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# Validation Case: Hollow Sphere, Release of Power

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

• Steady state heat transfer
• Volumetric heat source

The simulation results of SimScale were compared to the numerical results presented in [TPLV06]$$^1$$.

## Geometry

The geometry used for the case is as follows:

It represents a section of a hollow sphere with an internal radius of 1 $$m$$ and an external radius of 2 $$m$$. Face ABCD is the internal face and EFGH is the external face. Axis X passes through the centroid of both faces, making the volume symmetric across the XY and XZ planes.

## Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Heat transfer, linear, steady state.

Mesh and Element Types:

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

## Simulation Setup

Material:

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

Boundary Conditions:

• Constraints:
• Fixed temperature of 20 $$°C$$ on faces ABCD and EFGH (internal and external respectively)
• Volume heat source of 100 $$W/m^3$$ on the whole volume.

## Reference Solution

The reference solution is of the analytical type, as presented in [TPLV06]$$^1$$, originally from [VPCS]$$^2$$:

$$T = T_i + \frac{Q}{6\kappa} \Bigg[ \frac{ (R_e^2 – R_i^2 ) \Big[ \frac{1}{R_i} – \frac{1}{r} \Big] }{ \Big[ \frac{1}{R_i} – \frac{1}{R_e} \Big] } – ( r^2 – R_i^2 ) \Bigg]$$

$$T_i = 20\ °C$$

$$Q = 100\ W/m^3$$

$$\kappa = 1\ W/(m.K)$$

$$R_i = 1.0\ m$$

$$R_e = 2.0\ m$$

The reference solution will be taken at points $$r =$$ 1.25, 1.5 and 1.75 $$m$$:

$$T(r = 1.25) = 30.625\ °C$$

$$T(r = 1.5) = 32.500\ °C$$

$$T(r = 1.75) = 28.482\ °C$$

## Result Comparison

Comparison of temperatures at radii $$R =$$ 1.25, 1.5 and 1.75 $$m$$ with the reference solution is presented:

Illustration of the temperature distribution from the release of power simulation, case D:

Tutorial: Thermal Analysis of a Differential Casing

References

Note

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

Last updated: May 19th, 2021