Analytical Study of Wind Influence on In
J. Eng. Technol. Sci., Vol. 47, No. 3, 2015, 296-305 296 Analytical Study of Wind Influence on In-Flight Sprinkler Droplets Frank Agyen Dwomoh1,2*, Yuan Shouqi1 & Li Hong1 1 Research Centre of Fluid Machinery Engineering and Technology, Jiangsu University, 212013, Zhenjiang, Jiangsu, P. R. China 2 School of Engineering, Koforidua Polytechnic, P.O. Box 981 Ghana * Email: [email protected] Abstract. An analytical model to describe the dynamics of in-flight droplets is presented in this paper to augment information on wind influence on travel distance of in-flight sprinkler droplets. The model is ballistic-theory based. It employs a relatively simple, wide-range empirical relationship between drag coefficient and Reynolds’ number to replace the several sets of relations for a specified range of Reynolds numbers. The fourth-order Runge-Kutta numerical integration techniques were used to solve the trajectory equations. A modified exponential model for droplet size distribution was used during the simulation. Comparative analysis showed that agreement exists between the predictions of this model and that of earlier models. Droplets with a diameter smaller than 0.1 mm travelled farthest. Within the droplet range of 0.5 mm to 4.5 mm, as droplet diameter increased, travelled distance increased with increasing wind speed. The extent of drift increased sharply within the droplet range of 0.5 mm to 0.05 mm and increased mildly for droplet diameters greater than 0.5 mm. The model also attempts to identify droplets that are likely to contribute to drift loss and those that have a high probability of contributing only to distortion of the distribution pattern. Keywords: analytical model; droplet drift; distribution pattern; simulation; sprinkler droplets; traveled distance; wind influence. 1 Introduction 0B The influence of wind during sprinkler irrigation pose challenges that need attention especially in this era of water conservation towards a sustainable use of resources. Most sprinklers apply water to the ground by projecting water jets into the air at high velocity, which later fall down as water droplets. Under windy conditions, in-flight sprinkler water droplets may impact the ground or plant canopy, experience droplet evaporation or be wind drifted [1-3]. Sprinkler droplet travel under no-wind condition is undisturbed and thus a characteristic of the sprinkler nozzle for a given operation configuration. If droplets travel beyond their characteristic distances for the same sprinkler Received March 12th, 2014, 1st Revision October 16th, 2014, 2nd Revision November 27th, 2014, 3rd Revision March 17th, 2015, Accepted for publication May 11th, 2015. Copyright ©2015 Published by ITB Journal Publisher, ISSN: 2337-5779, DOI: 10.5614/j.eng.technol.sci.2015.47.3.5 Wind Influence on In-Flight Sprinkler Droplets 297 nozzle and pressure configuration, they are considered as drifted. These drifted droplets contribute to wind distortion of the distribution pattern [3,4]. Several simulation studies have been carried out to model various aspects of wind effect on sprinkler droplets over the years [3,5-9]. Several factors affect the trajectories and losses of in-flight water droplets that complicate adequate description and estimation of wind drift [10-12]. Studies that have simulated droplet drift loss are few. Notable among these are Edling ; Seginer, et al. ; Martin and Newman ; Thompson, et al. ; McLean, et al. ; Teske, et al. . Edling and Chowdhury  and Longley  presented theoretical models for estimating spray evaporation and wind drift from low-pressure spray sprinklers. Molle, et al.  also reported on evaporation and wind drift loss during sprinkler irrigation. Lorenzini and Saro  studied thermal fluid dynamic modeling of a water droplet evaporating in air by considering wind drift (but with uniform velocity field) by applying the Runge-Kutta integration method. This paper presents an analytical description of the dynamics of droplets from a single operated irrigation sprinkler to augment the pioneering works of earlier researchers on in-flight sprinkler droplets. Specifically, we seek to simulate the dynamics of wind influence on the travel distance of sprinkler droplets. 2 Materials and Method 2.1 Model of Droplet Motion Several models have already been developed by researchers that consider a sprinkler as a device emitting numerous droplets with diameter as a function of their travelled distances [5,6,8]. According to ballistic theory, droplets’ motions are influenced by the initial velocity vector, the gravitational force, the wind vector and the viscous drag force. Eqs. (1) to (3) were solved to compute the droplet trajectories. ̈ = 2 2 = 4 � − � ̈ = 2 2 = − 4 � � − ̈ = 2 2 � 3 � 3 = − 4 � − � � 3 (1) (2) (3) where , and z are the coordinates referring to the ground (with origin at the sprinkler nozzle); is the droplet diameter (mm); ̅ is the density ratio of air 298 Frank Agyen Dwomoh, et al. and water respectively; is time (s), and g is acceleration due to gravity. C is the air drag coefficient of the droplet moving at the speed . 2 2 = �[� − � + � − � + 2 ] (4) and are the horizontal and vertical components of the droplet velocity, respectively; , and are the , y and components of the wind velocity respectively. Since the logarithmic profile of wind speed is generally considered to be a more reliable estimator of the actual field conditions, the average wind speed ( ) at height r (cm) above the ground was calculated for all conditions as: ln[(−)/ ] = ln[(−)/0 ] (5) 0 U m = wind speed (m/s) measured at reference height m (cm) above the ground. and are roughness height (cm) and roughness parameter (cm) respectively, both are functions of crop height ℎ (cm), given by: 2.2 log = 0.997ℎ − 0.1536 log = 0.997 log ℎ − 0.883 (6) Boundary and Initial Conditions Height of sprinkler nozzle: 1.2 m (most sprinklers mounted on risers are within the range of 0.8 to 1.5 m); droplet diameter range considered: 0 < droplet diameter (mm) < 5; wind speeds: 0, 2.5, 3.5 and 4.5ms-1; operating pressure: 250,300,350 kPa. The fourth-order Runge-Kutta numerical integration techniques were used to solve equation (1), (2) and (3) for droplet movement with the specification of initial conditions as follows: ( = 0) = 0;̇ ( = 0) = 0, ; ( = 0) = 1.2 (height from ground to sprinkler nozzle = 1.2 m); ̇ ( = 0) = 0, . 0, = 0 cos ; 0, = 0 sin ; is the inclination of the sprinkler nozzle to the horizontal. The velocity of the sprinkler jet exiting from the nozzle was calculated as: 0 = Cd (2)0.5 (7) where () is the operating pressure head at the nozzle and Cd is the discharge coefficient, equal to 0.98. By setting = 0 (soil surface) or catch can elevation, each trajectory solution is constituted by the x and y coordinates. Two categories of simulations were conducted: no-wind and in-wind conditions. Droplet travel distances were simulated for both under-wind and no-wind conditions. The horizontal distance Wind Influence on In-Flight Sprinkler Droplets 299 between the nozzle exit and the droplet landing point was simulated as the droplet travel distance. 2.3 Empirical Model of the Drag Coefficient To determine the trajectory of the droplet projectiles in the air, a relatively simple, wide-range empirical relationship between the drag coefficient (C) and the Reynolds’ number ( ), proposed by Holterman , was employed to replace the several sets of relations for a specified range of Reynolds numbers, as displayed in Eqs. (7) and (8). = �� � = 1� + � (8) (9) where = 24; = 0.32; = 0.52; =droplet diameter (), v=velocity (ms-1) and =the kinematic viscosity of the air (2 −1). The adopted relationship compares very well with the well-known set of relations by Fukui, et al. . The model is applicable not only to the turbulent-flow regime, but also to the Stokes regime. However, it shows some deviation from the experimental data for > 104 . 2.4 Estimation of Droplet Size Distribution Several mathematical models and data have been published for drop size distribution for distinct types of sprinkler devices using different methods and operated at varying pressures, nozzle sizes and heights [21-25]. In this study, the simple exponential model used by Li, et al. , which was later modified by Kincaid, et al. , was used. The exponential model is given by Eq. (10). ) �� 50 = 100 �1 − �−0.693( (10) 50 = + and = + (11) Where is the percentage (%) of the total drops that are smaller than d; is the drop diameter (mm); 50 is the volume mean drop diameter (mm); is the dimensionless exponent. Kincaid, et al.  found out that Eq. (10) together with the following suggested adjustment factors gave reasonable predictions that cater for smaller diameter droplets. The regression coefficients used for estimating the drop size distribution parameters for the impact sprinkler with small round nozzle (3 mm) are: = 0.31; = 11, 900; = 2.04; = -1,500; is the ratio of the nozzle diameter to the pressure at the base of the sprinkler device. Sprinkler droplets were assumed to be spherical in shape (this is consistent with the photographic 300 Frank Agyen Dwomoh, et al. studies by Okaruma and Nakanishi ). It was also assumed that the volume of the droplet is invariant during its flight from the nozzle to the ground. The droplet sizes distributions derived from Eqs. (10) and (11) that were used in the analysis are shown in Figure 1. Volume cumulative frequency (%) 100 80 250kPa 300kPa 350kPa 60 40 20 0 0 2 4 Droplet diameter (mm) Figure 1 Droplet size distribution derived and used for model analysis. 3 Results and Discussion 3.1 Comparative Analysis of Models Travel Distance [m] A comparative analysis of droplet travel distance by the models of Molle, et al. , Fukui, et al. , von Bernuth and Gilley  and our model are presented in Figure 2. Agreement exists between our model and that of Molle, et al.  for droplets with diameters greater than 2.5 mm, while some differences exist for droplets with diameters smaller than 2.5 mm. 18 16 14 12 10 8 6 4 2 0 Molle, et al.  Fukui, et al.  Bernuth & Gilley  Our Model 0 2 3 Droplet Size [mm] 5 Figure 2 Comparison between other simulated travel distances. Wind Influence on In-Flight Sprinkler Droplets 301 Even though similarity exists in terms of the shape of the trajectory, the travel distances simulated by our model are longer than those of Fukui et al.  and von Bernuth & Gilley  but shorter than those of Molle et al. . The disparities can be attributed mainly to differences in the operating parameters and assumptions used in the simulation. 3.2 Effect of Wind Speed on Droplet Travel Distance Simulated droplet travel distances from the sprinkler for three wind speeds with downwind direction, and zero-wind condition at constant pressure (300 kPa) are compared in Figure 3. Droplets with diameters smaller than 0.1 mm travelled farthest, travelling beyond 24 m from the nozzle exit. This is in agreement with the work of Molle, et al. . Wind increased droplet travel distance downwind. Within the range of 0.5 mm to 4.5 mm, as droplet size and wind speed increased, travelled distance also increased. The extent of drift is defined here as the difference between the droplet travel distance under no-wind and inwind situations for the same sprinkler nozzle-pressure configuration. The extent of drift increased sharply within the droplet range of 0.5 mm to 0.05 mm and then increased mildly for droplets diameters greater than 0.5 mm (Figure 4). From Figure 1 (for 300 kPa), droplets with a mean diameter larger than 4.0 mm, representing a frequency of 0.92% of the total number of droplets, and droplets with a mean diameter smaller than 0.2 mm, representing a frequency of less than 3%, traveled beyond the wetted radius (Figure 3). Such categories of droplets, apart from contributing to distortion of the distribution pattern, also have a higher probability of contributing to wind drift for sprinklers that are located at the periphery of the irrigated field. 0[m/s] 3.5 [m/s] Travel distance (m) 30 2.5[m/s] 4.5 [m/s] 24 18 12 6 0 0 1 2 3 Droplet diameter (mm) 4 Figure 3 Comparison of droplet travel distance between no-wind and in-wind conditions at constant pressure (300 kPa). Frank Agyen Dwomoh, et al. Extent of Drift (m) 302 30 2.5m/s 3.5m/s 4.5m/s 25 20 15 10 5 0 0 2 4 Droplet diameter (mm) Figure 4 Comparison of extent of drift (m) as a function of droplet size (mm) at three wind speeds at constant pressure (300 kPa). The remaining droplets are more likely to contribute to distortion of the distribution pattern. Even though larger droplets (with diameter > 4 mm) represent a small percentage of the number of droplets in the droplet distributions considered (Figure 1) due to their high volume per droplet; if they are wind-drifted they will constitute a high percentage loss. For example, at a constant operating pressure of 300 kPa, at wind speeds of 3.5 m/s and 4.5 m/s, 20% and 32% of the total volume travelled beyond 17 m (the wetted radius), respectively. Of these percentages, 70-90% were larger drops (> 3.9 mm) representing 3.6% and 6.3% of the total number of drops in the distribution, respectively. Hence the percentage of large droplets in the distribution spectrum should not only be of interest for predicting water droplet impact , but can also be critical for estimating wind drift losses as well. The above observation is particularly important as it partially identifies droplets that are likely to contribute to drift losses and those that have a high probability of contributing only to distortion of the distribution pattern. 4 Conclusion The paper presented an analytical model to describe the dynamics of wind effect on in-flight sprinkler droplets. A comparative analysis showed that agreement exists between the predictions of this model and those of earlier models. Droplets with a diameter smaller than 0.1 mm travelled farthest. Within the droplet range of 0.5 mm to 4.5 mm, as the droplet diameter increased travelled distance increased with increasing wind speed. The extent of drift increased sharply within the droplet range of 0.2 mm to 0.05 mm and then increased gently for droplet diameters greater than 0.5 mm. The model also identified Wind Influence on In-Flight Sprinkler Droplets 303 droplets within the mean diameter ranges of 0.05 to 0.1 mm and greater than 3.9 mm as likely to contribute to both distortion of the distribution pattern as well as wind drift, especially for sprinklers located at the periphery of the irrigated area when wind speeds are greater than or equal to 3.5m/s. Acknowledgements The financial support provided by the program for National Hi-tech Research and Development (863 Program No. 2011AA100506 and 2011GB2C100015) of China is gratefully acknowledged. References  Heermann, D.F. & Kohl, R.A., Fluid Dynamics of Sprinkler Systems, In: Jensen, M.E., (ed.), Design and Operation of Farm Irrigation System, ASAE, St Joseph, Mich, pp. 58-61, 1981.  Solomon, K.H., Zoldoske, D.F. & Oliphant, J.C., Laser Optical Measurement of Sprinkler Drop Sizes, Center for Irrigation Technology, Standards Notes, 1996.  Silva, W.L.C. & Larry, G.J., Modeling Evaporation and Microclimate Change in Sprinkler Irrigation I, Model Formulation and Calibration, Transactions of ASAE, 31(5), pp. 1481-1486, 1988.  Martin, C.F. & Newman, J., Analytical Model of Water Loss in Sprinkler Irrigation, Applied Maths and Computation, 43, pp. 19-41, 1991.  Fukui, Y., Nakanishi, K. & Okamura, S., Computer Evaluation of Sprinkler Irrigation Uniformity, Irrigation Science, 2, pp. 23-32, 1980.  Vories, E.D., von Bernuth, R.D. & Mickelson, R.H., Simulating Sprinkler Performance in Wind, Journal of Irrigation and Drainage Engineering, 113(1), pp. 119-130, 1987.  von Bernuth, R.D. & Gilley, J.R., Sprinkler Droplet Size Distribution Estimation from Single Leg Test Data, Transactions of ASAE, 27, pp. 1435-1441, 1984.  Seginer, I., Kantz, D. & Nir, D., The Distortion by Wind of the Distribution Patterns of Single Sprinklers, Agricultural Water Management 19(4), pp. 314-359, 1991.  Teske, M.E., Bird, S.L., Esterly, D.M., Curbishley, T.B., Ray, S.L. & Perry, S.G., AgDRIFT: A Model for Estimating Near-Field Spray Drift from Aerial Applications, Environ. Toxicology and Chemistry, 21(3), pp. 659-671, 2002.  Lorenzini, G. & De Wrachien, D., Theoretical and Experimental Analysis of Spray Flow and Evaporation in Sprinkler Irrigation, Irrigation and Drainage Systems, 18(2), pp. 155-166, 2004. 304 Frank Agyen Dwomoh, et al.  De Wrachien, D. & Lorenzini, G., Modeling Jet Flow and Losses in Sprinkler Irrigation: Overview and Perspective of a New Approach, Biosystems Engineering, 94(2), pp. 297-309, 2006.  Derrel, L.M. & Kincaid, D.C., Chapter 16: Design and Operation of Sprinkler Systems in Design and Operation of Farm Irrigation Systems, 2nd Edition, 2007.  Edling, R.J., Kinetic Energy, Evaporation and Wind Drift of Droplets from Low Pressure Irrigation Nozzles, Transactions of ASAE, 28(5), pp. 1543-1550, 1985.  Thompson, A.L., Gilley, J.R. & Norman, J.M., A Sprinkler Water Droplet Evaporation and Plant Canopy Model: II, Model Application, Transactions of ASAE, 36(3), pp. 743-750, 1993.  McLean, R.K., Sri Ranjan, R. & Klassen, G., Spray Evaporation Losses from Sprinkler Irrigation Systems, Canadian Agricultural Engineering, 42(1), 2000.  Edling, R.J. & Chowdhury, P.K., Kinetic Energy Evaporation and Wind Drift of Droplets from Low Pressure Irrigation Nozzles, ASAE, paper No. 842084, 1984.  Longley, T.S., Evaporation and Wind Drift from Reduced Pressure Sprinklers, PhD Diss. Univ. of Idaho, 1984.  Molle, B., Tomas, S. & Hendawi, M., Evaporation and Wind Drift Losses During Sprinkler Irrigation Influenced by Droplet Size Distribution, Irrigation and Drainage, 61(2), pp. 240-250, 2012.  Lorenzini, G. & Saro, O., Thermal Fluid Dynamic Modeling of A Water Droplet Evaporating in Air, International Journal of Heat and Mass Transfer, 62(c), pp. 323-335, 2013.  Holterman, H.J., Kinetics and Evaporation of Water Drops in Air, IMAG report 2003-12. Wageningen UR July 2003.  Kohl, R.A., Drop Size Distribution from Medium-Sized Agricultural Sprinklers, Transactions of ASAE, 17(4), pp. 690-693, 1974.  Solomon, K.H., Kincaid, D.C. & Bezdek, J.C., Drop Size Distribution for Irrigation Spray Nozzles, Transactions of ASAE, 28(6), pp. 1966-1974, 1985.  Dadio, C. & Wallender, W., Droplet Size Distribution and Water Application with Low-Pressure Sprinkler, Transactions of the ASAE, 28, pp. 511-516, 1985.  Kincaid, D.C., Solomon, K.H. & Oliphant, J.C., Drop Size Distribution for Irrigation Sprinklers, Transactions of ASAE, 39(3), pp. 839-845, 1996.  Li, J., Kawano, H. & Yu, K., Droplet Size Distributions from Different Shaped Sprinkler Nozzles, Transactions of ASAE, 37(6), pp. 1871-1878, 1994. Wind Influence on In-Flight Sprinkler Droplets 305  Okaruma, S. & Nakanishi, K., Theoretical Study on Sprinkler Sprays (Part Four) Geometric Pattern Form of Single Sprayer under Wind Conditions, Transactions of Japanese Society of Irrigation Drainage Reclamation Engineering, 29, pp. 35-43, 1969.  von Bernuth, R.D. & Gilley, J.R., Sprinkler Droplet Size Distribution Estimation from Single Leg Test Data, Transactions of the ASAE, 27, pp. 1435-1441, 1984.