Page 170 - Adsorption Technology & Design, Elsevier (1998)

158 Design procedures

components, including inerts, and the process is isothermal. A non-
isothermal system will include an additional mass transfer zone which
propagates with the velocity of the temperature front because the adsorp-
tion equilibria are temperature dependent. A non-isothermal system of n
components therefore could be considered to be equivalent to an isothermal
system with (n+ 1) components.
As the number of components increases the solutions to the set of design
equations, with their appropriate boundary conditions, become increasingly
difficult and therefore it is usually necessary to make approximations. The
mathematical models which are thereby obtained differ in their generality
and in their applicability according to the extent and severity of the
approximations which have been made. As an example of the number and
type of assumptions and approximations which might need to be invoked for
a non-isothermal process, let us consider the relatively simple model of
Cooney (1974). In this case it is assumed that

(1) there is only one adsorbable component, and therefore there is
only one material balance equation;
(2) axial dispersion is negligible and hence the first term of equation
(6.19) is omitted;
(3) the adsorbate is dilute and hence u can be taken outside the
differential of the second term in equation (6.19); in association
with this simplification it is also assumed that the effect of
temperature on u is ignored;
(4) the equilibrium is described by the Langmuir isotherm for which
the van't Hoff equation is applicable;
(5) the local rate of adsorption is described by a linear driving force
expression in which the mass transfer coefficient is assumed to be
independent of temperature;
(6) axial conduction of heat is negligible and thus kL in equation (6.39)
is set equal to zero;
(7) the bed is initially free of the adsorbate and is at a uniform
temperature To throughout; and
(8) there is a step change at time zero to a feed of fixed concentration at
the inlet, and the feed is at To.

The fluid phase mass balance, fluid phase heat balance, adsorption rate, heat
transfer rate, isotherm and van't Hoff equations which describe the process
are as follows:

+ U~ + pp -- 0 (6.41)
t~t aZ t~t