Concept of Friction Factor in a pipe flow

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  • The friction factor in the case of a pipe flow was already mentioned in lecture 26. 
  • We will elaborate further on friction factor or friction coefficient in this section. 
  • Skin friction coefficient for a fully developed flow through a closed duct is defined as
(35.1)

where, is the average velocity of flow given by and are the volume flow rate through the duct and the cross-sectional area of the duct respectively.

From a force balance of a typical fluid element (Fig. 35.1) in course of its flow through a duct of constant cross-sectional area, we can write
(35.2)
FIG 35.1 Force Balance of a fluid element in the course of flow through a duct

where,  is the shear stress at the wall and  is the piezometric pressure drop over a length of and are respectively the cross-sectional area and wetted perimeter of the duct.
Substituting the expression (35.2) in Eq. (35.1), we have,
(35.3)
where,  and is known as the hydraulic diameter .

In case of a circular pipe, Dh=D, the diameter of the pipe. The coefficient Cdefined by Eqs (35.1) or (35.3) is known as Fanning's friction factor .
  • To do away with the factor 1/4 in the Eq. (35.3), Darcy defined a friction factor f (Darcy's friction factor) as
(35.4)
  • Comparison of Eqs (35.3) and (35.4) gives . Equation (35.4) can be written for a pipe flow as
(35.5)
  • Equation (35.5) is written in a different fashion for its use in the solution of pipe flow problems in practice as
(35.6a)
or in terms of head loss (energy loss per unit weight)
(35.6b)
where, hrepresents the loss of head due to friction over the length L of the pipe.
  • Equation (35.6b) is frequently used in practice to determine hf
  • In order to evaluate hf, we require to know the value of f. The value of f can be determined from Moody's Chart.