2-1

Chapter 2

Unsteady State Molecular Diffusion

2.1 Differential Mass Balance

When the internal concentration gradient is not negligible or Bi << 1, the microscopic or
differential mass balance will yield a partial differential equation that describes the concentration
as a function of time and position. For a binary system with no chemical reaction, the unsteady
state molecular diffusion is given by


A
c
t

= (D
AB
c
A
) (2.1-1)

For one-dimensional mass transfer in a slab with constant D
AB
and convective conditions of h
m

and c
A,
, equation (2.1-1) is simplified to


A
c
t

= D
AB

2
2
A
c
x

(2.1-2)

x=0
L -L
h , c
m A,inf
h , c
m A,inf

Figure 2.1-1 One-dimensional unsteady mass transfer in a slab.

Equation (2.1-2) can be solved with the following initial and boundary conditions

I. C. t = 0, c
A
(x, 0) = c
Ai

B. C.

x = 0,
0
A
x
c
x
=

= 0; x = L, D
AB
A
x L
c
x
=

= h
m
(c
Af
c
A,
)

In general, the concentration within the slab depends on many parameters besides time t and
position x.

c
A
= c
A
(x, t, c
A,i
, c
A,
, L, D
AB
, h
m
)

2-2
The differential equation and its boundary conditions are usually changed to the dimensionless
forms to simplify the solutions. We define the following dimensionless variables

Dimensionless concentration:
*
=
,
, ,
'
'
A A
A i A
c K c
c K c

c
A
=Kc
A,
+
*
(c
A,i
Kc
A,
)

Dimensionless distance: x
*
=
L
x
x = L x
*


Dimensionless time or Fourier number: t
*
= F
o
=
2
AB
D t
L
t =
2
AB
L
D
Fo

K is the equilibrium distribution coefficient. Substituting T, x, and t in terms of the
dimensionless quantities into equation (2.1-2) yields

(c
A,i
c
A,
)
1
AB
D
2
AB
D
L Fo

= (c
A,i
c
A,
)
2
1
L
2
* 2
* x




Fo

=
2
* 2
* x

(2.1-3)

Similarly, the initial and boundary conditions can be transformed into dimensionless forms


*
(x
*
, 0) = 1


0
*
*
*
=

x
x

= 0;
1
*
*
*
=

x
x

= Bi
m
*
*
(1, t
*
), where Bi
m
=
'
m
AB
h L
K D


Therefore
*
= f(x
*
, F
o
, Bi
m
)

The dimensionless concentration depends
*
only on x
*
, F
o
, and Bi
m
. The mass transfer Biot
number, Bi
m
, denotes ratio of the internal resistance to mass transfer by diffusion to the external
resistance to mass transfer by convection. Equation (2.1-3) can be solved by the method of
separation of variables to obtain


*
=

=1 n
n
C exp(
2
n
F
o
) cos(
n
x*) (2.1-4)

where the coefficients C
n
are given by

C
n
=
) 2 sin( 2
sin 4
n n
n

+


and
n
are the roots of the equation:
n
tan(
n
) = Bi
m
.
2-3

Table 2.1-1 lists the Matlab program that evaluates the first ten roots of equation
n
tan(
n
) = Bi
m

and the dimensionless concentrations given in equation (2.1-4). The program use Newtons
method to find the roots (see Review).

Table 2.1-1 Matlab program to evaluate and plot
*
=

=1 n
n
C exp(
2
n
Fo) cos(
n
x*)

% Plot the dimensionless concentration within a slab
%
% The guess for the first root of equation z*tan(z)=Bi depends on the Biot number
%
Biot=[0 .01 .1 .2 .5 1 2 5 10 inf]';
alfa=[0 .0998 .3111 .4328 .6533 .8603 1.0769 1.3138 1.4289 1.5707];
zeta=zeros(1,10);cn=zeta;
Bi=1;
fprintf('Bi = %g, New ',Bi)
Bin=input('Bi = ');
if length(Bin)>0;Bi=Bin;end
% Obtain the guess for the first root
if Bi>10
z=alfa(10);
else
z=interp1(Biot,alfa,Bi);
end
% Newton method to solve for the first 10 roots
for i=1:10
for k=1:20
ta=tan(z);ez=(z*ta-Bi)/(ta+z*(1+ta*ta));
z=z-ez;
if abs(ez)<.00001, break, end
end
% Save the root and calculate the coefficients
zeta(i)=z;
cn(i)=4*sin(z)/(2*z+sin(2*z));
fprintf('Root # %g =%8.4f, Cn = %9.4e\n',i,z,cn(i))
% Obtain the guess for the next root
step=2.9+i/20;
if step>pi; step=pi;end
z=z+step;
end
%
% Evaluate and plot the concentrations
hold on
Fop=[.1 .5 1 2 10];
xs=-1:.05:1;
cosm=cos(cn'*xs);
for i=1:5
2-4
Fo=Fop(i);
theta=cn.*exp(-Fo*zeta.^2)*cosm;
plot(xs,theta)
end
grid
xlabel('x*');ylabel('Theta*')

Bi = .5
Root # 1 = 0.6533, Cn = 1.0701e+000
Root # 2 = 3.2923, Cn = -8.7276e-002
Root # 3 = 6.3616, Cn = 2.4335e-002
Root # 4 = 9.4775, Cn = -1.1056e-002
Root # 5 = 12.6060, Cn = 6.2682e-003
Root # 6 = 15.7397, Cn = -4.0264e-003
Root # 7 = 18.8760, Cn = 2.8017e-003
Root # 8 = 22.0139, Cn = -2.0609e-003
Root # 9 = 25.1526, Cn = 1.5791e-003
Root # 10 = 28.2920, Cn = -1.2483e-003


Figure 2.1-2 shows a plot of dimensionless concentration
*
versus dimensionless distance x
*
at
various Fourier number for a Biot number of 0.5.

Figure 2.1-2 Dimensionless concentration distribution at various Fourier number.


For the roots of equation
n
tan(
n
) = Bi
m
, let

f = tan() Bi
m


Then f = tan() +(1 + tan()
2
);
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x*
T
h
e
t
a
*
Temperature distribution in a slab for Bi = 0.5
Fo=1
Fo=2
Fo=10
Fo=0.1
Fo=0.5
2-5
The differential conduction equation for mass transfer in the radial direction of an infinite
cylinder with radius R is


A
c
t

= D
AB
r
1
A
c
r
r r

| |
|

\
(2.1-5)

The differential conduction equation for mass transfer in the radial direction of a sphere with
radius R is


A
c
t

= D
AB
2
1
r
2 A
c
r
r r

| |
|

\
(2.1-6)

Equations (2.1-5) and (2.1-6) can be solved with the following initial and boundary conditions

I. C. t = 0, c
A
(r, 0) = c
A i

B. C.

r = 0,
0
A
r
c
r
=

= 0; r = R, D
AB
A
r R
c
r
=

= h
m
(c
Af
c
A,
)

The solution of equation (2.1-5) for the infinite cylinder is given as


*
=

=1 n
n
C exp(
2
n
F
o
) J
0
(
n
x*) (2.1-7)

where J
0
(
n
x*) is Bessel function of the first kind, order zero. The coefficient C
n
are not the same
as those in a slab. The solution of equation (2.1-6) for a sphere is given as


*
=

=1 n
n
C exp(
2
n
F
o
)
*
*) sin(
r
r
n
n

(2.1-8)

Since
0 *
lim
r *
*) sin(
r
r
n
n

=
0 *
lim
r
n
n n
r

*) cos(
= 1, it should be noted that at r* = 0


*
=

=1 n
n
C exp(
2
n
F
o
)

For one-dimensional mass transfer in a semi-infinite solid as shown in Figure 2.1-3, the
differential equation is the same as that in one-dimensional mass transfer in a slab


A
c
t

= D
AB
2
2
A
c
x

2-6
x
Semi-Infinite Solid

Figure 2.1-3 One-dimensional mass transfer in a semi-infinite solid.

We consider three cases with the following initial and boundary conditions

Case 1: I. C.: c
A
(x, 0) = c
Ai
B. C.: c
A
(0, t) = c
As
, c
A
(x , t) = c
Ai


Case 2: I. C.: c
A
(x, 0) = c
Ai
B. C.: D
AB

0
A
x
c
x
=

= N
A0
, c
A
(x , t) = c
Ai


Case 3: I. C.: c
A
(x, 0) = c
Ai
B. C.: D
AB
0
A
x
c
x
=

= h
m
(c
Af
c
A,
), c
A
(x , t) = c
Ai


All three cases have the same initial condition c
A
(x, 0) = c
Ai
and the boundary condition at
infinity c
A
(x , t) = c
Ai
. However the boundary condition at x = 0 is different for each case,
therefore the solution will be different and will be summarized in a table later.

2.2 Approximate Solutions

The summation in the series solution for transient diffusion such as equation (2.1-4) can be
terminated after the first term for F
o
> 0.2. The full series solution is


*
=

=1 n
n
C exp(
2
n
F
o
) cos(
n
x*) (2.1-4)

The first term approximation is


*
= C
1
exp(-
2
1
F
o
) cos(
1
x
*
) (2.2-1)

where C
1
and
1
can be obtained from Table 2.2-1 for various value of Biot number. Table 2.2-2
lists the first term approximation for a slab, an infinite cylinder, and a sphere. Table 2.2-3 lists
the solution for one-dimensional heat transfer in a semi-infinite medium for three different
boundary conditions at the surface x = 0. Table 2.2-4 shows the combination of one-dimensional
solutions to obtain the multi-dimensional results.
2-7
Table 2.2-1 Coefficients used in the one-term approximation to the series
solutions for transient one-dimensional conduction or diffusion

PLANE WALL INFINITE CYLINDER
SPHERE
Bi
m

1
(rad)

C
1

1
(rad)
C
1

1
(rad)
C
1

0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.15
0.2
0.25
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
20.0
30.0
40.0
50.0
100.0
500.0
1000.0

0.0998
0.1410
0.1732
0.1987
0.2217
0.2425
0.2615
0.2791
0.2956
0.3111
0.3779
0.4328
0.4801
0.5218
0.5932
0.6533
0.7051
0.7506
0.7910
0.8274
0.8603
1.0769
1.1925
1.2646
1.3138
1.3496
1.3766
1.3978
1.4149
1.4289
1.4961
1.5202
1.5325
1.5400
1.5552
1.5677
1.5692
1.5708
1.0017
1.0033
1.0049
1.0066
1.0082
1.0098
1.0114
1.0130
1.0145
1.0160
1.0237
1.0311
1.0382
1.0450
1.0580
1.0701
1.0814
1.0919
1.1016
1.1107
1.1191
1.1795
1.2102
1.2287
1.2402
1.2479
1.2532
1.2570
1.2598
1.2620
1.2699
1.2717
1.2723
1.2727
1.2731
1.2732
1.2732
1.2732
0.1412
0.1995
0.2439
0.2814
0.3142
0.3438
0.3708
0.3960
0.4195
0.4417
0.5376
0.6170
0.6856
0.7465
0.8516
0.9408
1.0185
1.0873
1.1490
1.2048
1.2558
1.5995
1.7887
1.9081
1.9898
2.0490
2.0937
2.1286
2.1566
2.1795
2.2881
2.3261
2.3455
2.3572
2.3809
2.4000
2.4024
2.4048
1.0025
1.0050
1.0075
1.0099
1.0124
1.0148
1.0173
1.0197
1.0222
1.0246
1.0365
1.0483
1.0598
1.0712
1.0932
1.1143
1.1346
1.1539
1.1725
1.1902
1.2071
1.3384
1.4191
1.4698
1.5029
1.5253
1.5411
1.5526
1.5611
1.5677
1.5919
1.5973
1.5993
1.6002
1.6015
1.6020
1.6020
1.6020
0.1730
0.2445
0.2989
0.3450
0.3852
0.4217
0.4550
0.4860
0.5150
0.5423
0.6608
0.7593
0.8448
0.9208
1.0528
1.1656
1.2644
1.3525
1.4320
1.5044
1.5708
2.0288
2.2889
2.4556
2.5704
2.6537
2.7165
2.7654
2.8044
2.8363
2.9857
3.0372
3.0632
3.0788
3.1102
3.1353
3.1385
3.1416
1.0030
1.0060
1.0090
1.0120
1.0149
1.0179
1.0209
1.0239
1.0268
1.0298
1.0445
1.0592
1.0737
1.0880
1.1164
1.1441
1.1713
1.1978
1.2236
1.2488
1.2732
1.4793
1.6227
1.7201
1.7870
1.8338
1.8674
1.8921
1.9106
1.9249
1.9781
1.9898
1.9942
1.9962
1.9990
2.0000
2.0000
2.0000

2-8
Table 2.2-2 Approximate solutions for diffusion and conduction (valid for Fo>0.2)

Fo =
2
AB
D t
L
=
2
0
AB
D t
r
,
*
=
,
, ,
'
'
A A
A i A
c K c
c K c

,
*
0
= C
1
exp(-
2
1
F
o
)
Diffusion in a slab
L is defined as the distance from the center of the slab to the surface. If one surface is insulated,
L is defined as the total thickness of the slab.

*
=
*
0
cos(
1
x
*
) ;
t
M
M

= 1
1
1
) sin(

*
0

Diffusion in an infinite cylinder
*
=
*
0
J
0
(
1
r
*
) ;
t
M
M

= 1
1
*
0
2

J
1
(
1
)
Diffusion in a sphere
*
=
*
1
1
r
*
0
sin(
1
r
*
) ;
t
M
M

= 1
3
1
*
0
3

[sin(
1
)
1
cos(
1
)]
If the concentration at the surface c
A,s
is known Kc
A,
will be replaced by c
A,s

1
and C
1
will be obtained from table at Bi
m
=

Notation:
c
A
= concentration of species A in the solid at any location at any time
c
A,s
= concentration of species A in the solid at the surface for t > 0
c
A,i
= concentration of species A in the solid at any location and at t = 0
c
A,
= bulk concentration of species A in the fluid surrounding the solid
Kc
A,
= c
A
*
= concentration of species A in the solid that is in equilibrium with c
A,

M
t
= amount of A transferred into the solid at any given time
M

= amount of A transferred into the solid as t (maximum amount transferred)

= erf
2
AB
x
D t
| |
|
|
\
; N
A0
= D
AB
0
A
x
c
x
=

=
,
( , )
AB A s A i
AB
D c c
D t

Constant Surface Flux: N
A
(x=0) = N
A0

c
A
(x, t) c
A,i
= 2N
A0
AB
t
D
2
exp
4
AB
x
D t
| |
|
\

0 A
AB
N x
D 2
AB
x
erfc
D t
| |
|
|
\

The complementary error function, erfc(w), is defined as erfc(w) = 1 erf(w)

Surface Convection: D
AB
0
A
x
c
x
=

= h
m
(c
Af
c
A,
)
,
, ,
'
A A i
A A i
c c
K c c

=
2
AB
x
erfc
D t
| |
|
|
\

2
exp
' '
m m
AB AB
h x h t
K D K D
( | |
| |
( + |
|
|
( \
\
' 2
m
AB AB
x h t
erfc
K D D t
( | |
+ ( |
|
(
\


Notation:
c
A
= concentration of species A in the solid at any location at any time
c
A,s
= concentration of species A in the solid at the surface for t > 0
c
A,i
= concentration of species A in the solid at any location and at t = 0
c
Af
= concentration of species A in the liquid at the solid-liquid interface at any time
c
A,
= bulk concentration of species A in the fluid surrounding the solid
Kc
A,
= c
A
*
= concentration of species A in the solid that is in equilibrium with c
A,

h
m
= k
c
= mass transfer coefficient
K = equilibrium distribution coefficient
D
AB
= diffusivity of A in the solid



L
L
c (r,x,t)
A

The concentration profiles for a finite cylinder and a parallelpiped
concentration profiles of infinite cylinder and slabs.
[ finite cylinder ] = [ infinite cylinder ] [ slab 2
[ parallelpiped ] = [ slab 2L
1
] [ slab 2

S(x, t)
,
Semi-infinite , ,
solid
( , ) '
'
A A
A i A
c x t K c
c K c

P(x, t)
,
Plane , ,
wall
( , ) '
'
A A
A i A
c x t K c
c K c

C(r, t)
,
Infinite , ,
cylinder
( , ) '
'
A A
A i A
c r t K c
c K c

2-10
Table 2.2-4 Multidimensional Effects

x
(r,x)
r
r
o
r
o
c (r,x,t)
The concentration profiles for a finite cylinder and a parallelpiped can be obtained from the
concentration profiles of infinite cylinder and slabs.
[ finite cylinder ] = [ infinite cylinder ] [ slab 2L ]
] [ slab 2L
2
] [ slab 2L
3
]
Semi-infinite


Infinite
cylinder

L
L

can be obtained from the