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4.2 Design of Adsorption and Ion-Exchange Processes 297

Using this equation, the ratio of the fluid volume to the solid mass needed to achiee a v
desired fluid-phase equilibrium concentration ( X ) can be calculated. v o r e a lw e e can achie W
v
liquid-phase concentration leel by using a lower V / m ratio, using, for example, a
higher amount of solid. Thus, equilibrium calculations result in the maximum V / m ratio
v
that should be used to achiee the desired equilibrium (final) concentration. But, ho w
v
much time do we need to achiee our goal in a batch reactor? This is a question to be
answered by kinetics.
By the definition of U ( t ), we ha v e

C t ()
C 1 Xt) (
Ut () o (4.124)
C o C e 1 X

where X ( t ) is gi en by v

Ct ()
Xt ()
C o (4.125)

Then, to achie v e C ( t ) equal to the equilibrium concentration for the specif ied V/m ratio,
U ( t ) has to be equal to 1. T this time for equilibrium is considerably high and
ypically
,
or that reason, impractical for most applications. F it is better to use V / m v alues lo wer than
eral v the maximum. By using se V / m v it is possible to cal- alues lo wer than the maximum,
culate the corresponding equilibrium concentration X . Then, U ( t ) could be calculated for
the desired X ( t ). Using these data, the kinetic model is applied to e aluate the time needed v
to achieve the desired U ( t ).
Note that X in ion exchange and adsorption is the ratio of the species concentration to
the initial concentration and is not equivalent to the conversion as defined in catalytic reac-
tions, which is

C C t ()
x o
C o (4.126)

and thus

C C t ()
x o X
1
C o (4.127)

Example 2
Meshko et al . (2001) studied the adsorption of a basic dye (Maxilon Goldberg GL EC
400% or MG-400) using zeolite in an agitated vessel of 0.5-L volume, under agitation (200
rpm) at ambient temperature (20 °C). The initial concentration of dye was 100 mg/L, the