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4.2 Design of Adsorption and Ion-Exchange Processes 315

Both models are good approximations, especially when we are interested in the first part
e,
of the breakthrough curv which is of great importance in industrial applications.
“Diffusion-type” models have been used for the adsorption of lead, copper, p -nitrophenol,
phenol, p -bromophenol, p -toluene sulfonate and dodec yl benzene sulfonate on activated car-
bon (Hashimoto et al ., 1977; Xiu and Li, 2000; Chen and Wang, 2004; Crittenden and Weber,
1978), and ion exchange of ammonia, lead, and other heavy metals on clinoptilolite
(Inglezakis and Grigoropoulou, 2003; Cincotti et al ., 2001; Semmens et al ., 1978; Coone y
et al ., 1999).
A look into the “constant pattern” condition
Under the assumption of plug flo w , the material balance in a f ed bed (eq. ix
(4.128)) can be written as follo ws:

X q Y X
u max 0 (4.156)
s b
z C t t
o
ati
v
In this equation, the partial deries with respect to time are
v
positive, whereas the one with respect to length is ne gati v e.
Furthermore, X C / C and Y q / q . From the above equation,
o max
we hae v
 
z   u s   (4.157)
t  q max b Y 
  C o  
X

where
z
0
t (4.158)

This partial derie is the velocity of the concentration front in the bed. The constant
v
ati
v
pattern assumption presupposes that this velocity is constant, or in other w is inde- ords,
pendent of the solution concentration. This means that all points on the breakthrough curve
are “traeling” in the bed under the same v and thus a constant shape of this curv , elocity e
v
is established (Wevers, 1959). this could happen only if According to the abo v e equation,
(Perry and Green, 1984)
Y
c onst. Y c X c Y X
X 1 2 (4.159)