Page 91 - Adsorption Technology & Design, Elsevier (1998)

88 Rates of adsorption of gases and vapours by porous media

(such as helium) across one face of the adsorbent material fabricated into a
cylindrical shape and sealed within the cell. A steady stream of pure inert gas
is allowed to pass through the detector volume at the obverse face of the
pellet where a thermistor detects the response signal arising from the gas
diffusing through the pellet from the original input pulse. Gaskets ensure
that no gas passes through to the detector volume except by diffusion
through the pellet. An unsteady state material balance for the adsorbate in
the direction z yields
c3c t3q c32c
8p ~ + pp = D~
t3t ~Z 2 (4.40)
where ep and pp are the porosity and density of the adsorbent pellet,
respectively. The net rate of adsorption may be represented by the
difference in rates of adsorption and desorption

aq = ka (c-q/K) (4.41)
at
where k~ is the adsorption rate constant and K the adsorption equilibrium
constant. At the face of the pellet there is a pulse input of adsorbate which is
represented by
z = 0 c = m t~ (t) (4.42)
where m is the magnitude of the concentration pulse and d; (t) is the Dirac delta
function. Within the detector perfect mixing is assumed, so if the cross-sectional
area of pellet is S and its volume is V the diffusive flux at the obverse face is

z = L, De (OC/aZ)z=L =- (V/S) (OCL/Ot) (4.43)
The initial condition, when the pulse is admitted to the pellet face, is
t=0 c=0 foraUz>-0 (4.44)
It is unnecessary to solve the above set of equations in the form c (t) because
the first absolute moment of the experimentally measured response curve
(see Figure 4.10) can be related, by means of Laplace transformation, to the
parameters D~, ep, K and V/SL. If a number of adsorbent pellets of differing
length are each subjected to the same magnitude of concentration impulse then
a plot of the first absolute moment (corrected for dead volume and the finite
time of injection) against V/SL yields a straight line of slope 1/D~ and intercept
- ep/2, the latter quantity also being determined independently by pyknometry.
For details of the technique of moments analysis the reader should consult a
text such as Wen and Fan (1975). Figure 4.10 illustrates the type of input
concentration pulse often used for this method and the corresponding output