We were discussing the concept of laminar
and turbulent flow, Reynolds
experiment and also frictional loss in pipes, in the subject of fluid
mechanics, in our recent posts.
Above equation of loss of head due to friction i.e. hf shows that there will be loss of head due to friction or intensity of pressure will be dropped in the direction of flow.
F1 = f ' x ПdL x V2
Where,
Let us write here the equation of equilibrium of forces
Now we will go ahead to find out the derivation of
expression for loss of head due to friction in pipes, in the subject of fluid
mechanics, with the help of this post.
Expression for loss of head due to friction in pipes
As we are quite aware that when liquid flows through
a pipe, velocity of the liquid layer adjacent to the pipe wall will be zero.
Velocity of liquid will be increasing from the pipe wall and therefore there
will be produced velocity gradient and shear stress in the liquid due to
viscosity of liquid.
This viscous action will cause the loss of energy
which will be termed as frictional loss or loss of head due to friction.
Now we will determine the expression for loss of head due to friction in pipes
Let us consider that fluid is flowing through a
uniform horizontal pipe with steady flow as displayed here in following figure.
Now we will assume two sections of pipe i.e. section
1-1 and section 2-2.
Let us consider the following terms to derive the
required expression of loss of head due to friction in pipe.
P1 = Pressure intensity at section 1-1
V1 = Velocity of flow at section 1-1
P2 = Pressure intensity at section 2-2
V2 = Velocity of flow at section 2-2
L = Length of pipe between section 1-1 and section
2-2
f ' = Frictional resistance per unit wetted area per
unit velocity
hf = Loss of head due to friction
A = Area of the pipe
d = Diameter of the pipe
Now we will apply the Bernoulli’s equations between section 1-1 and section 2-2.
Now we will apply the Bernoulli’s equations between section 1-1 and section 2-2.
Because,
Pipe is horizontal and hence, Z1 = Z2
Diameter of uniform pipe is same at both sections
and hence, V1 = V2
Above equation of loss of head due to friction i.e. hf shows that there will be loss of head due to friction or intensity of pressure will be dropped in the direction of flow.
Frictional resistance = Frictional resistance per
unit wetted area per unit velocity x wetted area x velocity 2
F1 = f ' x ПdL x V2
F1 = f ' x P x L x V2
Where,
P = Perimeter = Пd
Now we will consider the forces acting on the fluid between section 1-1 and section 2-2
Pressure force at section 1-1 = P1 x A
Pressure force at section 2-2 = P2 x A
Let us write here the equation of equilibrium of forces
Where,
P/A = Пd / (Пd2/4)
P/A = 4/d
And
f '/ρg = f/2, where f will be called as co-efficient
of friction
Above equation will be called as Darcy-Weisbach equation and commonly used to determine the loss of head due to friction in pipes.
There is one more expression of loss of head due to friction in pipes and this expression could be written as mentioned here.
Where, f is the friction factor
Further we will go ahead to derive the expression for coefficient of friction in terms of shear stress, in the subject of fluid mechanics, with the help of our next post.
Do you have any suggestions? Please write in comment box.
Above equation will be called as Darcy-Weisbach equation and commonly used to determine the loss of head due to friction in pipes.
There is one more expression of loss of head due to friction in pipes and this expression could be written as mentioned here.
Where, f is the friction factor
Further we will go ahead to derive the expression for coefficient of friction in terms of shear stress, in the subject of fluid mechanics, with the help of our next post.
Do you have any suggestions? Please write in comment box.
Reference:
Fluid mechanics, By R. K. Bansal
Image courtesy: Google
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