Prof. R. Shanthini 05 March 2013 1 CP302 Separation Process Principles Mass Transfer - Set 3

Prof. R. Shanthini 05 March 2013

1

CP302 Separation Process PrinciplesMass Transfer - Set 3


2

One-dimensional Unsteady-state Diffusion

Fick’s First Law of Diffusion is written as follows when CA is only a function of z:

(1)JA = - DABdCA

dz

JA = - DAB∂CA

∂z(39)

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:

Observe the use of ordinary and partial derivatives as appropriate.


3

JA, inJA, out

ΔCA

z+Δz

JA, in = - DAB∂CA

∂z at z

JA, out = - DAB∂CA

∂z at z+Δz

z

Mass flow of species A into the control volume

= JA, in x A x MA

where

Mass flow of species A out of the control volume

= JA, out x A x MA

where

A: cross-sectional area

MA: molecular weight of species A



4

JA, inJA, out

ΔCA

z+Δzz

A: cross-sectional area

MA: molecular weight of species A

= ∂CA

∂t

Accumulation of species A in the control volume

x (A x Δz) x MA

Mass balance for species A in the control volume gives,

JA, in= JA, out

A MA A MA

+∂CA

∂t (A Δz) MA



5

Mass balance can be simplified to

- DAB∂CA

∂z at z

- DAB∂CA

∂z at z+Δz

= ∂CA

∂t+ Δz

∂CA

∂t= DAB

(∂CA/∂z)z+Δz (∂CA/∂z)z-

Δz

(40)

The above can be rearranged to give



6

In the limit as Δz goes to 0, equation (40) is reduced to

= ∂2CA

∂z2DAB (41)

∂CA

∂t

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.



7

= ∂2CA

∂z2DAB (42)

Fick’s second law, applies to one-dimensional unsteady-state diffusion, is given below:

∂CA

∂t

= ∂

∂r

DAB(43)

Fick’s second law for one-dimensional diffusion in radial direction only for cylindrical coordinates:

∂CA

∂t r

∂CA

∂rr

= ∂

∂r

DAB(44)

Fick’s second law for one-dimensional diffusion in radial direction only for spherical coordinates:

∂CA

∂t r2

∂CA

∂rr2



8

= ∂2CA

∂x2DAB

Fick’s second law, applies to three-dimensional unsteady-state diffusion, is given below:

∂CA

∂t+ +

∂2CA

∂z2

∂2CA

∂y2

= ∂

∂r

DAB

Fick’s second law for three-dimensional diffusion in cylindrical coordinates:

∂CA

∂t r∂CA

∂rr

= ∂

∂r

DAB

Fick’s second law for three-dimensional diffusion in spherical coordinates:

∂CA

∂t r2

∂CA

∂rr2

+ ∂

∂θ

∂CA

r ∂θ+

∂

∂z

∂CA

∂zr (45b)

(45a)

∂

∂θ

∂CA

∂θsinθ+

1

sinθ

∂2

∂2Φ+

1

sin2θ

CA

(45c)

Three-dimensional Unsteady-state Diffusion


9

Unsteady-state diffusion in semi-infinite medium

z = 0 z = ∞

= ∂2CA

∂z2DAB (42)

∂CA

∂t

z ≥ 0

Initial condition:

CA = CA0 at t ≤ 0 and z ≥ 0

Boundary condition:

CA = CAS at t ≥ 0 and z = 0

CA = CA0 at t ≥ 0 and z →∞


10

Introducing dimensionless concentration change:

= ∂2Y

∂z2DAB

∂Y

∂t

Y = CA – CA0

CAS – CA0

Use

and transform equation (42) to the following:

where ∂Y

∂t=

∂CA / ∂t

CAS – CA0

∂2Y

∂z2=

∂2CA / ∂z2

CAS – CA0


11

Introducing dimensionless concentration change:

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 → ∞

Y = CA – CA0

CAS – CA0

Use

and transform the initial and boundary conditions to the following:


12

Solving for Y as a function of z and t:

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 →∞

= ∂2Y

∂z2DAB

∂Y

∂t

Since the PDE, its initial condition and boundary conditions are all linear in the dependent variable Y, an exact solution exists.


13

Non-dimensional concentration change (Y) is given by:

Y = CA – CA0

CAS – CA0

= erfc z

2 DAB t(46)

where the complimentary error function, erfc, is related to the error function, erf, by

erfc(x) = 1 – erf(x) = 1 – 2π

⌠⌡ 0

x

exp(-σ2) dσ

(47)


14

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:

erf(x) = 1 – π x

exp(-x2)


15

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

= 1 - 0.01 = 0.99 erf 100

2 0.1 x t

100

2 0.1 x t= 1.8214

t = 2.09 h


16

Get back to (46), and determine the equation for the mass flux from it.

JA = - DAB∂CA

∂z at z

DAB / π tJA = exp(-z2/4DABt) (CAS-CA0) (48)

DAB / π tJA = (CAS-CA0)

Flux across the interface at z = 0 is

at z = 0(49)


17

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 z

2 DAB t


18

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

Let us get the complete profile using MATLAB which has a built-in error function.


19

t = 1 min to 3 h


20


21

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


2.39 year.

In solid: DAB = 10-9 cm2/s


239 centuries.

Documents

Prof. R. Shanthini 05 March 2013 1 CP302 Separation Process Principles Mass Transfer - Set 3