# Configuration factor relations

 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}

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.