Environmental Engineering Reference
In-Depth Information
18.2 Linear Systems
A first, most simple approach comprises component mass fluxes as linear functions
of some state variable of the involved compartments. The procedure is best
demonstrated on a sequence of lakes, connected by streams, as depicted in
Fig.
18.5
. If some substance is introduced into the lake most upstream, it is subject
of mixing processes first. It is assumed that mixing is fast in comparison to
the
residence time
of the lake
1
. In the argumentation we assume all fluxes
Q
i
and
volumes
V
i
to be known, and the concentrations
c
i
being the unknowns to be
determined.
The concentration in the outflow of the lake is equal to the concentration
c
1
. The
total mass leaving the lake per time is thus
Q
1
c
1
(
t
), where
Q
1
denotes the mean
water flux. Neglecting further losses, that mass enters the next lake downstream.
Again the same argumentation can be used to set
Q
2
c
2
(
t
) as the flux out of the
second lake into the third. The procedure can be extended to the entire system
of lakes. Following the principle of mass conservation (Chap. 2), one obtains
a differential equation for each lake. For the
i
th lake holds:
V
i
@
c
i
@t
¼ Q
i
1
c
i
1
Q
i
c
i
(18.3)
where
V
i
denotes the mean volume of the lake. In systems theory such a set-up
is called
donor controlled
, as the input for the following compartment (lake) is
determined by the state variable (concentration) of the previous compartment.
The contrasting term is
recipient controlled
, i.e. the flux is determined by the
concentration of the receiving compartment (for the lake sequence that approach
does not make sense). Each equation can be divided by
V
i
.
Using matrix notation
the resulting equations can be written in one system, representing all lakes:
@
@t
c
¼
Bc
(18.4)
with elements
B
i
1
;i
¼ Q
i
1
=V
i
and
B
i;i
¼Q
i
=V
i
. The matrix
B
is a generalization
of the adjacency matrix. The off-diagonal locations of the zeroes in both matrices
Discharge Q
2
Recharge Q
0
Discharge Q
1
Lake 1
Lake 2
...
Volume V
1
Concentration c
1
Volume V
2
Concentration c
2
Fig. 18.5
Scheme for a sequence of lakes
1
Under steady state conditions (inflow
Q
i
1
¼
outflow
Q
i
) the residence time is given by
V
i
/Q
i
;
the notation is given in the text.
