Page 163 - Adsorption Technology & Design, Elsevier (1998)

Design procedures 151

(6.22)) and a finite mass transfer rate will prevent the formation of the
limiting condition which is a compressive shock transition (step change) that
travels through the bed with a velocity determined by a mass balance over
the MTZ:

dz u
-- = (6.25)
dt

1 + pp AC

A perfect shock front can only be attained if there are no dispersive effects,
i.e. if there is no axial dispersion, and if there are no mass transfer
resistances to the adsorption process. Such a situation is most unlikely, since
even if the mass transfer resistances were extremely low, the axial dispersion
effects would become significant as the step change or shock were
approached and the shape of the MTZ would tend to become fixed.
There can be no change of shape in the MTZ if the isotherm is linear
(equation (6.26)) because in this case all the terms on the right-hand side
of equation (6.24) would be independent of the fluid phase concentration
C.

q* = Kc (6.26)
The time taken for a point of given fluid concentration to move through the
bed can be obtained from equation (6.24). Consider the case of a linear
isotherm and an initially clean bed. Assume that breakthrough occurs when
the fluid phase concentration is just about to increase beyond zero. By
setting dq*/dc = K in equation (6.24), the time to breakthrough for a given
bed length can be obtained or, vice versa, a bed length for a given time to
breakthrough can be obtained:

t= Lu-~l +pp (~~-~)K] (6.27)

In the case of a Langmuir isotherm (see Section 3.3.1):

bqmc
q* = (6.28)
1 +bc
the slope of the isotherm is as follows:

dq* bqm
dc (1 + bc) 2 (6.29)