Page 798 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Biological Reactors 753

1.0=d(rock). Therefore, the number of rocks in a unit volume, Rearranging (23.A.17) to solve for qS=qZ,
say 1.0 m, is

qS 1 S pf 1
3
1:0 m ¼ ^ m X (23:A:18)
(23:A:8) qZ Y K m þ S HLR
d(rock)
N(rocks) ¼ d r
Substituting (23.A.8) in (23.A.7), which in finite-difference form is
3
2 1:0 m (23:A:9) DS 1 S pf 1
A(rocks) ¼ pd(rock) ¼ ^ m X (23:A:19)
d(rock) DZ Y K m þ S d r HLR
p
(23:A:10)
¼
d(rock) which in a form for numerical solution is
pf
(23:A:11)

¼ S ZþDZ S z 1 S pf 1
d(rock) ^ m X (23:A:20)
DZ ¼ Y K m þ S d r HLR
where
f is the packing ratio (number of rocks actual=number of or
spherical rocks with rectangular arrangement)
d r d(rock) (m) 1 S pf 1
^ m (23:A:21)
X
Y K m þ S d r HLR
S ZþDZ ¼ S z
Keeping in mind that the packing is not rectangular, a
factor, f, is introduced in (23.A.10) to give (23.A.11); in
Any model must be ‘‘calibrated’’; a calibration factor, F,is
which f 1.0.
inserted in (23.A.21),
Substituting (23.A.11) in (23.A.5),

qS qS 1 S pf 1

P A DZ ¼ v DZ P A ^ m X F DZ
qt qZ S ZþDZ ¼ S Z Y K m þ S HLR
o d r

1 S pf (23:A:22)

^ m X A DZ

Y K m þ S d r
The factor F is inserted in lieu of adjusting the kinetic coef-
(23:A:12)
ficients. This model can be depicted by means of a spread-
Divide both sides of (23.A.12) by PADZ, sheet.

qS qS 1 S X pf 23.A.1.2 Approximation Model by ‘‘Lumping’’
¼ v ^ m (23:A:13)
qt qZ Y K m þ S P d r Coefficients
Another approach in modeling the packed-bed biofilm reactor
Recall that is to ‘‘lump’’ the coefficients along with the S variable in the
denominator in the Monod relation. This approach is used
vPA ¼ HLR A (23:A:14) most often in other references. Repeating (23.A.18),
or
ds 1 S pf 1

^ m X (23:A:18)
dz Y K m þ S HLR
¼
HLR d r
(23:A:15)
v ¼
P
Regrouping terms in (23.A.18) and including F,
3
where HLR is the hydraulic loading rate (m water=s=m 2

filter bed). ds 1 pf X S
¼ ^m F (23:A:23)
Substituting (23.A.15) in (23.A.13) dz K m þ S d r Y HLR

qS HLR qS 1 S X pf Separating the variables,
^ m (23:A:16)
¼

qt P qZ Y K m þ S P d r
o

X
dS 1 pf F
Assuming steady state, ¼ ^m dZ (23:A:24)
S K m þ S d r Y HLR

qS 1 S pf S 1 pf X Z

0 ¼ HLR ^ m X (23:A:17) ln ¼ ^m F (23:A:25)
qZ Y K m þ S d r S o K m þ S d r Y HLR