Results

In this section, we will discuss the results of each case and calculate the error by comparing the values obtained with the exact values for each pair of faces in the geometry. Therefore, we will apply heat transfer radiation analysis in all these cases

1. first test

The dimensions of this geometry and explanation are provided in the following section: Case : Triangular Cavity After compiling and executing the code, the results are stored in this matrix:

\[\begin{pmatrix} 0 & 0.250018 & 0.748253 \\ 0.333427 & 0 & 0.665082 \\ 0.599727 & 0.399716 & 0 \end{pmatrix}\]

The exact view factor values are given by this matrix:

\[\begin{pmatrix} 0 & 1/4 & 3/4 \\ 1/3 & 0 & 2/3 \\ 3/5 & 2/5 & 0 \end{pmatrix}\]

The errors between the two results are presented in this table:

Face Pair	Exact Value	Approximated Value	Error
$F_{11}$	0	0	0
$F_{12}$	$\frac{1}{4}$	0.250018	$1.8 \times 10^{-5}$
$F_{13}$	$\frac{3}{4}$	0.748253	$1.747 \times 10^{-3}$
$F_{21}$	$\frac{1}{3}$	0.333427	$9.4 \times 10^{-5}$
$F_{22}$	0	0	0
$F_{23}$	$\frac{2}{3}$	0.665082	$1.585 \times 10^{-3}$
$F_{31}$	$\frac{3}{5}$	0.599727	$2.73 \times 10^{-4}$
$F_{32}$	$\frac{2}{5}$	0.399716	$2.84 \times 10^{-4}$
$F_{33}$	0	0	0

2. second test

The dimensions of this geometry and explanation are provided in the following section: Case : Rectangular Cavity After compiling and executing the code, the results are stored in this matrix:

\[\begin{pmatrix} 0 & 0.292696 & 0.413725 & 0.292696 \\ 0.292696 & 0 & 0.292696 & 0.413725 \\ 0.413725 & 0.292696 & 0 & 0.292696 \\ 0.292696 & 0.413725 & 0.292696 & 0 \end{pmatrix}\]

The exact view factor values are given by this matrix:

\begin{pmatrix} 0 & 1 - \frac{\sqrt{2}}{2} & \sqrt{2} - 1 & 1 - \frac{\sqrt{2}}{2} \\ 1 - \frac{\sqrt{2}}{2} & 0 & 1 - \frac{\sqrt{2}}{2} & \sqrt{2} - 1 \\ \sqrt{2} - 1 & 1 - \frac{\sqrt{2}}{2} & 0 & 1 - \frac{\sqrt{2}}{2} \\ 1 - \frac{\sqrt{2}}{2} & \sqrt{2} - 1 & 1 - \frac{\sqrt{2}}{2} & 0 \end{pmatrix}

The errors between the two results are presented in this table:

Face Pair

Exact Value

Approximated Value

Error

$F_{11}$

$F_{12}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{13}$

$\sqrt{2} - 1$

0.413725

$4.89 \times 10^{-4}$

$F_{14}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{21}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{22}$

$F_{23}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{24}$

$\sqrt{2} - 1$

0.413725

$4.89 \times 10^{-4}$

$F_{31}$

$\sqrt{2} - 1$

0.413725

$4.89 \times 10^{-4}$

$F_{32}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{33}$

$F_{34}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{41}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{42}$

$\sqrt{2} - 1$

0.413725

$4.89 \times 10^{-4}$

$F_{43}$

$1 - \frac{\sqrt{2}}{2}$

0.292696

$1.97 \times 10^{-4}$

$F_{44}$

3. Third test

The dimensions of this geometry and explanation are provided in the following section: Case : Cylindrical Cavity After compiling and executing the code, the results are stored in this matrix:

\begin{pmatrix} 0 & 0.0548874 & 0.945113 \\ 0.0412309 & 0 & 0.944355 \\ 0.0886256 & 0.117885 & 0.764229 \end{pmatrix}

The exact view factor values are given by this matrix:

\begin{pmatrix} 0 & 9 - \frac{\sqrt{320}}{2} & \sqrt{320} - 8 \\ 9 - \frac{\sqrt{320}}{2} & 0 & \frac{\sqrt{320}}{2} - 8 \\ \frac{1}{10} \left( \frac{\sqrt{320}}{2} - 8 \right) & \frac{3}{25} \left( \frac{\sqrt{320}}{2} - 8 \right) & 1 - \frac{11}{50} \left( \frac{\sqrt{320}}{2} - 8 \right) \end{pmatrix}

The errors between the two results are presented in this table:

Face Pair

Exact Value

Approximated Value

Error

$F_{11}$

$F_{12}$

$9 - \frac{\sqrt{320}}{2} \approx 0.0558$

0.0548874

$9.125 \times 10^{-4}$

$F_{13}$

$\sqrt{320} - 8 \approx 9.8882$

0.945113

$8.943087 \times 10^{0}$

$F_{21}$

$9 - \frac{\sqrt{320}}{2} \approx 0.0558$

0.0412309

$1.45691 \times 10^{-2}$

$F_{22}$

$F_{23}$

$\frac{\sqrt{320}}{2} - 8 \approx 0.9442$

0.944355

$1.55 \times 10^{-4}$

$F_{31}$

$\frac{1}{10} \left( \frac{\sqrt{320}}{2} - 8 \right) \approx 0.09442$

0.0886256

$5.7944 \times 10^{-3}$

$F_{32}$

$\frac{3}{25} \left( \frac{\sqrt{320}}{2} - 8 \right) \approx 0.113304$

0.117885

$4.581 \times 10^{-3}$

$F_{33}$

$1 - \frac{11}{50} \left( \frac{\sqrt{320}}{2} - 8 \right) \approx 0.792764$

0.764229

$2.8535 \times 10^{-2}$