Environmental Engineering Reference
In-Depth Information
k
∑
−
=
1
Q
=
∆
Q
−
∆
Q
5.15
f
i
,
f
j
,
f
j
1
Thus, each pipe
j
connected to the junction with the failed pipe
f
carries the flow
increment/decrement that can be correlated to the original flow of pipe
f
. In the above
network,
Q
f
=
Q
J7-J3
= 10.51 l/s during regular operation. After the closure of pipe
J7
-
J3
, in
node
J7
ΔQ
J2-J7,f
= -9.73 l/s and
ΔQ
J6-J7,f
= -0.78 l/s; both flows balance the value of
Q
f
because
ΔQ
J7,f
= 0. Equally, in node
J3
,
ΔQ
J3-J4,f
= -1.55 l/s and
ΔQ
J3,f
= -8.96 l/s, which also
balance the
Q
f
.
Eventually, the changes in flow rates and nodal demands in all the pipes and nodes can be
correlated to certain proportion
p
of the loss of demand and/or the flow of pipe
f
before the
failure. For instance, after combining Equation 5.9 and 5.15, the continuity of flows in two
junctions that connect the failed pipe yields:
k
−
1
−
p
(
)
(
)
∑
5.16
Q
=
−
p
Q
1
−
ADF
−
p
Q
⇔
Q
=
α
Q
1
−
ADF
;
α
=
i
f
i
tot
f
j
f
f
i
tot
f
i
k
−
1
∑
1
+
p
j
=
1
j
j
=
1
which resembles the format of Equation 5.2 and 5.7. Similar relations can be developed for
other junctions connected to these, or to the source(s), confirming the dependency of the loss
of demand after particular pipe fails on the flow it carries under regular conditions.
Equation 5.16 will be valid in the nodes where the demand reduction takes place i.e. the
nodal pressure is below the PDD threshold. Where
ΔQ
i,f
= 0, the sum ∑
p
j
= 0; in this case the
impact of the pipe closure is propagated towards surrounding nodes.
In the above network, the following values of
p
i
,
p
j
and
α
i
can be calculated by analysing the
flow continuity in each junction (also shown in Figure 5.10):
R1
:
p
R1-J2
= -9.73/9.63 = -1.0104,
p
R1-J6
= 0.10/9.63 = 0.0104
J7
:
p
J2-J7
= -9.73/10.51 = -0.9258,
p
J6-J7
= -0.78/10.51 = -0.0742
J2
:
p
R1-J2
= -1.0104,
p
J2-J7
= -0.9258,
α
J2
= -1.0104/-0.9258 = 1.09
J3
:
p
J3
= -8.96/9.63 = -0.9304,
p
J3-J4
= -1.55/10.51 = -0.1475,
α
J3
= 0.9304/0.8525 = 1.09
After combining the continuity equations for nodes
J6
and
J4
, the flow increment of 0.88 l/s
registered in pipe
J6
-
J4
will disappear. Hence:
J6
:
p
R1-J6
= 0.0104,
p
J6-J7
= -0.0742 and
J4
:
p
J4
= -0.67/9.63 = -0.0696,
p
J3-J4
= -0.1475
and consequently,
α
J4
= (0.0104 + 0.0742)/(-0.0696 + 0.1475) = 1.09
In all the cases, the continuity equations in junctions lead to the value of
α
i
of 1.09, which is
the ratio between the flow in pipe
J7
-
J3
during regular operation and the loss of demand after
that pipe has been closed. Thus, 10.51/9.63 = 1.09. Consequently, the bigger value of
α
i
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