**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

Handling of drugs for administration

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.

**Types of flow- **

Turbulent Flow

Laminar Flow

**Types Of Flows Based**

On Reynold Number –

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

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

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

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

U_{B}2 – U_{A} 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.

**ADVANTAGES**

For permanent installations

Power loss is less

Head loss is negligible

**DISADVANTAGES**

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.

**Advantages:**

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.

**Disadvantages:**

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

the reading

**Advantages**:

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.

**Disadvantages:**

Impact of gravity.

Accuracy of rotameter.

Uncertainty of the measurement