SOFTWARE

Model and Software for the Regulation of an Inclined Belt Sorter for Agricultural Products

Ing. Raúl Torres-Cepero^{I  *} 

Dr.Cs. Arturo Martínez-Rodríguez^I 

MSc. Ana Rosario^II 

^IUniversidad Agraria de La Habana (UNAH), Facultad de Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba.

^IIInstituto de Tecnología del Estado de Trujillo. Venezuela.

ABSTRACT

As a result of the development of a sorting table for agricultural products for the selection of tomatoes, potatoes or other products with approximately spherical shape, a mechanical-mathematical model and software were developed that make it possible to determine and adjust the design parameters of that sorting table. The model interrelates a set of parameters such as the coordinates of the drop point of the fruits or tubers on the conveyor belt and the angle of transverse inclination of the conveyor belt. In addition, it interrelates the rolling friction angle of the product with respect to the conveyor surface, the components of the falling speed and the linear speed of the conveyor belt, among others. The modeling was carried out applying the laws of Newtonian mechanics. The development of the software was carried out on the basis of the Mathcad 2000 Professional software.

Keywords: Potato; Tomato; Other Spherical Products; Design Parameters; Conveyor

INTRODUCTION

As part of a project financed by the Ministry of Higher Education and the Center for Plant Biotechnology of the Central University of Las Villas, the Center for Agricultural Mechanization of the Agrarian University of Havana was commissioned to develop an equipment for classification of mini potato tubers obtained from tissue culture and propagated to produce potato seeds. This equipment was developed and installed in the mini tuber classification line of the Plant Biotechnology Center.

Initially, a bibliographic review was made on the cultivation of potatoes and other products with approximately spherical form and the models used. (^{Sablikov, 1978}; ^{Mesemov, 1987}; ^{Paneque, 1988}; ^{McGarry et al., 1996}; ^{Peters, 1996}; ^{Minag-Cuba, 1997}; ^{Baritelle et al., 2000}; ^{Iglesias, 2002}; ^{Alvarado, 2004}; ^{Buitrago, 2004}; ^{Bentini et al., 2006}; ^{Montesdeoca et al., 2006}; ^{Polanco, 2007}; ^{Escalona y Elorza, 2008}; ^{Ramos et al., 2010}; ^{FAO, 2014}; ^{López et al., 2017}; ^{Infoagro, 2018}).

The versatility of this sorting machine makes it suitable for the classification of other agricultural products with a spherical or similar shape, among which are tomato, seed or consumption potatoes, various types of fruit, etc.

As a result, a mechanical-mathematical model and software were developed. They facilitate the adaptation of the design and regulation parameters of this machine for its use during the processing of other agricultural products.

Although at the international level, agricultural product sorting machines with different working principles are marketed according to ^{Sablikov (1978)} and ^{Paneque et al. (2018)}, it was considered convenient to facilitate the generalization of this machine developed in Cuba for the classification of different products.

THEORETICAL FOUNDATION

Machine Productivity Calculation

For the calculation of productivity, the daily production of the machine (Q, kg/day) in which the classification service will be provided must be taken into account.

From this datum and the hours of work in clean time allocated to the daily service of the machine (T, h/day), the productivity of the equipment W (kg/h) is determined according to the expression:

$W\equiv \frac{Q}{T};kg/h$ (1)

Classification table parameters

The main parameters of the classification Table (Fig. 1) are:

• The width of the conveyor belt b, m);

• The speed of the conveyor belt Vt, m/s;

• The transverse angle of the conveyor belt α, (°);

• The gaps of the sorting gates hi, mm;

• The length of the gates Li, m (Fig. 3);

• The angle of the exit ramps of the classified material β, (°) (Fig. 4).

The calculation and selection of these parameters will depend on the kinematic and dynamic relationships that occur during the tuber-sorting table interaction, and of course, the main physical-mechanical properties of the materials involved in the process will also intervene.

For the analysis, the free body of the tuber starts from its interaction with the conveyor belt (Figure 1).

FIGURE 1 Diagram of the analysis of interaction tuber-conveyor belt. Source: Authors’ elaboration. 

The system of forces acting on the tuber consists of the weight force (m∙g , N), the normal reaction of the conveyor surface (N, N) and the friction force between the tuber and the conveyor belt (F, N). For the analysis of the dynamics of movement in the direction of the X-axis, it is assumed that the transported fruits roll on the surface of the conveyor belt in this direction. While in the direction of the Y-axis, it is assumed that the fruit is transported without relative slippage with the conveyor belt.

Applying Newton's second law in the direction of the X-axis, it is obtain:

$mg\bullet \sin \alpha -F=m\bullet a_x$ (2)

Being

$F=N\bullet \tan {\varnothing }_r$ (3)

where

∅r:	Rolling friction angle between the tuber and the conveyor belt (^o).
ax:	x component of the absolute acceleration of the tuber in its movement on the conveyor belt, m/s^2

The normal reaction N is given by:

$N=m\bullet g\bullet \cos \alpha$ (4)

Substituting 4 and 3 into 2 and grouping together the acceleration in the X direction it is obtained:

$a_x=g(\sin \propto -\cos \propto \bullet \tan {\varnothing }_r)$ (5)

Since g, α and ∅_r are constant, a uniformly accelerated movement in the X direction will be obtained, and the velocity in X (V_x) and the X coordinate can be calculated as follows:

$V_x=V_{ox}+a_x\bullet t$ (6)

$x=x_o+V_{ox}\bullet t+\frac{1}{2}{a}_x\bullet t^2$ (7)

In the direction Y, the tubers move at constant speed, together with the conveyor. The acceleration, speed and displacement in this direction being given by:

$a_y=0$ (8)

$V_y=V_t$ (9)

$y=y_o+V_t\bullet t$ (10)

The evaluation of these expressions will allow calculating, based on the knowledge of the physical-mechanical properties of the tubers, the different parameters under study. For this evaluation, a program was developed in Mathcad 2000 Professional, whose description is shown below.

RESULTS AND DISCUSSION

Software Description

The input parameters to the program are:

• The coordinates of the tuber drop point on the conveyor belt: (x_o, y_o);

• The angle of transverse inclination of the conveyor belt: α;

• The rolling friction angle of the tuber with respect to the conveyor surface: ∅_r;

• The acceleration of gravity: g;

• The components of the speed of fall according to X and Y: V_x , V_oy;

• The linear speed of the conveyor belt: V_t (m/s).

As output parameters of the program, the components of the acceleration, speed and displacement of the tubers on the conveyor belt as a function of time are obtained, which makes it possible to obtain, graphically or tabulate, the trajectory of the tubers in the X coordinate system, X,Y .

Figure 2 shows the output graphs of the program, where the trajectories corresponding to different angles of friction, productivities and speeds of the conveyor belt can be seen.

FIGURE 2 Graphic output of the program developed in Mathcad.  

The linear speed of the conveyor belt (V_t) is determined from an analysis of load and capacity of the machine. For the analysis, it is considered that, to achieve satisfactory quality in the classification process, the tubers must be lined in a single row at the entrance to the gates so that there is no agglomeration of the product (Figure 3).

FIGURE 3 Diagram of the inclined belt sorter (top view).  

The number of tubers aligned per linear meter of the conveyor belt in the longitudinal direction (Y-axis) is given by:

$n_y=\frac{1}{\mathcal{l}}\mathrm{,\; tub/m}$ (11)

where l: is the average length of the tubers, m.

The tuber flow per second will be:

$q_t=n_y\bullet V_t=\frac{V_t}{\mathcal{l}}\mathrm{,\; tub/s}$ (12)

The mass flow (processing capacity) will be given by:

$q=q_t\bullet P_e=\frac{V_t}{\mathcal{l}}\bullet P_e\mathrm{,\; kg/s}$ (13)

where: $P_e$ is the average mass of the tubers, kg.

On the other hand, the load of the machine (productivity in clean time) is known:

$W=\frac{Q}{3600\bullet T}\mathrm{,\; kg/s}$ (14)

To obtain an optimal operating regime, it is necessary to equalize the load with the capacity, from which it is obtained:

$\frac{Q}{3600\bullet T}=\frac{V_t}{\mathcal{l}}\bullet P_e$

Clearing the expression to calculate the speed of the conveyor belt is obtained:

$V_t=\frac{Q\bullet \mathcal{l}}{3600\bullet T\bullet P_e}\mathrm{,\; m/s}$ (15)

Other parameters to be determined, such as α, B and the dead space Lo, are determined by evaluating the program with successive runs. The dimensions of the gates L₁, L₂ and L₃ are determined based on the probability distribution of frequencies of the classes to be classified. The product properties required for the evaluation of the equations, such as ∅_r , P_e and l are determined experimentally.

The regulation of the opening h of the gates is also determined on the basis of the standard dimensions for the classes to be classified, taking into account the consideration of a small opening angle in the direction of the longitudinal axis to avoid blocking of the tubers against the contact line of the gates.

The angle β (Figure 4) of the evacuation channels of the classified tubers is determined according to the condition:

$\beta >{\varnothing }_r$ (16)

FIGURE 4 Angle of the outlet channels of the classified product. 

Some screenshots during the program run for a specific case study are shown below.

CONCLUSIONS

• A mechanical-mathematical model is developed that makes it possible to calculate the main design and regulation parameters for agricultural product classification machines that use a conveyor belt with a transverse inclination;

• The model is complemented by software, developed in a Mathcad 2000 Professional environment, which facilitates the quick execution of the calculations.

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The mention of trademarks of specific equipment, instruments or materials is for identification purposes, there being no promotional commitment in relation to them, neither by the authors nor by the publisher.

Received: May 12, 2019; Accepted: December 04, 2020

^*Author for correspondence: Raul Torres-Cepero, e-mail: [e-mail]

Raúl Torres-Cepero, Especialista, Universidad Agraria de La Habana (UNAH), Facultad de Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Arturo Martínez-Rodríguez, Profesor e Investigador Titular, Universidad Agraria de La Habana (UNAH), Facultad de Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Anna Rosario, Profesora, Investigadora, Instituto de Tecnología del Estado de Trujillo, Venezuela, e-mail: [e-mail]

The authors of this work declare that they have no conflict of interest.