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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:

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:

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 $$W/(m.K)$$
• 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:

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

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