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:

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:

Case	Mesh Type	Number of Nodes	Element Type
A	Standard	1372	1st order tetrahedral
B	Standard	9608	2nd order tetrahedral

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

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:

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

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

$$ \epsilon = 0.6 $$

$$ T_0 = 500\ °C $$

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:

Case	Face	Compute Average \([K]\)	Compute Average \([°C]\)	Reference \([°C]\)	Error
A	INTERNAL	364.893	91.743	91.77	-0.03 %
	EXTERNAL	344.359	71.209	71.22	-0.16 %
B	INTERNAL	364.921	91.771	91.77	0.00 %
	EXTERNAL	344.37	71.22	71.22	0.00 %

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

Related Tutorials

• Thermal Analysis of a Differential Casing

Last updated: July 21st, 2021