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Flame propagation in a substance reacting at initial temperature
C O M B U S T I O N A N D F L A M E 3 9 : 2 1 9 - 2 2 4 (1980)
219
Flame Propagation in a Substance Reacting at Initial Temperature Y. Z E L D O V I C H Institute of Chemical Physics, Vorobyevskoye Chossee 2b, 117977 Moscow V-334, USSR A substance that reacts at initial temperature cannot sustain flame propagation with constant flame velocity, with temperature and composition being functions of a single variablx z = x - ut, u = const. Nevertheless, a solution almost similar to uniform flame propagation but with time-dependent velocity is a good approximation to the exact solution. In typical cases it is the correct intermediate asymptotic between the initial period, when the specific form of igniting impulse is smoothed out, and the final period, when adiabatic self-ignition of the substance occurs independent of the approach of the flame. The differential equation is found governing the distribution of temperature and concentration in the flame front as well as the momentary value of the flame propagation velocity. This is the equation for flame propagation with a modified source function. This source function permits one to find the exact solution of the differential equation, contrary to the nonmodified source function that prohibits the existence of the solution. Therefore a natural answer is given to the old question of how to calculate flame propagation with a smooth source function, which is positive at initial temperature. Practical application to flame propagation in a preheated substance is also of interest.
INTRODUCTION The propagation o f flame in a substance capable o f exothermic reaction is one o f a few basic phenom e n a in c o m b u s t i o n , One seeks to determine that given a u n i f o r m substance, with coordinate i n d e p e n d e n t properties, the theory must describe flame propagation with c o n s t a n t velocity U, at least in the idealized case of a substance at rest before c o m b u s t i o n and that, of a plane flame. The early theories, for example, that of Daniel1 [1], confirmed this hope. Daniell assumed that a definite ignition temperature, Ti, exists, so that the reaction rate ~ is identically zero below Ti, thatis, (~(T < Ti) - O. The initial temperature of the cold substance being low enough To < Ti, t h e n ~(To) -0. The ~ is c o n s t a n t inside the Ti < T < Tb region, The realistic flame propagation theory o f Zeldovich and F r a n k - K a m e n e t s k y [2] introduced the Arrhenius dependence o f the reaction rate on temperature, ~ ~ exp ( - A / R T ) . Therefore ~ is different from zero at every initial temperature; ~b(To) > 0, except for the unrealistic case To -- 0. Copyright © 1980 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017
It turns o u t that with ~ ( r o ) > 0 the u n i f o r m flame propagation is impossible. This is quite obvious w i t h o u t any calculations, because a substance reacting at initial temperature has a finite lifetime up to the c o m p l e t i o n o f reaction, to, even w i t h o u t any external influence. Therefore in a tube that is long enough, L > Uto, the substance in the far end will b u r n out before the approach o f the flame. The formal investigation o f the equation o f flame propagation confirms this qualitative result. With the source f u n c t i o n (heat p r o d u c t i o n rate) nonvanishing at the initial temperature, one cann o t construct a solution with the needed asymptotic T = T o, ~ T / S X = 0 at X -~ oo. The physical significance o f this mathematical result, just described, was treated as a nuisance in the formal development of the theory. Artificial m e t h o d s were used in order to make practical calculations. Zeldovich and F r a n k - K a m e n e t s k y [2] simply neglected the reaction rate at a temperature lower than some T', To >~ T' > To. It was easy to show that the results are insensitive to the choice o f T' unless T' -- To < const •
0010-2180/80/090219+06501.75
220
Y. ZELDOVICH
exp ( - A / R T o ) . At low To the Arrhenius exponent is tremendously small and the introduction of T' is obviously harmless. Another procedure, used by Hirschfelder and Curtiss [3], changed the boundary condition to dT/dx 4:0 at some x = x o, con-
and rewrite Eq. 3 in the form
sider a flame holder, applied to a porous surface at x = Xo with an out flowing substance. The "natural" flame velocity is obtained when dT/dx is small,
Putting n(T) into ~(n, T, x), we reduce it to F(T, x). To formulate the problem o f flame
These artificial changes of source function or
disturbing Eqs. (3) to (5), we use the x-dependence o f F. Suppose that somewhere, for example, at x < Xo, F is very large, there is a catalyst and the substance burns rapidly up to n = 0, T = Tb; thereafter, at x > x o, there is a definite reaction velocity F(T) independent of x. The partial differential equations are used to study the solution in the uncatalyzed region, ignited by the burnt-out hot products of reaction. The solution becomes the propagating type and the propagation velocity can be determined. Spalding [4] was the first to use this procedure (see also Ref. [5] ). In this case no difficulties are met if F(To) is nonzero but small enough so that the time span considered is sufficiently large to go through the ignition time, but yet smaller than the reaction time to of the substance at the initial temperature. However, finding the solution of an ordinary differential equation for propagating flame by a roundabout procedure o f solving a much more complicated partial differential equation is not commendable. I propose the direct approach; consider
boundary condition are unappealing, but at least they worked in the usual situation, with initial room temperature and activation energy A > 20 kcal/mol, In connection with combustion of preheated and compressed air-fuel mixtures the problem of initial reaction rate is more than unpleasant; this reaction is of practical importance. In the next paragraph it will be shown that there is a natural generalization o f the flame propagation theory which solves simultaneously both the aesthetic problem at low T O and the practical problem at moderate To.
FLAME PROPAGATION ON TIMEDEPENDENT BACKGROUND The most general description of combustion is given by nonstationary differential equations for concentrations, temperature, and so on. A usual approximation is: ~n a2n --3t = D 3X 2 -- ¢(n, T, x)
(1)
aT b2T + Q -bt- = D -bX - 2 C - ¢(n, T, x),
(2)
where T = To, c = c o at t = - oo. We obtain the correlation of T and n Q T = T O + ~ (n o --n).
n
no
T b -- T
Tb _ To.
propagation as a result of local ignition without
~T ~2T ~-t -- D --0x2 + F(T),
To = To +-cn°'
(6)
where the initial condition t = 0, T-+ T O at x ~ o% with F(Tb) = O, F(T < Tb) > 0. The solution is of the form
T = [T o -- 0(t)] ~o(z, t) + O(t)
(7a)
with (3) z = x --
It is useful to introduce Q
(5)
(4)
f
u(r) dr.
(7b)
The dimensionless function ~0 describes the flame propagation with velocity u (t) which is vari-
FLAME PROPAGATION
221
able; the variability of u(t) is connected with the dependence on time o f the conditions in the unburnt zone, reflected in the function O(t). The direct dependence of ~(z, t) on t is weak and negli-
with F[Tb-0)~+0l--(1--tp)F(0) ~ = Tb -- 0
gible. The function ~0 has the usual properties o f a propagating front, ~0 = 1 at z = -- oo and ~0 = 0 at z = + oo. The function O(t)describes the evolution of the substance before it is reached by the flame. In the corresponding region, at large z, the substance state does not depend on coordinates, therefore the 82T/ax 2 term can be neglected, so that 0 satisfies dO = F(O). (8) dt Now I proceed to the construction o f the equation determining ~0and u. The approximation uses the idea that both the function ~o(z, t) and the parameter u are timedependent, however weakly! This corresponds to ideas o f quasi-adiabatic changes o f state in thermodynamics or to the Wentzel-Kramers-Brillouin method in quantum mechanics. In practice, I neglect the partial derivative ~ / 3 t / z = const, The result is aT d~ dO dO = -- (T b -- O)u - - - - - - ~o+ - at dz dt dt d2tp+ =D(Tb-O)~z2 F(T) (9)
The nonlinear Eqs. (13a) and (13b), determining the form of the propagating front ¢(z, t), are obtained. It is well known that these equations do not have the needed solution for all values of the parameter u - t h e propagation velocity. Therefore the equations are used to find the actual value of u. The new point is the source function, if(e) see Eq. (13b) on the right-hand side of Eq. (13a). The normalization by the denominator Tb -- 0 is trivial. The nontrivial point is (1 - ¢)F(0). Due to the subtraction of (1 - ~0)F(0), the source is modified in its most important property. The primary source F(T) in Eq. (6) was nonzero, and positive for all T, including T = T o and T = 0(t); that is F 4:, 0 when ~0 = 0, namely, at the temperature of the substance far from the front. But the modification, i.e., the subtraction of the linear function (1 - ~o)F(0) identically equal to the source at ~ = 0 will cancel ~ at ~ = 0. Therefore the modified function is identically zero far from the front, ¢(~ = 0) = 0. On the other hand, the modification does not affect the other end, at ~ = 1 the (1 - ~0)F(0) term vanishes and the modified 4(1)--0 is equal to zero just as the primary F(Tb) = O. With the new modified ~(¢) from Eq. (13b) there are no more internal logical difficulties to find ~0(z, t) and the corresponding u(t) from Eq. (13a). The dependence on t comes into the equation through the O(t) dependence.
-
-
d~0 d2~o F ( T ) - (1 -~o)clO/dt - - u - ~ = D dZ---2+ Tb --0
(10)
(13b)
Using Eq. (8) for dO/dt, we obtain THE SOLUTION OF THE EQUATION FOR -- u d~o - - = Dd2~o -- + F(T)--(1--~o)F(0) dZ dZ 2 T b -- 0
(11)
One must express T as a function of ~0 (see Eq. (7 a))
T
= ( T b - - O)~p +
0
(12)
PROPAGATION VELOCITY Equations (13a) and (13b) are solved in two stages. First, the order is lowered by the standard procedure: One introduces d~o d p. . . . . . . dZ dZ
d p -d~o
(14)
+ ~0(~0)
(15)
dp in order to obtain the final equation
d~ d2 ~ + --u - - = D if(C), dZ dZ 2
u •p =D •p •~ (13a)
dp d¢
-
u
~b(~)
D
D.p
(16)
222
Y. ZELDOVICH
It is convenient to begin from the point ¢ = 1, p = 0 and to try several u values (Fig. 1). Start with u = 0. In this case p increases when ~o goes from right to left, from 1 to 0. Actually, p(0) = %/2/Df4)(~o)d~o. Therefore the condition p = 0 at ~o = 0 is not fulfilled. But taking u ~ 0 we diminish all p(~0) and in particular p(~0 = 0). At some u the needed value p = 0 at ~o = 0 is achieved. See Ref. [6] concerning the uniqueness of the procedure, This is the eigenvalue of the equation, that is, the momentary value u(t) at given t with known O(t). The function p(~0) is found simultaneously when finding u. Second, to obtain the form of the front, one integrates the equation d~o_
dZ
p(~0);
f 4d Z = - - . , Ph o) = Z(~o).
(17)
The inversion of z(~o) gives ~p(z)-the form of the front. Remember, u, p(~o), z(~o), and ~o(z) all depend on time through the O(t) dependence and through 0 being involved in Eq. (12) in the formulation of the modified source q)(~o). Therefore what has been found is actually u(t) and ~0(z, t). There is a degree of freedom, that is, a free parameter in the integration Eq. (17), giving z ( ~ o ) ,
Let us fix the lower limit of integration at a definite value ~ = a with 0 < a < 1, for example, a = 1/2. So we propose that Z = f~
(18)
d~o p (~0)"
This leads to z = 0 at ~o= a. Therefore the function ~0(z) is fLxed; ~0= a at z = 0. One side of this result is the exact definition of u(t) which is implied: u is the velocity of propagation of the temperature T(a) dividing the interval Tb -- 0 in the constant proportion T(a) -- 0 =
a(Ta --0). For example, a = 1/2 corresponds to T(a) = T a + 0/2, so that z = x - fudt = 0 is the middle point of the temperature curve and u is the velocity of this point. On the other hand, after the value of a is chosen we obtain a solution using pinpointed ~(z = 0, t) = a, so that Otp(z, t ' ) / "/z= = O. 0t o,~o=<~ This confirms our approximation. Deriving the approximate equation for ~o(z) we neglected
#)
."/ /
/
"-.", ~" \
o
Fig. 1.
_M--P <÷> /
~
tho
right u
FLAME PROPAGATION
223
a¢/Ot/z = const. This term is identically zero at z = +_oo that is at T = Tb and T = 0, but now we found the procedure nullifying O~o/at at a definite nontrivial point inside the interval, at T~ = o t T b + (1 -- a)O. The best choice is probably a = 1/2. A function equal to zero on both ends and in the middle of the interval is safely disregardedalthough not a rigorous statement, it is a feeling born from practical calculations.
SPECIFIC PROPERTIES OF THE SOLUTION The solution of one first-order ordinary Eq. (13a), even using as a trial one parameter, u, is a very simple task in the era of pocket programmable calculators. Still an analytical formula is valuable. If ~(~0) has a sharp maximum at ¢ near to 1 (inside the interval 1 > ~0 > 1 - e, with e ~ 1), one can use the procedure of Zeldovich and Frank-Kamenetsky [2] : the first approximation is [p(¢)] 2 2
1 1.1 D if(C) d~o
J,
r~ P(~ < 1 -- e) = 4 D
rl J0 ~ (¢) d~,
(19)
(20)
and the second approximation is P
t~(~p) d~0 - - D (1 -- ~).
The corresponding representation of ~ for a reaction of kth order is ~(tp) "~ (1--~o)ke - A / R T ( ~ ° ) --> ( 1 - - ¢ ) k e - r ( l - ¢ ) (24)
ff(~o) d e ~ K! • r - - ( K + 1).
But as already pointed out by Karman [7], this procedure is in some cases not exact enough; the original Eq. 22 is better, and further approximation is possible. This standard approach called "'T b dominance" is valid as long as the temperature dependence of the reaction rate is sufficiently strong and the temperature before the front 0 is low. We are investigating flame propagation superimposed on the adiabatic rise of the temperature before the flame, O(t). Therefore sooner or later the condition for "Tb dominance" will be broken and the reaction rate in all the temperature interval O(t) -- Tb is important. In this case we must study more carefully the structure of the modified source function, ff(~o), at the cold end, that is, near ¢ = 0. Only one property has been mentioned: ¢ = 0, 4 = 0 . By using Eq. (13b), we can easily establish the first term of the decomposition at ~ "~ 1. ~ =g
(21)
(25)
•
~'
g
=
dF
d-T
/T
=0
F(O)
+ ~Tb - - - ~-,-
(26)
The integration can be performed using the Frank-Kamenetsky method of treating the Arrheniusexponent [ 7 ] , t h a t is,
Therefore there is no finite interval of ~, where qJ = 0 exactly! The derivative d~/d~ is finite at = 0. This is just the case studied by outstanding mathematicians Kolmogorov, Petrovsky, and Piskunov [8] (known as KPP) as early as 1937. They solved a biological problem, the spreading of a new animal type, which is better fit for multiplication, compared with the old one already occupying the area. KPP introduced the idea of propagation velocity.* The equation coincides with that of flame propagation Eq. (13a). The source function was
e-A/R T
*
The condition ~ = 0, p = 0 gives u
=;i' 2D
~
~(~) d~0.
e - n / R T b "l- (T -- Tb)A RTb 2
(22)
(23)
Added in proof: An independent solution was given by R. Fisher Annals of Eugenics 1937, 7, 355-369.
224
Y. ZELDOVICH
taken by KPP in the form = a~0(1 -- ~)
(27)
The general idea was that reproduction is"autocatalytic," that is, proportional to the actual concentration ~0, but that growth is somehow limited by the overall food resources, which is reflected by the ( 1 - ~0)multiplier. The result of KPP analysis is that there is a continual spectrum of possible propagation velocities, limited from below u >~ Umin = 2x/a-D. (28) Moreover KPP have shown that all u > Umin are artificial. To obtain u > Umin one needs to make at the initial moment a definite distribution ~ z ) ensuring consecutive independent demographic explosions in different parts of the area. The local ignition, that is, mutation ¢ > 0, [ x [ < X o on the virgin background ~0- 0 at [ x I >
domain of applicability of this solution was already described in the abstract. A very general outlook on asymptotic solutions is given by Barenblatt [9] ; the methods described in Ref. [9] are of great importance for explosion and combustion problems. Our information o f the problem, that is, the solution in Eqs. (Ta) an (7b) led to simple ordinary differential equations. The exactitude of the approximation must be checked by comparison with much more tedious calculations of the nonsteady process, using the partial differential equation for T(x, t). The preliminary results are excellent. Apart from the practical side (the method o f calculation for the case of rapid reaction in initial mixture), it should be stressed that the new formulation gives natural logical foundation to the flame propagation theory in all cases, even using a s m a l l but still positive, nonzero-reaction rate in the initial mixture.
Xo, at the initial moment always leads to the establishment of minimum speed propagation, u --Umin, with corresponding form o f the front ~mm(Z). Returning to the combustion problem we anticipate that the growth of 0 will lead to a situation of the KPP type. Particularly, the propagation velocity at high 0 will be given by
The author is grateful to A. P. AMushin, G. J. Barenblatt, Yu. K Frolov, S. I. Khudyaev, and A. G. Merzhanov for discussions, helps, and encouragement. Also, appreciation is due the referees, who encouraged the author to rewrite a more detailed and organized paper.
u = 2X/~,
REFERENCES
g =
(29) o
It remains to investigate in detail how the transformation from Eqs (22) to (29) is fulfilled in the process of 0 growth. CONCLUSION
An approximate method of analyzing flame propagation in a substance, reacting at initial temperature, is worked out. First, the equations are formulated, followed by a discussion of the existence of the solution of the equation for propagation velocity, and then the properties of the solution are given.
As stated explicitly, and used in calculations, we are asking for an intermediate asymptotic approximate solution, and not an exact one. The
1. Daniell, P. J.,Proc. Roy. Soc. A126:393 (1930). 2. Zeldovich, Y., and Frank-Kamenetsky, D., J. Phys.
Chem. 12:100 (1938) (Russ.). 3. Hirschfelder, J., and Curtiss, H., Third Symposium (International) on Combustion, Baltimore, 1949, p. 124. 4. Spalding, D.,Proe. Roy. Soc. A240:83 (1957). 5. Zeldovich, Y., and Barenblatt, G., Comb. Flame 3: 61-74 (1959). 6. Zeldovich, Y.,J. Phys. Chem. 22(1):27 (1948)(Russ.).
7. Karman, Th. yon, Sixth Symposium (International) on Combustion, Reinhold, New York, and Chapman and Hall, London, 1957. 8. Kolmogorov, A., Petrovsky, I., and Piskunov, N., Bull. MGU, Moscow State University, USSR, Section A, t.
1, No. 6, 1937 (Russ.). 9. Barenblatt, G., The Similarity, the Automodelling, the Intermediate Asymptotics. Gidrometeoizdat, Leningrad, USSR, 1978 (Russ.) Plenum Press, 1979 (Engl.).
Received 18July 1978,'revised 1May 1979
219
Flame Propagation in a Substance Reacting at Initial Temperature Y. Z E L D O V I C H Institute of Chemical Physics, Vorobyevskoye Chossee 2b, 117977 Moscow V-334, USSR A substance that reacts at initial temperature cannot sustain flame propagation with constant flame velocity, with temperature and composition being functions of a single variablx z = x - ut, u = const. Nevertheless, a solution almost similar to uniform flame propagation but with time-dependent velocity is a good approximation to the exact solution. In typical cases it is the correct intermediate asymptotic between the initial period, when the specific form of igniting impulse is smoothed out, and the final period, when adiabatic self-ignition of the substance occurs independent of the approach of the flame. The differential equation is found governing the distribution of temperature and concentration in the flame front as well as the momentary value of the flame propagation velocity. This is the equation for flame propagation with a modified source function. This source function permits one to find the exact solution of the differential equation, contrary to the nonmodified source function that prohibits the existence of the solution. Therefore a natural answer is given to the old question of how to calculate flame propagation with a smooth source function, which is positive at initial temperature. Practical application to flame propagation in a preheated substance is also of interest.
INTRODUCTION The propagation o f flame in a substance capable o f exothermic reaction is one o f a few basic phenom e n a in c o m b u s t i o n , One seeks to determine that given a u n i f o r m substance, with coordinate i n d e p e n d e n t properties, the theory must describe flame propagation with c o n s t a n t velocity U, at least in the idealized case of a substance at rest before c o m b u s t i o n and that, of a plane flame. The early theories, for example, that of Daniel1 [1], confirmed this hope. Daniell assumed that a definite ignition temperature, Ti, exists, so that the reaction rate ~ is identically zero below Ti, thatis, (~(T < Ti) - O. The initial temperature of the cold substance being low enough To < Ti, t h e n ~(To) -0. The ~ is c o n s t a n t inside the Ti < T < Tb region, The realistic flame propagation theory o f Zeldovich and F r a n k - K a m e n e t s k y [2] introduced the Arrhenius dependence o f the reaction rate on temperature, ~ ~ exp ( - A / R T ) . Therefore ~ is different from zero at every initial temperature; ~b(To) > 0, except for the unrealistic case To -- 0. Copyright © 1980 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017
It turns o u t that with ~ ( r o ) > 0 the u n i f o r m flame propagation is impossible. This is quite obvious w i t h o u t any calculations, because a substance reacting at initial temperature has a finite lifetime up to the c o m p l e t i o n o f reaction, to, even w i t h o u t any external influence. Therefore in a tube that is long enough, L > Uto, the substance in the far end will b u r n out before the approach o f the flame. The formal investigation o f the equation o f flame propagation confirms this qualitative result. With the source f u n c t i o n (heat p r o d u c t i o n rate) nonvanishing at the initial temperature, one cann o t construct a solution with the needed asymptotic T = T o, ~ T / S X = 0 at X -~ oo. The physical significance o f this mathematical result, just described, was treated as a nuisance in the formal development of the theory. Artificial m e t h o d s were used in order to make practical calculations. Zeldovich and F r a n k - K a m e n e t s k y [2] simply neglected the reaction rate at a temperature lower than some T', To >~ T' > To. It was easy to show that the results are insensitive to the choice o f T' unless T' -- To < const •
0010-2180/80/090219+06501.75
220
Y. ZELDOVICH
exp ( - A / R T o ) . At low To the Arrhenius exponent is tremendously small and the introduction of T' is obviously harmless. Another procedure, used by Hirschfelder and Curtiss [3], changed the boundary condition to dT/dx 4:0 at some x = x o, con-
and rewrite Eq. 3 in the form
sider a flame holder, applied to a porous surface at x = Xo with an out flowing substance. The "natural" flame velocity is obtained when dT/dx is small,
Putting n(T) into ~(n, T, x), we reduce it to F(T, x). To formulate the problem o f flame
These artificial changes of source function or
disturbing Eqs. (3) to (5), we use the x-dependence o f F. Suppose that somewhere, for example, at x < Xo, F is very large, there is a catalyst and the substance burns rapidly up to n = 0, T = Tb; thereafter, at x > x o, there is a definite reaction velocity F(T) independent of x. The partial differential equations are used to study the solution in the uncatalyzed region, ignited by the burnt-out hot products of reaction. The solution becomes the propagating type and the propagation velocity can be determined. Spalding [4] was the first to use this procedure (see also Ref. [5] ). In this case no difficulties are met if F(To) is nonzero but small enough so that the time span considered is sufficiently large to go through the ignition time, but yet smaller than the reaction time to of the substance at the initial temperature. However, finding the solution of an ordinary differential equation for propagating flame by a roundabout procedure o f solving a much more complicated partial differential equation is not commendable. I propose the direct approach; consider
boundary condition are unappealing, but at least they worked in the usual situation, with initial room temperature and activation energy A > 20 kcal/mol, In connection with combustion of preheated and compressed air-fuel mixtures the problem of initial reaction rate is more than unpleasant; this reaction is of practical importance. In the next paragraph it will be shown that there is a natural generalization o f the flame propagation theory which solves simultaneously both the aesthetic problem at low T O and the practical problem at moderate To.
FLAME PROPAGATION ON TIMEDEPENDENT BACKGROUND The most general description of combustion is given by nonstationary differential equations for concentrations, temperature, and so on. A usual approximation is: ~n a2n --3t = D 3X 2 -- ¢(n, T, x)
(1)
aT b2T + Q -bt- = D -bX - 2 C - ¢(n, T, x),
(2)
where T = To, c = c o at t = - oo. We obtain the correlation of T and n Q T = T O + ~ (n o --n).
n
no
T b -- T
Tb _ To.
propagation as a result of local ignition without
~T ~2T ~-t -- D --0x2 + F(T),
To = To +-cn°'
(6)
where the initial condition t = 0, T-+ T O at x ~ o% with F(Tb) = O, F(T < Tb) > 0. The solution is of the form
T = [T o -- 0(t)] ~o(z, t) + O(t)
(7a)
with (3) z = x --
It is useful to introduce Q
(5)
(4)
f
u(r) dr.
(7b)
The dimensionless function ~0 describes the flame propagation with velocity u (t) which is vari-
FLAME PROPAGATION
221
able; the variability of u(t) is connected with the dependence on time o f the conditions in the unburnt zone, reflected in the function O(t). The direct dependence of ~(z, t) on t is weak and negli-
with F[Tb-0)~+0l--(1--tp)F(0) ~ = Tb -- 0
gible. The function ~0 has the usual properties o f a propagating front, ~0 = 1 at z = -- oo and ~0 = 0 at z = + oo. The function O(t)describes the evolution of the substance before it is reached by the flame. In the corresponding region, at large z, the substance state does not depend on coordinates, therefore the 82T/ax 2 term can be neglected, so that 0 satisfies dO = F(O). (8) dt Now I proceed to the construction o f the equation determining ~0and u. The approximation uses the idea that both the function ~o(z, t) and the parameter u are timedependent, however weakly! This corresponds to ideas o f quasi-adiabatic changes o f state in thermodynamics or to the Wentzel-Kramers-Brillouin method in quantum mechanics. In practice, I neglect the partial derivative ~ / 3 t / z = const, The result is aT d~ dO dO = -- (T b -- O)u - - - - - - ~o+ - at dz dt dt d2tp+ =D(Tb-O)~z2 F(T) (9)
The nonlinear Eqs. (13a) and (13b), determining the form of the propagating front ¢(z, t), are obtained. It is well known that these equations do not have the needed solution for all values of the parameter u - t h e propagation velocity. Therefore the equations are used to find the actual value of u. The new point is the source function, if(e) see Eq. (13b) on the right-hand side of Eq. (13a). The normalization by the denominator Tb -- 0 is trivial. The nontrivial point is (1 - ¢)F(0). Due to the subtraction of (1 - ~0)F(0), the source is modified in its most important property. The primary source F(T) in Eq. (6) was nonzero, and positive for all T, including T = T o and T = 0(t); that is F 4:, 0 when ~0 = 0, namely, at the temperature of the substance far from the front. But the modification, i.e., the subtraction of the linear function (1 - ~o)F(0) identically equal to the source at ~ = 0 will cancel ~ at ~ = 0. Therefore the modified function is identically zero far from the front, ¢(~ = 0) = 0. On the other hand, the modification does not affect the other end, at ~ = 1 the (1 - ~0)F(0) term vanishes and the modified 4(1)--0 is equal to zero just as the primary F(Tb) = O. With the new modified ~(¢) from Eq. (13b) there are no more internal logical difficulties to find ~0(z, t) and the corresponding u(t) from Eq. (13a). The dependence on t comes into the equation through the O(t) dependence.
-
-
d~0 d2~o F ( T ) - (1 -~o)clO/dt - - u - ~ = D dZ---2+ Tb --0
(10)
(13b)
Using Eq. (8) for dO/dt, we obtain THE SOLUTION OF THE EQUATION FOR -- u d~o - - = Dd2~o -- + F(T)--(1--~o)F(0) dZ dZ 2 T b -- 0
(11)
One must express T as a function of ~0 (see Eq. (7 a))
T
= ( T b - - O)~p +
0
(12)
PROPAGATION VELOCITY Equations (13a) and (13b) are solved in two stages. First, the order is lowered by the standard procedure: One introduces d~o d p. . . . . . . dZ dZ
d p -d~o
(14)
+ ~0(~0)
(15)
dp in order to obtain the final equation
d~ d2 ~ + --u - - = D if(C), dZ dZ 2
u •p =D •p •~ (13a)
dp d¢
-
u
~b(~)
D
D.p
(16)
222
Y. ZELDOVICH
It is convenient to begin from the point ¢ = 1, p = 0 and to try several u values (Fig. 1). Start with u = 0. In this case p increases when ~o goes from right to left, from 1 to 0. Actually, p(0) = %/2/Df4)(~o)d~o. Therefore the condition p = 0 at ~o = 0 is not fulfilled. But taking u ~ 0 we diminish all p(~0) and in particular p(~0 = 0). At some u the needed value p = 0 at ~o = 0 is achieved. See Ref. [6] concerning the uniqueness of the procedure, This is the eigenvalue of the equation, that is, the momentary value u(t) at given t with known O(t). The function p(~0) is found simultaneously when finding u. Second, to obtain the form of the front, one integrates the equation d~o_
dZ
p(~0);
f 4d Z = - - . , Ph o) = Z(~o).
(17)
The inversion of z(~o) gives ~p(z)-the form of the front. Remember, u, p(~o), z(~o), and ~o(z) all depend on time through the O(t) dependence and through 0 being involved in Eq. (12) in the formulation of the modified source q)(~o). Therefore what has been found is actually u(t) and ~0(z, t). There is a degree of freedom, that is, a free parameter in the integration Eq. (17), giving z ( ~ o ) ,
Let us fix the lower limit of integration at a definite value ~ = a with 0 < a < 1, for example, a = 1/2. So we propose that Z = f~
(18)
d~o p (~0)"
This leads to z = 0 at ~o= a. Therefore the function ~0(z) is fLxed; ~0= a at z = 0. One side of this result is the exact definition of u(t) which is implied: u is the velocity of propagation of the temperature T(a) dividing the interval Tb -- 0 in the constant proportion T(a) -- 0 =
a(Ta --0). For example, a = 1/2 corresponds to T(a) = T a + 0/2, so that z = x - fudt = 0 is the middle point of the temperature curve and u is the velocity of this point. On the other hand, after the value of a is chosen we obtain a solution using pinpointed ~(z = 0, t) = a, so that Otp(z, t ' ) / "/z= = O. 0t o,~o=<~ This confirms our approximation. Deriving the approximate equation for ~o(z) we neglected
#)
."/ /
/
"-.", ~" \
o
Fig. 1.
_M--P <÷> /
~
tho
right u
FLAME PROPAGATION
223
a¢/Ot/z = const. This term is identically zero at z = +_oo that is at T = Tb and T = 0, but now we found the procedure nullifying O~o/at at a definite nontrivial point inside the interval, at T~ = o t T b + (1 -- a)O. The best choice is probably a = 1/2. A function equal to zero on both ends and in the middle of the interval is safely disregardedalthough not a rigorous statement, it is a feeling born from practical calculations.
SPECIFIC PROPERTIES OF THE SOLUTION The solution of one first-order ordinary Eq. (13a), even using as a trial one parameter, u, is a very simple task in the era of pocket programmable calculators. Still an analytical formula is valuable. If ~(~0) has a sharp maximum at ¢ near to 1 (inside the interval 1 > ~0 > 1 - e, with e ~ 1), one can use the procedure of Zeldovich and Frank-Kamenetsky [2] : the first approximation is [p(¢)] 2 2
1 1.1 D if(C) d~o
J,
r~ P(~ < 1 -- e) = 4 D
rl J0 ~ (¢) d~,
(19)
(20)
and the second approximation is P
t~(~p) d~0 - - D (1 -- ~).
The corresponding representation of ~ for a reaction of kth order is ~(tp) "~ (1--~o)ke - A / R T ( ~ ° ) --> ( 1 - - ¢ ) k e - r ( l - ¢ ) (24)
ff(~o) d e ~ K! • r - - ( K + 1).
But as already pointed out by Karman [7], this procedure is in some cases not exact enough; the original Eq. 22 is better, and further approximation is possible. This standard approach called "'T b dominance" is valid as long as the temperature dependence of the reaction rate is sufficiently strong and the temperature before the front 0 is low. We are investigating flame propagation superimposed on the adiabatic rise of the temperature before the flame, O(t). Therefore sooner or later the condition for "Tb dominance" will be broken and the reaction rate in all the temperature interval O(t) -- Tb is important. In this case we must study more carefully the structure of the modified source function, ff(~o), at the cold end, that is, near ¢ = 0. Only one property has been mentioned: ¢ = 0, 4 = 0 . By using Eq. (13b), we can easily establish the first term of the decomposition at ~ "~ 1. ~ =g
(21)
(25)
•
~'
g
=
dF
d-T
/T
=0
F(O)
+ ~Tb - - - ~-,-
(26)
The integration can be performed using the Frank-Kamenetsky method of treating the Arrheniusexponent [ 7 ] , t h a t is,
Therefore there is no finite interval of ~, where qJ = 0 exactly! The derivative d~/d~ is finite at = 0. This is just the case studied by outstanding mathematicians Kolmogorov, Petrovsky, and Piskunov [8] (known as KPP) as early as 1937. They solved a biological problem, the spreading of a new animal type, which is better fit for multiplication, compared with the old one already occupying the area. KPP introduced the idea of propagation velocity.* The equation coincides with that of flame propagation Eq. (13a). The source function was
e-A/R T
*
The condition ~ = 0, p = 0 gives u
=;i' 2D
~
~(~) d~0.
e - n / R T b "l- (T -- Tb)A RTb 2
(22)
(23)
Added in proof: An independent solution was given by R. Fisher Annals of Eugenics 1937, 7, 355-369.
224
Y. ZELDOVICH
taken by KPP in the form = a~0(1 -- ~)
(27)
The general idea was that reproduction is"autocatalytic," that is, proportional to the actual concentration ~0, but that growth is somehow limited by the overall food resources, which is reflected by the ( 1 - ~0)multiplier. The result of KPP analysis is that there is a continual spectrum of possible propagation velocities, limited from below u >~ Umin = 2x/a-D. (28) Moreover KPP have shown that all u > Umin are artificial. To obtain u > Umin one needs to make at the initial moment a definite distribution ~ z ) ensuring consecutive independent demographic explosions in different parts of the area. The local ignition, that is, mutation ¢ > 0, [ x [ < X o on the virgin background ~0- 0 at [ x I >
domain of applicability of this solution was already described in the abstract. A very general outlook on asymptotic solutions is given by Barenblatt [9] ; the methods described in Ref. [9] are of great importance for explosion and combustion problems. Our information o f the problem, that is, the solution in Eqs. (Ta) an (7b) led to simple ordinary differential equations. The exactitude of the approximation must be checked by comparison with much more tedious calculations of the nonsteady process, using the partial differential equation for T(x, t). The preliminary results are excellent. Apart from the practical side (the method o f calculation for the case of rapid reaction in initial mixture), it should be stressed that the new formulation gives natural logical foundation to the flame propagation theory in all cases, even using a s m a l l but still positive, nonzero-reaction rate in the initial mixture.
Xo, at the initial moment always leads to the establishment of minimum speed propagation, u --Umin, with corresponding form o f the front ~mm(Z). Returning to the combustion problem we anticipate that the growth of 0 will lead to a situation of the KPP type. Particularly, the propagation velocity at high 0 will be given by
The author is grateful to A. P. AMushin, G. J. Barenblatt, Yu. K Frolov, S. I. Khudyaev, and A. G. Merzhanov for discussions, helps, and encouragement. Also, appreciation is due the referees, who encouraged the author to rewrite a more detailed and organized paper.
u = 2X/~,
REFERENCES
g =
(29) o
It remains to investigate in detail how the transformation from Eqs (22) to (29) is fulfilled in the process of 0 growth. CONCLUSION
An approximate method of analyzing flame propagation in a substance, reacting at initial temperature, is worked out. First, the equations are formulated, followed by a discussion of the existence of the solution of the equation for propagation velocity, and then the properties of the solution are given.
As stated explicitly, and used in calculations, we are asking for an intermediate asymptotic approximate solution, and not an exact one. The
1. Daniell, P. J.,Proc. Roy. Soc. A126:393 (1930). 2. Zeldovich, Y., and Frank-Kamenetsky, D., J. Phys.
Chem. 12:100 (1938) (Russ.). 3. Hirschfelder, J., and Curtiss, H., Third Symposium (International) on Combustion, Baltimore, 1949, p. 124. 4. Spalding, D.,Proe. Roy. Soc. A240:83 (1957). 5. Zeldovich, Y., and Barenblatt, G., Comb. Flame 3: 61-74 (1959). 6. Zeldovich, Y.,J. Phys. Chem. 22(1):27 (1948)(Russ.).
7. Karman, Th. yon, Sixth Symposium (International) on Combustion, Reinhold, New York, and Chapman and Hall, London, 1957. 8. Kolmogorov, A., Petrovsky, I., and Piskunov, N., Bull. MGU, Moscow State University, USSR, Section A, t.
1, No. 6, 1937 (Russ.). 9. Barenblatt, G., The Similarity, the Automodelling, the Intermediate Asymptotics. Gidrometeoizdat, Leningrad, USSR, 1978 (Russ.) Plenum Press, 1979 (Engl.).
Received 18July 1978,'revised 1May 1979