written 6.6 years ago by | modified 5.5 years ago by |
When a fluid flows steadily through a pipe of constant diameter, the average velocity at each cross section remains the same. This is necessary from the condition of continuity since the velocity V is given by,V = Q/A. The static pressure P drops along the direction of flow because the stagnation pressure drops due to loss of energy in over coming friction as the flow occurs.
Let, $P_1$ = intensity of pr. at section 1
$P_2$= intensity of pr. at section 2
L = length of the pipe, between section 1 and 2.
D = Diameter of the pipe
Cd = co-efficient of drag.
f = co-efficient of friction (whose value depends on type of flow, material of pipe and surface of pipe)
$h_f$ = loss of head due to friction.
Propelling pressure force on the flowing fluid along the flow = $ (P1 –P2)[πD]2/4 $
Frictional resistance force due to shearing at the pipe wall = $Cd. 1/2 \rho V^2. \pi D L$
Under equilibrium condition,
$\text{Propelling force = frictional resistance force}$
$(P1 - P2) \frac{[ \pi D ] 2}{4} = Cd. 1/2 \rho V^2. \pi D L$
$ \frac{(P1 - P2)1}{\rho g} = \frac{1}{(D.2g)}.Cd.LV^2$
Noting $(P1 –P2)1/ρg$ is the head loss due to friction, hf and d C equal the coefficient of friction.
$h_f = \frac{1}{(D.2g)} .4.LV^2$
This is known as Darcy-Weisbach equation and it holds good for all type of flows provided a proper value of f is chosen.