[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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