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.     

The back-calculation of specific rates of breakage and non-normalized breakage distribution parameters from batch grinding data

This paper describes a mathematical approach for calculating parameters relating specific rates of breakage and breakage products distribution from batch grinding a known feed for several grinding times. Knowledge of such parameters leads to improved equipment design criteria and consistent operating correlations. The approach involves the use of non-linear optimization with appropriate statistical tests and gives parameter estimates close to those gained from direct experimental measurements. Some results and experiences using the technique are summarized.     

A pore-throat model based on grain-size distribution to quantify gravity-dominated DNAPL instabilities in a water-saturated homogeneous porous medium

A pore-scale network model based on spherical pore bodies and cylindrical pore throats was developed to describe the displacement of water by DNAPL. The pore body size and the pore throat size were given by statistical distributions with user-specified values for the minimum, mean and maximum sizes. The numerical model was applied to a laboratory experiment conducted on a sand-filled glass column. The parameters relative to pore body size and pore throat size that were used in the construction of the equivalent network were derived from the discrete grain-size distribution of the real porous medium. The calculated arrival times of the DNAPL front were compared with those measured using optic fibre sensors placed at different points on the control section of the experimental device. Furthermore, the model simulated DNAPL pressure measured at the entrance section of the system. In general, the numerical results obtained with the model were in good agreement with the actual measurements.     

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.     

Modeling of particle breakage in a smooth double roll crusher

A model has been developed to describe the breakage behavior of particle breakage in a smooth double roll crusher. In addition to this, an empirical relationship has been obtained between Hardgrove Grindability Index (HGI) of material and breakage parameters. The model uses matrix method of breakage process analysis. Experiments were carried out in a laboratory scale smooth double roll crusher using different materials. The model has been verified experimentally and gives product size distribution after breakage which matches with the experimental distribution well within the satisfactory range.     

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.     

An elastoplastic constitutive model for frozen saline coarse sandy soil undergoing particle breakage

The mechanical property of frozen saline sandy soil is complicated due to its complex components and sensitivity to salt content and confining pressure. Thus, a series of triaxial compression tests were carried out on sandy samples with different Na₂SO₄ contents under different confining pressures to explore the effects of particle breakage, pressure melting, shear dilation and strain softening or hardening. The test results indicate that the stress–strain curves exhibit strain softening/hardening phenomena when the confining pressures are below or above 6 MPa, respectively. A shear dilation phenomenon was observed in the loading process. With increasing confining pressure, the strength firstly increases and then decreases. By taking into consideration the changes between the grain size distributions before and after triaxial compression tests, a failure strength line incorporating the influences of both particle breakage and pressure melting is proposed. In order to describe the deformation characteristics of frozen saline sandy soil, an elastoplastic incremental constitutive model is established based on the test results. The proposed model considers the plastic compressive, plastic shear and breakage mechanisms by adopting the non-associated flow rule. The breakage mechanism can be reflected by an index related to the initial, current and ultimate grain size distributions. The hardening parameters corresponding to compressive and shear mechanisms consider the influence of particle breakage. Then the effect of particle breakage on both the stress–strain and volumetric strain curves is analyzed. The calculated results fit well with the test results, indicating that the developed constitutive model can well describe the mechanical and deformation features of frozen saline sandy soil under various stress levels and stress paths. In addition, the strain softening/hardening, contraction, high dilation and particle breakage can be well captured.

     

Particle transport within water-saturated porous media: Effect of pore size on retention kinetics and size selection

The transport and filtration behaviour of fine particles (silt) in columns packed with sand was investigated under saturated conditions by using step-input injections. Three samples of different particle size distributions (coarse medium, fine medium and a mixture of both) were used in order to highlight the influence of the pore size distribution on particle retention and size selection of recovered particles. The main parameters of particle transport and deposition were derived from the adjustment of the experimental breakthrough curves by an analytical model. The higher particle retention occurs in the mixture medium, owing to its large pore size distribution, and the filtration coefficient decreases with increasing flow velocity. Particle size distribution of recovered particles shows a thorough size selection: (i) the first recovered particles are the coarser ones; (ii) the size of the recovered particles increases with increasing flow velocity and enlarger pore distribution of the medium.     

An enhanced constitutive model for crushable granular materials

Studies in the past have tried to reproduce the mechanical behaviour of granular materials by proposing constitutive relations based on a common assumption that model parameters and parameters describing the properties, including gradation of individual grains are inevitably linked. However successful these models have proved to be, they cannot account for the changes in granular assembly behaviour if the grains start to break during mechanical loading. This paper proposes to analyse the relation between grading change and the mechanical behaviour of granular assembly. A way to model the influence of grain breakage is to use a critical state‐based model. The influence of the amount of grain breakage during loading, depending on the individual grain strength and size distribution, can be introduced into constitutive relations by means of a new parameter that controls the evolution of critical state with changes in grain size distribution. Experimental data from a calcareous sand, a quartz sand, and a rockfill material were compared with numerical results and good‐quality simulations were obtained. The main consequences of grain breakage are increased compressibility and a gradual dilatancy disappearance in the granular material. The critical state concept is also enriched by considering its overall relation to the evolution of the granular material. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling the effect of wetting on the mechanical behavior of crushable granular materials

It is well known that the compressibility of crushable granular materials increases with the moisture content,due to the decrease of particle strength in a humid environment.An existing approach to take into account the effect of grain breakage in constitutive modeling consists in linking the evolution of the grain size distribution to the plastic work.But how the material humidity can affect this relationship is not clear,and experimental evidence is quite scarce.Based on compression tests on dry and saturated crushable sand recently reported by the present authors,a new non-linear relationship is proposed between the amount of particle breakage and the plastic work.The expression contains two parameters:(1)a material constant dependent on the grain characteristics and(2)a constant depending on the wetting condition(in this study,dry or saturated).A key finding is that the relationship does not depend on the stress path and,for a given wetting condition,only one set of parameters is necessary to reproduce the results of isotropic,oedometric,and triaxial compression tests.The relationship has been introduced into an elastoplastic constitutive model based on the critical state concept with a double yield surface for plastic sliding and compression.The breakage ratio is introduced into the expression of the elastic stiffness,the critical state line and the hardening compression pressure.Incremental stress-strain computations with the model allow the plastic work to be calculated and,therefore,the evolution of particle crushing can be predicted through the proposed non-linear relationship and reintroduced into the constitutive equations.Accurate predictions of the experimental results in terms of both stress-strain relationships and breakage ratio were obtained.     

Determination of the distribution of size of irregularly shaped particles from laser diffractometer measurements

The formal stereological transformation equation for particle sieve size distribution from measurements in lower dimensional spaces is applied to laser diffractometer measurements. The transformation function for iron ore particles is measured experimentally, and modeled. The solution is tested against the measured transformation function data as well as synthetic composite distributions of the original sample. The natural size distribution of a sample taken from a grinding circuit stream was measured by a combination of standard sieving and cyclosizer, and the result is compared to the transformed size distribution calculated from laser diffractometer measurements. The stereological transformation technique performed well in all cases.     

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.     

Bayesian Estimation of Earth’s Undiscovered Mineralogical Diversity Using Noninformative Priors

Recently, statistical distributions have been explored to provide estimates of the mineralogical diversity of Earth, and Earth-like planets. In this paper, a Bayesian approach is introduced to estimate Earth’s undiscovered mineralogical diversity. Samples are generated from a posterior distribution of the model parameters using Markov chain Monte Carlo simulations such that estimates and inference are directly obtained. It was previously shown that the mineral species frequency distribution conforms to a generalized inverse Gauss–Poisson (GIGP) large number of rare events model. Even though the model fit was good, the population size estimate obtained by using this model was found to be unreasonably low by mineralogists. In this paper, several zero-truncated, mixed Poisson distributions are fitted and compared, where the Poisson-lognormal distribution is found to provide the best fit. Subsequently, the population size estimates obtained by Bayesian methods are compared to the empirical Bayes estimates. Species accumulation curves are constructed and employed to estimate the population size as a function of sampling size. Finally, the relative abundances, and hence the occurrence probabilities of species in a random sample, are calculated numerically for all mineral species in Earth’s crust using the Poisson-lognormal distribution. These calculations are connected and compared to the calculations obtained in a previous paper using the GIGP model for which mineralogical criteria of an Earth-like planet were given.

     

考虑颗粒破碎效应的堆石料静动力本构模型

为合理反映颗粒破碎对堆石料力学特性的影响,基于试验结果分析,得出了堆石料在压缩和剪切作用下的颗粒破碎特性规律。通过引入压缩破碎和剪切破碎的相关参数,借鉴已有本构模型的合理定义,吸收临界状态理论和边界面理论的优点,发展了考虑颗粒破碎和状态相关的堆石料静动力统一弹塑性本构模型,并阐述了模型参数的确定方法。该模型不仅能够反映堆石料在静力荷载作用下的低压剪胀、高压剪缩、应变软化和硬化等特性,还能够反映在循环荷载作用下应力-应变的滞回特性和残余变形的累积效应。最后为验证模型的合理性,分别对堆石料的静力三轴和循环三轴试验进行了数值模拟预测,结果表明：模型预测与试验数据吻合良好,所提出的本构模型能够合理地描述颗粒破碎对堆石料静动力变形特性的影响。     

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.     

Validation of a model for impact breakage incorporating particle size effect

Investigation of a breakage probability model published by Vogel and Peukert [Vogel, L. and Peukert, W., 2004. Determination of material properties relevant to grinding by practicable labscale milling tests. Int. J. Miner. Process., 74S, 329–338.] has led to a modification of their model to describe the degree of impact breakage, t₁₀. The modified model takes a form similar to the JKMRC prior art breakage model, but with particle size and breakage properties incorporated explicitly in the model.     

Relationship between breakage parameters and process variables in ball milling — An industrial case study

The performance of the secondary ball mill at the New Broken Hill Consolidated Ltd. concentrator is analysed using the perfect mixing model and an ore-specific breakage distribution function. This function was determined from single-particle breakage tests using a computer-monitored twin pendulum apparatus.The ratio of the breakage rate to the normalized discharge rate, r/d^*, determined for the ball mill using the ore-specific breakage distribution function for a range of grinding conditions is related to the mill power consumption. The mill power consumption is related to the percentage of mill volume occupied by the ball charge and to the percentage of solids in the mill feed.     

脆性颗粒材料的动态多尺度模型研究

脆性颗粒材料的多尺度模型一般包含微观尺度的基本粒子、细观尺度的颗粒和宏观尺度的颗粒堆积体3个尺度。基于离散元方法（DEM）构建多尺度模型,并将该模型应用于动态加载。首先,对多尺度模型所涉及的两种接触模型和两种黏结模型的参数进行分析,详细讨论微细观模型参数与宏观材料常数之间的联系。然后,选用Hertz-Mindlin接触模型[1]和平行键黏结模型,建造石英砂的动态多尺度模型。通过选择合适的强度和局部阻尼参数发现,模型宏细观尺度上的动态压缩响应与对石英砂的相关试验结果吻合很好。利用多尺度模型和选定的参数,探讨与动态加载密切相关的局部阻尼机制对多尺度模型各个尺度上力学响应的影响。结果表明,阻尼越大则颗粒材料对波的衰减能力越强,但过高的阻尼会使团簇强度和模型的宏观压缩曲线都表现出异常的加载速度效应（后者实际是阻尼引起的微惯性效应）。另外,高阻尼会过度衰减颗粒破碎过程产生的应力波,从而阻碍颗粒破碎。最后,应用改进的动态多尺度模型,对脆性颗粒材料的动态破碎特性进行研究,发现该模型不但能给出与试验相吻合的颗粒级配曲线,还能揭示出颗粒破碎过程中微裂纹分布的空间不均匀性,即颗粒破碎过程中波的产生机制和衰减机制相互作用导致的微裂纹聚团分布的现象。     

基于分形理论研究土壤结构及其水分特征关系

为定量获得土壤结构对其水力性质的指示作用,室内实验选用华北平原子牙河流域原状土样为研究对象,用张力计法和激光粒度分析仪分别测定土壤水分特征曲线和样品粒度分布,基于分形理论计算土壤粒度分布的分形维数,采用实验测定与模型验证相结合的方法对水分特征曲线进行分析.结果表明,土壤颗粒粒度分布在[10 μm,50 μm]区间内的分段分维值是表征土壤粒度累积分布显著上升段特征的关键参数,与0~80 kPa吸力范围内的土壤水分特征曲线幂函数模型拟合参数(a、b)有极显著相关关系.研究区内土壤水分特征曲线以分形形式表达的幂函数模型为:θ=100.78×(3-D)S(D-3)/3,利用土壤结构分形特征能够有效指示其水力性质.