CHEMICAL ENGINEERING

268 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

PROBLEM 10.38

Based on the assumptions involved in the penetration theory of mass transfer across a
phase boundary, the concentrationCAof a soluteAat a depthybelow the interface at a
timetafter the formation of the interface is given by:

CA

CAi

Derfc

[

y

2

p

Dt

]

whereCAiis the interface concentration, assumed constant andDis the molecular diffu-
sivity of the solute in the solvent. The solvent initially contains no dissolved solute. Obtain
an expression for the molar rate of transfer ofAper unit area at timetand depthy,and
at the free surface (yD0).
In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continu-
ously fed to a continuous phase of lower density which is flowing vertically upwards, and
hence countercurrently with respect to the droplets. The resistance to mass transfer may
be regarded as lying wholly within the drops and the penetration theory may be applied.
The upward velocity of the liquid, which may be taken as uniform over the cross-section
of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid.
Occasionally, two droplets coalesce forming a single drop of twice the volume. What
is the ratio of the mass transfer rate (kmol/s) at a coalesced drop to that at a single droplet
when each has fallen the same distance, that is to the bottom of the column?
The fluid resistance force acting on the droplet should be taken as that given by Stokes’
law, that is 3duwhereis the viscosity of the continuous phase,dthe drop diameter
anduits velocity relative to the continuous phase.
It may be noted that:

erfcx

D

2

p



∫ 1

x

ex

2

dx.

Solution

CA

CAi

Derfc

y

2

p

Dt

D

2

p



∫ 1

y/ 2

p

Dt

ey

(^2) / 4 Dt
d

(

y

2

p

Dt

)

Differentiating with respect toyat constanttgives:

1

CAi

∂CA

∂y

D

2

p



∂

∂y

{

1

2

p

Dt

∫ 1

y/ 2

p

Dt

ey

(^2) / 4 Dt
dt

}

D

1

p

Dt

ey

(^2) / 4 Dt
Thus:NA (^) tDD

∂CA

∂y

DCAiD

√

1

Dt

ey

(^2) / 4 Dt
DCAi

√

D

t

ey

(^2) / 4 Dt
At the interface, whenyD0:NADCAi

√

D

t

Dbt^1 /^2