Ch. 19 Equation of change for multicomponent systems
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Transcript Ch. 19 Equation of change for multicomponent systems
고급전달공정
Advanced Transport Phenomena
(ch. 19)
Major: Interdisciplinary program of the integrated biotechnology
Graduate school of bio- & information technology
Young-il Lim (N110), Lab. FACS
phone: +82 31 670 5200 (secretary), +82 31 670 5207 (direct)
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Ch. 19 Equation of change for multicomponent systems
- Mass balance over an arbitrary differential fluid element Equation of
continuity in a multicomponent mixture.
- momentum/conduction/mass flux diffusion equations (2v, 2T, 2cA)
- Equation of change = equation of motion, equation of energy and equation of
continuity (=conservation laws)
19.1 the equations of continuity for a multicomponent mixture
- The law of conservation of mass in a finite volume of x, y, and z
The equation of continuity for species.
( n ) r
t
n v j
Ch. 19 Equation of change for multicomponent systems
19.1 the equations of continuity for a multicomponent mixture
( n ) r
t
n v j
- The equation of continuity for each species.
What assumption is
used for this equation
of continuity?
v j r
t
- equation of continuity for the mixture = equation of continuity.
v
t
0 v
Ch. 19 Equation of change for multicomponent systems
19.1 the equations of continuity for a multicomponent mixture
- The equation of continuity for each species in mass.
v j r
t
v j r
t
v
t
0 v
v j r
t
- The equation of continuity for each species in molar quantity.
c
c v* j* R
t
N
c
*
cv R
t
1
N
0 cv* R
1
x
c
c x v* j* R
t
N
x
c
cv* x j* R x R
t
1
Ch. 19 Equation of change for multicomponent systems
19.1 the equations of continuity for a multicomponent mixture
- Binary systems with constant mass diffusivity (DAB)
v j r
t
A
v A DAB 2 A rA
t
- Binary systems with constant mole diffusivity (cDAB)
N
x
c
cv* x j* R x R
t
1
x A
c
cv* x A cDAB 2 x A xB RA x A RB
t
- Binary systems with zero velocity and without reaction (v* = 0, RA=0, RB=0)
x A
c
cDAB 2 x A
t
c A
DAB 2 c A
t
Fick’s second law of
diffusion
Ch. 19 Equation of change for multicomponent systems
19.2 Summary of the multicomponent equations of change
- Three equations of change = three conservation laws
mass flux n
momentum flux
energy flux e
Ch. 19 Equation of change for multicomponent systems
19.3 Summary of the multicomponent fluxes
- Three equations of change = three conservation laws
mass molecular flux j A DAB A
2
momentum molecular flux [ v ( v )t ] ( )( v )
3
N
H
energy molecular flux q kT j
1 M
- Diffusion flux
- Viscous flux
- Conduction heat flux
- Diffusion thermo effect
19.4 Use of the equations of change for mixtures
Ex. 19.4.1: simultaneous heat and mass transport
(a) Mole fraction profile, xA(y)?
(b) Temperature profile, T(y)?
Assumption: steady-state, no reaction, no convection,
ideal gas of A, constant P, no radial heat transfer,
constant physical properties.
mass balance : 0 j A
dN Ay
dy
de y
energy balance : 0 e y
dy
N Ay
cDAB dx A 1 x A 1 x A
,
1 x A dy 1 x A0 1 x A0
y/
, N Ay
cDAB 1 x A
ln
1 x A0
C
N Ay P ,A y
k
T T0 1 e
dT
e y k
N Ay ( H A0 C p ,A ( T T0 )),
C
N Ay P ,A
dy
T T0
k
1 e
19.4 Use of the equations of change for mixtures
Ex. 19.4.2: Concentration profile in a tubular reactor
c A
1 c A
DAS
r
z
r r r
1 d dv z
momentum balance : 0 r
r
r dr dr
mass balance : 0 n A , v z
(a) Mole concentration profile, cA(y)?
Assumption: steady-state, isothermal, catalytic
reaction, parabolic velocity, diffusion of A,
constant P, no radial heat transfer, ignoring
product A & B.
2
r
v z ( r ) v z ,max ( 1 )
R
1 c A c A
r c
v z ,max ( 1 ) A DAS
r
,
r r r c A0
R z
2
0
0
e d
3
e d
3
19.4 Use of the equations of change for mixtures
Ex. 19.4.3: Catalytic oxidation of CO
mass balance : 0 j A ,
N1 z
dN iz
0
dz
(a) Mole concentration profile, cA(y)?
Assumption: steady-state, isothermal, catalytic
reaction, parabolic velocity, diffusion of A,
constant P, no radial heat transfer, ignoring
product A & B.
1
1
N 2 z N3z
2
2
dx3
N
1
3 z ( 1 x3 )
dz
cD13
2
N
N
dx1
3 z ( 1 3 x1 x3 ) 3 z ( 2 x1 x3 )
dz
2cD12
2cD13
N3 z
2cD13 x3 2
ln
x30 2
19.5 Dimensional analysis of the equations of change for binary mixtures
- Equation of continuity
v 0
- Equation of motion
Dv
2 v p g
Dt
- Equation of energy
DT
2T
Dt
- Equation of continuity of A
D A
DAB 2 A
Dt
- Dimensional analysis: dimensionless quantity, dimensionless group
19.5 Questions for discussion
1.
2.
Equation of change for reacting mixtures?
Flux equations for reacting mixtures?
3.
Under what conditions is divergence of v (v) zero?
4.
Mass and molar based equations of continuity (mass balance) are physically equivalent.
For what kinds of problems is there a preference for one form over the other?
5.
Interpret physically each term in the equations in Table 19.2.3?
vi vx v y vz
v
x y z
i 1 xi
3
Dv x vx
v
v
v
vx x v y x vz x
Dt
t
x
y
z
19.5 Questions for discussion
1.
2.
3.
Gradient p = p
Divergence v = v
Substantial time derivative (p 83) of c = Dc/Dt
vx v y vz
v
x y z
Dc c
c
c
c c
vx
vy
vz
v c
Dt t
x
y
z t