For FPCs with no reflectors,\u00a0C<\/em>\u00a0=\u00a01. For concentrators,\u00a0C<\/em>\u00a0is always greater than 1. Initially the maximum possible concentration ratio is investigated. Consider a circular (three-dimensional) concentrator with aperture\u00a0A<\/em>a<\/sub>\u00a0and receiver area\u00a0A<\/em>r<\/sub>\u00a0located at a distance\u00a0R<\/em>\u00a0from the center of the sun, as shown in\u00a0Figure 3.35. We saw in\u00a0Chapter 2\u00a0that the sun cannot be considered a point source but a sphere of radius\u00a0r<\/em>; therefore, as seen from the earth, the sun has a half angle,\u00a0\u03b8<\/em>m<\/sub>, which is the acceptance half angle for maximum concentration. If both the sun and the receiver are considered to be blackbodies at temperatures\u00a0T<\/em>s<\/sub>\u00a0and\u00a0T<\/em>r<\/sub>, the amount of radiation emitted by the sun is given by:<\/p>\n\n\n\n FIGURE 3.35<\/strong> <\/a>Schematic of the sun and a concentrator.<\/p>\n\n\n\n A fraction of this radiation is intercepted by the collector, given by:<\/p>\n\n\n\n Therefore, the energy radiated from the sun and received by the concentrator is:<\/p>\n\n\n\n A blackbody (perfect) receiver radiates energy equal to 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\u00a0Q<\/em>r\u2013s<\/sub>\u00a0=\u00a0Q<\/em>s\u2013r<\/sub>. Therefore, from Eqs\u00a0(3.90)\u00a0and\u00a0(3.91),<\/p>\n\n\n\n Since the maximum value of F<\/em>r\u2013s<\/sub> is equal to 1, the maximum concentration ratio for three-dimensional concentrators is [sin(\u03b8<\/em>m<\/sub>) = r<\/em>\/R<\/em>]:<\/p>\n\n\n\n A similar analysis for linear concentrators gives:<\/p>\n\n\n\n As was seen in\u00a0Chapter 2, 2\u03b8<\/em>m<\/sub>\u00a0is equal to 0.53\u00b0 (or 32\u2032), so\u00a0\u03b8<\/em>m<\/sub>, the half acceptance angle, is equal to 0.27\u00b0 (or 16\u2032). The half acceptance angle denotes coverage of one half of the angular zone within which radiation is accepted by the concentrator\u2019s receiver. Radiation is accepted over an angle of 2\u03b8<\/em>m<\/sub>, 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.<\/p>\n\n\n\n Equations\u00a0(3.93)\u00a0and\u00a0(3.94)\u00a0define the upper limit of concentration that may be obtained for a given collector viewing angle. For a stationary CPC, the angle\u00a0\u03b8<\/em>m<\/sub>\u00a0depends on the motion of the sun in the sky. For a CPC having its axis in an N\u2013S direction and tilted from the horizontal such that the plane of the sun\u2019s 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\u00a0h of useful sunshine collection, 2\u03b8<\/em>m<\/sub>\u00a0=\u00a090\u00b0 (sun travels 15\u00b0\/h). In this case,\u00a0C<\/em>max<\/sub>\u00a0=\u00a01\/sin(45\u00b0)\u00a0=\u00a01.41.<\/p>\n\n\n\n For a tracking collector, \u03b8<\/em>m<\/sub> is limited by the size of the sun\u2019s disk, small-scale errors, irregularities of the reflector surface, and tracking errors. For a perfect collector and tracking system, C<\/em>max<\/sub> depends only on the sun\u2019s disk. Therefore,<\/p>\n\n\n\n For single-axis tracking, C<\/em>max<\/sub> = 1\/sin(16\u2032) = 216.<\/p>\n\n\n\n For full tracking, C<\/em>max<\/sub> = 1\/sin2<\/sup>(16\u2032) = 46,747.<\/p>\n\n\n\n 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 \u03b8<\/em>m<\/sub> is very small. In practice, due to various errors, much lower values than these maximum ones are employed.<\/p>\n\n\n\n EXAMPLE 3.7<\/p>\n\n\n\n From the diameter of the sun and the earth and the mean distance of sun from earth, shown in\u00a0Figure 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\u00a0K. If the distance of Venus from the sun is 0.71 times the mean sun\u2013earth distance, estimate the solar constant for Venus.<\/p>\n\n\n\n Solution<\/p>\n\n\n\n The amount of energy emitted from the sun, Q<\/em>s<\/sub>, is:<\/p>\n\n\n\n or<\/p>\n\n\n\n From Eq.\u00a0(3.90), the solar constant can be obtained as:<\/p>\n\n\n\n The area of the earth exposed to sunshine is \u03c0d<\/em>2<\/sup>\/4. Therefore, the amount of energy received from earth\u00a0=\u00a0\u03c0(1.27\u00a0\u00d7\u00a0107<\/sup>)2<\/sup>\u00a0\u00d7\u00a01.363\/4\u00a0=\u00a01.73\u00a0\u00d7\u00a01014<\/sup>\u00a0kW. These results verify the values specified in the introduction to\u00a0Chapter 2.<\/p>\n\n\n\n The mean distance of Venus from the sun is 1.496 \u00d7 1011<\/sup> \u00d7 0.71 = 1.062 \u00d7 1011<\/sup> m. Therefore, the solar constant of Venus is:<\/p>\n\n\n\n <\/p>\n\n\n\n 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\u00a0Figure 3.36. It is a linear two-dimensional concentrator consisting of two distinct parabolas, the axes of which are inclined at angles \u00b1\u03b8<\/em>c<\/sub>\u00a0with respect to the optical axis of the collector. The angle\u00a0\u03b8<\/em>c<\/sub>, called the collector\u00a0half acceptance angle<\/em>, is defined as the angle through which a source of light can be moved and still converge at the absorber.<\/p>\n\n\n\n FIGURE 3.36<\/strong> <\/a>Construction of a flat receiver compound parabolic collector.<\/p>\n\n\n\n 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.<\/p>\n\n\n\n The receiver of the CPC does not have to be flat and parallel but, as shown in\u00a0Figure 3.5, can be bifacial, a wedge, or cylindrical.\u00a0Figure 3.37\u00a0shows 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\u00a0\u03b8<\/em>c<\/sub>\u00a0with 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 hardly affects the acceptance angle but\u00a0results in considerable material saving and changes the height-to-aperture ratio, the concentration ratio, and the average number of reflections.<\/p>\n\n\n\n FIGURE 3.37<\/strong> <\/a>Schematic diagram of a CPC collector.<\/p>\n\n\n\n These collectors are more useful as linear or trough-type concentrators. The orientation of a CPC collector is related to its acceptance angle (2\u03b8<\/em>c<\/sub>, in\u00a0Figures 3.36\u00a0and\u00a03.37). The two-dimensional CPC is an ideal concentrator, that is, it works perfectly for all rays within the acceptance angle, 2\u03b8<\/em>c<\/sub>. 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\u2013south or east\u2013west direction and its aperture tilted directly toward the equator at an angle equal to the local latitude. When oriented along the north\u2013south 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.<\/p>\n\n\n\n When the concentrator is oriented with its long axis along the east\u2013west direction, with a little seasonal adjustment in tilt angle, the collector is able to catch the sun\u2019s 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\u2013south 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\u00b0. This angle covers the declination of the sun from summer to winter solstices (2\u00a0\u00d7\u00a023.5\u00b0). 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\u00a0Pereira (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.\u00a0Thus, for\u00a0C<\/em>\u00a0\u2264\u00a03 the collector needs only biannual adjustment, while for\u00a0C<\/em>\u00a0>\u00a010 the collector requires almost daily adjustment; these systems are also called\u00a0quasi-static<\/em>.<\/p>\n\n\n\n Concentrators of the type shown in\u00a0Figure 3.5\u00a0have an area concentration ratio that is a function of the acceptance half angle,\u00a0\u03b8<\/em>c<\/sub>. For an ideal linear concentrator system, this is given by Eq.\u00a0(3.94)\u00a0by replacing\u00a0\u03b8<\/em>m<\/sub>\u00a0with\u00a0\u03b8<\/em>c<\/sub>.<\/p>\n\n\n\n The instantaneous efficiency, \u03b7<\/em>, of a CPC is defined as the useful energy gain divided by the incident radiation on the aperture plane; that is,<\/p>\n\n\n\n In Eq.\u00a0(3.95),\u00a0G<\/em>t<\/sub>\u00a0is the total incident radiation on the aperture plane. The useful energy,\u00a0Q<\/em>u<\/sub>, is given by an equation similar to Eq.\u00a0(3.60), using the concept of absorbed radiation as:<\/p>\n\n\n\n The absorbed radiation,\u00a0S<\/em>, is obtained from (Duffie and Beckman, 2006):<\/p>\n\n\n\n where<\/a><\/p>\n\n\n\n <\/a>\u03c4<\/em>c<\/sub> = transmittance of the CPC cover.<\/p>\n\n\n\n <\/a>\u03c4<\/em>CPC<\/sub> = transmissivity of the CPC to account for reflection loss.<\/p>\n\n\n\n The various radiation components in Eq.\u00a0(3.97)\u00a0come from radiation falling on the aperture within the acceptance angle of the CPC and are given as follows:<\/p>\n\n\n\n In Eqs\u00a0(3.98a)\u2013(3.98c),\u00a0\u03b2<\/em>\u00a0is the collector aperture inclination angle with respect to horizontal. In Eq.\u00a0(3.98c), the ground-reflected radiation is effective only if the collector receiver \u201csees\u201d the ground, that is, (\u03b2<\/em>\u00a0+\u00a0\u03b8<\/em>c<\/sub>)\u00a0>\u00a090\u00b0.<\/p>\n\n\n\n It has been shown by\u00a0Rabl et al. (1980)\u00a0that the insolation,\u00a0G<\/em>CPC<\/sub>, of a collector with a concentration\u00a0C<\/em>\u00a0can be approximated very well from:<\/p>\n\n\n\n <\/a>It is convenient to express the absorbed solar radiation, S<\/em>, in terms of G<\/em>CPC<\/sub> in the following way:<\/p>\n\n\n\n or<\/p>\n\n\n\n where<\/a><\/p>\n\n\n\n <\/a>\u03c4<\/em>cover<\/sub> = transmissivity of the cover glazing.<\/p>\n\n\n\n <\/a>\u03c4<\/em>CPC<\/sub> = effective transmissivity of CPC.<\/p>\n\n\n\n <\/a>\u03b1<\/em>r<\/sub> = absorptivity of receiver.<\/p>\n\n\n\n <\/a>\u03b3<\/em> = correction factor for diffuse radiation, given by:<\/p>\n\n\n\n The factor\u00a0\u03b3<\/em>, given by Eq.\u00a0(3.102), accounts for the loss of diffuse radiation outside the acceptance angle of the CPC with a concentration\u00a0C<\/em>. The ratio\u00a0G<\/em>D<\/sub>\/G<\/em>t<\/sub>\u00a0varies from about 0.11 on very clear sunny days to about 0.23 on hazy days.<\/p>\n\n\n\n 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):<\/p>\n\n\n\n The effective transmissivity, \u03c4<\/em>CPC<\/sub>, 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, \u03c1<\/em>, of the CPC walls and the average number of reflections, n<\/em>, expressed approximately by:<\/p>\n\n\n\n This equation can also be used to estimate\u00a0\u03c4<\/em>CPC,B<\/sub>,\u00a0\u03c4<\/em>CPC,D<\/sub>, and\u00a0\u03c4<\/em>CPC,G<\/sub>\u00a0in Eq.\u00a0(3.97), which are usually treated as the same. Values of\u00a0n<\/em>\u00a0for full and truncated CPCs can be obtained from\u00a0Figure 3.38. 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\u00a0Figure 3.38, the average number of reflections is a function of concentration ratio,\u00a0C<\/em>, and the acceptance half angle,\u00a0\u03b8<\/em>c<\/sub>. For a truncated concentrator, the line (1\u00a0\u2212\u00a01\/C<\/em>) 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\u00a0Figures 3.39\u00a0and\u00a03.40.\u00a0Figures 3.38\u20133.40\u00a0can be used to design a CPC, as shown in the following example. For more accuracy, the equations representing the curves of\u00a0Figures 3.38\u20133.40\u00a0can be used as given in\u00a0Appendix 6.<\/p>\n\n\n\n EXAMPLE 3.8<\/p>\n\n\n\n Find the CPC characteristics for a collector with acceptance half angle \u03b8<\/em>c<\/sub> = 12\u00b0. Find also its characteristics if the collector is truncated so that its height-to-aperture ratio is 1.4.<\/p>\n\n\n\n Solution<\/p>\n\n\n\n For a full CPC, from\u00a0Figure 3.39\u00a0for\u00a0\u03b8<\/em>c<\/sub>\u00a0=\u00a012\u00b0, the height-to-aperture ratio\u00a0=\u00a02.8 and the concentration ratio\u00a0=\u00a04.8. From\u00a0Figure 3.40, the area of the reflector is 5.6 times the aperture area; and from\u00a0Figure 3.38, the average number of reflections of radiation before reaching the absorber is 0.97.<\/p>\n\n\n\n For a truncated CPC, the height-to-aperture ratio\u00a0=\u00a01.4. Then, from\u00a0Figure 3.39, the concentration ratio drops to 4.2; and from\u00a0Figure 3.40, the reflector-to-aperture area drops to 3, which indicates how significant is the saving in reflector material. Finally, from\u00a0Figure 3.38, the average number of reflections is at least 1\u00a0\u2212\u00a01\/4.2\u00a0=\u00a00.76.<\/p>\n\n\n\n EXAMPLE 3.9<\/p>\n\n\n\n A CPC has an aperture area of 4 m2<\/sup> and a concentration ratio of 1.7. Estimate the collector efficiency given the following:<\/a><\/p>\n\n\n\n <\/a>Total radiation = 850 W\/m2<\/sup>.<\/p>\n\n\n\n <\/a>Diffuse to total radiation ratio = 0.12.<\/p>\n\n\n\n <\/a>Receiver absorptivity = 0.87.<\/p>\n\n\n\n <\/a>Receiver emissivity = 0.12.<\/p>\n\n\n\n <\/a>Mirror reflectivity = 0.90.<\/p>\n\n\n\n <\/a>Glass cover transmissivity = 0.90.<\/p>\n\n\n\n <\/a>Collector heat loss coefficient = 2.5 W\/m2<\/sup> K.<\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si194.png\")
(3.87)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si195.png\")
(3.88)<\/strong><\/a><\/p>\n\n\n\n![\"image\"\/](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030f03-35-9780123972705.jpg\")
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and a fraction of this reaches the sun, given by:<\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si199.png\")
(3.91)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si200.png\")
(3.92)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si201.png\")
(3.93)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si202.png\")
(3.94)<\/strong><\/p>\n\n\n\n![\"image\"\/](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si203.png\")
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<\/figure>\n\n\n\n3.6.1 Optical analysis of a compound parabolic collector<\/h3>\n\n\n\n
![\"image\"\/](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030f03-36-9780123972705.jpg\")
<\/figure>\n\n\n\n![\"image\"\/](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030f03-37-9780123972705.jpg\")
<\/figure>\n\n\n\n3.6.2 Thermal analysis of compound parabolic collectors<\/h3>\n\n\n\n
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(3.95)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si208.png\")
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(3.98c)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si212.png\")
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(3.100)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si214.png\")
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(3.102)<\/strong><\/p>\n\n\n\n![\"image\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9780123972705\/files\/images\/F000030si216.png\")
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(3.104)<\/strong><\/p>\n\n\n\n