Page 75 - Adsorption Technology & Design, Elsevier (1998)

72 Rates of adsorption of gases and vapours by porous media

Equations (4.9), (4.10) and (4.11) give the values of the Maxwellian (or
bulk), Knudsen and resultant diffusion coefficients, respectively, for
unconstrained transport in a single straight cylindrical pore. However, a
gaseous component diffusing to the interior of a porous material travels
tortuous pathways and is impeded by the volume fraction of solid
unavailable for diffusion so that the resultant flux would be less than would
otherwise be calculated by the above equations. Accordingly an effective
diffusivity De represents the net reduced diffusion coefficient in a porous
medium. It may be estimated by one of several methods. Experimental
methods (outlined in Section 4.3) are the most reliable way of obtaining the
effective diffusivity although an empirical method of estimation is often
used. If ep is the intrapellet void fraction and r a factor (termed the
tortuosity) accounting for the tortuous nature of the pores (diffusing
molecules have to travel longer distances than if the pores were straight)
then one may write
De = ~,p D/r (4.12)
Because ep < 1 and r > I then De < D always. For most porous materials of
known particle porosity, the value of r lies within the range 1.5 to 10. In the
field of catalysis, many related practical catalysts have a tortuosity factor
between 1.8 and 3. Satterfield (1970) has outlined the method of obtaining r.
The effective diffusion coefficient De is found by an appropriate experimen-
tal method (see Section 4.3) for particles of known porosity and the resultant
diffusivity, D, calculated from a knowledge of pore size distribution data.
can then be calculated directly from equation (4.12). Figure 4.1 shows a plot
of the ratio of the measured effective diffusivity to the calculated resultant
diffusivity, DJD, as a function of measured porosity ep. Most of the points
correlate with a line of slope 1.5 when a logarithmic scale is used for ordinate
and abscissa. Yang and Liu (1982) conclude that r is also a function of t~p and
show that for most porous structures De/D is approximately equal to/~pE/'t'.
The random pore model of Wakao and Smith (1962) for a bidisperse pore
structure may also be applied in order to estimate De. It was supposed that
the porous solid is composed of stacked layers of microporous particles with
voids between the particles forming a macroporous network. The
magnitude of the micropores and macropores becomes evident from an
experimental pore size distribution analysis. If Dm and Du are the
macropore and micropore diffusivities calculated from equations (4.9) and
(4.10), respectively, the random pore model gives the effective diffusivity as
De = •m 2 Dm + e/a 2 (1 + 3em) DJ(1 - em) (4.13)
where/~m and eu are the macro and micro void fractions, respectively.
Many adsorbents have a broad distribution of pore sizes and neither