Bernoulli's Equation Chapter 4.1-4.6

STATIONARY FLUID – FLUID STATICS – HYDROSTATIC EQUATIONpressure and weight force balance in vertical direction

ρgzp ρg,zp

=−=∂∂

szz

dssVV

dx ds dsspp

∂∂

+

∂∂

+

∂∂

+

[ ]

zV

dx ds p

Assumptions:steady flow1DinviscidaddiabaticW=0

s)f(n,ρcontantρ

=≠

MOVING FLUID – EULER and BERNOULLI EQUATIONSforce balance along a streamline

weightpressuremomentum FFF ∆−∆−=∆

ns

z

y

szz ds,

sVV dx, dsds

spp

∂∂

+∂∂

+

∂∂

+

[ ]

zV

dxdn pweightpressuremomentum FFF ∆−∆−=∆

( )

dsdzρg-

spx

sVρV

EQUATION STREAMLINE EULERS

dx dsdn dsdzρg-dxdn ds

spdx dsdn

sVρV

dx dsdn dsdzρg∆F

weight,ofcomponent s for the

dx dsdn g ρ∆F

dxdn dsspdxdn pds

spp∆F

dx dsdn sVρVdx dsdn /dtVds

sVVρρV∆∆F

weight

weight

pressure

momentum

∂∂

−=∂∂

∂∂

−=∂∂

=

=∂∂

=

−

∂∂

+=

∂∂

=

−

∂∂

+==

s)f(n,ρcontantρ

=≠

BERNOULLI’S EQUATION (1740)force balance along a streamline

dsdzρg-

spx

sVρV

s)f(n,constant,ρEQUATION STREAMLINE EULERS

∂∂

−=∂∂

=ρ≠−

dz g ρdpρVdV

dzdzdsdz

dVdssV

dpdssp

dsszg ρds

dsdpds

sVV ρ

dsby gmultiplyin

−−=

=

=∂∂

=∂∂

∂∂

−−=∂∂

BERNOULLI’S EQUATIONsteady1D – streamlineinviscid – T= constantconstant densityno heat transferno work

constantgz2

Vρp

densityconstant for 2

=++

STAGAT ION PRESSURE

2

1p

p

21

1O

0

0

211

0

20O

2

V ρ 21

ppC

C t,coefficien pressure2

Vρpp

0Vfor

z g2

Vρpz g

2V

ρp

constantz g2

Vρp

∞

−=

+=

=

++=++

=++

2SIPHON

2h1

z

( )

( )

3 2

2atm32

32atm2

322

3atm

3arm1

32

122322311

2

222

1

211

Vlower thehhigher thegpphh

hhgpp

hhg2gh2pgh0p

pppgh2VV,hhz,hz,0V

gz2

Vpgz2

Vp2point 1topoint from

ρ=+

+−ρ

=ρ

+++ρ

=++ρ

====+===

++ρ

=++ρ

−

3h3

( )21

33

23

3

3arm1

233

3atm

2311

3

233

1

211

gh2V

2Vgh

ppp

0g2

Vpgh0p

0z,hz,0V

gz2

Vpgz2

Vp3point 1topoint from

=

=

==

++ρ

=++ρ

===

++ρ

=++ρ

m 10.24m9.9999.2

101,325h

m hsecm

kgm9.9mkg999.2

mseckgm101,325

mseckgm1000

m1000NlkPa

hmsecm9.9

mkg999.2101.325kPa

mkg999.2ρ C, t20afor water

h g ρp

critical

critical3322

222

23

3

atm

=×

=

××=

==

××=

=

= ft 33.962.4

14414.7h

ft hftlb62.4

ftin144

inlb14.7

fthsec

ft32.2ft

slugs1.94ftin144

inlb14.7

ftslugs1.94ρ F, 50at for water

h g ρp

critical

critical3f

2

2

2f

critical232

2

2f

3

atm

=×

=

×=×

××=×

=

=atmospherep p=0

h

EULER NORMAL EQUATIONforce balance normal to a streamline

Assumptions:Steady flow1DinviscidaddiabaticW=0

weightpressuremomentum FFF ∆−∆−=∆

V

nzz

dxdn dnnpp

∂∂

+

∂∂

+

ns

z

x

[ ]z

dxdn p

EULER NORMAL EQUATIONforce balance normal to a streamline

0np effect, no hasgravity where

nzρg

np ,Rfor

\IONEQUAT NORMAL EULER RVρ

nzρg

np

nzg ρ-

np

RVρ-

dx dsdn nzg ρ-dxdn ds

np

RVρ-

dx dsdn nzg ρ∆F

dxdn dsnpdxdn pdn

npp∆F

RVρ∆F

2

2

2

weight

pressure

2

momentum

=∂∂

∂∂

−=∂∂

∞→

=∂∂

+∂∂

∂∂∂

−=

∂∂∂

−=

∂=

∂∂

=

−

∂∂

+=

−=

nzz

dxdn dnnpp

∂∂

+

∂∂

+

[ ]z

dxdn pV

First Law of ThermodynamicsExamples:Observations:

wheela brakingbearing wheelain friction togetherhandsyou rubbing

∑ ∑∝

=

δWδQheat more friction more

heat into ed transferrbecan work

Experiments:

( ) 0δWδQ

δWδQ

Cycle afor LawFirst m kg .427calorie 1

lbft 778BTU 1

δWCδQ

WCQ

f

=−

=

==

=

=

∫∫∫

∫∫∑∑

WW

QThe First Law is a fundamental observationof nature, an axiom, which can not be provedbut has never been found to be violated

JOULE EXPERIMENT

First Law of Thermodynamics

( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( ) WQδWδQEE

U(T).PEKE forms, allin energy as E Define property. a therefore

andpath oft independen is δWδQ

δWδQδWδQ

0δWδQδWδQ C and B Processes

0δWδQδWδQ CA Cycle

0δWδQδWδQ BA Cycle

0δWδQ

Cycle afor LawFirst δW δQ

1

221

CB

CB

CA

BA

−=−=−

++

−

−=−

=−−−

=−−−⇒

=−−−⇒

=−

=

∫

∫∫∫

∫∫∫∫∫∫

∫∫∫

AB

C

δWdEδQW∆EQ

Processes afor LawFirst

+=+=

1

2

( )

shaft

2

shaft1

2

2221

2

111

22211shaftout flowin flowshaft

22out flow

1111final 1initial 11in flow

Wρgh)2

Vpv(umQ

W)ρgh2

Vvpm(u)ρgh2

Vvpm(uQ

ρgh2

V(T)UPEKE(T)UE

vp mvp mWWWWWvp mW

vp mVpVVppdVW

LawFirstW∆EQ

++++∆=

++++−+++=

++=++=

−+=++==

==−==

+=

∫

Steady Flow Energy EquationOpen, steady flow thermodynamic system - a region in space

QshaftW

p1p2

v2

v1 h2

V

V

h1

BERNOULLI’S EQUATION from THE FIRST LAW

0Wρgz)2

Vpv(um

constantgz2

Vρp

0ρgz2

Vpv

0T0,0Tc u

ρgh)2

Vpv(u

0W,0Q,D1

Wρgz)2

Vpv(umQ

shaft

2

2

2v

2

shaft

2

=++++∆

=++

=++

=∆⇒=µ=∆=

+++∆

==

++++∆=