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Derive Darcys Weisbach equation for calculating loss of head due to friction in pipes.
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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.

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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.

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