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 centre 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, 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.

Drop vaporization frequency response: an approximate analytical solution for mixed injection regimes

“…The transfer time by thermal diffusion process is defined as , which will be called from now on, the thermal exchange ratio or more briefly the exchange ratio, as it is of the same order of magnitude as1/ L Pe , L Pe being the Péclet number of the liquid phase. The coefficient 9 permits to obtain later a simple expression of the transfer function and will be kept for comparison purposes with results obtained in previous studies [16,17]. During the vaporization, intrinsic or external pressure-related oscillations can cause departure from stabilized-state conditions.…”

“…In the same order, a reduced frequency u depending on the residence time v  and defined as 3 v u   will be considered. Here again, the coefficient 3 is kept for simplifying later expression of the transfer function and for comparison purposes with results obtained in previous studies [16,17].…”

“…The linearized equations for the liquid/gas interface initially presented in [15] and used in [16,17], are here briefly recalled. Introducing the harmonic perturbations, the ambient pressure is given…”

“…In [16], Anani and Prud'homme extended the study of this pure conduction model by taking into account the isothermal feeding process at the mean droplet centre. Recently, an approximate analytical model has been performed by Anani et al [17] where was relaxed the simplifying assumption of a negligible radial thermal convection effect inside the liquid phase. Nevertheless, only the two extreme cases of centre injection that are the adiabatic and the isothermal feeding regimes were considered.…”

“…The present paper aims at contributing to the linear analysis of subcritical combustion instabilities by analytical approaches based on the mean spherical droplet configuration as in [17]. In the following section, a brief description is given of the unperturbed state corresponding to the vaporization of the continuously fed spherical droplet in a stable environment.…”