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4.2 Design of Adsorption and Ion-Exchange Processes 293

Finally, the agitation rate does not af fusion controls e rate if the particle dif fect the uptak
we
v
er
,
the process. Ho the latter criterion may be not safe; the agitation in solution may
icienc
,
f
y
have attained its limiting hydrodynamic ef so that a change in the agitation rate
has no effect on the uptake rate een in film dif fusion–controlled systems. v
Shrinking core model There is a considerable difference between k and the other two
f
kinetic parameters D s and k ; the latter are independent of particle size while k is a func-
f
tion of agitation (and thus of Re p ) and is inversely proportional to particle size (Levenspiel,
1972). The time needed to achieve the same fractional conversion for particles of different
enspiel, v but unchanging sizes is gien by (Le 1972) v
t
r b (4.108)
o
where:
1.5 to 2, for film diffusion control (the exponent drops as Re p rises)
b 2, for solid dif fusion control
1, for chemical reaction control
This criterion can easily distinguish between reactions in which the chemical and
physical (diffusion) steps control. Furthermore, the chemical reaction step is usually more
Thus, fusion) steps. temperature sensitive than the physical (dif performing e xperiments at
different temperatures could be another relatively safe way to distinguish between the con-
trolling mechanisms.

Agitated vessels (liquid–solid systems) Below the off-bottom particle suspension state,
the total solid–liquid interfacial area is not completely or eficiently utilized. Thus, the
f
mass transfer coeficient strongly depends on the rotational speed below the critical rota- f
tional speed needed for complete suspension, and weakly depends on rotational speed
above the critical vWith respect to solid–liquid reactions, the rate of the reaction
alue.
increases only slowly for rotational speed aboe the critical value for two-phase systems v
where the solid–liquid mass transfer controls the whole rate. When the reaction is the rate-
controlling step, the oerall rate does not increase at all beyond this critical speed, i.e.
v
when all the surface area is available to reaction. The same holds for gas–liquid–solid sys-
tems and the corresponding critical rotational speed.
It should be noted that this is not a safe criterion for the controlling mechanism. It is
true that if the controlling mechanism is the reaction, then the increase of agitation
v
v
aboe its critical value will hae no effect on the oerall rate. But this effect cannot be
v
used safely as a criterion when searching for the controlling mechanism; there is a case
where the liquid film diffusion could be the controlling mechanism and yet the agita-
v
tion may hae no effect on the oerall rate. This could happen in the case where the
v
mass transfer coeficient reaches a (practically) constant maximum value by increasing
f
the agitation rate aboe a limit, and at the same time the other steps i.e. solid diffusion
v
and/or rate are very fast. In this case, the liquid-film diffusion could be still the con-
trolling mechanism.