Effect of thermal convection on frequency response of a perturbed vaporizing pastille-shaped droplet

Abstract

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.

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

“…The linearized equations for the liquid/gas interface [13,14,21], based on the classical theory of quasi-stationary spherical drop [16,17] will now be used. The unperturbed state is a stable situation for which any thermodynamic variable f of droplet has a uniform distributionf .…”

“…The loop once established, instability of release of heat and mass occurs. Those problems received a great deal of attention during the last decades and many papers have been published among which we can cite Bhatia and Sirignano [2], Delplanque and Sirignano [3], Yang and Anderson [4], DiCicco and Buckmaster [5], Dubois et al [6], Duvvur et al [7], Harrje and Reardon [8], Heidmann and Wieber [9], Heidmann [10], Prud'homme [11,12], Prud'homme et al [13], Anani et al [14].…”

“…The dynamic response of a vaporizing droplet to these pressure oscillations is typically computed by using drop-evaporation theory [15][16][17], on the basis of the Rayleigh criterion [18], with simplifying assumptions [9][10][11][12][13][14]. The classical model of a continuously fed spherical droplet submitted to acoustics oscillations was first formulated in 1960s by Heidmann ( [9] and references therein).…”

“…Also in this study, a numerical analysis of heat transfer inside the mean evaporating droplet, termed the multi-layer model, was reported. In their recent work, Anani et al [14] replace the spherical droplet by a cylindrical pastille with an impermeable and adiabatic lateral surface. This latter model, also based on the linear analysis of small harmonic perturbations in pressure, enables the analytical study of the response of the mass and the heat of the pastille-shaped droplet under quite all other assumptions.…”

“…Moreover analytical expressions of temperature field perturbation within the liquid-phase are derived for both isothermal and adiabatic feeding regimes in this new model. As in [11,13,14] we will look for the response factor defined as the ratio of evaporating mass or heat flow rate perturbation to the pressure perturbation. First, a frequency response of the vaporization process to small oscillations in pressure will be evaluated for the isothermal and adiabatic centre regimes.…”