Exchange Factor Calculation

The “exchange factor” Fij between two clusters is the fraction of the total radiation energy emitted from one of the clusters that is intercepted by other clusters. Hence, given a point j at the center of a cluster on the radiative surface, a radiation ray is fired from j for each one of the specified directions. Let i be the impingement point, starting from the origin j, the ray is traced to the i.

Then, the incident radiation at point i, Qiin, is calculated by adding up the contributions due to all the rays that reach point i:

the incident thermal radiation is defined as:

(1)
the incident solar radiation is defined as:

(2)

	(1)

	(2)

The radiation leaving from the i-th cluster, Qiout, is then computed as a sum of the reflected portion of Qiin and the emissive power of the surface:

for thermal radiation:

(3)
for solar radiation:

(4)

	(3)

	(4)

The radiation arriving at i-th cluster from solar sources, , is calculated as:

for the backward solar ray tracing method:

(5)
for the forward solar ray tracing method:

(6)

	(5)

	(6)

“Backward” means that rays are emitted from the radiative surfaces. Each solar radiation source produces one ray that follows the directional vector backward. After it reaches the outer boundary or the surface having appropriate radiation boundary condition, the exchange factor can be estimated as:

	(7)

The ISk is the solar radiation intensity of the k-th solar source. The value δki equals 1 if the i-th cluster is visible for the k-th solar source, and 0 if not.

“Forward” means that the rays are emitted from the radiation source. The radiation emitted by all solar radiation sources can be distributed evenly over all rays traced from radiation sources. The radiation transferred by each ray, , can be estimated by summing up the radiation emitted by all solar sources, , and then divided by the number of emitted rays, Nrays:

	(8)

The value Ni is the number of rays intercepted the i-th cluster. The forward method is recommended in case of refraction and reflection radiative surfaces.

The solar ray tracing method and the number of rays traced from radiation sources can be changed by the user via the Calculation Control Options dialog box.

Please note that for the sake of simplicity the set of equations described here defines the radiation heat transfer between clusters only and does not take into account the outer boundaries radiation and diffusive radiation sources. In the full set of equations these sources are also considered.