3.MASS-TRANSFER THEORIES
(1)Mass Transfer Coefficient
dyA
•For steady-state
mass
transfer
through a stagnant
J A  N A   Dv  M
db rate can be predicted by
layer of fluid , mass transfer
dyA
B
y
J A equations:
T N A   Dv  M A
following
dbA
N A  db   Dv  M  dy
B
(1)Equimolal
diffusion yyAiA
0T
N A  db   Dv  M  dyA
Dv  M
N A 0 J A 
(yyAi Ai  y A )
(17.19)
BT
Dv  M
NA  JA D
( y Ai  y A )
(17.19)
or N A  J A  vB(Tc Ai  c A )
(17.20)
BT
Dv
NA  JA 
(c Ai  c A )
(17.20)
BT
1
 
A
A



ln
0 vDvM M (D1vyMA ) y (1  y A ) 1  y Ai
D
(2)One-component
mass
Ai transfer (one-way diffusion)
BT
yA
N A db
N A B1T y
dyA
1 yA
Dv 
ln )
N
0 ADv  M BMDlnv 1M y A y (1  y A()17.24
1  y Ai
Ai
T
Ai
Dv  M y Ai  y A
NA 
(17.26)
BT (1  y A ) L
A
A
T
•More common used type of equations:
Analogous to heat transfer,
Heat transfer rate=(Heat transfer coefficient)  (Heat
transfer driving force)
Mass transfer rate=(Mass transfer coefficient) 
(Mass transfer driving force)
2
J AA  N AA  Dv v M M db
db
B
yA
•Definition
of mass transfer
coefficient: The rate of
BT T
yA
dbcper
DDvunit
M area
dy
massNNtransfer
A db
Aper unit concentration

A 
v M  dyA
yyAi on equal molal flows.
difference,
usually based
00
Ai
Dv  MDv  M
y Ak)y ( y Ai  y(A17
) .19)
NJ AA  J A  ( y Ai (yyAiA)dy
A
B
B
J A  N TA  T Dv  M
db
Dv
Dv
dy
B
y
c)Aikcc(AAc)Ai A c A ) (17.20)
JNJA ATJNA(cAic(AD
A B
A B
v M
T
T
db
N A  db   Dv  M
dy

Other forms of mass transfer equations:
N
db


D

dy


J  k (x  x )
B0T
A
A
0
x
Ai
v
A
A
yyAiA
M
A
y Ai
Dv
J AA  kJgA( PAi  P(Ac)Ai  c A )
N
BT
Dv
(17.20)
3
N A  J A yAi
( y Ai  y A )
(17.19)
y A BT
Dv  M
NNAA  db
J A  c  Dv (MyJAiA dy
 yAA )
(17.19)
Therefore,
kc BT
(17.36)
0
y Ai
(c Ai  c A )
dyA
J A Dv  M
kNc A=mass
17
J A  N A(coefficient

 J A transfer
y Ai(D
yv A.36
)M )based on
(17molal
.19)
(
c

c
)
BT driving forcedb
Ai
A
concentration
dyA
BT
yA
dy
J A  N A  kgm
Dv olM
A
J

N


D

D3v ]M [m
dy
[kc ]A [ AN2A  dbv Mdb
/ sA]

db
s  m  0kgm oly/A m
BT
y Ai
0
BT
BT
yA
NNA  db
cDv 
dydy
J
M
A

db
D

A
A 
v
M
A

k

(17.40)
0
y Ai
g
0
PAi  PAy
J A Dv  M
mass( transfer
coefficient
based
on the
k gN=gas
A  Jphase
(
17
.
40
)
D

y

y
)
(
17
.
19
)
v
A
Ai (c A  c )
N

J

(17.20)
P

P
A
A
Ai force
A
Ai pressure
A BT
partial
driving
BT
Dv ol
kgm
N[Ak g] J A[  2 (c Ai 
(17.20)
] cA )
4
s  mBT  kPa
Ai
BT
yA
dy
N A  db  JDv MN  dy
A
AD 


A
AJ
v M
A
0
y
db(17.37)
k y  Ai
BT y Ai dy
yA
 yA A
J
A  Dv  M dy
A 
kJ yAJ=gas
 NN
(AD
17v .37
) dyA
N
db


phase
mass
transfer
coefficient
based on the
db
A
M


D

D


A y
A
v
M
v
Ai  y
A 
N
J

(c Aiy 
cA )
(17.20)
db
BT
yAA
mole
fraction
A 0 differences
Ai
B
y BT
D
N
db
 Dvv 
dy
J
A 
M 
NN

J
(
c

(17.20)
Ac AA)
A
A
Ai
db

c

D

dy
k

0A 
BxT xv y AiM x A
0
Ai y A
JA
Dv
Dv  mass
k x =liquid

phase
transfer
coefficient based on
M
N A xAiJA xNA A  J A( yAi  y(cA )Ai  c A ) (17.19)(17.20)
the mole fraction
differences
BT
BT
Dv
N[Ak ] JA[k ]  [(c Ai  c A ) kgm ol (17.20) ]  [ J ]
y
x B
A
2
T s  m  unit m ole fraction
0
T
y Ai
A
Ai
5
•Relations between mass transfer coefficients:
 cA  y A  M
JA
ky 

y Ai  y A c Ai
JA
M

cA
M
JA
 M
c Ai  c A
kc P
 k y   M  kc 
(17.38)
RT
Similarly, in liquid phase,
 cA  xA  M
JA
JA
JA
 kx 

 M
c
x Ai  x A c Ai
c Ai  c A
 A
M
M
kc  x
 k x   M  kc 
(17.39)
M
6
c Ai N  c AcAD  dyA c Ai

c
x A cJAi
c

cA
A
Ai

A
 A v M
Here,
MM
M M db
kckc x x=density of liquid,
dyAkg/m3

J AA  db
N A(

D.v39
 N
17
) ) dyAJ A  N A   Dv  M dyA
MM
(D
17
.v 39
weight of liquid db
MM 0 =average molecular
y db
BT
yA
Ai
BT
yA
JA
JA B
JA y
N kA g 
db   Dv M  dyAN A db c  Dv  M dyA



PAi  PA yPy

Py
P
(
y

y
0
Ai
Ay )
Ai Ai
0A
T
A
Ai
k y Dkv c
D

v
M)
N kA gJ A  (c Ai  c A )N (A17
(
17
.
20
 .J41

( y Ai  y A )

)
A
BT
P BRT
T
Dv
D
Significance
N A  J A  of kc:
(c Aifrom
 c A )J A  v (c(17
.20)
Ai  c A )  k c (c Ai  c A )
BT
BT
Dv
kc 
BT
(17.42)
7
Dv
kc 
BT
(17.42)
•For steady-state equimolal diffusion in a stagnant
film, mass transfer coefficient kc is the molecular
diffusivity divided by the thickness of the stagnant
layer（B ）.
T
•When we are dealing with unsteady-state diffusion
or diffusion in flowing streams（对流）, Eq.(17.42) can
still be used to give an effective film thickness BT
from known values of kc and Dv.对于对流传质，
（17.42）式有效, BT为有效膜厚度
8
t
(2)Film Theory
q
•Analogous to convective heatTtransfer,
Heat
Tw transfer rate q:
Effective heat
boundary layer

t
t
qt
qT
TTw
t
q
T
q
T
Tw
TFluid
w
q
k
t
(T  Tw )  h(T  Tw )
kq  k (T  T )
Tqw  k (T  qT ) (T Tw ) w
t
w t
k t
q  (T  Tw )
Metal wall
t
Laminar
layer
9
膜理论基本概念是传质阻力相等于
停滞膜厚度
EffectiveB film
thickness T
Liquid
BT
c Ai
c Ai
BT c A
cA
BNT
c Ai
c Ai N A
A
cA
BT
Gas
NA
cA
N
InterfaceA
Laminar layer
thickness
dyA
•The
J A  basic
N A  concept
 Dv  M of the
db
film theory is that the
BT
yA
resistance to diffusion
N  db  c  Dv  M  dyA
canA be
considered
0
y Ai
equivalent to
that in
Dv of
M a
aNstagnant
film

J

( y Ai  y A
A
A
BT
certain thickness
Dv
•Then,
NA  JA 
( c A  c Ai )
BT
（相等于）
Dv
kc 
BT
(17.42)
Illustrational diagram
of wetted wall tower
10
Dv
kc 
BT
(17.42)
•The implication is that the coefficient kc varies with the
first power of Dv, which is rarely true, but this does not
detract from the value of the theory in many applications.
The film theory is often used as a basis for complex
problems of multi-component diffusion
or diffusion
plus chemical reaction.
（多组分扩散）
•The value of BT depends on the diffusivity Dv and not
just on flow parameters, such as Reynolds number. The
concept of an effective film thickness is useful, but values
of BT must not be confused with the actual thickness of
the laminar layer
.
（层流底层）
11
•Effect of one-way diffusion
•When only one component A is diffusing through a
stagnant film, the rate of mass transfer for a given
concentration difference is greater than if component
B is diffusing in the opposite direction.
NA
1
1


1
J A (1  y A ) L ( yB ) L
(17.43)
Where,
NAA =molal11flux of one-way
1 1 diffusion
N


(17.(43
17).43)
((11yy
) L) of equimolal
( y(By)BL ) diffusion
flux
JJ AA =molal
AA
L
L
12
•Some times the mass transfer coefficient for one-way
transfer is denoted by kc’ or ky’, then
kc k y
1
1



1
kc k y (1  y A ) L ( yB ) L
N A  k y ( y Ai  y A )
(17.45)
NA 
k y ( y Ai  y A )
(1  y A ) L
When
When y Ay A0.01.,1k, yk yk yk y
(17.44)
(17.46)
liquid
, take (1(1x xA )AL) L1 1
InIn liquid
, take
Because the correction is small compared to the uncertainty in
the diffusivity and the mass-transfer coefficient.
13
(3)Boundary Layer Theory
•When diffusion through a stagnant fluid film ,
k c  Dv
•Whenk  D
place in a thin boundary layer
k c  Dwhere
near a surface
the fluid is in laminar flow,
v
2/3
diffusion
takes
c
v
kc  D
2/3
v
•For boundary layer flows, no matter what the shape of
the velocity profile or value of the physical properties,
the transfer rate cannot increase with the 1.0 power of
the diffusivity, as implied by the film theory.
14
(4)Penetration Theory（渗透理论） and
Surface Renewal Theory表面更新理论
•When the boundary layer becomes turbulent or
separation occurs, penetration theory and surface
kc theory
Dv apply, and
renewal
kc  D
1/ 2
v
15
(5) Two-Film Theory
•Basic viewpoints:
•1)On two sides of the interface, there exist two
effective films of certain thickness, component A
passes through these two film by molecular diffusion.
•2)At the interface, the gas is in equilibrium with
liquid.
•3)The concentration gradients in the two bulk phases
equal to zero. （1） 接触的两相流体间存在相界面，界面两侧各有一个很
薄的停滞层，组分A以分子扩散方式通过此两膜层。（2） 相界面处，气、液两相
达到平衡（界面上不存在阻力）。 （3）两滞流膜外的气液相主体中，流体充分
湍动，物质浓度均匀。
16
BT 2
BT 1 film thickness
Effective
y
B
T 1A
of liquid
phase
B
T2
yA
y Ai
y

A
Bulk
x Ai
liquid
xA
Effective film
T1
thickness of gasBphase
BTy2Ai BT 1
 B
T1
y
yB
B
A
A T1
T2
B
T2
x
yB
y
Ai
AiT 2
A
yA

x
y AyAA y Ai
y Ai
x Aiy Ai y A

y
x Ay A xAAi
x Ai
x Ai x A
xA
Interface
xA
[xAi is in equilibrium with yAi]
BT 2
yA
y Ai

y
BulkAgas
x Ai
xA
r ( N A )
r  k y ( yA  y
17
•In the two-film theory, the rate of mass transfer to
the interface is set equal to the rate of the transfer
rfrom
( Nthe) interface:
A
r  k y ( y A  y Ai )  k x ( x Ai  x A )
(17.52)(17.53)
BT 1
BT 2
BT 1 film thickness
Effective
BTy1A
of liquid
phase
B
T2
yA
y Ai
y

A
Bulk
x Ai
liquid
xA
Effective film
thickness of gasBphase
T1
BTy2Ai BT 1
 BT 1
y
yB
A 1 BT 2
AT
BT 2
xAiAi
yB
T 2 yA
y
y xAyAA yAAi
y
y Ai yAiA
x Ai

 yA
x Ay A x Ai
x
x Ai xAiA
xA
Interface
x
BT 2
yA
y Ai

y
BulkAgas
x Ai
r ( NxAA)
r  k ( y  y )  k (x  x )
18
(17.52
r ( N A )
r (Let
N A ) r  K y ( y A  y A )
(17.54)

=overall
r  K y ( y A  y A )mass transfer
(17.54) coefficient in gas phase
NA)
K y ( y A  y A ) =overall
(17.mass
54) transfer driving force
BT 1
BT 2
BT 1 film thickness
Effective
BTy1A
of liquid
phase
B
T2
yA
y Ai
y

A
Bulk
x Ai
liquid
xA
Effective film
thickness of gasBphase
T1
BTy2Ai BT 1
 BT 1
y
yB
A 1 BT 2
AT
BT 2
xAiAi
yB
T 2 yA
y
y xAyAA yAAi
y
y Ai yAiA
x Ai

 yA
x Ay A x Ai
x
x Ai xAiA
xA
Interface
x
BT 2
yA
y Ai

y
BulkAgas
x Ai
r ( NxAA)
r  k ( y  y )  k (x  x )
19
(17.52
•To get Ky in terms of kx and ky,

A
y A  y Ai y Ai  y
1
yA  y



Ky
r
r
r

y A  y Ai
y Ai  y A
1


K y k y ( y A  y Ai ) k x ( x Ai  x A )

A
(17.55)
(17.56)

A
 y Ai  m xAi , y  m xA ,
1
1 m

 
K y k y kx
(17.57)
20
1 y Ai y AmxAiy,Aiy A  m xyAAi,  y A


(17.56)
K y 1k y ( y1A  ym
k x ( x Ai  x A )
Ai )

 
(17.57)

y m
y AixkAiy, y Ak
 yyAiK
,y A
x ym
A
Ai x
A


(17.56)
k y1( y A 1y Ai ) m k x ( x Ai  x A )

   resistance
17.57
)
=overall
to (mass
transfer

y AiyxAi , kyyA ykm
A ,y A
K
m
Ai 
Aix x

(17.56)
y A  1y Ai ) m k x ( x Ai  x A )
  =resistance to mass
(17.57
)
transfer
in the gas film
k xxA ,
xy Ai , ykAy  m
m

kx
(17to.57
) transfer in the liquid film
=resistance
mass
21
•Similarly,
r ( N A ) let

A
r  K x ( x  xA )
(Where
NA)
y

y
1

K
(
y

y
)
NA)


(17.54) coefficient
Ai
A
y  y in liquid phase

(17.56)
k y=overall
( y A  ymass
k x ( x Ai 
x
)
Ai ) transfer
A
driving force

=overall
mass Ai
transfer
x
A
AA
K y
K y ( xA  xA )
(17.54)
 ycan
m xAi , y
•We
Ai get

A
 m xA ,
1
1
1

 
K x k x m ky
22
1 y Ai ym
xAiy,Aiy A  m xyA Ai
,  yA
A 


(17.56)
K y 1 k y ( y1A  y Ai1) k x ( x Ai  x A )

 

y AikxAix , ym
 yyAiKA x m
kyymAixA ,y A
A


(17.56)
k y1( y A 1y Ai ) m k x ( x Ai  x A )

   resistance to (mass
17.57
)
=overall
transfer

K
k
k
x , yyA 
y AimxxAy,A
iym
Aix Ai

(17.56)
A  y1Ai ) mk x ( x Ai  x A )
 = resistance to mass
(17.57
)
transfer
in the liquid film

k
kxx ,
y , y x m
Ai
A
A
1

m ky
=resistance to mass transfer in the gas film
23

Ai
A
“controls”

y Ai film
 m x , y  m and
xA , Liquid film “controls”
•Gas
1
1 m

 
(17.57)

K y1 k y y kA x y Ai
y Ai  y A


K y 1 kmy ( y A1  y Ai
1 ) k x ( x Ai  x A )
•When
ky

,
kx K y

k y
A
(17.56)
 Gas film " controls"
 y Ai  m xAi , y  m xA ,
1
1
1

 
K x k x m ky
1
1
1
1
,
  Liquid film " controls"
•When  
k x m ky K x k x
24