Positioning and Moving Sprinkler Systems for Irrigation

In this paper, we present a model that minimizes the amount of time required to irrigate a field that is 80 meters by 30 meters under the given conditions. Firstly, by the empirical formula, we obtain the range of sprinkler (20m). Then by means of references, by analysis, we ascertain the shape of the wetting pattern is cone, according to which together with the conservation of the flow rate, it is facile to get the irrigation algorithm which provides us a continuous analytic function?hBpB function. Based on the cone model, we determine that the number of sprinklers is two and they are set on both ends of the pipe set respectively. When the pipe set has only two positions to place, considering the symmetry, we prove that the least amount of time required maintaining the irrigation is 40h, However, the problem becomes more complex as the number of the positions increases, which results in the advent of the panes, for the discretization of hBpB. We divide the rectangle (80×30) into 2400 small panes(1×1), then we can use the numerical solution of the intersection points to approach the analytic solution of min{hBp}.B In order to realize the programming easily searching the best positions to put the pipe set, we assume the pipe can only move along reticle. Under the Passumption, we find that when the coordinates of the midpoint of the pipe are determinate, then the coordinates of the sprinklers at the end of the pipe set have only two cases. Thus, we can arrange the pipe by arranging the midpoint to make it easier to implement it by program. But it is time-consuming to have a global search, we resort to the simulated annealing algorithm to search the approximate optimal solution, that is, the minimal time is 23 and 27 hours respectively when the pipe could be put on 3 and 4 positions. At last, we focus on the availability of water. We draw plan forms, surface plots and contour maps to help observe the distribution of the water used and the water wasted.