An environment geometry where the center of the paraboloidal reflector defines the ori­gin of Euclidean space and the z-axis is aligned with the optical axis of the mirror, (forcing the focus to be at the position [0 0 3] and the base of the paraboloidal reflector to be at the point [0 0 0]) was generated. The flux distribution on 200 planes about the focal point was calculated and the initial data points of flux representing 400 times the solar insola­tion were isolated.

The correction of the data cube for non-horizontal surfaces took only five iterations and rapidly converged to high tolerances. The resulting surface designed for even illumina­tions can be seen in Figure 3a. The coefficients of the surface fit to Equation 1 are k1 = -3.915e-02, k2 = -3.915e-02, k3 = -1.208e-01, k4 = 2.755e-11, k5 = 8.652e-07, k6 = — 8.590e-07, ky = -3.184e-05, k8 = 3.183e-05 and kg = 6.951e-01.

Removing the near zero coefficients from the best fit surface generated above, the sur­face can be approximated by the general equation for this application by

x2 + y2 + a(z-b)2 = r2, (2)

which is an ellipsoid where a = k3/k1, b = kg/(2k3) and r2 = kg2/(4k1k3) + 1/k1. Again using

the code from [12] the flux distribution onto the surface defined in Equation 2 was gener­ated the results of which can be seen in Figure 3b. The variation in the intensity across the surface shape in no greater than ±5% from the desired value of 400 suns, which is within the accepted tolerances of common high-concentration photovoltaic cells.

The optical efficiency, defined here as the ratio of the amount of energy striking the re­ceiver surface to that of the total energy collected through the aperture of the paraboloidal concentrator (including a 100% reflection coefficient) was calculated at only 74% and de­creases for surfaces with higher solar concentration. This represents a value much below what is required to be commercially competitive. This short fall is created from the inabil­ity of the least squares approximation of the quadric surface to mold to the steep sides of the desired surface as seen in Figure 3a.

Discussion

It is clear that by applying the method described, a surface can be generated where upon a constant flux is incident. For the case of a paraboloidal dish concentrator where a con­stant illumination of 400 suns on its surface was desired, homogeneity was achieved with a tolerance of ±5% of value under simulation only The optical performance of such a generated surface was 74% of the available energy.

While the tolerances in homogeneity is acceptable for photovoltaic applications the op­tical performance is far below the ideal. This value however, could be greatly increased by changing the order of the surface fitting algorithm to either a cubic or even a quartic function. This gives greater flexibility in the surface fitting algorithm to mold to the desired shape. Alternatively greater optical efficiency can be gained by using combinations of quadric surfaces in regions or rapid change in structure (such as combinations of ellip­soids and tubular structures in our example).

This method is ideal for creating an evenly illuminated surface under static solar condi­tions, such as for paraboloidal dish collectors. However, the incoming solar radiation is a dynamic system. Variations in the shape of the receiver surface and/or surface orien­tation may be required to allow for variations in the terrestrial spatial energy distribution, tracking errors in concentrating components, degradation in the performance of optical components over time and the dynamic nature or Fresnel mirror concentrators such as power towers and central receiver designs. Creating a static surface to allow for all of these changes presents technical difficulties. The tolerances that each of these surfaces has for dynamic conditions needs to be determined.

This paper has presented a method to produce homogeneous solar flux distributions on a generated receiver surface. This work is complimentary to the that of [10], using differ­ent approaches to generated the receiver surfaces, for which both methods achieve high levels of homogeneity