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100% solar water heating?
4 sep 2001
we might solar heat water all year simply and economically by collecting
enough sun to heat water on an average day in the worst-case month and
storing heat for a cloudy week. say we heat water from 60 to 110 f for 5
10 minute 3 gpm showers per day, ie 5x10x3x8(110-60) = 60k btu/day...

nrel says january is the worst-case month for solar house heating in phila.
a south wall gets 1000 btu/ft^2 of global sun on an average 30.4 f day, and
a 1-axis ew concentrator can gather 729 btu/ft^2-day of direct sun. let's
aim an ew fixed downward-reflecting solar trough at the south horizon with
a focal line at x=8' at dawn (y^2 = 32x), with y=16' and x=8', max, so the
upper edge of the trough will only begin to shade the 8'-wide target when
the sun reaches atn(2) = 64.4 degrees elevation.

it might look like this, in courier font:

             y --- ---
           . .        
         .   .           30 f
     r8 .    .                    the reflector and tank would be 
       .     . 16'       s ->     20' tall and 16' long, big for 
      .  tt  .                    a water heater, but small for
     .       . r1   20'           a greenhouse.
     . 163 f .                
(0,0)......... --- x -->          the tank would be 4' wide and 
     | | 4'| |  3'                3' deep and 14' long, made with
 r50 |  ---  | ---                a single 10'x20' folded piece of
------------------ ---            epdm rubber surrounded by strawbales 
   |   8'  |                      or 8" fiberglass under 2" styrofoam.

if the south side is glazed with a single layer of r1 polycarbonate with 90%
solar transmission, and the parabolic north part is 90% reflective, with
aluminized mylar greased to "r7.2" 1" double-foil polyisocyanurate board
under greenhouse poly film, and the target (an 8'x16'x2" water trough made
with 2 layers of uv greenhouse poly film laid flat over a dark-colored sip)
is 90% transmissive, 900 btu/ft^2-day of global sun enters the structure,
and at least 530 of that enters the target, over say, 6 hours...
we might move 5 gpm of water through the trough with a low-head pump during
the day and let it drain back into the tank at night, and keep the tank
full with a float valve and cold water from the house, and empty it via
the same hose and an rv pump and pressure tank. people in socks might use
the structure once in a while, with the trough empty.

a round tank would lose less heat, but it wouldn't easily serve as
the reflector's foundation...


materials: 500 ft^2 1" polyiso board $150
           500 ft^2 aluminized mylar   50
           200' 2x4s                   50
           256 ft^2 polycarbonate     400
           pump                        75
           controller                  50
           rv pressure pump            75
           60 strawbales              180...

10 ta=30'ambient temp (f)
20 tw=163'average day water temp (f)
30 x=8'width of trough (feet)
40 y=16'height of trough (feet)
50 length=16'length of trough (feet)
60 enda=2/3*x*y'endwall area (ft^2)
70 a=sqr(4*x^2+y^2)
80 paralen=(a+y^2/(2*x)*log((2*x+a)/y))/2'north wall curve length (feet)
90 gref=2*enda/8+length*paralen/8+length*y/1'envelope cond. (btu/h-f)
100 gtar=length*x/1'target to indoor air conductance (btu/h-f)
110 gsun=.9*1000*length*y/6'global sun entering envelope (btu/h)
120 dsun=.9*.9*.9*729*length*y/6'direct sun entering target (btu/h)
130 tth=tw+(gsun-dsun)/gref'thevenin equivalent target temp (f)
140 qtar=(tth-ta)/(1/gtar+1/gref)'heatflow out of envelope (btu/h)
150 tref=ta+qtar/gref'indoor air temp (f)
160 qu=dsun-(tw-tref)*gtar'useful heat gain (btu/h)
170 dgain=6*qu'useful daily heat gain (btu/day)
180 print dgain
190 w=4'tank width (feet)
200 d=3'tank depth (feet)
210 l=14'tank length (feet)
220 dload=60000'useful water heating (btu/day)
230 cap=62*w*d*l'tank capacitance (btu/f)
240 astore=2*w*d+2*(w+d)*l'tank surface (ft^2)
250 gstore=astore/50'tank conductance (btu/h-f)
260 avloss=24*(tw-ta)*gstore'average thermal loss from tank (btu/day)
270 print dload+avloss'average total loss from tank (btu/day)
280 c=gstore/cap'linear term in diff. equation
290 d=-dload/(24*cap)+ta*gstore/cap'constant in diff. equation
300 t=-log((110-d/c)/(tw-d/c))/c'cloudy supply time (h)
310 print t

74615.24'average heat gain (btu/day)
74044.8'average heat loss (btu/day)

186.0581'number of cloudy hours until the tank reaches 110 f.

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