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]$^1$.

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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 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:

Case	Mesh Type	Number of Nodes	Element Type
A	1st order hexahedral	245	Standard
B	2nd order hexahedral	515	Standard
C	1st order tetrahedral	2094	Standard
D	2nd order tetrahedral	14939	Standard

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 generated in SimScale with extrusion refinements to obtain a hexahedral mesh.

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:

CASE	R $[m]$	COMPUTED $[K]$	COMPUTED $[°C]$	REFERENCE $[°C]$	ERROR
A	1.25	303.660	30.510	30.625	-0.38 %
	1.5	305.540	32.390	32.500	-0.34 %
	1.75	301.568	28.418	28.482	-0.23 %
B	1.25	303.774	30.624	30.625	-0.00 %
	1.5	305.649	32.499	32.500	-0.00 %
	1.75	301.632	28.482	28.482	0.00 %
C	1.25	303.699	30.549	30.625	-0.25 %
	1.5	305.576	32.426	32.500	-0.23 %
	1.75	301.597	28.447	28.482	-0.12 %
D	1.25	303.776	30.626	30.625	0.00 %
	1.5	305.65	32.5	32.500	0.00 %
	1.75	301.632	28.482	28.482	0.00 %

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

TPLV06 – Dégagement de puissance dans une sphère creuse – Code Aster validation case
Guide de validation des progiciels de calcul de structures. Société Française des Mécaniciens, AFNOR 1990 ISBN 2-12-486611-7

Note

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Last updated: April 3rd, 2026

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