$3-1(40$ Points$)$: Radiation View Factors

Determine $F_{12}$ and $F_{21}$ for the following configurations

using the reciprocity theorem and other basic shape fac$-$

tor relations. Do not use tables or charts.

$($a$)$ Long duct

$($b$)$ Small sphere of area $A_1$ under a concentric

hemisphere of area $A_2=2A_1$

$($c$)$ Long duct. What is $F_{22}$ for this case?

$($d$)$ Long inclined plates $($point $B$ is directly above

the center of $A_1)$

$($e$)$ Sphere lying on infinite plane

$($f$)$ Hemisphere$-$disk arrangement

$($g$)$ Long, open channel

$($h$)$ Long concentric cylinders

$($b$)$

$($e$)$

$($c$)$

$($g$)$

$($d$)$