a frugal solar hot tub SOLAR.KnowledgePublications.com

a frugal solar hot tub
7 nov 2000
  the most serious mistake was making the outer container of the receiver
  of plywood. we thought that the plywood would be sufficiently insulated
  from the copper panel which was the receiver proper, that it would not
  get too hot. the copper panel was separated from the plywood by 4" of
  fiberglass insulation. nevertheless, the plywood caught fire and the unit
  was completely destroyed. we suppose this is a success, of sorts...

  from "a solar collector with no convection losses," (a downward-facing 
        receiver over a 4:1 concentrating parabolic mirror) written by
        h. hinterberger and j. o'meara of fermilab, ca 1976

a 6' square x 2' deep hot tub surrounded by us r15 insulation (obfrug: a
$20 plywood box with a $50 10'x14' folded epdm rubber liner and 3" ($75)
of latex-painted styrofoam screwed to the outside, underneath, and hewn
into a rigid cover) has 120 ft^2 of surface and a heat conductance of
120ft^2/r15 = 8 btu/h-f. at 104 f on a 30 f day, with the cover on,
it needs 24h(104f-30f)8btu/h-f = 14.2k btu/day (4.2 kwh.)

nrel says a square foot of 1-axis ew concentrator can collect 729 btu on
an average january day in phila. with a layer of glazing with 90% solar
transmission and an linear parabolic 90% up-reflector (nielsen's mylar,
$0.09/ft^2 in 4' wide rolls from http://www.snomo.com/mylar.html) draped 
from the upper north to the lower south wall, we need 14.2k/(0.9x0.9x729)
= 24 ft^2 of collection area, min, eg a 64 ft^2 side of an 8' cube with
a 2'x4'x8' water tank overhead that collects 0.9x0.9x8'x8'x729 = 37.8k
btu/day. the reflector might bounce sun onto the tank bottom, a 4' wide
x 8' long piece of dark-painted galvanized metal under the rubber liner,
with foil-faced foamboard extending below that around the perimeter. 

with a tank conductance 112ft^2/r15 = 7.47 btu/h-f, and 37.8k-14.2k btu
= 24h(t-30)7.47, its average day temp ta = 30+23.6k/(24x7.47) = 161.6 f.

the tank contains 2x4x8x64 = 4096 pounds of water, ie 4096 btu/f of
heat capacitance. at temp t on a cloudy 30 f day, it loses heat to
the outdoors at a rate of (t-30)7.47 btu/h and supplies 592 btu/h for
the tub, so t decreases at a rate dt/dt = -((t-30)7.47+592)/4096, which
is a first-order differential equation of the form dt/dt+ct = d, with
c = 0.0018229 and d = -0.089844, and solution t = d/c+(ta-d/c)exp(-ct)
= -49.3+(161.6+49.3)exp(-c120). after 120 hours, t = 120 f, and the tub
needs 592 btu/h of heat, ie an average 592/(120f-104f) = 37 lb/h (0.08
gpm) of 120 f water flowing through it (see appendage.)

we might add 24 gallons of fresh rainwater per day through an efficient
counterflow heat exchanger, eg 100' of 1/2" tubing inside 100' of hose.
ntu = au/cmin = pi(1/2"/12)100'x30btu/h-f-ft^2/8btu/h-f = 49, so e = 49/50
= 0.98, and heating 1 gallon/hour of 34 f rainwater adds 0.02x8(104-34)
= 12 btu/h to the tub loss. is 24 gallons per day enough water exchange
to avoid using any chemicals?

nick

10 dh=2'hot tub depth (feet)
20 wh=6'hot tub width (feet)
30 lh=6'hot tub length (feet)
40 ah=2*(dh*wh+dh*lh+wh*lh)'tub surface (ft^2)
50 rvh=15'us r-value of tub surface
60 gh=ah/rvh'thermal conductance of tub (btu/h-f)
70 eh=(104-30)*gh'hourly tub heating requirement (btu/h)
80 d=2'tank depth (feet)
90 w=4'tank width (feet)
100 l=8'tank length (feet)
110 ct=d*w*l*64'tank heat capacitance (btu/f)
120 vt=ct/8'tank volume (gallons)
130 at=2*(d*w+d*l+w*l)'tank surface
140 rvt=15'us r-value of tank surface
150 gt=at/rvt'thermal conductance of tank (btu/h-f)
160 hr=8'reflector height (feet)
170 dr=8'reflector depth (feet)
180 f=hr^2/(4*dr)'focal length (feet)
190 as=hr*l'solar aperture (ft^2)
200 es=.81*729*as'average solar gain (btu/day)
210 et=24*eh'daily tub heating requirement (btu/day)
220 ta=30+(es-et)/(24*gt)'average day tank temp (f)
230 t=ta
240 print f,vt,t,es
250 for h = 1 to 120'consecutive cloudy hours
260 t=t-((t-30)*gt+eh)/ct
270 if h mod 24 = 0 then print h,t
280 next
290 d=(30*gt-eh)/ct
300 c=gt/ct
310 print d/c+(ta-d/c)*exp(-c*120)'solution to dt/dt+ct = d

focal   tank          average
length  volume        day temp      solar gain
(feet)  (gallons)     (f)           (btu/day)

2       512           161.6036      37791.36

consecutive           tank temp
cloudy hours          (f)

24                    152.568
48                    143.9196
72                    135.6416
96                    127.7184
120                   120.1346 (by hourly simulation)

                      120.1686 (by differential equation)