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336 4. Adsorption and Ion Exchange

on the system evolution when compared to that of unfavorable equilibrium. In this case, the
equation of continuity reduces to (Perry and Green, 1999)
d Y d( ) FX
T T
d X d X (4.203)

where Y F ( X ) is the equilibrium relationship between Y and X . The limits of v alidity of
this equation can be found by setting X 0 (for T minimum) and X 1 (for T maximum).
Using the local equilibrium analysis, the isotherm of a system can be found from break-
wing equation: through experiments using the follo

1
Y ∫ T X d (4.204)
0
a
In the case of forable equilibrium, the local equilibrium analysis predicts that at T = 1
v
the concentration X will rise instantly from 0 to 1 (ideal step change). This situation is ideal
and does not correspond to real situations, as when a system exhibits forable equilib- v a
rium, the mass transfer is alays the controlling step. w
In the special case of ion exchange and unforable equilibrium, i.e. v a A B < 1, with A
originally in the solution, under the condition of suficiently long bed, W f s solution alter’
could be used. Ws equation is a special case of the Thomas model for arbitrary
alter’
isotherm and the kinetic law equialent to a reersible second-order chemical reaction v
v
(Helfferich, 1962):

R R T
X (4.205)
R 1

for T / R R . The dimensionless effluent concentration X is zero for T 1 R and equal to 1
for T
R . In this equation, R alue of the reciprocal separation f erage v v . is the a A B .
actor 1/
For a Langmuir isotherm, A B and La are related:

1
AB (4.206)
La

In this case, R 1 La . Finally, T is the throughput ratio defined in eq. (4.136).
Experimental methods for the determination of the controlling mechanism in a f ixed-bed
operation
For the determination of the controlling mechanism, in the case of mass transfer–controlled
systems, the following method can be used (Inglezakis, 2002b). This approximate method
, requires only the experimental bed data, and specifthe set of e ically xit concentrations and