Differential Conservation Equations Part 3
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Transcript of Differential Conservation Equations Part 3
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8/6/2019 Differential Conservation Equations Part 3
1/23
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 ::
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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
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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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8/6/2019 Differential Conservation Equations Part 3
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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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8/6/2019 Differential Conservation Equations Part 3
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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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8/6/2019 Differential Conservation Equations Part 3
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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
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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 :