As shown in Figure 4, the computational problem to be solved is that of a spherical sample of thermoplastic material exposed to heat flux uniformly over its surface. The amount of heat that actually enters the sample is reduced by reflective, convective, and radiative losses. At given time the sample may contain numerous bubbles of variety of sizes. Each bubble grows in time and migrates toward the surface, where it bursts and releases its gases. The presence of bubbles within the thermoplastic sample affects the transport of both heat and mass.
Figure 4: Bubbling thermoplastic sphere model.
Bubble development and behavior in heated polymer are very complex. Since the glass temperature is often well below the temperature of gasification, bubbles appear due to either homogeneous nucleation or, in the presence of impurities, heterogeneous nucleation. As bubble moves through the melt due to Marangoni (surface tension gradient) forces and forces due to gradients in viscosity resulting from the temperature gradient, it continues to grow by diffusion of the gases generated nearby. The bubble slows as it approaches the sample surface, where the thin film between the bubble and the gaseous environment outside the sample drains until the bubble bursts and delivers its gases to the surroundings. The release of condensed phase droplets that may accompany the bursting process [22], [23] is not considered in this model.
In order to include the details of bubble phenomena in a model whose main interest is in the combustion behavior of the thermoplastic sample as whole, the approach to this problem separates phenomena into two systems that undergo separate analysis but strongly interact. The bubble submodel follows the growth, migration, and eventual bursting of each individual bubble according to the local conditions at the bubble location. The mixture submodel studies the evolution of the heated thermoplastic sample as whole, including the temperature field and mass loss rate, by treating the bubble-containing polymer as mixture whose bulk material properties are an appropriately weighted average of gas and melt properties, with spatial dependence on radius alone.