# Simplified Solar Heating Model

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1 Simplified Solar Heating Model For SolarAttic PCS1 Pool Heater Prepared from notes of the 1994 Scoping Analysis as performed by Professor Ephraim M. Sparrow, University of Minnesota, Minneapolis, MN. Heat Supply To Attic Air: 1. Direct solar. Use 250 Btu/hr ft2 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 non-normal 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, low-pitch roof. Perhaps a more typical roof would be half shadowed. For such a roof, the absorbed solar would be 130,000 Btu/hr. 2. 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 Btu/hr 3. Total Range of solar absorpton. 175, ,500 Btu/hr Simplified Solar Model & Notes Page 1 of 6

2 Heat Losses From Attic Roof: 1. Convection losses. Assume roof outside surface temperature is 115o F and that the air ambient is 85 o F. Convective heat loss equation: Q conv = T/R, R= 1/hA where R is the thermal resistance without wind, h 0.7 Btu/hr-ft2 - o F. So that Q conv = (115-85) = 31,500 Btu/hr 1 (0.7)(1500) For moderate wind, use h 2 Btu/hr - ft 2 - o F Then, Q loss = (115-85) 1 (0.7)(1500) Q loss = 90,000 Btu/hr 2. Radiation losses. If the diffuse sky radiation is characterized by T sky, then, Q loss = EσA (T 4 -T 4 sky ) This equation is for infrared radiation. The emissivity E is about 0.85 for both black and white roof surfaces; σ = x 10-8 Btu/ hr ft 2 o R 4. Both T and T sky are in degrees Rankine, o R = o F Then, Q loss = (0.85)( x 10-8 ) (1500)( ) Simplified Solar Model & Notes Page 2 of 6

3 Q = 141,000 Btu/hr. loss Heat Balance: Max Q = 90, ,000 loss = 231,000 Btu/hr Min Q = 31, ,000 loss = 172,500 Btu/hr Range of losses: 172, ,000 Btu/hr Range of solar absorbed: (α s = 1) 175, ,500 Conclusion: There are clearly many cases where the PCS1 system will work (i.e. provide = or > 90,000 Btu/hr for heating a pool.)... Comment: On an outside sunny day with 85 F temperatures, the roof s surface temperature can easily reach F [At least on my own brown asphalt roof in Elk River, MN (EGP)]. However, the PCS1 will function as an internal heat sink drawing solar energy inwardly. Also, thermal air streams on the roof s surface could easily be drawn inwardly through the passive vents as heat energy travels from hot to cold. The interior of the attic is no longer building resistance to thermal conductivity as it would be without the PCS1 working to lower its internal temperature. Qconv Simplified Solar Model & Notes Page 3 of 6

4 Qconv = BTUs/Hour moderate wind C.2. Radiation Losses QLoss (.85 ). ( ). ( 1947 ). ( ) QLoss = BTUs per Hour D. Heat Balance Max QLoss = 116, ,855 = 299,675 BTUs/Hr Min QLoss = 40, ,855 = 223,742 BTUs/Hr Range of Losses : 223, ,675 BTUs/Hr Range of Solar Absorbed: 228, ,135 BTUs/Hour VI. Utility Co. Best Case Scenario: Max Absorption & Minimum Losses Maximum absorption = 399,135 Minimum loss = 223,742 Simplified Solar Model & Notes Page 4 of 6

5 399, ,742 = 175,393 BTUs per Hour Available to PCS1 VII. Utility Co. Worse Case Scenario: Min Absorption & Maximum Losses Minimum absorption = 228,772 Maximum loss = 299,675 Losses equal or exceed gain (Utility Co. Model) BUT, this obviously doesn t apply to hot attics! AND, the PCS1 install criteria calls for a hot attic. I.E. Attic Temp 100 F on 70 F Sunny Days. See #23 pg. 8 of Common Questions in Ref. section of Technical Manual. See specifications sheet for PCS1. VIII. Utility Co. Pool Requirements Degrees Btus Req Simplified Solar Model & Notes Page 5 of 6

6 IX. Conclusion The largest known daily temperature increase in current PCS1 installations is 8 F in a single day. This is the Orange Park, Florida installation. This pool owner s data would suggest, that, under his particular circumstances, the PCS1 is capable of operating around its sensible heat nominal design of 60,000 Btus per hour. Note: This is a large 20,000 gallon pool. This scoping analysis model indicates that the PCS1 could easily raise the pool from 8-10 F under optimum solar conditions. This is in direct conflict with the Utility Company s math model that suggests the PCS1 could not raise the pool 1.8 F. The main problem I see with this traditional model (scoping analysis) is the QLoss calculations which somehow need to reflect the presence of a much cooler attic with the PCS1 in operation. This simplified solar math model demonstrates that 175,393 Btus per hour could be available for a system that only seeks 20,000-60,000 Btus per hour to satisfy pool owners. Simplified Solar Model & Notes Page 6 of 6

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