CP302 Separation Process Principles
Mass Transfer - Set 3
Prof. R. Shanthini
05 March 2013
1
One-dimensional Unsteady-state Diffusion
Fick’s First Law of Diffusion is written as follows when CA is only a
function of z:
JA = - DAB
dCA
dz
(1)
Fick’s First Law is written as follows when CA is a function of z as
well as of some other variables such as time:
JA = - DAB
Prof. R. Shanthini
05 March 2013
∂CA
∂z
Observe the use of ordinary and
partial derivatives as appropriate.
(39)
2
One-dimensional Unsteady-state Diffusion
JA, out
JA, in
ΔCA
z
JA, in = - DAB
z+Δz
A: cross-sectional area
MA: molecular weight of
species A
Prof. R. Shanthini
05 March 2013
Mass flow of species A into the
control volume
= JA, in x A x MA
where
∂CA
∂z
at z
Mass flow of species A out of the
control volume
= JA, out x A x MA
where
JA, out = - DAB
∂CA
∂z
at z+Δz
3
One-dimensional Unsteady-state Diffusion
Accumulation of species A in the
control volume
JA, out
JA, in
ΔCA
z
z+Δz
A: cross-sectional area
MA: molecular weight of
species A
Prof. R. Shanthini
05 March 2013
∂CA
=
x (A x Δz) x MA
∂t
Mass balance for species A in the
control volume gives,
JA, in A MA = JA, out A MA
∂CA
(A Δz) MA
+
∂t
4
One-dimensional Unsteady-state Diffusion
Mass balance can be simplified to
- DAB
∂CA
∂z
= - DAB
at z
∂CA
∂z
∂CA
+
Δz
∂t
at z+Δz
The above can be rearranged to give
∂CA
= DAB
∂t
(∂CA/∂z)z+Δz - (∂CA/∂z)z
Δz
(40)
Prof. R. Shanthini
05 March 2013
5
One-dimensional Unsteady-state Diffusion
In the limit as Δz goes to 0, equation (40) is reduced to
∂CA
∂t
= DAB
∂2CA
∂z2
(41)
which is known as the Fick’s Second Law.
Fick’s second law in the above form is
applicable strictly for constant DAB and for
diffusion in solids, and also in stagnant liquids
and gases when the medium is dilute in A.
Prof. R. Shanthini
05 March 2013
6
One-dimensional Unsteady-state Diffusion
Fick’s second law, applies to one-dimensional unsteady-state
diffusion, is given below:
∂CA
= DAB
∂2CA
(42)
∂t
Fick’s second law for one-dimensional diffusion in radial
direction only for cylindrical coordinates:
∂CA
∂t
=
DAB
r
∂z2
∂
∂r
∂CA
r
∂r
(43)
Fick’s second law for one-dimensional diffusion in radial
direction only for spherical coordinates:
∂CA
∂t
Prof. R. Shanthini
05 March 2013
=
DAB
r2
∂
∂r
r2
∂CA
∂r
(44)
7
Three-dimensional Unsteady-state Diffusion
Fick’s second law, applies to three-dimensional unsteady-state
diffusion, is given below:
∂CA
= DAB
∂2CA
+
∂2CA
+
∂2CA
(45a)
∂t
∂x2
∂y2
∂z2
Fick’s second law for three-dimensional diffusion in cylindrical
coordinates:
∂CA
DAB
= r
∂t
∂
∂CA
∂ ∂CA
∂
∂CA
r
r
+
+
∂r
∂r
∂θ r ∂θ
∂z
∂z
(45b)
Fick’s second law for three-dimensional diffusion in spherical
coordinates:
∂CA
DAB
=
∂t
r2
Prof. R. Shanthini
05 March 2013
∂ 2 ∂CA
1
∂
∂CA
1
∂2 CA
r
sinθ
+
+ 2
∂r
∂r
sinθ ∂θ
∂θ
sin θ ∂2Φ
8
(45c)
Unsteady-state diffusion in semi-infinite medium
∂CA
∂t
∂2CA
∂z2
= DAB
(42)
z≥0
z=0
z =∞
Initial condition:
CA = CA0
at t ≤ 0 and z ≥ 0
Boundary condition:
Prof. R. Shanthini
05 March 2013
CA = CAS
at t ≥ 0 and z = 0
CA = CA0
at t ≥ 0 and z → ∞
9
Introducing dimensionless concentration change:
Use Y =
CA – CA0
CAS – CA0
and transform equation (42) to the following:
∂Y
∂t
= DAB
∂2 Y
∂z2
where
∂Y
∂t
∂2Y
∂z2
Prof. R. Shanthini
05 March 2013
=
=
∂CA / ∂t
CAS – CA0
∂2CA/ ∂z2
CAS – CA0
10
Introducing dimensionless concentration change:
Use Y =
CA – CA0
CAS – CA0
and transform the initial and boundary conditions to the
following:
Initial condition:
CA = CA0 becomes Y = 0
at t ≤ 0 and z ≥ 0
Boundary condition:
CA = CAS becomes Y = 1
at t ≥ 0 and z = 0
CA = CA0 becomes Y = 0
at t ≥ 0 and z → ∞
Prof. R. Shanthini
05 March 2013
11
Solving for Y as a function of z and t:
∂Y
∂t
= DAB
∂ 2Y
∂z2
Initial condition:
Y=0
at t ≤ 0 and z ≥ 0
Boundary condition:
Y=1
at t ≥ 0 and z = 0
Y=0
at t ≥ 0 and z → ∞
Since the PDE, its initial condition and boundary conditions
are all linear in the dependent variable Y, an exact solution
exists.
Prof. R. Shanthini
05 March 2013
12
Non-dimensional concentration change (Y) is given by:
Y=
CA – CA0
CAS – CA0
= erfc
z
(46)
2  DAB t
where the complimentary error function, erfc, is related to the
error function, erf, by
x
erfc(x) = 1 – erf(x)
=1–
⌠ exp(-σ ) dσ
⌡
2
π
2
0
Prof. R. Shanthini
05 March 2013
(47)
13
A little bit about error function:
- Error function table is provided (take a look).
- Table shows the error function values for x values up to 3.29.
- For x > 3.23, error function is unity up to five decimal places.
- For x > 4, the following approximation could be used:
exp(-x2)
erf(x) = 1 –
πx
Prof. R. Shanthini
05 March 2013
14
Example 3.11 of Ref. 2: Determine how long it will take for
the dimensionless concentration change (Y) to reach 0.01 at
a depth 1 m in a semi-infinite medium. Assume DAB = 0.1
cm2/s.
Solution:
Starting from (46) and (47), we get Y = 1 - erf
z
2  DAB t
Using Y = 0.01, z = 1 m (= 100 cm) and DAB = 0.1 cm2/s, we get
erf
100
2  0.1 x t
100
2  0.1 x t
Prof. R. Shanthini
05 March 2013
= 1 - 0.01 = 0.99
= 1.8214
t = 2.09 h
15
Get back to (46), and determine the equation for the mass
flux from it.
JA = - DAB ∂CA
∂z
at z
JA =  DAB / π t exp(-z2/4DABt) (CAS-CA0)
(48)
Flux across the interface at z = 0 is
JA
at z = 0
Prof. R. Shanthini
05 March 2013
=  DAB / π t
(CAS-CA0)
(49)
16
Exercise: Determine how the dimensionless concentration
change (Y) profile changes with time in a semi-infinite
medium. Assume DAB = 0.1 cm2/s. Work up to 1 m depth of
the medium.
Solution:
Starting from (46) and (47), we get
Y = 1 - erf
Prof. R. Shanthini
05 March 2013
z
2  DAB t
17
Let us get the
complete profile
using MATLAB
which has a builtin error function.
Prof. R. Shanthini
05 March 2013
clear all
DAB = 0.1;
%cm2/s
t = 0;
for i = 1:1:180
%in min
t(i) = i*60;
z = [0:1:100];
%cm
x = z/(2 * sqrt(DAB*t(i)));
Y(:,i) = 1 - erf(x);
end
plot(z,Y)
xlabel('z (cm)')
ylabel('Non-dimensional concentration, Y')
grid
pause
plot(t/3600,Y(100,:))
xlabel('t (h)')
ylabel('Y at z = 100 cm')
grid
18
t = 1 min to 3 h
Prof. R. Shanthini
05 March 2013
19
Prof. R. Shanthini
05 March 2013
20
Diffusion in semi-infinite medium:
In gas:
DAB = 0.1 cm2/s
Time taken for the dimensionless concentration
change (Y) to reach 0.01 at a depth 1 m is
2.09 h.
In liquid:
DAB = 10-5 cm2/s
Time taken for the dimensionless concentration
change (Y) to reach 0.01 at a depth 1 m is
2.39 year.
In solid:
DAB = 10-9 cm2/s
Time taken for the dimensionless concentration
change (Y) to reach 0.01 at a depth 1 m is
Prof. R. Shanthini
05 March 2013
239 centuries.
21