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    Validation Case: Design Analysis of a Spherical Pressure Vessel

    This design analysis of a spherical pressure vessel validation case belongs to thermomechanics. This test case aims to validate the following parameters:

    • Transient thermostructural analysis

    The simulation results of SimScale were compared to the analytical results presented in [Afkar]1.

    The geometry consists of 1/8th of a sphere, with an inner radius of 0.19 m and an outer radius of 0.2 m.

    sphere geometry for pressure vessel validation
    Figure 1: 1/8th sphere geometry used in the present validation project

    The coordinates for the points in the sphere are as tabulated below:

    ABCDEF
    x00.1900.200
    y0.1900.2000
    z00000.190.2
    Table 1: 1/8th sphere dimensions in meters

    Tool Type: Code_Aster

    Analysis Type: Transient thermomechanical, linear

    Mesh and Element Types: The mesh for cases A and B was created with the standard algorithm, with first order elements.

    The setup from cases A and B is the same, except for the thermal conductivity κ.

    CaseMesh TypeNodesThermal Conductivity κElement Type
    (A)1st order standard17283320 [Wm.K]Standard
    (B)1st order standard17283322 [Wm.K]Standard
    Table 2: Overview of the mesh, creep formulation, and element technology used for each case

    Find below the mesh used for both cases. It’s a standard mesh with first order tetrahedral cells.

    first order standard mesh for a sphere
    Figure 2: First order standard mesh used for cases A and B

    Material:

    • Steel (linear elastic)
      • E = 190 GPa
      • ν = 0.305
      • ρ = 7750 kg/m³
      • κ = 20 [Wm.K] and 22 [Wm.K] for cases A and B, respectively;
      • Expansion coefficient = 9.7e-6 1/K
      • T0 Reference temperature = 300 K
      • Specific heat = 486 Jkg.K

    Initial Conditions

    Temperature is 300 K in the entire pressure vessel.

    Boundary Conditions:

    • Constraints
      • dx = 0 on face ACFE;
      • dy = 0 on face BDFE;
      • dz = 0 on face ACDB.
    • Surface loads
      • Pressure boundary condition on face ABE. The pressure increases linearly from 0 MPa to 1 MPa according to formula P=(0.2e6).t, where t is time from 0 to 5 seconds;
      • Fixed temperature value boundary condition on face ABE. Temperature is increasing linearly, from 300 K to 500 K according to formula T=40.t+300, where t is time from 0 to 5 seconds;
      • Convective heat flux boundary condition on face CFD. The heat transfer coefficient is 90 WK.m2 and T0 reference temperature is 300 K.

    The analytical solution is given by the equations presented in [Afkar]1.

    Since no value for thermal conductivity κ was provided, the values of 20 Wm.K and 22 Wm.K were used. For the final time step, the SimScale results for von Mises stress [MPa] and temperature [K] over the edge EF are compared to those from [Afkar]1.

    spherical pressure validation result comparison
    Figure 3: Comparing temperature and von Mises stress results for cases A and B with those from [Afkar]¹.

    In Figure 4, we can see how temperature is changing in the sphere’s width, for the last time step:

    temperature contours sphere validation
    Figure 4: Temperature on the 1/8th sphere, for time = 5 seconds

    Last updated: November 7th, 2023