The view factors F₁₂ and F_21, for the sphere of diameter d and a cubical box of length l = d as shown in the figure, respectively, are

Concept:

Shape Factor is defined as the fraction of radiation energy leaving a surface that reaches another surface, for example, F12 means the fraction of energy leaving surface 1 reaches surface 2.

Summation Rule:

F11 + F12 = 1

Reciprocity theorem:

A1 F12 = A2 F21 (A is the surface area)

Calculation:

Given:

side of the cube = a = l = d, diameter of sphere = d.

Since the sphere is completely enclosed by the cube, therefore,

Area of the cube of side a = 6a2

Area of sphere of diameter d = 4π(d/2)²

F₁₂ = 1

By Reciprocal theorem:

A₁ F₁₂ = A₂ F₂₁

\(\begin{array}{l} 4\;π {\left( {\frac{d}{2}} \right)^2} \times 1 = 6 {d^2} \times{F_{21}}\\ {F_{21}} = \frac{π }{6} \end{array}\)