View Factor Calculator: Analytical and Monte Carlo Method

In this web page, radiation view factors (also know as configuration factors) for different geometrical configurations can be calculated by the analytical formula and Monte Carlo method. The Monte Carlo calculation uses WebGL 2.0. Depending on the browser and hardware configuration, this functionality might not work. The number of rays for the view factor calculation is 100,000. The seeds of the random number generator are fixed, so the results are reproducible. For details on the random number generation implementation in this tool, see: "MRG32k3a Random Number Generator on WebGL2".

Differential surface to circular disk

r ( r > 0 ) :
h ( h > 0 ) :
θ [°] ( 0 ≤ θ ≤ 180 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \frac{r^2}{r^2+h^2} \cos \theta, ~~ \mathrm{where}~~\theta \le \arctan \frac{h}{r} \\ F_{d1-2} = 0, ~~ \mathrm{where}~~\theta > \arctan \frac{r}{h} + \frac{\pi}{2} \end{gather}

Note: For detailed derivations and other configurations, see: "Disk View Factor from a Plate Element"

Differential surface to circular disk in parallel plane

r ( r > 0 ) :
h ( h > 0 ) :
a ( a ≥ 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \frac{1}{2} \left[ 1 - \frac{Z-2R^2}{\sqrt{Z^2 - 4R^2}} \right] \\ \mathrm{where}~~H = \frac{h}{a}, ~~ R = \frac{r}{a}, ~~ Z = 1 + H^2 + R^2 \\ \end{gather}

Note: For detailed derivations, see: "Disk View Factor from a Plate Element"

Differential surface to rectangular surface in parallel plane

a ( a > 0 ) :
b ( b > 0 ) :
c ( c > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \frac{1}{2\pi} \left( \frac{A}{\sqrt{1+A^2}} \arctan \frac{B}{\sqrt{1+A^2}} + \frac{B}{\sqrt{1+B^2}} \arctan \frac{A}{\sqrt{1+B^2}} \right) \\ \mathrm{where}~~A = \frac{a}{c}, ~~ B = \frac{b}{c} \end{gather}

Differential surface to rectangular surface in 90° angle

a ( a > 0 ) :
b ( b > 0 ) :
c ( c > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \frac{1}{2\pi} \left( \arctan \frac{1}{Y} - \frac{Y}{\sqrt{X^2+Y^2}} \arctan \frac{1}{\sqrt{X^2+Y^2}} \right) \\ \mathrm{where}~~X = \frac{a}{b}, ~~Y = \frac{c}{b} \end{gather}

Differential surface to sphere

r ( 0 < r < h ) :
h ( 0 < r < h ) :
θ [°] ( 0 ≤ θ ≤ 180 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \left( \frac{r}{h} \right)^2 \cos \theta, ~~ \mathrm{where}~~\theta \le \arccos \frac{r}{h} \\ F_{d1-2} = 0, ~~ \mathrm{where}~~\theta > \arcsin \frac{r}{h} + \frac{\pi}{2} \end{gather}

Differential surface to cylinder

r ( 0 < r < h ) :
h ( 0 < r < h ) :
l ( 0 < l ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \frac{L}{\pi H} \left[ \frac{1}{L} \arctan \frac{L}{\sqrt{H^2-1}} + \frac{X-2H}{\sqrt{XY}} \arctan \sqrt{\frac{X(H-1)}{Y(H+1)}} - \arctan \sqrt{\frac{H-1}{H+1}} \right] \\ \mathrm{where}~~L = \frac{l}{r}, ~~ H = \frac{h}{r}, \\ X = (1+H)^2 + L^2, ~~ Y = (1-H)^2 + L^2 \end{gather}

Differential surface to right triangle in parallel plane

h ( h > 0 ) :
l ( l > 0 ) :
θ [°] ( 0 < θ < 90 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{d1-2} = \frac{D}{2\pi A} \arctan \left( \frac{D \tan \theta}{A} \right), ~~ \mathrm{where}~~D = \frac{d}{h}, ~~ A = \sqrt{1+D^2} \end{gather}

Disk to parallel coaxial disk

h ( h > 0 ) :
r₁ ( r₁ > 0 ) :
r₂ ( r₂ > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{1-2} = \frac{1}{2} \left\{ X - \sqrt{X^2 - 4\left( \frac{R_2}{R_1} \right)^2} \right\} \\ \mathrm{where} ~~ X = 1 + \frac{1 + R_2^2}{R_1^2}, ~~ R_1 = \frac{r_1}{h}, ~~ R_2 = \frac{r_2}{h} \end{gather}

Note: For detailed derivations, see: "View Factors for Disks and Cylinder Walls"

Base disk to inside surface of cylincer

h ( h > 0 ) :
r₁ ( r > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{gather} F_{1-2} = 2H \left[ \sqrt{1+H^2} - H \right] \\ \mathrm{where} ~~ H = \frac{h}{2r} \end{gather}

Note: For detailed derivations, see: "View Factors for Disks and Cylinder Walls"

Identical, parallel, directly opposed rectangles

a ( a > 0 ) :
b ( b > 0 ) :
c ( c > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} F_{1-2} = &\frac{2}{\pi XY} \left\{ \ln \left[ \frac{(1+X^2)(1+Y^2)}{1+X^2+Y^2} \right]^{1/2} + X \sqrt{1+Y^2} \arctan \frac{X}{\sqrt{1+Y^2}} \right. \\ &\left. + Y \sqrt{1+X^2} \arctan \frac{Y}{\sqrt{1+X^2}} - X \arctan X - Y \arctan Y \right\} \\ &\hspace{70pt}\mathrm{where} ~~ X = \frac{a}{c}, ~~ Y = \frac{b}{c} \end{align}

Two rectangles with one common edge and 90° angle

h ( h > 0 ) :
w ( w > 0 ) :
l ( l > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} &F_{1-2} = \frac{1}{\pi W} \left( W \arctan \frac{1}{W} + H \arctan \frac{1}{H} - \sqrt{H^2 + W^2} \arctan \frac{1}{\sqrt{H^2 + W^2}} \right. \\ &\left. + \frac{1}{4} \ln \left\{ \frac{(1+W^2)(1+H^2)}{1+W^2+H^2} \left[ \frac{W^2(1+W^2+H^2)}{(1+W^2)(W^2+H^2)} \right]^{W^2} \left[ \frac{H^2(1+W^2+H^2)}{(1+H^2)(W^2+H^2)} \right]^{H^2} \right\} \right) \\ &\hspace{110pt}\mathrm{where} ~~ H = \frac{h}{l}, ~~ W = \frac{w}{l} \end{align}

Unit sphere to rectangle

h ( h > 1 ) :
l₁ ( l₁ > 0 ) :
l₂ ( l₂ > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} F_{1-2} &= \frac{1}{4\pi} \arctan \sqrt{\frac{1}{H_1^2+H_2^2+H_1^2H_2^2}} \\ &\mathrm{where} ~~ H_1 = \frac{h}{l_1}, ~~H_2 = \frac{h}{l_2} \end{align}

Unit sphere to coaxial disk

h ( h > 1 ) :
r ( r > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} F_{1-2} &= \frac{1}{2} \left[ 1 - \frac{1}{\sqrt{1+R^2}} \right] \\ &\mathrm{where} ~~ R = \frac{r}{h} \end{align}

Sphere to coaxial cone

h ( h > 0 ) :
r₁ ( r₁ > 0 ) :
r₂ ( r₂ > 0 ) :
θ [°] ( 0 < θ < 90 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} F_{1-2} &= \frac{1}{2} \left[ 1 - \frac{1+H+R/\tan\theta}{\sqrt{(1+H+R/\tan\theta)^2 + R^2}} \right] \\ &\mathrm{where} ~~ H = \frac{h}{r_1}, ~~ R = \frac{r_2}{r_1}, ~~ \theta \ge \arcsin \frac{1}{1+H} \end{align}

Interior of outer cylinder to exterior of coaxial inner cylinder

h ( h > 0 ) :
r₁ ( r₁ > 0 ) :
r₂ ( r₂ > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} &F_{2-1} = \frac{1}{R} \left( 1 - \frac{H^2+R^2-1}{4H} - \frac{1}{\pi} \left\{ \arccos \frac{H^2-R^2+1}{H^2+R^2-1} \right. \right. \\ &\left. \left. - \frac{\sqrt{(H^2+R^2+1)^2-4R^2}}{2H} \arccos \frac{H^2-R^2+1}{R(H^2+R^2-1)} - \frac{H^2-R^2+1}{2H} \arcsin \frac{1}{R} \right\} \right) \\ &\hspace{50pt} \mathrm{where} ~~ R_1 = \frac{r_1}{h}, ~~R_2 = \frac{r_2}{h}, ~~A = R_2 + R_1, ~~B = R_2 - R_1 \end{align}

Interior of cone to base disk

h ( h > 0 ) :
r ( r > 0 ) :

Analytical calculation	Calculate	0.0
Monte Carlo method	Calculate	0.0

\begin{align} &F_{1-2} = \frac{1}{\sqrt{1+H^2}}, ~~ \mathrm{where} ~~ H = \frac{h}{r} \end{align}