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]\(^1\).

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 \(\kappa\).

Case

Mesh Type

Nodes

Thermal Conductivity \(\kappa\)

Element Type

(A)

1st order standard

172833

20 \([\frac {W}{m.K}]\)

Standard

(B)

1st order standard

172833

22 \([\frac {W}{m.K}]\)

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.

Simulation Setup

Material:

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]\(^1\).

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

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