In this publication the Radiative Transfer Equation (RTE) (eq. (7)) contains the following emission term:


where $n_i$ is the refractive index of component $i$, $\kappa_{d,i}$ is the discrete-scale absorption coefficient and $L_{b,i}(\mathbf{r})$ is the blackbody discrete-scale intensity emitted in direction $\hat{\mathbf{s}}$. The wavelength subscripts have been dropped for brevity.

My question is, how does the $n_i^2$ part appear in this term?

The referenced publications include three textbooks; I have checked the first two which include full derivations of the RTE and they both agree the emission term takes the form $\kappa L_{b}\left(\mathbf{r}\right)$; I do not have access to the other book.

I'm guessing the answer has something to do with the fact that the radiative properties and intensity are labelled as 'discrete scale'. But google (scholar and web) has yielded no results for various searches including the terms discrete, scale, refract, and radiation. The objective of the paper is to spatially average the RTE, and for the mass, momentum, and energy transport equations, the starting point has always been the general (plain) form of the equation with all terms included. Here, the author is starting from a slightly different form and I would like to understand why.

I've contacted the author about it and I am waiting on a response.

I'm hoping someone here can riddle out how the emission term could contain the refractive index. Keep in mind assumption (iii):

the refractive index is constant in each phase


1 Answer 1


I have discovered the source of the discrepancy with help from the researcher (thanks Dr. Lipinski!).

It seems that many sources (including Wikipedia, the undergraduate heat transfer texts by Incropera & DeWitt and Lienhard & Lienhard) mention only breifly, or not at all, the dependence of the speed of light in the medium on the refractive index, $n$, i.e.:


where $c_0$ is the speed of light in a vacuum. The above resources use $c$ in Plank's Law, and carry that through the typical derivations for blackbody emissive power, Wien's Displacement Law, Wien's distribution, and calculations of total blackbody emissive power. They are, for the most part, correct, but they omit any further discussion of the subtle dependence on $n$. I suspect this is because, for most undergraduate heat transfer courses, the goal of instructors is to impart an understanding of surface-to-surface radiation heat transfer, where the medium between the surfaces is assumed to be non-participating.

When the dependence on $n$ is carried through, the Stephan-Boltzmann law becomes:

$$E_b \left(T\right) = n^2\sigma T^4$$

This is carried through in the development of the emission term of the RTE.


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