Chapter 3 Introduction to Mass Transfer
3.1 Basic Concepts
3.1.1 Diffusion and convection
Mass transfer refers to the transfer of a species in the presence of a
concentration gradient of the species. Under a concentration gradient
mass transfer can occur by either diffusion or convection.
Diffusion refers to the mass transfer that occurs a stationary solid or fluid
in which a concentration gradient exists. In contrast, convection refers to mass
transfer that occurs across a moving fluid in which a concentration gradient
exists.
Consider the dissolution of a sugar cube
in water, the concentration of sugar molecules
is significant only in the vicinity of the cube.
By stirring the water with a spoon to create
forced convection, sugar molecules are
transferred to the bulk water much faster.
1
1
3.1.2 Fick’s law of diffusion
3.1.2.1 One-dimensional
Consider the diffusion of species A through a thin sheet of thickness L,
as shown. Let wA be the mass fraction of species A in the sheet. A steady-state
concentration profile wA(y) is established in the sheet.
A diffusion flux, jAy, is defined as the amount of material diffused per unit
area per unit time and can be expressed by Fick’s law of diffusion
j Ay
dwA
  DA
dy
[3.1.1]
Where  is the mass density of the solution, DA the diffusion coefficient of
species A in the solution, and wA the mass (or weight) fraction of species A.
Minus sign and unit.
2
3.1.2.2 Three-dimensional
In the case of three-dimensional diffusion, the diffusion equation can be
expressed as follows
j Ax
dwA [3.1.2]
  DA
dx
j Ay   DA
dwA
dy
[3.1.3]
dwA [3.1.4]
j Az   DA
dz
Note that is has been assumed that the material is isotropic, that is,  and DA
are direction-independent.
Eqs. [3.1.2] through [3.1.4] can be expressed in a vector form as follows:
jA  DAwA
[3.1.5]
jA is the mass diffusion flux vector. If the mass density  is constant, Eq.[3.1-5]
can be written as follows:
jA   DA A
Where A =wA is the mass of species A per unit volume of the solution, or
the mass concentration of species A.
[3.1.6]
3
Fick’s law of diffusion can also be written as
j A  cDAxA
[3.1.7]
Where j A is the molar diffusion flux vector, c the molar density of the
solution, and xA the mole fraction of species A.
If the molar density of the equation is constant, Eq. [3.1-7] can be written as:
j A  DAcA
[3.1.8]
Where cA =cxA is the moles of species A per unit volume of the solution,
that is, the molar concentration of species A. In a dilute solution the molar
density of the solution c is essentially constant.
3.1.3 Thermal diffusion (thermal diffusion describes the tendency for species to
diffuse under the influence of a temperature gradient; this effect is quite small,
but devices can be arranged to produce very steep temperature gradients so
that separations of mixtures are effected)
In a nonisothermal system, spatial temperature variations can induce the socalled thermal diffusion, and Fick’s law of diffusion can be modified as follows:
jA   DA ( A 
1

T  A  B ln T )
[3.1.9]
4
and dividing by MA, the molecular weight of species A,
j A   DA (c A 
1

 T c AcB  ln T )
[3.1.10]
Where T and T are thermal diffusion factors based on mass and molar
concentrations, respectively. The two factors are related to each other
through T = MBT, where MB is the molecular weight of species B.
3.1.4 Diffusion boundary layer
Consider a fluid approaching a flat plate in the direction parallel to the
plate, as shown in Fig. 3.1-3. the plate is coated with a material containing
species A, which has a limited solubility in the fluid. The composition of the
approaching fluid is wA∞ and that of the fluid at the plate surface is wAS,
both of which are constant.
Because of the effect of diffusion, the concentration of the fluid in the
region near the plate is affected by the coating, varying from wAS at the
plate surface to wA∞ in the stream. This region is called the diffusion or
concentration boundary layer.
5
The thickness δc of the concentration boundary layer is taken as the
distance from the plate surface at which the dimensionless concentration
(wA-wAS)/(wA∞ -wAS) or (wAS-wA)/(wAS-wA∞) levels off to 0.99. In practice it is
usually specified that wA = wAS and wA / y  0 at y = δc .
With increasing distance from the leading edge of the plate, the effect of
diffusion penetrates farther into the stream and the boundary layer grows
in thickness. The effect of diffusion is significant only in the boundary layer.
Beyond it the concentration is uniform and the effect or diffusion is no longer
significant.
6
3.1.5 Mass transfer flux
Let vA and vB be the velocities of species A and B with respect to stationary
coordinates, respectively. These species velocities result from both the bulk
motion of the fluid at velocity v and the diffusion of the species super imposed
on the bulk motion. The mass flux of species A with respect to stationary
coordinates, specially, nA = AvA, can be considered to result from a mass
flux due to the bulk motion of the fluid, Av, and a mass flux due to the
diffusion superimposed on the bulk motion, jA. In other words,
n A   A v A   A v  jA
[3.1-11]
For a binary system consisting of species A and B, which is the focus of
the present chapter, the velocity v is a local mass average velocity defined by
 A v A + B v B  A v A + B v B n A +n B
v


 wA v A  wB v B
 A + B


[3.1-12]
Substituting Eq.[3.1-12] into Eq. [3.1-11]
n A  wA v  jA  wA (n A  n B )  jA
[3.1-13]
7
Similarly, the molar flux of species A with respect to stationary coordinates,
specially, n A  cA vA, can be considered to result from a molar flux, c A v ,
due to the bulk motion of the fluid at velocity v and a molar flux due to the diffusion
superimposed on the bulk motion, j A
. In other words,
n A  c A v A  cv  jA
[3.1-14]
Where the local molar average velocity
v
c A v A  cB v B cA v A  cB v B
n  nB

( A
)  xA v A  x B v B
c A  cB
c
c
[3.1-15]
Substituting Eq. [3.1-15] into Eq. [3.1-14], we have
n A  xAcv  jA  xA (n A +n B )  jA
[3.1-16]
8
3.1.6 Mass transfer coefficient
Consider fluid flow over a flat plate as shown in Fig. 3.1-3. The mass diffusion
flux across the solid/liquid interface is
jAy
y 0
wA
   DA
y
[3.1-17]
y 0
This equation cannot be used to calculate the diffusion flux when the
concentration gradient is an unknown. A convenient way to avoid this problem
is to introduce a mass transfer coefficient.
At the solid/liquid ( or liquid/gas) interface the mass transfer coefficient km is
defined by
nAy
y 0
 km   A0   A 
[3.1-18]
Where A0 and A∞ are the mass concentrations of species A in the fluid at the
interface and in the bulk (or free-stream) fluid, respectively. If  is constant,
Eq. [3.1-18] can be rewrite as
nAy
y 0
nAy
y 0
 km   wA0  wA 
 km   wA0  wA 
[3.1-19]
9
If the solubility of species A in the fluid is limited so that vy is essentially zero
at the interface, the following approximation can be made in view of Eq. [3.1-11]
nAy
y 0
 jAy
y 0
[3.1-20]
Substituting Eqs [3.1-17] and [3.1-19] into Eq. [3.1-20], we have
km 
j Ay
y 0
  wAS  wA 

 DA  wA y  y 0
[3.1-21]
 wAS  wA 
For fluid flow through a pipe such as that shown in Fig. 3.1-4, we can write
km 
j Ar
r R
  wAS  wA,av 

 DA  wA r  r  R
wAS  wA,av
[3.1-22]
Where the average concentration is defined as follows
10
wA,av
 w vdA 



  vdA
A
A
A
 wAvdA
m


A
A
 wAvdA
 vav A
[3.1-23]
Notice that the numerator is the species mass flow rate.
Similar equations on the molar basis can be written for flow over a flat plate
km 
jAy
y 0
c  xAS  xA 

 DA  xA y  y 0
xAS  xA
[3.1-24]
and for flow through a tube
km 
jAy
rR
c  x AS  x A,av 

 DA  x A r  r  R
x AS  x A,av
[3.1-25]
Where the average concentration is defined as follows
x A,av
cx vdA 



 cvdA
A
A
A
A
cx A vdA
m


A
cx A vdA
cvav A
[3.1-26]
11
3.1.7 Diffusion in solids
3.1.7.1 Diffusion mechanisms
Vacancy diffusion and interstitial diffusion are the two most frequently
encountered diffusion mechanisms in solids, although other mechanisms
have also been proposed.
In vacancy diffusion an atom in a solid
jumps from a lattice position of the
solid into a neighboring unoccupied
lattice site or vacancy, as illustrated in
Fig. 3.1-5a. At temperature above
absolute zero all solids contain some
vacancies; the higher the temperature,
the more the vacancies. The atom
can continue to diffuse through the
solid by a series of exchanges with
vacancies that appears to be adjacent
to it from time to time.
12
Vacancy diffusion is usually the diffusion
mechanism for substantially solid solutions.
In such materials the solute atoms, which are
comparable to the solvent atoms in size,
substitute the solvent atoms at their lattice
sites. Examples of substitutional solid
solutions are Cu-Zn alloy and Au—Ni alloy.
In interstitial diffusion an atom in a solid
jumps from an interstitial site of the lattice to
a neighboring one, as illustrated in Fig. 3.1-6a.
The atom can continue to diffuse through the
solid by a series of jumps to neighboring
interstitial sites that are unoccupied.
Interstitial diffusion is the diffusion mechanism for interstitial solid solutions.
In such materials the solute atoms, which are significantly smaller than the
solvent atoms, occupy the interstitial sites of the lattice. The most well know
example of interstitial solid solution is the iron-carbon alloy, specially, carbon
steel, in which the small carbon atoms occupy the interstitial sites of the iron
lattice.
13
3.1.7.2 Diffusion coefficients
In the so-called self-diffusion experiment, a solute A in the form of a radioactive isotope, such as 63Ni, is allowed to diffuse through the lattice of a nonradioactive solid of the same material, Ni. The diffusion coefficient DA is known
as the self-diffusion coefficient, in view of the absence of a chemical
composition gradient as the driving force for diffusion.
In practical situations, however, diffusion usually occurs under the influence
of a chemical composition gradient, such as the diffusion of carbon in steel
from a higher-carbon-concentration region to the lower one. This diffusion
coefficient DA is known as the intrinsic diffusion coefficient. The so-called
interdiffusion coefficient D is often used to describe situations involving the
interdiffusion of two different chemical species, such as Au into Ni and Ni into
Au as in an Au-Ni diffusion couple.
3.1.7.3 Effect of temperature
The diffusion coefficient has been observed to increase with increasing
temperature according to the following Arrhenius equation:
D  D0 e  Q / RT
[3.1-27]
Where D: diffusion coefficient; D0: a proportional constant; Q: the active energy
R: the gas constant; T: absolute temperature
14
As illustrated in Fig. 3.1-5b, a significant energy barrier has to be overcome
before an atom can jump from one lattice site to a neighboring one by vacancy
diffusion.
Similarly, as illustrated in Fig. 3.1-6b, a smaller but still significant energy
barrier has to be overcome before an interstitial atom can jump from one
interstitial site to a neighboring one by interstitial diffusion.
Tables 3.1-3 and 3.1-4 list the experimental data of D0 and Q for substitutional diffusion and interstitial diffusion in some materials. As shown, Q
is significantly lower for interstitial diffusion than for substitutional diffusion.
15
As shown in Table 3.1-3, Q is smaller for
substitutional self diffusion in body-centercubic iron than in face-center-cubic iron.
Since atoms are more loosely packed in
a bcc structure than an fcc structure, Q is
smaller in bcc iron than in fcc iron. For the
same reason, Q is also smaller for interstitial
diffusion of C, N, and H in bcc iron than in
fcc iron. Fig. 3.1-7 shows some diffusion
coefficients as a function of temperature.
16
3.2 Species overall mass-balance equation
3.2.1 Derivation
Consider an arbitrary stationary control volume Ω bounded by surface A
through which a moving fluid is flowing, as illustrated in Fig. 3.2-1. The control
surface A can be consider to consist of three different regions.
[3.2-1]
A A  A  A
in
out
wall
Consider the transfer rate of species A through dA shown in Fig. 3.2-2a.
As shown in Fig. 3.2-2b, the outward transfer and inward transfer rate are ja•ndA
and -ja．ndA, respectively.
Consider the mass conservation law for species A in the control volume shown in
Fig.3.2-1:
 Rate of
  Rate of
  Rate of


 
 

species
A

species
A
in

species
A
out

 
 

 accumulation   by mass inflow   by mass outflow 

 
 

(1)
(2)
 Rate of other species

  A transfer to system
 from surroundings

(4)
(3)
  Rate of species
 
  A generation
  in system
 


 [3.2-3]


(5)
17
Term 1: Mass of species A in the control volume

 Ad (int egral ) ; M A (overall )

The rate of change in the mass of species A in Ω

 A d


t
d
(int egral ) ;
M A (overall ) term (1)
dt
Terms 2&3: Mass flow rate of species A through a differential area dA
   A v  ndA (int egral );
A
( A vdA)in -(  A vdA)out +( A vdA) wall (overall ) terms (2 & 3)
A
A
A
Term 4: Species A goes into the control volume from the surrounding other than
terms (2) and (3):
  jA  ndA (int egral ) ; J A (overall ) term (4)
A
Term 5: The mass production rate of species A in Ω:


rAd (int egral ) ; R A (overall ) term (5)
18
Substituting the integral form of terms (1) through (5) into Eq. [3.2.3], we have

 A d      A v  ndA   jA  ndA   rA d 


A
A

t
[3.2-4]
Note that the mass convection and diffusion terms can be combined into one
through nA =Av+jA
A similar equation can be derived on the basis of the molar density cA, the
molar flux j A , the local molar average velocity v, and the molar production rate of
A per unit volume rA . This equation is

c A d     cA v  ndA   jA  ndA   rA d 


A
A
A
t
[3.2-5]
Note that the molar convection and diffusion terms can be combined into one
through n A  c A v  jA
19
Now substituting the overall form of terms (1) through (5) into Eq.[3.2-3] and with
A =wA.

 
dM A
   wAvdA    wAvdA
A
A
in
dt
 (mA )in  (mA )out  J A  RA

out
 J A  RA
Where MAΩ is the mass of species A in the control volume; that is
[3.2-6]


wAd
When the convective mass transfer at the wall has been neglected, substituting
Eq. [3.1-23] into Eq.[3.2-6], we obtain
dM A
   wA,av vav Ain    wA,av vav Aout  J A  RA
dt
 (mwA,av )in  (mwA,av )out  J A  RA
Where
[3.2-7]
MAΩ: mass of species A in control volume((=wAΩ=MwA if uniform wA)
M: mass flow rate at inlet or outlet (=vAVA)
RA:mass generation rate of species A in control volume (= rA )
JA: mass transfer rate of species A into control volume from
surroundings by diffusion
20
Equations similar to [3.2-6] and [3.2-7] can be derived on the molar basis, i.e.,

 
dM A
  cxA vdA   cxA vdA
A
A
in
dt
 (mA )in  (mA )out  J A  RA

out
 J A  RA
[3.2-8]
and
dM A
  cxA,av vav Ain   cxA,av vav Aout  J A  RA
dt
 (mxA,av )in  (mxA,av )out  J A  RA
Where
[3.2-9]
MAΩ: mass of species A in control volume(=cxAΩ=MxA if uniform cxA)
m: mass flow rate at inlet or outlet (=cvavA)
RA:mass generation rate of species A in control volume(= rA )
JA: mass transfer rate of species A into control volume from
surroundings by diffusion
21
Example 3.2-2 Diffusion through composite foil
Given: No convection nor chemical reactions
Steady-state
dM A
Basic Eq. :
   wA,av vav Ain    wA,av vav Aout  J A  RA
dt
 (mwA,av )in  (mwA,av )out  J A  RA
Sol.:
J H  ( AjH ) z   ( AjH ) z   0
1
2
J H  ( AjH ) z   ( AjH ) z   0
1
jH
z1
jH
A
 jH
 jH
z2
B
2
 jH
 jH
C
A
 constant
 jH =constant
dwH 
dwH 
dwH 





D



D



D
 JH
H
H
H






dz  A 
dz  B 
dz  C
22
wH 1  wH 2 
jH
( z2  z1 )
(  DH ) A
[3.2-26]
wH 2  wH 3 
jH
( z3  z2 )
(  DH ) B
[3.2-27]
wH 3  wH 4 
jH
( z4  z3 )
(  DH )c
[3.2-28]
Adding Eqs. [3.2-26] through [3.2-28], we have
wH 1  wH 4
 z2  z1
z3  z2
z4  z3 



 jH
 (  DH ) A (  DH ) B (  DH )C 
or
1
 z z
z z
z z 
jH   2 1  3 2  4 3  ( wH 1  wH 4 )
 (  DH ) A (  DH ) B (  DH )C 
1
 z2  z1
z3  z2
z4  z3 
1/ 2
1/ 2


jH  


K
(
p
)

(
p
)

p 
H 2 high
H 2 low 
 (  DH ) A (  DH ) B (  DH )C 
23
Example 3.2-3 Diffusion of gas through tube wall
Given: No convection nor chemical reactions
Steady-state
Basic Eq. : dM
A
  cxA,av vav Ain   cxA,av vav Aout  J A  RA
dt
 (mxA,av )in  (mxA,av )out  J A  RA
Sol.: no chemical reaction RA=0
consider a C.V. of length dz along the longitudinal direction
(mxA )in  m( xA  dxA )out  J A  0
 0  cAS
J A  ( Ddz ) DA 
 l

  ( Ddz ) DAk A x A / l

Substituting Eq. [3.2-35] into [3.2-34], we have
dxA
  DDA k A 
 d (ln xA )   
 dz
xA
 ml 
integrating
ln
xAL
xA0

 DDA k A L
ml
or xAL  xA0exp(-
 DDAk A L
ml
)
24
3.3 Species differential mass-balance equation
3.3.1 Derivation
The species integral mass-balance equation can be written as follows
 A
 t d   A  A v  ndA  A jA  ndA   rAd 
[3.3-1]
The surface integrals in Eq. [3.3-1] can be converted into volume integrals using the
Gauss divergence theorem

A

 A v  ndA    A vd 

j v  ndA   jAd 
A A

[3.3-2]
[3.3-3]
Substituting Eq. [3.3-2] and [3.3-3] into Eq. [3.3-1]
  A





v



j

r
A
A
A d   0
  t

[3.3-4]
The integrand, which is continuous, must be zero since the equation must hold for
any arbitrary region Ω. Therefore,
25
 A
    A v    jA  rA    n A  rA (variable properties)
t
Noting that n A   A v +jA from Eq. [3.1-11]
[3.3-5]
By following a similar approach, an equation can be derived based on the basis
of molar density cA (moles A per unit volume), the molar diffusion flux j A , the local
molar average velocity v , and the molar production rate of A per unit volume rA .
This equation is
c A
   c A v    jA  rA    n A  rA (variable properties)
t
[3.3-6]
Noting that n A  c A v + jA from Eq. [3.1-14]
Fick’s law of diffusion, according to Eqs. [3.1-5] and [3.1-7], is
jA    DAwA
[3.3-7]
jA  cDAx A
[3.3-8]
26
Substituting Eqs. [3.3-7] and [3.3-8] into Eqs. [3.3-5] and [3.3-6], respectively, the
following equations can be obtained;
 A
     A v       DAwA   rA
t
[3.3-9]
cA
    c A v      cDAx A   rA
t
[3.3-10]
Let us now consider the case of incompressible fluids. From the continuity
Equation  v = 0 and so   (  A v) = v A +A  v =v A. Since
for constant  and DA, Eq. [3.3-9] reduces to
 A
 v  A  DA 2  A  rA (constant  and D A )
t
c A
 v c A  DA 2 c A  rA (constant  and D A )
t
[3.3-11]
[3.3-12]
Eqs. [3.3-11] or [3.3-12] is the species differential mass-balance equation or the
species continuity equation.
27
3.3.2 Dimensionless form
Besides the dimensionless parameters listed in Sections 1.5.2 and 2.3.2, we
Includes a new parameter for concentration:
cA

c A  c A0

c A1  c A0
(dimensionless concentration)
[3.3.14]
Where (cA1 –cA0) is characteristic concentration difference.
If no chemical reaction occurred, Eq. [3.3.12] reduces to
c A
 v c A  DA 2 c A
t
[3.3.20]
Substituting Eq. [3.3-13] through [3.3-19] (please see the book for details) into
Eq. [3.3-20]
V 

  1
c c  c A0    Vv c A  c A1  c A0  
  A  A1
L t
L
1 2 
DA 2  c A  c A1  c A0 
L
[3.3.21]
28
Multiplying Eq. [3.3-21] by L/[V (cA1 –cA0)]
cA
DA 2 
  
 v  cA 
 cA

t
LV
[3.3.22]
By combining Eq.[3.3-22] with Eqs. [2.3-18] through [2.3-24], the following
equations can be obtained for mass transfer in forced convection:
Continuity:
Momentum:
  v   0
v
1 2  1

 
 
 v  v   p 
 v 
eg

t
Re
Fr
Energy:
T 
1
1 2 

 
2 
 v  T 
 T 
 T

t
Re Pr
P eT
Species:
cA
1
1 2 

 
2 

v

c


c

 cA
A
A

t
Re Sc
P eS
[3.3.23]
[3.3.24]
[3.3.25]
[3.3.26]
29
where Re, Fr, and Pr are defined in the previous chapter, the new dimensionless
parameters defined in this chapter are listed in the following:

viscous diffusivity 
[3.3.30]
Sc 
(Schmidt number = species diffusivity D )
DA
A
By combining Eq.[3.3-22] with Eqs. [2.3-18] through [2.3-24], the following
equations can be obtained for mass transfer in forced convection:
PeT  Re Pr 
LV

(thermal Peclet number =
LV
DA
(solutal Peclet number =
convection heat transport  C v V  T1  T0 
conduction heat transport k  T1  T0  / L
)
[3.3.31]
PeS  Re Sc 
convection species transport V  c A1  c A0 
diffusion species transport D A  c A1  c A0  / L
)
[3.3.32]
3.3.3 Boundary conditions
Boundary conditions commonly encountered in mass transfer are summarized
and listed:
1. At the plane or axis of symmetry, the concentration gradient in the transverse
direction is zero, (case 1)
30
2. A wall in contact with a fluid or the surface of a solid or fluid may be kept at a
given solute concentration, (case 2)
3. A wall in contact with a fluid or the surface of a solid or liquid may allow no
penetration, evaporation, or reactions, (case 3)
4. The free surface of a fluid may be
exposed to a gas of solute concentration wAf. A boundary conditions
consistent with Eq. [3.1-21] is as
follows: (case 4)
jAy

  DA
wA
 km  wA  wAf 
y
[3.3-33]
5. For two phase in perfect contact
with each other, the concentration
and the diffusion flux are both
continuous across the interface,
that is, they are the same
on both sides of the interface, (case 5)
31
32
3.3.4 Solution procedure
The purpose of the species equation is to determine the concentration distribution,
the step-by step procedure for solving the species continuity equation are listed in
Fig. 3.3-3
33
Example 3.3-1
Given: Initial A* (in moles):M (at t=0)
Find: cA*(x,t) and diffusion flux jA*x
Assume:Overall molar concentration c and
the self-diffusion coefficient DA*x are constant
Ananlysis: Stationary:vx=vy=vz, one-dimensional
problem (varies in x-dir only)
Sol:
cA*
t
 v cA*  DA*  2cA*  rA* (constant  and DA )
c A*
c A*
c A* 
 c A*
 vx
 vy
 vz
  DA*
t 
x
y
z 
  2c A*  2c A*  2c A* 

 rA*
 2 
2
2 
y
z 
 x
No chemical reaction rA* =0
The governing Eq. becomes
c A*
t
 DA*
 2 c A*
x 2
34
with initial condition
c A* ( x, 0)  0 and c A* ( , t )  0
and the mass conservation

M   c A* dx

35
Example 3.3-2
Given: Two interstitial alloys with concentration
of cA1 and cA2 at t=0
Find: Concentration profile cA(x,t)
Assume: Overall molar density c and intrinsic
diffusion coefficient DAare constant
Analysis: Symmetric concentration with
respect to the interface concentration cAS,
due to constant c and DA, cAS = (cA1+cA2)/2
=constant
2

c

cA
A
Sol: Basic Eq.
D
t
A
x 2
B.Cs.& I.Cs.:
cA ( x,0)  cA2
The solution
cA  cAS
 erf
cA2  cAS
cA (0, t )  cAS
cA (, t )  cA2
 x 


 4 DAt 
The solution can be used to determine the intrinsic diffusion coefficient
DA from experiment
36
Example 3.3-3
Given: Initial composition of cA1 and cA2 in
solids 1 and 2, respectively. The concentration
profile after annealing is shown
Find: Intrinsic diffusion coefficient DA
Analysis: Overall molar concentration c is constant
and no bulk motion and chemical reaction
Sol:
Basic Eq.
c A  
c 
  DA A 
t
x 
x 
Combine x and t so that cA can be expressed as a
x
function of a single variable (combination of variables), let   1
c A c A 
1 x c A

 3
t
 t
2 t 2 
c A c A 
1 c A

 1
x
 x t 2 
t2
Substituting the above two equations into governing Eq.
37

dc 
1
  dcA  d  DA A 
2
d 

B.Cs. c A  c A 2 at x      
c A  c A1 at x  
   
Eq. (3.3-57) can be integrated as follows:



dcA 
dc A 
dc A 
1 cA
   dcA   DA
   DA
   DA

c
A
1
2
d

d

d


 cA 
 cA1 
 cA
or
DA c  
A
Because dc A
dc A
1
2  dcA / d  c
d

dc A1
d

dc A 2
d
d

cA
c A1
 dcA
A
 0 at   
 0 at   
 cA  cA1  and
 cA  cA2 
38


dc 
dc 
1 cA
   dcA   DA A    DA A   0
2 cA1
d  c
d  c


A2
A1
Express Eq. 3.3-61 & 3.3-62 in terms of x rather than η
dc A dc A dx dc A 1/ 2


t
d
dx d dx
1/ 2
1/ 2

dc

t
xdc

t
A
 A 
 xdcA
DA c
and
A
1
1

2t (dc A / dx) c

cA 2
c A1
xdc A  0

cA
c A1
xdc A
[3.3-63]
A
[3.3-64]
Eqs. [3.3-63] and [3.3-64] were used to determine the diffusion coefficient,
and was called Boltzman-Matano Method.
39
Example 3.3-6
Problem: A chemical species A diffuses
from a gas phase into a porous catalyst
sphere of radius R in which it is
converted into species B.
Given: Concentration of A at the surface
of the sphere is cAS, A is consumed
according to rA = -k1acA, constant DA
Find: Steady-state concentration
distribution of A in the sphere
c A
0
Analysis: steady-state
t
no convection within the sphere v r  v = v  0
no concentration gradients in the θand ψdirections
cA  2cA

0
2


Sol: The species continuity eq. (Eq. [c] of Table 3.3-2] can be reduced to
1 d  2 cA  k1a
cA
r

2
r dr  r  DA
Concentration distribution
B.Cs
c A
 0 at r=0, c A  c AS at r=R
r
cA R  ebr  ebr 
ka
  bR bR  where b  1
cAS r  e  e 
DA
40