Simulation and control of ball mills under uncertainty conditions

The article considers a model for monitoring the grinding process in a ball mill, where the ground material itself is involved in grinding. The constructed model takes into account the uncertainties associated with the quality indicators of the processed ore at the beginning and in the process of grinding, the influence of the quality indicators of the ore in terms of particle size and their hardness on the kinetics of grinding. The non-linearity of the technological process and uncertainties are taken into account in the control system aimed at stabilizing the system. The model is presented in the form of four blocks describing the main links of the mill plant. At the output of the model, the participation of the mill, the volume of the sump are determined, and the particle size at the outlet of the hydrocyclone is estimated. The operation of the hydrocyclone control system is demonstrated, which controls the loading density and particle size depending on the change in ore quality.


Introduction
The extraction of rare and precious metals from ore is associated with its grinding.Ball, rod, semiautogenous and autogenous mills are used for grinding ore [1].In the latter type of mills the ore itself serves as the grinding media, while in semi-autogenous and ball mills special balls are used as grinding media along with the ore.
Technological schemes of grinding have different configurations.The technological process of grinding can be displayed in the form of mathematical models of various types [2].In order to control the technological process of grinding, block predictive models [3] [4] are often used, made in the form of modules, for example, for the mill itself, hydrocyclone and sump [5].The block principle of creating a model allows you to simulate the grinding process according to various schemes.In this case, technology nonlinearities and uncertainties are automatically taken into account by the control system.The control system of the grinding process pursues several objectives: firstly, stabilisation of the system, secondly, optimisation of the technological process [6].The economic goal consists of several components, each of which contributes to the achievement of a single goal: • improving product quality, which consists of stabilizing the fineness of grinding, and minimizing fluctuations in the size of the product; • performance maximization; • minimizing the number of steel balls used per tonne of produced product; • minimizing energy costs per ton of manufactured products, etc.All these goals are interrelated and require a certain compromise solution since, for example, productivity cannot be increased to certain values without compromising product quality.

Materials and methods
Let us take a closer look at the grinding process in a ball mill, where crushed material itself participates in grinding.The change in the state of the ore before and during grinding is a significant uncertainty factor in modelling and controlling the technological process of grinding.The composition of the ore in terms of particle size and their hardness and distribution in the crushed medium affects the kinetics of grinding, which also contributes to the treasury of uncertainty.Reducing the influence of these uncertainties, feedback schemes are used to control this process [7].For example, if the output particle size is taken as the determining parameter of the grinding technology, and it is not possible to conduct continuous automatic control over the particle size in the apparatus and the hardness of the grinding medium in the apparatus, this may lead to the impossibility of controlling the influence of disturbing influences and conforming to the uncertainty of the dynamic process [8].
The problem of controlling the grinding process is to take into account the increased correlation between factors, large time delays, uncontrolled perturbations, changes in parameters in time, nonlinearity of the technological process, as well as the suboptimal design of the grinding plant [9][10][11].This is achieved by introducing into the predictive model the description of uncertainties and cases of deviation from the technological process.The control system calculates the optimal sequence of control or develop certain feedback laws that optimize the given objective function.

Results
Let us build a dynamic mathematical model of a ball mill, shown in Figure 1.The mill installation consists of the mill itself, a sump and a hydrocyclone.The mill receives ore together with grinding balls, water and waste return flow, from the bottom of the hydrocyclone.
The crushed ore in the mill is mixed with irrigation water to form a slurry, which is discharged through an end-discharge screen.After being withdrawn from the mill, the slurry is sent to the sump.Before being pumped into a cyclone for classification, it is diluted with water.The hydrocyclone is designed to separate substandard ore unloaded from the sump.The bottom product is returned to the mill for further grinding of out-of-spec particles.The list of flows entering and exiting the technological process is presented in the table 1.To describe the grinding process as shown in fig. 1.One can use a relatively simple nonlinear dynamic model that uses a minimum number of refined parameters, but allowing one to obtain sufficiently accurate corresponding responses of the model.The model assumes the use of six forms of materials in the grinding circuit: stones, solids, coarse particles, fine particles, balls, water.
The model consists of four blocks describing the constituent parts of the mill plant.A description of the indices of measured costs V and those calculated using the model X is given in Table 2.

Lower index Description
The left index corresponds to the link in question and the second index shows the shape (stones, solids, coarse particles, fine particles, balls, water).The index spending shows what it is: an inflow, an overflow, or a decrease in flow.Below in the equations that represent the contour of a mill with continuous time in state space.where Xmw, Xmsp, Xmf, Xmst, Xmb -volume of mill water, solids, fines, stones and balls in the mill respectively, Xsw, Xssp, Xsf -respectively volumes of water, solids and fines in the sump, and Vfwg , Vfspg , Vffg these are respectively: irrigation water flow, particles that do not meet the required size and the finished product in the hydrocyclone.Since the solids include finely ground and coarse ore, a calculation of the change in solids and fines is required.The values of the model parameters are presented in Table 3.The model has three outputs: the pulveriser fraction, the ZUMFP volume, and an estimate of the dispersibility of the fraction at the outlet of the FPSO hydrocyclone, which are determined by the relations: where Vrmchg and Vrtchg -are the volumetric velocity of fine and solid particles at the outlet of the hydrocyclone.The equations are required for (1) determined by the equations: where φ is empirically determined rheological coefficient, Pmill is the power input of the mill, Zx -the effect of charge affecting power consumption, and Zr -the effect of charge rheology affecting power consumption.The intermediate equations are required for (1) and (2), derived for the cyclone, are determined by the equations: where: Vfwg is the water flow into the hydrocyclone; Vfspg-consumption of solid particles; Vfcpg flow rate of coarse particles and fine fraction at the bottom of the hydrocyclone; Vfsps and Vffs are respectively, the flow rates of solid and fine particles at the outlet of the sump; Flf-the proportion of solid particles in the lower flow of the hydrocyclone.
To check the model's operability, the parameter values indicated in Tables 3 and 5 had taken from.The working points of the grinding circuit are borrowed from the technological regulations of the GMZ-2 of the Navoi Mining and Metallurgical Combine.

Discussion
The results of many years of observations show that the qualitative changes occurring in the ore and changes in its physical properties are the main sources of uncertainties encountered in modelling, especially when the ore comes from different deposits.By increasing the capacity required to produce one ton of product (   ) by 50 percent in 10 minutes.It is possible to simulate change in the physical properties of the ore, and change of its composition is modeled by increasing the share of raw materials (   ) by 50% over a time of 100 minutes.The particle size of the crushed ore is controlled by changing the cyclone section.The hydrocyclone cross-section is changed by varying the flow rate of the circulating mixture (Fig. 2b) and the hydrocyclone loading density.The loading density of hydrocyclones (Fig. 2b) can be controlled by changing the mill unloading density, which is determined by the ratio of the loading density of hydrocyclones to the ore supplied for grinding (Fig. 2b), etc.
To raise the cyclone efficiency, the control system has to control the hydrocyclone loading density, crushed ore particle size and sump level, for which it is provided with three degrees of freedom (Figure 2a).The required product particle size and mill load are provided by the predictive control of a robust nonlinear model (PRNM) (Fig. 2a) regardless of the perturbing influences.In Figure 2b we can see a slight decrease in the average ore feed to grinding rate associated with the physical change in ore properties.Differences in the hardness of the feed ore have an impact on the process time in the mill.Large variations in hydrocyclone feed density are required to control the sump level to minimise the density of the slurry in the sump.Figure 2a illustrates the temporal fluctuations in throughput determined by the amount of product being moved downstream for further processing, the power consumption of the mill drive and the grinding rheology within the mill.
Lines drawn horizontally by a dotted line indicate constraints on the variables, and lines drawn vertically by a dotted line shows the moments of influence of perturbations.
The throughput of the mill is calculated taking into account the amount of ore that is fed for grinding.
In the case of a steady state, the productivity of the mill should be equal to its throughput.If the hardness of the feedstock increases, while maintaining the grinding particle size at the set value, the control system will reduce productivity, if the goal of stabilizing the productivity is pursued, the control system will keep it close to the operating point, and the grinding particle size will decrease.
Equation ( 5) determines the functional dependence of the rheological factor on the flow rate of water and solid particles inside the mill.
The conducted study allows us to conclude that the maximum productivity value can be achieved under optimal conditions for the course of the destruction process, achieved at an optimal value of the rheology factor equal to 0.51.To do this, the control system must regulate the inlet water flow to the mill and its throughput.The change in the index of physical properties of the feedstock is compensated by increasing the water flow rate relative to the solids flow rate fed into the mill.To reduce the density of the cyclone, the flow of water into the sump is increased.This ensures the stability of the particle size in the ground product.However, an increase in the rheological factor leads to a drop in the power of the mill.To prevent this event and maintain the necessary conditions for the fracture process, the control system increases the rate of steel feeding into the mill (Fig. 2b).

Conclusion
The results show that if in the case of particle size increases at the mill outlet, then in this case the mill capacity decreases, this leads to unstable grinding process.In order to increase the stability of the process, the control system will have to maintain the set operating point, which will lead to the reduction of the particle size and to maintain the productivity.From the equations, the functional dependence of the rheological coefficient on the amount of water and solids in the mill is determined.In this case, if the values will reach the optimum values, the grinding process will be equal to the optimum values of the rheology factor.To ensure particle size stability, the density of the cyclone is reduced.However, the increase of rheology factor leads to the decrease of mill power.To prevent the mill performance from decreasing, it is necessary to speed up the ball feeding into the mill, then the process will work in a balanced way.

Figure 1 .
Figure 1.Ball mill plant for grinding ore.

Figure 2 .
Figure 2. Temporary fluctuations of the main indicators of the mill: a) controlled variables, b) manipulated variables

Table 1 .
List of input and output streams

Table 4 .
Operating parameters of the grinding process

Table 5 .
Initial states of the shredder and zupf