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Design Analysis of 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:

Table 1: 1/8th sphere dimensions in meters

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Transient thermomechanical

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 \(\kappa\).

CaseMesh TypeNodesThermal Conductivity \(\kappa\)Element Type
(A)1st order standard17283320 \([\frac {W}{m.K}]\)Standard
(B)1st order standard17283322 \([\frac {W}{m.K}]\)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

Simulation Setup


  • Steel (linear elastic)
    • \(E\) = 190 \(GPa\)
    • \(\nu\) = 0.305
    • \(\rho\) = 7750 \(kg/m³\)
    • \(\kappa\) = 20 \([\frac {W}{m.K}]\) and 22 \([\frac {W}{m.K}]\) for cases A and B, respectively;
    • Expansion coefficient = 9.7e-6 \(1/K\)
    • \(T_0\) Reference temperature = 300 \(K\)
    • Specific heat = 486 \(\frac {J}{kg.K}\)

Initial Conditions

Temperature is 300 \(K\) in the entire pressure vessel.

Boundary Conditions:

  • Constraints
    • \(d_x\) = 0 on face ACFE;
    • \(d_y\) = 0 on face BDFE;
    • \(d_z\) = 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 \(\frac {W}{K.m^2}\) and \(T_0\) reference temperature is 300 \(K\).

Reference Solution

The analytical solution is given by the equations presented in [Afkar]\(^1\).

Result Comparison

Since no value for thermal conductivity \(\kappa\) was provided, the values of 20 \(\frac {W}{m.K}\) and 22 \(\frac {W}{m.K}\) 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
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