THETHEORY OF VLAME PROPAGATION
April, 1Y53
which is more than the minimum required, but the probability is low that whatever energy remains after dissociation be disproportionally distributed. Of course, a dissociation yesulting from a non-adiabatic process, involving a transition from a metastable excited electronic state to a lower repulsive state is not so limited in energy. Now consider the restrictions imposed by conservation of
403
momentum. It is clear that to yield products which show excessive rotational temperatures, the bond S-Y must break when the molecule is in a highly excited symmetric bending mode Y b . 8 . Furthermore, the J values for the two fragments must be roughly the same. This implies, unless their moments of inertia are comparable, that. the available energy is distributed with considerable disparity.
THE THEORY QF FLAME PROPAGATION.
IV’
BY J. 0. HIRSCHFELDER, C. F. CURTISS AND DOROTHY E. CAMPBELL Naval Research Laboraloi,y, Department of Chemistry, Universilu of Wisconsin, Madison, Wisconsin Received December 1. 1968
The equations and boundary conditions which describe the one dimensional steady state propagatioii of flames have been derived and discussed previously. In this paper three examples are discussed: (1) the unimolecular decomposition of hydrazine; (2) the bimolecular reversible decomposition of nitric oxide; and (3) the two step chain mechanism describing the decomposition of ozone. The experimental flame velocity for the ozone decomposition is 55 cm./sec. as compared to our calculated value of 47 cm./sec.; for hydrazine decomposition the experimental flame velocity is 200 cm./sec. as compared to our calculated value of 127 cm./sec.; and for nitric oxide decomposition although a n experimental value will soon be forthcoming there is a t present no data with which t o compare our calculations. One purpose of these calculations is to learn the relative importance of the detailed chemical kinetics, heat conductivity and diffusion, insofar as they affect flame propagation. Another purpose is to learn more about the detailed structure of the flame zone. For example, from the calculations for the ozone decomposition it becomes apparent that for most chain reacting flame systems there is an enormous concentration of free radicals in the vicinity of the hot boundary thousand8 of times greater than would be expected on the basis of thermal equilibrium. The difficulties of treating specific flame systems are of two sorts. First, there is only a limited number of flames for which the complete system of chemical kinetics including the reaction rate constants are known. Secondly, in those cases where the chemical and physical constants are known there are still the mathematical problems of integrating the differential equations. Some of these systems of equations can be integrated easily while others require new types of mathematical methods and cannot be integrated numerically by the old methods even if super-high speed calculating machines are available.
In this paper the theory of flames is applied to three examples: (1) the unimolecular decomposition of hydrazine; (2) the bimolecular reversible decomposition of nitric oxide; and (3) the two step chain mechanism describing the decomposition of ozone.
second as a result of chemical reactions. The mass rate of flow, M , is a constant determined by the boundary conditions. (2). Equations of Diffusion
A. The Equations Describing the Propagation of-a Here n is the total number of moles of gas per Steady State, One-dimensional Flame and the Dij are the coefficients of ordinary diffusion
The following relations apply to the propagation of steady state one-dimensional flames2+ in which the velocity of flow of the hot gases is small compared to the velocity of sound. (1). Equations of Continuity MdGi/dx =
m i K i ( l ~ j ,2‘)
in a binary mixture of substances i and j. Equation (2) is a generalization of Fick’s law to multicomponent systems. (3). Equation of Energy Balance ,
i = 1,2,. . .
(I) is the mass rate of flow of the gases, A4 =
Here M pu, where p is the gas density and u is the mass average velocity of flow; x is the distance from the flame-holder; G; is the fraction of the mass rate of flow which is contributed by the i-th chemical species; mi is the molecular weight of molecules of the i-th kind; y; is the mole fraction of molecules of the i-th kind; and Ki(yj, 5”) is the net number of moles of the i-th species which are formed per ~ 1 2 1 per .~ (1) This work was carried out under Contract KOrd 9938, Kavy Bureau of Ordnance. Paper 111, J. 0. Hirschfelder, C. F. Cnrtiss and D. E. Campbell, “Fourth Symposium on Flanies, Combustion and Detonations,” Cambridge, Mass., 1952, Williams and Wilkins, Baltiinore, Md. (2) (a) J. 0. Hirschfelder and C. F . Curtiss, J . Chem. Phua., 17, ’1076 (1949); (b) “Third Symposium on Combustion, Flames and Explosion Phenomena,” Williams and Wilkins, Baltimore, Md., 1949. (3) J. 0. Hirschfelder and C . F . Curtiss, THIS JOURNAL, 55, 774 (195 1). (4) J. 0. Hirsclifelder, C. B. Curtiss and D . E. Campbell, ref. 1. (5) J. 0. Hirschfelder, C. F. Curtiss, R . B. Bird and E . L. Spots, “The Molecular Theory of Gases and Liquids,” John Wiley and Sons, 1110 , New York, N . Y.. 1953.
Here X is the coefficient of thermal conductivity; hi is the enthalpy per gram of the i-th component; the subscript ‘(~0 ” indicates the conditions at the hot boundary. Usually the are approximated by the form
ai
e,
&i(T) = fii(0)
+
eiT‘
where the and the fi,(O) are constants. (4). The Equation of State p = nRT
(-1)
Here p is the pressure and R is the gas constant. This perfect gas equation of state applies to flames a t pressures below 50 to 100 atmospheres. At higher pressures, corrections for gas imperfections may be important. The boundary conditions which the solutions to the flame equations must satisfy are the following : Hot Boundary (designated by subscript “ ”), x = w.-At the hot boundary we assume coin-
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J. 0. I~IRSCHFELDER, C. D‘. CURTISSAND DOROTHY E. CAMPBELL
Vol. 57
plete chemical and thermal equilibrium. The yi a t perature is T, = 1933°K. Murray and Hall esthis temperature are determined by the condition timate that the enthalpy released by the decompothat the Ki(yj, T) = 0. The (Gi) = mi(yi) a / sition of the hydrazine is 1000 cal./g. Thus the Zmj(yj) a. The first derivatives of T, yi and Gi average specific heat is j with respect to x are then zero. The derivatives & = - T=m- -=- To 1510 ‘Oo0 0.6623 cal./g. dey. ( 9 ) of yi and Gi with respect to T however are not (usually) zero at the hot boundary. Murray and Hall further estimated that a t the flame Cold Boundary ((designated by the subscript temperature the thermal conductivity is A m = “O”), x = 0.-At the cold boundary, the Gi are 0.0067 cal./cm. deg. sec. specified. They are simply related to the composiThe reduced diffusion coefficient is defined as tion of the fuel gas mixture in the mixing chamber (Gib = ( t n i ( y i ) / C n l j y j ) a
mixing chamber
The values of the (y& are not known a priori because of back diffusion up to the flame-holder. The temperature a t the cold boundary is determined from eq. (3) by the requirement that (dT/ dr)o have a specified value corresponding to a specified heat transfer to the flame-holder. B. The Decomposition of Hydrazine Flames resulting from the decomposition of hydrazine have been observed and studied by Murray and H a L 6 We consider here a theoretical study of these flames. In the theoretical study of flame propagation the first question which arises is the nature of the kinetics of the decomposition. Murray and Hall found that in the decomposition of hydrazine the over-all reaction is 2N2H4 +2NH3
+ + H1 K2
(6)
However, Szwarc’ found that the first step in the decomp,osition is the unimolecular reaction N2H4
+2NH2
(10)
(5)
(7)
where ml = 32 and m2 = 16 are the molecular weights. The best estimate of the value of the diffusion constant consists of taking 6 = 0.75 and independext of temperature. The equations describing the propagation of the hydrazine flame are: (a) The Equation of Energy Balance d‘l’ldx = sg
where g =
(T
- T,)
+ (Z’,
s = e-P ill = 99
x
W
(11)
- To)Gi
&I cm.?
(12)
(13)
(b) The Equation of Continuity
where R
= 1.98i
cal./mole deg.
a = 4 X 1012nzlnX/^Cp= 8.164 X 106(g.2/cm.4sec.2) (16)
(c) The Diffusion Equation
with the rate constant k = 4 X 1 0 I 2 exp (-6O,OOO/RT)
(8)
The actual mechanism of the hydrazine decomposition is clearly a complicated chain reaction in which eq. (7) is the first st’ep. The chain reaction aspects of the combustion of hydrazine have been discussed rebently by Adams and Stocks.* The present treatment of the hydrazine flame is based on an idealization of the decomposition mechanism. The initial unimolecular decompositio;? step, eq. (7), is considered and the remainder of the reactions are neglected. Thus we consider only which we two molecular species, NzH4 and “2, designate by the subscripts “1” and “2,” respectively. This idealization forms a simple example of the flame theory, although it only remotely resembles the hydrazine decomposition. Now let us consider the values of the numerical constants entering into the equations describing the hydrazine flame. From the thermochemical considerations using eq. (6) for the over-all reaction, and from actual experimental measurements, Murray and Hall found that if the ambient temperature of the hydrazine is To = 423”K., the flame tem( 6 ) R. C. Murray and A. R. Hall, Trans. Faraday Soc., 47, 743 (1951) (7) 34. Szwarc, J . Chem. PhUs., 17, 505 (1949). (8)G . I
