The effect of ball size on mill performance

The specific rates of breakage of particles in a tumbling ball mill are described by the equation S_i = ax^α_i(Q(z), where Q(z) is the probability function which ranges from 1 to 0 as particle size increases. This equation produces a maximum in S, and the particle size of the maximum is related to ball diameter by x_m = k₁d². The variation of a with ball diameter was found to be of the form a = k₂/d^1.5. Both k₁ and k₂ vary with mill diameter, and simple power laws have been assumed, k₁ ∝ D^0.1, k₂ ∝ D^0.6. If it is also assumed that the mean overall values of S_i for a mixture of balls is the weighted mean of S_i values for each ball size, equations are derived for calculating this mean value. As an example, the results are used in a mill simulation to show the quantitative effect of different ball mixes in a two-compartment cement mill versus a uniform mix over the whole mill.     

Introduction to powder surface area : By S. Lowell,Wiley, New York, 1979; pp 189; £11.00.

The values of first-order specific rates of breakage, S_i, and primary daughter fragment distributions, B_i,j, of quartz were determined in a laboratory ball mill. It was concluded that B_i,j values were constant for all conditions and that the specific rates of breakage fitted the relation S_i = ax^α_i, x_i being sieve size. The value of α was 0.80 for normal filling conditions but decreased to 0.53 for values of U < 0.3, U being defined by f_c/0.4J, f_c and J being the fractional volume filling of the mill by powder and balls respectively (based on a formal porosity of 0.4 for powder and ball bed). At high values of U the grinding became non-first order. In the normal range of U values, the results could be fitted by the empirical equation

This gives maximum mill capacity at U values in the range 0.4 to 0.6.     

An analysis of ball-and-race milling part II. The babcock E 1.7 mill

A mathematical model was developed for a ball-and-race mill based on specific rates of breakage and primary fragment distributions. The model includes internal classification of particles falling back into the race and external classification due to the built-on classifier. It was demonstrated that the normalized primary fragment distribution produced in a pilot-scale Babcock E-type mill of 17 in. race diameter was the same as in the Hardgrove laboratory test mill and that the specific rates of breakage varied with particle size in the same manner. Steady-state continuous tests on the pilot-scale mill showed that breakage rates depended on the rate of feed, since the mill pulled less power at low feed rates. This effect plus the residence time effect gave coarser product size distributions at low and high feed rates than at a medium feed rate. Model simulations based on parameters measured in the Hardgrove mill correctly predicted the product size distribution from the E-type mill.     

Kinetics of fine wet grinding in a laboratory ball mill

The kinetics of batch wet grinding of quartz from a feed of 600×425 μm to a product of 80% less than 8 μm have been determined using sieving and laser diffractometer sizing for size analysis. A dispersing agent was added while proceeding to longer grinding times to prevent particle agglomeration in the mill. The specific rates of breakage (S_i) values obtained were higher than those of dry grinding of quartz at the same experimental conditions, but the primary breakage distribution (B_i,j) values were the same. Non-first order grinding was observed with continued decrease of the specific rates of breakage for finer grinding. The simulations of the product size distributions were in good agreement with the experimental data, providing the decrease in rates was included.     

The investigation of grinding kinetics of power plant solid fossil fuel in ball mill

The kinetics of batch dry grinding of power plant solid fossil fuel, from the feeds of sieve sizes −3.350 + 2.360, −2.360 + 1.700, −1.180 + 0.850, −0.425 + 0.300 and −0.212 + 0.150 mm have been determined using a Bond ball mill with a mixture of five ball sizes. The mill used has a diameter of 30.5 cm, length of 30.5 cm, providing a total mill volume of 22.272 cm³ with a total mass of 20.125 g steel ball mixtures of 38.10, 31.75, 25.40, 19.05 and 12.70 mm diameters. The balls occupied 22% of mill volume. The speed of rotation of the mill was chosen as 70 rpm. The specific rates of breakage (S_i) and primary breakage distribution (B_i,j) values, called as grinding breakage parameters, were determined for those feed size fractions to simulate the product size distributions for comparison to the experimentally obtained data. As the feed sizes increase, the S_i values also increase, that is, faster breakage values from higher to lower values were in the order of solid fossil fuel by comparing to its α values. Breakage distribution functions were found non-normalizable. It is dependent upon the initial feed particle sizes. In other words, the simulations of product size distributions for fossil fuels were in good agreement with the experimental data using a ball mill simulation program, called JKSimMet.     

Mathematical modeling of fluid energy milling based on a stochastic approach

In this study, the stochastic method is used to simulate the grinding process in a fluid energy mill: the product particle size distribution is regarded as the result of repeating elementary breakage events, i.e. M_p=M₀[T_m]^m, where M₀ is the row vector of the size distribution of feed particles, M_p is the row vector of the size distribution of product particles, m is the number of elementary steps, and T_m is the matrix of transition probabilities representing the elementary breakage event. The matrix of transition probabilities can be related to the breakage rate function and the breakage distribution function of the elementary breakage event. A specially designed apparatus, named single-event fluid mill, was employed to experimentally estimate those two breakage functions of the elementary breakage event with a breakage rate correction factor θ. The classification effect is taken into consideration by defining a cutting size under which the particle will not break any more. Using this strategy, the product particle size distribution is calculated. The good consistency between the simulation and the experimental results indicates that this model is valid to quantitatively estimate the grinding performance of the fluid energy mill.     

An analysis of wet grinding operation using a linearized population balance model for a pilot scale grate-discharge ball mill

Batch and continuous wet grinding experiments were carried out in a 40 × 40 cm grate-discharge ball bill. Well-distributed ?4 and ?20 mesh feeds of two complex pyritic ores were used. The vol. % solids in the pulp ranged from 30 to 40, the particle load from 5 to 8 kg, and the solids feed rate from 70 to 110 kg/h.Analysis of the experimental data showed that over the range of operating conditions investigated, a linearized model could be used to predict the size distributions of both the mill product as well as the mill hold-up of the solids, very accurately. The breakage rate parameters, S_i, were found to be dependent on the mode of operation, independent of the percent solids in the pulp, and approximately inversely proportional to the mill hold-up weight of the solids. In the case of the continuous mode of operation, with decrease in particle size the rate parameter values became increasingly smaller by comparison with the corresponding expected batch operation values.     

A comparison of dry and wet grinding of a quartzite ground in a small batch ball mill

Classical grinding models involve the selection function (S), which gives the rates of breakage of particles of each screen size fraction, and the breakage function (B), which describes the instantaneous size distributions of fragments produced when the particles of each fraction are broken. In order to investigate the differences between dry and wet grinding as far as the selection and breakage functions are concerned, batch grinding experiments were performed on both dry and wet bases, on the same material, a quartzite, in a small ball mill under similar experimental conditions.On a dry basis, the rates of breakage were found to be time invariant and independent of the size environment in the mill. It is logical to postulate a similar behavior for the breakage function. On a wet basis (65% solids), an increase of the rates of breakage was observed as grinding proceeds. This behavior is essentially due to the variation of the size environment within the mill. This increase in breakage rates was, however, less and less important as the particle size decreased and was not observed for the smallest particles tested. These points were confirmed by considering the disappearance kinetics of samples of different screen size fractions of quartzite injected in the mill during the batch grinding of a limestone. Moreover, it is not impossible that the breakage function could also vary with grinding time, giving rise to finer instantaneous size distributions of fragments as the size environment in the mill becomes finer. As an overall result, wet grinding has appeared more selective than dry grinding for coarse material, while it did not produce more schlamms.     

Note on conversion of discrete size interval values of breakage parameters S and B to point values and vice versa

When the breakage of material in a mill can be characterized as an approximate first-order breakage process: rate-of-breakage of size j = S_jw_jW where W is the weight of total charge in the mill, w_j is the fraction of this charge of size j, and S_j is a specific rate-of-breakage parameter. The values of S_j can be measured experimentally for a finite size interval, often a geometric sieve ratio. It is clear, however, that the value of S in general depends on the magnitude of the size interval chosen, e.g. whether it is a √2 progression in upper to lower size of the interval, a 4√2 progression, etc. This makes it impossible to compare values between workers using different size interval magnitudes. The same reasoning applies to the size distributions of the daughter fragments produced on breakage, called the cumulative breakage distribution.This note describes methods of converting data from one size of interval to point values or to another size of interval. It is concluded that when B is non-normalizable, the value of α in S(x)=ax^X will depend on the interval size used to determine the S values.     

Continuous grinding in a small wet rod mill Part I. Comparison with a small ball mill

Tests have been done with a small continuous wet rod mill in order to characterise its dynamic behaviour in terms of the type of model previously developed for a small continuous wet ball mill. The three functions studied were the distribution of residence time, the breakage function and the breakage rate as a function of size.Distribution of residence time data showed that flow through the rod mill was similar to that through the ball mill and the contents of the mill were very well mixed. The breakage function results showed that any breakage event statistically produced less fines than in a ball mill. Breakage rate data showed that the coarsest sizes were broken more selectively, thus resulting in less fines in the product than would have been obtained if the breakage had been a result of ball action. Nevertheless it was shown that, overall, breakage in the rod mill could be described in similar terms to breakage in the ball mill, and the models were very similar.     

The back-calculation of specific rates of breakage from continuous mill data

Computer programs for back-calculation of specific rates of breakage S_i from continuous mill data, or batch grinding data, are described. A series of statistical tests to determine the statistically acceptable ranges of values are presented. The most reliable test uses independent estimates of error variance calculated from replicated data. It is shown that the interval-by-interval method of calculating S_i values is especially subject to errors in the top sizes. Back-calculation methods which use only the circuit product in the calculation give different ranges of statistically acceptable values than those which use all the size distributions round the circuit. The statistically acceptable ranges of values from using only the circuit product often do not include the values determined by direct laboratory investigation, and tend to be wider than the ranges given by using all the size distributions; thus, using only circuit product is a non-preferred method. The use of a fully-mixed residence time distribution (RTD) instead of a true RTD leads to radically incorrect values, which cannot be detected by statistical analysis except in the presence of unrealistically low experimental error. It must be realized that a set of breakage parameters which reproduce the data used in their back-calculation with good accuracy are not necessarily real, and statistical analysis is essential to define the statistically acceptable ranges of the parameters.     

A model for continuous grinding in a laboratory hammer mill

The equations of fully mixed, steady continuous grinding with rapid ideal classification through a discharge screen have been modified to allow for non-rapid removal of material less than the screen size. Values of specific rates of breakage, daughter fragment distributions and the variation of specific rates with mill hold-up determined in prior batch tests on a hammer mill were used in the equations, to predict hold-up versus feed rate and product size distribution versus feed rate. The predictions were in reasonable agreement with the experimental values from continuous grinding. There is a maximum production rate which can obtained, and energy utilization is inefficient at low production rates.     

A study on the specific rate of breakage of cement materials in a laboratory ball mill

The specific rate of breakage (S_i) in the widely accepted first-order expression of grinding rate is one of the important factors required to evaluate a grinding process, particularly for the initial grinding stage in various mill types.In this study, the effects of ball diameter and feed size on the specific rate of breakage were investigated on limestone, trass and clinker samples at batch grinding conditions based on a kinetic model. Eight different monosize fractions were prepared between 1.7 and 0.106 mm, using a √2 sieve series. The specific rates of breakage (S_i) were determined from the size distributions at different grinding times, and the specific rates of breakage were compared for three different ball diameters (41, 25.4 and 9.5 mm).The results indicated that the variation of the specific rate of breakage with feed size of cement materials could be expressed. For the specific rate of breakage of each material, empirical equations were developed to express it as a function of feed size and ball diameter.     

A simulation model for an air-swept ball mill grinding coal

The conventional model for grinding in tumbling ball mills was modified to allow for air-sweeping, for the case where all the material is carried out of the mill in the air stream. It was shown that this type of mill can be treated as a single fully mixed reactor. The values of the internal classification numbers given by the air-sweeping were determined for a 1 m by 1.5 m pilot-scale mill grinding coal. The results showed that only 1 to 2% of the mill charge are exposed to the sweeping action per mill revolution. S and B values determined in a laboratory mill were scaled-up for use in the continuous mill model and the simulations gave product size distributions and mill capacities which agreed with the pilot-scale mill data within the experimental accuracy of the pilot-scale data.     

Use of the attainable region analysis to optimize particle breakage in a ball mill

Particle size reduction is one of the most widely encountered, yet least energy efficient, processes. Therefore, potentially significant energy and cost savings exist with even the slightest increase in milling efficiency. Often one would like to mill particles to a certain size, and no smaller, while minimizing energy use and milling time. We use the attainable region (AR) analysis to optimize the comminution of silica sand particles in a bench top laboratory ball mill. When the mill is loaded with a large number of grinding media (J=volume of media/mill volume=10.7%), the breakage profiles are indistinguishable over all rotation rates investigated. However, operation at lower grinding media fill level (J=1.5%) reveals separation between the grinding profiles for different rotation rates, suggesting more efficient breakage occurs at a lower grinding media fill level for a given rotation rate. Our results show that operation at multiple speeds, fast at first and then slower (φ_c=0.03), takes advantage of the initially overlapping grinding profiles and produces a similar particle size distribution with a decreased amount of processing time—less than half the time required for the single rotation rate milling. A natural extension of this idea is continuous milling, where the first mill can operate at a higher energy input for a shorter amount of time and the second mill can operate at a lower energy input for a longer amount of time.     

Forthcoming meetings

Using the back-calculation approach, simultaneous estimation of rate and breakage distribution parameters has been carried out for a complex pyritic ore from the batch ball mill size distribution data. The ore showed several deviations from the normal grinding behavior: (i) the slope of the Gaudin-Schumann plots in the fine size range varied with the feed size and the grinding time, (ii) the first-order plots showed marked curvature in the initial period of grinding, (iii) the rate parameters did not conform to the relationship S_i = Ax^α_i, even in the fine size range, and (iv) the breakage distribution parameters exhibited a high degree of non-normalizability. The main problem encountered was the choice of a suitable functional form for the breakage distribution function. This paper illustrates the limitations of the two existing functional forms and several other functional forms used in this work, and emphasizes the need for having a more general and practical functional form for the breakage distribution function.     

An analysis of fine dry grinding in ball mills

The kinetics of dry grinding of several cement clinkers and two coals were investigated in a laboratory tumbling ball mill. The kinetic process is first-order at first, but the rates of breakage decrease as fines accumulate in the bed. It was demonstrated that the slowing of the breakage rates applies to all sizes in the mill, indicating that the cushioning action of fines affects the whole breakage process, even though mill power remains constant. Tests on cleaning or non-cleaning the balls showed that the major factor was not the build-up of a coating on the balls. Radio-tracing tests showed that the effect was not due to pelletizing of fines into larger particles. The quantitative magnitude of the cushioning action was different for different materials. It is, therefore, postulated that cushioning is affected not only by air trapped in the bed of fine particles but also by the cohesive attraction of fine particles, which is a function of the material.     

Determination of parameters for wet-grinding model in perl mills

The paper presents the method of obtaining a model of the kinetic grinding process in perl mills.After carrying out several grinding tests with varying parameters in two laboratory mills of 1 × 10^?3 and 6 × 10^?3 m³ volume, the parameters of the specific rate of breakage S_i and the cumulative parameters of the primary breakage function B_t,j were calculated. Those values were then related to the particle size.Synten® dyestuff in water suspension having particles sized up to 22 μm was used for the grinding tests with addition of dispersing agents.Measurements of particle size were carried out by means of particle counter.     

Regrinding sulphide minerals — Breakage mechanisms in milling and their influence on surface properties and flotation behaviour

Changes in surface properties with grinding and regrinding play a key role in mineral flotation performance. Different particle breakage mechanisms in grinding mills may change the mineral surface properties in different ways, possibly leading to different mineral floatabilities depending upon the predominant breakage mechanism. The Magotteaux Mill^® and IsaMill were selected as representations of a tumbling and a stirred mill, respectively. The latter has a greater contribution to particle size reduction from the abrasion mechanism than the former which also has contributions from impact breakage.Mineral recovery decreased with size reduction through stirred mill regrinding (i.e., the IsaMill) employing ceramic media from 90%, achieved before regrinding (d₈₀ 80 μm regrind feed), to 71, 58, 20 and 5% achieved after regrinding to d₈₀ values of 60, 40, 20 and 10 μm, respectively. A similar trend of decreasing recovery was also observed with regrinding in the tumbling mill (i.e., Magotteaux Mill^®). Changes in mineral flotation behaviour were investigated with respect to (i) particle size, (ii) increase in surface area, and (iii) surface contamination with size reduction in the two different mills. The flotation of pyrrhotite with additional reagents illustrated that the total change in recovery through regrinding results mainly from the increase in surface area of the pyrrhotite afforded by size reduction. The effects of the predominating particle breakage mechanism on the change of mineral surface properties were studied through regrinding in the two different mills. In particular it was observed that the hydrophobicity/floatability of the coarse particles decreased to a greater extent with stirred mill regrinding than with tumbling mill regrinding at coarser regrind product sizes (d₈₀ 70 and 60 μm) presumably due to the greater contribution of the abrasion mechanism to size reduction afforded by the stirred mill. It was also observed that the difference in recovery for the same regrind product size from the two different mills decreased when approaching finer regrind sizes, which indicated that the particle breakage mechanisms of the different mills for fine regrind product size were not as influential as for coarse regrind product sizes.     

Retention of Liquid by Solids in Suspensions and Sediments

Abstract

The effects of associated immobile liquid on the settling behavior of particles in solid-liquid suspensions are investigated. The effects are studied with respect to such sedimentation parameters as suspension porosity ?, final sediment volume fraction u_s, internal void fraction of the flow unit u_i, Steinours parameter A, and McKay's packing factory which corrects for the effective volume and density of the flow units in a sedimenting suspension. It is shown that the degree of liquid association with the settling particles is a function of their physicochemical characteristics. Suspensions containing clays and carbonates, owing to their ionic nature and loose packing behavior, absorb large amounts of liquids and result in larger settled sediments of a compressible nature. On the other hand, systems like glass beads-liquid suspensions absorb little associated liquid and virtually result in noncompressible settled sediments due to their rigid spherical shapes and close packing. The addition of hydrocolloids (such as natural gums) provides ionic stability to suspensions by giving them low cmpressibility and low values of u_s, u_i, A, and p.