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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

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
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: ρ
    ρ

     

    = 7200 kg/m³, κ

    κ

     

    = 48.822 W/(m °C), Specific heat c

    c

     

    = 669 J/°C

Initial Conditions:

  • Initial Temperature Tinitial
    Tinitial

     

    = 20°C

Loads:

  • Convective heat flux on face ABCD with a temperature Touter
    Touter

     

    = 1000°C and a convection coefficient h

    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] (123) The results are provided by VPCS as stated in [TTLV01]
[TTLV01] (12345) TTLV01 – Sphère : échange de chaleur par convection
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