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re: inexpensive 100% solar-house heating with warm air collection
13 apr 2003
a 12'x16' shed with r20 walls and ceiling and 8 ft^2 of r2 windows and
15 cfm of natural air leaks has a thermal conductance of about 15 btu/h-f
for the air plus 12'x16'/r20 = 10 btu-h-f for the ceiling plus 8/2 = 4
for the windows plus 440/20 = 22 for the walls, a total of 51 btu/h-f. 
january is the worst-case month for solar house heating in phila, with 30 f
average days with an average daily max of 38. at an average 65 f indoors,
the shed needs 24h(65-30)51 = 42.8k btu. with no internal electrical usage
or solar heat gain, it needs 214.2k btu for 5 30 f cloudy days in a row. 

say we collect solar heat in warm air from a low-thermal-mass sunspace on
the south wall and store it in a 2'x4'x8' tall box with a 2'x4'x4' tall
water tank below and 12 2'x8' shelves above which act as an air-water heat
exchanger with a slow-moving air to water thermal conductance of about
12x2'x8'x1.5x2 = 576 btu/h-f...

say we drape a 100' length of 12" round poly film greenhouse duct ($38) in
a zigzag over 100' of 3' wide 2"x4" mesh welded-wire fencing ($20), with a
2x4 at each end to make the water depth 2", and a low-power pump to circulate
water between the shelves and the tank, which has plywood walls and a 10'x16'
piece of epdm rubber folded up like a chinese takeout box for a liner.

the shed needs (65-30)51 = 1785 btu/h on an average day. the shelf water
needs to be 1785/576 = 3.1 f warmer than the air in the box to provide this.
with natural convection and a 6' height difference between a 4 ft^2 supply 
ent with a 2-watt motorized damper and an 4 ft^2 return vent and a dt box-
to-room air temp diff, 1785 = 16.6x4ft^2'xsqrt(6)dt^1.5 makes dt = 4.9 f,
so the shelf water temp needs to be at least 70+3.1+4.9 = 78.0 f to keep
the shed 70 f. 

the 2'x4'x8'+2'x2'x8' = 96 ft^3 of water in the box has about 96x64 = 6,144
btu/f of thermal capacitance. if it stores heat for 5 cloudy days in a row,
214.2k = 6144(twa-78.0), and the water temp twa needs to be at least 112.9 f
on an average day. 

nrel says 1,000 btu/ft^2 of sun falls on a south wall on an average jan day
in phila, 420 falls on east and west walls, and 620 falls on a horizontal
surface. a 4' deep x 8' tall sunspace with r1 polycarbonate walls and roof
with 90% solar transmission would gain 0.9(2x4x8x420+8x16x1000+4x16x620)
= 175.1k btu on an average day. the 256ft^2 of glazing could supply 42.8k
btu at an average air temp ts, where 175.1k = 6h(ts-34)256ft^2/r1+42.8k,
so ts = 120 f. 

say the shed has a 30" diameter x 16' long poly duct containing t inches
of water resting on foil on a 4'x16' piece of welded wire fence above the
collar beams, and we keep the shed 70 f for 6 hours per day using 120 f air
from the sunspace which warms the overhead water enough to store 100% of
the overnight heat the shed needs on an average day, with the help of a
slow ceiling fan and a thermostat.

on an average day, the sunspace needs to supply 42.8k btu over 6 hours at
7133 btu/h. with natural convection and a 6' height difference between an
a ft^2 supply vent with a 2-watt motorized damper and an a ft^2 return vent,
7133 = 16.6asqrt(6')(120-70)^1.5 makes a = 0.2 ft^2. we might use a 1 ft^2
sunspace damper. 

on an average day, we need to store 30.6k btu in the overhead water with
a heat capacitance c and water-air conductance g = 30pi/12x16x1.5 = 188.4
btu/h-f. rc = c/188.4 = 0.00531c. with t (f) water at dusk and 60 f water
at dawn, c(t-60)=32.1k and t = 120+(60-120)e^(6/(rc)) = 120 - 60e^-(1131/c)
makes c = 535/(1-e^-1131/c). plugging in 535 on the right makes 608 on the
left, then 634, 643, 646, 647.5, 647.9 and 648 btu/h, with t = 109.5 f and
648 = 7.11(47.1t-t^2), and a water depth t = 2.02 inches.

on an average day, with water temp tc and r30 insulation, the 2'x8'x8' box 
in the sunspace loses about 18h(tc-30)192ft^2/r30 btu to the sunspace and
supplies no heat to the shed. if cloudy days are like coin flips, it needs
to supply 42.8k btu about half the time, ie about 21.4k btu/day at a rate
of about 3.6k btu/h, so the air near the shelves needs to be 3.6k/576 = 6.2 f
warmer than the shelf water during charging. with the shelves behind 64 ft^2
of their own glazing behind the sunspace glazing and a 4 ft^2 airflow path
in the shelf-heating loop, the air in the shelf heater needs to be dt f
warmer than the air near the shelves, where 3.6k = 16.6x4sqrt(6')dt^1.5, so
dt = 7.9. with 1000 btu/day of sun passing through two 90% r1 glazings, 
810x64ft^2 = 6h(tc+6.2+7.9-120)64ft^2/r1 + 18h(tc-30)192ft^2/r30 + 21.4k
makes ts = 137 f. this assumes (conservatively) that the shelves don't gain
any more heat on a clear day than an average day. 

the shelves would cool by about 42.1kbtu/2048btu/f = 20.5 f over a cloudy
day and might rewarm the following day. leaving them cool is desirable for
efficient solar heat collection. the circulating pump might run just enough
to keep the lower tank warm on an average day, and more on a cloudy day... 
if we turn on the pump when the shelf water is at least 5 f warmer than the
tank water, the pump needs to move p pounds of water per day, where 21.4k
= 5p, and p = 4280 pounds per day or 0.4 gpm. taco's $101 60 watt 006-bt4-1
pump (grainger item 5p429) can move 5 gpm with a 6' head, so it might only
run 1.8 hours per day, consuming 39 kwh per year at a cost of $3.90. 

the ashrae handbook of fundamentals says the 99% "winter design temp"
in phila is 10 f, which determines the required heating system capacity.
the shed needs (65-10)51 = 2805 btu/h to stay 65 f on that very cold day,
when the shelf water needs to be 2805k/576 = 4.9 f warmer than the air in
the box, and 2805 = 16.6x4ft^2'xsqrt(6)dt^1.5 makes dt = 6.7 f. at 5 gpm,
2805 = 5x8x60dtw, so dtw = 1.2 f, and the tank water temp needs to be at
least 65+4.9+6.7+1.2 = 77.8 f on that day. we might turn on the pump when
the shelf water temp is 5 f more than the tank water temp or the shelf
water temp is less than 77.8... 


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