ON THE FERTILIZER PARTICLE MOTION ALONG THE VANE OF A CENTRIFUGAL SPREADER DISC ASSUMING PURE SLIDING OF THE PARTICLE

: At present, spreading of mineral fertilizers in Europe is most commonly performed using centrifugal disk spreaders with attached vanes. The motion of an ideal spherical homogeneous fertilizer particle along the straight vane attached to a flat rotating disc was analyzed in this paper. The analysis was performed in the non-inertial reference coordinate system. From the assumptions introduced to enable analytical describing of the real particle motion as well as the general tools of solid mechanics, the ordinary in-homogenous second-order differential equation having constant coefficients arose. Its solution represents an approximation of the real relative motion of the fertilizer particle along the straight radial vane fixed to the flat horizontal disc rotating at a constant angular velocity. However, the solution of this kind can be very useful for the optimization of centrifugal spreader working parameters.


Introduction
Different kinds of solid fertilizer spreaders have been designed in the past. However, it seems that centrifugal disc spreaders (Figure 1) are the most popular fertilizing machines worldwide, as well as in Serbia. Their main advantages in comparison with other available designs lie in lower costs, simple design and maintenance, followed by large soil covering width per single pass (Villette et al., 2008), over 45 [m] (Cool et al., 2016). On the fertilizer particle motion along the vane of a centrifugal spreader disc 85 experimentally, applying the traditional simple test method, namely the "collection tray method". According to this method, while the machine passes the rows of collection trays (placed perpendicular to the travel direction) at a constant travel speed, the fertilizer particles are collected at different transverse positions. After weighing the amount of fertilizer in each tray of each row, the spatial distribution pattern over the experimental plot surface can be composed.
Further improvement of the fertilizer spreading models has been continued in the 21 st century: Aphale et al. (2003), Dintwa et al. (2004), Villette et al. (2005), Cool et al. (2014), Cool et al. (2016), etc. However, these models have been based on the assumption that particles do not interact mutually, which may cause prediction errors. In order to improve the predicting accuracy, an alternative approach has been introduced. Methods of this kind include the particle interactions in a modelling algorithm using the numerical mathematics -primarily discrete element methods: Tijskens et al. (2005), Van Liedekerke et al. (2009), etc. Unfortunately, quantitative simulations based on this method need physical fertilizer characteristics as input data, which are at present difficult to measure (Villette et al., 2008).
This work presents an analytical solution of the dynamic model that describes the motion of a fertilizer particle along the radial vane of the centrifugal spreader disc. This motion was analyzed in the non-inertial reference system fixed to the spinning disc ( Figure 3). Therefore, the carrying motion was accounted by introducing the appropriate inertial forces. To facilitate the problem related to the mathematical apparatus needed, a number of assumptions have been introduced in the mechanical model of particle motion. Consequently, the resulting equations and their solutions represent an appropriate approximation of the real motion, which enables the analysis of working parameters that govern the fertilizer spreading process.

Materials and Methods
Dynamics of a single fertilizer particle movement along a straight radial vane attached to the horizontal flat spinning disc of a centrifugal spreader was analyzed in this paper. Thus, only the first phase of a fertilizer spreading process was comprehended, i.e. the model considered the motion of the particle along the vane, until it dropped out. A particle motion model is based on the assumption of continuous motion of a single particle along the vane, and laws of theoretical mechanics (Beer et al., 2010;Hibbeler, 2010).
The focus of interest of this manuscript is the dynamic equation of the fertilizer particle motion. In this particular case, the non-homogeneous secondorder differential equation of motion was solved analytically following methods of mathematical analysis (Soare et al., 2007). The solution was applied on specific data, processed using MS Excel. Vera B. Cerović et al. 86 The last phase of a spreading process, the ballistic flight of the fertilizer particle through the atmospheric air until its lending, has not been analyzed. However, this model provides an additional insight in the fertilizer particle moving over the spinning disc and its velocity vector at the leaving point from the vane. The information on this velocity vector is important, because it influences the flight of the particle through the air and the fertilizer spreading pattern.

Results and Discussion
The analytical model of the fertilizer particle motion is based on various assumptions.
An individual fertilizer particle moves along the vane, mounted radially on the horizontal flat disk. Thus, particle interactions were neglected and achieved results approximated the real spreading process (Villette et al., 2008).
The spreading disc is rotating at a constant angular velocity ω D [s -1 ]. Fertilizer particles are ideally spherical homogeneous particles, having constant density. Dynamics of the spherical particle on the spreading disc is based on the approximation of pure sliding against the vane.
According to Olieslagers et al. (1996), Aphale et al. (2003), Dintwa et al. (2004), particles bouncing over the disc are negligible: tight continual contact between the observed particle and the vane is established along the whole particle travel over the disc.
The particle motion follows the well-known Newton's second law: where is the particle mass, (i=1,2,...,n) are forces acting on it and · is the absolute linear acceleration (i.e. the first derivative of the absolute velocity vector ) and the second derivative of the position vector of the particle in the inertial reference coordinate system Oxyz ( Figure 2). Figure 2. Inertial coordinate system.
On the fertilizer particle motion along the vane of a centrifugal spreader disc 87 Nomenclature: R 0 , R 1 the entrance radius of the particle and the external radius of the rotating disc [m]; the radial distance from the disc center in the rotating (relative) system Oξηζ [m]; x,y,z the coordinates of a fertilizer particle in the inertial coordinate system [m]. Nomenclature: , the friction forces between the particle and the disc and the vane, [N]; the centrifugal force exerted on the fertilizer particle [N]; the Coriollis force exerted on the fertilizer particle [N]; , the gravitational force [N] and acceleration [m · s -2 ] of the particle; m the mass of fertilizer particle [kg]; , the orthogonal forces of the disc and the vane exerted on the particle [N]; ξ, η, ζ the coordinates in the non-inertial relative system fixed to the disk [m]; ω D the angular velocity of the rotating disc [s -1 ].
However, the traveling of the particle is considered in the relative reference system Oξηζ (Figure 3), fixed to the center of a spreading disc and rotating together with the vane along which the observed fertilizer particle moves. Therefore, the coordinate system Oξηζ is a non-inertial reference system. This means that Newton's second law can be applied only by introducing the additional imaginary "inertial" forces: the centrifugal and Coriollis forces ( Figure 3). Figure 3. Forces exerted on the particle in the relative reference system.
The following real forces also act on the particle: the friction forces exerted by the rotating disc [N] and the vane [N], particle weight [N], orthogonal reactions of the disc [N] and the vane . Thus, the motion of a fertilizer particle can be described by the vector differential equation: Vera B. Cerović et al. 88 and represented in the relative coordinate system Oξηζ by three scalar equations: ξ ≡ r r : · ; (3) Equations (3), (4) and (5) govern the relative motion of the particle. To be solved, it is necessary to previously evaluate unknown intensities of forces exerting on the particle.
Particle weight is the multiple of the mass m [N] and the gravitational acceleration g [m · s -2 ]. It is vertically aligned, but oriented downward, opposite to the axis Oζ: The Coriollis force originates from the combined particle motion in the rotating coordinate system. It depends on the carrying angular velocity , which is the disk angular velocity , and the relative linear velocity of the particle along the vane (Figure 2): where is the unity ort-vector defining the direction of the axis Oη. The orthogonal vane reactive force on the particle is the reaction on the influence of the Coriollis force of the particle against the vane. Hence, these two forces are collinear but of opposite direction. It follows from equations (4) and (7) that this force ( Figure 2) is: Similarly, the vector is the orthogonal reaction of a spreading disc on the fertilizer particle, caused by the particle weight (active force). Therefore, these two forces are collinear but take opposite directions. Regarding equations (5) and (6), it follows that the vector of this force (Figure 2a) is: The centrifugal force, · · · , (10) On the fertilizer particle motion along the vane of a centrifugal spreader disc 89 depends on the particle mass m [kg], its current radius [m] along the vane (the distance from the axis of disc rotation ), and the disc angular velocity . The friction forces and are caused by simultaneous sliding of the particle over the spreading disc and vane. They are functions of the dynamic friction coefficients , , and normal reactions , Including equations (10), (11) and (12) in the equation (3) gives the secondorder linear inhomogeneous ordinary differential equation with constant coefficients, which describes the relative particle motion: A general solution of the inhomogeneous differential equation can be found as the sum of the solution of the homogeneous equation and a particular solution of the inhomogeneous equation . (14) The homogeneous part of this ordinary differential equation is having the solution: where and are the unknown integration constants.
A particular solution of the inhomogeneous differential equation (13) can be found by the method of an undetermined constant: .
After differentiation of the equation (17) and inclusion in the equation (13), the particular solution of the inhomogeneous differential equation arose: giving the general solution of the differential equation (13), which describes the relative radial position of the fertilizer particle: The derivation of this equation gave the relative radial velocity of the particle: The integration constants and were determined from the initial conditions, which arose from the design and operational conditions of the analyzed fertilizer disc spreader: Namely, each fertilizer particle starts the relative motion along the vane from its inner edge and accelerates starting from the zero radial velocity. Combining the formulas (19), (20), (21) and (22) gave the constants and . Their introducing in equations (19) and (20) provided the final expressions defining the relative position and velocity of the particle along the vane: The fertilizer particle ejection (launching) time represents the traveling time of a fertilizer particle along the whole length of a centrifugal disk spreader vane, : On the fertilizer particle motion along the vane of a centrifugal spreader disc 91 The absolute velocity vector of the particle is a vector sum of the carrying velocity , induced by vane and disc spinning, and the relative velocity along the vane (Figure 4): .
(26) Figure 4. Components of the absolute velocity vector of the fertilizer particle.
The relative velocity vector is aligned with the positive direction of the Oξ axis and the carrying velocity is aligned with the positive orientation of the axis Oη. The magnitude of the relative velocity, as a function of time, is defined by the formula (24). Thus, its magnitude at the particle ejection point can be calculated by substituting in the equation (24): where the ejection time is defined by the formula (25).
The magnitude of the carrying velocity vector is the multiple of the particle radius and the angular velocity . of a carrying motion (spreading disc rotation): · .
At the ejection point, a fertilizer particle reaches the outer disc radius . Therefore, at the ejection point, the magnitude of the carrying velocity has the value: Therefore, the magnitude of the absolute velocity vector (Figure 4) was evaluated using the well-known Pythagoras theorem: . (30) Numerical results All calculations, based on the presented analytical model, were performed using MS EXCEL and the input data given in Table 1. The radial position of the sliding fertilizer particle was calculated using the expression (23), as it is illustrated in Figure 5a. The total particle traveling time , along the vane, is defined by the formula (25) that cannot be solved analytically. Therefore, its solution was found numerically, using MS EXCEL and the "sweeping method" applied to the equation (25) Figure 6. The radial velocity of the fertilizer particle along the radial vane, with respect to its distance from the spinning axis.
As can be seen from the charts presented in Figures 5 and 6, the launching value of the absolute velocity vector magnitude (at the ejecting point) is

Conclusion
The dynamic model for evaluation of the ejection velocity vector of the fertilizer particle from the spinning disc of a centrifugal spreader, having the flat disk with straight radial vanes, is presented in the paper. This work will be continued in order to develop a ballistic model of the fertilizer particle flight, which is expected to enable the prediction of the landing distance and the fertilizer spreading pattern.
The influence of the inclination vane angle toward the spinning disc (inclined forward, inclined backward or orthogonal to a disk) has not been completely resolved yet, especially when combined with the influence of a friction coefficient. This phenomenon should be also included in the more advanced future models of solid fertilizer spreading by disk spreaders.