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4.2 Design of Adsorption and Ion-Exchange Processes 291

only for irreversible systems. Biot numbers have been also used by Ko et al . (2003) for the
irreversible adsorption process and by Yoshida et al . (1984) for irreversible adsorption and
ion-exchange systems (rectangular isotherms). Concluding, Bi and similar dimensionless
orable equilibrium,
numbers can be used for f while the criteria based on these numbers a v
are safer for irre ersible systems. v
Ion exchange The nature of the rate-determining step can be predicted by means of the
en belo w (Hellferich, v simple criterion gi 1962):

qD o s  2 
M  5  (4.106)
CD r o f o a  A B 

where:
C o the total counterion concentration in the liquid-phase
D f the liquid-phase dif icient f fusion coef
the film thickness
r o the particle radius
a A–B the separation f . actor
The criterion states that

• particle diffusion controls if M 1
• film diffusion controls if M 1

The use of a constant separation factor is a reasonable approximation if the actual f actor
ary signif does not vicantly. Film thickness for spherical particles is related to the Sh num-
ber via the relation

d p
Sh (4.107)

The film thickness is a fictitious quantity and cannot be measured directly. Its magnitude
is usually of the order 10 3 –10 2 cm, depending on agitation. This criterion is valid only
we
,
for counterions of equal mobility and infinite solution volume. Ho it is a useful
v
er
approximation despite its limited precision.
Generally, control by liquid-phase mass transfer is f v ored by a
• low liquid-phase concentration (small driving force in the liquid)
• high ion-exchange capacity (large driving force in the e xchanger)
• small particle size (short mass transfer distances in the bead)
• fusion in open structure of the exchanger, e.g. lo w cross-linking (little obstruction to dif
xchanger) the e
• ineffective agitation of the liquid (low contribution of convection to liquid-phase mass
transfer).