Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-1 2. Diffusion in Dilute Solutions 2.1 Diffusion across thin films and membranes 2.2 Diffusion into a semi-infinite slab (strength of weld, tooth decay) 2.3 Examples 2.4 Dilute diffusion and convection Graham (1850) monitored the diffusion of salt (NaCl) solutions in a larger jar containing water. Every so often he removed the bottle and analyzed it. Mass Transfer
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Mass Transfer 2. Diffusion in Dilute Solutions Transfer –Diffusion in Dilute Solutions_ Fick‘sLaws 2-1 2. Diffusion in Dilute Solutions 2.1 Diffusion across thin films and membranes
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Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-1
2. Diffusion in Dilute Solutions
2.1 Diffusion across thin films and membranes
2.2 Diffusion into a semi-infinite slab (strength of weld, tooth decay)
2.3 Examples
2.4 Dilute diffusion and convection
Graham (1850) monitored the diffusion
of salt (NaCl)
solutions in a larger jar containing
water. Every so often he removed the
bottle and analyzed it.
Mass Transfer
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-2
Initial salt
concentration,
Weight-% of NaCl
Relative Flux
1 1.00
2 1.99
3 3.01
4 4.00
He postulated that
a) The quantities diffused appear to be proportional to the salt
concentration.
b) Diffusion must follow diminishing progression.
Fick (1855) analyzed these data and wrote
“The diffusion of the dissolved material ... is left completely to the influence
of the molecular forces basic to the same law ... for the spreading of
warmth in a conductor and which has already been applied with such great
success to the spreading of electricity.”
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-3
Fick’s first law:dc
j Ddz
This is analogous to Newton’s lawdy
dvxyx
This is analogous to Fourier’s lawdx
dTqx
Tq cDj or
These equations imply no convection (dilute solutions !).
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-4
2.1 Diffusion across thin films and membranes
Example 2.1.1: Diffusion across a thin film
Dz
z z 0
1C
C10
Goal: concentration profile in the
film, and the flux across it at steady
state.
Mass balance across arbitrary thin layer Dz:
D
zz at
layer of out
diffusion of rate
z at layer the into
diffusion of rate
onaccumulati
solute
0
Steady state
)jj(A0 zz1z1 D
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-5
)jj(A0 zz1z1 D
Divide this equation by the film volume A‧Dz
D
D
z)zz(
jj0
z1zz1
21
2
1dz
cdD0j
dz
d0 0z DAs
Fick’s
first law(1)
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-6
If we solve this equation we have the concentration profile of c in
and then we can calculate the flux
from Fick’s first law 11
dcj D
dz (2)
by estimating the dz
dc10z zorat
The boundary conditions are 0z
z
10cc
1cc
Then the solution to eq.1 is bzac1
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-7
and using the boundary condition gives:
z)cc(cc 101101
1011010z1
1 ccD1
cc0Ddz
dcDj
101101z1 cc
D1cc0D
dz
dcD
or
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-8
Example 2.1.2: Membrane diffusion
Derive the concentration profile and the flux for a single
solute diffusing across a thin membrane.
The analysis is the same as before leading to
21
2
zz1z1dz
cdDjjA0 D
but the boundary conditions differ:
11
101
Hcc,z
Hcc,0z
where H is a partition coefficient (the concentration in the
membrane divided by that in the adjacent solution e.g. Henry’s or
Raoult’s law).
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-9
Then the concentration profile becomes:
z
ccHHcc 101101
10c
1c
The solute is more soluble in
the membrane than in the
adjacent solution
10c
1c
The solute is less soluble in
the membrane than in the
adjacent solution
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-10
Example 2.1.3: Concentration–dependent diffusion coefficient
The diffusion coefficient D can vary with concentration c.
(water across films and in detergent solutions)
Assumption:
cc1
ccc 1
ccc 1
slow diffusion (small D), DS
fast diffusion (large D), D
10c
sD
1CcD
1c
l-ZcZc
Consider two-films in series.
At steady state j1 =const in
both films.
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-11
In film 1: Large sD small dz
dc
c cz c
css dcDdzj
dz
dcDj
011
11
1
10
)cc(z
Dj c
c
s1101 (1)
In film 2: Smalldc
large Ddz
c cz
c
c
dcDdzjdz
dcDj
1
1
111
1 )cc(z
Dj 1c1
c1
(2)
)cc(D)cc(D
D)cc(z
1c1c110s
sc110c
)cc(D
)cc(D1
z
c110s
1c1c
(1) = (2)
The flux becomes then:
)cc(D)cc(Dj ccs 111101
2
1101
ccc c
2
DDD s then If
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-12
In the following film two compounds A and B diffuse from 1 to 2
through the film Dz.
Which one diffuses faster or which one has the largest
Diffusivity?
1
2
BD
ADAc1
Bc1Ac2
Bc 2zD
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-13
A compound diffuses through two films in series. When it
diffuses faster in film A than in film B, which concentration profile
best describes this process, 1,2 or 3 and why?
1
2
3
BA
1c
1c
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-14
2.2 Diffusion in a Semi-infinite Slab
Fick’s Second Law
Diffusion is the net migration (mass transfer-transport) of
molecules from regions of HIGH to LOW concentration.
jX: flux of particles in the x-direction
A
B
C
D
dx
dy
dz
j jx
xx
x
d
2 jjx
xx
x
d
2
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-15
Rate at which particles enter the
elemental volume dxdydz across the
left side of that volume
zy
2
x
x
jj XX dd
d
IN
IN - OUT =
zy2
x
x
jj xx dd
d
gradient of jx at the
center of dxdydz
Net rate of transport
into that element
x
jzyx x
ddd
A
B
C
D
dx
dy
dz
jj
x
xx
x
d
2j
j
x
xx
x
d
2
Similarly for the dxdz face:y
jzyx
y
ddd
z
jzyx z
dddand for dxdy face:
OUT
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-16
The rate of change of the number of particles per unit volume
(& size), n, in the elemental volume dxdydz is:
d d d
d d d c x y z
t
j
x
j
y
j
z
x y z x y z
jz
j
y
j
x
j
t
c zyx
From experimental observations:x
cDjx
(Fick’s first law without convection, dilute solutions).
Substituting it in the above gives Fick’s second law:
cDz
c
y
c
x
cD
t
c 2
2
2
2
2
2
2
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-17
Example 2.2.1:
Unsteady diffusion in a semi-infinite slab
Consider that suddenly the
concentration at the interface
changes.
Goal: To find how the
concentration and flux
varies with time.
Very important in diffusion in solids (tooth decay, corrosion of
metals). This is the opposite to diffusion through films. Everything
else in the course is in between.
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-18
At t ≤ 0: 11 cc but at t > 0: 101 cc
Mass balance:
D
Dzz at
layer the of out
diffusion of rate
z at
layer the into
diffusion of rate
z Avolume in
onaccumulati solute
)jj(A)czA(t
zzz DD
111
Divide by ADz:
D
D
z
jj
t
c zzz 111 0Dzz
j
t
c
11
Combine this with Fick’s first law gives:
21
21
z
cD
t
c
(1)
dz
dcDj
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-19
21
21
z
cD
t
c
(1)
Boundary Conditions: t = 0 all z: 11 cc
101 cc
11 cc
t > 0 z=0:
z=:How to solve Fick’s 2nd law?
Define a new variable (Boltzmann): Dt
z
4 (2)
(It requires the wild imagination of
mathematicians)
So eqn. (1) becomes:2
21
21
zd
cdD
td
dc
or using eqn. 2: 02 121
2
d
dc
d
cd (3)
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-20
The B.C. become: 1010 cc
11 cc
Set yd
dc
1 so eqn.(3) becomes: 02 y
d
dy
22 alnylndy
dyor:
integrate
)exp(ay 2
Resubstitution: )exp(ad
dc 21
0
21 kds)sexp(ac (4)
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-21
at 0
0
2100 kds)sexp(ac
kc 10
2
1
0
2101 ds)sexp(accat
so2
101
/
cca
in (4):
0
2101101 ds)sexp(
2/
)cc(cc
0
2
101
101 2erfds)sexp(
cc
cc
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-22
So the flux can be obtained as :
2z-
1 4Dt1 10 1
cj -D D / t e (c - c )
z
and the flux across the interface becomes (z=0) :
)cc(t
Dj z 11001
This is the flux at time t.
Total flux at time t
t
z dtj0
01
Mass Transfer – Diffusion in Dilute Solutions_ Fick‘s Laws 2-23