# Hollow sphere: Convection, Radiation¶

## Overview¶

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

- Steady state heat transfer
- Radiation boundary condition
- Convective heat flux

The simulation results of SimScale were compared to the results presented in [TPNV01]. The meshes used in (A) and (B) were created locally and uploaded to SimScale platform whereas, meshes used in (C) and (D) were created with the automatic-tetrahedralization-tool on the SimScale platform.

## Geometry¶

Only a portion of a hollow sphere with an internal and external radius of 0.3 and 0.392 meters was used for the analysis, as shown in the picture above. Face ABCD is the external face and face EFGH is the internal face.

## Analysis type and Domain¶

**Tool Type** : Calculix

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

**Mesh and Element types** :

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

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

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

(C) | linear tetrahedral | 1638 | 6460 | 3D isoparametric |

(D) | quadratic tetrahedral | 10735 | 6460 | 3D isoparametric |

## Simulation Setup¶

Material:

- isotropic: \(\rho\) = 1 kg/m³, \(\kappa\) = 40 W/(m K), Specific heat \(c\) = 1 J/(kg K)

Initial Conditions:

- Initial Temperature \(T_{initial}\) = 293.15 K

Heat flux:

- Convective heat flux on face ABCD with a temperature \(T_{outer}\) = 293.15 K and a convection coefficient \(h\) = 133.5 W/(m² K)
- Radiation boundary condition on face EFGH with a temperature \(T_{inner}\) = 773.15 K and an emissivity \(\epsilon\) = 0.6

## Results¶

Comparison of average temperature on internal edge (EH) and external edge (AD) obtained with SimScale with the results presented in Case A of [TPNV01]. The Error was calculated with respect to [VPCS].

Case | Edge | [VPCS] | [TTLV01] | SimScale | Error |
---|---|---|---|---|---|

(A) | EH | 364.92 K | 364.92 K | 364.07 K | 0.23% |

AD | 344.37 K | 344.37 K | 343.84 K | 0.15% | |

(B) | EH | 364.92 K | 364.92 K | 364.18 K | 0.20% |

AD | 344.37 K | 344.37 K | 343.85 K | 0.15% | |

(C) | EH | 364.92 K | 364.92 K | 364.17 K | 0.21% |

AD | 344.37 K | 344.37 K | 343.85 K | 0.15% | |

(D) | EH | 364.92 K | 364.92 K | 364.19 K | 0.20% |

AD | 344.37 K | 344.37 K | 343.85 K | 0.15% |