1994 Scoping Analysis
SolarAttic^{TM} Solar Pool Heater
From the handwritten notes of the 11/29/93 "scoping analysis"
performed by
Professor Ephraim M. Sparrow, University of Minnesota, Minneapolis, MN.
Heat Supply To Attic Air :
 Direct solar. Use 250 Btu/hr ft^{2} incident
normal to the sun's rays. The angle of incidence varies throughout a given
day and also depends on latitude and on the season. Use a factor of 0.7 to
account for the nonnormal incidence. As a best case, assume that all the
incident solar is absorbed (i.e., d s= 1). If a 1500 ft ^{2} roof
is everywhere incident by direct solar, then
250 x 0.7 x 1500 = 262,500 Btu/hr.
This would be valid for an unshaded, lowpitch roof. Perhaps
a more typical roof
would be half shadowed. For such a roof, the absorbed solar would be @
130,000
Btu/hr.
 Scattered Solar.
Use 30 Btu/hr  ft ^{2} as a typical value. For total absorption
on a 1500 ft^{2} roof
30 x 1500 = 45,000
 Total solar. Range:
175,000  307,500
Heat Losses From Attic Roof:
 Convection losses. Assume roof outside surface temperature
is 115 degrees F and that the air ambient is 85 degrees F.
 Convective heat loss equation:
Q conv = T/R,
R= 1/hA
where R is the thermal resistance without wind,
h @ 0.7 Btu/hrft^{2} ° F.
So that

(115 
 85) 

Q conv = 
____1____ 
= 31,500 Btu/hr 

(0.7) (1500) 

For moderate wind, use h @ 2 Btu/hrft^{ 2} °
F
Then,
Q_{ loss} = (115  85)/ (1/(2)(1500))
Q _{loss} = 90,000 Btu/hr
 Radiation losses. If the diffuse sky radiation is
characterized by T_{ sky}, then,
Q _{loss} = E s A
(T^{4} T_{ sky}4 )
This equation is for infrared radiation. The emissivity E is
about 0.85 for both
black and white roof surfaces;
s = 0.1712 x 10^{ 8} Btu/
hrft^{2 }° R^{4 }.
Both T and T _{sky} are in degrees Rankine, °
R = ° F+ 460. Then,
Q_{ loss} = (0.85) (0.1712 x 10^{ 8 }) (1500)
(575^{ 4}  460^{4} )
Q_{ loss} = 141,000 Btu/hr.
Heat Balance:
Max Q_{ loss} = 90,000 + 141,000
= 231,000 Btu/hr
Min Q_{ loss} = 31,500 + 141,000
= 172,500 Btu/hr
Range of losses: 172,500  231,000 Btu/hr
Range of solar absorbed: ( a s = 1)
175,000  307,500
Conclusion: There are clearly many
cases where the system will work (i.e.
provide = 90,000 Btu/hr), but the combination of a half or
more shadowed roof
and moderate wind will unable the [pool] water heating to be
accomplished.
If you want to "run the numbers," you can download the Acrobat PDF file.
