Environmental Engineering Reference
In-Depth Information
l
=
m
−
n
+
1
6.1
These are basic loops which, if mutually connected, also merge into larger loops, increasing
the number of alternative routes for water flow in case of pipe failure. For adjacent loops, the
total number of network loops
L
n
, which includes both the basic loops and complex loops
combined from the basic loops, will be:
(
) (
)
1
2
L
n
=
l
(
l
−
1
+
1
=
m
−
n
+
m
−
n
+
6.2
Equations 6.1 and 6.2 are also valid if pipes cross each other without having a connection,
which is rare but possible case in reality.
Thus, the number of loops will be unique for specified value of
a
=
m
-
n
. A
network grid
index
(NGI) can be defined taking into consideration the ratio between the number of basic
loops and the total number of loops, assuming that the basic loops are adjacent or are at least
sharing one node.
l
a
2
NGI
=
1
−
=
6.3
2
L
a
+
a
+
1
n
Equation 6.1 is correct for any network configuration. In case of serial- and branched
configurations,
a
= - 1. This makes the value of
L
n
in Equation 6.2 equal to 1, and
consequently the
NGI
in Equation 6.3 is also equal to 1. For looped configurations,
l
>
0, and
NGI
takes the values greater than 0 and lower than 1. To arrive at practical range, the
NGI
for
serial- and branched configurations is set at 0 instead of 1, to indicate the lowest connectivity;
this is also done in view of the fact that the result
L
n
= 1 leads to a false conclusion because
those configurations have no loops. Hence, the higher value of
NGI
should indicate the
networks of higher degree of connectivity, in general.
Parameter
a
denotes a unique value of
NGI
that can be achieved for various combinations of
m
and
n
. For example, either for
n
= 999 and
m
= 1001, or for
n
= 5 and
m
= 7, there will be
three basic loops, although not necessarily adjacent in case of larger networks. Due to the
difference in size, these two networks will never be compared for any viable reason. Still,
there is a problem that selected combination of
m
and
n
offers more than one possible layout
evaluated by the same value of
NGI
. Four networks shown in Figure 6.2, illustrate this point.
In all four cases,
m
= 13,
n
= 11,
l
= 3,
a
= 2,
L
n
= 7, and
NGI
= 0.571. Also, all four networks
have total 26 connections to the nodes, which are distributed in different way. In case of
network L1, 11 nodes are connected in the following scheme: two nodes with one connection
(R1 and N11), three nodes with two connections (N4, N5 and N10) and the remaining six
nodes with three connections (N2, N3, N6, N7, N8 and N9).
Reconnecting pipe P7 to node N11, instead of N8, yields configuration L2 (upper-right in the
figure); as a result, the supply reliability of N11 is likely to improve being now connected
from two sides. Thus, the connectivity scheme has changed, with only the source in R1
having one connection, five nodes having two connections and five nodes having three
connections. Alternatively, reconnecting P7 with N6 would result in the connectivity scheme
L3 (lower-left in the figure) consisting of two nodes with one connection, four with two, four
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