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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:

geometry model validation case hollow sphere power release
Figure 1: Only one portion of the hollow sphere is modeled

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:

CaseMesh TypeNumber of
Nodes
Element Type
A1st order hexahedral125Standard
B2nd order hexahedral425Standard
C1st order tetrahedral2094Standard
D2nd order tetrahedral14939Standard
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 release of power simulation project.

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

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:

CASER
\([m]\)
COMPUTED
\([K]\)
COMPUTED
\([°C]\)
REFERENCE
\([°C]\)
ERROR
A1.25303.61130.46130.625-0.54 %
1.5305.48432.33432.500-0.51 %
1.75301.53128.38128.482-0.35 %
B1.25303.7930.6430.6250.05 %
1.5305.64932.49932.500-0.00 %
1.75301.62128.47128.482-0.04 %
C1.25303.69930.54930.625-0.25 %
1.5305.57632.42632.500-0.23 %
1.75301.59728.44728.482-0.12 %
D1.25303.77630.62630.6250.00 %
1.5305.6532.532.5000.00 %
1.75301.63228.48228.4820.00 %
Table 2: Results comparison and computed errors for cases A through D

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

temperature plot validation case hollow sphere power release
Figure 4: Temperature distribution on the body from 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: July 30th, 2020

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