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.

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. 

Scaled generations of all-glass, glass-making solar furnaces

Specifications for scaled generations of all-glass, glass-making solar furnaces.

A classic science-kid's demonstration is making glass from sand with a solar furnace, and, in fact, the glass-making capacity of a solar furnace is prodigious. A solar furnace of any size, even if it is 100% glass, can make all the glass needed to replicate itself in a matter of weeks. The glass replication time of a small backyard solar furnace may be only a matter of days. More likely at large scale, the glass for a furnace would be produced by a smaller solar furnace that produces at a rate that does not outstrip the capacity to fabricate and assemble glass parts. That smaller furnace, in turn, may have been made by an still smaller furnace; and so on, through multiple generations of scale.

Starting at a very big scale (a solar furnace on 9 square miles of land) and working for the most part within areal units of the Public Land Survey System of the western United States, the table above shows specifications for six scaled generations. If these calculations can be taken seriously, the time from the completion of the room-sized 9-square meter furnace to the completion of its descendant quarter-township furnace is 6.3 years.

Reproduction times and mass flows for all-glass, glass-making solar furnaces

All-glass, glass-making solar furnaces can be built up starting from a small seed.

When we tap power from a solar furnace (which is really what CSP power towers are,) we are tapping power from a thing that can make other things. When the day comes that solar furnace energy is cost competitive with coal for boiler-temperature heat, it will already be a factor of two or three times cheaper than any other source of high-temperature heat for making things. The economics of certain materials will be revolutionized by the availability of cheaper high-temperature heat, none more so than glass.

Laurent Pilon et al. studied thermal transfer in a glass-making furnace that produced 3.35 kg/s of soda-lime glass, with a heat transfer from the combustion space to the melt of 8.3 MW. Thus the energy intensity of this portion of the glass making process is (8,300,000 J/s) / (3.35 kg/s) = 2.5 E6 J/kg. The room temperature density of soda-lime glass is 2600 kg/m3, so, ignoring additional heat requirements in the annealing, the energy intensity of finished glass is 6.4 E9 J/m3.

The solar generating unit Ivanpah 1, produces 126 MWe at a 32% capacity factor, thus averaging 40 MWe over the course of 24 hours. At a conversion efficiency of 0.40, Ivanpah 1 must be producing thermal power of 40 MWe / (0.40 MWe/MWth) = 100 MWth = 1.0 E8 J/s .

Therefore, Ivanpah 1, relieved of its power-generating duties, could make glass at the rate of (1.0 E8 J/s) / (6.4 E9 J/m3) = 0.016 m3/s or 16 liter/s.

Ivanpah 1 has 53,527 heliostats each with a mirror area of 15 m2, giving a total mirror area of 803,000 m2.

Now comes the guessing part: if a solar furnace is made entirely of glass, and the total volume of glass is pro-rated to the heliostat mirror area, how thick will the layer be? I'm going to say 5 cm (0.05 m.) On that basis, in making the glass for an Ivanpah 2, Ivanpah 1 must make (.05 m) * (803,000 m2) = 40,000 m3 of glass.

At 16 liter/s (0.016 m3/s), that task will occupy Ivanpah 1 for (40,000 m3) / (0.016 m3/s) = 2.5 E6 s = 700 h = 29 days.

Making the glass for two copies of itself, an all-glass Ivanpah 1 solar furnace would be offline for just two months! (Of course there's a lot more to it than just making glass, but that other stuff is not Ivanpah 1's responsibility.)

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Suppose each solar furnace makes the glass for two copies of itself and only afterwards gets about the business of full-time power generation. The glass for two units can be made in 2 months, but completing them will take longer, let's say it takes an additional 10 months. Setting the year clock to n = 0 when the first solar furnace starts making glass, and taking a census of solar furnaces annually, the census series will be:

1, 3, 7, 15, 31, 63, 127, 255, 511…  =  2ⁿ⁺¹ - 1

Replicating to 300 units takes less than 8 years.

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That said, it might make more sense to do things the old-fashioned way, using a special-purpose solar furnace sited near the best raw materials. An individual glass-making unit would need 300 months (25 years) to make glass for 300 units, so 3 specialized glass-making furnaces would be needed to make 300 units in about 8 years.

The quarter-township units proposed in an earlier post are considerably larger than Ivanpah 1, their thermal power is

(1000 MWe) * ( 0.75 capacity factor) / (0.40 MWe/MWth) = 1875 MWth,

which is comparable to the thermal power of a 1 GWe coal-fired plant.

That is 1875 / 100 = 19 times more thermal energy than Ivanpah 1 produces, so it yields 19 * 16 l/s  =  300 liters of glass per second. If the glass layer thickness (0.05 m) is the same, the reproduction time will be the same as calculated above (4 weeks), but the mass flows are huge: 2,800 tons of glass per hour, 24/7. Put in terms of the 120-ton capacity of one railroad coal car, that would be 570 cars—or about five, 120-car unit trains per day. That is a mass flux is needed in two directions, carrying materials in and glass parts out.

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Such big mass flows need to be as short as possible. A "growing from an acorn" motto seems more appropriate than "dividing like yeast." Since a small unit can make in three years the glass for a unit 36 times larger than itself; and, since solar resource quality varies gradually over wide areas; it makes sense to site a relatively small, specialized glass-making unit where raw material resources are excellent, and let it slowly build the larger specialized power-generating unit. When the local area is considered built-out, the glass-making unit might be disassembled and relocated, or retired in-place to making spares.

To make a quarter-township solar generating unit, a one-section (one square mile) solar glass-making unit would need 9 months to make the glass; a quarter-section solar glass-making unit would need 36 months. Since it usually takes 3 to 4 years to build a conventional power plant, a quarter-section glass-making unit is probably about right.

The quarter-section glass-making unit could itself be built in 16 months by a 10-acre glass-making unit. If fabrication and assembly proceed just-in-time with the glass production, a quarter-township generating unit would be completed in 16 + 16 + 36 = 68 months (5.7 years) after the 10-acre unit starts producing glass.

In this way, the maximum mass flow, which occurs during the 36 months when the quarter-section glass-making unit is making parts for the quarter-township generating unit, amounts to 570/36 = 16 rail cars per day, but these trips are from local quarries and out to the adjoining, under-construction, heliostat field.

If the 10-acre (40,470 m2) unit can be slimmed down to a 2.5 cm glass layer—say, by using half-scale heliostats—then its mass is

40,470 m2 * (0.70 mirror/land ratio) * (0.025 m) * (2600 kg/m3) = 1800 tons

amounting to 15 rail cars to deliver it ready-to-assemble.

As a check: the trucked-in plant would make that much glass (15 rail cars) in just 2 weeks (being a slimmed-down, half-thickness design), so it can make about 1 rail car per day of glass. After successive multiplications of 16x (going up to a quarter-section, or 160 acres) and 36x (going up to a quarter-township, or 9 sections) the yield of the quarter-township unit should be (1 railcar/day) * 576 = 576 railcars/day, which checks against 570 rail-cars calculated above.

If it is necessary to reduce the freight still more, another factor of 16, down to a 2500 m2 land area (50 m x 50 m,) would just add 8 more months (since slimmed-down units replicate in just half a month.) The trucked-in weight now down to 110 tons. Another factor of 16, down to a 160 m2 land area (13 m x 13 m,) would add 8 months and reduce freight down to one load for a medium duty truck, 7 tons.