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Wednesday, 5 September 2018

DERIVE AN EXPRESSION FOR THE LOSS OF HEAD DUE TO FRICTION IN PIPES

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. 

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. 
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. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

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