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.

Oculus de-concentrator

An oculus de-concentrator can be used to shield the boiler water wall from sunlight.

Shielding the boiler tubes from direct sunlight may require a de-concentrator, i.e., an optical device that decreases beam divergence. If this device is reflective, then it will be heated more by the sunlight than the thermal back-radiation. Being reflective, it can be actively cooled on its non-illuminated side. The heat balance is taken at the smaller, entry, aperture, so maximum concentration does not need to be increased. In the diagram the peak solar flux at the smaller entry aperture is 3960 kw/m2, 1960 kw/m2 at the exit.

Heating of an oculus concentrator

An oculus concentrator must stand off from the oculus to reduce the intensity of its heating.

Sunlight must pass through the oculus at high concentration because there is a high level of radiant flux inside the hot space which will be leaking out the same opening the sunlight enters. At the empty storage temperature of 1250 °C, the blackbody radiant flux inside the hot space is 305 kw/m2, and 72% of this is outside the 0.35-2.0 micron passband of glass; at the full storage temperature of 1800 °C, the radiant flux inside the hot space is 1047 kw/m2, and 49% of this is outside the passband of glass. Thus a hypothetical glass window covering the oculus would be absorbing between 220 and 513 kw/m2 of back-radiation depending on the hot space temperature. And the heating from the concentrated sunlight may be comparable. With 8000 kw/m2 of solar flux at the oculus, an absorption of just 2% in the optics would amount 160 kw/m2 of additional  heating.

Due to the heating of the optics, it is necessary to separate an oculus concentrator by some distance from the oculus itself. Mirror optics would have the advantage that special arrangements for cooling can be made on the dark face, transmissive optics are practically limited to radiant and air convection cooling. A tolerable heating flux on transmissive optics may be 10 kw/m2, which requires standing off till the flux is reduced to 1/75 the flux at the oculus. For example, an oculus concentrator that is a hemisphere of area 2πR^2 =75 * π *r^2, so R/r = sqrt(75/2) = 6, where R is the radius of the oculus concentrator and r is the radius of the oculus.

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.

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.

Thermal flux at the glass lake

Thermal flux on a glass lake solar thermal store.

Assuming an average annual rated power of 51 W_e per m² of mirror area, peak thermal flux will be about 4 x 51 x 0.71 / 0.4  = 357 W_th per m² land area. Concentrating this flux on a circle having diameter equal to the optics height, H, would give a land-area concentration factor of 28 x 28 = 784, producing a thermal flux of 784 x 0.357 suns = 280 suns. To discharge the lake at 250 suns requires a charging flux of 500 suns, and, since one third of the incident flux is diverted to power generation, a total flux on the lake of 750 suns. That implies a lake with a diameter of 0.61H .

To have the same 250-sun flux at the water wall of the boiler and at the surface of the discharging lake requires the two areas to be equal, and thus the height of the water wall must be one-quarter the diameter of the lake (see diagram below.)

For the area of a cylinder's wall to equal the area of one of its end caps, the height of the wall must be one-quarter the cylinder's diameter.

For an approximately 1 GWe target design, with R_f = 2660 m, and H = 190 m, the lake diameter will be 0.61 x 190 m = 116 m, and the height of the boiler's water wall will be 116 m /4 = 29 m. To accommodate 2.5 times the thermal flux that was modeled in the previous post's thermal simulations requires a lake 2.5 times as deep, and, to keep the same maximum temperature, 2.5² = 6.25 times the optical absorption length. However, that would give a 25m-deep lake with a absorption length of 63 m—so something's got to give.

Charging and discharging transparent glass-melt solar thermal storage

Molten glass thermal storage after 6 hours of solar charging from empty = 0.

Molten glass thermal storage after 6 hours of radiant discharging from full = 250.

The two simulations above were run in Energy2D for a 10-meter depth of molten glass with a 10-meter optical absorption length, charging at 200,000 W/m2 and discharging at 100,000 W/m2. Charging produces a stable thermocline in the melt, while discharging drives natural convection that maintains a nearly uniform temperature within the convecting layer. Surprisingly perhaps, despite the high viscosity of molten glass, the natural convection during discharge is fully turbulent. The discharge simulation above had to be run at 1/10 earth gravity to keep convective turbulence at a scale the model could handle.