To validate the model, some experimental data will be considered. Results will be obtained under different conditions, and some of them are indicated in Tables 1 and 2. Position, inclination, orientation and solar exposure provide angles that make possible the determination of the incidence angle during the direct radiation, в (Table 1). These angles, along with the material characteristics, permit to calculate the absorbed heat. Table 2 presents other important data to solve the equation set and to carry out the optimization process. Notice that the distances are Ax = Ay = 0.005m.

Table 1. Characteristics angles during the test Test No. Collector Inclination Latitude Ф Time Angle W Declination 5 Azimuth Angle Y Direct Irradiation Angle 0 1 38 38° 0° 20.73 0° 20.93° 2 28 38° 7.5° 21.27 0° 13.55° 3 48 38° 0° 21.43 0° 31.63°

Table ^ 2. Measured values during the test Test No. It (W/m2) Ta (°С) Tfi (°С) Tfo (°С) Tp ( °С ) Fluid flow mf (kg/h) 1 1,006.2 24.5 32.5 51.5 58.5 6.42 2 1,016.3 20.9 29 46 55 7.35 3 797 25.8 34 46 52.5 7.35

According to this, the determination of the optimal value of the parameters is carried out by means of parameter identification, making possible the coincidence between the real data and the data given by the bulb sensor Tp, and the thermometer, which measures the output fluid temperature Tfo. This produces satisfactory results, after slightly modifying the material parameters. Table 3 shows the optimized values.

Table 3. Parameter values before and after optimization Test No. ke -i mm ‘ N2 ae £p £g KiT W/mK kp W/mK Uc W/m3K h W/m2K Initial 0.4 1.526 0.801 0.039 0.88 0.029 402.4 11.8-103 11 1 0.402 1.566 0.759 0.039 0.88 0.0293 453.5 12.7-104 55.7 2 0.400 1.546 0.790 0.039 0.88 0.0291 430.7 11.9-104 49.6 3 0.402 1.555 0.765 0.039 0.88 0.0293 447.4 12.5-104 48.9

Results related with temperature distribution are shown in Figure 2. Fig. 3 presents the variation of temperature across different sections of the у-axis. Although this variation is insignificant, the highest temperatures are reached while the distance from the pipes increases, as found by other researchers [Duffie & Beckman, 1991]. The temperature evolution of the working fluid between the inlet and the outlet of the pipe is shown in Fig. 4. Extreme values are obtained from the model, and it can be seen that they are the same as the measured ones.

Results show that optimized values are very similar on each test. Some preliminary conclusions can be derived from this study. Although final values indicate that the material characteristics were adequately chosen, some little differences appeared, due to the presence of manufacturing defects. Final results related to crystal and plate emmitances, £g and £p, isolator conduction coefficient KiT, and extinction coefficient ke, showed very similar values to those initially considered. In the opposite, refraction index value was found to be higher than the initial one, due to the presence of impurities. Directional absorbance ao decreased around 6% compared to the initial value, because it was considered to be a constant. Cooper conduction coefficient of the absorber kp, was found to be 5% lower than the initially considered value, indicating that this material is not pure. In general terms, we can conclude from this field trial that the chosen initial values are adequate for a suitable approximation to reality. Nevertheless, convection factors suffered changes in a drastic manner. This can be attributed to the fact that the tube was considered as a flat element, for simplification purposes during the analysis, and also that the instability of the fluid flow through the system was obviated. The optimization process shows that there is a critical point in the model that must be considered.

As a result, only two parameters will change: Uc and h, depending on Kuc and Kh. Due to the dynamic complexity presents in the fluid under laminar regime, it is difficult to estimate the flow that passes through each riser, as well as the convection coefficient. This problem, together with the simplification of the pipes considered as plane surfaces, make necessary a serious study that accomplishes the determination of these factors as a function of physical, geometrical and dimensionless variables (see eq. (13)).

Kuc = fi(K, Tp, T€,TS, R,mf) Kh = f2(k2,T ,T€,Ts, R,mf);

with kl, k2 cons tan ts, and mf depending on collector geometry

At first sight, that is not enough, these factors are presented (see Fig. 5) depending on medium temperature of plate and inlet flow. For the study, data were collected every five minutes, during a two hours test, considering different inclinations. The high dispersion on the data made impossible an approximation of these factors with functions only depending on those temperatures.

Fig. 5.- Values of the factors Kuc and Kh and the relations between them

To prevent this problem, a parameter identification process must be made by using both factors and considering each experimental data of the study. These optimized values will be used as input data to validate the subsequent transient regime analysis.

SHAPE * MERGEFORMAT

Equations set (eq. (14) and (15)) are solved using different tests at different inclinations, every two hours. For each of them, solar radiation, temperature and flow mass fluid, material properties, convection factors and environmental conditions are known values. The temperature distribution on the absorber and the water are evaluated instantaneously. Final results of the modelled plate and the output flow temperatures are compared with the real values. The initial data and the results for one of the tests are shown in Fig. 6. It is to stress that both curves present the same tendency: real and modelled, for the plate and the output flow temperatures.

Results can be improved by decreasing the intervals of time, as well as by solving the system with another method [Ketkar, 1999]. However, the computing time would increase drastically.

3. — Conclusions

A simple method to approximate the temperature distribution in the absorber and the heating fluid of a solar flat plate collector is presented. Stationary and transient regimes are analyzed. For this purpose, a model for a bi-dimensional surface is described. Heat conduction and basic solar collector theories are used, defining equations systems for calculus by finite differences method. Then, accommodation of critical parameters of the described model is carried out by means of a parameter identification technique. This technique, based on optimization algorithms, compares model results and experimental data. The development of this process only requires a few measurement systems. So, the procedure can be applied to systems with relatively few sensors.

Results show that materials properties are adequate and close to the originally chosen. Temperatures distribution and variation on the absorber and the refrigerant fluid are qualitatively similar to those found by other researchers. Nevertheless, those approximations present a critical factor, caused by a hydraulic disequilibrium due to the laminar regime flow and to the geometry simplification of pipes. This can be solved by means of two proportionality factors associated to convection heat transfer, both of them identified the same time as the rest of the parameters. Once obtained, they are used as input data for the transient regime.

This study allows investigate the process which takes place in the system since the solar input pass through the working fluid. This can be accomplished with low computational requirements, because the equations system is linear, and various terms related to radiation and convection transfer are simplified by means of the general theory.

So, a correct fitting of those two parameters needs the knowledge of local loss factors during laminar regime. In this way, it would be possible to estimate the real flow of each riser, as well as the relationship with the collector geometry. This procedure supplies an adequate convection coefficient for simulations. However, a deeper analysis of the whole process is needed, i. e. by means of the finite element methods for fluid and convection heat transmission.

A preliminary application of this analysis could be the acquisition of the real values of the system parameters. Also, non tested conditions can be valuated, such as modifications in materials and geometry, optimum values of the risers diameter and the distance between them to maximize the captured energy, and so on.