Improved model of thermal flux on the glass lake thermal storage unit

Thermal flux values at glass lake thermal store.

This design reduces the thermal flux at the boiler's water wall to 210 suns, the discharging flux at the lake surface is smaller than that because the lake has 1.5 times the area of the water wall. A much wider range of temperatures is used this time, 550 °C (1250 °C - 1800 °C) but the thermal stratification that occurs during charging allows only about 52% utilization of the glass lake's theoretical storage capacity. The depth of the glass lake and its optical absorption length are both 8 m. Two thirds of the peak solar power of 7.5 GWth goes to storage, giving a plant capacity factor of about 0.75.

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.

Maximum temperatures in glass-melt solar storage

Temperature field in a glass-making furnace in °K. (1820 °K = 1550 °C.)  Image quoted from L. Pilon et al.

As calculated in a previous post, the minimum (i.e., empty) temperature of a glass-melt storage is about 1340 °C (1610 °K) because of the need to supply high radiant flux (about 250 suns) to the steam tubes in order to achieve rated output. The maximum temperature depends on the high temperature materials available for the tank, the furnace roof, and the shading elements that modulate the radiant flux on the stem tubes.

Properties of ultra-high-temperature ceramic insulation. Image quoted from Rath USA.

Being optimistic, I'll say 1800 °C is an acceptable maximum glass-melt temperature—that's 250 °C hotter than a glass making furnace.

The storage temperature range is 1800 °C - 1340 °C = 460 °K, and the mean temperature is 1470 °C.  From Pilon et al., the specific heat, c, of molten glass between 1000 °C and 2000 °C is about 1231 J/kgK, so a 460 °K storage range stores 460 * 1231 = 566,000 J/kg. The density of molten glass in this temperature range is about 2300 kg/m3, so the volumetric energy storage is 1.3 E9 Jthermal/m3.

From the previous post, we need 5.3 E6 Jthermal/m2-land to store 15 hours of heat for rated output, so, spread over the land area of the heliostat field, the storage glass would form a layer 5.3 E6 / 1.3 E9 = 4 mm thick. That is less glass than would be needed for the mirrors!

Again from the previous post, the total land area for 3.2 GWe Gemasolar plant is 71 E6 m2, so the total storage volume is 71 E 6 * .004 = 284,000 m3, equivalent to a hemisphere with radius, r:

pi * 2/3 * r^3 = 284,000

r = 51 m, a pool with a perimeter of 2*pi*r = 320 m

At a thermal efficiency of 0.46, the rated thermal power is 3.2 E9 / 0.46 = 7.0 E9 Wthermal.

At 250 suns, the furnace wall area is 7.0 E9 / 250,000 = 28,000 m2.

With furnace wall area arrayed around the 320 m perimeter of the pool, the furnace wall is 28,000 / 320 = 87 m high, that is close to the height of the actual furnace walls at Plant Bowen, which appear to be about 75 m tall in photos. But I think 87 m is too tall in relation to the pool diameter (102 m) to comport with the earlier assumption of a 0.7 view factor, so this calculation is going to need more iterations.

What does an adaptive primary do?

An adaptive primary is a concentrating mirror that redirects sunlight to a fixed focus—no matter where the sun appears in the sky. A mirror that accomplishes this task must adapt its curvature as the sun rises higher or sets lower in the sky. To a first approximation, the curvature of the mirror is toric, that is, the mirror is always approximately shaped like a small patch on the surface of a torus. The adaptation required is slight, the changes in curvature will be scarcely visible to a viewer looking sideways at the mirror.

In the animation above—rendered in POV-Ray using Mega POV—the zenith distance of the sun varies from 15°  to 75°, while the adaptive primary acquires a toric curvature by the thin-shell bending of an initially spherical mirror. The POV-Ray scene description file for the animation is here.

In addition to changing its curvature, an adaptive primary needs to follow the sun in two angular dimensions: turning to face the sun's azimuth while also tilting its normal to one half the sun's zenith distance. (At the most an adaptive primary only needs to be tilted 45° away from horizontal.)

Toric curvature of an adaptive primary

Bending a thin-shell, initially spherical, primary mirror to accommodate the changing angle of solar incidence, θ,  induces a toric curvature in the mirror.

The primary mirror of a two-mirror heliostat must function over a wide range of the solar incidence angle θ: from θ ≈ 0° at summer noon, to θ ≈ 45° at sunrise and sunset. A unchanging mirror profile cannot handle this wide range of incidence angle. It is necessary for the primary mirror to change shape only slightly. Luckily, simple thin-shell bending gives the correct range of profiles.

In the thin-shell bending of a surface, its Gaussian curvature—the product of the curvatures in the two principal planes (the planes that reveal respectively the maximum and minimum curvatures)—is constant. In a spherical mirror these to curvatures are equal, so as a thin-shell spherical mirror bends, one principal curvature increases while the other decreases in the inverse proportion.

Consider a spherical mirror reflecting sunlight. The plane that contains both the sun and the surface normal at the center of the mirror is called the meridional plane. The plane that likewise contains the surface normal at the center of the mirror, but is disposed perpendicular to the meridional plane is called the sagittal plane. When the primary mirror is adapted correctly to the changing angle of solar incidence, the meridional and sagittal planes are also the principal planes of curvature. In particular, as the angle of solar incidence, θ, increases, the radius of curvature in the meridional plane must increase as 1/cosθ, while the radius of curvature in the sagittal plane will, correspondingly, decrease as cosθ.

In the open-source ray-tracing program POV-Ray, a torus is specified by its major radius and its minor radius (as identified in the illustration above.) Thus, in this system, a sphere of radius 1 is also a torus with major radius 0, and minor radius 1.

For correct adaptation to a changing angle of solar incidence θ, we need the meridional radius of curvature (which is equal to the major toric radius plus the minor toric radius) to be proportional to 1/cosθ; and therefore, under the constraint of thin-shell bending, the sagittal radius of curvature (which is equal to the minor toric radius) will be proportional to cosθ.

Two-mirror heliostats

Optical diagram of a two-mirror, or telescopic, heliostat. All profiles are parabolic and confocal. The primary mirror is gimbaled and bendable to accommodate the changing angles of the sun. The secondary mirror is fixed.

Definition: In solar engineering, a two-mirror heliostat, or telescopic heliostat, is a heliostat composed of two off-axis, parabolic mirrors, arranged in the configuration of a Mersenne telescope.

The larger primary mirror of a two-mirror heliostat is gimbaled and thin-shell bendable to accommodate the apparent movement of the sun. The small secondary mirror, which redirects concentrated sunlight toward the target, is rigid and fixed. The advantage of two-mirror heliostats over conventional one-mirror heliostats is that they can be packed closely together without incurring blocking losses—even when aiming at a target of low angular elevation. The optics and tracking motions of two-mirror heliostats are identical over the entire field.

The telescopic heliostat as a Mersenne telescope

Optical diagram of a Mersenne telescope. This is Mersenne's illustration of 1636 reoriented to be observing the zenith. Image quoted from Fred Watson, "Stargazer: the life and times of the telescope."

I learned recently that the afocal, parabola/parabola Cassegrain telescope is more properly termed a Mersenne telescope. Marin Mersenne's 1636 design for a two-mirror, reflecting telescope is the basis of the telescopic heliostat. In a telescopic heliostat, both mirrors of the Mersenne telescope (the objective, or primary; and the eyepiece, or secondary) are off-axis parabolas. The objective of a telescopic heliostat also operates with varying off-axis angles of incidence and in consequence must be able to bend to adjust its profile.