Experimental Characterization of Water Droplet Dynamics in Sprinkler Irrigation Using High-Speed Photography

Abstract

A clear understanding of water droplet formation and distribution dynamics is fundamental to improving the hydraulic performance and operational efficiency of sprinkler irrigation systems. This study presents an experimental investigation of droplet characteristics using high-speed photography under controlled laboratory conditions. The objective was to analyze droplet diameter, ellipticity, frequency, and velocity at working pressures of 0.2, 0.25, and 0.3 MPa. Median droplet diameters measured at 6–8 m from the nozzle were 2.79 mm, 3.41 mm, and 3.68 mm at 0.2 MPa, with a reduction of up to 17.7% as pressure increased to 0.3 MPa. Smaller droplets were predominantly concentrated near the nozzle and decreased with radial distance, influencing water application uniformity. Morphological parameters such as uniformity (1.3), ellipticity (2.13), and circularity (0.81) were quantified. Cumulative frequency curves revealed 12% droplet fragmentation at 7–8 m under higher pressures, illustrating the dynamic nature of droplet breakup. A strong linear correlation between droplet diameter and calibrated reference diameter confirmed the reliability of the measurement technique. These findings demonstrate that high-speed photography is a robust method for droplet characterization and provides accurate, repeatable data essential for optimizing sprinkler designs to reduce water loss due to evaporation and wind drift. The study contributes to precision irrigation research by offering a detailed understanding of droplet behavior under varying operating pressures.

1. Introduction

Efficient water management is essential for achieving sustainable agricultural production, particularly in regions facing increasing water scarcity [1,2]. Among existing irrigation methods, sprinkler irrigation remains one of the most effective systems for ensuring uniform water application across various soil types and topographies [3]. However, the overall performance of sprinkler systems is largely determined by droplet characteristics such as size [4], velocity [5], and spatial distribution [6], which directly influence infiltration rate, evaporation losses [7], and water use efficiency [8,9,10]. Therefore, a precise understanding of droplet dynamics is crucial for optimizing sprinkler design and improving irrigation effectiveness. Conventional droplet measurement methods, such as laser diffraction, flour pattern analysis, and phase Doppler particle analysis, have long been applied in agricultural water research [11,12]. Although these methods have contributed valuable insights into droplet characterization [13], they often present limitations, including sensitivity to environmental fluctuations [14], restricted precision [15], and reliance on expensive instrumentation that complicates their practical use [16] in irrigation studies [17,18,19]. Efforts by researchers such as [20,21] to refine the flour method and improve mounting configurations have enhanced measurement accuracy to some extent; however, these approaches still fail to deliver high-resolution, real-time data under operational conditions. Recent technological advances in high-speed imaging and digital image processing offer new avenues for overcoming these limitations. High-speed cameras, capable of capturing 500 to 10,000 frames per second, enable researchers to observe transient droplet formation, breakup, and movement with exceptional temporal resolution [22,23]. When combined with backlighting and stroboscopic illumination, these systems provide a powerful platform for analyzing translucent water droplets with precise spatial clarity [24]. These tools allow for real-time observation and quantification of critical parameters such as droplet fragmentation, trajectory [25], and evaporation; all of which influence the uniformity and efficiency of water application in sprinkler irrigation systems [26,27,28,29]. Nevertheless, applying high-speed photography under actual field conditions continues to pose challenges due to variable lighting, wind, and droplet overlap, which can compromise image quality and measurement accuracy. Although sprinklers are primarily used in the field, controlled laboratory experiments are essential for quantitatively characterizing droplet formation, breakup, and velocity [30]. Laboratory studies provide baseline, reproducible data that can guide sprinkler design, optimize operating pressures, and enable predictive modeling of field performance [31,32,33].

The present study advances this approach by developing a high-speed imaging framework for the quantitative characterization of water droplet morphology and motion produced by a spiral sprinkler nozzle. The novelty of this research lies in its integrated image-based methodology that simultaneously quantifies droplet diameter, ellipticity, and circularity, while validating measurements through a calibrated reference diameter correlation. This combination of high-speed visualization and morphological analysis provides a robust means of understanding droplet fragmentation and its influence on water application uniformity.

The main objective of the study is to experimentally characterize droplet formation, breakup, and velocity under controlled conditions using a working pressure range of 0.20–0.30 MPa, which represents the typical operational range for medium-pressure sprinklers widely used in agriculture, in order to establish the relationship between operating pressure and droplet behavior parameters [34]. Studies report that operating below 0.20 MPa tends to produce oversized droplets and poor radial distribution, which increases runoff and reduces infiltration efficiency [35]. Conversely, pressures above 0.30 MPa generate excessively fine droplets that are highly sensitive to wind drift and evaporation, resulting in significant water loss and reduced irrigation efficiency [36,37]. Therefore, the selected range of 0.20–0.30 MPa reflects a practical balance between hydraulic performance, droplet stability, and field applicability supported by recent sprinkler-irrigation research. The resulting insights are expected to contribute to the optimization of sprinkler nozzle design and the development of water-efficient irrigation strategies, which are increasingly vital as global water scarcity intensifies due to climate change.

2. Materials and Methods

This study was conducted under controlled laboratory conditions to investigate water droplet characteristics in sprinkler irrigation systems using high speed photography. The methodology was designed to ensure precision, repeatability, and realism by replicating actual operating conditions of agricultural sprinklers while minimizing environmental interference [38].

2.1. Experimental Setup

The experimental system was designed to characterize the droplet characteristics produced by a spiral nozzle under different working pressures. The setup consisted of a spiral sprinkler, electromagnetic flow meter, pressure gauge, pressure-regulating valve, water hose, water pump, measuring tape, and a high-speed digital camera system. The test bench also included a black backdrop screen and a dedicated light source for backlighting (Figure 1a,b).

Figure 1. Photography and schematic of the experimental setup: (a) Photograph of the laboratory test arrangement showing the sprinkler head, pressure gauge, water pump, high-speed digital camera, black background screen, and computer; (b) Schematic diagram illustrating the layout and positioning of the sprinkler head, measuring instruments, camera setup, and background screen.

The sprinkler head used in this study operates within a pressure range of 10–800 kPa and delivers a full-circle (360°) spray pattern. It provides a spraying radius of 6.0–8.5 m and incorporates a single nozzle with an outlet diameter of 5 mm. The unit is fitted with a 19.05 mm (¾ inch) threaded connection and is constructed from durable zinc/brass materials, ensuring stability and corrosion resistance during prolonged testing.

The droplet imaging was carried out using a laboratory-grade high-speed digital camera operating at 7500 frames per second (7.5 × 10³ fps). The system features a sensitive CMOS sensor, microsecond-range shutter speed, and supports high-magnification lenses suitable for capturing fast-moving droplets with minimal motion blur. The camera exposure time was set to 5 µs, and the droplet displacement between consecutive frames provided a temporal resolution of 0.1 ms, enabling precise tracking of droplet motion and breakup. The camera position was optimized through preliminary trials to ensure clear visualization and accurate quantification of droplet size, shape, and motion. A height of 1.4 m above the ground aligned the imaging axis with the primary droplet trajectory, minimizing parallax and ensuring that measurements represented free-flight droplets unaffected by splash or rebound. The horizontal distance of 1.7 m from the sprinkler provided an optimal compromise between spatial resolution and field of view for the 100 mm micro-lens, ensuring that individual droplets remained within the focal plane during motion. The 1.5 m separation from the background minimized light reflections and improved threshold-based segmentation accuracy. The camera’s optical axis was adjusted to intersect the sprinkler jet perpendicularly to minimize perspective distortion and ensure that recorded droplet diameters represented true physical dimensions.

To ensure reliability and statistical robustness, each measurement was repeated three times, and only consistent datasets were retained for analysis. This configuration ensured uniform illumination, consistent image clarity, and stable imaging conditions across all operating pressures (0.20–0.30 MPa).

2.2. Image Acquisition and Preprocessing

The captured image sequences were processed to extract droplet characteristics. Raw grey-scale images were first converted into binary images using a threshold segmentation method. This transformation, defined by Equation (1), allowed for clearer distinction between droplets and background:

$$f(x,y)=\left\{\begin{array}{cc}255& g(x,y)\ge T\\ 0& g(x,y)<T\end{array}\right.$$

(1)

where (g (x, y)) represents the grayscale intensity of each pixel in the original image, (T) is the selected threshold value, and (f (x, y)) is the binary output image. Pixels with intensities greater than or equal to (T) are assigned a value of 255 (white), while those below (T) are assigned 0 (black). The accuracy of image segmentation, as well as subsequent shape description and analysis, is directly affected by the threshold selection. Bimodal, experimental, and maximum inter-class variance methods were evaluated to determine the optimal segmentation threshold.

The following Equations (2)–(8) were applied to compute the grayscale histogram and determine the bilateral filtering threshold for effective droplet edge enhancement:

$$N={\displaystyle {\sum }_{i=0}^{L-1}{n}_i}$$

(2)

$$P_i={\displaystyle \frac{n_i}{N}}$$

(3)

$$P_0={\displaystyle {\sum }_{i=0}^{T-1}{p}_i}$$

(4)

$$P_1={\displaystyle {\sum }_{i=T}^{L-1}{p}_i}$$

(5)

$${\mu }_0={\displaystyle \frac{1}{P_0}}{\displaystyle {\sum }_{i=0}^{T-1}i{p}_i}$$

(6)

$${\mu }_1={\displaystyle \frac{1}{P_1}}{\displaystyle {\sum }_{i=T}^{L-1}i{p}_i}$$

(7)

$${\sigma }_B^2=P_0{P}_1{({\mu }_0-{\mu }_1)}^2$$

(8)

Let the grey scale range of the image be [0, L−1], and $N_i$ denotes the number of pixels with grayscale in the image; $N$ represents the total number of pixels with a greyscale range of [0, L−1]; $P_i$ represents the probability that the grey value is 1; $T$ denote the initial threshold; $P_0$ is the probability that the grey value appears in [0, T−1], and $P_1$ represents the probability that the grey value lies within [T, L−1]. The mean grayscale levels of the two regions are denoted by ${\mathsf{\mu }}_O$ for [0, T−1] and ${\mu }_1$ for [T, L−1]. The total variance of these two grayscale regions is used to determine the optimal threshold. The initial threshold value is selected from the interval [0, L−1], and the value that maximizes the total variance is considered the best for segmenting the image into foreground and background regions. Three dimensionless morphological parameters, namely uniformity (η), ellipticity (ε), and roundness (ψ), were determined using Equations (9)–(12):

$$\eta ={\displaystyle \frac{R_{\max }-R_{\min }}{R_{\mathrm{mean}}}}$$

(9)

$$R_{\mathrm{mean}}={\displaystyle \frac{{\displaystyle {\sum }_{n=1}^{180}{R}_{\mathrm{n}}}}{180}}$$

(10)

$$\epsilon ={\displaystyle \frac{L_{\max }}{L_{\min }}}$$

(11)

$$\mathsf{\Psi }={\displaystyle \frac{P^2}{4\pi A^2}}$$

(12)

These parameters define a shape space, through which different shapes can be identified. The calculation formula can be found in Equation (9), where ƞ is the uniformity, $R_{max}$ and $R_{min}$ are the maximum and minimum distances from the centre of gravity to the contour, and mean is the average radius. P and A stand for the perimeter and area of the droplet profile, respectively. L_max and L_min are the length and the width of the rectangle closest to the image containing the droplet.

2.3. Data Cleaning and Outlier Detection

Droplet size data from high-speed images were preprocessed to enhance accuracy and consistency. Outliers were identified using the interquartile range (IQR) method, with values outside $Q_1-1.5\times \mathrm{IQR}$ and $Q_3+1.5\times \mathrm{IQR}$ removed. Measurements below the imaging system’s resolution limit (0.1 mm) were discarded as noise. The dataset was further checked for missing values and inconsistencies, which were resolved through cross-verification with raw images. Only physically plausible and optically confirmed droplets were retained for statistical analysis.

2.4. Statistical Fitting and Droplet Size Distribution

The results of the lognormal distribution of droplet diameter in Table 1 were obtained by Equations (13) and (14).

$$y=y_0+{\displaystyle \frac{A}{STD\sqrt{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{e}^{-{\displaystyle \frac{{(\ln d-\ln d_c)}^2}{2ST{D}^2}}}$$

(13)

where D is the diameter of the water drop, mm; d_c represents the mean value of the model; STD represents the standard deviation of the model; A and Y₀ represent model fitting coefficients.

$$P_L={\displaystyle \frac{(A_1-A_2)}{1+e^{(\frac{x-x_0}{d})}}}+A_2$$

(14)

where P_L represents the cumulative distribution frequency of water droplet diameter; d represents the diameter of the water droplet; A₁, A₂, and X₀ represent regression fitting coefficients.

Table 1. Calibration diameter of the sample droplet.

Diameter (mm)	1	2	3	4	5	6	7	8	9	10	11	12
Sample 1	1.78	3.52	5.03	6.95	8.84	10.46	11.71	14.05	15.05	16.68	18.42	20.30
Sample 2	1.72	3.53	5.10	6.79	8.52	10.33	12.15	13.81	15.32	16.86	18.53	20.35
Sample 3	1.83	3.41	5.31	6.93	8.73	10.10	11.7	13.92	15.42	16.98	18.55	20.26
Sample 4	1.67	3.51	5.01	7.01	8.45	10.14	11.72	13.67	15.33	16.90	18.39	20.12
Sample 5	1.74	3.51	5.36	7.04	8.35	10.51	12.04	13.65	15.52	16.73	18.60	20.21
Sample 6	1.73	3.38	5.28	6.88	8.75	10.02	11.74	13.94	15.50	16.80	18.44	20.04
Sample 7	1.79	3.37	5.19	6.87	8.62	10.28	12.25	13.39	15.23	16.83	18.51	20.33
Sample 8	1.80	3.47	5.22	7.21	8.61	10.10	12.25	13.43	15.34	17.01	18.60	20.32
Sample 9	1.70	3.59	5.09	6.69	8.85	10.56	12.18	13.46	15.01	16.72	18.45	20.29
Sample 10	1.75	3.42	5.18	7.15	8.68	10.41	11.77	13.62	15.78	16.97	18.47	20.09
Sample 11	1.74	3.49	5.37	7.17	8.68	10.29	12.14	13.94	15.13	16.89	18.62	20.21
Sample 12	1.77	3.39	5.21	7.10	8.82	10.27	12.04	13.92	15.08	16.80	18.45	20.01
Average	1.75	3.46	5.20	6.99	8.66	10.29	11.97	13.74	15.29	16.84	18.50	20.22

The parameters of centrality and droplet size dispersion were determined by the statistical analysis of the droplet data set—average volume diameter (d_v, mm), arithmetic mean diameter (d, mm) and median droplet size (d₅₀)—following Equations (15)–(17):

$$\overline{d}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{m}_i{d}_i}}{{\displaystyle \sum _{i=1}^n{m}_i}}}$$

(15)

$${{\displaystyle d}}_v={\displaystyle \frac{{\displaystyle {\sum }_n^n-=1{{\displaystyle {{\displaystyle d}}_i}}^4}}{{\displaystyle {\sum }_n^n=1{{\displaystyle {{\displaystyle d}}_i}}^3}}}$$

(16)

$$\mathrm{SD}=\sqrt{1/\left(n-1\right){\displaystyle {\sum }_{1=1}^n{{\displaystyle \left(t-\stackrel{-}{t}\right)}}^2}}$$

(17)

where SD is the standard deviation; t_i is the mean quadrant completion time through the ith quadrant in a complete rotation (360°); n is the number of replications, and $\overline{t}$ is the mean is the arithmetic mean of completion time through the four quadrants.

3. Results and Discussions

3.1. Image Acquisition and Preprocessing of Water Droplets

Figure 2 presents a water droplet image acquired by high-speed photography. From the figure, it can be seen that the internal grey value of the acquired two-dimensional imaging is unbalanced, and there is interference (white shadow in the lower left corner of Figure 2a). The grey value of the image reflects the brightness of the image. The value range is from 0 to 255. The grey value of 0 shows black in the image, and the 255 value shows white. Since the water droplet is brighter than the surrounding background—that is, the grey value of the target is larger than the background—the calculation formula of the subtraction method can be obtained from Equation (1). Figure 2b is an effect diagram of removing background noise. The smoothing of the water drop image is preprocessed by the bilateral filtering method and gamma enhancement method. The concept of bilateral filtering was proposed by [39]. It contains two Gaussian basis filtering functions. Therefore, when processing the grey value or colour information of adjacent pixel values, the bilateral filter considers the geometric proximity of each pixel and the similarity in brightness, which has a dual filtering effect.

Figure 2. Image of subtraction algorithm: (a) Before aberration and (b) Post-contrast.

3.2. Image Thresholding Methods for Droplet Segmentation

The bimodal method is based on the characteristics that the distribution of the interior of the image target object and the background area is relatively uniform, while the valley bottom between the two peaks is selected as the threshold. Figure 3a,b are the image grey histogram of the bimodal method. It can be seen that the grey-scale histogram has no bimodal characteristics, and the grey-scale histogram is only the statistics of each pixel. Its peak and peak valley characteristics may not necessarily represent the target and background. The experimental procedure involves using human vision to observe and repeatedly apply different thresholds until the image satisfies the predetermined criteria. The experimental technique to determine the threshold will result in an excessive effort due to a large number of collected photos, and subjective considerations have a significant impact on the segmented image’s quality. The maximum inter-class variance approach, which was first suggested by [40], is a widely used threshold selection process [41]. Its core idea is to select the optimal threshold and the variance between the average grey level of the target area, the average grey level of the background area, and the average grey level of the whole image [42]. Among the many threshold selection algorithms, the Otsu algorithm is the easiest to calculate and has the most stable performance and can be obtained from Equations (2)–(8).

Figure 3. Gray-scale histograms using the bimodal method: (a) histogram of the two apexes and (b) histogram after bilateral filtering.

3.3. Identification of Water Droplet Shape Parameters

Figure 4a–c show the identification results of the uniformity, ellipticity, and roundness of water droplets, respectively. Through observation, it was discovered that the imaging of the water droplets at the time was not round, and that enormous bright spots overlapped. The mean values of the uniformity, ellipticity, and roundness of water droplets in box 1 were 1.3, 2.13, and 0.81, respectively. The recognition results of uniformity, ellipticity, and roundness of water droplets in box 2 are 0.46, 1.49, and 1.51, respectively. The number n of water droplets is the number of connected regions in the image f (x, y). The parameters commonly used to describe the particle size of water droplets include track average diameter, volume average diameter, equal circle average diameter, and geometric average diameter [43]. The results above demonstrate that the recognitions observed are satisfactorily accurate compared to current optical dendrometers for measuring recognition.

Figure 4. Identification results of water droplet characteristics: (a) uniformity, (b) ellipticity, and (c) roundness.

3.4. Dimension Calibration

Table 1 presents the calibration of the water drop diameter. To determine the functional relationship between the two units since the target water droplet in the digital image produced by high-speed photography is in pixels and the length unit of the water droplet feature parameter needs to be retrieved in millimeters, manual calibration was used in this study. First, a standard calibration syringe was used to calibrate water droplets measuring 1 mm, 2 mm, and 3–12 mm. The digital photos of these calibrated water drops were captured at the same moment on the same high-speed photography test bench, with 12 samples being injected into each droplet’s diameter.

To reduce the error caused by the above-mentioned, a series of image processing on the water droplets was carried out to obtain the data in millimeters and the pixels of the droplet diameter in the samples, which are shown in Table 1. The average value of the samples corresponding to each droplet diameter is the average value of the 10 samples after the maximum and minimum values were discarded from the 12 samples.

Figure 5 shows the relationship between the true droplet diameter and the calibrated measurements. The plot demonstrates a strong linear and positive correlation, confirming that the calibration procedure reliably captures the actual droplet size. The close clustering of data points around the regression line indicates minimal measurement distortion and validates the accuracy of the optical calibration.

Figure 5. Relationship between water droplet diameter and calibration diameter.

Building on this calibration, a comparison between the calibrated and simulated droplet diameters shows excellent agreement, with deviations restricted to ±0.1 mm which aligns with [44,45]. Statistical analysis (R² = 0.9996, RMSE = 0.10 mm) further confirms the high fidelity of the simulation model in reproducing experimentally observed droplet behavior. Slight underprediction at smaller diameters and marginal overprediction at larger diameters are within acceptable limits and likely due to scale-dependent modelling assumptions. Overall, the strong alignment between calibrated and simulated values demonstrates the robustness of both the measurement protocol and the numerical model.

3.5. Morphological Characteristics of Water Droplets at the End of Spraying

Figure 6a–c shows the distribution of droplet diameter and shape parameters under different conditions. To explore the distribution of droplet diameter, working pressure was varied from 0.20 MPa at 6 m from the nozzle, 0.25 MPa at 7 m from the nozzle, and 0.30 MPa at 8 m from the nozzle. It can be seen from Figure 6a that under the pressure of 0.20 MPa, the frequency of water droplets with a diameter of more than 3 mm at 6 m from the nozzle is as high as 92%. Compared with other working conditions, there is no water droplet with a diameter of 1–2 mm. The maximal range of the water droplet diameter frequency is 3–4, and its frequencies are 32% and 50%, respectively, under the operating conditions depicted in Figure 6a,b. The number of water droplets with a diameter of 3–4 mm reaches around half of the total number of water droplets under the operating conditions depicted in Figure 6b. Under the working conditions shown in Figure 6c, the frequency of water droplet diameter in the range of 4–5 mm is 35%. With a diameter of 4–5 mm as the boundary, the diameter of water droplets on both sides of the boundary is not symmetrically distributed. The distribution of uniformity, ellipticity and roundness shows that 3 mm droplets are more circular, smoother, and less irregular, comparable to those produced by natural rainfall [46,47].

Figure 6. Distribution of droplet diameter and shape parameters at the spraying end under different conditions: (a) 0.20 MPa, 6 m from the nozzle; (b) 0.25 MPa, 7 m from the nozzle; (c) 0.30 MPa, 8 m from the nozzle.

3.6. Statistical Analysis of Droplets Dispersion

Figure 7 presents the probability density function distribution of droplet diameter under different working pressures. As shown in the figure, the water droplet diameter is unimodal, and the unimodal section is more than 25%. For instance, the frequency of water droplet diameters between 5 and 5.5 mm reached 25% at a pressure of 0.20 MPa. The droplet diameter ranges from 4 to 5 mm with a 25% frequency at the same pressure. The droplet diameter distribution range was narrowed to 2–4.5 mm with a frequency of 30% when the pressure was raised to 0.30 MPa. Significantly more small-diameter droplets are present, while fewer large-diameter droplets appear over time. This occurs as a result of the large-diameter droplets continuously splitting into smaller diameter droplets as the working pressure rises, intensifying the degree of spatial fragmentation of the droplets. These results are in line with previous research by [30,34].

Figure 7. Probability density function distribution of droplet diameter under different pressures: (a) 0.20 MPa, (b) 0.25 MPa, and (c) 0.30 MPa.

The fitting coefficients of the lognormal distribution for the different pressure conditions are shown in Table 2. The mean fitted droplet diameters were 3.57 mm, 3.24 mm, and 2.97 mm for 0.20, 0.25, and 0.30 MPa, respectively. The standard deviations increased slightly from 0.16 to 0.21 as pressure increased. The coefficient of determination (R²), which exceeded 0.93, confirms that the lognormal model appropriately represents the droplet size distribution under all tested pressures [48].

Table 2. Fitting coefficient table of the lognormal distribution of droplet diameter.

P (MPa)	y0	dc	STD	K	R2
0.20	0.02	3.57	0.16	0.34	0.93
0.25	0.01	3.24	0.18	0.38	0.93
0.30	0.01	2.97	0.21	0.44	0.97

Note(s): Where d_c represents the mean value of the model; STD represents the standard deviation of the model; K and y₀ represent model fitting coefficients.

The changes in droplet size distribution with pressure can be explained by the basic fluid dynamic processes that control jet breakup. At low pressures, the water jet leaves the nozzle at a low velocity. Under these conditions, surface tension stabilizes the liquid column and allows larger droplets to form. The breakup process is dominated by capillary forces that act to reduce surface area, so the droplets detach as relatively large units. When the pressure increases, the jet velocity rises and the ratio of inertial forces to surface tension forces becomes larger. This ratio is expressed by the Weber number, which increases sharply as pressure rises. Once inertial forces exceed surface tension forces, the jet becomes unstable. Disturbances caused by air movement and internal liquid fluctuations grow rapidly and cause the liquid column to break into finer structures. These structures detach as smaller droplets. Stronger shear forces at the nozzle exit and higher levels of turbulence inside the jet also contribute to faster breakup. As a result, the droplet size distribution becomes narrower, and the mean droplet size decreases. This behavior agrees with classical descriptions of liquid jet breakup and with earlier studies on sprinkler performance.

3.7. Droplet Diameter Distribution Along the Route

Figure 8 presents a frequency distribution which analyses the accumulated frequency distribution of droplets at different distances from nozzles under different working pressures. It was observed that under the working pressure of 0.2 to 0.3 MPa, a small droplet was found closer to the nozzle. With the increase of the distance, the number of small-diameter water droplets gradually decreases. It can be seen in Figure 8a that at 0.2 MPa pressure, the cumulative frequency curves of water droplets at 7 m and 8 m from the nozzle are almost coincident, and the cumulative frequency of the two curves at 5.25 mm is 88%; this indicates that 12% of the water droplets at 7 m and 8 m from the nozzle are not fragmented. From Figure 8b, it was discovered that when the pressure increased to 0.25 MPa, the cumulative distribution frequency curves of water droplets at 6 m and 7 m are very close; this indicates that the large-diameter droplets were not collected at 6 m and reached the shape of water droplets at 6 m. However, from Figure 8c, it was found that when the pressure is 0.3 MPa, the droplet diameter is 4 mm, and the cumulative frequency curves 6 m and 7 m away from the nozzle tended to be normal, while 8 m was still in the rising stage. The result further shows that the droplet diameter at 6 m and 7 m is mainly distributed below 4 mm. At a distance of 6 m from the nozzle, the cumulative frequency of the droplet diameter is below 3.75 mm under the pressure of 0.2 MPa and 0.25 MPa at 83% and 82%, respectively. The findings revealed that with the increase in pressure, the frequency decreases by 1%. This is possible because there are a few water droplets of the same order of magnitude that collide in space and fuse. When the pressure increases, the probability of collision increases. These results agree with the findings of [38,39,40].

Figure 8. Accumulated frequency distribution of droplets at different distances from the nozzle under varying working pressures: (a) 0.20 MPa, (b) 0.25 MPa, (c) 0.30 MPa.

Table 3 presents the results of the fitting coefficient of the cumulative frequency distribution of droplets under different working pressures. From the results, it was observed that the R² value is greater than 0.98, which indicates that the cumulative frequency distribution of droplets at 6, 7, and 8 m under different pressure conditions conforms to the Boltzmann law, which is a valid model that describes the cumulative frequency distribution of droplets under the specified pressure conditions from 0.20 to 0.30 MPa.

Table 3. Fitting coefficient of the accumulated frequency distribution of droplet.

P (Mpa)	L (m)	A1	A2	X0	d	R2
0.20	6	−0.07	0.99	2.47	0.39	0.994
	7	−0.16	0.97	2.94	0.81	0.995
	8	−0.04	0.97	3.52	0.61	0.991
0.25	6	−6.08	0.99	2.39	0.43	0.990
	7	−0.11	0.98	2.54	0.51	0.996
	8	−0.05	0.97	3.07	0.50	0.994
0.30	6	−0.87	1.00	1.28	0.73	0.999
	7	−0.08	0.99	2.4	0.4	0.992
	8	−0.06	1.00	2.77	0.47	0.997

Note(s): d represents the diameter of the water droplet; A1, A2, and X₀ represent regression fitting coefficients; L represents distance.

Table 4 shows the calculated results of arithmetic mean diameter, median diameter, standard deviation, skewness, kurtosis, and other parameters at 6, 7, and 8 m away from the nozzle under pressures of 0.20, 0.25, and 0.30 MPa. It was found that the mean droplet diameter and volumetric median diameter differ greatly from each other at a distance of 6 to 8 m from the nozzle. The variation range of the average droplet diameter ranged from 2.82 to 4.04 mm, and the variation range of the average droplet diameter was 3.36 to 5.12 mm. At the pressure of 0.3 MPa, the average diameter of droplets calculated by the two methods at a distance of 6 m from the nozzle was 4.12 and 3.07 mm, respectively, and the average diameter of water droplets was 34.2% larger than that calculated by the volume mean diameter. The reason for this difference could be that the number of large-diameter droplets accounts for a small proportion of the number of sampling water droplets, while the weight corresponding to large water droplets accounts for a large proportion of the total water weight corresponding to droplets at the sampling point. Similar results were published by [49]. Under the same pressure, the median diameter increases with the distance. Under the pressure of 0.2 MPa, the median diameter of droplets at 6-8 m from the nozzle is 2.79, 3.41, and 3.68 mm, respectively. When the pressure increases to 0.3 MPa, the median diameter of water droplets decreases by 1.8%, 16.1%, and 17.7%. At a distance of 6 m from the nozzle, the median diameter of droplets increases slightly with the increase of pressure. These increases could be attributed to the outlet of the rotating jet nozzle attached to the wall. With the increase of pressure, the burrs on both sides of the water column increase significantly, and the water droplets are mainly formed by the shedding of burrs. Under the working conditions shown in Table 4, the standard deviation of water droplet diameter varies from 0.64 to 1.34. At the working pressure of 0.2 MPa, the standard deviation 7 m away from the nozzle is 1.34, which indicates that the dispersion degree of the water droplet diameter collected at 7 m under the pressure of 0.2 MPa is higher than that of other groups. In Table 4, the deviation values of droplet diameters are all greater than 0.0, indicating that the distribution of droplet diameters is right-biased. The kurtosis ranges from 3.37 to 4.69, all of which are greater than 3.0; that is, the distribution of the droplet diameter has excessive kurtosis compared with the normal distribution [50,51,52].

Table 4. Statistical characteristics of droplet diameter.

Distance (m)	6			7			8
Pressure (MPa)	0.20	0.25	0.30	0.20	0.25	0.30	0.20	0.25	0.30
$\overline{d}$ (mm)	2.88	2.84	3.07	3.73	3.07	2.82	4.04	3.52	3.18
$d_{\mathrm{v}}$ (mm)	3.36	3.45	4.12	4.42	3.86	3.37	5.12	4.33	3.77
d50 (mm)	2.79	2.72	2.84	3.41	2.88	2.86	3.68	3.28	3.03
STD (mm)	0.64	0.72	1.01	1.34	0.85	0.68	1.16	0.92	0.76
$G_1$	0.87	0.97	0.93	1.31	1.10	1.06	0.93	1.03	0.87
$G_2$	3.88	3.96	3.37	4.55	3.92	4.69	4.01	4.07	3.73

Note(s): $\overline{d}$ = arithmetic mean diameter (mm), $d_v$ = volume mean diameter (mm), $d_{50}$ median diameter (mm), standard deviation (STD mm), $G_1=$ skewness, and $G_2=$ kurtosis.

4. Conclusions

This study underscores the significant potential of high-speed photography as a powerful tool for advancing irrigation technology. These advancements can reduce water waste, lower operational costs, and support sustainable agricultural practices. By providing detailed insights into droplet dynamics under varying pressures, the findings contribute to the development of more efficient sprinkler systems. At a working pressure of 0.2 MPa, the median droplet diameters at distances of 6–8 m from the nozzle were 2.79, 3.41, and 3.68 mm, respectively. When the pressure increased to 0.3 MPa, the median droplet diameter decreased by 1.8%, 16.1%, and 17.7%. These results reveal how pressure changes impact droplet fragmentation and size distribution, with smaller diameter droplets becoming more prevalent. The study also identified the uniformity, ellipticity, and circularity of the droplets as 1.3, 2.13, and 0.81, respectively, indicating a high degree of consistency in droplet formation.

Additionally, the findings show how the selected image processing and acquisition techniques can effectively measure a range of droplet sizes and velocities. These insights are directly applicable to improving sprinkler system design, reducing water waste, and supporting sustainable agricultural practices. The study’s results have practical significance in irrigation, especially in the context of global sustainability goals, by promoting water efficiency and reducing operational costs. Future research could extend these findings by incorporating other environmental variables, such as wind and temperature, to further enhance the practical applicability of the study. Integrating these insights into irrigation system design represents a crucial step toward achieving more sustainable agricultural practices.