Page 161 - Adsorption Technology & Design, Elsevier (1998)

Design procedures 149

tion equation may be obtained for step and impulse inputs of the adsorbate.
With isotherms of favourable shape in the region of interest, that is, type I
isotherms the MTZ tends to sharpen as it progresses through the bed until
the dispersive actions of axial dispersion and intraparticle diffusion cause the
MTZ to take on a constant shape, or pattern. Again, analytical solutions for
the constant pattern MTZ may be obtained. With isotherms of unfavourable
shape in the region of interest, the MTZ broadens as it passes through the
bed and analytical solutions are not generally possible.
Changes in bulk fluid velocity across the MTZ are negligible in the case of
a system comprising a very dilute adsorbable component in a non-adsorbing
carrier fluid. The second term in equation (6.19) can then be simplified by
taking the velocity u outside the differential. On the other hand, if the
adsorbable components are present at high concentration, such as in the
case of air separation, then the fluid velocity across the MTZ will decrease
from the trailing to the leading edge and the variation of u with distance must
be retained in the model.
The axial dispersion term, the first term in equation (6.19), can be omitted
if pure plug flow can be assumed. The conservation equation then reduces to
first-order hyperbolic form. If axial dispersion is significant, then the first
term in equation (6.19) must be retained, and the flow is known as axially
dispersed plug flow. It is generally undesirable to have radially dispersed
flow in an adsorption bed, and thus this aspect is generally not incorporated
into flow models for adsorption.
Instantaneous equilibrium can be assumed to occur at all points in an
adsorption bed only if all resistances to mass transfer, including those
outside and inside the adsorbent particles, are negligible. The equilibrium
assumption can rarely be invoked, and therefore it is necessary to
incorporate a specific rate expression for the fourth term of equation (6.19).
The simplest rate expression contains only one mass transfer resistance,
while the most complex may contain three.
Possibilities for a single resistance include a linear rate expression with a
lumped parameter mass transfer coefficient based either on the external
fluid film or on a hypothetical 'solid' film, depending on which film is
controlling the rate of uptake of adsorbate. A quadratic driving force expres-
sion, again with a lumped parameter mass transfer coefficient, may be used
instead. Alternatively, intraparticle diffusion, if the dominant form of mass
transfer, may be described by the general diffusion equation (Fick's second
law) with its appropriate boundary conditions, as described in Chapter 4.
Example models with two resistances include external fluid film resistance
plus intraparticle diffusion or two internal diffusional resistances such as
macropore and micropore. A complex three-resistance model might include
the external film resistance plus two intraparticle resistances. Such a