Radiation Heat Transfer in Spherical Cavity

This example shows the effect of surface-to-surface radiation in thermal analysis of nested annular spheres. Surface-to-surface radiation is the only mechanism of heat transfer between the spheres because they are separated by a spherical cavity.

This example uses SI units of measurement.

Geometry

Create a geometry consisting of two annular spheres, one inside the other.

ri = 0.01;
r1 = 0.02;
r2 = 0.03;
ro = 0.04;

gm = multisphere([ri,r1,r2,ro], ...
                  Void=[true,false,true,false]);

Plot the geometry.

pdegplot(gm,FaceLabels="on", ...
            FaceAlpha=0.25);

Thermal Analysis Model

Create an femodel object for thermal steady-state analysis and include the geometry.

model = femodel(AnalysisType="thermalSteady", ...
                Geometry=gm);

Specify a different thermal conductivity for each sphere.

model.MaterialProperties(1) = ...
    materialProperties(ThermalConductivity=2);
model.MaterialProperties(2) = ...
    materialProperties(ThermalConductivity=0.35);

Specify temperatures on the innermost face and the outermost face.

model.FaceBC(1) = faceBC(Temperature=300);
model.FaceBC(4) = faceBC(Temperature=1300);

Assign the emissivity values to faces forming the cavity.

model.FaceLoad(2) = faceLoad(Emissivity=0.5);
model.FaceLoad(3) = faceLoad(Emissivity=0.8);

Assign the initial temperature of 300 K to the entire geometry. Because this analysis includes radiation, use an absolute temperature.

model.CellIC = cellIC(Temperature=300);

To achieve faster convergence, specify initial temperature values on the enclosure faces that are close to the solution.

model.FaceIC(2) = faceIC(Temperature=500);
model.FaceIC(3) = faceIC(Temperature=1000);

Radiation heat transfer analysis requires the Stefan-Boltzmann constant. The toolbox has no default value for the Stefan-Boltzmann constant because its value depends on the system of units. Specify the constant in the SI system of units.

model.StefanBoltzmann = 5.670367E-8;

Generate a mesh.

model = generateMesh(model);

Define the radiation enclosure. Use the Monte Carlo method to compute view factors between the mesh facets in the enclosure faces.

model = setupRadiation(model, ...
                       EnclosureFaces=[2,3], ...
                       ViewFactorMethod="montecarlo");

Use the SolverOptions property of the model to specify the residual tolerance and maximal number of iterations, and to display internal solver statistics and a convergence report during the solution process.

model.SolverOptions.MaxIterations = 200;
model.SolverOptions.ResidualTolerance = 1E-5;
model.SolverOptions.ReportStatistics = 'on';

Solve the model.

R = solve(model);

Iteration     Residual     Step size  Jacobian: Full
   0          1.0000e+03
   1          3.8524e+02   1.0000000
   2          4.8260e-01   1.0000000
   3          6.6264e-07   1.0000000

The reference solution temperatures on the radiation enclosure faces are 578.225 K (on face 2) and 1035.024 (on face 3). Compare the solution to the reference solution.

nF2 = findNodes(model.Mesh,"region",Face=2);
nF3 = findNodes(model.Mesh,"region",Face=3);
T2mean = mean(R.Temperature(nF2))

T2mean = 574.6954

T3mean = mean(R.Temperature(nF3))

T3mean = 1.0260e+03

References

[1] Salazar, Giovanni, Justin C. Droba, Brandon Oliver, and Adam J. Amar. "Development, Verification and Validation of Enclosure Radiation Capabilities in the CHarring Ablator Response (CHAR) Code." NASA Lyndon B. Johnson Space Center, Houston, TX, 77058, USA, June 2016.