Page 520 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Adsorption 475

governs, which has been designated ‘‘advection kinetics’’ (see where C ¼ C(Z 0 ), and Z 0 is the ‘‘balance point’’ where
0
0
Hendricks 1973, 1980). These two mechanisms, i.e., particle (qX=qt) P ¼ (qX=qt) A ; also the distance Z is Z ¼ Z Z 0 . The
0
0
kinetics and advection kinetics, are distinguished symbolic- term l is an experimentally determined coefficient, which
ally by the terms, [qX=qt] P and [qX=qt] A , respectively. Stated depends upon v; the function is unique for a given porous
mathematically, when media. Taking first and second derivatives of Equation 15.29,

qX qX qC 0 lZ 0
< (15:26) ¼ lC e (15:30)
0
qt qt qZ
A P
¼ lC(Z ) (15:31)
0
then advection transport to the particle is rate controlling. The
numerical solution to Equation 15.24 involves a test of this
and
criterion at each slice, 1 i n, in the column and for each
Dt time pass in the iteration. 2
q C 2
¼ l C(Z ) (15:32)
0
qZ 2
15.2.3.2.1 Advection Kinetics Model
An expression for advective transport (i.e., advection kinetics) Substituting (15.31) and (15.32) into Equation 15.28 gives
can be derived starting with Equation 15.24. The premise is
that the uptake rate capacity by the particle, i.e., [qX=qt] P ,
qX 1 P
exceeds the transport rate to the particle by advection and ¼ (v þ Dl)lC(Z ) (15:33)
0
qt r 1 P
dispersion, i.e., [qX=qt] A . A
When the adsorption rate is limited by the transport of
adsorbate to an adsorbent particle (by advection and disper- Substituting (15.30) in (15.33) gives
sion) none of the adsorbate molecules making contact with
the external surface of an adsorbent particle is ‘‘rejected.’’ For
qX 1 P 0 lZ 0
this condition, the concentration profile in the zone where ¼ (v þ Dl)lC e (15:34)
0
qt r 1 P
advective-dispersion advection kinetics governs, is approxi- A
mately ‘‘steady state’’ and at any Z in this zone, the observed
[dC=dt] 0 0. Thus Equation 15.24 is Equation 15.33 or 15.34 has an interesting physical signifi-
cance. The terms v, D, and C(Z) are indicative of the transport
rate at the position Z, but the term l tells how many collisions
2
qC qC q C 1 P qX
¼ 0 ¼ v þ D r (15:27) will occur per unit distance. In other words, the expression
qt 0 qZ qZ 2 P qt given in (15.33) or (15.34) gives the probability that within
the time period Dt the solid-phase adsorbent concentration at
Solving for qX=qt gives Z will have increased by DX. At the same time, in the zone
where convective dispersion transport governs, i.e., Z > Z 0 ,
0
2 the concentration profile, C(Z ) t , may be computed by Equa-
qX 1 P qC q C
¼ r v þ D (15:28) tion 15.29. The transition point, i.e., where kinetics changes
qt P qZ qZ 2
A from particle kinetics being rate controlling to where advec-
tion=dispersion is rate controlling is the inflection point in the
Thus the uptake rate [qX=qt] A is dependent solely upon the concentration profile, i.e., the ‘‘wave front.’’
rate of transport of adsorbate by advection, (vqC=qZ), and
2
2
dispersion, (Dq C=qZ ). 15.2.3.2.2 Model Delineation
The task now is to find a C(Z) t function which will allow a
In a packed bed of granular adsorbent, Equation 15.30 is the
solution of Equation 15.28. To deduce a C(Z) t relationship for
‘‘mathematical model’’ which depicts the adsorbate concen-
advection kinetics, consider a porous medium with a given
tration profile in the column to the right of the inflection point,
flow velocity. An adsorbate molecule will have a probability
i.e., as seen in Figure 15.13. To the left of the inflection point,
of say 0.50 of making a collision with an adsorbent particle
particle kinetics governs and Equation 15.23 is the basis for
within a certain distance of travel (call it the ‘‘half distance’’
computation of C(Z, t). The C(Z) t curve in the advection–
if it is desired to see the analogy with radioactive disintegra-
dispersion kinetics zone is constant in shape. But then Z 0 0
tion with time). Now if we consider 100 particles in the fluid
moves downstream as X(Z ¼ 0) increases (shifting control to
stream initially, 50 will remain after one half distance, 25 after
particle kinetics). At the same time, the inflection point, i.e.,
the next half distance, and so on. This suggests a decay
where [qX=qt] P ¼ [qX=qt] A , starts to move downstream for
equation of the form,
t > 0. At some point in time, i.e., t 0, the C(Z) t profile
assumes a steady-state shape, such as in Figure 15.13, and
0 lZ
C(Z ) ¼ C e 0 (15:29) ‘‘translates’’ downstream.
0
0