fluid flow processes for technology

7/24/2019 Fluid Flow Processes for Technology

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Pressure and Fluid Statics

By:

Dr. Muhammad Farhan


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Pressure

Pressureis defned as a normal forceexerted by a uid per unit area.

Units o pressure are N/m2 !hich is

called a pascal "Pa#. Since the unit Pa is too small orpressures encountered in practicekilopascal "$ %Pa & $'(Pa# and

megapascal"$ MPa & $')

Pa# arecommonly used. *ther units include bar atm, kgf/cm2,

lbf/in2=psi.


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+,solute -a-e and acuumpressures

+ctual pressure at a -ie point is calledthe absolute pressure.

Most pressuremeasurin- deices arecali,rated to read 0ero in theatmosphere and thereore indicategage pressure P-a-e&Pa,s Patm.

Pressure ,elo! atmospheric pressureare called vacuum pressure Pac&Patm

Pa,s.


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+,solute -a-e and acuumpressures


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Pressure at a Point

Pressure at any point in a 1uid is thesame in all directions.

Pressure has a ma-nitude ,ut not aspecifc direction and thus it is ascalar uantity.

i i i h


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3ariation o Pressure !ithDepth

4n the presence o a-raitational feld pressureincreases !ith depth ,ecausemore 1uid rests on deeperlayers.

5o o,tain a relation or theariation o pressure !ithdepth consider rectan-ularelement

Force ,alance inzdirection -ies

Diidin- ,y xand rearran-in-

-ies

2 1

0

0

z zF ma

P x P x g x z

= =

=

2 1 sP P P g z z = = =

3 i i P i h


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3ariation o Pressure !ithDepth

Pressure in a 1uid at rest is independent othe shape o the container.

Pressure is the same at all points on ahori0ontal plane in a -ien 1uid.


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Scu,a Diin- and 6ydrostaticPressure


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Scu,a Diin- and 6ydrostaticPressure

( ),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

=

= =

Pressure on dierat $'' t7

Dan-er oemer-ency ascent7

100 ft

1

2

Boyles law

If you hold your breath on ascent, your lung

volume would increase by a factor of 4, which

would result in embolism andor death!


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Pascal8s 9a!

1 2 2 2

1 2

1 2 1 1

F F F AP P

A A F A= = =

Pressure applied to aconfned 1uid increasesthe pressure throu-hout,y the same amount.

4n picture pistons are at

same hei-ht:

atio +2/+$is calledideal mechanicaladvantage


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5he Manometer

1 2

2 atm

P P

P P gh

=

= +

+n eleation chan-e ozin a 1uid at restcorresponds to P/g.

+ deice ,ased on thisis called a manometer.

+ manometer consistso a Utu,e containin-one or more 1uids suchas mercury !ateralcohol or oil.

6eay 1uids such as

mercury are used ilar-e pressuredi;erences areanticipated.


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Mutli1uid Manometer

For multi1uid systems Pressure chan-e across a 1uid

column o hei-ht his P = gh.

Pressure increases do!n!ardand decreases up!ard.

5!o points at the same eleationin a continuous 1uid are at thesame pressure.

Pressure can ,e determined ,y

addin- and su,tractin- gh

terms.

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


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Measurin- Pressure Drops

Manometers are !ellsuited to measurepressure drops acrossales pipes heate


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5he Barometer

C atm

atm

P gh P

P gh

+ =

=

+tmospheric pressure ismeasured ,y a deicecalled a barometer= thusatmospheric pressure isoten reerred to as thebarometric pressure.

P#can ,e ta%en to ,e 0erosince there is only 6-apor a,oe point > and itis ery lo! relatie to Patm.

>han-e in atmosphericpressure due to eleation

has many e;ects: >oo%in-nose ,leeds en-ineperormance aircratperormance.


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Fluid Statics

Fluid Staticsdeals !ith pro,lems associated!ith 1uids at rest.

4n 1uid statics there is no relatie motion,et!een ad?acent 1uid layers.

5hereore there is no shear stress in the 1uidtryin- to deorm it. 5he only stress in 1uid statics is normal stress

Normal stress is due to pressure 3ariation o pressure is due only to the !ei-ht o

the 1uid @ 1uid statics is only releant in presenceo -raity felds.

+pplications: Floatin- or su,mer-ed ,odies!ater dams and -ates liuid stora-e tan%setc.


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6ooer Dam


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6ooer Dam


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6ooer Dam

Ahapter

"Bernoullieuation#.

6 d t ti F Pl


19/40

6ydrostatic Forces on PlaneSuraces

*n aplanesuracethe hydrostatic orcesorm a system oparallel orces

For many applications

ma-nitude andlocation o application!hich is called centerof pressure must ,edetermined.

+tmospheric pressurePatmcan ,e ne-lected!hen it acts on ,othsides o the surace.


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esultant Force

5he ma-nitude o $%actin- on a plane

surace o a completely su,mer-ed plate ina homo-enous 1uid is eual to the producto the pressure P#at the centroid o the

surace and the area&o the surace


21/40

>enter o Pressure

,xx C

p C

c

Iy y

y A= +

9ine o action o resultantorce $%=P#& does not passthrou-h the centroid o thesurace. 4n -eneral it liesunderneath !here thepressure is hi-her.

3ertical location o Centerof Pressureis determined,y euation the moment othe resultant orce to themoment o the distri,utedpressure orce.

E4ur ed


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6ydrostatic Forces on >uredSuraces

$%on a cured surace is more inoled since

it reuires inte-ration o the pressure orces

that chan-e direction alon- the surace. Aasiest approach: determine hori0ontal and

ertical components $'and $(separately.


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6ydrostatic Forces on >uredSuraces

6ori0ontal orce component on cured surace:$'=$x. 9ine o action on ertical plane -iesy

coordinate o center o pressure on curedsurace.

3ertical orce component on cured surace:$(=$y)* !here *is the !ei-ht o the liuidin the enclosed ,loc% *=g(. xcoordinate othe center o pressure is a com,ination o lineo action on hori0ontal plane "centroid o area#

and line o action throu-h olume "centroid oolume#. Ma-nitude o orce $%=+$'2)$(2/2

+n-le o orce is = tan!+$(/$'


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Buoyancy and Sta,ility

Buoyancy is due to the 1uiddisplaced ,y a ,ody. $-=fg(.

Archimedes principal: 5he,uoyant orce actin- on a ,odyimmersed in a 1uid is eual to the!ei-ht o the 1uid displaced ,y the

,ody and it acts up!ard throu-h thecentroid o the displaced olume.


25/40

Buoyancy and Sta,ility

Buoyancy orce $-iseual only to thedisplaced olume

fg(displaced.

5hree scenariospossi,le

1. bodyuid: Floatin- ,ody

2. body=uid: Neutrally

,uoyant

3. body"uidSin%in- ,ody

A


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A


27/40

A


28/40

A


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A


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A


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A


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Sta,ility o 4mmersedBodies

otational sta,ility o immersed ,odies dependsupon relatie location o center of gravity0and

center of buoyancy-. 0,elo! -: sta,le 0a,oe -: unsta,le

0coincides !ith -: neutrally sta,le.


33/40

Sta,ility o Floatin- Bodies

4 ,ody is ,ottom heay"0lo!er than -# it isal!ays sta,le.

Floatin- ,odies can ,esta,le !hen 0is hi-her

than -due to shit inlocation o center,uoyancy and creationo restorin- moment.

Measure o sta,ility isthe metacentric hei-ht1. 4 01G$ ship issta,le.


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i-idBody Motion

5here are special cases !here a ,ody o 1uid can under-ori-id,ody motion: linear acceleration and rotation o acylindrical container.

4n these cases no shear is deeloped.

Ne!tonCs 2nd la! o motion can ,e used to derie an

equation of motion or a 1uid that acts as a ri-id ,ody

4n >artesian coordinates:

P gk a + =

( ), ,x y xP P P

a a g ax y z

= = = +


35/40

9inear +cceleration

>ontainer is moin- on a strai-htpath

5otal di;erential o P

Pressure di;erence ,et!een 2points

Find the rise ,y selectin- 2 pointson ree surace P2& P$

0, 0

, 0,

x y z

x

a a a

P P Pa g

x y z

= =

= = =

xdP a dx gdz =

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

( )2 1 2 1x

s s s

az z z x x

g = =

otation in a >ylindrical


36/40

otation in a >ylindrical>ontainer

2

2

, 0

, 0,

r za r a a

P P Pr g

r z

= = =

= = =

>ontainer is rotatin- a,out the 0a


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Aro!n o 6iero 44 Hin- o


38/40

5he olden >ro!n o 6iero 44 Hin- oSyracuse

+rchimedes 2IJ2$2 B.>. 6iero (')2$ B.>.

6iero learned o a rumor !herethe -oldsmith replaced some o

the -old in his cro!n !ith siler.6iero as%ed +rchimedes todetermine !hether the cro!n!as pure -old.

+rchimedes had to deelop anondestructie testin- method

5he olden >ro!n o 6iero 44 Hin- o


39/40

5he olden >ro!n o 6iero 44 Hin- oSyracuse

5he !ei-ht o the cro!n andnu--et are the same in air:*c= c(c= *n= n(n.

4 the cro!n is pure -oldc&n!hich means that theolumes must ,e the same(c=(n.

4n !ater the ,uoyancy orceis -='2(.

4 the scale ,ecomesun,alanced this implies thatthe 3cK 3n !hich in turnmeans that the c K n

oldsmith !as sho!n to ,e araudL


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6ydrostatic Bodyat 5estin-

hat is the ,est !ay to

measure ,ody at7 6ydrostatic Bodyat 5estin-

usin- +rchimedes PrincipleL

Process

Measure ,ody !ei-ht*=body(

et in tan% e

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