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4.2 Design of Adsorption and Ion-Exchange Processes 275

For the whole U ( t ) range, the approximation of Vermeulen can be used (Helf ferich,
1962):

Ut ( )  1 e ) T 2   0.5 (4.40)
xp(

The error of this simplified equation is less than 5% for U ( t ) 0.1 compared to the e xact
solution (eq. (4.35)).

Infinite fluid volume and liquid diffusion contr ol General case: For the case of the
Langmuir isotherm and the solid phase initially free of solute, the solution is (Perry and
Green, 1999)

 1  3 kt C
L
La 1  Ut () ln 1 Ut () f o (4.41)
1
a

q 

e r o pe q
Linear equilibrium case: v The aboe equation, in the case of linear equilibrium ( La 1),
is reduced to
 3 kC t 
Ut ()  f o  (4.42)
p
x
1 e
 rq oe p 
The same equation is used in the case of isotopic exchange (Helf 1962): ferich,

 3 DC t 
Ut ()  f o  (4.43)
1 e
x
p
 rQ M p 
o
where:
k f the fluid-film mass transfer coef f icient
D f the dificient in liquid phase fusion coef f
the particle density .
p
The only difference is that Helfferich uses D f / instead of k , where is the film thick-
f
ness. For a well-stirred solution, the film thickness is about 10 3 ferich, cm (Helf 1962).
Note that for ion e the parameter xchange, Q M is used in the place of q . e
Infinite solution v olume—intermediate case (between solid and liquid diffusion con-
trol) and linear equilibrium For the case of infinite solution volume and linear equilib-
rium, the follo1962): ferich, wing equation can be used (Helf

6 2 ∞ A sin 2 m r( o )
Ut () 2 ∑ n n exp ( Dm t s n ) (4.44)
r m 4
o n 1 n