4. Radiation Plate to a Plate

A plate steadily radiates heat on a plate. This example demonstrates the view factor capabilities of B2000++. An analytical solution is available for the view factor in the text book by Siegel and Howell [34].

Two rectangular plates are meshed independently one of the other as shown in the figure below. If only radiation elements (Q4 elements) are used, the solution will consist of exchanged heat between the hot and the cold plate. Since all nodes have prescribed temperatures there are no unknowns. However, B2000++ has to solve for the non-linear essential boundary conditions imposed by the radiation. Special attention has to be brought to the directions of the normals of the surfaces (see figure). If the normals are not oriented properly the structures will not radiate in the desired directions.

Radiation of plate to plate: Mesh and normals

Figure 99. Radiation of plate to plate: Mesh and normals


Radiation problems are nonlinear as specified in the adir command below. The linear transient response is calculated with the default non-linear solver of B2000++, with the parameters defined in the case command (see input file radiation_plate_plate.mdl):

cases
   case 1
      ebc                      1
      component radiation_heat NBC "VISI_RAD"
      step_size_init           1
      step_size_min            1
      step_size_max            1
   end
end

adir
   analysis nonlinear
   case 1
end

By setting the initial step size, the minimum step size, and the maximum step size to 1, the solver will try to solve the problem for the final step with one step.

Radiation of plate to plate: Accumulated heat.

Figure 100. Radiation of plate to plate: Accumulated heat.


The approximate analytical value for the radiation configuration factor based on pure geometric considerations can be found in the graph of the text book by Siegel and Howell.



[34] R. Siegel, J. R. Howell; Thermal Radiation Heat Transfer, 4th ed.; Taylor and Francis, New York, 2001