Page 79 - Adsorption Technology & Design, Elsevier (1998)

76 Rates of adsorption of gases and vapours by porous media

The set of equations (4.17) to (4.21) inclusive was solved analytically by
Ruckenstein et al. (1971) who compared the mass of adsorbate adsorbed at a
given time, mt, with the amount adsorbed after an infinite lapse of time,
moo, (when the crystals were saturated with adsorbate). They expressed
the ratio mt[m~, as a function of time t. It should be noted that, for small
quantities of adsorbate introduced to the system the adsorption isotherm is
linear and q = Kp. The intercrystalline diffusivity can then be regarded as
independent of adsorbate concentration. The theoretical fractional ap-
proach to equilibrium was shown to be

__1 (n2tr2Dct .
~
I
Z
mt _1_~ 6_~ _
moo ,,=l n 2exp -- rc ] (4.22)
When the fractional uptake is greater than 70%, only the first term of the
summation is retained. When, on the other hand, the fractional uptake is
less than 30%
mt-6( Dot ),/5
- (4.23)
mm 7rre 2
is a good approximation.
The intracrystalline diffusion coefficient Dc was considered to be inde-
pendent of adsorbate concentration in the above analysis of Ruckenstein
et al. (1971). However, if the initial quantity of adsorbate admitted to the
adsorbent is such that the vapour phase concentration does not remain
constant, then account should be taken of the variation of the intracrystal-
line diffusivity with concentration. This dependence of diffusivity on
concentration may be derived by equating the Fickian flux to the thermo-
dynamically defined flux, the latter depending on the product of concentra-
tion and the gradient of chemical potential. It then follows (Ruthven 1984,
Yang 1987) that the diffusion coefficient is related to adsorbed phase
concentration by the equation

Dc = Do d In p/d In q (4.24)
in which Do is independent of concentration. If the adsorbate-adsorbent
system obeys a Langmuir relation then
De = Do/(1 - (q/qm)) (4.25)

The diffusion equation is then written
Oq _DoO{ i r 20q}
(4.26)
Ot - P Or 1-(q/qm)) Or