Discussion of “ Method to Cope with Zero Flows in Newton Solvers for Water Distribution Systems ” by

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The authors of the discussed paper show a possible strategy for dealing with zero flows in solving the nonlinear equations for water distribution systems when the Hazen-Williams equation is used. Recently, Elhay and Simpson (2011) presented a similar method for solution of the zero-flow problem when the Hazen-Williams model is used, but they also explain and give a solution for the possible problem with zero flow when the Darcy-Weisbach model is used. In this discussion, a few simple remarks on how to avoid the zero-flow problem in a network of pipes will be highlighted. Also, possible physical interpretation related to the problem will be explained.

Zero Flow in the Hazen-Williams Model
Both contributions, by the authors of the original paper and by Elhay and Simpson (2011), to the solution of the zero-flow problem when the Hazen-Williams model is used cannot be disputed. Mathematical interpretation of the problem from both papers stands, but at the same time, everybody has to be aware that the Hazen-Williams equation used in both papers is obsolete and hence should not be used (Liou 1998;Brkić 2012c;Simpson and Elhay 2012). Zero flow can occur when the Hazen-Williams formula is used because the coefficient is always independent of flow. The argument that the Hazen-Williams model can be used because it has been in common use for a very long time  simply does not stand. The fact that the Hazen-Williams model is used for calculation in EPANET is also avoidable because this software equally allows the use of the Darcy-Weisbach model Brkić 2012c). Because the Darcy-Weisbach model with the Colebrook formula for the friction factor is theoretically more sound (Brkić 2011c(Brkić , 2012b, the use of the Hazen-Williams equation is strongly discouraged. Finally, the Darcy-Weisbach model can also be used for calculation of gas distribution networks, whereas the Hazen-Williams model cannot be used in any circumstances (Brkić 2009(Brkić , 2011a.

Zero Flow in the Darcy-Weisbach Model
In contrast, the zero-flow problem can occur when the Darcy-Weisbach formula is used only if laminar flow takes place Simpson and Elhay 2011;Brkić 2012c). This is because the resistance is independent of flow when the Darcy-Weisbach formula is in use only in the case of a laminar flow regime. So, knowing that laminar flow can occur only rarely and only in a few pipes of a water distribution network, calculation for these pipes should be performed in the same way as for the other pipes in which turbulent flow takes place. Further calculation with this assumption will not introduce significant error in the final result. Existence of pipes with laminar flow only means that the model of the network is not rationally planned. This subsequently means that diameters of these pipes have to be changed. The network should be calculated for maximum possible nodal demands, which means that the network is rationally planned only if turbulent flow takes place in all pipes.

Analogy with Electrical Networks
It is true that laminar flow resistance in the Darcy-Weisbach interpretation is a constant for a single pipe Simpson 2011, 2012;Brkić 2012c). This means that flow resistance, r ≠ rðλÞ, in the laminar regime does not depend on the value of the Darcy friction factor, λ (for the laminar regime, the Darcy friction factor can be calculated as λ ¼ 64=R, where R is the dimensionless Reynolds number). In contrast, in the turbulent regime, flow resistance does depend on the Darcy friction factor, i.e., r ¼ rðλÞ (where the Darcy friction factor can be calculated using the well known Colebrook formula). To make a point, a clear analogy with electrical resistance exists in the case of resistance in laminar flow. So, knowing that electrical networks can be solved in a noniterative procedure using only Ohm's and two of Kirchhoff's laws, it can be concluded that hydraulic networks can be equally solved using some sort of Ohm's law rearranged for use in hydraulic networks and two of Kirchhoff's laws. Laminar flow resistance is independent of flow, but the whole calculation will be spoiled if even a single pipe of the hydraulic network has turbulent flow (a single pipe with turbulent flow renders a noniterative calculation of the whole network impossible). In such a network, in which in all pipes laminar flow takes place, pipes with zero flow will be treated simply as a break in the circuit (a connection with an infinitely large resistance) or as a totally choked pipe, which will not cause any problem because no iterative procedure is needed.

Division by Zero in the Computer Environment
Computers today use the IEEE standard for arithmetic precision, and therefore small numbers below a standard boundary will also be treated in the computer as zero, which also can lead to the singularity of matrices used in calculation of the water distribution network (Brkić 2012a;Sonnad and Goudar 2004). Also, use of software specialized only for matrix calculation (such as MatLab by MathWorks or even MS Excel) can be sometimes recommended as a better solution compared with the use of specially developed software for a water distribution network. In MatLab, it is possible to devise all parts of the calculation, whereas in a specialized software program for water networks, such as EPANET, the designer is more restricted because the calculation procedures are already incorporated in the program code.

Possible Physical Interpretation of Zero Flow
Although pipes with no flow in a real looped network of pipe can exist, it is more likely that a quite unrealistic model of a water distributive network is chosen if zero flow occurs (or the model does not accurately represent the system). Consider the network model from Fig. 1 of this discussion, which has a vertical axis of symmetry (symmetry in pipe diameters and nodal demands). Obviously, such a network is excellent for the examination of the zero-flow problem. Symmetric networks can be found in Elhay and Simpson (2011) and in Álvarez et al. (2011). A symmetric network was referred to in the discussed paper in the work of Elhay and Simpson (2011).
To further illustrate the point of the shown zero-flow problem, the nonzero demand of node 2 of the network from Fig. 1 is equal to the demand of node 3, node 4 is equal to node 5, and node 6 is equal to node 7. Also, it can be assumed that all pipes have the same diameter. In that way, symmetry of the network and symmetry of the node demands leads to the logical conclusion that zero flow takes place in pipes 2, 6, and 9. This subsequently leads to the conclusion that the consumer connected to pipes 2, 6, and 9 will suffer water shortage because water users are really located between junctions (Fig. 2).
In reality, the consumers connected to pipes 2, 6, and 9 will almost certainly have enough water because these pipes are supplied from two sides (two-way supplied pipes). In other words, the lowest pressure of water is somewhere between the two nodes (Brkić 2009). This situation is not allowed and cannot be calculated using any of the Hardy Cross type methods of s for calculation of looped  (Brkić 2011b). For example, the normal situation for pipe 5 is that water flows from node 3 toward node 5. This means that the pressure in node 3 is higher than the pressure in node 5 with a monotonically decreasing pressure through pipe 5. In contrast, the pressures in nodes 2 and 3 of the network from Fig. 1 are equalized, which means that flow through pipe 2 is logically impossible. This assumption can be disputed knowing that the point of the lowest pressure (lower than in nodes 2 and 3) in reality is somewhere between these two nodes. This situation produces simultaneous flow from node 2 toward node 3 and from node 3 to node 2 (two-way flow or simultaneous flow from two opposite directions). This is possible if the nodes in a model of the network are poorly spatially distributed. A good engineer should know that the real consumers are not concentrated in a node (Fig. 2). They are actually distributed between nodes. Consumption concentrated in a node is only a model of the real situation. Also, nodes are not necessarily the only junctions in a network (Fig. 3). In the network from Fig. 1, nodes should also be placed between nodes 2 and 3, between nodes 4 and 5, and between nodes 6 and 7 (nodes 9, 10, and 11 in Fig. 3 of this discussion). The actual situation of the demand pattern will in that way be modeled more realistically (Fig. 3). It also has to be noted that an initially poorly conditioned network has as the consequence a poorly conditioned Jacobian, which leads directly to a singularity in the related matrix.
The general recommendation is that the symmetry in a network should be avoided, and if the symmetry exists, nodes should at least always be placed at the axis of symmetry (in that case, a node should be placed at every point at which pipes and the axis of symmetry cross one another). Symmetry of node demands and pipe diameters also should be avoided.
To conclude, temporary zero flow rarely can occur in some of the pipes during the calculation of a looped network (virtual change of flow direction during the iterative procedure usually does not cause the zero-flow problem). However, if zero flow remains as is at the end of the calculation, this usually means that the modeled network is not a good image of the real situation in the field.