FLUID MECHANICS : Flow through pipes Notes

By Deepanshu Rastogi|Updated : April 1st, 2021



Flow-through Pipes


Major Loss: It is calculated by Darcy Weisbach formulas

Loss of head due to friction

 14-Head-losses (2)


L = Length of pipe,

v = Mean velocity of flow

d = Diameter of pipe,

f = Coefficient of friction

14-Head-losses (3) friction factor

14-Head-losses (4)

For turbulent flow, coefficient of friction

14-Head-losses (5)

Chezy’s Formula: In fluid dynamics, Chezy’s formula describes the mean flow velocity of steady, turbulent open channel flow.

14-Head-losses (6)v= c √mi, c= Chezy's Constant = √(8g/f)

i = Loss of head per unit length of pipe

 14-Head-losses (8) (hydraulic slope tan θ)

m = Hydraulic mean depth

   14-Head-losses (9)

Relation between Coefficient of Friction and Shear Stress

14-Head-losses (10)

We get 14-Head-losses (11)


f = Coefficient of friction

τ0 = Shear stress

Minor Loss:

The another type of head loss in minor loss is induced due to following reasons

Loss due to Sudden Enlargement

Head loss 14-Head-losses (12)

Loss due to Sudden Contraction

Head loss, hL = 0.5 v22/ 2g 

Remember v2 is velocity at point which lies in contracted section.

Loss of Head at Entrance to Pipe

Head loss, 14-Head-losses (14)

Loss at Exit from Pipe

Head loss, 14-Head-losses (15)


Note: In case 1 and 2, flow occurs between pipe to pipe, while in case 3 and 4, flow occurs between tank and pipe. We are taking entry or exit w.r.t. pipe. So, be careful.

Combination of Pipes: Pipes may be connected in series, parallel or in both. Let see their combinations.

Pipe in Series: As pipes are in series, the discharge through each pipe will be same.

14-Head-losses (16)

Q = A1v1 = A2v2 = A3v3

Total loss of head = Major loss + Minor loss

14-Head-losses (17)

Major loss = Head loss

due to friction in each pipe

14-Head-losses (18)

While, minor loss = Entrance loss + Expansion loss + Contraction loss + Exit loss

14-Head-losses (19)

If minor loss are neglected then,

14-Head-losses (20)

14-Head-losses (21)

Pipes in Parallel: In this discharge in main pipe is equal to sum of discharge in each of parallel pipes.

14-Head-losses (22)

Hence, Q = Q1 + Q2

Loss of head in each parallel pipe is same

14-Head-losses (23)

where, 14-Head-losses (24) and 14-Head-losses (25) are head loss at 1 and 2 respectively.

Equivalent Pipe: A compound pipe which consists of several pipes of different lengths and diameters to be replaced by a pipe having uniform diameter and the same length as that of compound pipe is called as equivalent pipe.

14-Head-losses (26)

(where, L = L1 + L2 + L3)

If f = f1 = f2 = f3



Hydraulic Gradient Line (HGL) and Total Energy Line (TEL)

14-Head-losses (28)

HGL → It joins piezometric head (p/ρg + z) at various points.

TEL → It joins total energy head at various points:

{(p/ρg + z) + v2/2g}

Note: HGL is always parallel but lower than TEL.

Power Transmission through Pipe (P)

14-Head-losses (32)

14-Head-losses (33)

Power delivered by a given pipe line is maximum when the flow is such that one third of static head is consumed in pipe friction. Thus, efficiency is limited to only 66.66%

Maximum efficiency, 14-Head-losses (34)

Water Hammer: When a liquid is flowing through a long pipe fitted with a vale at the end of the pipe and the valve is closed suddenly a pressure wave of high intensity is produced behind the valve. This pressure wave of high intensity is having the effect of hammering action on the walls of the pipe. This phenomenon is known as water hammer.

Intensity of pressure rise due to water hammer,

14-Head-losses (35)

When valve is closed gradually when valve closed suddenly with rigid pipe.

14-Head-losses (36)

When valve closed suddenly with plastic pipe

14-Head-losses (37)

If the time required to close the valve

14-Head-losses (38) Valve closure is said to be gradual.

14-Head-losses (39) The valve closure is said to be sudden.


L = Length of pipe

D = Diameter of pipe

C = Velocity of pressure wave produced due to water hammer 14-Head-losses (40)

v = Velocity of flow

K = Bulk modulus of water

E = Modulus of elasticity for pipe material.

t = Time required to choose the valve.

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