The first option proposed consists of two pairs of curved mirrors and a flat one. Figure 1 shows a sectional view of this concentrator (the flat mirror is not presented here). The bigger curved mirrors are parabolic and the smaller are elliptic and they have the following parametric equations, where the parameter т corresponds, in a polar coordinate system, to the angular coordinate of a point on a mirror. The semi angle of acceptance of the concentrator is 00, which has a value between 0 and 30° is a veiy important design parameter. The extreme value of 30° corresponds to a flat collector without MCC.

For the CPCi (truncated) mirror:

Where:

q = Tan 1

And the geometric solar concentration is:

Only a part of the beam radiation impinging on the concentrator aperture reaches the absorbers of the flat collectors. The acceptation function F (t ) is defined as the fraction of beam radiation, which reach the absorber for a given angle of incidence t.

if 0 <T<TC = — — 30o

Now, the thermal efficiency of a flat collector with the MCC is ingenuously approximate by:

(16)

where r|max, B and C have the same values that in equation 5, and the energy gathered per m2 of flat collector is now calculated from:

Qu = C Г+ F (t (t)) r (t) G (t)k (t) dt

ta

And the mean efficiency is given by:

Г + N F (t(t))r (t) G (t)k (t) dt

ta

ta + N

Г G (t) dt

ta

The calculated values for the energy gathered for flat-plate solar collectors with this MCC as compared to a the same collectors without the MCC shows that Qu increases as the acceptance half angle e0 decreases even though the overall efficiency can decrease for small values of Є0. This MCC makes a best job when high values temperature are needed. For temperatures about 80° C, for example, the rate of the flat solar collectors is as least doubled when the MCC is implemented in them.

For solar collectors whose absorbers cannot resist high temperatures, a second option is suggested. This variation consist in to substitute the parabolic mirrors with flat mirrors with the same acceptance half angle. The elliptic mirrors must be substituted by others parabolic (non-truncated) as it is shown in figure 2. The acceptation function changes, but the most significant difference consists of a geometric concentration Cg much smaller, so the benefits are reduced. But this option is still attractive for inexpensive arrays of flat-solar collectors.

For this second option, the coordinates of extreme points Pmax and Pi are the following:

And the geometric concentration is given by:

(22)

The new acceptation function is defined as:

Where

Equations 16, 17 and 18 are used to evaluate the performance of this arrays with the MCC solar concentrator for different operational temperature, acceptance half angles, reflectance of the mirrors, ambient temperature profiles, etcetera. Table 1 shows a comparison between two-flat collector arrays with and without MCC. It must be noted that the maximum values of the mean thermal efficiency and the useful heat for each temperature of operation correspond to different acceptance half angle. This occurs because the MCC increases the area of solar acceptation, but it shadows partially the absorbers of the flat collectors.

Therefore exists a trade-off between the energy collected, the mean thermal efficiency, the temperature of operation and, of course, the cost of the MCC and the flat collectors. Tables like table 1 can help to choose the most convenient option for a given application and budget. As an example, for an array of commercial solar collectors like the model used for build the table 1, a MCC with an half acceptance angle of 22° would deliver 9,75 % more useful heat if the application is at 30 ° C, but it would deliver 27% , 55,6 % or 111,6 more for operational temperatures of 70, 90 and 110 ° C, respectively. For 30° C the output energy would be slightly smaller.

Table 1 Output energy ratings ( MJ / m2 day) for a two flat-plate solar collectors system with and without MCC, for different operational temperatures. Collector Model EP-40-1.5, SunEarth, Inc. MCC with two flat and two parabolic mirrors, p = 0,85 in a typical Spring day in Mexico City Temp О о О СО сл о о О ■’J О о О СО О о О 110 ° C Qu MJ / m2day п (%) Qu MJ / m2 day п (%) Qu MJ / m2 day п (%) Qu MJ / m2 day п (%) Qu MJ/m2dy п (%) No MCC 12,660 58,67 10,342 47,92 7,917 36,69 5,548 25,71 3,365 15,59 & II ГО 00 о 12,849 53,09 10,872 44,93 8,742 36,12 6,585 27,21 4,502 18,60 о CD CM II s 12,869 47,90 11,205 41,70 9,368 34,87 7,446 27,74 5,520 20,54 & II ГО о 12,738 43,00 11,345 38,30 9,794 33,06 8,122 27,42 6,391 21,58 о CM CM II cfi 12,532 38,54 11,350 34,91 10,050 30,91 8,632 26,55 7,119 21,89 о о CM II cfi 12,140 34,08 11,139 31,27 10,047 28,21 8,868 24,90 7,599 21,33 & II 00 о 11,711 30,00 10,858 27,81 9,937 25,46 8,949 22,92 7,894 20,22 о CD II cfi 11,204 26,11 10,487 24,43 9,712 22,62 8,886 20,70 8,009 18,66 & II О 10,661 22,43 10,052 21,15 9,405 19,79 8,720 18,35 7,996 16,82

CONCLUSIONS

A simple multi-compound solar concentrator intended to improve the performance of arrays of flat-plate solar collectors have been developed in two options. Both of them improve the performance of the array in an economical way cause the cost of the added MCC is a small fraction of the cost of the system but the rate can be doubled or almost triplicate, if the required temperature of application is high enough. In this paper it is not described the effect of a fifth mirror placed at the bottom side of the array, nor the increase of the angle of inclination of the collectors. These two aspects have a very important role in boosting even more the system performance and will be presented in a future paper.

REFERENCES

[1] ANSI/ASHRAE 93-1986 (1986), Methods of testing to determine the thermal performance of solar collectors, ASHRAE Standard, USA.

[2] Duffie J. A, and Beckman W. A. (1980), Solar Engineering of Thermal Processes. 2nd Ed., John Wiley & Sons, Inc., USA.

[3] Fernandez Zayas J. y Estrada-Cajigal V. (1983), Calculo de la radiacion solar instantanea en la Republica Mexicana, Series del Instituto de Ingenieria No. 472, UNAM.