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,


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):


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)]








' 1


, C



' 1



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



S Gt Tcover TCPCar Y


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

Compound Parabalic

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 3.8

Find 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.


For 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

Cpc Compound Parabolic Concentrator

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

Compound Parabalic

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.


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):


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):


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.



Compound Parabolic Solar Collector

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


'Vr '



+ ln


Vr '

+ tan


. 2 .

. 2 .

. 2 .

where Hp = lactus rectum of the parabola (m). This is the opening of the parabola at the focal point.

As shown in Figure 3.38 for the same aperture, various rim angles are possible. It is also shown that, for different rim angles, the focus-to-aperture ratio, which defines the curvature of the parabola, changes. It can be demonstrated that, with a 90° rim angle, the mean focus-to-reflector distance and hence the reflected beam spread is minimized, so that the slope and tracking errors are less pronounced. The collector's surface area, however, decreases as the rim angle is decreased. There is thus a temptation to use smaller rim angles because the sacrifice in optical efficiency is small, but the saving in reflective material cost is great.

Rim Angle For Trough

Example 3.10

For a parabolic trough collector with a rim angle of 70°, aperture of 5.6 m, and receiver diameter of 50 mm, estimate the focal distance, the concentration ratio, the rim radius, and the length of the parabolic surface.


Therefore, f

4tan(tpr/2) 4tan(35) From Eq. (3.102), the concentration ratio is

C = Wa /nD = 5.6/0.05n = 35.7 The rim radius is given by Eq. (3.98):


The parabola lactus rectum, Hp, is equal to Wa at jr = 90° and f = 2 m. From Eq. (3.101),

Finally, the length of the parabola can be obtained from Eq. (3.105) by recalling that sec(x) = 1/cos(x):


'Vl '


Vl '

+ ln


Vl '

+ tan


^ 2 ,

2 ,

. 2 ,

. 2 J

optical efficiency

Optical efficiency is defined as the ratio of the energy absorbed by the receiver to the energy incident on the collector's aperture. The optical efficiency depends on the optical properties of the materials involved, the geometry of the collector, and the various imperfections arising from the construction of the collector. In equation form (Sodha et al., 1984), r\o = pTa^[(1 - Af tan(0))cos(0)]

where p = reflectance of the mirror. t = transmittance of the glass cover. a = absorptance of the receiver. ^ = intercept factor. Af = geometric factor. 9 = angle of incidence.

The geometry of the collector dictates the geometric factor, Af, which is a measure of the effective reduction of the aperture area due to abnormal incidence effects, including blockages, shadows, and loss of radiation reflected from the mirror beyond the end of the receiver. During abnormal operation of a PTC, some of the rays reflected from near the end of the concentrator opposite the sun cannot reach the receiver. This is called the end effect. The amount of aperture area lost is shown in Figure 3.39 and given by


Usually, collectors of this type are terminated with opaque plates to preclude unwanted or dangerous concentration away from the receiver. These plates result in blockage or shading of a part of the reflector, which in effect reduces the aperture area. For a plate extending from rim to rim, the lost area is shown in Figure 3.39 and given by

where hp = height of parabola (m).

It should be noted that the term tan(9) shown in Eqs. (3.107) and (3.108) is the same as the one shown in Eq. (3.106), and it should not be used twice. Therefore, to find the total loss in aperture area, Aj, the two areas, Ae and Ab, are added together without including the term tan(9) (Jeter, 1983):

48 f2

Compound Parabolic Trough

Finally, the geometric factor is the ratio of the lost area to the aperture area. Therefore,

The most complex parameter involved in determining the optical efficiency of a parabolic trough collector is the intercept factor. This is defined as the ratio of the energy intercepted by the receiver to the energy reflected by the focusing device, i.e., the parabola. Its value depends on the size of the receiver, the surface angle errors of the parabolic mirror, and the solar beam spread.

The errors associated with the parabolic surface are of two types: random and nonrandom (Guven and Bannerot, 1985). Random errors are defined as those errors that are truly random in nature and, therefore, can be represented by normal probability distributions. Random errors are identified as apparent changes in the sun's width, scattering effects caused by random slope errors (i.e., distortion of the parabola due to wind loading), and scattering effects associated with the reflective surface. Nonrandom errors arise in manufacture-assembly or the operation of the collector. These can be identified as reflector profile imperfections, misalignment errors, and receiver location errors. Random errors are modeled statistically, by determining the standard deviation of the total reflected energy distribution, at normal incidence (Guven and Bannerot, 1986), and are given by

0 a/ O" 2un + 4CT2lope + 0" mirror (3.111)

Nonrandom errors are determined from a knowledge of the misalignment angle error (3 (i.e., the angle between the reflected ray from the center of the sun and the normal to the reflector's aperture plane) and the displacement of the receiver from the focus of the parabola (dr). Since reflector profile errors and receiver mislocation along the Y axis essentially have the same effect, a single parameter is used to account for both. According to Guven and Bannerot (1986), random and nonrandom errors can be combined with the collector geometric parameters, concentration ratio (C), and receiver diameter (D) to yield error parameters universal to all collector geometries. These are called universal error parameters, and an asterisk is used to distinguish them from the already defined parameters. Using the universal error parameters, the formulation of the intercept factor, is possible (Guven and Bannerot, 1985):

= 1 + cos(9r) rErf I sin(^r)[1 + cos(y)][1 - 2d sin(y)] - n|3 [1 + cos^)] I 2sin(^r) { | V2nCT*[1 + cos(9r)] J

sin(^r)[1 + cos(^)][1 + 2d*sm(^)] + n3*[1 + cos(^r)] I d9

Erf i

where d* = universal nonrandom error parameter due to receiver mislocation and reflector profile errors, d* = d-/D. (* = universal nonrandom error parameter due to angular errors, (* = (C. ct* = universal random error parameter, ct* = ctC. C = collector concentration ratio, = Aa/Ar. D = riser tube outside diameter (m). dr = displacement of receiver from focus (m). (3 = misalignment angle error (degrees).

Another type of analysis commonly carried out in concentrating collectors is ray tracing. This is the process of following the paths of a large number of rays of incident radiation through the optical system to determine the distribution and intensity of the rays on the surface of the receiver. Ray tracing determines the radiation concentration distribution on the receiver of the collector, called the local concentration ratio (LCR). As was seen in Figure 3.37, the radiation incident on a differential element of reflector area is a cone having a half angle of 16 ' . The reflected radiation is a similar cone, having the same apex angle if the reflector is perfect. The intersection of this cone with the receiver surface determines the image size and shape for that element, and the total image is the sum of the images for all the elements of the reflector. In an actual collector, the various errors outlined previously, which enlarge the image size and lower the local concentration ratio, are considered. The distribution of the local concentration ratio for a parabolic trough collector is shown in Figure 3.40. The shape of the curves depends on the random and nonrandom errors mentioned above and on the angle of incidence. It should be noted that the distribution for half of the receiver is shown in Figure 3.40. Another, more representative way to show this distribution for the whole receiver is in Figure 3.41. As can be seen from these figures, the top part of the receiver essentially receives only direct sunshine from the sun and the maximum concentration, about 36 suns, occurs at 0 incidence angle and at an angle (, shown in Figure 3.40, of 120°.

Compound Parabolic Collector
Receiver angle p (degree) FIGURE 3.40 Local concentration ratio on the receiver of a parabolic trough collector.
Compound Parabolic Collector
FIGURE 3.41 a more representative view of LCR for a collector with receiver diameter of 20 mm and rim angle of 90°.

Example 3.11

For a PTC with a total aperture area of 50 m2, aperture of 2.5 m, and rim angle of 90°, estimate the geometric factor and the actual area lost at an angle of incidence equal to 60°.


As jr = 90°, the parabola height hp = f. Therefore, from Eq. (3.101), hp = f

4 tan




The area lost at an incidence angle of 60° is:

Area lost = A tan(60) = 3.125 X tan(60) = 5.41m2 The geometric factor Af is obtained from Eq. (3.110):


Aa 50

3.6.4 Thermal Analysis of Parabolic Trough Collectors

The generalized thermal analysis of a concentrating solar collector is similar to that of a flat-plate collector. It is necessary to derive appropriate expressions for the collector efficiency factor, F; the loss coefficient, UL; and the collector heat removal factor, FR. For the loss coefficient, standard heat transfer relations for glazed tubes can be used. Thermal losses from the receiver must be estimated, usually in terms of the loss coefficient, UL, which is based on the area of the receiver. The method for calculating thermal losses from concentrating collector receivers cannot be as easily summarized as flat-plate ones, because many designs and configurations are available. Two such designs are presented in this book: the parabolic trough collector with a bare tube and the glazed tube receiver. In both cases, the calculations must include radiation, conduction, and convection losses.

For a bare tube receiver and assuming no temperature gradients along the receiver, the loss coefficient considering convection and radiation from the surface and conduction through the support structure is given by

The linearized radiation coefficient can be estimated from hr = 4oeT? (3.114)

If a single value of hr is not acceptable due to large temperature variations along the flow direction, the collector can be divided into small segments, each with a constant hr.

For the wind loss coefficient, the Nusselt number can be used.

Estimation of the conduction losses requires knowledge of the construction of the collector, i.e., the way the receiver is supported.

Usually, to reduce the heat losses, a concentric glass tube is employed around the receiver. The space between the receiver and the glass is usually evacuated, in which case the convection losses are negligible. In this case, UL, based on the receiver area Ar, is given by

linearized radiation coefficient from cover to ambient estimated by Eq.

external area of glass cover (m2).

linearized radiation coefficient from receiver to cover, given by Eq. (2.74):

In the preceding equations, to estimate the glass cover properties, the temperature of the glass cover, Tc, is required. This temperature is closer to the ambient temperature than the receiver temperature. Therefore, by ignoring the radiation absorbed by the cover, Tc may be obtained from an energy balance:

Solving Eq. (3.118) for Tc gives

Arhr,r-cTr + Ac (hr.c-a + hwT Arhr,r-c + Ac (hr,c-a + hw)

The procedure to find Tc is by iteration, i.e., estimate UL from Eq. (3.116) by considering a random Tc (close to Ta). Then, if Tc obtained from Eq. (3.119) differs from original value, iterate. Usually, no more than two iterations are required.

If radiation absorbed by the cover needs to be considered, the appropriate term must be added to the right-hand side of Eq. (3.116). The principles are the same as those developed earlier for the flat-plate collectors.

Next, the overall heat transfer coefficient, Uo, needs to be estimated. This should include the tube wall because the heat flux in a concentrating collector is high. Based on the outside tube diameter, this is given by

Do hfD


Do = receiver outside tube diameter (m). D i = receiver inside tube diameter (m).

hfi = convective heat transfer coefficient inside the receiver tube (W/m2-K).

The convective heat transfer coefficient, hfi, can be obtained from the standard pipe flow equation:


kf = thermal conductivity of fluid (Wlm-K).

It should be noted that Eq. (3.121) is for turbulent flow (Re > 2300). For laminar flow, Nu = 4.364 = constant.

The instantaneous efficiency of a concentrating collector may be calculated from an energy balance of its receiver. Equation (3.31) also may be adapted for use with concentrating collectors by using appropriate areas for the absorbed solar radiation (Aa) and heat losses (Ar). Therefore, the useful energy delivered from a concentrator is

Note that, because concentrating collectors can utilize only beam radiation, GB is used in Eq. (3.122) instead of the total radiation, Gt, used in Eq. (3.31).

The useful energy gain per unit of collector length can be expressed in terms of the local receiver temperature, Tr, as

Aa noGB ArUL

In terms of the energy transfer to the fluid at the local fluid temperature, Tf (Kalogirou, 2004),

Do hfA

If Tr is eliminated from Eqs. (3.123) and (3.124), we have

where F' is the collector efficiency factor, given by


Ul hfD

As for the flat-plate collector, Tr in Eq. (3.122) can be replaced by T through the use of the heat removal factor, and Eq. (3.122) can be written as

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  • Martin Henderson
    What are the designing parameters of 2d compound parabolic solar collector?
    2 years ago

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