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Validation Case: Hollow Sphere, Convection and Radiation

This validation case belongs to heat transfer, with the case of a hollow sphere under convection and radiation. The aim of this test case is to validate the following parameters:

  • Nonlinear steady state heat transfer
  • Convection heat transfer condition
  • Radiation heat transfer through heat flux condition

The simulation results of SimScale were compared to the numerical results presented in [TPNV01]\(^1\).


The geometry used for the case is as follows:

geometry model validation case hollow sphere convection radiation
Figure 1: Only one portion of the hollow sphere is modeled.

It represents a section of a hollow sphere with an internal radius of 0.3 \(m\) and an external radius of 0.392 \(m\). Face ABCD is the external face and EFGH is the internal face. Axis X passes through the centroid of both faces, making the volume symmetric around the XY and XZ planes.

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Nonlinear heat transfer, steady state.

Mesh and Element Types:

CaseMesh TypeNumber of
Element Type
AStandard13721st order tetrahedral
BStandard96082nd order tetrahedral
Table 1: Mesh details for each case

The tetrahedral meshes were computed using SimScale’s standard mesh algorithm and automatic sizing:

tetrahedral mesh validation case hollow sphere convection radiation
Figure 2: Finite tetrahedral elements mesh used on cases A and B

Simulation Setup


  • Density \( \rho = \) 1 \( kg/m^3 \)
  • Thermal conductivity \( \kappa = \) 40 \( W/(m.K) \)
  • Specific heat \( C_p = \) 1 \( J/(kg.K) \)

Boundary Conditions:

  • Convective Heat Flux:
    • Applied on face ABCD
    • Reference temperature \(T_0 = \) 20 \(°C\)
    • Heat transfer coefficient of 133.5 \(W/(m.K)\)
  • Radiation Heat Flux:
    • Applied on face EFGH
    • Modeled as a temperature-dependent surface heat flux
    • Heat flux dependent on temperature, according to the Boltzmann equation:

$$ \varphi = \sigma \epsilon [ (T_0 + 273.15)^4 – (T + 273.15)^4 ] \tag{1} $$

$$ \sigma = 5.73×10^{-8} \ W/(m^3.K^4) $$

$$ \epsilon = 0.6 $$

$$ T_0 = 500\ °C $$


Equation 1 was used to compute a table for a temperature range of 0 to 100 \(°C\) and uploaded to the platform to model the radiation heat flux with the surface heat flux boundary condition. This temperature-dependent condition dictates the need of using nonlinear heat transfer analysis.

Reference Solution

The reference solution comes from analytical expressions solved numerically, as presented in [TPNV01]\(^1\). The reference solution is presented as the temperature at the internal and external faces:

\( T_{int} = 91.77\ °C \)

\( T_{ext}= 71.22\ °C \)

Result Comparison

Comparison of average temperatures at internal (EFGH) and external (ABCD) faces with the reference solution, for each case, is presented:

CaseFaceCompute Average \([K]\)Compute Average \([°C]\)Reference
AINTERNAL364.89391.74391.77-0.03 %
EXTERNAL344.35971.20971.22-0.16 %
BINTERNAL364.92191.77191.770.00 %
EXTERNAL344.3771.2271.220.00 %
Table 2: Results comparison and computed errors for average temperatures for both cases

Illustration of the temperature distribution from the sphere with convection and radiation simulation, case B:

temperature plot validation case hollow sphere convection radiation
Figure 3: Temperature distribution contours on the body from case B


If you still encounter problems validating you simulation, then please post the issue on our forum or contact us.

Last updated: July 21st, 2021