Radiative Transport Within an Ablating Body

Abstract

The emission and reabsorption of thermal radiation within a semitransparent material provides a mechanism which supplements ordinary thermal conduction in transporting energy from hotter to colder regions. A method has been developed for the calculation of net radiative flux and temperature distribution within a semi-infinite body which emits, absorbs, and scatters this radiation and which allows some radiation to escape from its surface. This method has been applied to the problem of calculating temperature distributions within bodies in steady-state ablation. The bodies are characterized by their refractive index, surface reflectivity, absorption and scattering coefficient, ablation velocity, and surface temperature as well as by their heat capacity and thermal conductivity. Numerical results are presented for the temperature distribution with various values of these parameters. As a result of this analysis, simple formulas are presented for the temperature distribution very near the surface of an ablating body which are particularly useful in predicting the temperature distribution in the “liquid layer” of a glassy ablating body. These simple formulas are presented for two limiting cases: Case I, in which the liquid layer is so thin that it is almost completely transparent to radiation, and Case II, in which the radiation mean free path is so short that radiative transport can be completely neglected in determining the temperature distribution in the liquid layer. Also resulting from this numerical analysis is a simple relation between the ablation rate and the emissive power of a body in steady-state ablation. The concept of effective conductivity is extended to scattering media. It is noted that this concept fails whenever the temperature or optical properties of the medium change appreciably within one radiation mean free path. In particular, the optical properties change discontinuously at any boundary. Thus, in general, the effective conductivity concept fails near a boundary and results in completely wrong answers for the temperature distribution in the liquid layer.