64 ENVIRONMENTALENGINEERING
This may be expressed mathematically as:d
dt-D(t) = k’,z(t) - kiD(t),
which can be solved to give:
(4.11)(4.12)where Do is the initial oxygen deficit in the stream at the point of wastewater discharge,
after the stream flow has mixed with the wastewater, in milligrams per liter (mg/L).
The deficit equation can also be expressed in common logarithms:
(4.13)since:
e-k’t = when k = 0.43k‘.
The initial oxygen deficit (Do) is calculated as a flow-weighted proportion of the initial
stream oxygen deficit and the wastewater oxygen deficit:
DSQS + DpQp
Do =
Qs+Qp ’(4.14)where D, is the oxygen deficit in the stream directly upstream from the point of
discharge, in milligrams per liter (mg/L); Q, is the stream flow upstream from the
wastewater discharge, in cubic meters per second (m3/s); Dp is the oxygen deficit in
the wastewater being added to the stream, in milligrams per liter (mgL); and Qp is the
flow rate of wastewater, in cubic meters per second (m3/s).
Similarly, the ultimate carbonaceous BOD (Lo) is:
LQs + LpQp
Lo =
Qs+Qp ’(4.15)where L, is the ultimate BOD in the stream immediately upstream from the point of
wastewater discharge, in milligrams per liter (mg/L); Qs is the stream flow upstream
from the wastewater discharge, in cubic meters per second (m3/s); L.,, is the ultimate
BOD of the wastewater, in milligrams per liter (mgiL); and Qp is the flow rate of the
wastewater, in cubic meters per second (m3/s).
The most serious water quality concern is the downstream location where the
oxygen deficit will be the greatest, or where the dissolved oxygen concentration is the