3.2 Energy Balance in Flat-Plate Collectors

A fundamental concept for thermal analysis of any thermal system is the conservation of energy, which can be analyzed through energy balance calculation under steady state conditions. In steady state, the useful energy output of the collector is the difference between the absorbed solar radiation and the total thermal losses from the collector

Useful energy = Absorbed solar energy - Thermal losses

Obviously, the higher the useful energy output from a particular design, the higher the expected efficiency. Thermal efficiency of the collector is an important parameter to consider in this kind of analysis as it creates the basis for comparison of different materials and modifications of collector systems. So many theoretical calculations presented in the books (as well as in this Lesson), are eventually aimed at evaluating efficiency.

Let us define the thermal efficiency (η) first, as it will be the focus and final destination of this chapter. 

\[\eta = \frac{{{Q_u}}}{{{A_c}{G_T}}}\]

where Q_u is the useful energy output from a collector, G_T is the incident solar radiation flux (irradience), and A­_c is the collector area. So the denominator here is the total energy input for the collector. In this formula the G_T is the parameter characterizing the external conditions, and it is usually known from practical measurements (with a pyranometer) or assumptions for a specific location. The collector area is a set technical characteristic. So the main question here is how to estimate the Q_u  - the useful energy.  

As was mentioned above, to find how much energy remains available for useful thermal work, we need to understand the energy balance within the collector: absorbed energy - losses.

The energy balance can also be expressed via the following key equation: 

\[{Q_u} = {A_c}[S - {U_L}({T_{plate}} - {T_{ambient}})]\]

where S is the absorbed solar radiation, U_L is the total losses, T_plate is the temperature of the absorbing plate, and T_ambient is the temperature of the air, and A_c again is the area of the collector surface.

This equation stands as a cornerstone of the energy balance analysis presented in Chapter 6 of Duffie and Beckman's textbook. To implement this question, we need to understand how the quantities S and U_L can be obtained. The most complete explanation can be found in the following reading. 

In a general case, when measurements of incident solar radiation (I_T) are available, the convenient approximation for the absorbed energy is given by: 

\[S = {(\tau \alpha )_{av}}{I_T}\]

where (τα)_av is the product of transmittance of the collector cover and absorptance of the plate averaged over different types of radiation. In fact, (τα)_av ≈ 0.96(τα)_beam based on practical estimattions.

Now let us see how the radiation losses can be determined. Please refer to the following reading. 

Now as the absorbed radiation and losses are defined, the useful energy gain can be determined via the energy balance equation given above.