Environmental Engineering Reference
In-Depth Information
For a description that takes more details into account concerning the bacterial
degradation, a formulation can be used in which the bacteria population
X
appears
as another variable. In addition to the differential equation for the chemical species,
another differential equation for the biological species has to be added. As exten-
sion of equation
(3.20)
it is possible to write:
R
@
c
c
cþ
b
@t
¼r
D
rð Þ
v
rc
a
X
@
X
@t
¼
g
X
c
cþ
b
d
X
n
(7.12)
with retardation factor
R
, dispersion tensor
D
, velocity
v
and parameters
a
;
b
;
g
. The degradation rate for the substrate is linearly dependent on
X
.
For high concentrations
c
, a maximum rate of
and
d
a
X
is reached. Half of that maximum
is given for the substrate concentration
C¼
b
. Such behavior is described by the
Monod term, the last term in the first equation of system (
7.12
). The same func-
tional dependency is used to describe the growth of the bacteria population where
the coefficient g includes the relation between bacteria population and substrate
concentration. The last term in the second equation of system (
7.12
) accounts for
the decline of the bacteria population or natural death of the bacteria, for which two
additional parameters,
and
n
, are introduced. Bacteria are assumed not to migrate
with flow; that's why the dispersion and advection terms are missing in the second
equation.
Approaches as in (
7.12
) are common in biogeochemical modeling. The use of
a linear term for the decline of
X
is a common approach used in biogeochemical
modeling (Lensing
1995
; Tebes-Stevens et al.
1998
). For
n ¼
d
2, the approach
coincides with the so-called logistic equation, which is most popular in the
biological and ecological sciences; see Chap. 9). Marsili-Libelli (
1993
) refers to
the given approach with a first order growth term in
X
and a degradation term with
a free exponent as
Richards dynamics
. Even more general approaches with free
exponents in growth as well as in decay terms are examined by Savageau (
1980
).
An alternative formulation of similar complexity is obtained when the second
equation of the system (
7.12
) is replaced by
@X
@t
¼
g
X
c
cþ
b
d
1
X
d
2
X
2
(7.13)
In (
7.13
) a linear and a quadratic decay term are included. In a discussion of
various different mortality terms in ecological models, Fulton et al. (
2003
) favour
such an approach stating that the linear term represents 'basal' mortality, while the
quadratic term is due to predators which are not explicitly represented in the model.
A further approach for modeling the bacteria population was suggested by
Sch
afer et al. (
1998
), following Kindred and Celia (
1989
), using an inhibition
term for the bacteria population:
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