The size range for all-glass, glass-making solar furnaces

LIMITS ON LARGE-SCALE, ALL-GLASS SOLAR FURNACES

At mega-scale, the height of the lamp when made of glass, is constrained by the specific strength of glass fibers, σ/ϱ, measured in Pa-kg/m3 = N-m/kg, which, when divided by the acceleration of Earth gravity, 9.81 N/kg, gives a characteristic breaking length. Using data for S-2 glass fibers , σ = 4.9 E9 Pa, ϱ = 2460 kg/m3, giving a breaking length of 200 km. This high value is only for pristine glass fibers, but the needed safety factor can be properly lumped-in with the yet unknown factor, very much less than 1, that, dependent on the lamp's structural design, converts the breaking length to the lamp height.

Another possible size-limiting factor is atmospheric turbidity. Mark Schmitz et al. in "Assessment of the potential improvement due to multiple apertures in central receiver systems with secondary concentrators," recommend an approximation for atmospheric attenuation, η_aa, between heliostat and receiver separated by distance d_hr in meters:

η_aa = exp(-.00011 * d_hr)

for d_hr > 1000 m, and visibility = 40 km.

A quarter-township heliostat field (about 4.8 km x 4.8 km) has a maximum d_hr of about (1.12) * (4,800/2) = 2,688 m, giving η_aa = 0.74; in other words, 26% of the redirected solar energy from the most distant heliostats would be lost to scattering on the way to the lantern. That would indicate quarter-township units are about the largest solar furnaces the turbidity of the atmosphere allows.

LIMITS ON SMALL-SCALE, GLASS-MAKING SOLAR FURNACES 

Markus Kayser with his SolarSinter glass-making solar furnace. Image quoted from www.creativeapplications.net .

A glass bowl produced by SolarSinter.

The most down-scalable solar glass-making technique is Markus Kayser's SolarSinter, in which sunlight is directly focussed onto sand. The thermal conductivity, κ, of sand, about 0.2 W/m-°K, limits how steep a thermal gradient can be created using a given thermal flux. The optical flux achievable in a solar furnace is the flux seen on the sun's surface as attenuated by Earth's atmosphere—about 40 E6 W/m2—multiplied by the square of the non-dimensional numerical aperture (NA) of the optics.

Microscope objectives of different NA. Image quoted from www.microscopyu.com

In air, numerical aperture can theoretically approach 1, but, more realistically, some headroom must be left between the optics and the melting sand. An NA of 0.87 (on the right of the image above), may be taken as a practical maximum. NA = 0.87 allows a solar furnace to achieve an optical flux of 40 E6 W/m2 * (0.87)² = 30 E6  W/m2. The albedo of dry sand is about 0.4, so the actual thermal flux, Φ, is (1.0 - 0.4) * 30 E6 W/m2 = 18 E6 W/m2. A ΔT of approximately 2000 °K is needed to melt sand that is initially at room temperature. In one-dimensional, steady-state flow, 18 E6 W/m2 can produce a 2000 °K ΔT in a layer of sand of thickness δ,

 δ = κ ΔT / Φ = (0.2 W/m-°K) * (2000 °K) / (18 E6 W/m2) = 0.022 mm .

This is a best-case scenario since transient or 3-dimensional heat flow would require even greater thermal flux to melt the sand. Assuming we can reduce the radius of the focal spot, r, down to r = δ without stopping the sand from melting, then, working backward from the geometric concentration factor, C,

C = (30 E6 W/m2) / (1 E3 W/m2) = 30,000X ,

indicates that the solar furnace must have an entry aperture with radius R,

R = r √C = 170 * δ = 3.8 mm

For comparison, the radius of SolarSinter's Fresnel lens appears to be around 500 mm, so in theory it should be possible to down-size SolarSinter by something like two orders of magnitude.

However, the finite size of sand grains may set the actual bound. Fine sand grains may have diameter 0.125 mm to 0.25 mm  which is an order of magnitude larger than the value for δ calculated above. That suggests SolarSinter, when working with fine sand grains, can be down-sized just one order of magnitude to around r = 50 mm, or roughly a square 0.1 m on a side.

The size range from 0.1 m x 0.1 m to 6.25 m x 6.25 m is about 6 molts or linear doublings (areal quadruplings,) from there, there are nine molts (the last being extra large) to quarter-township size. Whether we start at 0.1 m x 0.1 m (requiring 15 molts) or 6.25 m x 6.25 (requiring 9 molts) growing a maximal-size terrestrial solar furnace takes something like three years.

Sizes of fractally-grown solar furnaces: a table of molts within the Synthetic PLSS.

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).

Solar plants don't need us.

Beam-down optics for making glass in a solar furnace using an array of telescopic heliostats.

A common assumption is that the purpose of a solar plant is to produce electricity. That is incorrect. The purpose of a solar plant is to reproduce itself. Electricity is just the reward we symbionts get for helping reproduction along.

The easiest way to make a solar plant that can reproduce itself is to form it out of glass. The raw materials of glass are very cheap, except for one—high-temperature heat—and that is precisely the resource a solar furnace produces in abundance.

Overall view of a glass-making solar furnace employing an array of telescopic heliostats.

An order-of-magnitude calculation of the reproductive rate of an all-glass, solar glass-making plant can be attempted.

Direct normal insolation in Tucson averages just over 7 kw-hr/m2 per day. Assume one-fifth of this flux is collected by the glass-making plant per horizontal square-meter of its total land area, or about 1.4 kw-hr/m2 per day. Assume the total glass used in building such a plant is equivalent to a 2-cm thick layer of glass covering the land area of the plant, or about 50 kg/m2. Assume the thermal energy used in making glass this way is similar to the energy intensity of industrial glass-melting and refining, which is 6.5 MMBtu/ton, or about 2 kw-hr/kg of glass. Thus each square meter of land area needs to produce 2 kw-hr/kg x 50 kg = 100 kw-hr in the time it takes for a plant to make a copy of itself. Under these assumptions, the time required is 100 kw-hr/(1.4 kw-hr/day) = 71 days.

If glass-making is indeed the limiting factor, a single glass solar furnace can spawn a thousand glass solar furnaces in ten reproductive cycles, or about two years. If we don't need such a high rate of growth, we can parasitize the plants with radiant boilers and make some electricity on the side.