Calculation of the throwing angle of a fertilizer centrifugal device as functions of random coordinates of feeding point

Liquid fertilizers fed into centrifugal device are spread along angle and radius of feed under action of blades. This article describes how to calculate throwing angular characteristics using Mathcad. The package consists of four programs. Program Mf is intended for calculation of probability density of supply point coordinates under assumption of bivariant normal distribution of system r, γ, which are specified in the form of vectors. The result of the calculation is displayed as matrix Mf. The program Mα calculates the throwing angle for all combinations r, γ.. To calculate the throwing angle, the method of solving differential equations of particle movement along the blade of the device with input data was used: Radius of the disk R, angular speed ω, coefficient of friction of fertilizers on the blade f. The program Ms extracts from the matrix Mf elements corresponding to a throw angle less than a given number A. The program F (A) sums the elements of the matrix Ms. We obtained the values of the throw angle distribution function by multiplying the resulting sum by the intervals of vectors r and γ. The calculated throwing angle distribution function is approximated by the standard normal distribution function.


Introduction
Сentrifugal devices for distribution of mineral fertilizers are widely used because of their simplicity, high productivity, and easy loading. The theory of the operation of centrifugal devices has been significantly developed [1,2]. However, the theory considers the motion of single particles, which limits the use of these dependencies in design calculations. Flowing fertilizers fed to the distributing device get a spread in the angle and radius of the feed under the impact of the shovels. Therefore, programs for calculating the throwing angle should be updated so that the throwing angle is determined as a function of two random arguments -the polar coordinates of the feed points.
The throwing angle α (Figure 1) relative to the line of motion is determined by the formula where λ -an angular coordinate of the feed point of the particle, θ is the angle between the absolute velocity and the radius vector of the particle, ωt 1 is the angle of descent of the particle, i.e. the angular sliding of the particle in absolute motion until it leaves the shovel. The angle of descent of the particle is a function of the radius of the feed from the disk r 0 ωt l and is defined as the multiplication of the angular velocity of the disk by the time of the sliding of the particle along the shovel.
For a radial shovel, the equation has the form We substitute (2) into (1) and get the throwing angle α as a function of random arguments r 0 and λ.
where the angles α, λ are measured from the longitudinal coordinate axis in the direction of rotation of the disk.
The last equation in the polar coordinates r, λ gives a logarithmic spiral [1] of the form ρ = а·ехр(кλ), where a and k are the parameters of the spiral determined by the dependencies: . cos sin k )); ( cos Fertilizer feed to any point of the spiral provides a set constant value of the angle α. The shape of the spiral depends only on the angle of friction of the particles on the shovels, while its location depends on φ, α and R. The spiral, built at α = α1, divides the feed zone into two parts ( Figure 1) [1]. The combinations r 0 , λ, belonging to the area S1 located to the right of the spiral, provide an angle α less than α 1 , while at the combinations (r 0 , λ) not belonging to the area S1, we have α > α1.
The distribution function of a random variable α can be obtained by integrating the density of the system r 0 , λ over the area S 1 .
Let us consider the application of the method for calculating the throwing angle for a device with radial shovels.
2 Algorithm for calculating 1. We should record initial data: mathematical expectation of the radius of the feed Mr; standard deviation of the radius of the feed σr; mathematical expectation of the angular coordinate of the fertilizer feed point Mλ; standard deviation of the angular coordinate of the feed σλ; radius of the distribution disk R; angle of inclination of the shovel to the radius ψ; friction coefficient of fertilizers on the shovel f; angular speed of the disk ω. 2. We should create vectors r i ;λ j with the number of elements Nr=Nλ=12. So the intervals between the elements of the vectors are equal to 0,5·σr, 0,5·σλ.
3. We should create a probability density matrix for the bivariate distribution of the coordinates of feed r i ;λ j . We consider random variables to be independent, therefore, the probability density for all combinations is determined by Program Mf for a bivariate normal distribution. 4. We should create a matrix Mα of throwing angles for all combinations r i ;λ j . Program Mα (Figure 4) has two cycles where it changes the polar coordinates of the fertilizer feed points and calculates the throwing angle of the disk with radial shovels as a function of the polar coordinates of the feed points [2].  The matrix Mα can determine the mathematical expectation of the throwing angle. It is necessary for a two-disk device to obtain = 0.6 rad, for a one-disk device it has Mα =0. These values must be obtained for the center of the matrix. One can correct them by changing Mλ. 5. One should select from the matrix Mf the elements corresponding to a throwing angle less than that given by number A. By summing the selected matrix elements and multiplying the sum by the intervals of the vectors, we obtain the value of the distribution function of the throwing angle.
The selection of the elements of the matrix Mf is performed by Program Ms(A) ( Figure  6), the result of which is presented in Figure 7.   The approximation of the obtained distribution function of the throwing angle by the standard normal distribution function (Figure 9) turns out to be in a good agreement with the results. The mathematical expectation of the throwing angle, obtained by calculation, corresponds to the optimal value for a two-disk device. The standard deviation of the throwing angle is less than the optimum one being equal to 0.6 radians. It can be increased in two ways: by spreading the feed along the angle λ or by applying the disk shovels deflected forward to the direction of rotation. In the latter case, one should replace Program Mα [1,3] with Program Disk-.

Conclusions
1. The method for calculating the distribution function of the throwing angle of fertilizers by a centrifugal device as a function of the random coordinates of feed points was verified by a numerical illustration. 2. The distribution function of the throwing angle obtained by calculation is approximated by the standard normal distribution function. The mathematical expectation of the throwing angle is obtained as close to the optimal value for the two-disk device. The standard deviation of the throwing angle obtained by calculation is less than the optimal value. Therefore, it should be recommended to repeat the calculation with the shovels deflected forward to the direction of rotation.
3. An increase in the standard deviation of the throwing angle can be obtained by spreading the feed of fertilizers to the disk, for example, by applying feed through two fertilizer guides. In this case, the probability density of the system r, λ should be calculated by the superposition formula of bivariate normal distributions.