Should heat conduction not always be accompanied by heat radiation?

Question

Consider two solid elements exchanging heat due to the temperature gradient between them by heat conduction. Such problems are always modelled with one mechanism only. Example: The transient behaviour of a pipe shell is to be modelled and the shell is discretised radially in n element. Element 1 is the most inner element. It "receives" heat from the flow through the pipe by convective transfer and it conducts heat towards the next-outer element. Consider element 2. Should the energy balance contain conduction with the neighbours 1 and 3 only (first row), or should the pairs 1-2 and 2-3 also be connected through thermal radiation (2nd row)?

Note: I am applying an energy balance for a closed system in the form of dU = sum(heatflows) in a transient way here. Caloric equation of state of dU.

If applying the same thought to convection I believe I understand the reason why it is not done: the convection coefficient is mostly an empiric value which probably includes all relevant means of heat exchange.

Answer 1

Radiation is generally ignored within opaque materials because there's no temperature difference over any transparent line of sight. That is, the net radiative transfer between adjacent atoms in a solid is generally assumed to be zero.

The outcome is consistent with finite element model behavior: Upon mesh refinement (i.e., decreasing element size), for an opaque material, the apparent internal radiation term disappears because $T^4_1−T^4_2\to 0$, for example. (This is not the case with conduction because the numerator and denominator in $\frac{T_1−T_2}{\Delta r}$ both continue to decrease, corresponding to a nonzero temperature gradient.)

Expanding now to include non-opaque materials, I disagree with the premise "Such problems [i.e., heat transfer in condensed matter] are always modelled with one mechanism [namely, conduction] only." A quick look at Google Scholar reveals the following papers in the first few hits:

• "Heat transfer in semitransparent solids"

• "Interaction of radiation and conduction in glass"

• "Prediction of coupled radiative and conductive heat transfer in concentric cylinders with nonlinear anisotropic scattering medium by spectral collocation method"

Thus, when the circumstances warrant, radiation within condensed matter is indeed incorporated into heat-transfer models that would otherwise contain only conduction.