The approach is based on simple geometry and the optical characteristics of CPCs. The goals are (1) to fill the field of view of the CPC as much as possible and (2) to identify those design factors





that either limit the achievable concentration or, on the other hand, allow the ideal limit on concentration to be approached as closely as possible.
The following general simplifying assumptions were made. No preconceived constraint on the basic configuration was imposed. The intercept of the view cone of the CPC was used to define the envelope of the area containing heliostats. Within this area, the finegrained heliostat field losses were assumed to be represented by a field wide average. Only geometric losses were considered. Reflection losses were not taken into account. Blocking and shading losses were not calculated in detail. The angular distribution characterizing the incident solar radiation (i. e., sun size, slope, and specularity errors, etc.) was taken to be defined by a cone of a given halfangle (a “pillbox” distribution). The “maximum achievable concentration” was calculated by requiring that the optical intercept factor at the target for a particular configuration be 100%.
9.1.2.1 Surround Field Geometry. One can use simple geometry to determine the maximum solar image size for a particular configuration and for a particular halfangular subtense (including optical errors). From this, one can calculate the maximum achievable geometric concentration ratios C1 and C, which can be attained without intercept losses, respectively for one – and twostage configurations, and compare these with the ideal limiting concentration for the same angular subtense. These ratios, relative to the ideal limit, are plotted in Figure 9.2 for a circular surround field central receiver geometry as a function of the ratio of the tower height (H) to the diameter (D) of the circular field (Figure 9.1a). The concentration achievable by the field alone (C1) maximizes at a value corresponding to onefourth of the ideal limit at a tower height/diameter ratio of about 0.5. This is the same result that was found in Chapter 5 to apply to dishes, which have essentially the identical geometry to that shown in Figure 9.1a. If a CPC with a halfangle of acceptance just sufficient to “see” the entire field is placed in the target plane, it will achieve an additional secondary concentration of C2 = 1/sin2(0c), where 0c = tan1(D/2H). The product of this with the field concentration (C12 = CtC2), also plotted in Figure 9.2, will exceed that for the heliostat field alone and approach the ideal limit at large tower heights. Again, this is the same result as is well established for dishes (O’Gallagher and Winston, 1985). Note, however, that very tall towers are not needed, since even at tower height/diameter ratios near 1.0, the system concentration is over 80% of the ideal limit, more than three times the maximum achievable without a secondary. The CPC acceptance angle, в, for the corresponding twostage system is also indicated as a function of the tower height/diameter ratio in Figure 9.2.
9.1.2.2 Field Size. We would like to estimate the scale of the required solar plant (i. e., the actual size of the heliostat field (and tower height, etc.). To do this, we combine the effect of all the loss mechanisms in a single parameter, X, such that the total power delivered to the tower is XIA, where I is the direct
FIGURE 9.2: The maximum achievable geometric concentration relative to the thermodynamic limit for one – and twostage central receiver systems in a surround field geometry as a function of the tower heighttofield diameter ratio. Also indicated is the corresponding secondary acceptance angle. 
normal insolation and A is the total area within the secondary field of view. This efficiency factor, X, will of course be a further function of solar incidence angles, location within the heliostat array, and material properties, among others. But for purposes of illustration, we will assume that these can all be approximated by some “average” over the array.
For a solar plant that would deliver a net total thermal power P, we will need a total area A = P/XI. If we rather arbitrarily take X = 0.25 and assume for simplicity that I ~1000 W/m2, then, for a net solar thermal power of at least a megawatt, we will need a total ground area of about A = 4000 m2. We use this to define a baseline configuration for comparison with alternative designs. From Figure 2, we are led to choose a twostage central receiver design with a symmetric circular surround field geometry, H/D near 1.0, and a CPC with 0c = 25°. For an area of 4000 m2, this corresponds to a tower height of about 77 m, which is tall but not prohibitively so.
9.1.2.3 Nonsurround Field Designs. Although the baseline configuration defined earlier achieves 80% of the ideal limit for a given set of optical errors, it requires a relatively tall tower for the size of the heliostat field. One of the main purposes of this analysis is to explore the effect of deviating from this optimal axially symmetric configuration by tilting the CPC axis and aperture plane away from directly downward and pointing it at some “lookout angle” g off to the north. This alternative configuration is illustrated schematically in Figure 9.1b. The intercept of the view cone with the ground (assumed flat and horizontal) is an ellipse (a prototypical conic section). If one keeps the tower height constant, the area of the intercepted ellipse gets rapidly larger as g is increased.
For purposes of illustration, we rescale the geometry of the basic configuration as g is increased to keep the area constant, here, A = 4000 m2. This might be regarded as a rather arbitrary procedure since so many other things (e. g., obliquity effects, blocking, and shading, among others) are also likely to change as g is varied. However, these effects can be incorporated later and this assumption keeps the scale of all the configurations comparable, so that the effect on other geometric quantities is directly manifest and easily interpreted.
In Figure 9.3, the effect on a number of geometric quantities of gradually tilting the CPC axis northward is plotted as a function of the lookout angle g. The quantities plotted include the tower height, distance to the furthest heliostat, and the ellipse location. Also plotted is the maximum geometric concentration ratio achievable with the twostage system. This has been calculated by first determining C1 max, the maximum geometric concentration from the first stage, as determined directly from geometry and using the same approach as was used by Rabl (1976a). The maximum possible twostage concentration ratio is then determined simply from
C12 = C1 *C2 = C1 /sin2(0) (9.1)
12,max 1,max 2,max 1,max v c
This is the same approach used by O’Gallagher and Winston (1985) for point focus dishes and for the central receiver with a circular surround field plotted in Figure 2. In all these calculations, the geometric concentration ratio C = A /A where A is the effective collecting aperture of the system and Aabs is the area of the target. Furthermore, in nonimaging optics, when we say that the concentration is “achievable,” we mean that it applies to some directional distribution seen by Acol, such that the throughput to Aabs is 100%—that is, it is without any geometric throughput losses. This means that Aabs can be no smaller that the size of the sun’s image produced by the heliostat most distant from the target. Finally, we note that Acol is the effective area of the collecting aperture, including the effect of cosine factors. For configurations such as those schematically illustrated in Figure 1b, Acol is not simply the gross area of the ellipse. Instead, it can be seen to be equal to the area of a circle seen from the target that would fill the same field of view as that of the ellipse. This is the circle defined by the intersection of the CPC acceptance cone with the plane perpendicular to the cone axis at the point at which that axis meets the ground. Substituting in all the appropriate geometric quantities, we find the scale independent result that
c12>max = cos2(y +0c)/cos2(y) (9.2)
which is plotted as a function of g in Figure 3. Note that, in the limit of a small g, this quantity approaches the value for the circular surround field (as it must) and decreases steadily as one tilts the CPC plane away from directly downward and moves the heliostat field northward.
FIGURE 9.3: The maximum achievable geometric concentration as a percentage of the ideal limit is plotted as a function of lookout angle g for a baseline twostage concept with a 25° CPC (Cmax = 5.60x). The variation of tower height and some other dimensions are also plotted. The area of the elliptical “footprint” intercepted by the CPC view angle is held constant at A = 4000 m2 as g is varied from 0° (the surround field limit) to 90° – 0c. The length L is the distance from the target plane (the top of the tower) to the farthest point in the heliostat field, Xmin is the distance from the bottom of the tower to the southernmost boundary of the footprint ellipse, and Xmax is the distance from the bottom of the tower to the northernmost boundary of the ellipse. 
9.1.2.4 Obliquity (cosine) Effects. The main reason for locating central receiver heliostat fields mostly to the north of the tower is to minimize cosine losses from the mirror area. In order to estimate the importance of these effects in this study, we have calculated the cosine correction in the meridian plane (the northsouth plane containing the tower for a tower height of 60 m, a latitude of 45°, and for three noontime sun positions: winter solstice, equinox, and summer solstice). We find that the effect ranges from a maximum average cosine loss over the whole field of about 20% at winter solstice to about 5% at summer solstice. Note that the effect is not nearly as severe as it is for cases with a shorter tower relative to the field dimensions, such as most contemporary tower designs. It should also be noted, that although these effects are losses in costeffectiveness, they do not result in a real loss in throughput or in concentration.
For a given set of optical tolerances, Cmax is determined by simple geometry and the distance L. This determines the relative size of the image of the sun at the target, which in turn determines the maximum achievable geometric concentration. Here it can be seen that the achievable concentration is greatest—about 80% of the ideal limit—for the surround field (g = 0) and decreases steadily as the CPC is tilted to look out toward the north.