Page 37 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN001-13 May 7, 2001 12:29

Adsorption (Chemical Engineering) 259

of axial dispersion Wakao was able to correlate many of known concentrations. The same method has also been
the data from different laboratories for both gas and liquid adapted to the measurement of micropore diffusion in
systems in accordance with the following correlation for large crystals of certain zeolites.
the Sherwood number: Alternatively one can in principle derive both micro-
pore and macropore diffusivities from measurements of
2k f R p 1/3 1/2
Sh = = 2.0 + 1.1Sc Re (14) the transient uptake rate for a particle (or assemblage of
D m
crystals) subjected to a step change in ambient sorbate
where Sc is the Schmidt number and Re the Reynolds
pressure or concentration. The main problem with this ap-
number (based on particle diameter). However, it should
proach is that the overall uptake rate may be controlled by
be recognized that if this correlation is used to estimate
several different processes, including both heat and extra-
the film coefficient it is essential also to use a realistic
particle mass transfer as well as intraparticle or intracrys-
value for the axial dispersion coefficient. Otherwise, the
talline diffusion. The intrusion of such rate processes is
combined effects of external mass transfer resistance and
not always obvious from a cursory examination of the ex-
axial mixing will be underestimated.
perimental data, and the literature of the subject is replete
with incorrect diffusivities (usually erroneously low val-
D. Overall Mass Transfer Resistance ues) obtained as a result of intrusion of such extraneous
effects. Nevertheless, provided that intraparticle diffusion
It has been well established that the kinetics of a diffusion-
is sufficiently slow, the method offers a useful practical
controlled process can be approximately represented by a
alternative to the Wicke–Kallen bach method.
linearized rate expression of the form:
Chromatographic methods offer a useful alternative to
∂ ¯ q/∂t = k(q − ¯ q) (15) conventional batch uptake rate measurements. The advan-
∗
tage of these methods is that heat transfer effects can be
where the effective rate constant k is related to the diffu- greatly reduced and in most cases eliminated by the use
2
sional time constant by k ≈ 15D/r (r being the particle of a high carrier flow rate and a low sorbate concentra-
radius), and ¯ q is the value of q averaged over a parti- tion. The main disadvantage is that the broadening of the
cle. This approximation, due originally to Glueck, is at its response peak results from the combined effects of ax-
best for linear equilibrium systems and long adsorption ial dispersion and mass transfer resistance. It is therefore
columns, and it is at its worst when the isotherm is rectan- necessary either to eliminate or to allow for axial disper-
gular and for very short columns or single particles. When sion in the column, and this is often more difficult than it
several resistances to mass transfer are significant (as in may at first sight appear. Nevertheless, the method is quick
Fig. 4), the overall rate constant is given approximately and straightforward and requires no special equipment. It
by the reciprocal addition rule: is therefore especially useful for preliminary adsorbent-
1 R p R p 2 r c 2 screening studies when a rapid means of obtaining ap-
= + + (16) proximate kinetic and equilibrium data is required.
kK 3k f 15ε p D p 15KD c
In the zero length column (ZLC) method, which can be
where ε p is the macroporosity of the adsorbent particle and regarded as a derivative of the traditional chromatographic
D c the intracrystalline (micropore) diffusivity. These ap- method, a small sample of adsorbent is pre-equilibrated
proximations are especially useful in the modeling of ad- with the sorbate under well-defined conditions and then
sorption column dynamics for more complex nonisother- purged, at a constant flow rate, with an inert (nonadsorb-
mal and multicomponent systems, since the replacement ing) gas (usually He), monitoring continuously the com-
of a diffusion equation by a simple linearized rate expres- position of the effluent stream. From analysis of the ZLC
sion leads to a general reduction in mathematical complex- desorption curve both the adsorption equilibrium constant
ity and a corresponding reduction in the computer time and the internal diffusivity can be obtained. The method
requirement. The rigorous solution of diffusion equation retains the advantages of the traditional chromatographic
models is generally not practically feasible except for the method while eliminating the need to account for axial
simplest systems. dispersion.
A more sophisticated method which has found wide ap-
plication in the study of intracrystalline diffusion in zeo-
E. Measurement of Intraparticle Diffusivities
lites is the nuclear magnetic resonance (NMR) pulsed field
The customary way of measuring intraparticle macropore gradient self-diffusion method. The method, which is lim-
diffusivities is the Wicke–Kallenbach method, which de- ited to hydrocarbons and other sorbates with a sufficient
pends on measuring the flux through a pellet under steady- density of unpaired nuclear spins, depends on measuring
state conditions when the two faces are maintained at directly the mean square distance traveled by molecules,