Non-elastic matrix model for hydraulic networks
Introduction: In the water resources field, the unbalance between water supply and water demand obliges more and more elaborated solutions from the engineer. As countries develop, problems related to water, like cities supply, water transference among watersheds and mainly the lack and the difficulty to obtain more and more efficient. The operational control of hydraulic networks to attend population demands during the day is a problem that has been searched for many years and until nowadays the solutions are not always optimized, resulting in flaw risks for water supply. The operational control of hydraulic networks has many variables that must be controlled and optimized to obtain the best efficiency in operation, such as:
a) water level in reservoirs.
b) Pressure heads all over the hydraulic network.
c) Number of valve settings.
d) Supply discharge.
e) Booster set ups.
f) Operations to avoid hydraulic transients.
Literature review: Ormsbee and Wood proposed an algorithm that used a truncated expansion ofTaylor’s series to linearize the energy equations and the conservation of massequations (written in terms of pipe diameter and velocity) for all network pipesand nodes respectively. This method is a modified version of the linear methodapud Wood.
Jowitt and Xu developed an algorithm to determine the values of flow control valve settings to minimize leakage. The non-linear basic hydraulic equations of the network, which describe the node heads and the flow rates in the pipes, are augmented by terms that explicitly account for pressure-depended leakage by terms that model the effect of valve actions. These equations were linearized using the method apud Wood.
Filion and Karney developed a hybrid model that combines the modeling sophistication of a transient simulator and the time-stepping efficiency of a quasi-steady state model and can simulate steady and unsteady interactions in a system over an extended period. The model’s procedure consists of running water hammer simulations at the start and end of an extended time step to track the rate of filling of a system’s reservoirs and then use this information to update reservoir levels at the end of the time step. Extended period and worst-case simulations presented in a case study suggest that the hybrid model has a high routing accuracy and can be used to identify the critical state, which will produce the most severe transients in a system.
Method: The non-elastic matrix model for hydraulic networks calculation is based on amethod created by Nahavandi and Catanzaro. It is a method that calculatesthe discharges and the pressure heads distribution in hydraulic networks for thesteady state, for the extended period and for the transient state.
This method has great advantages concerning the Cross method, because the latter doesn’t allow the calculation of transient situations such as the settings of valves, the starting and stopping of boosters, the branch ruptures, etc.
Results: The non-elastic matrix model has already been tested to calculate manyhydraulic networks. Two of these hydraulic networks are located at Paulínia – SPBrazil. These hydraulic networks supply water to two neighborhoods ofPaulínia. One of them supplies the upper part of the neighborhoods andthe other one supplies the lower part of the neighborhoods. It was decided touse the hydraulic networks of Paulínia because the calculated pressure heads ofsome nodes of the hydraulic network were compared to the pressure headsgauged in situ on the same nodes of the hydraulic network. Although the comparison between the calculated pressure heads of some nodes and the pressure heads gauged in situ on the same nodes will be shown for both hydraulic networks, it was decided to show the results obtained for the steady state and for the extended period for the hydraulic network of the upper part of the neighborhoods schemed in fig. 1 for being more complex than the other one.
Conclusions: The matrix method is working well, because the values calculated by themodel for the hydraulic networks used as examples are similar to the onesobtained from the technical literature. The model was applied to calculate tworeal hydraulic networks. The calculated pressure heads of some nodes of the twohydraulic networks were compared to the pressure heads gauged in situ on thesame nodes of the hydraulic networks and the results were close. Theapplicability of the method created by Nahavandi and Catanzaro wasenhanced, because the programming and the input data to consider the presenceof valves, reservoirs or boosters in the hydraulic network were developed.Furthermore, the mathematical formulation and the programming to calculate theextended period and the transient state also were developed.