Crash Course Chapter2 Fluid Statics

8/8/2019 Crash Course Chapter2 Fluid Statics

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Chapter 2Chapter 2Chapter 2Chapter 2

Fluid StaticsFluid Statics


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INTRODUCTION

Fluid statics is the study of fluids in which there is no relativemotion between fluid properties

No shear stress existOnly normal stress exist - pressure - primary interest in fluid statics

The situation will be considered:Pressure variation through a fluidEffect of pressure on submerged surfaces

Manometer, Buoyancy


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Pressure in a Fluid at Rest

Acceleration, a = 0

y

x

z

There was no pressure variation in horizontal direction (x-y)

Pressure increased with depth

The equation is used to convert pressure to a height of liquid

equationcHydrostati...........gdzdp

dz

dp

V

K

!

!

hghp KV !!


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Pressure in an Incompressible Fluid at Rest

Liquids are incompressible under most

condition and is

called in

compressiblefluids

Incompressible fluid hasconstant density

Pressure varies linearly with depth

Pressure difference between 2 pointscan be specified by the distance, hsince,

Pressure at any depth below the free

surface is given by equation:

K

21 pph

!

atmosphereppphp !! 00 ....K

Figure 2.3: Notation for pressurevariation in a fluid at rest with a

freesu

rface.


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Pressure in a Compressible Fluid at Rest

Compressible fluid = atmosphere (gasessuch as air, oxygen, nitrogen)Density change significantly with changesin pressure and temperature,cannot be consideredindependent of height

Ideal Gas Law:

Where p is absolute pressure, Ris the gasconstant and Tis absolutetemperature. The relationship can be combined to give;

RTp V!

RT

gp

dz

dp!

Integrate, and we get;

-

!

0

1212

)(exp

RT

zzgpp

(Isothermalcondition)


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Measurement of Pressure

Pressure at a point within a fluid mass: designated either an absolute

pressure or a gage pressure.

Absolute pressure: measured relative to absolute zero pressure (apressure that would only occur in a perfect vacuum)

Gage pressure: measured relative to the local atmospheric pressure

Abs: always +ve

Gage: either +ve or ve

Depends on: pressure above atmospheric pressure (+ve)

: pressure below atmospheric pressure (-ve)

-ve gage pressure also referred assuction or vacuum pressure.


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Measurement of Pressure

ATMOSPHERIC PRESSUREThe measurement of Patm isusually accomplished with a mercury barometerasshown in Figure 2.5

Figure 2.5: Mercury barometer

The tube isinitially filled with mercury and turn upsidedown with the open endin the container of mercury.

Thecolu

mn of mercu

ry will

come to an eq

uilibr

ium po

sitionwhere its weight plus the force due to the vapor pressure

balances the force due to the atmospheric pressure. Thus,

Where is the specific weight of mercury.

For the most practical purposes, the contribution of thevapor pressure can be neglectedsince it is very small [formercury,pvapor=0.000023 1b/in2. (abs) at a temperature of680F]so that patm h.

It isconvenient to specify atmospheric pressure in term ofheight, h, in millimeters or inches of mercury.

vaporatm php ! K


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Manometer

Standard techniques for measuring pressure involves the use of liquid

columnsin vertical or inclined tube.

Involve column of fluids at rest

Three common types: piezometer tube, U-tube manometer, inclined-tubemanometer.

Figure 2.6: Piezometer tube

Piezometer Tube

Fundamental equation describing their use:

Remember!

Fluid at rest: pressure willincreased as we move downward,

decreased when move upward.

Apply the equation to piezometer tube gives;

Pressure at point 1, P1 = PA since the elevation issame.

0php ! K

11hpA K!


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U-tube Manometer

Figure 2.7: Simple U-tubemanometer

Fluidin manometer: gage fluid

To find PA , we start at point A and work around tothe open end by applying equation P = h + P0

Start at A, PA= P1, move downward to 2, pressureincreased. So,

PA + 1h1,

P2 = P3, From P3to upward, pressure decreased, sothe equation above become

12.2........02211 eqhhp ! KK

In U-tube manometer, gage fluidcan be different from theliquidin the container in which the pressure is to bedetermined. If Acontained of gas, PA P2 and eq. 2.12becomes,

22hp

K!


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The U-tube manometer is also widely used to measuredifference in pressure between two containers or two pointsin a given system.

Equation obtained:

BA phhhp ! 332211 KKK

and the pressure difference is:

113322 hhhpp B KKK !

Figure 2.8: Differential U-tube manometer


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Inclined-Tube Manometer

Used

to measu

resma

llpre

ssure

change

s.

Figure 2.9: Inclined-tube manometer

The pressure difference(PA-PB) can be expressed by:

113322

332211

si

si

hhpp

or

phhp

B

B

KKUK

KUKK

!

!

N

N

Noted that, the pressuredifference between points (1)and (2) isdue to the verticaldistance between the points,which can be expressed as2sin

h2

2


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The inclined-tube manometer is often used to measure small

differencesin gas pressures, so that if pipesA and B contain a gasthen,

Where the contributions of the gascolumn h1 and h3 have beenneglected.

UK

UK

si

si

2

2

22

B

B

ppor

pp

!

!

N

N


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Buoyancy, Flotation and Stability

Archimedes PrincipleWhen a body iscompletely submergedin a fluid, or floatingso that it is only partially submerged, the resultant forceacting on the body iscalled the buoyant force, FB

! KB

F

= specific weight of fluid= volume of the body @ displacedliquid

K


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Thus, the buoyant force has a magnitude equal to the weightof fluiddisplaced by the body andisdirected verticallyupward.

The resul

tis

referred

asA

rchime

des pr

inci

ple.

The buoyant force passes through the centroid of thedisplaced volume, and the point through which the buoyantforce actsiscalledcenter of buoyancy.


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Stability

A body issaid to be in a stable equilibrium positionif, when displaced, it returns to its equilibrium

position.If unstable equilibrium position, when displaced (evenslightly) it moves to a new equilibrium position.


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For submerged body:

If the center of gravity of the body is above the centroid(center of buoyancy) - a small angular rotation result in amoment that continue to increase - overturning (a)

If the center of gravity is below the centroid a small angularrotation provide restoring moment, - the body isstable

(b) shows the neutralstability center of gravity = centroid


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For floating body:

Asshown

in F

ig. 2.16, a f

loat

ing bo

dy

such a

sa barge that r

ideslow

inwater can be stable even though the center of gravity lies above the

center of buoyancy.

As body rotatesFB shifts to pass through the centroid of the newlyformeddisplaced volume andcombined with weight, W, to form couple,

whic

h will

caus

e the body to ret

urn to

its

origina

leq

uilibr

ium po

sition.

Figure 2.16: Stability of a floating body-stable configuration


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However, for the relatively tall, slender body shown in Fig.2.17, a small rotationaldisplacement can cause the buoyant

force and the weight to form an overtuning couple asillustrated.

Figure 2.17: Stability of a floating body-unstable configuraton.


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Crash Course Chapter2 Fluid Statics