Problem Description

This problem was taken from Thermal Radiation Heat Transfer, by Robert Siegel and John R. Howell.

Figure 1. Problem #1 from R. Siegel and J.R. Howell, Thermal Radiation Heat Transfer (Hemisphere Publishing Corporation, New York, 1981) pp726-727.

The problem involves two parallel plates of glass separated by a small distance, with solar energy being applied. The goal is to determine the fraction of heat transferred to the plates and to the environment. The fractions are:

Table 1: Fractional Components of Energy

Throughout the rest of this validation, we will use the same notation that is present in Figure 19-5 within Figure 1.

Geometry Description:

The geometry consists of three elements 1.0m X 1.0m, all separated by a 1.0 mm air gap. All elements are the same size and are parallel to each other. The distance of 1.0 mm was chosen to reduce end effects of solar radiation. The following is a diagram of the setup used in RadTherm.

 Figure 2. Geometric Configuration for RadTherm

The first and second elements are the pieces of glass (m and n), while the third element will be a black body that will be used to measure the amount of energy transmitted through both pieces of glass.

Model Conditions:

The top plate is Conventional Automotive Glass, and the second is Reflective Automotive Glass. The surface properties of these materials are in the table below.

Table 2: Problem #1 Surface Properties

The third element is the black body. This element is given a custom surface condition where the emissivity and the absorptivity equal 1.0. This ensures that any energy that is absorbed by the black body is only the energy incident from the previous two plates. There is no energy reflected back to the glass plates or environment.

Because the fraction of energy absorbed, transmitted, and reflected by the plates will not change over time, the time in which results are taken is irrelevant.  The numbers used here were taken at July 19, 1984 at 12:00.

Environment:

In order to eliminate any background reflections from the sun, the solar absorptivity was set to 1.0. Any energy that is incident on the background will all be absorbed.  

Simplifying Assumptions:

Within the Post Processor of RadTherm there are means to measure the absorbed energy in the two glass plates and the Black Body. There is no simple way of measuring the energy reflected back to the environment. This is not a problem because of Conservation of Energy.

We know the energy that was originally incident on plate m; this is the total solar energy available in the post processor. We also know the energy absorbed by the two plates and by the black body. Therefore, the amount reflected is: 

q_total=A_m+A_n+T_m+n+R_m+n

T_m+n=q_total-(A_m+A_n+R_m+n)

So the only equation we will not use directly is equation 19-10 in Figure 1. This will have no impact on assessing the accuracy because instead of using the derived equation from Siegel and Howell, we will be using the 1^st Law of Thermodynamics.

Objective:

To produce the same results using an analytical solution and using the numerical solution within RadTherm.

TAI Results:

In order to get the energy fraction absorbed by each plate, you must first know the total solar energy. Total solar energy is available in the Post Processor – Environment Tab. This is the total solar energy in the entire environment at that time step.

The net solar energy (Q Solar) absorbed by each plate is available in the Post Processor – Results Tab. In order to get the amount of solar energy absorbed by the glass plate, you must add the contributions on each side. Once you have the net solar absorbed, divide this by the total solar to get the absorbed fraction.

The solar energy transmitted through both plates is the solar energy absorbed by the black body.  This number (Q Solar) is again available in the Post Processor – Results Tab.

Table 3: Results for Problem #1

The relative error between RadTherm and the analytical solution is insignificant.

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