Environmental Engineering Reference
In-Depth Information
there are conditions required at both ends of the interval of the independent
variable. A special MATLAB
related textbook on modeling with odes is
published by Shampine et al. (
2003
).
®
9.1 Streeter-Phelps Model for River Purification
A relatively simple model concerning decreased concentration of oxygen down-
stream from a polluting discharge and the recovery to background level was
proposed by Streeter and Phelps already in
1925
. Although the application is
based on several assumptions, such type of modeling is used as part of regulations
for sewage facilities. Bacteria gradually degrade the organic matter contained in the
discharge along the course of the river downstream. The most relevant bacteria
prefer aerobic conditions, i.e. they also rely on oxygen (DO
dissolved oxygen).
The change of these two components (organic matter and DO) along the flow path is
simulated in the following.
In the model, the concentration of the organic pollutant is measured as biode-
gradable oxygen demand (BOD) as a proxy. Two parameters are connected to the
BOD behavior: the inflow rate
f
BOD
(kg/m
3
/s) and the degradation rate
k
1
(1/s).
Degradation concerns both BOD and DO, which is expressed in the system of two
ordinary differential equations
¼
@
c
BOD
@t
¼ f
BOD
k
1
c
BOD
@
c
DO
@t
¼ k
2
c
DO;sat
c
DO
k
1
c
BOD
(9.1)
The oxygen concentration
c
DO
is additionally determined by the reaeration
process. Reaeration brings oxygen back into the water, for which various processes
may be relevant. One of the most important is the contact with atmospheric air. If
the contact time is long enough, the equilibrium between partial pressure of oxygen
in air
p
O
2
and
c
DO
is established. According to Henry's Law
1
such equilibria are
characterized by a component specific ratio of dissolved concentration and partial
pressure. As in the earth atmosphere
p
O
2
is fixed with approximately 0.21 atm,
c
DO
can reach a value of 12.9 mg/l,
2
which is the saturation limit for oxygen in water.
However, the saturation limit in a natural river may be somewhat smaller as
a result of other processes that influence the oxygen balance. Aquatic plants
produce oxygen, aquatic fauna consumes oxygen. Moreover, oxygen is needed by
bacteria which are busy degrading natural organic matter at the bottom of the water
1
William Henry, 1775-1836, English chemist.
2
Henry's Law constant for the equilibrium between gaseous and aqueous phase oxygen at 5
Cis
61.2 mg/l/atm.
