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A Solar Hot Tub

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Posted by nicksanspam on July 13, 2007, 3:03 pm
Dan writes:

But a hot tub can't store any heat if its temperature never changes...

Perhaps you mean the heater initially turns off after the tub reaches 104,
then cycles as needed to maintain 104 F until 8 PM.

That would seem to make sense for people who like to jump into the tub
at any time, without waiting for it to reheat.

That kind of solar heating doesn't seem to save much energy.
Consider 4 scenarios:

1. "100% solar heat" with a secondary storage tank.
2. solar heat for 6 hours per day, with a constant 104 F tub temp
3. solar heat for 6 hours per day, with electric reheat on a sunny day
4. electric reheat on a cloudy day.

   Eon        Eoff          Esun          Elec          Delay
   (Btu/day)  (Btu/day)     (kWh/day)     (kWh/day)     (minutes)

1  5409.405   7496.244      3.782429      0             0
2  5409.405   7496.244      .4323834      3.350046      0
3  5390.759   7496.244      1.364025      2.41294       8.636646
4  5358.812   7496.244      0             3.767602      62.82312

In scenario 1 the sun provides all the heat for up to 5 cloudy days
in a row. No electricity is used for heat.

In scenario 2 the sun only provides 13% of the heat, keeping the tub
104 F with no reheat delay.

In scenario 3 the sun provides 56% of the heat, like a very good
direct gain (aka direct loss :-) house, with a small delay.

In scenario 4 the sun provides no heat, with a very large
inconvenient reheat delay.

The biggest heat loss (Eoff) happens with the cover off,
and scenarios 2-4 don't help at all with that.

If cloudy days are like coin flips, averaging scenarios 3 and 4, this
"solar heated" tub would only be 18% solar heated on an average day, using
3.1 kWh/day of electricity with an average wait time of 36 minutes.
Extremely lame, IMO.

I disagree. A 170 F tank with some evacuated tubes could 100% solar heat
this tub for 5 cloudy days with 5x3.76x3412/(170-110) = 1069 pounds of water
cooling from 170 to 110 F. That's 17.2 ft^3, eg a 2' tall x 3.3' diameter
well-insulated unpressurized tank, eg a single piece of EPDM rubber inside
a 10'x2' piece of aluminum coil stock roofing material inside 6" fiberglass
insulation inside a 4'x4'x2' tall foil-faced 2" polyiso board box.

Au contraire. If cloudy days are like coin flips, you'll have a hot tub
100(1-2^-5) = 97% of the time, without using the electric heater.


20 PI=4*ATN(1)
30 D=5'tub diameter (feet)
40 H=4'tub depth (feet)
50 ALID=PI*(D/2)^2'lid area (ft^2)
60 ATUB=PI*D*H-ALID'tub area, without lid (ft^2)
70 RV'R-value of tub and lid (ft^2-F-h/Btu)
80 PRINT"    Eon        Eoff          Esun          Elec          Delay"
90 PRINT"    (Btu/day)  (Btu/day)     (kWh/day)     (kWh/day)     (minutes)
110 '1. 100% solar heat
120 EON"*(104-45.3)*(ALID+ATUB)/RV'lid-on energy (Btu/22-hour day)
130 PW=EXP(17.863-9621/(460+104))'tub vapor pressure ("Hg)
140 PA).921/(1+.62198/.0087)'air vapor pressure ("Hg)
150 EOFF=2*((104-45.3)*ATUB/RV+100*ALID*(PW-PA))'lid-off energy (Btu/day)
160 SKWH1=(EON+EOFF)/3412'solar heat (kWh/day)
180 '2. solar heat for 6 hours per day
190 ESUN=6*(104-45.3)*(ALID+ATUB)/RV/3412'solar energy (kWh/day)
210 '3. solar heat with delay, sunny day
220 G=(ALID+ATUB)/RV'tub conductance (Btu/h-F)
230 C=ALID*H*62.33'thermal capacitance (Btu/F)
240 RC=C/G'time constant (hours)
250 T6E.3+(104-45.3)*EXP(-3/RC)'tub temp at 6 PM (F)
260 REHEAT=(104-T6)*C/3412'reheat energy (kWh/day)
270 PHEAT=1.5'heater power (kW)
280 DELAY`*REHEAT/PHEAT'reheat delay (minutes)
290 T9E.3+(104-45.3)*EXP(-13/RC)'tub temp at 9 AM (F)
300 EON=(104-T9)*C+6*(104-45.3)*G+REHEAT*3412
310 ESUN=((104-T9)*C+6*(104-45.3)*G)/3412
340 '4. solar heat with delay, cloudy day
350 T6E.3+(104-45.3)*EXP(-22/RC)'tub temp at 6 PM (F)
360 REHEAT=(104-T6)*C/3412'reheat energy (kWh/day)
370 DELAY`*REHEAT/PHEAT'reheat delay (minutes)
380 ESUN=0

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