Fluid mechanics is the branch of science which deals with the behaviour of fluids (liquids or gases) at rest as well as in motion.
This branch of science deals with the static, kinematic and dynamic aspects of fluids.
The study of fluids at rest is called fluid statics.
The study of fluids in motion, where pressure forces are not considered, is called fluid kinematics
and if the pressure forces are also considered for the fluids in motion, that branch of science is called fluid dynamics.
Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its volume. Thus mass per unit volume of a fluid is called density. It is denoted by the symbol ρ (rho). The unit of mass density in SI unit is kg per cubic metre, i.e., kg/m³.
The density of liquids may be considered as constant, while that of gases changes with the variation of pressure and temperature.
Mathematically,
Mass density (ρ) = Mass of fluid/ Volume of fluid
The value of the density of water is 1 g/cm³ or 1000 kg/m³.
Specific weight or weight density of a fluid is the ratio between the weight of a fluid to its volume. Thus weight per unit volume of a fluid is called weight density, and it is denoted by the symbol w.
The value of specific weight or weight density (w) for water is 9.81 × 1000 Newton/m³ in SI units.
Specific volume of a fluid is defined as the volume of a fluid occupied by a unit mass, or the volume per unit mass of a fluid is called specific volume. Mathematically, it is expressed as;
Thus, specific volume is the reciprocal of mass density. It is expressed as m³/kg. It is commonly applied to gases.
Specific gravity is defined as the ratio of the weight density (or density) of a fluid to the weight density (or density) of a standard fluid. For liquids, the standard fluid is taken as water, and for gases, the standard fluid is taken as air. Specific gravity is also called relative density. It is a dimensionless quantity and is denoted by the symbol S. Mathematically,
If the specific gravity of a fluid is known, then the density of the fluid will be equal to the specific gravity of the fluid multiplied by the density of water. For example, the specific gravity of mercury is 13.6, hence the density of mercury = 13.6 × 1000 = 13600 kg/m³.
Viscosity is defined as the property of a fluid which offers resistance to the movement of one layer of fluid over another adjacent layer of the fluid.
When two layers of a fluid, a distance ‘dy’ apart, move one over the other at different velocities, say u and u + du as shown in Fig. the viscosity together with relative velocity causes a shear stress acting between the fluid layers.
The top layer causes a shear stress on the adjacent lower layer, while the lower layer causes a shear stress on the adjacent top layer. This shear stress is proportional to the rate of change of velocity with respect to y. It is denoted by the symbol τ (Tau).
where μ (called mu) is the constant of proportionality and is known as the coefficient of dynamic viscosity, du or only viscosity. du/dy represents the rate of shear strain, or rate of shear deformation or velocity gradient.
It is defined as the ratio between the dynamic viscosity and the density of the fluid. It is denoted by the Greek symbol (v) called ‘nu’. Thus, mathematically,
In MKS and SI, the unit of kinematic viscosity is metre2/sec or m2/sec while in CGS units it is written as cm²/s. In CGS units, kinematic viscosity is also known as a stoke.
It states that the shear stress (t) on a fluid element layer is directly proportional to the rate of shear strain. The constant of proportionality is called the coefficient of viscosity. Mathematically,
Fluids which obey the above relation are known as Newtonian fluids, and fluids which do not obey the above relation are called Non-Newtonian fluids.
Temperature affects the viscosity. The viscosity of liquids decreases with the increase in temperature, while the viscosity of gases increases with the increase in temperature.
This is due to reason that the viscous forces in a fluid are due to cohesive forces and molecular momentum transfer. In liquids, the cohesive forces predominate the molecular momentum transfer, due to closely packed molecules and with the increase in temperature, the cohesive forces decrease the resulting in decreased viscosity.
But in the case of gases, the cohesive forces are small and molecular momentum transfer predominates. With the increase in temperature, molecular momentum transfer increases and hence viscosity increases.
The fluids may be classified into the following five types :
1. Ideal fluid,
2. Real fluid,
3. Newtonian fluid,
4. Non-Newtonian fluid, and
5. Ideal plastic fluid.
1. Ideal Fluid: A fluid which is incompressible and has no viscosity is known as an ideal fluid. Ideal fluid is only an imaginary fluid, as all the fluids which exist have some viscosity.
2. Real Fluid: A fluid which possesses viscosity is known as a real fluid. All the fluids, in actual practice, are real fluids.
3. Newtonian Fluid: A real fluid, in which the shear stress is directly proportional to the rate of shear strain (or velocity gradient), is known as a Newtonian fluid.
4. Non-Newtonian Fluid: A real fluid, in which the shear stress is not proportional to the rate of shear strain (or velocity gradient), known as a Non-Newtonian fluid.
5. Ideal Plastic Fluid: A fluid in which shear stress is more than the yield value and shear stress is proportional to the rate of shear strain (or velocity gradient) is known as an ideal plastic fluid.
Fluids consist of liquids or gases. But gases are compressible fluids, and hence thermodynamic properties play an important role. With the change of pressure and temperature, the gases undergo a large variation in density. The relationship between pressure (absolute), specific volume and temperature (absolute) of a gas is given by the equation of state as
Dimension of R. The gas constant, R, depends upon the particular gas. The dimension of R is obtained from the equation
If the change in density occurs at constant temperature, then the process is called isothermal and the relationship between pressure (p) and density (p) is given by
If the change in density occurs with no heat exchange to and from the gas, the process is called adiabatic. And if no heat is generated within the gas due to friction, the relationship between pressure and density is given by
One kilogram mole is defined as the product of one kilogram mass of the gas and its molecular weight.
Compressibility is the reciprocal of the bulk modulus of elasticity, K, which is defined as the ratio of compressive stress to volumetric strain.
Surface tension is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two immiscible liquids such that the contact surface behaves like a membrane under tension.
The magnitude of this force per unit length of the free surface will have the same value as the surface energy per unit area. It is denoted by the Greek letter σ (called sigma). In MKS units, it is expressed as kgf/m while in SI units as N/m.
Where,
Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is held vertically in the liquid.
The rise of the liquid surface is known as capillary rise, while the fall of the liquid surface is known as capillary depression.
It is expressed in terms of cm or mm of liquid. Its value depends upon the specific weight of the liquid, the diameter of the tube and the surface tension of the liquid.
A change from the liquid state to the gaseous state is known as vaporization.
The vaporization (which depends upon the prevailing pressure and temperature conditions) occurs because of the continuous escaping of the molecules through the free liquid surface.
When vaporization takes place, the molecules escape from the free surface of the liquid. These vapour molecules get accumulated in the space between the free liquid surface and the top of the vessel. These accumulated vapours exert a pressure on the liquid surface. This pressure is known as the vapour pressure of the liquid.
If the pressure at any point in this flowing liquid becomes equal to or less than the vapour pressure, the vaporization of the liquid starts. The bubbles of these vapours are carried by the flowing liquid into the region of high pressure, where they collapse, giving rise to high impact pressure.
The pressure developed by the collapsing bubbles is so high that the material from the adjoining boundaries gets eroded and cavities are formed on them. This phenomenon is known as cavitation.
It states that “the pressure or intensity of pressure at a point in a static fluid is equal in all directions”.
i.e Px = Py = Pz
This is proved as: The fluid element is of very small dimensions, i.e., dx, dy and ds.
which states that “the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point.”
(∂p/∂z) = ρ x g = w
So; p = ρgZ and
Z = P / (ρ x g)
Z is called the pressure head.
The pressure on a fluid is measured in two different systems.
In one system, it is measured above the absolute zero or complete vacuum, and it is called the absolute pressure, and in another system, pressure is measured above the atmospheric pressure, and it is called gauge pressure. Thus:
1. Absolute pressure is defined as the pressure which is measured with reference to absolute vacuum pressure.
2. Gauge pressure is defined as the pressure which is measured with the help of a pressure measuring instrument, in which the atmospheric pressure is taken as datum. The atmospheric pressure on the scale is marked as zero.
3. Vacuum pressure is defined as the pressure below the atmospheric pressure.
Note.
(i) The atmospheric pressure at sea level at 15°C is 101.3 kN/m² or 10.13 N/cm² in SI units. Inthe case of MKS units, it is equal to 1.033 kgf/cm².
(ii) The atmospheric pressure head is 760 mm of mercury or 10.33 m of water.
The pressure of a fluid is measured by the following devices:
1. Manometers
2. Mechanical Gauges.
Manometers are defined as devices used for measuring the pressure at a point in a fluid by balancing the column of fluid by the same or another column of fluid. They are classified as :
(a) Simple Manometers,
(b) Differential Manometers.
Mechanical gauges are defined as devices used for measuring pressure by balancing the fluid column with the spring or dead weight. The commonly used mechanical pressure gauges are :
(a) Diaphragm pressure gauge,
(b) Bourdon tube pressure gauge,
(c) Dead-weight pressure gauge, and
(d) Bellows pressure gauge.
A simple manometer consists of a glass tube having one of its ends connected to a point where pressure is to be measured, and the other end remains open to the atmosphere. Common types of simple manometers are:
1. Piezometer: It is the simplest form of manometer used for measuring gauge pressures. One end of this manometer is connected to the point where pressure is to be measured, and the other end is open to the atmosphere.
2. U-tube Manometer: It consists of a glass tube bent in a U-shape, one end of which is connected to a point at which pressure is to be measured, and the other end remains open to the atmosphere.
The tube generally contains mercury or any other liquid whose specific gravity is greater than the specific gravity of the liquid whose Pressure is to be measured.
3. Single Column Manometer: A single-column manometer is a modified form of a U-tube manometer in which a reservoir, having a large cross-sectional area (about 100 times) as compared to the area of the tube, is connected to one of the limbs (say, the left limb) of the manometer.
Due to the large cross-sectional area of the reservoir, for any variation in pressure, the change in the liquid level in the reservoir will be very small, which may be neglected, and hence the pressure is given by the height of the liquid in the other limb. The other limb may be vertical or inclined.
Also, known as micrometre, used to measure small pressures where high accuracy is required to measure.
Thus there are two types of single-column manometers:
1. Vertical Single Column Manometer.
2. Inclined Single Column Manometer.
Differential manometers are the devices used for measuring the difference in pressures between two points in a pipe or between two different pipes.
A differential manometer consists of a U-tube, containing a heavy liquid, whose two ends are connected to the points whose difference of pressure is to be measured. The most common types of differential manometers are :
1. U-tube differential manometer and
2. Inverted U-tube differential manometer: It consists of an inverted U-tube, containing a light liquid. The two ends of the tube are connected to the points whose difference in pressure is to be measured. It is used for measuring the difference in low pressures.
Total pressure is defined as the force exerted by a static fluid on a surface, either plane or curved, when the fluid comes in contact with the surface. This force always acts normally to the surface.
The centre of pressure is defined as the point of application of the total pressure on the surface.
There are four cases of submerged surfaces on which the total pressure force and centre of pressure are to be determined. The submerged surfaces may be:
1. Vertical plane surface,
2. Horizontal plane surface,
3. Inclined plane surface, and
4. Curved surface.
Note:
When a body is immersed in a fluid, an upward force is exerted by the fluid on the body. This upward force is equal to the weight of the fluid displaced by the body and is called the force of buoyancy or simply buoyancy.
other words, the upward force exerted by a liquid on a body when the body is immersed in the liquid is known a buoyancy or force of buoyancy
It is defined as the point through which the force of buoyancy is supposed to act. As the force of buoyancy is a vertical force and is equal to the weight of the fluid displaced by the body, the centre of buoyancy will be the centre of gravity of the fluid displaced.
other words, the point through which the force of buoyancy is supposed to act is called centre of buoyancy.
It is defined as the point about which a body starts oscillating when the body is tilted by a small angle. The meta-centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.
other words, the point about which a body starts oscillating when the body is tilted is known as the meta-centre.
The distance MG, i.e., the distance between the meta-centre of a floating body and the centre of gravity of the body, is called meta-centric height.
A submerged or a floating body is said to be stable if it comes back to its original position after a slight disturbance.
The relative position of the centre of gravity (G) and centre of buoyancy (B₁) of a body determines the stability of a submerged body.
(a) Stability of a Submerged Body: The position of the centre of gravity and centre of buoyancy in the case of a completely submerged body is fixed.
(b) Stability of Floating Body: The stability of a floating body is determined from the position of the Metacentre (M). In the case of a floating body, the weight of the body is equal to the weight of the liquid displaced.
Summary,
| Equillibrum | Floating Body | Sub-merged Body |
| (i) Stable Equillibrum | M is above G | B is above G |
| (iii) Neutral Equilibrium | M is below G | B is below G |
| (iii) Neutral Equilibrum | M and G coincide | B and G coincide |
Kinematics is defined as that branch of science which deals with the motion of particles without
considering the forces causing the motion.
The fluid motion is described by two methods. They are
In the Lagrangian method, a single fluid particle is followed during its motion and its
velocity, acceleration, density, etc., are described.
In the case of the Eulerian method, the velocity, acceleration, pressure, density, etc., are described at a point in the flow field. The Eulerian method is commonly used in fluid mechanics.
The fluid flow is classified as :
(i) Steady and unsteady flows: Steady flow is defined as that type of flow in which the fluid characteristics, like velocity, pressure, density, etc., at a point do not change with time. Unsteady flow is that type of flow in which the velocity, pressure or density at a point changes with respect to time.
(ii) Uniform and non-uniform flows: Uniform flow is defined as that type of flow in which the velocity at any given time does not change with respect to space (i.e., length of direction of the flow). Non-uniform flow is that type of flow in which the velocity at any given time changes with respect to space.
(iii) Laminar and turbulent flows: Laminar flow is defined as that type of flow in which the fluid particles move along well-defined paths or streamlines, and all the streamlines are straight and parallel. Thus, the particles move in laminas or layers, gliding smoothly over the adjacent layer. This type of flow is also called streamline flow or viscous flow.
Turbulent flow is a type of flow in which the fluid particles move in a zig-zag way. Due to the movement of fluid particles in a zig-zag way, the eddies formation takes place, which is responsible for high energy loss.
For a pipe flow, the type of flow is determined by a non-dimensional number VD/v called the Reynolds number,
where D = Diameter of pipe
V = Mean velocity of flow in pipe
and v = Kinematic viscosity of fluid.
If the Reynolds number is less than 2000, the flow is called laminar. If the Reynolds number is more
than 4000, it is called turbulent flow. If the Reynolds number lies between 2000 and 4000, the flow may
be laminar or turbulent.
(iv) Compressible and incompressible flows: Compressible flow is that type of flow in which the density of the fluid changes from point to point, or in other words, the density (ρ) is not constant for the fluid.
Thus, mathematically, for compressible flow
ρ ≠ Constant
Incompressible flow is a type of flow in which the density is constant for the fluid flow. Liquids
are generally incompressible, while gases are compressible. Mathematically, for incompressible flow
ρ = Constant.
(v) Rotational and irrotational flows: Rotational flow is that type of flow in which the fluid particles, while flowing along streamlines, also rotate about their own axis. And if the fluid particles while flowing along streamlines, do not rotate about their own axis, then that type of flow is called irrotational flow.
(vi) One, two and three-dimensional flows: One-dimensional flow is that type of flow in which the flow parameter, such as velocity, is a function of time and one space coordinate only, say x. For a steady one-dimensional flow, the velocity is a function of one spatial coordinate only. The variation of velocities in the other two mutually perpendicular directions is assumed to be negligible. Hence, mathematically, for one-dimensional flow u= f(x), v= 0 and w=0, where u, v and w are velocity components in x, y and z directions, respectively.
Two-dimensional flow is that type of flow in which the velocity is a function of time and two rectangular space co-ordinates, say x and y. For a steady two-dimensional flow, the velocity is a function of two space coordinates only. The variation of velocity in the third direction is negligible. Thus, mathematically, for two-dimensional flow. u=f1(x, y), v=f,(x, y) and w = 0.
Three-dimensional flow is a type of flow in which the velocity is a function of time and three mutually perpendicular directions. But for a steady three-dimensional flow, the fluid parameters are functions of three space co-ordinates (x, y and z) only. Thus, mathematically, for three-dimensional flow u=f(x, y,Z), V=fa(%, y, Z) and w = f3(x, y, Z).
It is defined as the quantity of a fluid flowing per second through a section of a pipe or a channel. For an incompressible fluid (or liquid), the rate of flow or discharge is expressed as the volume of fluid flowing across the section per second.
For compressible fluids, the rate of flow is usually expressed as the weight of fluid flowing across the section. Thus
(i) For liquids, the units of Q are m3/s or litres/s
(ii) For gases, the units of Q are kgf/s or Newton/s
Consider a liquid flowing through a pipe in which
A = Cross-sectional area of pipe
V = Average velocity of fluid across the section
Then discharge (Q) = A x V.
The equation based on the principle of conservation of mass is called the continuity equation. Thus, for a fluid flowing through the pipe at all the cross-sections, the quantity of fluid per second is constant.
Let V1 = Average velocity at cross-section 1-1
ρ1= Density at section 1-1
A1 = Area of pipe at section 1-1
and V2, ρ2, A2, are corresponding values at section 2-2.
Then, the rate of flow at section 1-1 = ρ1A1V1 &
Rate of flow at section 2-2 = ρ2A2V2
So, ρ1A1V1 = ρ2A2V2 ………………….(*)
Equation * applicable to the compressible as well as incompressible fluid.
If the fluid is in- a pipe compressible, then ρ 1 = ρ 2, and continuity reduces to A1V1 = A2V2
Vortex flow is defined as the flow of a fluid along a curved path or the flow of a rotating mass of fluid is known a ‘Vortex Flow’. The vortex flow is of two types, namely:
1. Forced vortex flow: Forced vortex flow is defined as that type of vortex flow in which some external torque is required to rotate the fluid mass. The fluid mass in this type of flow, rotates at constant angular velocity, ω. The tangential velocity of any fluid particle is given by v= ω x r.
e.g., Flow of liquid inside the impeller of a centrifugal pump & through the runner of a turbine.
2. Free vortex flow: When no external torque is required to rotate the fluid mass, that type of flow is called free vortex flow. Thus, the liquid in the case of a free vortex is rotating due to the rotation which is imparted to the fluid previously.
e.g., Flow of liquid through a hole provided at the bottom of a container, Flow of liquid around a circular bend in a pipe, A whirlpool in a river, Flow of fluid in a centrifugal pump casing.
An ideal fluid is a fluid which is incompressible and inviscid. Incompressible fluid is a fluid for which the density (ρ) remains constant. Inviscid fluid is a fluid for which viscosity (μ) is zero. Hence, a fluid for which the density is constant and the viscosity is zero is known as an ideal fluid.
The source flow is the flow coming from a point (source) and moving out radially in all directions of a plane at a uniform rate.
The sink flow is the flow in which fluid moves radially inwards towards a point where it disappears at a constant rate.
The flow patterns due to uniform flow, a source flow, a sink flow and a free vortex flow can be superimposed in any linear combination to get a resultant flow which closely resembles the flow around bodies. The resultant flow will still be potential and ideal. The following are the important Super-imposed flow :
(i) Source and sink pair
(ii) Doublet (special case of source and sink combination)
(iii) A plane source in a uniform flow (flow past a half body)
(iv) A source and sink pair in a uniform flow
(v) A doublet in a uniform flow.
Note:
The dynamic behaviour of the fluid flow is analysed by Newton’s second law of motion, which relates the acceleration to the forces. The fluid is assumed to be incompressible and non-viscous.
According to Newton’s second law of motion, the net force F, acting on a fluid element in the direction of x is equal to the mass m of the fluid element multiplied by the acceleration a, in the x-direction. Thus mathematically, Fx. =m.ax …..(i)
In the fluid flow, the following forces are present :
(i) Fg gravity force.
(ii) Fp the pressure force.
(iii) Fv force due to viscosity.
(iv) Ft force due to turbulence.
(v) Fc force due to compressibility.
Thus in equation (i), the net force
Fx = (Fg)x + (Fp) x + (Pv) x + (Ft) x + (Fc) x ………………..(**)
(i) If the force due to compressibility is negligible, the resulting net force Fx = (Fg)x + (Fp) x + (Pv) x + (Ft) x and the equation of motion are called Reynolds’ equations of motion.
(ii) For flow, where turbulence is negligible, the resulting equations of motion are known as the Navier-Stokes Equation.
(iii) If the flow is assumed to be ideal, the viscous force is zero, and the equations of motion are known as Euler’s equations of motion.
This is the equation of motion in which the forces due to gravity and pressure are taken into consideration. This is derived by considering the motion of a fluid element along a streamline.
(dp/ρ) + gdz + vdv = 0
Is known as Euler’s equation of motion.
Bernoulli’s equation is obtained by integrating the Euler’s equation of motion
If flow is incompressible, ρ is constant and
is a Bernoulli’s equation in which
ASSUMPTIONS
The following are the assumptions made in the derivation of Bernoulli’s equation :
Statement of Bernoulli’s Theorem. It states that in a steady, ideal flow of an incompressible fluid, the total energy at any point of the fluid is constant. The total energy consists of pressure energy, kinetic energy and potential energy or datum energy. These energies per unit weight of the fluid are :
Thus mathematically, Bernoulli’s theorem is written as
The Bernoulli’s equation was derived on the assumption that fluid is inviscid (non-viscous) and therefore frictionless. But all the real fluids are viscous and hence offer resistance to flow.
Thus, there are always some losses in fluid flows, and hence in the application of Bernoulli’s equation, these losses have to be taken into consideration.
Thus, the Bernoulli’s equation for real fluids between points 1 and 2 is given as
Bernoulli’s equation is applied in all problems of incompressible fluid flow where energy considerations are involved. But we shall consider its application to the following measuring devices:
A venturimeter is a device used for measuring the rate of a flow of a fluid flowing through a pipe. It consists of three parts: (i) A short converging part, (ii) the Throat, and (iii) a diverging part. It is based on the Principle of Bernoulli’s equation.
It is a device used for measuring the rate of flow of a fluid through a pipe. It is a cheaper device compared to a venturimeter. It also works on the same principle as that of a venturimeter.
It consists of a flat circular plate which has a circular, sharp-edged hole called an orifice, which is concentric with the pipe.
The orifice diameter is kept generally 0.5 times the diameter of the pipe, though it may vary from 0.4 to 0.8 times the pipe diameter.
It is a device used for measuring the velocity of flow at any point in a pipe or a channel. It is based on the principle that if the velocity of flow at a point becomes zero, the pressure there is increased due to the conversion of the kinetic energy into pressure energy.
Velocity at any point v = Cv √(2gh)
Where, Cv = Coefficient of Pitot-Tube
It is based on the law of conservation of momentum or on the momentum principle, which states that the net force acting on a fluid mass is equal to the change in momentum of flow per unit time in that direction. The force acting on a fluid mass ‘m’ is given by Newton’s second law of motion, F= m.a
Also can be written as F.dt = d(mv)
which is known as the impulse-momentum equation, states that the impulse of a force F acting on a fluid of mass m in a short interval of time dt is equal to the change of momentum d(mv) in the direction of force.
A free liquid jet is defined as the jet of water coming out from the nozzle in the atmosphere. The path travelled by the free jet is parabolic.
An orifice is a small opening of any cross-section (such as circular, triangular, rectangular, etc.) on the side or at the bottom of a tank, through which a fluid flows. it is used to measure rate of flow.
The orifices are classified based on their size, shape, nature of discharge and shape of the upstream edge. The following are the important classifications:
1. The orifices are classified as small orifice or large orifice depending upon the size of the orifice and the head of liquid from the centre of the orifice. If the head of liquid from the centre of the orifice is more than five times the depth of the orifice, the orifice is called a small orifice. And if the head of liquids is less than five times the depth of the orifice, it is known as a large orifice.
2. The orifices are classified as (i) Circular orifice, (ii) Triangular orifice, (iii) Rectangular orifice and (iv) Square orifice depending upon their cross-sectional areas.
3. The orifices are classified as (i) Sharp-edged orifice and (ii) Bell-mouthed orifice depending upon the shape of the upstream edge of the orifices.
4. The orifices are classified as (i) free-discharging orifices and (ii) drowned or submerged orifices depending upon the nature of discharge. The submerged orifices are further classified as (a) Fully submerged orifices and (5) Partially submerged orifices.
The vena contracta is the narrowest point of a jet of fluid after it passes through a constriction, such as a valve or an orifice. At this point, the velocity of the fluid is maximal, and the pressure is minimal, due to the conservation of mass and Bernoulli’s principle.
Q = A x V……(1)
V = √(2gh)……(2)
This is the theoretical velocity. Actual velocity will be less than this value.
a. Co-efficient of Velocity (Cv): is defined as the ratio between the actual velocity of a jet of liquid at vena-contracta and the theoretical velocity of the jet. It is denoted by Cv.
The value of Cv varies from 0.95 to 0.99 for different orifices, depending on the shape, size of the orifice and on the head under which flow takes place. Generally, the value of Cv = 0.98 is taken for sharp-edged orifices.
b. Co-efficient of Contraction (Cc): is defined as the ratio of the area of the jet at vena-contracta to the area of the orifice. It is denoted by Cc.
The value of Cc varies from 0.61 to 0.69 depending on the shape and size of the orifice and the head of liquid under which flow takes place. In general, the value of C. may be taken as 0.64.
c. Co-efficient of Discharge (Cd): is defined as the ratio of the actual discharge from an orifice to the theoretical discharge from the orifice. It is denoted by Cd.
The value of Cd varies from 0.61 to 0.65. For general purpose, the value of Cd is taken as 0.62.
Relation, Cd=Cc x Cv………(*)
DETERMINATION OF CO-EFFIEIENT(a) Coff. of velocity (Cv)
Where,
(b) Coff. of discharge (Cd)
Where,
(c) Coff. of contraction (Cc)
1. Flow through large rectangular orifice
2. Flow through fully submerged orifice
3. Flow through partially submerged orifice
Note:
A mouthpiece is a short length of a pipe which is two to three times its diameter in length, fitted in a tank or vessel containing the fluid. it is used for measuring the rate of flow of fluid.
1. The mouthpieces are classified as
(i) External mouthpiece
(ii) Internal mouthpiece, depending upon their position with respect to the tank or vessel to which they are fitted.
2. The mouthpiece are classified as
(i) Cylindrical mouthpiece
(ii) Convergent mouthpiece
(iii) Convergent-divergent mouthpiece, depending upon their shapes.
3. The mouthpieces are classified as
(i) Mouthpieces running full
(ii) Mouthpieces running free, depending upon the nature of discharge at the outlet of the mouthpiece.
This classification is only for internal mouthpieces which are known Borda’s or Re-entrant mouthpieces.
A mouthpiece is said to be running free if the jet of liquid after contraction does not touch the sides of the mouthpiece.
But if the jet after contraction expands and fills the whole mouthpiece it is known as running full.
Note:
a. The value of Cd, for mouthpiece is more than the value of Cd, for orifice, and so discharge through mouthpiece will be more.
b. Co-efficient of discharge for,
A mouthpiece is a short length of a pipe which is two or three times its diameter in length. If this pipe is fitted externally to the orifice, the mouthpiece is called external cylindrical mouthpiece and the discharge through orifice increases.
Discharge (Q) = = Cd x Area x Velocity = 0.855 x a x √(2gH)
If a mouthpiece converges upto vena-contracta and then diverges then that type of mouthpiece is called Convergent-Divergent Mouthpiece. As in this mouthpiece there is no sudden enlargement of the jet, the loss of energy due to sudden enlargement is eliminated. The coefficient of discharge for this mouthpiece is unity.
Q is given as Q = ac x √(2gH)
where ac = area at vena-contracta.
A short cylindrical tube attached to an orifice in such a way that the tube projects inwardly to a tank, is called an internal mouthpiece. It is also called Re-entrant or Borda’s mouthpiece.
If the length of the tube is equal to its diameter, the jet of liquid comes out from mouthpiece without touching the sides of the tube. The mouthpiece is known as running free.
But if the length of the tube is about 3 times its diameter, the jet comes out with its diameter equal to the diameter of mouthpiece at outlet. The mouthpiece is said to be running full.
a. Borda’s Mouthpiece Running Free
b.Borda’s Mouthpiece Running Full
A notch is a device used for measuring the rate of flow of a liquid through a small channel or a tank. It may be defined as an opening in the side of a tank or a small channel in such a way that the liquid surface in the tank or channel is below the top edge of the opening.
A weir is a concrete or masonary structure, placed in an open channel over which the flow occurs. It is generally in the form of vertical wall, with a sharp edge at the top, running all the way across the open channel.
The notch is of small size while the weir is of a bigger size. The notch is generally made of metallic plate while weir is made of concrete or masonary structure.
1. According to the shape of the opening :
2. According to the effect of the sides on the nappe :
Weirs are classified according to the shape of the opening, the shape of the crest, the effect of the sides on the nappe and nature of discharge.
The following are important classifications.
(a) According to the shape of the opening :
(b) According to the shape of the crest :
(c) According to the effect of sides on the emerging nappe :
A triangular notch or weir is preferred to a rectangular weir or notch due to following reasons :
Note:
5. Fransis’s formula for a rectangular weir is given by:
a. Q = 1.84 × [L − 0.2H] × H3/2 …….for two end contraction
b. Q = 1.84 × L × H3/2 …………………..if contraction are suppressed
c. Q = 1.84 × L × [(H + ha)3/2 − ha3/2] …….if velocity approach is considered
6. Bazins’s formula for a rectangular weir is given by:
a. Q = M × L × √2g × H3/2 …….without velocity approach
b. Q = M × L × √2g × [(H + ha)]3/2 …….without velocity approach
7. A trapezoidal weir with side slope 1:4 is called Cipoletii weir. Discharge through this weir is given by:
8. The discharge over a broad crested weir is given by:
9. The discharge over an ogee weir is given by :
10. The discharge over sub-merged or downed weir is given by:
H = height of water on the upstream side of weir
h = height of water on the downstream side of weir
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