A rooftop doubt dispeller

This fall, it looks like PGW may finally shut off the gas for 40,000 non-paying customers in Philadelphia. What would they do for space heat and showers and cooking?
An equilateral A-frame on a flat Phila roof with a $200 12'x16' Dynaglas polycarbonate south wall might collect 0.9x16(6x620+12cos(30)1000) = 203.2K Btu of sun and lose 6h(Tg-34)12x16/R1 = 1152Tg - 39.2K on an average January day, where Tg (F) is the A-frame air temp. If it collects Q Btu/day of useful heat, Tg = 210.4-Q/1152.
If a north wall reflects 0.9x16(6x620+0.9x12cos(30)1000) = 188.4K Btu into a polyethylene film Tw (F) water duct on the floor and it loses 6h(Tw-Tg)1.5x12x16 = 1728(Tw-Tg), Q = 188.3K-1728(Tw-(210.4-Q/1152)) = 691(319-Tw). Tw = 100 F makes Q = 152K Btu (45 kWh for PV fans :-) per day, with Tg = 79 F. How can we distribute this heat?
If ridge R of the A-frame reflects direct sun rays S from the horizon vertically down onto edge Q of a duct that extends 6' to the south from base B of the north wall, the ridge needs a 45 degree north wall slope. Two north slopes might join at J, like this, viewed in a fixed font:
R --- U . . 45 (xo,yo) J . . . . <-- S south --> . | y . . . . beta 10.4' = 12'cos(30) ^ . . . | L. | | . . . | . | 45 . beta alpha ----O----------B--------------Q--------------N--> x | 4.4' | 6' | (not to scale)
| 10.4' |
With a beta degree tilt at bottom B, reflected ray angle BJQ = beta, so alpha = 180-2beta. If x=y along line OR, tan(beta) = xo/(xo-4.4), so xo = 4.4tan(beta)/(tan(beta)-1) = 10.4tan(beta)/(tan(beta)+1), ie beta = 67.96 degrees, which makes the lower slope L = 7.97' and the upper slope U = 4.25'. We might make L = 8' and U = 4', with 4'x8' double-foil polyisocyanurate boards and a low vertical wall at B.
This would reduce the heat losing surface and raise the solar collection efficiency in times of direct sun, with 2 suns from B to Q, so we might produce 2 kW of electrical power with 1 kW of standard PV panels under that 6'x16' of duct. Polyethylene and water and glass have similar refractive indices, so the ducts might transmit most of the sun to the PVs. When I put a water-filled duct on a PV panel, the output current decreased 6%, in a crude experiment. Can we make 3 suns? :-)
A draindown EPDM bladder with an upper constraint (eg welded-wire fence) from Q to N might collect diffuse sun, with warm water naturally thermosyphoning up through the BQ duct when available.
Nick
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They will be off better paying their bill then following your tune [ your still out of key].
Hows your Kenner [Nick] easy bake oven [sauna] going, the one with the lightbulbs..
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