Page 567 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

522 Fundamentals of Water Treatment Unit Processes: Physical, Chemical, and Biological

16.2.3 KINETICS in which
2
j A is the flux density of species A (mol A=cm =s)
The theory of ion-exchange kinetics is virtually the same as 2
D A is the diffusion coefficient (cm =s)
the theory of adsorption kinetics. The three basic steps in each 3
C A is the concentration of A in (mol A=cm )
case are: (1) film diffusion, (2) particle diffusion, and (3)
grad is the mathematical term for ‘‘gradient,’’ i.e., concen-
attachment. For adsorption, the substances of interest are
tration gradient in the case at hand
molecules or small particles, and for ion-exchange they are
ions. A difference is that for the ion-exchanger, mass transport
The application of Equation 16.7 is difficult because usually,
of ions in one direction, e.g., out of the particle, must be
D A is not constant; it varies with temperature and with con-
balanced by mass transport in the other direction, e.g., into
centration. Then for the solid phase, its magnitude is unique
the particle, which preserves electroneutrality.
for a given ion-exchanger. Thus, as stated by Helfferich,
Equation 16.7 is essentially a definition of D A .
16.2.3.1 Rate-Determining Step
As with adsorption, the rate-determining step is either film 16.2.3.2.1 Particle Diffusion
diffusion or particle diffusion. Figure 16.6 illustrates succes- Equation 16.7 applies to particle diffusion as noted. For an ion
sive C(r)t curves for the two cases, respectively. In Figure species A , within an ion-exchanger, it has the form,
þ
16.6a the rate of particle diffusion is high, i.e., with no
substantial concentration gradient within the particle,
causing film diffusion to be rate controlling. Such a condi- j ¼ D A grad C A (16:8)
A
tion occurs when the particle structure is open as with low
degree of cross-linking and small particle size. In Figure in which
2
16.6b particle diffusion is rate controlling. Here film diffu- j A ¼ flux density of ion species A (mol A =cm =s)
þ
þ
2
sion is faster and the needed film concentration gradient is D A ¼ diffusion coefficient within ion-exchanger (cm =s)
slight to achieve the needed flux into the particle (to equal C A ¼ concentration of A in ion-exchanger (mol A =cm )
3
þ
þ
the particle flux).
The bars refer to the interior of the ion-exchanger.
16.2.3.2 Fick’s First Law
To evaluate the changes within the ion-exchanger as a
Whether the diffusion of an ion species, say ‘‘A,’’ is rate function of time, Fick’s second law is applicable, i.e.,
limiting in the liquid phase or in the solid phase, the random
motion of ions and molecules causes a flux which is propor-
tional to the concentration gradient (see also Sections 4.2.2.6). qC A ¼ div j A (16:9)
This is described by Fick’s First Law, i.e., qt
(16:7)
j A ¼ D A grad C A div ¼ divergence

t=∞ t=∞
t>>t(1/2)
t>>t(1/2) t=t(1/2)
Concentration of ion in solution C o t<<t(1/2) Solid phase concentration of ion Solution concentration of ion C o t <<t(1/2) Solid phase concentration of ion

t=0
r
r o t>0 o

Film Film
(a) (b)
FIGURE 16.6 Concentration profiles, C(r)t, of diffusing ions illustrating rate-controlling mechanisms. (a) Film diffusion rate controlling;
(b) particle diffusion rate controlling. (Adapted from Helfferich, F., Ion-Exchange, McGraw-Hill, New York, 1962, p. 254.)