The instantaneous efficiency of a solar collector is universally expressed by the following equation:

Qu

(Ac G )

According to the most utilized technical standards for solar flat-plate collectors, e. g. ASHRAE 96-1980 (RA 1989) standard [1], the forgoing instantaneous efficiency can be expressed (as an approximation to the actual values) as a quadratic function of the variable Z = (Ti-Ta)/ G as follows:

П = T|max — B Z — C Z (2)

Here r|max is the maximum efficiency, which corresponds to the ideal non-losses case, and B and C are coefficients, which depend on how big, are the conduction, convection and radiation thermal losses. These values are obtained by an experimental procedure described in the standard. For a good commercial flat-plate collector qo is about 0,70 ; A is about 3,5 and B is near 12,0. These values can be entered to the computer code developed to simulate the performance of the system. Ti is the fluid temperature in the entrance of the collector and Ta is the ambient temperature.

The total (global) irradiance G is a quasi-chaotic function of time due to the presence of clouds, dust and other non-deterministic factors which affect its value in a given instant. However, for a very clear day, G can be approximate by a deterministic function, taken into account the latitude, solar hour, day of the year, etcetera. In order to simplify the modeling the following very simple approximation [3] is employed:

G (t) = Gm ( sin 12 [ n (t — ta) / N ]) (3)

Here N is the length of a given day, ta is the dawning time and Gm is the maximum irradiance at this day. Obviously, N, ta and Gm varies day after day, but in the equinoxes N is 12 hours, ta is about 6 o’clock a. m. (solar time) and Gm is almost 1000W/m2.

The ambient temperature reach its minimum value just before dawning, it increases as the Sun rises and has its maximum value about two hours after noon, approximately. The temperature begins to descend till the dawn of the morning after. At the afternoon twilight the ambient temperature is generally higher than in the dawn due to the infrared radiation of the earth surface during the night, which cause a cooling effect. The ambient temperature is registered daily in meteorological stations, so its profile is easy to obtain for most important cities. This profile can be modeled by a curve adjusted with cubic splines from meteorological data. Here a few data is needed to create a profile for a given day. Other approximation is by means of a polynomial function as follows:

Ta (t) = c0 + c1 t + c212 + c313 +… cn tn (4)

Now it is possible to model not only the instantaneous efficiency of a flat-plate solar collector, but the useful heat in a given day of the year and the mean thermal efficiency

that day, among many other interesting parameters. This way, the instantaneous thermal efficiency, the useful energy gathered (per m2) in a day and the daily mean efficiency can be evaluate by the following formulae:

Qu = f П (t) G (t )k (t) dt

ta

where the integration limits correspond to the dawning instant ta and the Sun set hour ta+N, both in solar time. Of course, these limits vary from place to place and day by day in a deterministic and well-known way [2]. The angle modifier function is there:

where 0, is the incidence angle of beam solar rays with the normal to the cover of the collectors, which is a trigonometric function of time t that takes into account the reflection losses at this cover [3]. The coefficient of the parenthesis term is experimentally obtained and has a value of about 0,19 for a typical flat collector.

The forgoing formulae can be easily inserted in a computer code to determine the mean efficiencies and the gathered energy in one hour intervals of time, for inlet temperatures of 30, 60, 90, 120 and 140° C, for example.

It was observed that in the first and in the final hours of solar energy collection, the thermal efficiencies is too small to contribute to the energy gathering, so it is proposed to sacrifice these ineffective periods by setting mirrors which shadow the collector absorbers when the efficiency is too low, but increase the area of collection when the thermal efficiency of the collector is greater. The mirrors form a multi-compound solar concentrator, which is described below.