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312 4. Adsorption and Ion Exchange

15 DV
N s o
s 2
rQ o (4.139)
where t is the time, V o the total bed volume, the bed v oidage, Q the v w rate, olumetric flo
the bulk density of the bed, and q max the operating capacity (in mass of solute per unit
b
mass of the solid when the fluid-phase concentration is in mass of solute per unit v olume
of the fluid). The operating capacity is the solid-phase concentration at equilibrium with
the initial fluid-phase concentration for adsorption systems and the MEL for ion-exchange
systems. In general, the operating capacity is experimentally more accurately determined
ix
under dynamic conditions, i.e. in a fed-bed system. Subscript “f” refers to the fluid, “s”
to the solid, and “p” to the pore fluid-phase resistance.
The following equations constitute the approximate solutions of the fed-bed model ix
under the constant pattern and plug-flow assumption for the f orable Langmuir isotherm v a
and linear driving forces (Perry and Green, 1999):

ln X () La ln 1 ( X )
NT ( 1) 1
f (4.140)
1 La

1 La  X ln () ln 1 ( X ) 
NT ( s 1)  1 
S  1 La  (4.141)

where
0.894
(4.142)
S
1 0.106 La 0.5
Eq. (4.140) is for liquid-film diffusion control and eq. (4.141) for solid diffusion control.
The following equation is a solution of the fixed-bed model under the constant pattern and
plug-flow assumption, for fluid-film diffusion control and the forable Freundlich v a
isotherm (Fleck et al ., 1973):

 1 Fr 
Fr
NT ( ) 1 1 ln ( X ) ln 1  X Fr 
f (4.143)
1 Fr  
Fr Fr
∑
Fr 1 k k Fr 1 ( Fr)  (4.144)
k 1  

In Figure 4.23, the model results for solid dif ferent fusion control (eq. (4.141)) and tw o dif
values of the Langmuir constant ( La ) are presented. In Figure 4.24, the model results for
solid diffusion and liquid-film diffusion control (eq. (4.140)) for La 0.5 are presented.
From Figure 4.23, it is clear that the more forable the isotherm (lower value of a v La ),
e,
the steeper the breakthrough curv and thus, the better the performance of the operation.