View factor sum and residual view factor

The theoretical view factor sum for an element is 1. However, because of model geometry imperfections and approximations made during shadowing calculations, this condition is not always fulfilled.

where:

• n is the total number of surfaces within the enclosure.
• VF_ij is the view factor from surface i to surface j.

The residual view factor is an indication of the accuracy of the view factor calculations:

The ideal value of R_i is 0. Residual view factors between -0.05 to 0.05 are usually acceptable. Departures from this may occur because of the approximations in view factor calculations, or when an improper enclosure is defined, for example when i sees the back of another element. Very large departures are usually an indication of an improper model. The physical effect of R_i > 0 is to reduce the radiation leaving i, in effect making the surface more concave.

The thermal solver provides the following methods of compensating for R_i ≠ 0.

Creating a self-view factor

The effect of creating a self-view factor may be viewed as assuming the element is concave. Some of the radiation is incident on itself, which is then reflected to the elements it sees in proportion to the view factors times the reflectivity.

Setting the residual view factor to an element

When setting the residual view factor for an element, where the element number is greater than zero, if the element is connected to another, a view factor from the enclosure to that element is used if it already exists. If no such view factor exists, the residual view factor is set to the self-view factor. This prevents unwanted radiation leakage from inside the enclosure to an element that cannot see it.

A new, very large black element is created in the enclosure.

If the element number is greater than zero, for each element where the residual view factor is less than zero, the option to set the element number to zero is applied.

If the element number is 2E6, the space element number is substituted for the code 2E6.

Approximate proportional adjustment of the view factors

With this recommended option, only the shadowed view factors are adjusted iteratively to minimize the maximum R_i. The unshadowed view factors are assumed to be accurate, as they are presumed to have been calculated using the Exact Contour Integral Technique. During each iteration, each shadowed view factor is multiplied by the factor:

The R_i values are re-evaluated during each iteration. The iterations stop when the maximum R_i​ is minimized.

Generally, view factor calculation accuracy can be increased by specifying higher elemental subdivision for the deterministic method or by increasing the number of rays for Monte Carlo method.