Page 169 - Adsorption Technology & Design, Elsevier (1998)

Design procedures 157

the differential heat and mass balance equations for the individual com-
ponents. The analysis becomes even more complex when the rate of
adsorption of any individual species is affected by the presence of the others.
This could arise, for example, when individual diffusion coefficients are
found to be functions of composition. Clearly the complexity and hence the
degree of difficulty in obtaining solutions for the MTZ and breakthrough
curve increase rapidly as the number of components increases. Very few
analytical solutions are available and therefore considerable research effort
has been directed at the simultaneous numerical solution of the set of
coupled heat and mass balance equations, the set of rate expressions, and
the set of equilibrium expressions, all being subject to a set of initial and
boundary conditions.
Consider the case when the flow pattern can be assumed to be axially
dispersed plug flow. By not making the assumption that the system is dilute,
the differential fluid phase material balance for each component is given by:

+ + + pp ~---~-] -~ - = 0 (6.37)
-DL OZ 2 OZ Ot Ot
The particle phase mass balance which provides the adsorption rate
equation for each component may be written in generalized form as follows:

----- = f(q~, qj .... ct, cj...) ' (6.38)
0t
Rather than being a simple algebraic expression, the function fis more likely
to represent a set of diffusion equations with their associated boundary
conditions. The continuity equation must also be satisfied, and so in the
general case the mass balance equations may not all be independent. When
the process cannot be considered to be isothermal, then energy balances for
both the element of packed bed (equation (6.39)) and the adsorbent particle
(equation (6.40)) are also required:

- ~ + Cs + cf (6.39)
kL-LU 0;

(TI- rw)
= (- AHi) -~ ed
t
aTsat Z (- an~) ,~,~
c~ Rp3h (T f- Ts) =
~ Ot (6.40)
i
The response to a change in inlet conditions will generate a set of (n-l) mass
transfer zones which propagate through the column when there are n