IntroductionTop

Sorting, the separation of granular material into one or more fractions on the basis of some measurable physical property, is required in many industrial processes including the preparing of foods (Feistritzer et al., 1981; Fellows, 2000; Grandison, 2006). Size sorting, in which separation process is based on the measurable physical property being size, is physically and most economically realized by means of vibrating systems. Vibrating systems realizing the separation function involve vibrating screens. There are works in the literature in which different aspects of vibrating screens have been handled. Tan & Harrison (1987) have investigated the influence of several different planar and spatial drive mechanisms on the screening efficiency by means of the so called screening index. Shen et al. (2009) reported a novel vibration sieve mechanism based on a parallel mechanism, in which optimum design has been demonstrated with regard to structure, kinematic analysis and motion simulation. The motion of granular material has also been analyzed by Hongchang & Yaoming (2011) with respect to a performance parameter called throwing index, the most suitable value of which has been found corresponding to the best performance of vibrating screen.

The vibrating screen has found extensive applications in many different areas. For instance, in coal processing, vibrating screen with variable elliptical trace has been examined from the standpoint of its “ideal motion” with the purpose of improving processing capacity and efficiency through computer models (He & Liu, 2009). In the area of construction bulk material screening for civil engineering purposes, sieve vibrations have been analyzed by another work utilizing computer models involving an algorithm based on the so-called false-position method (Stoicovici et al., 2009). There have been some works related to the vibratory motion of the granular material using the so-called Discrete Element Method (DEM). For instance, Alkhaldi & Eberhard (2006) have analyzed screening phenomena in mechanical environment called vertical tumbling cylinder. A numerical study of the motion particulates along a circularly vibrating screen deck has been done using the three - dimensional DEM (Zhao et al., 2011) in which the effect of vibration amplitude, throwing index and screen deck inclination angle on the screening process has been determined for optimizing the design of screen separators.

A number of disadvantages associated with vibrating screens can be stated. In the first place, accelerations are unavoidable part of vibrating screens. This means large inertia forces in connection with large moving masses. These inertia forces do not only raise power consumption but also create material fatigue stresses. Vibrations cause at the same time disassembling forces on the connections. Additionally, unbearable noise levels are the results of vibrations.

Jiannan et al. (2017) developed and used rotary table-roller coating equipment to coat peanut seeds. Inspired by such studies and due to the above mentioned negative aspects of vibrating sieves, the idea of using cylindrical sieves has been formed in the classification of seeds.

In this study, a rotary cylindrical screen was designed based on the idea that sieving and grading processes can be performed without vibration. To ensure control and continuity over the flow of the sorting material, helicoid surfaces were placed inside the cylindrical screen. In order to free the motion of the sorting material off vibrations to induce a uniform flow, the screen was rotated at a constant speed. The movement of the kernels was modeled on the basis of a continuous layer of particles directed along circular and helical paths. Finally, the theoretical classification conditions were compared with the results obtained in the experiments.

 

 

Material and methodsTop

Theory

Based on the fact that one significant factor in causing inefficiency in the sorting of particulate material is the uncontrolled, irregular motion of the particulate material relative to the mechanical medium by which sorting is to be realized. Thus, in order for the efficiency of sorting to be improved, there is a need to regulate the motion of the material. To this end, the motion of the particulate material is considered to be regulated in two ways. The first one is to understand and formulate the motion of the material in a rotating cylinder and the second one is the understanding and formulating how this circular motion is affected by a helicoid wrapped inside the cylindrical surface.

In order to formulate the motion of the material Fig. 1a is considered. The basic assumptions in this formulation were that there is a single and continuous layer of average thickness of “e” supposed to slide only, excluding any rolling or tumbling motions. Practical means can be deployed to satisfy these assumptions (Fig. 1b). The cylinder which receives particulate material from the feeder is rotating at an appropriate constant speed w and has a radius of r. The coefficient of friction between surface of the cylinder and the granular material is µd. As the cylinder rotates, the particulate material assumes a height from the lowest point of the cylinder which means a gain of potential energy. Under suitable motion conditions this potential energy will be converted into a relative sliding motion of the particulate material with respect to the surface of the cylinder. Then the material inside the rotating cylinder will come to equilibrium under the influence of basically three forces, which are particulate material weight, friction force and inertial force.

Now, the equilibrium position of the layer of parti-culate material will be formulated referring to Fig. 2. In Fig. 2 an infinitesimal mass (dm) of layer of particulate material is considered at a position of α from the verti-cal axis, which is assumed to come to equilibrium under the influence of static weight (dm.g), where g represents gravitational acceleration, inertial force (dm.w2.r), inte-raction force (Nr) between the infinitesimal mass and in-ternal cylinder surface and friction force (µd.Nr), F force representing resistive effect of the rest of the layer below α position, F+dF representing the force responsible for the downward sliding motion, where dF is a force change due to a position change dα.

If equilibrium equation along the surface normal of the cylinder is considered, assuming outward to be positive the following equation is written:

 

Now, assuming downward sliding direction to be po-sitive, equilibrium equation is written along the sliding direction:

 

If the interaction force Nr is drawn from Eqn. (1) and substituted in to Eqn. (2) the following results:

 

Now designating the thickness of the particulate layer by e and layer density per unit length by ρ, infinitesimal mass dm can be expressed as follows (Fig. 2):

 

Substituting dm from Eqn. (4) into Eqn. (3) and inte-grating between α0 and β, where α0, β designate the angular position of the beginning and end of the layer, respec-tively, the following is obtained:

 

By Eqn. (5) it becomes possible to estimate the span (β-α0) of the layer which comes to equilibrium during the rotation of the cylinder. If α0 is assumed to be 0, mea-ning that the layer starts from the lowest position, Eqn. (5) turns out to be:

 

The importance of F function described by Eqns. (5) and (6) lies in the fact that it represents the net force acting on the layer along the motion direction, which is necessari-ly zero for equilibrium under stable steady-state conditions.

Eqn. (6) is a nonlinear algebraic equation in the unk-nown β which requires numerical solution. Nevertheless, for small β angle an initial solution to start a numerical iteration procedure can be found as follows:

 

In order to determine the location (θ*) of the center of gravity of the layer, it is necessary to take into account the external moment about cylinder axis that will cause the mass dm to have a potential energy of dm.g.r(1-cos α). Accordingly the center of mass is determined as follows;

 

Substituting relationship (4) in Eqn. (10) and carrying out integration, the following is obtained:

 

If α0 is assumed to be 0, meaning that the layer starts from the lowest position, Eqn. (11) becomes:

 

The significance of the center of mass lies in the fact that the whole layer of particulate can be imagined to be concentrated as a lumped mass at that point so that the motion of the layer can be better understood. For small angles θ*, α0 and β the following approximate relations-hip can be given:

 

In particular when α0 is set equal to 0 it can be shown that:

 

 

By means of Eqn. (14) it is possible to visualize the cir-cular motion of the layer as such: when the layer center of mass exceeds θ* value it means tendency to have sliding acceleration and when it comes to this angle the layer will assume equilibrium positon.

In order to let the layer of particulate material to be forwarded along the cylinder axis (Z-axis) a helicoid will be installed into its internal surface of the rotating cylin-der. Now the motion of the layer along helical path will be described mathematically. The net force on the layer of mass dm downwards in a circular cross section is con-ceived to be dm.g.sinα at a positon of α. This force will be decomposed into axial force component of dm.g.sinα.cosΨ and normal force component of dm.g.sinα.sinΨ where Ψ is the angle of helical path with the vertical di-rection (Fig. 3). Now equilibrium equations will be wri-tten along the radial, helicoid normal and helical sliding directions, respectively.

 

where Nh, µh are interaction force and coefficient of friction between helicoid surface and layer, respectively. If Nr and Nh are drawn from Eqns. (15) and (16) respecti-vely together with Eqn. (4) substituted into Eqn. (17) and integration is carried out between β0 and βh, the following equation results:

 

For a particular case of β0=0, Eqn. (18) becomes:

 

Both Eqns. (18) and (19) signify net forces acting on the whole layer, which are 0 under equilibrium condi-tions. If for small angles in Eqn. (7) is used in Eqn. (19) the following relationships are obtained:

 

If equilibrium angles for circular and helical motions represented by Eqns. (9) and (20) respectively are compa-red, it will be seen that under the condition

 

equilibrium angle (βh) for helical motion will be grea-ter than that (β) for circular motion. In order to estimate the mass flowrate (dm/dt) through the rotating cylinder, two elements are needed as depicted in Fig. 4.

 

where, from Fig 4-b

 

 

Figure 1.  Helical path in the screen (a) and the prototype sorting machine (b): 1, store; 2, feeder outlet; 3, transport channel; 4, cylindrical sieve; 5, helical paths; 6, large kernels output; 7, small kernels output; 8, electrical motor; 9, belt pulley mechanism; 10, brush.

 

 

Figure 2.  Forces acting on sliding mass (dm).

 

 

Figure 3.  Forces on developed plane

 

 

 

Conditions of screening

The basic principle underlying size sorting is that a parti-cle of a particular size falls into the relevant section by pas-sing through the holes of the screen. In order for the prin-ciple to be materialized, the control over particle velocity is essential. It is assumed that the holes are always open and ready to allow the passing of the particle without hindrance.

In order to derive the mathematical expressions related with the velocity control, a particle is shown in Fig. 5 at an initial position before entering the hole of the screen.

If Newton’s Second Law is written for the motion of the particle in the z and y axes (Fig. 5), the following equations result:

 

 

Assuming that the particle has velocity vz and zero displacement initially, the solutions of equation set (24) will be following:

 

In case t is eliminated in equation set (25), the following is obtained:

 

Now, the condition for the particle to pass through the hole without hitting its lower corner can be established as such:

 

where, c, e, lb are the sheet metal thickness, particle thick-ness, hole length, respectively. If conditions (27) are eva-luated in Eqn. (26), the following result is obtained:

 

 

 

It means that the particle velocity in the z direction should be restricted.

Similarly, by the same reasoning the tangential velocity (vθ) in the direction perpendicular to cylinder axis should also be limited through the following relationship:

 

where la is the width of hole.

Relationships (28) and (29) imply that the particle should have controlled velocities along the cylinder axis as well as perpendicular to it for effective screening.

 

Figure 4.  Mass flowrate formulation.

 

 

Figure 5.  Motion of a single particle by the sieve hole

 

 

 

Efficiency formulation

System efficiency (η) is formulated according to criteria of separating the input mass into two parts referred to as small-size group and large-size group as follows;

 

 

where K is small-size group mass content, B is large-si-ze group mass content of the inflowing mass, K’ is the small-size mass content, KB is small-size items mass con-tent in the large-size box, B’ is the large-size mass con-tent in the large-size box. Here in this formulation rK, rB designates the input ratios of small and large-size mass contents in the total inflowing mass respectively; r’K, r’B symbolize the ratio of small-size to the total mass in the small box, ratio of large-size mass in the large-size box respectively; ηK, ηB signify small-size and large-size effi-ciency respectively.

 

Screen design

Design considerations

The theoretical model, so far developed, essentially couples the physical properties of the granular mate-rial with the characteristics of the mechanical envi-ronment. The basic physical properties in question are density and dimensions (width, length and thickness) of the grain; coefficients of friction between granular and screen and helicoid materials. The characteristics of the mechanical environment are fundamentally the shape and dimensions of the screen, namely cylinder radius and length, screen hole sizes (width and length), helix angle of helicoid; angular, axial and tangential velocities. The other quantities which go into the de-sign procedure are the ones that depend on the interac-tion between granular material properties and screen characteristics. These are namely equilibrium angle, mass flowrate, limiting values of helix angle, axial and tangential velocities.

Another factor to be taken into consideration in des-cribing the mechanical characteristics of the screen is the dimensions of the helicoid surface to be placed inside the cylinder surface of the screen. For all practical purposes, given the outer diameter of helicoid piece (Dd), pitch (h) and the width a, internal (Di’), and external (Dd’) dia-meters of the planar ring that will form the helicoid were calculated as follows:

 

The design procedure is to accept the desirable mass f lowrate and physical properties of the granular material as the basic input and is to return the design quantities of the interest as the basic output.

 

Design algorithm

The mathematical model developed previously will be transformed into a number of series of consecutive steps to form an algorithm yielding the values of the design parameters. Series of steps are given below:

i. Desirable mass flowrate and the physical properties of the granular material are entered.

ii. The width of the oblong hole of the sheet metal is selected according to the classification table of granular material. For instance, the references (Guzel et al., 2005, 2007; Akcali & Guven, 1990; Akcali et al., 2006) are to be referred to when the granular material is peanut.

iii. Hole length is selected in relation to grain length.

iv. Taking into account the standards of sheet metal size, corresponding cylinder radius, length and thickness of sheet metal are chosen.

v. Limiting values of axial and tangential velocities are computed by (28) and (29) respectively.

vi. In order for the bulk not to stick on the internal surface of the cylinder angular speed should always be less than

vii. Tangential velocity of the granular material is computed by Eqn. (23).

viii. Tangential velocity is compared with its limiting value. If the limiting value is exceeded, the following actions are executed until satisfaction of the condition:

a. Steps are iterated from vi.

b. If action in step a is not sufficient, steps are iterated from iv.

ix. Helix angle is selected so as to satisfy inequality (21).

x. Axial velocity of the granular material is computed by Eqn. (22).

xi. Axial velocity is compared with its limiting value. If the limiting value is exceeded, steps are iterated from ix until satisfaction of the condition.

xi. Axial velocity is compared with its limiting value. If the limiting value is exceeded, steps are iterated from ix until satisfaction of the condition.

xiii. Mass flowrate is calculated by means of Eqn. (21) and the relevant data.

xiv. Mass flowrate is compared with its desirable value. Until the desirable mass flowrate is satisfied with allowable tolerance, the following steps are iterated:

a. Steps starting from number ix are repeated.

b. If the steps in the previous iteration (a) do not satisfy the basic mass flowrate criterion, steps starting from number vi are repeated.

c. In case the mass flowrate criterion is not satisfied in the previous step (b), then iteration is started from number iv thereafter.

xv. Helicoid dimensions are determined by Eqn. set (32).

Sorting machine was manufactured according to the above-mentioned design algorithm. The machine is shown in the Fig. 6. The basic feature of the machine is to allow different agricultural granular materials to be sorted out by the possibility of changing the cylindrical screen according to the fitting size of the granular material. In this particular machine, screen is intended to separate granular materials into two parts with diameter less or greater than 7 mm (Table 1). The dimensions and technical data, which belong to the cylindrical helical sieve, are presented in Table 1.

Limit and experimental speed values for the product to be moved on the sieve were calculated and given in Table 2. An examination of expressions (28) and (29) reveals that the limiting axial velocity (vz) and tangential velocity (vθ) are proportional with the dimensions of sieve hole along the axial and tangential directions, respectively. Axial hole dimension (i.e. hole length lb) can be more freely selected whereas tangential hole dimension (i.e. hole width la) is dependent on the size of the granular material.

When the thickness of the screen metal (c) and thickness of the granular material (e) are increased, both velocity limits are decreased at the same time.

 

Figure 6.  EHelical screen

 

Table 1. The dimensions and technical data of the helical screen

 

Table 2. Limit and experimental speeds on the sieve

 

 

 

Machine experiments

The parts that make up the prototype sorting machi-ne are generally as follows: store (1), feeder outlet (2), transport channel (3), cylindrical sieve (4), helical paths (5), large kernels output (6), small kernels output (7), electrical motor (8), belt pulley mechanism (9) and brush (10). At the outlet (2) of the feeder there is a gate con-trolling the amount of particulate material fed into sys-tem, which is driven by a cylindrical cam mechanism at a frequency four times that of the cylinder (4). The system is activated by an electrical motor (8) through a belt and pulley mechanism (9). The machine functions in such a way that upon activation, the outlet (2) of the feeder sto-re (1) filled with large and small sized kernels, forwards the particulate material through the transport channel (3) into the inlet part of the cylindrical screen (4). To prevent oblong holes on the screen from being blocked by kernels, a brush (10) extending along the cylinder is placed (Fig. 1b). In order to secure single and continuous layer, three helical paths (triple helicoid) have been deployed inside the cylinder.

Experiments on the machine are carried out in the following way: nine different large and small peanut mix-tures (10-90%, 20-80%, 30-70%, 40-60%, 50-50%, 60-40%, 70-30%, 80-20% and 90-10%) were prepared with a total mass of 2 kg. Each mixture was filled into the store (1) at the back of the machine and the separation work was done by operating the machine. This process was re-peated three times. The feeder outlet (2) was kept at the lowest level in order to enable the kernels to form a single layer in the cylinder. In the separation work, small sized peanuts fall down through the holes on the cylindrical screen (4) and are taken out from the small kernels outlet (7). The large kernels move along the cylinder and leave the machine from the large kernels outlet (6) at the end of the cylinder. Kernels from the outlets (6 and 7) were weighed on a sensitive scale with 1 g resolution and the resulting efficiencies of the mixture were determined.

The data were processed with the Excel 2016 package program. ANOVA was performed to determine the effect of large-small kernel mixture ratio on sorting success (SPSS v17.0).

 

 

Table 3. Experimental results