Velocity Distribution of Turbulent Flow in a Pipe
In case of turbulent flow Reynolds found that the loss in pressure head is proportional to V
n where V
is mean velocity and n varies from 1.75 to 20. Transition from laminar to turbulent flow depend upon
pipe diameter and the physical properties of the fluid also.
• The factors which affect the transition from laminar to turbulent flow are:
1. Turbulence prevailing in the incoming fluid.
2. The pressure gradient: For accelerated flow critical Reynolds number increases whereas
for retarded flow , critical Reynolds number decreases.
3. The roughness of boundary decrease the critical Reynolds number.
• Below a critical Reynolds number all disturbances are suppressed by viscous damping while above
the critical Reynolds number certain frequencies will be amplified and others damped.
• According to Darcy-Weisbach equation, the loss of head due to friction in the pipe is given by
where f = Friction factor
L = Length of pipe
V = Mean velocity of liquid
D = Diameter of pipe
• Major loss of head is due to friction.
• Minor losses of head included the following cases.
1. Loss of head due to sudden enlargement
he =
Since from the continuity equation V1A1 = V2A2
.
2. Loss due to sudden contraction hc =
where Cc = Coefficient contraction = .
3. Loss of head at the inlet (entry loss):
4. Loss of head at the outlet of pipe (exit loss)
5. Loss of head in pipe fittings
hL =
where KL
is loss coefficient.
• The line representing the sum of pressure head and datum head w.r.t. some reference line is called
hydraulic gradient line (H.G.L).
• The line representing the sum of pressure head, datum head and velocity head w.r.t. some reference
line is known as total energy line (T.E.L).
• The losses in pipe fittings are expressed is terms of an equivalent length which is a length of an
unobstructed straight pipe in which an equal loss of energy due to friction would occur for the
same discharge.
Leq = KL
• A pipe of length L1
, diameter d1 and friction factor f1
is equivalent to a pipe of length L2
, diameter
d2 and friction factor f2
, if = 
.
• The equivalent size of the pipes connected in series is given by
=
where L = L1 + L2 + L3 … = Equivalent length
d = Equivalent size
d1
, d2
, d3 … = Diameters of pipes connected
• The rate of discharge in the main pipe is equal to the sum of discharges in each of the parallel pipes.
Q = Q1 + Q2 + …
and the loss of head in each pipe is same hf1 = hf2
i.e. =
• A syphon is a long bent pipe which carries liquid from higher level to a lower level through an
intermediate high obstruction.
Water Hammer
• When the valve at the end of the pipe is suddenly closed a pressure wave of high intensity is
produced in the fluid which has the effect of hammering action on the walls of the pipe. This
phenomenon is known as water hammer.
• The magnitude of the water hammer depends upon:
1. The length of the pipe
2. The elastic properties of pipe material
3. The elastic properties of the liquid flowing
4. The speed at which valve is closed.
Thus,
pi = , if valve is closed gradually
= , if valve is closed suddenly
= , if valve is closed suddenly and pipe is elastic.
where L = Length of pipe
V = Velocity of flow
K = Bulk modulus of fluid
E = Modulus of elasticity of pipe material
D = Diameter of pipe
T = Time for closing valve.
• The valve closure is said to be gradual
If, t >
where C = velocity of pressure wave produced due to water hammer = .
Flow through Open Channels
• The flow in an open channel is characterized by the existence of a free surface and interfaces being
subjected to a constant pressure throughout its length and breadth.
• A sewer under ordinary condition of flow behaves as an open channel and a channel under pressure
when sewer runs full at times of heavy rains.
• The flow in open channels is at the expense of potential energy and the flow is caused by gravity
force provided by the sloping bottom.
• The open channels may have various shapes, like triangular, rectangular, trapezoidal circular or any
irregular.
The surface roughness in open channels varies over wide limits.
• The piezometric head in open flow is Z + y where y is depth of flow and Z is datum head where as
in pipe flow it is Z + .
• If Froude number in channel flow is
1. less than 1, the flow is said to be subcritical
2. equal to 1, the flow is known as critical
3. more than 1, the flow is called supercritical
• Manning’s formula used in open channel flow is
V =
where, V = Mean velocity in m/sec
R = Hydraulic radius in metre
S = Channel slope
n = Pugosity coefficient
• A relation between the Chezy’s C and Mannings n is
C =
• Most economical section of channel is the one which
1. Gives maximum discharge for a given cross-sectional area and bed slope.
2. Has minimum wetted perimeter.
3. Involves lesser excavation for the designed amount of discharge.
• For a trapezoidal channel to be most efficient hydraulic radius R is equal to half the depth of flow.
• In an efficient rectangular channel the flow depth is half the bed width B and hydraulic radius R is
half of flow depth.
• The optimum inclination of the sides of a channel is 60° to horizontal for a given area and flow
depth.
• A semicircular channel is theoretically the most efficient channel for getting maximum discharge for
a given cross-sectional area and bed slope.
Non-Uniform Flow
• The specific energy is the energy per unit weight of flowing fluid measured above the channel
bottom and is equal to the depth of flow plus the velocity head.
E =
• The depth of water in a channel corresponding to the minimum specific energy is known as critical
depth.
• At the critical state of flow, the specific energy in a rectangular channel is equal to 1.5 times the
depth of flow
• In case of a rectangular channel with discharge q per unit width, critical depth is
• When the flow is critical, the discharge per unit width is maximum for a given specific energy.
• A venturiflume is essentially an artificial construction in a channel which by producing a change in
velocity and depth, facilitates the measurement of flow rate. A venturiflume in its conventional
from consists of a bell mouthed entry, parallel throat and a diverging portion in the downstream.
• The coefficient of venturiflume, generally lies between 0.91 and 0.99.
• At the critical depth, both specific energy and specific force attain minimum values.
Flow Measurements
• For pipeline flow measurement, any of the following devices can be used:
1. Venturimeter
2. Nozzle meter
3. Orificemeter
4. Bend meter
• For measurement of flow in open channels is generally by means of weirs and gates.
• In large open channels and rivers the flow is estimated by dividing the flow section into a number of
smaller sections and determining the average velocity for each by means of current meter.
• In venturimeter if A1 and A2 are inlet and outlet sectional areas, h is the difference in the piezometric
head,a
Q =
=
where k is venturimeter constant for a particular meter.
• Nozzle meter is simply a construction with well rounded entrance placed in the pipeline. It is
simpler than a venturimeter.
• In its simplest and most familiar form, the orifice is a circular hole in a flat plate which is fixed
between the flanges at a joint in the pipeline with its plane at right angles to the axis of the pipe
and the hole concentric with the pipe . If plate is thicker, on the outlet side edge the hole is
champered. Measuring pressure on both sides of the gauge help in determining the flow.
• The bend meter utilizes the fact that in a curved flow pressure increases with radius and hence a
pressure difference exists between the outer and inner wall of the bend.
• A short pipe equal to the size of the orifice connected to orifice is known as mouthpiece. It usually
extends downstream by not more than 2.5 times the diameter of orifice. It may be cylindrical,
converging or diverging type.
1. For cylindrical mouthpiece, rounding the entrance greatly reduces the losses.
2. For converging and diverging type of mouthpieces coefficient of discharge depends upon the
angle of convergence/divergence.
3. For maximum discharge to pass through a diverging mouthpiece, its length should be equal to 9
times its diameter and the angle of divergence be equal to 5°.
• Weir is an obstruction in the channel which causes the liquid to rise behind it and then allows to
flow over it. By measuring the height of upstream liquid surface, the flow rate can be determined.
• The jet of liquid over the weir, so formed is known as ‘nappe’.
• If nappe springs freely leaves the upstream face, the weir is known as strap-crested weir.
• The broad crested weirs are those which support the falling nappe over their crest.
• The weirs may be rectangular, triangular, trapezoidal, etc.
• When the liquid level downstream of weir is above its crest level, the weir is said to be a
submerged weir.
• According to Francis formula, the discharge over a rectangular weir is
Q
(B – 0.1 nH)
where n = Number of contractions
B = Length of weir
• In case of a triangular notch discharge varies as H5/2
, i.e., Q
.
• Cipoletti weir is a trapezoidal weir in which the sides have a slope of 1 horizontal to 4 vertical.
Advantage of Cipoletti weir is that the decrease in discharge due to end contraction is balanced by
the discharge through the triangular portion and hence rectangular weir formula can be used to find
out discharge.
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