The full problem of flame stability for the two-reactant model, which takes into account thermal expansion effects for all disturbance wave lengths, is examined. It is found that the stability problem for the class of two-reactant flames is equivalent to the stability problem for the class of one-reactant flames with an appropriate interpretation of Lewis numbers

Jackson, T. L.

English

NASA Technical Reports Server

NASA Contractor Report 178168 
lCASE REPORT NO. 86-56 
leASE 
EFFECT OF THERMAL EXPANSION ON THE STABILITY 
OF TWO-REACTANT FLAMES 
T. L. Jackson 
Contract No. NASI-18107 
August 1986 
If A$/l c;e- /7;;"-/~.J 
NASA-CR-178168 
19860023553 
INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING 
NASA Langley Research Center, Hampton, Virginia 23665 
Operated by the Universities Space Research Association 
NI\SI\ 
National Aeronautics and 
Space Administration 
Langley Research Center 
Hampton, Virginia 23665 
111111111111111111111111111111111111111111111 
NF00195 
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r· ,"0'''' t" I,: }"'G U,_, I':" I~V 
.LANGLEY RESEARCH CENTER 
LIBRARY, NASA 
HM.i~Otlt VIRGINIA 
https://ntrs.nasa.gov/search.jsp?R=19860023553 2020-03-20T13:14:01+00:00Z
EFFECT OF l'HERHAL EXPANSION ON THE STABILITY 
OF 'l'WO-REACTANT FLAMES 
T. L. Jackson 
Institute for Computer Applications in Science and Engineering 
NASA Langley Research Center 
Hampton, Virginia 23665 
Abstract 
The full problem of flame stability for the two-reactant model, which 
takes into account thermal expansion effects for all disturbance wave lengths, 
is examined. It is found that the stability problem for the class of two-
reactant flames is equivalent to the stability problem for the class of one-
reactant flames with an appropriate interpretation of Lewis numbers. 
This work was supported under the National Aeronautics and Space 
Administration under NASA Contract NASI-IBI07 while the author was in 
residence at the Institute for Computer Applications in Science and 
Engineering, NASA Langley Research Center, Hampton, VA 23665. 
~ 
,,/.. -# 
/Y '%0-3:3025 

1. Introduction. Recently, Jackson and Kapila (1984, 1986a,b) reexamined the 
problem of the stability of a plane, premixed flame in the near-
equidiffusional (NEF) framework. The study was aimed at understanding the 
role of thermal expansion on flame behavior. Neutral stability curves were 
obtained in the Lewis number-wave number plane and contrasted with the 
classical results of Sivashinsky (1977), who had employed the Constant Density 
Approximation (CDA). 
The analysis was based on the simple kinetic model A + B which, as is 
well known, is also appropriate for a multi-reactant system when a single, 
deficient component controls the combustion process. Near stoichiometry, 
however, revisions are necessary since the number of controlling reactants is 
more than one. The first effort in this direction was due to Sivashinsky 
(1980) who examined the stability of a two-reactant model at EXACT 
stoichiometry, under the CDA; Joulin and Mitani (1981) later extended the 
result to near-stoichiometric mixtures. The essential point made by these 
studies is that the single-reactant results hold even near stoichiometry, 
provided the Lewis number is replaced by a suitably defined effective value. 
The purpose of this note is to show that even when the CDA is abandoned 
and thermal expansion fully taken into account, the notion of an ef fecti ve 
Lewis number persists. Thus, upon suitable interpretation, the results of 
Jackson and Kapila (1984, 1986a,b) can be applied to two-reactant systems. 
2. Governing Equations. Consider the two-reactant model 
v 1M1 + v2~ + Products, 
-2-
where Ml and M2 denotes the chemical symbols of the deficient and abundant 
reactants respectively. Here 
"j is the stoichiometric coefficient for 
species j. We take the reaction to be irreversible and of Arrhenius-type so 
that its rate expression is of the form 
2 In.. n (2.1) W = B P Y1 x2 exp (-E/RT), 
where p is the density, Y1 and Y2 are the mass fractions of the deficient and 
abundant components, respectively, and T the temperature. The parameters 
appearing in the rate expression are the activation energy E, the universal 
gas constant R, the pre-exponential factor B, and the reaction orders m and n 
with respect to the deficient and abundant reactants, respectively. 
The basic equations of the two-reactant system can be modelled by the 
following equations: 
(2.2) Dp + pV·v = 0 Dt , 
P = pRT 
W 
'" 
Dv 
(2.3) - 1 2 P Dt + VP = lJ{'3 V(V.v) + V v}, 
'" 
(2.4) DT DP 2 pep Dt - Dt = A V T + ~, 
DYj 
(2.5) 2 
- "jWjW ,j=1,2. p De = p Dj V Yj 
Besides Yj and T, the state of the system is determined by the density p , 
pressure P and velocity y. The other quantities appearing above are the 
specific heat at constant pressure Cp ' viscosity lJ ,thermal conductivity 
-3-
A , species diffusion coefficient Dj of species j, chemical heat release Q, 
molecular weight Wj of species j and molecular weight W of the inert. 
Finally, D/DT denotes the material derivative, defined by 
(2.6) D Dt 
a at + v·V 
Equations (2.1) - (2.5) are supplemental by appropriate boundary conditions 
and jump conditions across the reaction zone. 
For nondimensionalization of the governing equations, the state of the 
fresh mixture is chosen as the reference, defined by the constant values 
P , P , T and v , 
000 0 
while the initial concentrations are rendered 
dimensionless relative to the initial concentration of the deficient reactant 
Y10. The characteristic length and time scales are taken to be the adiabatic 
diffusion scales 
(2.7) R. = A/(p C v ) 
o 0 p 0 
2 t = A/(p C v ). 
o 0 p 0 
It is convenient to introduce the flame Mach number as M = v Ic , where Co 000 
is the frozen sound speed in the fresh mixture. Finally, pressure deviations 
from the ambient are referred to yM 2p , Y being the specific-heats ratio. 
o 0 
The governing equations then take the dimensionless form 
(2.8) Dp + p V·v = a , 1 = pT , Dt 
Dv 
(2.9) p Dt 1 2 + VP = Pr {3 V(V·v) + v v} 
,..., 
(2.10) DT V2T + an p Dt = 
-4-
(2.11 ) 
DY. 
J -1 2 
p Dt = Lj V Yj - a j S'2 j = 1,2 
(2.12) S'2 = D --2 "IW p ym+n-l M 1 0 10 
p2y~n -a/T 1 2 e • 
o 
The yet undefined nondimensional parameters appearing above are: 
Pr = II C I). p 
a = QYl0/CpTo"IWl 
a = E/RT 
o 
e: = (1 +a)2 /a 
Lj = )./PDj Cp 
2 D = ).B/p C c 
o p 0 
a. = ". W • l"IWl J J J 
Prandtl number, 
Heat release number, 
Activation energy, 
Reciprocal activation energy, 
Lewis number for species j, 
Damkohler number, 
Parameter involving stoichiometry. 
When a steady, planar, adiabatic flame is analyzed in the limit 
a+ eo , i.e., e: + 0, the following expression for the Mach number 
obtained (see, e.g., Sen and Ludford (1979) and Mitani (1980»: 
(2.13) 
where 
(2.14) 
D a m+n+1(1+a)2 
2= n 
Mo 2 a 2 G(m,n,A) 
e: 
-1-m-n I-m-n Y10 
m n 
"IWIPoLl L2 
a 
l+a 
e {l + O( e:)}, 
( feo m n -~ G m,n,A) = 0 ~ (~+A) e d~. 
Here, A is the normalized final concentration of the abundant, given by 
(2.15) A = ~ (cjl-l) 
e:L2 
M 
o 
is 
and cjl is the equivalence ratio ("IWIY201"2W2YI0). Equation (2.13) is valid 
for the entire range of mixture ratio. However, we are only interested in the 
-5-
mixture-ratio range ~ - 1 = O( e ); i.e., near-stoichiometric mixtures, since 
the reaction rates are then governed by the deficient component as well as the 
abundant component of the mixture. The special cases of very-near 
stoichiometry 2 ( ~ - 1 = 0 ( e ) ; i • e., A + 0) and away-from stoichiometry 
(~ - 1 = 0(1); i.e., A +~) can be considered as limiting cases of the near-
stoichiometric analysis, and will be discussed in the conclusions. Equation 
(2.13) also shows that the Mach number is exponentially small in the limit of 
large activation energy. 
The near-equidiffusional formulation (NEF) , in addition to large 
activation energy, is characterized by near-unity Lewis numbers and nearly-
uniform enthalpies, i.e., 
(2.16) -1 1.1 H = 1 + a + eh, L 1 =l-e(-) T + aYl :: a 
-1 1.2 T + ~ Y :: a L = 1 - d- ) G = 1 + a
2 
Y2f + ego 2 a a2 2 
(2.17) 
Here, Y2f = Y20/Y10 denotes the dimensionless mass fraction of Y2 in the 
fresh mixture. Then, to leading order in e ,equations (2.8), (2.9) remains 
unaltered, while equations (2.10) and (2.11) are replaced by 
(2.18a) 
(2.18b) 
(2.19a) 
(2.19b) 
DT _ V2T = 0 PDt in the burnt region, 
T = 1 + a in the unburnt region, 
1. 
Dh = V2h + (..l.) V2T P Dt a' 
~ 2 1. P = V g + (~) V2T. Dt a 
-6-
Equations (2.8), (2.9), (2.18) and (2.19) now constitute the basic set. 
The following analysis is by now standard, and the reader is referred to 
Jackson and Kapila (1984) for details, hereafter referred to as I. For a two-
dimensional flow, let the reaction-zone location be given by 
x = xf :: F(y,t). It is convenient to shift to a reference frame in which 
the reaction zone is stationary. Let a new coordinate ~ be defined by the 
transformation 
(2.20) x = F(y,t) + ~ , 
and let the unburnt and burnt regions be confined, respectively, to 
~ < 0 and ~ > O. Then one obtains equations (8)-(22) of I, except now the 
enthalpy equation (141) is replaced by the enthalpy equations for hand g, 
1 
p (ht + Sh~ + vhy) = V2h + ( ~) V2T, (2.21a) 
(2.21b) 
1 
p(gt + sg~ + Vgy) = V2g + ( !) V2T, 
while the boundary conditions (211) and (221) at ~ = 0 are replaced, 
respectively, by 
(2.22) 
(2.23) 
h/2 
6T = -ae 
~ I(1+F 2 ) G(m,n,A) y 
6\ = 11 (-a) 6T~ 
{J
o
m ~m (A+g-h+~)ne-~d~} 1/2 
6g~ 
12 . 
= - (-a) 6T~. 
Equations (81)-(221), together with (2.21)-(2.23) describe completely the NEF 
problem. 
-7-
3. Linearization. Small perturbations of the steady flame are governed 
by linearized equations, which can be obtained by setting 
(3.1) s -ljI = ljI (0 + oljl (~,y, t) 
for all variables and taking the limit 0 + O. We focus our attention only on 
the enthalpy equations (2.21) and the boundary conditions (2.22) and (2.23) 
since linearization of the other equations yields the identical equations of 
I. 
Linearization of (2.21) leads to 
(3.2a) s- - - s - - - s P h t + h~ + m h~ = h~~ + hyy - Fyyh~ 
1. 1 - - s 
+ (a-)(T~~ + Tyy - F yyT~ ), 
(3.2b) s- - s - - - s P g + g + m g = g + g - F g t ~ ~ ~~ yy yy ~ 
1. 2 - - - s 
+ (a-) (T~~ + Tyy - FyyT~ ) 
for hand g, respectively, while linearization of the jump conditions (2.22) 
and (2.23) at ~ = 0 yields 
(3.3) OT~ = - ~ {Ii + H (g - li)} , 
1. 1. 
- - - 2-o~ = - (-) oT , og = - (-) oT • 
a ~ ~ a ~ (3.4) 
Here, 
(3.5) 
-8-
H = n G(m,n-l,A) 
G(m,n,A) 
If we define a new function I by 
(3.6) I = h + H (g - h), 
then (3.2) reduces to a single equation for 7, namely 
(3.7) ~ - -s - - s p f t + f ~ + mf ~ = f ~ ~ + f yy - F y/ ~ 
1 - - s 
+ a (1 1 + H (R. 2 - 11» {T~~ + Tyy - Fyy T~ }, 
while (3.3) and (3.4) determines the jump conditions at ~ = 0, 
(3.8) o T~ a - 0 I = -'2 f , ~ 1 -= - a (1 1 + H (1 2-1 1» oT~. 
These equations are identical to the linearized enthalpy equation for the 
premixed flame (see (301),(361), and (421» with a reduced Lewis number 1 , 
c 
defined by 
(3.9) 1c = 11 + H (12 - 11). 
4. Discussion and Conclusions. The stability problem of the two-reactant 
flame consists of the linear system (241)-(421), with (301) and (361) replaced 
by (3.7), and the boundary condition (421) replaced by (3.8). In particular, 
we find that this system is governed only by the parameter 1, defined in 
c 
-9-
(3.9). It is only here that the mixture ratio appears through H, and thus we 
consider the following three limiting cases. For near-stoichiometric 
mixtures, H is given by (3.5) and can not be simplified further. For very-
near stoichiometry, H is given by 
(4.1a) H = n m+n + O(e:) cj> - 1 2 = 0 (e: ), 
and the corresponding reduced Lewis number becomes 
(4.1b) 1. = 
c 
m 1.1 + n 1.2 
m + n 
+ 0 (e:), 
i.e., the average value of the two Lewis numbers weighted by their respective 
reaction orders (Joulin and Mitani (1981». Finally, away from stoichiometry, 
H is given by 
(4.2a) e:n H = a (cj>-l) + 0 (e: 2) cj> - 1 = 0 (1), 
with corresponding 1.
c
' 
(4.2b) 1.
c 
= 1.1 + 0 (e:). 
Thus, for far-from stoichiometric mixtures, the stability problem reduces to 
that of a one-reactant flame governed by the Lewis number of the deficient 
component. 
-10-
Sivashinsky (1980) and Joulin and Mitani (1981) have shown that within 
the diffusional-thermal model, the stability problem for two-reactant flames 
at or near stoichiometry reduces to the stability problem for single-reactant 
flames with a reduced Lewis number. We have shown that the above result also 
holds for the more important case when thermal expansion is accounted for, and 
for the entire range of mixture ratio. The following general statement can 
now be made: The linear stability problem for the class of two-reactant 
flames is equivalent to the stability problem for the class of one-reactant 
flames with Lewis number L = 1 + e:(R. la). 
c 
gravity do not alter this conclusion. 
The effects of heat loss and 
Acknowledgements. The author would like to thank A. Kap1la for helpful 
comments in connection with this work. 
-11-
REFERENCES 
Jackson, T. L. and Kapi1a, A. K. (1984). "Effect of thermal expansion on the 
stability of plane, freely propagating flames," Comb. Sci. Tech., 41, pp 191-
207. 
Jackson, T. L. and Kapi1a, A. K. (1986a). "Effect of thermal expansion on the 
stability of plane, freely propagating flames. Part II: Incorporation of 
gravi ty and heat loss," Comb. Sci. Tech., in press. 
Jackson, T. L. and Kapila, A. K. (1986b). "Thermal expansion effects on 
perturbed premixed flames: A review," Lectures in Applied Mathematics, AMS, 
24, pp 325-347. 
Jou1in, G. and Mitani, T. (1981). "Linear stability analysis of two-reactant 
flames," Combust. Flame, 40, pp 235-246. 
Mitani, T. (1980). "Propogation velocities of two-reactant flames," Comb. 
Sci. Tech., 21, pp 175-177. 
Sen, A. K. and Ludford, G. S. S. (1979). "The near-stoichiometric behavior of 
combustible mixtures. Part I: Diffusion of the reactants," Comb. Sci. Tech., 
21, pp 15-23. 
Sivashinsky, G. I. (1977). "Diffusional-thermal theory of cellular flames," 
Comb. Sci. Tech., 15, pp. 137-146. 
Sivashinsky, G. I. (1980). "On flame propagation under conditions of 
stoichiometry," SIAM J. App1. Math, 39, pp 67-82. 
Standard Bibliographic Page 
1. Report No. NASA CR-178168 12. Government Accession No. 3. Recipient's Catalog No. 
ICASE Report No. 86-56 
4. Title and Subtitle 5. Report Date 
EFFECT OF THERAL EXPANSION ON THE STABILITY August 1986 
OF TWO-REACTANT FLAMES 6. Performing Organization Code 
7. Author(s) 8. Performing Organization Report No. T. L. Jackson 86-56 
!}:rtmrffi~O~ZeffiH~!Milfe'ations in Science 10. Work Unit No. 
and Engineering 
1~f§Y~f8foTrant No. Mail Stop 132C, NASA Langley Research Center 
Hampton, VA 23665-5225 
12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered 
Contractor Report 
National Aeronautics and Space Administration 14. Sponsoring Agency, Code 
Washington, D.C. 20546 505-31-83-01 
15. Supplementary Notes 
Langley Technical Monitor: Submitted to Combustion Science and 
J. C. South Technology 
Final Report 
16. Abstract 
The full problem of flame stability for the two-reactant model, which takes 
into account thermal eX'pansion effects for all disturbance wave lengths, is 
examined. It is found that the stability problem for the class of two-reactant 
flames is equivalent to the stability problem for the class of one-reactant 
flames with an appropriate interpretation of Lewis numbers. 
, 
i 
17. Key Words (Suggested by Authors(s» 18. Distribution Statement 
stability, two-reactant flames, 59 - Mathematical and Computer 
asymptotics Sciences 
Unclassified - unlimited 
U'i1.~r~Iiwlfrda.(of this report) rrl'cff3firfil~if.(Of this page) ¥3 No. of Pages 122. ISze 
For sale by the National Technical Information Service, Springfield, Virginia 22161 
NASA Lanlley Form 63 (June 1985) NASA-Langley, 1986 
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Effect of thermal expansion on the stability of two-reactant flames

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