### Flow in Non-circular Ducts

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

DH = 4(p/4)D2 / (pD) = D

For an annulus of outer dia Do and inner dia Di :

DH = 4 ( (pDo2 /4) - (pDi2 /4) ) / ( p(Do + Di) ) = (Do2 - Di2) / (Do + Di) = Do - Di

For a duct of rectangular cross-section Da by Db :

DH = 4 DaDb / ( 2(Da + Db) = 2DaDb / (Da + Db)

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

DH = 4 Da2 / (4Da) = Da

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.