# Radiation Heat Transfer Through a Cube with a Specular Interface

In this application, AcuSolve is used to simulate the heat transfer due to radiation through a specular interface within an absorbing, emitting, but not scattering solid cube. One of the cube’s walls is modeled with an isotropic external radiation source while the remainder of the cube is held at fixed temperature conditions and modeled with pure radiation, neglecting the effects of conduction.

The internal surface between the two equal solid portions of the cube is specified to be a completely specular boundary condition. AcuSolve results are compared with accepted numerically derived results of incident radiation, as described in Muresan (2004) and Łapka (2012). The close agreement of the AcuSolve results and the published data validates the solver’s ability to model cases with semi-transparent media with differing material properties using the discrete ordinates radiation model.

## Problem Description

The problem consists of a solid region with fixed material properties similar to glass separated into two equal size cuboids to form a symmetrical cube with a side length of 1.0 m. The left most wall (minimum z) receives an external radiation source at a fixed temperature of 450.0 K (177 ºC) while the right most wall (maximum z) is prescribed a fixed temperature of 375.0 K (102 ºC), as shown in the following image, not drawn to scale. The solid, semi-transparent material is specified for each glass cuboid with n1=1.5, κ1=0.2 for glass1 and n2=1.25, κ2=2.0 for glass2. Only the solution to the radiative transfer equation on a discrete number of ordinate directions is considered, thus neglecting conduction. The solid regions equilibrate to a steady radiative state and are compared with a semi-analytical solution for the incident radiation across the centerline of the cube.

The simulation was performed as a three-dimensional problem by constructing a volume mesh that contains all hexahedral elements with 20, 20, 40 cells in the x, y, z directions.

## AcuSolve Results

The AcuSolve solution converges to a steady state and the results reflect the mean radiative state within the cube. Since the left side of the cube receives the external radiation source, the incident radiation is highest at that location. The transmission of the radiation decreases linearly within the first cuboid until it reaches the interface between the two materials. The discontinuity in the radiation field is demonstrated at the interface location, resulting from the specular reflection of radiation as it tries to travel from one material to the other. As the incident radiation passes through the second glass material, the simulation demonstrates how the material’s higher absorption properties will produce a non-linear irradiance distribution. The following image shows the incident radiation contour within the two media along with the two-dimensional plot of the normalized incident radiation as a function of the non-dimensional cube length, where the incident radiation is normalized by $G*=\sigma T$ 4. The incident radiation is compared with the numerical solution as described in Muresan, et al. (2004).

## Summary

The AcuSolve solution compares well with numerical results for heat transfer due to radiation within a cube containing a specular interface. In this application, two arbitrary materials are subjected to a constant external radiative heat source and the media are defined to absorb, emit, but not scatter the irradiation. As a result of the prescribed radiative heat flux on the left most wall and the fixed temperature constraint across the entire domain, each media receives a non-trivial, varying amount of incident radiation. The AcuSolve solution for the incident radiation across the interface matches well compared to the numerical solution with only a minor difference at the far-right hand side of the domain. The discrete ordinate model requires a fairly large number of ordinate directions in order to obtain the highest accuracy solution. The published results are produced using the S14 quadrature, which is equivalent to 224 discrete ordinate directions.

## Radiation Heat Transfer Through a Cube with a Specular Interface

Global

• Problem Description
• Analysis type - Steady State
• Turbulence equation - Advective Diffusive
• Radiation equation - Discrete Ordinate
• Solver Settings
• Convergence tolerance - 1.0e-3
• Relaxation Factor - 0.3
• Flow - off
• Temperature - off
• Nodal_Initial_Condition
• Temperature
• Type - Constant
• Default Value - 300.0 K

Materials

• Glass_HM
• Density
• Type - Constant
• Density - 2500.0
• Specific Heat
• Type - Constant
• Conductivity - 830
• Conductivity
• Type - Constant
• Conductivity - 1.3
• Allow Participating Media Radiation - On
• Absorption Coefficient - 0.2
• Scattering Coefficient - 0.0
• Phase Coefficient - 0.0
• Refractive Index - 1.5
• Glass2_HM
• Density
• Type - Constant
• Density - 2500.0
• Specific Heat
• Type - Constant
• Conductivity - 830
• Conductivity
• Type - Constant
• Conductivity - 1.3
• Allow Participating Media Radiation - On
• Absorption Coefficient - 2.0
• Scattering Coefficient - 0.0
• Phase Coefficient - 0.0
• Refractive Index - 1.25
Emissivity Model
• BB05
• Emissivity - 0.5
• BB
• Emissivity - 1.0
Volumes
• S1
• Type - Solid
• Material model - Glass_HM
• S2
• Type - Solid
• Material model - Glass2_HM
Surfaces
• Wall_S1
• Simple Boundary Condition
• Type - Wall
• Temperature BC type - Value
• Temperature - 450.0 K
• Radiation interface type - external
• External emissivity model - BB
• Diffused fraction - 0.0
• Wall_S2
• Simple Boundary Condition
• Type - Outflow
• Type - Opening
• Opening temperature - 375.0 K
• Emissivity model - BB05
• Diffused fraction - 0.0
• Interface_S1
• Simple Boundary Condition
• Type - Auto Wall
• Radiation interface type - internal
• Diffused fraction - 0.0
• Interface_S2
• Simple Boundary Condition
• Type - Auto Wall
• Radiation interface type - internal
• Diffused fraction - 0.0
• Slip_1_S1
• Simple Boundary Condition
• Type - Slip
• Slip_2_S1
• Simple Boundary Condition
• Type – Slip
• Slip_3_S1
• Simple Boundary Condition
• Type - Slip
• Slip_4_S1
• Simple Boundary Condition
• Type - Slip
• Slip_1_S2
• Simple Boundary Condition
• Type - Slip
• Slip_2_S2
• Simple Boundary Condition
• Type - Slip
• Slip_3_S2
• Simple Boundary Condition
• Type - Slip
• Slip_4_S2
• Simple Boundary Condition
• Type - Slip

## References

Muresan, Cristian & Vaillon, Rodolphe & Ménézo, Christophe & Morlot, Rodolphe. (2004). Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation. Journal of Quantitative Spectroscopy and Radiative Transfer. 84. 551-562. 10.1016/S0022-4073(03)00271-1.

Łapka, Piotr & Furmański, Piotr. (2012). Fixed Cartesian grid based numerical model for solidification process of semi-transparent materials II: Reflection and refraction or transmission of the thermal radiation at the solid–liquid interface. International Journal of Heat and Mass Transfer. 55. 4953–4964. 10.1016/j.ijheatmasstransfer.2012.03.077.