Flow in pressurized conduits 61
N-number of branches; Equation 4.4). Thus, an equation based on the
energy conservation principle can be written for each loop of the network,
stating that the algebraic sum of the head losses along the branches of a
certain loop is zero. This is true when no pump or turbine is placed along
a pipe that would add or subtract energy from the loop. The loop equation
can also be explained by the fact that the total head loss between any two
junctions is independent of the path followed (Figure 4.3). Thus, for each
loop (l) the energy equation takes the form
σ m m m
m
R Q
*
2
0
(
)=
∑
(4.11)
where σ m
* is the directional sign of the flow in each branch forming the
loop (m is the number of branches comprising loop l). In the following, the
clockwise direction is taken as positive and the convention is retained for
all the loops of the network.
The total number of unknown discharges equals the number of N-branches
in the network. The total number of equations that can be written is (J – 1)number of junction equations (Equation 4.10) and M-number of loop equations (Equation 4.11). Since N = M + J – 1, the number of unknowns equals
the number of equations. The resulting non-linear system can also be solved
numerically by using some successive iteration algorithm.
– Q m=2
– Q m=1
+ Q m=4
– Q j+1
Junction
(j)
Junction
(j + 1)
+
Piezometric surface
p j /γ
p j+1 /γ
Loop
(l)
+ Q m=3
+ Q j
Figure 4.3 Energy balance within a loop.