Consider a cylindrical enclosure with \(A_{1}, A_{2}\), and \(A_{3}\) representing the internal base, top, and side surfaces, respectively. Using the length to diameter ratio, \(K=L / D\), determine (a) the expression for the view factor from the side surface to itself \(F_{33}\) in terms of \(K\) and \((b)\) the value of the view factor \(F_{33}\) for \(L=D\).

The view factor between two surfaces in the enclosure can be understood as the fraction of the total emitting or receiving power at one surface that reaches or leaves the other surface directly. In our case, we need to determine the view factor from the side surface to itself, denoted by \(F_{33}\).

First, let's write down the view factor relations for the case of the cylindrical enclosure with three surfaces. The relations are given by: $$F_{11}+F_{12}+F_{13}=1$$ $$F_{21}+F_{22}+F_{23}=1$$ $$F_{31}+F_{32}+F_{33}=1$$

We also know that a view factor obeys the reciprocity theorem, which can be stated as: $$A_{1}F_{12} = A_{2}F_{21}$$ $$A_{1}F_{13} = A_{3}F_{31}$$ $$A_{2}F_{23} = A_{3}F_{32}$$ Now, let's use these relations in our problem.

Given that \(K=L / D\), we can use this ratio to express the areas of the cylindrical enclosure as: $$A_1 = A_2 = \frac{\pi D^2}{4}$$ $$A_3 = \pi D L = \pi D^2 K$$ Now we need to find the value of \(F_{33}\). We can use the relation for \(F_{31}\): $$F_{31} = \frac{A_3}{A_1}F_{13}$$ Since \(F_{33}+F_{32}+F_{31}=1\), we can write the expression for the view factor \(F_{33}\) as: $$F_{33} = 1-F_{31}-F_{32}=1-\frac{A_3}{A_1}F_{13}-\frac{A_2}{A_3}F_{23}$$ From the reciprocity theorem, we can further simplify the expression, because \(F_{13}=\frac{A_3}{A_1}F_{31}\) and \(F_{23}=\frac{A_3}{A_2}F_{32}\): $$F_{33} = 1-\frac{A_3}{A_1}F_{31}-\frac{A_3}{A_1}(1-F_{31}-F_{33})$$ $$F_{33}\left(\frac{2A_3}{A_1}+1\right) = 1-\frac{A_3}{A_1}$$ Now we can express the view factor \(F_{33}\) in terms of \(K\): $$F_{33} = \frac{1-K}{1+2K}$$

Now, we need to find the value of \(F_{33}\) when the length equals the diameter, or \(K=1\). Plug \(K=1\) into the expression for \(F_{33}\): $$F_{33} = \frac{1-1}{1+2(1)} = \frac{0}{3} = 0$$ Thus, the value of the view factor \(F_{33}\) for \(L = D\) is 0.