Flow in Non-circular Ducts

For turbulent flow in a duct of non-circular cross-section, the hydraulic mean diameter may be used in place of the pipe diameter and the formulae for circular pipes can then be applied without introducing a large error. This method of approach is entirely empirical.

The hydraulic mean diameter D_H is defined as four times the hydraulic mean radius r_H. Hydraulic mean radius is defined as the flow cross-sectional area divided by the wetted perimeter: some examples are given. For circular pipe:

D_H = 4(p/4)D² / (pD) = D

For an annulus of outer dia D_o and inner dia D_i :

D_H = 4 ( (pD_o² /4) - (pD_i² /4) ) / ( p(D_o + D_i) ) = (D_o² - D_i²) / (D_o + D_i) = D_o - D_i

For a duct of rectangular cross-section D_a by D_b :

D_H = 4 D_aD_b / ( 2(D_a + D_b) = 2D_aD_b / (D_a + D_b)

For a duct of square cross-section of size D_a :

D_H = 4 D_a² / (4D_a) = D_a

For laminar flow this method is not applicable, and exact expressions relating the pressure drop to the velocity can be obtained for ducts of certain shapes only.