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4.2 Design of Adsorption and Ion-Exchange Processes 279

with La being the Langmuir constant and La is calculated using the same definition of
La , eq. (4.8), but with the equilibrium fluid phase C e instead of the initial fluid concentra-
tion C . Note that for ion e the parameter xchange, Q M is used in the place of q . e
o
olume,
Linear equilibrium case : For isotopic exchange and finite solution v Helf ferich
(1962) gives the follo wing solution:
 3 DM q V C t ) 
(
Ut ()  f s o o o 
p
1 e
x

 rQ V M o p  (4.63)
o
where is the particle density and as in all cases of ion and isotopic e , xchange, Q M is the
p
REC expressed per unit mass of the solid phase. Note that for adsorption, the parameter q e
is used in the place of Q M .
Finite solution volume-intermediate case (between solid and liquid diffusion contr ol)
and linear equilibrium The solution is the following (Perry and Green, 1999):
 
exp  pD 2 t 2 
 n s r 
Ut () ∑ o 2 4 (4.64)
1 6
n 1 w 9 (1 wp ) 2 w (5 1) (1 p n w ) p n
w
1 w n Bi Bi 2
m m
where p n v are the positie roots of the equation

1 w p 2
3 n
tan( ) p n w Bi m
(4.65)
p 1 ( w Bi 1) p 2
n m n
3+
w Bi
m
where Bi m is a modified Biot number defined as

kr fo
Bi (4.66)
DK
m
p s linear
Finally, K linear is the equilibrium parameter in the expression of the linear equilibrium:

q e K linear C e (4.67)

xc Ion ehange

ar
, So f the analysis has been restricted to adsorption and exchange of isotopes (isotopic
v
we,
exchange). Ho in most cases, the ion-exchange process ines the exchange of olv v
er