Heat flux from radiation

Heat flow from a body surrounded by a medium at a temperature $T_\infty$ is governed by the Stefan-Boltzmann Law

$$q(T, T_\infty) = \varepsilon \sigma (T^4 - T_\infty^4)$$ (2.27)

where $\varepsilon\in\langle 0, 1 \rangle$ represents emissivity between the surface and the boundary at temperature $T_\infty$. $\sigma=5.67\cdot 10^{-8}$ W/m$^{-2}$K$^{-4}$ stands for a Stefan-Boltzmann constant. Transport elements in OOFEM implement Eq. (2.27) and require non-linear solver.

Alternatively (not implemented), a linearization using Taylor expansion around $T_\infty$ and neglecting higher-order terms results to

$$q(T, T_\infty) \approx q(T=T_\infty) + \frac{\partial q(T,T_\infty)}{\partial T_\infty} (T_\infty-T) = 4\varepsilon \sigma T_\infty^3 (T-T_\infty)$$ (2.28)

leading to so-called radiation heat transfer coefficient $\alpha_{rad}=4\varepsilon \sigma T_\infty^3$. The latter resembles similar coefficient as in convective heat transfer. Other methods for Eq. (2.27) could be based on Oseen or Newton-Kantorovich linearization. Also, radiative heat transfer coefficient $\alpha_{rad}$ can be expressed as [#!Baehr:06!#, pp.28]

$$q(T, T_\infty) = \varepsilon \sigma \frac{T^4 - T_\infty^4}{T-T_\... ...ace{\varepsilon \sigma (T^2+T_\infty^2)(T+T_\infty)}_{\alpha_{rad}}(T-T_\infty)$$ (2.29)