ore dressing vibrating screen efficiency calculation

factors affect screening efficiency of vibrating screen - xinhai

Screening effect and efficiency of the vibrating screen can affect the output and production efficiency of the whole production line. Here are some reasons that affect the screening effect and efficiency of vibrating screen.

Experiments show that adding surface-active material to water containing material can increase the activity and dispersibility of materials, and improve the screening condition. Besides, using the vibrating screen deck that is made up of non-hydrophilic materials also can improve the screening efficiency.

The soil is easy to knot and lock the mesh in the screening process. Clayey materials and viscous materials can only be screened with large mesh in some special cases. Wet screening is spraying water on the material that is moving along the vibrating screen deck, and pre-desliming before screening.

The particle characteristic mainly refers to the content of various grained materials that have a specific influence on the vibrating screening process. When the difficult screening material is much more than other fractions, the screening process is better. The particle characteristics also include the shape of particles. It is easier to screening for those particles who have close three-dimensional sizes, such as spheres and polyhedrons. In other conditions, it is more difficult to screening for those particles with large different sizes, such as slice, strip and weird shape.

There is no effect on the screening effect if all particles are the same density. However, if the density of coarse and fine particles are different, the situation is totally different. If the coarse-grained density is small, then fine-grained density is large. In contrast, coarse-grained density is larger and fine-grained density is small.

Movement type of vibrating screen deck is related to the looseness of the oversize material layer and the speed, direction and frequency of vibrating screen motion. In the vibrating screen, movement type of screen deck is divided into several kinds of forms: circular vibration, linear vibration and elliptical vibration. Xinhai has developed a series of vibrating screens with high screening quality, such as circular vibrating screen, auto centering vibrating screen, linear vibrating screen.

In general, the screen width directly determines the processing capacity of the vibrating screen. The wider the screen, the larger the processing capacity. The screen length determines the screening efficiency. The longer the screen, the higher the screening efficiency. When the screen length reaches a certain size, the screening efficiency will be rarely increased, or even no longer increases. And if you increase the screen length now, only the size and quality of screen are increased.

Screen slope has a great influence on the screen strength. The larger the screen slope, the smaller particle size, the smaller the screen slope, the larger particle size. In order to exclude oversize material easier, sometimes the screen deck should be installed obliquely. The larger slope, the faster the material layer moves on the screen deck, the greater the productivity. But the time that material stays on the screen deck is shortened, which decreases the screen-penetrating probability and reduces the screening efficiency.

The larger screen, the higher unit area screening efficiency and the higher screening efficiency. In general, the larger the mesh size, the higher porosity of the screen. When the mesh size is certain, the larger porosity of screen is more beneficial to the screening of vibrating screen. However, but the porosity of screen is often limited by screen strength and service life.

For certain screen and raw materials, the operating condition mainly refers to the quantity and mode of feeding. Continuous and even feeding is the best feeding method. In addition, please timely clean and maintain screen deck, which is also conducive to screening effect of the vibrating screen.

screening - sciencedirect

Basic design aspects of different types of straight screens and curved screens are described and explained. Mathematical aspects of the design processes are illustrated. Methods of calculating mass balances, screening efficiencies and screen capacities are explained using practical examples and solutions of typical problems that frequently arise. Operating mechanisms of both straight single deck and multiple decked and curved screens are explained and discussed mathematically so that correct estimates of over-flow and under-flow products, grades and concentrations can be obtained.

dynamic modeling of a vibrating screen considering the ore inertia and force of the ore over the screen calculated with discrete element method

Manuel Moncada M., Cristian G. Rodrguez, "Dynamic Modeling of a Vibrating Screen Considering the Ore Inertia and Force of the Ore over the Screen Calculated with Discrete Element Method", Shock and Vibration, vol. 2018, Article ID 1714738, 13 pages, 2018. https://doi.org/10.1155/2018/1714738

Vibrating screens are critical machines used for size classification in mineral processing. Their proper operation, including accurate vibration movement and slope angle, can provide the benefits of energy savings and cost reductions in the screening process and the whole mining process. Dynamic models of the vibrating screen movement available in the literature do not simulate ore motion or its interaction with screen decks. The discrete element method (DEM) allows for the calculation of the dynamic of the ore. In this paper, two 2D three-degrees-of-freedom dynamic models for a vibrating screen are tested, using linear and nonlinear approaches for angular displacement. These models consider the inertia of the ore and the ore force calculated with DEM. A double-deck linear motion vibrating screen is simulated using the DEM software LIGGGTHS. DEM is used to obtain the ore parameters in the steady state and the force on the screen decks. Two cases are compared: Case 1 considers the ore as moving together with the vibrating screen, and Case 2 considers the ore force on the screen deck as calculated by DEM. Simulations are carried out with data for an industrial vibrating screen used in copper mining. The force over the screen is significantly different between the cases. Case 1 produces a force that is unrealistic because the ore cannot produce a high-amplitude adhesion force over the screen decks. In Case 2, no adhesion force acts between the ore and deck. It is concluded that the linear dynamic model used in Case 2 is adequate to evaluate the influence of the ore on the movement of the vibrating screen. The linear dynamic model considering the force as in Case 1 can be used to simulate a vibrating screen, as long as a correct calibration parameter is included to obtain an accurate motion amplitude.

Size classification of particulate materials is an important process in mineral processing [1], particularly in the copper industry. Vibrating screens are frequently used to separate granulated ore materials based on particle size for particles with diameters greater than 0.5mm [2]. In the copper industry, the most commonly used vibrating screens have linear motion and horizontal, sloped, or multisloped (banana) screens [3]. A vibrating screen has one or more screening surfaces (decks) with square or rectangular openings, and its vibratory movement depends on a system of unbalanced masses. The appropriate type of vibration allows for better movement of ore and stratification of minerals [4].

Vibrating screens are critical machines prone to successive failures that can result in huge economic losses [5] and must be constantly improved in order to meet the requirements of the mining industry [6]. Proper operation of this machine has important benefits for the whole mineral process [7], and for this reason, many studies have been conducted to investigate the behavior and operational parameters for successful operation. For example, a change in the slope of 1 can cause a decrease of 1-2% in the screening efficiency [1, 8, 9], whereas a change of 1mm in the vibration amplitude can cause a loss of between 5% and 10% of the screening efficiency [1, 9, 10]. These efficiency losses depend on the vibrating screen design, ore characteristics, and operational conditions. Inadequate classification produces undersize particles in an oversize stream (overflow) entering a comminution process, resulting in extra energy expenditure and obstruction of the grinding by packing of fines [11]. Inadequate classification in a closed circuit produces a short circuit of the undersize stream and results in recirculating a constant percentage of fine ore in the circuit. To ensure proper operation and high performance, it is necessary to have a dynamic model that is able to predict movement of the deck to maintain or increase throughput.

The literature presents several models and studies covering different aspects of vibrating screens. Models of screens using a probabilistic approach [12, 13], stratification and passage in the screening process [14], particle movement [1519], screen blocking [20], finite elements [5, 2124], crushing plants [7, 25], and phenomenological models [26] can be found in the literature. Dynamic models of vibrating screens can simulate motion of the vibrating screen structure and show good agreement with experimental measurements [27] and finite element method (FEM) results [22]. The linear model proposed by He and Liu [28] considers three degrees of freedom. The excitation force is circular, and the vibrating screen structure is supported by symmetrical damping springs with equal stiffness. Liu et al. [29] developed a linear model with three degrees of freedom in 3D that has an excitation force in the vertical direction and different damping spring stiffnesses in each support position. Their study focused on vibrating screen fault diagnosis by performing a dynamical simulation on when the supports lose stiffness. Liu et al. [30] proposed a linear model similar to that developed by He and Liu [28] by incorporating a quadruple excitation mechanism into the model. The model proposed by Baragetti and Villa [22] considers motion in a plane with three degrees of freedom. Their model was used to calculate the dynamical response and natural frequencies, and the optimization used FEM and experimental measurements. Using that model, Baragetti [31] patented a new design for a vibrating screen. For optimum screening performance, the angular displacement was set equal to zero and held constant over time, and the load eccentricity was also made null. Thus, the resulting equations have two degrees of freedom. Slepyan and Slepyan [32] and Zahedi and Babitsky [33] analyzed a vibrating screen operating with parametric resonance, using a linear model with damping and a tensile force [32]. They employed a nonlinear dynamic model with a system of autoresonant control [33]. Peng et al. [34] developed a model with a single degree of freedom for a large vibrating screen that considered the bending and random vibration in the design of these machines, and in which the ore was simulated as a random force. Jiang et al. [27] proposed a linear dynamic model of a single-deck equal-thickness vibrating screen. In their results, the simulated amplitudes of the vertical and horizontal motion deviated by less than 5% from the corresponding experimental results. Wang et al. [19] developed a nonlinear dynamic model of a planar reciprocating vibrating screen and employed a matrix method to derive the equation of motion in order to analyze the motion of a particle on the screen. With dynamic simulation software, Jiang et al. proposed a new design of a vibrating screen, where its screen decks are composed by rigid-flexible rods [35].

These linear models [2730] assume that the angular motion of the vibrating screen is low and implement linearization as and , which is useful under nominal operating conditions where angular displacement is not significant. However, in practice, significant angular motion occurs in vibrating screens during startup [31], shutdown, and under off-design operating conditions [3]. The nonlinear models [19, 33] have been developed for particular types of vibrating screens and thus are not always applicable to the vibrating screens normally used in the copper industry, due to its deck material or type of movement.

Rodrguez et al. [3] developed a 2D nonlinear dynamic model of a vibrating screen with three degrees of freedom that allows for significant angular motion and damping in which the nonlinearity is geometric due to angular displacement. They proposed a range of admissible loss of stiffness as a percentage of nominal stiffness in order to ensure proper operation using orbital analysis. The calculated stiffness range was found to be 38% on the feed side and 46% on the discharge side. Moncada and Rodrguez [36] used a nonlinear model [3] to calculate the effects of loss of stiffness in the supporting positions on the steady and transient responses of a vibrating screen. For the steady response, the change in orbital direction was analyzed, while the transient response analyzed the change in the natural frequencies due to loss of stiffness in the supporting positions.

Simulations carried out with full-ore loads [3, 19, 36] assume that the ore has the same movement as the vibrating screen, i.e., the ore is added to the decks, approaching this force to the ore inertial force , where is the mass of the ore and is the acceleration of the center of mass of the ore. This produces an unrealistic attractive force when the ore is in its highest position (ore cannot pull the vibrating screen deck) or in the free-fall phase or when throwing index is equal to zero [15, 17].

Because the force generated by the ore material over the decks is inaccurate owing to the impossibility of traction or a pulling force over the deck, this study aims to calculate the ore force over a vibrating screen deck by means of simulations using the discrete element method (DEM). This method considers the interaction of each particle with each other and with decks and allows forces for different simulation conditions to be obtained. Furthermore, this method is widely used in the literature. In the mining field, several machines and processes have been simulated using DEM, including cone crushers [37], mills [38], hopper discharge [39], jaw crushers [40], feed boxes [41], and vibrating screens [1, 810, 4251]. Cleary et al. [42, 43] performed a DEM analysis of an industrial double-deck banana screen for a range of peak accelerations and two feed size distributions. Dong et al. [44] conducted a numerical analysis of the particle flow on a banana screen and demonstrated the importance of operational parameters like the slope angle of each deck, vibration amplitude, and frequency. Zhao et al. [9] carried out a numerical study of the motion particulates follow along a circularly vibrating screen deck using 3D-DEM. They studied the effects of vibration amplitude, throwing index, and screen deck inclination angle on the screening process. Fernandez et al. [45] used a one-way coupled model of smoothed particle hydrodynamics (SPH) and DEM to simulate large banana screens. DEM was used for the coarse particulate flow, while SPH was employed to model the transport of the fine particle slurry. Delaney et al. [46] used DEM to investigate the flow of a granular material over a horizontal vibrating screen, and performed a quantitative comparison between laboratory scale experiments and the simulation results. Liu et al. [47] simulated the particle flow on a banana screen deck using DEM and investigated the effects of slope angle and deck length on the screening process. Dong et al. [48] simulated the screening of particles for different vibration modes (linear, circular, and elliptical) and studied the resulting motion and penetration of the particles on the screen deck. Li et al. [49] used DEM to optimize the design and operational parameters of a linear vibrating screen. Jahani et al. [10] investigated the screening performance of banana screens using the DEM solver LIGGGHTS. An industrial double-deck banana screen with five panels and two laboratory single-deck banana screens with three and five panels were simulated, and the effects of design parameters such as the slope angle of decks, vibration amplitude, and frequency were analyzed. Their results were validated with partition numbers obtained from the literature [44]. Zhao et al. [50] quantitatively compared DEM results and experimental data for a specially designed circularly vibrating screening model under a range of operating conditions. Jafari and Nezhad [8] studied the effects of different parameters on process efficiency and mesh wear using LIGGGHTS. Dong et al. [51] conducted a numerical analysis of the effects of aperture shape, length, and orientation on particle flow and separation in a vibrating screen process. Zhao et al. [1] analyzed the combined effects of vibration parameters on a circularly screening processes. With a DEM simulation, the effects of various design and operating variables on the efficiency of screen were investigated using open-source LIGGGHTS solver with spheral and irregularly shaped particles [52]. Particle velocity, mass of oversized material, screening efficiency, and impact force were studied to reflect the performance of vibrating screen [53].

From this review of DEM simulations, we can summarize the following: (i) there is significant evidence that vibratory parameters are important for proper screening performance [1, 810, 4244, 47, 52, 53]; (ii) DEM simulations do not consider the effects of ore mass flow on amplitude movement although mass flow affects amplitude [54, 55] and efficiency depends on amplitude [1, 9, 10, 42, 43, 52, 53]; and (iii) DEM simulations do not focus on vibration. Most models use circular motion even when there are experimental data showing that the motion is not circular. In addition, rotation is neglected [19], which significantly affects the movement described for the vibrating screen supports.

The dynamic simulations in this study make two different assumptions or considerations. In Case 1 [3, 36], the ore moves together with the screen deck, which means that physically the ore adheres to the screen and there is no relative displacement between them. For modeling, parameters such as the mass of the vibrating screen with the load and the position of center of mass must be calculated. In Case 2, interaction between the ore and the screen deck is represented by a time vector force for each degree of freedom (). Movement of the ore is simulated with DEM to obtain the force over the screen deck, and this force is then used in the calculation of the dynamic model.

In this study, a 2D linear model with three degrees of freedom that considers ore inertia and the ore force over the screen calculated using discrete element method is developed in order to determine the influence of the ore on the movement of a vibrating screen. For cases 1 and 2, DEM is used to simulate a double-deck vibrating screen. The DEM simulation is carried out in LIGGGHTS and is used to obtain ore parameters and the ore force on the screen over time. Finally, the two dynamical responses are compared.

The dynamic model for a linear motion vibrating screen has three degrees of freedom in the plane , as shown in Figure 1(a) in its equilibrium position. The excitation force of the system is time-dependent on amplitude, , has a constant direction, , with respect to the line linking points A and B, is applied at a distance, , from the center of mass, and has excitation frequency. The structure of the vibrating screen and all its components is defined as a rigid body inclined at an angle, , with a center of mass located at a distance, , perpendicular to segment . Supports are located at points A and B and consist of a horizontal and vertical spring and damper, each at a distance, , from a projection of the center of mass, and with a stiffness, , and damping, . Figure 1(b) presents the degrees of freedom of the modelhorizontal displacement, , vertical displacement, , and angular displacement, .

For the simulation cases in this study, two approaches for modeling the ore are used. In Case 1, the ore is modeled as a rigid body and vibrate together with the screen deck. Therefore, the mass, , inertia, , and position of the center of mass of the vibrating screen with a load are the sum of the empty vibrating screen plus the ore. In the second case, the ore movement is calculated by DEM simulation. The load is represented in the dynamic model by a time vector force with three components for each degree of freedom: , , and , applied at the center of mass of the vibrating screen.

Parameters for the ore particles in the DEM simulation must be known in order to perform the dynamical simulations in Case 1. For that purpose, the mass, , radius, , inertia, , and position of every particle in the particles system are used in equations (1a)(1f) to obtain the load parameters. These load parameters include the total mass, , inertia, , position of the ore center of mass relative to the axes , and position relative to the vibrating screen ( and ). Figure 2 shows a vibrating screen modeled as a rigid body, along with the upper and lower deck, feeder chute, particles, and variables used in these equations. It also illustrates the geometry to allow comparison of the dynamic modeling and DEM, and it should be noted that the variable in the dynamic modeling and in the DEM modeling have different directions. The subscript represents the center of mass, represents the ore load over the screen decks, represents the empty vibrating screen, represents the screen deck, and and represent the lower and upper decks, respectively.

With these load data, the parameters for the vibrating screen with a load can be obtained using equations (2a)(2g). Figure 2 also shows this situation, illustrating the position of vibrating screen center of mass without a load or empty, , with a ore load, , and the new position of the center of mass of the vibrating screen with a load, represented without subscripts. The average values of in steady state are used in the model.

The force calculated with DEM, , is the force of all the particles over the screen decks. Applying Newtons second law to the particles system results in the following:where is the force of the screen decks over the particles, is the mass of a particle, is the gravitational acceleration, and is the acceleration of each particle. Therefore, using Newtons third law, the force of the particles over the screen decks is

As includes the inertial force of the particles, Case 2 also represents the dynamic characteristics of the particles. This force is applied to the vibration screen center of mass and parameters for the vibrating screen in empty conditions are used.

The equations of motion are developed using Lagranges equations. Applying the linearization and , the resulting linear equations of motion are given in equations (5)(7). These three equations correspond to a second-order linear system of equations, and can be solved using the Newmark method with a constant time step. The nonlinear model used for simulations where the angular displacement is significant is presented in Appendix [3]. In these equations, nonlinearity is present in the terms , , and , among others. For the force direction, nonlinearity is present in terms that consider a change in , as can be seen in the term .

The equations of motion are developed in an equilibrium position with an inclination angle, . If the position of the center of mass or amount of mass changes, the vibrating screen will experience a variation in its equilibrium position, the slope . A free-body diagram calculation allows the new slope of to be determined before these changes occur. This is useful when a loss of stiffness exists in the support system [36].

The discrete element method was originally developed by Cundall and Strack in 1979 [56] and it has proven to be an effective numerical technique for calculating particle movement in granular material flows. It is also useful for obtaining the force acting on each particle, which is difficult to obtain on a similar scale with physical experimentation [57]. Since it is a Lagrangian method, Newtons second law is applied for each particle in the simulation domain. This explicitly determines the trajectories and kinematics of each particle at every time step by accounting for the interaction between particles and their environment with contact or field forces.

Using Newtons second law, the equations of motion for the translational and rotational motion of each particle, , in contact with a particle or wall, , are as follows:where is the particle mass, is its velocity, is the gravitational acceleration, is the contact force exerted by particle on , is the particle inertia, is its angular velocity, and is the contact moment.

By assuming that the particles are rigid, spherical, and of uniform material, the contact force, , for particle-particle and particle-wall interactions can be calculated with the HertzMindlin contact model [56]. This force is composed of an elastic and viscous component and is expressed by equations (9a)(9c) in the normal and tangential directions, being and the respective unit vectors. The tangential overlap is truncated to fulfil Coulomb friction criterion in tangential force. Overlap distance is composed of a normal and tangential component, as shown in equation (9d). The choice of a contact model depends primarily on the experimental comparison and type of results expected. A simple model, such as the Hooke linear model, can also produce good agreement in some situation [58].

A constant directional torque (CDT) model is used to calculate the rolling resistance [59, 60]. A constant moment, , is applied to a particle to represent the rolling friction, where and are the angular velocities of the particles in contact, is the rolling friction coefficient, is the effective radius, and is the contact normal force. The torque is always in the opposite direction to the relative rotation between the two contact particles.

Figure 3 shows the schematic of the contact model and rolling resistance model, illustrating two particles, and , with an overlap, , and with contact stiffness, , and damping, . Each particle is characterized by its Youngs modulus, , mass, , and radius, .

Model coefficients for equation (9) are calculated using equations given in the LIGGGHTS manual [61]. These coefficients depend on the material parameters, such as the restitution coefficient, , Youngs modulus, , and Poisson coefficient, .

To numerically solve the motion equations for each particle, the simulation is divided into time steps, , and the equations are solved to obtain a solution at the end of each time step. The velocity Verlet algorithm is used for these calculations, which can be derived by an approximation of the Taylor series [62].

DEM simulations are implemented in LIGGGHTS [63], an open source software based on LAMMPS used to perform massive granular simulation in parallel using MPI [64]. This software makes it possible to import CAD files for the wall geometries; assign different movements to these geometries, such as a 3D vibration with a different amplitude, frequency, and phase; use nonspherical particles [65]; perform stress and wear analysis [8]; and develop smoothed particle hydrodynamics (SHP) models [66] and CFD-DEM coupling simulations [67]. For these reasons, it has been used in the literature for vibrating screen DEM simulations [8, 10]. LIGGGHTS 3.6.0 was used to perform the simulations on a desktop computer.

The simulations in this study make use of data obtained from a vibrating screen in use at a copper mine [3]. This screen is a double-deck linear motion vibrating screen with a size of 3.66 7.32m and a 10.5mm nominal stroke. The CAD model of this vibrating screen is shown in Figure 4(a), where the upper and lower screen decks, feed chute, and lateral and rear walls that serve as boundaries of the particles are detailed. The lateral wall on the left side is not presented in this figure, but it is included in the simulation. Only the features necessary for DEM are geometrically modeled, as the details of the structure of the machine are not relevant; the screen decks are correctly modeled.

The screen deck is composed of 27 square modules in the longitudinal direction, with a side length of each square module of 305mm. Each deck has rectangular openings, slotted with the flow direction, of 60 20mm (Figure 4(b)) and 47 11mm (Figure 4(c)) for the upper and lower decks, respectively. The opening area represents 30% and 36 of the upper and lower decks, respectively.

To reduce the computational cost of the DEM simulation, the simulated vibrating screen is a 1/12th scale model in the cross direction [44] of the real machine, using the symmetry in the -plane corresponding to a row of modules, as shown in Figure 4. This simplification implies that the intensive variables, such as force, mass, and inertia, must be multiplied by 12. In spite of the influence of the particle shape on stratification and passing [46], spherical particles are adopted because of their lower computational cost. The particle size distribution corresponds to nominal data provided by the manufacturer, and a diameter threshold of 6mm is used in order to reduce the simulation time [68]. Youngs modulus is for the particles, and for the boundary walls. The common simplification to reduce [10] is not applied because it influences the contact force [69], and thus, the force between the particles and decks. With the period of vibration, , and steps per cycle, a time step is , or approximately . That time step is acceptable for this simulation [63] because it is smaller than the Rayleigh time step. Details of the geometrical conditions and material parameters are listed in Table 1 and are based on nominal data and previous studies [17, 37, 46]. These parameters correspond to the simplified vibrating screen, and thus, the mass flow and width are 1/12th the real value. The subscript refers to the particle.

To promote stratification and provide movement to the ore, a vibrating screen vibrates with a particular frequency, direction, and amplitude. In the DEM simulations, motion is imposed on both decks, with vibration in the x-axis and y-axis, as well as angularly an amount with respect to its center of mass. The vibration direction is ensured by the phase difference between the vibrations in x and y. The vibrational parameters are listed in Table 2. These parameters are obtained from dynamical simulations, and thus the frequency, , of the deck vibration movement is the same as that in the dynamic simulations. To the comparison of the cases be valid, both cases consider the same value of .

Table 3 lists the parameters used in the dynamical simulations for Cases 1 and 2 that correspond to the nominal data without load and geometrical conditions. The excitation frequency is calculated by .

Figure 5 presents snapshots of the DEM simulation of the vibrating screen in the steady state. The ore entering through the chute, distribution on the decks, stratification, and passing can be observed. In this figure, the red spheres have a larger diameter and the blue spheres have smaller diameters. The steady state is defined as when the ore mass on the vibrating screen remains approximately constant with time.

The total force and moment over the screen exerted by the ore are presented in Figure 6 for the directions. The vertical component, , has a greater amplitude than the horizontal component , and its peak amplitude is 7 times greater. Both amplitudes match when they have null values, as that corresponds to when the ore is detached from the screen. The value of exhibits an amplitude change every cycle, i.e., between peaks (1) and (2), that is related to the ore movement. On average, the first is equal to 122.4kN and the second is equal to 111.5kN. The moment, , has the same tendency as components x and y, and its maximum amplitude occurs when ore comes into contact with the screen deck.

The overall screening performance of both decks can be investigated with the partition curve of the overflow, which is defined as the ratio of the number of residue particles in the overflow to that of fed particles [44], and is shown in Figure 7. In terms of mass flow, the partition number is equal tofor each particle size, where refers to overflow stream and to feed. These results agree with the nominal screening efficiency of the vibrating screen, which is approximately 90%.

Figure 8 compares the forces in Case 1 and Case 2 with a frequency spectrum and orbit. Figure 8(a) shows the frequency spectrum of the signal in the vertical direction, y. For Case 1, only the 1X component is observed. For Case 2, a constant component at 0Hz is observed, which represents the mean time value, along with harmonics from 1X4X. The logarithmic spectrum shown in Figure 8(c) exhibits, in addition to these harmonics, nonsynchronous components at 0.5X and 1.5X, which are highlighted in the grass spectrum.

Figure 8(b) shows both forces in the -plane, allowing for a graphical understanding of the direction of each force. In Case 1, the orbit is elliptical with a nearly negligible semiminor axis, and thus, it approximately corresponds to a line. Case 2 has an ore force that is always negative in the direction of y because the ore cannot produce an adhesion force that lifts up the screen decks.

Arrows indicate the variation in each force over time. In Case 2, the orbit is a cycle, from O to O in clockwise direction, whereas the orbit of Case 1 moves between A and B. While the Case 1 orbit moves from B to A, the Case 2 orbit stays close to zero.

In analyzing these results, it can be concluded that Case 1 does not accurately physically represent the effect of the ore on the vibrating screen, because this force must be vertical and repulsive, as in the DEM simulation.

Movement in the two cases is compared by means of orbit analysis of the movement of support A. Figure 9 presents the frequency spectrum and displacement orbit for support A. Figure 9(a) shows the frequency spectrum of vertical movement and indicates that Case 1 only exhibits a 1X component, which differs by 12% from that of Case 2. Case 2 exhibits a low amplitude harmonic at 2X. For Case 2, owing to the transient condition of , a resonant zone appears near 1.875Hz. This corresponds closely to the values of the natural frequencies, and , which are 1.982Hz and 2.067Hz, respectively.

Figure 9(b) shows the orbits. In Case 1, an elliptical orbit is clearly observed, while in Case 2, the transient condition of the excitation force, , results in an elliptical trajectory that changes position in the vertical direction by an amount = 1.2mm. A difference in stroke is also observed, equal to = 9.230mm in Case 1 and = 10.592mm in case 2 on average. These values agree with commonly found in experimental measurements and in the literature [3, 10]. Furthermore, Figure 9(b) presents the nominal stroke length equal to = 10.5mm. Comparing nominal stroke with simulated stroke, case 2 provides a better agreement.

Angular displacement of the vibrating screen is obtained, and it is presented in Figure 10. Both cases present a low peak amplitude, less than 0.025. Figure 10(a) presents frequency spectrum showing that both have a 1X component that differs on 0.003. Case 2 presents a low amplitude 2X component and, as well as vertical movement, a resonant zone. Figure 10(b) shows waveform of angular displacement. It should be noted that Case 2 has nonzero mean value equal to . This value depends of , which is calculated based on and and the position of the center of mass of the ore with respect to the center of mass of vibrating screen.

Conclusions obtained in this study can be summarized as follows:(i)The proposed dynamic model allows for prediction of the behavior of the vibrating screen operating at both the nominal condition and high angular displacement with a nonlinear model [3]. This model simulated ore movement along with the screen deck (Case 1), as well as the ore force over the screen decks calculated with DEM (Case 2). In both cases, the movement of the vibrating screen supports was also obtained.(ii)DEM simulation of a double-deck vibrating screen was carried out using the open-source LIGGGHTS software. Movement of the ore center of mass and the force exerted by the ore over the screen deck were obtained. The force has the same frequency as the excitation force; high oscillations in the ore do not produce significant changes in the force exerted over the screen.(iii)The partition curve and stroke of the vibrating screen motion have very good agreement with the nominal data, validating the model results.(iv)In a comparison of the results for the proposed cases, where the ore is represented as moving together with the vibrating screen (Case 1), or the ore force is obtained from DEM (Case 2), it is observed that:(a)The force over the screen deck is completely different in both cases, both in terms of the magnitude (the peak-to-peak amplitude in Case 1 is more than twice that of Case 2) and the shape on the -plane. Case 1 produces an unrealistic force, because it includes a contact force of adhesion between the ore and screen deck. However, this is the hypothesis used in most of the existing dynamic models available in the literature. The DEM model allows calculation of a force closer to reality, because it calculates the interaction of each individual particle with the screen deck.(b)Case 1 available in the literature was successfully implemented. The computation of Case 1 is faster than Case 2, because only a linear ordinary differential of three-degrees-of-freedom equation is solved, in contrast to DEM simulation that simulates the movement of 42000 particles. This is the reason why Case 1 is commonly used in the mining industry.(c)Case 2 is a new simulation approach that allows the coupling of simulation results in DEM in dynamic models. This model is able to evaluate how the ore affects the movement of the vibrating screen.(d)Notwithstanding the clear inequality in the force calculated for cases 1 and 2, the approach used in Case 1 can be used to predict the movement of vibrating screen if a correct calibration parameter is included.(e)Comparing with the nominal data and the results of Case 2, frequency, direction, and inclination calculated with Case 1 are accurate. The amplitude obtained using Case 1 is not accurate and must be corrected. In this case, the parameter of mass of ore must be adjusted with experimental data, decreasing its value so that the amplitude increases. To decrease the value of is physically correct, since the model of Case 1 considers that all the ore is in contact with the screen decks, while there is ore in free fall that is not in contact.(v)Under nominal operating conditions, the angular response, , has low amplitude (0.014), whereas the steady responses obtained for the linear model, equations (5)(7), and the nonlinear model, equations (12)(14), have negligible differences. Consequently, for simplification and lower computational cost, the linear model can be used for the steady case or without deterioration in the supports. For transient signals or when the amplitude of the angular response is high, the linear model is not recommended.(vi)The proposed dynamic model allows for greater accuracy and validity in different operating conditions, which is useful for predicting the angle of operation and the vibratory amplitude, parameters that affect the screening efficiency.

The DEM raw data used to support the findings of this study have been deposited in the Raw data of DEM simulation of linear motion double deck vibrating screen repository (DOI: 10.17632/cc798dvdbn.2).

Copyright 2018 Manuel Moncada M. and Cristian G. Rodrguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

common & basic formulas for mineral processing calculations

The control of a milling operation is a problem in imponderables: from the moment that the ore drops into the mill scoop the process becomes continuous, and continuity ceases only when the products finally come to rest at the concentrate bins and on the tailing dams. Material in process often cannot be weighed without a disturbance of continuity; consequently, mill control must depend upon the sampling of material in flux. From these samples the essential information is derived by means of analyses for metal content, particle size distribution, and content of water or other ingredient in the ore pulp.

The following formulas were developed during a long association not only with design and construction, but also with the operation of ore dressing plants. These formulas are herein the hope that they would prove of value to others in the ore dressing industry.

Pulp densities indicate by means of a tabulation the percentages of solids (or liquid-to-solid ratio) in a sample of pulp. This figure is valuable in two waysdirectly, because for each unit process and operation in milling the optimum pulp density must be established and maintained, and indirectly, because certain important tonnage calculations are based on pulp density.

As used in these formulas the specific gravity of the ore is obtained simply by weighing a liter of mill pulp, then drying and weighing the ore. With these two weights formula (2) may be used to obtain K, and then formula (1) to convert to S, the specific gravity. A volumetric flask of one liter capacity provides the necessary accuracy. In laboratory work the ore should be ground wet to make a suitable pulp. This method does not give the true specific gravity of the ore, but an apparent specific gravity which is more suitable for the intended purposes.

A mechanical classifier often receives its feed from a ball mill and produces (1) finished material which overflows to the next operation and (2) sand which returns to the mill for further size-reduction. The term circulating load is defined as the tonnage of sand that returns to the ball mill, and the circulating load ratio is the ratio of circulating load to the tonnage of original feed to the ball mill. Since the feedto the classifier, the overflow of the classifier, and the sand usually are associated with different proportions of water to solid, the calculation of circulating load ratio can be based on a pulp density formula.

Example: A mill in closed circuit with a classifier receives 300 dry tons of crude ore per day, and the percentages of solid are respectively 25, 50, and 84% in the classifier overflow, feed to classifier, and sand, equivalent to L: S ratios of 3.0, 1.0, and 0.190. Then the circulating load ratio equals

A more accurate basis for calculation of tonnage in a grinding circuit is the screen analysis. Samples of the mill discharge, return sand, and the classifier overflow are screen sized, and the cumulative percentages are calculated on several meshes. Let:

The efficiency of a classifier, also determined by means of screen analyses, has been defined as the ratio, expressed as percentage, of the weight of classified material in the overflow to the weight of classifiable material in the feed. Overflow having the same sizing test as the feed is not considered classified material. Let:

When no other method is available an approximation of the tonnage in a pulp stream or in a batch of pulp can be quickly obtained by one of these methods. In the dilution method water is added to astream of pulp at a known rate, or to a batch of pulp in known quantity, and the specific gravity of the pulp ascertained before and after dilution.

In both cases Dx and D2 are dilutions (tons of water per ton of ore) before and after addition of water. These are found from the specific gravities of the pulp, by formulas (4) and (6) or directly by the use of the tabulation on these of Pulp Density Tables.

The Pulp Density Tables were compiled to eliminate the many complicated calculations which were required when using other pulp density tables. The total tank volume required for each twenty-four hour period of treatment is obtained in one computation. The table gives a figure, in cubic feet, which includes the volume of a ton of solids plus the necessary volume of water to make a pulp of the particular specific gravity desired. Multiply this figure by the number of dry tons of feed per twenty-four hours. Then simply adjust this figure to the required treatment time, such as 16, 30, 36, 72 hours.

In the chemical method a strong solution of known concentration of common salt, zinc sulphate, or other easily measured chemical is added to the flowing pulp at a known rate, or to a batch of pulp in known quantity. The degree of dilution of this standard solution by pulp water is ascertained by chemical analysis of solution from a filtered sample, and the tonnage of ore is then calculated from the percentage solid. This method is impractical for most purposes, but occasionally an exceptional circumstance makes its employment advantageous. It has also been suggested as a rapid and accurate method of determining concentrate moistures, but in this application the expense is prohibitive, since ordinary chemicals of reasonable cost are found to react quickly with the concentrate itself.

With the above chart the per cent solids or specific gravity of a pulp can be determined for ores where gravities do not coincide with those in the Pulp Density Tables.This chart can also be used for determining the specific gravity of solids, specific gravity of pulps, orthe per cent solids in pulp if any two of the three are known.

These are used to compute the production of concentrate in a mill or in a particular circuit. The formulas are based on assays of samples, and the results of the calculations are generally accurate as accurate as the sampling, assaying, and crude ore (or other) tonnage on which they depend.

The simplest case is that in which two products only, viz., concentrate and tailing, are made from a given feed. If F, C, and T are tonnages of feed r on-centrate, and tailing respectively; f, c, and t are the assays of the important metal; K, the ratio of concentration (tons of feed to make one ton of concentrate); and R, the recovery of the assayed metal; then

When a feed containing, say, metal 1 and metal z, is divided into three products, e.g., a concentrate rich in metal 1, another concentrate rich in metal z, and a tailing reasonably low in both l and z, several formulas in terms of assays of these two metals and tonnage of feed can be used to obtain the ratio of concentration, the weights of the three products, and the recoveries of 1 and z in their concentrates. For simplification in the following notation, we shall consider a lead-zinc ore from whicha lead concentrate and a zinc concentrate are produced:

The advantages of using the three-product formulas (20-25) instead of the two-product formulas (14-19), are four-fold(a) simplicity, (b) fewer samples involved, (c) intermediate tailing does not have to be kept free of circulating material, (d) greater accuracy if application is fully understood.

In further regard to (d) the three-product formulas have certain limitations. Of the three products involved, two must be concentrates of different metals. Consider the following examples (same as foregoing, with silver assays added):

In this example the formula will give reliable results when lead and zinc assays or silver and zinc assays, but not if silver and lead assays, are used, the reason being that there is no concentration of lead or silver in the second concentrate. Nor is the formula dependable in a milling operation, for example, which yields only a table lead concentratecontaining silver, lead, and zinc, and a flotation concentrate only slightly different in grade, for in this case there is no metal which has been rejected in one product and concentrated in a second. This is not to suggest that the formulas will not give reliable results in such cases, but that the results are not dependablein certain cases one or more tonnages may come out with negative sign, or a recovery may exceed 100%.

To estimate the number of cells required for a flotation operation in which: WTons of solids per 24 hours. RRatio by weight: solution/solids. LSpecific gravity, solution. SSpecific gravity, solids. NNumber of cells required. TContact time in minutes. CVolume of each cell in cu. ft.

Original feed may be applied at the ball mill or the classifier. TTons of original feed. XCirculation factor. A% of minus designated size in feed. B% of minus designated size in overflow. C% of minus designated size in sands. Circulating load = XT. Where X = B-A/A-C Classifier efficiency: 100 x B (A-C)/A (B-C)

Original feed may be applied at theball mill or the primary classifier. TTons of original feed. XPrimary circulation factor. YSecondary circulation factor. A% of minus designated size in feed. B% of minus designated size in primary overflow. C% of minus designated size in primary sands. D% of minus designated size in secondary overflow. E% of minus designated size in secondary sands. Primary Circulating Load = XT. Where X = (B-A) (D-E)/(A-C) (B-E) Primary Classifier Efficiency: 100 xB (A C)/A (B C) Secondary Circulating Load = YT. Where Y = (D-B)/(B-E) Secondary Classifier Efficiency: 100 xD (B-E)/B (D E) Total Circulating Load (X + Y) T.

Lbs. per ton = ml per min x sp gr liquid x % strength/31.7 x tons per 24 hrs.(26) Solid reagents: Lbs. per ton = g per min/31.7 x tons per 24 hrs.(27) Example: 400 ton daily rate, 200 ml per min of 5% xanthate solution Lbs. per ton = 200 x 1 x 5/31.7 x 400 = .079

Generally speaking, the purpose of ore concentration is to increase the value of an ore by recovering most of its valuable contents in one or more concentrated products. The simplest case may be represented by a low grade copper ore which in its natural state could not be economically shipped or smelted. The treatment of such an ore by flotation or some other process of concentration has this purpose: to concentrate the copper into as small a bulk as possible without losing too much of the copper in doing so. Thus there are two important factors. (1) the degree of concentration and (2) the recovery ofcopper.

The question arises: Which of these results is the most desirable, disregarding for the moment the difference in cost of obtaining them? With only the information given above the problem is indeterminate. A number of factors must first be taken into consideration, a few of them being the facilities and cost of transportation and smelting, the price of copper, the grade of the crude ore, and the nature of the contract between seller and buyer of the concentrate.

The problem of comparing test data is further complicated when the ore in question contains more than one valuable metal, and further still when a separation is also made (production of two or more concentrates entirely different in nature). An example of the last is a lead-copper-zinc ore containing also gold and silver, from which are to be produced. (1) a lead concentrate, (2) a copper concentrate, and (3) a zinc concentrate. It can be readily appreciated that an accurate comparison of several tests on an ore of this nature would involve a large number of factors, and thatmathematical formulas to solve such problems would be unwieldy and useless if they included all of these factors.

The value of the products actually made in the laboratory test or in the mill is calculated simply by liquidating the concentrates according to the smelter schedules which apply, using current metal prices, deduction, freight expense, etc., and reducing these figures to value per ton of crude ore by means of the ratios of concentration.

The value of the ore by perfect concentration iscalculated by setting up perfect concentrates, liquidating these according to the same smelter schedulesand with the same metal prices, and reducing theresults to the value per ton of crude ore. A simple example follows:

The value per ton of crude ore is then $10 for lead concentrate and $8.50 for zinc, or a total of $18.50 per ton of crude ore. By perfect concentration, assuming the lead to be as galena and the zinc as sphalerite:

The perfect grade of concentrate is one which contains 100% desired mineral. By referring to the tables Minerals and Their Characteristics (pages 332-339) it is seen that the perfect grade of a copper concentrate will be 63.3% when the copper is in the form of bornite, 79.8% when in the mineral chalcocite, and 34.6% when in the mineral chalcopyrite.

A common association is that of chalcopyrite and galena. In concentrating an ore containing these minerals it is usually desirable to recover the lead and the copper in one concentrate, the perfect grade of which would be 100% galena plus chalcopyrite. If L is the lead assay of the crude ore, and C the copper assay, it is easily shown that the ratio of concentration of perfect concentration is:

% Pb in perfect concentrate = K perfect x L.(30) % Cu in perfect concentrate = K perfect x C..(31) or, directly by the following formula: % Pb in perfect concentrate = 86.58R/R + 2.5.(32) where R represents the ratio:% Pb in crude ore/% Cu in crude ore Formula (32) is very convenient for milling calculations on ores of this type.

by (29) K perfect = 100/5.775+2.887 = 11.545 and % Pb in perfect concentrate = 11.545 x 5 = 57.7% and % Cu in perfect concentrate = 11.545 x 1 = 11.54% or, directly by (32), % Pb = 86.58 x 5/5 + 2.5 = 57.7%

Occasionally the calculation of the grade of perfect concentrate is unnecessary because the smelter may prefer a certain maximum grade. For example, a perfect copper concentrate for an ore containing copper only as chalcocite would run 79.8% copper, but if the smelter is best equipped to handle a 36% copper concentrate, then for milling purposes 36% copper may be considered the perfect grade.

Similarly, in a zinc ore containing marmatite, in which it is known that the maximum possible grade of zinc concentrate is 54% zinc, there would be no point in calculating economic recovery on the basis of a 67% zinc concentrate (pure sphalerite). For example, the following assays of two zinc concentrates show the first to be predominantly sphalerite, the second marmatite:

The sulphur assays show that in the first case all of the iron is present as pyrite, and consequently the zinc mineral is an exceptionally pure sphalerite. This concentrate is therefore very low grade, from the milling point of view, running only 77.6% of perfect grade.On the other hand, the low sulphur assay of concentrate B shows this to be a marmatite, for 10% iron occurs in the form of FeS and only 2.5% iron as pyrite. The zinc mineral in this case contains 55.8% zinc, 10.7% iron, and 33.5% sulphur, and clearly is an intermediate marmatite. From the milling point of view cencentrate B is high grade, running 93% of perfect grade, equivalent to a 62% zinc concentrate on a pure sphalerite.

grinding cylpebs

Our automatic production line for the grinding cylpebs is the unique. With stable quality, high production efficiency, high hardness, wear-resistant, the volumetric hardness of the grinding cylpebs is between 60-63HRC,the breakage is less than 0.5%. The organization of the grinding cylpebs is compact, the hardness is constant from the inner to the surface. Now has extensively used in the cement industry, the wear rate is about 30g-60g per Ton cement.

Grinding Cylpebs are made from low-alloy chilled cast iron. The molten metal leaves the furnace at approximately 1500 C and is transferred to a continuous casting machine where the selected size Cylpebs are created; by changing the moulds the full range of cylindrical media can be manufactured via one simple process. The Cylpebs are demoulded while still red hot and placed in a cooling section for several hours to relieve internal stress. Solidification takes place in seconds and is formed from the external surface inward to the centre of the media. It has been claimed that this manufacturing process contributes to the cost effectiveness of the media, by being more efficient and requiring less energy than the conventional forging method.

Because of their cylindrical geometry, Cylpebs have greater surface area and higher bulk density compared with balls of similar mass and size. Cylpebs of equal diameter and length have 14.5% greater surface area than balls of the same mass, and 9% higher bulk density than steel balls, or 12% higher than cast balls. As a result, for a given charge volume, about 25% more grinding media surface area is available for size reduction when charged with Cylpebs, but the mill would also draw more power.