Chapter 13, Problem 13.1P

Determine F₁₂ and F₂₁ for the following configurations using the reciprocity theorem and other basic shape factor relations. Do not use tables or charts.

1. Long duct

Small sphere of area A₁ under a concentric hemisphere of area $A_2=2A_1$

Long duct. What is F₂₂ for this case?

Long inclined plates (point B is directly above the center of A₁)

Sphere lying on infinite plane

Hemisphere−disk arrangement

Long, open channel

Long concentric cylinders

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 1 and 0.212.

Explanation of Solution

Formula used:

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{12}=\frac{4}{6\pi }{F}_{21}$

Here, $F_{12}$ is the shape factors of surface 2 with respect to surface 1 and $F_{21}$ , is the shape factors of surface 1 with respect to surface 2

Calculation:

The geometrical shape is given as,

  

Figure (1)

The above geometry shows that inner surface completely intercepted by the outer surface. Therefore, the value of shape factors can be calculated as,

  $\begin{array}{l}{F}_{12}=1\\ F_{12}=\frac{4}{6\pi }{F}_{21}\\ F_{12}=\frac{4}{6\pi }\times 1\\ F_{12}=0.212\end{array}$

Conclusion:

Therefore, the shape factors are 1 and 0.212.

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 0.5 and 0.25.

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}+F_{13}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\frac{1}{2}\times F_{12}$

Calculation:

The geometrical shape is given as,

  

Figure (2)

The above geometry is a convex geometry, therefore the values of shape factor are obtained as,

  $\begin{array}{l}{F}_{11}=0\\ F_{12}=F_{13}\end{array}$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{c}{F}_{11}+F_{12}+F_{13}=1\\ 0+F_{12}+F_{13}=1\\ F_{12}+F_{12}=1\\ F_{12}=0.5\end{array}$

Further obtaining the values as,

  $\begin{array}{l}{F}_{12}=0.5\\ F_{13}=0.5\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $\begin{array}{l}{F}_{21}=\frac{1}{2}\times F_{12}\\ F_{21}=\frac{1}{2}\times 0.5\\ F_{21}=0.25\end{array}$

Conclusion:

Therefore, the shape factors are 0.5 and 0.25.

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 1, 0.637 and 0.363.

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\frac{2}{\pi }\times F_{12}$

The summation rule is given as,

  $F_{21}+F_{22}=1$

Calculation:

The geometrical shape is given as,

  

Figure (3)

The above geometry have a flat geometry at bottom, therefore the values of shape factor are obtained as,

  $F_{11}=0$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{l}{F}_{11}+F_{12}=1\\ 0+F_{12}=1\\ F_{12}=1\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,

  $\begin{array}{l}{F}_{21}=\frac{2}{\pi }\times F_{12}\\ F_{21}=\frac{2}{\pi }\times 1\\ F_{21}=0.637\end{array}$

Further obtain the values of shape factor as,

  $\begin{array}{c}{F}_{21}+F_{22}=1\\ 0.637+F_{22}=1\\ F_{22}=0.363\end{array}$

Conclusion:

Therefore, the shape factors are 1, 0.637 and 0.363

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 0.5 and 0.707.

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}+F_{13}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\sqrt{2}\times F_{12}$

Calculation:

The geometrical shape is given as,

  

Figure (4)

The above geometry have a flat geometry at bottom, therefore the values of shape factor are obtained as,

  $\begin{array}{l}{F}_{11}=0\\ F_{12}=F_{13}\end{array}$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{c}{F}_{11}+F_{12}+F_{13}=1\\ 0+F_{12}+F_{12}=1\\ F_{12}=0.5\\ F_{13}=0.5\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,

  $\begin{array}{l}{F}_{21}=\sqrt{2}\times F_{12}\\ F_{21}=\sqrt{2}\times 0.5\\ F_{21}=0.707\end{array}$

Conclusion:

Therefore, the shape factors are 0.5 and 0.707.

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 0.5 and 0.

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}+F_{13}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\frac{A_1}{\infty }\times F_{12}$

Calculation:

The geometrical shape is given as,

  

Figure (5)

The above geometry have a flat geometry at bottom, therefore the values of shape factor are obtained as,

  $\begin{array}{l}{F}_{11}=0\\ F_{12}=F_{13}\end{array}$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{c}{F}_{11}+F_{12}+F_{13}=1\\ 0+F_{12}+F_{12}=1\\ F_{12}=0.5\\ F_{13}=0.5\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,

  $\begin{array}{l}{F}_{21}=\frac{A_1}{\infty }\times F_{12}\\ F_{21}=\frac{A_1}{\infty }\times 0.5\\ F_{21}=0\end{array}$

Conclusion:

Therefore, the shape factors are 0.5 and 0.

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 1 and 0.125.

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}+F_{13}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\frac{1}{4}\times F_{12}$

Calculation:

The geometrical shape is given as,

  

Figure (6)

The above geometry have flat disk surface, therefore the values of shape factor are obtained as,

  $\begin{array}{l}{F}_{11}=0\\ F_{33}=0\\ F_{13}=0\\ F_{31}=0\end{array}$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{c}{F}_{11}+F_{12}+F_{13}=1\\ 0+F_{12}+0=1\\ F_{12}=1\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,

  $\begin{array}{l}{F}_{21}=\frac{1}{4}\times F_{12}\\ F_{21}=\frac{1}{4}\times 1\\ F_{21}=0.25\end{array}$

Conclusion:

Therefore, the shape factors are 1 and 0.25.

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 0.5 and 0.637.

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}+F_{13}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\frac{4}{\pi }\times F_{12}$

Calculation:

The geometrical shape is given as,

  

Figure (7)

The above geometry have flat disk surface, therefore the values of shape factor are obtained as,

  $\begin{array}{l}{F}_{11}=0\\ F_{12}=F_{13}\end{array}$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{c}{F}_{11}+F_{12}+F_{13}=1\\ 0+F_{12}+F_{13}=1\\ F_{12}=0.5\\ F_{13}=0.5\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,

  $\begin{array}{l}{F}_{21}=\frac{4}{\pi }\times F_{12}\\ F_{21}=\frac{4}{\pi }\times 0.5\\ F_{21}=0.637\end{array}$

Conclusion:

Therefore, the shape factors are 0.5 and 0.637.

To determine

The shape factors.

Answer to Problem 13.1P

The shape factors are 1 and $\frac{D_1}{D_2}$ .

Explanation of Solution

Formula used:

The summation rule is given as,

  $F_{11}+F_{12}=1$

The expression for the shape factor for surface 2 with respect to surface 1 is given as,

  $F_{21}=\frac{D_1}{D_2}\times F_{12}$

Calculation:

The geometrical shape is given as,

  

Figure (8)

The above geometry have spherical surface, therefore the values of shape factor are obtained as,

  $F_{11}=0$

Further obtaining the values of shape factor by summation rule as,

  $\begin{array}{c}{F}_{11}+F_{12}=1\\ 0+F_{12}=1\\ F_{12}=1\end{array}$

The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,

  $\begin{array}{l}{F}_{21}=\frac{D_1}{D_2}\times F_{12}\\ F_{21}=\frac{D_1}{D_2}\times 1\\ F_{21}=\frac{D_1}{D_2}\end{array}$

Conclusion:

Therefore, the shape factors are 1 and $\frac{D_1}{D_2}$ .