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.

Transflection as a negative-one refractive index lens

Negative refractive index materials have not been found in nature so far, but the equivalent effect has been produced in artificially structured materials, or metamaterials. Substituting a negative refractive index into Snell's Law of Refraction shows that the angle of refraction also becomes negative: the refracted light will be bent back to the same side of the surface normal as the incident light. For solar power optics the case of a refractive index equal to negative one is of unique interest: such a lens can have an air-filled interior and exterior (the only structured material needed is at the interface) and the throughput is potentially 100% (at least the Second Law of Thermodynamics will not offended if it is.) Lens optics with negative-one refractive index (nori for short) is both a more precise and more concise description of what I have earlier described as transflection.

In talking about the divergence pattern of optical rays passing through a point, a useful homely analogy is to place the point in question at the center of a transparent earth: then incident rays can be specified by their geographic source, i.e., the geographic point they pass through on their way toward the earth's center; and the emergent rays can be described by their geographic sink, the geographic point they pass through on their way out. I further adopt an orientation of the globe such that the surface normal at the point of interest is a ray from the earth's center toward its north pole.

Using the analogy, we can describe the action of the four basic kinds of optical surfaces—window, mirror, retroreflector, and transflector—visually in terms of where they send rays sourced by a familiar geographic area, let's say the lower 48 American states.

Rays incident on a retroreflector completely reverse course and exit through the same geographic point they entered through:

Retroreflector

Rays incident on a window pass through undeflected, so each ray exits through the antipodes of its source (which, it turns out, is not Australia after all!)

Window

Rays reflecting from a mirror (having its surface normal pointing toward the north pole) will exit at the same latitude as their source, but 180° opposite in longitude.

Mirror

Mirror (with surface normal in center of view)

Rays exiting a transflector have the same longitude as their source, but their latitude is reflected to the opposite hemisphere.

Transflector

Note the similarity of the exit patterns for a transflector and a mirror when seen in the polar view (i.e., with the mirror normal in the center of the image.) For the mirror, the relationship between the incident and emergent ray distributions is a point reflection about the mirror's surface normal. This is equivalent to a 180° rotation about the point—left and right are not reversed. For the transflector, the emergent ray distribution is a reflection over a line (the equator)—and left and right are reversed. (Of course, in one sense, a mirror really does reverse left and right, but our way of describing ray directions is producing its own reversal as the plot for a window shows.)

Reflection about a point. (Image quoted from wikimedia.)

Reflection over a line. (Image quoted from http://geometry.freehomeworkmathhelp.com)

Reflection over a line can be equivalent to reflection about a point if the object has an axis of symmetry perpendicular to the line of reflection.

As diagrammed in the image above, if a divergence pattern has an axis of symmetry that perpendicularly intersects the mirror line, then we have a special case where reflection about the point of intersection is indistinguishable from reflection over the line. Therefore, given such a symmetry in the beam divergence, a Fresnel mirror with 90° facets can act as a nori lens. The radial symmetry of the heliostat field and the central optics about a common axis guarantees that the needed symmetry will exist.

Beam divergence at the lamp (diagrammed in pink) will approximate a segment of a circle of latitude. In this view, the Fresnel mirror's surface normal will lie somewhere on the 0° line of longitude, fulfilling the constraint that it lie on an axis of symmetry for the divergence pattern. Therefore, for this divergence pattern, a Fresnel mirror with 90° facets can precisely simulate a nori lens.

 Gunnar Dolling and Martin Wegener, and R.Varalakshmi have demonstrated that POV-Ray correctly ray-traces in media with negative index of refraction.

Direct solar absorption and storage in high-purity glass melts

Terrestial solar spectrum superposed on the absorption spectrum of low-loss glass fibers.

At 1570 °C (a candidate "empty" temperature for a heat-storage glass melt,) the viscosity of soda-lime glass is only about half that of room-temperature honey; at 1800 °C (a candidate "full" temperature for a heat-storage glass melt,) its viscosity has fallen nearly to that of room-temperature motor oil. In a large storage pond of molten glass, convection will easily be turbulent; unfortunately, convection cannot be effectively driven by heating from the top.

Using the relations for the effective conductivity of soda-lime glass melts in Pilon et al. (which likely give values too low for high-purity melts because of the importance of the radiative contribution) the effective thermal conductivity of molten glass at 1570 °C is 95 W/m-°K (nearly twice that of room-temperature steel,) and it increases with temperature: at 1800 °C the effective thermal conductivity is 160 W/m-°K (comparable to room-temperature magnesium.) However, with a thermal diffusivity around 0.00004 m²/sec, a six-hour pulse of solar heating would only travel about 1 meter into the glass melt by conduction/radiation.

The only option for charging the thermal storage is a glass melt that is so transparent to solar radiation that much of the energy is absorbed in the lower half of the pond's depth. If half of the solar energy is absorbed in traveling to a 20 m depth, that is a loss rate of 3 dB/0.02 km = 150 dB/km. Data for very pure glass fibers (see diagram above) show that such transparency can be far exceeded for wavelengths shorter than 2 microns (which account for more than 90% of the AM 1.5 Direct solar spectrum, see diagram below.) It remains to be seen if such high-purity, transparent glass melts can be economical for direct solar absorption and storage.

Solar spectral distribution for AM 1.5 Direct (green line). Image quoted from http://pveducation.org .

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.

Truth in labeling: Solar multiple vs. capacity factor

It is natural to discuss a solar plant in terms of its rated output (i.e., peak electrical power,) but this often gives a false impression of the plant's economic value. For example, a utility might expect a new coal-fired plant to annually generate electric energy equivalent to its producing at full power 20 hours per day (capacity factor = 20/24 = 83%.) A solar plant in a desert climate will, over the course of a year, only produce energy equivalent to its producing at full power about 6 hours per day (capacity factor = 6/24 = 25%)

With energy storage a solar plant can operate with a higher capacity factor (saving money on some aspects of the plant) and making its rated output more directly comparable with a conventional plant. In a simplistic calculation, a 6 hr/day solar plant can become an 18 hr/day solar plant if we divert 2/3 of its output to storage. That is, by later withdrawing the stored energy we can have two more 6 hr periods of full power operation, giving a total of 18 hr/day. Our hypothetical solar plant would be said to have a capacity factor of 75%, 12 hours of storage, and a solar multiple of 3 (i.e., the ratio of its rated power without storage to its rated power with storage.)

In the real world, the above calculation would require a computer simulation, but we can borrow some real world numbers from Gemasolar, a thermal solar plant in Spain, which has 15 hours of storage and a 75% capacity factor. For obvious reasons, solar plants with a significant amount of storage prefer to advertise their annual electricity production rather than their rated power. From Gemasolar's stated annual production of 110,000 MWh/yr, and stated capacity factor of 75%, we can calculate that its rated power is:

110,000 MWhr/yr * (1/0.75) * (1/8760) yr/hr = 16.7 MW

Gemasolar has 304,750 m² of mirrors, so its rated power comes to 55 w_e/m² of mirror area.

Gemasolar and the Ivanpah thermal solar plant differ in detail (and Ivapah has the better solar climate,) but it is relevant to note that Ivanpah has three units rated 123 MW + 133 MW + 133 MW = 389 MW total, no storage, and 1,079,232 MWh/yr annual output—a capacity factor of 32%. In total, Ivanpah has 2,600,000 m² of mirror area, so its rated output of 389 MW comes to 150 W_e/m² of mirror area. That suggests that Gemasolar's solar multiple is about 150/55 = 2.7

The conclusion here is that solar plants without storage should be de-rated by approximately a factor of three before being directly compared with conventional fossil-fueled plants on the basis of rated power.

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.

Radiative view factors in glass-melt solar storage

Radiative view factors between a cylinder and its endcaps. Diagram from Isidoro Martinez, "Radiative View Factors."

In radiative heat transfer, a view factor F_A→B is the proportion of the radiation which leaves surface A and (directly) strikes surface B. In some cases there may also be an adiabatic blackbody surface C which, at thermal equilibrium, must re-radiate all of the photons it receives. In this case we can define an effective view factor  F*_A→B that includes the re-radiated photons that reach B indirectly:

F*_A→B = F_A→B + (F_A→C)(F_C→B)

When all the surfaces are blackbodies (an assumption that greatly simplifies the math, and is approximately true in practice) the net radiative flux from A to B, averaged over  S_A, the area of A, is:

Φ_A = F*_A→B * σ(T_A⁴ - T_B⁴),

where σ is the Stefan–Boltzmann constant, σ = 5.7 E−8 W m−2 K−4.

The corresponding net flux at surface B, Φ_B, must be Φ_A multiplied by the ratio of the two areas:

Φ_B = Φ_A * (S_A/S_B) = F*_A→B * σ(T_A⁴ - T_B⁴) * (S_A/S_B).

When solar energy is stored in a glass melt, we can consider the free surface of the melt to be surface A (the lower cap of a cylinder,) the water wall tubes of the boiler to be surface B (the walls of the cylinder,) and the roof of the furnace to be the adiabatic surface C (the upper cap of the cylinder.)

Water wall tubes. Image quoted from http://tubeweld.com.

In the view factor diagram above, a reduced radius 'r' is related to the cylinder's radius, R, and its height, H, by:

r = R/H,

and a parameter ρ is defined to be:

ρ = (√(4 * r2 + 1) -1) / r.

The correspondences between our lettered surfaces and the diagram's numbered surfaces are:

A = 3, B = 1, C = 4.

From the diagram,

F_A→B = F_3→1 = ρ/2r

F_A→C = F_3→4 = 1 - ρ/2r

F_C→B = F_4→1 = F_3→1 = ρ/2r

So,

F*_A→B = F_A→B + (F_A→C)(F_C→B) =  ρ/2r + (1 - ρ/2r)(ρ/2r) = (ρ/2r) (2 - ρ/2r),

and the area ratio, S_A/S_B, is

S_A/S_B = πR² / 2πRH = R/2H = r/2.

The average flux on the water wall, Φ_B, is

Φ_B = F*_A→B * σ(T_A⁴ - T_B⁴) * (S_A/S_B) = (ρ/2r) (2 - ρ/2r) (r/2) * σ(T_A⁴ - T_B⁴)

Φ_B = (ρ/4) (2 - ρ/2r) * σ(T_A⁴ - T_B⁴)

With R = 51 m, H = 87 m, r = 0.58, ρ = 0.93, T_A = 1610 °K (1340 °C) , T_B = 730 °K (460 °C), as calculated in an earlier post,

Φ_B = (ρ/4) (2 - ρ/2r) * σ(T_A⁴ - T_B⁴) = 0.26 * 367 kw/m2 = 96 kw / m2

which is not nearly the 250 kW/m2 we need to see on a water tube wall.

We need to increase radiant transfer by upping the "empty" temperature of the glass melt. This will also incidentally decrease storage density, resulting in an increase in R and a decrease in H (as calculated on the basis of 250 kW/m2.)

A spreadsheet exploring radiant transfer in glass-melt thermal storage.

Exploring these relations in a Numbers spreadsheet shows that Tempty = 1570 °C 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.