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Heat Transfer of a sphere under convection

Overview

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

  • Transient heat transfer
  • Convective heat flux

The simulation results of SimScale were compared to the results presented in [TTLV01]. The meshes used in (A) and (B) were created with the automatic-tetrahedralization-tool on the SimScale platform.

Import validation project into workspace

Geometry

HeatTransferOfASphereUnderConvection-geometry

Geometry of the sphere

Only a portion of a sphere with a diameter of 0.2 m is used for the analysis, as shown in the picture above. M is the center of the sphere and the face ABCD represents the outer surface.

Analysis type and Domain

Tool Type : Calculix, Code_Aster

Analysis Type : Transient heat transfer

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) linear tetrahedral 3714 3D isoparametric
(B) quadratic tetrahedral 26494 3D isoparametric
HeatTransferOfASphereUnderConvection-mesh-a

Mesh used for the SimScale case (A)

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: \(\rho\) = 7200 kg/m³, \(\kappa\) = 48.822 W/(m °C), Specific heat \(c\) = 669 J/°C

Initial Conditions:

  • Initial Temperature \(T_{initial}\) = 20°C

Loads:

  • Convective heat flux on face ABCD with a temperature \(T_{outer}\) = 1000°C and a convection coefficient \(h\) = 232.5 W/(m² °C)

Results

Comparison of temperature at the center point M and at the outer surface obtained with SimScale with the results presented in [TTLV01]. The Error was calculated with respect to [VPCS].

Comparison of temperature at the center point M
Time [VPCS] [TTLV01] Case A - Calculix Error Case A - Code_Aster Error Case B - Calculix Error Case B - Code_Aster Error
400 s 334°C 340.56°C 336.48°C 0.74% 341.16°C 2.14% 336.48°C 0.74% 341.15°C 2.14%
600 s 500°C 493.15°C 487.89°C 2.42% 493.48°C 1.30% 487.86°C 2.43% 493.45°C 1.31%
800 s 618°C 610.27°C 604.86°C 2.13% 610.61°C 1.20% 604.82°C 2.13% 610.57°C 1.20%
1000 s 706°C 700.18°C 695.12°C 1.54% 700.66°C 0.76% 695.08°C 1.55% 700.61°C 0.76%
1200 s 774°C 769.35°C 764.77°C 1.19% 769.88°C 0.53% 764.72°C 1.20% 769.83°C 0.54%
1400 s 828°C 822.57°C 818.50°C 1.15% 823.1°C 0.59% 818.45°C 1.15% 823.05°C 0.60%
1600 s 872°C 863.33°C 859.96°C 1.38% 864.01°C 0.92% 859.92°C 1.39% 864.38°C 0.87%
1800 s 902°C 894.73°C 891.95°C 1.11% 895.45°C 0.73% 891.91°C 1.12% 895.42°C 0.73%
2000 s 923°C 918.91°C 916.63°C 0.69% 919.63°C 0.37% 916.6°C 0.69% 919.6°C 0.37%
2200 s 942°C 937.54°C 935.67°C 0.67% 938.22°C 0.40% 935.64°C 0.68% 938.19°C 0.40%
2400 s 956°C 951.89°C 950.37°C 0.59% 952.5°C 0.37% 950.34°C 0.59% 952.48°C 0.37%
Comparison of temperature at the outer surface
Time [VPCS] [TTLV01] Case A - Calculix Error Case A - Code_Aster Error Case B - Calculix Error Case B - Code_Aster Error
400 s 461°C 474.82°C 471.04°C 2.18% 474.88°C 3.01% 470.96°C 2.16% 474.8°C 2.99%
600 s 608°C 596.37°C 591.9°C 2.65% 596.37°C 1.91% 591.82°C 2.66% 596.28°C 1.93%
800 s 696°C 689.64°C 685.12°C 1.56% 689.71°C 0.90% 685.05°C 1.57% 689.63°C 0.92%
1000 s 774°C 761.24°C 757.05°C 2.19% 761.46°C 1.62% 756.98°C 2.20% 761.39°C 1.63%
1200 s 828°C 816.33°C 812.55°C 1.87% 816.63°C 1.37% 812.48°C 1.87% 816.56°C 1.38%
1400 s 868°C 858.7°C 855.37°C 1.46% 859.03°C 1.03% 855.31°C 1.46% 858.97°C 1.04%
1600 s 902°C 891.16°C 888.4°C 1.51% 891.63°C 1.15% 888.36°C 1.51% 891.58°C 1.16%
1800 s 923°C 916.17°C 913.9°C 0.99% 916.69°C 0.68% 913.85°C 0.99% 916.65°C 0.69%
2000 s 942°C 935.42°C 933.57°C 0.90% 935.96°C 0.64% 933.53°C 0.90% 935.92°C 0.65%
2200 s 956°C 950.26°C 948.74°C 0.76% 950.77°C 0.55% 948.71°C 0.76% 950.74°C 0.55%
2400 s 962°C 961.69°C 960.45°C 0.16% 962.15°C 0.02% 960.42°C 0.16% 962.13°C 0.01%

References

[VPCS](1, 2, 3) The results are provided by VPCS as stated in [TTLV01]
[TTLV01](1, 2, 3, 4, 5) TTLV01 – Sphère : échange de chaleur par convection