Browsing by Author "Govender, Indresan"

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    Open Access
    Axial segregation of granular flows in rotating drums
    (2017) Ahmed, Elbasher M E; Govender, Indresan
    A mechanistic model of axial segregation in rotating drums is presented for mixtures of granular material. We show that grains composed of different material properties which are heuristically argued to manifest as differences in frictional properties at the continuum scale-diffuse into axial bands as a consequence of concentration fluctuations in the free surface layer caused by friction- limited mobility. The model is composed of two new ingredients that success- fully recover the well-known phenomenon of axial banding and subsequent band coarsening in the long-time evolution: (1) A Bagnoldian stress assumption facilitated band formation at drum fill levels < 50% that have hitherto not been possible with a Newtonian shear stress ansatz, through the formation of an asymmetric free surface profile that is experimentally verified by the nuclear imaging technique of Positron Emission Particle Tracking. (2) The geometric slope corresponding to the gravity-driven axial flux between adjacent bands is (binomially) expanded to second order thereby ensuring band coarsening in the long-time limit for all boundary conditions investigated. Numerical implementation of the axial diffusion model is shown to be sensitive to the boundary conditions of the tumbling mill (particle size, volume concentration, mill speed, average friction, end wall friction, drum length and drum diameter). The explicit solutions for a binary-, ternary-, quaternary- and n-species granular mixture was developed with numerical implementation up to n = 4. Consistent with experimental observations in the literature, the 3-species mixture successfully captures bands within bands, while the 4-species mixture successfully recovers bands within bands, within bands.
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    Circulation rate modelling of tumbling mill charge using positron emission particle tracking (PEPT)
    (2013) Von Kallon, Daramy Vandi; Govender, Indresan; Mainza, Aubrey
    This research is focused on developing theoretical understandings of charge circulation trends as observed in tumbling mills at different operating conditions. Of particular interest is the underlying assumptions being made by many mill models that a particle imparts energy for potential breakage only once per revolution of the mill to the charge body – that is, that the circulation rate of mill charge can be assumed to be constant irrespective of the speed at which the mill is run. The trajectory data used in this thesis is derived from positron emission particle tracking (PEPT) experiments conducted at the University of Birmingham positron imaging centre and further experiments were conducted at the iThemba LABS in Cape Town. The experimental approach is highly suited to allow the effective examination of the assumption that the grinding charge in these mills circulates at a constant rate of unity.
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    Dense granular flow in rotating drums: a computational investigation of constitutive equations
    (2018) Povall, Timothy Mark; Govender, Indresan; McBride, Andrew; Reddy, B Daya
    The constitutive laws of dense granular flow are investigated. Simulations of a drum, with periodic boundary conditions, rotating at varying speeds are performed. From the resulting data, kinematic and kinetic fields are extracted and used to investigate the validity of constitutive relations proposed in the literature. Two key constitutive assumptions are (a) isotropy and (b) incompressibility. The rotating drum system is found to be largely isotropic for high rotational speeds. For low rotational speeds, anisotropy is observed in the bottom part of the system, where the particles are flowing upwards. A small degree of compressibility is observed in the downward-flowing layer. The friction coefficient for the granular constitutive relations is also investigated. An empirically-derived friction law has a better fit to the data when compared to other friction laws proposed in the literature. Lastly, two scaling laws are investigated: the scaling between the scaled flow-rate (flux) and the thickness of the downward- flowing layer and the scaling between the dynamic angle of repose of the bed and the flux through the downward- flowing layer. The thickness-flux scaling is measured by interpolating the flux over a number of slices through the flowing layer, this is done in a number of different ways. The size of the measured section through the flowing layer is varied. The orientation of the slices is also varied. Also investigated is whether the total velocity or the tangential velocity produce the same scaling. The size of the section of the flowing layer significantly changes the scaling, this shows that the scaling is not constant throughout the flowing layer. The dynamic angle of repose is determined using two methods, one which is determined unambiguously as the repose angle of the ellipse fitted to the equilibrium surface and the other which is the changing angle of the tangent to the equilibrium surface or free surface. The first repose angle is found to be highly dependent on the flux even in the limit of infinite drum length, which is modelled using axial periodic boundary conditions. The second definition results in two sets of repose angles with complex behaviour that may be due to inertial effects. An instability in the system is observed, this is conjectured to be due to a frictional threshold that is breached as the rotational speed of the drum increases. Algorithms for calculating field variables and features of the charge are presented.
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    Developing a mechanistic model for flow through a perforated plate with application to screening of particulate materials
    (2016) Ogunmodimu, Olumide; Mainza, Aubrey; Govender, Indresan; Franzidis, Jean-Paul
    Screening in mineral processing is the practice of separating granulated ore materials into multiple particle size fractions, and is employed in most mineral processing plants. Models of screening performance have been developed previously with the aim of improving process efficiency. Different methods have been used, such as physical modelling, empirical modelling, and mathematical modelling including the discrete element method (DEM). These methods have major limitations in practice, and experimental data to validate the models have been difficult to obtain. Currently, the design and scale-up of screens still relies on rules of thumb and empirical factor methods rather than a fundamentally based understanding of the behaviour of the granular system. To go beyond the current state-of-the-art in screen modelling requires a clear understanding of the particle motion along a dynamic (vibrating) inclined plane. Central to this understanding is the notion that granular systems exhibit a unique rheology that is not observed in fluids; i.e. neither Newtonian nor non-Newtonian. It is thus imperative to fully quantify the granular rheology, which is determined by the depth of the particle bed along the screen, the solids concentration, and the average velocity of the granular avalanche on the screen. The concept of granular rheology is important. Existing empirical models of vibrating screens tend to be extremely dependent on boundary conditions of a particular machine design. The concept of granular rheology is important because, akin to fluid flow, granular flow exhibits different flow regimes depending on the extent of energy input in the system. This work employed DEM to quantify the granular rheology of particles moving along a vibrating inclined screen in order to begin the development of a phenomenological model of screening. The model extends the visco-plastic rheology formation of Pouliquen et al. (2006) to capture the kinetic and turbulent stresses obtained in granular flow on an inclined vibrating screen. In general, DEM was employed to develop a mechanistic model of screening which includes a description of the rheology of granular flow on a vibrating screen. Microscopic properties of granular flow were used in DEM to simulate screening of particulate materials. Granular mixtures of two particle constituents (3 mm and 5 mm) were simulated on an inclined vibrating screen of 3.5 mm apertures. For the base case, frequency and amplitude are 4 Hz and 1 mm, respectively. While microscopic properties were employed for the simulation, the properties extracted from the simulations are macroscopic fields which are consistent with the continuum equations of mass, momentum and energy balance. From the continuum equations, a micro-macro transition method called the coarse-graining approach was employed to obtain the volume fraction and the tangential velocity as a function of the depth of flow along the inclined surface. This approach is suitable for this work because the produced fields satisfy the equations of continuum mechanics; even near the base of the flow. The continuum analysis of the flowing layer reveals a coexistence of flow regimes- (i) quasi-static, (ii) dense (liquid-like), and (iii) inertial. The regimes are consistent with the measured solids concentrations spanning these regimes on inclined vibrating screens. The quasi-static regime is dominated by frictional stress and corresponds to low inertial number (I). Beyond the quasi-static regime, the frictional stress chains break and the collisional-kinetic and turbulent stress begin to dominate. The variation of the effective frictional coefficient with the inertial number (I) characterises the flow. Finally, an effective frictional coefficient model that is based on frictional, collisionalkinetic and turbulent stress was developed. Data analyses for this model were done at a steady flow in the base case where a coexistence of three flow regimes were observed. It was observed that each regime of flow is dominated by corresponding shear stresses. While the quasi-static regime is dominated by frictional stress, the kinetic and the inertial regimes are dominated by kinetic and turbulent shear stresses, respectively. This model was tested by varying the intensity of vibration in the base case and it was observed that at higher frequencies and amplitudes, the quasi-static regime gradually disappeared. Furthermore, the inertial number at which transition occurs to different regimes varies in response to the intensity of vibration. This is an important step in developing a phenomenological model of screening. The model presents a fundamental understanding of the mechanisms governing transport of granular matter on an inclined vibrating screen.
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    Developing a methodology for characterising in-situ viscosity profiles in tumbling mills
    (2012) Mangesana, Nobathembu; Mainza, Aubrey Njema; Govender, Indresan
    Milling is the most expensive operation in a mineral processing circuit and accounts for the highest amount of energy utilized. The efficiency of the milling process is affected by the rate of breakage of bigger rocks and the rate of transport of the slurry within and out of the mill. The transport of material in the mill is dependent on the rheological properties of the slurry such as viscosity. Viscosity is therefore one of the important parameters that influences the transport of the material in the mill. Slurries in tumbling mills are known to exhibit non-Newtonian behaviour. This means slurry viscosity is not constant but is a function of shear rate. To characterize slurry transport in the mill it is necessary to study the viscosity distributions in order to determine regions of high and low resistance to flow in the mill. To determine the influence of slurry viscosity on the transport of particles it is important to quantify the typical shear rate ranges that occur inside the tumbling mill. The aim of this project w as to develop a methodology for characterising viscosity distributions inside a tumbling mill using in-situ shear rate distributions obtained from PEPT at different solids concentrations and mill speeds. The P EPT technique was used to study the in-situ mot ion of slurry particles with in tumbling mill charge using a single radioactive tracer. Rheology experiments were conducted using a U-tube rheometer. The rheometer experiments were performed to characterise the rheological behaviour of the El Soldado slurry at a wide range of shear rates. The PEPT results provided information about the typical shear rate ranges that occur inside the mill. The combination of these results enabled the quantification of viscosity distribution from the slope of the rheogram at typical shear rates found in the tumbling mill. The rheology results indicated that the Bingham model is the rheological model which gives the best description of the rheology of El Sol dado slurry. It had the highest R² adjusted values at all tested solids concentrations concentration ranges. The rheological behaviour of El Soldado slurry with a particle size fraction of -75+53 microns can be described as Newtonian. This means that the viscosity is constant and does not vary with shear rate. At low solids concentrations up to 30wt%, the viscosity values attained are equivalent to that of water at room temperature at 0.001 Pa.s. At higher solids concentrations up to 60wt%, the viscosity increases to 0.007 Pa.s for the shear rate range tested. The Bingham viscosity and yield stress increases in an exponential form with increasing solids concentration. PEPT experiments were conducted to quantify and characterise in-situ shear rates in a laboratory scale tumbling mill. The highest shear rates were obtained at the lowest solids concentration for both mill speeds. It was 30s-¹ at 60% critical mill speed and 36s-¹ at 75% critical mill speed. Mill shear rates decreased with increasing slurry solids concentration. This was attributed to increased particle-particle interactions and reduced voidage. There is less volume available for particle shearing. The magnitude of the maximum shear rate is higher for the 75% critical mill speed at all slurry solids concentrations compared to the shear rates at the lower mill speed. At a higher speed the mill charge is fairly dilated by the strong centrifugal effects that oppose the natural packing structure which results in an overall decrease in bulk density. The increased voidage allows more volume for relative motion, producing an overall increase in shear rate. The narrow shear rate range and the Bingham rheological behaviour of the slurry resulted in a constant viscosity value that could be used in a viscosity model provided the solids concentration is uniform across all regions of the mill. Viscosity is more significantly impacted by solids concentration than shear rate distributions in the tumbling mill. Work should be done to study the solids concentration profiles in the tumbling mill to provide better insight on areas of active transport.
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    Evaluating the influence of lifter face angle on the trajectory of particles in a tumbling mill using PEPT
    (2014) Takalimane, Motena; Mainza, Aubrey; Govender, Indresan
    The work performed in this thesis was aimed at evaluating the influence of lifter face angle on the charge kinematics for a laboratory scale mill. The study involved tracking a single particle representing the ensemble using the Positron Emission Particle Tracking (PEPT) to obtain the location of the particle with time. The particle was radiated with a radionuclide; ⁶⁸8Ga, which has a half-life of 68 minutes. The objectives of the study involved tests with different lifter face angles at different mill speeds and volumetric mill filling. After performing the tests the data was analysed to obtain probability density distributions for each test conditions from key charge descriptors. Charge descriptors such as the Centre of Circulation (CoC), shoulder angle, toe angle, the free surface and also kinematic information such as the velocity profile along a carefully chosen radial line from the centre of the mill that passes through the CoC were obtained. The time averaged velocity data was used when assessing the influence of the lifter face angle on the velocity profile. The results showed notable effects of lifter face angle on charge characteristics. No real definitive trend was observed for the CoC as the lifter face angle was altered at all mill speed and filling conditions. However, the CoC showed an outward shift toward the mill shell with an increase in mill speed but an inward shift toward the mill centre with increase in charge filling degrees. Mill speed is expected to cause a load expansion as the charge approaches centrifugation.
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    A GPGPU implementation of the discrete element method applied to modeling the dynamic particulate environment inside a tumbling mill
    (2013) Hromnik, Marius; Govender, Indresan; Wheaton, Spencer
    Tumbling mills have been an integral part of the comminution circuit for more than a century. With the advent of better computing, discrete element modeling (DEM) has taken on the challenge to model the dynamic particulate environment inside these devices in the search for understanding and hence improving the process of the size reduction of ore. This process represents a large percentage of the energy consumption of a mine. In this work, a discrete element modeling tool was built on a GPU-based platform to perform simulations on a single commodity hardware PC. With a view to elucidating the governing mechanisms inside such devices, the extreme capabilities of the GPU are utilised to provide performance and accurate simulation. The simulation environment offers control that can never be achieved in an experimental setup. Notwithstanding, when agreement with physical experiment is achieved, confidence can be gained in the computational results. In this work the foundations and framework for a large scale GPU based discrete element modeling tool have been built with an emphasis on strict physics requirements, rather than on performance or appearance. In this regard we demonstrate the validity of the GPU implementation of a Hertz-Mindlin-based contact model.
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    A granular flow model of an annular shear cell
    (2016) Bremner, Sherry; Govender, Indresan; Mainza, Aubrey Njema
    Machinery such as an IsaMillTM used in communition to produce fine particle sizes that allow minerals to be extracted are best modelled using granular flows. A single rheological description that captures all the features of granular flows has not yet been realised, although considerable progress towards a complete theory has been made. Existing models of such horizontally stirred mills are empirical, tend to be extremely dependent on boundary conditions and do not allow for confident extrapolation beyond their window of design. As a first step to understanding the dynamics inside the IsaMillTM,a constitutive stress model of a horizontal annular shear cell is developed. This shear stress model was used in an athermal energy balance to develop a description of the power dissipation, which drives the communition purpose of the IsaMillTM. The key ingredients (velocity, shear rate and volume fraction distributions) to the granular ow model are extracted from experiments using Positron Emission Particle Tracking (PEPT), as well as Discrete Element Method (DEM) simulations. 5mm glass beads were used to fill an annulus 51mm wide. In the PEPT experiments, two different surfaces of the driving wall (the inner cylinder of the shear cell) were used, over two shearing velocities. The effect of two friction coefficients over a range of shearing wall velocities were examined in the DEM simulations. The data were examined over 3 selected radial lines and utilised to calculate the shear stress distribution and the power dissipation from the developed models. It was found that even the usually simple relations describing the dynamics within a vertical shear cell are greatly modified by changing the orientation of the rotation axis.
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    Granular flow modelling of rotating drum flows using positron emission particle tracking
    (2015) Pathmathas, Thirunavukkarasu; Govender, Indresan
    Tumbling mills are characterized by a flowing granular mixture comprising slurry, ore and grinding media. Akin to fluid flow, a rheological description underpinning granular flow has long been expected and pursued by many researchers. Unfortunately, no single theory has hitherto been able to successfully describe all the peculiar features and flow phases of granular systems. Tumbling mills exhibit a rich coexistence of all known flow phases and is arguably the most complicated of the granular flow geometries. Not surprisingly, current comminution models are almost entirely empirical with limited predictive capability beyond their window of design. Using Positron Emission Particle Tracking (PEPT) data we recover the key ingredients (velocity, shear rate, volume concentration, bed depth) for developing, testing and calibrating granular flow models. In this regard, 5 mm and 8 mm glass beads are rotated within a 476 mm diameter mill, fitted with angled lifter bars along the inner azimuthal walls and operated in batch mode across a range of drum rotation speeds that span cascading and cataracting Froude regimes. After averaging the PEPT outputs into representative volume elements, subsequent continuum analysis of the flowing layer revealed a rich coexistence of flow regimes - (i) quasi-static, (ii) dense (liquid-like), and (iii) inertial - that are consistent with the measured volume concentrations spanning these regimes in rotating drums. Appropriately matched constitutive choices for the shear stresses then facilitated the derivation of a new granular rheology that is able to (smoothly) capture all phases of the tumbling mill flow at transition points that match leading experimental findings reported in the literature. Limiting our models to athermal boundary conditions, we then derive the power density for better understanding of flow dissipation that ultimately drives the comminution purpose of tumbling mills. The rheology and power density models were then applied to the 5 mm and 8 mm glass bead data to reveal that shear power density is an order of magnitude larger than the normal component. Notwithstanding, the effective friction coefficient - which is akin to viscosity in typical fluids - remains relatively constant across most of the flowing layer with notable exponential growth across the interface from dense-to-inertial that continued into the inertial regime.
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    The influence of slurry viscosity on hydrocyclone performance
    (2012) Waters, Jason G; Mainza, Aubrey; Govender, Indresan
    Many of the empirical hydrocyclone cut size models available consider the feed solids content to be an important variable in determining the efficiency and resultant classification. However, feed viscosity as a whole is admittedly a more accurate variable to consider as it can be affected by many factors, including solids content, particle size distribution, particle size, mineralogy, pulp chemistry, particle shape and carrier fluid temperature. Earlier theoretical models did try to incorporate slurry viscosity but were developed for very low solids content systems due to the difficulty in measuring slurry viscosity at higher concentrations. Current hydrocyclone models being applied in industry have difficulty in predicting rheological fluctuations at constant solids content, for example with an increase in feed clay content. Investigations, focussed on expanding the studies of previous researchers in this area, were conducted. Experiments involved the design and operation of two test rigs incorporating three hydrocyclone sizes (75,100, 165 mm), with two different ore types (platreef ore, copper ore) from a secondary stage operation used as the feed material. A change in relative viscosity was investigated by altering the viscosity of the carrier fluid (water) by the addition of sucrose, and modification of the slurry temperature. Hydrocyclone feed flow rate and solids content were also modified. A custom made on-line tube rheometer allowed viscosities of feed concentrations of up to 43% (by vol.) to be measured over a range of shear rates (200/s to 1500/s). The slurries under investigation were found to be settling in nature, and therefore a decision was made to exclude data below the critical settling velocity of the tube. Rheological characterisations revealed both ore types exhibited Bingham plastic behaviour. A concentration versus viscosity relationship was determined from the rheological data and the resultant viscosity values were then linked to hydrocyclone efficiency. The significant findings of this work included the following: * Increased pulp viscosity achieved by 1) sucrose addition and 2) decreased slurry temperatures resulted in a drop in hydrocyclone performance attributed to the combined effect on the partition curve parameters namely - an increased cut size (d50c), decreased water split to O/F (C) and reduced value of alpha. *Increased pulp viscosity achieved by 3) feed solids content had dissimilar effects on the partition curve parameters. An optimum viscosity point was reached for water split to O/F and alpha parameters, however cut size increased with increased pulp viscosity. * Rheological effects on the cut size parameter appeared more significant for the largest of the three cyclone body diameters used in the study. This can be attributed to the decreased tangential velocities inside the larger cyclone radius. The combined rheology and hydrocyclone data from this thesis will provide useful validation data for the new hydrocyclone models currently being developed as part of the P9 project. The models are to be incorporated in the JKSimMet simulation package and consider the effects of viscosity in their equations for cut-size and water split.
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    Investigations into a positron emission imaging algorithm
    (2012) Bickell, Matthew; Buffler, Andy; Govender, Indresan
    A positron emission imaging algorithm which makes use of the entire set of lines-of-response in list-mode form is presented. The algorithm parameterises the lines-of-response by a Cartesian mesh over the field-of-view of a Positron Emission Tomography (PET) scanner to find their density distribution throughout the mesh. The algorithm is applied to PET image reconstruction and Positron Emission Particle Tracking (PEPT). For the PET image reconstruction, a redistribution of the lines-of-response is employed to remove the discrete nature of the data caused by the finite size of the detector cells, and once the density distribution has been determined, it is filtered and corrected for attenuation. The algorithm is applied to static and dynamic systems of hard phantoms, biological specimens and fluid flow through a column. In the dynamic systems, timesteps as low as 1 second are achieved. The results from the algorithm are compared to the standard Radon transform reconstruction algorithm, and the presented algorithm is observed to produce images with superior edge contrast, smoothness and representation of the physical system.
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    Multiple particle tracking in PEPT using Voronoi tessellations
    (2016) Blakemore, Dylan; Govender, Indresan; McBride, Andrew Trevor
    An algorithm is presented which makes use of three-dimensional Voronoi tessellations to track up to 20 tracers using a PET scanner. The lines of response generated by the PET scanner are discretized into sets of equidistant points, and these are used as the input seeds to the Voronoi tessellation. For each line of response, the point with the smallest Voronoi region is located; this point is assumed to be the origin of the corresponding line of response. Once these origin points have been determined, any outliers are removed, and the remaining points are clustered using the DBSCAN algorithm. The centroid of each cluster is classified as a tracer location. Once the tracer locations are determined for each time frame in the experimental data set, a custom multiple target tracking algorithm is used to associate identical tracers from frame to frame. Since there are no physical properties to distinguish the tracers from one another, the tracking algorithm uses velocity and position to extrapolate the locations of existing tracers and match the next frame's tracers to the trajectories. A series of experiments were conducted in order to test the robustness, accuracy and computational performance of the algorithm. A measure of robustness is the chance of track loss, which occurs when the algorithm fails to match a tracer location with its trajectory, and the track is terminated. The chance of track loss increases with the number of tracers; the acceleration of the tracers; the time interval between successive frames; and the proximity of tracers to each other. In the case of two tracers colliding, the two tracks merge for a short period of time, before separating and become distinguishable again. Track loss also occurs when a tracer leaves the field of view of the scanner; on return it is treated as a new object. The accuracy of location of the algorithm was found to be slightly affected by tracer velocity, but is much more dependent on the distance between consecutive points on a line of response, and the number of lines of response used per time frame. A single tracer was located to within 1.26mm. This was compared to the widely accepted Birmingham algorithm, which located the same tracer to within 0.92mm. Precisions of between 1.5 and 2.0mm were easily achieved for multiple tracers. The memory usage and processing time of the algorithm are dependent on the number of tracers used in the experiment. It was found that the processing time per frame for 20 tracers was about 15s, and the memory usage was 400MB. Because of the high processing times, the algorithm as is is not feasible for practical use. However, the location phase of the algorithm is massively parallel, so the code can be adapted to significantly increase the efficiency.
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    A one-way coupled DEM-CFD scheme to mode free-surface flows in tumbling mills
    (2012) Malahe, Michael; Buffler, Andy; Govender, Indresan
    There is a great need in the minerals processing industry for accurate prediction of slurry flows in tumbling mills. This flow is currently best understood through empirical, mechanistic and computational models. In the case of computational models, a realistic description of the system requires two fluid phases (for slurry and air) and one solid phase (for the charge). Existing 3-phase work makes use of a purely particle-based description, coupling the discrete element method (DEM) for the solids with smoothed particle hydrodynamics (SPH) for the fluids. This work is the first presentation of a 3-phase approach for tumbling mills that combines a particle description for the solids, DEM, and a continuum description for the fluids, computational fluid dynamics (CFD). In this approach, the phase coupling is only one-way, in the sense that forces are applied from the solids to the fluids, but not the other way around. In the course of developing the approach, some computational geometry algorithms are presented for efficiently converting the particle description of DEM into an appropriate continuum description for CFD.
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    PHY1004W - Matter & Interactions
    (2014-09-18) Buffler, Andy; Fearick, Roger; Govender, Indresan; Peshier, Andre
    PHY1004W is a first-year, calculus-based introductory Physics course for Science students intending to continue with second-year Physics. MODERN MECHANICS: Matter and interactions, conservation laws, the momentum principle, atomic nature of matter, conservation of energy, energy in macroscopic systems, energy quantization, multiparticle systems, exploring the nucleus, angular momentum, entropy, kinetic theory of gases, efficiency of engines. ELECTRIC AND MAGNETIC INTERACTIONS: Electric fields, electric potential, magnetic fields, electric circuits, capacitance, resistance, magnetic force, Gauss' law, Ampere's law, Faraday's law, induction, electromagnetic radiation, waves and particles, semiconductor devices.
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    PHY2015S - Classical & Quantum Mechanics (mainstream physics 2nd year)
    (2014-09-18) Fearick, Roger; Govender, Indresan
    PHY2015S is the second-year second semester half course for science students in the MPSS programme. It is a mainstream physics course, and is also suitable for mathematicians, astronomers, chemists and computer scientists. CLASSICAL MECHANICS: Review of Newton’s laws, constraints, d’Aelmbert principle, Lagrangian formulation of mechanics, conservation laws, applications, central forces, planetary motion, small oscillations, normal co-ordinates. QUANTUM MECHANICS: The basic assumptions of quantum mechanics, solutions of Schrodinger's equation, properties of wave functions and operators, one-dimensional applications, angular momentum in quantum mechanics, three-dimensional applications, the hydrogen atom, approximate methods.
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    Using discrete element modelling (DEM) and breakage experiments to model the comminution action in a tumbling mill
    (2008) Kulya, Chisenga; Powell, Malcolm; Mainza, Aubrey; Govender, Indresan; Sarracino, Robert
    The Discrete Element Method (DEM) is a powerful modelling tool that characterises the system at the individual particle level. This makes it particularly well suited for simulating tumbling mills whose charge is principally individual particles (steel balls, rocks and fines). The use of DEM to simulate tumbling mills has proliferated since the early 1990s and been successfully employed to predict important milling parameters such as charge motion, power draw, liner wear and impact energy distribution. The ultimate aim of any model of the tumbling mill is to predict the product of the milling process. Current DEM simulations of the tumbling mill however do not simulate the breakage of the particles and as such can not directly predict the product. In order to predict the performance of industrial-scale tumbling mills, laboratory-scale mills are used to experimentally obtain data, which is then scaled up using black box mathematical models. In this thesis a tumbling mill model that utilises the power of DEM to provide the mechanical environment and the energies available for breakage is proposed. The incorporation of DEM eliminates the need to scale up because DEM is able to simulate the actual industrial-scale device. Data from breakage experiments on the ore being treated is also incorporated into the model to determine the breakage functions. Population balance techniques are applied in the mathematical framework of the model to predict the product of the comminution process. In order to test the proposed tumbling mill model, DEM simulations of a 1.695m diameter pilot SAG mill using charge based on actual operation data were performed and analysed. Results from the DEM simulation and Drop Weight Tester breakage experiments were then used in the proposed tumbling mill model to predict the evolution of the product size distribution.
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    Using positron emission particle tracking (PEPT) to investigate the motion of granular media in a laboratory-scale tumbling mill
    (2012) Morrison, Angus James; Govender, Indresan; Mainza, Aubrey Njema
    Positron emission particle tracking is a Lagrangian, single particle tracking technique in which the trajectory of a representative tracer particle is triangulated from the decay products of the positron-emitting radioisotope with which it is labelled. Although the trajectories of a tracer particle moving in a bulk of similar particles can be of interest, it is often more informative to employ the ergodic assumption and to thus convert trajectory data in the Lagrangian reference frame of the tracer particle into a fixed Eulerian reference frame. This has, in the past, been done by dividing 3D space into voxels and assigning a location probability density to each voxel based on the number of times that triangulated tracer particle locations fall into it- a process called simple binning. A major outcome of my work has been to develop an alternative probability density based on the cumulative time spent by the tracer particle in a given voxel. This method is called residence time binning, and the resultant probability distribution- which I argue is proportional to, among other things, the mass and solidicity distributions of the tracer particle - the residence time distribution (RTD). In this work I propose, implement and test the residence time binning method, and show that it significantly outperforms the simple binning method in all situations. A second thrust of my work has been to develop a suite of general analysis routines for positron emission particle tracking (PEPT) data, based on the RTD. This suite contains routines for the triangulation, optimisation and pre-processing of PEPT data, as well as for obtaining residence time probability and time-averaged kinematic distributions in 3D space, and for aggregating and visualising the results. I have also extended this general set of routines for the special case of cylindrical symmetry through the addition of routines for the further pre-processing of RTDs, as well as for the calculation of angular measures about an arbitrary axis in space. Finally, I further extended this set of routines for application to tumbling mills. My tumbling mill analysis includes the identification of charge features and regions, and the isolation of charge in each region so-defined for further analysis. These features, particularly the shape of the bulk free and equilibrium surfaces, the angular position of the centre of circulation (CoC) of the charge, and the position of its impact toe allow me to characterise the behaviour of the charge under a range of conditions. This characterisation, together with the shear rate distributions and power draughts that I also calculate, allow me to speak meaningfully about the evolution of grinding regions in tumbling mills- information that could be used to construct charge motion and grinding models to inform the use of tumbling mills in industry. In this work, I apply these analysis routines to a small subset of the experiments performed by the UCT Centre for Minerals Research (CMR) on laboratory-scale tumbling mills, and in so-doing elucidate the behaviour of charge in its different regions- and the evolution of such behaviour with mill operating parameters- and discuss the implications of these to grinding efficacy in tumbling mills.
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    X-ray motion analysis of charge particles in a laboratory mill
    (2005) Govender, Indresan; Powell, Malcolm
    The work reported herein is of an automated X-ray vision system used to track the 3D trajectories of a typical bulk charge particle within an experimental Perspex mill, the intention of which is to simulate the grinding motion of rock found in typical industrial mills The experimental rig is constructed through an optimisation scheme that determines the maximum allowed dimensions of the mill that can be imaged by the X-ray system, however, the optimisation principle is not limited to the current study and can be employed to maximise any volume being investigated with the X-ray system. The raw data for each position of the tracked particle comprises of two X-ray images of the tumbling mill generated orthogonal to each other, with a phase lag between them. The correction of the phase lag between the biplanar images is a new addition to the usual usage of the system resulting in an effective sampling rate of 100 frames per second, thereby ensuring that the resolution is sufficient to conduct detailed kinematic studies. The processing of the raw images are achieved with a dynamic template matching algorithm followed by a modified, and improved, implementation of the Canny edge detector, while the centering of the edge images are based on an adapted conic fitting routine, resulting in an overall subpixel centering accuracy. The processed images are then mapped to object space using the direct linear transformation (DLT), equipped with a physically valid variance model that is shown to improve the standard implementation even when robust solvers are employed. The final reconstruction accuracy of the tracked particle was shown to be 0.15 mm and is achieved without iteration due to the appropriateness of the variance model. The high accuracy data. was initially used to benchmark the discrete element method (DEM), providing the first numerical comparisons that surpassed the usual end-window snapshots employed by other investigators for the purpose of validation. The analysis of the data was not restricted to DEM verification, and in some sense, surpassed the initial objective by yielding trends useful to communication practices. Amongst these analyses was the finding that the circulation rate of the charge is not once per mill revolution but greater, depicting a linear trend with mill speed. The slip between charge layers was shown to follow a linear pattern, with the degree of slip increasing linearly with mill speed. The phenomenon of charge surging was quantified, resulting in a trend for the variation of the surge amplitude with mill speed. A mechanism for the surging phenomenon was also proposed in this study. A particularly useful outcome of the data analyses was the formulation of a power model through heuristic trends of the center of mass (CoM) and center of circulation (CoC) of the charge. The methodology outlined by the model was shown to be robust, providing a correct approach to obtaining a truly fundamental power model based on generally applicable principles.