Page 84 - Adsorption Technology & Design, Elsevier (1998)

Rates of adsorption of gases and vapours by porous media 81

1 'P'p" aq
1 a R2Dp ac ac
( ) = ~ + (4.27)
R E ag aR 0t at
where the average adsorbate concentration t] is expressed by equation
(4.21). The coupled equations (4.17) and (4.27) were first solved analytically
by Ruckenstein et al. (1971) who assumed a step change in concentration at
the gas-solid interface of a composite pellet containing micropores and
macropores. Ma and Lee (1976) extended the analysis to include the possi-
bility of a progressive change in concentration of the gas external to the
pellet as adsorption proceeds. The boundary conditions which then apply to
the coupled equations (4.17), (4.21) and (4.27) are, for the centre of the
microporous crystalline spheres,
aq
r - 0 ..... = 0 for all t -> 0 (equation (4.18))
Or
and at the periphery of the crystals

r = re q = Kc (4.28)
assuming a linear isotherm over the concentration range considered and
where c is the gas phase concentration in the macropore structure of the
whole pellet at a radial position r at a given time t. For diffusion in the
macropores the boundary conditions are, at the pellet centre
ac
R = 0 ........ = 0 for all t -> 0 (4.29)
aR
while at the interface between bulk gas phase and pellet
R = Rp c = Co for all t -> 0 (4.30)
Initial conditions apposite to the stated problem are
t=0 c=0=q forallR>-0 (4.31)

The analytical solution given by Ruckenstein et al. (1971) expresses the
dimensionless uptake of adsorbate mt/m~, as a function of time. The
solution is complex and involves two parameters defined by a = (Oc/rc2)]
(Dp/Rp 2) and fl = 3a (1 - ep) qo/e, pCo. When the resistance to diffusion is con-
trolled by diffusion in the micropores (fl ~ 0), the system is described by
equations (4.17) to (4.21) inclusive, the uptake of adsorbate being
represented by equation (4.22). When, on the other hand, macropore
resistance dominates the diffusion process (fl -, o0), then equations (4.27) to
(4.30) inclusive apply and the condition (4.18) is redundant because the
concentration throughout the crystal is uniform. The solution is then
identical to equation (4.22) with rp and Dp replacing re and D~, respectively.