An optimization model and waterhammer for sprinkler irrigation systems

There are many possible layouts for transporting water from its source to hydrants. That is why the most economical solution must be found. In practice, trial and error procedures are usually applied to determine the economical layout. The diameters of conveyance systems are then found by optimizing the network. However, such an approach does not guarantee an optimum solution. Nowadays, there are many models and related computer programs to solve this problem, but they also have many drawbacks. For example, dynamic programming procedures have limited usage due to the necessity of preparing new programs for each application. In sprinkler irrigation systems, which are pressurized branched piping systems, flow changes initiate pressure fluctuations called `waterhammer'. Such hydraulic transients must be investigated thoroughly for the safety of the system. For the economical design of a sprinkler irrigation system for a given field, a two-step procedure comprising 1. optimization of system layout, 2. optimization of the system, may be used, but it is usually misleading to separate these two steps. This is because the diameters must be known to optimize the layout, and the layout of the system is necessary to determine the diameters. That is why these two steps must be handled together for the optimum design. In this study, the shortest path algorithms of Graph theory are first used for determination of the layout and then the optimum diameters are determined. The economical solution is investigated by taking into account the cost of each pipe segment and by changing their lengths. The obtained networks are then optimized to arrive at the economical system. The dynamic programming models for optimizing the gravitational or pumping systems with one source are presented. The branched systems may have any configuration. Furthermore, a computer program is developed to analyze waterhammer by the use of the characteristics method with interpolation. The approach used is demonstrated in the case of a simple system and the results are evaluated.