The following polynomials are used for different ranges of thickness-to-diameter or thickness-to-length ratios. Coefficients are fit to published view factor data for furnace openings.

Polynomial Coefficients for View Factor Cases: Each polynomial is of the form where (TR) is the thickness-to-dimension ratio for the opening:
\begin{equation}\label{eq:opening-view-factor-2} F_\text{Case#} = \frac{C_0 + C_1 TR + C_2 TR^2 + C_3 TR^3 + C_4 TR^4 + C_5 TR^5}{100}\end{equation}
The following table lists the coefficients (C₀–C₅) for each polynomial used to approximate the radiative view factor for various opening geometries in the code:

Case	C₀	C₁	C₂	C₃	C₄	C₅
1	1.1	92.8571	-57.5893	15.625	0	0
2	29.5	26.8417	-4.35417	-8.33e-2	.104167e-1	-8.33e-3
3	3.5	89.5833	-50.0	10.4167	0	0
4	24.0	39.3917	-11.6042	1.85417	-.145833	4.17e-3
5	2.7	112.679	-70.9821	15.625	0	0
6	35.5	29.4583	-4.52083	-6.875e-1	.270833	-.208333e-2
7	13.0	123.75	-100.0	31.25	0	0
8	27.0	64.5667	-29.9167	7.14583	-8.33e-1	3.75e-2

Circular Openings

\begin{equation}\label{eq:opening-vf-case1} F_1(TR) = \frac{1.10 + 92.86TR - 57.59TR^2 + 15.62TR^3}{100}\end{equation}

\begin{equation}\label{eq:opening-vf-case2} F_2(TR) = \frac{29.50 + 26.84TR - 4.35TR^2 - 0.083TR^3 + 0.104TR^4 - 0.0083TR^5}{100}\end{equation}

Circular Opening View Factor Case Table: The following table shows which polynomial formula is used for each thickness ratio (TR) for circular openings:


TR ≤ 0.1	\(TR = 0.1\) \(F_1 * \frac{TR}{0.1}\)
0.1 < TR ≤ 6	\(F_1\)
6 < TR	\(TR = 6\) \(F_2\)

Where:

TR = thickness ratio (diameter/thickness)
For TR < 0.1, use TR = 0.1
For 6 < TR, use TR = 6

Rectangular Openings

\begin{equation}\label{eq:opening-vf-case3} F_3(TR) = \frac{3.50 + 89.58TR - 50.00TR^2 + 10.42TR^3}{100}\end{equation}

\begin{equation}\label{eq:opening-vf-case4} F_4(TR) = \frac{24.00 + 39.39TR - 11.60TR^2 + 1.85TR^3 - 0.146TR^4 + 0.0042TR^5}{100}\end{equation}

\begin{equation}\label{eq:opening-vf-case5} F_5(TR) = \frac{2.70 + 112.68TR - 70.98TR^2 + 15.62TR^3}{100}\end{equation}

\begin{equation}\label{eq:opening-vf-case6} F_6(TR) = \frac{35.50 + 29.46TR - 4.52TR^2 - 0.687TR^3 + 0.271TR^4 - 0.021TR^5}{100}\end{equation}

\begin{equation}\label{eq:opening-vf-case7} F_7(TR) = \frac{13.00 + 123.75TR - 100.00TR^2 + 31.25TR^3}{100}\end{equation}

\begin{equation}\label{eq:opening-vf-case8} F_8(TR) = \frac{27.00 + 64.57TR - 29.92TR^2 + 7.15TR^3 - 0.833TR^4 + 0.0375TR^5}{100}\end{equation}

Rectangular Opening View Factor Case Table: The following table shows which polynomial formula is used for each combination of thickness ratio (TR) and lateral dimension ratio (LDR) for rectangular openings:

	1 ≤ LDR ≤ 2	2 < LDR ≤ 9	9 < LDR
TR ≤ 0.1	\(TR = 0.1\) \(F_3 + (F_5 - F_3) * (LDR - 1)\)	\(TR = 0.1\) \(F_5 + (F_7 - F_5) * \frac{(LDR - 2)}{8}\)	\(F_7(TR=0.1)*(\frac{TR}{0.1})\)
0.1 < TR ≤ 0.9	\(F_3 + (F_5 - F_3) * (LDR - 1)\)	\(F_5 + (F_7 - F_5) * \frac{(LDR - 2)}{8}\)	\(F_7\)
0.9 < TR ≤ 6	\(F_4 + (F_6 - F_4) * (LDR - 1)\)	\(F_6 + (F_8 - F_6) * \frac{(LDR - 2)}{8}\)	\(F_8\)
6 < TR	\(TR = 6\) \(F_4 + (F_6 - F_4) * (LDR - 1)\)	\(TR = 6\) \(F_6 + (F_8 - F_6) * \frac{(LDR - 2)}{8}\)	\(TR = 6\) \(F_8\)

Where:

TR = thickness ratio (length/thickness or height/thickness)
LDR = lateral dimension ratio (length/height or height/length)
For TR < 0.1, use TR = 0.1
For 6 < TR, use TR = 6