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4.2 Design of Adsorption and Ion-Exchange Processes 283

of applications of each model, examining the effect of various parameters in y v The olv ed.
showed that any of the tested models could be applied provided that the values of w , and ,
U ( t ) were within specific limits. Specif Vs approximation can be used in ically , ermeulen’
any case of an ion-exchange process within certain limits for U ( t ), the lo wer limit depend-
ing on w and the upper limit depending on s approximation can be used for atterson’ . P
any value of w and up to a v alue of U ( t ) depending on . In both cases, must ho we v er be
less than 20.
Inglezakis and Grigoropoulou (2001) experimentally found that for the case of
Pb 2 –Na exchange on clinoptilolite, een for v w = 0.4, the results using V s ermeulen’
approximation could be acceptable.
Exchange of trace components : The equations for adsorption (diffusion) can be equally
applied in the case of isotopic exchange (exchange of isotopes) with minor changes. The
same equations can be also be used in the case of the exchange of trace components of dif-
ferent valences (Helf 1962). This is the case where the uptake or release of an ion
ferich,
takes place in the presence of a large amount of another ion in both the solid and liquid
phase. In such systems, the amounts remoed are so small that the concentrations in both v
phases are practically constant, and thus in turn the individual diffusion coeficients also f
v remain unaffected. Moreoer, the rate-controlling step is the diffusion of the trace ion.

Shrinking core model The shrinking core model has been deried for noncatalytic v
we
er
solid–fluid reactions (Le 1972). Ho it has been successfully used for spe-
v
enspiel,
v
,
cific ion-exchange systems—those using synthetic ion exchangers, mainly chelating resins
(Cortina et al ., 1998; Juang, 1999).
Consider the heterogeneous reaction between a solid and a fluid phase:
A (fluid) B (solid) products b
In ion exchange, A is the incoming ion and B is the ion originally found in the solid phase.
v The oerall rate is a combination of the diffusion rate of A in the fluid f the dif ilm, fusion
rate of A in the solid, and the chemical reaction rate. Note that in ion e the coef- xchange,
ficient b corresponds to the ion exchanged from the solid phase. The reaction occurs f irst
, at the outer skin of the solid particle. Then, progressithe reaction zone mo ely v es into the v
enspiel, v solid, leaat , ving behind completed concerted solid (ash) (Le 1972). Consequently
any time there exists an unreacted core of the solid, which shrinks in size during the reac-
tion (Figure 4.16). In this model, the particle size is unchanged during the reaction. In the
following, the case of elementary irre a single v ersible reactions is presented. Furthermore,
controlling mechanism is assumed.
In all equations, the fractional con ersion v X is used:
B

V r   3
1 X u u
B   (4.88)
V r  
o o
where V and r are respectiely the volume and radius of the unreacted core, and V and
v
u u o
r are the total volume and the radius of the particle, respecti . v ely
o