Counterflow spray combustion modelling

COMBUSTION A N D F L A M E 81:325-340 (1990) 325 Counterflow Spray Combustion Modeling G. C O N T I N I L L O * and W. A. S I R I G N A N O Department of MechanicalEngineering, University of California, Irvine, CA 92717 A fuel spray planar counterfiow flame has been modeled by means of a low-Mach-number boundary layer approach. The model considers density variations and full coupling between the two phases by means of a hybrid Eulerian-Lagrangian formulation, with droplet drag and vaporization. A spherically symmetric, unsteady droplet model with variable properties has been employed. Conditions for similarity in such a two-phase, particle-laden reacting boundary layer have been determined. Similarity solutions have been derived that are valid at any distance from the stagnation point and for any liquid fuel concentration and droplet number density within the limits of negligible particle-particle interactions. Results for the specific calculation of a monodisperse n-octane spray flame are reported and discussed. NOMENCLATURE B bi BM Br C1,C2 CD Cp Di Ea f fo g K L Le (m)i preexponential factor stoichiometric coefficient for species i Spalding transfer number for mass Spalding transfer number for energy constants in the Clausius-Clapeyron relationship drag coefficient specific heat mass diffusivity for species i activation energy similarity variable contribution to the stream function due to mass vaporization vector acceleration of gravity strain rate of the outer flow latent heat of vaporization Lewis number molecular weight, species i * On leave from Istituto Ricerche sulla Combustione CNRNapoli, Italy. Copyright ~) 1990 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010 m n P Pr Q ¢ R r Re (R s Sc Se, Sin, Sv t T u 13 V X x X mass vaporization rate droplet number density pressure Prandtl number heat of combustion energy flux into droplet interior droplet radius radial coordinate, droplet interior Reynolds number universal gas constant transformed x component of droplet vector position Schmidt number source terms in the gas equations due to droplets time coordinate temperature x velocity component y velocity component vector velocity source term due to chemical reaction x space coordinate droplet vector position species mass fraction 0010-2180/90/$3.50 326 Y Y G. CONTINILLO and W. A. SIRIGNANO y space coordinate species mass fraction Greek Symbols ,y ~7 0 P /) exponents in the Arrhenius expression for reaction rate ratio between specific heats Kroneker delta similarity independent variable nondimensional temperature heat conductivity viscosity kinematic viscosity transformed radial coordinate, droplet interior density transformed time coordinate Subscripts CX~ + boil crit f F i / O O P S X Y infinity positive y side negative y side boiling point critical point film fuel species liquid oxygen initial or outer flow products surface x component y component Superscripts t derivative along 7, gas equations; along z, droplet equations reference quantity dimensional variable INTRODUCTION The counterflow flame has been given much attention in the past decade both from an experimental and from a theoretical point of view. More than one reason accounts for such interest. The configuration of two impinging streams can encompass a complete spectrum of boundary conditions and relevant flame characters that, for a gaseous system, vary from the diffusion flame (fuel from one side, oxidant from the other) to partially premixed diffusion flames, to the symmetric twin premixed flames, to the simple premixed flame (fuel and oxidant from one side, inert and/or products from the other). All of the mentioned configurations appear to be one-dimensional in essence, at least in the neighborhood of the stagnation plane; this reflects the existence of similarity solutions to the boundary layer equations governing the problem, given the proper boundary conditions, and with a minimum amount of hypotheses valid in almost all of the cases of interest [1]. The flames are generally located in a defined region around the stagnation point, over a wide range for the values of the parameters. Without heat losses to an external body, such as a flame holder, the flame stability is due to a complex balance between convection, heat and mass diffusion and chemical reaction. For this last reason, the diffusion counterflow flame is often referred to as the "pure diffusion flame," in that it has, for sufficiently high DamkShler number, no cold zone in which partial mixing of the reactants takes place, and is therefore best suited for the analysis of certain fundamental processes, like diffusion flame extinction, which cannot be studied in the classical Burke-Schumann configuration, as noted by Tsuji [2]. The same reference provides an extensive review of the research work done for the counterflow gaseous diffusion flame configuration. Another reason to look into counterflowing reacting systems is illustrated by the work of Libby and Williams [3], in which strained laminar flamelets are studied as elements of a premixed turbulent flame. Experimental and theoretical work has been done for diffusion [1, 4, 5], partially premixed [6] and premixed [7-11] gaseous counterflow flames, flame extinction [6], often by means of large activation energy asymptotics, and inhibition [12] have been addressed. Less attention has been devoted to two-phase systems. Graves and Wendt [13] first used an op- COUNTERFLOW SPRAY COMBUSTION MODELING posed jet diffusion flame in an experimental study of laminar, pulverized coal flames, and by experimental evidence and theoretical arguments suggested that the system be still similar in essence even when particles do not follow gas streamlines. More recently, Puri and Libby [14] studied the behavior of single liquid fuel droplets in a counterflowing methane flame, comparing observed trajectories with predictions obtained by using a number of different models for droplet drag in an assumed flow field. The same reasons that make the counterflow configuration interesting in the study of homogeneous (gas) flames are valid as well for twophase (spray) flames. In particular, the counterflow spray configuration is seen here as a very interesting model problem, in which droplet vaporization, drag, and combustion can be followed in a well-defined region. The interaction of convection, diffusion, and chemical reaction phenomena determined the flame structure, and their relative influence can be studied by changing a few main parameters, especially the rate of strain of the flow. High-temperature differences and sharp gradients are present as in any practical flame, and boundary effects and heat losses are limited to a minimum. In order to study fully the system without having to confine attention to particular regimes like flame extinction or small liquid-tovapor fuel ratios, full coupling between the two phases and finite-rate chemistry are included in the model, as illustrated in the following section. Under certain conditions, not very restrictive and identified in the foregoing, this will not affect the similarity of the solution; it is clear, however, that a numerical approach, successively described, is required. Results are reported and discussed, and areas of interest are identified for future work. 327 and droplets, in any combination, can be fed from either side of the stagnation plane. In this work, the steady state is considered for the system, so that the effects of the inherent unsteadiness due to the discrete spray distribution is neglected. The analysis is confined to a singlecomponent, monodisperse spray; the extension to multicomponent, polydisperse sprays does not affect the following analysis other than to increase the number of equations for the species in both the gas-phase and the liquid phase, and the number of characteristics for the droplet motion. A hybrid Eulerian-Lagrangian formulation is adopted. The problem will be reduced to one space dimension via a similarity analysis. One droplet will represent all those lying on the same plane, perpendicular to the space coordinate. Among the assumptions made are constant specific heats, equal binary diffusion coefficients, and ideal gas mixture. More assumptions are mentioned where invoked. We use dimensional variables in the Eqs. 1-32; in the remaining equations, the same symbols will refer to the corresponding nondimensional variables, unless otherwise specified. Gas-Phase Equations The gas-phase boundary layer equations are O(pu) O(pt,) - + - Ox Oy (1) --So, ÷Sin, (2) M A T H E M A T I C A L MODEL Figure 1 illustrates the system configuration, identifying the stagnation plane, the symmetry plane (dot-and-dash), the gas-phase streamlines, and the droplet trajectories. The shaded area represents the high-temperature region as found in one of the particular cases studied and discussed in the foregoing. Air, fuel vapor, inert, hot products, -u +N _ + ix \ O y J XOT - Qfvp + Se, (3) 328 G. CONTINILLO and W. A. SIRIGNANO the mass vaporization rate, am ~-" ( dut ) - n ~mt ~ + rn(u - Ul) , S e = - - n { q ÷rh[CpF(T - T s ) + L ] } , O -0 ..° 0.. "'""0.... (10) where t~ is the total heat transferred to the, droplet interior. Other symbols are identified in the Nomenclature. In writing the equations we have already made the boundary layer assumptions. We also assume that o , C (9) ., /~ T I~o - (11) To' and that Cp, (r~), Le, and Pr are constant. If we now multiply Eq. 2 by u and add the result to Eq. 3 we obtain x Fig. 1. Schematic of the problem. ° (c,r + d- ~Vi -t- (t$iF -- Y i)Su, 6/ p = p,_--~z=,T tin) + pv@ O (CpT + (4) (5) where Pr is a constant of order 1. The terms (~pT + u2/2) can be rewritten as The expression for wi is given as fi'i = (m)ibiB e x p ( - E a /(RT)CF~Co (3, (6) where M 2 ---- U 2 / ' r R T . If we restrict our study to flame situations when the Mach number is low, the second term in parentheses can be neglected, and Eq. 3 can be substituted with where a chemical reaction bFF + boO --~ - bpP pCpv is taking place, and .Yi Ci = (m)i" (7) If the droplets are all represented by one group, the expressions for the source terms are S~ - n r n , (8) where n is the droplet number density and rn is = C-ZP t g POy r #-O-y +Se -QCep (3 t) Equation 3' can also be derived from Eq. 3 by neglecting the pressure work and the viscous dissipation term. COUNTERFLOW SPRAY COMBUSTION MODELING 329 Boundary Conditions with [15] The system of Eqs. 1-4 is subject to the boundary conditions th = 2 7 r o f D f R ~ d h ln(1 + B M ) , Sh - - 2 + [(1 + ReSc)U3[max(1, Re)] °'°77 - 1] y = +oc: u -----Uo+(X); T = To+; Y i = Yio+, (17) •F(BM), " y = -cx~: u = Uo_(X); T = T o - ; Y i -= Y i o - . (12) Our case is a counterflow, for which the outer flow is the potential flow of two impinging streams •(Fig. 1), described by Uo+ = K + x ; Vo+ = - K + y ; Y F s -- Y F ~ BM-- 1 -- Y F s (m)FXFs YEs (m)FXFs + (1 - X v s ) ( m ) ' 1 (13) If the reference frame is fixed to the stagnation plane, overall momentum balance requires, for the o<~boundary conditions, that p o _ K _ 2 __ po+K + 2, if we neglect the dynamic pressure contribution of the incoming droplets. Equating the static pressure at the stagnation plane for the potential field and assuming that the stagnation pressures are the same for the two streams, we have there, since v = 0 at the stagnation plane, OT 1 0 (r20T) Ot -- at ~ ~ \ (22) 07" 3 7 r=R(t) - 47rR2otlPlC pl ' 03T7 r=0 = o ; (23) that is ( O =m \ - rs) BT Droplet Equations CplSh We use a hybrid Eulerian~Lagrangian formula. tion. A single-component, spherically symmetric droplet, surrounded by a quasi-steady spherically symmetric film is considered; the transient droplet heat-up, vaporization, and motion is modeled• The free-stream conditions are those of the local gas phase surrounding the droplet. The equations are Balance of mass = -rr/, - L), (24) (14) BT =(1--BM) rR3ol (20) (21) Or J ' T ( r , O) = Tlo, where ~-~ (19) Balance of energy--conduction limit model Po + Oo+K+ 2X2 = r(O) = P o + Po - K - 2 X 2 , "Po+K +2 = # o _ K _ 2. (18) ' X F s • -- e x p ( - - C I F / T s + C2F). P Uo- = K_x, Vo- = K _ y . (16) (15) ~ -1, 1 4~ = C p / R u Le' (25) (26) /Xru ----2 + [(1 + RePr)U3[max(1, Re)] °'°°7 - 1] •F(Br), L = Lboil ( Tcrit-Ts ) 0"38 k.Tcrit -- Tboil (27) (28) In this study, we assume Re << 1; hence At'u = Sh = 2. 330 G. CONTINILLO and W. A. SIRIGNANO tions will be determined. We assume that similarity exists, leaving it to be proven at the end of the analysis. We first nondimensionalize the equations by introducing the following reference quantities: Droplet Motion The vector equation for droplet dynamics is 4R3 ~,r dVi Or-d-f = ,rR2 ~o(V - Vt)[V - VtlCn 4 3 + gTrR otg, (29) xt(O)=xlo, t* = 1/K_; L* = r* =.to_ P* = Po; O* = O o - ; Vt(O) = Vto. (30) For Cn we assume Stokes flow: 24 u* : ~ ; Under the assumptions made, the state Eq. 5 becomes 24v Cn - Re -- 2 I V - VtlR" Q* = ~_.pT*; M* = o*L .3. (31) All the cases studied considered situations in which droplet initial vector velocities match gas velocities at the boundary: slip occurs due to gas thermal expansion, and due to the curvature of the flow. The deviation from Stokes's drag has been evaluated to reach 12% in the worst case, and during a small extent of the droplet lifetime. However, the assumption of Stokes's drag must be relaxed if the droplet Reynolds number becomes of the order 1 or larger. A convective correction factor will have to be included at least; the influence of mass vaporization rate on the drag and a better consideration of droplet mechanics in shear flows will have to be addressed by subsequent studies. Droplet Number Density 1 O = ~, (33) where 0 = T/T* is the nondimensional temperature, and all the quantities are now nondimensional unless explicitly specified. The nondimensional equations are :so, cOu + (34) cOu = -OC ~ x +oL cOy (o°u \ cOy] + OSm, (35) where For a given particle velocity field Vt, at steady state, we can write, if particles are not generated or destroyed: c = po/(,o~o_r_) V.(nVt)=O. cO0 cOO 0 cO u-ff~ x + V~y -- Pr Oy (32) SIMILARITY ANALYSIS In principle, we now have all the equations to solve the problem. In this section, it will be shown that the problem is suitable for a similarity analysis--like the simple gas-phase counterflow flame--under certain conditions, and the condi- (ooo × ~, Oy,} + OSe - OQWF, (36) uaYi aYi = O I L (oOYi~ Ox +o f scoy \ + O05iF -- Yi)S~ + Ofvi, (37) COUNTERFLOW SPRAY COMBUSTION MODELING where Yi = Yi-oo; 0 = 1 S o - o*/t*' Sm Se = r/ = +c~z: f ' = -v/T+o~/T_~; O*u*/t*' Yi = Yi+oo; 0 = T+o~/T_o~ Se p*CpT*/t*' (~ Q - CpT*' gel fiPi o*/t*' and ^ denote dimensional quantities. We now introduce the similarity variable: = f Y dy ,7 331 (38) JoT' f = x d~ - (39) OS~d~ =-O(f+fv). t; = Ro2/Ott; L; = Ro; 3 M ; = -~rRo at; Assuming similarity, all x derivatives are zero, and from Eq. 34 we find v =-0 has a similarity solution only if So, Sm Ix, and Se are independent of x. In order to nondimensionalize the droplet equations, we start with the moving boundary problem constituted of Eq. 21 with the relevant initial and boundary conditions. The following reference quantities are introduced: 4 and the stream function-like quantity: (44) T; = Tto and the following transformation of variables is performed: r f - R(t); ~fotdZ T = f--7' (40) with z a dummy variable and fs = By using the Euler equation to express d p / d x as a function of the outer flow velocity Uo = x, we find the transformed momentum equation f ' " + ( f + f v ) f " = (f,)2 _ 0 -- --,0Sin (41) go OOI ( L dfs fXl OOt Or \ f s dr J Of 1 0 (2001) - f2 Of f ~ ' (46) where primes denote derivation in 7. Equation 36 becomes + ( f + fo)O' -----OSe + OQg'F (45) ' so that the problem transforms into X 10Pr R(t) (42) Or(f, 0) = 1, 001 ~=0 ~- = 0; (47) 001 ~ O ~=, -- 3~s' (48) and Eq. 37 becomes where q has been nondimensionalized as follows: 1 "_L_y~.,+ ( f + fv)Y[ = --(~iF Sc -- Yi)OSo - OWl. (43) 0 = m o-Ots BT - L (49) It is clear that the resulting problem, together with the boundary conditions: r/ = - o ~ : f = f - o ~ ; f ' = 1; rn with respect to MT/t 7 and L with respect to CptT 7. Equation 15, combined with Eq. 16 and 332 G. CONTINILLO and W. A. SIRIGNANO aondimensionalized, becomes d~s _ dr 27r M* L7 t7 3 M 7 L* t* pf__f_hDS ln(1 + BM), (50) where Of and D f are the nondimensional average film density and mass diffusivity, respectively. If the droplet velocities and accelerations are aondimensionalized with respect to the gas refer~nce quantities, the vector equation of the droplet motion becomes dEXtd72 ~sl \-d-rT(d~s_ aO) ~dxt = abOV + b2~s2g, (51) where _ M 7 =67r -ff/ * L t* tt t,*, . b t/* = -t* By projecting Eq. 51 on to the x and the ~ axis ~¢e obtain d2x, d7"2 1 ( d~s ) dXl ~s \ dr - aO dr not know about is (s, the instantaneous nondimensional radius of the droplet. It comes as a result of the whole integration, as solution to Eq. 50, which, in its turn, depends on BM, which is a function of the local gas-phase fuel vapor concentration and surface vapor pressure, which are functions of the temperature at the drop surface, which in turn results from the energy equation, which is coupled to the gas-phase equations through the boundary conditions, and so on. If we suppose that the solution to Eq. 53, with initial conditions that do not depend on x, is independent of x, then all the droplets will have the same ~/ trajectory and, since the gas-phase variables do not depend on x, the free-stream history will be the same for all the droplets. Hence, if the droplet initial radii and temperature distributions are all the same, the vaporization history will be the same, i.e., (s will change with r (and with ~lt(r)) regardless of the x location of the droplet. This constitutes one necessary condition for ~l(r) being independent of x, in that it ensures that Eq. 53 is the same for all the droplets. The other one is that initial conditions be obviously chosen independent of x, that is, ~t(O) = ~to, = a b O ~ x l +b2~s2gx, (52) 7/[(0) = 7/[o, (54) and, since dyt _ 0 d~lt ; dEyl dz -~r dr 2 _d % dO ( and together the two conditions are sufficient. The solution to Eq. 52, that is, the x trajectory, is obviously dependent on x. Droplets move in x and y, so determining the number density variation in the field. The initial conditions for Eq. 52 are the degree of freedom that we shall use in order to make the source terms for the gas-phase independent of x. To do that, we must calculate the number density as a function of the droplet velocity field. Equation 32 is also written as )2 we have dZm + l dO ( drtl' 2 ----e dT O t,-d; ) 1 (d s ~s \ d r - aO )d., dr V . (nVt) = Vt • V n + n V . V~ = 0, ~s 2 = ab[-O(f + f v ) ] + b2-ff-gy. (53) In Eq. 53 all the coefficients depend on variables that are a function of ~/ only, if similarity exists for the gas phase. The only quantity that we do (55) and, by definition, Dn II# • Vn = Dt dn dt (56) COUNTERFLOW SPRAY COMBUSTION MODELING Integrating To calculate V . 1"1 we first observe that dr# Dvt Ovt Ovt dt - D~ - Vt • Vvt = ut-~x + vt-~y 333 (57) nsvl = rloSoVlo, or else so Ovt Oy 1 dvt vt dt (58) since Ovt /Ox = 0 for similarity. To calculate Out/Ox, it is convenient to introduce the variable nsOn[ = nosoOonlo, that is, nosoOo~[o n - s -- Os~i (64) Xl (59) Ulo so that Eq. 52 becomes d2s l(d = dr 2 We are now ready to evaluate the source terms to the gas phase equations. Starting with Eq. 41, we have )as Oh f ^ x = abO~--~s + b2~s2gx, dftl ~mt--~- + ( ftt OSm ~s \ dr - aO -~r _~)~) (65) xp*u*/t* (60) By definition, with the initial conditions 3M 7 t7 ~ = ~ ' d&l _ 47r/~03~/3~s2 d~s s(O) - Xto _ So; s'(O) = 1. I11o (61) di 3 Since fxt- dxt ds Xto ds XI = U l o S, Ul -- _ _ dt - U t ° d t - s d t ' Yet ds s dt Ou, l dS 0 (lds~ Ox - s dt + X#° ox k s dt J " XI -- If we choose So to be constant, then s will not depend on x, being a solution to Eq. 60 with initial conditions 61 that are independent of x. Therefore we have L* 1 xt ds t; ~s s d r ' ds s dt" 1 1 dot n d-t + ~ d t - + = Uto = const, S dill di Yet d2s s di 2 (62) _ Note that So = Xto/Uto = 1 corresponds to matching gas and droplet velocity components along x at the starting location for the droplets, since, for the gas phase, in the outer flow, we have x / u = 1/K in dimensional variables. We now have expressions for all the terms in Eq. 55, which becomes 1 dn - and, remembering that we have Out Ox dt 1 ds s ~ -- 0. (63) L* 1 xl ['d2s t/.2 ~s2 S ~ d r 2 1 d~sds ~s d r d r ) and, finally, from Eq. 39 we know that L* ~ = - - t , f 'x • After substitutions into Eq. 65, with the aid of the expression for n just found as Eq. 64, we finally 334 G. CONTINILLO and W. A. SIRIGNANO obtain OSm - -X M,. ( , . ) 2 -- M* \-i-it] n°s°O°Tl[° dEs __ 3 t; ~ + ~s~ t* ~sf'S d~s d s 2~ X S21,1[ (66) independent of x. For Eq. 42 we have -- OSe = M* Oh L,3CpT*/t * X {q -l-m[CpF(7: --'Fs) -']-L]}, (67) which, remembering the expression for q given by Eq. 24, becomes - OS~ = OhDICpF(T M * C_.pT* L .3 7"s)1 + B r BT t* That is, CpF M ; t* . --OSe = C p M't--7 m x T; ° 1 + Br nosoOoTl:o 0 - T* Is) BT sTq[ (68) For Eqs. 43, we have ^ ." r/m OSo = Op, /t-----z. (69) That is, OSo = M~ t* nosoOo~[o 3~sd~z . M* 7; s~; (70) NUMERICAL SOLUTION PROCEDURE The equations for the gas phase with the relevant boundary conditions constitute a boundary- value problem, while the droplet equations, in their lagrangian form, constitute an initial value problem--with the exception of the liquid-phase energy equation, which is a parabolic partial differential equation. The gas-phase, droplet, and liquid-phase problems are coupled to each other through source terms and/or boundary conditions. An iterative procedure is employed. The gas-phase equations are solved first, followed by the droplet equations and the liquid-phase energy equation by using the just computed values of the gas-phase variables in the coupling terms. The source terms are then calculated and fed back to the gas-phase equations, which are solved again, so forth. The equations for the gas phase are discretized by means of finite-difference schemes, and a fictitious timelike independent variable is introduced. In this way, the equations are integrated in sequence, and the "steady-state" solution is sought that corresponds to the initial guess. The convergence criterion for this inner iteration loop involving the gas-phase equations is chosen not so severe as the overall criterion (10 -4 relative error between two iterations), since the droplet contributions are to be evaluated with the updated gas-phase variables, and there is no point in calculating an accurate solution with the incorrect source terms at each cycle of the outer, main iteration loop. Due to the nonlinearity of the problem, multiple solutions exist for the same set of boundary conditions. For example, both for a diffusion and a premixed configuration in laminar flow, the physical problem always has a stable "cold" solution with no flame, and may or may not have stable solutions with flames. In the affirmative case, the choice of the initial guess, and the numerical procedure itself, determine the convergence to either of the solutions. With the presence of droplets, more nonlinearity is added to the problem, and there appears to be no easy way to tell how many different solutions the problem has. In this study, no such analysis is attempted. When the influence of a parameter is studied, the parameter is varied by small increments from one case to the next, and the last computed solution is used as a first guess. COUNTERFLOW SPRAY COMBUSTION MODELING TABLE l Main Values o f P a r a m e t e r s U s e d in the Calculations B = 4 . 6 × 10 ]1 s - ] g - m o l -°'75 E a = 3.0 X 10 4 cal g - m o l - ~ K t~ = 0 . 2 5 8 = 1.5 Cp = 1.29 x 1 0 3 J k g - I K - I Po = 1.0133 x l 0 s Pa Q = 4.43504 x 107Jkg -l p / = 7.03 x 1 0 2 k g m -3 C p / = 2 . 4 2 x 1 0 3 J k g - I K -1 L~,il = 2 . 9 7 8 x l 0 s J k g -1 Tbon = 3 . 9 8 8 X 10 z K To = 3 x 102 K #o = 1.8 x 10 - S k g m - I s -I RESULTS A N D D I S C U S S I O N Three basic configurations have been considered in the present work. The first configuration is a symmetric stagnation flow field, generated by two impinging air streams at 300 K, perturbed by a n-octane spray injected from the left side in our representation. The second and the third are analogous to a simple gaseous counterflow premixed 2 / / o o > LC >" ] .1 ! l ! f i # i - ".... tl. O.C -15.0 ,,,--Ii ?d.. tPt . !! ",. . . . , . !1 " .54 ! ni 0.0 , ' y [mm] 15.0 Fig. 2. Results o f calculations for the base case. Profiles o f fuel v a p o r ( Y F , dotted) a n d o x y g e n ( Y O , broken) mass fractions, t e m p e r a t u r e (T, solid line) a n d log o f the reaction rate (dot a n d dash, a r b i t r a r y units). Initial droplet diameter: D o = 50 # m . Strain rate o f the outer flow: K = 55 s - l . 335 and diffusion flame, respectively. A cold stream containing the spray (at the left in our representation) and a hot stream form a stagnation flow. In the former, "premixed" case, the cold stream consists of air carrying a n-octane spray at 300 K, while the hot stream is inert at 1200 K. In the latter, "diffusion" case, inert and droplets come from the left at 300 K, hot air (1200 K) from the right. Equation 6 represents the simplest way of dealing with chemical kinetics (one-step overall reaction). The form for the reaction rate and the numerical values for the parameters are those suggested by Westbrook and Dryer [16] for a premixed flame of n-octane in air. In fact, it is likely that this kinetic model is inadequate for accurate predictions of flames in configurations other than that explored by the cited authors. Moreover, limit situations like those of flame extinction certainly cannot be adequately described by a global, onestep kinetic model, no matter what values are used for the parameters in the reaction rate formula. Table 1 presents the values of the kinetic parameters along with those of other properties used in the calculations. In all of the cases considered, the velocities of the droplets match the gas velocities at the left boundary, as the temperatures (300 K) do; the initial equivalence ratio is l; body forces are neglected; no fuel vapor is injected with gas flow. Figure 2 shows the y profiles of fuel vapor, oxygen, temperature, and the logarithm of combustion reaction rate for a case with K = 55 s - t and initial droplet diameter of 50 #m, which is the base case for the first configuration. The flame structure is interesting in that it consists of two main reaction zones separated by a region in which most of the vaporization occurs. The first flame, at left, has a predominant diffusion character, but some prevaporization occurs, as shown by the fuel vapor profile. It appears that the droplets cross the first flame at left and vaporize in the high-temperature region, closer to the stagnation plane (at y = 0). Due to both the curvature of the gas flow and the thermal expansion, the mixture becomes rich as the stagnation plane is approached, since the droplets do not follow the gas motion promptly. 336 G. CONTINILLO and W. A. SIRIGNANO ~r =o t.- .2 o 0 >- u." t.C >.- >,. / ... t -'--~ ,. I ", al 1.1 ; ; I O.Q ...... ~ "../l ; l[ ..~.,~ ~,~ -15.0 I 15.0 0.0 0.0 -15.0 , ir L...... .d."f , 0.0 15.0 y [mm] y [mm] (a) (c) Fig. 3. (Continued) =o £_ .2 o >,. u." >.. I i [ .:. ~".,1 O.G -15.0 I I ,I L_ I 0.0 I 15.0 y [mm] Co) Fig. 3. Influence of the strain rate. Profiles of fuel vapor and oxygen mass fractions, temperature, and log of the reaction rate. Initial droplet diameter: Do = 5 0 / z m . (a) K = 100 s - I . (b) K = 3 0 0 s - l . (c) K = 5 0 0 s -t. A second, pure diffusion flame stabilizes approximately at the stagnation plane. Figure 3e illustrates the influence of the strain rate of the flow, for a given droplet initial diameter. The strain rate is increased from that of the base case in Fig. 2 up to 500 s -1. It appears that increasing the strain rate causes the two react- ing zones to narrow and approach the stagnation plane. At the highest value, the two peaks in the reaction rate profile are very close to each other, while the temperature profile no longer shows two distinct maxima. This indicates that the two reaction zones will merge into the zone at high strain rates. Figure 4 illustrates the influence of the initial diameter of the droplets on the flame structure for the first configuration. It is seen that, as the droplets are chosen larger, they are able to penetrate the first flame, thus producing a peak in the vapor fuel mass fraction, corresponding to the droplet consumption, and higher vapor fuel mass fraction in between the two flames. As already noted, the particle-gas relative motion is responsible for the differences observed with respect to the known behavior of simple gaseous systems. Figure 5 illustrates the gas and particle velocity components perpendicular to the stagnation plane for the cases of Figs. 2 and 4c, respectively. The gas velocities (solid lines) do not differ much from the usual counterflow gas flame profile, with a velocity jump across the flame and the linear, potential outer flow towards the boundaries. A slight inflection can be detected corresponding to the point at which the droplets vaporize completely and cease to contribute to the gas balance equations. It is seen that, the larger COUNTERFLOW SPRAY COMBUSTION MODELING 337 .3 E "t ~o 0 o >u_ >- LI." >- ! O.O -20.0 i! J I # i ~ ....- " 0.0 ~ " I I 20.0 0.0 -20.0 .!~.~ j 20.0 0.0 y [mm] y [mm] (c) (a) Fig. 4. ( C o n t i n u e d ) E ,i- iii o .2 / / o> EL" >- ! , I ; \~ 0.0 -20. &- .....,,.~. , = ,~d 0.0 , , 20.0 y [rnm] (b) Fig. 4. Influence o f the initial droplet diameter. Profiles of fuel vapor and oxygen mass fractions, temperature, and log of the r e a c t i o n rate. Strain rate o f the o u t e r flow: K = 55 s - I (a) Do = 1 0 / ~ m . (b) Do = 3 0 # m . (c) Do = tO0 #m. are the droplets, the larger is the mismatch between the velocities of the two phases. The same phenomenon (not shown) occurs for the velocity component along the stagnation plane; the gas is always accelerating and so are the droplets, which are dragged by the gas and therefore move more slowly. By combining the two effects (droplet initial diameter and flow strain rate), it can be said that higher strain rate for larger droplets produces a more compact flame zone if compared with smaller droplets (Fig. 6). The second configuration (Fig. 7) differs from the first one in that it has no oxygen in the right stream, and therefore only one flame zone is observed. The temperature of the right, hot stream is "chosen lower than the maximum temperature reached in the flame; however, it appears to have little, if any, influence on the flame. In fact, the flame stabilizes itself out of the boundary layer, identified by the "s"-shaped part of the temperature profile across the stagnation plane. The left part of the structure is very similar to that found for the first configuration (Fig. 2). This result is obviously dependent on the particular set of the parameters chosen, first of all the strain rate; higher strain rates will make flame and boundary layer merge. The third configuration (Fig. 8) produces results that are most similar to those of a pure gaseous system. This can be explained by the fact that, since the carrier gas does not contain oxygen, the droplets vaporize almost completely in the preheating zone while no reaction takes place. Different behavior can be expected for larger and faster incoming droplets, having enough momen- 338 G. CONTINILLO and W. A. SIRIGNANO •~ 2.(1 l r i "--~ E ~., t. . . . . l .2 LU O j # I-'- O.C ~7 O >U-- .1 >- -2.0 -15.0 i I i i 0.0 -15.0 = 0.0 15.0 y [ram] = ' 0.0 y [ram] 15.0 Fig. 6. C o m b i n e d effect o f i n c r e a s e d droplet initial d i a m e t e r a n d strain rate. Profiles o f fuel v a p o r a n d o x y g e n m a s s fractions, temp°erature, a n d log o f the reaction rate. Strain rate o f the outer flow: K = 110 s - 1 . Initial droplet diameter: Do = 100 # m . (a) ~.o stream. Figure 9 shows the results for K = 25 s - t , wherein the flame zone is broadened. E CONCLUSIONS A similar solution for a planar counterflow spray flame with transient heating and vaporization of o.o .3 , , , E -2.0 "1 5m0 i I I , 0.0 ",t , 15. 0 y [ram] Co) Fig. 5. Influence o f the initial droplet d i a m e t e r on interphase slip. G a s - p h a s e (solid) a n d droplet (dotted) velocity c o m p o n e n t p e r p e n d i c u l a r to the s t a g n a t i o n plane (at y = 0). Strain rate o f the outer flow: K = 55 s - 1 . (a) D o = 5 0 # m . (b) Do = 100 /~m. wo .2 I-- o) . . uS >- I ,1 • tum to cross the stagnation plane and possibly the flame before complete consumption. The temperature of the right stream is important in this case, as opposed to the premixed case. In fact, a diffusion flame could not be stabilized at the strain rate of 55 s - l with cold (300 K) right "'" "*'"-..... !~ O.O -20.0 ~ "-.. ..,.-.-L ; M I 0.0 "...~ , y [ram] , 20.0 Fig. 7. Results o f calculations for the second configuration. Profiles o f fuel v a p o r a n d o x y g e n m a s s fractions, t e m p e r a t u r e a n d log o f the reaction rate. Initial droplet diameter: D o = 50 # m . Strain rate o f the outer flow: K = 55 s -~ . COUNTERFLOW SPRAY COMBUSTION MODELING .3 i i E / ll.- .2 \ uS >- 0.C -20., I I .... ~°'° I I I 0.0 20.0 y [mm] Fig. 8. Results of calculations for the third configuration. Profiles of fuel vapor and oxygen mass fractions, temperature and log of the reaction rate. Initial droplet diameter: Do = 50/zm. Strain rate of the outer flow: K = 55 s -1 . Temperature of the right stream: To+ = 1200 K. droplets has been obtained. The mathematical formulation is complete in that it accounts for interphase exchange of mass, momentum and energy. It is noteworthy that numerical solutions can be computed within 10 to 30 minutes on 1 MIPS com.3 i i i i i 339 puter. This means that parametric studies can be easily conducted. Various configurations of counterflow of air, droplets, and inert gas can be considered. In some configurations, two distinct flame zones are encountered. Droplets, in general, do not follow gas streamlines. Flames with both premixed-like and diffusion-like characters are found. Initial droplet size and flow strain rate are important parameters. An increase in initial droplet diameter tends to cause a separation of the flame into a premixed flame zone and a diffusion flame. Increase of the strain rate tends to make the flame more compact and to inhibit such separation. A comparison of the results with values coming from experimental measurements is highly desirable; however, no such a configuration of a spray flame is available to the authors' knowledge. The experimental setup is seen as one of the most interesting areas of development for future work. As mentioned earlier, particle drag is to be investigated more thoroughly both with theory and experiments. Finally, more detailed description of the combustion chemistry will be necessary in order to make model results more realistic, especially in the study of chemically controlled situations, like, for example, flame extinction. i REFERENCES 1. i'- .2 l >. 2. 3. i 4. 5. i .1 6. .... / 7. .......... O.C' -20.4 "f ~ I I 8. I 20.0 0.0 y [rnml Fig. 9. Results of calculations for the third configuration. Profiles of fuel vapor and oxygen mass fractions, temperature, and log of the reaction rate. Initial droplet diameter: Do = 50/~m. Strain rate of the outer flow: K = 55 s -1 . Temperature of the right stream: To+ = 300 K. 9. 10. 11. Krishnamurthy, L., Williams, F. A., and Seshadri, K., Combust. Flame 26:363 (1976). Tsuji, H., Prog. Ener. Combust. Sci. 8:93 (1982). Libby, P. A., and Williams, F. A., Combust. Flame 44:287 (1982). Lifian, A., Acta Astronaut. 1:1007 (1974). Hamins, A., and Seshadri, K., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1903. Seshadri, K., Puri, I., and Peters, N., Combust. Flame 61:237 (1985). Buckmaster, J., and Mikolaitis, D., Combust. Flame 47:191 (1982). Libby, P. A., and Williams, F. A., Combust. Sci. TechnoL 31:1 (1983). Libby, P. A., Lifian, A., and Williams, F. A., Combust. Sci. Technol. 34:257 (1983). Libby, P. A., and Williams, F. A., Combust. Sci. TechnoL 37:221 (1984). Libby, P. A., and Williams, F. A., Combust. Sci. TechnoL 54:237 (1987). 340 12. Niioka, T., Mitani, T., and Takahashi, M., Combust. Flame 50:89 (1983). 13. Graves, D. B., and Wendt, J. O. L., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, p. 1189. 14. Puri, I., and Libby, P. A., 1988 Spring Meeting of the Western States Section, The Combustion Institute, Paper no. 88-42. G. C O N T I N I L L O and W . A. S I R I G N A N O 15. Abramzon, B., and Sirignano, W. A., Second ASME-JSME Joint Thermal Engineering Conference, Hawaii, March 1987. 16. Westbrook, C. K., and Dryer, F. L., Prog. Ener. Cornbust. Sci. 10:1-57 (1984). Received 23 January 1989; revised 21 November 1989