HW/Tutorial Week #10
WWWR Chapters 27, ID Chapter 14
• Tutorial #10
• WWWR # 27.6 &
27.22
• To be discussed on
March 31, 2015.
• By either volunteer or
class list.
Unsteady-State Diffusion
• Transient diffusion, when concentration at a
given point changes with time
c A
NA 
 RA  0
t
• Partial differential equations, complex
processes and solutions
• Solutions for simple geometries and
boundary conditions
• Fick’s second law of diffusion
c A
 2c A
 DAB
2
t
z
• 1-dimensional, no bulk contribution, no
reaction
• Solution has 2 standard forms, by Laplace
transforms or by separation of variables
• Transient diffusion in semi-infinite medium
–
–
–
–
uniform initial concentration CAo
constant surface concentration CAs
Initial condition, t = 0, CA(z,0) = CAo for all z
First boundary condition:
at z = 0, cA(0,t) = CAs
for t > 0
Second boundary condition:
at z = , cA(,t) = CAo
for all t
Using Laplace transform, making the boundary
conditions homogeneous
  cA  cAo
– Thus, the P.D.E. becomes:
– with

 2
 DAB 2
t
z
• (z,0) = 0
• (0,t) = cAs – cAo
• (,t) = 0
– Laplace transformation yields
  0  DAB
d 2
dz2
which becomes an O.D.E.
d 2
s

 0
2
dz
DAB
– Transformed boundary conditions:
•  ( z  0) 
c As  c Ao
s
•  ( z  ) 0
– General analytical solution:
 s / DAB z
 s / DAB z
  A1e
 B1e
– With the boundary conditions, reduces to
(c As  c Ao )  s / DAB z
 
e
s
– The inverse Laplace transform is then


z

  (c As  c Ao ) erfc
2 D t 
AB 

– As dimensionless concentration change,
• With respect to initial concentration




c A  c Ao
z
z




 erfc
 1  erf




c As  c Ao
 2 DABt 
 2 DABt 
• With respect to surface concentration


c As  c A
z
  erf  
 erf 


c As  c Ao
2
D
t
AB


– The error function  
z
2 DABt
is generally defined by
erf   
2



0
e
 2
d
– The error is approximated by
• If   0.5
2 
3 
  
erf   
3

• If   1
erf    1 
1

e
 2
– For the diffusive flux into semi-infinite
medium, differentiating with chain rule to the
error function
cAs  cA0 
dcA
dz
z 0

DABt
and finally,
N A, z z  0 
DAB
cAs  cAo 
t
• Transient diffusion in a finite medium, with
negligible surface resistance
– Initial concentration cAo subjected to sudden
change which brings the surface concentration
cAs
– For example, diffusion of molecules through a
solid slab of uniform thickness
– As diffusion is slow, the concentration profile
satisfy the P.D.E.
c A
 2c A
 DAB
t
z 2
– Initial and boundary conditions of
• cA = cAo
• cA = cAs
• cA = cAs
at t = 0
at z = 0
at z = L
for 0  z  L
for t > 0
for t > 0
– Simplify by dimensionless concentration
change
c A  c As
Y
c Ao  c As
– Changing the P.D.E. to
Y
 2Y
 DAB 2
t
z
Y = Yo
Y=0
Y=0
at t = 0
at z = 0
at z = L
for 0  z  L
for t > 0
for t > 0
– Assuming a product solution,
Y(z,t) = T(t) Z(z)
– The partial derivatives will be
Y
T
Z
t
t
 2Y
2Z
T 2
2
z
z
– Substitute into P.D.E.
T
2Z
Z
 DABT 2
t
z
divide by DAB, T, Z to
T 1  2 Z

DABT t Z z 2
1
– Separating the variables to equal -2, the
general solutions are
T t   C1e
 DAB2t
Z z   C2 cosz   C3 sinz 
– Thus, the product solution is:


Y  C cos(z)  C sin(z) e
'
1
'
2
– For n = 1, 2, 3…,
n

L
 DAB2t
– The complete solution is:
c A  c As 2 
 nz  ( n / 2)2 X D
 nz 
Y
  sin
Yo sin
e
dz

c Ao  c As L n1  L 
 L 
0
L
DABt
XD  2
x1
where L = sheet thickness and
– If the sheet has uniform initial concentration,
c A  c As 4  1  nz  ( n / 2)2 X D
  sin
e
c Ao  c As  n1 n  L 
for n = 1, 3, 5…
– And the flux at z and t is

4 DAB
 nz  ( n / 2)2 X D
cAs  cAo  cos e
N A, z 
L
 L 
n 1
Example 1
Example 2
• Concentration-Time charts
Example 3