Table 13.2

Table 13.2 View factors for three-dimensional geometries.

Geometry	Relation
Aligned Parallel Rectangles	$\bar{X} = X / L, \bar{Y} = Y / L$ $\begin{matrix} F_{i j} & = & \frac{2}{π \bar{X} \bar{Y}} {\ln {[\frac{(1 + {\bar{X}}^{2}) (1 + {\bar{Y}}^{2})}{1 + {\bar{X}}^{2} + {\bar{Y}}^{2}}]}^{1 / 2} \\ + \bar{X} {(1 + {\bar{Y}}^{2})}^{1 / 2} \tan^{- 1} \frac{\bar{X}}{{(1 + {\bar{Y}}^{2})}^{1 / 2}} \\ + \bar{Y} {(1 + {\bar{X}}^{2})}^{1 / 2} \tan^{- 1} \frac{\bar{Y}}{{(1 + {\bar{X}}^{2})}^{1 / 2}} - \bar{X} \tan^{- 1} \bar{X} - \bar{Y} \tan^{- 1} \bar{Y}} \end{matrix}$
Coaxial Parallel Disks	$\begin{matrix} R_{i} & = & r_{i} / L, R_{j} = r_{j} / L \\ S & = & 1 + \frac{1 + R_{j}^{2}}{R_{i}^{2}} \\ F_{i j} & = & \frac{1}{2} {S - {[S^{2} - 4 {(r_{j} / r_{i})}^{2}]}^{1 / 2}} \end{matrix}$
Perpendicular Rectangles with a Common Edge	$\begin{matrix} H = & Z / X, W = Y / X \\ F_{i j} = & \frac{1}{π W} (W \tan^{- 1} \frac{1}{W} + H \tan^{- 1} \frac{1}{H} \\ - {(H^{2} + W^{2})}^{1 / 2} \tan^{- 1} \frac{1}{{(H^{2} + W^{2})}^{1 / 2}} \\ + \frac{1}{4} \ln {\frac{(1 + W^{2}) (1 + H^{2})}{1 + W^{2} + H^{2}} {[\frac{W^{2} (1 + W^{2} + H^{2})}{(1 + W^{2}) (W^{2} + H^{2})}]}^{W^{2}} \\ \times {[\frac{H^{2} (1 + H^{2} + W^{2})}{(1 + H^{2}) (H^{2} + W^{2})}]}^{H^{2}}}) \end{matrix}$