## Thermal Analysis of Compound Parabolic Collectors

The instantaneous efficiency, r|, of a CPC is defined as the useful energy gain divided by the incident radiation on the aperture plane; that is,

AaGt

In Eq. (3.85), Gt is the total incident radiation on the aperture plane. The useful energy, Qu, is given by an equation similar to Eq. (3.60), using the concept of absorbed radiation as

The absorbed radiation, S, is obtained from (Duffie and Beckman, 1991):

S GB,CPCTc,BTCPC,Ba B + GD,CPCTc,D TCPC,Da D + GG,CPCTc,G TCPC,GaG

where tc = transmittance of the CPC cover.

tcpc = transmissivity of the CPC to account for reflection loss.

The various radiation components in Eq. (3.87) come from radiation falling on the aperture within the acceptance angle of the CPC and are given as follows:

Gb cpc = GBn cos(0) if (P - 0c) < tan-1[tan($)cos(z)]

D,CPC

G,CPC

D,CPC

G,CPC

GD | |

C | |

GD |
' 1 |

2 |
, C |

0 | |

Gg |
' 1 |

2 |
C |

In Eqs. (3.88a)-(3.88c), (3 is the collector aperture inclination angle with respect to horizontal. In Eq. (3.88c), the ground-reflected radiation is effective only if the collector receiver "sees" the ground, i.e., (( + 9c) > 90°.

It has been shown by Rabl et al. (1980) that the insolation, GCPC, of a collector with a concentration C can be approximated very well from

It is convenient to express the absorbed solar radiation, S, in terms of GCPC in the following way:

S GCPCTcover TCPCttr

Gt Tcover TCPCttr

Gt Tcover TCPCttr

1 - |
[1 -1 |
Gd | ||||||||||||||||||||||||||||||||

C, |
S Gt Tcover TCPCar Y where Tcover = transmissivity of the cover glazing. tcpc = effective transmissivity of CPC. ar = absorptivity of receiver. = correction factor for diffuse radiation, given by Gd Gt The factor given by Eq. (3.92), accounts for the loss of diffuse radiation outside the acceptance angle of the CPC with a concentration C. The ratio GD/Gt varies from about 0.11 on very clear sunny days to about 0.23 on hazy days. It should be noted that only part of the diffuse radiation effectively enters the CPC, and this is a function of the acceptance angle. For isotropic diffuse radiation, the relationship between the effective incidence angle and the acceptance half angle is given by (Brandemuehl and Beckman, 1980): The effective transmissivity, tcpc, of the CPC accounts for reflection loss inside the collector. The fraction of the radiation passing through the collector aperture and eventually reaching the absorber depends on the specular reflectivity, p, of the CPC walls and the average number of reflections, n, expressed approximately by Tcover TCPCar 6 8 10 Concentration ratio FIGURE 3.34 Average number of reflections for full and truncated CPCs. (Reprinted from rabl (1976) with permission from Elsevier.) 6 8 10 Concentration ratio FIGURE 3.34 Average number of reflections for full and truncated CPCs. (Reprinted from rabl (1976) with permission from Elsevier.) This equation can also be used to estimate tcpcb, tcpc,d, and tcpcg in Eq. (3.87), which are usually treated as the same. Values of n for full and truncated CPCs can be obtained from Figure 3.34. As noted before, the upper ends of CPCs contribute little to the radiation reaching the receiver, and usually CPCs are truncated for economic reasons. As can be seen from Figure 3.34, the average number of reflections is a function of concentration ratio, C, and the acceptance half angle, 9C. For a truncated concentrator, the line (1 - 1/C) can be taken as the lower bound for the number of reflections for radiation within the acceptance angle. Other effects of truncation are shown in Figures 3.35 and 3.36. Figures 3.34 through 3.36 can be used to design a CPC, as shown in the following example. For more accuracy, the equations representing the curves of Figures 3.34 through 3.36 can be used as given in Appendix 6. ## Example 3.8Find the CPC characteristics for a collector with acceptance half angle 9c = 12°. Find also its characteristics if the collector is truncated so that its height-to-aperture ratio is 1.4. ## SolutionFor a full CPC, from Figure 3.35 for 9c = 12°, the height-to-aperture ratio = 2.8 and the concentration ratio = 4.8. From Figure 3.36, the area of the reflector is 5.6 times the aperture area; and from Figure 3.34, the average number of reflections of radiation before reaching the absorber is 0.97. Full CPC ffi CP Full CPC 6 8 10 Concentration ratio FIGURE 3.35 Ratio of height to aperture for full and truncated CPCs. (Reprinted from Rabl (1976) with permission from Elsevier.) 6 8 10 Concentration ratio FIGURE 3.35 Ratio of height to aperture for full and truncated CPCs. (Reprinted from Rabl (1976) with permission from Elsevier.) ra a ra a 6 8 10 Concentration ratio FIGURE 3.36 ratio of reflector to aperture area for full and truncated CPCs. (reprinted from rabl (1976) with permission from Elsevier.) 6 8 10 Concentration ratio FIGURE 3.36 ratio of reflector to aperture area for full and truncated CPCs. (reprinted from rabl (1976) with permission from Elsevier.) For a truncated CPC, the height-to-aperture ratio = 1.4. Then, from Figure 3.35, the concentration ratio drops to 4.2; and from Figure 3.36, the reflector-to-aperture area drops to 3, which indicates how significant is the saving in reflector material. Finally, from Figure 3.34, the average number of reflections is at least 1 - 1/4.2 = 0.76. Example 3.9 A CPC has an aperture area of 4 m2 and a concentration ratio of 1.7. Estimate the collector efficiency given the following: Total radiation = 850 W/m2. Diffuse to total radiation ratio = 0.12. Receiver absorptivity = 0.87. Receiver emissivity = 0.12. Mirror reflectivity = 0.90. Glass cover transmissivity = 0.90. Collector heat loss coefficient = 2.5 W/m2-K. Circulating fluid = water. Entering fluid temperature = 80°C. Ambient temperature = 15°C. Collector efficiency factor = 0.92. Solution The diffuse radiation correction factor, is estimated from Eq. (3.92): From Figure 3.34 for C = 1.7, the average number of reflections for a full CPC is n = 0.6. Therefore, from Eq. (3.94), tcpc = Pn = 0.9006 = 0.94 The absorber radiation is given by Eq. (3.91): s = GtTcoverTCPCarY = 850 X 0.90 X 0.94 X 0.87 X 0.95 = 594.3W/m2 The heat removal factor is estimated from Eq. (3.58): acUl UlF'Ac m cp 0.015 X 4180 0.015 X 4180 The receiver area is obtained from Eq. (3.77): Ar = Aa/C = 4/1.7 = 2.35m2 The useful energy gain can be estimated from Eq. (3.86): 0.86 Qu = Fr [SAa - ArUL (Ti - Ta)] = 0.86[594.3 X 4 - 2.35 X 2.5(80 - 15)] = 1716W The collector efficiency is given by Eq. (3.85): 3.6.3 Optical Analysis of Parabolic Trough Collectors A cross-section of a parabolic trough collector is shown in Figure 3.37, where various important factors are shown. The incident radiation on the reflector at the rim of the collector (where the mirror radius, rr, is maximum) makes an angle, jr, with the center line of the collector, which is called the rim angle. The equation of the parabola in terms of the coordinate system is y where f = parabola focal distance (m). For specular reflectors of perfect alignment, the size of the receiver (diameter D) required to intercept all the solar image can be obtained from trigonometry and Figure 3.37, given by where 9m = half acceptance angle (degrees). For a parabolic reflector, the radius, r, shown in Figure 3.37 is given by where j = angle between the collector axis and a reflected beam at the focus; see Figure 3.37. Receiver Receiver Solar radiation m FIGURE 3.37 Cross-section of a parabolic trough collector with circular receiver. Solar radiation FIGURE 3.37 Cross-section of a parabolic trough collector with circular receiver. As j varies from 0 to jr, r increases from f to rr and the theoretical image size increases from 2f sin(9m) to 2rr sin(9m)/cos( jr + 0m). Therefore, there is an image spreading on a plane normal to the axis of the parabola. At the rim angle, jr, Eq. (3.97) becomes Another important parameter related to the rim angle is the aperture of the parabola, Wa. From Figure 3.37 and simple trigonometry, it can be found that which reduces to The half acceptance angle, 9m, used in Eq. (3.96) depends on the accuracy of the tracking mechanism and the irregularities of the reflector surface. The smaller these two effects, the closer 9m is to the sun disk angle, resulting in a smaller image and higher concentration. Therefore, the image width depends on the magnitude of the two quantities. In Figure 3.37, a perfect collector is assumed and the solar beam is shown striking the collector at an angle 29m and leaving at the same angle. In a practical collector, however, because of the presence of errors, the angle 29m should be increased to include the errors as well. Enlarged images can also result from the tracking mode used to transverse the collector. Problems can also arise due to errors in the positioning of the receiver relative to the reflector, which results in distortion, enlargement, and displacement of the image. All these are accounted for by the intercept factor, which is explained later in this section. For a tubular receiver, the concentration ratio is given by By replacing D and Wa with Eqs. (3.96) and (3.100), respectively, we get The maximum concentration ratio occurs when jr is 90° and sin(jr) = 1. Therefore, by replacing sin(jr) = 1 in Eq. (3.103), the following maximum value can be obtained: The difference between this equation and Eq. (3.84) is that this one applies particularly to a parabolic trough collector with a circular receiver, whereas Eq. (3.84) is the idealized case. So, by using the same sun half acceptance angle of 16 ' for single-axis tracking, Cmax = 1Arcsin(16 ') = 67.5. In fact, the magnitude of the rim angle determines the material required for the construction of the parabolic surface. The curve length of the reflective surface is given by
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