3lecture conservation eqs
TRANSCRIPT
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Fundamental Considerations ofFundamental Considerations of
Continuum ConservationContinuum Conservation
January 2011
Suman Chakraborty
Professor
Mechanical Engineering Department
Indian Institute of Technology (IIT) Kharagpur, India
E-mail: [email protected]
http://www.stanford.edu/~sumancha/
mailto:[email protected]:[email protected] -
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General Philosophy of ConservationGeneral Philosophy of Conservation
Balance: IN OUT + GENERATED = CHANGE
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Reynolds Transport Theorem (RTT)Reynolds Transport Theorem (RTT)
Control mass system Control volume concept
Eulerian approach
Applicable for any property :-
mass, momentum, energy etc
Can be vectors or Scalars
Lagrangian approach
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I
II
IIIN=extensive property of a system
n= N per unit mass
Nt
= (NI
)t
+ (NII
)tNt+t = (NII
)t+t + (NIII
)t+t
t
NNLt
t
NNLt
dt
dN tIIttIIt
ttt
tsys
)()(
00
Rate of outflow -inflow
cv
ndt
cs
r dAVn ).(
cs
r
cvsys
dAVnndtdt
dN ).( RTT
where Vr
is the velocity of fluid relative to the CV
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Conservation of MassConservation of MassN=m (mass)n=1
Assuming stationary CV : Vr
=V
( . )sys cv cs
dm
d V dAdt t
=0
0).( dVdt
cvcv
Non-deformable CVUsing Gauss divergence theorem
0).(
V
t
For any arbitrary CV
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Conservation of Linear MomentumConservation of Linear Momentum
Assuming stationary CV :Vr
=V
N mV
n V
( )( . )
cv cssys
d mVVd V V dA
dt t
( )sys
sys
d mVFdt
Newtons second law
cvsys FFas t0
Surface forces
Body forces
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Surface forces: Concept of StressSurface forces: Concept of Stress
Definition of StressThe stress field is the distribution of internal "tractions" that balance a given set
of external tractions.Traction T
represents the force per unit area acting at
a given location on the body's surface. Traction T is abound vector, which means T cannot slide along its
line of action or translate to another location and
keep the same meaning. In other words, a traction
vector cannot be fully described unless both the force
and the surface where the force acts on has beenspecified. Given both Fand s, the traction Tcan be defined as
n
iTComponents ofT
are
designated by 2 indices
Direction normal to
the chosen area
Direction of action of the force component
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Stress tensor components are denoted by where iis the direction normal to
the face on which it is acting andjrepresents the direction of action of the
concerned force component
23
2221
33
3231
1312
11
3x
2x1x
Stress Tensor Components
ij
Our objective now is to express the traction vector at a point on any
arbitrary plane in terms of the stress tensor components at that
point
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1 1 2 2 3 3
n
i i i i ji jT n n n n
O
A
B
C
x1
x2
x3
Consider surfaces:
S1
: OAB with surface normal along x3S2
: OBC with surface normal along x2S3
: OAC with surface normal along x1& S: ABC with surface normal along
knjninn 321
Applying force balance:
where b
is the body force per unit volume
h is the perpendicular distance from O to ABC
Noting Si
=Sni
and taking Limit as h0
Generalised
to => Cauchys theorem
Traction Vector on an arbitrary surfaceTraction Vector on an arbitrary surface
1 11 11 1 21 2 31 3 1 1 13 3
nF S S S T S Shb Sha
3312211111 nnnTn
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Conservation of angular momentumConservation of angular momentum
Net moment w.r.t
centroidal
axis (OO) of the element=Io
as y, x0
valid when couple stresses (body
couples) are absent.
jiij
)( 222112 yxyxKyxxy
x21
x22
x22x21
y12
y11y
11
y12
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bodysurfcv FFF
cv
i
cs
icvidbdAnF .
cv
i
cv
i
cv
i
cv
icvidVudu
tdbdF ).()(.
assuming non-deformable control volume
(From RTT)
Since choice of CV is arbitrary,
( ) .( ) .i i i iu u V bt
Can be represented in tensor notation as:
i
j
ij
ji
j
i bx
uux
ut
)()(
Naviers
equation of
equillibrium
NavierNavierss equation of equilibriumequation of equilibrium1 2 3
i i i ii j k
where
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Substituting the Constitutive RelationshipSubstituting the Constitutive Relationship
where p is the thermodynamic pressure
(satisfies Eq. of state)
For a Newtonian fluid, is a linear function of eij
dev
ij ijkl klC e
dev
ij ij ijp
}]{}[{21
i
j
j
i
i
j
j
i
j
i
xu
xu
xu
xu
xu
a function of the rate of deformation
Symmetric part
Rate of deformation eij
Anti-Symmetric part
ij is a function ofeij
only
ij
A 4th
order tensor that maps a 2nd
order tensor onto a 2nd
order tensor
Note that
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lkjiijkl DCBACs
Thus, for isotropic fluid,
jkiljlikklijijklC
jiij
(Volume dilation coefficient)(Viscosity coefficient)
)(i
j
j
iij
k
kijij
x
u
x
u
x
up
Finally the stress tensor yields the form
Define
isotropic
scalar ).)(.().)(.().)(.( CBDADBCADCBA For isotropy
jjiijjiijjii CBDADBCADCBA
)( jkiljlikklijlkji DCBA
Using we can show
Noting
2dev k
ij ij ij
k
ue
x
Special case: Homogeneous + Isotropic Fluid
Position independent for homogeneous
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11 22 33
3
mp
k
km
x
upp
)(
32
Stokes hypothesis: pm
=p 32
automatically satisfied for incompressible fluid as:
and also for monoatomic
gas
0k
k
u
x
time scale of change needs to be larger compared to
the molecular relaxation time
The thermodynamic pressure (p) accounts for translational + rotational +
vibrational
modes of energy of the molecules
The mechanical pressure considers only the translational mode
32 known as the bulk viscosity of fluid
Towards the Stokes Hypothesis
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NavierNavier--Stokes equationStokes equation
i
j
ij
ji
j
i bx
uux
ut
)()( Navier
equation
LHS= ][)]([)()(j
ij
ij
j
iji
j
ixuu
tuu
xtuuu
xu
t
0 (from continuity)
Substituting the expression for stress in RHS we obtain:
i
k
k
ij
i
jij
ij
i bx
u
xx
u
xx
P
x
uu
t
u
]
3[][][
=0for incompressible
fluid
Navier-Stokes equation
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Conservation of EnergyConservation of EnergyN=E (Total energy )
= Internal energy (mi)+Kinetic
Energy (mv2/2)+Potential Energy (mgz)
n=e
Assuming stationary CV : Vr
=V
( . )sys cv cs
dEed e V dA
dt t
cv j
jd
x
eu
t
e]
)()([
using non deformable control volume and Gauss divergence theorem
cvcv j
j
j
j
sysdDt
De
dx
u
tex
e
ut
e
dt
dE
)}
)(
()({
=0
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First law of ThermodynamicsFirst law of Thermodynamics
cvcv
sys
WQdt
dE
dAnqdQQcscv
cv.
'''''
'''[ ]jjcv
qQ d
x
( )
n
cv i i i i
cv cs
ii i i
cv cs
W b u d T u dA
b u d u ndA
cv j
iij
ii dx
uub ])([
Q
is the rate of heat generation
q is the heat flux
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j
iij
ii
i
i
x
u
ubx
qQDt
De
)('''
For any arbitrary choice of control volume, combining the above two results we get,
Above is the statement of total energy conservation, however we are interested in
thermal energy only and hence we would subtract the mechanical energy from the
above equation, to get an equation in terms of specific internal
energy (i)
Multiplying ui
with the Naviers
equation:
iji
i i ij
Duu u b
Dt x
Mechanical Energy
And then subtracting we obtain,
j
i
ijj
j
x
u
x
q
QDt
Di
''' })(
{j
j
j
j
x
u
t
i
x
iu
t
i
Dt
Di
j
j
x
iu
t
i
)()(
j
iij
j
j
j
j
x
u
x
qQ
x
iu
t
i
''')()(
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Vpx
u
j
iij .
We need to find
j
iij
x
u
where
])()()[(
])()()[(
2
2
3
3
22
1
3
3
12
1
2
2
1
2
3
3
2
22
3
3
1
12
2
2
1
1
32
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
It is important to note that >0 (Viscous dissipation)
Using the stress tensor for the Newtonian and Stokesian
fluid we get,
Using the expression for and noting that
j
iij
x
u
we get, ''' j
j
q Dh DpQ
Dt Dt x
Generalised
thermal energy
conservation equation
ph i
Viscous heating because of the energy dissipation
due to work done against viscous shear
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h=h(T, p) for a simple compressible pure substance & assume no phase change
dPP
hdTCdP
P
hdT
T
hdh
T
p
Tp
again [ ] [ ] p T p P
s s s vdh TdS vdP T dT T v dP T dT T v dP
T P T T
andpdT
dv
v
1
gives(1 )p
Dh DT DPC v T
Dt Dt Dt
Substituting the above relation in''' j
j
q Dh DpQ
Dt Dt x
and assuming Fouriers law of heat conduction to be valid, we obtain:
).(''' TkQDt
DpT
Dt
DTCp
Express enthalpy gradient in terms of pressureExpress enthalpy gradient in terms of pressure
and temperature gradientsand temperature gradients
Governing differential equation
for T
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Species Conservation. . . .
, , , , (all for species )i in i out i gen i stored im m m m
.ii i iu r
t
ir
This principle can be applied to a differential control volume in a
manner very similar to the continuity relation. The only new
consideration is that the mass of speciesican be created or destroyedvia chemical reactions, so that :
Where Is the generation rate ofiper unit volume
with units of3
kg of i
s m
Species velocity = velocity of i w.r.t. ground (this
velocity is a continuum average that includes the
effects of bulk flow and diffusion)
iu
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There is a constraint on these variables, namely :
1
for N speciesN
i
i
Next, we define several variables. The mass averaged velocity
of the mixture is
1 mass averaged velocity
N
i i
i
u
u
The quantity
1
N
i i
i
u u
the local rate at which mass passes through a unitcross sectional area perpendicular to .u
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the drift velocity of species i causes a flux which can be
expressed as :
drift component of species i (w.r.t. )i i i J u u u
Therefore ,
i i i iu J u
iJ
.i i i i J u r t
Substituting this into the expression for species conservation :
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From Ficks law for binary systems where the drift velocity is
due only to diffusion, we can write :
. .i i ij i iu D C r t
. . .i i i iu J rt
or
i i J D C
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Diffusive
SV
t
2).()(
General form of the Conservation EquationsGeneral form of the Conservation Equations
General conservative form
Unsteady Source
S Eqn
1 0 0 continuity
u x-mom
v y-mom
w z-mom
T k/Cp energy
C D r species
xbx
V
x
P
).(
3
yby
VyP
).(
3
zbz
V
z
P
).(
3
p
gen
pp Cq
CDtDpT
C
Advective