152
6. Water Quality and Pollution
Considerations
Inventory constraints are the constraints that describe how the DO,
the BOD, and oxygen deficits vary from the beginning to the end of a
reach. The material balance constraints ensure that conservation of mass
is obeyed at each confluence for total water flow, DO, and BOD.
The total flow in link m is the sum of the flow in the previous link
(m — 1), Qi t m-i t t,
the tributary flow entering the link, QT imt ,
and the waste
water flow discharged into the link, QWi mt
:
Qimt - Qi.m-i.t + QT imt
+ QW im t
for all t, m, and t
(6.10)
Equation (6.10) corresponds exactly to Eq. (2.13a). Assuming complete
mixing, the BOD concentration at the beginning of each reach BB imi
is
equal to the sum of the BOD concentration at the end of the previous
reach, BEi, m -\,t,
that in the tributary, BT{ mi ,
and that from the waste
water plant(s), BWi mt ,
times their respective flows:
BBimt = {\/Qimt)[BEi tm -.\ % tQi t m-l,t
+ BT i mt QT i mt +
BWimtQWimi]
for all i f ra, and t (6.11)
In a similar fashion, the total DO at the beginning of link ra, CBi mt ,
may
be calculated:
CB imt
= (l/QimtKCEi^&i.rn-l.t
+ CT imt QT imt
+
CWimtQWimtl
for all i } m, and t (6.12)
The DO deficit at the beginning of each link, DB{ mt ,
is the difference between the saturation concentration, CSi mt ,
a constant, and the initial DO
concentration:
DB imt
= CSimt - CB imt
ΐοτ all i, m, and t
(6.13)
The BOD concentration at the end of link m, BE imt ,
is a linear function of
BBimt :
BEimt = VmBBimt + Mm
for all t, m, and t
(6.14)
Similarly the DO deficit at the end of link m, DE im t,
is a linear function of
DBimt and BB im t
:
DEimt = QmDBimt + jmBBimt
+ Pm
ior all, z, m, and t
(6.15)
The DO concentration at the end of each link, CEimt, is the difference
between the saturation concentration CSimt and the final deficit DEimt :
CE^t
= CSi„t - DE^t
ίοτ all i, m, and t
(6.16)