Diffusion Mass Transfer
Chapter 14
Sections 14.1 through 14.7
General Considerations
General Considerations
• Mass transfer refers to mass in transit due to a species concentration gradient
in a mixture.
 Must have a mixture of two or more species for mass transfer to occur.
 The species concentration gradient is the driving potential for transfer.
 Mass transfer by diffusion is analogous to heat transfer by conduction.
• Physical Origins of Diffusion:
 Transfer is due to random molecular motion.
 Consider two species A and B at the same T and p,
but initially separated by a partition.
– Diffusion in the direction of decreasing
concentration dictates net transport of
A molecules to the right and B molecules
to the left.
– In time, uniform concentrations of A and
B are achieved.
Definitions
Ci :
Definitions
Molar concentration  kmol/m  of species i.
3
i : Mass density (kg/m3) of species i.
Mi : Molecular weight (kg/kmol) of species i.
i  MiCi
J i* : Molar flux  kmol/s  m 2  of species i due to diffusion.
 Transport of i relative to molar average velocity (v*) of mixture.
N i : Absolute molar flux  kmol/s  m 2  of species i.
 Transport of i relative to a fixed reference frame.
ji :
Mass flux  kg/s  m  of species i due to diffusion.
 Transport of i relative to mass-average velocity (v) of mixture.
ni:
Absolute mass flux  kg/s  m 2  of species i.
2
 Transport of i relative to a fixed reference frame.
xi :
Mole fraction of species i  xi  Ci / C  .
mi :
Mass fraction of species i  mi  i /   .
Property Relations
Property Relations
• Mixture Concentration:
C   Ci   xi  1
i
i
• Mixture Density:
   i   mi  1
i
i
• Mixture of Ideal Gases:
p
Ci  i
iT
i 
pi
RiT
p   pi
i
xi 
Ci pi

C
p
Diffusion Fluxes
Molar and Mass Fluxes of Species A due to Diffusion
in a Binary Mixture of Species A and B
• Molar Flux of Species A:
 By definition:
J A  C A  v A  v 
v  xAvA  xB vB
 From Fick’s law (mass transfer analog to Fourier’s law):
J A  CDABxA
Binary diffusion coefficient or mass diffusivity (m2/s)
• Mass Flux of Species A:
 By definition:
jA   A  vA  v 
v = mAvA  mB v B
 From Fick’s law:
jA    DABmA
Absolute Fluxes
Absolute Molar and Mass Fluxes of Species A
in a Binary Mixture of Species A and B
• Molar Flux of Species A:
N A  CAvA  J A  CAv
N A  JA  C A  xA v A  xB v B 
N A  CDABx A  x A  N A  N B 
• Mass Flux of Species A:
nA   Av A  jA   Av
nA  j A   A  mA v A  mB v B 
nA    DABmA  mA  nA  nB 
• Special Case of Stationary Medium:


 v  0  N A  J A
 v  0  nA  jA
 Achieved to a good approximation for xA (or mA )
1 and N B (or nB )  0.
Conservation of Species
Conservation of Species
• Application to a Control Volume at an Instant of Time:
M A,in  M A,out  M A, g 
dM A
 M A,st
dt
M A,in , M A,out  rate of transport across the control surfaces
M A, g  rate of generation of A due to homogeneous chemical reactions
occurring in the control volume
M A,st  rate of accumulation of A in the control volume
• Application in Cartesian Coordinates to a Differential Control Volume for a
Stationary Medium of Constant DAB and C or  :
 Species Diffusion Equation on a Molar Basis:
 2 C A  2 C A  2C A N A
1 C A




x 2
y 2
z 2 DAB DAB t
 Species Diffusion Equation on a Mass Basis:
2 A 2 A 2 A n A
1  A




x 2
y 2
z 2 DAB DAB t
Conservation of Species (cont)
• Boundary Conditions (Molar Basis):
 Consider a Gas (A) / Liquid (B) or
Gas (A) / Solid (B) Interface.
Known surface concentration:
xA  0  xA,s
For weakly soluble conditions of a gas A in liquid B,
p
x A, s  A
(Henry’s law)
H
H  Henry's constant (Table A.9)
For gas A in a uniform solid B,
C A  0   Sp A
S  solubility  kmol/m3  bar  (Table A.10)
 Heterogeneous (surface) reactions (Catalysis)
dx
N A  0   N A  CDAB A
dx x  0
Special Cases
Special Cases for One-Dimensional , Steady-State Diffusion
in a Stationary Medium
• Diffusion without Homogeneous Chemical Reactions
 For Cartesian coordinates, the molar form of the species diffusion equation is
d 2 xA
0
2
dx
(1)
 Plane wall with known surface concentrations:
x A  x    x A,s ,2  x A, s ,1  x  x A, s ,1
L
dx A DAB  C A,s ,1  C A,s ,2 


N A, x  J A, x  CDAB

dx
L
N A, x  AN A, x 
Rm,diff 
DAB A
C A,s ,1  C A,s ,2 

L
L
DAB A
Results for cylindrical and spherical shells
Table 14.1
Special Cases (cont)
 Planar medium with a first-order catalytic surface:
Assuming depletion of species A at the catalytic surface (x = 0),
N A, x  0   N A  k1 C A  0 
Reaction rate constant (m/s)
 DAB
dx A
dx
x 0
 k1 x A  0 
Special Cases (cont)
Assuming knowledge of the concentration at a distance x=L from the surface,
xA  L   xA, L
Solution to the species diffusion equation (1) yields a linear distribution for x A  x  :

x A  x  1  xk1 / DAB

x A, L
1  Lk1 / DAB




Hence, at the surface,
xA  0 
1

x A, L
1  Lk1 / DAB


dx
N A  0   CDAB A
dx

x 0
Limiting Cases:
– Process is reaction limited:
k1  0
xA  0
1
xA  L 
k1Cx A, L

1  Lk1 / DAB

 Lk  / D 
1
AB
1
N A  0   k1CxA, L
Special Cases (cont)
– Process is diffusion limited:
xA  0   0
k1  
 Equimolar counterdiffusion:
 Lk  / D 
1
AB
1
CDAB x A, L
N A  0   
L
Occurs in an ideal gas mixture if p and T, and hence C, are uniform.
N A, x   N B, x
N A, x  DAB
C A,0  C A, L DAB p A,0  p A, L

L
T
L
Special Cases (cont)
• Diffusion with Homogeneous Chemical Reactions
For Cartesian coordinates, the molar form of the species diffusion equation is
d 2C A
DAB
 NA  0
dx 2
For a first-order reaction that results in consumption of species A,
N A  k1C A
and the general solution to the diffusion equation is
C A  x   C1emx  C2e  mx
m   k1 / DAB 
1/ 2
Consider diffusion and homogeneous reaction of gas A in a liquid (B) container
with an impermeable bottom:
Special Cases (cont)
Boundary conditions
CA  0   CA,0
dC A
dx
0
xL
Solution
CA  x   CA,0  cosh mx  tanh ml sinh mx 
N A, x  0   DABC A,0 m tanh ml
Column Evaporation
Evaporation in a Column: A Nonstationary Medium
 Special Features:
– Evaporation of A from the liquid interface  xA,0  xA,sat ( v)  xA, L 
– Insolubility of species B in the liquid. Hence downward motion by diffusion
must be balanced by upward bulk motion (advection) such that the absolute
flux is everywhere zero.
N B, x  0
– Upward transport of A by diffusion is therefore augmented by advection.
Column Evaporation (cont)
 Solution:
1  x A  1  x A, L 


1  A,0  1  x A,0 
N A, x 
x/L
CDAB  1  x A, L 
1n 

L
1

x
A,0 

Transient Diffusion
One-Dimensional, Transient Diffusion in a Stationary Medium
without Homogeneous Chemical Reactions
• Species Diffusion Equation in Cartesian coordinates
 2C A C A
DAB

2
x
t
• Initial and Boundary Conditions for a Plane Wall with Symmetrical Surface Conditions
C A  x , 0   C A ,i
C A  L, t   C A, s
C A
x
0
x 0
• Nondimensionalization
C  C A, s
  A
C A ,i  C A , s
x  x
L
tm 
DABt
 Fom
L2
Mass transfer Fourier number
Transient Diffusion (cont)
 Species Diffusion Equation
 2 
 

2
x
Fom
 Initial and Boundary Conditions
   x , 0   1
  1, Fom   0
 
x
0
x  0
• Analogous to transient heat transfer by conduction in a plane wall with symmetrical
surface conditions for which Bi  ,and hence Ts  T .
Hence, the corresponding one-term approximate solution for conduction may be
applied to the diffusion problem by making the substitutions
   
Fo  Fom
• Table 14.2 summarizes analogy between heat and mass transfer variables.