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*is the incident solar radiation flux (irradience), and

_{T}*A*is the collector area. So the denominator here is the total energy input for the collector. In this formula the

_{c}*G*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

_{T}*Q*- the useful energy.

_{u}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*is the temperature of the absorbing plate, and

_{plate}*T*is the temperature of the air, and

_{ambient}*A*again is the area of the collector surface.

_{c}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.

#### Reading Assignment

Look through the following section of the D&B textbook to understand the ways to estimate the absorbed radiation S on a collector surface

Duffie, J.A., and Beckman, W.A., Solar Engineering of Thermal Processes, Wiley and Sons, 2013, Chapter 5, Section 5.9 (3 pages).

Equations (5.9.1) and (5.9.3) in the above reading provide the basis for estimating absorbed radiation depending on what initial information on incident radiation is available.

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*)

*based on practical estimattions.*

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

#### Reading Assignment

Duffie, J.A., and Beckman, W.A., Solar Engineering of Thermal Processes, Wiley and Sons, 2013, Chapter 6, Sections 6.1-6.4 (18 pages).

These sections of the book explain the model and assumptions for flat-plate collector analysis. The thermal losses are specifically addressed in Section 6.4, and you are welcome to dig through the complete derivation and examples. Of practical interest are the charts in Figure 6.4.4 which describe the results of the model calculations of thermal loss coefficient versus plate temperature.

Another useful outcome from this chapter is the empirical equation (6.4.9), which offers an algebraic method of finding the losses from the top of the collector. You will have a chance to look closer at this equation and see how it works further in this lesson activity.

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