8/6/2019 Differential Conservation Equations Part 3

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Conservation of

mechanical energy

Conservation of

mechanical energy

start with the momentum equation :start with the momentum equation :

"change of kinetic energy is a result

of work done by external forces"

jj FdVuDtD

!V juv

forcesexternalbyworkchangeenergykinetic

2 ji

ij

j

jjudV

xdVf

uudV

Dt

D

x

x!

WVV

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Conservation of

mechanical energy

Conservation of

mechanical energy

work

energykinetic

2velocitydensity

-

-

-

21

21

21

j

i

ij

jj

V

jj

jj

uxuf

dVuu

uu

x

x

!

!

!

vv

W

V]

VJ

VNset :set :

j

i

ij

jj

i

jj

i

jj

ux

uf

uuu

x

uu

t

xx!

!

x

x

x

x

W

22

kinetic energy equationkinetic energy equation

for an elementary volumefor an elementary volume dVdV ::

8/6/2019 Differential Conservation Equations Part 3

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Conservation of

mechanical energy

Conservation of

mechanical energyintegrated forintegrated for

a final volumea final volume VV

bound by a surfacebound by a surface AA ::

ui

ni dA

dVV

A

forcessurfaceofwork

forcesbodyofwork

outletinletconvectionenergykinetic

changeenergykinetic

22

!

x

x

A

ijij

V

jj

Aii

jj

V

jj

dAnudVuf

dAun

uu

dV

uu

t

WV

VV

8/6/2019 Differential Conservation Equations Part 3

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work

ndeformatio

i

j

ij

work

kinetic

i

ij

j

work

total

ijj

i x

u

xuu

x x

x

x

x!

x

xW

WW

Conservation of

mechanical energy

Conservation of

mechanical energy

j

i

ij

jj

i

jj

i

jj

ux

uf

uuu

x

uu

t

x

x!

!

x

x

x

x

W

22

d

ij

d

ij

i

iijij

x

up [W[W

x

x!

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Conservation of

mechanical energy

Conservation of

mechanical energy

xx!

xx

outletinletconvectionenergykinetic

changeenergykinetic

22i

jj

i

jj

t VV

x

x

for ss rfaceofwortotalforcesbody

ofwor

jij

i

jj ux

uf WV

work

nde ormatiototal

work

nde ormatiosp erical-non

work

nde ormatiosp erical

d

ij

d

ij

i

ixp [Wx

x

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Conservation of internal

(thermal) energy

Conservation of internal

(thermal) energy

"change of internal (thermal) energy

is a result of the transferred heat "

sf rt trrd f rm ti

sf rt trdu tiri t r l

i

j

ij

ii

Tk

t x

x

x

x

x

x! WYV

ijiji

i

x

qdV

Dt

D[WYV

x

x!

qi

ni dA

dV V

Ai

ix

Tkq

x

x!

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Conservation of internal

(thermal) energy

Conservation of internal

(thermal) energy

transferheat-

-

-

i

j

ij

ii

V

x

u

x

Tk

x

dVcT

cT

x

x

x

x

x

x!

!

!

vv

W]

VJ

VN

energyinternal

etemperaturheatspeci icdensity

set :set :

i

jij

ii

i

i

xu

xTk

x

cTux

cTt

xx

xx

xx!

!x

x

x

x

W

VV

thermal energy equationthermal energy equation

for an elementary volumefor an elementary volume dVdV ::

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Conservation of internal

(thermal) energy

Conservation of internal

(thermal) energy

integrated forintegrated for

a final volumea final volume VV

bound by a surfacebound by a surface AA ::

ui

ni dA

dVV

A

sf rh t tr nn workd form tio

sf rh t tr nondu tion

outl tinl tonv tionn rgyint rn lh ngn rgyint rn l

x

x

x

x

!

x

x

V i

j

ij

A

i

i

A

ii

V

dVu

dAnT

k

dAucTndVcT

t

W

VV

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Conservation of

energy (total)

Conservation of

energy (total)"change of total energy is a result

of work done by external forces

and heat transferred"

!

changeenergytotal

2cT

uudV

Dt

D jjV

sfrh r nondu on

d

x

k

x ii

x

x

x

x

forcesexternalbywork

dVu

x

uf jiji

jj

x

x

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Conservation of

chemical species

Conservation of

chemical species" change of concentration of chemical

species is a result mass transfer

and chemical reaction "

]

reaction

chemi altodueksour e/sin

transfermassdiffusion

changeionconcentrat

s

i

s

s

i

sR

x

m

xdm

Dt

D

x

x

x

x!

DV

si

s

is Rx

qdVm

Dt

D

x

x!V

qis

ni dA

dV V

Ai

sss

ix

mq

xx

! D

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Conservation of

chemical species

Conservation of

chemical species

reactionchemical

aniffusion

massspecies

ionconcentratmasspecies

s

i

s

s

i

s

s

R

x

m

x

dVm

m s

x

x

x

x

D

set :set :

s

i

s

s

i

i

s

i

s

Rxm

x

umx

mt

x

xxx!

!x

x

x

x

D

mass transfer equationmass transfer equation

for an elementary volumefor an elementary volume dVdV ::

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Conservation of

chemical species

Conservation of

chemical speciesintegrated forintegrated for

a final volumea final volume VV

bound by a surfacebound by a surface AA ::

ui

ni dA

dVV

A

reactionchemicaltodueksource/sin

trans ermassdi usions ecies

outletinletconvections ecieschangeionconcentrats ecies

x

x

!x

x

V

s

A

i

i

s

s

A

ii

s

V

s

dVRdAnx

m

dAunmdVmt

D

VV

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SOME

SIMPLIFICATIONS

SOME

SIMPLIFICATIONS

of theof the

conservationconservation

(transport)(transport)eq ationseq ations

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Simplification of the

contin ity eq ation

Simplification of the

contin ity eq ation

for an incompressible fluidfor an incompressible fluidVV=const=const::

0!x

x

i

i

x

u

written for an elementary volumewritten for an elementary volume dVdV ::

ii

uxt

VVxx!

xx

const.!!xx

1

1

1

ux

u

for 1D flow:for 1D flow: uu22=u=u33=0=0

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Simplifications of the

moment m eq ations

Simplifications of the

moment m eq ations

x

x

x

x

x

x

x

x

x

x

!x

x

x

x

i

i

ji

j

ij

j

iji

j

x

u

xx

u

xx

pf

uuxut

QQ3

start with the Navierstart with the Navier--Stokes equation:Stokes equation:

assume the fluid is ideal (inviscid)assume the fluid is ideal (inviscid) QQ=0=0 ::

assume the fluid is incompressibleassume the fluid is incompressible

VV=const.=const.result is theresult is the EulerEulerequation:equation:

jj

i

j

i

j

x

p

fx

u

ut

u

x

x

!x

x

x

x

V

1

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Simplifications of the

moment m eq ations

Simplifications of the

moment m eq ationsassume the only body force is gravity:assume the only body force is gravity:

j

jjjxzgfgf

xx!! or

! pgz

xxuu

tu

ji

j

i

j

momentum equation for anmomentum equation for anideal incompressible fluid in theideal incompressible fluid in the

gravity field:gravity field:

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Simplifications of the

moment m eq ations

Simplifications of the

moment m eq ations

ifthe fluid is not movingifthe fluid is not moving uuii= u= ujj= 0= 0 ::

!

pgz

xx

uu

t

u

ji

j

i

j

.constpgz !V

0!V

o

o

ppzzg

result is the equation ofresult is the equation offluid staticsfluid statics::

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Simplifications of the

moment m eq ations

Simplifications of the

moment m eq ations

x

x!

x

x

x

x

V

pgz

xx

uu

t

u

ifthe movement ofthe fluidifthe movement ofthe fluid

along the streamline is followedalong the streamline is followed

in the directionin the direction xx, then, then uu ii= u= ujj= u= u::

for a steady statefor a steady state

0

2

2

!

V

pgz

u

x

.2

2

onst

p

gz

u

! V

integration givesintegration gives BernoulliBernoulli equation:equation:

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Simplification of the

total energy eq ation

Simplification of the

total energy eq ation

forcesexternalbyork

dVux

uf jiji

jj

xx! WV

!

dVcTuu

t

D jj V

energyinternalineticomass)unit(perchangetotal

s erheat tranconduction

dVT

ii

x

x

x

x

f hf h

8/6/2019 Differential Conservation Equations Part 3

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Simplification of the

total energy eq ation

Simplification of the

total energy eq ation

!

xx

xx

outl tinl tonve tionener

hangeenergylocal

22ii

jj

i

jj cTuuuux

cTuut

] forcessurfaceofworforcesbody

ofwor

1jij

ijj uxuf WV x

x

! sferheat tran

1

x

x

x

x

ii

T

V

QWUPEKE

UPEKE

!

x

x

iflown

outflown

t

which iswhich is 1st law of thermodynamics1st law of thermodynamicsfor a control volume (open system)for a control volume (open system)

f hf h

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Simplification of the

total energy eq ation

Simplification of the

total energy eq ation

1st law of thermodynamics1st law of thermodynamics

for a control mass (closed system)for a control mass (closed system)

assuming no kinetic energy (KE)assuming no kinetic energy (KE)

oror

potential energy (PE) changepotential energy (PE) change

QWU !

x

x

t

f hf h

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Simplification of the

total energy eq ation

Simplification of the

total energy eq ationifthere is noifthere is no

movementmovement

ofthe fluidofthe fluid

and noand no

deformationdeformationandandVV=const.=const.

andandk=const.k=const.

!

changeenergytotal

2

cTuu

dV

Dt

D jjV

s rt trd ti

d

x

Tk

x ii

x

x

x

x

forcesexternalby ork

du

x

f jiji

j

x

x! WV

2

2

ixct x

x

!x

x

V

result is theresult is the heat conductionheat conduction equation :equation :

Si lifi i f thSi lifi i f th

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s

i

s

s

i

i

s

i

s

x

m

x

umx

mt

x

x

x

x!

!x

x

x

x

D

Simplification of the

mass transfer eq ation

Simplification of the

mass transfer eq ation

ifthere is noifthere is no

movementmovement

ofthe fluidofthe fluid

no chemicalno chemical

reactionreactionandandDDss=const.=const.

2

2

i

ss

x

m

Dt

m

x

x

!x

x

result is theresult is the Fick second lawFick second law equation :equation :