i. The quantity of air passing through both ducts is same and

ii. The velocity of air flowing through both ducts is same

The friction factor remains same for both ducts

Subject:- Refrigeration and Air Conditioning

Topic:- Duct Design, Controls & Applications

Difficulty:- Medium

Where f = friction factor

L= length of duct in meter

v = velocity of air in duct in m/sec

D= diameter of duct in meter

Discharge through duct is given by, Q= A.v …….(2)

Where A = cross sectional area of duct

Expression for the equivalent diameter of a circular duct corresponding to a rectangular duct when the quantity of air passing through both ducts is same

Substituting equation (2) in equation (1),

$\frac{h_f}{L}=\frac{(f(Q/A)^2)}{2gD}$

$D^{5}=\frac{64(a,b)^{2}}{2\pi^{2}(a+b)}$

D=1.265$\frac{a^{3}b^{3}}{a+b}^{1/5}$

ii.Expression for the equivalent diameter of a circular duct corresponding to a rectangular duct when the velocity of air flowing through both ducts is same

We can write velocity from equation (1) as

$\frac{a,b}{2(a+b)}$=$\frac{\frac{\pi}{4}D^{2}}{\pi.D}$

$D=\frac{2ab}{a+b}$