Page 110 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Unit Process Principles 65

The advection and the dispersion terms are,
BOX 4.2 MATERIALS BALANCE EQUATION
advection in advection out FOR PACKED-BED REACTOR
q( vC) DZ q( vC) DZ
(b) In general, Equation 4.16 must be solved by a finite-
qZ 2 qZ 2 difference algorithm using a high-speed computer. The
vC in ¼ vC (a) vC out ¼ vC
(4:11) kinetic term is specific to the particular system being
dispersion in dispersion out modeled, for example, biological, chemical, adsorptive,
ion exchange, etc.
qj DZ qj DZ
in (a) out (b) Consider granular activated carbon as the media in a
qZ 2 qZ 2 packed bed. The reaction term is
j ¼ j j ¼ j þ
(4:12)

qC (1 P) qX
in which ¼ r (4:17)
qt r P qt
C is the concentration of given contaminant at center of the
3
infinitesimal slice (kg=m ) in which
j is the dispersion flux density at center of the infinitesimal P is the porosity of porous media
2

slice (kg=m =s) X is the concentration of contaminant in solid
phase (kg contaminant=kg adsorbent)
In words, the advection flux into the element is the advection r is the density of adsorbent particles (kg
3
flux in the center minus the rate of change of the advec- adsorbent=m )
tion flux with respect to Z times the distance of half the
slice, that is, DZ=2. The same idea holds for the advection Substituting Equation 4.17 in Equation 4.16 gives
flux leaving the element.
2

The dispersion flux into the element is the dispersion flux qC qC q C 1 P qX
¼ v þ D r (4:18)
in the center of the element minus the rate of change of the qt qZ qZ 2 P qt
0 r
dispersion flux with respect to Z times the distance of half the
slice, that is, DZ=2. The same idea holds for the dispersion Equation 4.18 cannot be solved mathematically. It must
flux leaving the element. be converted to finite difference form and then solved
Substituting Equations 4.11 and 4.12 into 4.10, and can- numerically by computer. Also a kinetic equation is

celing A, needed for the [qX=qt] r term. The computational scheme
is to divide the column into n slices and to solve the

qC q( vC) DZ q( vC) DZ materials balance for each slice starting at the top of the
qt DZ ¼ vC qZ 2 vC þ qZ 2 column. In doing this, the output of the ith slice equals
0 in out
the input to the i þ 1 slice and the computation yields C i
qj DZ qj DZ qC
DZ for time t þ Dt. Since the variables C i and X i change with
qZ 2 qZ 2 qt
þ j j þ
in out r both distance and time, millions of iterations are neces-
(4:13) sary for a ‘‘solution’’ to Equation 4.18. The form of the
solution would be the concentration profile at successive
Then, for the condition that v is constant, Equation 4.13 times, that is, C(Z) t . The solution is discussed in more
simplifies to detail in Chapter 15.

qC qC qj qC
v (4:14)
qt qZ þ qZ qt In words, Equation 4.16 says: the observed mass rate of change
0 r
of a given contaminant within the slice equals the net rate of
The dispersion transport rate is analogous to Fick’s law, and is advection for the slice plus the net rate of dispersion for the
expressed as slice minus the rate of reaction within the slice. As a note,
Equation 4.16 is written often in terms of volume rather than
qC
j ¼ D (4:15) time, for example, [qC=qV] 0 instead of [qC=qt] 0 and [qC=qV] r
qZ
instead of [qC=qt] r . Since the volume of fluid that has passed a
Substituting (4.15) into (4.14), certain point is, V ¼ Qt;then, dV ¼ Qdt.
In addition to packed-bed reactors, Equation 4.16 and
2
qC qC q C qC Figure 4.7 are applicable to fluidized-bed reactors in which
¼ v þ D (4:16)
qt qZ qZ 2 qt the concentration changes with distance. Examples include
0 r
streams which may receive a wastewater flow, in which
Equation 4.16 is a ‘‘final’’ form of the materials balance equa- concentrations of various contaminants change with distance,
tion for a nonhomogeneous reactor in which the infinitesimal a ‘‘plug flow’’ activated sludge reactor, an air-stripping tower,
element is a column ‘‘slice,’’ as shown in Figure 4.7 (Box 4.2). and a chlorination basin.