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]\(^1\).
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.
Figure 2: First order standard mesh used for cases A and B
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]\(^1\).
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:
Figure 4: Temperature on the 1/8th sphere, for time = 5 seconds
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