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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]$$^1$$.

## Geometry

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

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

## 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 $$\kappa$$.

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.
• 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:

Last updated: August 4th, 2020