# Surfaces with Varying Temperature, Radiative Flux, or Properties

Figure 1: Enclosure geometry for surfaces with varying properties and radiation variables.

One of the stipulations for use of the configuration factor is that the radiosity J must not vary across the surface described by the configuration factor. In many cases of finite surfaces, this requirement is not met; in particular, near the intersection of two surfaces, the irradiation G is likely to vary considerably, so that even if ε and T are constant across the surface, the radiosity $J = \varepsilon \sigma {T^4} + \rho G$ will vary.

In such a case, a surface must be divided into ever smaller increments so that each increment will meet the requirement of uniform radiosity as shown in Fig. 1. In the limit, the net radiation equations must be rewritten for a differential element dAk on surface k at location $({{\mathbf{r}}_k})$. The following equations

${J_k} = {\varepsilon _k}{E_{b,k}} + {\rho _k}{G_k} = {\varepsilon _k}\sigma T_k^4 + \left( {1 - {\varepsilon _k}} \right){G_k}$
q''k = JkGk

from Gray Surfaces for that element k are

${J_k}\left( {{{\mathbf{r}}_k}} \right) = {\varepsilon _k}\left( {{{\mathbf{r}}_k}} \right)\sigma T_k^4\left( {{{\mathbf{r}}_k}} \right) + \left( {1 - {\varepsilon _k}\left( {{{\mathbf{r}}_k}} \right)} \right){G_k}\left( {{{\mathbf{r}}_k}} \right)\qquad \qquad(1)$
${q''_k}({{\mathbf{r}}_k}) = {J_k}({{\mathbf{r}}_k}) - {G_k}({{\mathbf{r}}_k})\qquad \qquad(2)$

Because the radiosity may vary over each surface, the irradiation arriving at surface k from a given surface j is now in the form of an integral,

${G_{j \to k}}({{\mathbf{r}}_k}) = \int_{{A_j}} {{J_j}({{\mathbf{r}}_j})d{F_{dk - dj}}({{\mathbf{r}}_k},{{\mathbf{r}}_j})} \qquad \qquad(3)$

and Equation ${G_k} = \sum\limits_{j = 1}^N {{J_j}} {F_{k - j}}$ from Gray Surfaces must now sum the irradiation coming from each finite surface j in the enclosure to become

${G_k}({{\mathbf{r}}_k}) = \sum\limits_{j = 1}^N {\int_{{A_j}} {{J_j}({{\mathbf{r}}_j})d{F_{dk - dj}}({{\mathbf{r}}_k},{{\mathbf{r}}_j})} } \qquad \qquad(4)$

The result is that we now must solve 3N simultaneous integral equations rather than 3N simultaneous algebraic equations as for the uniform-radiosity case. The problem can again be reduced to eliminate the irradiation ${G_k}({{\mathbf{r}}_k})$ to give

${J_k}\left( {{{\mathbf{r}}_k}} \right) = \sigma T_k^4\left( {{{\mathbf{r}}_k}} \right) - \frac{{\left[ {1 - {\varepsilon _k}\left( {{{\mathbf{r}}_k}} \right)} \right]}}{{{\varepsilon _k}\left( {{{\mathbf{r}}_k}} \right)}}{q''_k}\left( {{{\mathbf{r}}_k}} \right)\qquad \qquad(5)$

And

${J_k}\left( {{{\mathbf{r}}_k}} \right) = {q''_k}\left( {{{\mathbf{r}}_k}} \right) + \sum\limits_{j = 1}^N {\int_{{A_j}} {{J_j}\left( {{{\mathbf{r}}_j}} \right)} } d{F_{k - j}}\left( {{{\mathbf{r}}_k},{{\mathbf{r}}_j}} \right)\qquad \qquad(6)$

Finally, eliminating the radiosity from eqs. (5) and (6) gives the set

$\begin{array}{l} \sum\limits_{j = 1}^N {\sigma \int\limits_{{A_j}} {\left[ {T_k^4\left( {{{\mathbf{r}}_k}} \right) - T_j^4\left( {{{\mathbf{r}}_j}} \right)} \right]} } d{F_{_{dk - dj}}}\left( {{{\mathbf{r}}_k},{{\mathbf{r}}_j}} \right) \\ {\rm{ }} = \frac{{{{q''}_k}\left( {{{\mathbf{r}}_k}} \right)}}{{{\varepsilon _k}\left( {{{\mathbf{r}}_k}} \right)}} - \sum\limits_{j = 1}^N {\int_{{A_j}} {\frac{{1 - {\varepsilon _j}\left( {{{\mathbf{r}}_j}} \right)}}{{{\varepsilon _j}\left( {{{\mathbf{r}}_j}} \right)}}{{q''}_j}\left( {{{\mathbf{r}}_j}} \right)d{F_{_{dk - dj}}}\left( {{{\mathbf{r}}_k},{{\mathbf{r}}_j}} \right)} } \\ \end{array}\qquad \qquad(7)$

## References

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

Howell, J.R., 2003, A Catalog of Radiation Configuration Factors, 2nd ed., http://www.me.utexas.edu/~howell/