Chapter 3 fluid mechanics ppt

SHROFF S. R. ROTARY INSTITUTE OF CHEMICAL TECHNOLOGY (SRICT)

DEPARTMENT OF MECHANICAL ENGINEERING.

Subject: Applied Thermal and Hydraulic Engineering (ATHE)

Prepared by: Mr. Mitesh Gohil

Chapter 3. Fluid Mechanics

Chapter 3. Fluid Mechanics

3.1 Introduction

3.2 Fluid Properties

3.3 Pressure measuring devices

3.4 Hydrostatic Law

3.5 Continuity Equation

3.6 Momentum Equation

3.7 Euler’s equation of motion along a stream line

3.8 Bernoulli’s equation

3.9 Flow Measuring Devices

3.10 Pitot tube

3.11 Summary

Outline

3.1 Introduction

Fluid mechanics is the branch of engineering science deals with the study

of fluid (Liquid and Gas) at rest as well as in motion.

Application in Engineering: Flow of river, Pump, Fan, Turbine, Air Planes

Jet etc.


1) Density:

It is the ratio of mass of fluid to it’s volume.

𝜌 =𝑚

𝑉=1

𝑣

Weight Density or specific weight (w)

w = 𝜌 𝑔

Specific gravity or Relative density (s)

For gas,

𝑆𝑔𝑎𝑠 =𝜌𝑔𝑎𝑠

𝜌𝑎𝑖𝑟

For liquid,

𝑆𝑙𝑖𝑞. =𝜌𝑙𝑖𝑞.

𝜌𝑤𝑎𝑡𝑒𝑟


2) Viscosity:

It appears that there is a property that represents the internal resistance of a fluid to

motion or the “fluidity,” and that property is the viscosity.

It is the property of fluid by virtue of which it offers resistance to the movement of one

layer of fluid over another adjacent layer of the fluid.

Increase in gas and decrease in liquid with increment of temperature.

Newton’s law of viscosity:

For most fluids the rate of deformation

(and thus the velocity gradient) is

directly proportional to the shear stress 𝜏,

𝜏 ∝𝑑𝑢

𝑑𝑦

𝜏 = 𝜇𝑑𝑢

𝑑𝑦𝜇 = Co-efficient of viscosity

= Dynamic viscosity( 𝑁 𝑠𝑒𝑐 𝑚2)Fig. 3.1 The rate of deformation (velocity

gradient) of a Newtonian fluid


Kinematic viscosity:

It is the ratio of dynamic viscosity to density of fluid.

𝛾 =𝜇

𝜌3) Surface tension:

It is defined as the tensile force acting on the surface of a liquid in contact with a gas

or on the surface between immiscible liquids such that the contact surface behave like

a membranes under tension.

This effect induced due to cohesive (intra molecular) attraction force and adhesion

(inter molecular) attraction force.

Spherical drop of water reveals the surface tension on its surface.

4) Capillary of Meniscus effect:

The phenomenon of liquid rise to fall in a small diameter tube is known as the capillary

or meniscus effect.

Surface tension at surface of liquid creates capillary or meniscus effect.


Capillary rise:

Fig. 3.2 Surface tension (i) Capillary rise (ii) Capillary fall

Surface tension force at surface is balanced by weight force of rise liquid.

2𝜋𝑅𝜎 cos 𝜃 = 𝜋𝑅2ℎ𝜌𝑔

ℎ =2𝜎 cos 𝜃

𝑅𝜌𝑔 Capillary fall

Surface tension force at surface is balanced by hydrostatic force due to above void.

Same way we can prove,

ℎ =2𝜎 cos 𝜃

𝑅𝜌𝑔


5) Pressure

Atmospheric Pressure:

It is pressure exerted by air at sea level (at 15 ºC) is called atmospheric pressure.

1 atm. = 101.325 𝑘𝑁 𝑚2 = 760 mm of Hg. = 10.33 m of water

Measured by mercury column barometer.

Gauge Pressure:

When pressure measured by taking atmospheric pressure as datum (reference) is

called as Gauge Pressure

Vacuum Pressure:

When pressure is below the atmosphere is called as vacuum pressure.

Absolute Pressure:

Pressure measured by taking perfect vacuum as datum is called as gauge pressure.

𝑃𝑎𝑏𝑠. = 𝑃𝑎𝑡𝑚 + 𝑃𝑔𝑎𝑢𝑔𝑒

𝑃𝑔𝑎𝑢𝑔𝑒 is positive while measuring above atmosphere and negative while

measuring vacuum pressure.


Manometer

It is a device used for measuring the pressure at a point in a fluid by balancing

the column of fluid with the same column or another column of fluid.

There are several type of Manometer are available based on their use.

Simple manometers are following explained.

1) Piezometer

Fig. 3.4 Piezometer

attached to a pipe.

A piezometer is a vertical tube, usually transparent, in which

a liquid rises in response to a positive gage pressure.

Pressure in the pipe pushes the water column to a height h,

and the gage pressure at the center of the pipe is which

follows directly from the hydrostatic equation

𝑃𝐴,𝑔𝑎𝑢𝑔𝑒 = 𝜌𝑔ℎ


2) U- tube manometer

A manometer, often shaped like the letter “U,”

is a device for measuring pressure by raising or

lowering a column of liquid.

One end of U-tube is connected to a point at

which pressure is to be measured and other end

open to the atmosphere.

The tube generally contains mercury or any

other liquid known as manometric liquid whose

specific gravity is higher than specific gravity

of liquid whose pressure is to be measured.

In the case shown, positive gage pressure in the

pipe pushes the manometer liquid up a

height ℎ.

We know the pressure at open end of U tube is

1 atm.

Fig. 3.5 The basic U tube

manometer


Fig. 3.5 The basic U tube manometer

By hydrostatic law at point 3,

𝑃3 = 𝑃𝑎𝑡𝑚 + 𝜌𝑚𝑔 ℎ2

In mercury at same level,

𝑃2 = 𝑃3

By hydrostatic law between point 2 & 1,

𝑃1 = 𝑃2 − 𝜌𝐴𝑔 ℎ1

From above three equations,

𝑃1 = 𝑃𝑎𝑡𝑚 + 𝜌𝑚𝑔 ℎ2 − 𝜌𝐴𝑔 ℎ1𝑃1,𝑔𝑎𝑢𝑔𝑒 = 𝜌𝑚𝑔 ℎ2 − 𝜌𝐴𝑔 ℎ1


3) Differential Manometer

The differential manometer is used to

measure the pressure difference between

any two points in a pipeline or in two

pipes or in two containers.

The differential manometer consists of U-

shape bent glass tube and it contains

heavy liquid, whose two ends are

connected to the points, whose pressure

difference is to be measured.

By the hydrostatic law we can find,

𝑃𝐴 + 𝜌𝐴 𝑔 ℎ1 − 𝜌𝑚 𝑔 ℎ2 − 𝜌𝐵𝑔 ℎ3 = 𝑃𝐵

𝑃𝐴 − 𝑃𝐵 = 𝜌𝑚 𝑔 ℎ2 + 𝜌𝐵𝑔 ℎ3 − 𝜌𝐴 𝑔 ℎ1

Fig. 3.6 Differential Manometer


Bourdon Tube Pressure Gauge

The bourdon tube gauge consists of an

elastic element bent in a circular arc, fixed

at one end and free at the other end.

Fixed end is open and attached with

measuring vessel.

Free end is connected with needle via

linkage and gears.

Cross area section of bourdon tube is

elliptical or oval and that will try to

become circular when subjected to

Fig. 3.7 Bourdon Tube Pressure Gauge

This displacement is transfer to the needle to show the pressure on calibration scale.

pressure whilst the circular shape tube will try to become straight and so cause

displacement at the end.

3.4 Hydrostatic Law

The rate of increase of pressure in vertically downward direction is equal to the weight

density of fluid (due to gravity) at that point

Consider a small element of fluid totally surrounded with fluid.

Fig. 3.8 Variation of pressure

vertically under the gravity

Force acting on the fluid element.

1. At surface AB,

𝐹1 = 𝑃1 𝑑𝐴 = 𝑃 𝑑𝐴

2. At surface CD,

𝐹2 = 𝑃2 𝑑𝐴 = 𝑃 +𝜕𝑃

𝜕𝑧𝑑𝑧 𝑑𝐴

3. At Centre of Gravity,

𝑤 = 𝑚𝑔 = 𝜌 𝑑𝐴 𝑑𝑧 𝑔

Since fluid is at rest the element must be in

equilibrium and the sum of all vertical force

must be zero.

𝐹2 − 𝐹1 − 𝑤 = 0

𝜕𝑃

𝜕𝑧= 𝜌 𝑔 Integrate equation, ∆𝑃 = 𝜌 𝑔 ∆𝑧

3.5 Continuity Equation

According to the principle of mass conservation, matter can be neither created nor

destroyed except nuclear processes.

Mass entering in control volume is sum of mass leaving and mass accumulation in

control volume.

𝑚1 = 𝑚2

𝜌1𝐴1𝑉1 = 𝜌2𝐴2𝑉2

For incompressible flow, 𝜌1 = 𝜌2,

𝐴1𝑉1 = 𝐴2𝑉2

Fig. 3.9 Continuous flow through a stream tube

Mass accumulation is zero at steady flow condition.

So, mass flow rate at section 1 is equal to mass leaving at section 2


In fluid mechanics, the momentum of a particle is defined as the product of its mass and

velocity.

Newton’s second law says that the net force acting on a fluid mass is equal to the

change in momentum of flow per unit time in that direction.

𝐹 =𝑑

𝑑𝑡𝑚 𝑉 = 𝑚𝑉

Consider a control volume

ABCD as shown in Fig.

The rate at which momentum enters the control volume across AB is

𝑚1𝑉1

Similarly the rate at which momentum enters the control volume across CD is

𝑚2𝑉2

Fig. 3.10 Momentum in a flowing fluid


Thus the rate of change of momentum across the control volume

𝐹 = 𝑚2𝑉2 − 𝑚1𝑉1

𝐹 = 𝑚(𝑉2 − 𝑉1)

This is the resultant force acting on the fluid element ABCD in the direction of motion.

(∵ 𝑚1 = 𝑚2 = 𝑚)


The following forces are present when the fluid is flowing

(1) Gravity force

(2) Pressure force

(3) Friction force due to Viscosity

(4) Force due to Turbulence force

(5) Force due to compressibility

In Euler’s equation of motion, the forces due to gravity and pressure are considered

and other forces are neglected.


This is derived by considering the motion of fluid element in S-direction along a

stream line

Fig. 3.11 Force on a fluid element

The forces acting on the cylindrical element are:

(1) Force due to pressure, (𝑃 𝑑𝐴)in the direction of flow

(2) Force due to pressure, 𝑃 +𝜕𝑃

𝜕𝑆𝑑𝑠 𝑑𝐴 opposite to the direction of flow

(3) Weight of element(𝜌 𝑔 𝑑𝐴 𝑑𝑆).


The resultant force on the fluid element in the direction of S must be equal to the mass

of fluid element × acceleration in the direction S.

𝑃𝑑𝐴 − 𝑃 +𝜕𝑃

𝜕𝑆𝑑𝑠 𝑑𝐴 − 𝜌 𝑔 𝑑𝐴 𝑑𝑠 cos 𝜃 = 𝜌 𝑑𝐴 𝑑𝑠 𝑎𝑠

𝑎𝑠 = Acceleration in direction of S.

𝑎𝑠 =𝑑𝑉

𝑑𝑡=𝜕𝑉

𝜕𝑠

𝑑𝑠

𝑑𝑡+𝜕𝑉

𝜕𝑡= 𝑉

𝜕𝑉

𝜕𝑠+𝜕𝑉

𝜕𝑡

If the flow is steady𝜕𝑉

𝜕𝑡= 0,

𝑎𝑠 = 𝑉𝜕𝑉

𝜕𝑠

Simplify above equation by putting 𝑎𝑠 & cos 𝜃

𝜕𝑃

𝜌+ 𝑉𝑑𝑉 + 𝑔𝑑𝑍 = 0

(Where V is a

function of S and t)

This is known as Euler’s equation of motion along a streamline.

cos 𝜃 =𝑑𝑧

𝑑𝑠&


Bernoulli’s theorem states as follows:

“In an ideal, incompressible fluid when the flow is steady and continuous, the sum of

pressure energy, kinetic energy and potential energy is constant along a stream line.”

Bernoulli’s equation is obtained by integrating the Euler’s equation of motion.

𝑃

𝜌𝑔= Pressure head

𝑉2

2𝑔= Kinetic head

𝑍 = Potential or Elevation head

𝜕𝑃


𝑃

𝜌+𝑉2

2+ 𝑔𝑍 = 𝑐𝑜𝑛𝑠𝑡.

𝑃

𝜌𝑔+𝑉2

2𝑔+ 𝑍 = 𝑐𝑜𝑛𝑠𝑡.


Assumption:

(1) Fluid is ideal, i.e. inviscid, incompressible

(2) Flow is steady, one-dimensional, uniform, irrational

(3) Forces that considered are only gravity and pressure.

All the real fluids are viscous and hence offer resistance to flow. So, Bernoulli’s by

considering losses,

𝑃1𝜌𝑔

+𝑉1

2

2𝑔+ 𝑍1 =

𝑃2𝜌𝑔

+𝑉2

2

2𝑔+ 𝑍2 + ℎ𝐿

ℎ𝐿 = Frictional head loss


1) Venturimeter

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

Three Part (1) Converging Prat (2) Throat (3) Diverging part

Fig. 3.12 Venturimeter


Flow rate in Venturimeter, Orifice and Nozzle meter are find by Bernoulli’s equation.

Continuity equation (conserve volume in the case of an incompressible fluid)

𝐴1𝑉1 = 𝐴2𝑉2

𝑉1 =𝐴2𝑉2𝐴1

Bernoulli’s principle with same elevation (𝑍1 = 𝑍2)

𝑃1𝜌𝑔

+𝑉1

2

2𝑔=𝑃2𝜌𝑔

+𝑉2

2

2𝑔

From Continuity equation & Bernoulli’s equation we have

𝑉2 =2(𝑃1 − 𝑃2)

𝜌 1 −𝐴2𝐴1

2


Now we know, 𝑄 = 𝐴1𝑉1 = 𝐴2𝑉2

In order to account for viscous effects the relations given above are multiplied by a

quantity Cd called the discharge coefficient and it varies between 0.96 to 0.97.

𝑄 = 𝐶𝑑𝐴22(𝑃1 − 𝑃2)

𝜌 1 −𝐴2𝐴1

2= 𝐶𝑑𝐴2

2𝜌𝑚 𝑔ℎ

𝜌 1 −𝐴2𝐴1

2

∵ 𝑃1 − 𝑃2 = 𝜌𝑚 𝑔ℎ

𝑄 = 𝐴2𝑉2 = 𝐴22(𝑃1 − 𝑃2)

𝜌 1 −𝐴2𝐴1

2


2) Orifice meter

It is cheaper device as compare to venturimeter works on the same principle and for

cross sectional area 1 and 2 we can derive same equation.

𝑄 = 𝐶𝑑𝐴22 ∆𝑃

𝜌 1 −𝐴2𝐴1

2

Fig. 3.13 Orifice meter

Pipe orifice is a device used for

measurement the rate of flow of a

fluid through a pipe. The orifice

meter consists of a thin circular

plate with a hole in it.

The hole diameter is generally

half of pipe diameter. The plate is

held in the pipeline between two

flanges.


It is actually a correction factor that recognizes that the pressure drop developed by

the variable area flow is not all due to change in area alone. We also introduce a

factor, co-efficient of contraction (𝐶𝑐) is a factor equal to the ratio of flow area at

vena contracta (𝐴𝑐) to the orifice area (𝐴2), in the case of orifice plate with the

downstream tap at the vena contracta.

So, Final equation,


𝜌 1 − 𝐶𝑐𝐴2𝐴1

2

In orifice meter, the value of 𝐶𝑑 is much smaller than that for a venturimeter.


3) Nozzle meter

A nozzle meter is shown in Fig.

It is similar like orifice meter but

smooth rounded entrance which

practically eliminates the vena

contracta (𝐶𝑐 = 1).

The non-recoverable loss is large

because there is no diffuser

provided for gradual expansion.

The discharge and other equations for the nozzle meter are the same as for the

venturimeter and can be derived in the same manner


𝜌 1 −𝐴2𝐴1

2

Fig. 3.14 Nozzle meter


Notches are devices used for measuring the rate of flow of a liquid through a small

channel or a tank.

It is an opening in the side of a measuring tank or reservoir extending above the free

surface.

A Weirs is a concrete or masonry structure, placed in an open channel over which the

flow occurs.

It is generally in the form of vertical wall, with a sharp edge.

The notch is of small size while the weir is of a bigger size.

The shape of opening of notch or weir may be rectangular, triangular or trapezoidal.

4) Notches and Weirs


Discharge over a rectangular notch or weir

Fig. 3.15 Side view of Rectangular notch with cross sectional at crest

Consider a sharp edged rectangular notch/weir with crest horizontal and normal direction

of flow as shown in figure.

Consider an elementary horizontal strip of water of given dimension.

Theoretical kinetic energy of flowing fluid through strip is equal to loss of its potential

head.

ℎ =𝑉2

2𝑔

𝑉 = 2𝑔ℎ


Discharge through the strip

𝑑𝑄 = 𝐶𝑑 × 𝐴𝑟𝑒𝑎 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝐶𝑑 𝑑ℎ 𝐿 2𝑔ℎ

Total Discharge (flow rate)

𝑄 = 0

𝐻

𝑑𝑄

= 𝐶𝑑𝐿 2𝑔 0

𝐻

ℎ 1 2𝑑ℎ

= 𝐶𝑑𝐿 2𝑔𝐻 3 2

32

=2

3𝐶𝑑𝐿 2𝑔(𝐻 3 2)


Discharge over a triangular notch or weir

Triangular notch is also called as V-notch.

Considering horizontal strip of water.

Fig. 3.16 Front view of Triangular notch

From figure we can see that area of strip = 2𝐴𝐶 × 𝑑ℎ = (𝐻 − ℎ) tan𝜃

2𝑑ℎ

Discharge through the strip

𝑑𝑄 = 𝐶𝑑 × 𝐴𝑟𝑒𝑎 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 2𝐶𝑑 𝐻 − ℎ tan𝜃

2𝑑ℎ 2𝑔ℎ


Total Discharge (flow rate)

𝑄 = 0

𝐻

𝑑𝑄

= 2 𝐶𝑑 tan𝜃

22𝑔

0

𝐻

𝐻 − ℎ ℎ 1 2𝑑ℎ


22𝑔

0

𝐻

𝐻ℎ 1 2 − ℎ 3 2 𝑑ℎ


22𝑔

𝐻𝐻 3 2

32

−𝐻 5 2

52


22𝑔

𝐻 5 2

32

−𝐻 5 2

52

=8

15𝐶𝑑 tan

𝜃

22𝑔 𝐻 5 2

3.10 Pitot tube

The Pitot tube is used to measure the velocity of a fluid stream and consists of a

simple L shaped tube facing into the incoming flow.

It is based on the principle that if the velocity of flow at a point become zero

(stagnation point), the pressure head is increased due to the conversion of the kinetic

energy into pressure energy.

Fig. 3.17 Pitot tube

By Bernoulli’s equation,

𝑃1𝜌𝑔

+𝑉1

2

2𝑔=

𝑃𝑠𝜌𝑔

+𝑉𝑠2

2𝑔

𝑃1𝜌𝑔

+𝑉1

2

2𝑔=

𝑃𝑠𝜌𝑔

∵ 𝑉𝑠=0

𝑉1 =2(𝑃𝑠 − 𝑃1)

𝜌=

2𝜌𝑚 𝑔ℎ

𝜌

3.11 Summary

Newton’s law of viscosity:

𝜏 = 𝜇𝑑𝑢

𝑑𝑦

Hydrostatic Law ∆𝑃 = 𝜌 𝑔 ∆𝑧

Continuity Equation 𝜌1𝐴1𝑉1 = 𝜌2𝐴2𝑉2

Momentum Equation 𝐹 = 𝑚(𝑉2 − 𝑉1)

Euler’s equation 𝜕𝑃


Bernoulli’s equation𝑃

𝜌𝑔+𝑉2

2𝑔+ 𝑍 = 𝑐𝑜𝑛𝑠𝑡.

𝜇 = Co-efficient of viscosity

Chapter 3 fluid mechanics ppt

Engineering

Transcript of Chapter 3 fluid mechanics ppt