result, a diffusion flame forms at the boundary where these excess reactants meet. ... In fact, three distinct flames can be identified: a fuel-rich premixed flame,.
Flame Propagation in a Nonuniform Mixture: The Structure of Anchored Triple-Flames J. W. Dold* University of Bristol, Bristol, England, United Kingdom
Abstract A model, which includes upstream conduction and diffusion, is used to describe the structure of flames in a nonuniform medium. Such flames typically consist of fuel-rich and fuel-lean premixed flames, followed by a diffusion flame which starts where the two premixed flames meet. Such formations have been observed experimentally and probably occur as laminar flamelets in turbulent nonpremixed combustion. The equations are solved in a limit of low heat release (so that hydrodynamic effects need not be considered) , while the Zel 'dovich number is considered to be large. In this context, it is found that freely propagating triple-flames have propagation speeds that are bounded above by the maximum adiabatic flame speed of the system. In the context of a diffusion flame behind a splitter-plate, the flames are considered for blowing velocities slightly less than the maximum flame speed. For greater blowing velocities, the structure r~an only be maintained if combustion is initiated by some means other than upstream propagation. Solutions are obtained for such "anchored" flames . As the blowing velocity is reduced towards the triple-flame propagation speed, the freely propagating triple-flame solution is recovered. Introduction Given a region of nonuniformly premixed fuel and oxidant, the adiabatic laminar flame speed, as determined by the local mixture properties of the gas, varies from point to point. In particular , for similar initial temperatures of fuel and oxidant, the path of greatest adiabatic laminar flame speed tends to lie close to any stoichiometric boundary in the system. Thus , one should expect that the propagation of a premixed flame wo uld tend to surge ahead primarily around this boundary. After the flame has passed, hot excess fuel remains unburnt on one side of the stoichiometric line, while excess oxidant remains on the other side. As a result , a diffusion flam e forms at the boundary where these excess reactants meet. Away from the stoichiometric boundary, the propagation of the premixed flame becomes slower and eventually stops as the Harne enters regions of weakening mixture strength. In fact, three distinct flames can be identified: a fuel-rich premixed flame , leaving unburnt fuel behind it ; a fuel-lean flame, leaving oxidant; and a diffusion Copyright© 1988 by the American Institute of Aeronautics and Astronautics , 1nc . All rights reserved. *Mathema tics Department.
240
ANCHORED TRIPLE-FLAMES
241
flame along the stoichiometric boundary, beginning where all three Hames meet. As seen in Fig. 1, such triple-flames have been observed experimentally [Phillips (1965), lshizuka (1986)] and probably play a role in the combustion of turbulent diffusion fl ames, where a possible extinction of a locally laminar flamelet may lead to diffusive mixing and subsequent reignition of a region of nonuniformly premixed gases [Peters (1986)). Until recently (Dold ( 1988)], modelling of flames in non uniform media has relied on neglecting upstream conduction and diffusion [Liiian and Crespo (1976), Dold and Clarke (1986)). Although simplifying the mathematics, this effectively rules out the possibility of upstream flame propagation and cannot be justified for flows comparable with adiabatic flame speeds. Combustion must therefore be initiated in some other way, such as an ignition through thermal runaway or perhaps a hot wire. Subsequent combustion then takes the form of transversely propagating premixed flames (which slow down and eventually extinguish through propagating into regions of weakening mixture ratio) , a nd a diffusion flame which is created as one of these flames crosses a stoichiometric boundary. Thus, some aspects of triple-flame structure are retained, without the property of upstream propagation. In this paper, we use a small heat-release model (developed elsewhere (Dold (1988)]) to examine the structure of the triple-flame in a slowly varying regime of solution. The results are valid for large Zel'dovich number f3, and for transverse mixture-fraction gradients (on the length scale of a typical preheat-zone thickness) which are small compared with {3- 1 • Results are considered in the context of a diffusion flame behind a splitter-plate, with a blowing velocity not much less than the maximum adiabatic flame speed (at which stage the assumption of slow variation is valid). For blowing velocities greater than this maximum speed, solutions are found only if combustion is initiated in some way besides upstream propagation (thermal runaway or a hot wire as before). These solutions are examined as the blowing velocity is increased and are compared with solutions obtained through neglecting upstream conduction and diffusion. As the blowing velocity is progressively reduced , the flame structure abruptly recovers its freely propagating form .
..
•
Fig. 1 Triple-flame propagating in a nonuniform medium. Note the presence of threedimensional effects. (British Crown Copyright: Reproduced from PhiHips (1965) by kind permission of H. Phillips, Health and Safety Executive, Buxton, U.K .)
J. W. DOLD
242
Model With Upstream Conduction and Diffusion \Ve consider, for simplicity, the one-step reaction vFF + v_x X ~ 11pP rn which llF fuel molecules and vx oxidant molecules produce lip product molecules. A low hecit-release model has been derived [Dold (1986)] to describe the evolution of temperature in this system, with a uniform flow of reactants at low Mach number and unit Lewis number. Using this, we consider the equations, Z:i - V' 2 Z
=0
and
where
R = f3vF
+ 11x (Z -
ST) 11F (1 - Z - (1 - S)T] 11X e-f3(l - T) .
(1)
The presence of the Laplacian operator V' 2 shows that this model does not neglect conduction or diffusion in any direction. In the model, fuel and oxidant streams are taken to be originally separated, containing mass fractions YFo and Yxo respectively of fuel and oxidant. The mixture fraction Z of fuel, is then defined as
Z = ( 11Fy;F
+ y~: ~;x)
I (11~~F + v:~x)
(2)
where W"" is the molecular weight of the species u. Thus Z varies between 0 and 1 as YF and Yx vary between 0 and YFo or Yxo, respectively. For unit Lewis number and constant specific heat Gp, the mass fractions of fuel and oxidant are given in terms of Z and the nondimcnsional temperature 1' such that YF YFo(Z - ST) and Yx Yxo[l - Z - (1 - S)T]. In these formulae, s YFollx w x /(YFollx wx + YxollF w1'' ) is the stoichiometric value of Z , at which both YF and Yx will be zero if T ·takes its upp~-limiting value of unity. In terms of T, the dimensional absolute t emperature T is given by T T0 + (Ts -T0 )T where T0 is the initial (and we assume, equal) absolute temperature of the fuel and oxidant streams. With heat of reaction Q, the upper bound for the dimensional temperature is
=
=
=
=
(3)
Nondimensional lengths are measured relative to a typical conduction lengthscale VCpp5 /3..s, where V istheblowingvelocityoftheincominggases,and Ps and As are the density and thermal conductivity determined at the temperature upper bound T5 , and at ambient pressure. The constant V is defined such that V2 =
v2a f3c-.!2T-sPs~ A.X 5 Qvp Wp
( _ {3
)llF ( _ f3 Ps YFo Ps Yx o
)llx eE/(RT - --5)
(4)
ANCHORED TRIPLE-FLAMES
243
where A is a pre-exponential factor of the reaction rate and is taken to be constant; E is the activation energy of the reaction; R is the universal gas constant; and a is the heat-release ratio (Ts - T0 )/T5 . It can be seen that V is directly proportional to the dimensional blowing velocity. The quantity V 2 , appearing in Eqs. (1) , behaves as an inverse Damkohler number for the system. The nondimensional temperature T is defined so that, well ahead of any flames, T takes the upstream value of zero. It is assumed that any slow reaction in the cool inflowing gases does not have time to significantly affect the temperature before a triple-flame is encountered. In deriving Eqs. (1) the heat-release ratio a is taken to be small. We shall also assume that the Zel'dovich number f3 = aE/(RTs) is large, so that an asymptotic form of analysis can be developed. In the context of a triple-flame "far" downstream from a splitter-plate, which ends at (.x, y) = (xo, !lo), the mixture fraction Z can be modelled asymptotically for large values of x - x 0 as
g1vmg
(5) Thus, Z 1 is small and varies slowly with y for large values of x - x 0 . Also, lines of constant Z have small slopes. Locally therefore, we can approximate Z
by
= S + By/{3 + O(B 2 /{3 2 ) that B / f3 = z., (calculated
(6)
Z
where B is defined so where the stoichiometric boundary Z = S meets a triple-flame), and the y axis has been normalised so that, locally, the stoichiometric line lies close to y = O. For large values of x - x 0 , the ratio B / f3 is small. For flows near the maximum adiabatic flame speed, it will be seen that B also turns out to be small; from (5) this requires that x - x 0 ::> {3 2 • Remembering that x is scaled on a (generally thin) typical conduction length or preheat zone thickness, the physical distance of the flame from the splitter-plate need not actually be very large. In order for a triple-flame to settle around a value of x - xo of order {3 2 or less, the flow (that is, the triple-flame propagation speed relative to the gas) must be significantly less than the maximum adiabatic flame speed. Under these conditions, the "slowly-varying" assumption employed below cannot be justified [Dold (1988)]. We do not attempt to consider such situations in this paper. If we now write x = X (y) as the position of the reaction zone of the premixed flames, Eqs. ( 1) can be recast in terms of orthogonal coordinates q and which follow the flame-path as sketched in Fig. 2. This is most easily done by defining
r / f3
and 11 =
x/{3,
leads to
2
(S1+z)11F[(l-Sh-z] 11xe-ld1
fr
)
Jr
V
8
(e)
where
(9) Matching the temperature gradient T'1 between the two solutions leads to a firstorder equation for the flame slope qS which is accurate to order B 2 and valid provided 0 / V > B : Bdef> = (1 dz
+ qS 2 ) 1f 2 n(z)/V
- 1
0
in which
02 = 2
f° (51 + z ) 11F [(1 Jr t:1( e)
S)'y - z]ll.X e-/ d1 .
(10)
For slowly varying fl ames (lOa) is an equation for the flame slope which incorporates both upstream conduction and curvature effects. It should be noted that the equation is valid for order one flame slopes
