Surface-to-Surface Radiation with Diffuse and Specular Reflection

This tutorial shows how to use the Surface-to-Surface Radiation interface to simulate radiative heat transfer with radiation between diffuse emitters and diffuse-and-specular reflectors. This application is separated in two parts. The first part focuses on a validation test for the radiative heat flux computed from the ray shooting algorithm, wherein results are compared to an analytical solution for two parallel plates at constant temperature. The second part introduces coupling with Heat Transfer in Solids.

Introduction

Opaque surfaces can reflect incident energy in a diffuse and/or specular way. The smoother the opaque surface, the more specular the reflection is expected to be.

Following Ref. 1, the hemispherical reflectivity ρ of an opaque surface reads

(1)

where ρs, ρd, and ε are the specular reflectivity, the diffuse reflectivity, and the diffuse emissivity of the surface, respectively.

For an irradiation G, the outgoing energy is the radiosity εEb + ρdG plus the energy reflected specularly ρsG, so that the net radiative heat flux reads

with Eb the blackbody emissive power

and G is the irradiation

The latter expression depends on the external irradiation

and on the differential specular view factor

For more details on the terminology see Ref. 1.

Model Definition

The model uses the ray shooting algorithm as the surface-to-surface radiation method to model radiative heat transfer. The heat flux at each boundary element on the surface is computed by sending rays outward from the surface to query the temperatures of other surfaces in the geometry.

When is selected in the section, the user can adjust three parameters to speed up the simulation and/or to get more accurate results, namely the , the , and the . These settings are described in the Settings for the Surface-to-Surface Radiation Interface section of the Heat Transfer Module User’s Guide. This benchmark model illustrates the importance of the radiation resolution.

The Opaque Surface feature is used to model surfaces that are diffuse emitters and diffuse-specular reflectors. The emissivity and the specular reflectivity are defined in the section. The diffuse reflectivity is then set from Equation 1.

The model is separated into two parts.

In the first part, the numerical result obtained with the Surface-to-Surface Radiation interface is compared to the exact analytical solution.

This computes the heat flux at the surfaces of two identical infinitely long (out-of-plane in Figure 1) parallel plates placed in cold surroundings with mixed diffuse-specular radiation at their surfaces.

The geometry is illustrated in Figure 1. For the benchmark model, the lower and upper plates have the same temperature, Tl = Tu = T, the same emissivity, εl = εu = ε, and the same probability of specular reflection, such that ρl = ρu = ρ.

Figure 1: Schematics of the problem. The width of the plates is w = 10 cm, their thickness th = w/20, and the distance between the plates set to d = 10 cm. The temperature, emissivity and probability of specular reflection are equal for both plates and respectively set to T = 300 K, ε = 0.6, and ρs = 0.32.

Using symmetries, it is possible to determine the heat flux (ql = qu = q) on the lower and upper plates using the following analytical solution (see Ref. 1):

Here ξ = x/d, W = w/d, and Ψ = q/Eb. The heat flux can be computed using numerical quadrature; a typical solution is presented in Figure 2.

The second part keeps the parallel plates arrangement (same geometry) but changes the surface properties and couple the radiation model developed above to the Heat Transfer in Solids interface. Table 1 displays the surface parameters used for this part of the application.

Table 1: Surface parameters.
Surfaces	ε	ρs
Lower plate	0.9	0.01
Upper plate	0.6	0.32

For this model, the upper plate, made of copper, is heated locally from the top. The lower plate, made of quartz, is heated by the radiation emitted from the upper plate. Both plates simultaneously loss heat by natural convection. The plates’ surrounding temperature is set to 300 K.

Results and Discussion

Figure 2 shows a comparison of the normalized heat flux at the plate’s surfaces for the exact and numerical solutions (benchmark model). A good agreement is observed; the relative error in the radiative heat flux is below one percent.

When the radiative heat flux computed using the Surface-to-Surface Radiation interface is coupled to the Heat Transfer in Solids interface, the temperature field shown in Figure 3 is obtained. Figure 4 displays the normalized heat flux at the top of the lower plate and at the bottom of the upper plate.

Figure 2: Normalized heat flux at the bottom of the upper plate for T = 300 K, ε = 0.6, ρs = 0.32 and w/d = 1. The black circles represent the exact solution obtained from numerical quadrature see Ref. 1.

Figure 3: Temperature of the inner surfaces of the plates for the coupled model.

Figure 4: Radiative heat flux at the surface of the plates for the coupled model.

Notes About the COMSOL Implementation

The first study could have been done using lines instead of rectangles to model parallel plates, and even making use of the Symmetry for Surface-to-Surface Radiation feature to reduce the number of degrees of freedom. The Radiation Settings to use in the second study for such a geometry are determined from the first (validation) study.

Reference

1. M.F. Modest, Radiative Heat Transfer, 2nd. ed., Academic Press, 2003.

Heat_Transfer_Module/Thermal_Radiation/parallel_plates_diffuse_specular_ray_shooting

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
w	10[cm]	0.1 m	Width of the plates
d	10[cm]	0.1 m	Distance between the plates
th	w/20	0.005 m	Thickness of the plates
T0	300[K]	300 K	Room temperature

Geometry 1

Lower plate

In the toolbar, click

In the window for , locate the section.

In the text field, type w.

In the text field, type th.

Locate the section. From the list, choose .

In the text field, type -(d+th)/2.

Locate the section. Select the check box.

From the list, choose .

In the text field, type Lower plate.

Upper plate

Right-click and choose .

In the window for , locate the section.

In the text field, type (d+th)/2.

In the text field, type Upper plate.

Definitions

Study 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type Study 1.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
epsilon_mat	0.6		Emissivity
rhod_mat	0.08		Diffuse reflectivity

Interpolation 1 (int1)

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file parallel_plates_diffuse_specular_data.txt.

Locate the section. In the table, enter the following settings:


Argument	Unit
t	1

In the table, enter the following settings:


Function	Unit
reference_solution	1

Locate the section. In the text field, type reference_solution.

Click the

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

Click .

Surface-to-Surface Radiation (rad)

In the window, under click .

In the window for , locate the section.

Find the subsection. From the list, choose .

The resolution is increased because rays are emitted from the plates in all direction so some of them do not even strike the plates.

From the list, choose .

Click to expand the section. Find the subsection. Select the check box. When specular radiation is present it is good practice to do this because it improves the accuracy of the specular radiation evaluation at a slight computational cost increase. Because the geometry of this model contains only flat surfaces, selecting the check box has no influence on accuracy but it makes the model ready for geometries with curved surfaces.

The feature Opaque Surface allows to handle mixed diffuse and specular reflection.

Opaque Surface 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the T text field, type T0.

Locate the section. In the Tamb text field, type 0.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the Tinit text field, type T0.

Materials

Plates, boundaries

In the window, under right-click and choose .

In the window for , type Plates, boundaries in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Surface emissivity	epsilon_rad	epsilon_mat	1	Basic
Diffuse reflectivity	rho_d_rad	rhod_mat	1	Basic

Mesh 1

Mapped 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

From the list, choose .

Click

Study 1

In the toolbar, click

Results

Table 1

In the toolbar, click

In the window for , locate the section.

Click .

Browse to the model’s Application Libraries folder and double-click the file parallel_plates_diffuse_specular_data.txt.

Validation

In the toolbar, click

In the window for , type Validation in the text field.

Click to expand the section. From the list, choose .

In the text area, type Dimensionless radiative heat flux.

Locate the section.

Select the check box. In the associated text field, type x/w.

Select the check box. In the associated text field, type q/(\varepsilon Eb).

Line Graph 1

Right-click and choose .

Select Boundary 5 only.

In the window for , locate the section.

In the text field, type reference_solution(x/w).

Locate the section. From the list, choose .

In the text field, type x/w.

Click to expand the section. Find the subsection. From the list, choose .

Find the subsection. From the list, choose .

From the list, choose .

Click to expand the section. From the list, choose .

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
theory

Line Graph 2

In the window, right-click and choose .

Select Boundary 5 only.

In the window for , locate the section.

In the text field, type rad.rflux/(rad.epsilon*rad.ebu).

Locate the section. From the list, choose .

In the text field, type x/w.

Locate the section. From the list, choose .

Locate the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
simulation

Validation

Click the

button in the toolbar.

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add Multiphysics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Quartz

In the window, under right-click and choose .

In the window for , type Quartz in the text field.

Select Domain 1 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	1.1	W/(m·K)	Basic
Density	rho	2200	kg/m³	Basic
Heat capacity at constant pressure	Cp	480	J/(kg·K)	Basic

Copper

Right-click and choose .

Select Domain 2 only.

In the window for , type Copper in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	400	W/(m·K)	Basic
Density	rho	8700	kg/m³	Basic
Heat capacity at constant pressure	Cp	385	J/(kg·K)	Basic