Environmental Engineering Reference
In-Depth Information
Fig. 9.1 Model results for
river purification
9
concentration
8
BOD
DO
7
6
5
4
3
0
5
10
15
20
25
traveltime
is in mg/(l
d). All concentrations are in mg/l. The steady state is given by
c
BOD
¼
f
BOD
=k
1
¼
5 mg/l.
Figure
9.1
shows the result for the increased input values with
c
BOD
¼
3
:
33 mg/l and
c
DO
¼ c
DO;sat
f
BOD
=k
2
¼
8
:
7.5 mg/l.
BOD concentration decreases within a time period of 15 days. The content of
oxygen decreases within the first 3 days if degradation exceeds reaeration. After
that follows a second time period in which reaeration is dominant leading to
a gradual recovery of the DO level back to the natural state.
The Streeter-Phelps model is based on several conditions. Diffusion and disper-
sion processes are neglected. There is no distinction of concentrations within the
river cross-section. The system of differential (
9.1
) is based on a Lagrangian
3
description, which is a formulation for the concentration along the flow path. In
the Lagrangian description the advection terms disappear, whereas they remain in
the alternative Eulerian description. The Eulerian approach, which was introduced
in Chaps. 3 and 4, is based on the conception of a fixed space and delivers the
following set of equations
@
c
BOD
@t
¼r
j
ðc
BOD
Þþf
BOD
k
1
c
BOD
k
1
c
BOD
@
c
DO
@t
¼r
j
ðc
DO
Þþk
2
c
DO;sat
c
DO
(9.2)
Also with respect to biogeochemistry the Streeter-Phelps model has to be
extended to capture a more detailed behavior. Vanrolleghem et al. (
2000
) present
and discuss several generalizations of the simple Streeter-Phelps approach. In order
to consider photosynthesis-respiration, the model contains four more variables:
ammonium, nitrate, phosphorus and algae. The mathematical description takes
3
Joseph-Louis Lagrange, 1763-1813, French mathematician.
