The topic of fluid mechanics

1. IN THE NAME OF ALLAH THE MOST
BENEFICENT, THE MOST MERCIFUL.

2. Dr. Ateeq-ur-Rauf
Fluid Mechanics-I
(CE-206)
Department of Civil Engineering,
U.E.T Peshawar (Bannu Campus)

3. Dr.Ateeq-ur-Rauf
Department of Civil Engineering U.E.T
Peshawar (Bannu Campus) 3
Lecture # 4
 Lecture # 11
Weirs & Notches
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
NOTCH:
A notch may be defined as an opening in one side of a tank or
a reservoir, like a large orifice, with the upstream liquid level
below the top edge of the opening.
Notches & Weirs
Upstream

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 The bottom edge, over which the liquid flows, is known
as sill or crest of the notch and the sheet of liquid flowing
over a notch (or a weir) is known as nappe or vein.
 A notch is, usually made of a
metallic plate and is used to
measure the discharge of liquids.
 Since the top edge of the notch above the liquid level serves
no purpose, therefore a notch may have only the bottom
edge and sides.
Sill or Crest
Nappe

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
TYPES OF NOTCHES:
1.Rectangular Notch
2.Triangular Notch
3.Trapezoidal Notch
4.Stepped Notch

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Engr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Notches molded in
Metal Plates

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
DISCHARGE OVER A RECTANGULAR NOTCH
Consider a rectangular notch in one side of tank over
which water is flowing as shown in fig;
Where;
H = Height of water above
sill of notch
b = Width or length of the
notch
Cd = Coefficient of
discharge

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Discharge through the strip,
dq = Cd x Area of strip x Theoretical velocity
= Cd x b.dh x √2gh
Now total discharge, over the whole notch, may be found out by
integrating them above equation

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
H
H

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Lecture # 11
Fluid/Flow Measurements
 Some times, the limits of integration, in the above equation, are
from H1 to H2 ( i.e the liquid level is at height of H1 above the
top of the notch and H2 above the bottom of the notch, instead
of 0 to H. Then the discharge over such a notch will be given by
the equation.

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Engr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
(Va is the velocity not yet reached to the notch)
 The additional head caused due to the placement of the notch in the
channel is given by the g
v
Ha 2
2
 , Total Head at the notch = H+ Ha

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
a
d
B
b
bn
0.1H 0.1H
H
  2
3
2
3
1
.
0
2
3
2
2
3
2
H
nH
b
g
Cd
Q
H
b
g
Cd
Q
act
n
act








 
   
 
2
3
2
3
1
.
0
2
3
2
Ha
Ha
H
Ha
H
n
b
g
Cd
Qact 







Where n = no of end
contractions (n=2)
 By considering the
effect of velocity of
approach and end
contraction
simultaneously
End Contraction or Franci’s Formula for Discharge
 As the width of the notch is kept smaller as compared to the
channel, so the streamlines will converge and the ends of the
nappe will get contracted. Experimentally it is found that this end
contracted from on side is H/10

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
TRINGULAR NOTCH
A triangular notch is also called V-notch.
Consider a triangular notch, in one side of the tank,
over which water is flowing as shown in fig

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
DERIVATION:
x x/2

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 Let us consider a narrow rectangular strip of water flowing through the
triangular notch at a height (H-h) from the apex of the triangle having
width x and depth dh.
 Therefore the Area of strip = x. dh
 Area of strip = 2(H – h) tan θ /2 .dh
We know that theoretical velocity of water through strip = (2gh)1/2
And discharge over the notch,
dq = Cd x Area of strip x Theoretical velocity
dq = Cd x 2(H – h) tan θ /2 .dh x ( 2gh)1/2
The total discharge, over the whole notch, may be found out by integrating
the above equation within the limits 0 and H

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 A trapezoidal notch is a combination of a rectangular
notch and two triangular notches.
 The discharge over such a notch will be the sum of
the discharge over the rectangular and triangular
notches.
Discharge over a Trapezoidal Notch

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Consider a trapezoidal notch ABCD as shown in
figure. For the purpose of analysis, split up the notch
into a rectangular notch BCFE and two triangular
notches ABE and DCF. The discharge over these two
triangular notches is equivalent to the discharge over
a single triangular notch of angle θ

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Two Stepped Notch
Discharge over a Stepped Notch

24. Stepped Notch
6/21/2022 24
Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Engr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Example

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
WEIRS
 Weirs are structures consisting of an obstruction
such as a dam or bulkhead placed across the open
channel with a specially shaped opening. The weir
results an increase in the water level, or head,
which is measured upstream of the structure. The
flow rate over a weir is a function of the head on
the weir.
Types
1.Rectangular weir 2. Cippoletti weir
3.Broad-crested weir 4. Narrow Crested weir
5.Sharp-crested weir 6.Ogee weir

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Consider a rectangular weir over which the water is
flowing as shown in figure -
Let,
H = Height of the water above the crest of the
weir
L = Length of the weir and
Cd = Coefficient of discharge
Discharge Over A Rectangular Weir

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 Let us consider a horizontal strip of water of
thickness at a depth from the water surface as
shown in figure.
 Area of the strip = L.dh
 We know that the theoretical velocity of water
through the strip = √2gh
 Discharge through the strip = Area of strip
x Theoretical Velocity Q= Cd. L.dh. √2gh
 The total discharge, over the weir, may be found
out by integrating the above equation within the
limits 0 and H.

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Discharge Over A Cippoletti Weir
The "Cippoletti" weir is a trapezoidal weir, having 1 horizontal to 4
vertical side slopes, as shown in figure. The purpose of the slope,
on the sides, is to obtain an increased discharge through the
triangular portions of the weir, which, otherwise would have been
decreased due to end contractions in the case of rectangular
weirs. Thus the advantage of a Cippoletti weir is that the factor of
end contraction is not required, while using the Francis' Formula.

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 Let us split up the trapezoidal weir into a rectangular weir and
a triangular notch.
Now discharge over a rectangular weir,
and discharge over a triangular notch
So total discharge,
 Since the main idea of Cippoletti was to avoid the factor of end
contraction, and as such he gave the formula for the discharge,

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Discharge over a Narrow Crested Weir
Let b is the width of the crest of the weir and
H is the height of the water about the crest of the weir
If 2b < H, the weir is called as a narrow-crested weir.
But if 2b > H, then the weir is called as broad-crested weir.
Relation for Discharge is same as that for the rectangular weir.

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Discharge over a Broad Crested Weir
▼
b
H
A B h
Where
H = Head of water on the upstream side of the weir
h = Head of water on the downstream side of the weir.
V = velocity of the water on the downstream side.
Cd = Co-efficient of discharge
L = Length of the weir
b = thickness of the weir

35.    
 
3
2
2
th
th
act
2
2
A
V
Q
h
Hh
g
h
L
C
h
h
H
g
h
L
C
V
h
L
C
d
d
d



















0+0+H=0+h+
6/21/2022 35
Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Applying the energy equation at A & B.
B
B
B
A
A
A
Z
γ
P
2g
V
Z
γ
P
2g
V
2
2





g
v2
2
h
H
2g
v2


From above Equation Qact is maximum when  
3
2
h
Hh 
is maximum. Therefore differentiating the equation and equating
to zero. Gives us h = 2/3 H.

36. 6/21/2022 36
Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Putting the valve of h in above equation
Qmax = 1.71Cd . L . H^3/2
Sharp Crested Weir b ˂ H/2
Ogee Crested Weir
▼

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 A pitot tube is a pressure measurement instrument
used to measure fluid flow velocity.
 The pitot tube was invented by the French early
18th century and was modified to its modern form in
the mid-19th century by French scientist Henry Darcy.
Pitot Tube
 Pitot tubes are another types of differential
pressure flowmeters.
They are named after Henry Pitot who came with
his invention in the year 1732

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
There are three types of pitot tube which are as
under;
 Simple pitot tube
 Static source
 Pitot static source
Types
 Pitot tube is employed in a variety of flow measurement
application like air speed in racing cars and Air Force
fighter jets. In industries, pitot tubes are invariably put
into use for measurement of;
 Air flow in pipes, ducts and stacks and
 Liquid flow in pipes, weirs and open channels

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
 In pitot-static tube, the kinetic energy of the
flowing fluid is transformed into potential
energy for measurement of fluid flow
velocity.
 Since pitot tubes are trouble free and offer
consistent performance.
 Pitot tubes ought to be used in applications
where the minimum Reynolds number is
more than 2ooo.

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Major advantages of pitot tubes include;
 Cost effective measurement
 No moving parts
 Simple to use and install
 Low pressure drop

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Department of Civil Engineering U.E.T
Peshawar (Bannu Campus)
Dr.Ateeq-ur-Rauf
Lecture # 11
Fluid/Flow Measurements
Problems

44. Engr.Ateeq-ur-Rauf
Department of Civil Engineering U.E.T
Peshawar (Bannu Campus) 44

45. T H E E N D