Uniformity of Linear-Move Irrigation with a Single Sprinkler of the Self-Propelled Hard Hose Traveler

Abstract

In this study, a self-propelled hard hose traveler is developed as a modification of the conventional design. The traveler demonstrated enhanced field applicability and intelligence level in Europe and central–eastern China. A parametric configuration scheme was attained through the irrigator’s computational modeling and experimental validation. This study proposed a uniform water distribution calculation model for single-sprinkler linear-move irrigation. The deviation rate between calculated and experimental values was 7.3%. The average application depth decreased with increased sprinkler motion speed and path spacing. The uniformity of water distribution (CU value) exhibited an oscillating trend as the path spacing changed. As the sprinkler rotation angle increased along a specific path, the CU value first rose from 69.2% to 80.0% and then declined to 68.7%. When irrigation and sprinkler motions were combined, the CU value at 1.5 R initially decreased from 92.1% to 72.9%, then increased to 84.2% as the sprinkler rotation angle increased. The combined sprinkler and irrigation motions showed a significantly better uniformity than the specific path irrigation. The highest CU value was 95.0%, with a nozzle diameter of 16.0 × 6.0 mm, a sprinkler rotation angle of 180°, and a path spacing of 1.6 R. This study introduces a novel approach for water-saving irrigation equipment and offers practical guidance for farmers on operating the self-propelled hard hose traveler.

1. Introduction

The traditional hard hose traveler is water-saving irrigation equipment. It continuously operates with motion and sprinkling capabilities [1]. It is suitable for large- and medium-sized field plots in Europe and central–eastern China [2,3,4,5]. To enhance the irrigator’s applicability in small field plots [6] and its intelligence [7], this paper proposed a self-propelled hard hose traveler with enhanced mobility. The self-propelled hard hose traveler uses an electric tracked vehicle to carry the sprinkler for slope climbing and directional irrigation. The proposed traveler adjusts the speed through a remote terminal at different levels depending on the application depth, providing a novel approach for water-saving irrigation equipment.

Sprinkler irrigation machines primarily operate by moving in a straight path for irrigation purposes [8,9,10]. However, studies comparing the application scenarios of single-sprinkler and double-sprinkler layouts remain limited. In the double-sprinkler layout, each sprinkler irrigates one side of the motion path. Each side may use a different sprinkler type or pressure, depending on how far the path is from the field edge or how far each sprinkler can spray. Xu et al. [11] used a double-sprinkler layout and developed a model to calculate water volume on one side of the motion path, where irrigation was limited to that side only. To optimize irrigation efficiency and minimize the sprinkler’s path within the field, the area covered in a single motion should be maximized. It requires the radiation length of the sprinkler to match its effective range. Additionally, the rotation angle of the sprinkler in the calculation model is set between the range of 90° and 180°. Conversely, a single-sprinkler layout irrigates both sides of the motion path, with the irrigation range extending symmetrically. According to the Chinese standards for rotating sprinklers [12], when the inlet flow rate is the same, the range of a single-sprinkler outlet (in a single-sprinkler layout) is greater than that of two sprinkler outlets (in a double-sprinkler layout). The irrigation area covered in a single move by a single-sprinkler layout is larger than that of a double-sprinkler layout. This can effectively reduce the proportion of paths within the same field and increase the planting area, thereby generating more income for farmers. Zhu et al. [13], Ge et al. [14], and Zhao et al. [15] have found that a longer sprinkler range can effectively enhance the irrigation area covered. To maximize the irrigation area covered in a single move by a single-sprinkler layout, the rotation angle of the sprinkler is selected within the range of 180–360°. Single-sprinkler and double-sprinkler layouts differ in their irrigation range and rotation angle of the sprinkler. Consequently, the water volume superposition calculation model for the single-sprinkler layout needs to be reconstructed.

Water application performance is crucial for the quality of the irrigation system [14]. The application depth and CU value are influential metrics for evaluating the water application performance of traditional hard hose travelers [16]. The application depth depends on the crop water requirement and significantly impacts crop yield [17]. The American Society of Civil Engineers recommends CU value as a key indicator for evaluating the distribution of sprinkler water application depth [18]. The technical code for sprinkler engineering in China defines the CU value as the ratio of the sum of the absolute deviations of the application depth at each measurement point from the average application depth to the total application depth [19,20]. Therefore, the application depth and CU value are crucial for evaluating the sprinkler performance of a self-propelled hard hose traveler with a single-sprinkler motion in a straight line.

Currently, research on water application performance mainly focuses on computational models [21,22,23] and experimental outcomes [5,24,25]. Wang et al. [26] proposed a jet-impact rotating sprinkler by combining jet and impact flows. They studied the performance of sprinkler water application through experimental methods. Gao et al. [27] used Teejet atomizing nozzles. They explored the influence of equivalent diameter, installation height, and working pressure on the water application performance of atomizing nozzles through experimental analysis and theoretical calculations. Gao et al. [28] investigated the impact of installation height and working pressure on the performance of water applications using atomizing nozzles, employing computational models and experiments.

Numerous studies have demonstrated the relationship between operating conditions and water application performance using both computational and experimental methods in the irrigator field. Wu [29] and Bittinger et al. [30] utilized computational models to simplify the radial water distribution lines of sprinklers into regular shapes, such as triangles and ellipses, with time as the dependent variable. The CU value can be calculated by integrating this formulation. However, this simplification method for evaluating radial lines lacks accuracy. Smith et al. [31] and Wiggington et al. [32] investigated the CU value of a single traditional hard hose traveler used in the field. They found an average CU value of 62%, indicating poor performance. Therefore, they researched the combination of irrigation with sprinkler motion using traditional hard hose travelers. Ge et al. [33,34] represented traditional hard hose travelers’ radial water distribution lines using cubic spline interpolation, Lagrange interpolation, and polynomial fitting. They analyzed multiple rotation angles and path spacings using computational models. They found that the optimal sprinkler rotation angle is 240°–320°, the combination spacings are 1.5–1.7 R, and the CU value is greater than 85%. Ge et al. measured the fixed-spray radial water distribution lines. They observed that the selected nodes were relatively sparse. Hills et al. [35] used computations and experimentation to find that when a linear-move sprinkler system operates at 10–100% of its maximum speed, the CU value ranges from 92 to 96%. When the CU value slightly decreases, there is a good correlation between average application depth and motion speed. Li et al. [36] observed that the CU value of a center-pivot sprinkler system slightly decreased with increased speed. The published studies on computational and experimental methods have shown that sprinkler rotation angle, path spacing, and sprinkler motion speed significantly impact the irrigator’s average application depth and CU value.

This study employed cubic spline interpolation curves to determine the CU value for single-sprinkler linear-move irrigation, considering several water applications of the motion sprinkler and the radial application depth per revolution. This study investigated the water application performance of a sprinkler as it moves along a specific path, combining irrigation with sprinkler motion. This study examined the effects of nozzle diameter, motion speed, and sprinkler rotation angle as design variables.

2. Structure and Operating Principle of the Self-Propelled Hard Hose Traveler

A remote terminal operates the self-propelled hard hose traveler. It primarily comprises the reel cart, polyethylene tube, electric-tracked vehicle, impact sprinkler, and an integrated motor–gearbox system, as shown in Figure 1. The electric-tracked vehicle employs a crawler structure with sprinkler heads. The electric-tracked vehicle uses a crawler structure to increase the traction with the sprinkler heads mounted on top of the vehicle. Figure 2 shows a prototype of the self-propelled hard hose traveler. This study utilizes the reel cart from the JP50-180 model of a traditional hard hose traveler. Due to the limitation of the maximum economic flow velocity in the tube, the maximum flow rate is 19.7 m³ h⁻¹.

The reel truck allows the polyethylene tube to unfold, store, and retract. The electric-tracked vehicle is characterized by its maneuverability and flexible walking. The vehicle loads impact sprinklers, changing the irrigation position, unfolding the polyethylene tube, and guiding. The motor–gearbox system provides power to retract the polyethylene tub. During sprinkling irrigation, the reel truck is stored at the water source. The electric-tracked vehicle moves away from the hose reel cart, pulls out the polyethylene tube, and performs mobile sprinkling irrigation. The polyethylene tube disintegrates from the electric-tracked vehicle at the end of the sprinkling irrigation operation. The motor–gearbox system drives the reel to rotate and retract the polyethylene tube. Then, the electric-tracked vehicle is transferred to the next sprinkling point. The self-propelled hard hose traveler achieves multi-directional irrigation by synchronizing the rotation of the two motors in the electric-tracked vehicle.

3. Uniformity Model for Single-Sprinkler Linear-Move Irrigation

3.1. Radial Water Distribution Curve

Figure 3 shows the radial water distribution curve for the 40PY₂ impact sprinkler in the laboratory, with a 19.7 m³ h⁻¹ flow rate measured. This value of 19.7 m³ h⁻¹ matches the maximum capacity of the self-propelled hard hose traveler. By following the GB/T 22999-2008 [12] standard, the nozzles were selected for the impact sprinkler with diameters of 16.0 × 6.0 mm and 15.0 × 6.0 mm. Both sprinkler ranges are about 30 m, and the average application rate is approximately 7.5 mm h⁻¹. The cubic spline curves are used to calculate interpolation for the radial water distribution of sprinklers [37]. The water distribution curve is developed using Equation (1). In the laboratory tests, Equation (2) is used to calculate the number of irrigation cycles for the sprinkler. The sprinkler rotates at an angle of 180°, with a total irrigation time of 0.5 h. The rotation period T is recorded for the sprinkler to complete a 180° rotation. Drawing on the verification methods for radial water distribution curves [38,39], the deviation between the experimental and calculated values of the average irrigation intensity (set as the quotient of the flow rate and the irrigation area) is within 4%.

$$\rho \left(d\right)=\left\{\begin{array}{l}{A}_1{d}^3+B_1{d}^2+C_{1}d+D_1,\left(d_0<d<d_1\right)\\ A_2{d}^3+B_2{d}^2+C_{2}d+D_1,\left(d_1<d<d_2\right)\\ ⋮\\ A_i{d}^3+B_i{d}^2+C_{i}d+D_i,\left(d_{i-1}<d<d_i\right)\end{array}\right.$$

(1)

$$K=INT\left({\displaystyle \frac{1800}{T}}\right)+1$$

(2)

where d is the distance from the sprinkler to the measuring point in m; ρ(d) is the application rate based on the distance between the sprinkler and the measuring point in mm/h; A_i, B_i, C_i, and D_i are the coefficients of Equation (1); d_i is polynomial nodes with integer values in m; K is the number of irrigation cycles of the sprinkler in the laboratory.

3.2. Superposition of Transferred Water Volume

The superimposed value of transferred water volume is calculated by considering the number of watering passes during the movement of the sprinkler. The radial application depth per revolution determines the total application depth at each measurement point. Figure 4 shows the mobile irrigation process of the self-propelled hard hose traveler and the water reception status at measurement point M. Figure 4a shows the spray rotation direction (solid line) and return direction (dashed line) of the sprinkler. Figure 4b shows the water reception at measurement point M and the radial movement of the sprinkler. Figure 4c shows the positional relationship between the position of measurement point M, the position of the sprinkler, and the sprinkler’s rotation angle (where 180° ≤ β < 360°). A single sprinkler sprays water toward both sides of the motion path. When the sprinkler rotation angle is 180 degrees or more, it provides symmetrical coverage, with each side of the motion path receiving at least 90° of spray. In this scenario, the effective coverage length of the sprinkler is equal to its spray range R (m), and the coverage area remains unchanged. When the rotation angle of the sprinkler is less than 180° and coverage is symmetrical on both sides of the motion path, the effective coverage length becomes shorter than the spray range R. As the rotation angle decreases further, both the coverage length and the coverage area continue to decline. For fields with the same area and shape, using a sprinkler rotation angle of less than 180 degrees requires more passes by the self-propelled hard hose traveler compared to angles of 180 degrees or more. In addition, the extended motion path reduces the field area available for planting. This study excludes cases where the rotation angle of the sprinkler is less than 180°. M_start and M_end are the starting and ending positions of water reception at point M, respectively, with a distance of L (m) between them. β (°) shows the rotation angle of the motion sprinkler.

Motion sprinkler irrigation involves two dynamic processes. Firstly, the sprinkler moves uniformly along a path, and secondly, it rotates periodically. During the calculation, these two dynamic processes are considered independently. Based on the rotation cycle of the sprinkler, the distance traveled by the sprinkler is divided into multiple units. The distance between the sprinkler and measurement point M is calculated for each unit position of the sprinkler. This distance, combined with Equations (1) and (2), allows for determining the application depth within a particular cycle. Upon completion of the sprinkler’s movement, the total application depth at point M gives the superimposed value of the application depth at each unit position.

The following three assumptions were used in this study:

(1) The sprinkler jet morphology and the water distribution in the range of water application remain constant during each rotation.

(2) The change in distance between the measured point and the sprinkler is negligible during each rotation.

(3) Wind direction, speed, temperature, and humidity do not affect water distribution under natural conditions.

The process of superimposed water application at point M consists of five steps:

Step 1: The distance l (m) from the sprinkler to the point where the projection of M onto the path intersects is calculated according to Equation (3).

$$l=\left\{\begin{array}{l}\sqrt{R^2-x_m^2},R\sin \left(\pi -{\displaystyle \frac{\beta }{2}}\right)<x_m<R\\ x_m\cot \left(\pi -{\displaystyle \frac{\beta }{2}}\right),0<xm<R\sin \left(\pi -{\displaystyle \frac{\beta }{2}}\right)\end{array}\right.$$

(3)

where x_m is the distance from measurement point M to the motion path, in m.

Step 2: The period t (h) of the rotating angle β of the motion sprinkler and the radial motion distance Δl (m) within one complete cycle are established. These are expressed in Equations (4) and (5).

$$t={\displaystyle \frac{\beta }{180°}}T$$

(4)

$$\Delta l={\displaystyle \frac{vt}{3600}}$$

(5)

where v is the speed of the sprinkler motion in m h⁻¹.

Step 3: The sprinkler motion length L (m) is determined. If the measurement point M moves radially and the sprinkler sprays water in a fan shape at an angle β, then point M receives water within the fan-shaped area. Therefore, within the boundary of the fan shape, the length of the line passing through point M and parallel to the movement path is the sprinkler’s motion length L. It is shown in Figure 4c and expressed in Equation (6).

$$L=\left\{\begin{array}{l}2\sqrt{R^2-x_m^2},R\sin \left(\pi -{\displaystyle \frac{\beta }{2}}\right)<x_m<R\\ \sqrt{R^2-x_m^2}+x_m\cot \left(\pi -{\displaystyle \frac{\beta }{2}}\right),0<x_m\le R\sin \left(\pi -{\displaystyle \frac{\beta }{2}}\right)\end{array}\right.$$

(6)

Step 4: The real-time positions of the sprinkler and measurement point M are determined. Length L is divided into n distance units (Equation (7)). After labeling the distance units Δl with j ∈ (0, n), the distance ΔL from the sprinkler starting at point M_start to the jth unit is obtained, as expressed in Equation (8). Distance d_j (m) from the sprinkler to point M within the jth distance unit is given by Equation (9) as follows:

$$n=INT\left({\displaystyle \frac{L}{\Delta l}}\right)+1$$

(7)

$$\Delta L=j\Delta l$$

(8)

$$d_j=\sqrt{{\left|l-\Delta L\right|}^2+x_m^2}=\sqrt{{\left|x_m^2{\cot }^2\gamma -j\Delta l\right|}^2+x_m^2}$$

(9)

Step 5: The depth of application h(d_j) measured in mm at point M during one complete rotation cycle of the sprinkler and the cumulative depth of application h_n (mm) after n rotation cycles are calculated using Equations (10) and (11), respectively.

$$h\left(d_j\right)={\displaystyle \frac{\rho \left(d_j\right)}{K}}={\displaystyle \frac{\rho \left(d_j\right)}{INT\left(\frac{1800}{T}\right)+1}}$$

(10)

$$h_n={\displaystyle \sum _{j=0}^{n}h\left(d_j\right)}={\displaystyle \sum _{j=0}^n\frac{\rho \left(d_j\right)}{INT\left(\frac{1800}{T}\right)+1}}$$

(11)

3.3. Average Application Depth and Uniformity of Water Distribution

Chinese national standards [40,41] typically employ a radial or grid layout for arranging measurement points with the application depth for a specific area. The average application depth and water distribution uniformity [42,43,44] are calculated considering the application depth from multiple measurement points. It is expressed in Equations (12) and (13).

$$h={\displaystyle \frac{1}{N}}{\displaystyle \sum _{m=1}^N{h}_n}={\displaystyle \frac{2}{N}}{\displaystyle \sum _{m=1}^N{\displaystyle \sum _{j=0}^n\frac{\rho \left(d_j\right)}{INT\left(\frac{1800}{T}\right)+1}}}$$

(12)

$$CU=\left(1-{\displaystyle \frac{{\displaystyle \sum _{m=1}^N\left|h_n-h\right|}}{{\displaystyle \sum _{m=1}^N{h}_n}}}\right)\times 100$$

(13)

where h is the average application depth for motion sprinkler irrigation in mm; CU is the % uniformity of water distribution; m is the serial number of the measurement point; N is the total number of measurement points.

4. Experimental Validation of the Average Application Depth

In this study, grid layout measurement points were used for the experiments, with water buckets arranged in a 1.5 m × 1.5 m formation, as shown in Figure 5. The experiment was conducted at the Lizhong Agricultural Machinery Cooperative in Huantai County, Zibo, China. The experiment utilized an electrically tracked vehicle with sprinklers to achieve a speed of 15 m h⁻¹. After the sprinkler irrigation experiment, the radial lines of the columns were averaged. This average was taken as an experimental value. The experimental parameters were considered using the reel cart model JP50-180, a 40PY₂ impact sprinkler with a nozzle diameter of 16.0 × 6.0 mm, a flow rate of 19.7 m³ h⁻¹, a sprinkler rotation angle of 240°, a sprinkler installation height of 1.3 m, a plastic bucket diameter of 200 mm, and a plastic bucket height of 170 mm. Environmental parameters were characterized by gusty winds with a speed range of 1.23–2.55 m s⁻¹, maintaining an air temperature of 12.3 °C and an air relative humidity of 24.82%. Figure 5 shows the comparison of the experimental and the calculated values.

Figure 6 shows the calculated and experimental application depth curves for motion sprinkler irrigation, with an average application depth of 23.3 mm and 25.3 mm, respectively. The deviation rate is 8.6%, indicating that the calculated values of application depth are mostly accurate. When comparing the two curves, experimental values are higher than the calculated values. A slight deviation on the curve was observed at the 12-meter point, where the values were 30.5 mm and 29.7 mm, resulting in a deviation rate of 2.7%. The maximum deviation was found at the 28.5-meter point, where the values were 3.4 mm and 2.0 mm, resulting in a deviation rate of 70.0%. The wind speed and direction influence significant deviations at points on the two curves. Xu et al. [45] and Chang et al. [46], through research on sprinkler combination irrigation, discovered that the evaporation and drift loss amounted to approximately 5% at an average wind speed of 1 m s⁻¹, while the deviation rate of the CU value under wind speeds ranged from 0 to 5.8%. Overall, the accuracy of experimental values is controllable.

5. Irrigation Performance of Single-Sprinkler Linear-Moving Irrigation

5.1. Motion Along a Specific Path

5.1.1. Different Nozzle Diameters

Table 1. Average application depth and CU values of various nozzles of a 40PY₂ impact sprinkler.

Nozzle Diameter (mm)	Average Application Depth (mm)	CU Value (%)
15.0 × 6.0	22.4	74.9
16.0 × 6.0	23.3	80.0

and Figure 7 provide details of the application depth and CU values for various nozzles with a sprinkler rotation angle of 240° and a motion speed of 15 m h⁻¹. In Figure 7, the nozzle with a 15.0 × 6.0 mm diameter has a spray range of 30 m, which is greater than the 28.5 m of the 16.0 × 6.0 mm nozzle. Table 1 shows that the average application depth is around 23.0 mm, and the CU values for the two nozzles are 74.9% and 80.0%, respectively. The optimal nozzle diameter for the 40PY₂ impact sprinkler in terms of CU value is 16.0 × 6.0 mm.

5.1.2. Different Sprinkler Motion Speeds

The application depth and CU values are given in Figure 8 and

Table 2. Average irrigation amount and CU values at different motion speeds.

Motion Speed (m h−1)	Average Application Depth (mm)	CU Value (%)
6	58.5	79.9
8	43.8	79.9
10	35.0	80.0
12	29.2	79.9
15	23.3	80.0
20	17.4	80.0
25	13.9	80.1
30	11.6	80.2

when the sprinkler rotation angle is 240° and the nozzle diameter is 16.0 × 6.0 mm. In Figure 8, with the increased motion speed, the average application depth decreases in a power function form. Equation (14) explains this with a fitting coefficient of 1. From Table 2, the CU value exhibits slight oscillatory changes with an increasing motion speed of around 80%. The fluctuation range of the CU value at 0.3% is insignificant and does not demonstrate the impact of motion speed on water distribution. Furthermore, fluctuations in CU values are caused by variations in the length of Δl. Therefore, the influence of sprinkler motion speed on CU values can be neglected.

$$h=355.25v^{-1}$$

(14)

5.1.3. Different Sprinkler Rotation Angles

Figure 9 and

Table 3. Average application depth and CU values for different sprinkler rotation angles.

Sprinkler Rotation Angle (°)	Average Application Depth (mm)	CU Value (%)
180	23.1	69.2
210	23.2	78.2
240	23.3	80.0
270	23.4	74.2
300	23.4	70.0
360	23.5	68.7

present the application depth and CU values at 15 m h⁻¹ and a nozzle diameter of 16.0 × 6.0 mm. In Table 3, the average application depth remains constant at around 21.3–21.5 mm, whereas the CU value varies from 68.7 to 80.0%. The relative deviation in the average application depth is due to discrepancies in value selection and model calculations.

Figure 9 shows that the initial value of the application depth curve decreases as the sprinkler rotation angle increases. The curve shows a normal distribution, with a peak value around 13.5 m, indicating an upward trend. However, the average application depth remained unchanged. The CU value initially increased from 69.2% to 80.0% and then decreased to 68.7%. The sprinkler irrigation effect of the 40PY₂ impact sprinkler was optimal when the rotation angle was within 210–270°, with the best CU value being greater than or equal to 80.0%.

5.2. Combined Irrigation with Sprinkler Motion

This study shows that a specific path movement of the self-propelled hard hose traveler does not follow the technical code for sprinkler engineering [19] with a CU value of 85%. Consequently, a lower value for the combined sprinkler movement was used in this study for irrigation. At a path spacing of 2.0 R, the sprinkling effect is identical when not combined.

5.2.1. Different Nozzle Diameters

For the sprinkler rotation angle of 240° and the motion speed of 15 m h⁻¹, the average application depth and CU values for different nozzles at path spacings of 1.0–1.9 R are given in

Table 4. Average application depth and CU values of different nozzle diameters.

Index	Nozzle Diameter (mm)	Path Spacing (m)
		1.0 R	1.1 R	1.2 R	1.3 R	1.4 R	1.5 R	1.6 R	1.7 R	1.8 R	1.9 R
Average application depth (mm)	15.0 × 6.0	44.8	40.7	37.3	34.4	32.0	29.8	28.0	26.3	24.9	23.6
	16.0 × 6.0	46.6	42.2	38.5	35.4	32.8	31.6	29.5	27.7	26.1	24.6
CU value (%)	15.0 × 6.0	77.9	68.9	65.2	66.8	72.1	79.7	85.3	89.7	90	84.1
	16.0 × 6.0	80.0	69.6	64.4	65.5	71.5	75.4	82.2	88.1	92.0	86.3

. At the same path spacing, there are deviations in the average application depth and CU values at different nozzles, as shown in Table 4. As the path spacing increases, the average application depth constantly decreases for the same nozzle, while the CU value demonstrates a wavy trend. It is challenging to investigate the influence of nozzle diameter on the average application depth. However, the optimal CU values are above 90%. The highest CU value of 92.0% is achieved with a nozzle diameter of 16.0 × 6.0 mm at a path spacing of 1.8 R. The technical code for sprinkler engineering for the selected path spacings of the 40PY₂ impact sprinkler is 1.6–1.9 R.

5.2.2. Different Sprinkler Motion Speeds

Table 5. Average application depth and CU value at different sprinkler motion speeds.

Index	Motion Speed (m h−1)	Path Spacing (m)
		1.0 R	1.1 R	1.2 R	1.3 R	1.4 R	1.5 R	1.6 R	1.7 R	1.8 R	1.9 R
Average application depth (mm)	6	117.0	105.8	96.6	88.9	82.3	79.4	74.1	69.5	65.4	61.7
	8	87.7	79.3	72.4	66.6	61.7	59.5	55.5	52.1	49.0	46.3
	10	70.1	63.4	57.9	53.3	49.3	47.6	44.4	41.6	39.2	37.0
	12	58.3	52.8	48.2	44.3	41.1	39.6	36.9	34.6	32.6	30.8
	15	46.6	42.2	38.5	35.4	32.8	31.6	29.5	27.7	26.1	24.6
	20	34.9	31.6	28.8	26.5	24.5	23.7	22.1	20.7	19.5	18.4
	25	27.9	25.2	23.0	21.2	19.6	18.9	17.6	16.5	15.6	14.7
	30	23.1	20.9	19.1	17.6	16.3	15.7	14.7	13.7	12.9	12.2
CU value (%)	6	80.0	69.5	64.4	65.5	71.5	75.4	82.2	88.1	91.9	86.2
	8	80.0	69.5	64.4	65.5	71.5	75.4	82.2	88.1	91.9	86.3
	10	80.0	69.6	64.4	65.5	71.5	75.4	82.2	88.1	92.0	86.3
	12	80.0	69.5	64.4	65.5	71.5	75.4	82.2	88.1	92.0	86.3
	15	80.0	69.6	64.4	65.5	71.5	75.4	82.2	88.1	92.0	86.3
	20	80.1	69.6	64.5	65.5	71.5	75.4	82.2	88.1	92.1	86.4
	25	80.2	69.7	64.5	65.5	71.5	75.3	82.2	88.1	92.1	86.4
	30	80.3	69.8	64.5	65.6	71.5	75.4	82.2	88.1	92.1	86.5

presents the application depth and CU values for a sprinkler rotation angle of 240° and a nozzle diameter of 16.0 × 6.0 mm. Table 5 shows that at the same path spacing, the average application depth constantly decreases with an increased motion speed while the CU value remains unchanged. At the same motion speed, the average application depth continuously decreases as the path spacing rises, and the CU value demonstrates an oscillatory trend. The CU values for the 40PY₂ impact sprinkler fluctuate at 64.4–92.1%, and the highest CU value is 92.0 at a path spacing of 1.8 R. The technical code for sprinkler engineering for the selected path spacings for the 40PY₂ impact sprinkler is 1.7–1.9 R.

5.2.3. Different Sprinkler Rotation Angles

Table 6. Average application depth and CU value of different sprinkler rotation angles.

Index	Sprinkler Rotation Angle (°)	Path Spacing (m)
		1.0 R	1.1 R	1.2 R	1.3 R	1.4 R	1.5 R	1.6 R	1.7 R	1.8 R	1.9 R
Average application depth (mm)	180	46.1	41.8	38.1	35.1	32.5	31.3	29.2	27.4	25.8	24.4
	210	46.4	42.0	38.3	35.2	32.6	31.5	29.4	27.5	25.9	24.5
	240	46.6	42.2	38.5	35.4	32.8	31.6	29.5	27.7	26.1	24.6
	270	46.8	42.3	38.6	35.5	32.9	31.7	29.6	27.8	26.1	24.7
	300	46.9	42.4	38.7	35.6	33.0	31.8	29.7	27.8	26.2	24.8
	330	47.1	42.6	38.9	35.8	33.1	32.0	29.8	28.0	26.3	24.9
CU value (%)	180	83.9	79.0	78.5	82.3	88.7	92.1	95.0	90.6	83.2	76.1
	210	83.2	75.0	71.1	72.5	78.3	82.1	88.1	91.6	92.4	85.7
	240	80.0	69.6	64.4	65.5	71.4	75.4	82.2	88.1	92.0	86.3
	270	74.3	64.3	60.0	61.9	68.7	72.9	79.9	85.2	85.7	80.0
	300	70.7	62.5	60.1	63.6	71.3	75.0	80.7	84.3	81.3	75.9
	330	73.6	68.5	69.2	73.8	81.3	84.2	86.3	86.1	80.9	75.0

details the average application depth and CU values for a nozzle diameter of 16.0 × 6.0 mm and a motion speed of 15 m h⁻¹. In Table 5, at the same sprinkler rotation angle, the average application depth gradually decreases with increased path spacing, while the CU value exhibits a wavy trend. For the same path spacing, the average application depth fluctuates within ±1 mm as the sprinkler rotation angle increases. Meanwhile, the CU value first decreases from 92.1% to 72.9% at 1.5 R, then rises to 84.2%, with an overall fluctuation range of 60.1–95.0%. The highest CU value occurs at 95.0% when the sprinkler rotation angle is 180° and the path spacing is 1.6 R. The technical code for sprinkler engineering for the selected path spacing of the 40PY₂ impact sprinkler is 1.6–1.9R, and the selectable sprinkler rotation angles are 180°, 210°, and 240°.

6. Results and Discussion

When the sprinkler moves along a specific path and is used in combined irrigation with sprinkler motion, the optimal CU values are approximately 75% and 90%. It provides good evidence supporting the conclusions drawn by Smith and Wigginton et al. [31,32] (62%), as well as the findings of Ge et al. [33,34] (which exceeded 85%). The sprinkling effect of the sprinkler moving along a specific path is shown in Figure 10, with a nozzle diameter of 16.0 × 6.0 mm, a moving speed of 15 m h⁻¹, and a sprinkler rotation angle of 180°. Figure 11 shows the effect of combined irrigation with sprinkler motion with a path spacing of 1.6 R. Figure 11 shows improved application depth uniformity as compared to Figure 10, which is attributed to the overlapping of the trough values in the moving irrigation water volume curve. Combining irrigation with sprinkler motion suits conventional sprinkling operations, while motion along a specific path is used for irrigation in irregular areas. This study indicates that the nozzle diameter and sprinkler motion speed have no significant effects on the CU value, with the optimal range being 180–240° for the sprinkler rotation angle. It aligns with the conclusion of Hills and Li et al. [35,36], which showed that slight changes in movement speed alter the CU distribution, but differs from the findings of Ge et al. [33,34], who identified 240° to 320° as the optimal range for the sprinkler rotation angle. This is mainly caused by different radial water distribution curves produced by various sprinklers. The distances from the peaks and troughs of these varying radial water distribution curves to the sprinklers differ, and so do the differences between the peak and trough values.

From the technical code for sprinkler engineering, the highest CU value is approximately 95%, with a nozzle diameter of 16.0 × 6.0 mm², a sprinkler rotation angle of 180°, and a path spacing of 1.6 R.

Table 7. Summary of hydraulic parameters for self-propelled hard hose traveler.

Number	Hydraulic Parameters	Data
1	Flow rate (m3 h−1)	19.7
2	Sprinkler model	40PY2
3	Number of sprinklers	1
4	Nozzle diameter (mm)	16.0 × 6.0
5	Working pressure (kPa)	450
6	Sprinkler elevation angle (°)	25
7	Sprinkler range (m)	28.5
8	Motion speed (m h−1)	6~50
9	Variation in motion speed (m h−1)	1
10	Sprinkler rotation angle (°)	180, 210, 240
11	Path spacing (m)	1.6~1.9 R

summarizes the hydraulic parameters of the self-propelled hard hose traveler when the CU value is greater than or equal to 85%, showing key guidance for farmers in operating this sprinkler.

7. Conclusions

This study investigated the hydraulic performance of a self-propelled hard hose traveler with a single-sprinkler arrangement (40PY₂ impact sprinkler). The impact of nozzle diameter, sprinkler motion speed, sprinkler rotation angle, and path spacing on average application depth, as well as the CU values, are investigated through numerical modeling and experimentation. This study developed parameter configuration schemes to guide field operations and help fill the gap in the water volume superposition calculation model for the single-sprinkler layout. The specific conclusions of this study are as follows:

(1) A uniformity model for single-sprinkler linear-move irrigation was established. The deviation rate between calculated and experimental values was 7.3% in experimental validation.

(2) The CU value is affected by nozzle diameter and motion speed, exhibiting an oscillating trend with changes in path spacing. When moving along a specific path, the CU value first increased from 69.2% to 80.0% and then decreased to 68.7% as the sprinkler rotation angle increased. When irrigation and sprinkler motion are combined, the CU value at 1.5R initially decreased from 92.1% to 72.9%, then increased to 84.2% with increased sprinkler rotation angle. The average application depth decreases with increased sprinkler motion speed and path spacing and remains unaffected by the sprinkler rotation angle. Defining the variation in average application depth with nozzle diameter is challenging.

(3) The effect of combined irrigation with sprinkler motion is significantly better than that of moving along a specific path, with optimal CU values around 90% and 75%. This study provided parametric configurations for self-propelled hard hose travelers using combined irrigation with sprinkler motion. The highest CU value is 95.0%, with a parameter configuration of a nozzle diameter of 16.0 × 6.0 mm, a sprinkler rotation angle of 180°, and a path spacing of 1.6 R.

The irrigation performance of self-propelled hard hose travelers has been thoroughly studied. Future research will focus on path planning for cooperative irrigation among multiple sprinkler irrigators to achieve comprehensive irrigation (CU ≥ 85%) across complex and diverse field layouts, thereby reducing operational and management costs.

In future work, drone aerial photography will be used to create an image-based model. Key information about field plots will be extracted based on the spectral and local texture characteristics of the image model. Image features will be classified using the Bag-of-Words (BoW) model, and the resulting data will be used to train a Support Vector Machine (SVM) classifier for automatically constructing a multi-agent operating space.

Secondly, using the constraints and aiming to optimize factors will be applied to continuously link the existing static paths in the fields. Heuristic search algorithms will be used to develop decision-making strategies for the dynamic paths of multiple agents. An effective static path network for collaborative sprinkling by sprinkler robots (self-propelled hard hose travelers) will be established to achieve global path planning and enhance the intelligence level of the sprinkler system and equipment.

Finally, leveraging a multi-agent distributed consensus-based cooperative scheduling algorithm, a dynamic scheduling strategy will be formulated. Under multi-objective constraints, global and local optimal consistency in field plot sprinkler operations will be achieved, thereby improving the operational efficiency and comprehensive performance of the agent system.