Configuration factor relations

From Thermal-FluidsPedia

Factor	Geometry	Relation
1) Differential strip dA1 of any length to parallel cylindrical surface A2 of infinite length.		${{F}_{d1-2}}=\frac{1}{2}\left( \sin {{\theta }_{2}}-\sin {{\theta }_{1}} \right)$
2) Plane element dA1 to plane parallel rectangle A2: normal to element passes through corner of rectangle		X=a/c Y=b/c $\begin{align} & {{F}_{d1-2}}=\frac{1}{2\pi }(\frac{X}{\sqrt{1+{{X}^{2}}}}{{\tan }^{-1}}\frac{X}{\sqrt{1+{{Y}^{2}}}} \\ & \text{ +}\frac{Y}{\sqrt{1+{{Y}^{2}}}}{{\tan }^{-1}}\frac{Y}{\sqrt{1 {{X}^{2}}}}) \\ \end{align}$
3) Infinitely long directly opposed parallel plates of equal width		H = h/W ${{F}_{1-2}}=\sqrt{1-{{H}^{2}}}-H$
4) Identical parallel directly opposed rectangles		X=a/c Y = b/c $\begin{align} & {{F}_{1-2}}=\frac{2}{\pi XY}\{\ln {{[\frac{(1+{{X}^{2}})(1+{{Y}^{2}})}{1+{{X}^{2}}+{{Y}^{2}}}]}^{1/2}} \\ & \text{ +}X\sqrt{1+{{Y}^{2}}}{{\tan }^{-1}}\frac{X}{\sqrt{1+{{Y}^{2}}}} \\ & \text{ +}Y\sqrt{1+{{X}^{2}}}{{\tan }^{-1}}\frac{Y}{\sqrt{1+{{X}^{2}}}} \\ & \text{ }-X{{\tan }^{-1}}X-Y{{\tan }^{-1}}Y \\ \end{align}$
5) Infinitely long plates having a common edge with angle of 90o		H = h/w ${{F}_{1-2}}=\frac{1}{2}(1+H-\sqrt{1+{{H}^{2}}}$
6) Infinitely long enclosure formed of three plane areas		${{F}_{1-2}}=\frac{{{A}_{1}}+{{A}_{2}}-{{A}_{3}}}{2{{A}_{1}}}$
7) Infinitely long plates of equal width with a common edge and included angle.		${{F}_{1-2}}=1-\sin \left( \alpha /2 \right)$
8) Parallel circular disks of unequal radius with common axis		$\begin{align} & {{R}_{1}}={{r}_{1}}/h\text{ }{{R}_{2}}={{r}_{2}}/h \\ & X=1+\frac{1+R_{2}^{2}}{R_{1}^{2}} \\ & {{F}_{1-2}}=\frac{1}{2}\left[ X-\sqrt{{{X}^{2}}-4{{\left( \frac{{{R}_{2}}}{{{R}_{1}}} \right)}^{2}}} \right] \\ \end{align}$
9) Infinitely long plate of finite width to parallel infinitely long cylinder		${{F}_{1-2}}=\frac{r}{b-a}\left( ta{{n}^{-1}}\frac{b}{c}-ta{{n}^{-1}}\frac{a}{c} \right)$
10) Infinitely long parallel cylinders of same diameter		$\begin{align} & X=1+s/2r \\ & {{F}_{1-2}}=\frac{1}{\pi }\left(\sqrt{{{X}^{2}}-1}+{{\sin }^{-1}}\frac{1}{X}-X \right) \\ \end{align}$
11) Concentric cylinders of infinite length		$\begin{align} & {{F}_{1-2}}=1 \\ & {{F}_{2-1}}=\left( {{r}_{1}}/{{r}_{2}} \right) \\ & {{F}_{2-2}}=1-\left( {{r}_{1}}/{{r}_{2}} \right) \\ \end{align}$
12) Sphere to disk; normal to disk center passes through sphere center		R2 = r2/h ${{F}_{1-2}}=\frac{1}{2}\left( 1-\frac{1}{\sqrt{1+R_{2}^{2}}} \right)$
13) Concentric spheres		$\begin{align} & {{F}_{1-2}}=1 \\ & {{F}_{2-1}}={{\left( \frac{{{r}_{1}}}{{{r}_{2}}} \right)}^{2}} \\ & {{F}_{2-2}}=1-{{\left( \frac{{{r}_{1}}}{{{r}_{2}}} \right)}^{2}} \\ \end{align}$