Phyllotaxis-based packing of secondaries and primaries

A phyllotaxis-based radial packing of ellipses.

This figure, generated in Processing, shows the packing of ellipses with 36:1 aspect ratio. In an arrangement of secondaries the white areas would represent loss of brightness, the darker gray areas would represent blocking loss. Processing source code, based on a program by Jim Bumgardner, is here.

A center-coordinated packing of circles and ellipses. Both the primaries and secondaries of telescopic heliostats can be efficiently packed along the same phyllotaxis spiral.

Several thousand center-coordinated circles and ellipses arrayed along the same phyllotaxis spiral.

Application of phyllotaxis patterns to arrangements of conventional heliostats is described by Noone, Torrilhon and Mitsos in "Heliostat Field Optimization: A New Computationally Efficient Model and Biomimetic Layout."

6000 centroid-coordinated circles and 36:1 ellipses arranged along a phyllotaxis spiral.

The ellipses in these diagrams represent the secondary mirrors projected onto the plane of their center (a physically cast shadow on that plane would only be half an ellipse.) The fact that all the ellipses have the same aspect ratio in these examples means that the secondaries are targeted to a constant beaming angle, not aimed toward a single target point, which would result in more elongated ellipses in the periphery of the field.

When two adjacent ellipses overlap, it means that only one of them is visible from their target: in other words, there is blocking loss. When white can be seen between two ellipses, it means there is a loss of radiance at the focus relative to an ideal concentrator, and therefore a loss of thermodynamic efficiency relative to the ideal.