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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)?

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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.

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  • $\begingroup$ Could you add more details for what you mean by Such problems are always modelled with one mechanism only.? Otherwise, it is not obvious that radiation is excluded and that all the energy is passed via heat conduction.) $\endgroup$
    – Roger V.
    Dec 5, 2022 at 16:14
  • $\begingroup$ If a heat transfer problem was solved without considering radiation, either the solver doesn't know about that mode or decided (correctly or incorrectly) that it's irrelevant. We have no idea which is the case, because no examples are provided. The question is currently too vague to be answerable except to say: Yes, materials radiate. $\endgroup$ Dec 5, 2022 at 17:41
  • $\begingroup$ Thank you for providing an example. Is the pipe material opaque? I’m checking if a transparent line of sight exists between elements 1 and 3, which is necessary for radiative heat transfer. $\endgroup$ Dec 5, 2022 at 19:50
  • $\begingroup$ We are getting to the core of my question! It is not opaque. However, an infinitesimal small element should radiate heat, and the neighbouring element should absorb it as a grey/black body, shouldnt it? There must not be "empty" (not really empty, filled with gas e.g.) space inbetween the radiating bodies, right? $\endgroup$
    – Felix
    Dec 5, 2022 at 20:38
  • $\begingroup$ As you refine the mesh (i.e., decrease the element size), for an opaque material, apparent internal radiation disappears because $T_1^4-T_2^4\to 0$, for example. In other words, the net radiative transfer between adjacent atoms is generally assumed to be zero. (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.) But you say the pipe is not opaque? What pipe material are you considering? $\endgroup$ Dec 5, 2022 at 21:05

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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:

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

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