DEVELOPMENT OF A MODEL FOR TEMPERATURE IN A GRINDING MILL Edgar Kapakyulu MSc 2007 i DEVELOPMENT OF A MODEL FOR TEMPERATURE IN A GRINDING MILL Edgar Kapakyulu A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfillment of the requirements for the degree of Master of Science in Engineering. Johannesburg, January 2007 ii Declaration ________________________________________________________ I declare that this dissertation is my own, unaided work. It is being submitted for the Degree of Master of Science in the University of the Witwatersrand, Johannesburg. This thesis, to the best of my knowledge, has not been submitted before for any degree or examination in any other University. ________________________________ Edgar Kapakyulu _______________ day of ___________________ , ___________ iii Abstract Grinding mills are generally very inefficient, difficult to control and costly, in terms of both power and steel consumption. Improved understanding of temperature behaviour in milling circuits can be used in the model-based control of milling circuits. The loss of energy to the environment from the grinding mill is significant hence the need for adequate modeling. The main objectives of this work are to quantify the various rates of energy loss from the grinding mill so that a reliable model for temperature behaviour in a mill could be developed. Firstly models of temperature behaviour in a grinding mill are developed followed by the development of a model for the overall heat transfer coefficient for the grinding mill as a function of the load volume, mill speed and the design of the liners and mill shell using the energy balances in order to model energy loss from the mill. The energy loss via convection through the mill shell is accounted for by quantifying the overall heat transfer coefficient of the shell. Batch tests with balls only were conducted. The practical aspect of the work involved the measurement of the temperatures of the mill load, air above the load, the liners, mill shell and the environmental temperature. Other measurements were: mill power and sound energy from the mill. Energy balances are performed around the entire mill. A model that can predict the overall heat transfer coefficient over a broad range of operating conditions was obtained. It was found that the overall heat transfer coefficient for the grinding mill is a function of the individual heat transfer coefficients inside the mill and outside the mill shell as well as the design of the liners and shell. It was also found that inside heat transfer coefficients are affected by the load volume and mill speed. The external heat transfer coefficient is affected by the speed of the mill. The values for the overall heat transfer coefficient obtained in this work ranged from 14.4 ? 21W/m2K. iv List of Publications The author has published the following papers based on the contents of this dissertation as follows: Published conference abstract Kapakyulu, E., and Moys, M.H., 2005. Modelling of energy loss to the environment from the grinding mill, Proceedings of the Mineral Processing 2005? Conference, SAIMM, Cape Town, South Africa, 4-5 Aug. pp 65-66 - SP03 Research Papers: Accepted for publication and currently in press in Minerals Engineering: Kapakyulu, E., and Moys, M.H., 2006. Modelling of energy loss to the environment from a grinding mill, Part I: Motivation, Literature Survey and Pilot Plant Measurements, (Currently in press in Minerals Engineering) Kapakyulu, E., and Moys, M.H., 2006. Modelling of energy loss to the environment from a grinding mill, Part II: Modeling the overall heat transfer coefficient, (Currently in press in Minerals Engineering) v To the loving memory of my late Grandmother, Mrs Belita Kalenge Yengayenga vi Acknowledgements I would like to extend my gratitude to the following people and institutions that made it possible for me to complete this dissertation. ? Professor Michael Moys, for his role as supervisor and mentor throughout the execution of this work ? Dr. Hongjung Dong for his assistance with the instrumentation ? Mr. Godfrey Monama for providing many useful suggestions which enhanced the quality of this work. ? Mr. Theo Prassinos, the workshop manager for the assistance received from the work shop ? Eskom, Anglogold Ashanti and NRF (National Research Foundation) for their financial assistance and support ? I would also like to thank my family for the moral and spiritual support during my studies. ? Finally, I would like to thank the Almighty God my creator, for strength and comfort and also for this opportunity and ability to pursue this research work. vii Table of Contents DECLARATION ii ABSTRACT iii LIST OF PUBLICATIONS iv ACKNOWLEDGEMENTS vi TABLE OF CONTENTS vii LIST OF FIGURES xii LIST OF TABLES xvii NOMENCLATURE xix CHAPTER 1: INTRODUCTION 1 1.1 Importance of milling 2 1.2 Model-based mill control 3 1.3 Research objectives 6 1.4 Summary of the dissertation 7 CHAPTER 2: LITERATURE REVIEW 9 2.1 Introduction 10 2.2 Energy Balances and review of work in the energy balances 10 2.3 Mill power 11 2.3.1 Theory of mill power 11 2.3.2 Effect of speed on mill power 14 2.3.3 Effect of mill filling on mill power 15 viii 2.4 Energy and temperature in milling 16 2.5 Energy consumed in milling 19 2.6 The discrete element method 20 2.7 Energy losses in a milling system 22 2.8 Analysis of conduction and convection heat transfer 23 2.8.1 Conduction Heat Transfer 23 2.8.2 Convection Heat Transfer 24 2.8.3 The Practical Significance of the Overall Heat Transfer Coefficient 31 2.8.4 Estimation of parameters- Least squares filtering 32 2.9 Conclusions 33 CHAPTER 3: DEVELOPMENT OF A MATHEMATICAL MODEL 35 3.1 Introduction: Principles of mathematical modeling 36 3.1.1 The energy balance 37 3.1.2 Assumptions made in deriving the model 38 3.1.3 Model derivation 39 3.2 Energy balance models for the mill temperature 41 3.3 A model for the overall energy loss as a function of load volume and mill speed 44 3.3.1 Modeling of two parallel paths 44 3.4 Dynamic and steady state models 47 3.4.1 Steady state models 50 3.5 Conclusions 51 CHAPTER 4: EXPERIMENTAL APPARATUS, METHODOLOGY, MEASUREMENTS AND DATA ACQUISITION 52 4.1 Introduction ? Experimental setup 53 4.1.1 Description of the milling setup 53 ix 4.2 Measurements 54 4.3 Temperature probes 55 4.3.1 Choice of probe and material of construction 57 4.3.2 Expected temperature range 57 4.3.3 Insertion points 58 4.4 Temperature probe calibration 60 4.4.1 Calibration curves for the temperature probes 61 4.5 Torque measurement 63 4.5.1 Torque calibration 63 4.6 Measurement of Sound Energy 64 4.7 Experimental Procedure 65 4.8 Difficulties encountered 66 4.9 Summary and Conclusions 67 CHAPTER 5: ANALYSIS OF THE MILL TEMPERATURE DATA 68 5.1 Introduction 69 5.2 Results of the milling process 69 5.3 Analysis of temperature probe data 73 5.4 Variation of process temperatures with speed and load volume 74 5.5 Transient behaviour of grinding mill temperature with time 74 5.6 Effect of sudden water addition to mill on temperature and power 76 5.7 Effect of change of ambient conditions on mill temperature 77 5.8 Conclusions 78 CHAPTER 6: RESULTS OF POWER AND SOUND ENERGY DATA 80 6.1 Results of the Torque data 81 6.2 Results of the Sound energy data 83 6.3 Conclusions 84 CHAPTER 7: APPLICATION OF THE TEMPERATURE MODELS TO ESTIMATE AND QUANTIFY PARAMETERS 85 x 7.1 Introduction: Analysis of steady state results 86 7.2 Analysis of steady state temperature 88 7.3 Quantifying the Overall Heat Transfer Coefficient 88 7.3.1 Model Validation 88 7.3.2 Determination of the ball to air and ball to metal surface heat transfer coefficient 89 7.3.3 Validation of the Overall Heat Transfer Model 105 7.4 Conclusions 106 CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS 108 8.1 Introduction 109 8.2 Summary 109 8.3 Overall findings 109 8.4 Recommendations 110 REFERENCES 112 APPENDICES 116 APPENDIX A - Calibration of the Temperature probes: Calibration Equations and Constants 117 APPENDIX B - Torque calibration 122 APPENDIX C - Specific Heat Capacities, density and thermal conductivities of Steel balls, air mild steel and cast steel 124 APPENDIX D ? General Procedure for analytical solution for dynamic model 125 xi APPENDIX E ? Determination of the sound energy from the sound measurements 131 APPENDIX F ? Summary of steady state temperature data 133 xii List of Figures Chapter 1 Figure 1.1: The Sump energy balance output compared with the mass balance and manual samples 4 Chapter 2 Figure 2.1 Illustration of forces of friction and gravity for turning moment about the center of a ball mill (Austin et al., 1984) 13 Figure 2.2 The effect of critical speed on mill power 15 Figure 2.3 DEM contact force law diagram 21 Figure 2.4 Temperature profile in a grinding mill 26 Chapter 3 Figure 3.1 Batch grinding mill: energy balance measurements 38 Figure 3.2 Schematic diagram of the orientation of the load, air above the load, liner and mill shell 42 Chapter 4 xiii Figure 4.1 The schematic representation of the experimental mill set up 54 Figure 4.2 Electric circuit diagram for the thermistor 56 Figure 4.3 Measurement of the temperatures of the load and the air above the load inside the mill grinding chamber 59 Figure 4.4 Measurement of the liner and shell temperatures 59 Figure 4.5 Laboratory temperature probe calibration 61 Figure 4.6 Calibration of the torque load beam 64 Chapter 5 Figure 5.1 (a) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 20% and speed of 65% 69 Figure 5.1 (b) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 20% and speed of 75% 69 Figure 5.1 (c) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 20% and speed of 95% 70 Figure 5.1 (d) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 20% and speed of 105% 70 Figure 5.2 (a) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 25% and speed of 75% 70 Figure 5.2 (b) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 25% and speed of 95% 70 Figure 5.3 (a) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 30% and speed of 30% 71 xiv Figure 5.3 (b) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 30% and speed of 50% 71 Figure 5.3 (c) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 30% and speed of 80% 71 Figure 5.3 (d) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 30% and speed of 120% 71 Figure 5.4 (a) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 40% and speed of 75% 72 Figure 5.4 (b) Variation of mill power, mill process temperatures and ambient temperature with time at a mill filling of 40% and speed of 95% 72 Figure 5.4 (c) Variation of mill power, mill process temperature and ambient temperature with time at a mill filling of 40% and speed of 105% 72 Figure 5.5 Transient variations of mill process temperatures with time 75 Figure 5.6 Effect of water addition on mill power and process temperatures 77 Figure 5.7 Effect of ambient temperature on mill process temperatures 78 Chapter 6 Figure 6.1 Variation of mill power with mill speed at a mill filling of 20% 82 Figure 6.2 Variation of mill power with mill speed: load volume 30% 83 xv Chapter 7 Figure 7.1 Temperature profiles of the grinding mill at different mill speed at a constant mill filling of 20% 86 Figure 7.2: Temperature profiles of the grinding mill at different mill speed at a constant mill filling of 25% 87 Figure 7.3: Mill Power and Steady state mill process temperature distribution, J=30%, N=80% and t=3 hours 88 Figure7.4: Variation of ball temperature with time for ball with thermistor covered in pile of heated balls and for ball swung in air at various velocities 90 Figure 7.5: Ball to air heat transfer determination: Variation of hBall-Air with velocity 92 Figures 7.6: Load behaviour from DEM simulations at varying mil speeds and load volumes 94 Figure 7.7: Profile of a cataracting ball in a grinding mill 96 Figure 7.8: Variation of model and experimental values of )( LLDhA ? with mill speed 101 Figure 7.9: Variation of model and experimental values of )( AIRLDhA ? with mill speed 102 Figure 7.10: Variation of model and experimental values of )( LAIRhA ? with mill speed 102 Figure 7.11: Variation of model and experimental values of exth with mill speed 103 Figure 7.12: Variation of model with measured various rates of energy loss from the grinding mill 104 xvi Figure 7.13: Variation of measured with model steady state temperature values 104 Figure 7.14: Variation of the model and experimental values of the overall heat transfer coefficient with mill speed at various mill fillings 106 xvii List of Tables Chapter 4 Table 4.1 Calibration of the temperature probes: Table of temperature and voltage for the probes for the liner, shell, load and air above the load 62 Table 4.2 Table of temperature and voltage for the ambient thermistor calibration and values from the manufacturers 62 Chapter 6 Table 6.1 Variation of sound intensity from the mill with the mill speed 84 Chapter 7 Table 7.1: Table of DEM velocities and the corresponding values of the heat transfer coefficients from the ball to the air at various mill conditions 96 Table 7.2: Number of balls in contact with the air and liner surface form DEM simulations at varying mill conditions 97 Table 7.3: Approximate total number of balls in a 2D and 3D Mill 97 Table 7.4: Experimental and model values of )( AIRLDhA ? , )( LAIRhA ? , )( LLDhA ? , exth and extU at various mill conditions 99 xviii Appendix A Table A1: Temperature versus Resistance values for the Bead Thermistor 118 Table A2: Resistance-Temperature characteristics for the thermistors from manufacturer?s data 119 Table A3: Calculation of Power applied to the thermistor and the thermistor self heat 121 Appendix B Table B1: Calibration of the Torque load beam 122 Appendix C Table C1: Thermal properties of substances 124 Table F1: Summary of temperature results at steady state at various mill conditions 133 xix Nomenclature General: P = Mill power draw (W) T = Torque (N.m) Nc = Critical speed of the mill N = Mill speed (Fractional of the mill speed) J = Load volume (fraction of the mill volume) f = Fractional mill speed ?? ??? ? = Nc Nf t = Milling time (s) D = inside lining mill diameter (m) L = interior mill length (m) Pr = Prandtl number (Dimensionless) Q = Rate of energy loss (J/s or W) hLD-L = heat-transfer coefficient between the load and the liners (W/m2.K) hLD-A = heat-transfer coefficient between the load and the air above the load (W/m2.K) hA-L = heat-transfer coefficient between the air above the load and the liners (W/m2.K) xx hext = heat-transfer coefficient between the shell and the environment (W/m2.K) U = Overall heat transfer coefficients (W/m2 oK) CPair = specific heat capacity of the air above the load at constant pressure (J/kg.K) CpLD = specific heat capacity of the load (steel balls) at constant pressure (J/kg.K) CpL = specific heat capacity of the liners (cast steel) at constant pressure (J/kg.K) CpS = specific-heat capacity of the shell (mild steel) at constant pressure (J/kg.K) Kg = thermal conductivity of the air (W/m.K) kS = thermal conductivity of the shell (W/m.K) kL = thermal conductivity of the lifter (W/m.K) ? = viscosity of the fluid (Kg/m.s or Pa.s) TLD = Temperature of the load (grinding media) (oC) TAIR = Temperature of the air above the load (oC) TL = Temperature of the lifter/liner (oC) TL1 = Lifter temperature at surface 1 (oC) TL2 = Lifter temperature at surface 2 (oC) TS = Temperature of the shell (oC) xxi TS1 = Shell temperature at surface 1 (oC) TS2 = Shell temperature at surface 2 (oC) TA = Ambient temperature (oC) mLD = Mass of the load (Kg) mS = Mass of the shell (Kg) mL = Mass of the liners (Kg) A = Surface area (m2) U = Overall heat transfer coefficients (W/moK) v = Velocity (m/s) 1 Chapter 1 Introduction _________________________________ 2 1.1 Importance of milling Milling is an important and indispensable part of mineral processing. Rotary grinding mills are used in mineral processing for particle size reduction. A ball mill is a system composed of a number of interacting and interdependent elements working together to accomplish the grinding or breakage of ore (Radziszewski, 1999). The elements necessary in a ball mill are the cylindrical mill, the mill liners, the ball charge, the ore to be ground as well as the control system needed to govern the ore feed and water rates and the mill speed. These consist of a rotating cylindrical shell of length usually greater than the diameter. These elements interact in such a way as to continually lift the ball charge which then falls and breaks the ore nipped between the balls. Lifter bars and sacrificial plates are bolted to the inside of the mill shell. Rock generally arrives from the crusher or perhaps a semi-autogenous (SAG) mill depending on the design of the milling circuit and the mineral being processed and enters the feed end of the mill. Grinding media consisting of steel balls are present in the mill along with any rock that has not yet overflowed through the discharge end. Comminution or size reduction is effected by ore-to-ore, ore-to-media and ore-to-mill wall interaction. The importance of milling in mineral processing is to liberate or release the valuable minerals from the host rock or ore, to a size suitable for subsequent metallurgical processes. In mineral processing it is normally done wet and the resulting slurry is immediately passed on to the subsequent material separation stage. This is very often froth flotation, and grinding and flotation are then carried out in one combined section of the concentrator. Dry grinding is also widely applied in mineral processing and power generation. The milling process however is very inefficient, difficult to control and costly, in terms of both steel and power consumption. A typical comminution plant accounts for a significant proportion of the total operational costs. It is an indispensable operation of mineral processing and size reduction circuits expend significant amounts of energy (Stamboliadis, 2002). Fuerstenau (1999) reported that 3 comminution consumes up to 70% of all energy required in a typical mineral processing plant. It is also generally recognized that the actual energy required to reduce the particle size of a brittle material is a small percentage, about 4% or less, (Tavares and King, 1998) of the total energy consumed by grinding equipment. Stairmand (1975) indicated that about 99% of the energy supplied to an industrial grinding system appears as heat and there is very little evidence that any appreciable proportion is locked up as surface energy or strain energy. The loss of energy to the environment is significant and needs to be adequately modeled. These factors amplify the need to assess, control and improve the efficiency of milling circuits. The behaviour of temperature in milling circuits can be obtained from an energy balance methodology. Model-based control of milling circuits using temperature measurements could then provide substantial benefits. An example of a model- based control process is given in section 1.2. 1.2 The Model-Based Mill Control The energy balance models thus provide a tool for the computation of the heat transfer coefficients, the overall heat transfer coefficient and therefore estimates of the thermal energy loss from the grinding mill to the environment. The use of temperature measurements in the milling process may lead to improved understanding of temperature behaviour and can be used to control milling circuits. One method used to control grinding circuits involves continuous online matching of a dynamic model for the process to the measurements that are made on the process (Herbst et al, 1989). This is usually done using a Kalmin Filter. This allows the estimation of variables which cannot be measured, eg rate of grinding of solids in the mill, percent solids in the mill discharge, etc. This should enable more precise control of the milling circuit. 4 Briefly, a dynamic model of the process is formulated; based on certain assumptions, eg perfect mixing in each compartment of a 2-compartment model of the mill, perfect mixing in the discharge sump, first-order rate of grinding, three size classes, etc. Generally it is a substantially simpler model than is used for steady-state simulation of the process. Very accurate temperature measurements (?0.01?C) are made on key streams entering and leaving the mill in addition to all the other conventional measurements that are normally made. Since temperature measurements can be made far more accurately than other process measurements such as flowrate and density, much more precise knowledge of process behaviour becomes available. By applying laws of energy conservation it becomes possible to estimate several variables relating to mill load behaviour which have not been available before. Van Drunick and Moys (2002) applied this technique to the estimation of mill discharge percent solids. They used mass and energy balances around the mill discharge sump in order to estimate the flowrate and percent solids in the discharge from the mill. Their results are illustrated in Figure 1.1, which shows the variation of relative density (RD) as the rate of feed water addition to the mill is changed. The output of their estimator is labelled ?Dynamic Energy Balance? Mill Discharge Density of the Mass and Energy Balances Versus the Manual Samples (Step Changes Made in the Mill feed Water) 1 1.2 1.4 1.6 1.8 2 10:42:31 11:12:31 11:42:31 12:12:31 12:42:31 13:12:31Time Mi ll D isc ha rg e R ela tiv e D en sit y t/m ^3 0 50 100 150 200 250 300 350 400 Mi ll F ee d W ate r t /hr Mill Discharge RD from OCS Mass Balance Dynamic Energy Balance Manual RD Samples Mill FeedWater PV Mill Feed Water PV Manual RD Samples OCS RD Output Dynamic Energy Balance Figure 1.1: The Sump Energy Balance Output compared with the Mass Balance and Manual Samples Mass Balance estimations 5 and the experimentally measured values are labelled ?Manual RD samples?. These are compared to that produced by the similar algorithm based on mass balances only (labelled ?Mass Balance estimations?) and produced a substantially better fit, partly because of the accuracy of the temperature measurements and partly because the mass balance estimator is based on the assumption that the mill is perfectly mixed, which is by no means correct. It is estimated conservatively that more accurate control of the load behaviour will lead to substantial improvements in milling efficiency (eg a 3 - 5 % reduction in kWh/ton) and/or a greater stability in the circuit product size distribution leading to an average increase of 3-5 percent ?75 micron material (for a gold ore example). If desired, it should be possible to increase circuit capacity by a similar order of magnitude. Any of these improvements will lead to project pay-backs of the order of months. The model-based approach is becoming advantageous for several reasons. Firstly, modern processing plants are highly integrated with respect to the flow of both material and energy (Serbog et al., 1989). This integration makes plant operation more difficult. Secondly, there are economic incentives for operating plants closer to limiting constraints to maximize profitability while satisfying safety and environmental restrictions. There are a number of major steps involved in designing and installing a control system using the model-based approach. The first step involves formulation of the control objectives. In formulating the control objectives, process constraints must also be considered. After the control objectives have been formulated, a dynamic model of the process is developed. The process can have a theoretical basis, for example, physical and chemical principles such as conservation laws and rates of heat transfer, or the model can be developed empirically from experimental data. The model development usually involves computer simulation. The next step in the control system design is to devise an appropriate control strategy that will meet the control objectives while satisfying process constraints. Computer simulation of the controlled process is used to screen alternative control 6 strategies and to determine preliminary estimates of suitable controller settings. Finally, the control system hardware is selected, ordered, and installed in the plant. Then the controllers are tuned in the plant using the preliminary estimates from the design step as a starting point. Controller tuning usually involves trial and error procedures. Controller self-tuning techniques are also available. The use of models for on-line decision making represents the next logical step in mineral processing control system design. The key elements required for model- based control are good on-line models and accurate estimation procedures based on precise measurements. Models can be used in various ways in a control strategy as showed by Herbst et al, 1992. When the objective is to minimize the variance of the outputs of a process, then the dynamic models of the process can be used to find the optimal inputs. When the objective is to find the operating conditions that maximize profit or minimize cost, then from steady state models the desired level of process outputs is found. In both cases it is up to the control system to ensure that the set points provided by the model are achieved; the models act as a supervisor for the control system. The model can be used to estimate variables too costly or difficult to measure. In other words, a model contains missing information about the process. By building in such a model, ?well informed? responses to disturbances can be made and, ultimately, truly ?optimal? control performance can be achieved. 1.3 Research objectives The main issue in this research is the development of mathematical models of temperature behaviour in the grinding mill. The model is derived from the energy balance methodology and is based on a horizontal batch grinding mill. Due to difficulties in direct estimation of heat transfer coefficients in milling, considerable effort is made on modeling temperature in grinding mills. The objective of this research is to quantify the various rates of energy loss from the mill under a wide range of conditions, so that a reliable model for temperature 7 behaviour in a mill could be developed. The project is also aimed at developing a model for energy loss from the mill to the environment as a function of important mill operating parameters such as mill speed, load volume and mill shell/liner design from the energy balances. By using the models, important operating variables such as load volume, flow rate (as in continuous mills), % solids, etc, can be controlled and optimized and this would lead to improved control of a milling process. 1.4 Summary of the Dissertation This dissertation is organized into 8 chapters which includes the introduction. The first chapter presents an introduction to the project and most importantly the background and the objectives of this research. Chapter 2 provides a review of published literature on energy balances, mill power and temperature in milling, as well as related work in this field of study. Chapter 3 forms the core of this project, the development of mathematical models for temperature behaviour in a grinding mill. The chapter involves the development of a model for the overall heat transfer coefficient in terms of the individual heat transfer coefficients and the mill liner and shell design using the energy balances so as to model energy loss from the mill. Chapter 4 introduces the reader to the experimental equipment, instrumentation, data acquisition and methodology employed in this project. The analysis of data collected in this chapter is described in chapter 5 with particular emphasis on temperature data. In chapter 6, the analysis of data on torque and sound energy from the mill is done. Due to limited data on sound from the mill, not much analysis was done. Moreover, the energy converted into sound was found to be negligible compared to other forms of energy. In this chapter though, the analysis of torque data is 8 much more important as it was considered to be the only source of energy input to the mill. The analysis of steady state results are reported in chapter 7. This analysis involves evaluating such important parameters as the overall heat transfer coefficient of the mill shell as well as application or validation of the model for the overall heat transfer coefficient. Most of the analysis of the work done is presented in this chapter. Chapter 8 completes the thesis by summarizing the main conclusions, the overall findings and makes recommendations for further research. 9 Chapter 2 Literature Review _________________________________ 10 2.1 Introduction This chapter is concerned with review of energy balances and their applicability to the milling system. Milling is a very inefficient process and a lot of effort has been made on improving the current state of operation of milling systems. The energy balance methodology could prove to be a competent tool in the optimization of mill control and operation as compared to other balances such as the mass balance. A lot of work has been done on mass balances in milling circuits with important measurements being the flow rate and density. What follows is a review of literature on published work on the energy balances as well as related and earlier work done on the energy balances. With particular emphasis is the application of the energy balances to the milling system. 2.2 Energy Balances and a review of work done with energy balances Energy can be defined as the capacity to do work. Energy assumes many forms. It can assume the form of mechanical energy, chemical energy, electrical energy, heat energy, atomic energy, sound energy and many others. Energy can be converted form one form into another, but its total amount within a system remains the same, as energy can neither be created nor destroyed. The subjects of thermodynamics and energy transfer are highly complementary. The subject of heat transfer treats the rate at which heat is transferred and therefore can be viewed as an extension of thermodynamics (Incropera and Dewitt, 2002). Conversely, for many heat transfer problems, the first law of thermodynamics (the law of conservation of energy) provides a useful, often essential, tool. According to the time basis, first law formulations that are well suited for heat transfer analysis may be stated over a time interval (?t) as follows: 11 The amount of thermal energy and mechanical energy that enters a control volume, plus the amount of thermal energy that is generated within the control volume, minus the amount of thermal energy and mechanical energy that leaves the control volume must equal the increase in the amount of energy stored in the control volume (Incropera and Dewitt, op. cit). In recent work done in the milling system, the energy balance has been found to have advantages over the mass balance (Van Drunick and Moys, 2002). The energy balance relies on temperature data which can be obtained to a higher degree of accuracy than the flow data required by the mass balance. In addition to that, the temperature probes are quite cheap and affordable and this adds to the advantages. Since the initial results were published from the earlier work by Van Drunick and Moys (2002), it has now been decided to further develop the modeling techniques in order to focus on balancing the energy around the entire mill. Van Drunick and Moys (2003) developed the modeling techniques in order to focus on balancing the energy around the entire mill circuit. The energy balance around a mill is given by: Energy into the mill=Rate of energy loss from mill+Rate of energy accumulation in mill The equation above can be expanded as follows: E in = E out(shell) + E out(discharge) + E accumulated (2.1) 2.3 Mill Power 2.3.1 The Theory of Mill Power for Tumbling Mills Austin et al. (1984) from data given by numerous investigators (White, 1904; Davis, 1919; Rose and Sullivan, 1958; Hogg and Fuerstenau, 1972) pointed out 12 that there are two main approaches in the derivation of equations describing the power required to drive tumbling mills. One approach calculates the paths of balls tumbling in the mill and integrates the energy required to raise the balls over all possible paths. The other approach (Hogg and Fuerstenau, 1972) treats the ball- powder aggregate as shown in Figure 2.1, assuming that the turning moment of the frictional force must balance the turning moment of the centre of gravity of the bed around the mill center. The frictional force presumably arises from between the case and the bed plus friction of ball-on-ball as balls move up through the bed. In both cases, the energy used to turn a well-balanced empty mill is small since it consists only of bearing friction. In their approach Hogg and Fuerstenau (op. cit.) considered only the rate at which potential energy was gained by particles as they rose up the mill in a locked manner. Once they reach the upper-most point of their upward motion they were assumed to roll down the inclined charge surface and re-enter the charge lower down. By integrating the rate of potential energy gain over all paths they obtained the following equation: 5.23sinsin LDKPower qfra= (2.2) where: K = constant, ? = fraction of critical speed, L = interior mill length, D = interior mill diameter , ? = charge angle of repose, ? = angle related to mill filling and ? = mean bulk density of the charge 13 Figure 2.1: Illustration of forces of friction and gravity for turning moment about the center of a ball mill (Austin et al., 1984) Austin et al. (1984) also pointed out that the relation for net power for a grinding mill as given by Rose and Sullivan (1958) for dry grinding is: FJULDm bcbp )/4.01)()()(10)(8.2( 5.24 rsr +?= ? , kW (2.3) where the empirically determined constant (2.8)(10-4) is for units of ball density ?b in lb/tf3 and L and D in feet. For L and D in meters and ?b in kg/m3, the constant is (1.12)(10-3). In this equation, it is assumed that the power to the mill is proportional to ?c, from ?c = 0 to 0.8, so that the equation should not be extrapolated beyond ?c = 0.8. Liddell and Moys (1988) reported that an equation for mill power which is in common use was developed by Bond, who based it on the torque principle and modified it by the use of empirical results: ( )( )ffr 1093.2 2/1.01937.01262.12 ???= JJLDP b (2.4) where J is the fractional filling of the load. Rotation Centre of Gravity Gravitational Force 14 2.3.2 Effect of speed on mill power Power consumption in a milling process generally increases with increasing mill speed within operating limits (Taggart, 1927). The higher the speed the more power transmitted and therefore the more energy transfer to the load. He also showed that power consumption increases with increase in ball load until the load reaches a point at or slightly above the axis, after which further loading by ore or balls results in the decrease of power. The power drawn by the mill is a function of mill speed. At very low mill speeds, the charge (slurry & balls) moves over one another in concentric layers resulting in poor grinding. As the speed increases, the power draw increases, but it then decreases at supercritical speeds due to centrifuging. Critical speed is the speed at which the mill charge will centrifuge. Nc indicates fraction of critical speed. It is expressed as: DNc /3.42= rpm (2.5) where: D = mill diameter (m) For optimum charge movement, mills should operate at a narrow range of about 75-95% of critical speed. This is represented by the relationship between power drawn and %critical speed shown in Figure 2.2. 15 50 55 60 65 70 75 80 85 90 95 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Fraction critical speed % Po we r d ra wn Figure 2.2: The effect of critical speed on power drawn 2.3.3 Effect of mill filling on mill power Power drawn by the mill is a function of load volume. As the volume (and hence mass) of the load increases, the power draw increases, but it then decreases again after a certain point. This occurs when load volume exceeds 45-50%. Power draw also decreases when some of the load components begin to centrifuge, thereby reducing the active mass of the load and reducing the effective diameter of the mill. The charge volume has a significant effect on the power drawn by the mill. Thus, it is feasible to operate the mill as full as possible between 30 - 45%, since at this filling the power drawn is maximized. 16 2.4 Energy and Temperature in milling Grinding is directly related to energy absorbed in the mill chamber. Simply, the more energy you can input into a batch, the more grinding action is achieved. During milling for liberation of minerals the power drawn by the mill at any time is a direct index of the lifting action on the balls at the instant of time, and the rate of breakage is expected to be directly proportional to the lifting action (with all other variables constant). The power to the mill is used to produce the mechanical action, so the total efficiency involves (Austin et al, 1984): 1 Efficient conversion of input energy to mechanical action 2 Efficient transfer of the mechanical action to the particles 3 Matching the stress produced by the mechanical action to the failure stress of the particles For example, if a tumbling mill is operated with too small a loading of particles, the impact will be of steel balls and part 2 will clearly be very inefficient. Again, if the ball density, ball size, and mill diameter are all low and if the particles are large and strong, the mill might not break the particles at all (except by slow abrasion) and part 3 will be inefficient (Austin et al, op cit.). The energy input to mills is obviously linked to the specific energy of grinding, which is defined from a given feed size distribution to a desired product size distribution (Austin et al., op. cit.). Much of the power consumed in ball milling is transformed into heat and becomes apparent in rise of temperature of the load (i.e. raising the mill discharge temperature) as it passes through the mill (Taggart, op. cit). The mill speed, milling time and mill filling affect this temperature increase. Mill process temperatures are affected by mill conditions critical to mill operation, namely, mill speed and load volume. At higher mill speeds, more lift is imparted 17 to the load and therefore more cataracting, which means more energy generation due to friction and impacts. This will lead to higher mill temperatures than at lower speeds. At supercritical speed, the outermost layers of the balls are centrifuged, which results in a decrease in mill diameter and reduction in the active mass of the load; this produces a rapid loss of power as mill speed increases (Moys and Skorupa, 1992). At the speed at which the load centrifuges completely, no power is drawn by the mill. This loss of thermal energy increases due to the increase in the overall heat transfer coefficient (which increases with mill speed) on the outside of the mill. Only when there is substantial slip between the load and liner will there be power increase and therefore temperature rise. It has been noted that liner design determine speed of the mill at which cataracting and centrifuging occur. If higher lifter bars are employed, centrifuging will occur at relatively lower mill speeds. Generally, load behaviour of rotary grinding mills determines the level of temperatures attained and is obviously fundamental to mill efficiency. Mill temperature is also a function of milling time. Longer milling times will result in more energy being transferred to the load and this will lead to higher mill temperatures. This situation applies only for a batch mill. At steady state, when there is no further increase in mill temperature with time, power supplied to the mill will be equal to the rate of energy loss from the mill. Viscosity, particularly in wet mills, is affected by temperature. The viscosity or percentage of solids in the mill affects most of the parameters within the mill, and also has a large effect on the dynamic behaviour of the load. Variations in temperature affect the slurry viscosity by affecting the viscosity of the water in the slurry (Kawatra and Eisele, 1998). An increase in temperature decreases the viscosity of the carrier fluid and thus the viscosity of the slurry. Small changes in temperature of the slurry in the mill, however, would not produce dramatic changes in the slurry rheology. Large variations in pulp 18 temperatures are needed to observe significant changes in slurry rheology. In view of this, it can be said that temperature has little effect on the milling process in particular on the rheology of the slurry. Changes in particle size and percent change in solids produce far more significant changes in the slurry rheology and such changes in these variables are more realistic than large changes in pulp temperature. The viscosity of the slurry affects load behaviour and density. It also has a strong influence on the grinding performance of the mill media. An excessively high viscosity implies thick slurry inside the mill. The slurry effectively separates the grinding media such that the grinding performance decreases. On the other hand, when the viscosity is too low, the grinding media are allowed to come into direct contact and this results in excessive wear of grinding media. Therefore any shift from the operating mill temperature will affect the viscosity and therefore mill load behaviour. However it is worth stating that the effect of temperature on slurry viscosity may not be significant compared to the effect of percent solids. Van Nierop and Moys (2001) showed that the temperature in a 1.3MW gold mill increases from the feed end to the discharge end. They showed that the temperature difference along the length of the mill was approximately 5.5?C. This indicated that the mill is not perfectly mixed. When there is no flow through the mill the temperature of the middle of the mill stayed warmer than both the feed and discharge probe temperatures. This is because of a lower rate of energy transfer from the middle of the mill (as expected) than at the two ends of the mill. Energy is lost through the ends of the mill as well as through the shell. Van Nierop and Moys (1997) investigated the changes of temperatures in the mill in relation to other variables, principally mass and power. It was indicated that when the solids feed is switched off and the mass decreases, then the temperature increases, tending towards a steady state. The absence of new cold feed no longer keeps the temperature in the mill down. Where the feed is switched on, the mill load mass increases, and the temperature initially decreases, due to the cold feed, and then increases to a new steady state. When the mill is stopped, the temperature slowly decreases. This investigation showed that load temperature measurement, 19 together with other temperatures and flows, makes the calculation of the axial mixing coefficient possible. The amount of material present in a mill at a particular time and the magnitude of in and out-flows thus influences temperature in the mill. The higher the mill filling, the higher the temperature of the milling system. Mill filling is an important parameter in the control of temperature rise in the grinding mill. 2.5 Energy consumed in milling Grinding is a very inefficient process and it is important to use energy as efficiently as possible. Unfortunately, it is not easy to calculate the minimum energy required for a given size reduction process, but some theories have been advanced which are useful. Austin et al (1984) from data given by Rose (1967), showed by careful measurement of the energy balance in a mill, that the surface energy is only a very small fraction of the energy input to the mill. Within the limits of experimental error, he found that all energy to the mill appears as heat, sound, or the energy of phase transformation. Austin et al (1984) from data reported by Austin and Klimpel (1964) discussed the fracture process from the point of view of the utilization of energy in creating new surface. In any fracture process the solid must be raised to a state of strain to initiate the propagation of fracture cracks. Creation of this state of strain requires energy, greater than or equal to the stored strain energy. Whether fracture initiates at low strain energy or at high strain energy depends on a number of factors in addition to the value of the specific surface energy of the material. This includes: (i) The presence of pre-existing cracks or flaws; (ii) Whether plastic flow can occur in the solid or whether it is completely brittle; and (iii) The geometry and rate of stress application. 20 This picture applied to compressive or impact fracture in a milling machine immediately explains why the input energy appears primarily as heat. In a tumbling ball mill, for example, the mill energy is used to raise balls against the force of gravity, and when the balls fall the energy is converted to kinetic energy. The balls strike particles, nipping them between balls, and the kinetic energy is converted to strain energy in the particles. If the impact force is sufficient, a particle is rapidly stressed to the fracture point and it breaks via propagating branching fractures, giving rise to a suite of fragments. Each fragment converts the remainder of its stored strain energy to heat after fracture (Austin et al, op cit). The fraction of mill energy not converted to heat is very small. As additional confirmation, it is known that the heat rise of a material flowing in a continuous mill can be calculated quite accurately by direct conversion of the energy input rate to the mill to the sensible heat content of the material. It follows that the utilization of energy in a mill is extremely inefficient. Fuerstenau and Abouzeid (2002) reported, with regard to energy of comminution, that the well-known laws of comminution given by Kick and Bond have been the basis for assessing comminution energy, and indirectly, comminution efficiency for ball mills only. Rittinger considered that the required energy for comminution is proportional to the new surface produced. Kick considered that energy for comminution is determined by the energy required to stress the particle to failure. Fuestenau and Abouzeid, (op cit), defined the comminution efficiency as the ratio of the energy of the new surface created during size reduction to the mechanical energy supplied to the machine performing the size reduction. In terms of this concept, the energy efficiency of the tumbling mill is as low as 1%, or less. 2.6 The Discrete Element Method The Discrete Element Method (DEM) is a numerical method that is used to track the motion of individual balls in grinding mills. The DEM simulations can be used 21 to model the behaviour of both fluids and contacting particles. The simulations have wide applications in both mining and mineral processing (Zhang and Whiten, 1996). The discrete element method simulates the mechanical response of the systems by using discrete elements. In this method, the forces between assumed discrete components are calculated and used to determine the motion of discrete components thus giving a dynamic simulation. During the simulation process, the simulation time is discretized into small time steps. The motion of each particle boundary is calculated. The positions of these particles and boundaries are updated at each time step. There are a number of possible contact force models available in the literature that approximate collision dynamic. Cleary (2001) used the linear spring-dashpot model shown in Figure 2.3 below. Figure 2.3: Contact force law consisting of springs, dashpots and frictional sliders used for evaluating forces between interacting particles and boundaries ?Soft Particle? contact model Fn Ft Dash pot Spring Frictional sliders 22 x? is the particle overlap, nk and tk are the normal and tangential spring constants, nv and tv are the normal and tangential velocities, nC and tC are the normal and tangential damping coefficients, and m is the friction. Normal Force: nnnn vCxkF +??= Tangential Force ( )? += ttttnt vCdtvkFF ,min m Cleary (op. cit.) suggested that the normal force consists of a linear spring to provide the repulsive force and a dashpot to dissipate a proportion of the relative kinetic energy. The maximum overlap between particles is determined by the stiffness nk of the spring in the normal direction. In the case of a tangential force, the force vector tF and velocity tv are defined in the plane tangent to the surface at the contact point. The integral represents an incremental spring that stores energy from the relative tangential motion and models the elastic deformation of the contacting surfaces, while the dashpot dissipates energy from the tangential motion and models the tangential plastic deformation of the contact (Cleary, op. cit.). This DEM code has been used extensively for studying load behaviour in grinding mills. It can provide information on velocity profiles of each ball. It can be used to simulate load behaviour and therefore energy transfer in the grinding mill. In this study the DEM has been used to provide ball velocities and estimates of the number of balls in contact with the liners as well as the number of balls in contact with the air that is above the load. 2.7 Energy losses in a milling system There are considerable energy losses in any practical milling processing of which some of the most important are as follows (Stairmand, 1975): 23 (i) Energy converted into sound, heat lost in the motor and mechanical losses in the mill bearings and gear box. (ii) The actual mechanical efficiency of energy transmission to the grinding elements (e.g. the balls or beaters in a mill) will be less than 100%; losses are likely in the motors and in the transmission system. (iii) Some energy is used to wear the liners by impact and abrasion due to friction and some is lost in inter-particle friction. (iv) Not all of the energy transmitted to the grinding elements will reach the particles; a considerable part may be lost in inter-particle friction in the grinding zone. (v) Not all of the energy reaching the particles will be usefully employed in comminution. In a ball-mill, for example, the balls may be so small that they possess insufficient energy to break the particles to be comminuted, or so large that part of their energy is wasted in re-aggregating crushed particles. (vi) The conditions in the mill may be such that stresses are not applied in the most efficient manner e.g. stressing of semi-plastic materials may be too slow, so that energy is lost in plastic deformation. (vii) Auxiliaries such as air-sweeping systems and classifiers may be employed, in attempts to improve grinding conditions in the mill; while no doubt these will increase the actual grinding efficiency, they consume power, which must be debited to the process. 2.8 Analysis of conduction and convective heat transfer 2.8.1 Conduction Heat Transfer Consideration is now given to the analysis of conduction heat transfer in solids and fluids. The general theoretical analysis of conduction-heat-transfer problems involves (1) the use of (a) the fundamental first law of thermal dynamics and (b) the Fourier law of conduction (particular law) in the development of a 24 mathematical formulation that represents the energy transfer in the system; and (2) the solution of the resulting system of equations for the temperature distribution. Once the temperature distribution is known, the rate of heat transfer is obtained by use of Fourier law of heat conduction (Thomas, 1980). A simple practical approach to the analysis of basic steady state conduction- energy-transfer problems has been developed which involves the use of an equation developed from the fundamental and particular laws. This practical equation for conduction heat transfer takes the form: ( )21 TTL kAq ?= (2.6) Where q is the rate of heat conducted from a surface at temperature T1 to a surface at temperature T2, A is the heat transfer area and L is the thickness of a surface. The practical approach to the analysis of conduction-heat-transfer will be developed for multidimensional systems as in a mill under consideration in the following sections. 2.8.2 Convection Heat Transfer Convection is the transfer of heat from a surface to a moving fluid or vice versa. The conduction heat transfer mechanism plays a primary role in convection. In addition, the thermal radiation-heat transfer mechanism is also sometimes a factor. The theoretical analysis of convection requires that the fundamental laws of mass, momentum, and energy and the particular laws of viscous shear and conduction be utilized in the development of mathematical formulations for the fluid flow and energy transfer. The solution of these provides predictions for the velocity and temperature distributions within the fluid, after which predictions are developed for the rate of heat transfer into the fluid by the use of Fourier law of conduction. The practical approach to the analysis of convection heat transfer from surfaces has been developed which employs an equation of the form: 25 ( )Fss TThAq ?= (2.7) Where q is the rate of heat transferred from a surface at uniform temperature Ts to a fluid with reference temperature TF, As is the surface area, and h is the mean coefficient of heat transfer. Equations (2.6) and (2.7) will be useful in the next section and in the derivation of the steady state temperature models. The most important aspect to this study is the energy loss through the mill shell. This loss of energy is a function of mill speed, mill filling and the liner/lifter design. Most of the energy is lost from the mill in the form of thermal energy through the mill shell by conduction, convection and radiation, and also as sound energy (noise) from the mill. Energy lost in the form of heat can be modeled as a combined form of energy transfer in most cases by conduction and convection, and also radiation. Energy loss by radiation is applied where higher temperatures are involved and therefore is not considered in this Dissertation. Figure 2.4 below describes the physical situation. 26 Figure 2.4: Temperature profile in a grinding mill. The temperature profile on a liner with a lifter will be generally different from the temperature profile on liner position. The energy lost through the liner position of the mill lining would encounter a resistance due to the inside resistance, flat metal sheet lining, air gap (or cork), mill shell and the resistance from the air around the Ambient Mill Grinding Chamber Liner Shell Air-gap Liner Mill Shell Ambient Mill Grinding Chamber Mill radius (m) T 27 mill shell. It has however been found that the temperature of the shell behind the lifter and that of the shell behind the liner were the same. The energy lost through a liner with a lifter encounters a load-lifter resistance, transfer across the mill liners, transfer across the air-gap under the liners, transfer across the mill shell and transfer across the air on the outside of the shell. These boundaries will provide resistance in series to the flow of energy. The total resistance is the sum of these resistances in series, which will be equal to the overall resistance and therefore an overall heat transfer coefficient for the mill wall and fluid boundary layers can be defined. The energy flow leaving the mill must be equal to the energy entering the environment, since steady state precludes energy storage. The thermal energy will flow radially through the mill wall. On the outside shell of the mill, the energy transfer coefficient is influenced by the speed of the mill. Inside the mill, the thermal energy transfer coefficient is likely to be affected by the mill load mass or volume, the mill speed and the flow pattern of the air in the mill. The energy transfer is possibly also a function of the voidage which increases with increase in mill speed. Figure 2.4 shows the expected temperature profile of a grinding mill at steady state. The temperature of the mill decreases with increasing mill radius. For the entire boundary, R TQ ?= , where R is the total resistance and R = Rpar + R2 + R3 + R4 + R5 (2.8) RPAR, RL, Rg, RS and Rext are unequal as a result of differing conductivities and thickness; the ratio of the temperature difference across each layer to its resistance must be the same as the total temperature difference is to the total resistance, i.e. extSgLPAR R T R T R T R T R T R TQ 54321 ?=?=?=?=?=?= (2.9) 28 The individual heat transfer coefficients are a function of a number of variables, such as the fluid velocity, density, viscosity, mill speed, the load volume and the flow pattern of the air inside the mill. On the outside of the mill shell however, the outside heat transfer coefficient is a function of the mill speed only. If the mill is not enclosed in a building, wind speed would also be a variable. Generally, heat transfer by convection occurs as a result of the movement of fluid on a macroscopic scale in the form of circulating currents (Coulson and Richardson, 1999). If the currents arise from the heat transfer process itself, natural convection occurs. In forced convection, circulating currents are produced by an external agency such as an agitator in a reaction vessel or as a result of turbulent flow in a tube. In general, the magnitude of circulation in forced convection is greater, and higher rates of heat transfer are obtained than in natural convection. In most cases where convective heat transfer is taking place from a surface to a fluid, the circulating currents die out in the immediate vicinity of the surface and a film of fluid, free of turbulence, covers the surface (Coulson and Richardson, op. cit.). In this film, heat transfer is by conduction and, as the thermal conductivity of most fluids is low, the main resistance to transfer lies there. Thus an increase in the velocity of the fluid over the surface gives rise to improved heat transfer mainly because the thickness of the film is reduced. As a guide, the film coefficient increases as (fluid velocity)n, where 0.6??? ? ??? ? t g p r r where Ret is the Reynold?s number based on tube diameter, Dt, while kg, ?p, ?g, ?g, and dp are the gas thermal conductivity, particle density, gas density, gas viscosity, and mean particle diameter, respectively. One of the most reliable equations used to correlate the heat transfer coefficients as given by Coulson and Richardson (1991) was obtained from data given by Dow and Jakob. Their equation is: ( ) 80.025.017.065.0 155.0 ??? ???? ? ?? ????? ??? ??? ??? ??? ??? ? = m r r r tc p ssttt du ce ce d d l d k hd (2.19) where: h is the heat transfer coefficient, k is the thermal conductivity of the gas, d is the diameter of the particle, dt is the diameter of the tube, l is the depth of the bed of solids in the tube, e is the voidage of the bed, ? is the density of the gas, ?s is the density of the solids in the bed, cs is the specific heat of the solid, cp is the specific heat of the gas at constant pressure, ? is the viscosity of the gas, and ?c is the superficial velocity based on the empty tube. 2.8.3 The Practical Significance of the Overall Heat Transfer Coefficient The overall heat transfer coefficient, the total surface area of heat transfer and the temperature difference between the load temperature and the environment (ambient) will predict the rate of energy loss to the environment. The overall heat transfer coefficient, U, is a measure of heat conductivity for steady state heat transfer through the convective heat transfer coefficient on the inside of the mill, across the liners, through the small air gap between liner/lifter and the shell, across the mill shell and through the convective film of air on the outside of the shell. Knowledge of the overall heat transfer coefficient is as important for the examination of heat transfer processes, as is an understanding of the Ohm?s Law for electrical engineering. It embraces all heat transmission processes through plane or slightly curved walls consisting of one or more layers (Schack, 1965). 32 The concept of the overall heat transfer coefficient is best understood by analogy with the components of electrical conductance. According to Ohm?s Law, current intensity in a given cross-sectional area of a conductor is proportional to the potential and inversely proportional to the resistance. In the above statement, the resistance signifies the sum of the individual resistances. This relationship is analogous to the basic equation of heat transmission. The overall heat transfer coefficient is almost equal to the smallest heat transfer coefficient, h. The reason is purely numerical. The reciprocal of the largest h is smaller than that of the smallest h. The reciprocals are the heat transfer resistances. 2.9 Estimation of model parameters using parameter estimation (Least squares filtering) The technique of ?least squares filtering involves the use of measurements, together with model predictions of these values, to calculate the difference between what is measured and what is predicted. This difference is then squared, and the sum of the squared errors for each of the measurement-prediction pairs is then obtained. This total ?sum of squared errors? is then minimized by manipulation of the variables. The minimization procedure is carried out until the desired convergence is achieved. The use of a computer in this application is essential. In particular, Microsoft?Excel is used to perform the sum of squares minimization. The solver function is employed to minimize the sum of squares, subject to constraints that are described below. Parameter estimation is a common problem in many areas of process modeling, both in on-line applications such as real time optimization and in off-line applications such as the modeling of reaction kinetics and phase equilibrium. The goal is to determine values of model parameters that provide the best fit to measured data, generally based on some type of least squares filtering. In this study, mathematical models of temperature behaviour in a batch-grinding mill have been developed. The parameters within the model are found which 33 minimize the sum of squares of the errors between the temperature predicted by the mathematical model and the measured temperatures. The expression used for finding this minimum of errors is: ? = ?= n i measTTS 1 2 mod )( (2.20) Where Tmod and Tmeas are the steady state model and measured temperatures respectively. The parameters of the mathematical model for temperature are found such that ?S? is minimized. In other words parameters are found by fitting the mathematical model. In this estimation the parameters estimated are heat transfer coefficient from which the overall heat transfer coefficient is evaluated. The other parameter such as power, sound, ambient temperature and initial temperature are taken from measurements. 2.11 Conclusions The first law of thermodynamics encompasses the three fundamental laws; the conservation of energy, mass and momentum. The conservation of energy is fundamental in the study of energy systems in milling particularly when used together with Fourier?s law of heat conduction. The energy balance could prove to be a competent tool in the optimization of mill control and operation. Mill power is an important variable in energy modeling in a grinding mill. The power transmitted to the mill is proportional to the lifting action. Load behaviour of rotary grinding mills determines the level of temperature attained and therefore is important as far as mill efficiency is concerned. In conclusion, the energy going into the mill is used to lift the load, break the particles and creating new surfaces, raising the mill discharge temperature, and raising the temperature of the mill and the air in the mill. However, energy is also lost in the form of sound but mainly as heat. Some energy is used to wear the 34 liners by impact and abrasion due to friction. Losses are also likely to occur in the motor, gear box and bearings, and generally in the transmission system. 35 Chapter 3 Development of a Mathematical Model _________________________________ 36 3.1 Introduction: Principles of mathematical modeling A mathematical model can be a very powerful tool in the development of new and existing metallurgical processes when in combination with careful experimental measurements (Szekely et. al, 1988). It is needed to describe how a process reacts to various inputs. However, it is not the only tool that the process engineer will need to employ in process analysis because a measurement program is normally necessary as well. These are measurement of concentration, temperature, and fluid flow or mixing (tracer tests) in an existing process. A mathematical model offers many advantages on the objectives of a study: (i) It can increase understanding of a process in fundamental terms. (ii) A model can be used for the purpose of process control and optimization. (iii) It can be an invaluable aid in scale-up and design. (iv) It can assist in the evaluation of results from in-plant trials, for example, the conversion of temperature measurements to heat fluxes as in this research. (v) A model can assist in making decisions on trial conditions in an experimental campaign in-plant. A mathematical model in general, is an equation or set of equations, algebraic or differential, together with initial and/or boundary conditions, that represents the important chemical and/or physical phenomena in a process. Mathematical models can be classified as Fundamental or Mechanistic, Empirical and Population- Balance models. Fundamental models are based on mechanisms involving transport phenomena and chemical reaction rates in a process. Chemical and physical laws are applied to obtain mathematical equations relating independent and dependent variables. Examples of the laws involved are conservation of mass, energy and momentum; Fourier?s law (for conduction heat transfer), and Fick?s law (for molecular diffusion). Frequently, empirical constants are required in the model to characterize phenomena that are very complicated and defy rigorous mathematical treatment. Empirical models rely on the determination of mathematical equations linking dependent and independent variables using 37 measurements of the variables in an operating plant. The fitting of the relationships is usually based on statistical analysis; but fundamental knowledge of the process may influence the form of the equations. However, unlike the fundamental model, the empirical model relies much less (often hardly at all) on an understanding of the mechanisms involved in the process. This type of model has been used extensively by mineral engineers to describe such processes as flotation, crushing and grinding. The population-balance model has been used extensively in the mineral-processing field in recent years. It has been developed to describe processes having distributed properties, such as particle size in a ball mill. In this study, fundamental or mechanistic models are employed to develop the mill temperature model. 3.1.1 The Energy Balance The temperature profile within a body depends upon the rate of internal generation of energy, its capacity to store some of this energy, and its rate of thermal transfer by conduction and convection to its boundaries (where the heat is transferred to the surrounding environment). This can be written in the form of an energy balance as follows: Rate of energy into reactor = rate of energy out of reactor + rate of energy accumulation in the reactor (3.1) The behaviour of temperatures in a mill can be obtained from an energy balance methodology. The state variables of interest for the mill temperature model are the temperatures of the load, the air above the load, the liners and the shell. By definition a state variable is a variable that arises naturally in the accumulation term of a dynamic energy or material balance. It is a measurable quantity that defines the state of a system. The overall energy balance model considers an energy balance problem wherein energy coming into the batch mill is only in the form of power, P transmitted to 38 the mill (refer to figure 3.1). This energy is transferred to the load at the load-liner interface. Due to the tumbling action (cataracting and cascading leading to impacts and collisions), the load gains thermal energy which is accompanied by heat dissipation to its environment through the mill wall and attains a temperature of T(t) after time t. Some of the energy is also converted to sound. The energy loss through the mill wall occurs following Fourier?s theory of heat transfer with the temperature on the outer wall being cooled by air draft. Work is also done between the mill wall and the surrounding air due to friction. Some of the energy is also lost in the motor as well as mill bearings and gearbox. These energy losses are catered for in the no load power in section 4.5.1 of chapter 4 during load beam torque calibration. At steady state when there is no further increase in mill process temperatures, the energy input to the mill equals the energy loss from the mill. The temperatures become constant at steady state. Figure 3.1: Batch grinding mill: Energy balance measurements 3.1.2 Assumptions made in the derivation of the model The following assumptions were made in the derivation of the model: Pin Qout Sout 39 (i) It is assumed that all the power P is transmitted to the load; ignoring the fact that some frictional energy is actually dissipated at the liner surface (ii) The thickness of the mill wall, ?r, is small compared to the radius of the mill. (iii) The process reaches steady state after milling for some time. (iv) The overall heat transfer coefficient is constant at steady state and so are the individual heat transfer coefficients on the inside and out of the mill wall. (v) Heat dissipation occurs through the mill wall by conduction and convection. 3.1.3 Model Derivation The technique used in this study utilizes an energy balance that is applied to the load, the air above the load, liners and mill shell. The temperature measurements made in the mill included the temperatures of the load inside the mill, the air above the load, the liners, the shell, and ambient temperature. These temperatures (with exception of the ambient as it is not a state variable in this case) are measured and modeled, and therefore estimates of the heat transfer coefficients, the overall heat transfer coefficient and an estimate of the rate of thermal energy losses from the mill can be quantified. The steady state temperature models are of value in this study as our design variable, the overall heat transfer coefficient, are evaluated at steady state temperature. Another important measurement considered in this study is the power transmitted to the mill. This is considered as the only source of energy going into the mill. This is determined from the load beam calibrations as given in section 4.5.1 of chapter 4. The measurement of this variable is explained more in chapter 4. The model is based on a horizontal batch grinding mill shown in Figure 3.1. The overall energy balance is constructed around the entire grinding mill as follows: 40 dt tdEtStQtP )()()()( ++= (3.2) Where: Q is the rate of thermal energy loss through the mill wall S is the rate of energy conversion into sound P is the net power drawn by the mill and transmitted to the grinding media. This is the main term responsible for the temperature increase of the whole milling system. noloadloadnet PPP ?= The no load power incorporates the energy lost in the motor, mill bearings and gear box, as well as work done due to friction between the mill and the air as discussed earlier. This is measured by running an empty mill and determining the power drawn by an empty mill. dt dE and Q are functions of temperature, i.e, ( )[ ] dt TTCmd dt dE irefiipiN i , 1 ? = ? = (3.3) Where: Ti is the temperature of the component under consideration, i.e., the temperature of the load (i=1), air-above lifter (i=2), liner (i=3) and shell (i=4). Tref is the reference temperature and is taken to be zero. Equation (3.3) is a lumped model for all the mill components under consideration. These being: the load, air above the load, the liners and the shell. Therefore, the term mCp incorporates the mass and specific heat capacity of the ore, balls, the air above the load, water (in the case of a wet mill), liners and the mill shell. In this project, dry balls only experiments are employed. Of interest was the modeling of energy loss to the environment and not the energy consumed in comminution. 41 3.2 Energy balance models for the mill temperature The dynamic models are derived on the basis of Figure 3.2 below. The figure shows the schematic representation of the cross-section of the mill in a horizontal position. The presence of the air-gap between the liners and the shell offers resistance to the flow of energy and therefore a temperature drop exists. The energy balances are performed between the active load, the air above the load, the liners and the environment. The objective is to determine a model for the overall heat transfer coefficient as a function of the load volume, mill speed and the design of the liners and shell so as to quantify energy loss from the mill. The heat transfer coefficients governing transfer between these phases and the overall heat transfer coefficient U was correlated with mill speed and load volume. The data gathered in the milling experiments together with an energy balance model for temperatures in the mill was used to obtain estimates of interphase heat transfer coefficients and the overall heat transfer coefficients. The energy loss via convection through the mill shell is accounted for by quantifying the overall heat transfer coefficient of the shell. The ambient temperature allows the model to account for convective loss. U increases with both load volume and mill speed. 42 Figure 3.2: Schematic diagram of the orientation of the load, air above load, liner/lifters and mill shell. From Figure 3.2, it can be observed that thermal energy leaving the mill follows two paths. The first path involves energy leaving the load and passing through the liners, air gap and mill shell to the environment. In the second path, the energy is transferred from the load to the air above the load. From the air above the load the energy enters the liners through the mill shell to the environment. In other words two parallel paths of energy transfer are established in this system. The modeling of the parallel path is performed in the following section. The following is the mechanism of energy transfer in the grinding mill. (1) The direct transfer of energy from the load to the liner by convection and conduction. The model is given by: )()( LLDLLDLLD TThAQ ?= ?? (3.4) (2) Thermal energy transfer from the load to the air above the load by convection: LOAD AIR ABOVE LOAD QLD-AIR Q LD - L Q ext QAIR-L Q ext Environment Mill Shell Liner 43 )()( AIRLDAIRLDAIRLD TThAQ ?= ?? (3.5) (3) Transfer of energy from the air above the load to the liner by convection: ( )LAIRLAIRLAIR TThAQ ?= ?? )( (3.6) (4) Transfer of thermal energy across the liner by conduction: L L LLL L Ax TTkQ )( 21 ?= (3.7) (5) Transfer of energy from the liner to the mill shell across an air-gap (or across a gasket between the liner and the shell) by conduction: g SLgg g x TTAkQ )( 12 ?= (3.8) (6) Transfer of thermal energy across the mill shell: s SSss S x TTAkQ )( 21 ?= (3.9) (7) Transfer of thermal energy from the mill shell to the environment by forced convection: )( 2)( AmbSextext TThAQ ?= (3.10) It is assumed that the temperature of the shell TS2 is assumed to be constant for the entire shell. This will not hold in a continuous mill because the temperature will be changing from the feed end to the discharge end. Since at steady state, extsgL QQQQ === (3.11) In terms of the overall heat transfer coefficient, 44 ( )AmbLDext TTUAQ ?= )( (3.12) 3.3 Model for the overall heat transfer coefficient as a function of important variables: mill speed, load volume and mill liner /shell design The overall heat transfer coefficient is a function of the individual heat transfer coefficient both in the inside of the mill as well as outside the mill shell. The heat transfer coefficient is influenced by a number of factors: these are: the speed of the mill, the load volume and the physical properties of the fluid involved. The heat transfer coefficient from the load to liner, load to air and air to liner is affected by the velocity of the air inside the mill as well as the mill load volume. On the outside of the mill shell, the heat transfer coefficient is a function of the peripheral velocity of the mill and the thermal physical properties of the air in contact with the outside of the mill shell. It has been given in section 2.8 of chapter 2 that generally, the film heat transfer coefficient on a surface increases as (fluid velocity)n, where 0.6