Fick's Second Law

SHR §3.3.2
Fick’s Second “Law”
Species mole
balance
equation:
@c i
@t = r · N i + S i
@c i
@t = r · (c iv M ) r· J i + S i
Assume:
• no bulk flow (vM = 0)
• no reaction (Si = 0)
• Constant molar concentration, c.
• Fick’s law (binary) with constant DAB
Fick’s Law
(binary):
J A =
cD AB rx A
c @x A
@t
= cD AB r · (rx A )
Gradient
(cartesian
coordinates)
Laplacian
(cartesian
coordinates)
r = @
@x ~ i + @ @y ~ j + @ @z ~ k
r · r = r 2 = @2
@x 2 + @2
@y 2 + @2
@z 2
@x A
@t
@c A
@t
= D AB r 2 x A equivalent
since we
= D AB r 2 assumed c
c A is constant
Fick’s Second Law
Really just some assumptions on top of
the governing equations & Fick’s Law.
Fick’s Second Law in
cartesian coordinates:
@x A
@t
= D AB
✓ @ 2 x A
@x 2
+ @2 x A
@y 2
+ @2 x A
@z 2
◆
21 MassTransfer.key - February 5, 2014

Example: Loschmidt Tube
@c i
@t = r · (c iv M ) r· J i + S i
@x A
@t
What
assumptions
= D AB
@ 2 x A
xA+
xA-
z=
z=0
@z 2 t=0 t→∞
How many intial & boundary conditions do
we need What form should the BCs take
z=-
"
(x A x A )
(x A+ x A ) = 1
2 + 1 ⇡
1X
k=0
1
⇣ m⇡z
⌘ ✓ m 2
m sin ⇡ 2 ◆ #
exp Dt
`
`2
1
0.8
time
m = k + 1 2
Reminder: solutions to PDEs are heavily
influenced by initial and boundary conditions!
Mole Fraction
0.6
0.4
0.2
time
0
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2
z
22 MassTransfer.key - February 5, 2014

SHR §3.3.3
Example: Ground Contamination
A benzene spill occurs on the ground. Determine how long
it will take for the contamination to diffuse 1 meter deep.
@x A
@t
= D AB
@ 2 x A
@z 2
c A = c Ao t =0
c A = c As t>0, z=0
c A =0 t>0, z!1
✓
c A c Ao
=erfc
c As c Ao
t =
apple
z2
4D AB
Assumptions:
• the spill is “large” in x and y
directions (relative to z).
• D AB =10 -8 cm 2 /s.
• Fick’s second law holds.
• Soil density is 2 g/cm 3 .
• Benzene density is 0.8765 g/cm 3
z
2 p D AB t
◆
c Ao
µg Benzene per kg soil
✓ ◆ 2
erfc 1 cA
= 4.2 years
c As c Ao
4
3
2
1
x 10 8
EPA suggests less than 0.3 μg
benzene (A) per kg soil (B).
c max
A
=3⇥ 10 7 gA/kgB ⇢ B
M A
=3⇥ 10 7 gA/kgB 2000kgB /m 3
78 gA /molA
=7.69 ⇥ 10 6 molA/m 3
Each curve is a different
time (in years!)
1
2
4
8
16
32
64
0
0 0.5 1 1.5 2
z (m)
Disclaimer: there is actual flow (not just
diffusion) in such a spill. Contamination
could occur much faster...
23 MassTransfer.key - February 5, 2014

Example: Steady-State Diffusion & Reaction
A + B ! C
S A = S B = S C = kc A c B
0
@c A
@t
@c B
@t
@c C
@t
= D A
@ 2 c A
@z 2
= D B
@ 2 c B
@z 2
= D C
@ 2 c C
@z 2
+ S A
+ S B
+ S C
cA,i-1
cA,i cA,i+1
Δz
http://www.che.utah.edu/~sutherland/page_id=867
24 MassTransfer.key - February 5, 2014

Length & Time Scales in Diffusion
Fick’s Second Law:
@x A
@t
= D AB
@ 2 x A
@z 2
Let’s nondimensionalize this equation:
nondimensional time: t ⇤ = t /⌧
nondimensional length: z ⇤ = z /`
nondimensional diffusivity: D ⇤ = D⌧ /`2
If we choose and τ
associated with diffusion,
then we expect D*=1,
⌧ = `2
D
` = p D⌧
Characteristic length and time
scales for diffusion!
25 MassTransfer.key - February 5, 2014