On the Spherically Symmetric Phase Change Problem

Using the energy integral method, an analysis of the spherically symmetric phase change (moving boundary) problems is presented. The expression derived for the temperature gradient shows the effects of both the interfacial area change and the curvature. The results show that the thermal boundary layer from a growing sphere is thinner than that corresponding to a collapsing one. The solution of the energy equation is then used to analyze the thermal diffusion-controlled bubble (or droplet) growth or collapse problem. The expression derived for the radial velocity takes into account the effects of (a) the Jakob number, (b) the interfacial area change and (c) the curvature. It is shown that (a) for large values of the Jakob number, the radius depends upon the first power of the Jakob number, (b) for small values of the Jakob number it is a function of the square root of this number, and (c) although widely used in the literature, an application of the "thin thermal boundary layer" model to cavitation, i.e., to collapsing bubbles, is incorrect, particularly at low Jakob numbers. Finally, it is demonstrated that analyzes of bubble dynamics based on the energy integral method are in error because the assumed temperature profile was incorrect. It is shown that this erroneous temperature distribution can result in a 100% error when computing the temperature gradient at the interface. Although the analysis presented in this paper is formulated by considering the thermal diffusion of energy, the same analysis and results with an appropriate redefinition of property terms can be applied to a spherically symmetric mass diffusion problem with a moving boundary.