Now recall the control volume momentum relation for the steady incompressible flow,

(5.4.3)

In the present case, LHS of Eq. (5.4.3) may be considered as pressure force, gravity and shear force.

(5.4.4)

Till now, no assumption is made, whether the flow is laminar or turbulent. It can be correlated to the shear stress on the wall . In a general sense, the wall shear stress can be assumed to be the function of flow parameters such as, average velocity , fluid property , geometry and quality of the pipe.

(5.4.5)

By dimensional analysis, the following functional relationsip may be obtained;

(5.4.6)

The desired expression for head loss in the pipe can be obtained by combining Eqs (5.4.4 & 5.4.6).

(5.4.7)

The dimensionless parameter f is called as Darcy friction factor and Eq. (5.4.7) is known as Darcy-Weisbach equation. This equation is valid for duct flow of any cross-section, irrespective of the fact whether the flow is laminar or turbulent. In the subsequent part of this module, it will be shown that, for duct flow of any cross-section the parameter d refers to equivalent diameter and the term vanishes for laminar flow.