View Factor Computation

The computation of view factors is central when performing heat transfer simulations using the radiosity method to account for surface-to-surface radiation. This benchmark demonstrates how to compute geometrical view factors for two concentric spheres that emit and receive radiation from each other. It compares simulation results to exact analytical values, and illustrates how to use the symmetry of the geometry to reduce the computational cost.

Introduction

The surface-to-surface radiation method in the Heat Transfer module relies on the radiosity method. When surface-to-surface radiation is activated, operators are available to compute the view factors between diffuse surfaces that irradiate each other. For some standard configurations one can determine the view factors analytically, but in engineering applications this is rarely possible. A benchmark example that compares exact analytical values with simulation results shows the accuracy of the numerical method. Here, two concentric spheres are used, for which the analytical view factors are known.

Figure 1: Model geometry.

Model Definition

The model consists of two concentric radiating spheres (Figure 1) acting as perfect emitters (surface emissivities both equal to 1). The radiation direction of each sphere is such that they irradiate each other.

Analytical View Factors

A detailed introduction on view factors is given in (Ref. 1). An arbitrary view factor F1 → 2 is defined as the proportion of radiation leaving surface S1 and intercepting surface S2, that is

If the radiosity is the same on each surface, it only depends on the geometrical configuration and thus can be calculated from geometrical properties. For two concentric spheres, labeled ext for the outer sphere and int for the inner sphere, the view factors are:

Refer to Table 1 for numerical values of these quantities.

View Factor Computation in COMSOL Multiphysics

The view factor computation is performed at each evaluation point on the boundaries. A boundary has an upside and a downside and both can be exposed to radiation. In this model the outer sphere radiates inward, which is the downward direction and the inner sphere radiates outward, which is the upward direction. Two operators are available for evaluation of the mutual surface irradiation. These are:

•

radopu(expression_upside,expression_downside)

•

radopd(expression_upside,expression_downside)

Both are available on all boundaries where surface-to-surface features are active. The radopu(,) and radopd(,) operators perform the computation on the upside and downside of the boundary where they are evaluated. The expression_upside and expression_downside arguments are the radiosity expression on the upside and downside of the boundaries that irradiate the boundary where the operators are evaluated.

To compute the geometrical view factor, e.g. Fext → int, in COMSOL Multiphysics, the following integration needs to be defined:

where Sint and Sext denote interior and exterior surfaces, Aint and Aext are corresponding surface areas.

The integration at the numerator is defined over the inner sphere, which radiates only on the upside of its surface. Hence, radopd(,) evaluates to zero and the integrand reduces to radopu(0,expression_upside). Moreover since the outer sphere radiates only on the downside of its surface the expression_downside argument should be ext.

Results and Discussion

The view factors computed with COMSOL Multiphysics, by using or not the symmetry in the geometry, are listed in Table 1 together with analytical values and absolute errors.

Table 1: View Factor Computation.
View Factor	Analytical Value	Computed Value	Error (no symmetry)	Error (symmetry)
Fint → int	0	0	0	0
Fint → ext	1	1	2.4·10-5	6.2·10-5
Fext → ext	0.91	0.91	2.2·10-6	5.6·10-6
Fext → int	0.09	0.09	5.8·10-16	2.5·10-16

In this table, analytical and computed values are slightly different but rounded to hundredths. The very small errors are listed in the last columns and reflect accurate results of the view factors provided by the simulation.

Reference

1. Michael F. Modest, Radiative Heat Transfer, 3rd ed., Academic Press, 2013

Heat_Transfer_Module/Verification_Examples/view_factor

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

You are only interested in the view factors, hence Surface-to-Surface Radiation is the only needed interface.

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Define the radii of the spheres and the analytical values for the view factors as parameters.

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
r_int	0.3[m]	0.3 m	Radius, interior sphere
r_ext	1[m]	1 m	Radius, exterior sphere
F_ext_ext	1-(r_int/r_ext)^2	0.91	Analytical view factor from exterior sphere to exterior sphere
F_ext_int	(r_int/r_ext)^2	0.09	Analytical view factor from exterior sphere to interior sphere
F_int_int	0	0	Analytical view factor from interior sphere to interior sphere
F_int_ext	1	1	Analytical view factor from interior sphere to exterior sphere

Geometry 1

No domain information are required, define the geometry as surface objects.

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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

Create a selection of this sphere to easily access its entities throughout the modeling process.

Locate the section. Find the subsection. Click .

In the dialog box, type Inner sphere in the text field.

Click .

Sphere 2 (sph2)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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

Locate the section. Find the subsection. Click .

In the dialog box, type Outer sphere in the text field.

Click .

Thanks to the symmetries of the model, only one eighth of the geometry is needed. Remove the unnecessary parts.

Delete Entities 1 (del1)

In the window, right-click and choose .

On the object , select Boundaries 1–7 only.

In the window for , click

Add a material to the boundaries to define their emissivities.

Materials

Blackbody

In the toolbar, click

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

Click to expand the section. In the tree, select .

Click

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


Property	Variable	Value	Unit	Property group
Surface emissivity	epsilon_rad	1	1	Basic

Surface-to-Surface Radiation (rad)

Diffuse Surface 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Diffuse Surface 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Implement a symmetry condition for the view factor computation.

Symmetry for Surface-to-Surface Radiation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Define new variables and nonlocal integration couplings to evaluate the view factors directly after running the study. First, define variables that are used to identify the surfaces.

Definitions

Identifiers, Inner Sphere

In the toolbar, click

and choose .

In the window for , type Identifiers, Inner Sphere in the text field.

Locate the section. From the list, choose .

From the list, choose .

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


Name	Expression	Unit	Description
ext	0		Exterior surface indicator
int	1		Interior surface indicator

Identifiers, Outer Sphere

In the toolbar, click

and choose .

In the window for , type Identifiers, Outer Sphere in the text field.

Locate the section. From the list, choose .

From the list, choose .

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


Name	Expression	Unit	Description
ext	1		Exterior surface indicator
int	0		Interior surface indicator

Now, define nonlocal integration couplings for both spheres.

Integration, Inner Sphere

In the toolbar, click

and choose .

In the window for , type Integration, Inner Sphere in the text field.

In the text field, type intop_int.

Locate the section. From the list, choose .

From the list, choose .

Integration, Outer Sphere

In the toolbar, click

and choose .

In the window for , type Integration, Outer Sphere in the text field.

In the text field, type intop_ext.

Locate the section. From the list, choose .

From the list, choose .

Adjust the mesh size for both spheres manually to get approximately the same number of elements on each sphere. This is an efficient mesh for view factor computation.

Mesh 1

Free Triangular 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type r_int/5.

Size 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type r_ext/5.

Click

The view factors are computed automatically, before the actual study runs. To evaluate the geometrical view factors, it is sufficient to obtain the initial values.

Study 1

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Results

View Factors

In the window, expand the node.

Right-click and choose .

In the window for , type View Factors in the text field.

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


Expression	Unit	Description
intop_int(comp1.rad.radopu(int,0))/intop_int(1)	1	Interior to interior view factor
intop_ext(comp1.rad.radopd(int,0))/intop_int(1)	1	Interior to exterior view factor
intop_ext(comp1.rad.radopd(0,ext))/intop_ext(1)	1	Exterior to exterior view factor
intop_int(comp1.rad.radopu(0,ext))/intop_ext(1)	1	Exterior to interior view factor

Absolute Error, View Factors

In the toolbar, click

In the window for , type Absolute Error, View Factors in the text field.

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


Expression	Unit	Description
abs(intop_int(comp1.rad.radopu(int,0))/intop_int(1)-F_int_int)		Absolute error, interior to interior view factor
abs(intop_ext(comp1.rad.radopd(int,0))/intop_int(1)-F_int_ext)		Absolute error, interior to exterior view factor
abs(intop_ext(comp1.rad.radopd(0,ext))/intop_ext(1)-F_ext_ext)		Absolute error, exterior to exterior view factor
abs(intop_int(comp1.rad.radopu(0,ext))/intop_ext(1)-F_ext_int)		Absolute error, exterior to interior view factor