STATISTICAL AND THEORETICAL APPROACHES The Statistical Approach The Physical-Mathematical Approach

1. 34rd International Symposium Actual Tasks on Agricultural EngineeringOpatija, Croatia, 21-24 February 2006MODELLING SPRAY FLOW AND WASTE IN SPRINKLER IRRIGATION PRACTICE : AN OVERVIEWDe Wrachien D.* &Lorenzini G.** (*) Department of Agricultural Hydraulics, State University of Milan, Italye-mail: daniele.dewrachien@unimi.it(**) Department of Agricultural Economics and Engineering, Alma Mater Studiorum, University of Bologna, Italye-mail: giulio.lorenzini@unibo.it

2. STATISTICAL AND THEORETICAL APPROACHES The Statistical Approach The Physical-Mathematical Approach A NEW MODEL FOR SPRINKLING SPRAY FLOW AND WASTE Modelling Droplet Dynamics Modelling Droplet Evaporation EXPERIMENTAL STUDIES ON SPRAY FLOW AND WASTE CONCLUDING REMARKS

3. THE STATISTICAL APPROACH The mutual interactions of all the factors affecting the air path and waste of a water droplet (among which are worth mentioning dimension of the droplet, air temperature, air friction, relative humidity, solar radiation, wind velocity) leaving the sprinkler’s nozzle make it very hard to work out a proper description and assessment of the phenomena. Resorting to statistical (empirical) formulae becomes so often the only way to skip the difficulties, not to say the impossibilities, that analytical procedure would imply. An important work along this line is that of Frost and Schwalen (1955), resulting in a monogram relating spray losses to air relative humidity, air temperature, wind spread, nozzle diameter and nozzle pressure.

4. Yazar (1984), testing with lateral of sprinklers, obtained: where: E is the percentage of discharged flow lost due to evaporation (%);W is the wind speed (m/s); es and ea are the saturation vapour pressure and the actual vapour pressure of the air (kPa), respectively. Tarjuelo et al. (2000), on the basis of a set of experimental investigations for estimating drift and evaporation losses during sprinkler irrigation events, proposed the following linear statistical model for water losses prediction in sprinkler practice: where: Los are evaporation and drift losses (%); P is the working pressure (kPa); a, b and c are fit coefficients, e is the experimental error. The model proved to be a useful tool to determine the irrigation timing, as a function of environmental and operational conditions in order to minimise evaporation and drift losses.

5. THE PHYSICAL-MATHEMATICAL APPROACH Ranz and Marshall (1952), studied the evaporation of droplets in connection with spray drying and presented an equation for molecular transfer rate during evaporation along the flight path of the droplet. Goering et al. (1972), starting from the Marshall’s equation, arrived at the following formula for computing the change in droplet diameter D due to evaporation, based on heat and mass transfer analogy: where: Mv is the molecular weight of vapour; Mm is the molecular weight of air; K is the diffusivity of vapour in air in m2s-1; ρis the density of air in Kgm-3; ρd is the density of the droplet in Kgm-3 ; ΔP is the difference in Pa between the saturation pressure at the wet bulb temperature of air and the vapour pressure at the dry bulb temperature; Pf is the partial pressure of air in Pa and Nu’ is a specially defined Nusselt number for mass transfer.

6. Edling (1985) developed a model, based on the impulse momentum principle, to estimate kinetic energy, evaporation and wind drift of droplet from low pressure irrigation sprinkler. The model showed a rapid depletion of evaporation and drift losses when the drop diameter increases, as well as a high dependency of losses on wind speed and riser height. Edling inferred from his experiences that drop evaporation in sprinkler irrigation is almost negligible from a droplet diameter of 1.5-2mm. Thompson et al. (1993, 1997), proposed a model suitable for assessing water losses during sprinkler irrigation of a plant canopy under field conditions. The procedure combines equations governing water droplet evaporation-based on the heat and mass transfer analogy- and droplet ballistics (three-dimensional droplet trajectory equations) with a plant-environment energy model. Ranz and Marshall (1952), who based their investigations on Fröessling’s(1938) boundary layer equations and the equation for heat and transfer analogy, analysed the behaviour of very small droplets ( order of magnitude of μm).

7. A NEW MODEL FOR SPRINKLING SPRAY FLOW AND WASTE • Modelling Droplet Dynamics The sprinkler droplet air path is described by means of the Second Principleof Dynamics: where is the total force acting on the droplet and equal to the vectorial sum of the weight of the droplet of mass m diminished by its buoyancy force and of the friction force acting during the flight on the droplet of acceleration .

8. The friction factor f used in the model is that according to Fanning’s definition: (a) f= for Reynolds number Re 0.1; (b) f= for 2 Re 500; (c) f= 0.44 for 500 Re 200000. Case (a) expresses the conditions of a Stokes’ (laminar) flow law; case (b) of an intermediate flow law, and case (c) of a Newton’s (turbulent) flow law.

9. Modelling Droplet Evaporation Spray evaporation is assessed on the basis of the analytical procedure previously described (Lorenzini, 2004). The model accounts only the effects of air friction on the droplet evaporation and neglects all other parameters. Despite its limits, the model could improve the understanding of the sprinkler evaporation phenomenon. This approach has not been found elsewhere, probably because air friction, which is relevant, mainly, in the Newton’s law region, was considered as a factor of minor relevance in the process.

10. Three additional assumptions have been introduced: • Evaporation is obtained by the total work of the resulting force (sum of weight, buoyancy and friction force) acting on the droplet along its trajectory; • Droplet evaporation happens just at the end of the flight path; • The droplet is considered as a material point.

11. The second hypothesis entails a limitation to the results, as the final kinetic energy of the droplet is calculated by means of the initial mass: this implies that the evaporation losses are somehow over-estimated. A sort of “upper limit” of the “force-induced” droplet evaporation is so worked out in the present approach. The passage from the “upper limit” to the “actual value” entails the discretization of the aerial droplet path in as many intervals as possible, and the application of the same procedure to each interval ,using the output of the previous step as input of the next one.

12. EXPERIMENTAL STUDIES ON SPRAY FLOW AND WASTE Measurement problems exist when trying to quantify evaporation losses and validate flow and waste models in sprinkler irrigation practice. Evaporation loss is taken as the difference between the amounts of water leaving the nozzle and measured with a grid network of catch cans. Accurate measurement of water that reaches the ground is very tough with high wind because drift greatly increases the area where measurement is needed. Moreover, evaporation from the collection units is very hard to assess. Investigators have applied corrections to account for these errors, but accurate evaluations are difficult to achieve (Jensen, 1980). Related to this issue, interesting results were obtained by Zanon and Testezlaf (1995) and Zanon et al(2000), who studied problems of experimental techniques for automatic systems of water collection at ground level and the methods of measurement of the water collected, in order to reduce experimental errors.

13. Some investigations focused on radial or square-grid distribution of the catch cans in different environmental conditions. Bilanski and Kidder(1958) studied the effects of various sprinkler components, including pressure and nozzle size, on the pattern shape and radius. Seginer(1963) developed standardized patterns and related the pattern radius to the pressure head for certain nozzle sizes. Solomon et al. (1980) used a clustering algorithm to group pattern test data into typical standards shapes and used pattern radius to define a relative distance from the sprinkler. Kincard (1982) proposed an analytical approach suitable to describe the combined effects of nozzle size, pressure and nozzle discharge on sprinkler pattern radius.

14. The sizes of water drops from spray nozzles bear on important areas of irrigation experimental study, including the extent of wind drift and evaporation losses, distortions of spray patterns by the wind and the reduction of the soil’s infiltration rate due to drop impact on the soil surface. Moreover, knowing the droplets’ distribution within the jet and along a radius could help anticipate their path related to environmental conditions. Large droplets may lead to a reduced soil infiltration rate through soil surface disruption caused by droplet impact (Mohammad and Kohl, 1986). Nozzle configuration and water pressure are both important factors in determining droplet size as well as the field distribution pattern (Hills and Gu, 1989). All these studies share the awareness of the difficulty of being able to comprehend clearly the process of spray evaporation in sprinkler irrigation. The problem is due to the very many parameters that mutually affect each other.

15. Concluding Remarks • Scientific literature has treated the problem of spray evaporation losses as one of minor relevance, mainly, attributing to wind drift the global water mass reduction occurring during the air path of the droplet. The experimental tests, on the contrary, have clearly showed that aerial spray evaporation is a relevant cause of water sink. • A thorough understanding of the factors affecting spray flow and waste in sprinkler irrigation practice is important for developing appropriate water conservation strategies. To properly tackle this problem relevant theoretical and experimental studies have been carried out during the second half of the 20th century.

16. Among the analytical studies, the heat and mass transfer analogy, linked with particle ballistics offers a well –established approach to assess jet flow and waste . The procedure describes the event of a droplet travelling from the sprinkler nozzle to the ground as a combination of environment parameters such as pressure gradient, vapour concentration, air relative humidity, resulting in very elaborate formulae and strongly condition-dependent. • A new mathematical model for irrigation sprinkler droplet ballistics and evaporation, based on a simplified dynamic approach to the phenomenon, has been presented and validated. The tool can be considered as an indicator of the system performance and has proved to fully match the kinematic results obtained by more complicated procedures. Furthermore, the model made it possible to work out ready-to-apply formulae suitable to assess the “upper limit” of the contribution given to the droplet evaporation by the friction force during the droplet’s aerial flight. Results showed that air friction is of relevance in spray waste in the Newton’s law region.

17. Notwithstanding all these efforts, a full comprehension of sprinkler jet flow waste has not been reached yet; so a deepening of both the theoretical and experimental activities is needed to allow the scientific community futher steps toward a thorough understanding of the phenomenon.