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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.

Import validation project into workspace

Geometry

HollowSphereConvectionRadiation-geometry

Geometry of the hollow sphere

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
HollowSphereConvectionRadiation-mesh-caseB

Mesh used for the SimScale case (B)

HollowSphereConvectionRadiation-caseD

Mesh used for the SimScale case (D)

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].

Comparison of average temperature on internal edge (EH) and external edge (AD)
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%

References

[VPCS](1, 2) The results are provided by VPCS as stated in [TTLV01]
[TPNV01](1, 2) TPNV01 - Sphère creuse : convection, rayonnement