The natural draft force will be balanced to the major and minor loss in ducts , inlets and outlets. The major and minor loss in the system can be expressed as
dp = λ (l / d h ) ( ρ r v 2 / 2) + Σξ 1/2 ρ r v 2 (3)
dp = pressure loss (Pa, N/m 2 , lb f /ft 2 )
λ = Darcy-Weisbach friction coefficient
l = length of duct or pipe (m, ft)
d h = hydraulic diameter (m, ft)
Σ ξ = minor loss coefficient (summarized)
Equation (1) and (3) can be combined to express the air velocity through the duct
v = [(2 g ( ρ o - ρ r ) h ) / ( λ l ρ r / d h + Σ ξ ρ r )] 1/2 (4)
Equation (4) can also be modified to express the air flow volume through the duct
q = π d h 2 /4 [(2 g ( ρ o - ρ r ) h ) / ( λ l ρ r / d h + Σ ξ ρ r )] 1/2 (5)
q = air volume (m 3 /s)
The calculator below can be used to calculate the air flow volume and velocity in a duct similar to the drawing above. The friction coefficient used is 0.019 which is appropriate for normal galvanized steel ducts.
Calculate the air flow caused by natural draft in a normal family house with two floors. The height of the hot air column from ground floor to outlet air duct above roof is approximately 8 m . The outside temperature is -10 o C and the inside temperature is 20 o C .
A duct of diameter 0.2 m goes from 1. floor to the outlet above the roof. The length of the duct is 3.5 m . Air leakages through the building are neglected. The minor coefficients are summarized to 1.
The density of the outside air can be calculated as
ρ o = (1.293 kg/m 3 ) (273 K) / ((273 K) + (-10 o C))
= 1.342 kg/m 3
The density of the inside air can be calculated as
ρ r = (1.293 kg/m 3 ) ( 273 K) / ((273 K) + (20 o C))
= 1.205 kg/m 3
The velocity through the duct can be calculated as
v = [(2 (9.81 m/s 2 ) ((1.342 kg/m 3 ) - (1.205 kg/m 3 )) (8 m)) / ( 0.019 (3.5m)(1.205 kg/m 3 )/(0.2 m) + 1 (1.205 kg/m 3 ) )] 1/2
= 3.7 m/s
The air flow can be calculated as
q = (3.7 m/s) 3.14 (0.2 m) 2 / 4
= 0.12 m 3 /s
that these equations can be used for dry air, not for mass flow and energy loss calculations where air humidity may have vast effects.
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