Chapter 03(Thai)

8/12/2019 Chapter 03(Thai)

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Chapter 3: Pressure and Fluid Statics

Eric G. PatersonDepartment of Mechanical and Nuclear Engineering

The Pennsylvania State University

Spring 2005


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Pressure

PressureN/m2

Pascal (Pa)

practice, kilopascal (1 kPa = 103 Pa) and megapascal(1 MPa =

6 .

Other units include bar, atm, kgf/cm2, lbf/in2=psi.

For example, a 70 kg person with a total foot imprint area of

0.03 m2

exerts a pressure of (70x9.807/0.03x1000)kPa =22.9kPa.

Chapter 3: Pressure and Fluid StaticsIE255300 : Fluid Mechanicals2


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Absolute, gage, and vacuum pressures

abs

Most pressure-measuring devices are calibrated to

read zero in the atmosphere, and therefore indicate

= -gage a s atm

Pressure below atmospheric pressure are called

vacuum pressure, Pvac

=Patm

- Pabs

.



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Absolute, gage, and vacuum pressures



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Pressure at a Point

,

and if gives the impression of being a vector.

Pressure at any point in a fluid is the same in all

directions.



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Variation of Pressure with Depth

In the presence of a gravitationalfield, pressure increases with

depth because more fluid restson eeper ayers.

To obtain a relation for the,

consider rectangular elementForce balance in z-direction ives

2 1

0

0

z zF ma

P x P x g x z

= =

=

gives

= = =


s


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Variation of Pressure with Depth

Pressure in a fluid at rest is inde endent of theshape of the container.

plane in a given fluid.



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Scuba Diving and Hydrostatic Pressure



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Scuba Diving and Hydrostatic Pressure

Pressure on diver at100 ft?

( ),2 3 2 1998 9.81 100gage kg m mP gz ft = =

1

.

1298.5 2.95

101.325

atmkPa atm

kPa

= =

100 ft

Danger of emergency

,2 ,2 . .abs gage atm a m a m a m= = =

ascent?

1 1 2 2PV PV =

2

Boyles law1 2

2 1

3.954

1

V P atm

V P atm= = If you hold your breath on ascent, your lung

volume would increase by a factor of 4, which


wou resu n em o sm an or ea .


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Pascals Law

Pascals

.

In picture, pistons are at sameheight:

1 2 2 2

1 2

1 2 1 1P P A A F A= = =

a o 2 1 s ca e eamechanical advantage

Using a hydraulic car jack with a

piston area ratio of IMA = 10, For

example, a person can lift a 1000-kg


car y app y ng a orce o us g


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The Manometer

Manometer

Gravitational effects of gases

are negligible, the pressure

anywhere in the tank and at

pos on as same va ue1 2

2 atmP P gh

=

= +



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Mutlifluid Manometer

For multi-fluid systemsPressure change across a fluid

column of height h is P = gh. ,

decreases upward.

Two points at the same elevation in a

continuous fluid are at the samepressure.

adding and subtractinggh terms.

2 1 1 2 2 3 3 1P gh gh gh P + + + =



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Measuring Pressure Drops

Manometers are well--suited to measurepressure drops across

exchangers, etc.

Relation for pressure

drop P1-P2 is obtained bystarting at point 1 and

terms until we reach point

2.If fluid in pipe is a gas,2>>1 and P1-P2= gh



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The Barometer

Barometer

P

Ccan be taken to be zero since

there is only Hg vapor above

point C, and it is very low relative

to Patm

.

Note that the length and the

cross-sectional area of the tube

column of barometerC atm

P g P

P h

+ =

=



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Fluid Statics

Fluid Statics In fluid statics, there is no relative motion between adjacent

fluid layers.

Therefore, there is no shear stress in the fluid trying to

deform it.

The only stress in fluid statics is normal stress

Variation of pressure is due only to the weight of the

Fluid statics is generally referred to as hydrostatics when


.


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Hoover Dam



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Hoover Dam



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Hoover Dam

Exam le of elevationhead z converted to

velocit head V2/2 .We'll discuss this inmore detail in Cha ter

5 (Bernoulli equation).



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Hydrostatic Forces on Plane Surfaces

On a plane surface, thehydrostatic forces form asystem of parallel forces

,magnitude and location ofapplication, which is

called center ofpressure, must be

Atmospheric pressure

Patm can be neglectedwhen it acts on both sidesof the surface.



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Resultant Force

R completely submerged plate in a homogenous fluid

C centroid of the surface and the areaA of the



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Center of Pressure

Line of action of resultant force

R C the centroid of the surface. In

general, it lies underneath .

Vertical location of Center ofPressure is determined by

resultant force to the momentof the distributed pressure

.,xx C

p C

c

y yy A

= +

$Ixx,C is tabulated for simplegeometries.



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Hydrostatic Forces on Curved Surfaces

FR on a curved surface is more involved since itrequires integration of the pressure forces that

change direction along the surface.Easiest approach: determine horizontal andvertical com onents F and F se aratel .



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Hydrostatic Forces on Curved Surfaces

Horizontal force component on curved surface:H= x. ne o ac on on ver ca p ane g ves y

coordinate of center of pressure on curved.

Vertical force component on curved surface:F =F +W where W is the wei ht of the li uid in

the enclosed block W=gV. x coordinate of thecenter of pressure is a combination of line of

line of action through volume (centroid of

volume .Magnitude of force FR=(FH

2+FV2)1/2

An le of force is = tan-1 F /F



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Buoyancy and Stability

Buoyancy

Buoyancy force FB is

volume fgVdisplaced.

1. bodyfluid: Sinking body



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Example: Galilean Thermometer

Galileo's thermometer is made of a sealedglass cylinder containing a clear liquid.

Suspended in the liquid are a number of,with colored liquid for an attractive effect.

As the li uid chan es tem erature it chan es

density and the suspended weights rise andfall to stay at the position where their density is.

If the weights differ by a very small amount and

ordered such that the least dense is at the toand most dense at the bottom they can form atemperature scale.



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Example: Floating Drydock

Auxiliary Floating Dry Dock Resolute(AFDM-10) partially submerged

u mar ne un ergo ng repa r wor onboard the AFDM-10

Using buoyancy, a submarine with a displacement of 6,000 tons can be lifted!



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Example: Submarine Buoyancy and Ballast

Submarines use both static and dynamic depthcontrol. Static control uses ballast tanks

between the pressure hull and the outer hull.Dynamic control uses the bow and stern planesto generate trim forces.



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which damaged fore ballast tanks



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Dama e to SSN 711

(USS San Francisco)after running aground on8 January 2005.



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Ballast Control Panel: Im ortant station for controllin de th of submarine



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Stability of Immersed and Floating Bodies

Stable :

Neutrally stable :

Unstable :

For an immersed or floating body in static equilibrium,

balance each other and such bodies are inherently

stable in the vertical direction



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Stability of Immersed Bodies

Rotational stability of immersed bodies depends uponre at ve ocat on o center o grav ty an center obuoyancy B.

G above B: unstable

G coincides with B: neutrall stable.



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Stability of Floating Bodies

(G)

(B)

M (Metacenter)

GMG M

GM>0 is Stable

GM


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Rigid-Body Motion

There are special cases where a body of fluid can undergo rigid- ,

container.

In these cases, no shear is developed.

ew on s n aw o mo on can e use o er ve an equa on omotion for a fluid that acts as a rigid body

r

r

In Cartesian coordinates:

( ), ,x y xP P P

a a g ax y z

= = = +



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Linear Acceleration

Container is moving on a straight path0, 0

, 0,

x y z

x

a a a

P P P

a g

= =

= = =

Total differential of P

Pressure difference between 2 points

xP a x g z =

( ) ( )2 1 2 1 2 1xP P a x x g z z =

free surface P2 = P1

xaz z z x x = =


g


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Fluid in Rigid-Body Motion

Special Case 2 : Free Fall of a Fluid Body

Acceleration on Straight Path



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Rotation in a Cylindrical Container

Container is rotating about the z-axis2

2

, 0

, 0,

r za r a a

P P P

r g

= = =

= = = Total differential of P

On an isobar dP = 0

dP r dr gdz =

2 2

2

1

2

isobarisobar

dz rz r C

dr g g

= = +

Equation of the free surface2

2 22z h R r

=


4g


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Examples of Archimedes Principle

The Golden Crown of Hiero II King of


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The Golden Crown of Hiero II, King of

Archimedes 287-212 B.C.

Hiero, 306-215 B.C.

the goldsmith replaced some of

.

Hiero asked Archimedes to

pure gold.

rc me es a o eve op anondestructive testing method


The Golden Crown of Hiero II King of


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The Golden Crown of Hiero II, King of

The weight of the crown andc

cVc = Wn = nVn.

If the crown is pure gold, =which means that the volumesmust be the same, Vc=Vn.

,

B=H2OV.If the scale becomes unbalanced,c n,

which in turn means that the c

nGoldsmith was shown to be afraud!


H d t ti B d f t T ti


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Hydrostatic Bodyfat Testing

What is the best way to

Hydrostatic Bodyfat Testing

using Archimedes Principle!Process

Measure body weightW=bodyV

Get in tank, expel all air, and

measure apparent weight WaBuoyancy force B = W-Wa =

H2O . s perm scomputation of body volume.

Body density can be

body .Body fat can be computedfrom formulas.


H d t ti B d f t T ti i Ai ?


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Hydrostatic Bodyfat Testing in Air?

Same methodology asHydrostatic testing in water.

What are the ramifications ofusing air?Density of air is 1/1000th of

.

Temperature dependence ofair.

Measurement of small volumes.

Used by NCAA Wrestling (there

.


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