Validation Case: Radiation in a Closed Ring

This validation case belongs to fluid dynamics. The aim of this test case is to validate the following parameter on the radiative surfaces of a ring:

• Net radiative heat flux, $Q_r$.

The simulation results obtained from SimScale were compared to the results presented in the reference${}^1$.

Geometry

Based on the reference literature ${}^1$, the closed ring geometry is a simplified model of the vacuum vessel of a Fusion reactor.

In the figure below the different surfaces and the nomenclature of the different areas, from A1 – A7, are shown. The geometry contains a small surface (marked in red, A5) representing the upper surface of a specific component maintained at a lower temperature.

The dimensions of the geometry are:

• Inner Radius: – 6 $m$
• Outer Radius: – 12 $m$
• Height: – 11 $m$

Analysis Type and Mesh

Tool type: OpenFoam

Analysis type: Incompressible convective heat transfer with radiation

Turbulence model: k-ω SST turbulence model

Time dependency: Steady-state

Mesh and element types: The mesh was created using the standard mesher on the SimScale platform. It has 7.9 m cells and was created in 27 minutes.

Simulation Setup

Material:

• Air
  • Kinematic viscosity $\nu$ = 1.529e-5 $m^2/s$

Boundary conditions:

• No-slip walls
  • Fixed lower temperature of 323 $K$ on the small surface (A5).
  • Fixed higher temperature of 373 $K$ for the rest of the surfaces (A1, A2, A3, A4, A6, A7)
• All surfaces have a pure black-body behavior, with an emissivity equal to 1.

Reference Solution

The analytical solution for the net radiative heat flux $Q_r$ makes use of the view factor method as mentioned under reference${}^1$.

Result Comparison

While comparing the analytical results with SimScale results, the following points were taken into account:

1. While the analytical solution takes into account only thermal radiation, in the SimScale platform, radiation is a feature of convective heat transfer. This means that the entire heat exchange will not only happen between the walls, but part of it will heat the enclosed fluid volume as well. For this case, as the temperatures are not very high, radiation cannot be assumed to be the dominant mode of heat transfer. Hence, convective losses cannot be completely neglected.
2. The quantity evaluated is the net radiative heat flux ( $Q_r$ ) in Watts [ $W$ ], that a surface  emits (or absorbs). The user can easily calculate it by assigning an “Area Integral” to every surface under Result control > Surface data.

The table below summarizes the results with different radiation resolutions along with the analytical results:

• Overall the results obtained from the SimScale platform are in good agreement with the analytical solutions, and hence this serves as a good validation of the radiation feature.
• Unlike the analytical solution where the summation of the net radiative heat transfer from all the surfaces adds up to zero, the results from the platform do not. This is because, apart from surface to surface radiation, heat exchange between fluid and surfaces also takes place.