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, ηaa, between heliostat and receiver separated by distance dhr in meters:
ηaa = exp(-.00011 * dhr)
for dhr > 1000 m, and visibility = 40 km.
A quarter-township heliostat field (about 4.8 km x 4.8 km) has a maximum dhr of about (1.12) * (4,800/2) = 2,688 m, giving ηaa = 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
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| Markus Kayser with his SolarSinter glass-making solar furnace. Image quoted from www.creativeapplications.net . |
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| 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.
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| 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.
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.
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| Sizes of fractally-grown solar furnaces: a table of molts within the Synthetic PLSS. |
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