Running Waters - Mathematics and Statistics for the Environment - page 567

aquatic plants, primarily blue-green algae, slows down or ceases at night, creating a diurnal or
daily fluctuation in DO levels in streams. The amount of DO a stream can retain increases as water
temperatures cool and concentration of dissolved solids diminishes.

19.8 STREAM PURIFICATION: A QUANTITATIVE ANALYSIS *
Before sewage is dumped into a stream it is important to determine the maximum BOD loading for
the stream to avoid rendering it septic. The most common method of ultimate wastewater disposal
is discharge into a selected body of water. The receiving water, stream, lake, or river is given the
final job of purification. The degree of purification that takes place depends on the flow or volume,
oxygen content, and reoxygenation ability of the receiving water. Moreover, self-purification is a
dynamic variable, changing from day to day and closely following the hydrological variations char-
acteristic to each stream. Additional variables include stream runoff, water temperature, reaeration,
and the time of passage down the stream.
The purification process is carried out by several different aquatic organisms. As mentioned,
during the purification of the waste, a sag in the oxygen content of the stream occurs. Mathematical
expressions help in determining the oxygen response of the receiving stream. One must keep in
mind, however, that since the biota and conditions in various parts of the stream change (that is,
decomposition of organic matter in a stream is a function of degradation by microorganisms and
oxygenation by reaeration, which are competing processes that are working simultaneously), it is
difficult to quantify variables and results.
Streeter and Phelps first described the most common and well-known mathematical equation for
oxygen sag for streams and rivers in 1925. The Streeter-Phelps equation is presented as follows:

kL
kk e

×
−

( ) +

1
2

a − kt − kt − kt

D = − e De (19.11)

1 2 2

a

1

where
D = Dissolved oxygen deficit (ppm).
k 1 = BOD rate coefficient (per day).
L a = Ultimate BOD of the stream after the waste enters.
k 2 = Reaeration constant (per day).
t = Time of flow (days).
D a = Initial oxygen deficit (before discharge) (ppm).
Note: The deoxygenation constant, k 1 , is the rate at which microbes consume oxygen for aerobic
decomposition of organic matter. The following equation is used to calculate k 1 :

( )

= −

log 1

5

− yL /

( )

kt
=−

−
110 yL or k (19.12)

1

t

where
y = BOD 5 (5 days BOD).
L = Ultimate or BOD 21 .
k 1 = Deoxygenation constant.
t = Time in days (5 days).

* The important concepts presented in this section are excerpted from Spellman, F.R., Stream Ecology and Self-
Purification, , Technomic, Lancaster, PA, 1996.