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HW/Tutorial Week #10WWWR 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
• Partial differential equations, complex processes and solutions
• Solutions for simple geometries and boundary conditions
0
AA
A Rt
cN
• Fick’s second law of diffusion
• 1-dimensional, no bulk contribution, no reaction
• Solution has 2 standard forms, by Laplace transforms or by separation of variables
2
2
z
cD
t
c AAB
A
• Transient diffusion in semi-infinite mediumuniform 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
AoA cc
– Thus, the P.D.E. becomes:
– with(z,0) = 0(0,t) = cAs – cAo
(,t) = 0
– Laplace transformation yields
which becomes an O.D.E.
2
2
zD
t AB
2
2
0dz
dDAB
02
2
ABD
s
dz
d
– Transformed boundary conditions:•
•
– General analytical solution:
– With the boundary conditions, reduces to
– The inverse Laplace transform is then
s
ccz AoAs )0(
0)( z
zDszDs ABAB eBeA /1
/1
zDsAoAs ABes
cc /)(
tD
zcc
AB
AoAs2
erfc)(
– As dimensionless concentration change,• With respect to initial concentration
• With respect to surface concentration
– The error function
is generally defined by
tD
z
tD
z
cc
cc
ABABAoAs
AoA
2erf1
2erfc
erf2
erf
tD
z
cc
cc
ABAoAs
AAs
tD
z
AB2
deerf 0
22
– The error is approximated by• If 0.5
• If 1
– For the diffusive flux into semi-infinite medium, differentiating with chain rule to the error function
and finally,
3
2erf
3
211erf
e
tD
cc
dz
dc
AB
AAsz
A
0
0
AoAsAB
zzA cct
DN 0,
• 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.
2
2
z
cD
t
c AAB
A
– Initial and boundary conditions of• cA = cAo at t = 0 for 0 z L
• cA = cAs at z = 0 for t > 0
• cA = cAs at z = L for t > 0
– Simplify by dimensionless concentration change
– Changing the P.D.E. to
Y = Yo at t = 0 for 0 z L
Y = 0 at z = 0 for t > 0
Y = 0 at z = L for t > 0
AsAo
AsA
cc
ccY
2
2
z
YD
t
YAB
– Assuming a product solution,
Y(z,t) = T(t) Z(z)– The partial derivatives will be
– Substitute into P.D.E.
divide by DAB, T, Z to
t
TZ
t
Y
2
2
2
2
z
ZT
z
Y
2
2
z
ZTD
t
TZ AB
2
211
z
Z
Zt
T
TDAB
– Separating the variables to equal -2, the general solutions are
– Thus, the product solution is:
– For n = 1, 2, 3…,
tDABeCtT2
1
zCzCzZ sincos 32
tDABezCzCY2
)sin()cos( '2
'1
L
n
– The complete solution is:
where L = sheet thickness and – If the sheet has uniform initial concentration,
for n = 1, 3, 5…– And the flux at z and t is
dzL
znYe
L
zn
Lcc
ccY
n
L
oXn
AsAo
AsA D
sinsin2
1 0
)2/( 2
21x
tDX ABD
1
)2/( 2
sin14
n
Xn
AsAo
AsA DeL
zn
ncc
cc
DXn
nAoAs
ABzA e
L
zncc
L
DN
2)2/(
1, cos
4
Example 1
Example 2
• Concentration-Time charts
Example 3
