notch final.pdf

K L E’s C.B. Kolli Polytechnic Haveri.
Mechanical Engineering Department Page 1
ABSTRACT
Now a day in modern industries and open channel very precise and accurate
notches are used to measure the rate of flow for large and small flow. Notches are
classified according to shape, in this experimental investigation different types of
notches are used to measure the coefficient of discharge at different rate of flow.
Experimental investigation shows that geometry of notches widely affects the
discharge co-efficient. Notches of triangular and rectangular are used in this
investigation and compare the results of both to find the best notch.

CHAPTER - 1
INTRODUCTION
To measuring the flow of liquids in different channels and tanks, various types
of notches are used. Today in modern era of engineering, different types of notches
are used for measuring the flow rate in different channels and tanks. Conventional
trapezoidal, rectangular and inverted triangular notches having different angle
(450
,600
,900
most commonly) among the oldest notches. The triangular notches are
most commonly devices used for measuring the flow of liquids in open channel. Due
to its simplicity in design and calculations. Proper care should be taken while
measuring the readings. 1% of error in the measurement may produce 2.5% error in
result.
In this paper experiment are conducted on two types of notches .i.e. “V”
notches (600
&450
) and rectangular notches. Due to high accuracy at low flow
triangular notches are preferred over the rectangular notches.

CHAPTER - 2
LITERATURE REVIEW
Keshava Murthy and Giridhar [1] have proposed the use of Inverted V-Notch
(IVN) for flow measurement for nearly 72 % of the depth of IVN within a prefixed
permissible error of ±1.5%. They developed two optimization procedures to obtain
the linear head-discharge relationship through the notches. Using the same
optimization procedures, they improved the linearity range by introducing rectangular
notches at an optimum height over the IVN.
Later, Keshava Murthy and Shesha Prakash [2,3] presented a general
numerical optimization procedure to obtain the proportionality range for any sharp
crested notches to develop any type of head-discharge relationship. Subsequently, the
same authors presented a general algebraic optimization procedure to obtain the linear
characteristics of inverted semi-circular notches [4].
James Thomson, Professor of Civil Engineering at Queens College, Belfast,
Ireland, was among the earliest experimental investigators of the triangular-notch
notches (Thomson, 1858, 1861).the experiment is conducted at a pond in open field.
The purpose of investigation was to achieve the experimental evidence to
support his proposal that, for the measurement of discharge included a very small
flow the triangular notches are used instead of rectangular notches. Professor
Thomson's experimental equipment was crude by modern standards, and the range of
conditions covered by his experiments was small. Experiment is conducted on two
types of “V” notches i.e. 900
& 1270
where the discharge co-efficient is taken as the
0.593 & 0.617 respectively. The result is calculated as for a smaller flow two or more
900
angle notches are used instead of a single large notch angle.

CHAPTER - 3
FLUIDS AND IT’s TYPES
3.1 Definition of Fluids
The substance that has a tendency to flow is called as fluid. Generally, fluid is
defined as a substance which is capable of spreading and changing its shape,
according to is surroundings, without offering internal resistance.
Based on how the property of viscosity of fluid changes in various fluids, they
are divided into 5 types.
These have been described in details as follows:
1) Ideal fluid
2) Real fluid
3) Newtonian fluid
4) Non-Newtonian fluid
5) Ideal plastic fluid

 Ideal Fluid
The fluid, which is incompressible and has no viscosity or no friction, is
known as an ideal fluid. Ideal fluid is only an imaginary fluid.
 Real Fluid
A fluid, which possesses viscosity or friction, is known as real fluid.
 Newtonian Fluid
A real fluid, in which the shear stress is directly proportional to the rate of the
shear strain, is known as a Newtonian fluid. Fluids obeying Newton’s law where the
value of viscosity (μ) is constant are known as Newtonian fluids.
 Non-Newtonian Fluid
A real fluid, in which the shear stress is not proportional to the rate of the
shear strain, is known as a Non-Newtonian fluid. Fluids in which the value of
viscosity (μ) is not constant are known as non-Newtonian fluids.
 Ideal Plastic Fluid
A fluid, in which the shear stress is more than the yield value and shear stress
is proportional to the rate of shear strain, is known as ideal plastic fluid.
3.2 Properties of Fluids
The main properties of fluids are listed below:-
1. Density
2. Specific Volume
3. Specific Weight
4. Specific Gravity
5. Viscosity
6. Surface Tension
7. Capillarity
8. Compressibility
 Density
The density of a fluid is defined as the mass per unit volume at a standard
temperature and pressure and it is denoted by ‘𝜌’.
Mathematically density,
𝜌 =
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
=
𝑚
𝑉
𝐾𝑔
𝑚3
⁄
The density of water is 1000
𝐾𝑔
𝑚3
⁄ and is taken as a standard for the
measurement of density of the other fluids.

 Specific volume
The specific volume of fluid is defined as the volume per unit mass and it is
denoted by ‘v’.
Mathematically specific volume,
𝑣 =
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
=
𝑉
𝑚
𝑚3
𝐾𝑔
⁄
 Specific weight /Weight density
The specific weight of a fluid may be defined as the weight per unit volume at
the standard temperature and pressure and it is denoted by ‘ω’.
Mathematically specific weight,
𝜔 =
𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑
=
𝑊
𝑉
𝑁
𝑚3
⁄
𝜔 =
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 × 𝐴𝑐𝑐𝑒𝑙𝑎𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦
=
𝑚 × 𝑔
𝑉
= 𝜌 × 𝑔 𝑁
𝑚3
⁄
The specific weight of water is taken as 9.81 𝐾𝑁
𝑚3
⁄
 Specific gravity
The specific weight is defined as the ratio of the density or weight density of a
fluid to the density or weight density of standard fluid. The specific gravity is also
called as relative density and is denoted by ‘S’, mathematically
𝑆 =
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
=
𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑
𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
Specific gravity of water is taken as unity.
 Viscosity
Viscosity of a fluid is defined as the property by which it offers resistance to
the movement of one layer of fluid over another adjacent layer of fluid.
Viscosity or dynamic viscosity is basically defined as the resistance provided
to a layer of fluid when it will move over another layer of fluid.
Kinematic viscosity of fluid is basically defined as the ratio of dynamic
viscosity to the density of fluid.
Kinematic viscosity = Dynamic viscosity / Density of fluid
ν = μ/ρ

Unit of kinematic viscosity
Unit of kinematic viscosity in S.I system will be m2
/s
Unit of kinematic viscosity in C.G.S system will be stokes or cm2
/s
Unit of kinematic viscosity in F.P.S system will be inch2
/s
1 Stokes = 10-4
m2
/s
 Surface tension
Surface tension is basically defined as the tensile force acting on the surface of
a liquid in contact with gas such as air or on the surface between two immiscible
liquids. Surface tension is basically the tensile force per unit length of the surface of
liquid and therefore unit of surface tension will be N/m.
 Capillarity
Capillarity is a phenomenon of rise or fall of liquid surface relative to the
adjacent general level of liquid. This phenomenon is due to the combined effect of
cohesion and adhesion of liquid particle. The rise of liquid level is known as Capillary
Rise whereas the fall of liquid surface is known as Capillary Depression. It is
expressed in terms of cm or mm of liquid.
 Compressibility
The compressibility of the fluid may be defined as the ratio of volumetric
strain to the applied pressure and is denoted by ‘β’ mathematically,
𝛽 =
𝑉𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝑠𝑡𝑟𝑎𝑖𝑛
𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
3.3 Pascal's Law
Pascal's law states that the intensity of pressure at a point in a static fluid is
equal in all the directions (x, y and z directions).
Figure shows the transmission of fluid through the pipe from one container to
another in a hydraulic press. When a force F is applied at one end the same pressure
will be transmitted to the other end.

CHAPTER - 4
FLUID KINEMATICS
4.1 Types of Fluid Flow
The fluids can be classified into different types based on the variation of the
fluid characteristics like velocity, density etc. Depending on the type of flow, the
analysis method varies in fluid mechanics.
The different types of fluid flow are:
1. Steady and Unsteady Flow
2. Uniform and Non-Uniform Flow
3. Laminar and Turbulent Flow
4. Compressible and Incompressible Flow
5. Rotational and Irrotational Flow
6. One, Two and Three -dimensional Flow
 Steady and Unsteady Flow
A flow is defined steady when its fluid characteristics like velocity, density,
and pressure at a point do not change with time. A steady flow can be mathematically
expressed as:
(
𝑑𝑉
𝑑𝑡
)
𝑥0,𝑦0,𝑧0
= 0 (
𝑑𝑃
𝑑𝑡
)
𝑥0,𝑦0,𝑧0
= 0 (
𝑑𝜌
𝑑𝑡
)
𝑥0,𝑦0,𝑧0
= 0
Where, (𝑥0, 𝑦0,𝑧0) is a fixed point in a fluid field.
Where, V is the velocity of the fluid, ‘P’ is the pressure and ‘𝜌’ is the density.
A flow is defined unsteady, when the fluid characteristics velocity, pressure
and density at a point changes with respect to time. This can be mathematically
expressed as
(
𝑑𝑉
𝑑𝑡
)
𝑥0,𝑦0,𝑧0
= 0 (
𝑑𝑃
𝑑𝑡
)
𝑥0,𝑦0,𝑧0
= 0 (
𝑑𝜌
𝑑𝑡
)
𝑥0,𝑦0,𝑧0
= 0
 Uniform and Non-Uniform Flow
Uniform flow is the type of fluid flow in which the velocity of the flow at any
given time does not change with respect to space [Along the length of direction of
flow].A uniform flow can be mathematically expressed as:
(
𝑑𝑉
𝑑𝑆
)
𝑡=𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
= 0

Where, dV = Change of velocity.
dS = Length of flow in the direction ‘S’.
A non-uniform flow is a type of fluid flow in which the velocity of the flow at
any given time changes with respect to space. Mathematically, a non-uniform flow
can be expressed as
(
𝑑𝑉
𝑑𝑆
)
𝑡=𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
≠ 0
 Laminar and Turbulent Flow
Laminar flow is defined as a type of flow in which the fluid particles move
along a well-defined streamline or paths, such that all the streamlines are straight and
parallel to each other. In a laminar flow, fluid particles move in laminas. The layers in
laminar flow glide smoothly over the adjacent layer. The flow is laminar when the
Reynolds number is more than 4000.
Turbulent flow is a type of flow in which the fluid particles move in a zig-zag
manner. The movement in zig-zag manner results in high turbulence and eddies are
formed. This results in high energy loss. The flow is turbulent when the Reynolds
number is greater than 4000.
A fluid flow in a pipe, that has a Reynolds number between 2000 and 4000 is
said to be in transition state.
 Compressible and Incompressible Flows
A compressible flow is that type of flow in which the density of the fluid
changes from one point to another point. This means the density is not constant.
𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Incompressible flow is that type of flow in which the density of the fluid is
constant from one point to another. Liquids are generally incompressible and gases
are compressible.
𝜌 ≠ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

 Rotational and Irrotational Flows
A type of flow in which the fluid particles rotate about their own axis while
flowing along the streamlines is called a rotational flow.
If the fluid particles while flowing along the streamline do not rotate about
their own axis, then the flow is called irrotational flow.
 One, Two and Three Dimensional Flows
One dimensional fluid flow is a fluid flow in which, the flow parameter such
as velocity is expressed as a function of time and one space coordinates. That is,
u = f(x, y), v=0; w=0;
In this type, the velocity along y and z directions i.e. v and w are considered
negligible.
Two-dimensional flow is that type of flow in which the velocity is a function
of time and two rectangular space co-ordinates. The velocity of flow along the third
direction is considered negligible. That is,
u = f(x, y); v = g(x, y); w = 0;
Three-dimensional flow is the type of flow in which the velocity is a function
of time and three mutually perpendicular rectangular space coordinates (x, y, and z).
That is,
u = f(x, y, z); v = g(x, y, z); w = h(x, y, z)
4.2 Rate of Flow or Discharge (Q)
It is defined as the quantity of a fluid flowing per second through a section of a
pipe or channel.it is generally denoted by ‘Q’ and it is expressed in m3
/sec or liter/sec.
Consider a liquid flowing through a pipe
Let, A= cross section area of pipe in m2
V= average velocity of the liquid in m2
/sec
Then discharge, Q=A x V
Note: 1 m3
= 1000 liters

4.3 Continuity Equation
The equation based on law of conservation of mass is called the continuity
equation. This law states that “if an incompressible fluid flowing continuously
through a pipe or channel ,whose cross sectional area may or may not be constant,
then the continuity of fluid passing per second is constant at all sections”.
Consider a tapering pipe through which fluid is flowing as shown in fig and
consider two cross sections 1-1 and 2-2.
Let,
V1 = Average velocity at cross section 1-1.
A1 = Area of pipe at cross section 1-1.
And A2, V2 are corresponding values at section 2-2.
Then, Rate of flow at section 1-1 = A1V1
Rate of flow at section 2-2 = A2V2
According to the law of conservation of mass
Rate of flow at section 1-1 = Rate of flow at section 2-2
Therefore,
A1V1 = A2V2
The above equation is called the continuity equation.

CHAPTER - 5
FLUID DYNAMICS
5.1 Energy Present In Fluid Flow
The energy can be defined as the capacity to do work.it exists in various forms
and it can change from one form to another. Following are the important forms of
energies of a flowing liquid.
1. Potential energy.
2. Kinetic energy.
3. Pressure energy.
 Potential energy of a fluid particle in motion.
It is an energy possessed by liquid particles by virtue of its position. If a liquid
particles Z meters above the horizontal datum, the potential energy of the particle will
be Z m Kg/Kg of liquid.
The potential head of liquid, at that point, will be z meters of the liquid.
 Kinetic energy of a fluid particle in motion.
It is an energy possessed by liquid particles by virtue of its motion or velocity. If a
liquid particle flowing with a mean velocity of V m/sec, then the kinetic energy of the
particle will be
𝑉2
2𝑔
m Kg/Kg of liquid.
Velocity head of the liquid at that velocity will be
𝑉2
2𝑔
m of liquid.
 Pressure energy of a fluid particle in motion.
It is an energy possessed by liquid particles by virtue of its existing pressure.
If the liquid particle is under a pressure of P KN/m2,
then pressure energy of liquid
particle will be P/ω m Kg/Kg of liquid.
Where, ω = specific weight of the liquid.
Pressure head of the liquid under that pressure will be P/ω m of liquid.
5.2 Total Energy and Total Head.
 Total energy
The total energy of a liquid, in motion is the sum of its potential energy,
kinetic energy and pressure energy.
Mathematically,

𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦, 𝐸 = 𝑍 +
𝑉2
2𝑔
+
𝑃
𝜔
m of water
 Total head
The total head of a liquid, in motion is the sum of its potential head, kinetic
head and pressure head.
Mathematically,
𝑇𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑑 , 𝐻 = 𝑍 +
𝑉2
2𝑔
+
𝑃
𝜔
𝑚 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
5.3 Bernoulli’s Theorem.
It states that “In a steady flow of frictionless incompressible fluid flow system,
the total energy of a particle remains constant”.
That is “In a steady flow of frictionless incompressible fluid flow system, the
sum of potential energy, kinetic energy and pressure energy remains constant at all
sections”.
This statement is based on the assumption that there are no losses due to
friction in the pipe. Mathematically,
𝑍 +
𝑉2
2𝑔
+
𝑃
𝜔
= 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Where,
Z = Potential energy
𝑉2
2𝑔
= Kinetic energy
𝑃
𝜔
= Pressure energy
Consider a perfect incompressible liquid, flowing through a non-uniform pipe
as shown in fig.

We know that loss of potential energy + work done by pressure = gain in
kinetic energy.
𝑊(𝑍1 − 𝑍2) +
𝑊
𝑤
(𝑃1 − 𝑃2) =
𝑊
2𝑔
(𝑉2
2
− 𝑉1
2
)
(𝑍1 − 𝑍2) +
𝑃1
𝑤
+
𝑃2
𝑤
=
𝑉2
2
2𝑔
−
𝑉1
2
2𝑔
𝑍1 +
𝑉1
2
2𝑔
+
𝑃1
𝑤
= 𝑍2 +
𝑉2
2
2𝑔
+
𝑃2
𝑤
I.e. energy at section 1 = energy at section 2
5.4 Application of Bernoulli’s Equation.
Following are the applications of Bernoulli’s equation:
 Bernoulli’s equation is applied to all problems of incompressible fluid flow.
 The Bernoulli’s equation can be applied to the following measuring devices
such as venturi meter, Nozzle meter, Orifice meter, Pitot tube and its
applications to flow measurement from takes, within pipes as well as in open
channels.
 Bernoulli’s theory is used to study the unstable potential flow used in the
theory of ocean surface waves and acoustics.
 It is also employed for the estimation of parameters such as pressure and fluid
speed.
 Bernoulli’s principle can be applied in an aero plane. For example, this theory
explains why aero plane wings are curved upward and the ships have to run
away from each other as they pass.
5.5 Assumptions.
The following are the assumptions made in the derivation of Bernoulli’s equation:
 The fluid is ideal or perfect, that is viscosity is zero.
 The flow is steady (The velocity of every liquid particle is uniform).
 There is no energy loss while flowing.
 The flow is incompressible.
 The flow is irrotational.
 There is no external force, except the gravity force, is acting on the liquid.

5.6 Limitations
The following are the main limitation of Bernoulli’s equation:
 The velocity of the fluid particle in the center of a pipe is maximum and due to
the friction, it gradually decreases towards the pipe walls. Thus while using
Bernoulli’s equation, only the mean velocity of the liquid should be taken into
account because the velocity of the liquid particles is not uniform. As per
assumption, it is not practical.
 There are always some external forces acting on the liquid, which affects the
flow of friction. Thus while using Bernoulli’s equation all such external forces
are neglected which has not happened in actual practice. If some energy is
supplied to or extracted from the flow, the same should also take into account.
 In turbulent flow, some kinetic energy is converted into heat energy and in
viscous flow; some energy is lost due to shear forces. Thus while using
Bernoulli’s equation all such losses should be neglected, which has not
happened in actual practice.
 If the fluid is flowing by a curved path, the energy due to centrifugal forces
must also be taken into account.
5.7 Hydraulic Coefficients.
Following four coefficients are known as hydraulic coefficients or orifice coefficients.
1. Coefficient of contraction Cc
2. Coefficient of velocity Cv
3. Coefficient of discharge Cd
4. Coefficient of resistance Cr
 Coefficient of contraction (Cc)
It is defined as the ratio of area of water jet at vena-contracta to the area of the
orifice. mathematically,
𝐶𝑐 =
Area of water jet at vena contracta
Area of the orifice
The average value of Cc is 0.64

 Coefficient of velocity (Cv)
It is defined as the ratio of velocity of water jet at vena-contracta to the
theoretical velocity of the jet. Mathematically,
𝐶𝑣 =
Velocity of water jet at vena contracta
Theoretical velocity of the jet
The average value of Cv is 0.97
 Coefficient of discharge (Cd)
It is defined as the ratio of actual discharge through an orifice to the
theoretical discharge. Mathematically,
𝐶𝑑 =
𝐴𝑐𝑡𝑢𝑎𝑙 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒
=
𝑄𝑎𝑐𝑡
𝑄𝑡ℎ𝑒
The average value of Cd is 0.62
 Coefficient of resistance (Cr)
It is defined as the ratio of loss of kinetic energy to the actual kinetic
energy. Mathematically,
𝐶𝑟 =
Loss of KE
Actual KE
=
𝐿𝑜𝑠𝑠 𝑜𝑓 ℎ𝑒𝑎𝑑
𝐴𝑐𝑡𝑢𝑎𝑙 ℎ𝑒𝑎𝑑

CHAPTER - 6
NOTCHES
6.1 Introduction.
A notch means an opening provided in the side of a tank, such that the opening
extends even above the free surface of the liquid in the tank. It is in a way, a large
orifice having no upper edge. A notch is generally meant to measure the flow of water
from a tank. A weir is also a notch but it is made on a large scale. The weir is a notch
cut in a dam to discharge the surplus quantity of water.
Water flows over a notch or weir while water passes through an orifice. While
the stream of water discharged by an orifice is called a jet, the sheet of water
discharged by a notch or weir is called a nappe or vein. The upper surface of the notch
or weir over which the water flows is called the Crest or Sill.
A notch or a weir is a convenient device for the measurement of discharge in
an open channel. A notch or a weir is an obstruction provided in a channel that causes
the water to rise behind it so that the water is made to flow through it or over it. The
rate of flow can be determined by measuring the height of the upstream water level.
Basically there is no difference between a notch and a weir, except that a
notch is of small size while a weir is of large size. A notch is usually made of metal
plate whereas a weir is made of masonry or concrete.
The bottom edge over which the water flows is called the sill or the crest of
water the notch. The common shapes of weirs are, rectangular, triangular, trapezoidal,
composite, parabolic and proportional. The sheet of water which springs free from the
crest is called the nappe.
If the sheet of water springs free as it leaves the crest, the weir is called a sharp
crestel weir. In the case of a broad crested weir, there is a support for the falling nappe
over the crest in the direction of flow. As the water approaches the crest there is a fall
in the water surface forming a convex curve called the draw down. The draw down at
the weir crest is about 0.15 H where H is the head of water surface above the crest.

6.2 Classification of Notches.
The Notches are classified as:
1. According to the shape of opening.
 Rectangular notch.
 Triangular notch.
 Trapezoidal notch.
 Stepped notch.
2. According to the effect of sides on the nappe.
 Notch with end contraction.
 Notch without end contraction or suppressed notch.
6.3 Discharges over a Rectangular Notch.
Consider a rectangular notch in one side of a tank over which water is flowing
as shown in fig.
Let,
H= Height of water above sill of notch.
B= Width or length of the notch.
Cd= Coefficient of discharge.
Let us consider a horizontal strip of water of thickness dh at a depth of h from
the water level as shown in fig.
Therefore,
Area of the strip = b.dh (i)
We know that the theoretical velocity of water through the strip
=√2𝑔ℎ (ii)
Discharge through the strip,
dq = Cd × Area of strip × Theoretical velocity

= 𝐶𝑑 × 𝑏. 𝑑ℎ√2𝑔ℎ (iii)
The total discharge, over the whole notch, may be found out by integrating the
above equation within the limits 0 and H.
𝑄 = ∫ 𝐶𝑑. 𝑏. 𝑑ℎ. √2𝑔ℎ
𝐻
0
𝑄 = 𝐶𝑑. 𝑏. 𝑑ℎ√2𝑔 ∫ ℎ
1
2 𝑑ℎ
𝐻
0
𝑄 = 𝐶𝑑. 𝑏√2𝑔 [
ℎ
3
2
3
2
]
𝐻
0
𝑄 =
2
3
𝐶𝑑. 𝑏√2𝑔 [ℎ
3
2]
𝐻
0
𝑄 =
2
3
𝐶𝑑. 𝑏√2𝑔(𝐻)
3
2
6.4 Discharges over a Triangular Notch
A triangular notch is also called as a V- notch. Consider a triangular notch, in
one side of the tank, over which water is flowing as shown in fig.
Let,
H= Height of the liquid above the apex
of the notch.
θ= Angle of the notch and
Cd = Coefficient of discharge.
From the geometry of the figure, we find that the width of the notch at the
water surface.
= 2𝐻 tan
𝜃
2
Therefore, Area of the strip
Area of the strip = 2(𝐻 − ℎ) tan
𝜃
2
. 𝑑ℎ

We know that the theoretical velocity of water through the strip
= √2𝑔ℎ
And discharge over the notch,
dq = Cd ×Area of strip × Theoretical velocity
= 𝐶𝑑 × 2(𝐻 − ℎ) tan
𝜃
2
𝑑ℎ √2𝑔ℎ
The total discharge, over the whole notch, may be found out by integrating the
above equation within the limits 0 and H.
𝑄 = ∫ 𝐶𝑑. 2 (𝐻 − ℎ tan
𝜃
2
) 𝑑ℎ√2𝑔ℎ
𝐻
0
𝑄 = 2 𝐶𝑑 √2𝑔 tan
𝜃
2
∫(𝐻 − ℎ)
𝐻
0
√ℎ 𝑑ℎ
𝜃
2
∫ (𝐻ℎ
1
2 − ℎ)
3
2
𝐻
0
𝑑ℎ
𝜃
2
[
𝐻. ℎ
3
2
3
2
−
ℎ
5
2
5
2
]
𝐻
0
𝑄 =
8
15
𝐶𝑑 √2𝑔 tan
𝜃
2
× 𝐻
5
2
6.5 Discharges over a Trapezoidal Notch.
A trapezoidal notch is a combination of a rectangular notch and two triangular
notches as shown in fig.it is, thus obvious that the discharge over such a notch will be
the sum of the discharges over the rectangular and triangular notches.

Consider a trapezoidal notch ABCD as shown in fig for analysis purpose, split
up the notch into a rectangular notch BCFE and two triangular notches ABE and
DCF.the discharges over these two triangular notches is equivalent to the discharges
over a single triangular notch of angle θ.
Let,
H= Height of the liquid above the sill of the notch.
Cd1 = Coefficient of discharge for the rectangular portion.
Cd2 = Coefficient discharge for the triangular portion.
b = Breadth of the rectangular portion of the notch.
𝜃
2
= Angle, which the sides make with vertical.
Discharge, over trapezoidal notch,
Q=Discharge over rectangular notch + Discharge over the triangular notch.
=
2
3
𝐶𝑑1 .𝑏 √2𝑔 𝐻
3
2 +
8
15
𝐶𝑑2 √2𝑔 tan
𝜃
2
𝐻
3
2
6.6 Discharges over a Stepped Notch.
A stepped notch is combination of rectangular notches as shown in fig.it is,
thus obvious that the discharge over such a notch will be the sum of the discharges
over the different rectangular notches.
Consider a stepped notch as shown in fig.for analysis purpose, let us split up
the notch into two rectangular notches 1 and 2.the total discharge over the two
rectangular notches.
Let,
H1 = Height of the liquid above sill of the notch 1.
b1 = Breadth of notch 1.
H2 and b2 = Corresponding values for notch 2.
Cd = Coefficient discharge for both the notches.
From the geometry of the notch, we find that the discharge over the notch 1,

𝑄1 =
2
3
𝐶𝑑 . 𝑏1 √2𝑔 × 𝐻1
3
2
And discharge over notch 2,
𝑄2 =
2
3
𝐶𝑑 . 𝑏2 √2𝑔 [𝐻2
3
2 − 𝐻1
3
2]
Now the total discharge over the notch,
𝑄 = 𝑄1 + 𝑄2
6.7 Advantages of a Triangular Notch over Rectangular Notch.
A triangular notch has certain advantages over the rectangular notch when
used as a gauging device in a hydraulic laboratory.
 The coefficient of discharge for a triangular notch is practically
independent of the head. This is because, for all heads the ratio of the head
to the wetted length or crest is constant. But in a rectangular notch the ratio
of the head to the wetted length crest is not constant. Hence for a
rectangular notch the coefficient of discharge is not actually a constant but
is a function of the head over the notch.
 When the discharge rate is small a triangular notch provides a greater head
than the rectangular notch. Hence head measurement can be done more
accurately over the triangular notch than over the rectangular notch.
 When the discharge rate is small, there are chances of a clinging nappe to
be formed when a rectangular notch is used. But for the same discharge
over the triangular notch the head will be greater and the clinging nappe
will be avoided.
 When a triangular notch is provided, there will be no need for any special
arrangement for ventilating the nappe.

CHAPTER – 7
CALIBRATION OF TRIANGULAR NOTCH
7.1 Aim
 To determine the characteristics of open channel flow over a V- notch.
 To determine values of the discharge co-efficient.
7.2 Introduction
Different type of models are available to find discharge in an open channel as
notch, weir etc. for calibration of either rectangular notch, trapezoidal notch or V
notch some flow is allowed in the flume,. Once the flow becomes steady and uniform
discharge coefficients can be determine for any models.
In general, sharp crested notch are preferred where highly accurate discharge
measurement are required, for example in hydraulic laboratories, industry and
irrigation pilot schemes, which do not carry debris and sediments.
Notches are those overflow structure whose length of crest in the direction of
low is accurately shaped. There may be rectangular, trapezoidal, V notch etc. the V -
notch is one of the most precise discharge and head over the weir can be developed by
making the following assumptions as to the flow behavior.
 Upstream of the weir, the flow is uniform and the pressure varies with depth
according to the hydrostatic equation p = ρ g h.
 The face surface remains horizontal as far as plane of the weir, and all
particles passing over the weir move horizontally.
 The pressure throughout the sheet of liquid or nappe, which passes over the
crest of the weir, is atmospheric.
 The effect of viscosity and surface tension are negligible.
 The velocity in the approach channel is negligible.
A V-notch is having a V-shaped opening provided in its body so that water is
discharged through this opening. This line which bisects the angle of notch should be
vertical and the same distance from both sides of the channel. The discharge
coefficient cd of a v -notch may be determined by formula.

𝐶𝑑 =
𝑄
8
15
√2𝑔 𝐻
5
2
⁄
tan
𝜃
2
Where,
Q = the discharge over triangular notch.
H = the head over the crest of the notch.
g = acceleration due to gravity.
7.3 New Terms.
 Nappe/vein: The sheet of water flowing through a notch is called nappe/vein.
 Crest/sill: The bottom edge of a notch over which water flows is known as the
sill/ crest.
7.4 Observations.
 Type of notch= V notch
 Angle of notch = 90˚
 Cross sectional area of collecting tank =__________ m2
 Volume of water collected tank V = A x H =___________m2
 Height of the water collected tank = _______m
 Least count of hook gauge = _________
7.5 Specifications.
 A constant steady water supply tank (Notch tank) with baffles wall.
 Pointer gauge.
 Triangular notch angle.
 Dimension of measuring tank.
7.6 Experimental set-up.

The experiment set up consists of a tank whose inlet section is provided with 2
nos. of baffles for stream line flow. While at the downstream portion of the tank one
can fit rectangular notch, trapezoidal notch, V notch. A tank hook gauge is used to
measure the head of water over the model. A collecting is used to find the actual
discharge through the notch.
7.7 Tabular Column.
Sl.
No
Initial
Height
H1 in mm
Final
Height
H2 in mm
Time
in
Sec
Difference
head
H = H2-H1
Theoretical
discharge
in m3
/Sec
Actual
discharge
in m3
/Sec
Co-efficient
of discharge
Cd
1
2
3
4
5
7.8 Formula.
 Initial height H1 = ___________m
 Final height H2 = ___________ m
 Difference Head H = H2- H1 = ____________m
 Actual discharge Q(act) =
𝑉
𝑡
= _________ m3
/Sec
 Theoretical discharge Q(the) =
8
15
𝐶𝑑 √2𝑔 tan
𝜃
2
× 𝐻
5
2
 Co-efficient of discharge = Cd =
𝑄𝑎𝑐𝑡
𝑄𝑡ℎ𝑒
Where, θ = 90˚, g = 9.81, H = H2 – H1

7.9 Experimental Procedure
 The notch under test is positioned at the end of the tank, in a vertical plane and
with the sharp edge on the upstream side.
 The tank is filled with water up to crest level and subsequently note down the
crest level of the notch by the help of a point gauge.
 The flow regulating valve is adjusted to give the maximum possible discharge
without flooding the notch.
 Conditions are allowed to steady before the rate of discharge and head H were
recorded.
 The flow rate is reduced in stages and the reading of discharge and H were
taken.
 The procedure is repeated for other type of notch.

CHAPTER - 8
CALIBRATION OF RECTANGULAR NOTCH
8.1 Aim.
 To determine the characteristics of open channel flow over a rectangular
notch.
 To determine values of the discharge co-efficient.
8.2 Introduction.
Different types of models are available to find discharge in an open channel as
notch, venturi meter notch etc. for calibration of either rectangular notch, trapezoidal
notch some flow is allowed in the flume. Once the flow becomes steady and uniform
discharge coefficients can be for any model.
In general, sharp crested notches are preferred where highly accurate
discharge measurement is required, for example in hydraulic laboratories, industries
and irrigation pilot schemes, which do not carry debris and sediments. Notches are
those overflow structure whose length of crest in the direction of flow is accurately
shaped. There may be rectangular, trapezoidal, V notch etc. the relationship between
discharge and head over the notch can be developed by making the following
assumptions as to the flow behavior.
 Upstream of the notch, the flow is uniform and the pressure varies with depth
according to the hydrostatic equation P=  g h.
 The free surface remains horizontal as far as the plane of the notch, and all
particles passing over the notch move horizontally.
 The pressure throughout the sheet of liquid which passes over the crest of the
notch is atmospheric.
 The effect of viscosity and surface tension are negligible.
 The velocity in the approach channel is negligible.
A rectangular notch, symmetrically located in a vertical thin plate, which is
placed perpendicular to the side and bottom of a straight channel, is defined as a
rectangular sharp-crested notch. The discharge coefficient Cd of a rectangular notch
may be determined by applying formula.

𝐶𝑑 =
𝑄
2
3
√2𝑔𝐵 𝐻
3
2
⁄
Where,
Q = the discharge over a rectangular notch,
B = the width of notch,
H = the head over the crest of the notch
g = acceleration due to gravity.
8.3 Assumptions.
 The flow is constant, steady and uniform.
 The roughness of the wetted surface of the channel and thus the friction
coefficient is constant.
 The channel is sufficiently wide so that the end effects are negligible.
 Velocity of the fluid approaching the weir is small so that kinetic energy can
be neglected.
 The velocity depends only on the depth below the free surface.
8.4 Observation.
 Length of the collecting tank = …….
 Breadth of the collecting tank = ……..
 Length of rectangular notch, L = ……
 Number of end contractions, n =…
 Theoretical discharge, Qth= ………………
 Actual discharge, Qact = ………….
8.5 Experimental Set-up
The experiment setup consists of a tank whose inlet section is provided with –
2 nos. of baffles for stream line flow. While at the downstream portion of the tank one
can fix a notch of rectangular notch, trapezoidal notch or V-notch. A hook gauge is
used to measure the head of water over the model. A collecting tank is used to find the
actual discharge through the notch.

8.6 Experimental Procedure
The notch under test is positioned at the end of the tank, in a vertical plane,
and with the sharp edge on the upstream side.
 Ensure that the hydraulic bench is positioned so that its surface is horizontal.
 Mount the rectangular notch plate into the flow channel and position the
stifling baffle as shown in the diagram.
 Position the instrument carrier the opposite way around from that shown in the
diagram.
 Then lower the gauge until the point is above the notch base and lock the
coarse adjustment screw.
 After that, use the fine adjustment screw to position the gauge until the point
just touches the notch bottom and then takes the readings. Pay attention to the
notch so that it does not get destroyed.
 Mount the instrument carrier as shown in the diagram and then locate it
approximately just in between the stifling baffle and the notch plate.
 Then open the bench control valve and admit water to the channel, adjust
valve to give approximately 10mm depth above the notch base. The
coefficient of discharge values can be determined from measurements of the
height of the free surface of water above the notch base and corresponding
volume flow rate.
 For accurate height gauge readings, fine adjustment is used to lower the gauge
until the point just touches its reflection.

 Determine the volume flow rate by measuring the time required to collect a
known volume of water in the volumetric tank.
 Take the readings twice to ensure consistency.
 Repeat the procedure for different flow rates increasing depth by
approximately 10 mm until the water level reaches the top of the notch.
8.7 Tabular and Calculations.
Sl.
No
Initial
Height
H1 in mm
Final
Height
H2 in mm
Time
in
Sec
Difference
head
H = H2-H1
Theoretical
discharge
in m3
/Sec
Actual
discharge
in m3
/Sec
Co-efficient
of discharge
Cd
1
2
3
4
5
8.8 Formula.
 Initial height H1 = ___________m
 Final height H2 = ___________ m
 Difference Head H = H2- H1 = ____________m
 Actual discharge Q(act) =
𝑉
𝑡
= _________ m3
/Sec
 Theoretical discharge Q(the) =
2
3
√2𝑔 𝐵 (𝐻)
3
2 m3
/Sec
 Co-efficient of discharge = Cd =
𝑄𝑎𝑐𝑡
𝑄𝑡ℎ𝑒
Where, g = 9.81, H = H2 – H1

8.9 Discussion.
The results above show the calculations performed for rectangular and
V notch; the volume collected remains the same. For rectangular notch, as time for
collection was decreasing the water level was increased. As the head of a weir
increases the flow rate were decreasing, it can be observed from the graphs
computed above. As the flow rate increases, the discharge coefficient increases
however when calculating the theoretical discharge coefficient, the values are the
same as the one of the experimental and the percentage error is 0 which actually does
not make sense because in the experiment there are factors that contributes or
influence inaccuracy in readings. The results might be affected by maybe the
theoretical discharge coefficient equation used or the time when calculating the head
of the weir. These results of percentage error obtained, may results from the
theoretical equation used or the line of the graph when drawing the best fit to obtain
the linear equation.
8.10 Recommendations.
For the experiment to be successful and to obtain accurate readings or results,
the following should be considered: The apparatus should be checked before starting
with the experiment. More type of weirs should be introduced in order to see more
variations in terms of shapes, discharge coefficient and depth of the water. Ensure that
the apparatus is not vibrating that could cause inaccuracy of the results obtained.
Ensure the needle point is in zero positioning before setting the initial depth of water.
When changing the weir, the screw must be tight to avoid any leakage while the water
flows.

CHAPTER – 9
CONCLUSION
As a conclusion for this experiment, it is concluded that the discharge
coefficient for the triangular weir is higher/lower than rectangular weir. This is due to
the shape of the weir itself and the size of the space for the water to flow through the
weir. The relationship between the head of the weir and the discharge of the water
over the weir is directly proportional. However, the calculation for percentage error
shows no difference between the theoretical and experimental values. The
experiment was successfully conducted, and the aim/objectives of the experiment was
achieved but due to other results obtained such as percentage error still needs
investigation.

REFERENCES
1. Applied fluid mechanics lab manual, (n.d). “EXPERIMENT #9: FLOW
OVER
WEIRS”.Availablefrom:https://uta.pressbooks.pub/appliedfluidmechanics/cha
pter/experiment-9/.Accessed date: 05 October 2020.
2. Figure 3: c and k vs notch angle (n.d). available from;Accessed date: 26
October 2020
3. Tutorbin. (n.d).” LAB #5 DISCHARGES OVER WEIRS”. Available from:
https://tutorbin.com/static/img/qs_ans/LAB%20REPORT%20-
%20Discharge%20over%20Weirs.pdf. Accessed date: 26 October 2020.
4. Syahiirah, N. (2015). “CHE241 - Lab Report Solteq Flow Over Weirs
FM26(2015)”.Available from:
https://www.academia.edu/18747051/CHE241_Lab_Report_Solteq_Flow_Ov
er_Weirs_FM26_2015_. Accessed date: 25 October 2020

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