Page 797 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

combined with kinetics. The stationary bed could be a where
3
traditional trickling filter, about 2 m depth, or any deep-bed X is the cell concentration (kg=m filter volume)
nitrification biofilm reactor. The approach is generally Y is the yield coefficient (kg cells synthesized=kg substrate
applicable. degraded)
^ m is the maximum specific rate of reaction (cells synthesi-
23.A.1.1 Mathematics zed=cell mass=s)
3
K m is the half saturation coefficient (kg substrate=m )
The analysis of a biofilm reactor is described starting with the
definition sketch, Figure 23.A.1. The approach is the same as
The kinetics described by Williamson and McCarty (1976a,b)
the ‘‘plug-flow’’ model seen in Section 23.2.2.5 for activated
utilizes the Monod model but incorporates it into the diffusion
sludge.
process. This depiction does not consider diffusion and
For the infinitesimal element shown in Figure 23.A.1 the
focuses on a macroscopic model.
materials balance is
Substituting (23.A.2) in Equation 23.A.1,

qS qS qS
P A DZ ¼ v DZ P A P A DZ qS qS 1 S
qt qZ qt P A DZ ¼ v ^ m X A DZ
o r qt qZ DZ P A Y K m þ S
(23:A:1) o
(23:A:3)
where
3 The concentration of cells in the bulk volume is
S is the substrate concentration (kg=m water volume)
t is the elapsed time (s)

P is the porosity X ¼ XA(rocks) (23:A:4)
2
A is the area of filter bed (m )
Z is the depth of filter bed (m) where

DZ is the depth of finite element (m) X is the concentration of cells per unit of surface area (kg
2
3
X is the cell concentration (kg=m filter volume) cells=m rock surface)
A is the surface area of rocks or other media per unit of
Y is the yield coefficient (kg cells synthesized=kg substrate
3
2
degraded) bulk volume in filter bed (m rock surface=m bulk
filter volume)
As seen, Equation 23.A.1 is common to any kind of fixed-
bed reactor, such as used with granular activated carbon. Substituting (23.A.4) in (23.A.3) gives
The difference is with the kinetics equation. The Monod
equation is used, but arguably it is an ‘‘artifice.’’ The rate
qS qS
of substrate utilization, [dS=dt] r , may be limited by diffusion P A DZ ¼ v DZ P A
qt qZ
transport (Chapter 18) of the carbon substrate or another o

reactant such as a nutrient to the microbial cell for reaction. 1 S
As will be seen, the constants in the Monod equation will be Y ^ m K m þ S XA(rocks) A DZ
‘‘lumped’’ into an empirical coefficient so it does not make
(23:A:5)
much difference as far as final outcome is concerned
whether the rate is described actually by diffusion, i.e.,

The specific area, A, can be estimated by assuming the rocks
Fick’s first law, or the Monod equation. For reference, the
are spheres and that they have a rectangular packing.
Monod equation is

A(rocks) ¼ A(rock) N(rocks) (23:A:6)
qS 1 S
^ m X (23:A:2)
2
qt Y K m þ S ¼ pd(rock) N(rocks) (23:A:7)
¼
r
where
– 2
S – ΔZ vAP A(rock) is the surface area of one rock (m )
Z
2
N(rocks) is the specific number of rocks (number of
3
ΔZ rocks=m bulk filter volume)
d(rock) is the diameter of one rock (m)
–
S + ΔZ vAP
Z
2
The minimum number of rocks per unit volume may be
FIGURE 23.A.1 Definition sketch for finite element materials estimated by assuming that they have a rectangular packing.
balance. The number of rocks along a given side in a cube of 1.0 m is