Modelling of a micro-irrigation network
Micro-irrigation is recommended for use in arid and semi-arid countries such as Algeria. This method consists of accurately providing the right amount of water and mineral nutrients to the plant's root area. The goal is to provide water efficiently by applying it at the correct rate. However, irrigation efficiency is clearly a function of the uniformity of water application. Micro-irrigation is a technique in which a delicate instrument known as an emitter (a terminal element of the network) operating with low pressure is used. The emitter, designed and manufactured with high precision, is a system with hydraulic laws and norms considered as a black box model that discharges water at atmospheric pressure. The emitter is an element of a network that constitutes a unit called a system or physical model. Water and mineral elements are delivered to a localized place, to the level of each plant by the emitters whose discharge is a function of lateral pressure. The precision of the dosage of irrigation, which must exactly satisfy the requirement for cultivation, depends fundamentally on the design of the network. It takes into account the pressure variations, which are due not only to head loss in the lateral branches of the network but also to the land slope and to the characteristics of the emitters. Water and air temperature and the possible plugging of the emitter orifice also influence the discharge of an emitter. The network is designed to satisfy the water needs of all the plants. Uniformity of water distribution is a main criterion for network design. To understand the variations in water distribution with more precision, we based the present work on a hydraulic analysis focussed on the outflow. This approach yields differential equations in which the pressure and the velocity of water in the pipeline network are unknown; the uniformity of water distribution is largely dependent on these variables. The differential second-order equations obtained are non-linear and analytical resolution is impossible, due to the empirical relation of the discharge emitter and the energy loss relation. Thus, the solution is obtained by numerical methods using the Runge Kutta integration method. The conditions in the limit equation modelling the outflow in the lateral pipes are different from those for the submain pipe. For the lateral pipes, the velocity of water at the extremity of the downstream region is inevitably minimal, as the whole region of discharge in the last pipe section is delivered by the last emitter where the pressure is minimal (Hmin). The velocity and pressure are calculated step by step along the lateral pipe until the entrance of water into the network where the pressure is maximal (Hsmax). The algorithm developed to simulate the emitter discharge distribution from the lateral pipes is called the "RK" model, and when it includes the discharge in the submain pipe it is called the "RS" model. These two models are transcribed in Fortran language by a computer program that automates iterations and calculations. Twelve parameters are changed in turn, or per group according to the cases studied, and the choice of the optimal solution of the parameters includes: emitter coefficients (a, y and C vf ); length and diameter of lateral pipes; the submain and main network (L r , D r , L s , D s , L p , D p ); the roughness of the pipes (C); the spacing between the emitters (Δxr); the spacing between the lateral pipes (Δxs); and the water temperature. From these data of discharge and available pressure to the level of the parcel, the model precisely describes the distribution of the pressure and the discharges to all network emitters. In this case, the total discharge and the total required pressure, the uniformity of pressure and discharges are determined for each pattern of design. The combination of structural, functional and environmental factors is applied to guarantee an optimal exploitation taking into account the limits imposed by the specific norms for the micro-irrigation and the technical limits of velocity and pressure tolerance. Parameters that influence variations in uniformity are numerous and variable, which is why it is not easy to integrate them into this phase of study. The proposed model has merit as it avoids the complex numerical method of finite elements, recommended by some researchers (BRALTS et al., 1993; KANG and NISHIYAMA, 1994). The finite element method based on matrix structuring requires an important volume of iterations and calculations that could constitute a major constraint in the case of a large network. The model of BRALTS et al. (1993) is of particular interest in this regard ; our results have been confronted with those obtained with their model. Thus, the models presented in this study permit the simulation of water dynamics in micro-irrigation networks and offer the opportunity to determine the optimal design for such networks. Optimization is based on the variation of twelve classical parameters plus the associated geometric structure of the network, which was shown to be a non-negligible parameter. Optimization would not only reduce irrigation water volumes, but also fertilizer use and pumping energy. The example illustrated in Table 4 shows that although the networks deliver the same total discharge and have some similar design characteristics, the consumption of pumping energy changes from one geometric structure to another. Once a network is installed, it is impossible to change its design, so it is important to assure precision of the design calculations. This work shows promise in the simulation of the optimal design of micro-irrigation networks and also constitutes an economic means of making decisions. Moreover, the modelling results can guide field experimentation to explore other methods. Micro-irrigation can potentially solve many water shortage problems, but it requires further research in the safe reuse of low quality water and wastewater, the development of long term sustainability and the minimization salt accumulation and drainage problems.