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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Email: limyi@hknu.ac.kr, homepage: http://webmail.hknu.ac.kr/~limyi/index.htm
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)
 A

 v   A  DAB 2  A  rA
t


 v      j  r
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  cD AB 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
 cD AB 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
m ass flux  n
m om entumflux  
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
m ass m olecular flux  j A  DAB A
2
m om entumm olecular flux    [ v  ( v )t ]  (    )(   v ) 
3
N
H
energy m olecular flux  q   kT    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.
m ass balance: 0  j A 
dN Ay
dy
dey
cDAB dxA 1  x A  1  x A
energy balance: 0  e y 
N


,
 
Ay
dy
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 dvz
m om entumbalance: 0   r  
r
r dr dr
m ass 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
vz ( r )  vz ,max ( 1    )
R
1  c A c A
 r  c
vz ,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
m ass balance: 0  j A ,
N1 z 
dNiz
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  3x1  x3 )  3 z ( 2 x1  x3 )
dz
2cD12
2cD13
N3z  
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
Dvx 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