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Theory of laminar flame propagation with non-normal diffusion
THEORY
OF LAMINAR FLAME PROPAGATION NON-NORMAL DIFFUSION
WITH
V. K. JAIN AND R. N. KUMAR Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 12, India A study has been made of the problem of steady, one-dimensional, laminar flame propagation in premixed gases, with the Lewis number differing from (and equal to) unity. Analytical solution, using the method of matched asymptotic expansions, have been obtained for large activation energies. Numerical solutions have been obtained for a wide range ofthe reduced activation temperature parameter (n - E / ~ T b ) , and the Lewis number & The studies reveal that the flame speed eigenvalue is linear in Lewis number for first order and quadratic in Lewis number for second order reactions. For a quick determination of flame speeds, with reasonable accuracy, a simple rule, expressing the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function, is proposed. Comparisons have been made with some of the earlier works, for both first and second order reactions.
1 latrodaetioa THE problem of laminar flame propagation in premixed gases is of fundamental importance and has been extensively studied ~-4. An excellent survey of the early theoretical studies on this problem has been made by Evans ~. A more recent study has been made by Williams 2. The propagation characteristics of steady, adiabatic, one-dimensional, laminar flames in premixed gases, depend, primarily, on two properties of the combustion system ....the activation energy of the rate-controlling step, and the Lewis number. A majority of workers in combustion make the assumption of unity Lewis number (normal diffusion) since this brings about considerable simplification in the mathematical analysis. Several approximate methods of solution, available for this case, have been compared by Spalding s and by de Sendagorta et al. 4. Zeldovich and FrankKamenetsky 5 assumed that the whole reaction occurs at the highest temperature and they were able to give a closed form solution for the flame speed. For the case of unity Lewis number most of the useful results have been incorporated in the "centroid rule' proposed by Spalding 3. Later, Rosen 6 used his 'action principle" to give a plausible derivation of the centroid rule. Hirschfelder and co-workers 7 performed numerical integrations to obtain the flame speed.
Jain ~ obtained the numerically exact solutions for temperature-explicit reaction rate functions, and compared his results with some of the earlier ones. The assumption of unity Lewis number is, however, not valid for many combustion gases. and particularly so for gases like hydrogen and propane. Approximate methods for considering non-normal diffusion, mostly by the Karman Pohlhausen profile methods, have been suggested by some investigators 3'4'9-11 For first order reactions, Spalding 3 used the profiles suggested by Wilde 9, and modified the method slightly by bringing in the centroid rule. de Sendagorta10 used an exponential approximation for the species flux fraction and demonstrated its good accuracy by comparing his results with numerically exact values, which he obtained 4 earlier with Millan and Da Riva, for one value of the reduced activation temperature, E/~Th. von Kfirmfin and Penner 1~ obtained solutions by using zeroth, first and second approximations for the species flux fraction profiles. For second order reactions, Adler 12 obtained some approximate solutions, using Wilde's profiles, for temperature-explicit reaction rate functions. Semenov (see ref. 13) suggested that, for non-normal diffusion, the flame speed, as obtained by Zeldovich and Frank-Kamenetsky. 285
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V. K. JAIN AND R. N. KUMAR
should be multiplied by the reciprocal of the square root of the Lewis number for first order. and by the reciprocal of the Lewis number for second order reactions. However, this analysis is strictly valid for infinite activation energies. In the present work, analytical solutions employing the method of matched asymptotic expansions have been obtained for the limit of large reduced activation temperatures. Numerical solutions have been obtained for a wide range of values of the reduced activation temperature and the Lewis number, for both first order and second order reactions. Based on the numerical studies, a simple, yet reasonably accuraterule, expressing the flame speed eigenvalue 2 as a function of the Lewis number and the centroid of the reaction rate function is developed. The present study reveals that the flame speed eigenvalue is linear in Lewis number for first order, and quadratic-in Lewis number for second order reactions. Comparisons have been made with the results of Spalding a, Adler ~2 and de Sendagorta et ai. 4. Present numerically exact values for first order reactions show that the approximate theory o f Spalding 3 gives results with good accuracy. The results of the present work compare well with those of de Scndagorta et al. 4, for the only value of the reduced activation temperature for which they have presented their results. There is large disagreement between the present numerically exact solutions and the approximate solutions of Adler ~2 for second order reactions. This, we believe, is due to the fact that the p-profile Adler uses, does not satisfy the boundary condition at the hot end. The main contributions of the present work are thought to be: (i} A systematic, analytical derivation of the dependence of the flame speed or.. the Lewis number and the reduced activation temperature parameter has been made "in the large activation energy range, using the methods of "inner" and 'outer' expansions. t ii!, A simple rule, expressing the flame speed eigc~walue as a function of the Lewis number and the centroid of the reaction rate function (similar to Spalding's centroid rule for unit) Le~ is number) is proposed.
13
2 The Formulation 2.1 The equations
The governing equations for steady, one-dimensional, adiabatic, laminar flames, are available in many of the earlier works 1-4. The steady flow energy and species conservation equations, considering single-step chemical reactions with the usual assumptions, are a d.x
d
-G ~
[1]
[cT] = - Hrh~.'
D dms] - G dms = ~ '
P dx J
[2]
dx
with the boundary conditions: x=-~:
T=T~,
x=
T=
+ ~"
ms=ms, . [3]
T~, m y = m s , b ( - O )
The significance of the symbols is given under Nomenclature, following this text. We now introduce the following non-dimensional transformations: J = Dpc/k ~
nlf/ltlf,
-
( T
u
-
-
dy -= c(G/k}dx *
st*
/
[4]
R - k H m sik,,
R - .~ g(~, r)drlt5 - l 2 -- k ,,,q/( Tb -
T~) c 2 G 2
p = dz/dy
J
It can be shown that equations 1 and 2 (on assuming specific heat at constant pressure to be constant) can be written as p(dp/dz) - p = - 2 ~ , r)
1 + t~d~ldz)= (~t + z - 1)tp
[5] [6]
with the boundary conditions" z=0;
p=0,
ct=l O, ~ = 0
z=l;
p
[7]
The p-transformation (first used by Klein ~4} reduces second order differential equations 1
June 1969
FLAME PROPAGATION
and 2 to first order, and we have the added advantage that the domain of integration has been reduced from - o o to + ~ , to 0 to 1. The differential equations 5 and 6 subject to the boundary conditions 7 must be solved to obtain the flame speed eigenvalue 2. 2.2 The reaction rate function ck(~, T) We have used the temperature-explicit reaction rate functions as given below: First order reactions ~,
z ) = (n + 1)(n + 2)az"
Second order reactions
~)(~, z)
= ~(n + l)(n + 2)(n + 3) t~2"r"
These temperature-explicit reaction rate functions can be made to fit experimental data at least as closely as do the classical Arrheniustype reaction rate functions. Rosen ~5 has established a correspondence between the two systems.
3 Solutions Singular perturbation solutions 3A.1 First order reactions: Nature of the solutions When we examine the system 5-6- 7, we notice that in the limit as n --, oo, the RHS of equation 5 tends to zero. In this limit, an infinite gradient appears in the p-profile (Figure 1), if the boundary condition at the hot end is
287
(0-1). The problem belongs to the class of singular perturbation problems and is effectively handled by the method of matched asymptotic expansions. We first obtain solutions in a region away from z = 1. In this "outer' region, the RHS of equation 5 is taken to be zero, because these are exponentially small terms. (This procedure is consistent with the theory of asymptotic expansions.) Next, we obtain solutions in the 'inner' region (flame region) near z ~ !. This is performed by imposing the requirement of diffusive-reactive balance. 3A.2 Asymptotic expansions: Outer expansions--Defining e - 1/n and substituting
p°(e, 3, z) = p°(3, z) + ep°(6, z) + g2p°(6, Z) + . . .
[8]
+ ~20~2(6, T) + . . .
[9]
2(&6) = 20{5) + e2t(6) + ~222(6) + . . .
[10]
into equations 5 and orders: 0(1)" PoIPo o o, 0(~): Poet _o_o, +
[11]
6, we get to various 1) = 0 o o, -- pO = 0 PlPo
[12]
and O(1): O(e):
, 0 = - P o° + % 6%po po +
+z-
1 , op,] o = - po + =,
[13] [142
where the prime denotes differentiation w.r.t.r. Inner expansions--For 6-,. 1, it can be shown1 ~ that the appropriate variables are:
~
/
n tncreas~g
1 7"
FIGURE I. Profile for p/~ plane (m = 1)
to be satisfied. Hence, it is seen that, because of this singularity, solutions obtained by perturbing n will not be uniformly valid in the z domain
T
pi---- p
[ 15]
/a = ~/'e
[16]
r/ = ( 1 -- r)/e
[ 17]
Substituting the inner expansions:
p,~, & ~?) = Pi, o(6, q) + ~Pi. t(6, ri) + ez pi,2(6, rl) +
...
[18]
#(t, & n) =/~0(& rt) + t/~l(& n) + e2/A2(6, i~) + . . .
[19]
Voi 13
V. K. JAIN AND R. N. KUltt.R
288
[20]
,~(¢~, £ ) = ,~O(¢~} + £~1(6) d" £2/L2(t~} + . - -
into equations 5 and 6 we get, to various orders. O(1):p~,oPi, o = 2oPo e-'~
[21]
O(8):Pi, lP~',o + Pi.oP~,t + Pi, o -- e-~l[~.o],/0 (--~172 "l- 3) + 2t/~ o + #t20]
For matching, we use the condition that (Outer solution),_.t
(Inner solution)~_.®
It is readily seen from equations 27b and 28 that for proper matching K = - 1. Hence at =
[22]
=
1 -
z t/6
[27]
Expressing po° (from equation 26) in inner variables, we have
and
po° = (1 - UT)
0(I)" --3Pi.o/~ + Pi.o = 0, i.e. p~ = I//~
[23]
By matching equations 29 and 30, we get
O(8~" --t~[Pi.o//I + Pi, lffO] + Pi, t
;to = 6/2
= Po - r/
[24]
where a prime (in the inner region) denotes differentiation with respect to r/. The boundary conditions are: z=0:p °=0,~=
[30]
1
~ = l ' r / = 0"p~ = 0 , # = 0
[25]
3A.3 Solutions: (a) Zeroth order solutions~ Outer solutions" Equation 11, subject to boundary conditions 25, has the non-trivial solution d = r
[26]
Using this 26 in. equation 13 we can show that [27a]
• o = 1 + K x l/a
where K is the constant of integration that remains undetermined from the outer boundary condition on a. However, by matching, this K can be shown to be equal to - 1 (see below). Expressing this outer solution for a in inner variables, we have • o = 1 + K(I - e . q ) t / 6 = 1 + K [ 1 -
[311
3A.3 (b): First order solutions~ Outer solutions: It is readily seen that equation 12 has the solution pO= 0
[32]
It can be shown that equation 14 has the solution = (e/6)(1/6 - 1) r/2/2!
[33]
Inner solutions and matching: With the lower order solutions already obtained, it may be seen that equation 24 has, in the limit as ~ / ~ oo, the solution /~t = (1/6)(1/6 - 1)F/2/2!
[33a]
This is seen to match equation 33 exactly. Equation 22 leads, in the limit when ~/~ o~, to P'i, 1 + Pi.l
½~e -~
= -1 +
le-'~r/dr/
r/e-~ {[ e-~r/dr/} ~
(q/3)e
+ (r/2/6)(1/6 - I)e2/21- + ...] [27b]
[341
Inner solutions" Equation 23 subject to boundary conditions 25 has the solution
An examination of the solution of equation 34 in the limit as ~/-, ~ reveals that p~. t - ' - r/for proper matching with the outer solution for p. This is possible only when
Po
= r//6
i.e.(0eo)me ~ = r/e/6
[28]
Solution of equation 21 can be shown to be ~Pi, 0)2 -- I 2o07/b)e-"d o
= (2o/6)]
o
e - " d [29]
2, = - ~ 3
[353
3A.3 (c) Higher order solutions~It can be shown ~6 that the higher order 2s emerge as functions of the lower order ones. Hence we
June 1969
289
FLAME PROPAGATION
get a series for 2 as
Summing this to all orders, we get
2 = ½6n2/(n + 1)(n + 2)
[36]
3 B Second order reactions--Employing methods similar to the above (section 3A) it can be shown that 2 = ½62 n3/(n + 1Xn + 2)~n + 3)
[37]
(Details of Section 3 are available in ref. 16).
Numerical solutions 3C First order reactions--Equations 5 and 6 subject to the boundary conditions 7 are solved by the procedure of iteration, to yield the eigenvalue 4. For convenience, the integration is started from the hot end. To start the integration, we need the initial gradients of p and • at z = 1. When we substitute the boundary condi~dons 7 into the equations 5 and 6 we obtain indeterminate forms for (dp/dr)~= ~ and (d~/,:lz)~= ~. Application o ; the L'H6pital rule, however, leads to L -= (dp/dz),= t
= [ 1 - { 1 + 4 3 2 ( n + IXn+2)}}]/26 M = (dx/dz),= 1 = (1 - L)/(bL - 1)
near z = 1. The R K G method, which is basically a Taylor's series expansion procedure, leads to large errors due to the neglect of these higher derivative terms. Hence, a finite-difference method, having a very small step size (0.0002), has been used to obtain the value of 2 by iteration. Step size checks were made to ensure that the value of 2 obtained is independent of the step size. The iterations on 2 were stopped when the values of 2 obtained in two consecutive integrations, with halved step size, did not differ by more than 0-4 per cent. The problems are programmed in F O R T R A N and executed on a CDC-3600-160A computer. 4 Results The analytical expressions for the flame speed eigenvalue are available in equations 36 and 37. The eigenvalues 2 predicted by these equations are plotted in Figures 2 and 4. Numerically 1-2
First order reactions" singular perturbation sotutions
0"8
[38] [39]
The Runge-Kutta method with the Gill modification is employed for numerical integration of equations 5, 6 and 7. For an assumed value of 2, numerical integrations are carried out to get p at • = 0. The method of Regula falsi is used to find the correct value of 2, which satisfies the boundary condition p,--0 = 0, to the desired accuracy (0.0001). 3D Second order reactionsmIntegration cannot be started right from the hot end (z = 1) because the first gradient of p is zero at z = 1. The integration is started from a neighbouring point, = ,o, where the initialjump on z is ~ - ( 1 - %). A value of 10 -4 on ~ has been used in the present work. The standard Runge-Kutta-Gill method used earlier (section 3C) cannot be used here, as the higher order gradients ofp are very large ~6
04
Iv
I
I
I
0
1
2
3
FIGURE 2. V a r i a t i o n
35
o f ) , w i t h 6 for v a r i o u s n
exact solutions of the flame equations, treated under sections 3C, 3D have been obtained for n = 4, 7, 11, 15, 20 and 25 for both first and
v . K . JAtrN AND R. N. KUMAR
290
1-2
First order reactions: exact values
Vol. 13
"Exact" solutions o o o o Ad|er 12 Second order reactions, exact solutions
25 /
0-8
/
n=L.-
/// / /
/
s
s
s
1 .,,,.O " S
s s
s
s p
0.L
2
s
~
.-
7
n=/. i
i
2 0
1
3
2
8
4
3"5
i
~2
I
6
I
8
tO
FIGURE 5. Variation of 2 with 6 2 for various n
FIotatE 3. Variation of 2 with 6 for various n
. Second order reactions: singular perturbation solutions
Singular perturbation
-- -- -'Exact'
o o o Spalding 3 S=3 1.3 Compar,sons.first order reactions 116-f v Sendagorta f 1"1 - i / / / / / o 0"9
//
// //o
0"7
/
/
1"5
/
0.5 ~ / ~ ~/ ~ .
~ 1 " 1
,1.0
0.9 0-3 r..........D... ~
~
_0._
j
m
_0.--
~mm--O--
~
~
0.5 ,,
0"I
L
3
6
z,
i
i
i
~
i
8
12
16
20
24
n
FIGURE 4. Variation of 2 with f 2 for various n
FIGURE 6. Variation of 2 with n for various 6 (for order reactions)
291 FLAME PROPAGATION June 1969 second order reactions and 6 = 0-1, 0.2, 0.5, plotted together for comparison, again demon0.8, 0-9,1.0,1' 1,1.2,1.5, 2, 3 and 5. The numerically strates that the singular perturbation solutions exact values of 2 are plotted in Figures 3 and 5 generally overestimate 2 for 6 > 1 and underfor first order and second order reactions respecestimate 2 for 6 < 1. However, the error detively. creases as n increases. 5.3 The proposed rule
5 Discussion 5.1 First order reactions
The analytical expression (equation 36) for large n predicts a linear variation of 2 with 6. We note from the 'exa~ ~ solutions in Figure 3 that this linearity is confirmed, even for low n. For the sake of comparison, the results of singular perturbation theory and the 'exact' solutions are plotted on the same graph and presented in Figure 6. We notice that the results of singular perturbation theory generally overestimate the value of 2 when 6 is greater than one, and underestimate the value of 2 when 6 is less than one; however, the error decreases as n increases. This behaviour is understandable, as the singular perturbation analysis of section 3 is strictly valid only for large n and 6 of order unity. 5.2 Second order reactions
The analytical solution for large n (equation 37) predicts a linear variation of 2 with 6 z. We note from the 'exact' values in Figure 5 that this characteristic is nearly true except for small values of 6. Figure 7, where the :results of singular perturbation analysis and the 'exact' values are Second order reactions SinguLar perturbation .....
Exact
0
5
Sendagorta~.s s s "
~ ~ S~'
//$tt
.1
-"
"'"
~'* '~' ~
6=3
,<2
1.5
10 ~--i . . . . . .
5
]
~.-----~---'"--'
15
10
- --r . . . . . .
20
n
FIGURE 7. Variation o f 2 with n for various 3
2~
After an extensive study of the numerical solutions, the following rule was developed for predicting the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function. ~. = A6m + B~m-1 + (m - 1)C
where A ~ 0"5 - 1.17(1 - z*) (1-71T*)m-1
1) B = 0.548 %*"~ l - z * ) - ( m × [0-915 - 2"367 z* + 1"452 r~'2] C = 0-184{1 -- z*) It is found that the rule predicts the value of 2, with an error less than four per cent over most of the r*/6 domain (viz. ,~' > 0.625 and 0-1 ~< 6 ~< 5.0). It is felt that this accuracy is sufficiently good. considering the usual degree of uncertainty of data in combustion experiments. 5.4 Comparisons
Comparisons have been made with :(i) Spalding 3, for first order reactions, (ii) Adler 1: for second order reactions, and (iii) de Sendagorta et al. 4 for both first and second order reactions. For first order reactions, the approximate results of Spalding are plotted in Figure 6. We note that his results predict the values of ). with good accuracy. The error is within two to three per cent over most of the region, though maximum error of about seven per cent is observed at low n and large 6. For second order reactions, the approximate results of Adler tz are plotted along with the "exact' results of the present work in Figure 5. There is considerable disagreement between the present "exact" solutions and the approximate solutions of Adler. Although (because of his use of Spalding's centroid rule) the errors are small around unity 6, errors of about fifteen to twenty per cent are observed when i3 differs
292
Vol. 13
V. K. JAIN AND R. N. KUMAR
significantly from unity on either side (i.e. 6 X 1). This is thought to be due to the fact that the p-profile used by Adler zz does not satisfy the boundary condition at the hot end. de Sendagorta et al. +have performed numerical integrations to obtain flame speeds, for one value of the reduced activation temperature, 0o - E/~rrb. Since similar numerical integrations have been performed in the present work also (though for much wider ranges of n and 6) the two corresponding results must agree for a satisfactory comparison. But comparisons with their results are not very straightforward as they have used Arrhenius-type reaction-rate functions. Besides, there is no precise conversion from their O= = E/~Tb to our n. It can be shown t6 that their eigenvalue A bears the relation A = ~/IZo to the eigenvalue of the present work: where 1
I - ~ co dz (co is their + reaction rate function) and Z o = 0"2 c
;o"k
(1 -
T./Tb). ,
,
+
, ',0"02
\
-]1o.2
of n ~ (1 -- TJTb) (E/,~Tb), we have established correspondence between the Arrhenius-type reaction-rate functions and the temperatureexplicit reaction-rate functions through their centroids. Consequent upon Spalding's centroid rule+ values of n and 0o are taken to correspond with each other when they give the same value of the centroid for the temperature-explicit and the Arrhenius-type reaction-rate functions respectively. Values of the eigenvalue A of de Sendagorta et al. + converted thus into the of the present study, are plotted in Figures 6 and 7. Some approximations are involved in the conversion because of: (i) errors in Spalding's centroid rule, (ii)evaluation through exponential integrals, the conversion integral /, and by neglecting some terms, (iii) evaluating the centroids of Arrhenius-type reaction-rate functions, through exponential integrals, and by neglecting small terms. The disagreement is less than about two per cent (except for 6 = 2-0 for first order reactions) between the present 'exact" values and the 'exact' values of de Sendagorta et al. +. Hence the comparisons should be considered satisfactory. 18 Rosen ts
"},:'ro,on, wo,,,
2 nd order reactions
h~
X
.
12
¢.~
C
.9 10"z U}
X
/
Ist order " ~ ~ , :.."~reactions'~~
4
-. 10-3
tO
> tO
at 8a = 20
16 3
2
t
t
A
6
10
14
I 110"~.
18
e0
FIGURE 8. C o n v e r s i o n from A to').
This integral I is presented in Figure 8 for both first and second order reactions. Instead of using Rosen's t s approximate value
I
J
6
12 Oo
FIGURE 9. C o r r e s p o n d e n c e of 0. with n
Figure 9 shows the correspondence between n and 0,, = El~lTb calculated through thecentroids. Rosen's t5 values are also presented there. It
June |969
293
FLAME PROPAGATION
can be seen that there is a close agreement between the two. Conclusions (i) The flame speed eigenvalue is linear in Lewis number for first order and quadratic in Lewis number for second order reactions. (ii) The proposed new rule can be used to determine with an accuracy of three or four per cent the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function.
average reaction rate universal gas constant absolute temperature of gas distance in the flow direction
R T X
G(c/k) dx
dy Zo
1 - TgTb
mass fraction of reactants = my~my., Lewis number -~ Dpc/k II expansion parameter in inner and outer expansions ~/ stretched independent variable in the inner region - (1 - ,)/e
0 The authors are grateful to Mr H. S. Mukunda for valuable discussions and to Mr N. Ramani for programming. The authors are thankful to the authorities of the Tata Institute of Fundamental Research, Bombay, for having allowed the use of their CDC-3600-160A Computer facilities. One of the authors (RNIO thanks the National Aeronautical Laboratory, Baru3alore, for the award of a Council of Scientific and Industrial Research Fellowship, during the course of this work.
T/Tb
go E/ K A
flame speed eigenvalue of de Sendagorta ~ 2 flame speed eigenvalue /~ stretched ~ in inner region -= ~/~ independent variable around hot boundary --- (1 - z) p gas density z reactedness = (T - T,,)/(Tb - T~) z* centroid of reaction rate function 1
(Received August 1968 ;amended November 1968)
A, R C ¢
D E G H K k L M m
mf
vh}' n
Nomenc!ature constants in the proposed rule specific heat of gases at constant pressure diffusion coefficient of gas activation energy Mass flowrate per unit area heat ofcombustion per unit mass of fuel constant of integration in outer equations coefficient of thermal conductivity ef gas see equation 38 see equation 39 order of reaction concentration of fuel mass of fuel consumed per unit volume per unit time reduced activation temperature parameter in temperature-explicit reaction rate functions
p dz/dy R
reaction rate function
¢
!
non-dimensional reaction rate function reaction rate function of de Sendagorta
O)
Subscripts b.u denote burnt, unburnt conditions i inner variables
Superscript 0
outer variable
References t EVANS, M. W. Chem. Rev. 51,363 (1952) 2 WILLIAMS.F. A. Combustion Theorr. pp 95-136. Addison Wesley: Palo Alto, Calif. ~1965) 3 SPALDING. D. B. Combustion & Flame. !. 287 a,ad 296 (t957) 4 DE SENDAGORTA, J. M., MILLAN, G. and DA R~vA, I.
U.S.A.F. Rep. No. A.F. 61 (514), 997 (1957) 5 [,lso, MILLAN, G. and DA RIVA, I. U.S.A.F. Rep. No. A.F. 61 (514), 997 (1958) 5 ZELDOVICH. Y. B. and FRANK-KAMENETSKY. D. A. J
phys. Chem., Moscow, 12, 100 (1938) 6 ROSEN, G. Seventh Symposium (lnternatwnal) on Combustion, p 339. Butterworths: London (1958)
294
v . K. JAIN A N D R. N.
7 Hm~cHvla.m~ J. O., Ctnt'r~,, C. F. and C ~ B ~ L , D. E. Univ. Wisconsin Rep. No. CM-756, 24 (November 1952) s j;d~, V. K. Proceedings o f Sununer Seminar on Fluid Mechanics, p 191. LI.Sc. (1967) 9 WnDv., IC A. J. chem. Phys. 22, 1788 0954) to D~ SENDAGORTA,J. M. Combustion & Flame, 5, 305 (1961) 1~ VON ~ , T. and Pl~Nt,~-ex,S. S. Selected Combustion Prob~ms, AGARD Vol. I. Butterworths: London (1954)
VoL 13
12 ADLER,J. Combustion & FLame, 9, 273 (1965) is SoKOUX, A. S. ['Self-ignition, flame and detonation in gases'] (Translated from Russian), Jerusalem (1963) 14 KLeiN, G. Phil. Trans. A, 249, 389 (1957) Is ROSEN, G. Jet Propulsion, 28, 839 (1958) 16 Kut~p,, R. N. M.E. Project Report, Department of Aeronautical Engineering, l.l.Sc. (1968)
OF LAMINAR FLAME PROPAGATION NON-NORMAL DIFFUSION
WITH
V. K. JAIN AND R. N. KUMAR Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 12, India A study has been made of the problem of steady, one-dimensional, laminar flame propagation in premixed gases, with the Lewis number differing from (and equal to) unity. Analytical solution, using the method of matched asymptotic expansions, have been obtained for large activation energies. Numerical solutions have been obtained for a wide range ofthe reduced activation temperature parameter (n - E / ~ T b ) , and the Lewis number & The studies reveal that the flame speed eigenvalue is linear in Lewis number for first order and quadratic in Lewis number for second order reactions. For a quick determination of flame speeds, with reasonable accuracy, a simple rule, expressing the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function, is proposed. Comparisons have been made with some of the earlier works, for both first and second order reactions.
1 latrodaetioa THE problem of laminar flame propagation in premixed gases is of fundamental importance and has been extensively studied ~-4. An excellent survey of the early theoretical studies on this problem has been made by Evans ~. A more recent study has been made by Williams 2. The propagation characteristics of steady, adiabatic, one-dimensional, laminar flames in premixed gases, depend, primarily, on two properties of the combustion system ....the activation energy of the rate-controlling step, and the Lewis number. A majority of workers in combustion make the assumption of unity Lewis number (normal diffusion) since this brings about considerable simplification in the mathematical analysis. Several approximate methods of solution, available for this case, have been compared by Spalding s and by de Sendagorta et al. 4. Zeldovich and FrankKamenetsky 5 assumed that the whole reaction occurs at the highest temperature and they were able to give a closed form solution for the flame speed. For the case of unity Lewis number most of the useful results have been incorporated in the "centroid rule' proposed by Spalding 3. Later, Rosen 6 used his 'action principle" to give a plausible derivation of the centroid rule. Hirschfelder and co-workers 7 performed numerical integrations to obtain the flame speed.
Jain ~ obtained the numerically exact solutions for temperature-explicit reaction rate functions, and compared his results with some of the earlier ones. The assumption of unity Lewis number is, however, not valid for many combustion gases. and particularly so for gases like hydrogen and propane. Approximate methods for considering non-normal diffusion, mostly by the Karman Pohlhausen profile methods, have been suggested by some investigators 3'4'9-11 For first order reactions, Spalding 3 used the profiles suggested by Wilde 9, and modified the method slightly by bringing in the centroid rule. de Sendagorta10 used an exponential approximation for the species flux fraction and demonstrated its good accuracy by comparing his results with numerically exact values, which he obtained 4 earlier with Millan and Da Riva, for one value of the reduced activation temperature, E/~Th. von Kfirmfin and Penner 1~ obtained solutions by using zeroth, first and second approximations for the species flux fraction profiles. For second order reactions, Adler 12 obtained some approximate solutions, using Wilde's profiles, for temperature-explicit reaction rate functions. Semenov (see ref. 13) suggested that, for non-normal diffusion, the flame speed, as obtained by Zeldovich and Frank-Kamenetsky. 285
286
Vol.
V. K. JAIN AND R. N. KUMAR
should be multiplied by the reciprocal of the square root of the Lewis number for first order. and by the reciprocal of the Lewis number for second order reactions. However, this analysis is strictly valid for infinite activation energies. In the present work, analytical solutions employing the method of matched asymptotic expansions have been obtained for the limit of large reduced activation temperatures. Numerical solutions have been obtained for a wide range of values of the reduced activation temperature and the Lewis number, for both first order and second order reactions. Based on the numerical studies, a simple, yet reasonably accuraterule, expressing the flame speed eigenvalue 2 as a function of the Lewis number and the centroid of the reaction rate function is developed. The present study reveals that the flame speed eigenvalue is linear in Lewis number for first order, and quadratic-in Lewis number for second order reactions. Comparisons have been made with the results of Spalding a, Adler ~2 and de Sendagorta et ai. 4. Present numerically exact values for first order reactions show that the approximate theory o f Spalding 3 gives results with good accuracy. The results of the present work compare well with those of de Scndagorta et al. 4, for the only value of the reduced activation temperature for which they have presented their results. There is large disagreement between the present numerically exact solutions and the approximate solutions of Adler ~2 for second order reactions. This, we believe, is due to the fact that the p-profile Adler uses, does not satisfy the boundary condition at the hot end. The main contributions of the present work are thought to be: (i} A systematic, analytical derivation of the dependence of the flame speed or.. the Lewis number and the reduced activation temperature parameter has been made "in the large activation energy range, using the methods of "inner" and 'outer' expansions. t ii!, A simple rule, expressing the flame speed eigc~walue as a function of the Lewis number and the centroid of the reaction rate function (similar to Spalding's centroid rule for unit) Le~ is number) is proposed.
13
2 The Formulation 2.1 The equations
The governing equations for steady, one-dimensional, adiabatic, laminar flames, are available in many of the earlier works 1-4. The steady flow energy and species conservation equations, considering single-step chemical reactions with the usual assumptions, are a d.x
d
-G ~
[1]
[cT] = - Hrh~.'
D dms] - G dms = ~ '
P dx J
[2]
dx
with the boundary conditions: x=-~:
T=T~,
x=
T=
+ ~"
ms=ms, . [3]
T~, m y = m s , b ( - O )
The significance of the symbols is given under Nomenclature, following this text. We now introduce the following non-dimensional transformations: J = Dpc/k ~
nlf/ltlf,
-
( T
u
-
-
dy -= c(G/k}dx *
st*
/
[4]
R - k H m sik,,
R - .~ g(~, r)drlt5 - l 2 -- k ,,,q/( Tb -
T~) c 2 G 2
p = dz/dy
J
It can be shown that equations 1 and 2 (on assuming specific heat at constant pressure to be constant) can be written as p(dp/dz) - p = - 2 ~ , r)
1 + t~d~ldz)= (~t + z - 1)tp
[5] [6]
with the boundary conditions" z=0;
p=0,
ct=l O, ~ = 0
z=l;
p
[7]
The p-transformation (first used by Klein ~4} reduces second order differential equations 1
June 1969
FLAME PROPAGATION
and 2 to first order, and we have the added advantage that the domain of integration has been reduced from - o o to + ~ , to 0 to 1. The differential equations 5 and 6 subject to the boundary conditions 7 must be solved to obtain the flame speed eigenvalue 2. 2.2 The reaction rate function ck(~, T) We have used the temperature-explicit reaction rate functions as given below: First order reactions ~,
z ) = (n + 1)(n + 2)az"
Second order reactions
~)(~, z)
= ~(n + l)(n + 2)(n + 3) t~2"r"
These temperature-explicit reaction rate functions can be made to fit experimental data at least as closely as do the classical Arrheniustype reaction rate functions. Rosen ~5 has established a correspondence between the two systems.
3 Solutions Singular perturbation solutions 3A.1 First order reactions: Nature of the solutions When we examine the system 5-6- 7, we notice that in the limit as n --, oo, the RHS of equation 5 tends to zero. In this limit, an infinite gradient appears in the p-profile (Figure 1), if the boundary condition at the hot end is
287
(0-1). The problem belongs to the class of singular perturbation problems and is effectively handled by the method of matched asymptotic expansions. We first obtain solutions in a region away from z = 1. In this "outer' region, the RHS of equation 5 is taken to be zero, because these are exponentially small terms. (This procedure is consistent with the theory of asymptotic expansions.) Next, we obtain solutions in the 'inner' region (flame region) near z ~ !. This is performed by imposing the requirement of diffusive-reactive balance. 3A.2 Asymptotic expansions: Outer expansions--Defining e - 1/n and substituting
p°(e, 3, z) = p°(3, z) + ep°(6, z) + g2p°(6, Z) + . . .
[8]
+ ~20~2(6, T) + . . .
[9]
2(&6) = 20{5) + e2t(6) + ~222(6) + . . .
[10]
into equations 5 and orders: 0(1)" PoIPo o o, 0(~): Poet _o_o, +
[11]
6, we get to various 1) = 0 o o, -- pO = 0 PlPo
[12]
and O(1): O(e):
, 0 = - P o° + % 6%po po +
+z-
1 , op,] o = - po + =,
[13] [142
where the prime denotes differentiation w.r.t.r. Inner expansions--For 6-,. 1, it can be shown1 ~ that the appropriate variables are:
~
/
n tncreas~g
1 7"
FIGURE I. Profile for p/~ plane (m = 1)
to be satisfied. Hence, it is seen that, because of this singularity, solutions obtained by perturbing n will not be uniformly valid in the z domain
T
pi---- p
[ 15]
/a = ~/'e
[16]
r/ = ( 1 -- r)/e
[ 17]
Substituting the inner expansions:
p,~, & ~?) = Pi, o(6, q) + ~Pi. t(6, ri) + ez pi,2(6, rl) +
...
[18]
#(t, & n) =/~0(& rt) + t/~l(& n) + e2/A2(6, i~) + . . .
[19]
Voi 13
V. K. JAIN AND R. N. KUltt.R
288
[20]
,~(¢~, £ ) = ,~O(¢~} + £~1(6) d" £2/L2(t~} + . - -
into equations 5 and 6 we get, to various orders. O(1):p~,oPi, o = 2oPo e-'~
[21]
O(8):Pi, lP~',o + Pi.oP~,t + Pi, o -- e-~l[~.o],/0 (--~172 "l- 3) + 2t/~ o + #t20]
For matching, we use the condition that (Outer solution),_.t
(Inner solution)~_.®
It is readily seen from equations 27b and 28 that for proper matching K = - 1. Hence at =
[22]
=
1 -
z t/6
[27]
Expressing po° (from equation 26) in inner variables, we have
and
po° = (1 - UT)
0(I)" --3Pi.o/~ + Pi.o = 0, i.e. p~ = I//~
[23]
By matching equations 29 and 30, we get
O(8~" --t~[Pi.o//I + Pi, lffO] + Pi, t
;to = 6/2
= Po - r/
[24]
where a prime (in the inner region) denotes differentiation with respect to r/. The boundary conditions are: z=0:p °=0,~=
[30]
1
~ = l ' r / = 0"p~ = 0 , # = 0
[25]
3A.3 Solutions: (a) Zeroth order solutions~ Outer solutions" Equation 11, subject to boundary conditions 25, has the non-trivial solution d = r
[26]
Using this 26 in. equation 13 we can show that [27a]
• o = 1 + K x l/a
where K is the constant of integration that remains undetermined from the outer boundary condition on a. However, by matching, this K can be shown to be equal to - 1 (see below). Expressing this outer solution for a in inner variables, we have • o = 1 + K(I - e . q ) t / 6 = 1 + K [ 1 -
[311
3A.3 (b): First order solutions~ Outer solutions: It is readily seen that equation 12 has the solution pO= 0
[32]
It can be shown that equation 14 has the solution = (e/6)(1/6 - 1) r/2/2!
[33]
Inner solutions and matching: With the lower order solutions already obtained, it may be seen that equation 24 has, in the limit as ~ / ~ oo, the solution /~t = (1/6)(1/6 - 1)F/2/2!
[33a]
This is seen to match equation 33 exactly. Equation 22 leads, in the limit when ~/~ o~, to P'i, 1 + Pi.l
½~e -~
= -1 +
le-'~r/dr/
r/e-~ {[ e-~r/dr/} ~
(q/3)e
+ (r/2/6)(1/6 - I)e2/21- + ...] [27b]
[341
Inner solutions" Equation 23 subject to boundary conditions 25 has the solution
An examination of the solution of equation 34 in the limit as ~/-, ~ reveals that p~. t - ' - r/for proper matching with the outer solution for p. This is possible only when
Po
= r//6
i.e.(0eo)me ~ = r/e/6
[28]
Solution of equation 21 can be shown to be ~Pi, 0)2 -- I 2o07/b)e-"d o
= (2o/6)]
o
e - " d [29]
2, = - ~ 3
[353
3A.3 (c) Higher order solutions~It can be shown ~6 that the higher order 2s emerge as functions of the lower order ones. Hence we
June 1969
289
FLAME PROPAGATION
get a series for 2 as
Summing this to all orders, we get
2 = ½6n2/(n + 1)(n + 2)
[36]
3 B Second order reactions--Employing methods similar to the above (section 3A) it can be shown that 2 = ½62 n3/(n + 1Xn + 2)~n + 3)
[37]
(Details of Section 3 are available in ref. 16).
Numerical solutions 3C First order reactions--Equations 5 and 6 subject to the boundary conditions 7 are solved by the procedure of iteration, to yield the eigenvalue 4. For convenience, the integration is started from the hot end. To start the integration, we need the initial gradients of p and • at z = 1. When we substitute the boundary condi~dons 7 into the equations 5 and 6 we obtain indeterminate forms for (dp/dr)~= ~ and (d~/,:lz)~= ~. Application o ; the L'H6pital rule, however, leads to L -= (dp/dz),= t
= [ 1 - { 1 + 4 3 2 ( n + IXn+2)}}]/26 M = (dx/dz),= 1 = (1 - L)/(bL - 1)
near z = 1. The R K G method, which is basically a Taylor's series expansion procedure, leads to large errors due to the neglect of these higher derivative terms. Hence, a finite-difference method, having a very small step size (0.0002), has been used to obtain the value of 2 by iteration. Step size checks were made to ensure that the value of 2 obtained is independent of the step size. The iterations on 2 were stopped when the values of 2 obtained in two consecutive integrations, with halved step size, did not differ by more than 0-4 per cent. The problems are programmed in F O R T R A N and executed on a CDC-3600-160A computer. 4 Results The analytical expressions for the flame speed eigenvalue are available in equations 36 and 37. The eigenvalues 2 predicted by these equations are plotted in Figures 2 and 4. Numerically 1-2
First order reactions" singular perturbation sotutions
0"8
[38] [39]
The Runge-Kutta method with the Gill modification is employed for numerical integration of equations 5, 6 and 7. For an assumed value of 2, numerical integrations are carried out to get p at • = 0. The method of Regula falsi is used to find the correct value of 2, which satisfies the boundary condition p,--0 = 0, to the desired accuracy (0.0001). 3D Second order reactionsmIntegration cannot be started right from the hot end (z = 1) because the first gradient of p is zero at z = 1. The integration is started from a neighbouring point, = ,o, where the initialjump on z is ~ - ( 1 - %). A value of 10 -4 on ~ has been used in the present work. The standard Runge-Kutta-Gill method used earlier (section 3C) cannot be used here, as the higher order gradients ofp are very large ~6
04
Iv
I
I
I
0
1
2
3
FIGURE 2. V a r i a t i o n
35
o f ) , w i t h 6 for v a r i o u s n
exact solutions of the flame equations, treated under sections 3C, 3D have been obtained for n = 4, 7, 11, 15, 20 and 25 for both first and
v . K . JAtrN AND R. N. KUMAR
290
1-2
First order reactions: exact values
Vol. 13
"Exact" solutions o o o o Ad|er 12 Second order reactions, exact solutions
25 /
0-8
/
n=L.-
/// / /
/
s
s
s
1 .,,,.O " S
s s
s
s p
0.L
2
s
~
.-
7
n=/. i
i
2 0
1
3
2
8
4
3"5
i
~2
I
6
I
8
tO
FIGURE 5. Variation of 2 with 6 2 for various n
FIotatE 3. Variation of 2 with 6 for various n
. Second order reactions: singular perturbation solutions
Singular perturbation
-- -- -'Exact'
o o o Spalding 3 S=3 1.3 Compar,sons.first order reactions 116-f v Sendagorta f 1"1 - i / / / / / o 0"9
//
// //o
0"7
/
/
1"5
/
0.5 ~ / ~ ~/ ~ .
~ 1 " 1
,1.0
0.9 0-3 r..........D... ~
~
_0._
j
m
_0.--
~mm--O--
~
~
0.5 ,,
0"I
L
3
6
z,
i
i
i
~
i
8
12
16
20
24
n
FIGURE 4. Variation of 2 with f 2 for various n
FIGURE 6. Variation of 2 with n for various 6 (for order reactions)
291 FLAME PROPAGATION June 1969 second order reactions and 6 = 0-1, 0.2, 0.5, plotted together for comparison, again demon0.8, 0-9,1.0,1' 1,1.2,1.5, 2, 3 and 5. The numerically strates that the singular perturbation solutions exact values of 2 are plotted in Figures 3 and 5 generally overestimate 2 for 6 > 1 and underfor first order and second order reactions respecestimate 2 for 6 < 1. However, the error detively. creases as n increases. 5.3 The proposed rule
5 Discussion 5.1 First order reactions
The analytical expression (equation 36) for large n predicts a linear variation of 2 with 6. We note from the 'exa~ ~ solutions in Figure 3 that this linearity is confirmed, even for low n. For the sake of comparison, the results of singular perturbation theory and the 'exact' solutions are plotted on the same graph and presented in Figure 6. We notice that the results of singular perturbation theory generally overestimate the value of 2 when 6 is greater than one, and underestimate the value of 2 when 6 is less than one; however, the error decreases as n increases. This behaviour is understandable, as the singular perturbation analysis of section 3 is strictly valid only for large n and 6 of order unity. 5.2 Second order reactions
The analytical solution for large n (equation 37) predicts a linear variation of 2 with 6 z. We note from the 'exact' values in Figure 5 that this characteristic is nearly true except for small values of 6. Figure 7, where the :results of singular perturbation analysis and the 'exact' values are Second order reactions SinguLar perturbation .....
Exact
0
5
Sendagorta~.s s s "
~ ~ S~'
//$tt
.1
-"
"'"
~'* '~' ~
6=3
,<2
1.5
10 ~--i . . . . . .
5
]
~.-----~---'"--'
15
10
- --r . . . . . .
20
n
FIGURE 7. Variation o f 2 with n for various 3
2~
After an extensive study of the numerical solutions, the following rule was developed for predicting the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function. ~. = A6m + B~m-1 + (m - 1)C
where A ~ 0"5 - 1.17(1 - z*) (1-71T*)m-1
1) B = 0.548 %*"~ l - z * ) - ( m × [0-915 - 2"367 z* + 1"452 r~'2] C = 0-184{1 -- z*) It is found that the rule predicts the value of 2, with an error less than four per cent over most of the r*/6 domain (viz. ,~' > 0.625 and 0-1 ~< 6 ~< 5.0). It is felt that this accuracy is sufficiently good. considering the usual degree of uncertainty of data in combustion experiments. 5.4 Comparisons
Comparisons have been made with :(i) Spalding 3, for first order reactions, (ii) Adler 1: for second order reactions, and (iii) de Sendagorta et al. 4 for both first and second order reactions. For first order reactions, the approximate results of Spalding are plotted in Figure 6. We note that his results predict the values of ). with good accuracy. The error is within two to three per cent over most of the region, though maximum error of about seven per cent is observed at low n and large 6. For second order reactions, the approximate results of Adler tz are plotted along with the "exact' results of the present work in Figure 5. There is considerable disagreement between the present "exact" solutions and the approximate solutions of Adler. Although (because of his use of Spalding's centroid rule) the errors are small around unity 6, errors of about fifteen to twenty per cent are observed when i3 differs
292
Vol. 13
V. K. JAIN AND R. N. KUMAR
significantly from unity on either side (i.e. 6 X 1). This is thought to be due to the fact that the p-profile used by Adler zz does not satisfy the boundary condition at the hot end. de Sendagorta et al. +have performed numerical integrations to obtain flame speeds, for one value of the reduced activation temperature, 0o - E/~rrb. Since similar numerical integrations have been performed in the present work also (though for much wider ranges of n and 6) the two corresponding results must agree for a satisfactory comparison. But comparisons with their results are not very straightforward as they have used Arrhenius-type reaction-rate functions. Besides, there is no precise conversion from their O= = E/~Tb to our n. It can be shown t6 that their eigenvalue A bears the relation A = ~/IZo to the eigenvalue of the present work: where 1
I - ~ co dz (co is their + reaction rate function) and Z o = 0"2 c
;o"k
(1 -
T./Tb). ,
,
+
, ',0"02
\
-]1o.2
of n ~ (1 -- TJTb) (E/,~Tb), we have established correspondence between the Arrhenius-type reaction-rate functions and the temperatureexplicit reaction-rate functions through their centroids. Consequent upon Spalding's centroid rule+ values of n and 0o are taken to correspond with each other when they give the same value of the centroid for the temperature-explicit and the Arrhenius-type reaction-rate functions respectively. Values of the eigenvalue A of de Sendagorta et al. + converted thus into the of the present study, are plotted in Figures 6 and 7. Some approximations are involved in the conversion because of: (i) errors in Spalding's centroid rule, (ii)evaluation through exponential integrals, the conversion integral /, and by neglecting some terms, (iii) evaluating the centroids of Arrhenius-type reaction-rate functions, through exponential integrals, and by neglecting small terms. The disagreement is less than about two per cent (except for 6 = 2-0 for first order reactions) between the present 'exact" values and the 'exact' values of de Sendagorta et al. +. Hence the comparisons should be considered satisfactory. 18 Rosen ts
"},:'ro,on, wo,,,
2 nd order reactions
h~
X
.
12
¢.~
C
.9 10"z U}
X
/
Ist order " ~ ~ , :.."~reactions'~~
4
-. 10-3
tO
> tO
at 8a = 20
16 3
2
t
t
A
6
10
14
I 110"~.
18
e0
FIGURE 8. C o n v e r s i o n from A to').
This integral I is presented in Figure 8 for both first and second order reactions. Instead of using Rosen's t s approximate value
I
J
6
12 Oo
FIGURE 9. C o r r e s p o n d e n c e of 0. with n
Figure 9 shows the correspondence between n and 0,, = El~lTb calculated through thecentroids. Rosen's t5 values are also presented there. It
June |969
293
FLAME PROPAGATION
can be seen that there is a close agreement between the two. Conclusions (i) The flame speed eigenvalue is linear in Lewis number for first order and quadratic in Lewis number for second order reactions. (ii) The proposed new rule can be used to determine with an accuracy of three or four per cent the flame speed eigenvalue as a function of the Lewis number and the centroid of the reaction rate function.
average reaction rate universal gas constant absolute temperature of gas distance in the flow direction
R T X
G(c/k) dx
dy Zo
1 - TgTb
mass fraction of reactants = my~my., Lewis number -~ Dpc/k II expansion parameter in inner and outer expansions ~/ stretched independent variable in the inner region - (1 - ,)/e
0 The authors are grateful to Mr H. S. Mukunda for valuable discussions and to Mr N. Ramani for programming. The authors are thankful to the authorities of the Tata Institute of Fundamental Research, Bombay, for having allowed the use of their CDC-3600-160A Computer facilities. One of the authors (RNIO thanks the National Aeronautical Laboratory, Baru3alore, for the award of a Council of Scientific and Industrial Research Fellowship, during the course of this work.
T/Tb
go E/ K A
flame speed eigenvalue of de Sendagorta ~ 2 flame speed eigenvalue /~ stretched ~ in inner region -= ~/~ independent variable around hot boundary --- (1 - z) p gas density z reactedness = (T - T,,)/(Tb - T~) z* centroid of reaction rate function 1
(Received August 1968 ;amended November 1968)
A, R C ¢
D E G H K k L M m
mf
vh}' n
Nomenc!ature constants in the proposed rule specific heat of gases at constant pressure diffusion coefficient of gas activation energy Mass flowrate per unit area heat ofcombustion per unit mass of fuel constant of integration in outer equations coefficient of thermal conductivity ef gas see equation 38 see equation 39 order of reaction concentration of fuel mass of fuel consumed per unit volume per unit time reduced activation temperature parameter in temperature-explicit reaction rate functions
p dz/dy R
reaction rate function
¢
!
non-dimensional reaction rate function reaction rate function of de Sendagorta
O)
Subscripts b.u denote burnt, unburnt conditions i inner variables
Superscript 0
outer variable
References t EVANS, M. W. Chem. Rev. 51,363 (1952) 2 WILLIAMS.F. A. Combustion Theorr. pp 95-136. Addison Wesley: Palo Alto, Calif. ~1965) 3 SPALDING. D. B. Combustion & Flame. !. 287 a,ad 296 (t957) 4 DE SENDAGORTA, J. M., MILLAN, G. and DA R~vA, I.
U.S.A.F. Rep. No. A.F. 61 (514), 997 (1957) 5 [,lso, MILLAN, G. and DA RIVA, I. U.S.A.F. Rep. No. A.F. 61 (514), 997 (1958) 5 ZELDOVICH. Y. B. and FRANK-KAMENETSKY. D. A. J
phys. Chem., Moscow, 12, 100 (1938) 6 ROSEN, G. Seventh Symposium (lnternatwnal) on Combustion, p 339. Butterworths: London (1958)
294
v . K. JAIN A N D R. N.
7 Hm~cHvla.m~ J. O., Ctnt'r~,, C. F. and C ~ B ~ L , D. E. Univ. Wisconsin Rep. No. CM-756, 24 (November 1952) s j;d~, V. K. Proceedings o f Sununer Seminar on Fluid Mechanics, p 191. LI.Sc. (1967) 9 WnDv., IC A. J. chem. Phys. 22, 1788 0954) to D~ SENDAGORTA,J. M. Combustion & Flame, 5, 305 (1961) 1~ VON ~ , T. and Pl~Nt,~-ex,S. S. Selected Combustion Prob~ms, AGARD Vol. I. Butterworths: London (1954)
VoL 13
12 ADLER,J. Combustion & FLame, 9, 273 (1965) is SoKOUX, A. S. ['Self-ignition, flame and detonation in gases'] (Translated from Russian), Jerusalem (1963) 14 KLeiN, G. Phil. Trans. A, 249, 389 (1957) Is ROSEN, G. Jet Propulsion, 28, 839 (1958) 16 Kut~p,, R. N. M.E. Project Report, Department of Aeronautical Engineering, l.l.Sc. (1968)