# Simulation-based study of the influence of particle physical properties on fertilizer spreading ability

Tien-Thinh Le, Emmanuel Piron, Denis Miclet, Sylvain Villette

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## To cite this version:

Tien-Thinh Le, Emmanuel Piron, Denis Miclet, Sylvain Villette. Simulation-based study of the influence of particle physical properties on fertilizer spreading ability. Computers and Electronics in Agriculture, 2025, 229, pp.109753. 10.1016/j.compag.2024.109753. hal-04841294

HAL Id: hal-04841294

https://institut-agro-dijon.hal.science/hal-04841294v1

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# Simulation-based study of the influence of particle physical properties on fertilizer spreading ability

Tien-Thinh Leª, Emmanuel Pironª, Denis Micletª, Sylvain Villetteª,ᵇ,ᶜ

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

### Keywords:

- Particle size distribution

- Drag coefficient

- Density

- Throw distance

- Working width

- Spread pattern

## ABSTRACT

Despite spreader performance, different fertilizers are known to have inherent limitations that prevent them from achieving the same working width at a targeted spreading quality. Nevertheless, there is currently no mathematical relationship to express the maximum working width achievable by a fertilizer as a function of its physical properties. The aim of this paper is to develop a numerical method to provide such a mathematical relationship. Since the maximum working width $W_{WCV10}$ ensuring a satisfying uniformity (CV≤10%) is a complex parameter (resulting from different stages of the spreading process), the issue has been decomposed into two steps. First, a dataset is computed by numerical simulations. 3800 virtual fertilizers are designed and feed into numerical simulations based on a Monte Carlo process. For each fertilizer, twin disk spread patterns are computed for a set of virtual spreader settings. Then, $W_{WCV10}$ is deduced from simulations of adjacent passes and overlaps. The mean radius $R_m$ of the spread pattern is also computed. Second, based on this consistent dataset, multivariate polynomial regressions are established to express $W_{WCV10}$ and $R_m$ as mathematical functions of 5 independent physical characteristics of the fertilizer: the drag coefficient, the specific density and three particle size distribution parameters. Thus, for the first time, the contribution of each variable (to the ability of the fertilizer to achieve a working width) is quantified and the sensitivity of $W_{WCV10}$ and $R_m$ is analyzed by using partial derivatives. The correlation between $W_{WCV10}$ and $R_m$ is also demonstrated.

## 1. Introduction

Spreading uniformity directly affects the efficiency and environmental impact of fertilizers (Gyldengren et al., 2020; Miller et al., 2009; Richards and Hobson, 2013; Tissot et al., 2002). To achieve maximum work output, farmers aim to utilize the largest possible working width (i. e. distance between successive tractor tracks) while maintaining an acceptable level of application uniformity. This uniformity is typically evaluated calculating the coefficient of variation (CV) of adjacent, overlapping transverse distributions and a CV less than 15 % indicates correct application (EN 13739-2, 2011).

For applying granular fertilizers, centrifugal spreaders are the primary method across Europe. This process consists of two main stages: the motion of the particles on a spinning disk and the ballistic flight of the particles.

Concerning the first stage, the characteristics of the disk influence the motion of the particles on its vanes, and ultimately affect the particle

ejection. Numerous works studied on-disk motion and expressed outlet parameters as functions of the disk geometry and revolution speed using analytical approaches (Hofstee, 1995; Olieslagers, 1997; Villette et al., 2005). Experimental works also studied the influence of disk characteristics such as vane shape and profile (Fulton et al., 2020; Yildirim, 2008), cone angle (Yildirim, 2006) or revolution speed (Villette et al., 2012; Yildirim, 2006). These works demonstrated that target ejection characteristics (velocity and mass flow distribution around the disk) can be achieved by designing appropriate spreader hardware components. Thus, although particle ejection results from interactions between the particles and the spinning disk, it has been established that the particle ejection characteristics can be controlled by designing suitable parts and adjusting mechanical parameters of the spreader. These ejection characteristics, controlled by the spreader, are the initial parameters of the fertilizer ballistic flight.

On the opposite, once the particles left the spinning disk, the second stage of motion (i.e. ballistic flight) does not depend on spreader

<footnote>ª Université Clermont Auvergne, INRAE, UR TSCF, AgroTechnoPôle, Domaine des Palaquins, 40 route de Chazeuil, F-03150 Montoldre, France</footnote>

<footnote>ᵇ Institut Agro, 26 bd Docteur Petitjean, F-21000 Dijon, France</footnote>

<footnote>* Corresponding author at: Université Clermont Auvergne, INRAE, UR TSCF, AgroTechnoPôle, Domaine des Palaquins, 40 route de Chazeuil, F-03150 Montoldre, France.</footnote>

<footnote>E-mail address: sylvain.villette@institut-agro.fr (S. Villette).</footnote>
performances. It only depends on fertilizer physical properties. Size, specific density and drag coefficient have been known to be properties that affect the motion of fertilizer particles through the air (Hofstee and Huisman, 1990). Historically, the effects of these properties have mainly been studied in the context of preventing ballistic segregation during blended fertilizer spreading. Most experimental studies have highlighted segregation issues (Hoffmeister et al., 1964; Thaper et al., 2021; Virk et al., 2013; Yule, 2011), particularly due to size variability between fertilizer components, but they have not established any mathematical relationship between physical properties and spreading performance indicators. Miserque et al. (2008) were the only authors to mathematically related fertilizer physical characteristics to a ground distribution characteristic. They related the particle properties to the spreading width, which is the distance between the extreme left and right points where the fertilizer lands on the ground. Thus, they developed a multivariate regression model ($R^2$ = 0.60) to predict spreading width as a function of particle median diameter of the blend and density of the heaviest component. This study has interestingly highlighted the impact of particle characteristics on their ability to be projected up to a given distance. However, it focused on fertilizer mixtures and the risk of segregation between the components of the mixture.

Studying the influence of fertilizer physical properties is challenging through experimental measurements due to two main difficulties: first, separating the spreading process into on-disk and off-disk steps and second, isolating the impact of individual properties. Another way is to study the impact of fertilizer characteristics using theoretical models to compute particle ballistic flights. Using approximating equations for ballistic trajectories, Pitt et al. (1982) studied the influence of particle size variability on the variability in throw distance for a given initial velocity and height. They concluded that variation in particle size has only a mild effect on the lateral distribution of particles. They also modelled the transverse spread pattern assuming simple uniform distributions of particle drop point onto the spinning disk and derived U-shaped transverse spread patterns. Nevertheless, the effect of particle size distribution on the transverse spread pattern were not studied and adjacent overlaps were not considered. Based on the study of transverse spread patterns for granulated urea, prilled urea, and ammonium nitrate, Parkin et al. (2005) suggested calculating the theoretical stop distance as a characteristic parameter to compare the ability of fertilizers to reach the same spreading width. Nevertheless, this approach has not been explored in subsequent studies. Antille et al. (2013) suggested the particle size range of new fertilizers could be designed to meet a target working width and proposed to compute the ballistic flight to assess whether particle physical properties suited the spreading objective. The model predicted the throw distance of individual particles but did not simulate the spread pattern and did not address the working width. Grafton et al. (2015) also proposed using a ballistic model to investigate how fertilizer characteristics influence working width. The authors compared working widths deduced from experimental measurements with throw distances computed for fertilizer mean particle size but no mathematical relationship has been established.

Thus, no general relationship has ever been established in literature between the maximum achievable working width and other spreading characteristics such as the mean radius of the spread pattern. Furthermore, no mathematical relationship has ever been established between the fertilizer physical properties and the maximum working width. Consequently, there is a lack of models that can evaluate and compare fertilizers based on their ability to achieve a target working width with uniform distribution. Additionally, no model makes it possible to evaluate the respective contribution of the different physical properties. In practice, this leads fertilizer producers to still wonder about which product properties to modify to improve fertilizer spreading ability.

The objective of the study is to fill these gaps by demonstrating that the maximum achievable working width is linearly related to the spread pattern mean radius, and by expressing these spreading parameters as functions of fertilizer physical properties. The maximum working width

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Side view (top) and top view (bottom) of the counterclockwise S8 disk: $r_1, \alpha_1, \Omega_1$ are the radius, pitch angle and vertical angle respectively for the long vane and $r_2, \alpha_2, \Omega_2$ for the short vane; $v_{out}, \theta_{out}$ and $\Omega_{out}$ are the outlet velocity vector, the horizontal outlet angle and vertical outlet angle respectively; $\theta_{traj}$ is the horizontal angle of the trajectory; $\theta_{vane}$ is the angular location of the vane; $\mathbf{i}, \mathbf{j}, \mathbf{k}$, vectors of a right handed Cartesian coordinate system centered on the disk axle. The particle ejection is illustrated when a particle leaves the long vane.</figcaption></figure>

is a complex parameter, influenced by particle ballistic behavior, spreader adjustment and pass overlap. Consequently, the experimental study of the maximum working width as a function of fertilizer characteristics is unrealistic, while the theoretical study is impossible by classical mechanical approaches. The originality of this study consists in using numerical simulations to compute the working widths and the corresponding CVs for a large number of virtual fertilizers in order to obtain a database, which then feeds a regression analysis to establish the mathematical relationships relating the maximum working width to the physical characteristics of the fertilizer.

The study has four main contributions:

i) the development of a numerical method to provide a consistent set of maximum working width values computed for a wide range of fertilizer physical properties;

ii) the establishment of a mathematical model relating the maximum working width and the spread pattern mean radius to five independent variables describing fertilizer physical properties;

iii) the analysis of the relative contribution of each physical property on the ability of the fertilizer to achieve a targeted working width;

iv) the demonstration of the correlation between the maximum working width and the mean radius of the corresponding spread pattern.

## 2. Materials and methods

In this study, the fertilizer spreading ability was studied by computing two spreading performance indicators: the maximum working width $W_{WCV10}$ ensuring a satisfying uniformity ($CV \le 10\%$) and the mean radius $R_m$ of the spread pattern. To isolate the impact of the physical properties of fertilizers on these two indicators, independently of spreader characteristics, the study assumed the same initial conditions of ballistic flight for all fertilizers and only simulated the second stage of the spreading process: from the disk ejection to the particle distribution on the ground after adjacent overlaps.
<table><caption>**Table 1**
Characteristics of the S8 disk.</caption><thead><tr><th>Parameters</th><th>Nomenclature</th><th>Long vane<br>(i = 1)</th><th>Short vane<br>(i = 2)</th></tr></thead><tbody><tr><td><strong>Vane characteristic</strong></td><td></td><td></td><td></td></tr><tr><td>Radius of the vane</td><td>r<sub>i</sub></td><td>415</td><td>375</td></tr><tr><td>Pitch angle of the vane</td><td>α<sub>i</sub></td><td>5.4°</td><td>9.7°</td></tr><tr><td>Vertical angle of the vane</td><td>Ω<sub>i</sub></td><td>9.8°</td><td>8.8°</td></tr><tr><td><strong>Mass flow proportion</strong></td><td>p</td><td>43 %</td><td>57 %</td></tr><tr><td><strong>Revolution speed</strong></td><td>ω</td><td></td><td>900 rpm</td></tr><tr><td><strong>Disk spacing<sup>*</sup></strong></td><td>d<sub>D</sub></td><td></td><td>1.4 m</td></tr></tbody></table>

* distance between the axle of the right and left disks on the spreader AXIS 50.1.

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Free rotating wind vane used to measure the horizontal outlet angle.</figcaption></figure>

The measurement devices used to provide experimental data are presented in section 2.1. The characteristics of the actual and virtual fertilizers used for experimental and simulated data are presented in section 2.2. Models and simulations developed to compute $R_m$ and $W_{WCV10}$ are described in section 2.3. The regression model developed to $R_m$ and $W_{WCV10}$ as function of the fertilizer properties is described in section 2.4. Obtaining experimental values for $R_m$ and $W_{WCV10}$ is explained in section 2.5.

## 2.1. Experimental devices

### 2.1.1. Spinning disks

In this study, simulation results were compared to experimental data obtained with a marketed spinning disk, which was the S8 disk, commonly used with AXIS 50.1 spreaders supplied by RAUCH Company (Germany). This was a concave disk equipped with two straight pitched vanes of two different lengths. The revolution speed was 900 rpm. Fig. 1 presents the design of the counterclockwise disk. Table 1 provides the parameter values. The proportion $p$ of the mass flow collected by each vane was deduced from the angular location of the vanes on the disk.

For specific needs of certain stages of the study (described in following sections), three other custom-made counterclockwise disks had also been punctually used. The first one, called S8L had the same characteristics than the S8 disk except that it is equipped with two long vanes. The second one, called S8S had the same characteristics than the S8 disk except that it is equipped with two short vanes. The last one, called SD was a simplified concave disk equipped with two radial vanes with the following characteristics: the radius was 395 mm and the vertical angle of the vanes was 13.5°.

### 2.1.2. Horizontal outlet angle measurement

The horizontal outlet angle $\theta_{out}$ (Fig. 1) of the particles (when they leave the vane) was measured using a free rotating wind vane forced in direction by the particle flow movement (Fig. 2). This measurement principle has already been used by (Heppler, 1993; Persson, 1996). The

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Collecting elbow pipe used to measure the vertical outlet angle.</figcaption></figure>

narrow extremity of the wind vane was fixed to the upper end of a rotation shaft mounted on ball bearings. A single-turn rotary potentiometer was mounted at the lower end of the shaft and was used to provide an analog signal from 0 to 10 V linearly related to the rotation angle (0 to 360°). The analog signal was sampled and recorded with a frequency of 10 Hz. Previous to measurements, a geometrical setting was carried out so that the value was 90° when the vane axis is aligned with the radius of the spinning disk and is 0° when the vane axis is orthogonal to the radius of the disk. Since the rotation axle of the free vane cannot be placed just on the outer radius of the centrifugal disk, a geometrical correction was established to convert the angle value measured in the vicinity of the disk into the angle value at the disk edge. The mean value $\mu\theta_{out}$ and the standard deviation $\sigma\theta_{out}$ of the horizontal outlet angle were deduced from 15 successive measurements.

### 2.1.3. Vertical outlet angle measurement

The vertical outlet angle $\Omega_{out}$ (Fig. 1) of the particles (when they leave the vane) was measured using a collecting elbow pipe with inlet opening placed in front of the spinning disk and with outlet opening feeding a container mounted on load cell (Fig. 3). The inlet opening was 36 mm in diameter. The vertical motion of the collecting device was obtained by means of an electrical motor and a rotating screw. The rotation of the screw was transformed into linear motion of a movable nut on which is fixed the collecting device. The vertical motion of the device was controlled at 2 mm/s. The vertical location was measured with a laser telemeter. During the motion of the elbow pipe, the vertical location of the opening and the particle collected weight were simultaneously recorded (with a frequency of 5 Hz). This provide the vertical distribution of the mass flow at the measurement location. The mean value $\mu\Omega_{out}$ and the standard deviation $\sigma\Omega_{out}$ of the vertical outlet angle were deduced from the measured distribution compared to the vertical location of the outer vane extremity.

### 2.1.4. Spread patterns measurements

Stationary spread patterns were measured with the rotating test bench (EN 13739-2, 2011) developed by Piron and Miclet (2005) and called CEMIB. This measurement device consisted of a rotating carrier and a motion less row of 80 collection trays. Each trays has a collection area of 0.5 × 0.5 m and is mounted on load cell. For the measurements, the revolution speed of the spreader carrier was 3°/s. During the rotation, the spread pattern passed over the row of trays, and the cumulated mass collected by each tray was recorded. The collected mass and the angular orientation of the carrier were simultaneously recorded with the acquisition frequency of 10 Hz. A specific algorithm had been developed to process the recorded data and to provide the 2-dimensional static spread pattern with a sampling interval of 0.25 × 0.25 m in Cartesian
<figure><img src="image_8.png" /><figcaption>Fig. 4. Particle diameter distribution: example when $D_m = 3$, $s_1 = 0.9$ and $s_2 = 1.3$. From the right to the left, vertical dashed lines are placed at $D_m-s_1$, $D_m$ and $D_m + s_2$ respectively.</figcaption></figure>

coordinates with respect to the disk centre.

### 2.1.5. Particle size distribution measurement

Particle size distributions were measured using a videogranulometer. The device consisted of an inclined plate fed in its upper part by a flow of fertilizer particles. The plate was transparent and illuminated from the background. The feeding system provided a flow of separated particles crossing the plate and observed by a digital camera. The diameter of each particle was obtained using image processing so that the particle distribution of the fertilizer is deduced. Details concerning the measurement method are given in Le et al. (2018).

## 2.2. Actual and virtual fertilizers

### 2.2.1. Particle size distribution model

To describe particle size distributions, various models had been used in literature. Two-parameter distributions such as log-normal distribution (Gardner, 1956) or Rosin-Rammler distribution (Perfect et al., 1998) are widely used equations while more refined models had also been suggested, such as three- and four-parameter log-normal equations (Wagner and Ding, 1994) or five-parameter log-normal equation (Fredlund et al., 2000). In practice, the choice of the model is determined by striking a balance between the goodness of fit and the complexity of the model (i.e. number of parameters). Furthermore, the choice is determined by the possibility of easily interpreting the parameters in terms of distribution shape. Regarding this aspect, logarithmic models introduce dimensional and scaling changes, which complicate the parameter interpretation. Consequently, in this study, the probability density function $f(d)$ of particle diameters was modelled by a combination of two Gaussian membership functions as follows:

$$f(d) = \begin{cases} \frac{1}{(s_1 + s_2)} \sqrt{\frac{2}{\pi}} \exp\left(-\frac{1}{2}\left(\frac{d - D_m}{s_1}\right)^2\right) & \text{for } d \le D_m \\ \frac{1}{(s_1 + s_2)} \sqrt{\frac{2}{\pi}} \exp\left(-\frac{1}{2}\left(\frac{d - D_m}{s_2}\right)^2\right) & \text{for } d > D_m \end{cases} \quad (1)$$

where $D_m$ is the modal value of the distribution (diameter value at which the probability density function attains its maximum value), $s_1$ and $s_2$ are the standard deviations of the right and left Gaussian membership functions respectively.

As illustrated in Fig. 4, the advantage of this model is to provide three

<figure><img src="image_9.png" /><figcaption>Fig. 5. Distribution histograms of the characteristic parameters measured for 47 commercially available fertilizers.</figcaption></figure>

meaningful parameters to describe independently the mode of the distribution, the dispersion of diameters smaller than the modal value and the dispersion of diameters higher than the modal value.

### 2.2.2. Actual fertilizers

Forty-seven commercially available fertilizers had been considered in this study. Their particle size distributions were analyzed using the videogranulometer presented in section 2.1.4. The measured values of $D_m$ ranged from 1.9 to 3.7 mm, $s_1$ from 0.26 to 0.80 and $s_2$ from 0.39 to
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Particle size distribution parameters ($s_1$ and $s_2$) measured for 47 commercially available fertilizers (·) and discretized pairs of values chosen to simulate virtual fertilizers (○).</figcaption></figure>

1.01. The fertilizer specific density $ρ$ was derived from weighing a fertilizer bulk volume and weighing the same bulk volume after completing with a liquid of known density, so that the particle mass and volume was deduced. The 47 measured values ranged from 1250 to 2320 kg m⁻³.

The fertilizer drag coefficients $C_D$ was deduced from spread patterns measured with the simplified disk SD (see section 2.1.1). For each fertilizer, the drag coefficient value was estimated by minimizing the difference between the measured spread pattern and computed spread patterns. This method had already been used by Villette et al. (2017) and Abbou-ou-cherif et al. (2017). The 47 measured values ranged from 0.45 to 0.62.

Fig. 5 presents the distribution histograms of the measured values for the five characteristic parameters and Fig. 6 presents the pairs of values ($s_1$, $s_2$) measured for each fertilizer.

### 2.2.3. Virtual fertilizers

In order to study the influence of fertilizer physical properties, a set of virtual fertilizers (combining different characteristics) was defined.

Concerning the particle size distribution, the investigated values for $D_m$ were chosen from 1 to 5 in steps of 0.5 and the investigated values for $s_1$ and $s_2$ were chosen from 0.1 to 1.3 in steps of 0.2. To limit the number of virtual fertilizers, only certain paired values ($s_1$, $s_2$) were considered. These paired values ($s_1$, $s_2$) are in the triangle delimited by (0.1, 0.1), (0.5, 1.5) and (1.5, 0.5) as depicted in Fig. 6. Furthermore, to avoid unrealistic truncated distributions, the design of virtual fertilizers only considered pairs of values ($s_1$, $s_2$) for which $s_1 < D_m/3$. The combination of these discretized ranges of values and restrictions provided 152

<caption>**Table 2**</caption>

Values of parameters measured for urea and used to initiate ballistic flight simulations.

<table><thead><tr><th>Parameters</th><th>Nomenclature</th><th>Long vane</th><th>Short vane</th></tr></thead><tbody><tr><td><strong>Horizontal outlet angle</strong></td><td></td><td></td><td></td></tr><tr><td>Mean value</td><td>μθ<sub>out</sub></td><td></td><td>43.6°</td></tr><tr><td>Standard deviation</td><td>σθ<sub>out</sub></td><td></td><td>1.1°</td></tr><tr><td><strong>Vertical outlet angle</strong></td><td></td><td></td><td></td></tr><tr><td>Mean value</td><td>μΩ<sub>out</sub></td><td>7.8°</td><td>7.6°</td></tr><tr><td>Standard deviation</td><td>σΩ<sub>out</sub></td><td>1.0°</td><td>1.2°</td></tr></tbody></table>

triplets of values ($D_m$, $s_1$, $s_2$) describing 152 particle size distributions.

Accordingly to the distribution of the values obtained for actual fertilizers, the investigated range of values for $C_D$ and $ρ$ were chosen from 0.4 to 0.6 in steps of 0.05 and from 1200 to 2000 kg m⁻³ in steps of 200 kg m⁻³ respectively.

Combining 152 particle size distributions with 5 drag coefficient values and 5 density values, this provided 3800 virtual fertilizers with different physical characteristics, each defined by a 5-tuple of values ($C_D$, $ρ$, $D_m$, $s_1$, $s_2$).

## 2.3. Models and simulations

### 2.3.1. Virtual spreader

The study considered a virtual twin disk spreader composed of two disks with the right disk rotating counterclockwise and the left disk rotating clockwise. Both disks had the same revolution speed and were fed by the same fertilizer mass flow. In order to consider a realistic disk design, the characteristics of the virtual disk were taken from the S8 disk. Thus, the characteristics of the virtual disk were those of the S8 disk as specified in Table 1, including disk spacing set at 1.4 m.

### 2.3.2. Initial conditions of ballistic flight

In centrifugal spreading process, the initial conditions of the ballistic flight of the particles are defined by their locations when they leave the vane, their outlet velocities and directions.

In the Cartesian coordinate system (O, i, j, k), the location of a particle ($x_{out}$, $y_{out}$, $z_{out}$) when it leaves the vane (Fig. 1) is as follows:

$$ \begin{cases} x_{out} = r_i \times \cos\theta_{vane\ i} \\ y_{out} = r_i \times \sin\theta_{vane\ i} \\ z_{out} = h_i \end{cases} \qquad (2) $$

where i is 1 for long vane and 2 for short vane, $r_i$ is the radius of the vane, $θ_{vane\ i}$ is the angular location of the vane, $h_i$ is the height of the outer extremity of the vane.

According to Fig. 1, for a counterclockwise rotating disk, the horizontal angle of the particle trajectory is:

$$ \theta_{traj} = \theta_{vane\ i} - \theta_{out} + \pi/2 \qquad (3) $$

where $θ_{out}$ is the horizontal outlet angle.

The components of the outlet velocity ($vx_{out}$, $vy_{out}$, $vz_{out}$) are as follows:

$$ \begin{cases} v x_{out} = v_H \times \cos\theta_{traj} \\ v y_{out} = v_H \times \sin\theta_{traj} \\ v z_{out} = v_H \times \tan\Omega_{out} \end{cases} \qquad (4) $$

where $Ω_{out}$ are the vertical outlet angle and $v_H$ is the horizontal component of the outlet velocity defined as follows:

$$ v_H = \frac{r_i \omega}{\cos\theta_{out} + \sin\theta_{out} \tan\alpha_i} \qquad (5) $$

where $ω$ is the revolution speed of the disk and $α_i$ is the pitch angle of the vane.

More details concerning the demonstrations of above equations can be found in Villette et al. (2008) and Villette et al. (2017).

To focus on the spreading ability of fertilizers, the study considered the same initial conditions of the ballistic flight for all fertilizers. This aligned with the assumption that spreader components can be designed and adjusted to achieve the desired ejection characteristics.

To ensure realistic conditions for simulations, the disk outlet characteristics were based on data measured with urea. Since the virtual disk was modeled from the S8 disk (section 2.1.1), outlet characteristics for the short and long vanes were required for spread pattern simulations. These data were deduced from measurements carried out with the S8S disk (equipped with short vanes) and the S8L disk (equipped with long
<figure><img src="image_11.png" /><figcaption>Fig. 7. Angular mass flow distribution measured with urea for the short vane (continuous line) and the long vane (dashed line). Mass flow distributions used to initiate ballistic flight simulations.</figcaption></figure>

vanes) respectively. For all measurements, the spinning disks were mounted on the AXIS 50.1 spreader, the revolution speed was 900 rpm, the drop point index was 5 and the fertilizer mass flow was 1 kg/s.

The horizontal outlet angle was measured as described in section 2.1.2 for the disks S8S and S8L. As measured values were not significantly different, the same values were assumed for short and long vanes. This is in accordance with (Colin, 1997; Olieslagers, 1997; Villette et al., 2005). The mean value $\mu\theta_{out}$ and the standard deviation $\sigma\theta_{out}$ of the horizontal outlet angle are provided in Table 2.

The vertical distribution was measured as described in section 2.1.3 for the S8S and S8L disks. The mean values $\mu\Omega_{out}$ and the standard deviation $\sigma\Omega_{out}$ of the vertical outlet angle obtained for short (S8S) and long vanes (S8L) are provided in Table 2.

The angular distributions of the mass flow around the disks S8S and S8L were determined by analyzing stationary spread patterns measured for both disks as described in section 2.1.4. Fig. 7 presents the distributions obtained for short and long vanes. Details concerning the transformation of spread pattern into mass flow distribution can be found in Villette et al. (2017) and Abbou-ou-cherif et al. (2017).

### 2.3.3. Ballistic flight model

In this study, the ballistic flight model considers the gravity force and the drag force due to the motion of the particle through the air. This approach had been widely used by other researchers (Abbou-ou-cherif et al., 2017; Aphale et al., 2003; Griffis et al., 1983; Mennel and Reece, 1963; Olieslagers et al., 1996; Pitt et al., 1982; Villette et al., 2017), and described the motion of the particle by the following differential equations:

$$ 
\begin{cases}
\frac{d^2 x}{dt^2} = -\frac{1}{2m} C_D A_p \rho_{air} v x \sqrt{v x^2 + v y^2 + v z^2} \\
\frac{d^2 y}{dt^2} = -\frac{1}{2m} C_D A_p \rho_{air} v y \sqrt{v x^2 + v y^2 + v z^2} \\
\frac{d^2 z}{dt^2} = -g - \frac{1}{2m} C_D A_p \rho_{air} v z \sqrt{v x^2 + v y^2 + v z^2}
\end{cases}
\quad (6) $$

where x, y, z are the coordinates of the particle, m is the particle mass, $C_D$ is the drag coefficient, $A_p$ is the projected surface area of the particle, $\rho_{air}$ is the air density (set at 1.21 kg m$^{-3}$), $v x$, $v y$, $v z$ are the velocity components of the particle, g is the acceleration due to gravity.

In the model, the particle mass was related to the particle density $\rho$ and diameter d as follows:

<figure><img src="image_12.png" /><figcaption>Fig. 8. Example of spread patterns simulated for the virtual right disk and three different virtual fertilizers. The spread pattern SP1 (top) was computed with $C_D = 0.5$, $\rho=1400$ kg m$^{-3}$, $D_m = 3$ mm, $s_1 = 0.5$ mm, $s_2 = 0.5$ mm; SP2 (middle) was computed with the same values except for $D_m = 2$ mm; SP3 (bottom) was computed with the same values except for $D_m = 2$ mm and $\rho = 1800$ kg m$^{-3}$. The cross (+) is the location of the disk center. The graduation of the color scale reflects the fertilizer mass (kg) lying on the sampling area (0.5 × 0.5 m). (For interpretation of the reference to color in this figure legend, the reader is referred to the web version of this article).</figcaption></figure>

$$ m = \rho \frac{4}{3} \pi \left(\frac{d}{2}\right)^2 \quad (7) $$

Thus, considering the same initial conditions for the ballistic flight, the fertilizer spread pattern depends on the aerodynamic drag coefficient $C_D$, the specific density $\rho$ and the particle size distribution.

### 2.3.4. Spread pattern simulations for one disk and $R_m$ calculation

The simulations used a Monte Carlo process to compute the motion of a large number of fertilizer particles. These particles were assigned several characteristics that were randomly drawn from pre-established statistical distributions. The algorithms were developed with Matlab (2015) software and utilized its built-in random number generator. For
variables following a normal distribution, the values were obtained using the randn function. In the case of other non-standard distributions, the selection of random values was performed in two steps. First, random numbers were generated with a uniform distribution using the rand function within the range of 0 to 1. Second, the final random values were deduced from these numbers by inversing the cumulative frequency function of the specified distribution.

For each virtual fertilizer defined by the 5-tuple ($C_D, \rho, D_m, s_1, s_2$), the corresponding stationary spread pattern was computed for the right disk of the virtual spreader. This was a counterclockwise disk modeled from the S8 disk as presented in Fig. 1 with parameters values as specified in Table 1. The landing locations of the particles were successively computed for each vane. Regarding the mass flow proportion $p$ collected by each vane (Table 1), the number $N$ of particles considered in the simulations was 100,000 particles ejected by the short vane and 75,439 by the long one. Therefore, for each vane, the simulations consisted of the following steps.

First, a collection of $N$ virtual particles was generated by randomly selecting $N$ values of diameter $d$ from the size distribution defined by Eq. (1) and the assigned values of $D_m, s_1$ and $s_2$.

Second, the initial conditions for the ballistic flight were assigned to each particle, regardless of its diameter. For each particle, the values of the different variables were assigned as follows:

(1) The angular location of the vane $\theta_{vane}$ at the particle ejection time was randomly selected using the mass flow distribution (Fig. 7).

(2) The coordinates of the ejection point ($x_{out}, y_{out}, z_{out}$) were deduced from Eq. (2).

(3) The horizontal outlet angle $\theta_{out}$ was drawn from the normal distribution parametrized by $\mu\theta_{out}$ and $\sigma\theta_{out}$ (Table 2).

(4) The vertical outlet angle $\Omega_{out}$ was drawn from the normal distribution parametrized by $\mu\Omega_{out}$ and $\sigma\Omega_{out}$ (Table 2).

(5) The 3D-components of the outlet velocity ($vx_{out}, vy_{out}, vz_{out}$) were deduced from $\theta_{vane}, \theta_{out}$ and $\Omega_{out}$ using Eqs. (3) to (5).

Third, the mass of each particle was computed (Eq. (7)) and the coordinates of the landing point of each particle were computed by solving Eq. (6) with the initial conditions of flight $x_{out}, y_{out}, z_{out}$ and $vx_{out}, vy_{out}, vz_{out}$.

Cumulating all landing points for both vanes provided the simulated spread pattern for the right disk of the virtual spreader. Then, the mean radius $R_m$ was calculated as the mean distance from the disk center to the landing points of all the particles ejected by the disk.

For example, Fig. 8 presents spread patterns computed for three different virtual fertilizers when the spreader travel direction is along the Y-axis. This illustrates how the spread pattern can be affected by the reduction of $D_m$ (from SP1 to SP2) or the increase of $\rho$ (from SP2 to SP3). This also illustrates how certain parameters (e.g. $D_m$ and $\rho$) can compensate for each other to produce close spread patterns (comparing SP1 to SP3).

### 2.3.5. Twin disk spread pattern simulation and $W_{WCV10}$ calculation

The spread pattern resulting from left disk was deduced from the right spread pattern by symmetry with respect to the spreader travel direction.

The setting of the virtual spreader consisted in rotating the right and left spread patterns around each disk from the same angle $\alpha_{set}$ counterclockwise for the right disk and clockwise for the left one. This simulated modifying the angular location of the feeding point on each disk. Finally, the twin disk spread pattern was deduced from associating the left and right spread patterns after translating the particle coordinates by half the disk spacing (1.4 m) in the left or right direction. The values of the setting angle ranged from 0° to 60° in steps of 1°.

For each spreader setting (i.e. each value of $\alpha_{set}$), the transverse distribution is deduced from the twin disk spread pattern by considering a virtual row of collection trays aligned perpendicularly to the travel

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Setting maps corresponding to the spread patterns SP1, SP2 and SP3 presented in Fig. 8 and obtained for three different virtual fertilizers. The spread pattern SP1 (top) was computed with $C_D = 0.5$, $\rho=1400 \text{ kg m}^{-3}$, $D_m = 3$ mm, $s_1 = 0.5$ mm, $s_2 = 0.5$ mm; SP2 (middle) was computed with the same values except for $D_m = 2$ mm; SP3 (bottom) was computed with the same values except for $D_m = 2$ mm and $\rho = 1800 \text{ kg m}^{-3}$. The color scale corresponds to the value of the CV expressed in percentage. The largest working width respecting $CV \le 10\%$ is plotted (white circle). (For interpretation of the reference to color in this figure legend, the reader is referred to the web version of this article).</figcaption></figure>

path of the virtual spreader. For a given working width $W_W$, several identical spread patterns were shifted right and left by multiples of $W_W$ representing virtual adjacent passes and reproducing spread overlap. Depending on the transverse position of each particles of the spread patterns, the particles were affected to the corresponding virtual collection trays ($0.5 \times 0.5 \text{ m}$). Then, the transverse distribution of the fertilizer mass was determined by summing the mass collected by each tray, and the transverse coefficient of variation (CV) was calculated (EN 13739-2, 2011). For each virtual fertilizer, the setting map (Fig. 9) was finally created by storing the CV values in a matrix. In this matrix, rows represent different spreader settings $\alpha_{set}$, and columns represent achievable working widths $W_W$. In order to assess the fertilizer ability to be spread on large width with a good uniformity, the indicator chosen to
evaluate the performance was the maximum working width respecting CV≤10%, named W<sub>WCV10</sub>. The choice of limiting the CV to 10 % to ensure good uniformity complies with standard EN 13739-1 (2011) which requires the CV does not exceed 15 %. The corresponding working width W<sub>WCV10</sub>, was deduced from the setting map. First, the discretized value of the working width respecting the condition was deduced from the row value of the matrix and second, the refined value W<sub>WCV10</sub> was estimated by linear interpolation between the two neigh-

Thanks to this model, the sensitivity analysis was performed by calculating the partial derivatives of R<sub>m</sub> and W<sub>WCV10</sub> with respect to the independent variables. The total derivative of R<sub>m</sub> and W<sub>WCV10</sub> are provided in Eqs. (12) and (13) respectively.

$$dR_m = \frac{\partial R_m}{\partial D_m} dD_m + \frac{\partial R_m}{\partial s_1} ds_1 + \frac{\partial R_m}{\partial s_2} ds_2 + \frac{\partial R_m}{\partial \rho} d\rho + \frac{\partial R_m}{\partial C_D} dC_D \quad (12)$$

$$dW_{WCV10} = \frac{\partial W_{WCV10}}{\partial D_m} dD_m + \frac{\partial W_{WCV10}}{\partial s_1} ds_1 + \frac{\partial W_{WCV10}}{\partial s_2} ds_2 + \frac{\partial W_{WCV10}}{\partial \rho} d\rho + \frac{\partial W_{WCV10}}{\partial C_D} dC_D \quad (13)$$

boring discrete values.

Fig. 9 illustrates the setting maps corresponding to the three virtual fertilizers for which single disk spread patterns were presented in Fig. 8. Fig. 9 shows that W<sub>WCV10</sub> is 36.7 m when fertilizer properties are C<sub>D</sub> = 0.5, ρ=1400 kg m<sup>-3</sup>, D<sub>m</sub> = 3 mm, s<sub>1</sub> = 0.5 mm and s<sub>2</sub> = 0.5 mm. W<sub>WCV10</sub> decreases to 30.3 m when D<sub>m</sub> is 2 mm. When D<sub>m</sub> is maintained at 2 mm but ρ is increased to 1800 kg m<sup>-3</sup>, W<sub>WCV10</sub> increases anew to 35.2 m.

## 2.4. Regression model

The values of R<sub>m</sub> and W<sub>WCV10</sub> obtained from numerical simulations were utilized to create a model that directly relates these values to the physical characteristics of the particles: D<sub>m</sub>, s<sub>1</sub>, s<sub>2</sub>, ρ, and C<sub>D</sub>. The same model form was applied to both R<sub>m</sub> and W<sub>WCV10</sub>. First, a multivariate polynomial regression was performed for each of the 25 pairs of values (C<sub>D</sub>, ρ) to determine the coefficients C<sub>Ri</sub> and C<sub>Wi</sub> (with i ranging from 1 to 7) for the models defined by Eqs. (8) and (9). As a result, 25 values were obtained for each of the seven C<sub>Ri</sub> and C<sub>Wi</sub> coefficients. Subsequently, each of these coefficients was modeled using multivariate linear regressions as described in Eqs. (10) and (11).

$$R_m = \begin{pmatrix} C_{R1}(\rho, C_D) \\ C_{R2}(\rho, C_D) \\ C_{R3}(\rho, C_D) \\ C_{R4}(\rho, C_D) \\ C_{R5}(\rho, C_D) \\ C_{R6}(\rho, C_D) \\ C_{R7}(\rho, C_D) \end{pmatrix}^T \times \begin{pmatrix} D_m^2 \\ D_m \\ D_m s_1 \\ D_m s_2 \\ s_1 \\ s_2 \\ 1 \end{pmatrix} \quad (8)$$

$$W_{WCV10} = \begin{pmatrix} C_{W1}(\rho, C_D) \\ C_{W2}(\rho, C_D) \\ C_{W3}(\rho, C_D) \\ C_{W4}(\rho, C_D) \\ C_{W5}(\rho, C_D) \\ C_{W6}(\rho, C_D) \\ C_{W7}(\rho, C_D) \end{pmatrix}^T \times \begin{pmatrix} D_m^2 \\ D_m \\ D_m s_1 \\ D_m s_2 \\ s_1 \\ s_2 \\ 1 \end{pmatrix} \quad (9)$$

$$C_{Ri}(\rho, C_D) = c_{Ri1}\rho + c_{Ri2}C_D + c_{Ri3} \quad (10)$$

$$C_{Wi}(\rho, C_D) = c_{Wi1}\rho + c_{Wi2}C_D + c_{Wi3} \quad (11)$$

The analytical descriptions of the partial derivatives of R<sub>m</sub> are provided in Eqs. (14) to (18), as follows:

$$\frac{\partial R_m}{\partial D_m} = 2C_{R1(\rho, C_D)}D_m + C_{R2(\rho, C_D)} + C_{R3(\rho, C_D)}s_1 + C_{R4(\rho, C_D)}s_2 \quad (14)$$

$$\frac{\partial R_m}{\partial s_1} = C_{R3(\rho, C_D)}D_m + C_{R5(\rho, C_D)} \quad (15)$$

$$\frac{\partial R_m}{\partial s_2} = C_{R4(\rho, C_D)}D_m + C_{R6(\rho, C_D)} \quad (16)$$

$$\frac{\partial R_m}{\partial \rho} = C_{R11}D_m^2 + C_{R21}D_m + C_{R31}D_m s_1 + C_{R41}D_m s_2 + C_{R51}s_1 + C_{R61}s_2 + C_{R71} \quad (17)$$

$$\frac{\partial R_m}{\partial C_D} = C_{R12}D_m^2 + C_{R22}D_m + C_{R32}D_m s_1 + C_{R42}D_m s_2 + C_{R52}s_1 + C_{R62}s_2 + C_{R72} \quad (18)$$

Similarly, partial derivatives of W<sub>WCV10</sub> were obtained by substituting R<sub>m</sub> with W<sub>WCV10</sub> and C<sub>Ri</sub> with C<sub>Wi</sub> in Eqs. (14) to (18).

## 2.5. Experimental data for R<sub>m</sub> and W<sub>WCV10</sub>

Single disk spread patterns were measured for the 47 marketed fertilizers described in section 2.2.2. All spread patterns were measured with the rotating bench CEMIB as described in section 2.1.4 and with the counterclockwise S8 disk described in Table 1 and mounted on the AXIS 50.1 spreader. The revolution speed of the disk was 900 rpm. These 47 measured spread patterns were used to compute the experimental values of R<sub>m</sub> and W<sub>WCV10</sub> for the 47 fertilizers.

In order to simplify the use and processing of each spread pattern, it was considered as a 2-dimensional probability density function. Thus, the Cartesian coordinates of a set of 300,000 identical particles was randomly drawn from this 2D probability density function to model the fertilizer mass distribution on the ground. The major advantage of this method is to avoid complex interpolation and resampling on square grids when rotation or translation of spread patterns is required (to reproduce working width setting, overlapping adjacent passes or transverse distributions).

Using this approach, the mean radius R<sub>m</sub> of the single disk spread pattern was simply computed as the mean distance between the disk centre and the coordinates of all the particles representing the spread pattern.
<table><caption>**Table 3**
Numerical values of the model coefficients for R<sub>m</sub>.</caption><thead><tr><th>i</th><th>C<sub>Ri1</sub></th><th>C<sub>Ri2</sub></th><th>C<sub>Ri3</sub></th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>1</td><td>-1,0153E-04</td><td>3,2029E-01</td><td>-1,8406E-01</td><td>0,991</td></tr><tr><td>2</td><td>1,4628E-03</td><td>-4,4746E+00</td><td>4,3079E+00</td><td>0,995</td></tr><tr><td>3</td><td>3,1722E-05</td><td>-1,2102E-01</td><td>5,7649E-03</td><td>0,898</td></tr><tr><td>4</td><td>-2,9331E-04</td><td>8,6892E-01</td><td>-8,3262E-01</td><td>0,994</td></tr><tr><td>5</td><td>-4,2673E-04</td><td>1,3983E+00</td><td>-9,8010E-01</td><td>0,988</td></tr><tr><td>6</td><td>1,6275E-03</td><td>-4,9057E+00</td><td>5,9312E+00</td><td>0,992</td></tr><tr><td>7</td><td>2,2784E-03</td><td>-7,1467E+00</td><td>4,1132E+00</td><td>0,988</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Modeled values of R<sub>m</sub> with respect to values computed through numerical simulations (·) or experimental values (+). The continuous line is the identity line.</figcaption></figure>

The maximum working width $W_{WCV10}$ was computed from the measured single spread pattern by reproducing the setting of the AXIS 50.1 spreader. For this spreader, the setting of the working width is obtained by adjusting the angular location of the drop point onto the disk. As rotating the drop point rotates the spread pattern around the disk, the setting principle is exactly the same that the setting of the virtual spreader used in this study. Using the description of the spread pattern as a set of identical particles, the maximum working width $W_{WCV10}$ was computed from the measured spread pattern following the method described in section 2.3.5.

### 3. Results and discussion

The mean radius R<sub>m</sub> of spread patterns and the maximum working widths $W_{WCV10}$ respecting CV≤10%, have been computed for 3800 virtual fertilizers combining 152 particle size distributions (defined by $D_{mv}$, $s_1$, $s_2$), 5 drag coefficient values ($C_D$) and 5 density values ($\rho$). Models for R<sub>m</sub> and $W_{WCV10}$ have been fitted to simulated data (section 3.1), compared to experimental data (section 3.2) and used to assess how the physical properties of particles influence the spreading performances of fertilizers (section 3.3).

#### 3.1. Data fitting and model coefficients

##### 3.1.1. Model coefficients for R<sub>m</sub>

According to Eq. (8), the coefficients $C_{Ri}$ (with i ranging from 1 to 7) were calculated using the least squares method and fitting R<sub>m</sub> to the values obtained from numerical simulations. These coefficients were calculated for each pair of values ($C_D$, $\rho$). In all 25 regressions, the coefficients of determination ($R^2$) were found greater than 0.997.

<table><caption>**Table 4**
Numerical values of the model coefficients for $W_{WCV10}$.</caption><thead><tr><th>i</th><th>C<sub>Wi1</sub></th><th>C<sub>Wi2</sub></th><th>C<sub>Wi3</sub></th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>1</td><td>-1,7777E-04</td><td>5,7200E-01</td><td>-3,3673E-01</td><td>0,985</td></tr><tr><td>2</td><td>3,1687E-03</td><td>-9,9546E+00</td><td>9,5001E+00</td><td>0,995</td></tr><tr><td>3</td><td>-1,4341E-04</td><td>4,7779E-01</td><td>-4,6515E-01</td><td>0,984</td></tr><tr><td>4</td><td>-6,6163E-04</td><td>2,0399E+00</td><td>-1,9029E+00</td><td>0,989</td></tr><tr><td>5</td><td>0</td><td>0</td><td>0</td><td></td></tr><tr><td>6</td><td>3,9057E-03</td><td>-1,1921E+01</td><td>1,4189E+01</td><td>0,992</td></tr><tr><td>7</td><td>5,2146E-03</td><td>-1,5929E+01</td><td>9,9869E+00</td><td>0,988</td></tr></tbody></table>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Modeled values of $W_W$ with respect to values computed through numerical simulations (·) or experimental values (+). The continuous line is the identity line.</figcaption></figure>

Each of the 7 coefficients $C_{Ri}$ was subsequently fitted as a function of $\rho$ and $C_D$ according to Eq. (10). The values for coefficients $C_{Ri1}$, $C_{Ri2}$, $C_{Ri3}$ (with i ranging from 1 to 7) and the corresponding determination coefficients are provided in Table 3. The coefficients of determination are all greater than 0.98, except for the fitting of $C_{R3}$. To improve the modeling of $C_{R3}$, the inclusion of the quadratic terms $\rho^2$, $C_D^2$, and the interaction term $\rho C_D$, could be considered. With these three additional terms, the determination coefficient would exceed 0.98. However, due to the satisfactory predictive quality (Fig. 10) of R<sub>m</sub> obtained from Eqs. (8) and (10), this improvement was deemed unnecessary and is not accounted for in this analysis. In Fig. 10, the modeled values of R<sub>m</sub> are presented with respect to the values obtained from the numerical simulations with the 3800 virtual fertilizers. The coefficient of determination is then 0.997. The mean of the relative differences between modeled and simulated values is 0.95 % and 90 % of the differences are lower than 2.0 %.

##### 3.1.2. Model coefficients for $W_{WCV10}$

The modeling of $W_{WCV10}$ was conducted in a similar manner as explained earlier for R<sub>m</sub>. However, due to a low coefficient of determination obtained for the expression of $C_{W5}$, the variable $s_1$ was not considered in the modeling of $W_{WCV10}$. To maintain a consistent and general form shared by the modeling of both R<sub>m</sub> (Eq. (8)) and $W_{WCV10}$ (Eq. (9)), the coefficient $C_{W5}$ was therefore set to 0. Under these conditions, the coefficients of determination obtained for modeling $W_{WCV10}$ with each of the 25 pairs of values ($C_D$, $\rho$) are all greater than 0.997. The coefficients $C_{Wi1}$, $C_{Wi2}$, $C_{Wi3}$ (with i ranging from 1 to 7) and the corresponding determination coefficients are provided in Table 4. The values of $C_{W51}$, $C_{W52}$ et $C_{W53}$ were also set to zero, consistent with the imposed zero value for $C_{W5}$. Fig. 11 presents the values of $W_{WCV10}$ derived from
<figure><img src="image_16.png" /><figcaption>**Fig. 12.** $R_m$ (top row) and $W_{WCV10}$ (bottom row) as functions of $s_1$ and $s_2$, from three values of $D_m$ (from left to right). The white triangle delimits the area of parameter values covered by simulations. The white dotted line delimits the area where $s_1 < D_m/3$.</figcaption></figure>

the model (Eqs. (9) & (11)) with respect to the values computed by numerical simulations, with a coefficient of determination of 0.997. The mean of the relative differences between modeled and simulated values is 0.96% and 90% of the differences are lower than 2.0%.

### 3.1.3. Limit of use of the coefficients

The coefficients provided in Tables 3 & 4 were derived from simulations performed with fixed disk characteristics. These voluntary fixed spreader parameters were chosen to isolate the effect of fertilizer properties during the spreading process independently from initial ballistic flight conditions that can be adapted by modifying disk design. This ensures unbiased comparisons of fertilizer abilities. Thus, the predicted values calculated with the coefficient matrices given in Tables 3 & 4 suite for comparing fertilizer spreading ability and evaluating the respective contribution of the different physical properties. They are not intended to predict the absolute working width achievable with different spreaders.

## 3.2. Comparison to experimental data

Spreading performances ($R_m$ and $W_{WCV10}$) were also calculated using the models (Eqs. (8) and (9)) with the coefficients determined in Tables 3 and 4, and the physical characteristics measured for the 47 actual fertilizers described in section 2.2.2. The calculated values have been represented in Figs. 10 and 11 according to the experimental values obtained for $R_m$ and $W_{WCV10}$ (section 2.5).

The figures show greater disparities between modeled and experimental values for actual fertilizers compared to the disparities observed between modeled and simulated values. For $R_m$, the mean of the relative differences between modeled and simulated values is 4.3% and 90% of

the differences are lower than 8.0%. For $W_{WCV10}$, the mean is 4.1% and 90% of the differences are lower than 7.6%.

These greater differences highlight that the model aims to evaluate the ballistic abilities of the product and not to model its behavior over the entire centrifugal spreading process (on- and off-the-disk). Thus, the simulations were deliberately performed by considering identical initial ballistic flight conditions for all fertilizer particle size distributions. This ensures the assessment of fertilizer spreading performances regardless of spreader characteristics. On the opposite, when experimental spread patterns are used, the values deduced for $R_m$ and $W_{WCV10}$ are not only affected by the behavior of the fertilizer particles during ballistic flight but also by their interaction with the spinning disk. Thus, the dispersion obtained from data derived from actual measurements is not surprising taking into account differences in outlet disk conditions (i.e. outlet angles and mass flow distribution) for actual tests. This also illustrates the interest of numerical simulations to study the ability of fertilizer to be spread without bias induced by machine parameters.

Additionally, although the model does not take into account the specific disk outlet conditions of each actual fertilizer, it is worth noting that the values of $R_m$ and $W_{WCV10}$ deduced from the experimental data remain consistent with the model.

## 3.3. Fertilizer spreading performance analysis

### 3.3.1. Influence of particle size distribution

Considering a density $\rho$ of 1500 kg/m3 and a drag coefficient $C_D$ of 0.5, Fig. 12 illustrates the influence of particle size distributions on $R_m$ and $W_{WCV10}$. In order to improve the visibility of trends according to $s_1$ and $s_2$, the values were represented over the entire range of values going from 0 to 1.5 mm. The area covered by the pairs of values ($s_1, s_2$) used in
<figure><img src="image_17.png" /><figcaption>**Fig. 13.** $R_m$ (left) and $W_{WCV10}$ (right) as functions of $\rho$ and $C_D$, for a particle size distribution defined by $D_m = 3$, $s_1 = 0.4$ and $s_2 = 0.6$.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Sensitivity to the density $\rho$ of $R_m$ (left) and $W_{WCV10}$ (right) as functions of $D_m$ and $s_2$ for a value of $s_1$ set at 0.4 mm.</figcaption></figure>

simulations is delimited by a triangle. This triangle corresponds to the choice of the study area in accordance with usual particle size distributions (Fig. 6). Fig. 12 shows that the trends observed for $R_m$ and $W_{WCV10}$ are similar. First, $R_m$ and $W_{WCV10}$ increase with the $D_m$. This trend is consistent since larger particle diameters result in greater mass, higher energy acquired during ejection, and longer ballistic flight length (assuming other parameters remain constant). Second, it shows that an increase in $s_2$ leads to an increase in $R_m$ and $W_{WCV10}$, while an increase in $s_1$ results in a decrease in $R_m$ and $W_{WCV10}$. This trend is linked to the increase in the number of large particles ($d > D_m$) projected far from the disk when $s_2$ increases, and the increase in the number of small particles ($d < D_m$) projected closer to the disk when $s_1$ increases. Third, the figure demonstrates that the influence of $s_2$ is much more significant than that of $s_1$. This is explained by the asymmetric effect of $s_1$ and $s_2$ in terms of mass of particles projected closer or farther from the disk. Thus, the same increase in particle count produces a greater mass if it concerns larger diameters than smaller diameters. Fourth, the higher the initial values of $R_m$ and $W_{WCV10}$, the less impact the same absolute variations of

$s_1$ and $s_2$ have on $R_m$ and $W_{WCV10}$. Since the elevated values of $R_m$ and $W_{WCV10}$ result from higher values of $D_m$, the relative impacts of $s_1$ and $s_2$ in relation to $D_m$ are consequently diminished.

### 3.3.2. Influence of density and drag coefficient

Fig. 13 presents the evolution of $R_m$ and $W_{WCV10}$ as a function of the density and the drag coefficient for an example of particle size distribution defined by: $D_m = 3$, $s_1 = 0.4$ and $s_2 = 0.6$. The variations of $R_m$ and $W_{WCV10}$ should be compared to those presented in Fig. 12 concerning the impact of particle size distribution. The variations of $R_m$ and $W_{WCV10}$ as functions of $\rho$ and $C_D$ demonstrate the strong influence of these two parameters on spreading performances.

Fig. 14 presents the sensitivity of $R_m$ and $W_{WCV10}$ to density (Eq. (17)) as a function of predominant particle size parameters affecting spreading performance, namely $D_m$ and $s_2$. Similarly, Fig. 15 demonstrates the sensitivity of $R_m$ and $W_{WCV10}$ to the drag coefficient $C_D$ (Eq. (18)). The observed trends for $R_m$ and $W_{WCV10}$ are similar. Bulk density has a positive effect on $R_m$ and $W_{WCV10}$ ($dR_m/d\rho > 0$ and $dW_{WCV10}/d\rho > 0$).
<figure><img src="image_19.png" /><figcaption>Fig. 15. Sensitivity to the drag coefficient $C_D$ of $R_m$ (left) and $W_{WCV10}$ (right) as a function of $D_m$ and $s_2$ for a value of $s_1$ set at 0.4 mm.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 16. Variations of $R_m$ (top) and $W_{WCV10}$ (bottom) related to 10 % of variation for each parameter when the initial values are $D_m = 3$ mm, $s_1 = 0.4$ mm, $s_2 = 0.6$ mm, $\rho = 1500$ kg/m3, and $C_D = 0.5$.</figcaption></figure>

0), and the increase in $R_m$ and $W_{WCV10}$ resulting from an increase in $\rho$ is more significant when $D_m$ and $s_2$ are larger. On the other hand, the drag coefficient has a negative effect on $R_m$ and $W_{WCV10}$ ($dR_m/d\rho < 0$ and $dW_{WCV10}/d\rho < 0$). The impact of the drag coefficient in terms of reduction on $R_m$ and $W_{WCV10}$ is more significant as $D_m$ increases. The impact also increases with $s_2$. The influence of $s_2$ on the sensitivity of $R_m$ and $W_{WCV10}$ to $C_D$ is attenuated when $s_2$ decreases in proportion to $D_m$.

This also illustrates how the sensitivity of the maximum working width to fertilizer properties can be easily visualized thanks to the models derived from simulated data.

### 3.3.3. Analysis and visualization of each parameter contribution

As illustrated above, Eqs. (14) to (18) or equivalent forms for $W_{WCV10}$, provide the sensitivity of $R_m$ and $W_{WCV10}$ to each parameter. In addition, Eqs. (12) and (13) decompose the overall variation of $R_m$ or

$W_{WCV10}$ based on the variation of each model parameter. For instance, let's consider a 10 % increase in the value of each of these parameters compared to an initial situation defined by $D_m = 3$ mm, $s_1 = 0.4$ mm, $s_2 = 0.6$ mm, $\rho = 1500$ kg/m3, and $C_D = 0.5$. Fig. 16 presents the consequences of this increase on $R_m$ and $W_{WCV10}$ by analyzing the contribution of each parameter. A 10 % increase in the drag coefficient results in a reduction of approximately 2 m in the working width $W_{WCV10}$ that can be counterbalanced by a 10 % increase in the density, which also results in an increase of approximately 2 m in $W_{WCV10}$. Regarding the particle size distribution parameters, the figure once again demonstrates that $D_m$ is the most significant parameter. The impact of $s_2$ is less pronounced, while the impact of $s_1$ is negligible. The 10 % variation, applied to the input variables in this example, is used for illustrative purposes independently of the maximum amplitude of variation possible for each parameter and the practical difficulties of modifying the values of these
<figure><img src="image_21.png" /><figcaption>Fig. 17. Simulated values (·) and experimental values (+) of $W_{WCV10}$ with respect to $R_m$.</figcaption></figure>

parameters. Nevertheless, this example highlights the value of the model in assisting fertilizer producers in making the choice of product physical characteristics to compare or improve fertilizer spreading ability. For instance, this finding demonstrates that screening fertilizers to reduce the particle size dispersion has no benefit in improving fertilizer spreading ability, since $s_1$ has negligible effect and $s_2$ has a beneficial effect.

Since the numerical simulations have been carried out beforehand, the use of the established model does not require computing power and can be easily integrated into any type of software, such as spreadsheet. Moreover, the use of such an application is not limited to fertilizers but concerns all granular products spread using centrifugal process.

### 3.3.4. Relationship between $R_m$ and $W_{WCV10}$

Using numerical simulation results, Fig. 17 demonstrates $W_{WCV10}$ is directly correlated to $R_m$. The linear regression (with $R^2 > 0.998$) is as follows:

$$W_{WCV10} = 2.305R_m + 0.87 \quad (19)$$

Actual values obtained for 47 commercially available fertilizers have also been plotted. Despite a more limited range of explored values and scatted results, the experimental values align with the simulated results and also support that $W_{WCV10}$ is correlated to $R_m$ (with $R^2 > 0.968$). These findings demonstrate the necessity of a large mean radius of the spread pattern to achieve a wide working width while maintaining uniform spreading (CV≤10%).

## 4. Conclusions

The study demonstrates that the use of computer simulations is a relevant way to assess the ability of fertilizer to be spread on large working width. Considering 3800 virtual fertilizers, the advantage of the approach is to provide large amount of data, from which models can be derived. Another advantage is to isolate the influence of fertilizer properties from the effects of fertilizer-spreader interactions and spreader performance. Thanks to the large amount of simulated data, models are developed for the first time to relate the maximum working width and the mean radius of the spread pattern to 5 fertilizer physical properties: drag coefficient, specific density and three parameters describing the diameter distribution. Thereby, the study provides an analytical model to quantify the contribution of each of these five variables using partial derivatives. The drag coefficient, the specific density and the modal diameter of particle size distribution appear as key parameters.

The correlation between the maximum working width and the mean radius of the spread pattern is also demonstrated.

The model derived from numerical simulations provides fertilizer producers with an efficient tool. This tool enables them to make informed decisions about the particle physical properties required to achieve a specific level of fertilizer spreading performance. This is of practical use to guide the physical property design of new fertilizers, which are constantly being developed to reach more efficiency and reduce impacts on the environment, such as the development of granular organic fertilizer. In addition, the findings are not limited to fertilizers but concern all granular materials spread using centrifugal processes such as seeds for broadcast sowing.

Further researches will include the study of how fertilizer properties influence the robustness of the spread pattern, this means how fertilizer properties affect the spread pattern's ability to maintain acceptable uniformity as the working width deviates from its optimal value.

## CRediT authorship contribution statement

Tien-Thinh Le: Validation, Software, Methodology, Investigation.
Emmanuel Piron: Writing – review & editing, Validation, Methodology, Funding acquisition, Conceptualization.
Denis Miclet: Methodology, Investigation.
Sylvain Villette: Writing – original draft, Visualization, Formal analysis.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The work was supported by YARA Company (Norway) and the French Research National Institute for Agriculture, Food and Environment (INRAE). The authors acknowledges the support received from the Agence Nationale de la Recherche of the French government through the program "Investissements d'Avenir" (16-IDEX-0001 CAP 20-25). The authors gratefully acknowledge RAUCH Company (Germany) for providing the spreader and the spinner disks.

## Data availability

The authors do not have permission to share data.

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