There are limited number of theoretical and numerical models of materials that bubble and foam when they are heated. Many of these models simplify by solving one-dimensional heat transfer problem with an incident heat flux at one surface.
Wichman studied the effect of in-depth bubbles on the steady-state transport of volatile gases from thermoplastic material subjected to conductive incident heat flux theoretically [8]. Bubble nucleation, growth, and convection were incorporated into the bubble number distribution function. Writing the conservation equations for mass, momentum, species, and energy in terms of this distribution function enabled the determination of steady-state regression rate.
Several one-dimensional models have been developed to model the foaming behavior of intumescent fire-resistant coatings. These coatings provide protection to an underlying surface through the production of bubbles in charring thermoplastic medium. An endothermic gasification reaction generates bubbles in the molten thermoplastic, and the gas trapped in the final swollen char provides an insulating barrier to the transport of heat. The expansion and expansion rate due to bubble growth were shown to be of critical importance to the transport of heat by single-layer [9], [10] and two-layer models [11].
The development of two- and three- layer frontal models was based on the observation that intumescence takes place within a thin region, with solidification freezing the geometry in place once the front has passed. Anderson et al. [12] used an equivalent thermal resistance model to estimate the effective thermal conductivity of intumescent chars. In each of these models, swelling due to the formation of bubbles was provided as an input parameter rather than being determined from first principles. The goal of these models was to understand the heat transfer mechanisms protecting the underlying substrate from excessive temperature rises. The migration and bursting of bubbles at the surface was not addressed.
Bubble nucleation and growth in melting solid have also been studied in the field of coal pyrolysis. When most coals are heated, they swell to much larger volume until they reach critical final swelling temperature. During the swelling, the coal behaves like highly viscous liquid, and gas-producing chemical reactions generate bubbles. Attar modelled the mechanism of bubble nucleation theoretically [13]. Oh wrote mathematical model to predict volatile yields, plasticity, and swelling during softening coal pyrolysis and compared the results with experiment [14], [15]. This model treated the coal particles as spherical and isothermal, with spatially uniform bubble concentration. Gas diffusion, chemical reactions, coalescence, and bubble rupture upon contact with the particle surface were included. Because of the highly viscous nature of softening coal, the model neglected bubble movement.
A numerical model of high-energy, strained-molecule fuel droplets by Schiller et al. [16] considered the bubbles generated by the fuel vaporizing in depth. The model solved time-dependent equations in three-dimensions representing continuity, energy and species conservation, and radial bubble growth. Exothermic chemical reactions and local average mixture thermal conductivity and bubble densities were included. Bubbles burst, releasing fuel vapor and shrinking the droplet size, when the void fraction of the bubbles reached unity in the region close to the droplet surface. The intermittent bursting of bubbles caused the droplet size to alternately expand and contract with time. Migration of the bubbles in space was not considered in this model.
A three-dimensional, time-dependent numerical model of burning thermoplastic materials with in-depth bubble formation developed at NIST included the dynamics of bubble growth and migration, heat transfer through the material, and the chemistry of gasification [17]. However, this model was unsuccessful in incorporating thermal effects of the bubbles.