Page 35 - 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) 257

d ln p/d ln c may be very different from unity. Such con-
siderations apply equally to diffusion in liquids or gases
as well as to diffusion in an adsorbed phase. However,
for gaseous systems the deviations from ideality are gen-
erally small, and even for liquid systems the deviations
from Henry’s law are often modest over substantial ranges
of concentration. By contrast, for an adsorbed phase the
relationship between activity and concentration (the equi-
librium isotherm) is almost always highly nonlinear.
The factor d ln p/d ln q approaches unity in the Henry’s
law region and infinity in the saturation region of the
isotherm, so a strong concentration dependence of the
Fickian diffusivity (D increasing with q) is to be expected.
For example, for a Langmuir system,
FIGURE 4 Schematic diagram of a biporous adsorbent pellet
showing the three resistances to mass transfer (external fluid film,
d ln p 1 D 0
macropore diffusion, and micropore diffusion). R p pellet radius; r c = ; D = (10)
crystal radius. d ln q 1 − q/q s 1 − q/q s
In principle the mobility B and therefore the cor-
fluid film, the diffusional resistance associated with trans- rected diffusivity D 0 are also concentration-dependent,
port through the macropores, and the intracrystalline or so Eq. (12) does not necessarily predict quantitatively
micropore diffusional resistance (Fig. 4). Depending on the concentration dependence of D even for a system
the particular system and the conditions, any one of these where the isotherm obeys the Langmuir equation. Nev-
resistances may be rate controlling, or the rate may be de- ertheless, the concentration dependence of B is generally
termined by the combined effects of more than one mass modest compared with that of the thermodynamic factor,
transfer resistance. so a monatonic increase in diffusivity with adsorbed-phase
concentration is commonly observed (Fig. 5). Clearly in
any attempt to relate transport properties to the physical
A. Micropore Diffusion properties of the system it is important to examine the cor-
It is convenient to correlate transport data in terms of a rected, diffusivity D 0 (or the mobility B) rather than the
diffusivity defined according to Fick’s first equation, Fickian diffusivity, which is in fact a product of kinetic
and thermodynamic factors.
∂c
J =−D(c) (7) Micropore diffusion differs in several important re-
∂z spects from diffusion in macropores or in bulk fluids since
where J is flux, D diffusivity, c fluid-phase concentration, the diffusing molecule never escapes from the force field
and z the distance. The true driving force for any transport of the solid. Under these conditions repulsive interactions
process is, however, the gradient of the chemical potential are important, and relatively large differences in diffusiv-
rather than the concentration gradient, so one can write, ity may therefore occur between different stereoisomers,
more generally, reflecting differences in molecular shape. Furthermore,
small changes in pore diameter can affect the diffusivity
∂µ
J =−Bc (8) by orders of magnitude, and on this basis a suitable adsor-
∂z
bent may sometimes be tailored to provide a high kinetic
where B is mobility and µ the chemical potential. By con- selectivity between similar molecules. The most important
sidering equilibrium with an ideal vapor phase, it may be practical example is the separation of oxygen and nitrogen
shown that the Fickian diffusivity (D) and the thermody- on a carbon molecular sieve.
namic mobility (B) are related by: Micropore diffusion is an activated process, and the
d ln a d ln p temperature dependence can generally be correlated ac-
D = BRT = D 0 (9) cording to an Eyring equation,
d ln c d ln c
where c is the absorbed phase concentration, p the par- −E/RT
D 0 = D ∗ e (11)
tial pressure, and the limiting diffusivity D 0 = BRT.Itis
evident that for an ideal system (activity proportional to where D ∗ is a pre-exponential factor and E the diffusional
concentration) Eq. (9) reduces to the Fickian formulation activation energy. The diffusional activation energy is a
with D = D 0 . However, for a nonideal system the factor useful property which for a given sorbate–sorbent system