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Radiative Cooling of a Glass Plate
Introduction
In glass production, the glass melt is cooled down mainly by radiation. To avoid stresses it is important to evenly cool the glass body.
Numerical treatment of radiative heat transfer helps to optimize this cooling process. The governing equation — the Radiative Transfer Equation (RTE) — is an integro-differential equation that requires a lot of computational resources to be solved. Therefore, COMSOL Multiphysics offers three common methods to solve the RTE along with the heat transfer equation.
This tutorial is intended to show the typical set up of all methods computing the heat transfer by radiation inside a gray medium.
Figure 1: Cylindrical glass plate.
Model Definition
The model geometry is shown in Figure 1. It is a cylinder of radius r = 5 cm and height h = 1.5 cm. The radiative cooling starts from an initial temperature of 600°C due to radiation into an ambient surrounding at 20°C. Convective cooling is neglected, which is reasonable for high temperatures as in this model.
The material properties for the glass body are summarized in Table 1.
Table 1: Material Properties for Glass.
Material Property
Value
Thermal conductivity
1.2 W/(m·K)
Density
2200 kg/m3
Heat capacity at constant pressure
850 J/(kg·K)
Scattering coefficient
0
Scattering effects are neglected. For the absorption coefficient, a parametric sweep is used to get results for k = 5, 70, 120. The boundaries of the glass body have a surface emissivity, ε, equal to 1.
Thermal Analysis
A detailed explanation about the discrete ordinates method and the P1 method can be found in Radiative Heat Transfer in Finite Cylindrical Media or Radiative Heat Transfer in Finite Cylindrical Media — P1 Method respectively.
The Rosseland approximation is a simplified method that results in an additional nonlinear term for the thermal conductivity. Hence, this method has almost no impact on the computational cost.
For large optical thickness where the integral of the absorption coefficient along a typical path is large, radiation effects only spread at its close surrounding and does not travel far through the medium before being absorbed or scattered. This leads to a diffusion-like equation for the radiative heat flux (Ref. 1):
For a gray medium (after integration over all wave numbers) the radiative heat flux depends on the temperature gradient and can be expressed as:
where kr is a highly nonlinear coefficient, considered as a conductivity, for radiative transfer of the form
with βr, the Rosseland-mean extinction coefficient, σ the scattering coefficient and n the refractive index. Thus the Rosseland approximation method is also called the diffusion method.
Results and Discussion
The figures below compare the results for a low and a high absorption coefficient. The P1 method provides a very good approximation for lower absorption coefficients (Figure 2), but with increasing absorption coefficients the results differ increasingly.
Figure 2: Vertical temperature distribution at the center of the cylinder for k = 5.
The Rosseland approximation provides a fast and satisfying solution for the temperature field when a very high absorption coefficient or a rather high optical thickness is considered (Figure 3).
Figure 3: Vertical temperature distribution at the center of the cylinder for k = 120.
Both methods need to be used carefully and it is recommended to validate their applicability. If they can be used, they provide a very fast solution compared to the highly accurate discrete ordinates method.
Reference
1. M.F. Modest, Radiative Heat Transfer, 2nd ed., Academic Press, 2003.
Application Library path: Heat_Transfer_Module/Thermal_Radiation/glass_plate
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Heat Transfer > Radiation > Heat Transfer with Radiation in Participating Media.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
nr
1.45
1.45
Refractive index
T0
600[degC]
873.15 K
Initial temperature
T_amb
20[degC]
293.15 K
Ambient temperature
k
5[1/m]
5 1/m
Absorption coefficient
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose cm.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 5.
4
In the Height text field, type 1.5.
5
In the Geometry toolbar, click  Build All.
Create a user-defined material for the glass body.
Global Definitions
Glass
1
In the Model Builder window, under Global Definitions right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Glass in the Label text field.
Materials
Material Link 1 (matlnk1)
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose More Materials > Material Link.
Material Link 2 (matlnk2)
1
Right-click Materials and choose More Materials > Material Link.
2
In the Settings window for Material Link, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose All boundaries.
Global Definitions
Glass (mat1)
1
In the Model Builder window, under Global Definitions > Materials click Glass (mat1).
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
1.2[W/(m*K)]
W/(m·K)
Basic
Density
rho
2200[kg/m^3]
kg/m³
Basic
Heat capacity at constant pressure
Cp
850[J/(kg*K)]
J/(kg·K)
Basic
For radiative heat transfer, the absorption and scattering coefficients are also needed. Add these properties to the material.
4
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Absorption coefficient
kappaR
k
1/m
Basic
Scattering coefficient
sigmaS
0[1/m]
1/m
Basic
The boundary condition for radiative cooling requires a surface emissivity. One way to define this coefficient is to add a material for the boundaries.
5
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Surface emissivity
epsilon_rad
1
1
Basic
Start with the discrete ordinates method to compute the radiative cooling of the glass plate. This method provides the most accurate solution for arbitrary radiation models and is therefore the default method for radiation in participating media.
Radiation in Participating Media (rpm)
1
In the Model Builder window, under Component 1 (comp1) click Radiation in Participating Media (rpm).
2
In the Settings window for Radiation in Participating Media, locate the Participating Media Settings section.
3
Find the Radiation settings subsection. In the nr text field, type nr.
Heat Transfer in Solids (ht)
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Heat Transfer in Solids (ht) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the T text field, type T0.
Set the initial temperature to 600°C.
Surface-to-Ambient Radiation 1
1
In the Physics toolbar, click  Boundaries and choose Surface-to-Ambient Radiation.
2
In the Settings window for Surface-to-Ambient Radiation, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Surface-to-Ambient Radiation section. In the Tamb text field, type T_amb.
Mesh 1
Build a suitable mesh manually. First, mesh the surface with a free triangular mesh and then add a swept mesh.
Free Triangular 1
1
In the Mesh toolbar, click  More Generators and choose Free Triangular.
2
Select Boundary 4 only.
Size 1
1
In the Mesh toolbar, click Size Attribute and choose Extra Fine.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Edge.
4
Select Edges 4, 5, 8, and 11 only.
Size 2
1
In the Mesh toolbar, click Size Attribute and choose Fine.
2
In the Settings window for Size, click  Build Selected.
Swept 1
In the Mesh toolbar, click  Swept.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 8.
4
Click  Build Selected.
The mesh should look like that in the figure below.
Study 1: DOM
Next, rename the study node to identify the studies and the related solutions easily.
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1: DOM in the Label text field.
Step 1: Time Dependent
The model only compares the results after 10 seconds. To keep the file size small, let COMSOL Multiphysics store only this time step in the file. The computational time step is chosen automatically.
1
In the Model Builder window, under Study 1: DOM click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type 10.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
k (Absorption coefficient)
5 70 120
1/m
When a model involves the discrete ordinates method, the memory requirement during computation is generally lead by the radiative intensities since there are as many dependent variables as the number of directions accounted for. By default with the discrete ordinates method, a Direct, radiation variables solver is used to solve for the radiative intensities since it is generally the fastest. If it requires too much memory, the AMG, radiation variables iterative solver is an alternative that is more economical in memory. Follow these steps to select the linear solver used for the computation of radiative intensities.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Solution 1 (sol1) > Time-Dependent Solver 1 node.
4
In the Model Builder window, expand the Solution 1 (sol1) > Time-Dependent Solver 1 > Segregated 1 node, then click Radiative Intensities 1.
This segregated step controls the computation of all radiative intensities. If you have more than 10 GB of RAM, keep the default Direct, radiation variables solver in the Linear solver field. If you have less, switch to the AMG, radiation variables iterative solver.
5
In the Settings window for Segregated Step, locate the General section.
6
From the Linear solver list, choose either Direct, radiation variables or AMG, radiation variables.
7
In the Study toolbar, click  Compute.
Results
Temperature (ht)
Next, run the same model but with the P1 method. The only modification to do is to change the Radiation discretization method in the Heat Transfer with Radiation in Participating Media settings window.
Add Component
In the Model Builder window, right-click the root node and choose Add Component > 3D.
Geometry 2
1
In the Settings window for Geometry, locate the Units section.
2
From the Length unit list, choose cm.
Mesh 2
Import 1
1
In the Mesh toolbar, click  Import.
2
In the Settings window for Import, locate the Import section.
3
From the Source list, choose Meshing sequence.
4
Click Import.
Materials
Material Link 3 (matlnk3)
In the Model Builder window, under Component 2 (comp2) right-click Materials and choose More Materials > Material Link.
Material Link 4 (matlnk4)
1
Right-click Materials and choose More Materials > Material Link.
2
In the Settings window for Material Link, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose All boundaries.
Heat Transfer in Solids (ht), Radiation in Participating Media (rpm)
1
In the Model Builder window, under Component 1 (comp1), Ctrl-click to select Heat Transfer in Solids (ht) and Radiation in Participating Media (rpm).
2
Right-click and choose Copy.
Heat Transfer in Solids (ht2)
In the Model Builder window, right-click Component 2 (comp2) and choose Paste Multiple Items.
Heat Transfer in Solids (ht2), Radiation in Participating Media (rpm2)
In the Messages from Paste dialog, click OK.
Multiphysics
Heat Transfer with Radiation in Participating Media 2 (htrpm2)
In the Physics toolbar, click  Multiphysics Couplings and choose Domain > Heat Transfer with Radiation in Participating Media.
Radiation in Participating Media (rpm2)
1
In the Model Builder window, under Component 2 (comp2) click Radiation in Participating Media (rpm2).
2
In the Settings window for Radiation in Participating Media, locate the Participating Media Settings section.
3
Find the Radiation settings subsection. From the Radiation discretization method list, choose P1 approximation.
To compare the results, a second study is used to compute the same setup with the P1 method. Add an empty study and copy the settings from the first one.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Empty Study.
4
Click the Add Study button in the window toolbar.
Study 2: P1
In the Settings window for Study, type Study 2: P1 in the Label text field.
Study 1: DOM
Parametric Sweep, Step 1: Time Dependent
1
In the Model Builder window, under Study 1: DOM, Ctrl-click to select Parametric Sweep and Step 1: Time Dependent.
2
Right-click and choose Copy.
Study 2: P1
In the Model Builder window, right-click Study 2: P1 and choose Paste Multiple Items.
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
2
In the Solve for column of the table, under Component 1 (comp1), clear the checkboxes for Heat Transfer in Solids (ht) and Radiation in Participating Media (rpm).
3
In the Solve for column of the table, under Component 1 (comp1) > Multiphysics, clear the checkbox for Heat Transfer with Radiation in Participating Media 1 (htrpm1).
Study 1: DOM
1
In the Model Builder window, under Study 1: DOM click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 2 (comp2), clear the checkboxes for Heat Transfer in Solids (ht2) and Radiation in Participating Media (rpm2).
4
In the Solve for column of the table, under Component 2 (comp2) > Multiphysics, clear the checkbox for Heat Transfer with Radiation in Participating Media 2 (htrpm2).
Study 2: P1
1
In the Study toolbar, click  Compute.
The same procedure applies for solving with the Rosseland approximation.
Add Component
In the Model Builder window, right-click the root node and choose Add Component > 3D.
Geometry 3
1
In the Settings window for Geometry, locate the Units section.
2
From the Length unit list, choose cm.
Mesh 3
Import 1
1
In the Mesh toolbar, click  Import.
2
In the Settings window for Import, locate the Import section.
3
From the Source list, choose Meshing sequence.
4
Click Import.
Materials
Material Link 5 (matlnk5)
In the Model Builder window, under Component 3 (comp3) right-click Materials and choose More Materials > Material Link.
Material Link 6 (matlnk6)
1
Right-click Materials and choose More Materials > Material Link.
2
In the Settings window for Material Link, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose All boundaries.
Heat Transfer in Solids (ht2)
In the Model Builder window, under Component 2 (comp2) right-click Heat Transfer in Solids (ht2) and choose Copy.
Heat Transfer in Solids (ht3)
In the Model Builder window, right-click Component 3 (comp3) and choose Paste Heat Transfer in Solids.
Heat Transfer in Solids (ht2)
In the Messages from Paste dialog, click OK.
Heat Transfer in Solids (ht3)
Solid 1
In the Model Builder window, under Component 3 (comp3) > Heat Transfer in Solids (ht3) click Solid 1.
Optically Thick Participating Medium 1
1
In the Physics toolbar, click  Attributes and choose Optically Thick Participating Medium.
2
In the Settings window for Optically Thick Participating Medium, locate the Optically Thick Participating Medium section.
3
In the nr text field, type nr.
Add Study
1
Go to the Add Study window.
2
Find the Studies subsection. In the Select Study tree, select Empty Study.
3
Click the Add Study button in the window toolbar.
4
In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Solids (ht).
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3: Rosseland
In the Settings window for Study, type Study 3: Rosseland in the Label text field.
Study 2: P1
Parametric Sweep, Step 1: Time Dependent
1
In the Model Builder window, under Study 2: P1, Ctrl-click to select Parametric Sweep and Step 1: Time Dependent.
2
Right-click and choose Copy.
Study 3: Rosseland
In the Model Builder window, right-click Study 3: Rosseland and choose Paste Multiple Items.
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
2
In the Solve for column of the table, under Component 2 (comp2), clear the checkboxes for Heat Transfer in Solids (ht2) and Radiation in Participating Media (rpm2).
3
In the Solve for column of the table, under Component 2 (comp2) > Multiphysics, clear the checkbox for Heat Transfer with Radiation in Participating Media 2 (htrpm2).
Study 1: DOM
1
In the Model Builder window, under Study 1: DOM click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 3 (comp3), clear the checkbox for Heat Transfer in Solids (ht3).
Study 2: P1
1
In the Model Builder window, under Study 2: P1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 3 (comp3), clear the checkbox for Heat Transfer in Solids (ht3).
Study 3: Rosseland
In the Study toolbar, click  Compute.
Results
Incident Radiation (rpm2)
To compare the results, add a temperature plot along the centerline of the glass plate. Therefore create a cut line dataset for each parametric solution.
Cut Line 3D 1
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets and choose Cut Line 3D.
3
In the Settings window for Cut Line 3D, locate the Data section.
4
From the Dataset list, choose Study 1: DOM/Parametric Solutions 1 (sol2).
5
Locate the Line Data section. In row Point 2, set X to 0, and z to 1.5.
6
Click  Plot.
Copy the dataset twice and assign the solutions from the other studies.
Cut Line 3D 2
1
Right-click Cut Line 3D 1 and choose Duplicate.
2
In the Settings window for Cut Line 3D, locate the Data section.
3
From the Dataset list, choose Study 2: P1/Parametric Solutions 2 (6) (sol7).
Cut Line 3D 3
1
Right-click Cut Line 3D 2 and choose Duplicate.
2
In the Settings window for Cut Line 3D, locate the Data section.
3
From the Dataset list, choose Study 3: Rosseland/Parametric Solutions 3 (12) (sol12).
Now, add a 1D plot group for the temperature.
Temperature at Central Line for k = 5
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Temperature at Central Line for k = 5 in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Temperature at Central Line.
Line Graph 1
1
In the Temperature at Central Line for k = 5 toolbar, click  Line Graph.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Cut Line 3D 1.
4
From the Parameter selection (k) list, choose First.
5
From the Time selection list, choose Last.
6
Click to expand the Legends section. From the Legends list, choose Manual.
7
Select the Show legends checkbox.
8
In the table, enter the following settings:
Legends
DOM
9
In the Temperature at Central Line for k = 5 toolbar, click  Plot.
Line Graph 2
1
Right-click Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Cut Line 3D 2.
4
Locate the Legends section. In the table, enter the following settings:
Legends
P1
Line Graph 3
1
Right-click Line Graph 2 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Cut Line 3D 3.
4
Locate the Legends section. In the table, enter the following settings:
Legends
Rosseland
5
In the Temperature at Central Line for k = 5 toolbar, click  Plot.
Temperature at Central Line for k = 5
Duplicate the plot group and change the dataset to k = 120.
Temperature at Central Line for k = 120
1
In the Model Builder window, right-click Temperature at Central Line for k = 5 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Temperature at Central Line for k = 120 in the Label text field.
Line Graph 1
1
In the Model Builder window, expand the Temperature at Central Line for k = 120 node, then click Line Graph 1.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Parameter selection (k) list, choose Last.
Line Graph 2
1
In the Model Builder window, click Line Graph 2.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Parameter selection (k) list, choose Last.
Line Graph 3
1
In the Model Builder window, click Line Graph 3.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Parameter selection (k) list, choose Last.
4
In the Temperature at Central Line for k = 120 toolbar, click  Plot.
The plots are shown in Figure 2 and Figure 3. For high optical thicknesses, the Rosseland approximation better represents the overall temperature distribution, whereas the P1 approximation is appropriate for lower optical thicknesses.