Validation Case: Radiation in Cylinder

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 cylinder:

• Net radiative heat flux, \(Q_r\).

The simulation results obtained from SimScale were compared to the results presented in [1].

Geometry

The following cylinder was used as the computational domain for this simulation. Faces A1, A3, and A2 mark the bottom flat face, top flat face, and the curvature respectively.

The height, \(h\), and radius, \(r\), of the cylinder are 150 \(mm\) and 25 \(mm\) respectively.

Analysis Type and Mesh

Tool type: OpenFoam

Analysis type: Incompressible convective heat transfer with radiation

Turbulence model: Laminar and 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 2.5 million tetrahedral cells.

Simulation Setup

Material/Fluid:

• Air
  • Kinematic viscosity \(\nu\) = 1.2784e-5 \(m^2/s\)

Boundary Conditions:

• No-slip walls:
  • Fixed temperature of 300.15 \(K\) on the top surface (A3).
  • Fixed temperature of 1773 \(K\) on the rest of the faces ( A1 and A2).
• 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

When comparing radiative analytical results and SimScale results, there are a few points that need to be considered:

• 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 some part of it will heat the enclosed fluid volume as well. However as radiation is the dominant heat transfer mode with high temperatures, the difference is not significant. Also, laminar and turbulent behaviors were evaluated which provided the same results.
• 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 for different mesh resolutions along with the analytical results:

Surface	\(Q_r\) [\(W\)] Analytical	\(Q_r\) [\(W\)] SimScale Coarse radiation resolution	\(Q_r\) [\(W\)] SimScale Moderate radiation resolution	\(Q_r\) [\(W\)] SimScale Fine radiation resolution	Relative Error (for finest resolution) [ (%) ]
A1	-28.95	-28.846	-28.847	-28.847	-0.357
A2	-1070.36	-1073.730	-1073.730	-1073.188	0.264
A3	1099.31	1099.619	1099.619	1099.619	0.028
Balance	0	-2.975	-2.975	-2.416

• The results for area A2 are the least accurate and SimScale predicts higher heat loss value for this surface. This is due to the fact that A2 being the largest surface is in more contact with the fluid and thus loses more heat. This affects the heat balance, which analytically should be zero.
• 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.

Last updated: September 4th, 2023