# Release of power in a hollow sphere¶

## Overview¶

The aim of this test case is to validate the following functions:

- Steady state heat transfer
- Volume heat source

The simulation results of SimScale were compared to the results presented in [TPLV06]. Four mesh cases were considered; linear and quadratic tetrahedrals, linear and quadratic hexahedrals.

## Geometry¶

Only a portion of a hollow sphere with an internal and external radius of 1 and 2 meters respectively was used for the analysis, as shown in the picture above. Face AECG is the internal face and face BFDH is the external face.

## Analysis type and Domain¶

**Tool Type** : Calculix, Code_Aster

**Analysis Type** : Steady-state heat transfer

**Mesh and Element types** :

Case | Mesh type | Number of nodes | Number of 3D elements | Element type | Tool type |
---|---|---|---|---|---|

(A) | linear hexahedral | 125 | 64 | 3D isoparametric | CalculiX |

(B) | quadratic hexahedral | 425 | 64 | 3D isoparametric | CalculiX |

(C) | linear tetrahedral | 1730 | 7115 | 3D isoparametric | CalculiX |

(D) | quadratic tetrahedral | 11547 | 6460 | 3D isoparametric | CalculiX |

(E) | linear hexahedral | 125 | 64 | 3D isoparametric | Code_Aster |

(F) | quadratic hexahedral | 425 | 64 | 3D isoparametric | Code_Aster |

(G) | linear tetrahedral | 1730 | 7115 | 3D isoparametric | Code_Aster |

(H) | quadratic tetrahedral | 11547 | 6460 | 3D isoparametric | Code_Aster |

## Simulation Setup¶

Important

All temperature dependent data is given as a function of *°C*. This also applies for the data in the different simulations on the SimScale platform
(although it says *K*)!

Material:

- isotropic: \(\rho\) = 1 kg/m³, \(\kappa\) = 1 W/(m °C), Specific heat \(c\) = 1 J/(kg °C)

Initial Conditions:

- Initial Temperature \(T_{initial}\) = 0 °C

Temperature:

- Temperature \(T\) = 20 °C on face AECG and BFDH

Heat flux:

- Volume heat source on whole volume with \(q_{v}\) = 100 W/m³

## Results¶

The figure below shows the result of case A with marked temperature values along the edges at distance of 1.25, 1.5 and 1.75 meters from origin.

Comparison of average temperature along the edges at distance of 1.25, 1.5 and 1.75 meters from origin obtained with SimScale and the results presented in [TPLV06]. The Error was calculated with respect to [TPLV06] ¹.

Distance [m] | [TPLV06] ¹ [°C] | [TPLV06] ² [°C] | Case A [°C] | Error [%] | Case B [°C] | Error [%] | Case C [°C] | Error [%] | Case D [°C] | Error [%] | Case E [°C] | Error [%] | Case F [°C] | Error [%] | Case G [°C] | Error [%] | Case H [°C] | Error [%] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1.25 | 30.625 | 30.467 | 30.471 | 0.50 | 30.638 | 0.04 | 30.475 | 0.49 | 30.625 | 0.00 | 30.471 | 0.50 | 30.638 | 0.04 | 30.474 | 0.49 | 30.625 | 0.00 |

1.5 | 32.500 | 32.346 | 32.331 | 0.52 | 32.498 | 0.01 | 32.449 | 0.16 | 32.500 | 0.00 | 32.331 | 0.52 | 32.498 | 0.01 | 32.449 | 0.16 | 32.500 | 0.00 |

1.75 | 28.482 | 28.381 | 28.368 | 0.40 | 28.471 | 0.04 | 28.401 | 0.28 | 28.482 | 0.00 | 28.368 | 0.40 | 28.471 | 0.04 | 28.401 | 0.28 | 28.482 | 0.00 |

## References¶

[TPLV06] | (1, 2, 3, 4, 5, 6, 7) TPLV06 - Dégagement de puissance dans une sphère creuse |

¹ Results as mentioned in [TPLV06]: Guide de validation des progiciels de calcul de structures. Société Française des Mécaniciens, AFNOR 1990 ISBN 2-12-486611-7

² Code_Aster results as mentioned in [TPLV06]