Steady-state flows in straight pipes regular head loss

Steady-state flows in straight pipes regular head loss 

Consider a straight pipe of length L, in which a fluid of density p and kinematic viscosity v flows. The surface finish of the pipe is characterized by the roughness size, denoted by e (its dimension is a length). In the most generic case, the cross-sectional geometiy of the pipe is arbitrary (rectangular, circular, etc.). A size representative of its diameter is defined by the hydraulic diameter Dh, which is based on the ratio of the cross-section s area, S, to its perimeter P  

The notion of a hydrauhc diameter originates from the momentum theorem (Chapter 2, section 2.5). The factor 4 in [4.4] equates the hydraulic diameter with the physical diameter for a pipe of circular or square cross-section. For a rectangular pipe whose sides have lengths a and b, we have Dh = lab a+b). 

Dimensional analysis (described in Chapter 3) showed that the flow velocity U in the pipe is related to the regular head loss, AP/L, via a functional involving the following three dimensionless numbers  

The velocity U is the area-averaged streamwise velocity in the pipe. We therefore write  

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