Chapter 6
Principles of Diffusion
and Mass Transfer
Between Phases
1
1.THEORY OF DIFFUSION
•
Diffusion is the movement, under the influence of a
physical stimulus, of an individual component
through a mixture
• The most common cause of diffusion is a
concentration gradient of the diffusing component.
• E.g., The process of dissolution of ammonia into
water: (1)A concentration gradient in the gas phase
causes ammonia to diffuse to the gas-liquid
interface; (2)Ammonia dissolves in the interface;
(3)A gradient in the liquid phase causes ammonia
to diffuse into the bulk liquid.
2
•
A concentration gradient tends to move the
component in such a direction as to equalize
concentrations and destroy the gradient.
• If the two phases are in equilibrium with each other,
diffusion, or mass transfer fluxes is equal to zero.
• Other causes of diffusion: activity gradient (reverse
osmosis); temperature gradient (thermal diffusion);
application of an external force field (forced
diffusion, e.g., centrifuge, etc).
• Two kinds of diffusion caused by concentration
gradient : molecular diffusion（分子扩散） and
eddy diffusion（涡流扩散）. [e.g., diffusion process
of ink in the stagnant or agitated water…]
3
•
•
Mass transfer driving forces
E.g., absorption or stripping process: Gas-liquid
VVaa,,yyaa
Vphases
,
y
are not in equilibrium with each other.
a
a
VLa , ,yxa
a
a
LVa , ,xya
b
LLaa,,xxaa
VVbb,,yybb
VL
b , ,yx
b
b
b
LVb ,, xyb
LLbb,,xxbb
VV,,yy
VL, ,yx
VLa, ,xya
VLLa,,,xxya
La , xa
La , xa
Vb , yb
Vb , yb
Lb , xb
b
Absorption
,y
L , x Vcolumn
b
b
A
Ay
Driving
y
forcey

yx
x
x

x
A
A
yy
yy
xx


Equilibrium
A
curve
y
y
Driving

force x
x



xx

Driving forces: ( y  y )

( x  x)
4
Va , ya
VLa , ,yxa
a
VVaa,,yyaa
LLaa,,xxaa
LVa , ,xya
b
b
VL
b , ,yx
b
VVbb,,yybb
LLbb,,xxbb
LVb ,, xyb
VL, ,yx
VV,,yy
VLLa,,,xxya
VLa, ,xya
La , xa
La , xa
Vb , yb
Vb , yb
Lb , xb
a
ya min  m xa
yb
xb max 
m
b
b
Absorption
,y
L , x Vcolumn
b
V, y
ya min  m xa
yb
xb max 
m
b
L, x
5
Va , ya VVaa,,yyaa
LLaa,,xxaa
VL
a , ,yx
a
a
a
LVa , ,xya VVbb,,yybb
Equilibrium
A
curve
La, ,xya
V
La , xa
La , xa
Vb , yb
A
Ap
Drivin
y
y
g force

px
x
x

x
Vb , yb
Lb , xb
Driving forces:
b
b
LLbb,,xxbb
VL
b , ,yx
b
b
b
LVb ,, xpb VV,,yy
LLa,,,xcya
VL, ,yx V
Absorption
,y
Lb , xb Vcolumn
A
A
yy
yy
cx
xx


y
y
Drivin
g forcex


c

(p p )

(c  c )
•Question:
LCan
, x we use (p-c) or (y-x) as mass transfer
V, y
driving
L, x force? Compare mass transfer driving forces
with heat transfer driving force?
6
(1)Comparison of diffusion and heat transfer
du
Momentum transfer    
dy
dT
Heat transfer q   k
dy
Mass transfer J A   D AB
dcA
db
q  h(Th  Tw )
J A  kc (c Ai  c A )  k g ( PA  PAi )
7
(2)Diffusion quantities（扩散通量）
1.Velocity u, length/time.
2.Flux across a plane N, mol/area•time.
3.Flux relative to a plane of zero velocity J,
mol/area•time.
4.Concentration c and molar density M, mol/volume
(mole fraction may also be used).
5.Concentration gradient dc/db, where b is the length
of the path perpendicular to the area across which
diffusion is occurring.
Appropriate subscripts are used when needed.
8
(3)Velocities in diffusion（扩散速率）
•Velocity without qualification refers to the velocity
relative to the interface between the phases and is that
apparent to an observer at rest with respect to the
interface.
9
(4)Molal flow rate, velocity , and flux
N   M u0
(17.1)
•Where
M=molar density of the mixture
N= total molar flux in a direction perpendicular to a
stationary plane
u0= volumetric average velocity
10
•For components A and B crossing a stationary plane,
the molal fluxes are
N A  cAu A
(17.2)
N B  cBuB
(17.3)
•By definition there is no net volumetric flow across
the reference plane moving at the volume-average
velocity u0, although in some cases there is a net
molar flow or a net mass flow.
J A  N A  cAu0  cA (u A  u0 )
(17.4)
J A =Diffusion flux of component A in the mixture
11
J B  N B  cBu0  cB (uB  u0 )
(17.5)
J B =Diffusion flux of component B in the mixture
•Fick’s first law of diffusion for a binary mixture（二元混合物）:
dcA
J A   D AB
(17.6)
db
dcB
J B   DBA
(17.7)
db
dcA
  D AB =diffusivity
(17
)
of.6component
A in its mixture with
db
component
B, m2/s
dcB
dc
A
gradient, mol/m4
DAB DBA =molar
(17.concentration
6()17.7)
db db
12
(5)Relations between diffusivities
•For ideal gases, and for diffusion of A and B in a gas
at constant temperature and pressure,
P
c A  cB   M 
RT
dcA  dcB  d M  0
dcA
dcB


db
db
PV
and  n A  nB  n 
 Const  dnA  dnB
RT
dcA
dcB
 J A  J B  0  J A   J B   DAB
 DBA
db
db
 DAB  DBA
(17.11)
13
•For liquid with same mass density [kg/m3],
c A M A  cB M B    const
M A dcA  M B dcB  0
(17.13)
P
 cB M
 flow
 M across the reference plane, the
•ForMnoAc Avolume
B
RT
J


J


0
A
B
sum of
the volumetric
flows due to diffusion is zero.


c A M A dc
 cAB
Mdc
 dconst

0 the molar flow rate times
B 
B
M  is
The volumetric flow rate
dcA MM dc
dcB M B(17.13)
A
M
dc

0
dc
A dc
BD
the
M/
and
 DAmolar
volume

0
(17.14)
AB 
B
AB
BA


db

M A db M Bdb db 
JA 
 JB 
0


PV
and  n A  nB  n 
 Const  dnA  dnB
dcA M A
dcB RT
MB
 DAB

 DBA

0
(17.14)
db 
db 
dcA
dcB
 J A  J B  0  J A   J B   DAB
 DBA
•Substituting(代入) Eq.(17.13) into Eq.(17.14) gives
db
db
 DAB  DBA
(17.15)
14
•A common form of the diffusion equation gives the
total flux of component A:
dcA
N A  c Au0  Dv
(17.16)
db
dcA
[ From J A   Dv
and Eqs.(17.2), (17.4)]
db
Where Dv=volumetric diffusivity, m2/h, cm2/s
  M y A , u0  N /  M , Eq.(17.16)becom es
ForM
gases, c A M
JA  A  JB  B  0
dyA


N A  y A N  Dv  M
(17.17)
dcA M A
dcB M B db
For
DABliquid, if M
D

0
(17.14)
=Constant,
BA
cdb   x , u  db
N /   , Eq.(17.16)becom es
A
M
A
0
M
dxA
N A  x A N  Dv  M
db
(17.17b)
15
(6)Interpretation of diffusion equations扩散方程
•The vector nature of the fluxes and concentration
gradients must be understood, since these quantities
are characterized by directions and magnitudes.
•The sign of the gradient is opposite to the direction of
the diffusion flux, since diffusion is in the direction of
lower concentrations. 物质A的浓度梯度在方向上的变
化与扩散通量相反，表示扩散是沿物质浓度降低方向
进行的
16
(7)Equimolal diffusion(等分子扩散)
Zero convective flow and equimolal
counterdiffusion（ 等分子反扩散 ） of A and B, as
occurs in the diffusive mixing of two gases and in
the diffusion of A and B in the vapor phase for
distillations that have constant molal overflow.
17
N
N AA  JJAA
N
NB  JJB
Liquid
B
B
Gas
BT
y
Interface
Mole fraction
yi
.B
p
m
Co
Co
mp
.A
Distance from interface
Fig.17.1(a)
Component A and B
diffusing at same
molal (equimolal)
rates in opposite
directions [Like the
case of diffusion of A
and B in the vapor
phase for
distillations that
have constant molal
overflow].
Note that for
equimolal diffusion,
18
NA=JA.
•Assuming a constant flux NA and zero total flux
(N=0), integrating Eq.(17.17) over a film thickness BT,
dyA
J A  N A   Dv  M
db
dy
J A BTN A   Dv  M y A A
db
N ABT db   Dv  M y A dyA
N A 0db   Dv  M y AidyA
Dv  M y Ai
0
NA  JA 
( y Ai  y A )
DvBTM
NA  JA 
( y Ai  y A )
or N  J  DBv T(c  c )
A
A
Ai
A
B
DvT
NA  JA 
(c Ai  c A )
BT
(17.19)
(17.19)
(17.20)
(17.20)
19
N
N AA  JJAA
N
NB  JJB
Liquid
B
BT
y
Interface
Mole fraction
yi
B
Gas
BT
yi
y
.B
p
m
Co
BT
yi
Co
mp
.A
Distance from interface
y
•The concentration
gradient for A is
linear in the film,
and the gradient
dyA
Jfor
 Dvsame
M
A BNhas
A  the
db
magnitude
but the
BT
yA
opposite
sign,
as dy
N
db


D

A 
v M 
A
shown
in
Figure
0
y Ai
17.1a.
Dv  M
NA  JA 
( y Ai  y
[From Eq.(17.17),
BT
dyA
NA

 const
db DV  M
20
Interface
Mole fraction
(8)One-component mass transfer (one-way diffusion)
Fig.17.1(b)
Component A
diffusing,
Gas
Liquid
NA
N  NA
component B
NB  JB B
N B  J B stationary with
T
respect to interface.
BT
yi
[Like the case of
Conc. diffusing comp. A
yi
y
diffusion of solute A
B
T from gas phase into
y
yi liquid phase in
absorption process.]
Conc. of stagnant comp.B
Distance from interface
y
21
(8)One-component mass transfer (one-way diffusion)
•When only component A is being transferred, the
total flux to or away from the interface N is the same
as NA, and Eq.(17.17) becomes
dyA
N A  y A N A  Dv  M
(17.21)
db
dyA
N A (1  y A )   Dv  M
db
N A db
dyA

Dv  M
(1  y A )
N A db N A BT
dyA
1 yA
0 Dv  M  Dv  M   y (1  y A )  ln 1  y Ai
Ai
BT
yA
D
1 y
22
N A db
dyA

Dv  M
(1and
 y A integrating,
)
•Rearranging
we have
N A db N A BT
dyA
1 yA
0 Dv  M  Dv  M   y (1  y A )  ln 1  y Ai
Ai
BT
Dv  M 1  y A
NA 
ln
BT
1  y Ai
yA
(17.24)
•Equation(17.24) can be rearranged by using the
logarithmic mean of 1-yA for easier comparison with
Eq.(17.19) for equimolal diffusion. The logarithmic
mean of 1-yA is
(1  y A )  (1  y Ai )
y Ai  y A
(1  y A ) L 

ln[(1  y A ) /(1  y Ai )] ln[(1  y A ) /(1  y Ai )]
(17.25)
23
A T
A
A




ln
0 Dv  M Dv  M y (1  y A ) 1  y Ai
Ai
Combining Eqs(17.24) and
(17.25) gives
Dv  M y Ai  y A
NA 
(17.26)
BT (1  y A ) L
A
[Comparing one-way diffusion in the Chinese textbook,
D
P
NA 

( p A1  p A2 )
RTz pBm
Here
P
D  Dv , z  BT ,  M 
RT
1
P


1
(1  y A ) L pBm
P
 drift factor漂流因数]
pBm
24
•Comparing Eq.(17.26) with Eq.(17.19),the flux of
component A for a given concentration difference is
therefore greater for one-way diffusion than for
equimolal diffusion.
•[Example17.1.]
25
N A db N A BT
dyA
1 yA


 ln

Dv  M inDliquid,
(1  y A )
1  y Ai
v M
•For0 diffusion
y Ai
Dv  M x Ai  x A
NA 
BT (1  x A ) L
26
2.PREDICTION OF DIFFUSIVITIES
•Diffusivities are physical properties of fluids.
•Diffusivities are best estimated by experimental
measurements, or from published correlations.
•The factors influencing diffusivities are temperature,
pressure, and compositions for a given fluid.
(1)Diffusion in gases
•Fick’s first law of diffusion for a binary mixture:
dcA
J A   D AB
(17.6)
db
dcB
J B   DBA
(17.7)
db
27
c A1
•Assume concentrations of
component A in the two layers
c A1 distance cofA1the molecular
with
mean
a binary
c Aof
c A2 free path
mixture are cA1 2and cA2,
b diffusion

b  
respectively,
the
flux is
c A1
c A2
c A1
c A2
b  
c A2
b  
b   J   1
1 A
3
J A   1 u (c A1 
J A   3 u (c A11
c A1
1
1
1  dcA  1  dcA3Dv  u
J A   u (c AJ1Ac A2 ) u (c A1uc A2 )  
 u 1   3
c A2 3
1 u 
3
3  db  3Dv db
Dv  3 u 
3
1
1
D
b

•Therefore,
u  Dv  (17
u .27)
(17.27)
v 
3
3
1
1  dcA 
=average molecular velocity, m/s
J•Where

A   u (c A1  c A 2 )   u 
3
3  db 
28
1.5
T
T
  , u  T 0.5  Dv 
P
P
•A more rigorous approach based on modern kinetic
theory allows for the different sizes and velocities of the
molecules and the mutual interactions as they approach
one another [Eq.(17.28) for binary diffusion.]
•The collision integral D decreases with increasing
temperature, which makes DAB increase with more than
the 1.5 power of the absolute temperature.
•For diffusion in air,
Dv  T
1.7~1.8
[ T
1.75
] [T  300 ~ 1000K ]
29
•In general, influence of concentrations for diffusion
in gases can be neglected, and
M A , or M B  Dv 
T  Dv 
P  Dv 
4
5
In gases, Dv  10 ~ 10 m / s
9
10
2
•[*Diffusion
in, D
small
pores](自学)
In liquids
~ 10 m
v  10
2
/s
•Example 17.2. [p.520] (自学)
30
•(2)Diffusion in liquids
•Diffusivities in liquids are generally 4 to 5 orders of
magnitude smaller than in gases at atmospheric
pressure.
4
5
In gases, Dv  10 ~ 10 m / s
9
10
2
In liquids, Dv  10 ~ 10 m / s
2
•Influence of pressure for diffusion in liquids can be
neglected.
•Diffusivities for dilute liquid solutions can be
calculated from Eq.(17.31)[Empirical correlation].
Dv 
T

, [T  Dv ,   Dv 
31
•Other empirical correlations for diffusivities:
•For dilute aqueous solutions of non-electrolytes, using
Eq.(17.32).（自学）
•For dilute solutions of completely ionized univalent
electrolytes,using Nernst equation(17.33). （自学）
•(3)Schmidt number Sc
•Sc is analogous to the Prandtl number.


Sc 

Dv Dv
cp



Pr  


  [k /( c p )] k
32
•For gases, Sc is independent of pressure when the
ideal gas law applies, since the viscosity is independent
of pressure, and the effects of pressure on  and Dv
cancel. Temperature has only a slight effect on Sc
because  and Dv both change with about T0.7~0.8.
•For liquids, Sc decreases markedly with increasing
temperature because of the decreasing viscosity and
the increase in the diffusivity.
•Unlike the case for binary gas mixtures the diffusion
coefficient for a dilute solution of A and B is not the
same for a dilute solution of B in A. i.e., DAB≠DBA
[Comparing with eq.(17.15)?]
33
•EXAMPLE 17.3. [p.522]
•From eq.(17.31), it is apparent that unlike the case
for binary gas mixtures, the diffusion coefficient for a
dilute solution of A and B is not the same for a dilute
solution of B in A.
•But, from EXAMPLE 17.3., the diffusivities of
benzene in toluene and toluene in benzene have only
a slight difference, and in this case the conclusion of
Eq.(17.15) DAB=DBA is still effective
34
(4)Turbulent diffusion
•In a turbulent stream the moving eddies transport
matter from one location to another, just as they
transport momentum and heat energy.
dc dc
dc
17(..17
34.))34)
JJAA,t,Jt A,tN
((17
34
NN
dbdb
dc
db
J



(
17
.
34
)
A
,
t
N
=molal
flux
of
A,
relative
to
phase
as a
Where JJAA,t,Jt A,t
db
whole, caused by turbulent action
JN A,t diffusivity
NN =eddy
The total molal
dctodcthe entire phase,
 N flux, relative
dc
D
)
(
17
.
35
)
v(v D

)
(
17
.
35
)
JJAA JA((D
v
)
(
17
.
35
)
N
N
becomes
N
dbdb
dc
db
J A  ( Dv   N )
db
(17.35)
35
•The eddy diffusivity is not a parameter of physical
property, it depends on the fluid properties but also
on the velocity and position in the flowing stream.
36