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Documentation

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 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.

sphere geometry dimensions
Figure 1: Wireframe model of the sphere section used as the simulation domain

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.

CaseMesh typeNumber of nodesElement type
(A)1st order tetrahedral5286Standard
(B)2nd order tetrahedral38271Standard
Table 1: Characteristics of the two meshes

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

mesh first order elements tetrahedral
Figure 2: The 1st order tetrahedral mesh used for the SimScale Case (A)

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

Simulation Setup

Be Aware

All temperature dependent data in this cases is given as a function of °C (although it says °K in the Workbench)!
This is because of the equivalency between \(\frac {W}{°K}\) and \(\frac {W}{°C}\).

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\(^1\)

Time
\([s]\)
VPCS\(^1\)
\([K]\)
Case A
\([K]\)
Error
[%]
Case B
\([K]\)
Error
[%]
400 334341.182.15341.1532.14
600 500493.494-1.30493.453-1.31
800 618610.619-1.19610.572-1.20
1000 706700.66-0.756700.613-0.76
1200 774769.88-0.532769.835-0.538
1400 828823.093-0.593823.052-0.598
1600 872864.002-0.917863.964-0.87
1800 902895.450.726895.417-0.73
2000 923919.626-0.366919.598-0.369
2200 942938.212-0.402938.188-0.405
2400 956952.5-0.366952.48-0.368
Table 2: Comparison of the results for the center point temperature at different time steps

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

Time
\([s]\)
VPCS\(^1\)
\([K]\)
Case A
\([K]\)
Error
[%]
Case B
\([K]\)
Error
[%]
400 461474.8893.013474.82.99
600 608596.363-1.914596.28-1.93
800 696689.704-0.905689.628-0.92
1000 774761.458-1.62761.39-1.63
1200 828816.619-1.375816.56-1.38
1400 868859.024-1.03858.973-1.04
1600 902891.624-1.15891.58-1.16
1800 923916.685-0.684916.648-0.69
2000 942935.951-0.642935.92-0.65
2200 956950.762-0.548950.736-0.55
2400 962962.1480.015962.127-0.013
Table 3: Comparison of the results for the outer surface temperature at different time steps

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.

temperature distribution heat transfer sphere
Figure 3: The temperature distribution across the sphere for case A, during the final timestep (2400s)

Last updated: August 20th, 2020

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