Flow in pressurized conduits 63
the recently ‘corrected’ one) no longer satisfy the energy balance. Thus
a new correction must be made for all the loops.
Step 3—The procedure is repeated over all loops as many times as needed
until it satisfies a convergence criterion, that is, the maximum value of
correction ΔQ l for all loops of the network is less than a predefined
small number.
During the solution process it should be emphasized that (1) after each correction along a loop the volumetric continuity principle is maintained on all
junctions of the network, and (2) in the case that a discharge of a certain
branch changes sign due to the imposed correction, then care should be taken
for changing the sign of the discharge in the subsequent corrections. This
iterative procedure can be conveniently coded for computer programming,
provided that appropriate information is given describing the structure of the
network. This is accomplished by the synthesis of the connectivity matrix
having (M) rows and (N) columns equal to the M-number of loops and the
N-number of branches, respectively. Each element of the matrix is either 0, +1
or –1. If the branch is not part of the loop, then the element is zero (0); if the
branch is part of the loop with a positive flow direction, then it is positive one
(+1); and if the branch participates to the loop with a negative flow direction,
then it is negative one (–1). In the following, the Hardy Cross method and the
modelling process of its algorithm are explained by an example.
Example 4.1
Consider a pipe network comprised of nine (J = 9) junctions, thirteen
(N = 13) branches and four (M = 4) loops (Figure 4.4). The data provided
include: the external inflows (Q in ) and outflows (Q out ) at all network
junctions, and the resistance coefficients (R i ) and pipe diameters (D i ) for
all branches. Although not shown explicitly, the resistance coefficients
were estimated beforehand from known values of the friction coefficient
(f i ), pipe length (L i ), and pipe diameter (D i ), according to Equation 4.4.
In order to initiate the process, discharge values and directions were
assumed for all branches in such a way that the continuity principle
was maintained at all junctions (Figure 4.4 and Table 4.1). The inflows
to the junction were considered as positive and the outflows as negative. Once the discharges were selected, the connectivity matrix for
each of the four loops was established, keeping the convention of positive flows following a clockwise movement within the loop (Table 4.2).
After that step, all of the required information was available for initiating the iteration process. The convergence criterion for the maximum
correction value of ΔQ l was set equal to 0.001.
The algorithm converged very rapidly, and after seven iterations
the correction ΔQ l was less than 0.001 (Figure 4.5). The computed
discharges are shown in Figure 4.4. The algorithm also estimates the
velocities (u) and the head drop at each branch (dh). In this example,
the connectivity matrix remained unchanged during the iterations.
the recently ‘corrected’ one) no longer satisfy the energy balance. Thus
a new correction must be made for all the loops.
Step 3—The procedure is repeated over all loops as many times as needed
until it satisfies a convergence criterion, that is, the maximum value of
correction ΔQ l for all loops of the network is less than a predefined
small number.
During the solution process it should be emphasized that (1) after each correction along a loop the volumetric continuity principle is maintained on all
junctions of the network, and (2) in the case that a discharge of a certain
branch changes sign due to the imposed correction, then care should be taken
for changing the sign of the discharge in the subsequent corrections. This
iterative procedure can be conveniently coded for computer programming,
provided that appropriate information is given describing the structure of the
network. This is accomplished by the synthesis of the connectivity matrix
having (M) rows and (N) columns equal to the M-number of loops and the
N-number of branches, respectively. Each element of the matrix is either 0, +1
or –1. If the branch is not part of the loop, then the element is zero (0); if the
branch is part of the loop with a positive flow direction, then it is positive one
(+1); and if the branch participates to the loop with a negative flow direction,
then it is negative one (–1). In the following, the Hardy Cross method and the
modelling process of its algorithm are explained by an example.
Example 4.1
Consider a pipe network comprised of nine (J = 9) junctions, thirteen
(N = 13) branches and four (M = 4) loops (Figure 4.4). The data provided
include: the external inflows (Q in ) and outflows (Q out ) at all network
junctions, and the resistance coefficients (R i ) and pipe diameters (D i ) for
all branches. Although not shown explicitly, the resistance coefficients
were estimated beforehand from known values of the friction coefficient
(f i ), pipe length (L i ), and pipe diameter (D i ), according to Equation 4.4.
In order to initiate the process, discharge values and directions were
assumed for all branches in such a way that the continuity principle
was maintained at all junctions (Figure 4.4 and Table 4.1). The inflows
to the junction were considered as positive and the outflows as negative. Once the discharges were selected, the connectivity matrix for
each of the four loops was established, keeping the convention of positive flows following a clockwise movement within the loop (Table 4.2).
After that step, all of the required information was available for initiating the iteration process. The convergence criterion for the maximum
correction value of ΔQ l was set equal to 0.001.
The algorithm converged very rapidly, and after seven iterations
the correction ΔQ l was less than 0.001 (Figure 4.5). The computed
discharges are shown in Figure 4.4. The algorithm also estimates the
velocities (u) and the head drop at each branch (dh). In this example,
the connectivity matrix remained unchanged during the iterations.