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    Validation Case: Hollow Sphere, Convection and Radiation

    This validation case belongs to heat transfer, with the case of a hollow sphere under convection and radiation. The aim of this test case is to validate the following parameters:

    • Nonlinear steady state heat transfer
    • Convection heat transfer condition
    • Radiation heat transfer through heat flux condition

    The simulation results of SimScale were compared to the numerical results presented in [TPNV01]1.

    Geometry

    The geometry used for the case is as follows:

    geometry model validation case hollow sphere convection radiation
    Figure 1: Only one portion of the hollow sphere is modeled.

    It represents a section of a hollow sphere with an internal radius of 0.3 m and an external radius of 0.392 m. Face ABCD is the external face and EFGH is the internal face. Axis X passes through the centroid of both faces, making the volume symmetric around the XY and XZ planes.

    Analysis Type and Mesh

    Tool Type: Code_Aster

    Analysis Type: Nonlinear heat transfer, steady state.

    Mesh and Element Types:

    CaseMesh TypeNumber of
    Nodes
    Element Type
    AStandard13721st order tetrahedral
    BStandard96082nd order tetrahedral
    Table 1: Mesh details for each case

    The tetrahedral meshes were computed using SimScale’s standard mesh algorithm and automatic sizing:

    tetrahedral mesh validation case hollow sphere convection radiation
    Figure 2: Finite tetrahedral elements mesh used on cases A and B

    Simulation Setup

    Material:

    • Density ρ= 1 kg/m3
    • Thermal conductivity κ= 40 W/(m.K)
    • Specific heat Cp= 1 J/(kg.K)

    Boundary Conditions:

    • Convective Heat Flux:
      • Applied on face ABCD
      • Reference temperature T0= 20 °C
      • Heat transfer coefficient of 133.5 W/(m.K)
    • Radiation Heat Flux:
      • Applied on face EFGH
      • Modeled as a temperature-dependent surface heat flux
      • Heat flux dependent on temperature, according to the Boltzmann equation:

    (1)φ=σϵ[(T0+273.15)4(T+273.15)4]

    σ=5.73×108 W/(m3.K4)

    ϵ=0.6

    T0=500 °C

    Note

    Equation 1 was used to compute a table for a temperature range of 0 to 100 °C and uploaded to the platform to model the radiation heat flux with the surface heat flux boundary condition. This temperature-dependent condition dictates the need of using nonlinear heat transfer analysis.

    Reference Solution

    The reference solution comes from analytical expressions solved numerically, as presented in [TPNV01]1. The reference solution is presented as the temperature at the internal and external faces:

    Tint=91.77 °C

    Text=71.22 °C

    Result Comparison

    Comparison of average temperatures at internal (EFGH) and external (ABCD) faces with the reference solution, for each case, is presented:

    CaseFaceCompute Average [K]Compute Average [°C]Reference
    [°C]
    Error
    AINTERNAL364.89391.74391.77-0.03 %
    EXTERNAL344.35971.20971.22-0.16 %
    BINTERNAL364.92191.77191.770.00 %
    EXTERNAL344.3771.2271.220.00 %
    Table 2: Results comparison and computed errors for average temperatures for both cases

    Illustration of the temperature distribution from the sphere with convection and radiation simulation, case B:

    temperature plot validation case hollow sphere convection radiation
    Figure 3: Temperature distribution contours on the body from case B

    Note

    If you still encounter problems validating you simulation, then please post the issue on our forum or contact us.

    Last updated: July 21st, 2021

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