Unsteady upstream interactive combustion of two stoichiometric planar methane premixed flames that propagate towards each other as they consume the finite ...
Upstream Interactions between Planar Symmetric Laminar Methane Premixed Flames C. L. CHEN and S. H. SOHRAB* Department of Mechanical Engineering, Northwestern University Evanston, IL 60208
Unsteady upstream interactive combustion of two stoichiometric planar methane premixed flames that propagate towards each other as they consume the finite slab of combustible gas in between them is numerically investigated. The onset of hydrodynamic, thermal, and diffusional interactions between the merging flames that is governed by the relative magnitude of the diffusivity for momentum, heat, and mass are discussed in terms of the dimensionless groups Prandtl, Lewis, and Schmidt numbers. The average propagation velocity of laminar methane flames is found to first increase from 40.5 to 41.9 cm/s during their thermodiffusive interactions and subsequently to 282 cm/s during the chemical interactions associated with the merging of the reaction zones. The inlluence of the Lewis number of the fuel and the oxidizer on the nature of upstream flame-flame interactions is determined. The results show that under appropriate values of the transport coefficients the sharing of reactive radical species may lead into local homogeneous explosion of the reactive mixture.
INTRODUCTION Under certain range of Damkijhler and Reynolds numbers, turbulent premixed flames are modeled as an ensemble of randomly moving flamelets [l-3]. Such a model necessitates the occurrence of regions of combustible gas that are separated from neighboring regions of combustion products by flamelet surfaces that are either closed or open. Because of the large number of flamelets, interactions between the flamelets are inevitable. Therefore, many recent investigations [4-91 have been concerned with the interactive combustion of counterflow premixed flames. In turbulent premixed systems under strong stratification of the fuel concentration, triple flame interactions [5,6] as well as diffusion to premixed flame transition [lo] may also occur. Clearly, the hydrodynamic, the thermal as well as the diffusional (concentration) interactions between the neighboring flamelets will influence the local burning intensity and hence the global behavior of the reactive field. In premixed systems, two distinguishable types of flame-flame interactions were referred to as the downstream and the upstream flame interactions [6]. Downstream flame interaction occurs when two premixed flames propagate away from each other as they are
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further separated by a layer of combustion products as shown in Figs. la and b. Steady combustion under downstream interaction is possible in the back-to-back counterflow premixed flames [5-91. The second type of flame-flame interaction is the upstream flame interaction that is inherently unsteady and occurs when two premixed flames propagate towards each other as they consume the slab of the combustible gas in between them as shown in Figs. lc and Id [ill. Because of the turbulent fluctuations as well as the hydrodynamic or thermodiffusive flame instabilities, the flame surfaces are generally corrugated and may assume cellular structure [l, 12, 131. Recent investigations have shown that the preferential diffusion of heat versus mass have appreciable influence on turbulent burning rates [14, 151. However, interactions between flamelets are expected to have a substantial influence on the nature of their thermodiffusive instability [161. Also, in the spark-ignition internal combustion engines turbulent combustion is believed to occur under flamelet burning regime [3]. It is known that at higher engine speeds small pockets or islands of combustible gas that are surrounded by the combustion products occur [17]. As a result, upstream interactive combustion may occur under both spherically imploding [18] as well as planar [12] flame geometry. Because of its transient nature, the experimental investigation of upstream flame interactions is difficult. The present study is thereCOMBUSTIONAND FLAME 101: 360-370 (1995) Copyright 0 1995 by The Combustion Institute Published by Elsevier Science Inc.
LAMINAR
METHANE
PREMIXED
361
FLAMES
general aspects of hydrodynamic, thermal, diffusional, and chemical interactions between flamelets in turbulent premixed flames is first discussed. Next, the problem of interactions of two one-dimensional stoichiometric methane premixed flames and the method of numerical calculation is described. The results of the calculations for various Lewis numbers of the fuel and the oxidizer are then presented. The principal findings of the study are summarized in the concluding remarks.
(a
WI
Fig. 1. Schematic diagram of (a) in-phase downstream (b) out-of-phase downstream (c) in-phase upstream Cd) outof-phase upstream interactions between two premixed
flames. fore focussed on numerical investigation of the upstream interactions between two planar stoichiometric premixed methane flames. The reduced kinetic scheme for methane combustion by Seshadri and Peters [19] is employed with the numerical procedure developed for the solution of one-dimensional laminar premixed flames [20-261. It is found that the nature of the flame interactions is strongly dependent on the Lewis number of the fuel and the oxidizer. For example, depending on the value of the transport coefficients, the local burning intensity could substantially increase (decrease) thus leading to local homogeneous explosion (extinction). Because of the assumption of constant pressure made in the study, the hydrodynamic interactions between the planar flames is not considered. In the following section, the
HYDRODYNAMIC, THERMAL, DIFFUSIONAL, AND CHEMICAL INTERACTIONS BETWEEN TWO PREMIXED FLAMES The propagation of the laminar premixed flame involves four distinguishable length scales associated with the hydrodynamic, thermal, diffusional, and chemical aspects of the flamefront as schematically described in Fig. 2. First, is the flame hydrodynamic thickness 1, = V/U~ representing the transition layer within which the gas velocity (u_,, u,) changes. The parameters v and uf are the kinematic viscosity and the flame propagation velocity while u_, denotes the upstream gas velocity which is kept equal to the flame velocity uf = u_, to maintain a stationary flame front. Another characteristic length is the flame thermal thickness 1, = cr/uf, which is the thickness of the temperature CT_cy,I”.‘.)transition layer (Fig. 2) when LY= A/PC, is the thermal diffusivity and h, p, cP are the thermal conductivity, the density and the specific heat at constant pressure. Next is the flame diffusional thickness lDf = Di/uf
Fig. 2. Schematic diagram of the hydrodynamic I,, the thermal I,, the diffusional I,, and the chemical length scales associated with a laminar premixed flame.
362 associated with the thickness of the transition layer of the partial density of specie (i) across the flame ( pi_ m, p,,> when Di is the coefficient of mass diffusivity for binary diffusion of this specie within the background mixture. Finally, there is the thickness associated with the much thinner chemical reactions zone 1, = lT/P when /3 is the Zeldovich number p = EU’, - To)/RTF, E the activation energy, To and Tf are the upstream and the flame temperature [l]. Clearly, the chemical thickness 1, depends on the detailed kinetics and is generally much smaller than the other three length scales. It is reasonable to view the hydrodynamic, thermal and diffusional length scales (l,, l,, lo) defined above and schematically shown in Fig. 2 as three external length scales. On the other hand, the chemical length scale associated with the much thinner reaction zone 1, -e 1, = 1, = 1, could be referred to as the flame internal length scale. It is emphasized that within the thin reaction zone, a much thinner region 1, -s 1, may be identified which is associated with the radical production zone [l]. The occurrence of the four length scales associated with four physical attributes of the laminar flame suggests that four corresponding types of interactions could be defined when two premixed flames are brought in close spatial proximity of each other. Hence, hydrodynamic interaction between two neighboring premixed flames refers to the separation distance of 21, associated with the onset of merging of the velocity profiles of the flames (Fig. 2). Similarly, as the separation distance between flames reaches 21, or 21,, the thermal or the diffusional interactions between flames will begin corresponding to the merging of the temperature and concentration profiles. Finally, at the much smaller separation distance of 21,, the merging of the reaction zones of the two flames occurs which is identified as the onset of chemical interactions. The definition of the three length scales (I,, I,, lo) suggests that their respective values depend on the magnitude of the corresponding diffusivity (v, cy, D). Therefore, the relative magnitude of the Prandtl number Pr = v/a, the Lewis number Le = a/D, and the Schmidt number SC = v/D will determine the order of the onset of
C. L. CHEN AND S. H. SOHRAB hydrothermodiffusive interactions between two approaching premixed flames (Fig. 2). In the limit of vanishing dissipations when v = (Y= D = 0, the thickness of the transition layers collapse into delta functions (I,., = 1, = 1, = 0) and the flame may be considered as a mathematical surface of discontinuity. Because of the turbulent fluctuations or the intrinsic hydrodynamic or thermodiffusive flame instabilities interacting flamelets are expected to be corrugated. According to the schematic diagram in Figs. lc-ld, the relative phase between the corrugations of the two interacting flames will influence the nature of their interactions. In cellular premixed flames, it is known that the local flame temperature at the trough is higher than that at the crest [12, 271. Clearly, the heat loss/gain induced by upstream interactions between flamelets (Fig. 1) will influence their thermo-diffusive stability [16]. For example, out-of-phase interaction between two-dimensional flames (Fig. 3a) will result in the suppression of the flame corrugations and the eventual formation of planar slab of imploding combustible gas as shown in Fig. 3b. On the other hand, in-phase interaction between the flames (Fig. 34 results in the growth of the flame corrugations and eventual generation of cylindrical volumes of imploding combustible gas surrounded by combustion products as shown in Fig. 3d. If cylindrically symmetric flames rather than two-dimensional flames were to be considered, the interactions
w
(a
Fig. 3. Schematic diagram of the evolution of premixed flame topology under (a)-(b) in-phase (c)-(d) out-of-phase upstream interactions.
LAMINAR
METHANE
PREMIXED
shown in Figs. 3a and 3c would result in the formation of cylindrical and spherical volumes of combustible gas surrounded by combustion products. It is therefore clear that the geometry of flamelets may alter as a result of their interactions. All turbulent fields are known to involve an ensemble of spatiotemporal scales. Therefore, in turbulent reactive flows, the geometrical waves of flame surface would actually represent four types of waves respectively associated with the velocity, the temperature, the concentration as well as the chemical aspects of the flame (Fig. 2). The transient nonlinear interactions between these various wave fronts are exceedingly complex. Therefore, in the following section only the simultaneous thermal and diffusional interactions between stoichiometric premixed flames of methane will be investigated. The hydrodynamic interaction between the two flames is suppressed by the assumption of small Mach number that results in a constant pressure across the flame front. Also, only interactions between planar flames are considered and curvature effects such as are shown in Fig. 3 are not addressed. It will become evident that even in the absence of the hydrodynamic and the curvature effects, the unsteady thermodiffusive interactions between the merging flames are quite complex. INTERACTIONS OF TWO PLANAR METHANE PREMIXED FLAMES The upstream interactions of two stoichiometric planar premixed methane flames will be considered that are simultaneously ignited at the opposing ends of the computational domain and allowed to merge with each other at the plane of symmetry in the middle. The governing equations for the conservation of mass, energy, and species concentration are [l]
a( ~ = x+ dP
pu)
dx
363
FLAMES
to be considered with the relation between species mass fractions and the ideal gas law j-l
y,=l-
cyI-Y,z,
(4)
i=l
P = pl?T f: y/u/;. i= 1
(5)
The reduced four-step mechanism for oxidation of methane by Seshadri and Peters [19] is used: I
CH, + 2H + H,O --) CO + 4Hz,
II
(6)
CO + H,O ++ CO, + H,,
(7)
III
H+H+M-tH,+M,
(8)
IV
0, + 3Hz f) H,O + 2H,
(9)
where M denotes the third body, which may be CH,, H,O, CO,, H,, CO, O,, N,, etc. The above four reaction steps represent two reversible and two irreversible reactions among seven species. The rate constant for each reaction step is expressed as K, = A,T”exp(
-E,/RT)
where the preexponential factors A,, and the activation energy E,, are given in Ref. 19. The unsteady one-dimensional problem has been treated in prior investigations [20-261 some of which have considered methane combustion [23, 241. The problem is rendered convectionfree by using the Lagrangian coordinate transformation [20] * = j-;p
fix.
(10)
Under the assumption that PA/C,, ph, and as discussed by Beldjian equation is automatically satisfied and the problem reduces to the following system of equations.
p2D are constants [22], the continuity
0, dT -= dt au,
_=_-at
-@$-$‘$$$, ph
(11)
1 a’y,
+!_! c,, Le, &+G2 p’
i = 1,2,...6.
(12)
364
C. L. CHEN AND S. H. SOHRAB
The adiabatic boundary conditions dT
G-
-
0;
JY a+
- 0,
i=1,2
)...
7 6
(13)
are imposed at both ends of the calculation domain defined as 1I’= (-1.0
X lop4 to 1.0 X 10m4 gm/cm*). (14)
The initial conditions on the mass fractions and the temperature are expressed as
a.14
0.1
-0.05
o.02
0.02
0.06
0.1
0.14
X(cM)
Fig. 4. The initial temperature and concentration profiles for two stoichiometric premixed flames. (1) T X 10e3 (2) YH X 1000 (3) Yo, x 10 (4) YcH, X 10 (5) YHZX 100 (6) YHZo x 10 (7) Yco x 10 (8) Yco, x 10.
Ycn, = {(0.05518305), (0.05518305 x F)); Yn, = ((0.01, (2 x 1o-4 x P)}, Yn, = ((0.01, (0.025 x F)), V LO2
= Ilfi‘7WMAAfKQ\ ,\“.~~“l-r-r”d”,,
tn 73nl AAIXQ \“.~~“l-r-r”J”
” A
IT\\ 1’ ,, )
Yco = I(O.01, (0.05 x P>}, Yco* = {(0.0),(0.05
x F)),
T = T,,{(l.O), (1 + 5.666667 x F)}, YN, = 0.72467637. The values within the brackets { } in the above expressions respectively correspond to the two computation domains given as
and the function F has the form introduced by Margolis [25]
(15) The quantities T,, = 300 K, J& = 46, I,&,= &./ 50, I,!+ is half the computational domain which is 10m4 gm/cm’. The temperature and concentration profiles corresponding to the above initial conditions are shown in Fig. 4. Several initial conditions for the temperature were tested and it was found that the results were not sensitive to this initial condition as long as the total thermal energy imposed was above the critical minimum ignition energy.
By discretizing the second order spatial derivative but keeping the temporai derivative continuous in Eqs. 11 and 12, seven coupled ordinary differential equations (six species and one energy equation) are obtained which are solved by means of a fourth-order-accurate O[(Atj4, (AY)~] Runge-Kutta method with automatic error control during the time marching steps [26]. The time step and the local relative error are respectively 0.5 ps and 10e7. In the calculations, the value of heat of formation for species (i> at 300 K is taken from JANAF thermodynamic data and the values of ph/c, and cp appearing in Eqs. 11 and 12 are assumed to be 1.2 x 10e7 gm2/cm4/s and 0.325 cal/gm-K, respectively. The Lewis numbers of the species Len = 0.18, Leco = 1.11, and Leco* = 1.39 are kept at these constant values which are the same as those reported by Seshadri and Peters [19]. However, two sets of values for the fuel and oxidizer
Lewis numbers
are considered in order to illustrate their effects on the nature of flame interactions. Some of the Lewis numbers used in the numerical experiments do not represent realistic values for methane-air premixed system but are chosen to better reveal the nature of the interaction process. Several number of grid points were tested and it was found that 201 grid points are sufficient to render accurate results. The computational time on IBM 4381 under double precision mode is around 5 h.
LAMINAR
METHANE
PREMIXED
FLAMES
RESULTS AND DISCUSSIONS In the calculations, two premixed methane flames are initiated within a stoichiometric mixture of methane-air at the ends of the computational domain and allowed to propagate towards each other until their complete burnout at x = 0 (see Fig. 4). First, the influence of the atomic hydrogen concentration on the flame ignition, steady propagation and burnout is explored. For the calculations the initial concentrations are set at 2 x 1O-4 and 2 x 10m6 while the Lewis numbers are set at Le, = 0.97 and Leo* = 1.1. The results are expressed in terms of the fuel rate of mass consumption as a function of time over the half calculation domain as shown in Fig. 5. The area under the curves represents the total heat release within the volume and should be constant for the two calculations. The results in Fig. 5 clearly show that the higher initial concentration of atomic hydrogen results in shorter ignition delay times while leaving the steady flame burning rate unaltered. This is to be expected since the additional hydrogen atoms contribute to the chain branching reactions leading into faster development of the radical pool. Therefore, 1500 /_Gafter the initiation of ignition, effects associated with the initial conditions become nearly undetectable. It is also noted that the reduced ignition time results in shorter overall burnout time which is reached after the interactive combustion has been completed. Such augmentation of the ignition pro0.6
,
I
Fig. 5. The influence of atomic hydrogen concentration on the mass burning rate of stoichiometric premixed methane flames (1) Y, = 2.0 X 10e4 X F (2) YHzo = 2.0 X 1O-6 x F and F is defined in Eq. 15.
365 cess is in fact similar to the plasma-ignition devices which provide high-energy atomic species during the development of the flame kernel. In the remaining calculation to be discussed in the following, the initial concentration of atomic hydrogen is fixed at the value of 2 x 1O-4 x F. The results of calculations showing the evolution of the flame initiation and interactions are shown in Fig. 6a-6h. In these calculations, the Lewis numbers Le, = 0.97 and Leo = 1.1 used by Seshadri and Peters [19] are employed. Also, the transformed coordinate I,!J is converted to the real spatial coordinate z for better appreciation of the flame separation distances during the interaction process. The temperature as well as the mass fraction for seven species during the steady propagation are shown in Fig. 6a. The steady propagation velocity of the methane-air flame is found to be uf = 40.5 cm/s in close agreement with prior observations [28-301. The onset of thermal interaction is at t, = 1443 ps when the upstream edges of the two preheat zones of the two flames first meet. This point may also be defined as the time when the temperature of the gas at the line of symmetry at x = 0 (see Fig. 6b) rises by 1% from the initial value of 300 K. Therefore, at the point of initial flame interaction the separation distance between the two flames is found to be 2 x 0.0313 = 0.0626 cm. The flame location is defined herein as the spatial position of the maximum fuel mass consumption rate. The position for the onset of diffusional interactions between the two flames is defined as the time when the concentrations of either the fuel or the oxidizer at the plane of symmetry x = 0 changes by 1%. This point in time is also associated with the first merging of the diffusional length scales of either the fuel I, = DF/uf or the oxidizer I, = D,/uf. The times for the onset of diffusional interactions of the oxidizer versus fuel are found to be respectively given by t, = 1548 and to = 1586 pus, as shown in Figs. 6c and 6d. Finally, another length scale called radical-interaction length scale 1, is defined as the minimum distance between the interacting premixed flames at which the concentration profiles of the radical species first begin to merge with one another.
366
C. L. CHEN AND S. H. SOHRAB
According to Fig. 6e, the radical interaction associated with the merging of the concentration of atomic hydrogen t, = 1727 ps when the separation distance between the flames 1, = 2 X 0.0194 = 0.0388 cm. After the onset of radical interaction, the reaction proceeds very rapidly and at t = 1790 pus, the maximum fuel consumption rate occurs at x = 0 with very small amount of fuel left (Fig. 6f). The examination of the above results show that from the onset of the thermal interaction between the two flames at t, = 1443 ps to the onset of the radical interaction at t, = 1727 ps, the average flame propagation velocity is 41.91 cm/s. Also, from the onset of radical interactions to the fuel bum-out time t, = 1796 ps the average propagation velocity is about 282 cm/s. Such mean propagation speeds are calculated by dividing the separation distance
between the two flames by the corresponding duration of time of the interaction process. Clearly, these higher rates of fuel consumption are in part responsible for the higher burning rates generally associated with turbulent premixed flames as opposed to the laminar ones [l]. We note that the adiabatic flame temperature for stoichiometric methane-air is 2329 K which is the value obtained under steady buming in Fig. 4. However, incomplete combustion in the presence of flame interaction could lead to the formation of some CO in the products of combustion and a lower flame temperature. Also, at the end of the combustion process some oxygen remains as seen in Fig. 6h. According to the above results, the total interaction time is 353 ps from onset of thermodiffusive interactions at t, = 1443 ps to the complete burnout time at t, = 1796 pus. Of
A.3
3 1 2
2.0 -
1.5
I.5 -
1
1.0 -
a5
0.5 -
0 -0.14
-0.1
-0.06
.a02 0.02 X(CM)
0.M
0.1
0.14
0.0 3 -0.14
-0.1
-0X.5
(a)
-0.02 0.02 X(CM)
0.05
0.1
0.14
0.06
0.1
t
(b) 3
-0.14
-0.1
-0.06
-0.02 0.02 X(CM)
Cc)
0.06
0.1
0.14
0.0T
-0.14
-0.1
-0.06
-0.02 0.02 X(CM)
Cd)
Fig. 6. The evolution of the temperature and concentration profiles for two interacting stoichiometric premixed methane flames with species: (l)-(S) same as in Fig. 4 and Le, = 0.97, L.eo = 1.10, time t = (a) 1200 (b) 1443 (c) 1548 (d) 1586 (e) 1727 (f) 1790 (g) 1796 (h) 1900 ps.
LAMINAR
METHANE
0.0. ”
I
-0.14 -0.1
PREMIXED
367
FLAMES
A -0.06 -au2 0.02 X(CM)
0.06
0.1
0.14
-0.14 -0.1
-0.06 -0.02 0.02 X(CM)
(e)
0.06
0.1
t3.14
0.06
0.1
014
(0 25
15
1 -
2.0
1.5-
1.5
20
1.0-
1.0
2
0.5-
OS
.I...,.. 0.07.. I -0.14 -0.1 -0.06 -002
! 0.02
X(CM)
0.06
0.1
0.14
0.0 -0.14 -0.1
6.06
(8)
-0.02 0.02
X(CM) (h)
Fig. 6. (Continued).
this total interaction time, only 20% corresponds to the radical interactions and the remaining time is for thermodiffusive interactions. To examine the effects of the fuel versus the oxidizer Lewis numbers on the nature of the thermo-diffusive interactions, two sets of combinations of oxidizer and fuel Lewis numbers under the same initial conditions are explored. The two sets of parameters that are considered are {Leo = 1.1; Le, = 0.97, 0.7, 1.3) and {Leo = 1.1, 0.7, 1.4; Le, = 0.97) and the results for the times associated with the onset of
various types of interactions for each set of calculations are summarized in Table 1. In this table, t,, t,, and t,, respectively, refer to the time for the onset of interactions of profiles of the concentration of species i, of the temperature, and the burnout time measured from the ignition time t = 0. According to the data in Table 1, for the case (I) t, starts first and is followed by tCH4, to>, and finally t,. The order for the onset of diffusional interactions between the various species change as the Lewis numbers associated with various species are altered. For ex-
TABLE 1 Influence CASE I
II III IV V
of the Fuel and the Oxidizer hfmo
0.97/1.10 0.70/1.10 1.30/1.10 0.97/0.70 0.97/1.40
Lewis Numbers
on the Time ( ps) for the Onset of Various
Types of Interactions
‘T
‘02
'CH,
'H
th
1443 1450 1461 1508 1453
1586 1586 1611 1506 1626
1548 1444 1652 1630 1550
1727 1715 1762 1873 1706
1796 1789 1830 1964 1770
368
C. L. CHEN AND S. H. SOHRAB
ample, with smaller Le, the higher diffusivity of the fuel molecules DF leads into longer fuel diffusion length 1, = DF/uf and hence earlier fuel interaction times tCH, seen in case II of Table 1. Clearly, the onset of the diffusional interactions between species i and j will depend on the relative value of their corresponding Lewis numbers Le,/Lej. To further explore the influence of the fuel versus the oxidizer Lewis numbers on the dynamics of flame interactions, the mass rates of consumption of the fuel w (CH,) and the oxidizer w (0,) are calculated from t = 0 to the burnout time and the results are shown in Figs. 7 and 8, respectively, for the first and second sets of Lewis numbers defined above. Again, the areas under the curves in Figs. 7 and 8 are the same since they represent the total chemical enthalpy of the fixed quantity of fuel or oxidizer present in the calculations. Here, it is better to define the relative Lewis
(a)
^_
‘. 0.6
0,.
,
I
0
..,....,....,.... MO
mm
1Ym
a a
Time.sec(xl.oE-06)
(b) Fig. 8. The mass consumption rates of the (a) fuel wcr4 and (b) oxidizer wo2 during interaction between two stolchiometric premixed methane flames. Le, = 0 97, Leo = (I) 1.10 (IV) 0.70 (V) 1.4.
0
5aJ
rmo
1500
axa
Time.sec(x1.0E-cw
(a)
Fig. 7. The mass consumption rates of the (a) fuel wcH., and (b) oxidizer wo2 during interaction between two stolchiometric premixed methane flames. Leo = 1.1, Le, = (I) 0.97 (II) 0.70 (III) 1.3.
number LoF = Leo/I-e, = D,/D,. In Fig. 7a, it is noted that decreasing L,, by increasing Le, reduces the ignition time and hence promotes the combustion process. This behavior is to be expected since because of the stoichiometry, methane flame requires more oxygen than fuel for its combustion on the molar basis. Hence, the reduction in diffisivity of fuel DF leads to improved burning conditions under same oxidizer molar concentrations since closer approach to the stoichiometric conditions are achieved within the reaction zone. This phenomenon is related to the external versus internal equivalence ratio discussed in our previous study of diffusion flames 1311. From the results in Fig. 8, on the other hand we note that improved burning rates are achieved for smaller Leo, again resulting in reduced Lo, which is consistent with the above phenomenological description. Since all hydrocarbon fuels require more oxygen than fuel for their stoi-
LAMINAR
METHANE
PREMIXED
369
FLAMES
chiometric combustion, it is expected that transport effects which increase (decrease) concentration of oxidizer (fuel> in the flame zone will tend to enhance (diminish) the local burning rates. As the flames approach each other either the thermal or the diffusional interaction begins first depending on the relative size of the characteristic thermal I, = a/uf versus diffusional I, = D/uf lengths. Once the flames begin to merge, their propagation accelerates (decelerates) when the Lewis number of the deficient rate controlling specie is Le > 1 (Le < 1). It is found that the stoichiometric methane-air flame propagation velocity increases from uf = 40.5 cm/s for a steady isolated laminar flame to an average value of about 41.9 cm/s during the thermo-diffusive interactions. When the concentration profiles of the radical species merge the onset of radical interactions results into a more substantial increase of the mean propagation velocity to a value of 282 cm/s. Now the sharing of chain branching radicals such as the atomic hydrogen results in accelerative combustion of the gas between the merging flames. This mode of combustion can result in the onset of homogeneous explosion of the combustible gas depending on the Lewis number of the mixture (Figs. 7, 81 parallel to that obtained for imploding spherical premixed flames [18]. Clearly, the larger flame propagation velocity induced by the flame interaction effects is in part responsible for the higher burning rates of turbulent premixed flames. It is emphasized that the enhancement in turbulent burning rate induced by upstream interactions is in addition to that caused by the increased total flame surface area due to flame wrinkling. CONCLUDING REMARKS The concept of hydrodynamic, thermal, diffusional, as well as chemical interactions between premixed flames were described. The upstream interaction between two planar laminar stoichiometric methane premixed flames was numerically investigated. It was found that with large separation distance the propagation velocity of the two flames was that of the laminar methane flame 40.5 cm/s. As the
flames approached each other, the thermodiffusive interactions between them resulted in the average value of flame propagation speed of 41.9 cm/s. Upon closer approach, the merging of the two reaction zones resulted in the onset of chemical interactions accompanied by further acceleration of the flames to the mean propagation speed of 282 cm/s. The influence of the Lewis numbers of both fuel and oxidizer on the nature of flame interactions were determined. The accelerative (decelerative) burning found under Le, > Le, (Le, < Le,) was phenomenologically described in terms of the relative diffusivity of the fuel versus the oxygen D,/Do > 1 (< 1) which tends to alter the composition of the combustible gas towards (away from) the stoichiometric composition. Depending on the Lewis number of the mixture, small pockets of unburned reactants between merging flames could experience either homogeneous explosion or decelerating burning rates. The results may help the local modeling of the combustion processes in turbulent premixed flames. This research was supported by the National Science Foundation Grant No. CTS-8820077. C.L.C. was supported by the Taiwan Power Company. We also thank W. J. Sheu for his assistance in the numerical calculations.
REFERENCES 1. Williams, F. A., Combustion Theory, 2nd ed., Addison-Wesley, 1985, p. 412. 2. Peters, N. Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1231. 3. Liiian, A., and Williams, F. A., Fundamental Aspects of Combustion, Oxford University Press, Oxford, 1993, p. 118. 4. Tsuji, H., and Yamaoka, I., First Specialists Meeting (International) of The Combustion Institute, Bordeaux, 1981, p. 1111. 5. Tsuji, H., and Yamaoka, I. Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, p. 1533. 6. Sohrab, S. H., Ye, Z. Y., and Law, C. K., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1957. 7. Libby, P. A., and Williams, F. A., Combust. Sci. Technol. 37:221-252
8.
(1984).
Sohrab, S. H., Ye, Z. Y., and Law, C. K., Cornbust. Sci. Technol. 45~27-45 (1986).
370 9.
C. L. CHEN AND S. H. SOHRAB Chung, S. H., Kim, J. S., and Law, C. K., Twenty-First
20.
Symposium (International) on Combustion, The Com-
bustion Institute, Pittsburgh, 1986, p. 1845. 10. Lin, S. H., and Sohrab, S. H., Combust. 68:73-79
Flame
(1987).
11. Chen, C. L., and Sohrab, S. H., Upstream interactions between two laminar premixed methane flames, Fourth International Conference on Numerical Combustion, SIAM, St. Petersburg Beach, Florida, 1991, p. 190. 12. Markstein, G. H., Nonsteady Flame Propagation, Pergamon, London, 1964. 13. Buckmaster, J. D., and Ludford, G. S. S., Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982. 14. Wu, M. S., Kwon, S., Driscoll, J. F., and Faeth, G. M., Combust. Sci. Technol. 78:69-96
(1991).
15. Wu, M. S., Kwon, S., Driscoll, J. F., and Faeth, G. M., Combust. Sci. Technol. 83:187-202
(1992).
16. Sohrab, S. H., and Chao, B. H., Combust. Sci. Technol. 38:245-265 (1984). 17. Heywood, J. B., Internal Combustion Engine Fundamentals, McGraw-Hill, New York, 1988, p. 400. 18. Sivashinsky, G. I., Int. J. Heat Mass Transf. 17:14991506 (1974).
19. Seshadri, K., and Peters, N., Cornbust. Flame 81:96118 (1990).
21. 22. 23.
Spalding, D. B., Stephenson, P. L., and Taylor, R. G., Combust. Flame 17:55-64 (1971). Wilde, K. E., Combust. Flame 18:43-52 (1972). Bledjian, L., Cornbust. Flame 20:5-17 (1973).
Smoot, D. L., Hecker, W. C., and Williams, G. A., Combust. Flame 26:323-342
(1976).
24. Tsatsaronis, G., Combust. Flame 33:217-239 (1978). 25. Margolis, S. B., .I. Comput. Phys. 27:410-427 (1978). 26. Ramos, J. I., Int. .I. Numerical Methods Fluids 4:915-930
(1984).
27. Jacobi, A. N., and Sohrab, S. H., Cornbust. Sci. Technol. 69:17-32 (1990). 28. Andrews, G. E., and Bradley, D., Combust. Flame 18:133 (1972).
29. Tsuji, H., and Yamaoka, I., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1883. 30. Wu, C. K., and Law, C. K., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1941. 31. Chen, L. C., and Sohrab, S. H., Combust. Flame 86:383-393
(1991).
Received 10 March 1994; revised 22 August 1994
