# Matching the Liquid Atomization Model to Experimental Data Obtained from Selected Nozzles
## 2 Materials and Methods
### Development of A New Model of Liquid Amount Distribution  
In their work, scientists devoted significant attention to adjusting the distribution of droplets produced by nozzles to real spraying [17; 25; 26]. In order to develop a new model, the primary function of the liquid distribution for a single nozzle was defined according to the formula:  
\[f(x)=\left\{ \begin{array}{cc}b\left(\frac{x-0.5}{0.5}\right)^{p},&x\in(0.5;1 )\\ a-(a-b)\left(\frac{1-2}{0.2}\right)^{q},&x\in<1;1.2)\\ a-(a-b)\left(\frac{x-1-2}{0.2}\right)^{q},&x\in<1.2;1.4)\\ b\left(\frac{2-x}{0.6}\right)^{p},&x\in<1.4;2)\\ 0,&x\in(-\infty;0.5>\cup<2;+\infty)\end{array} \right. \tag{1}\]  
where: \(x\)--is the real variable with values from the spray range (0.5; 2) [m], \(0<a,b\)--are shape parameter dependent of specific properties of histograms, and \(1<p,q\)- are shape parameters. More exactly, the coordinates \((1;b)\), \((1.2;a)\), and \((1.4;b)\) on the real plane characterize the empiric distribution of the accumulation of liquid amounts during the spraying process. The method for construction of the density function presented above ensures semi-symmetry of distribution. The aim of the optimization process was to obtain the appropriate value of the liquid distribution coefficient on the basis of the selection of p and q parameters. The optimization process was performed in Microsoft Excel 2013 Solver Online. Microsoft Excel 2013 Solver has been used to optimize the use of fertilizers [27], crop structure to increase yields [28], and estimation of evapotranspiration [29; 30]. Then, the model described above was transformed into one that describes spraying from the entire sprayer boom. Of course, this was done on a test stand that reflects real working conditions. Hence, the formula of the function describing the spraying from the nozzle is as follows:  
\[g(x)=\sum_{n=0}^{24}f(x-0.5n) \tag{2}\]  
The arguments of the function \(g(x)\) runs over are numbers, each of which represents a single container from the measuring table. Subsequently, to reflect the real conditions, data from 25 identical tables are superimposed in such a way that they are 0.5 m apart, linearly. The amount of liquid that will be collected in the measuring containers is the same as the volume of liquid coming from the entire sprayer boom. Bearing in mind the ISO 16122-2:2015(E) guidelines, when calculating the coefficient of variation, it is necessary to take into account the measurements from the middle of the second nozzle to the second from \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name of Nozzle**} & **Name of** & **Type of Nozzle** & **Flow Rate** & **Size of Nozzle** & **Spray Angle** \\  & **Manufacturer** & **Type of Nozzle** & **[L min\({}^{-1}\)]** & **Size of Nozzle** & **Spray Angle** \\ \hline XR & Teejet, Spraying & Standard & 0.8 & 02 & \(110^{\circ}\) \\  & Systems Co. & & & & \\ EZK & MMT Polarol & Air-induction & 0.99 & 025 & \(110^{\circ}\) \\ RSMM & MMT Poland & Standard & 1.18 & 03 & \(110^{\circ}\) \\ AZMM & MMAT Poland & Anti-drift & 1.18 & 03 & \(110^{\circ}\) \\ IDK & Lechler GMBH & Air-induction & 1.58 & 04 & \(120^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nozzle characteristics. the end [31]. Therefore, the sum range in Equations (2) and (3) corresponds to containers 36 to 256. Using the parameters introduced to define the model, i.e., a, p, b, and q, the formula for the coefficient of variation takes the following form:  
\[V=\frac{S}{\overline{g}}\cdot 100\% \tag{3}\]  
where the formulas for the mean and variance, corresponding to the derived model, are as follows:  
\[\overline{g}=\frac{1}{221}\sum_{i=36}^{256}g(0.05i) \tag{4}\]  
\[s^{2}=\frac{1}{221}\sum_{i=36}^{256}[g(0.05i)-\overline{g}]^{2} \tag{5}\]  
and \(\overline{g}\) and \(s\) are the mean and standard deviation, respectively, of the values of function \(g(x)\).