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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 \(m\).

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

A | B | C | D | E | F | |

x | 0 | 0.19 | 0 | 0.2 | 0 | 0 |

y | 0.19 | 0 | 0.2 | 0 | 0 | 0 |

z | 0 | 0 | 0 | 0 | 0.19 | 0.2 |

**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 \(\kappa\).

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 |

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

**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 \(K\).

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

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: November 7th, 2023

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