Second law analysis
Tr. The remaining fraction Qo represents the collector - ambient heat loss:
Qo = Q* - Q (46)
For imaging concentrating collectors Qo is proportional to the receiver-ambient temperature difference and to the receiver area as:
Qo = UrAr(Tr - To) (47)
The analysis presented here is based on Bejan’s work [105,106]. The analysis however is adapted to imaging collectors because entropy generation minimisation 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 Fig. 17. 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 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 steady state, i. e.
Q* = q* Aa (44)
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. (46) and (47) it is apparent that the maximum receiver temperature occurs when Q = 0, i. e. when the entire solar heat transfer Qp is lost to the ambient. The maximum collector temperature is given in dimensionless form by:
Combining Eqs. (45) and (48):
For concentrating systems q* is the solar energy falling on the reflector. In order 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 the above errors. For the concentrating collectors, Eq. (33) can be used. The radiation falling on the receiver is:
qo = no q
The incident solar radiation is partly delivered to a power cycle (or user) as heat transfer Q at the receiver temperature
Considering that C = Aa/Ar, then:
По Ur To
As can be seen from Eq. (50), Umax is proportional to C, i. e. the higher the concentration ratio of the collector the higher is Umax and Tr, max. The term Tr, max in Eq. (48) is also known as the stagnation temperature of the collector, i. e. the temperature that can be obtained at no flow condition. In dimensionless form the collector temperature U = Tr/TO will vary between 1 and Umax, depending on the heat delivery rate Q. The stagnation temperature Umax is the parameter that describes the performance of the collector with regard to collector-ambient heat loss as there is no flow through the collector and all the energy collected is used to raise the temperature of the working fluid to stagnation temperature which is fixed at a value corresponding to the energy collected equal to energy loss to ambient. Thus the collector efficiency is given by:
The exergy inflow coming from the solar radiation falling on the collector surface is:
Ex, in = Q*( 1 - TO) (52)
where T, is the apparent sun temperature as an exergy source. In this analysis the value suggested by Petela  is adopted, i. e. T, is approximately equal to 3/4Ts, where Ts is the apparent black body temperature of the sun, which is about 6000 K. Therefore, T, considered here is 4500 K. It should be noted that in this analysis T, is also considered constant and as its value is much greater than TO, Ex, in is very near Q. The output exergy from the collector is given by:
Ex, out = q( 1 - To) (53)
whereas the difference between the Ex, in — Ex, out represents the destroyed exergy. From Fig. 18, the entropy generation rate can be written as:
By subtkutmg Umax by Tr, max=To and Uopt by Tr, opt=To; Eq. (60) can be written as:
Tr, opt VTr, max To
This equation states that the optimal collector temperature is the geometric average of the maximum collector (stagnation) temperature and the ambient temperature. Typical stagnation temperatures and the resulting optimum operating temperatures for various types of concentrating collectors are shown in Table 8. The stagnation temperatures shown in Table 8 are estimated by considering mainly the collector radiation losses.
As can be seen from the data presented in Table 8 for high performance collectors, like the central receiver, it is better to operate the system at high flow rates in order to lower the temperature around the value shown instead of operating at very high temperature, in order to obtain higher thermodynamic efficiency from the collector system.
By applying Eq. (60) to Eq. (59), the corresponding minimum entropy generation rate is:
Sgen, min 2Г Га 14 Umax 1 /624
= 2( u™x 2 1) 2 —^ (62)
where U* = T*=To. It should be noted that for flat-plate and low concentration ratio collectors, the last term of Eq. (62) is negligible as U* is much bigger than Umax — 1, but it is not for higher concentration collectors, like the central receiver and the parabolic dish ones, which have stagnation temperatures of several thousands of degrees.
3.3.3. Non-isothermal collector
So far the analysis was carried out by considering an isothermal collector. For a non-isothermal one, which is a more realistic model particularly for the long PTC, and by applying the principle of energy conservation:
q* = Ur (T — T0 ) + mcp d - (63) where x is from 0 to L (the collector length). The generated entropy can be obtained from:
Tout Q* Qo
Sgen = mcp ln - TfT - 2 - Tf + - ZT
Tin T T o
From an overall energy balance, the total heat loss is: Qo = Q* — mcp (Tout — Tin) (65)
Substituting Eq. (65) into Eq. (64) and performing the necessary manipulations the following relation is obtained:
Ns = ln Up — Uout + Un) — d + 1 (66)