Shape factor algebra and concentric spheres

(a) Shape factor and algebra

The shape factor $F_{ij}$ is the fraction of radiation leaving surface $i$ that strikes surface $j$ directly. It depends only on geometry and orientation.

• Reciprocity: $A_iF_{ij} = A_jF_{ji}$.
• Summation rule: for an enclosure of $N$ surfaces, $\sum_{j=1}^{N}F_{ij} = 1$.
• Superposition: if surface $j$ is made of parts $j_1, j_2$, then $F_{i(j_1+j_2)} = F_{ij_1}+F_{ij_2}$ and $A_iF_{i\to j}$ adds similarly.
• For a plane or convex surface, $F_{ii} = 0$; for a concave surface $F_{ii} > 0$.

(b) Concentric spheres

      .------------.
    /    .------.    \
   |    |  1     |  2 |
   |    |  T1    |  T2|
    \    '------'    /
      '------------'

Areas: $A_1 = 4\pi(0.1)^2 = 0.1257$ m$^2$, $A_2 = 4\pi(0.2)^2 = 0.5027$ m$^2$.

Shape factors. All radiation from the inner (convex) sphere reaches the outer one, so $F_{12} = 1$ and $F_{11} = 0$. By reciprocity:

$$F_{21} = \frac{A_1}{A_2}F_{12} = \frac{0.1257}{0.5027} = 0.25$$

By the summation rule for surface 2: $F_{22} = 1 - F_{21} = 0.75$.

Heat exchange. For a small body enclosed by a large one:

$$Q_{12} = \frac{\sigma A_1\left(T_1^4 - T_2^4\right)}{\dfrac{1}{\varepsilon_1}+\dfrac{A_1}{A_2}\left(\dfrac{1}{\varepsilon_2}-1\right)}$$ $$\begin{aligned} \sigma A_1(T_1^4-T_2^4) &= 5.67\times10^{-8}(0.1257)(600^4 - 300^4) = 865.7\ \text{W}\\ \text{denominator} &= \frac{1}{0.6}+0.25\left(\frac{1}{0.4}-1\right) = 1.667 + 0.375 = 2.042\\ Q_{12} &= \frac{865.7}{2.042} = 424\ \text{W} \end{aligned}$$

Answer: $F_{12} = 1$, $F_{21} = 0.25$, $F_{22} = 0.75$; $Q_{12} \approx 424$ W from the inner to the outer sphere.