22 Computational Modelling in Hydraulic and Coastal Engineering
2.2 WATER QUALITY MANAGEMENT IN A LAGOON
A second application of the numerical solution of an ODE involves the
water quality management in a lagoon connected through a small inlet
to a vast water body (the open sea). The water volume of the lagoon, V, is
renewed by means of a time-dependent discharge, Q(t). This inflow and
(equal) outflow discharge is driven by various natural causes such as the
wind, tide, watershed drainage and direct precipitation, and also human
activities (i.e. pumping). Considering a time-average renewal discharge, Q,
then the renewal (or flushing) time of the lagoon is defined as T
V
Q
f = .
The water quality in the lagoon could be described by knowing the concentration levels of various substances, such as dissolved oxygen (DO), biochemical oxygen demand (BOD), nutrients, heavy metals and toxic organic
compounds. In the following, for simplicity and without loss of generality, a
single pollutant will be considered. Thus, the water quality will be determined
by the mean concentration value (C) of a contaminant uniformly distributed
over the entire lagoon area. Contaminants may enter the lagoon through point
sources (industrial or municipal pipe outfalls) or distributed (agricultural or
urban runoff). Considering a point source with flow rate q and contaminant
concentration c, it is assumed that once discharged, the contaminant spreads
and mixes instantaneously and uniformly through the lagoon.
In order to estimate the contaminant concentration, C(t), in the lagoon,
a mass balance (continuity) equation can be derived for that particular
substance. Using the volumetric approach, over a finite time interval (Δt),
the volume of inflowing contaminant is cqΔt and that of the outflowing is
C(Q + q)Δt, assuming that there is not any pollution inflow from the open
sea. The difference between the inflow and outflow contaminant fluxes
defines the change of the contaminant concentration within the lagoon,
VΔC. Thus, the continuity equation reads
VΔC = cqΔt – C(Q + q)Δt
(2.14)
After taking the limits of ΔC and ΔQ, Equation 2.14 results in an ODE:
dC
dt
q
c
V
Q q
C
V
=
− +
(
)
(2.15)
This is a first-order inhomogeneous ODE analytically solvable for constant
q and Q, but requiring a numerical solution in the case of varying Q(t),
or q(t) (or c(t)). If both q and Q are constant, after a considerable time the
analytical solution for the concentration C stabilizes to
C
cq
Q q
= +
(2.16)
2.2 WATER QUALITY MANAGEMENT IN A LAGOON
A second application of the numerical solution of an ODE involves the
water quality management in a lagoon connected through a small inlet
to a vast water body (the open sea). The water volume of the lagoon, V, is
renewed by means of a time-dependent discharge, Q(t). This inflow and
(equal) outflow discharge is driven by various natural causes such as the
wind, tide, watershed drainage and direct precipitation, and also human
activities (i.e. pumping). Considering a time-average renewal discharge, Q,
then the renewal (or flushing) time of the lagoon is defined as T
V
Q
f = .
The water quality in the lagoon could be described by knowing the concentration levels of various substances, such as dissolved oxygen (DO), biochemical oxygen demand (BOD), nutrients, heavy metals and toxic organic
compounds. In the following, for simplicity and without loss of generality, a
single pollutant will be considered. Thus, the water quality will be determined
by the mean concentration value (C) of a contaminant uniformly distributed
over the entire lagoon area. Contaminants may enter the lagoon through point
sources (industrial or municipal pipe outfalls) or distributed (agricultural or
urban runoff). Considering a point source with flow rate q and contaminant
concentration c, it is assumed that once discharged, the contaminant spreads
and mixes instantaneously and uniformly through the lagoon.
In order to estimate the contaminant concentration, C(t), in the lagoon,
a mass balance (continuity) equation can be derived for that particular
substance. Using the volumetric approach, over a finite time interval (Δt),
the volume of inflowing contaminant is cqΔt and that of the outflowing is
C(Q + q)Δt, assuming that there is not any pollution inflow from the open
sea. The difference between the inflow and outflow contaminant fluxes
defines the change of the contaminant concentration within the lagoon,
VΔC. Thus, the continuity equation reads
VΔC = cqΔt – C(Q + q)Δt
(2.14)
After taking the limits of ΔC and ΔQ, Equation 2.14 results in an ODE:
dC
dt
q
c
V
Q q
C
V
=
− +
(
)
(2.15)
This is a first-order inhomogeneous ODE analytically solvable for constant
q and Q, but requiring a numerical solution in the case of varying Q(t),
or q(t) (or c(t)). If both q and Q are constant, after a considerable time the
analytical solution for the concentration C stabilizes to
C
cq
Q q
= +
(2.16)