Radiation in a Cavity

The following example shows how to build and solve a radiative heat transfer problem using the Heat Transfer with Surface-to-Surface Radiation interface. This example’s simple geometry allows a comparison of results from COMSOL Multiphysics with a theoretical solution.

Model Definition

The geometry consists of three rectangles with lengths 3 m, 4 m, and 5 m, and all have a width of 1 m. The rectangles are situated such that they form a triangular cavity (see Figure 1).

Figure 1: Three domains form a triangular cavity to study heat radiation.

The rectangles are made of copper, which is a good thermal conductor, and they transfer heat internally by conduction. Further, the three inner boundaries that form the cavity can exchange heat only by means of surface-to-surface radiation. The surface emissivities are ε1 = 0.4 on the bottom boundary, ε2 = 0.6 on the vertical boundary, and ε3 = 0.8 on the inclined boundary.

The system holds the temperature on the (outer) lower boundary at T1 = 300 K. Further, the system experiences an inward heat flux of q2 = 2000 W/m2 on the outer boundary of the vertical rectangle and q3 = 1000 W/m2 on the outer boundary of the tilted one. All remaining outer boundaries are insulated.

This example examines the steady state of the cavity’s interior boundaries. Specifically, you know the value of the heat flux on the inclined and vertical boundaries, and you want to find the temperature along these boundaries. Conversely, you know the temperature of the bottom boundary and want to determine the heat flux that leaves through it.

Results and Discussion

Figure 2 shows the temperature field resulting from a simulation. If you run the model as described later in this section you can see from the color display that the temperatures are fairly constant within the three rectangles. The theoretical results predict that a heat flux Q1 = −11,000 W/m leaves the cavity through the bottom boundary, that the vertical boundary has a temperature of T2 = 641 K, and that the inclined boundary has a temperature of T3 = 600 K. The results obtained are very close to these theoretical values. The section Comparing Results with Theory provides more details on the verification of these results.

Figure 2: Temperature as a function of position.

Figure 3 details the temperature distribution along the inclined boundary of the cavity. Note that while the temperature is not constant, it varies by only a few percent. The lowest temperature appears on the bottom left, and the highest temperature is on the right, which seems reasonable because that portion of the boundary is located adjacent to the vertical boundary, which has the highest inward heat flux.

Figure 3: Temperature distribution along the inclined boundary of the cavity.

Figure 3 plots the radiosity along the inclined boundary (in other words, the total heat flux that leaves the boundary into the cavity). The radiosity is the sum of the heat flux the boundary emits plus the heat flux it reflects.

Figure 4: Radiosity along the inclined boundary of the cavity.

Like temperature, the radiosity is fairly constant along the boundary. The small variations have roughly the same distribution as the temperature in Figure 3.

Comparing Results with Theory

As mentioned at the start of this section, you can compare the results from this tutorial to theoretical values, which are based on the sketch in Figure 5 and whose theory appears in the section “Enclosures with More Than Two Surfaces” in Ref. 1.

Figure 5: Problem sketch for theoretical analysis of radiation in a triangular cavity.

The theoretical model considers only the three boundaries that form the cavity. On these boundaries, the model either holds the temperature at a constant value or specifies a heat flux. Note that this approach differs somewhat from the model, which sets temperatures and heat fluxes on the rectangles’ outer boundaries.

The COMSOL Multiphysics model shows that the temperatures and heat fluxes are nearly equal on the inner and outer boundaries of the rectangles. Therefore, the COMSOL Multiphysics and the theoretical model show good agreement.

Using the notation from Figure 5 with the same assumptions as for the model, you obtain the following lengths and emissivities:

L1 = 4 m, ε1 = 0.4

L2 = 3 m, ε2 = 0.6

L3 = 5 m, ε3 = 0.8

The boundary conditions define either the temperature or the heat flux on each boundary. Now apply the same boundary conditions as in the problem where you set values for T1, Q2, and Q3. You must, however, make a small adjustment for T1 because the theoretical configuration sets it on the outer boundary and not on the cavity side. T1 is therefore slightly higher than 300 K in the theoretical analysis:

You describe the heat flux from a boundary with the following two equations:

For a triangular cavity, the following equation gives the view factor between surface i and surface j:

Substituting this expression for the view factor into the second equation for the heat fluxes results in a linear system of six equations. The six unknowns for this particular problem are J1, J2, J3, Q1, T2, and T3, and you would like to compare the three last to those from the solution. Solving the linear system yields the following values: Q1 = −11,000 W/m, T2 = 641 K, and T3 = 600 K.

To compare these results with the model, you must find the total heat flux through the cavity’s bottom (horizontal) boundary and also calculate the average temperatures on the other two boundaries.

The following table compares the results from COMSOL Multiphysics with the theoretical values:


quantity	COMSOL Multiphysics	theory
Q1	-10,965 W/m	-11,000 W/m
T2	645 K	641 K
T3	601 K	600 K

The differences are quite small. Also note that the theoretical model includes some simplifications. For example, it assumes that heat fluxes and temperatures are constant along each boundary. It also assumes a constant view factor for each pair of boundaries, whereas COMSOL Multiphysics computes a local view factor at each point of the cavity.

Reference

1. A. Bejan, Heat Transfer, John Wiley & Sons, 1993.

Heat_Transfer_Module/Verification_Examples/cavity_radiation

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

Geometry 1

The geometry consists of three rectangles positioned such that they create a triangular cavity.

Rectangle 1 (r1)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 4.

Locate the section. In the text field, type -1.

Rectangle 2 (r2)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 3.

Locate the section. In the text field, type 4.

Rectangle 3 (r3)

On the toolbar, click and choose .

In the window for , locate the section.

In the text field, type 5.

Locate the section. In the text field, type atan(3/4)[rad].

Form Union (fin)

In the window, under click .

In the window for , locate the section.

From the list, choose .

On the toolbar, click .

Click the button on the toolbar.

This completes the geometry modeling stage.

Materials

Material 1 (mat1)

On the toolbar, click .

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

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


Property	Name	Value	Unit	Property group
Thermal conductivity	k	400	W/(m·K)	Basic
Density	rho	8700	kg/m³	Basic
Heat capacity at constant pressure	Cp	385	J/(kg·K)	Basic