*3.7. Kinetic Energy per Unit Volume Radial Profiles*

Based on the kinetic energy distribution trend of droplets per unit volume, the kinetic energy per unit volume at different measuring points was calculated. Figure 17 shows the radial distribution of kinetic energy per unit volume of droplets sprayed from different nozzles under different pressure levels. As it can be seen in Figure 17, the higher the working pressure, the lower the kinetic energy per unit volume of the droplets at the same position, the smaller the increase of kinetic energy, and the less the damage to crops and soil surface. At 2 m, the kinetic energy per unit volume of droplets from each nozzle was not significantly different under all working pressures. According to Figure 17a–c, the maximum kinetic energy per unit volume of the elliptical nozzle was the lowest among the nozzles with equal flow and the same inlet cone angle and outlet diameter under each working pressure. At 100 kPa, the maximum kinetic energy per unit volume of the circular nozzle was the highest. According to Figure 17c–e, the larger the *L/D*, the smaller the wetted radius. The maximum kinetic energy per unit volume of the E1 nozzle (smallest *L/D*) at 100 kPa was the lowest. At 150–300 kPa, the E3 nozzle (largest *L/D*) had the lowest kinetic energy per unit volume. In Figure 17b,f,g, it can be observed that, the smaller the outlet diameter, the lower the kinetic energy per unit volume.

**Figure 16.** Relationship between droplet velocity and diameter sprayed from diamond nozzles with different aspect ratios. (D1, D2, and D3 refer to the diamond nozzles with aspect ratios of 1.54, 1.32, and 1.11, respectively).

In order to further investigate the radial distribution characteristics of the droplet kinetic energy per unit volume of the non-circular nozzles, in this paper, regression analysis on the kinetic energy per unit volume of each nozzle under various pressures was performed, and a distribution model of the kinetic energy based on the distance from the nozzle was established [30]. The relationship is given in the following equation:

$$E\_{\rm ks} = \rm al + b\_{\prime} \tag{8}$$

where a and b are fitting coefficients, *Eks* is the droplet kinetic energy per unit volume, and l is the distance from the nozzle position.

After fitting the measured data, it was found that the fitting correlation coefficient R2 of the kinetic energy per unit volume for the non-circular nozzles was between [0.89, 0.99], indicating that the fitting accuracy was high.

The data of the droplet kinetic energy per unit volume of the three elliptical nozzles with different aspect ratios under different working pressures were uniformly regressed, and the kinetic energy per unit volume was further analyzed. A mathematical model of the relationship between droplet kinetic energy per unit volume *Eks*, distance from the nozzle l, aspect ratio *β*, and working pressure *P* for non-circular nozzles was established. The functional relationship is as follows:

$$E\_{\rm ks} = 2.775P^{-0.318}l + 0.0926\beta + 0.932 \text{ (R}^2 = 0.924\text{)},\tag{9}$$

The fitting coefficient of the droplet kinetic energy per unit volume for the elliptical nozzles with different aspect ratios was 0.92, indicating that the fitting accuracy was high. According to this regression equation, we can obtain the influence of the change of aspect ratio on the kinetic energy of water droplets under different working pressures.

**Figure 17.** Kinetic energy per unit volume as a function of the distance from each nozzle under different working pressure levels. (C2 refers to the circular nozzle with a diameter of 5 mm; D1, D2, and D3 refer to the diamond nozzles with aspect ratios of 1.54, 1.32, and 1.11, respectively; E1, E2, and E3 refer to the elliptic nozzles with aspect ratios of 1.43, 2, and 2.58, respectively).

#### *3.8. Uniformity Coefficient of Kinetic Energy Intensity Distribution of Combined Sprinkler*

In general, the sprinklers in sprinkler irrigation systems are arranged in square and triangular shapes. In this study, the working pressure of the sprinkler was 100 kPa, 150 kPa, and 200 kPa, and a square arrangement was simulated. The uniformity coefficient of the kinetic energy intensity of combined sprinkler irrigation was simulated and calculated by using the MATLAB software. The combination spacing was selected as 1.0 R, 1.1 R, 1.2 R, and 1.3 R, where R is the wetted radius. The calculation results are listed in Table 7.

**Table 7.** Uniformity coefficient of the kinetic energy intensity distribution for each nozzle under different combination spacing.


It can be observed that the uniformity coefficients of the combined kinetic energy intensity of the nozzles under the three pressure levels were different under different spacing. The results indicated that the best combination spacing for the C2 nozzle was 1.0 R, and the best kinetic energy intensity distribution uniformity coefficients at 100 kPa, 150 kPa, and 200 kPa were 40.07%, 49.85%, and 55.93%, respectively. The best kinetic energy intensity distribution coefficient among the diamond nozzles was exhibited by the D2 nozzle. When the outlet diameter was 5 mm, the working pressure was 100 kPa, and the best combination spacing was 1.1 R and 1.2 R, the optimal kinetic energy intensity distribution uniformity coefficient was 67.59%. The optimal combination spacing for the elliptical nozzles was 1.0 R. The optimal uniformity coefficient of the kinetic energy intensity distribution was exhibited by the E3 nozzle (largest *L/D*) at the working pressure of 100 kPa (57.78%). In addition, it was found that the uniformity coefficient of the kinetic energy intensity distribution of the combined sprinkler increased gradually with increasing pressure. The 3D distributions of the combined sprinkler irrigation kinetic energy intensity of the nozzles with the best uniformity coefficient per shape type are exhibited in Figure 18.

**Figure 18.** Distribution of kinetic energy intensity under combined sprinkler irrigation for each nozzle shape with the best uniformity coefficient. (C2 refers to the circular nozzle with a diameter of 5 mm; D2 refers to the diamond nozzle with an aspect ratio of 1.32; E3 refers to the elliptic nozzle with an aspect ratio of 2.58).

#### **4. Discussion**

Most of the research on the distribution characteristics of water droplets in non-circular nozzles focuses on the influence of the shape and pressure on the jet shape. In this study, a video raindrop spectrometer is used to supplement the research on the influence of the shape and pressure of non-circular nozzles on the distribution characteristics of water droplets, such as diameter, velocity, and kinetic energy. The diameter of water droplets increases in the radial direction and decreases with the increase of pressure. The larger the diameter of the outlet is, the greater the speed of water droplets increases with the diameter of water droplets. When the diameter of water droplets is the same, the larger the diameter of outlet, the smaller the velocity of water droplets. The impact kinetic energy per unit volume of droplets at the same position and its growth range decrease with the increase of pressure.

In addition, the influence of the aspect ratio of a non-circular nozzle on spraying hydraulic performance and water droplet distribution characteristics is studied in this paper. The shape coefficient of the non-circular nozzle increases with the increase of aspect ratio, and the range decreases with the increase of aspect ratio. Under five working pressures, the diameter of water droplets in the diamond nozzle increases the most along the radial direction. The larger the aspect ratio is, the greater the speed of water droplets increases with the diameter of water droplets. With the increase of droplet diameter, the growth rate in droplet velocity of the elliptical nozzle is the smallest, while that of circular nozzle is the largest. The nozzle with the outlet diameter of 5 mm has the smallest average droplet diameter, and the droplet diameter decreases the most along the longitudinal direction.

#### **5. Conclusions**

A prediction model which can accurately reflect the droplet diameter, velocity, and kinetic energy distribution per unit volume of three types of nozzles was developed, and a fitting function of the relationship between *L/D* and kinetic energy per unit volume for non-circular nozzles was established. The relationship between droplet diameter and kinetic energy and the distance from the nozzle is an exponential and a linear function, respectively, and the fitting correlation coefficients were between [0.92, 0.99] and [0.89, 0.99], respectively. The distribution of droplet velocity and diameter was found to be logarithmic, and the fitting coefficient was between [0.86 and 0.98]. The relationship between *L/D* and kinetic energy was linear, and the fitting coefficient was 0.924. In all cases, the fitting accuracy was high.

Among the tested nozzles, the sprinkler with the circular nozzle generated the largest *CV*, while the elliptical and diamond nozzles generated similar *CV*s. When operating at the same working pressure, the diamond nozzle exhibited the smallest peak application rate value and the largest radially increasing trend in droplet diameter, while the elliptical nozzle had the lowest maximum droplet kinetic energy per unit volume. The more the nozzle shape deviated from a circle, the larger the radial increase of the kinetic energy intensity. For elliptical nozzles with equal flow and area, the smaller the *L/D*, the smaller the average droplet diameter and velocity, the lower the peak sprinkler water application rate, the more uniform the water distribution, the larger the wetted radius, and the smaller the radial increase of the kinetic energy intensity. The water distribution uniformity was the best when the *L/D* was the smallest.

Under a working pressure of 200 kPa and a combination spacing of 1.0 R, the uniformity coefficients of combined sprinkler irrigation and combined kinetic energy intensity distribution with circular nozzles were the highest, i.e., 65.26% and 55.93%, respectively. The uniformity coefficient of combined sprinkler irrigation with diamond nozzles was the highest (72.28%) when the *L/D* was 1.32, the working pressure was 200 kPa, and the combination spacing was 1.1 R. The combined kinetic energy intensity distribution of the diamond nozzle was the most uniform (67.59%) when the outlet diameter was 5 mm, the working pressure was 100 kPa, and the combination spacing was 1.1 R and 1.2 R. Among the three elliptical nozzles with the same flow rate and different *L/D*, the one with the largest *L/D* had the highest uniformity coefficient (68.72%) when the working pressure was 200 kPa and the combination spacing was 1.1 R. When the combination spacing was 1.0 R, the distribution uniformity coefficient of combination kinetic energy intensity was also the highest (57.78%). In general, the larger the *L/D* of the nozzle, the lower the maximum droplet kinetic energy per unit volume.

**Author Contributions:** Conceptualization, Y.J. and H.L. (Hong Li); methodology, Y.J., Z.W. and J.L.; validation, Z.W.; formal analysis, Y.J., Z.W. and J.L.; investigation, Y.J., H.L. (Hong Li) and H.L. (Hao Li); data curation, J.L.; writing—original draft preparation, J.L.; writing—review and editing, Y.J. and Z.W.; visualization, J.L.; supervision, Y.J. and H.L. (Hao Li); project administration, Y.J. and H.L. (Hong Li). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Postgraduate Research & Practice Innovation Program of Jiangsu Province, grant number SJCX22\_1870, the Jiangsu Province and Education Ministry Co-sponsored Synergistic Innovation Center of Modern Agricultural Equipment, grant number XTCX2018, the Changzhou Key Research and Development Program, grant number CE20222024, Zhenjiang Key Research and Development Program, grant number CN2022003, and the Youth Talent Development Program of Jiangsu University.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** Thanks for grateful to the Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX22\_1870), the Jiangsu Province and Education Ministry Co-sponsored Synergistic Innovation Center of Modern Agricultural Equipment (XTCX2018), the Changzhou Key Research and Development Program (No. CE20222024), Zhenjiang Key Research and Development Program (No. CN2022003) and the Youth Talent Development Program of Jiangsu University.

**Conflicts of Interest:** The authors declare no conflict of interest.

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