ch2 fluid properties

8/18/2019 Ch2 Fluid Properties

1/55

CE 351

Engineering Fluid Mechanics

CHAPTER 2Fluid Properties

Prof. Maed A!u"#reig


2/55

$ntroduction

Field of Fluid Mechanics can !e di%ided into 3

!ranches&

Fluid 'tatics& (echanics of fluids at rest )ine(atics& deals *ith %elocities and

strea(lines *+o considering forces or energ,

Fluid -,na(ics& deals *ith the relations !et*een%elocities and accelerations and forces eerted

!, or upon fluids in (otion


3/55

$ntro/con0t

Mechanics of fluids is etre(el, i(portant in (an,areas of engineering and science. Ea(ples are&

io(echanics lood flo* through arteries Flo* of cere!ral fluid

Meteorolog, and cean Engineering

Mo%e(ents of air currents and *ater currents Che(ical Engineering

-esign of che(ical processing euip(ent


4/55

$ntro/con0t

Mechanical Engineering-esign of pu(ps4 tur!ines4 air"conditioning

euip(ent4 pollution"control euip(ent4 etc. Ci%il EngineeringTransport of ri%er sedi(ents

Pollution of air and *ater -esign of piping s,ste(sFlood control s,ste(s


5/55

-i(ensions and nits

efore going into details of fluid

(echanics4 *e stress i(portance of units

$n .'4 t*o pri(ar, sets of units are used&1. '$ 6',ste(e $nternational7 units

2. English units


6/55

nit Ta!le

8uantit, '$ nit English nit

9ength 697 Meter 6m) Foot 6ft 7

Mass 6(7 )ilogra( 6kg 7 'lug 6slug7 :lb*sec 2 /ft

Ti(e 6T7 'econd 6s7 'econd 6sec 7

Te(perature 6 7 Celcius 6oC 7 Farenheit 6oF 7

Force ;e*ton(N)=kg*m/s2

Pound 6l!7

θ


7/55

-i(ensions and nits con0t

1 Newton < Force reuired to accelerate a

1 kg of (ass to 1 m/s2

1 slug < is the (ass that accelerates at 1

ft/s2 *hen acted upon !, a force of 1 lb

To re(e(!er units of a ;e*ton use F:(a

6;e*ton0s 2nd 9a*7 =F> : =(>=a>: kg*m/s2 = N


8/55

More on -i(ensions

To re(e(!er units of a slug also use

F:(a :? ( : F + a

=(> : =F> + =a> : lb / (ft / sec 2 ) = lb*sec 2 / ft

1 lb is the force of gra%it, acting on 6or

*eight of 7 a platinu( standard *hose

(ass is @.535B23 kg


9/55

eight and ;e*ton0s 9a* of Dra%itation

eightDra%itational attraction force !et*een t*o !odies

;e*ton0s 9a* of Dra%itation F : D (1(2+ r 2

D " uni%ersal constant of gra%itation

(14 (2 " (ass of !od, 1 and !od, 24 respecti%el, r " distance !et*een centers of the t*o (asses F " force of attraction


10/55

eight

(2 " (ass of an o!ect on earth0s surface (1 " (ass of earth

r " distance !et*een center of t*o (asses

r 1 " radius of earth

r 2 " radius of (ass on earth0s surface

r 2 r 14 therefore r : r 1r 2 G r 1

Thus4 F : (2 6D (1 + r 27


11/55

eight

eight 67 of o!ect 6*ith (ass (27 on surface of earth6*ith (ass (17 is defined as

: (2g I g :6D(1+r 27 gra%itational acceleration

g : B.J1 m/s2 in '$ units

g : 32.2 ft/sec 2 in English units

'ee !acK of front co%er of tet!ooK for con%ersion ta!les!et*een '$ and English units


12/55

τ =

F

A s h e a r s t r e s s

x

( )

L 'hear stress and pressure

p F

A n o r m a l s t r e s s p r e s s u r e

z= ( ( ) )

L 'hear stress and pressure at a point

τ =

→

F

A

x

Al i m 0

p F

A

z

A

=

→

l i m 0


13/55

[ ]

[ ] ( )

F

A

N

m P a P a s c a l i n S I u n i t s= =2

L nits of stress 6shear stress and pressure7

[ ]

[ ]

( ) F

A

l b

i n

p s i p o u n d s p e r s q u a r e i n c h i n E n g l i s h u n i t s= =2

[ ]

[ ] ( )

F

A

l b

f t p o u n d s p e r s q u a r e f o o t E n g l i s h u n i t s= =2


14/55

Properties of Fluids Con0t

Fluids are either liuids or gases

9iuid& A state of (atter in *hich the (olecules

are relati%el, free to change their positions *ithrespect to each other !ut restricted !, cohesi%e

forces so as to (aintain a relati%el, fied %olu(e

Das& a state of (atter in *hich the (olecules

are practicall, unrestricted !, cohesi%e forces. A

gas has neither definite shape nor %olu(e.


15/55

More on properties of fluids

Fluids considered in this course (o%e

under the action of a shear stress4 no

(atter ho* s(all that shear stress (a, !e6unliKe solids7


16/55

-efinition of a Fluid

hen a shear stress is applied& Fluids continuousl, defor(

'olids defor( or !end


17/55

Continuu( %ie* of Fluids

Con%enient to assu(e fluids are continuousl, distri!uted

throughout the region of interest. That is4 the fluid is

treated as a continuu(

This continuu( (odel allo*s us to not ha%e to deal *ith

(olecular interactions directl,. e *ill account for such

interactions indirectl, %ia %iscosit,

A good *a, to deter(ine if the continuu( (odel is

accepta!le is to co(pare a characteristic length of theflo* region *ith the (ean free path of (olecules4

$f 4 continuu( (odel is %alid

( ) Lλ

L < < λ


18/55

-ensit, and specific *eight

-ensit, 6(ass per unit %olu(e7& ρ = m

V

[ ]

[ ]

[ ] ( ) ρ = =m

V

k g

m i n S I u n i t s3nits of densit,&

'pecific *eight 6*eight per unit %olu(e7&

[ ] [ ] [ ] ( )γ ρ = = = g k g

m

m

s

N

mi n S I u n i t s3 2 3

nits of specific *eight&

γ ρ = g


19/55

'pecific Dra%it, of 9iuid 6S7

'ee appendi A of tet!ooK for specific

gra%ities of %arious liuids *ith respect to

*ater at @ oF

ater

liquid

ater

liquid

ater

liquid

g

g S

γ

γ

ρ

ρ

ρ

ρ ===


20/55

Niscosit, 6 7

Niscosit, can !e thought as the internal sticKiness of a fluid

Representati%e of internal friction in fluids

$nternal friction forces in flo*ing fluids result fro( cohesionand (o(entu( interchange !et*een (olecules.

Niscosit, of a fluid depends on te(perature& $n liuids4 %iscosit, decreases *ith increasing te(perature 6i.e.

cohesion decreases *ith increasing te(perature7

$n gases4 %iscosit, increases *ith increasing te(perature 6i.e.

(olecular interchange !et*een la,ers increases *ith te(perature

setting up strong internal shear7

µ


21/55

More on Niscosit,

Niscosit, is i(portant4 for ea(ple4 in deter(ining a(ount of fluids that can !e

transported in a pipeline during a specificperiod of ti(e

deter(ining energ, losses associated *ith

transport of fluids in ducts4 channels and

pipes


22/55

;o slip condition

ecause of %iscosit,4 at !oundaries 6*alls7

particles of fluid adhere to the *alls4 and

so the fluid %elocit, is Oero relati%e to the*all

Niscosit, and associated shear stress (a,

!e eplained %ia the follo*ing& flo*!et*een no"slip parallel plates.


23/55

'o(e 'i(ple Flo*s

Flo* !et*een a fied and a (o%ingplate

Fluid in contact *ith the plate has the sa(e

%elocit, as the plate

u : x "direction co(ponent of %elocit,

u=V Mo%ing plate

Fied plate

y

x

V

u=@

B ! "

V !u =)( Fluid


24/55

'o(e 'i(ple Flo*s

Flo* through a long4 straight pipe

Fluid in contact *ith the pipe *all has the

sa(e %elocit, as the *all

u : x "direction co(ponent of %elocit,

x

!

−=2

1)(

#

r V r u

V Fluid


25/55

The %elocit, induced !, (o%ing top plate can !e sKetched as follo*s&

!

u !( )

$

%

u !%

$ !( ) =

The %elocit, induced !, top plate is epressed as follo*s&

u !( )= =0 0

u ! $ % ( )= =


26/55

For a large class of fluids4 e(piricall,4 F A %

$ ∝

More specificall,4 F A % $

= µ ; µ i s c o e f f i c i e n t o f & i s i t !c o s

'hear stress induced !, is F τ µ = = F

A

%

$

Fro( pre%ious slide4 note thatd u

d !

%

$ =

Thus4 shear stress is τ µ = d ud !

$n general *e (a, use pre%ious epression to find shear stress at a point

inside a (o%ing fluid. ;ote that if fluid is at rest this stress is Oero !ecause

d u

d ! = 0


27/55

;e*ton0s euation of %iscosit,

τ µ = d u

d !

µ " %iscosit, 6coeff. of %iscosit,7

Fied no"slip plate

u ! & e l o c i t ! p r o f i l e( ) ( )

'hear stress due to %iscosit, at a point&

fluid surface

e.g.& *ind"dri%en flo* in ocean

ν µ ρ =

" Kine(atic%iscosit,

!


28/55

As engineers4 ;e*ton0s 9a* of Niscosit, is %er, useful to us as *e can use it to

e%aluate the shear stress 6and ulti(atel, the shear force7 eerted !, a (o%ing

fluid onto the fluid0s !oundaries.

τ µ a t b o u n d a r !d u

d !a t b o u n d a r !

=

;ote is direction nor(al to the !oundar, !


29/55

Flo* !et*een 2 plates

u=V Mo%ing plate

Fied plate

y

x

V

u=@

B ! "V !u =)( Fluid Force acting; the plate

21

21

222111

τ τ

τ τ

=

=

===

A A

F A A F

2211 τ µ µ τ === d!du

d!

du

Thus4 slope of %elocit,

profile is constant and

%elocit, profile is a st. line

Force is sa(e on top

and !otto(


30/55

Flo* !et*een 2 plates

u=V Mo%ing plate

Fied plate

y

x

V

u=@

B ! "V !u =)(

"

V

d!

du

µ µ τ ==

'hear stress an,*here

!et*een plates

τ

τ

'hear on fluid

m "

smV

' SAE m s N o

02.0

/3

)38@30(/1.0 2

=

=⋅= µ

2

2

/15

)02.0

/3)(/1.0(

m N

m

smm s N

=

⋅=τ


31/55

Ea(ple& ournal earing

Di%en Rotation rate4 ω : 15@@ rp(

d : c(

l : @ c(

" : .@2 c(

S#oil : @.JJ

νoil : @.@@3 (2+s

Find& Torue and Po*erreuired to turn the !earing

at the indicated speed.


32/55

Ea(ple& cont.

Assu(e& 9inear %elocit, profile in oil fil(

2/1242/)0002.0(

)2/06.0(1500*60

2

)003.0*998*88.0(

2/)(

)2/( StressShear

mkN

d (

d

d!

dV

=

=

−==

π

ω µ µ τ

m N

d l

d )

⋅==

=

2812

06.0)4.0*

2

06.0*000124*2(

2

)

2

2(!or"#e

π

πτ

k* sm N ) P 1.44/100441.15$*281%o&er =⋅=== ω


33/55

Nisco(eter

Coefficient of %iscosit, can !e (easured e(piricall, using a %isco(eter µ

Ea(ple& Flo* !et*een t*o concentric c,linders 6%isco(eter7 of length

Mo%ing fluid

Fied outer

c,linder

Rotating inner

c,linder

ω +

r

#

h

x

z

!

r " radial coordinate

,

L


34/55

$nner c,linder is acted upon !, a torue4 4 causing it to

rotate a!out point at a constant angular %elocit, and

causing fluid to flo*. Find an epression for

+ + k = ω ,

ecause is constant4 is !alanced !, a resisti%e torue

eerted !,

the (o%ing fluid onto inner c,linder

ω + + k =

+ + k r e s r e s= −( )

The resisti%e torue co(es fro( the resisti%e stress eerted !, the

(o%ing fluid onto the inner c,linder. This stress on the inner c,linder leads

to an o%erall resisti%e force 4 *hich induces the resisti%e torue a!out

point

x

!

z

,

+ + r e s=

+

τ r e s

⇒+

F r e s

τ r e s

# ⇒+ + r e s

F

r e s

+


35/55

+ + F #r e s r e s= =

F A # Lr e s r e s r e s= =τ τ π ( )2 6;eglecting ends of c,linder7

Ho* do *e get Q This is the stress eerted !, fluid onto inner

c,linder4 thus

τ r e s

τ µ r e s

a t i n n e r c ! l i n d e r r #

d u

d r =

=( )

$f 6gap !et*een c,linders7 is s(all4 thenh

r

u r ( )

# ω

r # h= +r #=

d u

d r

#

ha t i n n e r c ! l i n d e r r #( )=

= ω


36/55

Thus4 τ µ ω

r e s #

h=

+ + F #r e s r e s= =

+ + A # # L #r e s r e s r e s= = =τ τ π ( )2

=

µ ω

π #

h # L #( )2

+ # Lh

=3

2 µ ω π

Di%en pre%ious result (a, !e used to find of

fluid4 thus concentric c,linders (a, !e used as a %isco(eter

+ # L h ω µ


37/55

Ea(ple& Rotating -isK

Assu(e linear %elocit, profile& dV/dy :V/y :ω /y

Find shear stress


38/55

'urface Tension

elo* surface4 forces act euall,

in all directions

At surface4 so(e forces are

(issing4 pulls (olecules do*n andtogether4 liKe (e(!rane eerting

tension on the suf$ce

$f interface is cur%ed4 higher

pressure *ill eist on conca%e side

Pressure increase is !alanced !,surface tension4 σ

σ : @.@3 ;+( 6S 2@oC7

*ater

air

;o net force

;et force

in*ard

$nterface


39/55

'urface tensionL Consider inserting a fine tu!e into a !ucKet of *ater&

h

σ σ

Meniscus

x

!

σ " 'urface tension %ector 6acts unifor(l, along contact peri(eter !et*een liuid and tu!e7

Adhesion of *ater (olecules to the tu!e do(inates o%er cohesion !et*een

*ater (olecules gi%ing rise to and causing fluid to rise *ithin tu!eσ

θ θ

r " radius of tu!e


40/55

h

σ σ θ θ

*

x

! σ σ θ θ = +[ s i ' ( ) c o s ( ) ]i -

* * -= −( ) 6*eight %ector of *ater7

Euili!riu( in y%diection ,ields& σ θ π c o s ( ) ( ) ( ) 2 0r - * - -+ − =

Thus σ π θ = * r 2 c o s

*ith * r h a t e r = γ π 2


41/55


42/55

Co(pressi!ilit,

L All fluids co(press if pressure increases resulting in an

increase in densit,

L Co(pressi!ilit, is the change in %olu(e due to a

change in pressure

L A good (easure of co(pressi!ilit, is the !ulK (odulus

6$t is in%ersel, proportional to co(pressi!ilit,7

E d p

d υ υ

υ = −

υ ρ = 1

( ) s p e c i f i c & o l u m e

p i s p r e s s u r e

ρ ρ // ∆∆=

∆∆−= p

V V p E&


43/55

Co(pressi!ilit,

L Fro( a!o%e epression4 increasing pressure !, 1@@@ &si *ill co(press

the *ater !, onl, 1+32@ [email protected] of itsoriginal %olu(e

E p s iυ = 3 2 0 0 0 0

L Thus4 *ater (a, !e treated as inco(pressi!le 6densit, is constant7 ( ) ρ

L $n realit,4 no fluid is inco(pressi!le4 !ut this is a good approi(ation for certain fluids

L For *ater ' : 2.2 DPa4

1 MPa pressure change : @.@5 %olu(e change


44/55

Napor pressure of liuidsL All liuids tend to e%aporate *hen placed in a closed container

L NaporiOation *ill ter(inate *hen euili!riu( is reached !et*eenthe liuid and gaseous states of the su!stance in the container

i.e. U of (olecules escaping liuid surface : U of inco(ing (olecules

L nder this euili!riu( *e call the call %apor pressure the saturation pressure

L At an, gi%en te(perature4 if pressure on liuid surface falls !elo* the the saturation pressure4 rapid e%aporation occurs 6i.e. !oiling7

L For a gi%en te(perature4 the saturation pressure is the !oiling pressure


45/55

Napor Pressure

Pressure at *hich a liuid*ill !oil for gi%en te(p.

Napor pressure increases

*ith te(perature $ncreasing te(perature of

*ater at sea le%el to 212 oF4increases the %aporpressure to 1. psia and!oiling occurs

oiling can occur !elo* 212 oFif *e lo*er the pressure in the*ater to the %apor pressure ofthat te(perature

Vapor Press. vs. Temp.

0

20

40

60

80

100

120

0 10 20 30 40 50 60 0 80 !0 100

Temperature "o#$

V a p r o

P r e s s u r e " % P a $

L At 5@ oF4 the %aporpressure is @.1J psia

L $f ,ou reduce thepressure in *ater at thiste(perature4 !oiling *illoccur 6ca%itation7


46/55

Ta&le '.2PHV'$CA9 PRPERT$E' F DA'E' AT 'TA;-AR-

ATM'PHER$C PRE''RE A;- 15WC 65BWF7


47/55

Ea(ple

Di%en& ;atural gas

Ti(e 1& 1:1@oC4 &1:1@@ KPa

Ti(e 2& 2:1@oC4 &2:2@@ KPa

Find& Ratio of (ass at ti(e 2 to that at ti(e 1

$deal gas la* 6 & is a!solute pressure7

V #+

pV ) == ρ

2

1

2

1

2

1

p

p

V

#+

p

V #+

p

)

) ==

5.1200

300

1

2 ==kPa

kPa

)

)


48/55

Ea(ple

Esti(ate the (ass of 1 (i3 of air

in slugs and Kgs

Assu(e ρ $i : @.@@23 slugs+ft34

the %alue at sea le%el for standard

conditions

kg x )

slugskg slugs x )

slugs x )

V ) air

9

8

8

3

1009.5

/59.14*1049.3

1049.3

)2805(*0023$.0

=

=

=

== ρ


49/55

Ea(ple

(iven) Pressure of 2 MPa is

applied to a (ass of *ater that

initiall, filled 1@@@"c(3 %olu(e.

*ind) Nolu(e after the

pressure is applied.

+olution) E : 2.21@B Pa

"Ta!le A.57

3

3

3

9

6

01.999

909.01000

909.0

1000102.2

102

/

cmV

V V V

cm

cm Pa x

Pa x

V E

pV

V V

p E

final

final

&

&

=

−=∆+=

−=

−=

∆−=∆

∆∆−=


50/55

Capillar, Rise

(iven) ater S 2@oC4d : 1. ((

*ind) Height of *ater +olution) 'u( forces in

%ertical

Assu(e θ s(all4 cosθ 1

0)4

)((cos

0

2

=∆−=−

d hd

* F z π

γ θ σπ

σ

mmh

x

d h

6.18

106.1*9$90

0$3.0*4

4

3

=∆

=

=∆

−

γ

σ

σ F

W


51/55

Ea(ple 62.517

*ind) Capillar, rise !et*een t*o

%ertical glass plates 1 (( apart.

σ : .31@"2 ;+(

l is into the page

+olution)

t

σ σ

θ

mmh

mh

x

t h

hlt l

F &ertical

9.14

0149.0

9810*001.0103.$*2

2

02

0

2

==

=

=

=−=∑

−

γ

σ

γ σ


52/55

Ea(ples of 'urface Tension


53/55

Ea(ple

*ind) The for(ula for the gage

pressure *ithin a sperical

droplet of *aterQ

+olution) 'urface tensionforce is reisited !, the force

due to pressure on the cut

section of the drop

r p

r r pσ

σ π π 2

2)( 2

=

=


54/55

Ea(ple 62.J7 (iven) 'perical !u!!le4

inside radius 4 fil( thicKness

t 4 and surface tension σ .

*ind) For(ula for pressure

in the !u!!le relati%e to that

outside.

+olution)

Pa p

x p

r

p

r r p

F

0.$3

004.0

103.$*4

4

0)2(2

0

2

2

=∆

=∆

=∆

=−∆

=∑

−

σ

σ π π


55/55

ug Pro!le(

σ

Cross"section

of !ug leg

σ

F

ch2 fluid properties

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