Simulating water distribution patterns for fixed spray plate sprinkler using the ballistic theory

Ballistic simulation of the spray sprinkler for self-propelled irrigation machines requires the incorporation of the effect of the jet impact with the deflecting plate. The kinetic energy losses produced by the jet impact with the spray plate were experimentally characterized for different nozzle sizes and two working pressures for fixed spray plate sprinklers (FSPS). A technique of low speed photography was used to determine drop velocity at the point where the jet is broken into droplets. The water distribution pattern of FSPS for different nozzle sizes, working at two pressures and under different wind conditions were characterized in field experiments. The ballistic model was calibrated to simulate water distribution in different technical and meteorological conditions. Field experiments and the ballistic model were used to obtain the model parameters (D50, n, K1 and K2). The results show that kinetic energy losses decrease with nozzle diameter increments; from 80% for the smallest nozzle diameter (2 mm) to 45% for nozzle diameters larger than 5.1 mm, and from 80% for the smallest nozzle diameter (2 mm) to 34.7% for nozzle diameters larger than 6.8 mm, at 138 kPa and 69 kPa working pressures, respectively. The results from the model compared well with field observations. The calibrated model has reproduced accurately the water distribution pattern in calm (r = 0.98) and high windy conditions (r = 0.76). A new relationship was found between the corrector parameters (K1’ and K2’) and the wind speed. As a consequence, model simulation will be possible for untested meteorological conditions. Additional key words: sprinkler irrigation; ballistic model; center-pivot; kinetic energy losses. * Corresponding author: [email protected] Received: 22-12-13. Accepted: 07-07-14. This work has 1 supplementary table that does not appear in the printed article but that accompanies the paper online. Abbreviations used: C (aerodynamic drag coefficient); D (drop diameter, mm); D50 (mean drop diameter, mm); FSPS (fixed spray plate sprinkler); IDm (catch can values of measured irrigation depth, mm); IDm —— (average measured irrigation depths, mm); IDs (catch can values of simulated irrigation depth, mm); IDs —— (average simulated irrigation depths, mm); K1, K2 (empirical parameters); K1’, K2’ (new empirical parameters); n (dimensionless exponent); P (operating pressure, kPa); Pm (minimum probability for drops smaller than D50); R (coefficient of correlation); RMSE (root mean square error); RSPS (rotating spray plate sprinkler); Sm (standard deviation of measured ID); Ss (standard deviation of simulated ID); U (absolute drop velocity, m s); V (relative drop velocity in the air, m s); W (wind speed, m s); Wd (dominant wind speed, m s); α (angle formed by the vectors V and W); β (angle formed by vectors V and U); γ (probability for drops smaller than D); ρ (objective function); Φ (nozzle diameter). Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria (INIA) Spanish Journal of Agricultural Research 2014 12(3): 850-863 http://dx.doi.org/10.5424/sjar/2014123-5507 ISSN: 1695-971X eISSN: 2171-9292 RESEARCH ARTICLE OPEN ACCESS Current center-pivot models are based on the overlapping of experimental sprinkler application pattern (Omary & Sumner, 2001; Delirhasannia et al., 2010). Based on semi-empirical considerations and using a combination of beta functions (free from any ballistic consideration), Le Gat & Molle (2000) and Molle & Le Gat (2000) developed a model to simulate the application pattern of a single spray sprinkler, and to describe its performance in both windy and no-wind conditions. Ballistic simulation (Fukui et al., 1980) has been successfully applied to impact sprinklers (Montero et al., 2001; Playán et al., 2006). Ballistic sprinkler simulation models require information on drop diameter distribution to estimate the landing point and terminal velocity of drops resulting from a certain irrigation event. Procedures have been developed to estimate drop diameter distribution from the sprinkler application pattern using inverse simulation techniques (Montero et al., 2001; Playán et al., 2006). The percentage of the irrigation water collected at each landing distance can be used to estimate the percentage of the irrigation water emitted in drops of a given diameter. Ballistic theory requires the characterization of the drop diameter distribution. Li et al. (1994) proposed an exponential model to characterize the drop diameter distribution for circular and no circular nozzle. Kincaid et al. (1996) used this model to fit the drop diameter (D) distribution curve for different type of emitters according to the following equation: D n p n [–0.693 × (——) ] 0.693 × n × (——) × e D50 D50 γ = ——————————————————— [1] D where γ is the probability for drops smaller than D, D50 the mean drop diameter, and n is a dimensionless exponent. This empirical model permits to establish a functional relationship between the drop diameter and the sprinkler discharge. The estimation of the parameters of this equation permits to characterize the drop diameter distribution resulting from a given sprinkler, nozzle diameter and operating pressure. In order to reproduce the deformation of the circular water application area produced by the wind, Seginer et al. (1991) and Tarjuelo et al. (1994) reported on the need to correct the aerodynamic drag coefficient following this expression: C' = C (1 + K1 sinβ – K2 cosα) [2] where C’ is the corrected aerodynamic drag coeff icient; C is the aerodynamic drag coefficient which can be expressed as a function of the Reynolds number of a spherical drop and the kinematic viscosity of the air (Fukui et al., 1980; Seginer et al., 1991); β is the angle formed by vectors V (relative drop velocity in the air) and U (absolute drop velocity); α the angle formed by the vectors V and W (wind velocity); and K1 and K2 are the empirical parameters determined for each wind velocity conditions. The combination of both parameters has led to significant improvements in the simulation of wind distorted water distribution patterns (Tarjuelo et al., 1994). According to Montero et al. (2001), K2 is much less relevant than K1. Dechmi et al. (2004) confirmed this extreme and reported that K1 and K2 narrows and displaces, respectively, the water distribution pattern respect to the wind direction. However, ballistic simulation of the new emitter for self-propelled sprinkler irrigation machines requires the incorporation of the effect of the jet impact with the deflecting plate (Sánchez-Burillo et al., 2013). The case of the spray sprinklers commonly used in pivot or linear move irrigation machines differs from impact sprinklers. In this case, the jet produced at the nozzle immediately undergoes an inelastic shock as it frontally hits a plate. Although most spray sprinkler models include certain curvature in the plate and grooves designed to create a number of small jets, the energy lost at the plate is sufficiently large to create uncertainty about the initial velocity of the drops. As a consequence, ballistic models have rarely been applied to the two main designs of spray plate sprinklers. The shape, ridges and curvature of the deflecting plate determine the number of secondary jets, the vertical initial angle and the drop initial velocity (DeBoer et al., 1992). The pressure head at the nozzle, the nozzle diameter and the sprinkler design and manufacturing determine droplet kinetic energy (King & Bjorneberg, 2010). This energy is directly related to drop diameter and velocity (Kincaid, 1996). In kinetic energy analyses of sprinkler irrigation, the drop trajectory and velocity is commonly simulated using an estimation of initial velocity and ballistic simulation models (Kincaid, 1996). Several researchers have characterized the drop kinetic energy in irrigation machines (King et al., 2010; King & Bjorneberg, 2012), which are mainly focused on the hydraulic characteristic impact of the soil. Sánchez-Burillo et al. (2013) have characterized the drop initial velocity for fixed spray plate sprinkler in order to simulate the effect of the jet impact and incorporate it to the ballistic theory. In this research the water distribution pattern of FSPSs working at different technical (working presSimulating water distribution patterns for fixed spray plate sprinkler using the ballistic theory 851