X.S. Bai Transport of mass, momentum and heat
Mean speed of molecules: microscopic scale
1/ 28 B
ii
k Tsmπ
⎛ ⎞= ⎜ ⎟⎝ ⎠
kB is Boltzmann’s constant, mi is the mass of molecule i.
X.S. Bai Transport of mass, momentum and heat
Mean speed of molecules: microscopic scale
1 1
1
1
N N
i i i i Ni i
i iNi
ii
v vv Y v
ρ ρ
ρρ
= =
=
=
= = =∑ ∑
∑∑
Define a mass averaged mean velocity
X.S. Bai Transport of mass, momentum and heat
Mass transfer
• A mixture containing N species, each species has different velocity
ii Vvv +=
Velocityof
species iMass averages
velocity ofmixture
Diffusionvelocity ofspecies i
01
1
=
=
∑
∑
=
=N
iii
N
iii
VY
vYv
X.S. Bai Transport of mass, momentum and heat
Mass transfer
•
– diffusion velocity due to concentration gradient– diffusion velo. due to temperature gradient (Soret)– diffusion velocity due to pressure gradient
• the last two effects can be neglected• diffusion by concentration gradient is modeled by Fick’s Law
iV
iiNii YDVY ∇−=
Binary diffusivityNegative sign indicate flux fromhigh concentration to low
X.S. Bai Transport of mass, momentum and heat
A control volume CV for mass conservation
: reaction rate of i: mass fraction of i
density of the mixture: velocity of i
i
i
i
Y
v
ω
ρ :
A control volume CV for mass conservation: the boundary is open,species can enter or leave the CV; it can also be formed or consumed in the CV
X.S. Bai Transport of mass, momentum and heat
Law of mass conservation
Law of mass conservation says that the increase of the total mass of species i in the CV is equal to the net mass flow into the CV + the formation rate of the species in the CV.
X.S. Bai Transport of mass, momentum and heat
Mass transfer
• Transport of species i in a mixture of N species, in a control volume ...
iiii vY
tY ωρρ
+⋅−∇=∂∂
Change of mass ofspecies i
= Net massflux in
+Formation
rate ofspecies i
Known fromchemical kinetics
X.S. Bai Transport of mass, momentum and heat
Mass transfer
• Using Fick’s Law one has the transport equation for species i
• This equation is not closed without reaction kinetics rate information !
iiiNii YDvY
tY ωρρρ
+∇⋅∇=⋅∇+∂∂
Convectivetransfer
diffusivetransfer
X.S. Bai Transport of mass, momentum and heat
Mass transfer
• Summation over all species we have
∑∑∑
==
= +⋅−∇=∂
∂ N
ii
N
iii
N
ii
vYt
Y
11
1 ωρ
ρ
=1v =0
Law ofmassaction
0=⋅∇+∂∂ v
tρρ Hi, I know it !
It’s the continuityequation !
X.S. Bai Transport of mass, momentum and heat
Momentum conservation
We consider the momentum conservation of the mixture.
The law of momentum conservation is the Newton’s second law applied to Fluid Mechanics, which states that the change of the total momentum in the CV (Fig.4.1) + the exchange of momentum through the boundary of the CV is equal to the forces acting on the boundary + the body forces on the CV.
X.S. Bai Transport of mass, momentum and heat
Momentum transfer
( )
( )
, or ( 1 2 3)
or , or
i j iji
j i j
iji ij
j i j
ρu u τρuv pvv pI i , ,t t x x x
τu uv pv v pI ρ ρut t x x x
ρ ρ τ
ρ ρ τ
∂ ∂∂∂ ∂+∇ ⋅ = ∇ ⋅ + + = − + =
∂ ∂ ∂ ∂ ∂
∂∂ ∂∂ ∂+ ⋅∇ = ∇ ⋅ + + = − +
∂ ∂ ∂ ∂ ∂
23
ji kij ij
j i k
uu uτ μ δ μx x x
⎛ ⎞∂∂ ∂= − −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠
X.S. Bai Transport of mass, momentum and heat
A control volume CV for energy conservation
Energy flow outWork done to the
environment
Heat transfer from the environment
: enthalpy of i: mass fraction of i
density of the mixture: velocity of i
i
i
i
hY
vρ :
Energy flow in
the boundary is open, species can enter or leave the CV; it can also be formed or consumed in the CV
X.S. Bai Transport of mass, momentum and heat
Total energy change per unit mass
2
2
1
2
1
2
12
12
12
12
total
N
i iiN
i ii
e e v
Y e v
pY h v
ph v
ρ
ρ
=
=
= +
= +
= − +
⎛ ⎞= − +⎜ ⎟⎝ ⎠
∑
∑
X.S. Bai Transport of mass, momentum and heat
The energy flux at an open boundary
• Energy flux due to mass flow of species through the CV boundaries
• Energy flux due to temperature gradient through the CV boundaries
• Energy flux due to concentration gradient at the boundaries (Dufour effect, often negligible)
X.S. Bai Transport of mass, momentum and heat
Energy flux due to flow motion
( ) ( )2 21 12 2
1 1
2 21 12 2
1 1
2 21 12 2
1 1 1 1
212
N N
i i i i i ii i
N N
i i i i ii i
N N N N
i i i i i i i ii i i i
pv e v Y v V h v
p pv Y h v YV h v
p pv Y h v Y v YV h YV v
pvh v v
ρ ρρ
ρ ρρ ρ
ρ ρ ρ ρρ ρ
ρ ρρ
= =
= =
= = = =
⎛ ⎞+ = + − +⎜ ⎟
⎝ ⎠⎛ ⎞ ⎛ ⎞
= − + + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞= + − + + + − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎛
= + − +⎝
∑ ∑
∑ ∑
∑ ∑ ∑ ∑
1
N
i i ii
YV hρ=
⎞+⎜ ⎟
⎠∑
X.S. Bai Transport of mass, momentum and heat
Energy flux due to temperature gradients
q Tλ= − ∇
heat conductivity
X.S. Bai Transport of mass, momentum and heat
Total energy flux
• Through the boundary of a control volume
212
1
N
total i i ii
pq q vh v v YV hρ ρ ρρ =
⎛ ⎞= + + − + +⎜ ⎟
⎝ ⎠∑
conduction convection diffusion
X.S. Bai Transport of mass, momentum and heat
work per unit area done to the environment
( )pI vτ− + ⋅
X.S. Bai Transport of mass, momentum and heat
First law of thermodynamics
The first law of thermodynamics states that the energy increase in a system is equal to the heat transfer from the environment to the system - work done to the system. In differential equation form one can write
( ) ( )total total re q Q pI vtρ τ∂
+∇ ⋅ = +∇ ⋅ − + ⋅⎡ ⎤⎣ ⎦∂
X.S. Bai Transport of mass, momentum and heat
Energy transport equation – conventional forms
( )
( )( )
212
212
1
N
i i ii
r
h p vt
q vh v p v YV h
Q pI v
ρ ρ
ρ ρ ρ
τ=
∂− +
∂⎛ ⎞
+∇ ⋅ + + − + +⎜ ⎟⎝ ⎠
= + ∇ ⋅ − + ⋅⎡ ⎤⎣ ⎦
∑
X.S. Bai Transport of mass, momentum and heat
Energy transport equation – conventional forms
( )( )
( ) ( )
( ) ( ) ( )
2122 1
2
2122 1
2
:
:
vv v v v pItv v v pI v pI vtv vv pI v pI v
t
ρ ρ τ
ρ ρ τ τ
ρρ τ τ
∂⎛ ⎞⋅ + ⋅∇ = ⋅ ∇ ⋅ − +⎜ ⎟∂⎝ ⎠∂
⇒ + ⋅∇ = ∇⋅ − + ⋅ − − + ∇⎡ ⎤⎣ ⎦∂∂
⇒ +∇⋅ = ∇ ⋅ − + ⋅ − − + ∇⎡ ⎤⎣ ⎦∂
X.S. Bai Transport of mass, momentum and heat
Energy transport equation – conventional forms
( ) ( )
( )1
:
N
i i ii
r
h p q vh v p YV ht
Q pI v
ρ ρ ρ
τ=
∂ ⎛ ⎞− + ∇ ⋅ + + − +⎜ ⎟∂ ⎝ ⎠
= + − + ∇
∑
jj
D v uDt t t x
∂ ∂ ∂= + ⋅∇ ≡ +∂ ∂ ∂
1:
N
i i i ri
Dh Dp q YV h Q vDt Dt
ρ ρ τ=
⎛ ⎞− = −∇ ⋅ + + + ∇⎜ ⎟
⎝ ⎠∑
X.S. Bai Transport of mass, momentum and heat
Energy transport equation – conventional forms
1
11 :N
i i ri i
Dh Dp ρα h ρα h Y Q vDt Dt Le
ρ τ=
⎛ ⎞⎛ ⎞− = ∇ ⋅ ∇ − − ∇ + + ∇⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
ii p i
α λLeD ρc D
≡ =
Lewis number
X.S. Bai Transport of mass, momentum and heat
Energy transport equation – conventional forms
1
11 :N
i i ri i
Dh Dp ρα h ρα h Y Q vDt Dt Le
ρ τ=
⎛ ⎞⎛ ⎞− = ∇ ⋅ ∇ − − ∇ + + ∇⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
( )Dh ρα hDt
ρ =∇⋅ ∇
X.S. Bai Transport of mass, momentum and heat
Energy transport equation – conventional forms
( ) ( )1
p
N
i pi i i i ri
DT DpcDt Dt
T D c Y T h Q v
ρ
λ ρ ω τ=
−
= ∇ ⋅ ∇ + ∇ ⋅∇ − + + : ∇∑
( ) ( )1
N
p i pi i i ii
DTc T Dc Y T hDt
ρ λ ρ ω=
=∇⋅ ∇ + ∇ ⋅∇ −∑
X.S. Bai Transport of mass, momentum and heat
Energy transfer
• in species transport equations, there is a source term due to chemical reaction, why there is no source term due to chemical reactions in the energy equation?
• chemical energy is released to thermal energy, however the system’s energy is not changed due to chemical reactions !
• If energy equation is expressed as temperature transport equation, one should have the reaction source term
X.S. Bai Transport of mass, momentum and heat
Summary of governing equations
• Unkowns– 3 velocity
components– pressure– temperature or
enthalpy– N species– density
• total unknowns– N+6
• Governing equations– continuity 1– N-S equations 3– temperature or
enthalpy1
– equation of state 1– species transport N
• total equations– N+6
X.S. Bai Transport of mass, momentum and heat
Momentum:
0xu
j
j =+∂ρ∂
∂ρ∂t
j
ij
ij
jiixx
px
uutu
∂
∂+
∂∂
−=∂
∂+
∂∂ τρρ
Pri
ri j j
u hh h Qt x x x
ρρ μ⎛ ⎞∂∂ ∂ ∂+ = +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
ij
i
jj
ijixY
ScxxYu
tY ω
∂∂μ
∂∂
∂ρ∂
∂ρ∂
+⎟⎟
⎠
⎞⎜⎜
⎝
⎛=+
∑=i i
iWYTRp ρ0
Mass:
Energy:
Species:
Equation of state:
Simplified governing equations (unity Lewis number)
X.S. Bai Transport of mass, momentum and heat
Chemical reaction rates
( )
∏ ∏
∑
= =
=
−=
−=≡
N
i
N
iijbijfj
j
L
jijij
ii
ijij CkCkq
qdt
dC
1 1
1
'''
''' νν
ννω
LjMM i
N
iiji
N
iij ,...,1
1
''
1
' =⇔∑∑==
ννReactions:
Reactionrates:
X.S. Bai Transport of mass, momentum and heat
A comparison of mass diffusivity, heat conductivity and viscosity
Using kinetic theory it can be shown (cf. Turns book):
12/32
2/1
3
3
32
31 −∝⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛== PT
PT
mTkcD
ij
Bij
σπξ
Mean diffusion coefficient for species i in the mixture
12/3
,1
/
1 −
≠=
∝−
=
∑PT
DX
YD N
ijjijj
ii
X.S. Bai Transport of mass, momentum and heat
A comparison of mass diffusivity, heat conductivity and viscosity
Using kinetic theory it can be shown (cf. Turns book):
2/12/1
43
3
21 T
mTkc
Vnkk B
B ∝⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠
⎞⎜⎝
⎛=σπ
ξ
Thermal diffusion coefficientpc
kρ
α ≡
Lewis numberi
iiip
i LeD
DDckLe αα
ρ=∴=≡ ,
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