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.