Beamshaping is necessary

A representation of beam divergence that is wide in "longitude"

If the divergence of the sunlight reaching the lamp is too wide in "longitude," then, short of beamshaping, there is nothing a rotationally symmetric lens can do to fix it. It is therefore important to accurately model the divergence of this light.

Relations for the 180° and the general locus near the beam down optics. R is the field radius and r is the radius of the 180-degree locus. Theta is the beam angular radius. 

At a radius r = Rsinθ, the converging sunlight fills a full 180° of longitude. At a larger radius x, the sunlight fills an angle 2φ, where φ = sin^-1(r/x).

For θ = 1.5°, r/R = 0.026. For R = 2600m, r = 68 m.

For φ = 10°, x/r = 5.8

For R = 2600m and x = 120m, φ = 35°; x = 190m, φ = 21°.

Since the width of the beam in the latitude dimension is not likely to be more than 4 degrees, the beam aspect ratio is 10 or greater. Thus beamshaping is necessary. 

Veselago lens optics

A Veselago lens is a refractive lens with a refractive index of negative one. Though Veselago lenses, like all refractive lenses, obey Snell's Law of Refraction, they are also closely related to mirrors. The chart below summarizes mirror optics.

A short review of mirror optics. Rotating any light ray 180° about the point where it strikes the optical surface yields a diagram of a Veselago lens. In mirror optics the heterogeneous object/image pairs (i.e., real/virtual, virtual/real) are hyperbolic, and the homogeneous pairs are elliptical. In Veselago optics the opposite is the case. Underlying diagram quoted from "Mirrors" by Mike George.

Observe that just two kinds of mirror profile arise in practical cases: elliptical or hyperbolic (that is, conics of eccentricity e<1 or e>1.) There is not a practical need to consider parabolic profiles (e = 1), since there is no practical need to translate an object or image point "all the way" to infinity.

Of the four possible object/image cases, e.g., real/virtual, real/real, etc., only two classes are actually distinct. In two heterogeneous cases, real/virtual and virtual/real, we get the other case by simply reversing the direction of the light rays, and no new case is presents itself when we use mirror's other side. In the two homogeneous cases, real/real and virtual/virtual, we get the other case by simply using the mirror's other side.

A Veselago lens is like a transmissive mirror or "transflector." A mirror optics diagram can be transformed into a Veselago optics diagram by simply rotating one light ray 180° about the point where it intersects the optical surface. Additionally rotating the other light ray 180° would just bring us back to a mirror diagram—one that happens to use the other side of the mirror. The procedure is general: an odd number of 180° ray rotations about the same point carries us into the other domain, an even number of 180° ray rotations carries us back.

Rotating a ray flips the nature of one of the foci, either from real to virtual or from virtual to real, and therefore also flips the object/image type from heterogeneous to homogeneous or vice versa. Thus we these two domains of optics can be identified by the curvatures of their homogeneous realms: mirrors are homogeneous-elliptical, Veselago lenses are homogeneous-hyperbolic. That is, one would use an elliptical mirror to transport light from one point to another; a hyperbolic Veselago lens can accomplish the same end.

Any of the elliptical profiles would serve to transfer light between the two foci by mirror. Any of the hyperbolic profiles would serve to transfer the light by Veselago lens. Note that a planar disk is a degenerate ellipsoid—though not a very useful one; a plane is a degenerate hyperboloid, and a very useful one indeed.

It is sometimes surprising that a flat Veselago lens (i.e., a planar n/-n interface) yields a real image of a real object, but this is just the homogeneous counterpart of a flat mirror yielding a virtual image of a real object. Perhaps we do not take the virtual images that are all around us seriously enough.

A starting point for the design of the beam-down optics for a field of telescopic heliostats would be to place one of the foci on the perimeter of the heliostat field, and the other focus on the farther edge of the oculus, and then find the smallest hyperbola that works. Rotating this hyperbola around the rotational symmetry axis of the oculus give a candidate Veselago lens for the beam-down.

Ronian Siew has shown that amazing things can be accomplished in lens design using negative index metamaterials or NIMs and more than one interface. The two examples reproduced below, though they use refractive indices more negative than minus one, suggest what can be accomplished with multiple-interface Veselago lenses.

A negative refractive index (-1.517) singlet designed by Ronian Siew.

A negative refractive index (-2.0) singlet designed by Ronian Siew.

Uniform illumination: the inverse problem of solar beam-down optics

The design of solar beam-down optics is closely related to the following lighting problem:

Given a light source that is a horizontal, incandescent disk, design a luminaire to uniformly illuminate an annulus-shaped parking lot.

This problem is the ray-reversed version of collecting light uniformly from a heliostat field and concentrating all of it onto a circular target. Though we might like to add some more constraints to correspond more exactly to the properties of a field of telescopic heliostats, getting to an optical design that simply provides uniform illumination would be a big first step, illuminating in more ways than one.

The distribution of light from an incandescent disk is Lambertian, which means the disk appears just as bright no matter what angle we observe it from, but we do not have to utilize all of this light. Placing over the incandescent disk an oblate ellipsoidal mirror that images the edge of the disk back onto itself will return nearly all of the emitted light back to the disk. Since there is such an oblate ellipsoid profile passing through every point in space, we can truncate our luminaire wherever need be, switching at that point to the profile of an oblate ellipsoidal mirror, and thus preventing any unnecessary loss of light.

Looking at the inverse problem makes it obvious that we do indeed have here the degenerate case where lens and object share an axis of rotational symmetry—the object we are imaging is the oculus. So we are free to design with Veselago lenses and then use Fresnel mirror optics to precisely emulate them.

Fresnel-mirror Veselago lenses in solar beam-down optics

The vine above is of a POV-Ray ray-tracing of a spherical Veselago lens, that is, of a sphere with internal refractive index -1. Veselago lenses are well known in metamaterial optics, but a negative refractive index may sound impractical at large scale. As pointed out in an earlier post, Fresnel mirrors with 90-degree facets, under certain symmetry conditions possess exactly the same optics as Veselago lenses. Those particular Fresnel mirrors also have other attractive properties such as zero chromatic aberration, no thermodynamically imposed loss of radiance, identical facet angles, analytical lens profiles (the same as for any refractive lens) and easy ray tracing (in POV-Ray you use "interior { ior -1.0 }".) POV-Ray scene description file here.

One way to describe the optical action at a point on a surface is to center a small sphere on that point and speak in terms of where the incident and exiting rays cross the sphere. If the sphere is infinitesimally small, the surface intersects it in a great circle that we may call its equator. The action to  of an n/-n refractive interface, and thus the action of a Veselago lens, is now easy to describe: an incident ray that enters the sphere at a point A, will exit the sphere at a point A' that is the reflection of A about the equator. A Fresnel mirror facet mounted on the same surface at the same given point can now be characterized by the point M where its outward surface normal exits the sphere. The action of the Fresnel mirror facet is also easily described: an incident ray that enters the sphere at a point A, will exit the sphere at a point A' that is the reflection of A about the point M (i.e., the 180-degree rotation of point A around point M.) Since reflection about a line (mirror reflection) is different from reflection about about a point (180-degree rotation about a point,) no arrangement of Fresnel mirrors is equivalent to a Veselago lens. However, symmetry can create a degenerate case where these two different operations cannot be distinguished. That case is when both the lens and the object being imaged share an axis of rotational symmetry.

In the degenerate case, where both the Veselago lens and the object being imaged share the same axis of rotational symmetry, reflection about a point and a line become indistinguishable, and a Fresnel mirror with 90-degree facets can precisely emulate a Veselago lens.

For example, the yellow circle in the animation above shares an axis of rotational symmetry with the Veselago lens above it, therefore its image is behaving exactly as it would under a spherical Fresnel mirror having 90-degree facets.

Is beam-shaping necessary?

A Fresnel Veselago lens, without any beam-shaping, can produce an annular focal zone of modest concentration from a solar highbeams field.

Beam-shaping (reshaping divergence) at the Fresnel Veselago lens can potentially produce very high concentration at the oculus—and thus reduce thermal losses from the hot space—but this would come at a cost of complexity and reflection loss. Is this a marginal improvement we could postpone for later?

Since reflection losses in beam-shaping with lenticular lens arrays will probably subtract at least of 5% of the total power, and improving concentration at the oculus by a factor of four would reduce the radiant heat percentage, q, by the same factor, the improvement offered by beam-shaping is something like

0.75q - 0.05.

Assuming we would put off for now a 10% improvement in system power in the interest of simplification, that is, if:

0.75q - 0.05 < 0.1,

or q < 20% .

Radiant flux in the hot space at the 3/4-full temperature of 1936 °K is 800 suns (calculations included in the figure above too-generously assume radiant transfer at the empty temperature of 1250 °C = 1523 °K) this becomes a 20% loss when the incident flux is 4,000 suns. From the diagram above it looks like that is probably about four times more concentration than can be achieved at an annular focal zone without divergence reshaping. Adding a 2D CPC concentrator to the annulus or 2D radiation traps for thermal radiation escaping at wide angles might effectively double the concentration at the annulus to 2,000 suns, but that would still leave us a factor of two away, and we would really rather be at 8,000 suns so that heat loss is reduced to 10%.

A better approach is to bring light directly to a circular focal zone. Calculations in the diagram suggest we can get above 3500 suns without beam-shaping, and additionally we can  block much of the radiant heat loss by suspending an elliptical mirror over the oculus. (Sunlight is shown directly striking the boiler tubes, but this could be avoided by slanting the tubes, arranging them on the inside of a cone rather than a cylinder.)

A Fresnel Veselago lens, without any beam-shaping, can produce a circular focal zone with high concentration.

Is beam-shaping necessary? Likely not.

Visualizing telescopic heliostat beaming angles

Beaming angle and beam diameter visualized in comparison with the angular diameter of the full moon.

The nominal condition for a telescopic heliostat is 6x linear magnification (producing a beam diameter of six solar or lunar diameters—they are nearly the same) beamed at an angular elevation of 3° (also equal to six lunar or solar diameters.)

Solar Highbeams Target Design compared with Crescent Dunes

The Solar Highbeams Target Design compared with the Crescent Dunes Solar Energy Project. Underlying image quoted from Google Maps.

Table comparing the Solar Highbeams Target Design with the Crescent Dunes Solar Energy Project.

The ratings assume that the annual average power per mirror area of Crescent Dunes (51 W/m2) can also be achieved in the target design. Maximum boiler temperature at Crescent Dunes is 565 °C (1050 °F).

Advanced steam turbine technology: plant efficiency vs. steam temperature. Image quoted from EPRI, "Materials Technology to Enable High-Efficiency Advanced Ultrasupercritical (A-USC) Steam Power Plants"

The plot of plant efficiency vs. steam temperature above suggests that increasing steam temperature from Crescent Dunes' 565 °C (1050 °F) to an A-USC (advanced ultra supercritical) turbine's 760 °C (1400 °F), would increase plant output by about a factor of 46.5/42.3 = 1.10. That gives some leeway in rating the Solar Highbeams Target Design at 1 GW though its atmospheric turbidity losses will be greater than Crescent Dunes', and possibly its rather different shading and blocking losses will be greater as well. Eventually, thermophotovoltaic (TPV) conversion may offer even higher efficiencies.

Fractal growth of all-glass, glass-making solar furnaces

An all-glass glass-making solar furnace built through a sequence of five molts from a much smaller furnace.

Probably the fastest way to grow a solar furnace is to build three squares equal in area to the glass-making unit and then re-aim the heliostats of the glass-making unit (and dismantle its lamp) to constitute the fourth square. That way, the time, t_m, to build the next molt is just 3x the glass replication time, t_r, times an additional factor, i, accounting for the increase of glass thickness between molts. Generalizing from m = 4 to any molt area factor, m:

t_m = (m - 1) * t_r * i


Earlier it was assumed that the glass thickness would double in the course of a linear scale increase of 16, which corresponds to 4 stages (molts) of linear doubling. Therefore, in each molt, the glass thickness should increase by the fourth root of 2, or 1.189. The first molt requires 3 x 1.189 = 3.57 glass replication times to complete. For each later molt, the glass replication time increases by a factor of 1.189 on account of the increased glass thickness.

The fourth root of two is a rather small increase in glass thickness for a doubling in linear scale, but the value is based on the fact that heliostat size does not need to increase in proportion to field size. At larger scales, the lamp—being all of one piece and not scaled to the heliostats—becomes a larger portion of the total mass, so the glass thickness factor must increase somewhat. Also, when going down to much smaller scales the thickness factor must in the limit approach 2 (the factor for geometric similarity) to prevent mechanical interferences. Another way to think of it is that the field cannot be composed of fewer than one heliostat.

Starting with a small molt 0 unit having replication time t_r, m = 4, and molt thickness multiplier i, growing to molt n requires build time, t_b:

t_b = t_r * 3 * i * (1 + i + i² + i³ … + i^n-1)

t_b = t_r * 3 * (i + i² + i³ … + iⁿ)

For i = 1.189, the ratio t_b / t_r  to reach successive generations is

1     3.6
2     7.8
3   12.9
4   18.9
5   26.0
6   34.5
7   44.5
8   56.5

For example, growing a molt 0 unit that is 6.25 m square up to a unit 1600 m square requires 8 molts. If the glass replication time of the molt 0 unit is ten days, growing from that unit to a unit approximately one mile square takes 565 days, or a bit more than a year and a half.

The final molt will probably use a molt area factor, m, larger than 4, because it makes little sense to gather the forces to build a large project in a matter of a few months unless they can move directly on to something else. Using a final molt area factor, m_f, of 9 in the last stage effectively becomes 1.59 molts since

m_f = 9 = 4^1.59 .

The time, t_mf , required for the final molt is

t_mf = (m_f - 1) * (t_r * iⁿ) * ln(m_f / m)

where t_r is the glass replication time for molt 0, i is the glass thickness factor for the earlier molts, n is the number of the next-to-final molt, and m is the molt area factor for the earlier molts.

Here,  m = 4, m_f = 9,

t_mf = 8 * (t_r * iⁿ) * 1.59 = 12.7 * (t_r * iⁿ)

For the growth sequence of eight molts mentioned above, n = 8, and the final molt with m_f = 9 has

t_mf / t_r = 12.7 * i⁸

For i = 1.189,

t_mf / t_r = 12.7 * (1.189)⁸ = 50.7

The total 9-molt build from a 6.25 m square to a quarter-township takes (56.5 + 50.7) * t_r  = 107 * t_r .

If the glass replication time for the 6.25 m x 6.25 m solar furnace is ten days, it can grow to occupy a quarter-township in 1070 days, a little less than three years.

Starting from a smaller scale would only add a few days to the total. In fact, in the range where i = 2, the time for each molt is the sum of the times for all previous molts. In the case considered that would mean that all molts preceding our molt 0 would total less than 10 days, no matter how small we start.

Therefore, a quarter-township all-glass, glass-making solar furnace can be grown in a matter of a few years no matter how small the seed. 

Fitting a circular heliostat field in a square plot

Diagram of an oversized, circular heliostat field in a square plot of land.

A heliostat field wants to be circular, but plots of land are square.  Determining how much of a square plot to cover with heliostats would require a complex optimization, and it seems unlikely that filling the square all the way out to its corners would be the result.

An initial, plausible solution is an oversized circle with energy loss (land in the corners not covered by heliostats) that is twice as big as brightness loss (portions of the oversized circle that will be truncated by the square.) The reason to prefer energy loss, is that it is cheaper: we've already paid for the lamp to handle the radiation that brightness loss causes to go missing. By comparison, un-built-upon land is cheap.

By symmetry, this problem can be solved by looking at a single octant of the square and circle. From the diagram, the area of the full square exceeds the area of the full circle by 8*A. Assuming the circle is a unit circle, area A is:

A = (π*θ/2π)  - (sinθ cosθ)/2

= θ/2 - (sinθ cosθ)/2

Since the square, having side length s, exceeds the area of the circle by 8*A,

π + 8*A = s²,

π + 4θ - 4sinθ cosθ = s².

Also, s = 2cosθ, so

π + 4θ - 4sinθ cosθ = 4cos²θ

which is solved by

θ ≈ 0.421 radians, or 24.1° .

The circle diameter, d, is s/cosθ = 1.096 s .

Nine heliostat fields on square plots with d/s = 1.096.

---

Getting a handle on the value of land is difficult because its value changes when we build on it; also, once a property is hemmed in by other claims, land is the one resource of which no more can be trucked in. The figure above, intuitively, looks a little under-filled to me. One factor to consider is that the diameter of the circle (being visible only where it touches already-purchased, use-it-or-lose-it land in the corners of the plot) should be larger than might be expected from the height of the lamp: it should the diameter of a theoretical system built on free land.

If unused land is equally as valuable as unused lamp, the calculation is simpler. Then the tangerine and cyan areas in the top diagram are equal, so the full area of the square is equal to the full area of the circle.

s² = π/4 * d²


So d/s = √(4/π) = 1.128

Nine heliostat fields on square plots with d/s = 1.128.

As above, the area truncated from the circle is:

8 * A = 8 * ( θ/2 - (sinθ cosθ)/2 ) = 4θ - 4sinθ cosθ

since the area of the full unit circle, π,  is also the area of the full square, the covering factor is

1 - (4θ - 4sinθ cosθ)/π

s = d/1.128 = 2cosθ 

d = 2.256 cosθ

but d = 2, so cosθ = 0.887, θ = 0.48 or 27.6°, giving a covering factor of 0.91.

Scaled generations of all-glass, glass-making solar furnaces

Specifications for scaled generations of all-glass, glass-making solar furnaces.

A classic science-kid's demonstration is making glass from sand with a solar furnace, and, in fact, the glass-making capacity of a solar furnace is prodigious. A solar furnace of any size, even if it is 100% glass, can make all the glass needed to replicate itself in a matter of weeks. The glass replication time of a small backyard solar furnace may be only a matter of days. More likely at large scale, the glass for a furnace would be produced by a smaller solar furnace that produces at a rate that does not outstrip the capacity to fabricate and assemble glass parts. That smaller furnace, in turn, may have been made by an still smaller furnace; and so on, through multiple generations of scale.

Starting at a very big scale (a solar furnace on 9 square miles of land) and working for the most part within areal units of the Public Land Survey System of the western United States, the table above shows specifications for six scaled generations. If these calculations can be taken seriously, the time from the completion of the room-sized 9-square meter furnace to the completion of its descendant quarter-township furnace is 6.3 years.

Ten things we know about telescopic heliostats

They're necessary. Ground loss—the quantity of sunlight falling between heliostats—is atrocious in current generation CSP's. Obvious in a satellite view is the fact that far more sunlight is reaching the ground than the mirrors. Land requirements are being tripled! The Second Law of Thermodynamics decrees that a heliostat without ground loss must act like a telescope, i.e., it must increase the divergence of the reflected beam while maintaining collimation (parallel rays map to parallel rays.)

Telescopic heliostats must have two mirrors. No fewer than two lenses (objective and eyepiece) compose a telescope.

Optimum power is about 6X. Magnifying the sun's disk, which is 0.5° in diameter, six times makes an intensified, but still collimated beam (or highbeam) that can be aimed at an elevation angle as small as its own diameter, 3.0°. That condition minimizes the height of the central, beam-down optics or lamp.

Central, beam-down optics are necessary. At 3.0° divergence, the highbeams are simply too spread to form a high quality focus without another optical stage.

The lamp (central beam-down optics)—shaped something like an overturned apple—will be only 60% as tall as a power tower on the same field. Larger heliostat fields are thus made practical.

The objective needs to move, the eyepiece doesn't. The objective (primary) mirror can redirect sunlight vertically to a fixed focus: the much smaller eyepiece (secondary) mirror gets to sit right there.

The objective needs to have its optical profile continuously fine-tuned to the sun's changing zenith distance. To first order, the adaptation needed is simply a thin-shell bending of the mirror.

Telescopic heliostats move in concert. The only real difference between two telescopic heliostats in a field is the azimuth aiming of their eyepieces. All the objectives could be mechanically ganged.

At at 0.70 mirror/land ratio, telescopic heliostats can be sited with negligible blocking. Surprisingly, the presence of the eyepieces adds no complication at all to heliostat siting when a phyllotaxis-based algorithm is used. The packing density achieved is more than three times that of a conventional heliostat field.

These improvements in field size and field packing make it practical to design a replicable 1 GW, 75% capacity-factor, standard solar plant covering one quarter-township (9 square miles) in the southwest U.S. The central lamp, about 190 m high, would be only 20% taller than current-generation power towers. Goodbye coal!

Quarter-township: the natural size of a replicable solar plant in the U.S.

The natural size of a replicable solar plant in the western United States is a quarter-township, or 9 square miles.

A solar power plant worthy of replication needs to be a good fit to its circumstances. In the western United States (Texas excepted) one part of those circumstances is the Public Land Survey System (PLSS,) a rectilinear grid to which property lines conform. Surveyed from reference points named for their meridians (e.g. San Bernardino Meridian, Gila and Salt River Meridian) the Public Land Survey System is a grid of six-mile by six-mile squares called townships.

In most of the American West, property lines conform to a national grid of 6 mi x 6 mi townships.

Another pre-existing circumstance for a solar thermal plant is the technology of steam-electric generating equipment. For many decades, electric utilities have preferred to build coal plants with multiple, separately-fired, turbine-generator units that are rated in the range of 600 MW to 1200 MW (see the chart below of the latest advanced high-temperature turbines installed by Siemens.)

Siemens' advanced steam turbines are commonly manufactured in the range of 600 to 1200 MW

New coal plants might operate at 80% capacity factor—i.e., their annual output is equivalent to running full power for 80% of the time—a statistic dependent not only on equipment reliability, but also on steady demand for electricity at a price that exceeds fuel costs. Capacity factors tend to come down in competition with wind turbines and other sources that do not pay for fuel, but a utility would certainly expect a new generating plant, solar or not, to be running full power most of the time. The Gemasolar plant in Spain has demonstrated that a solar plant with thermal storage can achieve a 75% capacity factor.

Taking advantage of an excellent solar climate, California's Ivanpah Solar Electric Generating System Unit #1 achieves a capacity factor of 0.32 without any energy storage at all. It has a rated power of 126 MW from a land area of 3,800,000 square meters or 33 W/m2 of land area. The mirror-area/land ratio for this plant is only 0.21. Upgrading to telescopic heliostats would bring the mirror-area/land ratio to 0.70, boosting the rated power to 33 W/m2 * 0.70/0.21 = 110 W/m2 of land area.

Based on Google imagery, the heliostat field of Ivanpah 1 is very nearly a square, 1995 m on a side, with three corner truncations, giving a heliostat field land area of about 3,800,000 square meters. This unit has 53,527 heliostats each with a mirror area of 15 square meters, giving a total mirror area of 803,000 square meters. The mirror/land ratio is 0.21.

Raising the 0.32 capacity factor of Ivanpah to 0.75 requires some amount of thermal storage (half-a-day, roughly) and a de-rating of the plant to 110 W/m2 * 0.32/0.75 = 47 W/m2 .

A quarter-township plant occupies a square of land 3 miles (4,828 m) on a side, having a land area of 9 square miles or 23,000,000 m2. Therefore, in Ivanpah's excellent solar climate, a quarter-township solar highbeams plant would be rated about 47 W/m2 * 23,000,000 m2 = 1.1 GW at a capacity factor of 75%.

Of course, the precise rating of a quarter-township plant would depend on many details including the quality of the local solar resource, but the size constraint will be the same throughout the West: a quarter-township.

In round metric numbers, a quarter-township solar plant is 5 km x 5 km. The central optics will be about 190 m tall— not that much taller than the 160 m power tower at Crescent Dunes, Nevada, or the 169 m (all-masonry) Washington Monument.

Five advantages of the Solar Highbeams Project

For the same central height, a solar highbeams plant harvests than three times the land area.  (Underlying image of  Ivanpah Solar Electric Generating System quoted from the Washington Post.)

A solar highbeams plant catches more than three times the sunlight from an acre of land. (Aerial photo of  Ivanpah Solar Electric Generating System quoted from Google Maps.)

Solar high beams heliostats all move identically; this allows them to be mechanically ganged.

The focal zone in a solar highbeams plant is at ground level; this makes large-scale storage, power conversion, and industrial uses practical. (Glass furnace image quoted from Mirion Technologies.)

The rabbit advantage: because each solar highbeams power plant is a glass-making furnace as well, it can make the glass for additional plants. Making the glass for two more plants would delay power operations only a matter of months.

Solar re-powering Georgia's Plant Bowen in sunny Spain

Solar re-powering Georgia Power's Plant Bowen to 3.2 GWe (75% capacity factor) in Andalucia's climate would require a storage/boiler compartment about 60% as tall as one of its cooling towers—and 130% its diameter. The central optics would be about 110% the height of its smokestacks.

In Andalucia, Spain, the location of the Gemasolar plant, only 15 hours of thermal storage (plus whatever solar multiple, or thermal down-rating, Gemasolar is using) is needed to achieve an annual capacity factor of 75% in a solar thermal generating plant.

Plant Bowen (3.2 GWe and 82% capacity factor) in Euharlee, GA, USA, is the largest coal-fired plant in the United States, and the country's largest point source of CO₂ pollution. This post contemplates what Plant Bowen would look like if re-powered to run on the sun in Andalucia's sunny climate. Of course, it would be more interesting to see what Plant Bowen would look like solar-powered up to its own 82% capacity factor in northwest Georgia's own, somewhat less sunny, climate, but that would involve a detailed simulation. By figuratively moving Bowen to Spain, and adopting Gemasolar's 75% capacity factor, we can just steal data.

Quoting an earlier post:

Here are some statistics for Gemasolar gleaned from the National Renewable Energy Laboratory's site. 

Projected annual output: 110,000 MWhr/yr = 12.6 MWe  annual average.
Rated output  (calculated from the claimed 75% capacity factor): 16.7 MWe rated.
Output per mirror area (304,750 m2) :
       41 We/m2 annual average,
       55 We/m2 rated.
Land yield (1,950,000 m2; mirror/land ratio = 0.156):
        6.4 We/m2-land annual average,
        8.6 We/m2-land rated 

The 15 hours of storage based on 40% thermal efficiency is:
15 hrs * 3600 s/hr * 8.6 We/m2-land * 1/0.40 = 1.2 E6 Jthermal/m2-land 

A plant with telescopic heliostats and glass-melt storage would have some advantages over Gemasolar. Telescopic heliostats can be packed much more closely, increasing the mirror/land ratio to around 0.70, thus increasing land yield about 4.5 times that of Gemasolar. Also, because the glass melt transfers its heat to hotter steam (608°C vs. 565°C) the steam cycle efficiency can be greater, about 46% thermal efficiency as compared with 40%, a factor of 1.15 . 

So here are the Gemasolar statistics if it were rebuilt on the same plot of land with telescopic heliostats and glass-melt storage: 

Projected annual output: 110,000 MWhr/yr * 4.5 * 1.15 = 65 MWe  annual average.
Rated output  (calculated from the claimed 75% capacity factor): 87 MWe rated.

Output per mirror area (304,750 m2 * 4.5 = 1,370,000 m2) :
       41 We/m2 * 1.15 = 47 We/m2 annual average,
       55 We/m2 * 1.15 = 63 We/m2 rated. 

Land yield (1,950,000 m2; mirror/land ratio = 0.156):
        6.4 We/m2 * 4.5 * 1.15 = 33 We/m2-land annual average,
        8.6 We/m2 * 4.5 * 1.15 = 45 We/m2-land rated. 

The 15 hours of storage for the rebuilt plant becomes:
15 hrs * 3600 s/hr * 45 We/m2-land * 1/0.46 = 5.3 E6 Jthermal/m2-land 

Plant Bowen is a plant belonging to Georgia Power in Euharlee, Georgia. It is the largest coal-fired plant in the USA. It has four 800 MWe units, giving an aggregate rating of about 3.2 GWe. A telescopic heliostat / glass-melt power plant in Andalucia with 15 hours of thermal storage, having the same rated output of Plant Bowen, would occupy: 

3.2 GWe-rated / 45 We/m2-land rated = 71 E6 m2,
which is equivalent to a circle 4.8 km in radius.

The height of the central optics will be about 1/14 the field radius, or 340 m, or about 11% taller than Plant Bowen's 305 m smokestacks.

The heat flow to drive rated output is

3.2 GWe-rated * 1/0.46 = 7.0 GWthermal-rated

It remains to calculate storage and boiler dimensions. A boiler's water tube walls usually receive a thermal flux of around 250 kw/m2. Taking that value as a given, the total area of the water tube wall, S_B, will be

S_B = 7.0 GWthermal-rated / 250 kw/m2 = 28,000 m2.

The volumetric storage density in molten glass is

ΔT * 2300 kg/ m3 * 1231 J/kgK = ΔT * 2.8 E6 J/m3-K

Thermal storage needed for 15 hours of rated output is

7.0 GWthermal-rated * 15 hr * 3600 s/hr = 3.8 E14 J

So thermal storage volume V is

V = 1.35 E8/ΔT  m3

For a hemisphere

V = 2/3 π R³

so,

R =  (3/2π V)^0.33 m

R = (3/2π * 1.35 E8/ΔT)^0.33 m

R = (6.4 E7/ΔT)^0.33 m

The height, H, of the water tube wall can be calculated from

2πR * H = S_B


H = S_B * 1/2πR = 28,000 m2 * 0.159 / R = 4,460/R m

Exploring these relations in a Numbers spreadsheet shows that T_empty = 1570 °C (1840 °K) gives a consistent solution with R = 63 m, H = 71, and the flux on the water wall tubes = 250 kw/m2. The glass temperature range from empty to full is just 230 °C. T_empty is approximately the temperature of a glass-making furnace, so it is fair to say that the thermal storage is accomplished by overheating a soda-lime glass-making furnace by about 230 °C.

By comparison, Plant Bowen has four cooling towers that are 47 m in radius and 116 m tall—so the volume of the storage/furnace compartment of a solar-fired Plant Bowen would be comparable in volume to one of its current cooling towers.

Capacity factor and hours of solar storage

The Gemasolar plant in Andalucia, Spain operates at an annual capacity factor of 75% using just 15 hours of thermal storage.

The 17 MWe Gemasolar power tower in Fuentes de Andalucía, Spain is designed to operate at an annual capacity factor of 75%, and has run continuously for as long as 36 consecutive days. This remarkable accomplishment is achieved with just 15 hours of thermal storage. Clearly 15 hours of thermal storage is about the right amount for a solar plant!

Here are some statistics for Gemasolar gleaned from the National Renewable Energy Laboratory's site.

Projected annual output: 110,000 MWhr/yr = 12.6 MWe  annual average.
Rated output  (calculated from the 75% capacity factor): 16.7 MWe rated. 

Output per mirror area (304,750 m2) :
       41 We/m2 annual average,
       55 We/m2 rated.  

Land yield (1,950,000 m2; mirror/land ratio = 0.156):
        6.4 We/m2-land annual average,
        8.6 We/m2-land rated

The 15 hours of storage based on 40% thermal efficiency is:

15 hrs * 3600 s/hr * 8.6 We/m2-land * 1/0.40 = 1.2 E6 Jthermal/m2-land

A plant with telescopic heliostats and glass-melt storage would have some advantages over Gemasolar. Telescopic heliostats can be packed much more closely, increasing the mirror/land ratio to around 0.70, increasing land yield about 4.5 times that of Gemasolar. Also, because the glass melt transfers its heat to hotter steam (608°C vs. 565°C) the steam cycle efficiency can be greater, about 46% thermal efficiency as compared with 40%, a factor of 1.15 .

Now, the same Gemasolar statistics if rebuilt on the same land with telescopic heliostats and glass-melt storage:

Projected annual output: 110,000 MWhr/yr  * 4.5 * 1.15 = 65 MWe  annual average.
Rated output  (calculated from the 75% capacity factor): 87 MWe rated. 

Output per mirror area (304,750 m2 * 4.5 = 1,370,000 m2) :
       41 We/m2 * 1.15 = 47 We/m2 annual average,
       55 We/m2 * 1.15 = 63 We/m2 rated. 

Land yield (1,950,000 m2; mirror/land ratio = 0.156):
        6.4 We/m2 * 4.5 * 1.15 = 33 We/m2-land annual average,
        8.6 We/m2 * 4.5 * 1.15 = 45 We/m2-land rated.

The 15 hours of storage for the rebuilt plant becomes:

15 hrs * 3600 s/hr * 45 We/m2-land * 1/0.46 = 5.3 E6 Jthermal/m2-land

Plant Bowen in Euharlee, Georgia, is the largest coal-fired plant in the USA. It has four 800 MWe units, giving an aggregate rating of about 3.2 GWe. A telescopic heliostat / glass-melt power plant in Andalucia with 15 hours of thermal storage, having the same rated output of Plant Bowen, would occupy:

3.2 GWe-rated / 45 We/m2-land rated = 71 E6 m2,

or a circle 4.8 km in radius. The height of the central optics will be about 1/14 the field radius, or 340 m. This is about 11% higher than Plant Bowen's two 305 m smokestacks.

Plant Bowen, 3.2 GWe,

Direct absorption and storage of solar energy in glass melts

In a glass-making solar furnace, solar energy is directly absorbed in the semi-transparent melt.

Contrary to popular belief, renewable power does not "need" energy storage. When a GW of wind or solar power is brought online, the electric utility's least fuel-efficient 1 GW of conventional generating capacity is forced into semi-retirement. That is, those particular generating plants no longer have a job when the wind is blowing or the sun is shining. Since we have about 4 TW of conventional generating capacity to semi-retire in this way, renewable power will not be hurting for energy storage anytime soon. 

That said, in a thermal power plant some energy storage comes free—or at least at no additional cost—in the form of thermal inertia. The larger the plant, the more running time is extended by thermal inertia—and the cheaper it is to deliberately increase. Any process served by a solar furnace may benefit from this inexpensive form of energy storage. Since an all-glass, glass-making solar furnace will be first and foremost occupied in making its own glass parts, it is reasonable to look at the thermal inertia in the glass melt itself. 

A 2002 paper by L. Pilon, G. Zhao, and R. Viskanta looked at the thermophysical properties of glass melts. A melt of soda-lime glass is substantially transparent to both sunlight and high-temperature thermal radiation, so molten glass effectively has high thermal conductivity when it absorbs solar radiation directly or cools radiatively from high temperatures. For example, at 1400 °C (1700 °K,) a soda-lime glass melt has an effective thermal conductivity (phonic conduction + radiation) of 58 W/m-°K—that's more than the thermal conductivity of steel at room temperature.

Pilon et al. also give some representative numbers for industrial glass-making. They considered a glass melting tank approximately 16 m long, 7 m wide, and 1 m deep heated from above with a total heat input of 8.3 MW which averages to 72 suns (i.e., kw/m2) over the free surface of the melt. They estimate a maximum heat flux of 134 suns near the center. At melt surface temperatures around 1500 °C (1800 °K) they associate the maximum flux with vertical temperature gradients of about 1200 °C/m. At about 8 MWth, such an industrial glass-making furnace is only a small-scale model of a GW-scale solar glass-making furnace.

A coal-fired furnace for a 800 MWe generating unit might be 20 m x 20 m x 100 m high, corresponding to an average thermal flux per unit wall area of about 250 suns.

T-s diagram for a supercritical power plant. According to L & T Power, typical temperatures at points E and G for current technology are 565°C and 593°C, respectively; efficiency = 42%.

Heat balance for an advanced 800 MW power plant. Image quoted from Song Wu et al., "Technology options for clean coal power generation with CO2 capture." Mean temperature in the first heat (596 + 293)/2 = 445°C; second heat (608 + 342)/2 = 475°C; efficiency = 46%.

The steam tubes absorb heat over a range of temperatures, but 460°C may be taken as representative for the advanced supercritical cycle in the diagram above. At 460°C, a blackbody radiator emits about 16 suns (20 kw/m2.) The molten glass will need to be significantly hotter at its "empty" temperature in order to transfer a flux 250 suns to the furnace's steam tubes (in order to drive operation at rated power.) If the product of the emissivities and the view factor is about 0.7, the glass melt must be at a temperature where a blackbody emits about 380 suns, that is, around 1340°C. Using the thermophysical properties of soda lime glass melts quoted in Pilon et al., and the modified Rayleigh number, Ra*, defined in Bolshov et al., 1340°C is well within the range of turbulent convection for soda-lime glass. If we assume the pool of molten glass is a hemisphere 100 m in diameter, and that the average volumetric heating rate is 30 kw/m3, we have Ra* = 3E13 when the glass melt is at 1340°C.

Thermophysical properties of molten glass as calculated from the relations in Pilon et al.

Convection flow patterns and isotherms in a hemispherical pool with isothermal walls and top. Image quoted from Bolshov et al. The modified Rayleigh number, Ra* = 1E8 above, and Ra* = 1E9 below—much lower than the Ra* = 3E13 estimated for a GW-scale energy store.

Observed turbulent convection at Ra = 6.8 E8. Image quoted from X. D. Shang, X. L. Qiu, P. Tong, and K.-Q. Xia, Phys. Rev. Lett. 90, 074501 (2003).

Modeling parabolic primary mirrors in Povray

Simulation of a bendable parabolic primary mirror in Povray.

This povray file ray-traces the sunlight reflected from a bendable parabolic primary at any chosen time and latitude. Actual bending of the mirror is not simulated, but the focal length of the parabola adjusts to the sun's zenith distance.

I am rather uncertain of the ray-trace (photons) settings in the .pov file, but it works. 

Divergence rotation

Every centroid-coordinated tessellation of circles and ellipses (left) can be affine-transformed into a centroid-coordinated tessellation of ellipses that are identical but rotated 90° (right.)

A special case of divergence reshaping occurs when the new divergence pattern actually has the same shape, but is oriented differently.

Any coordinated tessellation of circles and ellipses can be transformed by affine transformation into a design for a divergence reshaper that rotates an incident elliptical beam 90°. If the ellipses in the coordinated tessellation have aspect ratio A, an affine shrinking of both circles and ellipses in the direction of the major axis of the ellipses by √A, yields a coordinated tessellation of identically shaped ellipses, all of aspect ratio √A, lying at right angles to each other.

From the previous result that hexagonally-packed coordinated tessellations of circles and ellipses exist for aspect ratios drawn from the central polygonal numbers, {1, 3, 7, 13, 21, 31, 43, 57, 73, 91…}, the magic aspect ratios for a tessellation of hexagonally-packed ellipses that can centroid coordinate with a 90° rotation of itself are the square roots of the central polygonal numbers, {√1, √3, √7, √13, √21, √31, √43, √57, √73, √91…}.

Another way to rotate the divergence pattern of a beam by 90° is to reflect it in a planar Rabl mirror, placing the dihedral line of the mirror at 45° to the major axis of the incident and rotated beams.

Cross-section of a planar Rabl mirror (3M prismatic film.) Image quoted from K.G. Kneipp, "Use of prismatic films to control light distribution."

A divergence rotating mirror: the divergence of a beam of light is rotated 90° by normal reflection from a Rabl prismatic mirror when the prisms'  dihedral lines lie at a 45° angle to the principal axes of the incident—and, as well, the reflected—beams.

Divergence reshapers

A pair of lens arrays can constitute a divergence reshaper.

Arrangements of lens arrays (a.k.a., micro lenses, mini-lenses, lenslets, fly's eye lenses, lenticular arrays) like the one in the image above are known in beam homogenization and integral field spectroscopy. Though the relation between the two lens arrays is actually reciprocal, the first-encountered array is usually termed the field lens array, and the second-encountered array is usually termed the pupil lens array. Each lens array lies in the focal plane of the other, and the reciprocal system works equally well in both directions.

In some applications both arrays have lenses of the same planform, usually either rectangular or hexagonal. However, it is not necessary for both arrays to have lenses of the same planform: the centroid-coordinated tessellations of the plane indicate an infinity of other possibilities. When the two arrays differ in planform, the arrangement can be used to reshape the divergence of a beam of light.

A micro array of lenses having hexagonal planform. Image quoted from Anteryon.com.

In solar energy applications, in order to reduce energy losses, it is advantageous to make both lens arrays plano-convex and cement the two plano surfaces together, or to simply mold the whole apparatus out of one piece of glass or plastic. Since the focal lengths of both arrays are the same, the radius of curvature of the lenses in both arrays must be the same. In effect, all the lenslets are portions of spheres of the same radius.

The lenslets in the two arrays need not have the same planform for their centers to align. Here, lenslets of circular planform align with lenslets of elliptical planform.

For perfect imaging, the curvature of the field should also have the same radius. A well-known solution to this problem exists, one that is used in the design of spherical retroreflectors. When spheres have refractive index 2, they focus light from a distant source onto their own rear surface. This arrangement has spherical aberration, but the f-numbers needed for divergence rotation in solar energy applications are small. The lenslets on each face have dimensions that are small compared to the focal length, so the effect of spherical aberration is slight.

An integrated array of cat eye lenses (refractive index = 2) can serve as a divergence reshaper if the planforms of the lenses on each face differ. 

The small f-numbers also mean that refractive indices less than 2 can also be used with tolerable results, and that defocussing due to chromatic dispersion over the wide solar spectrum is not too severe.

Optical diagram of a divergence reshaper having refractive index 1.5. Incident light strikes the new side. If the incident beam has a maximum divergence of 15°, lenslets on the old side will be shaped in planform like the divergence of the original beam, and will have an f-number  of 5.8 based on their maximum dimension. 

In a divergence reshaper, the light strikes the new side—the side whose lenses are shaped in planform like the new divergence pattern. Light exits from the old side—the side whose lenses are shaped in planform like the old divergence pattern.

Cooling the thermal mirrors

The thermal mirrors (thermal cap, yellow; thermal wall, orange) are exposed to intense thermal radiation from the furnace opening or oculus (black line.) The black fringes represent 0 suns, 1000 suns, 2000 suns, etc., of back radiation when the the furnace is at the temperature of the sun.

The thermal mirrors—the cap and the wall—are exposed to intense thermal radiation. If the oculus were at the temperature of the sun, the cap would see about 9,000 suns of flux, and the wall about 1,000 suns. Of course, for the sake of efficiency, the furnace will actually operate at a much lower temperature, reducing these fluxes by about a factor of ten. The mirrors will reflect most of this heat, but perhaps about 5% will be absorbed. This absorbed heat needs to be dissipated from these surfaces to keep the mirrors cool—a thermal flux amounting to about 45 suns on the cap and 5 suns on the wall. The thermal wall may use passive cooling, but active convection is needed for the cap. For comparison, a 2-kw stove element heating a 20-cm diameter pot produces a thermal flux of about 64 suns.

Water is too dangerous to use directly above the furnace opening, so this significant cooling must be obtained by convected air. If the cap is segmented into smaller mirrors, each shingled over the other, a chimney extending to the top of the lamp may draft enough air between the mirrors to keep the mirrors cool.