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an enclosed hot tub with an air-water heat exchanger
30 mar 1999
the worst-case month for solar heating in philadelphia is january, with
a long-term average daily air temperature of 30.4 f and an average daily
max of 37.9. the average amount of solar heat that falls on a horizontal
surface is 580 btu/ft^2 per day, and 1,000 falls on a south wall. 
we might heat an 8' diameter x 4' tall circular pool set on 1" styrofoam
board on the ground inside a docecahedron made with 25 $3 10'x2" pvc pipes.
cut 55 50 cent 6" stiff galvanized straps with holes in them, drill a hole
in each pipe a few inches from the end, bolt a strap inside each end with a
protruding tab, then use one more bolt to connect groups of 5 tabs (with 1
bolted through the tread of a tire filled with dirt on the ground) to make
11 hubs in a 17' o.d. x 13.9' id x 13' tall dome with 15 triangular sides.
(11' i.d. if made with 25 8' 2x3s instead of 10' pipes.)
cover it with one layer of r1 plastic film with 90% sun transmission,
using some pipes ripped in thirds with sheet metal screws for cap strips,
and hang some shadecloth or paint the northerly triangles darkish so it
gathers approximately 20'x13'x1000x0.9 = 234k btu of south sun (or more,
with a ground reflector on the south) and 5x43.3ft^2x580x0.9 = 112k btu
from overhead, totaling 346k btu (~100 kwh) on an average january day.

how much sidewall insulation does the pool need to be 104 f on an average
day, if it has a 3" styrofoam cover and the enclosure is warmer than the
pool for 6 hours? we lose about 18h(104f-30f)50ft^2/r15 = 5k btu via the
cover and about 6h(120f-34f)15x43.3ft^2/r1 = 335k btu through the gazebo
walls, if it's 120 f inside. not much left for sidewall heat loss.

let's try 2 layers of film for the 5 roof and northerly triangles, and
fill the spaces between the 5 northerly films with foam peanuts or pellets
or dry leaves to make them about r8. the gazebo conductance then becomes
5x43.3ft^2/r1 = 215 btu/h-f for the single film panels plus 108 for the
5 r2 panels plus 27 for the 5 r8 panels, a total of 350 btu/h-f, so it
loses 6h(120-34)350 = 160k btu/day, and the pool cover still loses 5k,
leaving 346k-181k-5k = 160k for the sidewall, which has about 100 ft^2
of surface, so 18h(104-30)100ft^2/rs = 160k, and we need r-value rs = 1
or greater. let's wrap the tub with 4 layers of 1/2" styrofoam to make
an r10 sidewall.

with 2,500 gallons of water and 14 btu/h-f, rc = 20kbtu/f/14btu/h-f
= 1,400 hours. after 5 cloudy 30 f days in a row, the water might be
30+(104-30)exp(-5x24/1400) = 98 f, still pretty warm. a 24' pool with
an r30 cover and r10 side holds about 200,000 pounds of water with a
conductance of 45 btu/f, so rc = 200k/45 = 4,400 hours, and it cools
to 30+(104-30)exp(-5x24/4400) = 102 f in 5 days. we could keep the
temperature more constant with a separate reservoir of hotter water.
a very well-insulated 4x4x8' tall box full of 128 f water might store
4x4x8x64(128-108) = 164k btu of useful heat and keep the 8' tub at
exactly 104 f for 8 cloudy days in a row.

on an average day, the tub needs to gain 21k btu of heat at a rate
itub = 21k/6h = 3.5k btu/h. (one manufacturer's large spa with a good
cover consumed 169 kwh per month at 102 f in 73 f air, about 20k btu/d,
when used 1.5 hours per day in 1995 in florida. same ballpark.) with
2 gpm of waterflow, ie 960 btu/h-f, we need an waterflow temperature
difference of 3500/960 = 4 f to make 3,500 btu/h flow into the tub. 

magicaire's copper $150 2'x2' shw 2347 duct heat exchanger with a fan
(or a $35 used 1984 dodge omni auto radiator with its attached 12 v fan)
might have an effective thermal conductance of about = 800 btu/h-f, so
it takes a temperature rise of 3.5k/800 = 4.3 f to make 3.5k btu/h of
air-water heatflow. the fan might pull hot air down through a duct near
the north of the gazebo peak through a heat exchanger near the ground
with hot tub water moving through the heat exchanger as needed, when a
thermostat turns on a small pump.

if the tub temperature is 104 f and the output of the heat exchanger is
108 and the average water temp in the exchanger is 106, we have something
like this equivalent dc circuit (use courier font):

                             1/350    |    1/800
                    34 f -----www-----*-----www-----106 f
                                      |     --->
                              isun    |     itub
                         |--| --> |---

where isun is 346kbtu/6h = 58k "amps," so ith = isun-itub = 54.5k btu/h
with a simpler thevenin equivalent circuit:
                              ith     t     rt
                         |--| --> |---*----www------tt 

where the parallel combination rt = 0.00087, and t with the current
source removed is tt = 84.1 "volts," so t = tt + ith x rt = 131.5 f,
a lot warmer than the 110.3 f we needed to make 3.5k of heatflow. 

we might make this gazebo thermally more efficient and more comfortable in
the summer by hanging a horizontal piece of 80% dark greenhouse shadecloth
(made from recycled soda bottles) from the 5 vertices of the upper pentagon,
with a vertical duct formed from an ew piece of film hanging from the ne and
nw vertices of the upper pentagon to let the fan pull hot air down through
the heat exchanger near the floor (below the tub water level), and let that
air flow up through another duct to be warmed again by the hot shadecloth
for the return trip. this would tend to keep the solar-warmed air near the
better-insulated gazebo panels.

if the upper 5 triangles are covered with 2 layers of r1 polycarbonate
glazing, with an average of 100 btu/h-ft^2 of overhead winter sun, the
stagnation (no heating load) temperature in the upper space would be
about 200 f more than the outdoor temperature.

the gazebo might look like this from above (use courier font):

                     ..heat . down => ef   up   .
                    . .store. duct => xa shower? .
                   .  ....... . . . . cn . . . .  .
                  .                                .
                 .                                  .  10'
                .                                    .
               .          overhead shadecloth         .
              .                 . . .                  .
             .               .         .                .
                            .    hot    .            
                  .         .           .          .
                            .    tub    .
                        .    .         .      .
              shallow           . . .  
                 pond?       .           .
to use this round tub or a conventional spa, we might unlock and lift
the (30 pound?) foamboard cover up into the roof pentagon via 3 ropes
that join another vertical rope above that runs through a pulley under
the gazebo peak and down through another at a roof corner to a
counterweight with a handle. 

in pennsylvania, the roof might supply an average of about 16 gallons
per day of rainwater for the tub to help keep it clean without chemicals,
along with a copper dishwashing sponge (thanks, raul :-) in a $75 sand
filter and a $100 ozonator from heliotrope general, (800) 552-8838.

a shallow pond to the south might attract interesting fauna and flora,
add more solar heat to the structure when frozen, treat graywater, and
cast enchanting rippling solar reflections inside the gazebo.

this might make an interesting $500 24' circular pool under a 34'
dodecahedron made with 20' pipes, with a winch to raise the cover.
greenhouse poly film comes in folded rolls up to 40' wide...

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