CHEMICAL ENGINEERING

258 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

The total surface is: 1DK

∫ 1

0

ekt

(^2) / 2
dtDK

√

2

k

∫ 1

0

ekt

(^2) / 2
d
p
k/ 2
t
Putting

√

k

2

tDX:

1 DK

√

2

k

∫ 1

0

eX

2

dXDK

√

2

k

p



2

DK

√



2 k

,

then: KD

√

2 k



and the age distribution function is:

√

2 k



ekt

(^2) / 2
PROBLEM 10.33
Explain the basis of thepenetration theoryfor mass transfer across a phase boundary.
What are the assumptions in the theory which lead to the result that the mass transfer rate
is inversely proportional to the square root of the time for which a surface element has
been expressed? (Donotpresent a solution of the differential equation.) Obtain the age
distribution function for the surface:
(a) On the basis of the Danckwerts’ assumption that the probability of surface renewal
is independent of its age.
(b) On the assumption that the probability of surface renewal increases linearly with
the age of the surface.
Using the Danckwerts surface renewal model, estimate:
(c) At what age of a surface element is the mass transfer rate equal to the mean value
for the whole surface for a surface renewal rate (s)of0.01 m^2 /m^2 s?
(d) For what proportion of the total mass transfer is surface of an age exceeding
10 seconds responsible?
Solution
(a)Danckwerts age distribution function
Dividing the total unit surface into elements each of duration dt, then:
dt
t − dt tt + dt t +2dt
dt dt
If the fraction of surface in age bandttotCdtis ft
dt, then:
the fraction of surface in age bandtdttotwill be ftdt
dt.