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**CHAPTER 2 MILLING THEORY AND MODELLING**

**INTRODUCTION**

Controlling a process usually requires a mathematical model of the process for design and simulation purposes. Model-based control further relies on an internal model of the process to calculate the control moves. A mathematical model of the ROM ore milling circuit is therefore essential in designing and simulating control strategies. There are two approaches to modelling grinding circuits, firstly a process design and optimisation approach and sec-ondly a control and estimation approach.

Process engineers want as much information as possible about the operation of the mill and grinding circuit as a whole. Simulation models provide insights into the mechanisms of breakage and material flow inside the mill (Hinde, 2007). The simulation models consist of a large number of states and parameters in order to model the size distributions and breakage distributions.

Control engineers have only limited measurements available from the mill and circuit (Apelt et al., 2002), that limit the number of states and parameters that can be estimated. Control engineers therefore prefer simple models with only a small number of states and parameters, while still capturing the essential dynamics for control purposes.

**THEORY OF MILLING**

**Introduction**

Milling or grinding reduces the size of ore by allowing the ore to tumble freely in a rotating or gyrating container, which causes a breaking action to be applied to the ore. The product size is determined by the size of the feed to the mill and by relative probabilities of breakage of the individual size fractions inside the mill (Stanley, 1987).

In gold ores, the valuable gold usually constitutes a very small part of the volume of the ore. The rest of the ore is mostly worthless. In order to extract the gold from the ore, the ore has to be ground down to a fine powder. This action or process of reducing the size of ore to minute particles or fractions is called comminution (Stanley, 1987).

**Comminuting ore therefore**

makes it more usable because of the reduced particle size, and liberates the different components in the ore from each other, which aids in the subse-quent separation of the components by down-stream processes.

The separation of the valuable material from the gangue, or worthless material, is achieved through chemical or physical mineral recovery processes. It is reasonable to assume that the closer the maximum discharge particle size from the milling process is to the grain size of the valuable material, the more efficient the downstream recovery processes will be, because the valuable material will be better liberated from the ore (Stanley, 1987).

The desired fineness of the particle size obtained from the milling process is also determined by the economics of the process. The gain in recovery is weighed against the cost of finer comminution (Stanley, 1987).

**Mills**

Tumbling mills are usually cylindrically shaped machines made of steel (Figure 2.1). The cylindrical container is rotated about its horizontal axis by some form of drive, usually elec-trical. Both ends of the cylinder are closed off with either cast iron or steel, with the centres of the ends extruding to from hollow trunnions that allow ore to enter the mill and pulp to exit. The trunnions are supported by trunnion bearings that allow the mill to rotate, while the bearings are mounted on some form of foundation (Stanley, 1987).

The ore enters the mill through the inlet trunnion by some form of feeder, which is a device that can continuously feed ore into the inlet trunnion from external sources. It can be a static device that allows the ore to flow by means of gravity (such as sprout and hopper feeders) or it may be attached to the mill that allows it to rotate and scoop the feed ore into the inlet trunnion (such as scoop and drum feeders) (Stanley, 1987).

The inside surfaces of the mill are protected against wear by mill liners. The mill liners are important for two reasons. Firstly, the mill liners are consumables that influence the opera-tional cost of the mill, because they wear away over time. The operational costs associated with mill liners can be minimised by maximising the time before the liners need to be re-placed, as well as minimising the cost of relining. Secondly, it is partly responsible for the effectiveness of the mill. The shell liners have different shapes that aid in lifting the ore and thus transferring energy to the ore for grinding. Ensuring that the lifters remain efficient over the lifetime of the liners is therefore also important (Chandramohan and Powell, 2006, McBride and Powell, 2006).

The pulp exits the mill through the outlet trunnion and there are several methods to facilitate this. There is a simple overflow mill (Figure 2.1) where the pulp overflows into the outlet trunnion aided by the rotation of the mill. Secondly, a screen-and-discharge mechanism can be employed, where the pulp passes through a screen that limits the particle size to the screen aperture size and lifter bars lift the pulp to the outlet trunnion (Figure 2.2). A peripheral discharge and open-ended discharge can also be employed to eliminate the need for lifting pulp through the outlet trunnion. The peripheral discharge (Figure 2.3) consists of grates on the side of the mill at the outlet end through which the pulp flows. With an open-ended discharge, the mill outlet end is not closed off by a solid side, but rather by a screen or grate, that allows pulp to exit through the whole outlet opening. The open-ended mill cannot be supported by an outlet trunnion, but rather by a roller system (Figure 2.4) or slipper bearing (Figure 2.5) (Stanley, 1987). The different discharge mechanisms influence the performance of the mill. If a slurry pool forms inside the mill, the grinding performance is degraded (Latchireddi and Morrell, 2003a, b).

There are different mill types:

Ball Mills use steel balls ranging from 100 mm down to 50 mm as grinding medium. Their primary role is for primary milling after crushing if the run-of-mine material has in-sufficiently sized pebbles to support autogenous milling.

Rod Mills use steel rods with diameters of about 100 mm and a length that is about 100 mm shorter than the length of the mill. Rod mills can handle coarser material than other

types of milling and produce finer particles than crushing, and are therefore ideal as intermediary between crushing and other types of milling.

Autogenous Mills use only the ore as grinding medium. There are two types of autogenous (AG) mills. The first type uses the ROM ore directly as grinding medium and is only suitable for primary milling. The advantage is that no crushing is required, except for very large pieces to facilitate handling. The second type uses pebbles rather than balls or rods as grinding medium and is therefore suitable for any stage of milling.

Semi-autogenous Mills use steel balls together with ore as griding medium. The power draw of the mill is a function of the bulk density of the load. The bulk density of AG mills is lower than that of same-sized ball and rod mills, which affects the power draw and the grinding capacity of the AG mills. The AG mills have to be larger than ball or rod mills to have the same grinding capacity. The SAG mills increase the bulk density of the mill compared to AG mills and therefore have increased grinding capacity for the same size as AG mills. SAG mills can still be used to grind run-of-mine ores.

### Hydrocyclones

Hydrocyclones (Figure 2.6) are centrifugal classifiers. They work by injecting a feed slurry tangentially into the cylindrical section of the device. The centrifugal forces that result force the solid particles through the suspended water to the cylinder wall. The particles experience drag forces as they move through the suspended water, causing heavier material to move preferentially to the cylinder walls, while lighter material does not move all the way to the cylinder wall. Most of the liquid is forced to the centre of the device owing to displacement taking place. The liquid together with the lighter material is ejected through the central vortex finder to the overflow. The heavier particles are displaced by newer particles entering the device that cause axial as well as tangential forces. The heavier particles cannot escape at the overflow, because they are stopped by the diaphragm and forced to the conical section where they are guided to the underflow opening or spigot. The cut size of the cyclone is highly dependent on the feed density and less on the cyclone dimensions. The cyclone cut can therefore be controlled by changing the feed density to the cyclone (Stanley, 1987).

**Sump**

The mill usually discharges into a sump from the top, where extra water is added to dilute the pulp. The sump acts as a buffer to disturbances in the feed to the cyclone. In smaller sumps, the pulp is assumed to be fully mixed, while in bigger sumps some settling may occur that could affect the discharge density at the bottom of the sump (Hulbert, 2005).

**Process of breakage**

Most comminution processes apply compression to ore particles. Elastic bodies distort when compressed by flattening in the direction of compression and bulging at right angles to the compression force. This bulging of the particle induces tensile stresses inside the particle. The tensile stresses are concentrated at the edges of flaws in the particles. The greater the flaw area is, the greater the concentration of force. If the tensile stress at the flaw edges reaches a critical value, the intermolecular bonds break. As the bonds break, the area of the flaw increases and the concentration of force at the edge of the flaw is increased, which causes more molecular bonds to break. The flaw is almost instantaneously converted to a crack. As the crack propagates, other flaws are activated and start to crack. This results in the particle being covered in a network of cracks that divide it into roughly equal fragments (Kelly and Spottiswood, 1990, Stanley, 1987).

The crack directs compressive strain energy in equal amounts to the two parts on either side of it. If the energy is sufficiently large, it can cause the pieces to break further. The two fragments are unlikely to be equally large in terms of mass, and the smaller fragment is therefore more likely to break owing to the greater energy per mass concentration of that fragment. The breakage process is concentrated on smaller and smaller particles. Every time a fragment splits, the energy is divided between the fragments, with the smaller fragment being more likely to break, until the energy levels fall below the threshold to support further breakage (Kelly and Spottiswood, 1990, Stanley, 1987).

The comminution process needs to apply enough energy to reach the critical level to cause the initial crack network to form. Once breakage starts to occur, most of the energy is then converted to heat in the form of vibrations within the particle fragments (Kelly and Spottiswood, 1990, Stanley, 1987).

#### Size distribution

Brittle fracture breakage produces a product with a spread of differently sized fragments. This final product consists of a particle distribution, where this distribution is an important factor in the efficiency of further processing. Determining this size distribution is done by sieving a sample of the final product. The aperture area of each successive sieve is half that of the previous sieve (Stanley, 1987). The sieve apertures are usually square and the size is expressed as the length of one of the aperture sides. The size distribution of the final product is expressed in discrete sizes by either stating the differential distribution, which is the percentage of the total sample mass in each size fraction in decreasing aperture sizes, or the cumulative distribution, which is defined as the combined mass of all the size classes starting from either the coarsest or the finest sieve to the sieve in question and expressing that mass as a percentage of the total sample mass.

1 INTRODUCTION

1.1 MILL CIRCUIT DESCRIPTION

1.2 OBJECTIVES IN MILL CONTROL

1.3 AIMS AND OBJECTIVES

1.4 ORGANISATION

2 MILLING THEORY AND MODELLING

2.1 INTRODUCTION

2.2 THEORY OF MILLING

2.3 MILLING MODELLING

2.4 CONCLUSION

3 MODEL PREDICTIVE CONTROL

3.1 INTRODUCTION

3.2 HISTORICAL BACKGROUND

3.3 STABILITY OF MPC

3.4 ROBUST MPC – STABILITY OF UNCERTAIN SYSTEMS

3.5 ROBUST NONLINEAR MPC FORMULATIONS

3.6 NONLINEAR MODEL PREDICTIVE CONTROL

3.7 ROBUST NONLINEAR MODEL PREDICTIVE CONTROL

3.8 STATE OBSERVERS

3.9 CONCLUSION

4 PID CONTROL

4.1 INTRODUCTION

4.2 PI CONTROL WITH ANTI-WINDUP .

4.3 LINEARISED MODELS FOR SIMC TUNING METHOD

4.4 SIMC TUNING METHOD

4.5 IMPLEMENTATION

4.6 SUMMARY

5 MILLING CIRCUIT CONTROL SIMULATION STUDY

5.1 INTRODUCTION

5.2 PERFORMANCE METRICS

5.3 SIMULATION RESULTS

5.4 DISCUSSION

5.5 SIMULATION SUMMARY

6 CONCLUSIONS AND FURTHERWORK

6.1 SUMMARY AND EVALUATION

6.2 FURTHER WORK

REFERENCES

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