## Materials and Methods
### Nozzle Configuration for the Center Pivot  
Center pivots use the central trestle tower as a pivot, and span tower carriages support and drive the sprinkler pipes to rotate around the central trestle tower in a circular movement. The moving sprinkler pipes can be seen as the sprinkling radius of the center pivot. Thus, sprinklers located near the end of the sprinkling radius have the highest moving speed relative to the ground; in contrast, those located near the pivot have quite a low moving speed. Moreover, pressurized water enters the sprinkler pipes from the central trestle tower. The pressure declines along the sprinkling radius due to the effects of head losses [45; 46; 47]. Therefore, the discharge of sprinklers should be increased gradually along the sprinkling radius in order to provide a uniform application volume over the full circular irrigated area. To this end, a popular sprinkler configuration method in the center pivot is an equal sprinkler interval but with various flow rates (increasing nozzle diameters) [13; 48]. To obtain a precise nozzle configuration along the sprinkling radius, a nozzle configuration model was used to deploy nozzles of varying diameters along the sprinkling radius [13]. The model was established as following: First, we can decide the expected flow rate and pressure of the end sprinkler in the center pivot. We take the expect values as the actual values for the ending sprinkler. Then, the nozzle diameter of the ending sprinkler (\(d_{ZN}\), mm) is defined as follows:\[\left\{ \begin{array}{l}q_{N}=2\pi\mathrm{s}R_{N}\frac{\omega p}{360}\\ q_{N}=c\sqrt{2 gH_{N}}d_{ZN}^{2}\frac{\pi}{4}\end{array} \right. \tag{5}\]  
where \(q_{N}\) is the actual flow rate of the ending sprinkler (m\({}^{3}\cdot\)s\({}^{-1}\)); \(H_{N}\) is the pressure of the ending sprinklers (m); \(R_{N}\) is the distance from the pivot to the ending sprinkler (m); and \(g\) is gravitational acceleration (m\(\cdot\)s\({}^{-2}\)). The pressure of the other sprinklers is decided by the head losses, which is calculated as follows:  
\[\left\{ \begin{array}{l}H_{\omega(i,i+1)}=f\frac{s\left(\sum\limits_{k=i+1}^ {N}q_{k}\right)^{m}}{d^{k}}+h_{j(i,i+1)}\\ H_{i}=H_{i+1}+H_{f(i,i+1)}\end{array} \right. \tag{6}\]  
where \(H_{i}\) and \(H_{i+1}\) is the pressure of the \(i\)-th and (\(i+1\))-th sprinklers (m); \(H_{\omega(i,i+1)}\) is the head losses between the \(i\)-th and (\(i+1\))-th sprinklers (m); \(h_{j(i,i+1)}\) is the local head losses between the \(i\)-th and (\(i+1\))-th sprinklers (m);\(f\) is friction coefficient; \(d\) is internal diameter of water pipe (mm); \(b\) is coefficient of pipe diameter; \(m\) is flow index; \(N\) is number of sprinklers; and \(\sum\limits_{k=k+1}^{N}q_{k}\) is flow rate in the section between \(i\)-th and (\(i+1\))-th sprinklers (m\({}^{3}\cdot\)s\({}^{-1}\)). Then, the nozzle diameters of the other sprinklers are defined as following:  
\[\left\{ \begin{array}{l}q_{i}=c\sqrt{2 gH_{i}}d_{Zi}^{2}\frac{\pi}{4}\\ d_{Zi}=\min\left|\frac{q_{i}-q_{i}}{q_{i}}\right|\end{array} \right. \tag{7}\]  
where \(d_{Zi}\) is nominal diameter of selected nozzles (mm); and \(q_{i}\)e is expected flow rate of the \(i\)-th sprinkler (m\({}^{3}\cdot\)s\({}^{-1}\)). In addition, the expected flow rate of the \(i\)-th sprinkler (\(q_{i}\)e) is calculated as following:  
\[q_{i}=2\pi\mathrm{s}R_{i}\frac{\omega p}{360} \tag{8}\]  
where \(s\) is sprinkler interval (m); \(R_{i}\) is the distance from the pivot to the \(i\)-th sprinkler (m); \(\omega\) is angular velocity of the operating center pivot (\({}^{\circ}\cdot\)s\({}^{-1}\)); and \(p\) is water application depth per unit time (mm\(\cdot\)h\({}^{-1}\)). For application of the above model we obtained nozzle configurations for the sprinkling radius, under conditions of 1.5-, 3.0-, and 4.5-m sprinkler intervals.