UTP - Fluid Mechanics Course - September 2012 Semester - Chap 3 Bernoulli Equations

8/9/2019 UTP - Fluid Mechanics Course - September 2012 Semester - Chap 3 Bernoulli Equations

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Fluid MechanicsMohd Faizairi Mohd Nor

Dept. of Mechanical Engineering,

Universiti Teknologi PETRN!"


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Elementary Fluid Dynamics

Reading: M$nson, et al., %hapter &


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&

'(ectives

1. comprehend the concepts necessary to analyse fluids inmotion.

2. identify differences between steady/unsteady,

uniform/non-uniform and compressible/incompressibleflow.

3. construct streamlines and stream tubes.

4. appreciate the Continuity principle through Conservation

of ass and Control !olumes.". derive the #ernoulli $energy% e&uation.

'. familiarise with the momentum e&uation for a fluid flow.


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)

Inviscid Flow

* +n this chapter e consider -ideal fl$id /otion knon

as inviscid flow0 this tpe of flo occ$rs hen either

12 µ →

3 4onl valid for 5e near 3 62, or#2 visco$s shearing stresses are negligi'le

* The inviscid flo ass$/ption is often valid in regions

re/oved fro/ solid s$rfaces0 it can 'e applied to /an

pro'le/s involving flo thro$gh pipes and flo over

aerodna/ic shapes


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7

Unifor/ Flo, "tead Flo

uniform flow: flow velocity is the same magnitude and direction at everypoint in the fluid.

non-uniform: (f at a given instant, the velocity is not the same at everypoint the flow. $(n practice, by this definition, every fluid thatflows near a solid boundary will be non-uniform - as the fluid

at the boundary must ta)e the speed of the boundary, usually*ero. +owever if the si*e and shape of the of the cross-section of the stream of fluid is constant the flow isconsidered uniform.%

steady: steady flow is one in which the conditions $velocity,pressure and cross-section% may differ from point to point but 0 change with time.

unsteady: (f at any point in the fluid, the conditions change with time,the flow is described as unsteady . $(n practice there is alwaysslight variations in velocity and pressure, but if the averagevalues are constant, the flow is considered steady .%


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Unifor/ Flo, "tead Flo 4cont.2

8

Steady uniform flow:

Conditions do not change with position in the stream or with time.

ample the flow of water in a pipe of constant diameter at constant velocity.

Steady non-uniform flow:

Conditions change from point to point in the stream but do not change with time.

ample flow in a tapering pipe with constant velocity at the inlet-velocity willchange as you move along the length of the pipe toward the eit.

Unsteady uniform flow:

t a given instant in time the conditions at every point are the same, but willchange with time.

ample a pipe of constant diameter connected to a pump pumping at a constantrate which is then switched off.

Unsteady non-uniform flow: very condition of the flow may change from point to point and with time at

every point.

ample waves in a channel.


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9a/inar and T$r'$lent Flo

:

Laminar flow

all the particles proceed along smooth parallel

paths and all particles on any path will follow itwithout deviation.

+ence all particles have a velocity only in thedirection of flow.

Tpical

particles

path

igure 3.1a 5aminar flow


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Turbulent Flow

;

Turbulent Flow

the particles move in an irregular manner through the flow field.

ach particle has superimposed on its mean velocity fluctuating velocitycomponents both transverse to and in the direction of the net flow.

Transition Flow

eists between laminar and turbulent flow. (n this region, the flow is very unpredictable and often changeable bac) and forth

between laminar and turbulent states.

odern eperimentation has demonstrated that this type of flow may compriseshort 6burst7 of turbulence embedded in a laminar flow.

Particle paths


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Representation of la/inar < T$r'$lent

=


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13


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11


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>ind T$nnel Testing

%E 173 1#


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1&

Newton’s 2nd Law for a Fluid article


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!he "ernoulli E#uation

* The /ost $sed and the /ost a'$sed e?$ation in fl$id /echanics.

* $%& Newton’s 'econd Law: F = ma

* * +n general, /ost real flos are &@D, $nstead 4 x, y, z, t; r,θ , z, t;etc2

* * 9et consider a #@D /otion of flo along -strea/lines, as shon 'elo.

1)


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Newton’s 2nd Law for a Fluid article

* The e?$ation of /otion for a fl$id particle in a stead inviscid floB

* >e consider force co/ponents in to directionsB along a strea/line

4 s2 and nor/al to a strea/line 4n2B

g p

ext g p

F F dt V d m

F F F F m

δ δ δ

δ δ δ δ δ

+=

++==∑

θ δ δ δ

θ δ δ δ

cos

sin

#

! F V

m

! F ds

dV mV

pn

ps

−=ℜ

−=


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1:

Newton’s 2nd Law (long a 'treamline

* Noting that

e haveB

, and

,sin ,

V ds

dp F

ds

dz V g ! V m

ps

δ δ

δ ρ θ δ ρδ δ

−=

==

ds

dp

ds

dz g

ds

dV V −−= ρ ρ


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1;

Formulation of "ernoulli e#uation


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1=

Newton’s 2nd Law (long a 'treamline

* +ntegrating along the strea/lineB

* +f the fl$id densit ρ re/ains constant

* This is the "erno#lli e$#tion

constant##1 =++∫ gz V

dp

s ρ

strea/lineaalongconstant

or

strea/lineaalongconstant

#

#1

#

#1

=++

=++

gz V p

gz V p

ρ ρ

ρ


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#3

Newton’s 2nd Law (cross a 'treamline

* ! si/ilar analsis applied nor/al to the strea/line

for a fl$id of constant densit ields

* This e?$ation is not as $sef$l as the Cerno$lli

e?$ation 'eca$se the radi$s of c$rvat$re of

the strea/line is seldo/ knon

constant#

=+ℜ

+ ∫ gz dnV p

n

ρ ρ

24ℜ


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#1

hysical Inter)retation of the "ernoulli E#uation

* !cceleration of a fl$id particle is d$e to an i/'alance

of press$re forces and fl$id eight

* %onservation e?$ation involving three energ

processesB

kinetic energ potential energ

press$re ork

strea/lineaalongconstant##1

=++ gz V p ρ ρ


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##

(lternate Form of the "ernoulli E#uation

* Press$re head 4 p/ ρ g 2 @ height of fl$id col$/n needed to prod$ce a press$re p

* Aelocit head 4V % /%g 2 @ vertical distance re?$ired for fl$id

to fall fro/ rest and reach velocit V

* Elevation head 4 z 2 @ act$al elevation of the fl$id .r.t. a

dat$/

strea/lineaalongconstant#

#

=++ z g

V

g

p

ρ


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#&

"ernoulli E#uation Restrictions

* The folloing restrictions appl to the $se of the 4si/ple2

Cerno$lli e?$ationB

12 fl$id flo /$st 'e inviscid

#2 fl$id flo /$st 'e stead 4i.e., flo properties are not f(t)at a given location2

&2 fl$id densit /$st 'e constant

)2 e?$ation /$st 'e applied along a strea/line 4$nless flo

is irrottionl 272 no energ so$rces or sinks /a eist along strea/line

4e.g., p$/ps, t$r'ines, co/pressors, fans, etc.2


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#)

*sing the "ernoulli E#uation

* The Cerno$lli e?$ation can 'e applied 'eteen an

to points, 412 and 4#2, along a strea/lineB

* Free (ets @ press$re at the s$rface is at/ospheric, or

gage press$re is zero0 press$re inside (et is also

zero if strea/lines are straight

* %onfined flos @ press$res cannot 'e prescri'ed

$nless velocities and elevations are knon

#

#

##

1

#1

#

1#

1

1 gz V p gz V p ρ ρ ρ ρ ++=++


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#7

Mass and +olumetric Flow Rates

* Mass flo rateB fl$id /ass conveed per $nit ti/e kgGsH

here V n I velocit nor/al to area /GsH

ρ I fl$id densit kgG/&H

& I cross@sectional area /#H

if ρ is $nifor/ over the area & and the average velocit

V is $sed, then

* Aol$/etric flo rate /&GsHB

∫ = & nd&V m ρ

&V m ρ =

&V ' =


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#8

,onservation of Mass

* ss cn neit*er +e creted nor destroyed

* For a control vol$/e $ndergoing stead fl$id

flo, the rate of /ass entering /$st e?$al the rateof /ass eitingB

* +f ρ I constant, then

###111

#1

V &V &

mm

ρ ρ =

=

#1##11 or ''V &V & ==


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#:

!he "ernoulli E#uation in !erms of ressure

* Each ter/ of the Cerno$lli e?$ation can 'e ritten

to represent a press$reB

* pgh B this is knon as the *ydrosttic press#re0hile not a real press$re, it represents the possi'le

press$re in the fl$id d$e to changes in elevation

24constant##1

- p gz V p =++ ρ ρ


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#;


* p B this is knon as the sttic press#re and represents the

act$al ther/odna/ic press$re of the fl$id


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#=


* The static press$re at 412 in Fig$re &.) can 'e

/eas$red fro/ the li?$id level in the open t$'e as

pg*

* B this is knon as the dynmic press#re0 it is

the press$re /eas$red ' the fl$id level 4 pg. 2 in

the stagnation t$'e shon in Fig$re &.) min#s thestatic press$re0 th$s, it is the press$re d$e to the

fl$id velocit

2

2& V


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&3


* The stgntion press#re is the s$/ of the static and

dna/ic press$resB

the stagnation press$re eists at a stagnation point, here

a fl$id strea/line a'r$ptl ter/inates at the s$rface of a

stationar 'od0 here, the velocit of the fl$id /$st 'e

zero

* -otl press#re 4 p- 2 is the s$/ of the static, dna/ic,

and hdrostatic press$res

#

1#

1

1# V p p ρ +=


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+elocity and Flow Measurement

* Pitot@static t$'e @ $tilizes the static and stagnation

press$res to /eas$re the velocit of a fl$id flo 4$s$all

gases2B

ρ G24# )& p pV −=


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* rifice, Nozzle, and Aent$ri /eters @ restriction devices

that allo /eas$re/ent of flo rate in pipesB


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&&


* Cerno$lli e?$ation analsis ields the folloing

e?$ation for orifice, nozzle, and vent$ri /etersB

Theoretical florateB

!ct$al florateB

H2G 41F

24##

1#

#1#

& &

p p &'idel

−

−

=

ρ

214


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&)


* "l$ice gates and eirs @ restriction devices that allo flo

rate /eas$re/ent of open@channel flosB


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UTP - Fluid Mechanics Course - September 2012 Semester - Chap 3 Bernoulli Equations

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