The analysis presented here is based on Bejan's work (Bejan et al., 1981; Bejan, 1995). The analysis, however, is adapted to imaging collectors, because entropy generation minimization is more important to high-temperature systems. Consider that the collector has an aperture area (or total heliostat area), Aa, and receives solar radiation at the rate Q* from the sun, as shown in Figure 3.44. The net solar heat transfer, Q*, is proportional to the collector area, Aa, and the proportionality factor, q (W/m2), which varies with geographical position on the
Parabola or heliostat of projected area, Aa
Parabola or heliostat of projected area, Aa
FIGURE 3.44 Imaging concentrating collector model.
earth, the orientation of the collector, meteorological conditions, and the time of day. In the present analysis, q is assumed to be constant and the system is in a steady state; that is,
For concentrating systems, q is the solar energy falling on the reflector. To obtain the energy falling on the collector receiver, the tracking mechanism accuracy, the optical errors of the mirror, including its reflectance, and the optical properties of the receiver glazing must be considered.
Therefore, the radiation falling on the receiver, qo, is a function of the optical efficiency, which accounts for all these errors. For the concentrating collectors, Eq. (3.106) can be used. The radiation falling on the receiver is (Kalogirou, 2004):
The incident solar radiation is partly delivered to a power cycle (or user) as heat transfer Q at the receiver temperature, Tr. The remaining fraction, Qo, represents the collector ambient heat loss:
For imaging concentrating collectors, Qo is proportional to the receiver ambient temperature difference and to the receiver area as
where Ur is the overall heat transfer coefficient based on Ar. It should be noted that Ur is a characteristic constant of the collector.
Combining Eqs. (3.133) and (3.134), it is apparent that the maximum receiver temperature occurs when Q = 0, i.e., when the entire solar heat transfer
Q is lost to the ambient. The maximum collector temperature is given in dimen-sionless form by
Considering that C = AaIAr, then:
As can be seen from Eq. (3.137), 9max is proportional to C, i.e., the higher the concentration ratio of the collector, the higher are 9max and Tr,max. The term Tr,max in Eq. (3.135) is also known as the stagnation temperature of the collector, i.e., the temperature that can be obtained at a no-flow condition. In dimen-sionless form, the collector temperature, 9 = TrITo, varies between 1 and 9max, depending on the heat delivery rate, Q. The stagnation temperature, 9max, is the parameter that describes the performance of the collector with regard to collector ambient heat loss, since there is no flow through the collector and all the energy collected is used to raise the temperature of the working fluid to the stagnation temperature, which is fixed at a value corresponding to the energy collected equal to energy loss to ambient. Hence, the collector efficiency is given by n = Q* = 1 - a138)
Therefore r|c is a linear function of collector temperature. At the stagnation point, the heat transfer, Q, carries zero exergy, or zero potential for producing useful work.
The minimization of the entropy generation rate is the same as the maximization of the power output. The process of solar energy collection is accompanied by the generation of entropy upstream of the collector, downstream of the collector, and inside the collector, as shown in Figure 3.45.
The exergy inflow coming from the solar radiation falling on the collector surface is
VEout= Q FIGURE 3.45 Exergy flow diagram.
where T* is the apparent sun temperature as an exergy source. In this analysis, the value suggested by Petela (1964) is adopted, i.e., T* is approximately equal to 3ATs, where Ts is the apparent blackbody temperature of the sun, which is about 5770 K. Therefore, the T* considered here is 4330 K. It should be noted that, in this analysis, T* is also considered constant; and because its value is much greater than To, Ein is very near Q*. The output exergy from the collector is given by
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