The two expressions for p(9) are equivalent for the point P in Figure 5.5, where 9 = 9c + -rc/2. The curve is generated by incrementing 9 in radians, calculating p, then calculating the coordinates, X and Y, by
Figure 5.5 shows a full, untruncated curve, which is the mathematical solution for a reflector shape with the maximum possible concentration ratio. The reflector shape shown in Figure 5.5 is not the most practical design for a cost-effective concentrator because reflective material is not effectively used in the upper portion of the concentrator. As in the case of the compound parabolic collector, a theoretical cusp curve should be truncated to a lower height and slightly smaller concentration ratio. Graphically, this is done by drawing a horizontal line across the cusp at a selected height and discarding the part of the curve above the line. Mathematically, the curve is defined to a maximum angle 9 value less than 3tc/2 - 9c. The shape of the curve below the cutoff line is not changed by truncation, so the acceptance angle used for the construction of the curve, using Eq. (5.16), of a truncated cusp is equal to the acceptance angle of the fully developed cusp from which it was truncated.
A large acceptance angle of 75° is used in this design so the collector can collect as much diffuse radiation as possible (Kalogirou, 1997). The fully developed cusp, together with the truncated one, is shown in Figure 5.7. The receiver radius considered in the construction of the cusp is 0.24 m. The actual cylinder used, though, is only 0.20 m. This is done in order to create a gap at
the underside of the receiver and the edge of the cusp in order to minimize the optical and conduction losses.
The final design is shown in Figure 5.8. The collector aperture is 1.77 m2, which, in combination with the absorber diameter used, gives a concentration ratio of 1.47 (Kalogirou, 1997). It should be noted that, as shown in Figure 5.8, the system is inclined at the local latitude in order to work effectively.
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