Fundamentals of Mass Transfer
When a system contains two or more components whose
concentrations vary from point to point, there is natural
tendency for mass to be transferred, minimizing the
concentration differences within the system.
Mechanisms of mass transfer
Molecular Transfer ~ random molecular motion in quiescent fluid
Convective Transfer ~ transferred from surface into a moving fluid
Analogy between heat and mass transfer
Mass transfer
Heat transfer
Molecular transfer
Convective transfer
Heat conduction
Heat convection
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Molecular Mass Transfer(Molecular Diffusion)
Fick’s law
j Az  D AB
d A
dz
J Az  CD AB
dx A
dz
j A  D AB A
In vector form
J A  CD ABx A
Mass diffusivity
D AB


Prandtl number
Pr 
Schmidt number
Sc 



D AB D AB
Lewis number
Le 
k


C pD AB D AB
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Mass Concentration
Molecule of
species A
a) mass concentration  A 
the mass of A per unit volume of mixture
- Total mass concentration(density)
   i
i
- mass fraction
dV in multicomponent
muxture
A 
A A


 i
i
 i  1
i
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Molar Concentration
b) molar concentrationC A  ~ the number of moles of A present per unit volume
of mixture
CA 
A
MA
M A - molecular weight of A
PAV  n A RT
For ideal gas
CA 
n A PA

V RT
PA - partial pressure of A
C   Ci
i
n
P
for ideal gas C  total 
V
RT
- Total molar concentration
- Mole fraction
CA
 xi  1
C
i
for ideal gas mixture
C
P RT PA
yA  A  A

C
P RT
P
for liquid
xA 
for gas
yA 
CA
C
 yi  1
i
Dalton’s law of gas mixture
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Mass Average and Molar Average Velocity
a) mass average velocity
v
 i v i
i
 i

 i v i
i

  i v i
v i - the absolute velocity of species i
relative to stationary coordinate axes
i
i
b) molar average velocity
V
 Ci v i
i
 Ci

 Ci v i
i
C
  xi v i
i
i
c) diffusion velocity - the velocity of a particular species relative to the mass
average or molar average velocity
vA  v
vB
vA
v
vB  v
vi  v
and
vi  V
A species can have a diffusion velocity
only if gradients in the concentration exist
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Mass Diffusivity
D AB
 L2
 JA
1
 M 

  

dC A dz  L2t  M L3 1 L  t
D AB  f Temp ., Pr essure , Compositio n 
Gases
5x10-6 ~ 10-5
Liquids
10-10 ~ 10-9
Solids
10-14 ~ 10-10
m2/s
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Molecular Mass and Molar Fluxes
Molar flux
J A  CA v A  V   CD ABxA
D AB
JA
Mass flux
mass diffusivity(diffusion coefficient) for component A
diffusing through component B
the molar flux relative to the molar average velocity
j A   A v A  v  D ABA
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Convective Mass and Molar Fluxes
Molar flux
J A  CA v A  V   CD ABxA
CA v A  CD ABxA  CAV
V
 Ci v i
i
 Ci

 Ci v i
i
C
  xi v i
i
i
C A v A  CD ABx A  x A  Ci v i
i
define
N A  CA v A
the molar flux relative to a set of stationary axes
N A  CD ABx A  x A  N i  J A  x A  N i
i
concentration
gradient
contribution
i
bulk motion
contribution
Mass flux relative to a fixed spatial coordinate system
n A  v A  D AB A   A v  D AB A   A  ni  j A   A  ni
i
i
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Related Types of Molecular Mass Transfer
According to 2nd law of thermodynamics, system not equilibrium will tend
to move toward equilibrium with time
Driving force = chemical potential
Nernst-Einstein Relation
v A  V  u A c  
uA
D AB
 c
RT
Mobility of component A, or the resultant velocity of
the molecule under a unit driving force
J A  C A v A  V   C A
D AB
 c
RT
in homogeneous ideal solution at constant T & P
c  0  RT ln C A
J A  D ABC A
Fick’s Equation
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Mass Transfer by Other Physical Conditions
~ produce a chemical potential gradient
Thermal diffusion(Soret diffusion)
Mass transfer by applying a temperature gradient to a multicomponent system
Small relative to other diffusion effects in the separation of isotopes
Pressure diffusion
Mass fluxes being induced in a mixture subjected to an external force field
Separation by sedimentation under gravity
Electrolytic precipitation due to an electrostatic force field
Magnetic separation of mineral mixtures by magnetic force
Component separation in a liquid mixtures by centrifugal force
Knudsen diffusion
If the density of the gas is low, or if the pores through which the gas
traveling are quite small
 the molecules will collide with the wall more frequently than each other
(wall xollion effect increases)
 the molecules are momentarily absorbed and than given off in the
random directions
 gas flux is reduced by the wall collisions
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Differential Equations of Mass Transfer
Conservation of Mass(overall)
A v  n dA 

V dV  0
t
Conservation of Mass of Component A
n A ,x yz xx  n A ,x yz x  n A ,y zx y y  n A ,y zx y

 n A ,z xy z  z  n A ,z xy z  A xyz  rA xyz  0
t
  nA 
 A
 rA  0
t
Equation of Continuity for Component A
 N A 
C A
 RA  0
t
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Differential
Equations of Mass
Transfer-1
In binary mixture
 A
 rA  0
t

  n B  B  rB  0
t
 A   B 
  n A  n B  
 rA  rB   0
t
n A  n B   A v A   B v B  v
  nA 
 A  B  
rA   rB
  v 

0
t
Equation of Continuity for the Mixture
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Differential
Equations of Mass
Transfer-2
n A  j A   Av
  n A    j A     Av 
 A
 rA  0
t
 A
 jA 
    A v   rA  0
t
  nA 



 A

  A  v   A   A  v
t
t
D A
   j A  rA  0
Dt
=0
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Differential
Equations of Mass
Transfer-3
in terms of molar unit
C A
 RA  0
t
C
  N B  B  RB  0
t
C A  CB 
  N A  N B  
 RA  RB   0
t
N A  N B  C A v A  C B v B  CV
 N A 
C A  CB  C
But RA is not always –RB
for only stoichiometry reaction ( A
  CV 
B)
RA   RB
C
 RA  RB   0
t
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Differential Equations
of Mass Transfer-4
N A  CD ABy A  y A  N A  N B 
or
N A  CD ABy A  C AV
n A  D AB A   A n A  n B 
or
n A  CD AB A   A v
D A
   j A  rA  0
Dt
  A

    A v   D AB 2  A  rA

 t


If

 and D AB are constant
 A

 A
  A   v   v   A  D AB 2 A  rA
t
t
  A

 v   A   D AB 2 A  rA
 t


C A
 V  C A  D AB  2 C A  R A
t
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
 A    n A  rA
t
n A  j A   vA
 

 v       g
t
     vv

   1 2
  U  2 v     e   v  g
t 

 T , A    

T

e  q    v   v  U 

T , A



T  T  



  

v2 


 A 
A T , A

     T  T    A   A

A
1
2




Dv
  p   g      g  T  T   g  A   A
Dt

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Boundary Conditions
Initial Conditions
at t  0, C A  C A0 or  A   A0
Boundary Conditions
Case 1) The concentration of the surface may be specified
C A  C A1 ,
y A  y A1 , x A  x A1 ,  A   A0 ,
A  A1 ,
Case 2) The mass flux at the surface may be specified
j A  j A1 , N A  N A1 ,
j A  0( impermeable ), j A ,z surface  DAB
d A
dz
z 0
Case 3) The rate of heterogeneous chemical reaction may be specified
N A ,z surface  k nC An
surface
Case 4) The species may be lost from the phase of interest by convective mass
transfer

N A ,z surface  kc C A ,surface  C A ,

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Steady State Diffusion of A through Stagnant B
Mass Balance Equation
Flowing of gas B
z  z2
SN A ,z
N A z z
z
z  z
 SN A ,z z  0
dN A ,z
0
dz
dN B ,z
0
dz
At z=z1, gas B is insoluble in liquid A
~ Component B is stagnant
NA z
N A ,z  CD AB
z  z1
Liquid A
S
 N B ,z  0
dy A
 y A N A ,z  N B ,z 
dz
N B ,z  0
N A ,z  
CD AB dy A
1  y A dz
at z  z1, y A  y A1
at z  z 2 , y A  y A 2
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Steady State
Diffusion of A
through Stagnant B-1
N A ,z zz dz  CD AB yy 
2
A2
1
N A, z 
A1
dy A
1 yA
CD AB 1  y A 2 CD AB y A1  y A 2
CD AB
y  y 
ln


 z2  z1  y B ,ln A1 A2
z 2  z1 1  y A1 z 2  z1 y B ,ln
yB ,ln 
For ideal gas
C
1  y A2   1  y A1 
yB 2  yB1

ln yB 2 y B1 ln 1  y A2  1  y A1 
n
P
P

and y A  A
V RT
P
N A, z 
D AB P PA1  PA 2
RT  z 2  z1  PB ,ln
Steady state diffusion of one gas through a second gas
~ absorption, humidification
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Film Theory
~ a model in which the entire resistance to diffusion from the liquid surface
to the main gas stream is assumed to occur in a stagnant or laminar film of
constant thickness d.
Flowing of gas B
zd
N A, z
z0
Main fluid stream
in turbulent flow
Slowly moving gas film
Liquid A
N A, z 
Convective Mass Transfer
kc 
D AB P PA1  PA 2
RTd PB ,ln
N A, z  kc C A1  C A2  
D AB P
PB ,ln d
actual
kc
PA1  PA2 
RT
0.5~1.0
kc  D AB
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Concentration Profile
dN A ,z
0
dz
N A ,z  
If C and
d  CD AB dy A 

0
dz  1  y A dz 
CD AB dy A
1  y A dz
D AB are constant under isothermal and isobaric conditions
d  1 dy A 

0
dz  1  y A dz 
at z  z1, y A  y A1
 ln 1  y A   C1 z  C2
1  y A  1  y A2 

 
1  y A1  1  y A1 
yB 
 y B dz
 dz

at z  z 2 , y A  y A 2
 zz   z z 
1
2
1
y B 2  y B1
 y B ,ln
ln y B 2 y B1
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Pseudo Steady State Diffusion through a
Stagnant Gas Film
the length of the diffusion path changes a small amount over a long
period of time ~ pseudo steady state model
Flowing of gas B
CD AB 1  y A2 CD AB y A1  y A2
ln

z2  z1 1  y A1 z2  z1 y B ,ln
 A, L dz

M A dt
N A ,z 
N A, z
Molar density of A in the liquid phase
N A, z
z  z0
z  zt
Liquid A
 A, L dz CD AB y A1  y A2

M A dt
z
y B ,ln
at t  t , z  zt
 A, L y B ,ln M A  zt2  z02 


t
CD AB  y A1  y A 2   2 
D AB
S
at t  0, z  z0
 A, L y B ,ln M A  zt2  z02 



C  y A1  y A 2 t  2 
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Diffusion through a Isothermal Spherical Film
T2  T1
Temperature
Temperature T1
d 2
r N A,r   0
dr
N A ,r  CD AB
N B ,r  0
dx A
 x A N A ,r  N B ,r 
dr
Since B is insoluble in liquid A
d  2 CD AB dxA 
r
0
dr  1  x A dr 
for constant temperature
Gas film
r1
r2
CD AB is constant
1 r 1 r 
 1 r 1 r 
1  x A  1  x A2

 
1  x A1  1  x A1 
WA  4r12 N A,r
r  r1

1
1
2
 1  x A2 
4CD AB

ln
1 r1   1 r2   1  x A1 
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Diffusion through a Nonisothermal Spherical Film
Temperature
Temperature T1
r 
T2  T1  2 
 r1 
D AB  r 
 
D AB,1  r1 
3n 2
d  2 CD AB dxA 
 r
  0
dr  1  x A dr 
if n  2
Gas film
n
r1
d  2 PD AB,1 RT1  r 
 
r
dr 
1  x A  r1 
WA  4r12 N A,r
r  r1

3n 2
dxA 
0
dr 
4PD AB,1 RT 1  n 2
 1  x A2 


ln
1  n 2 
1  n 2  n 2
1

x
1 r1 
 1 r2 
r1

A1 


r2
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Diffusion with Heterogeneous Chemical Reaction
Solid catalyzed dimerization of CH3CH=CH2
Gas A
Gas A & B
2A  B
N B , z   12 N A, z
N A, z
CD dxA
  1 AB
1 2 x A dz
d  1 dxA 
 1
  0
dz  1  2 x A dz 
Sphere with coating
of catalytic material
xA0
Z=0
Z
xA
A
Edge of hypothetical
Stagnant gas film
xB
B
Z=d
 2 ln1  12 x A   C1 z  C2  2 ln K1 z  2 ln K 2 
Catalytic surface
at z  0, x A  x A0
at z  d, x A  0
Irreversible and instantaneous rxn
1  12 x A   1  12 x A0 1 z d 
N A, z 
2CD AB  1 

ln 1
d
 1  2 x A0 
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Diffusion with Slow Heterogeneous Chemical Reaction
2A  B
N A, z  
N B , z   12 N A, z
d  1 dxA 
 1
  0
dz  1  2 x A dz 
CD AB dxA
1 12 x A dz
 2 ln1  12 x A   C1 z  C2
at z  0, x A  x A0
at z  d, x A 
N A, z
N A, z  k1C A  assume pseudo 1st order rxn
k1C
zd
1  12 x A   1  1 N Az  1  12 x A0 1 z d 
 2 k1C 
2CD AB d  1 
 k1 large 
N A, z 
ln 1
1  D AB k1d  1  2 x A0 
Da  k1d D AB
Damkohler No.: the effect of the surface reaction on the
overall diffusion reaction process
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Diffusion with Homogeneous Chemical Reaction
A  B  AB
First order reaction
N A, z z S  N A, z
z  z
S  k1C A Sz  0
dN A, z
 k1C A  0
dz
N A, z  D AB
d 2C A
D AB
 k1C A  0
dz 2
  C A C A0
ζ z L
d 2 2
   0
d 2
dC A
dz
at z  0, C A  C A0
at z  L, N A, z  0or dC A dz  0 
  k1 L2 D AB Thiele modulus
at   0,   1
at   1, d d  0
Sogang University
Diffusion with
Homogeneous
Chemical Reaction-1
  C1 cosh   C2 sinh ζ

cosh  cosh   sinh  sinh ζ cosh1   

cosh 
cosh 


C A cosh k1L2 D AB 1   z L 

C A0
cosh k1L2 D AB
C A,avg
C A0
N A, z
z 0

L
0 C A C A0 dz
 D AB
L
0 dz
dCA
dz
z 0

tanh 

C D 
  A0 AB  tanh 
 L 
Sogang University
Gas Absorption with Chemical Reaction
in an Agitated Tank
effect of chemical reaction rate on the
rate of gas absorption in an agitated
tank
Liquid B
Surface area
of the bubble
is S
Volume of liquid
phase is V
Gas A in
Example) The absorption of SO2 or
H2S in aqueous NaOH solutions
Semiquantative understanding can
be obtained by the analysis of a
relatively simple model
Sogang University
Gas Absorption with
Chemical Reaction
in an Agitated Tank-1
Assumptions:
1) Each gas bubble is surrounded by a stagnant liquid film of thickness d, which is
small relative to the bubble diameter
2) A Quasi-steady state concentration profile quickly established in the liquid film
after the bubble is formed
3) The gas A is only sparingly soluble gas in the liquid, so that we can neglect the
convection term.
4) The liquid outside the stagnant film is at a concentration CAd, which changes so
slowly with respect to time that it can be considered constant.
Gas in Liquid film
bubble
Main body of liquid
Without reaction
CAd
Liquid-gas
With reaction
interface
Z=0
Z=d
Sogang University
Gas Absorption with
Chemical Reaction
in an Agitated Tank-2
d 2C A
D AB
 k1C A  0
2
dz
at z  0, C A  C A0
  C1 cosh   C2 sinh ζ
at z  d, C A  C Ad
  C A C A0
ζ z d
  k1d 2 D AB
C A sinh  cosh   B  cosh  sinh ζ

C A0
sinh 
Assumption (d)
- SD AB
B
dC A
dz
z d

 Vk1C Ad
1
cosh   V Sd  sinh 
The total rate of absorption with chemical reaction
 N A ,z z 0 d

 
1
 cosh  

N

C A0D AB
sinh  
cosh   V Sd  sinh  
Sogang University
Gas Absorption with
Chemical Reaction
in an Agitated Tank-3
Chemical rxn is fast enough to keep
the bulk of the solution almost solute
free, but slow enough to have little
effect on solute transport in the film
for very slow reactions,
the liquid is nearly
saturated with dissolved gas
for large value of , dimensionless surface
mass flux increases rapidly with  and
becomes nearly independent of V/Sd.
Sogang University
Diffusion and Chemical Reaction inside a Porous
Catalyst
We make no attempt to describe the diffusion inside the
tortuous void passages in the pellet. Instead, we describe the
“averaged diffusion” of the reactant ~ effective diffusivity
N A,r r 4r 2  N A,r
lim
 0
r
r
2
N A ,r

r  r

2


4

r


r

R
4

r
r  0
A
r  r
 r 2 N A ,r
r
2

r
 r RA
2
d 2
r N A,r   r 2 RA
dr
Sogang University
Diffusion and Chemical
Reaction inside a
Porous Catalyst-1
dC A
dr
1 d  2 dC A 
DA 2
r
   RA
dr 
r dr 
N A  -D A
effective diffusivity
depends on P, T
and pore structure
First order chemical reaction,
a is the available catalytic surface per unit volume
DA
1 d  2 dC A 
r
  k1aC A
2
dr 
r dr 
CA 1
 f r 
C AR r
at r  R , C A  C AR
at r  0, C A  finite
d 2 f  k1a 
f

dr 2  D A 
C A C1
ka C
ka
 cosh 1 r  2 sinh 1 r
C AR r
DA
r
DA
C A  R  sinh k1a D A r
 
C AR  r  sinh k1a D A R
Sogang University
Diffusion and Chemical
Reaction inside a
Porous Catalyst-2
WA  4R 2 N A,R  4R 2D A
dC A
dr
r R

k1a
k1a 

WA  4RD AC AR 1 
R coth
R
D
D
A
A 

If the catalytically active surface were all exposed to the stream, then the
species A would not have to diffuse through the pores to a reaction site.
WAR,0 
Effective factor
A 

4
3

R3 a  k1C AR 
WAR
3
 2  coth   1
WAR,0 
  k1a D A
Thiele modulus
For nonspherical catalyst particles
 Vp 
Rnonsph  3 
 Sp 
 
WAR  V p ak1C AR A
A 
1
3 coth 3 1
32

  k1a D A V p S p

generalized
modulus
Sogang University
Diffusion and Chemical
Reaction inside a
Porous Catalyst-3
Sogang University
Textbook p567
Diffusion in a Multi-Component System
Liquid water(species 1) is evaporating into air, regarded as a binary mixture of
nitrogen(2) and oxygen(3) at a given T & P.
dN ,z
0
dz
Maxwell Stefan equation
N
x x 
 1
D 
x  
  1, 2, 3
for multicomponent diffusion in gases at low density
1
v  v   
x N  x N     1,2,3,..., N
C
D
 1

N
Since species 2 and 3 are not moving,
N
dx2
 1,z x2 ;
dz CD12
N 2 ,z  N3,z  0
N
dx3
 1,z x3
dz CD13
Sogang University
x2 L
x2
N1,z L
dx2

dz;
x2 CD12 z
 N L  z  
x2
;
 exp  1,z
x2 L
CD12 

x3 L
x3
N1,z L
dx3

dz
x3 CD13 z
 N L  z  
x3

 exp  1,z
x3L
CD13 

 N L  z  
 N L  z  

  x3L exp  1,z
x1  1  x2 L exp  1,z
CD12 
CD13 


At z=0
 N L
 N L
x10  1  x2 L exp  1,z   x3L exp  1,z 
 CD12 
 CD13 
Sogang University
Textbook p612
Transient 1-D Diffusion of a Finite Slab
~ negligible surface resistance
C A
 CA
 D AB
t
z 2
2
t
C As
C As
C A0 z 
Let Y 
z=L
at t  0, 0  z  L
CA  CAs
at z  0, t  0
CA  CAs
at z  L, t  0
CA  CAs
C A0  C A s
Y
 Y
 D AB 2
t
z
2
z=0
C A  C A0  z 
Y 1
Y 0
Y 0
at t  0, 0  z  L
at z  0, t  0
at z  L, t  0
by Separation variables method
Y  z , t   Z  z G t 
Sogang University
2005. 05. 30
Transient 1-D
Diffusion of a
Finite Slab -1
2   nz   n 2  X L
nz
Y   sin 
dz
0 Y0  z sin
e
n

1
L
L
 L 
X D  D ABt L 2
2
2
D
if Y0  z   Y0 ,
C A  C A s 4  1  nz   n 2  X
  sin 
e
n

1
C A0  C A s  n  L 
2
D
n  1,3,5

dC A 4D AB
C A s  C A0   cos nz e n 2  X
 D AB

n 1
dz
L
 L 
2
N A, z
D
n  1,3,5
Sogang University
Transient Diffusion in a Semi-Infinite Medium
C A
 2C A
 D AB
t
z 2
C A  C A0
at t  0, all z
CA  CAs
at z  0, t  0
C A  C A0
as z  , t  0
C A  C A0
 z t 
 t 
 f  , 2   f  2 
C A s  C A0
z 
 z1 z1 
Let   z 2 t , Y  C A  C A 0  C A s  C A 0  ~ Similarity Solution Technique

C A  C A0
z 

z 
 or C As  C A  erf 
 1  erf 


C A s  C A0
2
D
t
C

C
2
D
t
AB 

As
A0
AB 

2   2
e d

0

erf 0  0, erf    1
erf  
Sogang University
Textbook p613
Unsteady State Evaporation of a Liquid
V z
0
z
Continuity equation
Vz  Vz 0 t 
Flowing of gas B
In the binary system
N B ,z 0  0
Vz  
N A, z
z 0
Liquid A
x A
t
N A,z 0  N B ,z 0
Vz 
C
Gas B is insoluble in Liquid A
D AB x A
1  x A0 z
 D AB x A
  
 1  x A0 z
z 0
 x A
 2 xA

 D AB
z 2
z 0  z
at t  0, xA  0
S
at z  0, xA  xA0
at z   , xA  0
Sogang University
x A x A0  X ,
Zz
Unsteady State
Evaporation of a
Liquid-1
4D ABt
at Z  0, X  1
d2X
dX
 2Z  
0
at Z  , X  0
2
dZ
dZ
1 x A0 dX
Dimensionless molar
x A0   Vz t D AB  
2 1  x A0 dZ Z 0 average velocity
Y

dX
 C1 exp  Z  2
dZ



Z
2
X  C1  exp  Z   d Z  C2


0


 Z  
exp
0 
X Z   1 
 Z  
exp
0 
Z
X Z   1 
 d Z 
 1
 d Z

 
exp W dW
2
Z 
2



exp  W 2 dW
erf Z    erf 1  erf Z  

erf  erf
1  erf
2
Textbook p616
Fig. 20.1-1
Sogang University
1 x A0 dX
  x A0   
2 1 x A0 dZ
x A0 
Unsteady State
Evaporation of a
Liquid-2
 
1 x A0 erf  2
 x A0  
 1  x A0 1  erf
Z 0
1

1
 1  erf exp 2

1
Rate of production of vapor from surface of area S
dVA N A0 S
D

 S AB
dt
C
t
VA: the volume of A evaporation up to time t
VA  S 4D ABt
x A
 2 xA
 D AB
t
z 2

VAFick  Sx A0
VA  Sx A0
4D ABt

4D ABt


    x A0
Sogang University
Deviation from the Fick’s second law caused
by the nonzero molar average velocity
Textbook p617
Gas Absorption with Rapid Reaction
Gas A is absorbed by a stationary liquid solvent S containing solute B
Gas A
Liquid-Vapor Interface
ZR
Solute B
aA + bB
C A
 2C A
 D AS
t
z 2
C B
 2C B
 D BS
t
z 2
products
for 0  z  z R t 
for z R t   z  
at t  0,
C B  CB
at z  0,
C A  C A0
at z  z R t ,
for z  0
C A  CB  0
1
C A
1
C B
at z  z R t ,  D AS
  D BS
a
z
b
z
at z  ,
C B  C B
Sogang University
Gas Absorption with
Rapid Reaction-1
CA
 C1  C2erf
C A0
z
4D AS t
for 0  z  z R t 
CB
 C3  C4erf
C B
z
4D BS t
for zR t   z  

erf z
CA
 1
C A0
erf zR

4D AS t

1  erf z
CB
 1
CB
1  erf zR

from B. C. at z  ,
1  erf
C B  C B

4D AS t

4D BS t
for 0  z  z R t 

4D BS t

for zR t   z  
C A
C B
1
1
instead of B. C. at z  z R t ,  D AS
  D BS
a
z
b
z

aC
 B
D BS bC A0
 
D BS

 

erf
exp

D AS
D AS
 D AS D BS 
  z R2 4t  cons tan t
ZR increases as
t
Sogang University
Gas Absorption
with Rapid
Reaction-2
N A,z 0  D AS
C A
z

z 0
C A0
erf  D AS

D AS
t

The average rate of absorption up to t
N A,z 0 ,avg
C A0
1 t
  N A,z 0 dt 2
t 0
erf  D AS


D AS
t
Sogang University
Gas Absorption with Rapid Reaction-2
Sogang University
Textbook p.558
Diffusion into a Falling Liquid Film(Gas Absorption)
x
y
z
x=0
x
x=d
  x 2 
vz x   vm ax1    
  d  
N A,z z Wx  N A,z
CA0
Wx  N A,x x Wz  N A,x
x  x
Wz  0
N A,z N A,x

0
z
x
z
V(x)
z  z
N A ,z  -D AB
CA0
N A ,x  -D AB
C A
 x A N A ,z  N B ,z   C Av z x 
z
C A
C A
 x A N A ,x  N B ,x   D AB
x
x
C A
 2C A
vz
 D AB
z
x 2
Sogang University
2
Diffusion into a
Falling Liquid
Film(Gas
Absorption)-1
at z  0, C A  0
  x  C
 CA
vm ax1     A  D AB
x 2
  d   z
2
at x  0, C A  C A0
at x  d ,
C A
0
x
The substance A has penetrated only a short distance into the film~penetration model
at z  0 , C A  0
C A
 2C A
vmax
 D AB
z
x 2
at x  0, C A  C A0
at x   , C A  0
CA
2
 1
C A0

CA
 1  erf
C A0
N A ,x
WA0  0W 0LN A, x
x 0
x
0
4D AB z vm ax
 
exp   2 d
x
x
 erfc
4D AB z vm ax
4D AB z vm ax
 D AB
dzdy  WC A0
x 0
C A
x
x 0
 C A0
D ABvm ax
z
D AB vmax L 1
4D AB vmax
dz

WC
0
A0

z
L
Sogang University
Gas absorption from rising bubbles
Rybczynski-Hadamard circulation
Gas bubbles rising in liquids free surface –active
agents undergo a toroidal circulation
N A ,x
x 0
C A
x
 D AB

x 0
t exp osure 
For creeping flow
N A ,x
x 0

4D ABv t
C A0
D
vt
D
4D ABv t
C A0
3 D
Trace of surface-active agents cause a marked decrease in absorption rates from small bubbles
By preventing internal circulation
N A ,x
1/3
x 0
 D AB
Sogang University
Textbook p.562
Diffusion into a Falling Liquid Film(Solid Dissolution)
gd 2
vz 
2
d
  y  2  gd 2
1  1    
2
  d  
At end adjacent to the wall
Parabolic
Velocity
Profile of
Fluid B
y
  y   y 2 
2     
  d   d  
 y d2   y d
vz  gd y  ay
C A
 2C A
ay
 D AB
z
y 2
z
at z  0, C A  0
at y  0, C A  C A0
at y   , C A  0
CA0 =saturation
concentration
CA=0
Slightly soluble
Wall made of A
Similarity Solution Technique
C A C A0  f  ,
η  ya 9D AB z 1 3
d2 f
2 df

3

0
2
d
d
at   0 , f  1
at    , f  0
Sogang University

3
f  C1  exp   d   C2
0



CA
C A0
N A ,y
y 0
  3 d 
exp
  
 

3

0 exp   d 
C A
y
 D AB
y 0

W A0  
0

L
0

3
exp   d 


 43 
d C
 D ABC A0   A
 d  C A0
 exp   3
 D ABC A0 
 43 

W

N A, y

13
 a 


 9 D AB z 
  
 
 y  y 0

D C

 AB 4 A0
 3 

y 0
13
 a 


 9 D AB z 
2D AB C A0WL  a 


dzdx 
7
y 0
 3 
 9D AB L 
WA0  D AB L 
23
13
 43   0.8930
73   43 43   1.1907
Sogang University
Textbook p.585
Diffusion, Convection and Chemical Reaction
dC A
d 2C A
v0
 D AB
 k1C A
2
dz
dz
Liquid B with
small amounts
of A and C
at z  0, C A  C A0
A
C
by first order
reaction
z
Porous plug A
(slightly soluble
in B)
at z   , C A  0





CA
 exp  1  4k1D AB v02  1 v0 z 2D AB 
C A0
Liquid B
Sogang University
Analytical Expressions for Mass Transfer Coefficients
Mass Transfer in Falling Film on Plane surfaces
The absorption of a slightly soluble gas into a falling liquid film
WA0 
4D AB vmax
WL C A0  0  kc0,m AC A
L
kc0,m L
4 Lvmax
4  Lvmax   
  1.128Re Sc 1 2


Shm 


D AB
D AB
    D AB 
The dissolution of a slightly soluble solid into a falling liquid film
13
W A0
2D AB  a 



7
 3   9D AB L 
WL C A0  0  kc0,m AC A
a  gd   2vmax d
kc0,m L

2vmax dL2
2
1 16  L  Lv    
 L
  1.073   Re Sc 1 3
Shm 
 7 3
 7 3   max 
D AB  3 
9D AB
 3  9  d    D AB 
d
Sogang University
Analytical Expressions for
Mass Transfer
Coefficients-1
Mass Transfer for Flow Around Spheres
The gas absorption from a gas bubble surrounded by liquid in creeping flow
N A0 ,avg 
4 D ABv
0
C A  0  kcm
C A
3 D
kc0,m D
4 Dv
4  Dv   
  0.6415Re Sc 1 2


Shm 


D AB
3 D AB
3    D AB 
Re  0
Shm  2  0 .6415 Re Sc 
1 2
Shm  2
Creeping flow around a solid sphere with slightly soluble coating
N A0 ,avg
2

32 3 2D AB 3 D AB
v
C

27 3 73 
D2
kc0,m D 32 3 Dv

32 3
3
Shm 
 73 7 3

D AB
2  3  D AB 27 3 73 
Re  0
Shm  2
A
 0  kc0,m C A
 Dvmax   
  0.991Re Sc 1 3


   D AB 
Shm  2  0 .991 Re Sc 
13
Sogang University
Sogang University
Blasius’s Solution
Governing Equations
Boundary Conditions
v x v y

0
x y
v x
v x
 2vx
vx
 vy
 2
x
y
y
C A
C A
 2C A
vx
 vy
 D AB
x
y
y 2
vx v y
C A-C As
at y  0,

 0,
0
v v
C A -C As
as y  ,
if Sc 
vx
 1,
v
C A-C As
1
C A -C As

 1 (thermal boundary layer = hydrodynamic boundary layer)
D AB
v
C  C As
f2 x 2 A
v
C A   C As

y v
y v x y


Re,x
2 x 2 x 
2x
Sogang University
Blasius’s Solution-1
d 2vx v 
df
 f 0 
d 0
d  y 2 x  Re, x

dC A
dy
y 0


d 2 C A  C As  C A  C As 

d  y 2 x  Re, x
 0
D AB
C A
C A s  C A  y
Nu AB 
- Pohlhausen
 1.328
 0
 0.332 1 2 
 C A   C A s 
Re, x 
 x

N Ay  kc C A s  C A     D AB
kc  


y 0
C A
y
y 0
0.332D AB
Re,1x 2
x
kc x
 0.332 Re,1x 2
D AB
d
 Sc1 3
dM
if x  x0 , CA  C A and x  x0 , CA  C A s
Nu AB , x  0.332 Re,1x 2 Sc1 3
Nu AB 
0.332
3
1   x0 x 
34
Re,1x 2 Sc1 3
Sogang University
Mean Nusselt Number
C A
y
y 0
 0.332 1 2 1 3 
 C A  C A s 
Re,x Sc 
x


kc x
 Nu x ,AB  0.332 Re,1x 2 Sc1 3
D AB
WA  k c AC A s  C A     kc C A s  C A  dA
A
0.332D AB Re,1x 2 Sc1 3dA
 k cWL C A s  C A    C A s  C A  
A
x
kcL
 0.664 Re,1L 2 Sc1 3
D AB
Sogang University
Mass, Energy and Momentum Transfer Analogy
At steady state

if Sc 
1
D AB
d  vx 
d  C  C A,s 
 
  A
dy  v  y 0 dy  C A,  C A,s 
y 0

d  vx 
d  C  C A,s 
 
 D AB  A
dy  v  y 0
dy  C A,  C A,s 
N A,y  kc C A,s  C A,    D AB
kc 
 dv x
v dy
Cf 
y 0
kc C f

v
2
y 0
C A  C A,s 
y
y 0
w
2 dvx

v2 2 v2 dy
 0
Cf
h

vC p
2
Reynolds Analogy (1) Sc=1, and (2) no form drag ~ only skin drag
Sogang University
Chilton-Colburn Analogy
The Schmidt number is other than unity
Nu AB
Nu AB
0.332
kc x
12
13


 0.332 Re, x Sc
Re, x Sc1 3 Re,1x 2
D AB
Cf
Nu AB
Nu AB
23

Sc

2
Re, x Sc1 3 Re, x Sc
 k c x    D AB  2 3 k c Sc 2 3 Cf



Sc 

v
2
 D AB  xv     
Chilton-Colburn Analogy
jD 
(1) 0.6<Sc<2500, and (2) laminar flow
kc 2 3 C f
Sc 
v
2
Colburn j factor for mass transfer
jD  jH 
Cf
2
k
h
Pr 2 3  c Sc 2 3 (1) 0.6<Pr<2500
v C p
v
(2) 0.6<Sc<100
Sogang University
Mass Transfer to Non-Newtonian Fluids
Sogang University
Mass Transfer to Non-Newtonian Fluids
Mass Transfer to a Power-Law Fluid
Flowing on an Inclined Plate
fully developed flow
incompressible power-law fluid,
mass transfer rate is small
no chemical reaction
the solid surface consist of
a soluble material of length L
C*
C(x,z)

B.C. 1
B.C. 2
B.C. 3
 2c
c
D A 2  Vz ( x )
x
z
at z  0 , c  co
at x  0 , c  c * solubility
c
at x  d,
0
x
no mass transfer
Sogang University
Mass Transfer to a Power-Law Fluid
Flowing on an Inclined Plate-1
The stress distribution in the film
 xz  g( x  d ) cos 
The mass transfer is confined to a region near the solid surface
 xz   gd cos 
B.C. 3
at x  d,
c
0
x

B.C. 3’ at x  , c  co
for power-law fluids
1
n
n
dV
m z  gd cos 
dx
or
 gd cos  
Vz  
 x
m


the shear rate at the plane surface
 gd cos  

m


w  
1
n
Sogang University
Mass Transfer to a Power-Law Fluid
Flowing on an Inclined Plate-2
 2c
c
DA 2  w x
x
z
By using Laplacce transform
c( s )  0 exp(  sz )(c  co )dz
 2 c( s )
DA
 w xsc( s )
2
x
a form of the Bessel equation
 2  DA 

c( s )  (c  co )(3 ) 
 3  w s 
*
o
1
3
1
6
x  I  1 ( t )  I 1 ( t )
 3

3
1
2
 2  w s  16
 x
t   
 3  DA 
I  1 ( t ), I 1 ( t ) the modified Bessel functions of the first kind
3
3
of order -1/3 and 1/3, respectively
Sogang University
Mass Transfer to a Power-Law Fluid
Flowing on an Inclined Plate-3
c
    s 
 (c *  co )(3 ) 4  w 
x x  0
  3   DA 
2
3
1
6
1
3
c
3   
 (c *  co ) 1  w 
x x  0
 3   DA z 
1
3
1
3
the average(over the length L) mass transfer coefficient
ka 
1 L
0
L
c
4
2
1
dz
3
3
3 DA  w  3
x x  0

 
1
( c *  co )
2 3   L 
 DA
Sogang University
Mass Transfer to Non-Newtonian Fluids
Mass Transfer to a Power-Law Fluid
in Poiseuille Flow
completely analogous to convective heat transfer in Poiseuille flow
y c
 2c
Vo
 DA 2
R z
y
at z  0 , c  co
B.C. 1
inlet concentration
at y  0 , c  c *
B.C. 2
solubility
at y  , c  co
B.C. 3
B.C. 3 is valid only for the case of short contact time for which
the mass transfer, or diffusion, proceeds in the vicinity of the wall
Dimensionless variables
C
c * c
,
c *  co

y
,
R
 
DA z
V  R 2
Sogang University
Mass Transfer to a Power-Law Fluid in
Poiseuille Flow-1
The velocity profile for power-law fluids
1 1
Vz
Vz 3  n1 
n
 Q 
1

(
1


)

V  R 2 1  n1 
1 Q


Vo   w R   3   2
n  R

1 Q
 dV 

w   z    3   3
n  R

 dy  y  0
1  C  2 C


 3  
n    2

Similarity solution technique
   ( )
d 2C
2 dC

3

0
2
d
d

B.C. 1
B.C. 2
B.C. 3
at   0 , C  1
at   0 , C  0
at   , C  1
 9 
  

 3  (1 n) 
1
3
C( 0 )  0 and C()  1
1   3
 e d
4
 3 
Sogang University
Mass Transfer to a Power-Law Fluid in
Poiseuille Flow-2
The local mass transfer coefficient
c
 DA
 k loc (cb  c*)
y y  0
cb is the bulk solute concentration in the fluid
For short contact time, cbco
local Sherwood number(Nusselt number for mass transfer)
 c 
2 R 
 y  y  0
2k loc R
C
Shloc 

2
DA
(co  c*)
   0
Shloc  2
C

2 3
  
  4
    3 
 0 
1

2
 3   V  R
n

9 DA z
Sogang University
Mass Transfer to a Power-Law Fluid in
Poiseuille Flow-3
average Sherwood number
Sha 
2k a R 1 L
3 3
 0 Shloc dz  4
DA
L
 3 
1

2
3


 V  R
n

9 DA L
1
3
1

1
3  n 
Sha  1.75
Gz 3

 4 


Gz 
R 2 V 
DA L

Gz 
R 2 V 
DA L
  2 RV      R  
 R
   ReSc 


2

 L
  D A   L  2
Shear thinning(as n decreases) enhances the mass transfer rate
Sogang University
Concentration Distributions in Turbulent Flows
v x  v x  x , y , z   vx  x , y , z ,t 
C A  C A x , y , z   C A x , y , z ,t 
C A
 v  C A  DAB 2C A  k nC An
t
k1C A
C A

 v  C A  D AB  2 C A  v   C A  
 C 2A  C 2A 
t
k
 2


t 
J A  vCA
D
   v  0
Dt



v 
t 
Dv
  p        g
Dt
k1C A


v
t
DC A

     J A  J A    
2
2
  k2  C A  C  A 
 
Dt

 
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Concentration
Distributions in
Turbulent Flows-1
t 
t 
J A  D AB
dC A
dy
Turbulent Prandtl Number
Sc
t 

 t 
t 
D AB
1
Prandtl’s mixing length ~ normal to the direction of bulk flow
in the y direction
t 
J A  v y  C A   l 2
dv x dC A
dy dy
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TRANSFER COEFFICIENTS IN ONE PHASE
the rates of mass transfer across phase boundaries to the
relevant concentration differences, mainly for binary systems
Fig. 22.1-1. Example of mass transfer across a
plane boundary: 포화 평판의 건조
Fig. 22.1-2. Two rather typical kinds of
membrane separators, classified here
according to a Peclet number, or the flow
through the membrane. The heavy line
represents the membrane, and the arrows
represent the flow along or through the
membrane.
선택적 투과막
진짜 상 경계를 가짐
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유체역학적 물성의 급격한 변화
Fig. 22.1-3. Example of mass transfer
through a porous wall: transpiration cooling.
Fig. 22.1-4. Example of a gas-liquid contacting
device: the wetted-wall column. Two chemical
species A and B are moving from the downwardflowing liquid stream into the upward-flowing gas
stream in a cylindrical tube
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N
미분 면적에 정의되는 국부 전달계수
e  kT   H N 
 1
Partial molar enthalpy
J A* 0  J B* 0 ,
x A  xB
 k xA ,loc  k xB ,loc
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if the heat of mixing is zero (as in ideal gas mixtures)

~
H A  C pA ,0 T0  T o

T o : reference temperature
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Partial molar enthalpy
 H
H  
 n


T ,P ,n
H kn1 , kn2 , kn3 ,  kH n1 , n2 , n3 ,
H   n H
By Euler Theorem
Enthalpy ~ Extensive property
Homogeneous of degree 1

~
 H
~
H A  H  xB 
 x
 B
 H 


 n A  nB
 H

 xB



n
~
 H
~
HB  H  x A 
 x
 B
H x  H 

 H A   B 
n
n  xB  n

 H   n A 
 H
  
 
  
 n  n A  nB  xB  n  nB

H   n H  n AH A  nB H B




n
~
H  H / n A  nB   H / n
n A  1  xB n ;
  nB
 
 nA  xB
nB  x B n

  H A  n   H B  n 
n

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mass transfer coefficient
"apparent" mass transfer coefficient
The superscript 0 indicates that these quantities are applicable only for small mass-transfer
rates and small mole fractions of species A.
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TRANSFER COEFFICIENTS IN TWO PHASES
Interphase Mass Transfer
N A0
gas
 N A0
liquid
 N A0
N A0  k y0,loc  y Ab  y A0   k x0,loc x A0  x Ab 
Assuming equilibrium across the interface
y A0  f  x A0 
평형곡선
N A0  K y0,loc  y Ab  y Ae   K x0,loc x Ae  x Ab 
Overall concentration difference
y Ae Gas phase composition in equilibrium with a liquid at composition x Ab
x Ae Liquid phase composition in equilibrium with a gas at composition y Ab
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Fig. 22.4-2. Relations among gas- and
liquid phase compositions, and the
graphical interpretations of mx, and my.
N A0  K 0y ,loc  y Ab  y Ae   K x0 ,loc  x Ae  x Ab 
K 0y ,loc
the overall mass transfer coefficient "based on the gas phase’
K x0 ,loc
the overall mass transfer coefficient "based on the liquid phase"
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If k x ,loc mk y ,loc  1 the mass-transport resistance of the gas phase has little effect,
and it is said that the mass transfer is liquid-phase controlled. In practice, this means that
the system design should favor liquid-phase mass transfer.
0
0
If k x ,loc mk y ,loc  1 then the mass transfer is gas-phase controlled. In a practical
situation, this means that the system design should favor gas-phase mass transfer.
0
0
If 0 .1  k x0 ,loc mk 0y ,loc  10 , roughly, one must be careful to consider the interactions
of the two phases in calculating the two-phase transfer coefficients.
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Mean two phase mass transfer coefficient
bulk concentrations in the two adjacent phase do not change significantly over
the total mass transfer surface
K 0x ,approx 
1
1 k 0xm   1 mx k 0ym 
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Penetration model holds in each phase
k 0x ,loc  k x ,loc  c l
4 D AB
t exp
k 0y ,loc  k y ,loc  c g
4 D AC
t exp
Total molar concentraion in each phase
k 0x ,loc
cl

mk 0y ,loc c g
D AB 1
 1
D AC m
Only liquid phase resistance is significant
an oxygen stripper, in which
oxygen(A) from the water(B)
diffuses into the nitrogen gas(C)
bubbles.
Absorption or desorption of sparingly soluble
gases is almost always liquid phase controlled
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the law of conservation of mass of
chemical species  in a multicomponent
macroscopic flow system
the instantaneous
total mass of  in
the system
the mass rate of
addition of
species  to the
system by mass
transfer across
the bounding
surface
the net rate of
production
of species  by
homogeneous
and heterogeneous
reactions within
the system
in molar units
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Disposal of an unstable waste product
t  V w
When the tank is full
d
mA ,tot  w A0  k1''' mA ,tot
dt
w A0

mA ,tot 
1  expk1''' t 
'''
k1
t 0
mA ,tot  0
mA ,tot  wt A
 A 1  expk1''' t 

 A0
k1''' t
 AF 1  e  K

 A0
K
K  k1''' V w  k1''' V Q
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t  V w
d
 AV   w A0  w A  k1''' V
dt
d A
 1  K  A   A0
d
 1
  w V t
 A   AF
 A   A0 1  K 
 e 1 K  1 
 AF   A0 1  K 
 A 
 A0
 A0

1  K 1  k1''' V w 
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Binary Splitters
zF  yP  xW
F  P W
 P F
Cut
z  y  1   x
Separation factor
Y 
Mole ratio
y 

Y  X
y
1 y
,
X 
x
1 x
x
y
or x 
1    1x
    1y
Gas-liquid Splitters(equilibrium distillation)

Relative volatility
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Dirac 분리용량(separation capacity) 및 가치함수(value function)
서로 다른 분리공정의 유효성을 비교하기 위한 기준 수립
 
y 1  y 
y- x
 1
x 1  x 
x1- y 
  1 
x    1x

1    1x
약간 농축되는 계
y
 x    1x 1  x 
Dirac 분리용량(separation capacity)
  Pv (y )  Wv (x )  Fv (z )
Binary Splitters
zF  yP  xW
F  P W
Cut
z  y  1   x
 P F
or
z - x   y - x 
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
F
 v (y )  1   v (x )  v (z )
v (y )  v (x )  y  x v x  
1
2
v (z )  v (x )  z  x v x  

F
1
2
y  x  v x   
z  x  v x   
2
2
 21  1   y  x  v (x )
2
x    1x
y 
 x    1x 1  x 
1    1x

F
 21  1     1 x 2 1  x  v (x )
2
계의 분리용량은 실질적으로 농도에 무관하다고 가정

F
x 2 1  x  v (x )  1
2
 21  1     1
2
d v (x )
1

2
dx 2
x 2 1  x 
적분 상수를 구하기 위한 조건 선택
v (21 )  0, v (21 )  0
2
2
v (x )  2 x  1ln
x
1  x 
v (x )  2 x  1ln
 2  C 1x  C 2
x
1  x 
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Compartmental Analysis
A complex system is treated as a network of perfect mixers, each constant
volume, connected by ducts of negligible volume, with no dispersion occurring
in the connecting ducts
N
d  n
Vn
 Q mn m  n   V n rn
dt
m 1
Q mn
Hemodialysis
The volumetric flow rate of solvent flow from unit m to unit n
during dialysis period
d 1
 Q 1  2   G
dt
d
V 2 2  Q 1  2   D 2
dt
V1
Initial condition
t0
1  2  0
d 2 2  Q
Q
D  d 2 Q D
Q G








2
V

V1 V 2
V1 V 2
dt 2
 1 V 2 V 2  dt
d 2
D 0
t  0 2  0 and

dt
V2
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For the recovery period
 V1  V2
d 2 2


Q
2
dt 
 V1V2
 d2
QG


V1V2
 dt
 V V  
2
t   C4
2 ,cf  C3 exp Q 1
V
V
  1 2  
Initial condition
2 ,pi 
Gt 
V1  V2
t  0
 2   2
1  1
d2
d2 D2


dt
dt
V2
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during dialysis period
For the recovery period
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Tour du Mont Blanc
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Two or Three Dimensional Diffusion
Y
At steady state  2C  0
A
C A  C x
L
CA  0
0
 2C A  0
Infinite plate in z direction
 2C A  2C A

0
x 2
y 2
CA  0
CA  0 W
X
at x  0,
all y, C A  0
at x  W,
all y, C A  0
at y  0,
all x, C A  0
at y  L, all x, C A  C  x 
by Separation variables method
C A  x , y   X  x Y  y 
1 d 2 X 1 d 2Y
2




X dx 2 Y dy 2
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Two or Three
Dimensional Heat
Conduction-1
d2X
d 2Y
2
 X 0
 2Y  0
and
2
2
dx
dx
General solution of temperature distribution

C A x , y    A cos x  B sin x  Cey  De  y
at x  0,
all y, CA  0

A0
at y  0 , all x, C A  0
C  D
at x  W, all y, CA  0
DB sin W  0
W  n   n W
at y  0,
Bn 
4T1
n
all x, T  Ts  T1
for n  1, 3, 5
 n 
T1   Bn sin 
x
 L 
n 1
Bn  0 for n  2, 4, 6

2n  1x
4T1
e 2 n 1y L
T x, y   Ts 
sin
 2n  1

L
n
n   L y

T x, y   Ts   Bn sin  x e
 L 
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