Encyclopedia of Environmental Science and Engineering, Volume I and II

BIOLOGICAL TREATMENT OF WASTEWATER 139

or

Nt^ Noekot

where:

N 0 = Number of viable microorganisms per unit volume at

time t = 0

N t = N = Number of viable microorganisms per unit volume

at time t

and

k = Logarithmic growth rate constant, time^1.

In wastewater treatment practices, the growth pattern

based on mass of microorganisms has received more atten-

tion than the number of viable microorganisms. If each

microorganism is assumed to have an average constant mass,

then N in Eq. 1 can be replaced with X, the mass of active

microorganisms present per unit volume to obtain:

(^)
d
d
X
t
kX 0.
(2)
The growth of bacterial population may become limited
either due to exhaustion of available nutrients or by the accu-
mulation of toxic substances. The growth rate of bacteria
starts slowing down, and Eq. 1 changes to the form:
(^)
d
d
N
t
kNt
(3)
where growth rate factor k t , varies with time and becomes a
function of temperature, T, pH, substrate concentration, S,
and concentration of various nutrients, C n 1 , C n 2 , etc., i.e.:
k t = V 1 ( T, pH, C s , C n 1 , C n 2 , ... ).
Figure 4 shows variation in growth rate k t with change in
nutrient concentrations, assuming that T and pH are held con-
stant and substrate concentration, S, is greater than the critical
substrate concentration, S , above which k t , is independent
of S. Several interesting observations are made from these
curves.^8 First, the maximum value of k t is essentially constant.
Second, the shape of the curve and the limiting concentration
is different for each nutrient. Third, k t is shown to be zero
when any of the nutrients is missing. Fourth, as the biological
reaction proceeds, all nutrients are consumed. Thus, even if
all nutrients are initially in excess, the growth may eventually
become limited. Finally, as the concentration drops to zero, a
stationary phase is reached, i.e., d N /d t becomes zero.
In case of a substrate limited system, rate of growth is
given by:
(^)
d
d
N
t
mN
(4)
or
d
d
X
t
X.
The following simple relationship between specific growth
rate of microorganisms, μ , and substrate concentration, S,
was developed by Monod^9 and has been widely accepted:
(^)
  


d
d
d
d
N
Nt
X
Xt
S
KS
mmax
(5)
where K is a constant called half velocity coefficient and μ (^) max
is maximum specific growth rate.
It is postulated that the same amount of substrate is incor-
porated in each cell formed. Therefore, the rate of increase
in number or mass of microorganisms in logarithmic growth
phase, d N /d t, or d X /d t, is proportional to the rate of substrate
consumption, d S /d t, or d L /d t, if the substrate concentration
is measured in terms of its BOD, L, and the following rela-
tionship can be stated:
(^)
d
d
d
d
X
t
Y
S
t

(6)
or
∆ X = Y ∆ S
where Y is called the growth yield coefficient, ∆ X is the
cell mass synthesized in a given time, and ∆ S is substrate
removed in the same time. The substrate utilization rate, q,
per unit biomass has been defined as:
(^)
q
S
Xt

d
d^
(7)
0
0
Cn 2 CCnn 11
Cn 1 + Cn 2

• 
  


• 
  

  kmax
  k (C
  n^1
  , C
  n^2
  )
  k vs Cn 2 (Cn 1 > Cn 1 )
  k vs Cn 1 (Cn 2 > Cn* 2 )
  FIGURE 4 k vs nutrient concentration.
  C002_001_r03.indd 139C002_001_r03.indd 139 11/18/2005 10:15:49 AM11/18/2005 10:15:49 AM