Page 396 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Rapid Filtration 351

Simplifying, the v CAe terms drop out, the DZAe terms 12.3.3.6.3 Finite Difference Form of Materials
cancel, and dispersion is neglected: Balance Equation
Equation 12.29 cannot be solved analytically. A finite differ-

qC qC qC ence form is required, which can be obtained by replacing the
¼ v þ (12:26)
qt qZ qt infinitesimal designation (i.e., partial differential), q, with
o r
finite the symbol, D, that is,
Equation 12.26 describes what occurs in an infinitesimal slice
of the reactor, that is, the ‘‘the observed rate of change of DC Z ¼ v DC t þ 1 Ds (12:30)
suspension in the slice’’ equals the ‘‘net rate of advection’’ to Dt o DZ e Dt
and from the slice plus the ‘‘rate of concentration change in
the slice due to uptake to the solid phase.’’ which can be expressed as

12.3.3.6.1 Kinetics
C tþDt, Z C t, Z (C ZþDZ, t C Z, t ) 1 Ds
The ‘‘reaction’’ term in Equation 12.26 represents the rate of ¼ v
Dt o DZ e Dt Z, t
depletion of solids from the suspension that is deposited on
(12:31)
the filter media, that is,
where

qC qs
DZA (12:27) C tþDt,Z C t,Z ¼ DC Z , the change in concentration of the
DZAe ¼
qt qt
r interstitial suspension between time t and time t þ Dt,
3
at a given Z (kg solids in suspension=m suspension)
where [qs=qt] is the rate of increase of solids deposit on C ZþDZ,t C Z,t ¼ DC t , the change in concentration of the
filter media (kg suspended solids deposited=m 3 of bed interstitial suspension between slice Z and slice Z þ DZ,
3
volume). at a given time, t (kg solids in suspension=m suspension)
Note that in Equation 12.27, the left side is in terms of pore
water concentration which requires multiplication by e, while A second equation is required for the term representing
the right side is in terms of solids concentration for the filter as solids uptake rate. Adin and Rebhun (1977) developed such
a whole. Equation 12.27 simplifies to an equation (reviewed in the next section).

qC qs
(12:28) 12.3.4 SYNTHESIS OF A MODEL
e ¼
qt r qt
Modeling has, in general, followed Ives approach which has
been to consider the effect of solids deposits on the filter coef-
Substituting Equation 12.28 in Equation 12.26 gives
ficient, l and to compute the entire curve from the Iwasaki
::
kinetic relation, with the correction for l. The approach given

qC qC 1 qs
¼ v (12:29) here is that of Adin and Rebhun (1977) which provides an
qt qZ þ e qt
o expression of the solids uptake rate, which, in turn, is inserted
into the materials balance relation, Equation 12.29.
12.3.3.6.2 Discussion 12.3.4.1 Solids Uptake Rate
The materials balance expression as given in Equation 12.29 Adin and Rebhun (1977) have noted that the filtration process
is a common starting point for modeling of the filtration falls within a class of packed bed reactor problems (e.g.,
process. The equation says merely that the observed rate of filters, granular activated carbon, ion exchange) involving
change of suspended solids concentration within the pore materials balance and kinetics. They formulated a kinetics
volume of an infinitesimal column slice equals the net advec- expression as a second order relation for uptake with a scour
tion rate minus the uptake rate of solids by adsorption on term for solids depletion as
collectors. Note that the gradient, qC=qZ is usually negative,
that is, concentration decreases as Z increases. qs
¼ [k 1 vC(F s)] [k 2 si] (12:32)
To relate back to the Iwasaki materials balance equation, qt
Equation 12.11, assumes that the left side of Equation 12.29
is zero, that is, [qC=qt] o ¼ 0. Others, such as Ives have done where
2
this also. Later, Horner et al. (1986) called attention to the k 1 is the accumulation coefficient (m water=kg suspended
fact that the left side, that is, [qC=qt] o , term had been matter)
3
neglected in filtration modeling over the decades since k 2 is the detachment coefficient (m bed volume= kg sus-
Iwasaki’s work. pended matter=s)