Validation Case: Convective Heat Transfer of a Sphere

This case belongs to thermodynamics. The aim of this test case is to validate the following parameter during a convective heat transfer — governed by the Nusselt number — in a sphere for different time steps:

• Temperature at the center of the sphere (M).
• Temperature on the outer surface of the sphere (ABCD).

The simulation results of SimScale were compared to the results presented by VPCS in [TTLV01]\(^1\) .

Geometry

Only a portion of a sphere with a diameter of 0.2 \(m\) is used for the analysis, as shown in the figure below. M is the center of the sphere and the face ABCD represents the outer surface.

Analysis type and Domain

Tool type: Code_Aster

Analysis type: Heat transfer, linear

Time dependency: Transient

Mesh and element types: Two meshes, used in cases (A) and (B), were created with the standard meshing algorithm on the SimScale platform.

The mesh for case A was created using 1st order tetrahedral elements:

For case B, a similar mesh with 2nd order tetrahedral elements was used.

Simulation Setup

Material/Solid:

• Isotropic
  • Density (\(ρ\)) = 7200 \(kg \over \ m³\) ,
  • Thermal conductivity (\(\kappa\)) = 48.822 \(W \over \ (m \times \ K\ ) \),
  • Specific heat = 669 \(J \over \ kg \times \ K\)

Initial Conditions:

• Initial Temperature \(T_{initial}\) = 20 \(K\)

Loads:

• Convective heat flux on face ABCD
  • Reference Temperature \(T_{0}\) = 1000 \(K\)
  • Heat transfer coefficient = 232.5 \(W \over \ (K \times \ m² \ ) \)

Results Comparison

In the table below are the temperature values at different time steps at the center of the sphere. The SimScale results are compared against the results from VPCS\(^1\)

And at the outer surface the following comparison stats for the temperature were obtained:

Both cases are in a good agreement with the reference results with maximum error < 3%, which showcases the ability of SimScale to successfully analyze the heat transfer of a sphere.