Clustering and Ray Tracing

In order to reduce memory requirements, the problem of determining the leaving and net heat radiation fluxes is solved using a discrete ray Monte-Carlo approach consisting of the following main elements:

To reduce the number of radiation rays and, therefore, the required calculation time and resources, the computational mesh cells containing faces approximating the radiative surfaces are joined in clusters by a special procedure that takes into account the face area and angle between normal and face in each partial cell. The cells intersected by boundaries between radiative surfaces of different emissivity are considered as belonging to one of these surfaces and cannot be combined in one cluster. This procedure is executed after constructing the computational mesh before the calculation and after each solution-adaptive mesh refinement, if any.
For each of the clusters, the hemisphere governed by the ray’s origin and the normal to the face at this origin is evenly divided into several nearly equal solid angles generated by several zenith (latitudinal) angles Θ (at least 3 within the 0...90° range, including the zero zenith angle of the normal to the face) and several azimuth (longitudinal) angles Φ (at least 12 within the 0...360° range).

A radiation ray is emitted in each of the solid angles in a direction that is defined randomly within this solid angle. Each ray is traced through the fluid and transparent solid bodies until it intercepts the computational domain’s boundary or a cluster belonging to another radiative surface, thus defining a “target” cluster. Since the radiation heat is transferred along these rays only, their number and arrangement govern the accuracy of calculating the radiation heat coming from one radiative surface to another (naturally, the net heat radiated by a radiative surface does not depend on number of these rays).

Figure 1. Definition of rays emitted from cluster: P - polar ray, Θ - zenith (latitudinal) angle, Φ - azimuth (longitudinal) angle.

The total number of rays emitted from a cluster is calculates as:

(1)

where m is the number of different latitude values for the rays (including the polar ray), n is the number of different longitude values (n = 2 for 2D case).

The value of m is defined directly by the View factor resolution level which can be changed by the user via the Calculation Control Options dialog box. The value of n depends on m as follows:

(2)

The higher the View factor resolution level, the better the accuracy of the radiation heat transfer calculation, but the calculation time and required computer resources increase significantly when high values of View factor resolution level are specified.

To increase the accuracy of heat radiation calculation, the number of radiation rays emitted from each cluster can be increased automatically during the calculation, depending on the surface temperature and emissivity, to equalize the radiation heat emitting through the solid angles.
When a radiation ray intercepts a cluster of other radiative surfaces, the radiation heat carried by this ray is evenly distributed over the area of this cluster. The same procedure is performed if several radiation rays hit the same cluster. To smooth a possible non-uniformity of the incident radiation heat distribution on a radiative surface, a fraction of the radiation heat arriving with rays at a cluster can be transferred to the neighboring clusters also. In addition, small fluctuations are smoothed by the heat conduction in solid regions.