Low-aiming, target points, and optimization

Cross-section of an array of telescopic heliostats showing the target point for all heliostats lying at the same azimuth to the field center. The field radius is 14 times the height of the beam-down optics—about twice the comparable ratio for conventional heliostats.

Aiming slightly low in the outer portion of the heliostat field, and slightly high in the inner portion of the field, offers the advantage of making the focal zone more compact. This variable aiming can be idealized as aiming all the heliostats that lie on the same azimuth from the origin toward a single target point behind the focal zone. When heliostats are targeted in this way, the locus of the geometric constraint—the ideal shape of the beam-down optics—is no longer a family of confocal parabolas, instead it is a family of ellipses having as their foci the target point and the oculus center.

Whatever happens near the inner radius of the field is of minor importance due to the relatively small number of heliostats involved. On the other hand, what happens with heliostats near the outer radius of the field is important because there are the most numerous rows of heliostats.

Low-aiming of heliostats at the periphery can be carried too far. With constant aiming, the lowest ray in the beam from a secondary rises at only beta/2 (1:36 with 6x telescopic heliostats.) Lowering aim by a mere beta/2 would send the lowest rays traveling horizontally—guaranteeing that they would hit the back of another secondary. In fact, any lowering of the aiming angle will entail either some blocking by other secondaries (should we keep the primaries close-packed) or a decrease in land utilization (if we space the primaries out.) An optimized system will always have a bit of every possible kind of loss: all the possible losses come out to play in the big trade-off game. For the purpose of a simple baseline design we can be arbitrary: we will insist on zero blocking loss, but we will permit the land utilization in the outermost row to fall to 2/3.

The circles are the primary mirrors viewed from above; the ellipses are the shadows cast on a horizontal plane by the secondaries when illuminated from the direction of the lowest ray in there beam. 

In a system with 6x telescopic heliostats, the beam spread is 3°, and the nominal beaming angle is 3.1°. That leaves the lowest rays in the beam directed 1.6° above horizontal, or a run-over-rise of 36:1. The ellipses in the figure above are 9:1, they would need to be elongated to 36:1, but still with the same surface area, to represent the design condition. Aiming lower while keeping blocking at zero means that the ellipses must be further elongated, but this time the dimension of the minor axis is kept constant and the surface area is allowed to increase. The surface area of the circles (which represent the primary mirrors) are not allowed to change when we aim lower because we are using the same design of heliostats over the whole field. Therefore, the area increase of the ellipses associated with low-aiming is balanced by extra empty space between primaries.

At the last row we allow 2/3 land utilization, meaning that low-aiming is permitted to increase the surface area of the ellipses by 50%: their aspect ratio can increase from 36:1 to 54:1. So, at the last row, the lowest ray can have a run-over-rise of 54:1, which is an elevation angle of 1.1°. Recalling that the beam spread is 3°, the middle ray will have an elevation angle of 2.6°, or a run-over-rise of 22:1; the top ray will have an elevation angle of 4.1°, or a run-over-rise of 14 (putting that in a short hand, the outer heliostat operates at: 14-22-54, in run-over-rise numbers, which are also the reciprocals of the elevation angles measured in radians.) By comparison, a heliostat at its design condition operates at 12.4-18.5-36.

The unique row of heliostats that operates with both design condition aiming and chock-a-block packing should be located somewhere near the median radius of the field (outer radius / sqrt 2.) The top image diagrams such a system. The target point turns out to be little bit higher than the beam-down optics, and about one field radius behind the center of the field.

Calculation of target position, where γ is the ratio of design beaming angle, β, to the beaming angle, β_e at the outer radius, r_o; and r is the target point's radial distance from the center of the field. Heliostats operating at the design condition are assumed to lie at the median radius (0.707 * r_o) from the center of the field. In this example γ = 22/18.5 = 1.19, so r/r_o = 0.84. The height of the target point is then r_o * (1 + 0.84) / 22 = .084 r_o .

The run-over-rise number for the top ray of the outermost heliostats—in this example, 14—is also the ratio of the field radius to the tower height (or height at the top of the beam-down optics.) Telescopic heliostats give twice the value for conventional heliostat fields, but with much better land utilization.

Back radiation from the target

Far-field approximation of the radiant flux from a horizontal disk. The region closer than one disk radius from the center of the disk has been blacked out. If the disk is at the temperature of the sun, the remotest black fringe is at 0 suns of flux, the next at 1000 suns, etc.

In a perfect solar concentrator, the target reaches thermal equilibrium (radiative balance) upon reaching the temperature of the sun's surface. Radiant flux is then net zero throughout the system due to the presence of thermal radiation radiated back from the target. In the forward direction, sunlight uniformly illuminates the target; in the reverse direction, back radiation from the target is collimated and beamed toward the sun. When a perfect optical system is in thermodynamic equilibrium, these two fluxes cancel out everywhere.

At any point in empty space, the divergence pattern of the beam of sunlight will be exactly mimicked by the back radiation traveling in the opposite direction. At an optical surface—where light might be redirected and its divergence pattern reshaped—the fluxes on the two sides of the surface must still sum to zero.

Flux on a surface is proportional to cosθ, the cosine of the angle the incident rays make with the surface normal. This is Lambert's Law. The divergence pattern of light at a particular point, P, on a surface can be described as the interior of region drawn on a unit sphere. In geometrical terms, Lambert's Law says that the flux at the surface is not the area of this region on the unit sphere, but rather it is the area of this region after it has been projected orthogonally onto the surface at point P. The orthogonal projection supplies the factor of cosθ.   Our requirement for net zero flux through the optical surface is then that the projected areas for both sides of the optical surface are equal.

There is one part of the optical design we have no control over: the pattern of thermal radiation that will be emitted from the target. In the case of a solar glass-melting furnace, the target is the opening of the furnace, the oculus, which is effectively a horizontal disk. Calculating the flux of back radiation from this disk is merely a problem in visual perspective because it is a perspective view of the disk that determines the shape and area of the region on the unit sphere. According to Lambert's Law, we should follow upon that exercise in visual perspective with an orthogonal projection onto some surface (whose orientation would be somewhat unclear at this point,) but, if we assume the angles of incidence to be the same on both sides of the optical surface (a desirable condition for both efficiency and simplicity,) that step is unnecessary. We will make that assumption.

The problem in visual perspective gets a bit complicated close to the disk, but in a far-field approximation the horizontal disk appears as an ellipse, one angular dimension being proportional to 1/r, and the other to cosθ/r.

More precisely, taking the radius of the oculus = 1, and r = sqrt(x² + y² + z²),

the angular extent of the ellipse in the horizontal (non-foreshortened) dimension = 2 * arctan(1/r), or approximately 2/r,

and the angular extent of the ellipse in the vertical (foreshortened) dimension, is approximately cosθ * 2/r.

Therefore, our perspective view of the disk translates approximately into an elliptical region on the unit sphere having an area, measured in steradians, of cosθ * 4/r² * π/4 = πcosθ/r².

The angular diameter of the sun is .01 radians, so a flux of 1000 suns corresponds to an image of the sun covering 1000 * π * (.005)²  =  .079 steradians.

The graphic above shows the far-field approximation of the thermal flux from the oculus. When the disk is at the sun's blackbody temperature, each fringe represents 1000 suns, that is, the remotest black fringe is at 0 suns, the next black fringe is at 1000 suns, etc. To give a sense of scale, points closer to the center of the disk than one disk radius have been blacked out, but the far-field approximation is not actually good enough to go in that close.

Un-silvered mirrors

A solar concentrator with prismatic reflectors. Rabl & Rabl US4120565.

In making an all-glass solar furnace it would be desirable to avoid the use of silvered or aluminized mirrors. The use of prismatic reflectors, is a possibility for telescopic heliostats. The primary mirror tracks the sun precisely in azimuth, and the secondary mirror sees a stationary light source of only 3° divergence. Both of these situation are ideal for prismatic reflectors...

...on second thought, I don't think this can work. The optics of these surfaces depend on the dihedral line rather than the surface normal. For example, in the image of the trough concentrator above, if the prisms ran in the longitudinal direction there would be no concentration because there would be no curvature of the dihedral lines. This technique seems to need radial symmetry to form an image.