transport phenomena lec3 7

D EPA RTM EN T OF C IV IL EN GIN EER IN G

BITS PILA N I , R A J A S THA N

BY

D R . S HIBA N I KHA N R A J HA

A U GU S T 2 0 1 3

Fluids at Rest:

Pressure and its Effect

Lectures 3, 4, 5, 6, 7

Course: CE F212 Transport Phenomena 3 0 3

1


Topics to be covered

Pressure at a Point

Basic Equation for Pressure

Field

Pressure Variation in a Fluid

at Rest

Incompressible fluid

Compressible fluid

Standard Atmosphere

Measurement of Pressure


2

Manometry

Piezometer Tube

U-Tube Manometer

Inclined Tube Manometer

Hydrostatic force on a Plane Surface

Buoyancy, Floatation and Stability

Archimedes Principle

Stability

Pressure Variation in a Fluid with Rigid-Body Motion

Linear Motion

Rigid-Body Rotation

Fluids at Rest - Pressure and its Effects


3

The study of fluids that are at rest or moving in such a manner that there is no relative motion between adjacent particles is what we are going to discuss in these lectures.

There will be no shearing stress in the fluid and the only forces that develop on the surfaces of the particles will be due to pressure.

Our first concern is to investigate pressure and its variation throughout a fluid and the effect of pressure on submerged surfaces.

Image of hot air balloon

An example1: use of fluid static


4

heated air, which is less

dense than the surrounding

air, is used in hot air balloon

to create an upward buoyant

force. According to

Archimedes Principle, the

buoyant force is equal to the

weight of the air displaced

by the balloon.

Image of hurricane Allen

An example2: use of fluid static


5

Although there is

considerable motion and

structure to a hurricane, the

pressure variation along

vertical plane is

approximated by pressure-

depth relationship for a

static fluid

Fluid Pressure


6

A liquid or gas cannot sustain a

shearing stress - it is only restrained

by a boundary. Thus, it will exert a

force against and perpendicular to

that boundary.

The force F exerted by a fluid on the

walls of its container always acts

perpendicular to the walls.

Water flow

shows F

Fluid Pressure


7

Fluid exerts forces in many

directions. Try to submerge

a rubber ball in water to see

that an upward force acts on

the float.

F

Fluids exert pressure in all

directions.

Common units used for Pressure (ML-1T-2)


8

2

5

5

2

1 1 / [SI]

1 1.01325 10

1 10 1

1 760 760

1 14.7 / .

pascal N m

atmosphere Pa

bar Pa atm

atm mm of Hg torr

atm lb in

Pressure at a Point

The term pressure is used to indicate the normal force per unit area at a given point acting on a given plane within the fluid mass of interest.

(P=F/A)

There are no shearing stresses present in a fluid at rest

The pressure at a point in a fluid at rest or in motion is independent of direction as long as there are no shearing stresses present

This important result is known as Pascals law named in honor of Blaise Pascal (1623-1662), a French mathematician who made important contributions in the field of hydrostatics


9

Pressure at a Point (Pressure is a scalar quantity)

Force balance in the x-direction:


10

How the pressure at a point varies with the orientation of the plane

passing through the point?

Forces on arbitrary wedge

shaped element of fluid

Force balance in the z-direction:

Vertical force on DA Vertical force on

lower boundary Total weight of wedge element

= specific weight

Course: CE F212 Transport Phenomena 3 0 3 11

From last slide:

Divide through by to get the following

Now reduce the element size to a point such that: which finally leads to

This can be done for any orientation , so that following mathematical statement can be given

Course: CE F212 Transport Phenomena 3 0 3 12

0sin2

1D lpp zn

This mathematical statement shows that the pressure is

independent of direction (or it is a scalar quantity)

Basic equation for pressure field

Consider a small rectangular element or differential fluid

element of size dxdydz

There are two types of forces acting on this element:

Surface forces due to the pressure

Body forces due to the weight of the element

Other possible body forces (like magnetic fields) are not

taken into consideration

Pressure may vary across a fluid particle


13


Body Force can be given as

Where

g=acceleration due to gravity

dm=differential mass

=density of fluid

=differential volume of fluid element of size


14

Objective is.

How does the pressure in a fluid in which there are no shearing

stresses vary from point to point?

d dzdydx


Surface Force can be given as

Force balance along y-direction

Force balance along y-direction

With

Substituting the above relations, we get the following

expression for force balance along y-direction


15

RRLLy dApdApFd

RL dAdxdzdA

dxdydzy

pjFd y


Surface Force can be given as

Similarly for x and z directions

Total surface force=

Or finally we can write the following

Total force=body force +surface force

OR


16

dxdydzz

pkFd

dxdydzx

piFd

z

x

zyxS FdFdFdFd


Now applying Newtons Second Law

(for a fluid at rest a=0)

(equation of motion for fluid at rest)

For fluid in motion with acceleration a, general equation of

motion can be written as below


17

ak

p

Pressure Variation in a Fluid at Rest

We begin with general equation of motion

Or in component form we can write

Independent of x-y plane; so pressure in x-y plane remains same; it only depends on z;

Thus the general motion of fluid now can be written as ordinary differential equation as:

The above equation give the Pressure height relation

Pressure only depends on z and decreases as we move upward in a fluid at rest

For liquids or gases at rest the pressure gradient in the vertical direction at any point in a fluid depends only on the specific weight of the fluid at that point


18

ak

p

,,0,0

z

p

y

p

x

p


Incompressible Fluid- CONSTANT DENSITY FLUID

Integration the above equation between two points 1 and 2

h= pressure head

Such distribution of pressure in space is called hydrostatic distribution

If reference pressure is pressure at free surface (atmospheric pressure p0)

Incompressible Fluid Manometers


19

hpp 21


Incompressible Fluid- CONSTANT DENSITY FLUID


20

The following figure helps us to understand how the pressure is same at all points along the line AB even though the container may

have a very irregular shape.

Thus, pressure only depends on depth h, surface pressure p0, specific weight of the liquid in the container,

independent of size or shape of the container


21

Pressure in = Pressure out

Ideal mechanical advantage=

Transmission of Fluid Pressure: Pascals Law

Fout Fin Aout Ain

in out

in out

F F

A A

Pascals Law: An external

pressure applied to an enclosed

fluid is transmitted uniformly

throughout the volume of the

liquid.

in

out

A

A

Transmission of Fluid Pressure: Pascals Law


22

The transmission of pressure throughout a stationary fluid

is the principle upon which many hydraulic devices are

based

Operation of hydraulic jacks

Lifts and presses

Hydraulic controls on aircraft

Other types of heavy machinery

HYDRAULIC JACKS OR COMPRESSED AIR IS USED ON THE LIQUID

SURFACE DIRECTLY AS IS DONE IN HYDRAULIC LIFTS COMMONLY

FOUND IN SERVICE STATIONS


Compressible Fluid-VARYING DENSITY FLUID


23

A compressible fluid is one in which the fluid density changes

accompanied by changes in pressure and temperature.

We mostly think of gases as being compressible; the specific

weights of gases are small so the pressure gradient in the vertical

direction is small and even over greater distances the pressure

remains constant.

This way, attention must be given to the variation in specific

weight.

In order to analyze compressible fluids we use the equation of

state for an ideal gas and combine it with the equation for

pressure variation in fluids at rest:


Compressible Fluid-VARYING DENSITY FLUID

We begin with equation of motion

Since, density is varying; we use ideal gas law

Using ideal gas law in equation of motion

Need additional information, e.g., T(z) for atmosphere

For isothermal condition (T constant over z1 to z2)

This equation provides the desired pressure elevation relationship for an isothermal layer

2

1

2

11

2ln

z

z

p

pT

dz

R

g

p

p

p

dp

0

1212 exp

RT

zzgpp


24

Standard Atmosphere

Standard atmosphere is an idealized representation of mean conditions in the earths atmosphere

For measurement of pressure we need pressure versus altitude over the specific range for the specific conditions (temperature, reference pressure). However, this type of information is not available

Thus, a standard atmosphere has been determined that can be used in the design of aircraft, missiles and spacecraft and in comparing their performance under standard conditions

Currently accepted U.S Standard atmosphere is: Idealized representation of middle-latitude, year round mean conditions of the earths atmosphere


25

Standard Atmosphere

Several important properties of standard atmospheric

conditions at sea level are listed


26

Standard Atmosphere

Temperature decreases with altitude in the region nearest the

earths surface (troposphere), then becomes essentially

constant in the next layer (stratosphere) and subsequently

starts to increase in the next layer


27

Measurement of Pressure

Pressure at a point in the fluid mass is designated as either absolute pressure or gage pressure

Absolute pressure is measured relative to a perfect vacuum (absolute zero pressure)

Gage pressure is measured relative to the local atmospheric pressure

Zero gage pressure corresponds to local atmospheric pressure

Absolute pressures are always positive

Gage pressure can be either positive or negative depending on whether the pressure is above atmospheric pressure (a positive value) or below atmospheric pressure (a negative value)

A negative gage pressure is also referred to as a suction or vacuum pressure


28

Measurement of Pressure:

gage pressure and absolute pressure


29

Measurement of Pressure:

how to measure pressure - mercury barometer

A mercury barometer is used to measure atmospheric

pressure

The column of mercury will come to an equilibrium position

where its weight plus the force due to the vapor pressure

(which develops in the space above the column) balances the

force due to the atmospheric pressure.

For most practical purposes the contribution of the vapor

pressure (0.000023 lb/in2)can be neglected since it is very

small


30

31 2014/8/25 http://zimp.zju.edu.cn/~xinwan/

Mercury Barometer

E. Torricelli (1608-47)

How high will water rise?

No more than h = patm/g = 10.3 m

atmatm

PP gh h

g

If water was used in place of mercury, the

height of column would have to be

approximately 34 ft rather than 29.9 in

(760 mm Hg or 1 atm) mercury for an

atmospheric pressure of 14.7 psi

For mercury, h = 760 mm.

C atm

atm

P gh P

P gh

Manometry

The standard techniques are to use Manometers which are vertical or inclined liquid columns to measure pressure

The mercury barometer is an example of one type of manometer, but there are many other possible configurations

Three common types of manometers include: The Piezometer tube

The U-tube manometer

The inclined-tube manometer

The Mercury Barometric is mostly used to measure atmospheric pressure


32

Piezometer Tube

To determine pressure from a manometer, simply use the fact that the pressure in the liquid columns will vary hydrostatically

The simplest manometer consists of a vertical tube, open at the top and attached to the container in which the pressure is desired

Since manometers involve columns of fluids at rest, the fundamental equation describing their use is as follows

This gives the pressure at any elevation within a homogeneous fluid in terms of a reference pressure and the vertical distance h between p and .

In a fluid at rest, pressure will increase as we move downward and will decrease as we move upward.

Manometry:

Piezometer or Simple Manometer

0php

0p0p


33

Piezometer Tube - Limitations

It is only suitable, if the pressure in the container is greater

than atmospheric pressure, otherwise air would be sucked

into the system

The pressure to be measured must be relatively small so the

required height of the column is reasonable

Also, the fluid in the container in which the pressure is to be

measured must be a liquid rather than a gas

Manometry:

Piezometer or Simple Manometer


34

U-Tube Manometer

To overcome the difficulties aroused in the Piezometer, another type of manometer consisting of U shaped tube is widely used

The fluid in the manometer is called the gage fluid

Better for higher pressures.

Possible to measure pressure in gases.

Manometry:

U-Tube Manometer


35

Manometry:

U-Tube Manometer


36

Find pressure at center of pipe:

Can start either at open end or inside

pipe.

Here we start at open end:

If the fluid in the pipe is gas, the

contribution due to the gas column is

almost negligible and we get the

pressure at 4 as

p at open

end Change in p

from 1 to 2

Change in p

from 3 to 4 p in pipe

hp mp D

Manometry:

U-Tube Manometer


37

The U-tube manometer is also widely used to measure the difference in pressure between two containers or two points in a given system.

Consider a manometer connected between containers A and B as is shown in Fig.

The difference in pressure between A and B can be found by again starting at one end of the system and working around to the other end.

Differential U-tube manometer

Differential Manometer is

Used for measuring pressure differences between desired

points along a pipe

Manometry:

Differential U-Tube Manometer


38

Inclined-Tube Manometer

Inclined-Tube manometers can be used to measure small pressure variation accurately

or

Manometry:

Inclined-Tube Manometer

BA phlhp 332211 sin

113322 sin hhlpp BA


39

If the container A and B

contains the gas, then

or

sin22lpp BA

sin22

BA ppl

Fluid statics: Hydrostatic Force


40

The only stress in fluid statics is normal stress Normal stress is due to pressure

Variation of pressure is only due to the weight of the fluid fluid statics is only relevant in presence of gravity fields.

Applications: Floating or submerged bodies, water dams and gates, liquid storage tanks, etc.

Example of elevation head z converted to

velocity head V2/2g.

We'll discuss this in

more detail in coming

lectures (22-25)

Significance of Hydrostatic Pressure in

Scuba Diving


41

100 ft

1

2

Pressure on diver at 100 ft?

Danger of emergency ascent?

,2 3 2

,2 ,2

1998 9.81 100

3.28

1298.5 2.95

101.325

2.95 1 3.95

gage

abs gage atm

kg m mP gz ft

m s ft

atmkPa atm

kPa

P P P atm atm atm

1 1 2 2

1 2

2 1

3.954

1

PV PV

V P atm

V P atm

Boyles law

If you hold your breath on ascent, your lung

volume would increase by a factor of 4, which

would result in embolism and/or death.

Hydrostatic Force on a Plane Surface

In the determination of the resultant force on an area, the effect of atmospheric pressure often cancels

For a submerged surface in a fluid, forces develop on the surface due to the fluid

Pressure and resultant hydrostatic force developed on the bottom of an open tank


42


Determination of these forces is important in the design of

storage tanks, ships, dams and other hydraulic structures

For fluids at rest, the force must be perpendicular to the

surface since there are no shearing stresses present

Also pressure vary linearly with depth, if the fluid is

incompressible


43

Pressure and resultant hydrostatic force

developed on the bottom of an open tank

Hydrostatic Force on a Plane Surface-

Analysing forces on different surfaces

For a horizontal surface

such as the bottom of a liquid filled tank, the magnitude of the resultant force is simply

Where p is the uniform pressure on the bottom and A is the area of the bottom

For the open tank ,

If atmospheric pressure acts on both sides of the bottom, the resultant force on the bottom is simply due to the liquid in the tank

Since the pressure is constant and uniformly distributed over the bottom, the resultant force acts through the centroid of the area

Pressure on the sides of the tank is not uniformly distributed

pAFR

hp


44

Pressure and resultant

hydrostatic force developed

on the bottom of an open

tank

Hydrostatic Force on a Plane Surface-

Analyzing forces on different surfaces


45

Pressure on the sides of the

tank is not uniformly

distributed

CP

C

Pressure distribution on the sides

of an open tank


For inclined submerged plane surface:

We assume that the fluid surface is open to the atmosphere

Let the plane in which the surface lies intersect the free surface at 0 and make an angle with this surface

The x-y coordinate is defined so that 0 is the origin and y is directed along the surface


46


For inclined submerged plane surface:

The area can have an arbitrary shape

We need to determine the direction, location and magnitude of the resultant force acting on one side of this area due to the liquid in contact with the area


47


At any depth, h, the force acting on dA (the differential area) is

Thus, the magnitude of the resultant force can be found by summing

these differential forces over the entire surface as given below

Where

For constant and

The integral appearing in above equation, is the first moment of the

area with respect to the x-axis, so we can write

hdAdF

AA

R dAyhdAdFF sin

sinyh

A

R ydAF sin

AyydA cA


48


Where is the y coordinate of the centroid of area A measured from the x-axis which passes through 0. Thus the resultant force can be written as

or more simply as

Where is the vertical distance from the fluid surface to the centroid of the area

Note:

The magnitude of the force is independent of angle of inclination and depends only on the specific weight of the fluid, the total area, and the depth of the centroid of the area below the surface

The magnitude of the resultant force is the pressure at the centroid multiplied by the total area

cy

sincR AyF

AhF cR


49

ch


The y coordinate, , of the resultant force can be determined by

summation of moments around the x-axis. That is the moment of the

resultant force must equal the moment of the distributed pressure

force, or

And ,therefore

The integral in the numerator is the second moment of the area (moment of inertia), Ix, with respect to an axis formed by the

intersection of the plane containing the surface and the free surface (x-

axis). Thus, we can write

dAyydFyFAA

RR

2sin

Ry

sincR AyF

Ay

dAy

yc

AR

2

Ay

Iy

c

xR


50


Use can now use the parallel axis theorem to express Ix as

Where , is the second moment of the area with respect to an axis passing through its centroid (C) and parallel to the x-axis. Thus,

The above relation shows that the resultant force does not pass through the centroid but is always below it, since .

The x coordinate, , for the resultant force can be determined in a similar manner by summing moments about the y-axis. Thus,

and, therefore, where is the product of inertia with respect to the x and y axes

2

cxcx AyII

xcI

c

c

xcR y

Ay

Iy

0/ AyI cxc

Rx

xydAxFA

RR sin

Ay

I

Ay

xydA

xc

xy

c

AR

xyI


51


Using the parallel axis theorem, we can write

where is the product of inertia with respect to an orthogonal

coordinate system passing through the centroid of the area and formed

by a translation of the x-y coordinate system

The point through which the resultant force acts is called the center of

pressure (CP) (xR, yR)

c

c

xyc

R xAy

Ix

xycI


52



53

The following diagram shows the geometric properties of some common shapes:



In the figure, we see that the

difference between the weight in AIR

and the weight in WATER is 3 lbs.

This is the buoyant force that acts

upward to cancel out part of the

force. If you were to weigh the water

displaced; it also would weigh 3 lbs.


54

B fluid fluid

object

F Vg m g

mg Vg




55

Buoyant force: A resultant body force that is generated when a stationary body is completely or partially submerged in a fluid.

Archimedes principle: A body wholly or partly immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. The buoyant force can be considered to act vertically upward through the center of gravity of the displaced fluid.

FB = buoyant force = weight of displaced fluid



A net upward vertical force results because pressure increases with depth and the pressure forces acting from below are larger than the pressure forces acting from above

Consider the forces F1, F2, F3 and F4 are simply the forces exerted on the plane surfaces of the parallelepiped, W is the weight of the shaded fluid volume, and FB is the force the body is exerting on the fluid.

The forces on the vertical surfaces, such as F3 and F4 are all equal and cancel, so the equilibrium equation of interest is in the z direction and can be expressed as

WFFFB 12Course: CE F212 Transport Phenomena 3 0 3

56



If the specific weight of the fluid is constant, then

Where A is the horizontal area of the upper (or lower) surface of the parallelepiped, can be written as

Simplifying, we arrive at the desired expression for the buoyant force

Where is the specific weight of the fluid and is the volume of the body.

The direction of the buoyant force is directed vertically upward and the magnitude is equal to the weight of the fluid displaced by the body

This result is commonly referred as Archimedes principle in honour of Archimedes (287-212 B.C.), a Greek mechanician and mathematician who first enunciated the basic ideas associated with hydrostatics

AhhFF 1212

AhhAhhFB 1212

BF


57



The location of the line of action of the buoyant force can be determined by summing moments of the forces shown on the free body diagram wrt some convenient axis

And on substitution for the various forces

Where is the total volume

The right hand side of above equation, is the first moment of the displaced volume w r t x-z plane so that is equal to the y coordinate of the centroid of the volume

Buoyant force passes through the centroid of the displaced volume. The point through which the buoyant force acts is called the center of buoyancy

21112 WyyFyFyF cB

T Ahh 12

cy


58

21 yyy TTc


Stability

The stability of a body can be determined by considering what happens when it is displaced from its equilibrium position

A body is said to be in a stable equilibrium position if when displaced it returns to its equilibrium position

It is said to be in unstable equilibrium position if when displaced, even slightly, it moves to a new equilibrium position

Stability considerations are particularly important for submerged or floating bodies since the centers of buoyancy and gravity do not necessarily coincide.


59


Stability

A small rotation can result in either a restoring or overturning couple

For the completely submerged body, which has a center of gravity below the center of buoyancy, a rotation from its equilibrium position will create a restoring couple formed by the weight W and the buoyant force FB, which causes the body to rotate back to its original position

If center of gravity falls below the center of buoyancy, the body is in stable equilibrium position

If center of gravity falls above the center of buoyancy, the body is in unstable equilibrium position


60


Stability

For floating bodies the stability problem is more

complicated, since as the body rotates the location of the

center of buoyancy (which passes through the centroid of the

displaced volume) may change.


61


Even though a fluid may be in motion,

if it moves as a rigid body there will be

no shearing stresses present

The general equation of motion

(discussed earlier)

A general class of problems involving

fluid motion in which there are no

shearing stresses occurs when a mass

of fluid undergoes rigid body motion

ak p

z

y

x

az

p

ay

p

ax

p


62


Linear Motion

We first consider an open

container of a liquid that is

translating along a straight path

with a constant acceleration

Since , it follows that the

pressure gradient in the x-

direction is zero .

z

y

agz

p

ay

p

a

0xa

0/ xp


63

Linear acceleration

of a liquid

With free surface


Linear Motion: In the y and z directions

The change in pressure between two closely spaced points

located at y, z and y+dy, z+dz can be expressed as

or in terms of acceleration

Along a line of constant pressure, , and therefore

from the above equation it follows that the slope of this line

is given by the relationship

dzz

pdy

y

pdp

dzagdyadp zy

0dp

z

y

ag

a

dy

dz


64


Linear Motion

Along a free surface the pressure is constant, so that for the accelerating mass, the free surface will be inclined if

Additionally, all lines of constant pressure will be parallel to the free surface

For the special circumstances in which , which corresponds to the mass of fluid accelerating in the vertical direction

indicates that the fluid surface will be horizontal

The pressure distribution is not hydrostatic, but is given by the equation

zagdz

dp

z

y

ag

a

dy

dz

0ya

0,0 zy aa


65


Linear Motion

For fluids of constant density this equation shows that the

pressure will vary linearly with depth, but the variation is

due to the combined effects of gravity and the externally

induced acceleration

What is the pressure gradient for a freely falling fluid

mass???

Example: the pressure throughout a blob of a juice

floating in an orbiting space shuttle (a free fall) is ???

The only force holding the liquid together is ???


66 zagdz

dp


Rigid-Body Rotation of a liquid in a tank

A fluid contained in a tank that is rotating with a constant angular velocity about an axis will rotate as a rigid body

In terms of cylindrical coordinates the pressure gradient can be expressed as

Thus, in terms of this coordinate system


67

p

zrz

pp

rr

pp eee

1

0

0

2

z

rr r

a

a

ea


Rigid-Body Rotation

Thus,

This type of rigid body rotation shows that the pressure is a

function of two variables r and z and therefore differential

pressure is

or


68

z

p

p

rr

p

0

2

dzdrrdp 2

dzz

pdr

r

pdp

It is greater than ZERO. Thus, the pressure increases in the radial direction because of

centrifugal acceleration.


Rigid-Body Rotation

Along a surface of constant pressure, such as the free surface, , so that we can write

The equation for surface of constant pressure is

Integrating the pressure equation, we get

or Surfaces of constant pressure are parabolic.

They are curved rather than flat.

This result shows that the pressure varies with the distance from the axis of rotation, but at a fixed radius, the pressure varies hydrostatically in the vertical direction


69

0dp

tconsg

rz tan

2

22

dzrdrdp 2

tconszr

p tan2

22

g

r

dr

dz 2

Examples of Rigid Body Motion


70

Summary of the lectures 3-7

Pascals law

Surface force

Body force

Incompressible fluid

Hydrostatic pressure distribution

Pressure head

Compressible fluid

U.S standard atmosphere

Absolute pressure

Gage pressure


71

Vacuum pressure

Barometer

Manometer

Centre of pressure

Buoyant force

Archimedes principle

Centre of buoyancy

Rigid body motion

transport phenomena lec3 7

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