On the Theory of Flame Propagation

Abstract

A new theory of flame propagation in slow inflammation is proposed which considers that the highly energized atoms or radicals formed in the flame front play a more important rôle than heat conductivity in bringing the unburned gas to the reacting state. A small number of atoms or radicals diffuse over into the unburned phase, there initiating the chemical reaction. This and other considerations allow one to propose that the sum of thermal and chemical energy per unit mass of gas in all layers from the burned phase to the unburned phase is constant. The fundamental equation for the velocity V of the flame is derived simply from the fact that the number of molecules of combustible gas entering the reaction zone (flame front) in unit time equals the number reacting in the zone in the same time. If the flame front extends from xu to xb and the unburned gas enters it at a rate just sufficient to maintain the flame front stationary, then for $1{\overline{\mathrm{cm}}}^2$ cross section the equation reads: $${\mathrm{VN}}_{\mathrm{c(u)}}={\displaystyle {\int }_{\mathrm{Tu}}^{\mathrm{Tb}}}-\frac{\mathrm{\partial Nc}}{\mathrm{\partial t}}\frac{\mathrm{dx}}{\mathrm{dT}}\mathrm{dT}$$ where V is the flame speed which is identical with the volume of unburned gaseous mixture (under initial conditions) entering the reaction zone per sec.; N_c(u) is the concentration of combustible gas in molecules/cc in the unburned phase; and Tb and Tu are temperatures of the completely burned and unburned gases at xb and xu, respectively. This equation, in addition to the postulates that the sum of thermal and chemical energy per unit mass remains constant and that the rate of change of concentration of the various gas constituents in an elementary layer due to mass flow, diffusion and chemical reaction is zero, allows a solution of the problem. The theory is applied to explosions of mixtures of ozone gas and oxygen. Ozone decomposes to oxygen, with evolution of heat. With the help of the postulate that the total energy per unit mass is constant, the terms ∂Nc/∂t and dx/dT can be expressed as functions of temperature alone, and the above equation can be solved. In view of certain simplifications the calculated velocities are considered to be in satisfactory agreement with the experimental velocities. The theory furnishes knowledge of the structure of the flame front. A particular explosion is analyzed. The width of its flame front is shown to be about 10^‐3 cm. The temperature distribution, the distribution of the concentrations of various molecular and atomic species, and the rate of reaction throughout the flame front are given. The authors add an appendix, which includes a critical analysis of the essential differences between the present theory and previous theories of other investigators.