Modelling the performance of industrial ball mills using single particle breakage data

Ball milling is an energy-intensive unit operation and usually consumes a major proportion of the power drawn by a typical mineral processing plant. Hence, substantial economic benefits can be achieved by optimal design and by operating ball milling circuits under optimum process conditions. This requires an accurate ball mill modelling technique.In the multi-segment ball mill model, the size-dependent material transport within the mill varies systematically with the amount of coarse particles present in each segment. The ore-specific breakage distribution function can be determined from single particle breakage tests conducted using a computer-monitored twin pendulum apparatus. When the ore-specific breakage distribution function is used in the multi-segment, a constant relationship between the breakage rate parameters and mill diameter is observed. Thus, the performance of an industrial ball mill can be adequately described using the ore-specific breakage distribution function together with the systematic variation of the material transport and the breakage rate functions with process conditions and mill diameter, respectively.This ball mill modelling technique is illustrated using a case study on the design of a ball milling circuit for a particular grinding requirement and another case study on modelling the performance of an industrial ball milling circuit.     

Kinetics of fine wet grinding of zeolite in a steel ball mill in comparison to dry grinding

Batch wet grinding of zeolite was studied with emphasis on a kinetic study in a laboratory size steel ball mill of 200 mm diameter. The breakage parameters were determined by using the single sized feed fractions of − 850 + 600 µm, − 600 + 425 µm and − 425 + 300 µm for the zeolite samples. The S_i (specific rate of breakage) and B_i,j (primary breakage distribution) values were obtained for those feed size fractions in order to predict the product size distributions by simulation for comparison to the experimental data. The specific rates of breakage values for wet grinding in the first-order breakage region were higher than the dry values reported previously by a factor 1.7 at the same experimental conditions, but the primary breakage distribution (B_i,j) values were approximately the same. The simulations of the product size distributions of zeolite were in good agreement with the experimental data using a standard ball mill simulation program. The wet grinding of zeolite was subjected to slowing-down effect in the mill at 2 min of grinding, corresponding to an 80% passing size of about 400 µm. On the other hand, the slowing down effect in the dry grinding of zeolite was also seen at 4 min of grinding. In addition, effects of some operational parameters on dry and wet grinding of zeolite were determined by simulation using the breakage parameters obtained experimentally.     

Scale up of lifters in ball mills

Liner design and lifter geometry in tumbling mills are very important in comminution kinetics and energy consumption. In terms of the parameters of the batch grinding model, the breakage rate function decreases as lifter size increases while the breakage distribution function is independent of the lifter size. However, the specific energy consumption is not a function of lifter size. Reducing the lifter height from 2.54 cm to 0.32 cm increases the production from this mill by 37%. When the mill is operated without any lifters, the charge apparently slips inside the shell resulting in lower breakage kinetics and higher energy consumption.     

Modelling mineral size reduction in the closed-circuit ball mill at the Pine Point Mines concentrator

Using plug flow material transport and a cumulative-basis rate-of-breakage parameter, overall size reduction through the closed-circuit ball mills at the Pine Point and Gibraltar concentrators was simulated over a wide range of operating conditions. The rate-of-brakage parameter was related to particle size by a power law, the exponent (n) being: Pine Point, n = 1.043 ± 0.026, and Gibraltar, n = 0.747 ± 0.020. The success of this approach probably stems from the high (> 1.5) circulating load ratios encountered.By analogy individual mineral size reduction at Pine Point was examined. A similar rate-of-breakage parameter versus size relationship was found. Pyrite was the hardest mineral, but fine galena was equally resistant. However, the approximation that mineral and overall rates of breakage were the same gave an adequate fit to the mineral size reduction. This was emphasized by combining with a cyclone model to simulate cyclone overflow mineral size distribution. A more accurate cyclone model is shown to be more important in simulating mineral deportment at Pine Point.Complementary laboratory batch grinding tests were conducted on rod mill discharge and ball mill feed samples. Sufficient agreement with the first-order hypothesis was observed to analyse the rate-of-breakage parameter. The kinetics was similar for both samples and in turn similar to the plant-derived kinetics in terms of relative mineral rates-of-breakage and the relationship of the rate-of-breakage parameter with particle size.     

Wet grindability of an industrial ore and its breakage parameters estimation using population balances

The grindability of a diasporic ore was studied by wet grinding in a laboratory ball mill and its breakage parameters were determined based on the population balance model (PBM). Four different feeds with unnatural size distributions were designed to save grinding and sieving work. It was found that three size-intervals fitted the first-order breakage, but the coarsest interval did not. The non-first order breakage was most probably caused by the heterogeneity of the material. The heterogeneity was then confirmed by experiments. The breakage rate and the breakage distribution parameters were back calculated by treating the non-first order breakage as two linear segments according to the grinding time. The back-calculated breakage rates were in good agreement with the values calculated directly from the experimental data, and non-normalizable breakage distribution was observed. The model with the obtained parameters simulated the experimental product size distributions with good accuracy. These findings are valuable to the simulation and optimization of the industrial grinding processes of diasporic ores.     

DEM analysis of angular ballast breakage under cyclic loading

Railway ballast particles undergo significant amount of breakage under repeated train load. Breakage of ballast particles, especially highly angular fresh ones, causes an increase in settlement, contributing to track degradation. The quantitative analysis of the influence of breakage on the stress-strain properties of ballast can be performed either experimentally or numerically. Numerical modeling has the advantage of simulating ballast breakage subject to various types of loading and different boundary conditions for a range of material properties. In this paper, ballast breakage under cyclic loading is simulated using a 2D discrete element method (DEM) utilizing the software PFC^2D . A new subroutine is developed and incorporated in the PFC^2D analysis to study ballast breakage and to quantify breakage in relation to particle size distribution. The influence of confining pressure on both breakage and permanent deformation is also studied and compared with laboratory observations. The findings of this paper provide an insight into the true ballast behavior under cyclic loading and are expected to assist railway practitioners in developing suitable design criteria for track stability.     

Particle breakage of uniformly graded carbonate sands in dry/wet condition subjected to compression/shear tests

The behaviour of a granular material is primarily affected by its particle size distribution (PSD), which is not necessarily a soil constant as assumed in traditional soil mechanics. The PSD may change over time due to mechanical as well as environmental actions. In this study, a series of ring shear tests and one-dimensional compression tests were completed on carbonate sand, in both dry and saturated conditions. Samples were prepared with different initial uniform gradings, to investigate: (1) the influence of the saturation state and initial grading on mechanical and deformational behaviour of carbonate sands and (2) the evolution of the PSD as a result of breakage. The ring shear tests show that the residual friction angle remains almost constant, but dilatancy reduces with increasing saturation degree. In the one-dimensional compression test, the yield stress decreases with increasing saturation degree, but the compressibility (as defined by C_c) remains almost constant, irrespective of the saturation state. Moreover, saturated samples suffer more breakage than dry samples during ring shear tests, while there is no obvious effect of saturation state on particle breakage in one-dimensional compression. A recently proposed PSD model with only two parameters (λ_p and κ_p) is employed to model the evolution of PSD, as it can more broadly capture the whole PSD throughout the breakage process than existing breakage indices. Test results demonstrate that parameter λ_p is linearly related to Einav’s breakage index \( B_{\text{r}}^{*} \) and is dependent on initial grading, but independent of test mode. Parameter κ_p is in power relationship with \( B_{\text{r}}^{*} \) and is independent of initial grading or test mode. The evolution of parameters λ_p and κ_p is related to the input work for both ring shear and compression tests, with λ_p being hyperbolically related to input work and κ_p in power relationship with input work. Using such an evolution law provides an alternative approach to capture the effects of particle breakage in constitutive models.

     

Some breakage characteristics of ultra-fine wet grinding with a centrifugal mill

A centrifugal mill is a high-power intensity media mill that can be used for ultra-fine grinding, employing centrifugal forces generated by gyration of the axis of the mill tube in a circle. The mill charge motion is quite different depending on the ratio of the gyration diameter to the mill diameter (G / D ratio), varying from a motion similar to that of a conventional tumbling media mill to that of a vibration mill. In this study, a centrifugal mill was constructed with an arrangement where the gyration diameter could be readily adjusted. The batch grinding characteristics of three different minerals (limestone, talc and illite) in water with dispersing agent were investigated at various G / D ratios. It was found that the optimum G / D ratio in terms of the specific energy consumption to give a desired fineness of product was different for the three minerals. This was due to their different reactions to the breakage mechanisms provided by the mill charge motion at varying G / D ratios. The size distributions became progressively narrower at increased grinding times, and particles finer than about 0.1 μm were not detected even for prolonged grinding times. Measurement of specific surface areas indicated that this was not due to an artifact of the size measurements by laser diffractometry. This implies that there is a limitation in which particles finer than 0.1 μm are not produced under the conditions tested in this type of mill, but further investigation is needed for experimental verification of this limit of comminution.     

A comparison of linear and nonlinear models for open-circuit ball mill grinding

A detailed comparison is made of the capability of population balance models to predict steady-state product size distributions of a pilot-scale ball mill. The mill was operated at 60% solids with feed rates of crystalline limestone ranging from 90 to 450 kg/hr (200 to 1000 lb/hr). Two types of lumped parameter models are compared: a linear model in which size reduction parameters are independent of size consist and a nonlinear model in which these parameters are dependent on size consist. The nonlinear model is based on an empirical correlation between rate of breakage and size consist in the mill. Results indicate that the nonlinear model gives the most accurate predictions of product size distributions, however, at the cost of significantly more complex computations.     

The Kinetics of the Interaction Between Iron(III)-Ethylenediaminetetraacetate and Peroxynitrite

The kinetics of the formation of the purple-colored species between Fe^III-EDTA and peroxynitrite were studied as a function of pH (10.4–12.3) at 22°C in aqueous solutions using a stopped-flow technique. A purple-colored species was immediately formed upon mixing, which had an absorbance maximum at 520 nm. The increase in absorbance with time could be fit empirically by a power function with two parameters a and b. The power equation determined was absorbance = a*t ^b , where a increased linearly with pH and the concentration of peroxynitrite, while b almost remained constant with a value of ~0.25. The molar extinction coefficient ε_520 nm for the colored species was determined as 13 M⁻¹cm⁻¹, which is much lower than ε_520 nm = 520 M⁻¹ cm⁻¹ for the [Fe^III(EDTA)O₂]³⁻, a purple species observed in the Fe^III–EDTA–H₂O₂ system. The results of kinetics and spectral measurements of the present study are briefly discussed and compared with those of the reaction between Fe(III)-EDTA and hydrogen peroxide.     

Use of 234U and 238U isotopes to evaluate contamination of near-surface groundwater with uranium-mill effluent: a case study in south-central Colorado, U.S.A.

The ²³⁴U/²³⁸U alpha activity ratio (AR) was determined in 47 samples of variably uraniferous groundwater from the vicinity of a uranium mill near Cañon City, Colorado. The results illustrate that uranium isotopes can be used to determine the distribution of uranium contamination in groundwater and to indicate processes such as mixing and chemical precipitation that affect uranium concentrations. Highly to moderately contaminated groundwater samples collected from the mill site and land immediately downgradient from the mill site contain more than 100?μg/l of dissolved uranium and typically have AR values in the narrow range of 1.0–1.06. Other samples from the shallow alluvial aquifer farther downgradient from the mill contain 10–100?μg/l uranium and plot along a broad trend of increasing AR (1.06–1.46) with decreasing uranium concentration. The results are consistent with mixing of liquid mill waste (AR≈1.0) with alluvial groundwater of small, but variable, uranium concentrations and AR of 1.3–1.5. In the alluvial aquifer, the spatial distribution of wells with AR values less than 1.3 is consistent with previous estimates of the probable distribution of contamination, based on water chemistry and hydrology. Wells more distant from the area of probable contamination have AR values that are consistently greater than 1.3 and are indicative of little or no contamination. The methodology of this study can be extended usefully to similar sites of uranium mining, milling, or processing provided that local geohydrologic settings promote uranium mobility and that introduced uranium contamination is isotopically distinct from that of local groundwater.     

Microstructure characterization of mechanically activated hematite using XRD line broadening

The effect of dry milling in a vibratory mill on the structural changes and microstructural characteristics of hematite using different methods was investigated. We have described the line profile analysis (LPA) to extract the size of coherently diffracting domains and the lattice strain of activated hematite in a vibratory mill. The Warren–Averbach and Williamson–Hall methods were used as the main tools for characterization. The changes in the particle size, surface area and new phase formation of hematite concentrate were also investigated. It was concluded that the breakage and agglomeration of particles take place mainly at lower and higher levels of specific energy input, respectively. The pores in agglomerates remain accessible for the nitrogen gas. Milling of hematite increased specific surface area up to 18.4 m²/g. The hematite milled under various levels of specific energy input did not undergo a significant reaction or phase transformation during milling. The Williamson–Hall method confirms its merit for a rapid overview of the line broadening effects and possible understanding of the main causes. The anisotropic character of line broadening for deformed hematite as a function of specific energy input was revealed. Higher level of specific energy input favors the generation of small crystallite size, higher microstrain, BET surface area, amorphization and line breadth. The Warren–Averbach method suggested that the nanocrystalline hematite with grain sizes of 73.5–12.2 nm was formed by mechanical treatment using different milling intensities in the vibratory mill. The root mean square strain (RMSS) at L = 10 nm varies between 1.7 × 10^− 3 and 4.0 × 10^− 3 depending on the level of energy input. Limits in the applicability of Williamson–Hall method and reliability of the results are discussed in detail.     

Magnitude–frequency relations in debris flow

Debris flow occurs frequently in mountainous regions in China. Because of the difficulties involved in predicting and catching live debris flows, an assessment of the potential for debris flow is crucial in hazard mitigation. Magnitude–frequency (MF) relations are of special significance in such assessments. In previous studies, MF relations have been inferred by analyzing environmental factors and historical records and using empirical relations. This paper is concerned with the derivation of MF relations at regional and valley scales, using a large database of statistics. At the regional scale, it is represented by the distribution of the valley area, because the area is often taken to indicate the potential magnitude of debris flow. Statistics from over 5,000 debris flow valleys in various provinces in China show that a power law holds for the distribution, i.e., p(A) ∼ A ⁻ⁿ , where p(A) is the percentage of valleys with area A and n varies with region and thus describes regional differences. At the valley scale, a case study focusing on Jiangjia Gully (JJG) was conducted, and the MF relations derived from it were expressed by the distributions of discharge and runoff (i.e., the total volume) of living debris flows observed over the last 40 years. The distributions can be expressed as exponential functions where the exponents vary with the events. These MF relations provide not only a potential quantitative reference for practical purposes but also hint at the intrinsic properties of the debris flow.     

Steady-state simulation of a cement-milling circuit

The conversion factor for specific rates of breakage, to change from results in a 200-mm diameter laboratory mill with 25.4-mm diameter balls, at Bond filling conditions, to a 4-m diameter production cement mill was found to be 2.75. The residence time distribution of the 10-m long, 2-compartment production mill was equivalent to 10 equal fully-mixed reactors in series, with a mean overall residence time of 5.58 min. A mathematical simulation of the mill circuit using experimental separator selectivity values gave a reasonably close match between simulated and experimental size distributions at various points in the circuit.The steady-state simulation model was used to predict how design schemes for different operation conditions would influence the product size distributions and output rates, assuming that the discharge diaphragm could be redesigned to preserve design mill filling conditions at different circulating loads. It was concluded that the increase of circuit output from the breakage view-point due to separation and rejects recycle was slight, and that improved separator efficiency to produce the same product would not result in significant increase in output, primarily because the size distribution of the finish product is close to that predicted from open-circuit operation.     

Implications of the fault scaling law for the growth of topography: mountain ranges in the broken foreland of north-east Tibet

A fault scaling law suggests that, over eight orders of magnitude, fault length L is linearly related to maximum displacement D. Individual faults may therefore retain a constant ratio of D/L as they grow. If erosion is minor compared with tectonic uplift, the length and along‐strike relief of young mountain ranges should thus reflect fault growth. Topographic profiles along the crests of mountain ranges in the actively deforming foreland of north‐east Tibet exhibit a characteristic shape with maximum height near their centre and decreasing elevation toward the tips. We interpret the along‐strike relief of these ranges to reflect the slip distribution on high‐angle reverse faults. A geometric model illustrates that the lateral propagation rate of such mountain ranges may be deciphered if their length‐to‐height ratio has remained constant. As an application of the model, we reconstruct the growth of the Heli Shan using a long‐term uplift rate of ～1.3 mm yr^?1 derived from ²¹Ne and ¹⁰Be exposure dating.     

The influence of pulp composition and feed rate on hold-up weight and mean residence time of solids in grate-discharge ball mill grinding

The results of experiments on a 40 cm × 40 cm grate-discharge ball mill have been analysed for variation of mill hold-up weight of solids with solids feed rate, weight percent solids, mean feed particle size, material specific gravity and work-index. It is shown that mill hold-up weight is independent of the material specific gravity and mean feed particle size, and it varies linearly with solids feed rate and weight percent solids, at least over the range of practical interest. The variations in the transport behaviour of different materials have been attributed to the differences in the size distribution of the mill hold-up solids. It is shown that work-index can be used as the material characteristic for the development of an empirical correlation. Variation of mean residence time of solids with solids feed rate and weight-percent solids is also discussed.     

Combined mechanochemical and thiosulphate leaching of silver from a complex sulphide concentrate

The intensification of the thiosulphate leaching of silver from a silver-bearing complex sulphide concentrate (Casapalca deposit) using mechanochemical alkaline leaching as the pretreatment step was investigated. The leaching of “as-received” concentrate with the thiosulphate solution afforded only 6% recovery of Ag into leach. The mechanochemical pretreatment resulted 85% amorphization of tetrahedrite as a silver-bearing mineral in the concentrate as well as an increase in specific surface area from the original value of 0.26 m² g⁻¹ to a maximum value of 15.7 m² g⁻¹. The pretreatment was performed in a stirred ball mill using variable milling times and sample weights. The physico-chemical changes in the concentrate as a consequence of mechanochemical pretreatment had a pronounced influence on the subsequent silver extraction. The optimum results from mechanochemical pretreatment and subsequent leaching of the concentrate with ammonium thiosulphate were achieved by using a milling time of 60 min and a weight of sample of 60 g. In this case, 99% recovery of Ag was achieved after only 3 min of leaching.     

Discrete numerical modelling of rockfill dams

The aim of this study is to obtain quantitative information on the behaviour of rockfill used in embankment dams, and particularly on the influence of block breakage on the displacement field, from a numerical analysis using the Distinct element method. A methodology is set up to define the resistance of the 2D particles so that the same probability of breaking blocks may be reproduced as in a 3D material. The model uses the discrete element code PFC^2D (Itasca Consulting Group Inc., PFC2D (Particle Flow Code in Two Dimensions), Version 3.0, 2002) and considers breakable clusters of 2D balls. The different parameters are determined from experimental data obtained from laboratory tests performed on rock blocks. The model is validated by comparing the results of the simulation of shearing tests with actual triaxial tests on rockfill material published in the literature. The numerical analysis of block crushing in an actual dam is proposed in the last part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Sr,Cd, Mn and Co distribution coefficients in calcite as a function of calcite precipitation rate

Distribution coefficients, as a function of precipitation rate, were determined for the metals Sr²⁺, Co²⁺, Mn²⁺ and Cd²⁺in calcite. A pH-stat was used to maintain a constant degree of-saturation, and hence precipitation rate, during each coprecipitation run. The precipitation rate was proportional to the degree of supersaturation and the mass of seed crystal introduced. Distribution coefficients (λ) as a function of rate were determined using radioactive isotopes for solutions with saturations $\text{Ω = 1}$ to $\text{Ω = 5.5}$. Strontium distribution coefficients increased with increasing precipitation rate, while Co, Mn and Cd distribution coefficients decreased with increasing precipitation rate. The following rate expressions (at 25°C) were derived: logλ_Sr = 0.249 log R ?1.57logλ_Mn = ?0.266 log R + 1.35logλ_Co = ?0.173 log R + 0.68logλ_Cd = ?0.194 log R + 1.46 where R is the observed precipitation rate in nmoles CaCO₃ per mg seed crystal per min.In separate experiments the uptake of radioactive isotopes was monitored during the recrystallization of calcite seed crystals. Rates of recrystallization were from 100 to 10, 000 times slower than the pH-stat experiments, but yielded distribution coefficients consistent with the above rate expressions.Using gross estimates of biogenic crystal growth rates, aragonite to calcite transformation rates, and the above Sr rate expression, biogenic calcite and diagenetic calcite Sr contents are estimated. These experiments indicate that in addition to solution composition, precipitation rate is a significant factor influencing the trace metal content of naturally occurring calcite.     

Slope units-based flow susceptibility model: using validation tests to select controlling factors

A susceptibility map for an area, which is representative in terms of both geologic setting and slope instability phenomena of large sectors of the Sicilian Apennines, was produced using slope units and a multiparametric univariate model. The study area, extending for approximately 90 km², was partitioned into 774 slope units, whose expected landslide occurrence was estimated by averaging seven susceptibility values, determined for the selected controlling factors: lithology, mean slope gradient, stream power index at the foot, mean topographic wetness index and profile curvature, slope unit length, and altitude range. Each of the recognized 490 landslides was represented by its centroid point. On the basis of conditional analysis, the susceptibility function here adopted is the density of landslides, computed for each class. Univariate susceptibility models were prepared for each of the controlling factors, and their predictive performance was estimated by prediction rate curves and effectiveness ratio applied to the susceptibility classes. This procedure allowed us to discriminate between effective and non-effective factors, so that only the former was subsequently combined in a multiparametric model, which was used to produce the final susceptibility map. The validation of this map latter enabled us to verify the reliability and predictive performance of the model. Slope unit altitude range and length, lithology and, subordinately, stream power index at the foot of the slope unit demonstrated to be the main controlling factors of landslides, while mean slope gradient, profile curvature, and topographic wetness index gave unsatisfactory results.