Dynamic response of a vaporizing spray to pressure oscillations: Approximate analytical solutions

Abstract

In this work, we study thermal conduction and convection combined effects on frequency response to pressure oscillations of a spray of repetitively injected drops in a combustion chamber. The theoretical model is based on Heidmann analogy of the so called “mean droplet” which is a single spherical vaporizing droplet with constant average radius, given that this droplet is continually fed at a stationary flow rate. The feeding comes from a source point placed at the mean spherical droplet center in such a way that the injection process can be assumed to be isothermal (isothermal feeding regime) or adiabatic (adiabatic feeding regime). Drawing upon the linear decomposition of the energy conservation equation, approximate analytical solutions for the perturbed temperature field inside the droplet are obtained from some derived double confluent Heun equations. Frequency response factor of the evaporating mass is then evaluated on the basis of the Rayleigh criterion by means of the linearized equations of the gas phase. Compared to the results obtained for the previous pure conduction model of the same “mean droplet”, frequency response factor curves seem to be similar with reference to each feeding regime. Moreover, due to the radial thermal convection effect introduced in the present work, a frequency response factor curve with the same characteristic times ratio may exhibit a relatively larger frequency range for the instability domain. Data are found to be correlated in terms of period of pressure oscillations, of vaporization characteristics times and of fuel thermodynamic coefficients. In the isothermal feeding regime in particular, due to some possible values that can be taken by a certain thermodynamic coefficient, high and non-linear frequency responses may appear in the system.