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.

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.

We study the dynamic response to small acoustic oscillations of a vaporizing droplet in shape of a pastille (a small liquid cylinder, called "pastille" in the sequel, the height of which being smaller than the radius of the base). Contrary to some previously proposed models, where the thermal convection effect inside the droplet is often neglected, the continuously fed pastilleshaped model takes into account the effects of both thermal convection and conduction. Curves related to different heat exchange coefficients are presented for the frequency response of the vaporization rate. The case where the feeding process at the bottom of the pastille is assumed isothermal (isothermal bottom regime) is compared to the one where the feeding process at the bottom of the pastille is adiabatic (adiabatic bottom regime). The response factor curves for the pure conduction model of the spherical droplet and for the present model of the "equivalent pastille" are also compared. The temperature field perturbation is then examined. As well as for the evaporation mass flow rate perturbation, comparisons are made between the regime with an isothermal bottom and the one with an adiabatic bottom. We find that, in spite of some divergences observed between the various cases, the frequency response of a droplet submitted to acoustic oscillations presents also some common points. It is shown that the life time (or residence time), the thermal diffusion time, and the period of the harmonic perturbation do intervene strongly in the behaviour of the vaporizing pastille. The liquid propulsion is a possible application of this basic study conducted as part of a thesis.

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On examine la réponse dynamique d'une goutte en évaporation en forme de pastille sous l'effet de petites oscillations acoustiques. La pastille est alimentée de façon continue avec un coefficient d'échange thermique h et il est tenu compte des effets de la convection de la chaleur et de la conduction sur le facteur de réponse en fréquence du système et sur le champ thermique dans la gouttelette. L'effet du coefficient d'échange thermique h est étudié.Mots clés : pastille, évaporation, oscillations harmoniques, facteur de réponse, perturbations de température Abstract: The dynamic response of a drop-shaped evaporative drop under the effect of small acoustic oscillations is examined. The pellet is fed continuously with a heat exchange coefficient h and the effects of convection of heat and conduction on the frequency response factor of the system and on the thermal field in the droplet are taken into account. The effect of the heat exchange coefficient h is studied.