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.     

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.     

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.     

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.     

Energy distribution during the quasi-static confined comminution of granular materials

A method is proposed to calculate the distribution of energy during the quasi-static confined comminution of particulate assemblies. The work input, calculated by integrating the load-displacement curve, is written as the sum of the elastic deformation energy, the breakage energy and the redistribution energy. Experimental results obtained on samples subjected to compression stresses ranging between 0.4 and 92 MPa are used to calibrate the model. The elastic energy stored in the samples is obtained by simulating the compression test on the final particle size distributions (PSDs) with the discrete element method and by extracting the contact forces. A PSD evolution law is proposed to account for particle breakage. The PSD is related to the total particle surface in the sample, which allows calculating the breakage energy. The redistribution energy, which comprises the kinetic energy of particles being rearranged and the friction energy dissipated at contacts, is obtained by subtracting the elastic energy and breakage energy from the work input. Results show that: (1) at least 60% of the work input is dissipated by particle redistribution; (2) the fraction of elastic deformation energy increases, and the fraction of redistribution energy decreases as the compression stress increases; (3) the breakage energy accounts for less than 5% of the total input energy, and this value is independent of the compressive stress; (4) the energy dissipated by redistribution is between 14 and 30 times larger than the breakage energy.     

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.     

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.     

固体颗粒破碎的分维演化规律

固体颗粒破碎后的粒径分布符合分形模型,颗粒破碎程度采用粒径分布的分维表示。本文采用垃圾炉渣的颗粒破碎试验,统计计算炉渣颗粒破碎的分维。收集总结了岩石颗粒破碎分维随破碎应力和破碎能量的变化规律,颗粒破碎的分维是变化的,随着破碎应力和破碎能量增加而增加,颗粒完全破碎后的粒径分布的分维趋于定值,为2.6左右。将颗粒破碎过程中变化的分维称为颗粒破碎的“似分维”,简称分维;分维的极限值称为“真分维”。结合颗粒破碎概率的Weibull统计理论,建立了颗粒破碎“似分维”与破碎应力和破碎能量的相关关系。     

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.     

粗粒土的破碎耗能计算及影响因素

粗粒土的颗粒破碎直接改变了土体本身结构,对粗粒土的剪胀和内摩擦角都会产生影响。在土体剪切过程中,体积应力和剪切应力在体积应变和剪切应变上做功,这部分能量在剪切过程中转化为颗粒的弹性储能、颗粒间的摩擦耗能、颗粒剪胀时对外做功和颗粒破碎耗能4部分。准确计算剪切过程中粗粒土破碎耗能的目的是：从能量角度分析颗粒破碎对土体本构关系的影响,为建立考虑颗粒破碎的粗粒土本构关系创造条件。通过分析粗粒土的常规三轴试验数据,计算得到了剪切过程中的粗粒土破碎耗能。计算结果表明,常规三轴试验条件下粗粒土破碎耗能主要受固结应力、土体摩擦系数M等因素的影响。     

堆石粗粒料颗粒破碎试验研究

利用室内大型三轴试验,对堆石等粗粒料的颗粒破碎进行了分析。结果表明,颗粒破碎率随围压的增加而增加,呈非线性状态,二者之间的关系可以用双曲线表示。颗粒破碎的增加将导致粗粒料的抗剪强度降低,峰值内摩擦角与颗粒破碎率之间呈幂函数关系,不论颗粒的岩性、强度、大小、形状、级配和初始孔隙比等情况如何,试验资料都落在一个狭窄的区域,如果围压和材料的试验参数已知,则可估计颗粒破碎率。     

DEM investigation of the effect of intermediate principle stress on particle breakage of granular materials

This study investigates the influence of the intermediate principle stress on the particle breakage of granular materials. The crushable agglomerate method is applied to model soil particles and numerical true triaxial tests were carried out. The results show that particle breakage increases with increasing b value, the relationship of which follows an exponential function and agrees well with previous experimental results. More importantly, the study found that the relationship between particle breakage and total energy input is independent of the intermediate principle stress, which provides a good basis for the constitutive modeling of granular materials.     

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.     

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.     

考虑颗粒破碎的冻结砂土非线性本构模型研究

颗粒破碎是粒状材料在高应力状态下的一种基本现象。为了研究冻结砂土中颗粒破碎对应力应变关系的影响,将冻结砂土视为复合颗粒材料,忽略冰的压融,考虑内摩擦角随应力状态的变化,构建一个适用于冻结砂土的考虑颗粒破碎的非线性本构模型。构建过程分为三步,首先是基于三轴剪切前后颗粒分析对冻结砂土颗粒破碎模式和产生机理进行探讨;其次是基于考虑颗粒破碎的能量平衡方程,对冻土在三轴剪切试验过程中的颗粒破碎耗能进行分析,结果表明颗粒破碎耗能随轴向应变呈双曲线变化趋势;最后应用考虑颗粒破碎的剪胀方程修正沈珠江三参数非线性模型中的体积切线模量ν_t,得到一个考虑颗粒破碎的非线性本构模型,模型参数可以通过单轴压缩试验和常规三轴试验确定。将原模型和修正后模型的计算结果与控制温度为-6℃,围压为1 MPa、4 MPa、6 MPa、8 MPa和10 MPa时冻结砂土的试验结果进行对比,结果表明该模型能够较好的模拟冻结砂土从低围压到高围压的应变软化特征与剪胀特征。      

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.     

珊瑚砂大剪切应变下的剪切特性和分形维数

利用Wille Geotechnik环向剪切仪对3种不同粒径的珊瑚砂进行了大剪切位移的环向剪切试验,探讨了颗粒破碎对珊瑚砂强度和残余应变发展的影响。试验结果表明：颗粒破碎随着剪切位移的增加逐渐增加,且初始粒径较大的均匀级配珊瑚砂粒径越大颗粒破碎越多。颗粒破碎对峰值强度和残余强度无影响,但对珊瑚砂的体积应变有显著影响,颗粒破碎较多的试样其体积应变也较大。建立了考虑长宽比、球形度和凹凸度的分形维数计算公式。由于颗粒破碎后颗粒形状在全粒径范围内的自相似性和无尺度性,考虑颗粒形状与不考虑颗粒形状的公式计算得到的珊瑚砂分形维数基本一致。     

南海钙质砂宏细观破碎力学特性

针对取自我国南部某海域的钙质砂样本,做了以下两方面工作：一是通过电子显微镜获取了钙质砂颗粒的几何投影图像,利用图像处理技术对图形进行黑白二值化处理,获取单元颗粒形状轮廓边界,使用圆度和粗糙度2个参数对钙质砂的颗粒形状进行定义和量化。二是通过不同围压下的三轴固结排水剪切试验及试验前后的颗分测量对比,研究了颗粒破碎对钙质砂的变形、强度、能量耗散等特性的影响。研究表明,大粒径钙质砂（粒径大于2.0 mm）和小粒径钙质砂（粒径小于0.5 mm）形态比较接近圆形、颗粒表面相对光滑;相比而言,中间粒径（粒径介于0.5～2.0 mm之间）钙质砂形状较不规则,表面棱角较多。钙质砂在三轴排水剪切过程中发生颗粒破碎,试样向着级配均匀的方向发展。随着初始围压的增大,颗粒破碎程度加大,土样整体剪胀趋势减小,而破碎引起的能量耗散增加。而在高围压（初始围压为600 kPa）剪切过程中,仅考虑摩擦耗散,以及同时考虑摩擦、体积耗散两种情况下,计算得到的最大颗粒破碎耗散分别可达土样总输入塑性功的25%和18%。     

建筑废渣-黏土混合料压实特性试验研究

将建筑废渣按一定比例与黏土掺合形成的渣土混合料用作填筑材料,具有显著的社会经济与环保效益。为了揭示渣土混合料的压实特性,利用击实试验研究渣料含量和初始级配对渣土混合料压实性能及颗粒破碎特性的影响。研究结果显示,与黏土相比,渣土混合料的最优含水率较低而最大干密度较高,尽管渣料的初始级配有所不同,渣土混合料的最优含水率随掺渣量的增加而减小,最大干密度随掺渣量的增加而先增后减,并存在着一个约为30%的最优掺渣量;击实作用下渣料颗粒的相对破碎率随掺渣量的增加而增加,与初始级配均匀的渣料相比,相同掺渣量下初始级配不均匀渣料的相对破碎率较小,但受渣料初始颗粒大小的影响不大;与初始较小颗粒渣料的混合料相比,初始较大颗粒渣料的混合料的最大干密度较大,表明宽级配的渣土混合料压实性能较好。渣土混合料的压实特性通过结合掺渣量、颗粒破碎特性与渣土混合料结构响应分析得到合理解释。     

Study on mechanochemical effect of silica for short grinding period

Silica was ground in an oscillating mill at various grinding period to study the mechanochemical effect in fine grinding process. The ground particles exhibited massive size reduction where the volume moment diameter of 5.56 μm was reached within 600 s. Aggregation of fine particles was very pronounced when it was ground for 600 s due to high surface energy. Aggregation of fine particles caused the ground particles to exhibit poly-modal particle size distribution. Line broadening and reduction of diffractogram peak intensity were observed. Amorphization rate up to 16.9% was exhibited by the particle ground for 600 s. Preferential breakage of plane was observed where (101) and (111) was easily distorted compared to (110) and (200). Rapid reduction of crystallite size was observed at early stage of grinding until it reached a plateau at 5 nm at 600 s whilst the change in lattice strain was 0.5%.