Question

Consider an enclosure consisting of 13 surfaces. How many view factors does this geometry involve? How many of these view factors can be determined by the application of the reciprocity and the summation rules?

Short answer

Answer: For a geometry involving 13 surfaces, 78 view factors can be determined by applying reciprocity and summation rules.

Step-by-step solution

Calculate the Total Number of View Factors

To find the total number of view factors in a geometry involving 13 surfaces, we can use the formula \(N\times(N-1)\) where N is the number of surfaces. In our case, N = 13.

Plug in Values to Find the Total Number of View Factors

Applying the formula \(N\times(N-1)\), we get: \(13\times(13-1) = 13\times12 = 156\) So, the total number of view factors for this geometry is 156.

Calculate View Factors using Reciprocity Rule

The reciprocity rule states that \(A_iF_{ij} = A_jF_{ji}\). Since there are N surfaces, we need to apply the reciprocity rule \((N-1)\) times for each surface, which would result in \(N\times(N-1)\) factors. But, each factor appears twice, once for each surface, we get: \(\frac{N\times(N-1)}{2}\) view factors from the reciprocity rule.

Calculate View Factors using Summation Rule

The summation rule states that sum of view factors leaving a surface is equal to 1. Mathematically, \(\sum_{j=1}^N F_{ij} = 1\). Since there are N surfaces, we can find \((N-1)\) factors for each surface using the summation rule. Thus, there are a total of: \(N\times(N-1)\) view factors from the summation rule.

Combine Reciprocity and Summation Rule Results

The number of view factors determined by using both the reciprocity and summation rules can be found by adding the result of steps 3 and 4: \(\frac{N\times(N-1)}{2} + N\times(N-1)\)

Plug in Values to Find Total View Factors determined by Both Rules

Now, substituting the given value of N (13) in the equation, we get: \(\frac{13\times(13-1)}{2} + 13\times(13-1) = 78 + 156 = 234\) However, we should notice that we have used the summation rule to get each element of the row while we already have them from the reciprocity rule (except for the diagonal elements). So we counted them twice, and we need to subtract the double counted elements. We will subtract \(N(N-1)\) from 234: \(234 - 13\times(13-1) = 234 - 156 = 78\) So, the number of view factors that can be determined by applying reciprocity and summation rules is 78.