Gas dynamic calculation of detonation in variable cross-section ducts

Formulas of gas dynamic calculation of detonation parameters in variable cross-section ducts are presented and a design detonation diagram is given. The diagram shows the detonation characteristics of super-compressed detonation and under-compressed detonation as the function of shock wave specific speed depending on the intensity of temperature of the ideal gas in a subsonic one-dimensional flow behind the shock wave propagating in a chemically active air-fuel mixture and on the ratio of geometrical expansion (convergence) of the duct. The propagation of a stationary shock-wave the static pressure of which in the output cross-section of the expanded duct is equal to atmospheric pressure is referred to as design detonation. This means that all the energy of the shock wave at the output of the duct can be converted into polytropic work function of gas expansion in a detonation engine. Otherwise, if the flow takes place in the mode of overexpansion due to the separation of the compressive shock wave inside the duct or in the case of insufficient expansion part of the shock wave energy will be lost. The total impulse equation for a geometrically expanding duct is solved by replacing the integral describing the thrust force with the average integral value of the curve of the static pressure acting on the side wall of the expanding duct. The frictional force is neglected due to its insignificant value. It is shown that the presence of an insufficiently compressed shock wave is not possible as the shock wave moving at the supersonic speed in the convergent duct will be decelerated to the sonic speed. To stabilize it additional heat should be supplied to transform the convergent duct behind the compressive shock wave into a semi-permanent cross-section duct wherein thermal crisis stabilizing the shock wave can be achieved. The minimum value of the detonation pipe diameter of 50 mm is substantiated. Below that value sharp reduction of combustion efficiency takes place. The results of the work can be used for the computation of detonation engine thermodynamic cycle parameters.