CHEMICAL ENGINEERING

234 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

Considering the diffusion of soluteAaway from the interface in they-direction this
equation becomes:
∂CA
∂t

DD

∂^2 CA

∂y^2

The boundary conditions are:

tD 00 <y< 1 CADCAo

t> 0 yD 0 CADCAi

t> 0 yD1 CADCAo

whereCAois the concentration in the bulk of the phase andCAiis the equilibrium
concentration at the interface.
The instantaneous rate of mass transfer per unit areaNAat timetis given by:

NA (^) tDCA

√

D/t

Higbie assumed that every element of surface is exposed to the gas for the same length
of time)before being replaced by liquid of the bulk composition.
Amount absorbed in time):

QD

∫)

0

NA (^) td)D

∫)

0

CA

√

D/)d)D 2 CA

√

D)/

The average rate of absorption:Q/)D

(

2 CA

p

D)/

)

/)D 2 CA

p
D/)
Danckwerts suggested that each element would not be exposed for the same time but
that a random distribution of ages would exist. It is shown in Section 10.5.2 that this age
distribution may be expressed ft
Dsest. The average rate of absorption is the value of

NA (^) taveraged over all elements of the surface having ages between 0 and 1 is then
given by:
NADs

∫ 1

0

NA (^) tes)d)DCA

√

D/

∫ 1

0

es)/

p

)

d)DCA

p

Ds

PROBLEM 10.17

A solute diffuses from a liquid surface at which its molar concentration isCAiinto a
liquid with which it reacts. The mass transfer rate is given by Fick’s law and the reaction
is first order with respect to the solute. In a steady-state process, the diffusion rate falls at
adepthLto one half the value at the interface. Obtain an expression for the concentration
CAof solute at a depthyfrom the surface in terms of the molecular diffusivityDand
the reaction rate constantk. What is the molar flux at the surface?

Solution

As in Problem 10.13, the basic equation is: d^2 CA/dy^2 Da^2 CA (i)
whereaD

p

k/D

Then: CADAcoshayCBsinhay ii