Page 397 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

F is the theoretical ﬁlter capacity, or amount of retained Ds em
¼ k 1 vC Z,t (F s Z,t ) k 2 v
material per unit of bed volume which could clog the Dt Z,t r g
w
3
pores completely (kg suspended matter=m bed volume) s Z,t
i is the hydraulic gradient (m headloss per m of ﬁlter bed) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 (12:36)

s Z,t =F
K o 1
The ﬁrst term on the right side is an accumulation term and
depends on the particle ﬂux density, vC, and on the available
which becomes
capacity for solids deposit at any instant, (F s). The term is
a ‘‘second-order’’ kinetic expression, that is, it is proportional
s Z,tþDt s Z,t em
to both the advective ﬂux density and the capacity to hold ¼ k 1 vC Z,t (F s Z,t ) k 2 v
Dt r g
ﬂoc, (F s). The second term is the expression for the rate of w
detachment which is proportional to the concentration of s Z,t (12:37)

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3
solids previously attached, s, and the hydraulic gradient, i,a s Z,t =F
K o 1
surrogate for the shear. At some point, based on the pore
space occupied by adsorbed ﬂoc, the rate of detachment due
to shear equals the rate of accumulation (at a given Z and t). where (s tþDt,Z s t,Z ) is the change in concentration of solids
Observations with an endoscope by Ives (1989, p. 864) con- deposited on the collectors between time t and time t þ Dt,ata
3
ﬁrmed the detachment of particles, that is, as proposed by given Z (kg solids in deposited=m ﬁlter bed).
Professor D.M. Mintz (1966), with redeposit lower in the ﬁlter Equation 12.37 is the basis for calculating the s(Z, t), that
and conﬁrmed also by Cleasby (1969); Tien and Payatakes is, the accumulation of solids deposits within the ﬁlter bed.
(1979, p. 755); Saatci and Halilsoy (1987). Finally, Equation 12.37 when substituted in Equation 12.31
Returning to Equation 12.32, recall Darcy’s law, that is, permits calculation of C(Z, t).
v ¼ ( Kr w g=em)i, keeping in mind that i is a negative quan-
12.3.4.1.2 Computational Protocol
tity, and substitute for i, to give
Equation 12.37 may be used in conjunction with the materials
balance equation, Equation 12.31, which permits calculation
qs vem
¼ k 1 vC(F s) k 2 s (12:33) of C tþDt,Z . The ﬁrst step is to calculate C Z,t ¼ 0 by the Iwasaki
qt Kr g
w relations. These values of C are the initial input values for all Z.
The constants, k 1 , k 2 , F, and K o are determined as outlined
The hydraulic conductivity term is reduced from its clean- by Adin and Rebhun (1977). The second step is to calculate
bed value, K o , by the ratio of solids deposit, s, to capacity, F Ds=Dt as given by Equation 12.36, which then goes into
(Adin and Rebhun, 1977), Equation 12.31. At the same time, s tþDt,Z is calculated by
Equation 12.37.
3

s 0:5
(12:34)
K ¼ K o 1 12.3.4.2 Conditions at Equilibrium
F
The equilibrium condition within the ﬁlter is deﬁned as the
where zone where the rate of attachment is equal to the rate of
K is the intrinsic permeability of porous media as clogged detachment, thus, qs=qt ¼ 0; by deﬁnition, this is the ‘‘satur-
2
with solids (m ) ated zone,’’ that is, Z Z(sat). For such zone, qC=qZ ¼ 0,
r w is the mass density of water at a given temperature C ¼ C o , s ¼ s u , and there is no net change in the suspension
3
(kg=m ) concentration, or [qC=qt] o ¼ 0 (see also Ives, 1982, p. 4).
2
m is the viscosity of water at a given temperature (N s=m ) Therefore, for Equation 12.35, the left side equals zero, to
n
give, after canceling s,
Now substitute Equation 12.34 in Equation 12.33, to give,

after grouping terms, em s u
0 ¼ k 1 C o (F s u ) k 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 (12:38)

r g p
w
K o 1 s u =F
qs em s
¼k 1 vC(F s) k 2 v (12:35)
qt r g p ﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3
w
K o 1 s=F where
C o is the concentration of suspended solids entering the
3
ﬁlter bed (kg suspended matter=m water volume)
s u is the operational storage capacity for suspended solids
12.3.4.1.1 Finite Difference Form of Solids within the pores of ﬁlter (kg suspended matter=m bed
3
Uptake Rate volume)
Equation 12.35 applies to a particular depth, Z, and time, t.
Applying these subscripts and expressing the left side as a In other words, at equilibrium, the rate of attachment
ﬁnite difference, equals the rate of detachment.

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