Self-replicant fractal growth patterns

Fractal growth of a pentagonal annulus. Underlying image from Bowers and Stephenson, "A 'regular' pentagonal tiling of the plane."

Same image as above with some of the un-needed boundaries erased.

Peripheral and non-simple fractal growth.

By self-replicant fractal growth I mean growth that incorporates at each molt or growth stage an increment that makes the shape a larger version of its earlier self. It is easier to think about fractal growth time-reversed as a repeated subdivision of a large tile into smaller tiles. The algorithm is simple: subdivide the large tile in a way that leaves a sub-tile that is geometrically similar to the large one; reapply the subdivision to the similar tile; repeat step two. (To completely specify the subdivision procedure it may be necessary to mark a point on the boundary of the large tile, and the corresponding point on the boundary of the similar tile.)

These patterns are either central or peripheral according to whether the intersection of the boundaries of the large tile and the similar tile is empty or not. The peripheral patterns can be simple or non-simple according to whether or not the boundaries intersection is connected. The pentagonal example above is peripheral and non-simple.

Peripheral simple fractal growth of a triangle.

Peripheral simple fractal growth of a triangle.

Peripheral non-simple fractal growth of a triangle.

Central fractal growth of a triangle.

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.

Fractal growth of all-glass, glass-making solar furnaces

An all-glass glass-making solar furnace built through a sequence of five molts from a much smaller furnace.

Probably the fastest way to grow a solar furnace is to build three squares equal in area to the glass-making unit and then re-aim the heliostats of the glass-making unit (and dismantle its lamp) to constitute the fourth square. That way, the time, t_m, to build the next molt is just 3x the glass replication time, t_r, times an additional factor, i, accounting for the increase of glass thickness between molts. Generalizing from m = 4 to any molt area factor, m:

t_m = (m - 1) * t_r * i


Earlier it was assumed that the glass thickness would double in the course of a linear scale increase of 16, which corresponds to 4 stages (molts) of linear doubling. Therefore, in each molt, the glass thickness should increase by the fourth root of 2, or 1.189. The first molt requires 3 x 1.189 = 3.57 glass replication times to complete. For each later molt, the glass replication time increases by a factor of 1.189 on account of the increased glass thickness.

The fourth root of two is a rather small increase in glass thickness for a doubling in linear scale, but the value is based on the fact that heliostat size does not need to increase in proportion to field size. At larger scales, the lamp—being all of one piece and not scaled to the heliostats—becomes a larger portion of the total mass, so the glass thickness factor must increase somewhat. Also, when going down to much smaller scales the thickness factor must in the limit approach 2 (the factor for geometric similarity) to prevent mechanical interferences. Another way to think of it is that the field cannot be composed of fewer than one heliostat.

Starting with a small molt 0 unit having replication time t_r, m = 4, and molt thickness multiplier i, growing to molt n requires build time, t_b:

t_b = t_r * 3 * i * (1 + i + i² + i³ … + i^n-1)

t_b = t_r * 3 * (i + i² + i³ … + iⁿ)

For i = 1.189, the ratio t_b / t_r  to reach successive generations is

1     3.6
2     7.8
3   12.9
4   18.9
5   26.0
6   34.5
7   44.5
8   56.5

For example, growing a molt 0 unit that is 6.25 m square up to a unit 1600 m square requires 8 molts. If the glass replication time of the molt 0 unit is ten days, growing from that unit to a unit approximately one mile square takes 565 days, or a bit more than a year and a half.

The final molt will probably use a molt area factor, m, larger than 4, because it makes little sense to gather the forces to build a large project in a matter of a few months unless they can move directly on to something else. Using a final molt area factor, m_f, of 9 in the last stage effectively becomes 1.59 molts since

m_f = 9 = 4^1.59 .

The time, t_mf , required for the final molt is

t_mf = (m_f - 1) * (t_r * iⁿ) * ln(m_f / m)

where t_r is the glass replication time for molt 0, i is the glass thickness factor for the earlier molts, n is the number of the next-to-final molt, and m is the molt area factor for the earlier molts.

Here,  m = 4, m_f = 9,

t_mf = 8 * (t_r * iⁿ) * 1.59 = 12.7 * (t_r * iⁿ)

For the growth sequence of eight molts mentioned above, n = 8, and the final molt with m_f = 9 has

t_mf / t_r = 12.7 * i⁸

For i = 1.189,

t_mf / t_r = 12.7 * (1.189)⁸ = 50.7

The total 9-molt build from a 6.25 m square to a quarter-township takes (56.5 + 50.7) * t_r  = 107 * t_r .

If the glass replication time for the 6.25 m x 6.25 m solar furnace is ten days, it can grow to occupy a quarter-township in 1070 days, a little less than three years.

Starting from a smaller scale would only add a few days to the total. In fact, in the range where i = 2, the time for each molt is the sum of the times for all previous molts. In the case considered that would mean that all molts preceding our molt 0 would total less than 10 days, no matter how small we start.

Therefore, a quarter-township all-glass, glass-making solar furnace can be grown in a matter of a few years no matter how small the seed.