chapter 2 fliud flow - 12 marks fluid flow.pdf · fluid mechanics & machinery mr. r. r. dhotre...

Fluid Mechanics & Machinery

MR. R. R. DHOTRE (8888944788) 7

Chapter 2

FLIUD FLOW - 12 MARKS

Course Contents

1. Introduction

2. Types of Fluid flow

2.1 Steady and Unsteady Flow

2.2 Uniform and non-uniform flow

2.3 Laminar and turbulent flow

2.4 Compressible flow and incompressible flow

2.5 Rotational and irrotational flow

2.6 One, two and three dimensional flow

3. Rate of Flow or Discharge

4. Continuity Equation

5. Energy of fluid motion and total energy

6. Bernoulli’s Theorem

7. Venturimeter

8. Orifficemeter

9. Venacontracta

10. Hydraulic Coefficient

11. Pitot tube


MR. R. R. DHOTRE (8888944788) Page 18

1. Introduction

- Kinematics is defined as that branch of science which deals with motion of fluid particles

without considering the forces causing the motion. The velocity at any point in a flow field at

any time is studied in this branch of fluid mechanics.

The fluid motion is described by two methods. (1) Langrangian method (2) Eulerian Method

1. Langrangian Method

-In Langrangian method a single fluid particle is followed during its motion and its velocity,

acceleration, density etc., are described.

2. Eulerian Method

-In Eulerian Method, the velocity, acceleration, density etc., are described at point in flow field.

2. Types of Fluid flow

2.1 Steady and Unsteady Flow

-Steady flow is defined as that type of in which the fluid characteristics like Velocity, pressure,

Density etc. at a point do not change with time.

Mathematically,

,

,

-Unsteady flow is defined as that type of in which the fluid characteristics like Velocity,

pressure, Density etc. at a point changes with respect to with time.

Mathematically,

,

,

2.2 Uniform and Non-uniform Flow

-Uniform flow is defined as that type of in which the velocity at any given time does not change

with space (length of direction of flow).

Mathematically,

.

-Non-Uniform flow is defined as that type of in which the velocity at any given time changes

with space (length of direction of flow).

Mathematically,

.

2.3. Laminar and Turbulent Flow

-Laminar flow is defined as that type of in which the fluid particles moves along well defined

paths or stream line 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

laminar flow or stream line flow or viscous flow.


MR. R. R. DHOTRE (8888944788) Page 19

-Turbulent flow is defined as that type of in which the fluid particles moves in zig-zag way.

Thus eddies formation takes place which are responsible for high energy loss.

For pipe flow, the type of flow can be determined by Reynolds number (Re)

Reynolds number is given by,

Re =

Where, ρ = Density of fluid

V = Velocity of fluid

D = Diameter of pipe

μ = Dynamic viscosity of fluid

If the Reynolds number is less than 2000, the flow is called laminar.

If the Reynolds number is more than 2000, the flow is called turbulent.

If the Reynolds number lies between 2000 and 4000 the flow is called transition flow.

24. Compressible and Incompressible Flow

-Compressible flow is defined as that type of in which the density of fluid changes from point to

point or density is not constant for the fluid. i.e ρ Constant (for gases).

-Incompressible flow is defined as that type of in which the density of fluid is constant for the

fluid. i.e ρ Constant (for liquids).

2.5 Rotational and Irrotational Flow

-Rotational flow is defined as that type of in which the fluid particles while flowing along

stream lines, also rotates about their own axis.

-Irrotational flow is defined as that type of in which the fluid particles while flowing along

stream lines, do not rotates about their own axis.

2.6 One, Two and Three Dimensional Flow

-One dimensional flow is defined as that type of in which the fluid parameter such as velocity is

function of time and one space co-ordinate only say x.

Mathematically, u = f (x), v = 0 and w = 0.

e.g. Flow of fluid in pipe

-Two dimensional flow is defined as that type of in which the fluid parameter such as velocity is

function of time and two rectangular space co-ordinate only say x and y .

Mathematically, u = f1(x, y), v = f2(x, y), and w = 0.

e.g. The viscous flow between parallel plate, flow over a long spillway, flow below long weirs.


MR. R. R. DHOTRE (8888944788) Page 20

-Three dimensional flow is defined as that type of in which the fluid parameter such as velocity

is function of time and three mutually perpendicular directions say x, y and z.

Mathematically, u = f1(x, y, z), v = f2(x, y, z), and w = f3(x, y, z).

e.g. Flow of fluid in converging and diverging pipes, flow of water in open channel.

3. Rate of Flow or Discharge (Q)

Rate of flow or discharge is defined as the quantity of fluid flowing per second through a

section of a pipe. It is denoted by „Q‟. SI unit is m3/s.

Q =

Q = A.V

4. Continuity Equation

-The equation based on the principle of conservation of mass is called continuity equation. Thus

for fluid flowing through the pipe at all the cross sections, the quantity of fluid per second is

constant.

Consider two cross section of pipe as shown in Fig.

Let, ρ1 = Density at section A-B

A1 = Area of pipe at section A-B

V1 = Average velocity at cross section A-B

And ρ2, V2, A2 are corresponding values at section C-D.

Volume flow rate = A.V

Density ρ = Mass/Volume

Mass = ρ. Volume

Mass flow rate = ρ. Volume/ time

Mass flow rate = ρ. A. V

Then rate of flow at section A-B = ρ1 A1 V1


MR. R. R. DHOTRE (8888944788) Page 21

Rate of flow at section C-D = ρ2 A2 V2

According to law of conservation of mass,

Rate of flow at section A-B = Rate of flow at section C-D

Since Mass flow rate at any section is constant, Hence, ρ. A. V = Constant.

ρ1 A1 V1 = ρ2 A2 V2 = Constant

Above equation is applicable to compressible as well as incompressible fluid and is known as

Continuity equation.

If the fluid is incompressible then ρ1= ρ2 and continuity equation reduced to,

A1 V1 = A2 V2 = Constant

5. Energy of fluid motion and total energy

Energy is capacity to do work.

Three forms of energy are important for fluid in motion. These are

1) Potential energy- Represent the energy possessed by liquid due to its position. It is also

called datum energy. It is equal to datum head or potential head and denoted by Z meters.

2) Pressure energy- Represent the energy possessed by the fluid particle due to existing

pressure.

Pressure head =

3) Kinetic energy- Represent the energy possessed by liquid due to its velocity or motion.

Kinetic head or velocity head=

4) Total Energy- E= Potential energy + Pressure energy + Kinetic energy

Total head = potential head + Pressure head + Kinetic head

=

+

+ Z meters

6. Bernoulli’s Theorem

“In a steady, ideal and incompressible fluid flow, the total energy at any point of the fluid is

constant.”

The total energy consists of pressure energy, kinetic energy and potential energy.

Assumptions-

1. The flow is steady and continuous.

2. The fluid is an ideal fluid i.e viscosity is zero.

3. The flow is incompressible and laminar.

4. The flow is irrotational.

5. Gravity and pressure forces are only considered.


MR. R. R. DHOTRE (8888944788) Page 22

Bernoulli’s Theorem for ideal and real fluid

Consider the flow of an ideal fluid in a pipe line.

Consider two sections 1-1 and 2-2.

Let P, V, Z are the pressure, velocity and datum

head at respective section.

1. For ideal fluid

Thus the Bernoulli‟s equation for ideal fluids

Between point 1 and 2 is given as-

+

+ Z1 =

+

+ Z2

2. For real fluid

All the viscous fluids are viscous and hence offer resistance to flow.

Thus there is always some loss in fluid flows and hence in the application of Bernoulli‟s

equation, these losses have to be considered.

Thus the Bernoulli‟s equation for ideal fluids

Between point 1 and 2 is given as-

+

+ Z1 =

+

+ Z2 + hL

Where, hL is loss of energy between points 1 and 2.

Application of Bernoulli’s Theorem

Sr.

No. Application Use

1 Venturimeter To measure discharge (Q)

2 Orificemeter To measure discharge (Q)

3 Pitot tube To measure velocity (V)

7. Venturimeter

Use- It is a device used for measuring the rate of a flow of a fluid flowing through a pipe.

Principle- It based on Bernoulli‟s theorem. In a fluid flow, by reducing cross section area, the

velocity increases and pressure decrease. As a result, pressure difference is created. This

pressure difference is helpful to determine discharge.

Construction- It consists of three parts 1. A short converging part 2. Throat 3. Diverging part


MR. R. R. DHOTRE (8888944788) Page 23

It consist of a short converging conical tube which has total included angle of 210 ± 1

0

leading the cylindrical portion, called throat. Diameter of throat varies between

to

of main

pipe diameter. It followed by a long diverging section know as diffuser has included angle is 50

to 70. The length of divergent section is 3 to 4 times length of convergent section, to minimize

flow separation.

The pressure difference measured by using U tube manometer between the entry at

section 1 and throat at section 2.

Derivation-

Let, d1= diameter at inlet or at section 1,

P1 = pressure at section 1,

V1 = velocity of fluid section 1,

a1 = Area at section 1,

Z1 = Datum head at section 1.

And d2, P2, V2, a2 and Z2 are the corresponding values at section 2.

Applying Bernoulli‟s equation at section 1 and 2, we get

+

+ Z1 =

+

+ Z2

Since venture is horizontal Z1 = Z2

+

=

+

-

=

-


MR. R. R. DHOTRE (8888944788) Page 24

-

is the difference between pressure head at section 1 and 2, and it is denoted by h m odf

fluid passing through pipe.

=

-

----------- (1)

Now apply continuity equation at section 1 and 2,

Substituting this value of V1 in equation (1) and solving, we get,

Now Discharge,

Substituting value of v2 in above equation,

Q is the theoretical discharge under ideal conditions. Actual discharge will be less than

the theoretical discharge. The actual discharge is given by the formula

Where Cd is the coefficient of venturimeter and its value is less than 1. It‟s range from

0.9 to 0.99.

8. Orificemeter

Use- It is a device used for measuring the rate of a flow of a fluid flowing through a pipe.

Principle- It based on Bernoulli‟s theorem. In a fluid flow, by reducing cross section area, the

velocity increases and pressure decrease. As a result, pressure difference is created. This

pressure difference is helpful to determine discharge.


MR. R. R. DHOTRE (8888944788) Page 25

Construction- It consists of flat circular plate which has a circular sharp edge hole called orifice,

which is concentric with the pipe. The orifice meter generally kept 0.5 times diameter of the

pipe, though it may varies from 0.4 to 0.8 times the pipe diameter. Consider an orifice meter is

fitted in a horizontal pipe through which a fluid is flowing as shown fig.

Derivation-

Let, d1= diameter at inlet or at section 1,

P1 = pressure at section 1,

V1 = velocity of fluid section 1,

a1 = Area at section 1,

Z1 = Datum head at section 1.

And d0, P0, V0, a0 and Z0 are the corresponding values at orifice plate.


+

+ Z1 =

+

+ Z0

Since venture is horizontal Z1 = Z0

+

=

+

-

=

-

-

is the difference between pressure head at section 1 and 0, and it is denoted by h m odf

fluid passing through pipe.

=

–

----------- (1)

Now apply continuity equation at section 1 and 2,

a1V1 = a0V0

V1 =


MR. R. R. DHOTRE (8888944788) Page 26

Substituting this value of V1 in equation (1) and solving, we get,

V2 =

√

√

Now Discharge,

Q = a0V0

Substituting value of V2 in above equation,

Qth =

√

√

Q is the theoretical discharge under ideal conditions. Actual discharge will be less than

the theoretical discharge. The actual discharge is given by the formula

Qact = Cd

√

√

Where Cd is the coefficient of Orificemeter and its value is less than 1. It‟s range from

0.60 to 0.65.

Let, h = Head of fluid in meter of fluid,

Let the differential manometer contains a liquid which is heavier than the liquid flowing through

the pipe. (Horizontal Venturimeter)

h = x (

– 1)

Where,

Sh = Specific gravity of heavier liquid

So = Specific gravity of liquid flowing through pipe

X = Difference of heavier liquid in column in U-tube

Comparison between venturimeter and orificemeter-

1. In venturimeter losses are less therefore coefficient of discharge is high, whereas in

Orificemeter thee is no convergent and divergent cones so that losses are more and

coefficient of discharge is less.

2. Venturimeter requires less space for installation compared to Orificemeter.

3. Orifficemeter is simple in construction than venturimeter.

4. Orificemeter is less expensive (costly) than venturimeter.

Application of venturimeter & orificemeter- The flow measurement of

1. Water 2. Industrial waste 3. Slurries and dirty liquids


MR. R. R. DHOTRE (8888944788) Page 27

9. Venacontracta

Fig. shows the flow pattern of liquid. Let 0-0

represent the orifice section and 2-2 represent the

Venacontracta a2 is less than a0.

The liquid flowing from orifice forms a jet whose

area is less than area of orifice at section 2-2 called as

Venacontracta.

Then we define Coefficient of contraction (Cc), as the

ratio of area of Venacontracta to the area of orifice.

Cc =

10. Hydraulic coefficient

The hydraulic coefficients are discussed below:

1. Coefficient of velocity - It is defined as the ratio between the actual velocity of a jet of liquid

at Venacontracta and the theoretical velocity of jet.

Mathematically it can written as,

Cv =

=

=

√

Value of Cv ranges from 0.95 to 0.99.

2. Coefficient of contraction - It is defined as the ratio of the area of the jet at Venacontracta to

the area of the orifice.


Cc =

=

Value of Cc range from 0.615 to 0.64.

3. Coefficient of discharge - It is defined as the ratio of the actual discharge from an orifice to

the theoretical discharge from the orifice.


Cd =

=

Value of Cd ranges from for i) Venturimeter -0.9 to 0.99 ii) Orificemeter – 0.6 to 0.68.

Note: Relation between hydraulic coefficients is given by,

Cd = Cc.Cv


MR. R. R. DHOTRE (8888944788) Page 28

11. Pitot tube

Use- The Pitot tube is used for measure the velocity of flow at any point in a pipe or a channel.

Principal- It is based on the principle that if the velocity of the flow at a point becomes zero, the

pressure head is increase due to velocity head is zero.

In its simplest form, the Pitot tube consists of a glass tube, bent at right angles as shown in

figure.

Diameter of tube is kept larger to minimize capillary effect. One leg called as body is inserted in

pipe horizontally and other leg called stem is vertical and open to atmosphere.


+

+ Z1 =

+

+ Z2

Consider,

Z1 = Z2 as they are at same level,

H = Pressure head at point 1,

V1 = Velocity of fluid point 1,

H + h =Pressure head at point 2,

V2 = Velocity of fluid point 2 = 0 because come to rest at point 2,

Substitute above value in Bernoulli‟s equation,

H +

=h+ H

V1 = √

It represents the theoretical velocity.

Actual velocity V = Cv V1 V = Cv √

Where, Cv = Coefficient of velocity

chapter 2 fliud flow - 12 marks fluid flow.pdf · fluid mechanics & machinery mr. r. r. dhotre...

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