60 Computational Modelling in Hydraulic and Coastal Engineering
discharges of all pipes connected at the junctions. For this purpose the continuity and energy equations are employed (Lencastre 1995).
4.2.1 Continuity (junction) equation
Due to the incompressibility of water, at every junction (j) the sum of all
discharges – inflowing or outflowing and those from the connected pipes
(k-number) – should be zero:
(
)
σ k k
j
k
Q
Q
± =
∑
0
(4.10)
where Q k are the flows of all connecting pipes at joint j, σ k = (±) accounts for
the sign of each pipe flow (positive for inflowing and negative for outflowing) and Q j is the inflowing (+) or outflowing (–) discharge at the junction
(j = 1 to J-number of junctions; also see Figure 4.2). Since both the direction
and magnitude of the discharge of the connecting pipes are unknowns, it
is important that the discharges selected during the first approximation
satisfy the continuity equation (Equation 4.10), and also the convention of
positive and negative flows should be applied consistently throughout the
network.
4.2.2 Energy (loop) equation
As it has been presented before, for each branch of the network (i), the
energy loss h fi depends on the resistance coefficient R i and Q i
2
(where i = 1,
+ Q j
– Q k=3
+ Q k=1
+ Q j + Q k=1 – Q k=2 – Q k=3 = 0
Junction
(j)
– Q k=2
Figure 4.2 Continuity of flows at a junction.
discharges of all pipes connected at the junctions. For this purpose the continuity and energy equations are employed (Lencastre 1995).
4.2.1 Continuity (junction) equation
Due to the incompressibility of water, at every junction (j) the sum of all
discharges – inflowing or outflowing and those from the connected pipes
(k-number) – should be zero:
(
)
σ k k
j
k
Q
Q
± =
∑
0
(4.10)
where Q k are the flows of all connecting pipes at joint j, σ k = (±) accounts for
the sign of each pipe flow (positive for inflowing and negative for outflowing) and Q j is the inflowing (+) or outflowing (–) discharge at the junction
(j = 1 to J-number of junctions; also see Figure 4.2). Since both the direction
and magnitude of the discharge of the connecting pipes are unknowns, it
is important that the discharges selected during the first approximation
satisfy the continuity equation (Equation 4.10), and also the convention of
positive and negative flows should be applied consistently throughout the
network.
4.2.2 Energy (loop) equation
As it has been presented before, for each branch of the network (i), the
energy loss h fi depends on the resistance coefficient R i and Q i
2
(where i = 1,
+ Q j
– Q k=3
+ Q k=1
+ Q j + Q k=1 – Q k=2 – Q k=3 = 0
Junction
(j)
– Q k=2
Figure 4.2 Continuity of flows at a junction.