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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.

Import validation project into workspace

Geometry

ReleaseOfPowerInAHollowSphere-geometry

Geometry of the hollow sphere

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
ReleaseOfPowerInAHollowSphere-mesh-caseB

Mesh used for the SimScale case (B)

ReleaseOfPowerInAHollowSphere-mesh-caseD

Mesh used for the SimScale case (D)

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.

ReleaseOfPowerInAHollowSphere-result

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.

Comparison of the average temperature at distance of 1.25, 1.5 and 1.75 meters from origin [°C]
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]