# FLOW OF FLUIDS

## FLUID FLOW

A fluid is a substance that continually deforms (flows)
under an applied shear stress.

Fluids are a subset of the phases of matter and include
liquids, gases.

Fluid flow may be defined as the flow of substances that do
not permanently resist distortion

The subject of fluid flow can be divided into fluid static’s
and fluid dynamics.

## FLUID STATICS

Fluid static’s deals with the fluids at rest in equilibrium

Behavior of liquid at rest

Nature of pressure it exerts and the variation of pressure
at different layers

## FLUID DYNAMICS

Fluid dynamics deals with the study of fluids in motion

This knowledge is important for liquids, gels, ointments
which will change their flow behavior when exposed to different stress
conditions

MIXING ———  FLOW
THROUGH PIPES ———–  FILLED IN
CONTAINER

### Importance of Fluid Dynamics

Identification of type of flow is important in-

Manufacture of dosage forms

The flow of fluid through a pipe can be viscous or turbulent
and it can be determined by Reynolds number.

## REYNOLDS EXPERIMENT

Prof. Osborne Reynolds conducted the experiment in the year
1883.

This   was   conducted
to   demonstrate   the
existence   of   two
types   of   flow
:-

1. Laminar Flow
2. Turbulent Flow

Glass tube is connected to reservoir of water, rate of flow
of water is adjusted by a valve,

A reservoir of colored solution is connected to one end of
the glass tube with help of nozzle. Colored solution is introduced into the
nozzle as fine stream through jet tube.

Turbulent Flow

Laminar Flow

### Types Of Flows Based On Reynold Number –

If Reynold number, RN < 2000 the flow is laminar flow.

If Reynold number, RN > 4000 the flow is turbulent flow.

### REYNOLDS NUMBER

In Reynolds experiment the flow conditions are affected by-

Diameter of pipe

Average velocity

Density of liquid

Viscosity of the fluid

This four factors are combined in one way as Reynolds number

Re= D u ρ
INERTIAL FORCES

ƞ                                VISCOUS FORCES

Inertial forces are due to mass and the velocity of the
fluid particles trying to diffuse the fluid particles

Viscous force if the frictional force due to the viscosity
of the fluid which make the motion of the fluid in parallel.

Reynolds number have no unit

### APPLICATIONS

Reynolds number is used to predict the nature of the flow

Stocks law equation is modified to include Reynolds number
to study the rate of sedimentation in suspension

### BERNOULLI’S THEOREM

When the principals of the law of energy is applied to the flow
of the fluids the resulting equation is a Bernoulli’s theorem

Consider a pump working under isothermal conditions between
points A and B

Bernoulli’s theorem statement, “In a steady state the
total energy per unit mass consists of pressure, kinetic and potential energies
are constant”

At point a one kilogram of liquid is assumed to be entering
at point a, Pressure energy = Pa /g ρA

Where Pa = Pressure at point a

g = Acceleration due to gravity

ρA = Density of the liquid

Potential energy of a body is defined as the energy
possessed by the body by the virtue of its position-

Potential energy = XA

Kinetic energy of a body is defined as the energy possessed
by the body by virtue of its motion, kinetic energy = UA2 / 2g

Total energy at point A = Pressure energy + Potential energy
+ K. E

Total energy at point A = PaV + XA + UA2 / 2g

According to the Bernoulli’s theorem the total energy at
point A is constant

Total energy at point A = PAV +XA + (UA2 / 2g) = Constant

After the system reaches the steady state, whenever one kilogram
of liquid enters at point

A, another one kilogram of liquid leaves at point B

Total energy at point B = PBV +XB + UB2 / 2g

PAV +XA + (UA2/2g) + Energy added by the pump = PBV +XB +
(UB2/2g) V is volume and it is reciprocal of density.

During the transport some energy is converted to heat due to
frictional Forces

Energy loss due to friction in the line = F

Energy added by pump = W

Pa /ρ A +XA + UA2 / 2g – F + W = PB /ρ B
+XB + UB2 / 2g

This equation is called as Bernoulli’s equation

### ENERGY LOSS –

According to the law of conservation of energy, energy
balance have to be properly calculated. Fluids experiences energy losses in
several ways while flowing through pipes, they are

Frictional losses

Losses in the fitting

Enlargement losses

Contraction losses

### Application of BERNOULLI’S THEOREM

Used in the measurement of rate of fluid flow using flow
meters

It applied in the working of the centrifugal pump, in this
kinetic energy is converted in to pressure.

## MANOMETERS

Manometers are the devices used for measuring the pressure difference.
Different type of manometers are

Simple manometer

Differential manometer

Inclined manometer

## Simple manometer

This manometer is the most commonly used one

It consists of a glass U shaped tube filled with a liquid

A- of density ρA kg /meter cube and above A the arms are
filled with liquid B of density ρB

The liquid A and B are immiscible and the interference can
be seen clearly

If two different pressures are applied on the two arms, the
meniscus of higher than the other

Let pressure at point 1 will be P1 Pascal’s and point 5 will
be P2 Pascal’s

The pressure at point 2 can be written as

=P1+ (m + R )ρB g

since ∆P = ∆ h ρ g (m + R ) = distance from 3 to 5

Since the points 2 and 3 are at same height the pressure

Pressure at 3 =P1+ (m + R ) ρ B g

Pressure at 4 is less than pressure at point 3 by R ρA g

Pressure at 5 is still less than pressure at point 4 by mρ B
g

This can be summarise as

P1 + (m + R ) ρ B g – R ρA g – mρ B g= P2

∆P= P1-P2=R (ρ A- ρ B )g

### Application

Pressure difference can be determined by measuring R

Manometers are use in measuring flow of fluid.

## DIFFERENTIAL MANOMETERS

These manometers are suitable for measurement of small
pressure differences

It is also known as two – Fluid U- tube manometer

It contains two immiscible liquids A and B having nearly
same densities

The U tube contains of enlarged chambers on both limbs,

Using the principle of simple manometer the pressure
differences can be written as

∆P =P1 –P2 =R (ρc – ρA)g

## INCLINED TUBE MANOMETERS

Many applications require accurate measurement of low pressure
such as drafts and very low differentials, primarily in air and gas
installations.

In these applications the manometer is arranged with the
indicating tube inclined,

This enables the measurement of small pressure changes with
increased accuracy.

P1 –P2 = g R (ρ A – ρ B) sin α

To measure small pressure differences need to magnify Rm some
way.

## ORIFICE METER

### Principle

Orifice meter is a thin plate containing a narrow and sharp
aperture.

When a fluid stream is allowed to pass through a narrow constriction
the velocity of the fluid increase compared to up stream

This results in decrease in pressure head and the difference
in the pressure may be read from a manometer

### CONSTRUCTION

It is consider to be a thin plate containing a sharp
aperture through which fluid flows

Normally it is placed between long straight pipes

For present discussion plate is introduced into pipe and
manometer is connected at points A and B

### Working

When fluid is allowed to pass through the orifice the velocity
of the fluid at point B increase, as a result at point A pressure will be
increased.

Difference in the pressure is measured by manometer

Bernoulli’s equation is applied to point A and point B for
experimental conditions

Total energy at point A = Pressure energy + Potential energy
+ K. E Total energy at point A = PaV + XA + UA2 / 2g

Bernoullis eqn… Pa /ρ A +XA + UA2 / 2g – F + W = PB /ρB
+XB + UB2 / 2g

### Assumptions

Pipeline is horizontal A and B are at same position
Therefore XA=XB

Suppose friction losses are negligible F=0

As liquid is incompressible so density remain same,
Therefore ρ A=ρ B=ρ

No work is done on liquid therefore w=0

After applying assumptions Bernaoulis eqn…

PA /ρ A +XA + UA2 / 2g – F + W = PB /ρ B +XB + UB2 / 2g

Change to—

PA /ρ + UA2 / 2g = PB /ρ + UB2 / 2g

UA2 / 2g – UB2 / 2g = PB /ρ – PA /ρ

Multiply both sides by -2g

UB2 – UA 2= 2g.PA /ρ – 2g.PB/ρ

√UB2 – UA2 = √2g/ρ . (PA – PB)

√UB2 – UA2 = √2g∆H         …….. as (PA – PB)/ρ=∆H

√UB 2 – UA2 = √2g∆H

Diameter of vena contracta is not known practically

There are friction losses so above equation is modified to—

√U02 – UA2 =C0 √2g. ∆H

If the diameter of orifice is 1/5th of the diameter of pipe
then UA 2 is negligible

The velocity of the fluid at thin constriction may be
written as –

U0 = C0 √ 2g ∆H

∆H = Difference in pressure head, can be measured by
manometer

C0 = constant c-oefficient of orifice (friction losses)

U0 = velocity of fluid at the point of orifice meter

### Applications

Velocity at either of the point A and B can be measured

Volume of liquid flowing per hour can be determined by
knowing area of cross section.

## VENTURI METER

### Principle

When fluid is allowed to pass through narrow venturi throat
then velocity of fluid increases and pressure decreases

Difference in upstream and downstream pressure head can be
measured by using Manometer

U v = C v √ 2g . ∆H

Why Venturi meter if Orifice meter is available?

Main disadvantage of orifice meter is power loss due to
sudden contraction with consequent eddies on other side of orifice plate

We can minimize power loss by gradual contraction of pipe

Venturi meter consist of two tapperd (conical section)
inserted in pipeline

Friction losses and eddies can be minimized by this
arrangement.

For permanent installations

Power loss is less

Expensive

Need technical export

Not flexible it is permanent

## PITOT TUBE

A pitot tube is a pressure measurement instrument used to
measure fluid flow velocity.

The pitot tube was invented by the French engineer Henri
Pitot in the early 18th century and was modified to its modern form in the
mid-19th century by French scientist Henry Darcy.

It is widely used to determine the airspeed of an aircraft,
water speed of a boat, and to measure liquid, air and gas velocities in
industrial applications.

The pitot tube is used to measure the local velocity at a
given point in the flow stream and not the average velocity in the pipe or
conduit

### CONSTRUCTION

It is also known as insertion meter

The size of the sensing element is small compared to the
flow channel

One tube is perpendicular to the flow direction and the
other is parallel to the flow

Two tubes are connected to the manometer

2g∆Hp = U2

### WORKING

A pitot tube is simply a small cylinder that faces a fluid so
that the fluid can enter it.

Because the cylinder is open on one side and enclosed on the
other, fluid entering it cannot flow any further and comes to a rest inside of
the device.

A diaphragm inside of the pitot tube separates the incoming
pressure (static pressure) from the stagnation pressure (total pressure) of a
system.

The difference between these two measurements determines the
fluid’s rate of flow.

In industry, the velocities being measured are often those
flowing in ducts and tubing where measurements by an anemometer would be difficult
to obtain.

In these kinds of measurements, the most practical
instrument to use is the pitot tube.

The pitot tube can be inserted through a small hole in the
duct with the pitot connected to a U-tube water gauge or some other
differential pressure gauge for determining the velocity inside the ducted wind
tunnel.

One use of this technique is to determine the volume of air
that is being delivered to a conditioned space.

Pitot tubes measure pressure levels in a fluid.

They do not contain any moving parts and routine use does
not easily damage them.

Also, pitot tubes are small and can be used in tight spaces
that other devices cannot fit into.

Foreign material in a fluid can easily clog pitot tubes and
disrupt normal readings as a result.

This is a major problem that has already caused several
aircraft to crash and many more to make emergency landings

## ROTAMETER

### PRINCIPLE

It is a variable area meter which works on the principle of
upthurst force exerted by fluid and force of gravity

### Construction

It consists of vertically tapered and transparent tube
generally made of glass in which a plummet is centrally placed with guiding
wire.

Linear scale is etched on glass

During the flow the plummet rise due to variation in flow

The upper edge of the plummet is used as an index to note

No external power or fuel.

Manufactured of cheap materials.

Since the area of the flow passage increases as the float
moves up the tube, the scale is approximately linear.