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4.2 Design of Adsorption and Ion-Exchange Processes 289

where q e and q t are the solid-phase concentration at equilibrium and at time ( t ), respec-
tively. Here, k is the reaction constant in s 1 .

Pseudo second-order kinetic model In this model, the kinetic rate in dif ferential form
and its analytical solution can be expressed as

d q
t ( kq q ) 2 (4.98)
d t e t

t 1 1
t (4.99)
q t kq e 2 q e

where q e and q are the solid-phase concentration at equilibrium and at time t t , respecti . v ely
Here, the units of k are g/(mol s) provided that q is in mol/g.

Finding the rate-controlling mechanism
General As analyzed in Section 3.1.2, among the various steps that are part of a
process, there is frequently one that is much slower than the others, thus controlling the
rate of the whole mechanism. Hence, the slow step is called the “rate-limiting step” or the
”
“rate-controlling step. The principle of the rate-limiting step is often applied since it
greatly simplifies the models used, but we should keep in mind that it is valid for
processes in series. Most of the criteria that will be presented can be equally used in batch
ix and fed-bed operations.

Adsorption According to Fernandez and Carta (1996), who studied mass transfer in
agitated reactors, the relatie importance of external and intraparticle mass transfer resist- v
ances is strongly dependent on the solution composition. They used the following dimen-
sionless number:

kd C
fp o
(4.100)
10 Dq
sma x
where C is the initial solute concentration and q the saturation capacity of the adsor-
o max
bent. This number is dimensionless, and thus the concentration units are the same for both
the liquid and solid phase (e.g. g/g). If this number is small compared to unity the e , xter-
nal film resistance is dominant. Conversely, if this number is large compared to unity, then
intraparticle mass transfer resistance is dominant. According to this study this criterion
,
v holds because of the rectangular shape of the isotherm (irre which means ersible system),
a
v extremely forable uptake of solute by the adsorbent. Ho it is not valid for v ery
,
er
v
we
short times when external film resistance is alays predominant, and for very long times w
when the adsorbent is nearly saturated and intraparticle mass transfer becomes very slo . w
The same criterion has been used for fed-bed operation by Fernandez ix et al . (1996).