T.P. Meloy

Analysis and optimization of mineral processing and coal-cleaning circuits — Circuit analysis

Abstract In mineral processing and coal cleaning operations, there are two classes of unit operations used in the separation processes. The first type — crushing, grinding, pelletizing and settling — modifies the size distribution of the feed. The second type — heavy media, sizing, flotation, magnetic, electrostatic and jigging — separates the particles. In these latter unit operations, the size distribution is not changed; rather, the feed stream is split into a product and waste stream based on subtle physical and physicochemical differences between the different types of feed particles. In separation unit operations, the mathematical functions, representing the behavior of the unit operations, are easy to manipulate. Thus, a number of theorems, useful to both the plant designer, and the plant engineer, can be proven. The work presented in this paper makes the fundamental assumption of linearity as do all current plant design analysis systems. The linearity assumption states that in a separation operation, there is no particle-particle interaction effecting the probability that a particle will be selected for the product or waste stream. In other words, if one doubles or triples the feed rate to a unit operation, the fraction of particles of a given characteristic selected to appear in the product stream remains unchanged. In actuality, this assumption is not true because higher feed rates affect the behavior of separation operations. However, when designing a plant, one can calculate the feed rate to a given unit operation and then select a piece of equipment large enough to handle the computed tonnage. Thus, the designed circuit behaves linearly.

Dynamic model of flotation cell banks — circuit analysis

Abstract While all mineral industry flotation circuits are stable, they are sensitive to low-frequency perturbations in the feedrate. In both countercurrent and cocurrent circuits, the lead cell is more sensitive to feed variations. The frequency response predicts the amount of extra cell capacity needed to handle the maximum feed due to a sinusoidal forcing function. Feedback loops are more significant than sump delays. The countercurrent 4 × 4 circuit floating quartz, with a 200-sec retention time, requires 75 minutes for the concentration of quartz in the input to the first cell to reach 95% of its steady-state value. Countercurrent circuits were found superior to cocurrent circuits in all respects.

Liberation theory — Eight, modern, usable theorems

Abstract Eight, simple theorems are derived for locked particles. For the locked particles: the volume or mass size distribution curve slope is steep — 1.6 to 2.0, the surface area is constant, the total volume is proportional to the fineness of grind, and the fraction of particles locked at a given size decreases with particle size.

Measuring the particle shape mix in a powder with the cascadograph

Abstract Particle shape has been recognized as one of the important properties of the particles present in powders, particularly in the fluid-solid reaction system. The currently available technique of Fourier transformation to characterize particle shape is cumbersome and can be used only by a skilled operator. Recently developed, the sieve cascadograph is the first particle chromatograph that characterizes the shape of particles in the 0.1–1.0 mm size range. Working on the principle that the residence time of a particle on a sieve is a function of the particle shape, the cascadograph consists of a stack of sieves of identical size in a shaking device. A small monosized powder sample is placed in the topmost sieve, and shaking commenced. The weight of powder leaving the lowest sieve is measured as a function of time, and is used to produce a signature representative of the particle shape distribution in the powder sample. The sieve cascadograph is theoretically described and the results of preliminary experiments are presented.

Particle shape effects due to crushing method and size

Abstract Using Fourier shape analysis, differences in the shape mix of glass particles crushed under open and choke flow conditions were demonstrated. Particles crushed in open flow are more angular, have larger asperities and more complex shapes. Smaller particles are more elongate, complex and angular. Since it was found that smaller particles are more complex, irregular and have higher aspect ratios than larger ones, it is suggested that a major factor in lung damage from small particles may be due to the particles shape. Because of their irregular shape, fine particles do not pack well and have a low bulk density. The particle shape distribution of feed, product and waste streams of mineral processing unit operations needs to be characterized to learn how shape affects the recovery of minerals, so that equipment and circuits may be designed to optimize recovery. Experiments were reproducible and the method of Fourier analysis for characterizing shape distributions was found to be a good, reproducible research tool for comminution shape studies.

Industrial modeling of spirals for optimal configuration and design : spiral geometry, fluid flow and forces on particles

Abstract Gravity, centrifugal, drag, lift and friction forces act on a particle during its passage down the curvilinear path of the spiral. The resultant force function dictates the separation of particles on the spiral deck by their size and density. Except for gravity force, all other forces depend on the hydrodynamics prevailing on the spiral. The flow of fluid, in turn, is determined by the spiral geometry. In order to develop a tractable and working model for simulation and design of industrial spirals, we describe the spiral geometry in detail, test the suitability of various empirical power laws for different flow regimes that are available in fluvial hydrology and estimate the magnitude of forces acting on particles. It is shown that of the four laws examined, the transitional or mixed flow power law is seemingly most appropriate for simulating the flow indices, such as the water line profile, local flow velocity, flow depth and flow rate along the spiral trough. The power law involves relatively simple computations and mimics the principal features and broad trends of flows as measured experimentally or simulated by highly computationally intensive solution of the Navier–Stokes equations for spiral geometry. The magnitude of individual forces acting on a particle in most instances is less than 10−4 N, with considerable overlap between the forces. As a consequence, in general no force overwhelms the other forces and apparently it is the rate of change of forces with size, density, velocity and radial location which drives the separation.

Assessment of numerical solution approaches to the inverse problem for grinding systems: Dynamic population balance model problems

Abstract The solution of the grinding Inverse Problem using the dynamic population balance model (PBM) comminution model is problematic. The fact that the dynamic PBM model equation Inverse problem solution for grinding systems is degenerate or underspecified is demonstrated. Different solution approaches to the same PBM equations, per se, without transport, are provided. It is shown that there is a solution to the Inverse Problem if one of two numerical solution approaches is used. These are: (1) providing additional constraints on breakage selection functions or (2) performing the Arbiter—Bhrany (or other) normalization of the selection functions. Actual experimental anthracite grinding data is used to demonstrate the non-unique functionality of the dynamic mill selection and breakage functions for a real physical system. The Levenberg—Marquardt algorithm for systems of constrained non-linear equations is used to solve the dynamic PBM grinding equations to obtain the grinding selection and breakage rate functions. Two different solutions were obtained depending on the numerical solution approach. The severity of the non-uniqueness problem for dynamic grinding is demonstrated. Each solution approach to a dynamic PBM, while giving the same prediction for a given grinding time interval, gives drastically different solutions or predictions for mill composition for other grinding times. The fact that the constraint solution approach gives a solution may suggest that normalization is not necessary. This fact makes dynamic nodal analysis and control problematic.

Inclusion shape, mineral texture and liberation

Abstract The effect of an ores texture on liberation was explored using simplified texture models composed by a continuous phase and an inclusional phase with different geometric aspects of the inclusions. Unexpectedly, a parameter called the “textural rank” was developed that may be useful for a quantitative characterization of the rocks textures. Among the conclusions: the shape of the inclusions leads to the definition of a measurable textural rank, τ, that ranges continuously from 1, for predominantly platy inclusions, to 3, for predominantly blocky inclusions. When the reduction size is small compared to size of the inclusions the values for different textures of the degree of liberation of the two phases practically coincide while, as the reduction size increases, these values differ becoming dependent on the textural rank. The textural rank is a prime predictor of the degree of liberation of a given mineral in an ore. Finally, in the special case of cubic inclusions (as well as other regular polyhedra and spheres), the derived equation for calculating the degree of liberation of the inclusive phase reduces to Gaudins equation.

Optimizing for grade or profit in mineral processing circuits — Circuit analysis

Abstract Circuit analysis is applied to the general optimization of mineral processing circuits. Four functions are defined: feed, selectivity, composition and criteria. After defining the criteria function that is to be optimized, the general method of optimization is shown in closed form. Two examples are given. That optimum grade and recovery do not coincide is proven.

On the definition and separation of fundamental process functions

Abstract The three fundamental process functions performed by the unit operations in processing separation circuits — roughing, cleaning and scavenging — can be accurately defined. In our age of engineering specialization, the trend is to separate process functions from one another since different functions require specialized operating and design conditions. Separating different functions allows the engineer to control more exactly the operating conditions in each unit. Seven combinations of the three functions are possible in any unit operation. A circuit analysis function separation methodology has been developed for readily redesigning circuits to separate the three functions when they are combined in a single unit operation. Function separation was found to lead to lower unit operation feed loadings which in most instances can be expected to lead to improve circuit performance. This methodology may become a significant new tool in mineral circuit design.