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Expression for loss of head due to friction in pipes or Darcy –Weisbach Equation.

**Darcy-Weibach Equation**

**Expression for loss of head due to
friction in pipes or Darcy –Weisbach Equation.**

Consider a uniform
horizontal pipe, having steady flow as shown figure. Let 1 -1 and 2-2 is two
sections of pipe.

Let P_{1} =
pressure intensity at section 1-1. Let P_{2} = Velocity of flow at
section 1-1.

L = length of the pipe
between the section 1-1 and 2-2 d = diameter off pipe.

f^{1} =
Frictional resistance per unit wetted area per unit velocity. h_{f} =
loss of head due to friction.

And P_{2},V_{2} = are the values of
pressure intensity and velocity at section 2-2.

Applying Bernoulli’s - 1 equation &2-2 between section 1.1,1.2(previous)

Total head 1-1 = total
head at 2-2 + loss of head due to friction between 1-1&2-2 (P_{1}/?g)
_{1} ^{2}+/2g)(V+Z_{1} = (P_{2}/?g)_{2}
^{2} / +2g) +(VZ_{2}+h_{f} ------------(1)

but Z_{1} = Z_{1} [ pipe is
horizontal ]

V_{1}=
V_{2} [ diameter of pipe is same at 1-1 & 2-2]

(1)
becomes,

(P_{1}/ ?g)_{2}/?g)+h=_{f}(P
h_{f} = (P_{1}/ ?g)-(P_{2}/?g)

frictional resistance = frictional
resistance per unit wetted area per unit velocity X wetted area X velocity ^{2}.

F = f^{1} x ?d _{l}
x V^{2} [ Wetted area = ?d x L, and Velocity V = V_{1} = V_{2}]
F_{1} = f^{1}xPxLxV^{2} ----------- (2). [?d = wetted
perimeter = p]

The forces acting on the fluid between section 1-1
and 2-2 are,

1) Pressure
force at section 1-1 = P_{1}X A

2) Pressure
force at section 2-2 = P_{2} X A

3). Frictional
force F_{1}

Resolving all forces in the horizontal direction.,

P_{1} A –P_{2}A –F_{1} = 0

(P_{1}-P_{2})A
= F_{1 }= f^{1}xPxLxV^{2}

(P_{1}-P_{2})
= (f^{1}xPxLxV^{2} / A ).

But from (1) we get

P_{1} –P_{2}
= ?ghh_{f}

Equating the values of
(P1 –P2) we get

?gh_{f} = h(f^{1}xPxLxV^{2} / A ).

hf = (f^{1} /
?g) X (P/A)
X LX V^{2}

(P/A) = (?d / (?d^{2}/4))
= (4/d)

Hence, h_{f} =
( f^{1} / ?g) x
(. 4/d) x
LxV^{2}

h_{f} = 4 fLV ^{2}
/ 2gd

This equation is known
as Darcy –Weisbach equation. This equation is commonly used to find loss of
head due to friction in pipes.

**Problem**

**Water flows through a
pipe AB 1.2m diameter at 3 m/s and then passes through a pipe BC 1.5 m diameter
at C, the pipe branches. Branch CD is 0.8m in diameter and carries one third of
the flow in AB. The flow velocity in branch CE is 2.5 m/s. Find the volume rate
of flow in AB, the velocity in BC, the velocity in CD and the diameter of CE.**

Tags : Civil - Mechanics Of Fluids - Flow Through Pipes

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