As we have seen in Section 3.2, concentrating collectors work by interposing an optical device between the source of radiation and the energy-absorbing surface. Therefore, for concentrating collectors, both optical and thermal analyses are required. In this book, only two types of concentrating collectors are analyzed: compound parabolic and parabolic trough collectors. Initially, the concentration ratio and its theoretical maximum value are defined.

The concentration ratio (C) is defined as the ratio of the aperture area to the receiver-absorber area; that is,

For flat-plate collectors with no reflectors, C = 1. For concentrators, C is always greater than 1. Initially the maximum possible concentration ratio is investigated. Consider a circular (three-dimensional) concentrator with aperture Aa and receiver area Ar located at a distance R from the center of the sun, as shown in Figure 3.31. We saw in Chapter 2 that the sun cannot be considered a point source but a sphere of radius r; therefore, as seen from the earth, the sun has a half angle, 0m, which is the acceptance half angle for maximum concentration. If both the sun and the receiver are considered to be blackbodies at temperatures Ts and Tr, the amount of radiation emitted by the sun is given by

(4nr 2)oTs4

A fraction of this radiation is intercepted by the collector, given by

Concentrator

FIGURE 3.31 Schematic of the sun and a concentrator.

Concentrator

FIGURE 3.31 Schematic of the sun and a concentrator.

Therefore, the energy radiated from the sun and received by the concentrator is

4nr2 r2

A blackbody (perfect) receiver radiates energy equal to AtT4, and a fraction of this reaches the sun, given by

Under this idealized condition, the maximum temperature of the receiver is equal to that of the sun. According to the second law of thermodynamics, this is true only when Qr^s = Qs-r. Therefore, from Eqs. (3.80) and (3.81),

Ar r2

Since the maximum value of Fr^s is equal to 1, the maximum concentration ratio for three-dimensional concentrators is [sin(9m) = r/R]:

sin2(0m)

A similar analysis for linear concentrators gives

sin(em)

As was seen in Chapter 2, 29m is equal to 0.53° (or 32'), so 9m, the half acceptance angle, is equal to 0.27° (or 16'). The half acceptance angle denotes coverage of one half of the angular zone within which radiation is accepted by the concentrator's receiver. Radiation is accepted over an angle of 29m, because radiation incident within this angle reaches the receiver after passing through the aperture. This angle describes the angular field within which radiation can be collected by the receiver without having to track the concentrator.

Equations (3.83) and (3.84) define the upper limit of concentration that may be obtained for a given collector viewing angle. For a stationary CPC, the angle 9m depends on the motion of the sun in the sky. For a CPC having its axis in a N-S direction and tilted from the horizontal such that the plane of the sun's motion is normal to the aperture, the acceptance angle is related to the range of hours over which sunshine collection is required; for example, for 6 h of useful sunshine collection, 29m = 90° (sun travels 15°/h). In this case, Cmax = 1/sin(45°) = 1.41.

For a tracking collector, 9m is limited by the size of the sun's disk, small-scale errors, irregularities of the reflector surface, and tracking errors. For a perfect collector and tracking system, Cmax depends only on the sun's disk. Therefore,

For single-axis tracking, Cmax = 1/sin(16 ' ) = 216. For full tracking, Cmax = 1/sin2(16 ' ) = 46,747.

It can therefore be concluded that the maximum concentration ratio for two-axis tracking collectors is much higher. However, high accuracy of the tracking mechanism and careful construction of the collector are required with an increased concentration ratio, because 9m is very small. In practice, due to various errors, much lower values than these maximum ones are employed.

From the diameter of the sun and the earth and the mean distance of sun from earth, shown in Figure 2.1, estimate the amount of energy emitted from the sun, the amount of energy received by the earth, and the solar constant for a sun temperature of 5777 K. If the distance of Venus from the sun is 0.71 times the mean sun-earth distance, estimate the solar constant for Venus.

Solution

The amount of energy emitted from the sun, Qs, is

Qs/As = oTs4 = 5.67 X 10~8 X (5777)4 = 63,152,788 w 63MW/m2 or

Qs = 63.15 X 4n(1.39 X 109/2)2 = 3.82 X 1020 MW From Eq. (3.80), the solar constant can be obtained as

0s- = 4< = (L39 X 109/2)2 63152,788 = 1363W/m2 Aa R2 s (1.496 X1011)2

The area of the earth exposed to sunshine is %d2/4. Therefore, the amount of energy received from earth = -k(1.27 X 107)2 X 1.363/4 = 1.73 X 1014 kW. These results verify the values specified in the introduction to Chapter 2.

The mean distance of Venus from the sun is 1.496 X 1011 X 0.71 = 1.062 X 1011 m. Therefore, the solar constant of Venus is

0s- = 4 < = (L39 X 109f2 63152,788 = 2705W/m2 Aa R2 s (1.062 X1011)2

3.6.1 Optical Analysis of a Compound Parabolic Collector

The optical analysis of CPC collectors deals mainly with the way to construct the collector shape. A CPC of the Winston design (Winston and Hinterberger,

1975) is shown in Figure 3.32. It is a linear two-dimensional concentrator consisting of two distinct parabolas, the axes of which are inclined at angles ±9C with respect to the optical axis of the collector. The angle 9C, called the collector half acceptance angle, is defined as the angle through which a source of light can be moved and still converge at the absorber.

The Winston-type collector is a non-imaging concentrator with a concentration ratio approaching the upper limit permitted by the second law of thermodynamics, as explained in previous section.

The receiver of the CPC does not have to be flat and parallel but, as shown in Figure 3.5, can be bifacial, a wedge, or cylindrical. Figure 3.33 shows a collector with a cylindrical receiver; the lower portion of the reflector (AB and AC) is circular, while the upper portions (BD and CE) are parabolic. In this design, the requirement for the parabolic portion of the collector is that, at any point P, the normal to the collector must bisect the angle between the tangent line PG to the receiver and the incident ray at point P at angle 9c with respect to the collector axis. Since the upper part of a CPC contributes little to the radiation reaching the absorber, it is usually truncated, forming a shorter version of the CPC, which is also cheaper. CPCs are usually covered with glass to avoid dust and other materials entering the collector and reducing the reflectivity of its walls. Truncation affects little the acceptance angle but results in considerable material saving and changes the height-to-aperture ratio, the concentration ratio, and the average number of reflections.

Aperture

Aperture

FIGURE 3.33 Schematic diagram of a CPC collector.

These collectors are more useful as linear or trough-type concentrators. The orientation of a CPC collector is related to its acceptance angle (29c, in Figures 3.32 and 3.33). The two-dimensional CPC is an ideal concentrator, i.e., it works perfectly for all rays within the acceptance angle, 29c. Also, depending on the collector acceptance angle, the collector can be stationary or tracking. A CPC concentrator can be oriented with its long axis along either the north-south or east-west direction and its aperture tilted directly toward the equator at an angle equal to the local latitude. When oriented along the north-south direction, the collector must track the sun by turning its axis to face the sun continuously. Since the acceptance angle of the concentrator along its long axis is wide, seasonal tilt adjustment is not necessary. It can also be stationary, but radiation will be received only during the hours when the sun is within the collector acceptance angle.

When the concentrator is oriented with its long axis along the east-west direction, with a little seasonal adjustment in tilt angle, the collector is able to catch the sun's rays effectively through its wide acceptance angle along its long axis. The minimum acceptance angle in this case should be equal to the maximum incidence angle projected in a north-south vertical plane during the times when output is needed from the collector. For stationary CPC collectors mounted in this mode, the minimum acceptance angle is equal to 47°. This angle covers the declination of the sun from summer to winter solstices (2 X 23.5°). In practice, bigger angles are used to enable the collector to collect diffuse radiation at the expense of a lower concentration ratio. Smaller (less than 3) concentration ratio CPCs are of greatest practical interest. These, according to Pereira (1985), are able to accept a large proportion of diffuse radiation incident on their apertures and concentrate it without the need to track the sun. Finally, the required frequency of collector adjustment is related to the collector concentration ratio. Thus, the C < 3 needs only biannual adjustment, while the C > 10 requires almost daily adjustment; these systems are also called quasi-static.

Concentrators of the type shown in Figure 3.5 have an area concentration ratio that is a function of the acceptance half angle, 0c. For an ideal linear concentrator system, this is given by Eq. (3.84) by replacing 0m with 0c.

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