Environmental Engineering Reference
In-Depth Information
X
n
T
i;in
¼
f
ij
þ
z
i
j¼
1
where
z
i
is the boundary flow into i. The outflow from i can be expressed as:
X
n
T
i; out
¼
f
ji
þ
y
i
j¼
1
where
y
i
is the boundary outflow from i. At steady state, a necessary condition for
the network flow analysis,
T
i,in
=
T
i,out
and one compartmental throughflow vector
can be written as T =(
T
i
). The total system throughflow (TST) is given by the sum
of the compartmental throughflows:
X
n
TST
¼
T
i
j¼
1
The motivation for flow partitioning begins with nondimensional flow intensities
(i.e., throughflow-specific flows) which result when flows are divided
by throughflows of originating compartments:
g
ij
=
f
ij
/
T
j
. The elements of matrix
G =(
g
ij
) give the dimensionless transfer efficiencies corresponding to each direct
flow,
f
ij
. Powers G
m
of this matrix give the indirect flow intensities associated with
paths of lengths m = 2, 3,
. Due to dissipation, flow along these indirect paths
...
so that the power series
P
m¼
0
G
m
representing the sum
of the initial, direct, and indirect flows converges to an integral flow intensity
matrix, N:
approaches zero as m
!1
G
2
G
3
G
m
integral
¼
N
initial
þ
I
direct
þ
G
þ
þ
...
þ
þ
...
|{z}
|{z}
|{z}
|
{z
}
indirect
Þ
1
¼ð
I
G
N maps the steady-state input vector z into the steady-state system throughflow
vector:
G
2
G
3
G
m
T
¼
Nz
¼ð
I
þ
G
þ
þ
þ ::: þ
þ :::Þ
z
Term by term, flow intensities G
m
of different orders m are propagated over
paths of different lengths m. The first term, I, brings the input vector z across the
system boundary as input
z
j
to each initiating compartment, j. The second term, G,
produces the first-order direct transfers from each j to each i in the system. The
remaining terms where m
1 define m
th
order indirect flows associated with length
m paths. As stated before, these go to zero in the limit as m
>
, which is
necessary for series convergence. This demonstrates that each “direct” flow
f
ij
at
!1
