[Greenbuilding] Air heater algebra

nick pine nick at early.com
Fri Oct 14 14:11:32 CDT 2011


An air heater can be modeled with 2 temperatures, eg 70 F house air and a T 
(F) C cfm air heater output, with an average Ta = (T+70)/2 air temp in the 
heater. If a 20' wide x 24' tall 480 ft^2 patch of R2 glazing transmits 80% 
of 250 Btu/h-ft^2 full sun on a 14 F average December day in Edmonton, 
Alberta and the solar energy that flows into the heater equals the heat loss 
from the glazing to the outdoors plus the house heat gain, 0.8x480x250 = 96K 
Btu/h = (Ta-14)480ft^2/R2 +(T-70)C = 120T +5040 +(T-70)C, approximately, so 
91K = 120T +(T-70)C.

With a T = 100 F outlet temp, 91K Btu/h = 120x100 + 30C, so C = 2633 cfm, 
with about 2633(100-70) = 79K Btu/h of heatflow and a 100x79K/(96K/0.8) = 
66% solar collection efficiency. Warm air rises, and one empirical chimney 
formula says C = 16.6Asqrt(HdT) for 2 A ft^2 vents with an H’ height 
difference and a dT (F) temp diff. With Ta = (100+70)/2 = 85 F and H = 24’, 
C = 2633 cfm = 16.6Asqrt(24'(85-70)) makes A = 8.4 ft^2.

With A = 4 ft^2, C = 16.6x4sqrt(24(Ta-70)) = 325sqrt(Ta-70), and (T-70)C = 
230(T-70)^1.5 makes 91K = 120T +230(T-70)^1.5, ie T = 70+(396-0.522T)^(2/3). 
Plugging in T = 120 F on the right makes T = 117.9 on the right. Repeating 
makes T = 118.0, then 118.0, with Ta = 94 and C = 325sqrt(94-70) = 1592 cfm 
of airflow and about 1592(118-70) = 76.4K Btu/h of heatflow and a 
100x76.4K/(96K/0.8) = 64% solar collection efficiency.

Nick 





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