Computing View Factors with the Heat Transfer Module

In the past, I’ve received regular requests for the ability to check the view factors used by COMSOL Multiphysics. How accurate are they? What is the impact of a given parameter (mesh size, radiation resolution, etc.) on their accuracy? Good news: Version 5.0 provides new operators for postprocessing that correspond to the operators used to generate surface-to-surface equations. Allow me to demonstrate how to compute geometrical view factors.

About View Factors

Heat transfer computation often needs to include surface-to-surface radiation to reflect reality with accuracy. The numerical tools used to simulate surface-to-surface radiation differ significantly from those used for conduction or convection. Whereas the latter are based on local discretization of partial differential equations (PDEs), surface-to-surface radiation relies on non-local quantities — the view factors between diffuse surfaces that emit and receive radiation.

When surface-to-surface radiation is activated, the heat interface creates a set of operators that are evaluated like the irradiation variables in surface-to-surface radiation. Thanks to these operators, it is possible to retrieve the irradiation variable values and to compute the geometrical view factor in a given geometry.

In this blog post, I’ll explain how to compute geometrical view factors in a simple 3D geometry where analytical values of the view factor are available.

Operators

The new operators in COMSOL Multiphysics version 5.0 offer full access to all of the information used to generate surface-to-surface radiation equations. This is true for even the more advanced configurations, such as radiation on both sides of a shell and multiple spectral bands with different opacity properties.

Let’s start with the simplest case where we assume that surfaces behave like gray surfaces. In this case, we don’t need to distinguish between the spectral bands. We have two operators, one for each face (up or down) of the surfaces. They are as follows:

• radopu(expr_up,expr_down)
• radopd(expr_up,expr_down)

These two operators are designed to be evaluated on a boundary where the surface-to-surface radiation is active. Assuming that the heat interface tag is ht, ht.radopu(ht.Ju,ht.Jd) returns the mutual surface irradiation, ht.Gm_u, that is received at the evaluation point on the upside of the boundary.

Note that ht.Ju and ht.Jd define the radiosity on the up- and downsides of the boundaries, respectively. Similarly, ht.radopd(ht.Ju,ht.Jd) returns the mutual surface irradiation, ht.Gm_d, on the boundary downside.

When multiple spectral bands are considered, a given boundary can be opaque for a spectral band and transparent for another band. Hence, one pair of operators is needed per spectral band. They work exactly like the operator for gray surface and are named as follows:

• The first spectral band: ht.radopB1u(expr_up,expr_down) and ht.radopB1d(expr_up,expr_down)
• The second spectral band: ht.radopB2u(expr_up,expr_down) and ht.radopB2d(expr_up,expr_down)
• The third spectral band: ht.radopB3u(expr_up,expr_down) and ht.radopB3d(expr_up,expr_down)
• etc.

View Factor Definition

Let’s consider two diffuse gray surfaces: S1 and S2. We’ll assume that radiation occurs only on the upside of these surfaces. From a thermal perspective, the view factor between S1 and S2, F_{S1-S2}, is the ratio between the diffuse energy leaving S1 and intercepted by S2 and the total diffuse energy leaving S1.

Using the operators described above, we have

(1)

Note that for clarity, the ht. prefix has been removed.

Assuming that radiosity is the same value on all surfaces, the above definition can be simplified and no longer depends on J. In that case, F_{S1-S2} depends only on the geometrical configuration and no longer on the thermal configuration. Let’s call this a geometrical view factor to distinguish it from the view factor based on thermal radiation.

We now have

(2)

where S1 represents either the surface name or its area and I_{\textrm{S1}} is the function indicator of the surface S1, which returns 1 when it is evaluated on S1 and 0 elsewhere.

Benchmark Case: Two Concentric Spheres

In order to get used to the new operators and check their accuracy, I chose a simple configuration. The geometry consists of two concentric spheres of radius R_int and R_ext (with R_int < R_ext), as shown below:

The radiation occurs between the external side of the small sphere and the internal side of the large sphere. The geometrical view factors are:

• F_{S_{\textrm{ext}}-S_{\textrm{ext}}}=1-\left(\frac{R_{\textrm{int}}}{R_{\textrm{ext}}}\right)^2
• F_{S_{\textrm{ext}}-S_{\textrm{int}}}=\left(\frac{R_{\textrm{int}}}{R_{\textrm{ext}}}\right)^2
• F_{S_{\textrm{int}}-S_{\textrm{int}}}=0
• F_{S_{\textrm{int}}-S_{\textrm{ext}}}=1

Here, S_int and S_ext represent the interior and exterior sphere, respectively.

Verification in COMSOL Multiphysics

To compute the geometrical view factor in the COMSOL Multiphysics simulation software, we need to add a heat interface with surface-to-surface radiation activated, then draw the geometry and build the mesh.

Then, we don’t really need to run a heat transfer simulation since we are interested in the geometrical view factor. To have access to the radopu and radopd operators is enough to get initial values.

Before doing that, though, we’d better prepare a few tools that will help us with the postprocessing.

Surface Indicator

In the geometrical view factor expression, we have used the surface indicators I_{\textrm{S1}} and I_{\textrm{S2}}. These are defined as Variables in COMSOL Multiphysics and use the Geometric Entity Selection, so that it is 0 everywhere except on the corresponding surface where it is 1. Let’s name them ext and int.

Screenshots of the Geometric Entity Selection settings for the surface indicators.

Integration Over S_int and S_ext

Next, we define the integration operators intop_ext and intop_int. They will make it easy to compute surface integrals; for example, the surface of S_ext can be evaluated as intop_ext(1).

Screenshots of the settings for the integration operators.

Verifying the Radiative Side of Boundaries

We have seen that radiation may occur on the upside, downside, or on both sides of the boundaries. The radiation operators are designed to be able to distinguish the radiation coming from each side. Therefore, we need to check the sides on this model.

We can do this easily via the Diffuse Surface feature, where the radiation direction can be set to “Negative normal direction” (downside) or “Positive normal direction” (upside). Using this option prompts arrows to be automatically displayed to show the direction of radiation leaving the surface. In our example, the radiation occurs on the downside for S_ext and the upside for S_int.

Geometrical View Factor Computation

With all these tools available to us, evaluating the view factor using an expression similar to (1) is direct. For example,

(3)

is evaluated in COMSOL Multiphysics syntax with intop_ext(comp1.ht.radopd(0,ext))/intop_ext(1).

Note that the use of radopd is due to the fact that the radiation occurs on the downside of S_ext. The first argument of radopd is 0 for the same reason.

(4)

is evaluated with intop_ext(comp1.ht.radopd(int,0))/intop_int(1), where radopd is due to the fact that the radiation occurs on the downside of S_ext. But this time, the second argument of radopd is 0 because radiation occurs on the upside of S_int.

I’ve gathered all the results in a table:

Thanks to the radiation operators, we were able to retrieve the geometrical view factor.

Conclusion

With COMSOL Multiphysics 5.0, it is possible to compute geometrical view factors between diffuse surfaces, thanks to dedicated operators. This offers a solution for the requests we received since the surface-to-surface features have been released. But, these operators can do much more. They are flexible enough to provide all terms of the surface-to-surface radiation equation. They may also be used to formulate equations for other quantities.

Download the Model Shown Here

• Geometrical View Factor Computation