Spherical Harmonic Solution to the Radiative Transfer Equation : Masters Thesis Work

Computational Heat Transfer and Fluid Dynamics Laboratory, Mechanical Engineering, Ohio State University, Columbus, OH. ( January 2008 - May 2009 )

• Finite-Volume Formulation and Solution of the P3 Equations of Radiative Transfer on Unstructured Meshes:

  The P3 Equations represent an approximation to the Spherical Harmonics Solution of the Radiative Transport Equation ( a 7-dimensional integro-differential equation ) which result in 4 strongly coupled second-order differential equation with elliptic boundary conditions. It is a higher-order approximation compared to the more popular P1 approximation. A research code, written in FORTRAN, was developed to solve these equations. The overall scheme used is as follows.

  1. The P3 equations and Boundary conditions are discretized using Finite-Volume Approximation with an unstructured mesh as the stensil.
  2. An unstructured mesh is generated using the commercial software CFD-GEOM. The connectivity information is written into data files, read by the code at run-time.
  3. The solution domain is split into sub-domains. The idea is to solve or the unknowns within each sub-domain implicitly, and across domains explicitly. An completely implicit solution would be very efficient in terms of CPU time, but not in its memory requirements. Depending on resources available, the size of the sub-domains can be varied to achieve good performance.
  4. Solving the equations within a sub-domain boils down to solving an linear-algebraic equation of the form AX=B, where A is a block-sparse matrix, X is the vector of unknows and B is the source vector. A pre-conditioned GMRES solver is ideal for such situations and open-source solvers, like the one written by Yousef Saad, are pretty efficient.
  5. Repeated sweeps over the entire domain give a convergent solution
  6. The output is written into data files, formatted to be read into TECPLOT.

• Using Modified-Differential Approximation to simulate the process of Radiative Heat Transfer in a 2D or 3D enclosure

  The Modified Differential Approximation is a method suggested by Olfe. Here the Intensity of Radiation at all points of the solution domain are split into ballistic and diffusive componenets. The Ballistic Component is evaluated by assuming that there is no participating media in the enclosure, and walls interact directly. Then, the solution of this is used to calculate the diffusive component. We are interested in using MDA to solve for domains with or without cavities. The basic approach of MDA is explained in detail in Radiative Heat Transfer by Modest. The approach used by us is described below.

  1. An unstructed 2D/3D mesh is generated in CFD-GEOM and connectivity information is stored in data files. These files are used by the code at run-time.
  2. The view factors of all the boundary faces with respect to each other need to be calculated to solve for the ballistic component. This involves two steps.
     • Using ray-tracing approach, it is determined if the two faces in question have a line-of-sight ( i.e., the LOS is not blocked by a cavity )
     • Having accertained that there is an LOS, the value of the view factor is calculated from theorectical expressions between arbitrarily oriented traingles/quadrilaterals described here.
     This turns out to be an expensive computation in 3D. For a enclosure discretized into nb boundary faces, the number of computations is of the order nb² × (no. of sides per face)² × (no. of ops to compute a vector cross-product ) . The memory requirement to hold all the view factors would be nb² × (size of floating point number ) . This is one of the major unsolved challenges.
  3. Having evaluated the view factors the wall-component or ballistic componenet of radiative intensity at the walls is computed. This is done by solving a set of simultaneous linear algebraic equations using the pre-conditioned GMRES solver mentioned above.
  4. Solving the diffusive component is similar to solving the P1 approximation to the RTE on an unstructured mesh. An approach similar to the one described above was used.
  5. The sum of the ballistic and diffusive componenets of radiation, gives the actual value of radiation at any given point