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]$^1$.

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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

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

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:

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 %

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

Thermal Analysis of a Differential Casing

References

TPNV01 – Sphère creuse : convection, rayonnement – Code Aster validation case

Note

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

Last updated: July 21st, 2021

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