4.1.3 Energy Equation

For a one-fluid description of a multi-component fluid mixture, the components are assumed to be in thermal equilibrium, so that T_p = T_g = T. This is a reasonable assumption for the problem at hand since the ratio of gas and polymer thermal capacities, ρ_p (c_p)_p=ρ_g(c_p)_g is large. Other simplifying assumptions are:

• No internal radiation; all heat losses due to radiation occur at the surface
• The variation of chemical potential with temperature can be ignored
• Energy dissipation due to friction is neglected
• Polymer and gas are incompressible fluids; expansivity β = 0 and effects of pressure are not included.

Appendix A demonstrates how an equation for temperature can be derived directly from the internal energy equation using the principles of irreversible thermodynamics. Using the above assumptions, the energy equation for this spherically symmetric geometry is

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where k* is the effective thermal conductivity, H_v is the (positive) heat of vaporization and mass loss rate is given by the Arrhenius expression in equation (1).

The solution must satisfy the appropriate initial and boundary conditions. Initially, the temperature throughout the thermoplastic sample is the ambient temperature:

At the surface of the sphere, the heat flux into the sample is given by the incident heat flux minus losses due to reflectivity, convection, and radiation:

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where q₀ is the incident heat flux, ε is the surface emissivity, σ_SB is the Stefan-Boltzman constant, and h_c is the convective coefficient. The surface emissivity is equal to one minus the surface reflectivity, and is the fraction of the incident radiation that enters the bulk of the sample. At the center of the sphere, the temperature must be physically attainable; the values must be finite and the field must be smooth. Therefore,

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