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cifically, there may be a nonequipartition in the internal modes of these chemiluminescent species. A case in point is that of electronically excited OH radicals in the oxyacetylene flame. Analysis of the OH emission spectra of this flame shows a rotational "temperature" of about 3000°K. and a vibrational "temperature" of about 3750"K.,indicating the formation of OH* in a nonequilibrium distribution by some specific chemical reaction.
S. W. BENSON(University of Southern California) : The presentation omits an important term in the integral equation: namely, the conduction of heat to the walls of the vessel. If the heat change in the reaction amounts to 10 kcal./mole of reactant, then in the interval required for 1 per cent of the reaction to be over, 100 cal. have been liberated. If there is no conduction to the walls, the temperature will go up about 10" or 2O"C., assuming average heat capacities. If this effect is negligibly small, then conduction and convection to the walls must be significant and an important term has been omitted from the formulation. It is my opinion that one of the major reasons for the larger experimental errors inherent in gas reactions as opposed to solution reactions lies precisely in the temperature and concentration gradients present in gas reactions, which are not levelled out by a solvent. In our work at the University of Southern California we have run into a number of heterogeneous gas-solid systems in which the kinetics is controlled by the rate of heat conduction to and from the system.
0. K. RICE (University of North Carolina): I believe that the effect of conduction of heat to the walls is taken care of in the theory of thermal explosions. What is dealt with here seems to involve a different problem. I t has to do with local effects of individual hot molecules, and thermal conductivity can only be involved in the transfer of heat from these individual hot spots to the body of the gas.
T H E THEORY OF FLAME PROPAGATION. II"* J. 0. HIRSCHFELDER A N D C . F. CURTISS University of Wiseonsin Naval Research Laboratory, Madison, Wisconsin Received .4ugust 10, 1960
There is need for a fundamental theory of flame propagation relating the practical performance of burners to the basic properties of the gases,-diffusion, thermal conductivity, and chemical kinetics. A complete theory would describe I Presented before the Symposium on Anomalies in Reaction Kinetics which was held under the auspices of the Division of Physical and Inorganic Chemistry and the hlinneapolis Section of the American Chemical Society at the University of Minnesota, June 1!4-
21, 1950.
* This work was carried out under Contract S O r d 9938 with the Bureau of Ordnance, United States Navy Department.
THEORY OF FLAME PROPAGATION
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the detailed nature of a t,hree-dimensional flame, including ignition and stability phenomena. However, it is possible to approach the problem stepwise. I n the present paper we consider the steady-state solution of the one-dimensional flame equations. Later we shall consider questions related to stability and to practical three-dimensional flames as modifications of the results obtained here. I. THE BASIC EQUATIONS
In a previous paper (2) a set of first-order nonlinear ordinary differential equations was derived to explain the steady-state propagation of one-dimensional flames. These equations are the hydrodynamical equations for a compressible gas mixture including diffusion, heat conductivity, and an arbitrary system of chemical kinetics. The basic relations are expressed in terms of the following quantities:
x
distance from the flame holder, temperature at a point, pressure at a point, mole fraction of the ithchemical species, total mass rate of flow of the gases, fraction of the mass rate of flow which is contributed by the ith chemical species, = molecular weight of the ich chemical species, = total number of moles of gas per cubic centimeter = P / R T (here R is the gas constant), = mean molecular weight of gas = Z,miyi, = the enthalpy per mole of the ithspecies, = coefficient of thermal conductivity, = rate a t which moles of the ichchemical species are formed per cubic centimeter, = ordinary coefficients of binary diffusion, = mass average velocity = M / p , = number of kinds of chemical species, = number of kinds of atomic species, = number of atoms of the kch kind in molecules of the ithkind, and = total number of chemical reactions (and their reverse reactions) considered
= = P = yi = Af = miZi =
T
m, n p/n
Hi X K&j,
T) Dij
u s g Uik
p
Under steady-state conditions the mass rate of flow, M, is a constant independent of z,and the pressure, P,is almost constant. Variations in the pressure across the flame front are negligible (- 0.01 mm. of mercury) for one-dimensional problems, but even this small change would be important in threedimensional problems in establishing the flow pattern. The rates of formation, IC&,, T),are taken to be those applying to each of the chemical reactions occurring under the same conditions of temperature, pressure, and composition in a static system. It remains to be seen whether this assumption is valid. For example, it might turn out that in a practical flame, the successive reactions occur so rapidly that the internal degrees of freedom of the intermediate products
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do not have sufficient time to adjust to the local temperature. We shall denote quantities a t the flame holder by the subscript 0 and quantities a t the hot boundary by the subscript m. The equations for steady-state flame propagation can then be written as follows: 1. The equations of continuity:
M
dZ. --.! dx
K
