Radiation: ray tracing

View factor together with the n-band spectral approach can compute the radiative heat transfer in media which are either fully opaque or fully transparent in each spectral band.

A new ray tracing model is implemented into CrysMAS. It treats the heat transfer in participating media with a finite absorption coefficient.

The ray tracing based radiation model implemented in CrysMAS in order to compute the heat transfer by radiation in semitransparent media is based on the so called backwards ray tracing method.

To solve the Radiation Heat Transport Equation in the semitransparent media one has to know in a given point in volume or on a boundary semitransparent/transparent or semitransparent/opaque also the incident Radiative Heat Flux. This is done by computing the intensity of the radiation incident in that point coming from a given spatial direction. The radiation transport equation is integrated directly for the unknown intensity propagated along the direction.

To compute the intensity one has to know the point where the radiation originates and the path followed until the incident point was reached.

To compute the path we are using ray tracing. First of all we discretize the directions based on a given algorithm which takes into account also the axisymmetrical geometry of the problem. This allows us later to use only little more directions than the half of the total ones, giving us some important speedup.

Discrete spatial directions and the solid angle associated with each direction are computed. The discretization of the space angle is based on the recursive refinement algorithm. Additionally angle integrals of the specular reflection coefficient and the scalar product of the normal vector and the considered space direction are computed and stored for each radiating edge at the interface of the semitransparent volume.

By generating the mesh in CrysMAS one gets already the numerical discretization of the geometry. The starting target points for ray tracing algorithm are the vertices resulting from unstructured CrysMAS mesh. The rays are traced from some location outside towards the position of the numerical node.

There are two categories of such points:

If the target point lays on a boundary of the semitransparent volume then we trace a ray from this point in the direction opposite to the fixed spatial direction we use and follow this ray until the total attenuation of the radiation intensity along the ray path achieves the prescribed threshold value. The intensity will be reduced due to absorption along the ray path and due to the reflexion losses at the boundary. The backwards tracing procedure is interrupted if only very small fraction of the start intensity is remained.

Whenever the ray intersects a geometrical object a reflection is done, the new direction is computed and the intersection point is stored. In this way we end up having a ray path (the path of the ray inside our semitransparent media).

By doing this for every spatial direction we take into account, we end up having in every node of the unstructured mesh a so called ray bundle (a bundle of ray paths) stored for later application in the thermal computation.

The geometrical preprocessing depends on the type of the target node. If the starting point lays in volume and corresponds to the inner node of the semitransparent volume (semitransparent CrysMAS region), then we trace a ray from this point in every spatial direction and we stop the tracing after the first intersection with the boundary of the semitransparent volume. The intersection point is stored. Note that this intersection point lays always between two boundary points! We will end up with ray paths similar to that of the following figure.

Ray paths for computation of the radiation intensity. For the target node located in the inner part of the semitransparent volume the ray path is terminated after the first intersection with the boundary. For the target node at the boundary the ray is traced several reflections at the boundary until the required attenuation is achieved. 

Figure 12. Ray paths for computation of the radiation intensity. For the target node located in the inner part of the semitransparent volume the ray path is terminated after the first intersection with the boundary. For the target node at the boundary the ray is traced several reflections at the boundary until the required attenuation is achieved.

In computing the intensities we treat differently the point laying on the boundary from the points laying in volume.

Reccurent expressions for computation of the radiation intensity along the ray path. A series of reflections at the boundary is considered. 

Figure 13. Reccurent expressions for computation of the radiation intensity along the ray path. A series of reflections at the boundary is considered.

For the points laying on the boundary we compute the intensity according to the equations (1a), (1b) and (1c)

Where In represents the intensity at the last receiver and I0 is the start intensity (the intensity leaving the farthest point along the ray path - where the tracing was stopped) and Ibb is the intensity emitted by a black body at temperature T in the spectral band "gamma".

If the point lays at the boundary between semitransparent and transparent media then an additional term (which represents the intensity of the radiation coming from the transparent cavity) is added.

Computation of intensities for volume points 

Figure 14.  Computation of intensities for volume points

If the point lays in volume than we compute the intensity according to equations (2a) and (2b) were we take advantage of the already known values for the intensities in the points starting on the surface and we interpolate between those values to find out the start intensity I0. The interpolation parameter μ in equation (2b) is computed and saved during the geometrical preprocessing of the ray tracing.

The resulted intensity in equations (1a) and (1b) has 3 terms. The accuracy of the numerical evaluation of the term corresponding to the volume emission and absorption can be influenced by changing of the parameter ray subdivision density factor in the Computation -> Numerical Parameters Dialog in the Radiation tab.

Knowing the intensity incident from a given direction one can compute the total irradiation according to the equation (3) and then compute the net flux and use it as a source term to solve the heat transfer equation.

Computation of the irradiation in a given spectral band. The total irradiation is resulted from the sum over all semitransparent bands with different optical lengths. 

Figure 15.  Computation of the irradiation in a given spectral band. The total irradiation is resulted from the sum over all semitransparent bands with different optical lengths.