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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]\(^1\).

Geometry

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
Nodes
Element Type
A1st order hexahedral125Standard
B2nd order hexahedral425Standard
C1st order tetrahedral1372Standard
D2nd order tetrahedral9608Standard
Table 1: Mesh details for each case

The tetrahedral meshes were computed using SimScale’s standard mesh algorithm and automatic sizing. The hexahedral meshes were computed locally and uploaded into the simulation project.

hexahedral mesh validation case hollow sphere convection radiation
Figure 2: Finite hexahedral elements mesh used on cases A and B
tetrahedral mesh validation case hollow sphere convection radiation
Figure 3: Finite tetrahedral elements mesh used on cases C and D

Simulation Setup

Material:

  • 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:

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

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

$$ \epsilon = 0.6 $$

$$ T_0 = 500\ °C $$

Note

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:

CASEFACECOMPUTED
AVERAGE \([K]\)
COMPUTED
AVERAGE \([°C]\)
REFERENCE
\([°C]\)
ERROR
AINTERNAL364.80591.65591.77-0.13 %
EXTERNAL344.37271.22271.220.00 %
BINTERNAL364.91691.76691.770.00 %
EXTERNAL344.37271.22271.220.00 %
CINTERNAL364.89391.74391.77-0.03 %
EXTERNAL344.35971.20971.22-0.16 %
DINTERNAL364.92191.77191.770.00 %
EXTERNAL344.3771.2271.220.00 %
Table 2: Results comparison and computed errors for average temperatures for cases A through D

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

temperature plot validation case hollow sphere convection radiation
Figure 4: Temperature distribution contours on the body from case D

Tutorial: Thermal Analysis of a Differential Casing

Note

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

Last updated: August 4th, 2020

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