Environmental Engineering Reference
In-Depth Information
@X
@t
¼
g
X
c
cþ
b
e
Xþ
e
d
X
(7.14)
where
e
denotes one additional parameter. The inhibition factor
e
= Xþ ð Þ
has also
to be included in the decay term of the substrate equation.
If the one-dimensional formulation of the differential (
7.12
) is sufficient, one
may write:
v
@
c
R
@c
@t
¼
@
@x
a
L
v
@
c
c
cþ
b
@x
a
X
@x
@X
@t
¼
g
X
c
cþ
b
d
X
n
(7.15)
Concerning the formulation (
7.15
) it is assumed that dispersion dominates over
diffusion (see Chap. 3): molecular diffusivity is omitted. If the velocity and also the
dispersivity are constants, as for example in column experiments, the coefficients
a
L
and
v
can be taken out of the brackets of the first term of the right hand side.
The simulation of the transient change of concentration and/or population of
biological species, described by a set of 1D equations, can be performed by using
the MATLAB
pdepe
solver that was already described in chap. 4. In the sequel, as
another MATLAB
®
application, we determine the degradation characteristics by
evaluating the steady-state solution.
®
7.5 Steady States
In order to determine the degradation rate it may be sufficient to examine the steady
state. From the 1D formulation for the unsteady situation a set of ordinary differen-
tial equations emerges. Two equations result for the system described by the (
7.15
):
@x
a
L
v
@
@
v
@
c
cþ
b
¼
@x
c
@x
c
a
X
0
c
cþ
b
d
X
n
g
X
¼
0
(7.16)
If bacteria populations remain above zero, a steady-state value for
X
, in depen-
dence of
c
can be extracted from the second equation:
1
n
1
X ¼
g
d
c
cþ
b
(7.17)
