Page 87 - Adsorption Technology & Design, Elsevier (1998)

84 Rates of adsorption of gases and vapours by porous media

uptake of adsorbate by an adsorbent particle when only a small step change
in adsorbate concentration is introduced to the particle periphery includes
unsteady state balances for both mass and heat transfer. The transient mass
balance is

1 a [r2Dcaql = aq
(equation (4.17))
r 2 ar ~ ---~-r] cOt
with the average adsorbed phase concentration given by
re
0 = _.-25 qr 2 dr (equation 4.21))
rc 0
The unsteady state heat transfer equation is

d o dT
= ~ + ha (T-- Tg) (4.36)
(-AH)-d~ Cs dt
in which c~ represents the heat capacity of the solid porous adsorbent
particle. Note that equation (4.36) is an ordinary differential equation (as
opposed to a partial differential equation) because the temperature within
the particle depends only on the average adsorbed phase concentration in
the particle, the temperature being uniform throughout the particle because
all the heat transfer resistance is external to the particle. At the surface of the
crystal a linear equilibrium relationship is assumed to exist

q--qo
=l+laq*l [T--T~ atr-rc (4.37)
] ~ Tqoo -- qo
q~ -- qo ~ ] a p
where qo and To are the initial values of the adsorbed phase concentration
and the temperature, respectively, and qoo is the final adsorbed phase
concentration after adsorption has ceased. The term (Oq*/aT)p is the
gradient of the equilibrium isobar for the essentially constant partial
pressure at which the small step change in concentration occurs. Boundary
conditions apposite to this problem as it has been posed are
aq/ar = 0 at r = 0 for all t -> 0 (equation (4.18))

and
q=0 att=0 forallr>--0 (4.38)
The solution to the above set of equations, commencing with the restated
equation (4.18) through to equation (4.38), is given in a paper by Ruthven et
al. (1980). The principal features of the uptake curves are illustrated by
Figure 4.7. Two parameters ), (= har~2/csD~) and S (= AH (aq*/aT)/c~)
are required to describe the behaviour of the solution. The symbol c~