# Radiative heat transfer between two areas

Consider first a surface of area A, temperature T , and total emissivity ε. The rate of radiative energy emission from this surface per unit area is $E = \varepsilon \sigma {T^4}.$ Let the surroundings in all directions be at temperature T. If the surroundings are far from surface A or are black, then little of the energy emitted from A will be reflected from the surroundings, and the rate of energy incident on A per unit area from the surroundings is denoted by the symbol G, and is given by

$G = \sigma T_\infty ^4\qquad \qquad(1)$

The net radiative heat flux on surface A is the difference between the emitted and absorbed radiative fluxes, or

$q'' = E - \alpha G = \varepsilon \sigma {T^4} - \alpha \sigma T_\infty ^4\qquad \qquad(2)$

This can be simplified if some further approximations or assumptions are made. First, if the surface is gray, then Kirchhoff's Law gives $\alpha = \varepsilon .$ If TT, comparing the following two equations:

$\varepsilon (T) = \frac{{E(T)}}{{\sigma {T^4}}} = \frac{{\int_{\lambda = 0}^\infty {{E_\lambda }(\lambda ,T)d\lambda } }}{{\sigma {T^4}}} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }(\lambda ,T){E_{\lambda b}}(\lambda ,T)d\lambda } }}{{\sigma {T^4}}}$

$\alpha (T) = \frac{{\int_{\lambda = 0}^\infty {{\alpha _\lambda }} {G_\lambda }d\lambda }}{G} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }} {G_\lambda }d\lambda }}{G}$

from Opaque Surface Property Definitions also indicates that $\alpha \approx \varepsilon .$ In either of these two cases, eq. (2) reduces to

$q'' = \varepsilon \sigma \left( {{T^4} - T_\infty ^4} \right)\qquad \qquad(3)$

This is an extremely useful relation, but it involves some implicit assumptions in addition to those explicitly mentioned above. For example, surface A and the surroundings are assumed to each be isothermal, and the value of G does not vary across A so that the local heat flux at every point on A is the same.

Equation (3) is often used as part of a boundary condition for use in mixed mode problems, and it is clearly a simple way for handling radiation when the conditions and assumptions are satisfied. A further simplification of eq. (3) can be made by invoking the same approximation used in the following equations (see solar thermal energy collectors):

$q = - \bar hA\left( {\overline {{T_s}} - {T_\infty }} \right) - \varepsilon \sigma A\left( {\overline {T_s^4} - T_{sky}^4} \right)$

$\left( {{T^4} - T_\infty ^4} \right) = \left( {{T^2} + T_\infty ^2} \right)\left( {{T^2} - T_\infty ^2} \right) = \left( {{T^2} + T_\infty ^2} \right)\left( {{T^{}} + T_\infty ^{}} \right)\left( {{T^{}} - T_\infty ^{}} \right)\qquad \qquad(4)$
Comparison of heat fluxes

If the absolute temperatures T and T are not too far apart, then each can be replaced in the first two parentheses by their average, $\bar T = \frac{{T + {T_\infty }}}{2}$, resulting in

$\begin{array}{l} \left( {{T^4} - T_\infty ^4} \right) = \left( {{T^2} + T_\infty ^2} \right)\left( {T + T_\infty ^{}} \right)\left( {T - T_\infty ^{}} \right) \\ = 2{{\bar T}^2} \times 2\bar T\left( {T - T_\infty ^{}} \right) = 4{{\bar T}^3}\left( {T - T_\infty ^{}} \right) \\ \end{array}\qquad \qquad(5)$

and eq. (3) becomes

$q'' = 4\varepsilon \sigma {\bar T^3}\left( {T - {T_\infty }} \right) = {\bar h_{rad}}\left( {T - {T_\infty }} \right)\qquad \qquad(6)$

where ${\bar h_{rad}} = 4\varepsilon \sigma {\bar T^3}$ is the radiation heat transfer coefficient. The linearized form of eq. (6) makes possible the simple addition of radiation and convection terms in some combined-mode problems, if the loss in accuracy caused by the many assumptions and simplifications is justified. Note also that both the surface and surrounding temperatures must be known in order to determine $\bar T$; otherwise, an iterative solution must be invoked.

When T= 27oC, and vary T from 300 to 1000K. The error in q’’ obtained by eq. (6) as compared with that predicted by eq. (3). is shown in the figure on the right. The error is less than 6 percent up to surface temperatures of 500 K, which is within the range of many combined mode problems. It should be noted that the result is independent of the value of emissivity that is chosen.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.