# View Factor: Definition and Derivation

View factor is used in calculation of energy transfered by radiation.

## Definition of View Factor

Suppose there are plane $$i$$ and plane $$j$$ in vacuum space.

The area of plane $$i$$ and plane $$j$$ are $$A_i$$ and $$A_j$$.

Here, the energy tranfered from plane $$i$$ can be written as:

この時、面$$i$$から放射されるエネルギーは、放射強度を$$i_i$$（単位立体角あたりの放射エネルギー）とすると、

$Q_i = \pi i_iA_{i}$

where $$i_i$$ is the radiation strength (radiation energy per unit solid angle).

Of total energy radiated from plane $$i$$, the ratio of energy transfered to plane $$j$$, $$F_{ij}$$ should be:

$F_{ij} = \frac{Q_{ij}}{Q_{i}} = \frac{Q_{ij}}{\pi i_iA_{i}}$

This $$F_{ij}$$ is called view factor.

## Derivation of View Factor

Suppose there are infinitesimal areas $$dA_i$$ and $$dA_j$$ on plane $$i$$ and plane $$j$$, and define the distance between them as $$r$$.

Also, define the angle between normal vector of planes and the distance $$r$$ ad $$\phi_i$$ and $$\phi_j$$ as shown below.

Suppose that $$dQ_{ij}$$ denotes the energy from $$dA_i$$ to $$dA_j$$,

$dQ_{ij} = i_{i}dA_{i}\cos{\phi_{i}}d\omega$

where $$d\omega$$ is solid angle,

$d\omega = \frac{dA_{j}\cos{\phi_{j}}}{r^2}$

Thus, $$dQ_{ij}$$ can be written as:

$dQ_{ij} = i_{i}\frac{\cos{\phi_{i}}\cos{\phi_{j}}}{S^2}dA_{i}dA_{j}$

Integrating this equation, the following equation can be obtained:

$F_{ij} = \frac{Q_{ij}}{Q_{i}} = \frac{1}{\pi A_{i}}\int_{A_{i}}\int_{A_{j}}\frac{\cos{\phi_{i}}\cos{\phi_{j}}}{r^2}dA_{i}dA_{j}$