Optimizing telescopic heliostat arrays

Whoops! Looks like most of the sunlight is hitting the ground.

Conventional heliostat arrays have about 33% land utilization—most of the available sunlight strikes the ground! The angular elevation of the top of the tower as viewed from the farthest heliostat is about 7.4 degrees, or a run-over-rise of about 7.8. Telescopic heliostats can do better on both accounts.

Economic optimization of a heliostat array is a Godzilla of a problem requiring full cost data, full knowledge of the operation of the rest of the plant, and full knowledge of the solar resource. A simpler option is available for telescopic heliostats: we can simply design for negligible loss of available sunlight. (This is not an option with conventional heliostats because flat mirrors start blocking each others' view of the target long before they are packed closely enough to collect most of the available light.)

"Designing for negligible loss of available sunlight" in practice means packing the primaries closely and designing the secondaries on the basis of their lowest ray (i.e., the ray most likely to be intercepted by the back of another secondary.)

Closely packed primary mirrors shade about 90% of the land area in the worst case (sun at zenith.) Utilization improves as the sun lowers and the primary mirrors tilt up.

It is also necessary to avoid blocking (redirected light hitting the back of secondaries) if we want to avoid wasting light. We will consider a particular case.

Consider a telescope of linear magnification six. It intensifies sunlight 36 times while increasing its angular divergence by a factor of six. The beam spread of natural sunlight is 0.5°, so the sunlight emerging from the eyepiece of a 6x telescope will have a beam spread of 3°. To accomplish that increase in divergence requires the secondary mirror to be six times closer to the focussed image of the sun than the primary mirror that formed the image. The secondary mirror is tilted about 45° to redirect vertically upward sunlight to nearly horizontal, so it will have an oblong planform, but viewed from directly above (or from the distant central tower), it appears nearly circular, and this circle has one sixth the diameter of the primary mirror.

The now familiar figure below can represent this situation if the ellipses (which as-drawn have 9:1 aspect ratio) are redrawn to a 36:1 aspect ratio while keeping their area the same. Recall that the ellipses in this diagram represent the shadows cast on the ground when the secondaries are illuminated by an imaginary light source located at a design point on the beam-down optics. Since we are interested in minimizing blocking, this design point must be where the lowest rays from the secondaries converge—not their central rays. The 36:1 shadows cast by essentially circular objects are evidence that the imaginary light source is at a run-over-rise of 36:1, or an angular elevation of 1.6°. To this must be added the full 3° beam spread of the light emitted by the secondaries, placing the top of the beam-down optics at an angular elevation of 4.6°, a run-over-rise of 12.4.

Thus, for a given tower height, a telescopic heliostat array has about 1.59 times the radius and 2.5 times the area of a conventional heliostat array. It also has about three times the land utilization, giving overall a plant rating about 7.5 times that of a conventional system.

Primary mirrors can be close packed in a telescopic heliostat array without causing the secondaries to block each other.