Losses and Friction Factors

• Over a finite length l , the head loss                           (26.11)
  Combining Eqs (26.10) and (26.11), we get

or	(26.12)

• On the other hand, the head loss in a pipe flow is given by Darcy-Weisbach formula as

(26.13)

where "f" is Darcy friction factor . Equations (26.12) and (26.13) yield

which finally gives , where is the Reynolds number.

• So, for a fully developed laminar flow, the Darcy (or Moody) friction factor is given by

(26.14a)

Alternatively, the skin friction coefficient for Hagen-Poiseuille flow can be expressed by

With the help of Eqs (26.9b) and (26.9c), it can be written

(26.14b)

The skin friction coefficient C_f is called as Fanning's friction factor . From comparison of Eqs (26.14a) and (26.14b), it appears

• For fully developed turbulent flow, the analysis is much more complicated, and we generally depend on experimental results. Friction factor for a wide range of Reynolds number (10⁴ to 10⁸) can be obtained from a look-up chart . Friction factor, for high Reynolds number flows, is also a function of tube surface condition. However, in circular tube, flow is laminar for Re ≤ 2300 and turbulent regime starts with Re ≥ 4000.
• The surface condition of the tube is another responsible parameter in determination of friction factor.
• Friction factor in the turbulent regime is determined for different degree of surface-roughness of the pipe, where is the dimensional roughness and D^_h is usually the hydraulic diameter of the pipe .
• Friction factors for different Reynolds number and surface-roughness have been determined experimentally by various investigators and the comprehensive results are expressed through a graphical presentation which is known as Moody Chart after L.F. Moody who compiled it.
• The hydraulic diameter which is used as the characteristic length in determination of friction factor, instead of ordinary geometrical diameter, is defined as

(26.15)

where A_w is the flow area and P_w is the wetted perimeter .

• Kinetic energy correction factor , The kinetic energy associated with the fluid flowing with its profile through elemental area and the total kinetic energy passing through per unit time .
• This can be related to the kinetic energy due to average velocity(), through a correction factor, α as
  

or

• Here, for Hagen-Poiseuille flow,

(26.16)