CHEMICAL ENGINEERING

MASS TRANSFER 259

The surface not going fromtdt/ttot/tCdtDftdt
dtft
dt
Df^0 tdt
dtdt

This will be surface destroyed in time dt

Ddestruction rateðarea of surfaceðtime interval

Ds[ftdt

dt]dt

Thus:f^0 tdt

dtdtDs[ftdt

dt]dt

As dt!0 then: f^0 t

Csft

D 0

Using the Danckwerts model:sDconst.

estf^0 t

Csestft

D 0

estft

DK

ft

DKest

There is no upper age limit to surface

Thus total surfaceD 1 DK

∫ 1

0

estdtDK

[

est

s

] 1

0

DK/sor:KDs

ft

Dsest.

(b)sDatwhereais a constant.

Thus: eat

(^2) / 2
f^0 t
Cateat
(^2) / 2
ft
D 0
eat
(^2) / 2
ft
DK^0
and: ft
DK^0 eat
2 / 2
For unit total surface: 1DK^0

∫ 1

0

eat

2 / 2

dtDK^0

√

2

a

∫ 1

0

eat

2 / 2

d

(√

a

2

t

)

DK^0

√

2

a

p



2

and:K^0 D

√

2 a



and ft

D

√

2 a



eat

2 / 2

(c)regarding surface renewal as random

For unit total surface, mass transfer (mol/area time)D

∫ 1

0

kt^1 /^2 sest dtCA

DCAks

∫ 1

0

t^1 /^2 estdt.

whereCAis the concentration driving force in moles per unit volume

PuttingstDx^2 ,then:sdtD 2 xdx

The mass transfer rateDksCA

∫ 1

0

x^1

p

sex

2

2 xdx.

1

s

DCA 2 k

p

s

∫ 1

0

ex

2

dxDCA 2 k

p

s

p



2

DCAk

p

s