Environmental Engineering Reference
In-Depth Information
the same network parameters as discussed in the previous paragraph. Mathematical proof of
correlation between the pipe flow and the loss of demand after the pipe failure is however
more difficult to derive in case of fully looped 'non-optimised' networks. The resistance of
pipes and the head at supply point(s) will be factors of influence i.e. whether there will be any
loss of demand and how significant it will be in terms of the network coverage and intensity.
The changes in governing equations after the pipe failure occurs are described in the
following sections.
5.4.1 The Law of Continuity in Each Junction
After the failure of pipe f , the change of flow ( ΔQ j,f ) in k pipes connected to junction i will
match the change of demand/supply ( ΔQ i,f ) in that junction:
k
k
(
)
Q
+
Q
=
Q
+
Q
Q
=
Q
5.8
i
i
,
f
j
j
,
f
i
,
f
j
,
f
j
=
1
j
=
1
If the demand of node i has been reduced, ΔQ i,f will be negative. If the flow rate of pipe j has
decreased and/or its direction has been reversed, ΔQ j,f will also be negative. If the flow rate
has increased and has kept the same direction, ΔQ j,f becomes positive.
5.4.2 Total Loss of Demand from Failure of Pipe
For l sources and n nodes, the loss of demand as a result of the failure of pipe f will be:
1
l
1
n
1
ADF
=
Q
=
Q
5.9
f
s
,
f
i
,
f
Q
Q
s
=
1
i
=
1
tot
tot
l
n
Q
=
Q
=
Q
5.10
tot
s
i
s
=
1
i
=
1
Having in mind that the intended- i.e. design demand can be reduced even at no-failure
conditions, it will be more accurate to assume that Q i is the target demand ( Q i,t ), which is to
be supplied with sufficient head at the sources.
5.4.3 Relation Between Nodal Demand and Pressure
For given range of pressures 0 < p i < p min , the p min being the PDD threshold, the relation
between the nodal demand and pressure is usually assumed to be exponential, as follows:
0
p
Q
=
k
i
5.11
i
i
ρ
g
Above the threshold, the targeted demand ( Q i,t ) will be supplied 100 %. Consequently, the
loss of demand in node i caused by the failure of pipe f can be expressed as:
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