CHEMICAL ENGINEERING

250 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

N^0 ADD^0

CT

CN 2 CH 2

 lm

ð

CCO 2

L

DD^0 ð

1

yN 2 CH 2

 lm

ð

CTyCO 2

L

yCO 2 D 0. 35

yN 2 CH 2

 lD 1. 0 ,yN 2 CH 2

2 D 0. 65

and: N^0 AD 2. 4 ð 10 ^5 ð

ln 1 / 0. 65



 1  0. 65

ð 0. 35

C

L

D 1. 033 ð 10 ^5 CT/L

∴The ratio of mass transfer ratesD 1. 033 ð 10 ^5 / 4. 6 ð 10 ^6 D 2. 25

PROBLEM 10.28

Given that from the penetration theory for mass transfer across an interface, the instan-
taneous rate of mass transfer is inversely proportional to the square root of the time of
exposure, obtain a relationship between exposure time in the Higbie model and surface
renewal rate in the Danckwerts model which will give the same average mass transfer
rate. The age distribution function and average mass transfer rate from the Danckwerts
theory must be derived from first principles.

Solution

Given that the instantaneous mass transfer rateDKt^1 /^2 , then for the Higbie model, the
average mass transfer rate for an exposure timeteis given by:

1

te

∫te

0

Kt^1 /^2 dtD 2 Kte^1 /^2

For the Danckwerts model, the random surface renewal analysis, presented in Section
10.5.2, shows that the fraction of the surface with an age betweentandtCdtis a function
oftDft
dtand that ft
DKestwheresis the rate of production of fresh surface per
unit total area.
For a total surface area of unity:
∫ 1

0

KestdtD 1 DK

[

est

s

] 1

0

DK/s

∴ KDsand ft
Dsestdt

The rate of mass transfer for unit area is:
∫ 1

0

Kt^1 /^2 ðsestdtDKs

∫ 1

0

t^1 /^2 estdt

Substitutingˇ^2 DstandsdtD 2 ˇdˇ, then:

RateDKs

∫ 1

0

p

s

ˇ

eˇ

22 ˇ

s

dˇDK

p

sð 2

∫ 1

0

eˇ

2

dˇDK

p

sð 2 ð

p

/ 2 DK

p

s