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solar spa heating arithmetic
21 oct 1996
the comfort spa (109 woodbine downs, unit 3 & 4/etobicoke, ontario m9w 6x1,
(416) 213-9929) is octagonal and weighs "70 to 100 pounds." their ad has a
picture of 2 thin people lifting one up on top of a car. the comfy 1 model
fits 3 people, the comfort spa fits 6, and the turbo spa features "all of
the latest design innovations."
the ad says it has a marine grade vinyl exterior, and an interior liner of
poly tri-ply pvc reinforced with a nylon scrim, and a schedule 40 pvc frame,
and polyethylene foam insulation, and a limited 5 year warranty.
what would it take to solar heat this thing, assuming it's about 5' diameter,
with r20 sides and cover? say, sit it on top of 4 plastic 55 gallon drums
inside some r10 insulation (2" of styrofoam, with latex paint on the outside),
and add a 4' tall x 16' long polyethylene film window w on the front:
2'
-----
/ d \
/| d d |\ 5'
/r g d g r\
/ w w w w w w w w w \
16'
reflecting pool
the inside wall marked r might be reflective, with polycarbonate glazing g
over the 5 south sides of the drum enclosure. this could be a direct gain
system, with sun shining in on the drums. if there were a reflecting pool in
the front, the drums might receive 1300 btu/ft^2/day x 64 ft^2 x 0.9 x 0.9
= 67k btu/day of sun where i live, with an average outdoor temperature of 36 f
in december. the thermal conductance from the drums to the outside air would
be about 20ft^2/r20 = 1 for the top, 24ft^2/r10 = 2.4 for the 3 north sides,
and 40ft^2/r2 = 20 for the other 5 sides, for a total of 23.4 btu/hr-f-ft^2,
ignoring the spa for the moment, so we might find the steady-state drumwater
temp t from
24hr(t-36)23.4 = 67k, or t = 36 + 67k/5612.6 = 155f.
assuming we could make the water that temperature (unlikely, with radiation
losses) what happens after a cloudy week? rc = 4x500/23.4 = 85 hours, so
t(168 hours) = 36 + (155-36)exp(-168hours/85 hours) = 52 f. storing heat
for a cloudy week is difficult with a direct gain system. should we replace
the $3 piece of r1 poly film with $2500 worth of r8 windows? ("of course,"
window manufacturers might say :-)
or make it an indirect gain system, with an air heater on the sunny side, and
no airflow at night, with a conductivity of 1 for the top and 64ft^2/r10
= 6.4 for the sides, so rc = 4x500/7.4 = 270 hours? then, if the water started
at 136 f, after a cloudy week it would be about 36 + (136-36)exp(-168/270)
= 89.7 f. better, but still not warm enough. let's add more insulation: with
r20 sides we have rc = 2000/4.2 = 476 hours and 36 + exp(-168/476) = 106 f.
but each 55 gallon drum has a surface area of about 25 ft^2 and a still air
surface film conductance of about 40 btu/hr-f-ft^2, and we have 64 ft^2 of
glazing, for a thermal mass area/glazing area ratio of less than 2... so to
get 400 btu/hr x 64 ft^2 = 83 k btu/hr of full sun with reflection into the
drums, we need air around them that is about 83k/(4x40) = 519 f warmer than
the water inside. how can that be improved?
increase the surface area by placing 3 8x8x16" concrete blocks with 3 holes
in them under each 3' tall drum, and put 2 1 liter soda bottles full of water
in each hole? each block adds about 6 ft^2 of surface, and bottles add about
1 ft^2 each, for a total of about 4drumsx3blocksx12ft^2 = 144 ft^2 more area.
and let's fill up the 1ft^2 x 4 ft tall hole between the drums with another
13x4 1 and 2 liter bottles, for a total mass surface of 300 ft^2, reducing
the required air temp to 184 f warmer than the drum water, in full sun. but
only about 80% of the sun will pass through the glazing, reducing the delta t
to 148 f. making the drum surfaces rougher and moving air at 4 mph past the
drums should increase their thermal conductivity to 2+4/2 = 4 btu/hr-f-ft^2
and decrease the delta t to a more reasonable 83k/(300x4) = 69 f.
or, we could just reduce the solar glazing, since we are collecting about
10 times the solar power needed to keep the tub warm. just glazing the 3
south sides gives 24 ft^2 of solar aperture, about 4 times more than needed:
2'
-----
/ d \
| d d | 5'
g d g
g g g
reflecting pool
now the solar glazing/thermal mass area ratio is about 12:1, and full sun is
about 5760 btu/hr, and delta t = 5760/(300x4) = 5 f. about 28 k btu/day of
sun shines in through the glazing, and about 6 hr(t-36)24 ft^2/r1 of heat is
lost through the glazing each day, making the linear water temp prediction
approximately 36 + 28k/(6x24) = 231 f. still warm enough :-)
the spa itself might have a thermal conductivity of about 1 for the top and
3 for the sides, losing 24(105-36)4 = 6624 btu on an average december day.
the lower part would have a thermal conductivity of about 64ft^2/r20 = 3. the
55 gallon drums have a thermal mass of 4x55x8 = 1760 btu/f, and the bottles
add about 60 x 4.2 = 252, for a total thermal mass of about 2000 btu/f, so
a cloudy week might look like this:
day drum temp drum loss spa temp spa loss total loss
0 136 f 7200 btu 105 f 6624 btu 13824 btu
1 129.1 6703 105 6624 13327
2 122.4 6223 105 6624 12847
3 116.0 5758 105 6624 12382
4 109.8 5314 105 6624 11938
5 103.8 4884 103.8 6509 11393
6 98.1 4471 98.1 5962 10433...
on a typical day, we might supply 6624/24 = 276 btu/hr to the spa by pumping
276/20 = 13.8 pounds per hour or 0.029 gpm from a drum up into the spa, with
a 12 v 4 gpm pump running less than 1% of the time, while letting some spa
water flow back into the drum. on a sunny day, we might pump some pondwater
into the spa, and let it overflow on the ground, using no chemicals.
nick
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