLE-FLAME PROPAGATION

Transcription

1 Proceeding of the Combution Intitute, Volume 29, 2002/pp EFFECT OF VOLUMETRIC HEAT LOSS ON TRIPLE-FLAME PROPAGATION R. DAOU, J. DAOU and J. DOLD Department of Mathematic, UMIST, Mancheter M60 1QD, United Kingdom We preent a numerical tudy of the effect of volumetric heat lo on the propagation of triple flame in the trained mixing layer formed between two oppoed tream of fuel and air. The propagation peed of the triple flame i computed for a wide range of value of two non-dimenional parameter: a normalized flame thickne e, proportional to the quare root of the train rate, and a heat-lo parameter j. It i hown that, for relatively mall value of j, the propagation peed U i decreaed by heat lo, and it dependence on e i imilar to the adiabatic cae, known in the literature; in particular, a monotonic decreae in the peed from poitive to negative value i oberved a e i increaed. However, for j larger than a critical value, thi monotonic behavior i lot. It i hown that the more complex behavior obtained i mainly aociated with the fact that, in the preence of heat lo, the trailing planar diffuion flame i extinguihed both for ufficiently large and ufficiently mall value of the train rate. Moreover, for ufficiently mall value of e, the dependence of U on j i imilar to that of the non-adiabatic planar premixed flame, with total extinction occurring for a finite poitive value of U. On the other hand, for larger value of e, negative peed, correponding to extinction front, appear before total extinction i brought about by an increae in j. A ummary of the main reult i provided by delimiting the different combution regime oberved in the j e plane. Introduction The importance of triple flame i now well etablihed, in application involving combution phenomena, uch a flame pread over olid or liquid fuel urface, flame propagation in mixing layer, dynamic extinction of diffuion flame, and flame tabilization in reactive tream. Early experimental obervation of thi tructure wa made by Phillip [1] and an early analytical decription appear in Ohki and Tuge [2]. Detailed analyi of triple flame and their propagation regime wa undertaken by Dold and collaborator [3,4]. Several apect of the problem have ince been invetigated, including the effect of ga expanion, the influence of non-unit Lewi number and the tability of triple flame (ee Ref. [5 11] and reference therein). The aim of thi work i to extend current knowledge of triple flame by taking into account the influence of volumetric heat lo. Thi apect of the problem eem to have received no attention, at leat a far a the prototypical counterflow configuration i concerned. The aim of thi paper i to invetigate how triple flame, and their propagation regime, are affected by volumetric heat lo in thi configuration. The paper i tructured a follow: the problem i firt formulated in the context of a thermo-diffuive approximation, with contant denity and contant tranport propertie and a ingle Arrheniu reaction; thi i followed by preentation and dicuion of the numerical finding, in term of two main parameter related to the train rate and the rate of heat lo. Formulation The tudy i carried out in the familiar counterflow configuration, illutrated in Fig. 1, with the upper tream carrying oxidizer and the lower tream carrying fuel. The flow component are given by v X 0, v Y ay and v Z az, in the X,Y, and Z direction, repectively, with a denoting the train rate. We hall addre the teady propagation of triple flame in the mixing layer along the X axi, with the propagation peed Û being poitive if the front are moving along the negative X direction. The Fig. 1. The counterflow configuration. 1559

2 1560 LAMINAR FLAMES Partially-Premixed Flame chemitry i modeled by a ingle irreverible onetep reaction of the form F Ox r P q, where F denote the fuel, Ox the oxidizer, and P the product. The quantitie and q repreent the proportion of oxidizer conumed, to fuel conumed, and the heat releaed per unit ma of fuel. The reaction rate, ˆx, i aumed to follow an Arrheniu law of the form ˆx Bq 2 Y F Y O exp(e/rt), where B, q, Y F, Y O, and E/R repreent the pre-exponential factor, the (contant) denity, the ma fraction of fuel and oxidizer, and the activation temperature, repectively. The tretching of the flow in the Z direction tend to make the ytem uniform in Z, o that, in a reference frame attached to the flame, the governing equation become T T T T q ˆx Û D X X Y cp q T ay ĵ(t T 0) (1) Y 2 F F F F 2 Y Y Y Û D X X Y ˆx Y ay F (2) q Y 2 O O O O 2 Y Y Y Û D X X Y ˆx Y ay O (3) q Y Here, D F, D O, and D T are contant diffuion coefficient. The lat term on the right of equation 1 i included to account for a linear volumetric heat lo with coefficient ĵ, the temperature in both incoming tream being T 0. The boundary condition for equation 1 3, given in non-dimenional form below, correpond to the planar Y-dependent frozen olution a X r or Y r, and to vanihing X derivative a X r. The non-dimenional formulation of the problem follow Ref. [9] with the caled dependent variable being defined by YF YO T T0 yf, yo, and h YF,t YO,t Tad T0 Here the ubcript t refer to value at (X, Y Y t ), where Y t i the location of the uptream toichiometric urface defined by Y O Y F,or S erf(y /(2D /a) ) t F erf(y t/(2d O/a) ) S 1 with S (Y F y)/(y O y). The quantity T ad i defined by T ad T 0 qy F,t /C p. A unit length, we elect L/b, where L (2D T /a) i the (thermal) mixing layer thickne and b E(T ad T )/RT 2 i 0 ad the Zeldovich number; the unit of length i then a typical radiu of curvature of a triple flame. A unit peed, we adopt the laminar peed of a toichiometric planar flame, or more preciely it aymptotic value for large b under adiabatic equidiffuional 0 S L condition, namely (4b 3 Y O,t qd T B exp (E/RT ad )). In term of the coordinate y b(y Y t )/L and x bx/l, equation 1 3 now aume the non-dimenional form 2 2 h h h 1 U e e x x x y 1 2e y h e g jh (4) b b y b F F F 1 F y e y y U e x x Le x y 2e y y g F (5) b b y O O O 1 O y e y y U e x x Le x y 2e y y g O (6) b b y Here, the parameter e, the quare of which i the invere of the Damköhler number, i defined by 0 Fl T 0 S L l b(d /2) e a L/b It repreent the thickne of the laminar toichiometric flame lfl D T/S L, meaured in term of our 0 0 tandard unit of length L/b. The Lewi number of fuel and oxidizer are Le F D T /D F and Le O D T /D O, and g Y t /(2D T /a) characterize the location of the uptream toichiometric urface. The non-dimenional heat lo coefficient i j 02 b(d /S )ĵ and x i given by T L 3 b b(h 1) x y y exp (7) 4 1 (h 1) F O with (T ad T 0 )/T ad. In term of the new variable, the boundary condition a x r or y r are h 0 1 erf[(g y/b)le F ] y F 1 erf(g Le ) F O 1 erf((g y/b)le ) yo (8) 1 erf(g Le ) O

3 VOLUMETRIC HEAT LOSS IN TRIPLE FLAMES 1561 Fig. 2. Contour of the reaction rate (left) and temperature (right) for the cae j 0 with e 0.2 (top), e 1.2 (middle), and e 2.7 (bottom). and a x r h yf yo 0 (9) x x x In olving thi problem, the main aim i to determine the (caled) propagation peed U in term of the nondimenional parameter e, j, Le F, Le O, and g (a well a b and ). In thi tudy, we provide detailed numerical reult in term of the parameter e and j. Reult The problem coniting of equation 4 6 with the boundary condition 8 and 9 i olved numerically. The numerical method i the ame a the one ued in Ref. [9] and i baed on a finite volume dicretization combined with an algebraic multigrid olver [12]. The computational domain dimenion are typically 10 time the mixing layer thickne in the y direction and 100 time the planar laminar flame thickne in the x direction. The grid i non-uniform with typically 200,000 point. We report reult decribing the dependence on the parameter e and j, with the other parameter being aigned fixed value, namely b 8, 0.85, g 0, and Le F Le O 1. We begin by preenting a reference cae correponding to the familiar adiabatic ituation j 0. Shown in Fig. 2 are reaction rate contour (left) and correponding temperature contour (right). The ubfigure correpond to e 0.2, 1.2, and 2.7, from top to bottom, with the lat value characterizing near-extinction condition. The contour are equiditributed between zero and the maximum value of the field, which i indicated in each ubfigure. The dimenionle leading front become thicker for larger value of e (or train rate) which i accompanied by a decreae in the propagation peed from poitive to negative value (ee Fig. 4). Of coure, in the limit e r 0, correponding to large Damköhler number, the temperature of the trailing diffuion flame increae to unity, it adiabatic value, while the correponding reaction rate x decreae to zero (due to a vanihing rate of upply of the reactant to the reaction zone); a e i increaed thi trend i revered, at leat up to near-extinction condition (obtained for ufficiently high train rate). To illutrate the influence of heat lo on the triple flame, Fig. 3 depict the ame ituation a Fig. 2, with j 0.04, for e 0.2, 1.2, and 2.4; the final value again characterize near-extinction condition and i maller than in the adiabatic cae. An important feature aociated with the preence of heat lo i the extinction of the trailing diffuion flame for mall value of e, a can be oberved in the top ubfigure. Otherwie, the behavior of the triple flame a e i increaed i imilar to the adiabatic cae, with the front evolving continuouly from propagating front to retreating front until total extinction occur. However, it i important to note that the lat remark i valid only for ufficiently mall value of j. For j larger than ome critical value, more complex behavior i obtained. Thi i bet illutrated by plotting the propagation peed U veru e for elected value of j, a done in Fig. 4. The curve labeled j 0 in thi figure i the well-known adiabatic cae. Thi curve ha a vertical lope for a critical value of e which characterize the total extinction of the triple-flame tructure. A explained in Ref. [13], thi critical value i aociated with the quenching of the planar diffuion flame by an exceively high train rate. We note that the curve labeled j 0.04 diplay a imilar trend, in line with the obervation above, but that the cae correponding to higher value of j exhibit a markedly different behavior. In particular, the dependence of U on e i no longer monotonic and, in fact, the e-domain of exitence of the flame front eparate into two dijoint interval; thi i clearly een in the curve correponding to j 0.05 and For yet larger value of j, no

4 1562 LAMINAR FLAMES Partially-Premixed Flame Fig. 5. Plot of U veru j for elected value of e. Fig. 3. Contour of the reaction rate (left) and temperature (right) for the cae j 0.04 with e 0.2 (top), e 1.2 (middle), and e 2.4 (bottom). olution are found for mall value of e, a een in the curve for j Thi can be explained by the fact that, a e r 0, the propagation peed tend to that of the toichiometric planar flame, but the latter z Fig. 4. Plot of U veru e for elected value of j. z Fig. 6. Maximum temperature veru e for elected value of j for the planar diffuion flame. ceae to exit for exceive heat loe even though a narrower flame edge can till exit at higher value of e (according to numerical experiment). For larger value of j, approximately j 0.1 in thi example, we found no burning olution for any value of e. Another intructive way of examining the reult jut preented i to plot U veru j for elected value of e, a in Fig. 5. For mall value of e, the dependence of U on j i imilar to that of the nonadiabatic planar flame, with extinction occurring at a finite poitive peed. Thi can be confirmed by an aymptotic analyi in the limit e r 0, which i not included here due to pace limitation. From the figure, we can alo conclude that retreating triple flame (or extinction front, having U 0) can be obtained by increaing the intenity of the heat lo only if e (or the train rate) i above a critical value. To undertand better the dependence of U on e, in the preence of heat lo, it i ueful to compare it with a numerical decription of the planar

5 VOLUMETRIC HEAT LOSS IN TRIPLE FLAMES 1563 ȳf ȳo 1 d d b r From the lat three equation, it follow that d r b 2/3 e 2/3, and r z Fig. 7. Regime of triple-flame propagation, with and without an aociated diffuion flame, in the preence of heat lo. diffuion flame over the ame range of parameter. The latter i preented in Fig. 6, where the maximum temperature of the planar diffuion flame i plotted againt e for everal value of j. We note that for any non-zero value of j, there are two extinction value of e, a fact that i known in the literature (ee, for example, Ref. [14 17]). The larger extinction limit, which i alo encountered in the adiabatic cae, i due to flame quenching by an exceively high train rate. The lower extinction limit i partly aociated with the fact that the rate of heat generation by the chemical reaction decreae a the train rate (or reactant upply) i decreaed, leading to extinction for any non-zero value of j; moreover, the total ize of the region of hot ga alo increae, which increae the total heat lo and lower the flame temperature. The fact that extinction mut occur can be een from the following imple, order-of-magnitude argument. From the diffuive-reactive balance in the thin reaction zone, of typical (non-dimenional) thickne d r, ay, we have from the one-dimenional y-dependent verion of equation 5 and 6 ȳf ȳo e x dr dr where the bar indicate typical value in the reaction zone. From equation 7, uppoing that h i cloe to unity 3 x b ȳ F ȳ O Since the order of magnitude of the gradient of y F and y O in the reaction zone, ȳ F /d r and ȳ O /d r, repectively, are the ame a in the mixing layer (whoe non-dimenional thickne i b, given our choice of unit length) we may write 1/3 4/3 x b e Now, uing the temperature equation 4, we ee that the effective rate of heat generation (accounting for heat lo) i given, in order of magnitude, by x j/b. Thi quantity become negative in the limit e r 0, indicating that the temperature decreae, leading to extinction, whenever j 0. It alo indicate that extinction mut occur at leat when j O(b 2/3 e 4/3 ), or larger. For the adiabatic cae, j 0, extinction i impoible a e r 0, ince the net rate of heat generation remain poitive, although it become vanihingly mall. Thi i the claical Burke Schumann limit. We now return to Fig. 4 and 6, where it i een, by comparing, for example, the curve labeled j 0.06, that the more complex dependence of U on e when j 0 i directly linked to the behavior of the planar diffuion flame. Thi explain, when j i not too mall, both the monotonic variation of U with e and the fact that the e-domain of exitence of the flame front conit of two dijoint interval. For mall value of j, for example j 0.04, the lower extinction limit of the diffuion flame occur at value of e, which are ufficiently mall to have a negligible effect on the propagation peed. Thi i becaue, for ufficiently mall value of e, the leading premixed front i negligibly affected by the propertie of the field downtream, and hence by the trailing diffuion flame. Finally, a ummary of the main reult i preented in Fig. 7 in the pace of j and e. The dahed line characterize the extinction limit of the planar diffuion flame extracted from the previou figure. The quare correpond to the complete extinction of the triple-flame tructure, and are (partially) extracted from Fig. 5. The triangle decribe condition with zero-propagation peed. Four combution regime can thu be delimited in the j e plane. 1. In the domain labeled A, to the right of the quare, the triple-flame tructure i extinguihed. We note that the extinction in thi cae i dictated by the extinction of the diffuion flame in ituation where the quare lie on the dahed line. Thi occur for e larger than a critical value e* which i een to be cloe to 0.7. For mall value of the train rate (more preciely for e e*), the triple-flame tructure urvive in ituation where the planar diffuion flame i extinguihed. 2. In the domain labeled B, to the left of the quare and below the lower branch of the dahed curve, the triple flame have poitive peed and no trailing diffuion-flame tail (far downtream), a

6 1564 LAMINAR FLAMES Partially-Premixed Flame exemplified in the top ubfigure of Fig. 3. We could call thee taille triple flame. More generally, uch tructure arie in ituation where the flame behind a flame edge i extinguihed but the edge itelf continue to urvive. Such tructure hare ome imilaritie with thoe encountered in low Lewi number triple-flame tudie (ee Ref. [18]) where ocillatory propagation arie. 3. In domain C, below the triangle and above domain B, triple flame with poitive peed and trailing diffuion-flame tail are encountered. Thee may be referred to a ignition front of the diffuion flame, a in the familiar adiabatic ituation. 4. In the remaining domain, D, negatively propagating triple flame (retreating front) are obtained, again a found in the adiabatic ituation. Concluion We have preented a numerical decription of triple-flame propagation in a trained mixing layer under nonadiabatic condition. The reult indicate that variou combution regime arie, due to the difference in enitivity to heat lo of the premixed leading front and trailing diffuion flame. In particular, a ynthei of the main finding ha been given in term of the heat-lo intenity and the train rate. It i worth pointing out that the olution preented have been obtained by olving the teady-tate-governing equation. Their tability, which ha not been addreed in thi work, will be checked in future tudie by olving the correponding time-dependent problem. Acknowledgment The author are grateful to the EPSRC for financial upport. REFERENCES 1. Phillip, H., Proc. Combut. Int. 10:1277 (1964). 2. Ohki, Y., and Tuge, S., in Dynamic of Reactive Sytem, Part I (J. R. Bowen, J. C. Leyer, and R. I. Soloukhin, ed.), Progre in Atronautic and Aeronautic, Dold, J. W., Combut. Flame 76:71 88 (1989). 4. Hartley, L. J., and Dold, J. W., Combut. Sci. Technol. 80:23 (1991). 5. Liñán, A., in Combution in High Speed Flow (J. Buckmater, T. L. Jackon, and A. Kumar, ed.), Kluwer Academic, Boton, 1994, p Kioni, P. N., Rogg, B., Bray, C., and Liñán A., Combut. Flame 95: (1993). 7. Buckmater, J., and Matalon, M., Proc. Combut. Int. 22: (1988). 8. Ruetch, G. R., Vervich, L., and Liñán, A., Phy. Fluid 7(6): (1995). 9. Daou, J., and Liñán, A., Combut. Theory Model. 2: (1998). 10. Shay, M. L., and Ronney, P. D., Combut. Flame 112:171 (1998). 11. Short, M., Buckmater, J., and Kochevet, S., Combut. Flame 125:893 (2001). 12. Ruge, J., and Stüben, K., in Proceeding of Multigrid Conference, Britol, Dold, J. W., Prog. Atronaut. Aeronaut. 173:61 72 (1997). 14. Sohrab, S. H., Liñán, A., and William, F. A., Combut. Sci. Technol. 27: (1982). 15. Liu, F., Smallwood, G. J., Gülder, Ö. L., and Ju, Y., Combut. Flame 121: (2000). 16. T ien, J. S., Combut. Flame 65:31 34 (1982). 17. Chao, B. H., Law, C. K., and T ien, J. S., Proc. Combut. Int. 23:523 (1990). 18. Thatcher, R. W., and Dold, J. W., Combut. Theory Model. 4: (2000). COMMENTS Ihwar K. Puri, Univerity of Illinoi at Chicago, USA. I applaud your motivation; your reult are intuitive and ome are well known, namely, reduction in flame propagation peed with heat lo and extinction, and the formation of a flame nub with increaing tretch. However, I have ome reervation. You aume a uniform global volumetric heat lo, wherea a real radiating flame would loe heat according to it non-uniform product ditribution. Further, wherea you have aumed a contant train rate, a realitic flame experience non-uniform tretch along it topology that depend on flame curvature, hydrodynamic training, and flame thickne. Author Reply. The objective of the article i to etablih qualitative rather than quantitative accuracy in decribing the effect of heat lo on trained triple flame. Thi i a much a can be expected when, for example, uing a onetep model for the chemitry. For thi purpoe, uing a imple model ha numerou advantage. A more ophiticated model could certainly be adopted, and the likelihood i that exactly the ame form of behavior would be oberved, differing only in detail. The model adopted here, including that for heat lo, i choen to fit in with other tudie which, for example, have indeed hown a reduction in the flame peed of a planar premixed flame with heat lo, a you ay. However, we are aware of no exiting analytical or numerical tudie that decribe the combination of phenomena that we have oberved, including the appearance of a taille propagating flame edge. We have recently ubmitted another article that examine the phenomena from a more analytical perpective, which i mot eaily done uing a imple linear model for global heat lo.