Theoretical Analysis of Thermal Conduction Effect on Frequency Response of a Perturbed Vaporizing Spherical Droplet

Abstract

In this work, model results of the effect of thermal conduction on frequency response of a perturbed vaporizing spherical droplet are presented and discussed. The linear analysis of dynamic response to small acoustics oscillations are performed on the basis of the Rayleigh criterion for a mean spherical droplet representing the spray of repetitively injected droplets in the combustion chamber. Curves related to different heat exchange coefficients are presented for the frequency response of the vaporization rate. The not-yet-solved case of imposed temperature at the centre of the spherical droplet (isothermal centre regime or isothermal injection regime) is taking into account here. The case is now compared to the case where the feeding process at the centre of the spherical droplet is assumed adiabatic (adiabatic centre regime or adiabatic injection regime). Each feeding case here considered represents a specific boundary condition controlling the whole injection process. The temperature field perturbation inside the droplet is then examined. Comparisons are also made between the adiabatic and the isothermal injection regimes and differences are analysed. It is shown that the characteristic times of the evaporation process, the period of the harmonic perturbation and a particular parameter depending on fuel physical properties do intervene strongly in the behaviour of the vaporizing droplet. Especially, in the isothermal injection regime, due to this particular parameter, high and non-linear frequency responses may appear in the process. The results of this theoretical study may be applied in establishments of combustion systems stability limits.

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

“…These equations are briefly recalled in subsection 2.3 for a free vaporizing droplet at its unperturbed or stabilized state. For such a free vaporizing droplet, that is a single injected droplet during its course of evaporation, this state corresponds to a state of rest in an infinite atmosphere and is characterized by [15,16]: constant average evaporation rate, uniform temperature of saturated vapour at the transient droplet surface, and equal velocities of the ambient gas flow and of the droplet. Below, these assumptions will be discussed in connection to the stabilized mean spherical droplet.…”

“…Our objective is to build an approximate analytical model of reference for the harmonic perturbations in pressure. To this end, we will supplement the previous developments of the models of the continuously fed spherical droplet [13][14][15][16] by taking into account the not yet studied radial thermal convection effect on the process, for both adiabatic and isothermal feeding cases.…”

“…We will now recall the linearized equations for the liquid/gas interface used in [15,16]. At the stabilized state, any thermodynamic variable f of the gas-phase has a uniform distribution f .…”

“…We now briefly recall, concerning the pure conduction model of the spherical mean droplet [15,16], that the complex transfer function Z has the same expression as laid in Eq. (19).…”

“…Explicit analytical expression was then derived for the droplet mass transfer function in that pure conduction case and the results were discussed. With an aim of extending the analytical expressions of the perturbed evaporating rate and to the temperature field in a spherical symmetry configuration, Anani and Prud'homme (2016) have recently studied the pure thermal conduction effect on the perturbed mean spherical droplet and have pointed out the specificity of the isothermal injection regime [16]. The results derived from this case were then compared to that of the adiabatic feeding case.…”