Modeling of a grinding circuit with a swing-hammer mill and a twin-cone classifier

A model was formulated for a continuous, air-swept milling system, assuming the mill to be fully mixed, with all particles leaving the mill in the air stream. The air-sweeping effect was treated as an internal classification which allows fine particles to leave and returns coarse particles for regrinding. The kinetic parameters for the continuous model were estimated using experimental data from a continuous pilot-plant air-swept swing-hammer mill. Functional forms that provide information regarding the power, the internal classification action of the mill, and the action of the external classifier, were deduced for different plant conditions. It was found that the specific rates of breakage could be assumed to vary with particle size according to S_i = a(x_i/x_o)^α, where the value of α was 1.3 for the coal used in the tests. The values of a varied with mill hold-up W, giving a maximum of aW as W became large. The primary breakage distribution values were assumed to be normalized, giving a value of characteristic slope of γ = 0.60.     

Drag of non-spherical solid particles of regular and irregular shape

The drag of a non-spherical particle was reviewed and investigated for a variety of shapes (regular and irregular) and particle Reynolds numbers (Re_p). Point-force models for the trajectory-averaged drag were discussed for both the Stokes regime (Re_p ? 1) and Newton regime (Re_p ? 1 and sub-critical with approximately constant drag coefficient) for a particular particle shape. While exact solutions were often available for the Stokes regime, the Newton regime depended on: aspect ratio for spheroidal particles, surface area ratio for other regularly-shaped particles, and min-med-max area for irregularly shaped particles. The combination of the Stokes and Newton regimes were well integrated using a general method by Ganser (developed for isometric shapes and disks). In particular, a modified Clift-Gauvin expression was developed for particles with approximately cylindrical cross-sections relative to the flow, e.g. rods, prolate spheroids, and oblate spheroids with near-unity aspect ratios. However, particles with non-circular cross-sections exhibited a weaker dependence on Reynolds number, which is attributed to the more rapid transition to flow separation and turbulent boundary layer conditions. Their drag coefficient behavior was better represented by a modified Dallavalle drag model, by again integrating the Stokes and Newton regimes. This paper first discusses spherical particle drag and classification of particle shapes, followed by the main body which discusses drag in Stokes and Newton regimes and then combines these results for the intermediate regimes.     

Dispersion of carbon nanotubes in ethanol by a bead milling process

Carbon nanotubes (CNTs) were dispersed in ethanol by bead milling to form a CNT suspension. The size and shape of the CNTs were not changed after bead milling. The Brownian motion of the CNTs was observed by an optical microscope. It was shown by analysis off the trajectory of individual CNTs that the diffusion coefficient of their translational Brownian motion was 4.5 × 10⁻¹³ m² s⁻¹. The ratio of the particle mobilities due to drift flow and diffusion flow was 2.4, which is almost the same as for a high-molecular-weight protein. Shear thinning occurred in the CNT suspension, which means that the CNTs were oriented in the suspension under a high shear rate at low viscosity. The intrinsic viscosity for shear rates of 0 and ∞ agreed with those calculated using Simha’s equation and Leal and Hinch’s equation, respectively. Based on the experimental results, the CNTs were found to be well dispersed as isolated and nanosized rod-like particles in the suspension. CNT-dispersed Si₃N₄ ceramics were fabricated using the CNT suspensions. As a result, the bead milling improved the bend strength of the CNT Si₃N₄ ceramics compared with the wet ball milling because granules of CNTs were eliminated by the bead milling.     

Local Turbulent Energy Dissipation Rate in an Agitated Vessel: Experimental and Turbulence Scaling

The hydrodynamics and the flow field in an agitated vessel were measured using 2-D time resolved particle image velocimetry (2-D TR PIV). The experiments were carried out in fully baffled cylindrical flat bottom vessels 300 and 400 mm in inner diameter. The 300 mm inner diameter tank was agitated by a Rushton turbine 100 mm in diameter, and the 400 mm inner diameter tank was agitated by a Rushton turbine 133 mm in diameter. Three liquids of different viscosities were used as the agitated liquid: (i) distilled water (ν = 9.35 × 10^–7 m²/s), (ii) a 28 vol % aqueous solution of glycol (ν = 2 × 10^–6 m²/s), and (iii) a 43 vol % aqueous solution of glycol (ν = 3 × 10^–6 m²/s). The velocity fields were measured at an impeller rotation speed in the range from 300 to 850 rpm, which covers the Reynolds number range from 50000 to 189000. This means that fullydeveloped turbulent flow was reached. The experiments were performed to investigate the applicability of the following relations: ε* = ε/(u⁴/ν) = const, vK/u = const, Λ/ηK = const, τ_Λ/τ_K = const, ε* = ε/((Nd)4/ν) = const, Λ/d ∝ Re^–1, ηK/d ∝ Re^–1, vK/(Nd) = const, Nτ_Λ ∝ R^–1, Nτ_K ∝ Re^–1, and ε/(Nq) ∝ Re. These formulas were theoretically derived in our previous work, using turbulence theory, in particular, using turbulence spectrum analysis. The correctness of the proposed relations is investigated by statistical hypothesis testing.     

Experimental evidence supporting kinematic modelling of the flow of granular media in the absence of air drag

Experiments have been carried out with closely graded, coarse glass ballotini of various sizes to investigate the effect of varying the mean particle diameter on the value of the kinematic constant, B. A strong linear dependence is found between the two except for the case of the smallest ballotini (≈600 μm diameter). For this material, significantly different flow patterns were obtained and this is attributed to flow retardation by the interstitial fluid medium.Based on experimental evidence, a dimensionless, empirical constant, k, is proposed to replace the kinematic constant, B, for the flow of coarse granular media in the absence of air drag.     

Tetrahedrites synthesized via scalable mechanochemical process and spark plasma sintering

In this study, we demonstrate the use of elemental precursors (Cu,Sb,S) to synthesize tetrahedrite Cu₁₂Sb₄S₁₃ using an industrial eccentric vibratory mill. Mechanochemical synthesis of tetrahedrite leads to the formation of covellite (CuS), skinnerite (Cu₃SbS₃) or famatinite (Cu₃SbS₄) in dependence on milling time. However, the composite product can be modified in favour of prevailing tetrahedrite when Spark Plasma Sintering (SPS) treatment is applied after milling. The as-synthesized and sintered products are composed of polydisperse nanosized particles with dimensions up to 250 nm. The thermoelectric measurements reveal a maximum value of figure-of-merit ZT = 0.67 @ 700 K, as a consequence of a relatively high power factor (1.07 mW m⁻¹ K⁻²) and a low thermal conductivity (1.12 W m⁻¹ K⁻¹). The obtained zT values of products prepared in an industrial mill are comparable to the ones reported for compounds synthesized using laboratory mills. The synthesis of ternary and quaternary sulphides by a scalable and industrializable milling process represents a prospective route for mass production of thermoelectric materials.     

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.     

A model on attrition of quartzite particles as a bed material in fluidized beds

In this paper, a model on attrition of quartzite particles as an inert bed material in fluidized beds has been established on the particle-particle collision. For the convenience of describing the attrition of quartzite particles in fluidized beds, we chose the attrition rate constant (k_ARC) as one main characteristic parameter to develop the model.In order to verify the validity of the developed model, an attrition experiment of quartzite particles has been carried out in a lab-scale circulating fluidized bed. The predicted results from the population model were close to the experimental data as far as the engineering use is concerned. Finally, a sensitivity analysis was performed by using the developed model to examine effects of initial particle diameter, attrition time, and fluidization number on k_ARC.     

Limestone particle attrition and size distribution in a small circulating fluidized bed

Limestone particle attrition was investigated in a small circulating fluidized bed reactor at temperatures from 25 to 850 °C, 1 atm pressure and superficial gas velocities from 4.8 to 6.2 m/s. The effects of operating time, superficial gas velocity and temperature were studied with fresh limestone. No calcination or sulfation occurred at temperatures ?580 °C, whereas calcination and sulfation affected attrition at 850 °C. Increasing the temperature (while maintaining the same superficial gas velocity) reduced attrition if there was negligible calcination. Attrition was high initially, but after ∼24 h, the rate of mass change became constant. The ratio of initial mean particle diameter to that at later times increased linearly with time and with (U_g − U_mf)², while decreasing exponentially with temperature, with an activation energy for fresh limestone of −4.3 kJ/mol. The attrition followed Rittinger’s surface theory [Beke B. Comminution. Budapest: Akademiai Kiado, 1964; Ray YC, Jiang TS, Wen CY. Particle attrition phenomena in a fluidized bed. Powder Technol 1987a; 49:193-206]. The change of surface area of limestone particles was proportional to the total excess kinetic energy consumed and to the total attrition time, whereas the change of surface area decreased exponentially with increasing temperature. At 850 °C, the attrition rate of calcined lime was highest, whereas the attrition rate was lowest for sulfated particles. When online impact attrition was introduced, the attrition rate was about an order of magnitude higher than without impacts.     

Effect of the milling process on the properties of CoFe2O4 pigment

In this study, CoFe₂O₄ pigments were synthesised using both co-precipitation and conventional ceramic methods. Pigment particles prepared using the conventional ceramic method were subsequently milled to submicron size. The effects of the solvent, dispersant and milling type in the milling process were investigated. This study showed that planetary milling in a diethylene glycol (DEG) medium with sodium tripolyphosphate (STPP) was an effective method for producing submicron-sized pigment powders from pigments synthesised using the conventional method. With this method, submicron-sized pigment particles (approximately 190 nm) were obtained after milling for 4 h. Planetary milling was more efficient in reducing particle size compared to attrition milling. Co-precipitated pigment had a more intense black colour, due to the nanoscale particle size (<100 nm). However, conventional ceramic pigments also had an adequately intense black colour that increased after milling compared to unmilled conventional pigments. When considering production of industrial scale submicron-sized pigments, the milling of these pigments to submicron size can be a good alternative method for the production of ink colourants.     

Impact attrition of particulate solids. Part 2: Experimental work

Single particle impact testing has been carried out to evaluate the attrition by chipping of ionic single crystals of MgO, NaCl and KCl, used as model materials having a semi-brittle failure mode. The results have verified the theoretical predictions presented in Part 1. The dimensionless attrition propensity parameter, η=ρv²lH/K²_c, is shown to account for the dependence of the material loss per impact on the material properties, impact velocity and particle size. The ‘chipping’ mechanism of impact attrition is further confirmed by the observations of the impact damage by the use of scanning electron microscopy. In addition, the influence of rigidity of target materials on attrition rate is also addressed.     

A numerical study of particle wall-deposition in a turbulent square duct flow

The deposition of dense solid particles in a downward, fully developed turbulent square duct flow at Re_τ = 360, based on the mean friction velocity and the duct width, is studied using large eddy simulations of the fluid flow. The fluid and the particulate phases are treated using Eulerian and Lagrangian approaches, respectively. A finite-volume based, second-order accurate fractional step scheme is used to integrate the incompressible form of the unsteady, three-dimensional, filtered Navier-Stokes equations on an 80 × 80 × 128 grid. A dynamic subgrid kinetic energy model is used to account for the unresolved scales. The Lagrangian particle equation of motion includes the drag, lift, and gravity forces and is integrated using the fourth-order accurate Runge-Kutta scheme. Two values of particle to fluid density ratio (ρ_p/ρ_f = 1000 and 8900) and five values of dimensionless particle diameter (d_p/δ × 10⁶ = 100, 250, 500, 1000 and 2000, δ is the duct width) are studied. Two particle number densities, consisting of 10⁵ and 1.5 × 10⁶ particles initially in the domain, are examined.Variations in the probability distribution function (PDF) of the particle deposition location with dimensionless particle response time, i.e. Stokes number, are presented. The deposition is seen to occur with greater probability near the center of the duct walls, than at the corners. The average streamwise and wall-normal deposition velocities of the particles increase with Stokes number, with their maxima occurring near the center of the duct wall. The computed deposition rates are compared to previously reported results for a circular pipe flow. It is observed that the deposition rates in a square duct are greater than those in a pipe flow, especially for the low Stokes number particles. Also, wall-deposition of the low Stokes number particles increases significantly by including the subgrid velocity fluctuations in computing the fluid forces on the particles. Two-way coupling and, to a greater extent, four-way coupling are seen to increase the deposition rates.     

Behavior of Compact Nonspherical Particles in the TSI Aerodynamic Particle Sizer Model APS33B: Ultra-Stokesian Drag Forces

The aerodynamic behavior of aggregates consisting of uniform polystyrene latex (PSL) spheres and unaggregated cuboidal Natrojarosite particles in a TSI aerodynamic particle sizer (Model APS33B) has been studied. In initial tests, monodisperse PSL micro-spheres ranging from 0.3 to 7 μm in geometric diameter were generated from aqueous suspensions using a Lovelace nebulizer. APS33B responses for these uniform-sized particles showed multiple peaks. The major (primary) peak, which resulted from the smallest particle, corresponded to the unaggregated single spheres (singlets); the second, third, and fourth peaks were identified as doublets, triangular triplets, and tetrahedral quadruplets, respectively. Both doublets and triplets moved with their long axes in perpendicular (maximum drag) orientation to the flow direction in the APS33B. In contrast, the tetrahedral particles were isometric and had the same dynamic shape factor (drag resistance) for all three primary orientations. The particle Reynolds numbers (Re _p) for these particles were calculated and ranged from 0.2 to 30 in the sensing volume of the APS33B detector (i.e., ultra-Stokesian conditions). Ultra-Stokesian drag forces for all three types of aggregates were, therefore, estimated and expressed as a function of an empirical factor (1 + aRe ^b _p) to the Stokesian drag force. The ultra-Stokesian drag of a Natrojarosite particle was measured in the range 20 Re _p < 50 and could be described with a similar expression. This approach facilitates the study of the dynamic behavior of nonspherical particles and yields new information about the characteristics of drag forces in the ultra-Stokesian regime     

Impaction of spherical particles on cylinders at moderate Reynolds numbers

Inertial and interceptive impaction of spherical particles on circular cylinders was investigated theoretically. The particles were considered to be suspended in a fluid moving steadily through a random array of parallel cylinders. Fluid flow fields around the cylinders were obtained by numerically solving the Navier-Stokes Equations subject to Kuwabara's zero vorticity boundary condition. These solutions were subsequently utilized in calculating particle trajectories and impaction efficiencies. The latter are presented as functions of Reynolds number (0·2 ? Re_c ? 40), particle inertial parameter (0 ? P ? 1000), particle to cylinder size ratio (0·001 ? K ? 1·0) and cylinder concentration (10^?4 ? c ? 0·111).The impaction efficiencies and critical inertial parameters differ from earlier theoretical predictions. The discrepancies are primarily attributable to the inaccurate flow field representations used by previous authors. The agreement between Subramanyam and Kuloor's experimental work and present theory is satisfactory.     

Heat transfer and flow pattern in vertical liquid-solids flow

Wall-to-bed heat transfer in hydraulic transport of spherical glass particles of diameter 1.20, 1.94 and 2.98 mm and in single-phase flow regime was studied. Experiments were performed by transporting the spherical glass particles with water in a 25.4 mm I.D. copper tube equipped with a steam jacket.In the runs without particles, the tube Reynolds number varied between 2280 and 21,300, while in hydraulic transport runs, the tube Reynolds number varied between 3300 and 20,150. The loading ratio (G_p/G_f) was between 0.07 and 0.328, and the fluid superficial velocity was between 0.29·U_t and 2.86·U_t, where U_t represents the single particle terminal velocity. For these ratios, the voidage ranged from 0.715 to 0.895.The data for the heat transfer factor (j_H) in single-phase flow are correlated using a general form j_H=f(Re). The data for wall-to-bed heat transfer in the hydraulic transport of particles show that an analogy between heat and momentum transfer exists. The data were correlated by treating the flowing fluid-particle suspension as a pseudofluid, by introducing a modified suspension-wall friction coefficient (f_w) and a modified Reynolds number (Re_m).     

The flow of non-Newtonian slurries through fixed and fluidised beds

Measurements have been taken of the flow rate, pressure drop and bed height characteristics when non-Newtonian slurries flow through fixed and fluidised beds of uniformly sized spherical particles.In the case of fixed beds, the pressure drop-flow rate data has been interpreted using the capillary model of a porous medium together with rheological data for the slurries obtained from a tubular viscometer. The resulting friction factor-Reynolds number relationship is

This correlation was used to satisfactorily predict the minimum fluidisation velocity for a given solid/liquid system by equating the pressure drop to the net weight per unit area of particles in the bed. However, the correlation was not adequate for the prediction of bed expansion in the fluidised state. For systems which have a Reynolds number at minimum fluidisation, Re′_mf′ less than 40 an effect of particle diameter to bed diameter was observed. For systems having Re′_mf >40 the velocity, υ, and voidage, ?, were related to their values at minimum fluidisation by

It is therefore clear that, in the fluidised state, the capillary model does not present an adequate basis for the prediction of bed expansion.     

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.     

Homogeneous drag models in gas–solid fluidization: Big data analytics and conventional correlation

The drag force model is vital for capturing gas–solid flow dynamics in many simulation approaches. Most of the homogeneous drag models in the literature are expressed as a function of phase fraction (ε) and particle Reynolds number (Re_s). In this work, we use a “big data” approach to analyze ~10⁸ data points for drag coefficient (F_d) for Geldart Group A particles at atmospheric pressure and find that the contribution of Re_s on F_d is much less than ε based on the Maximal information coefficient analysis. Thus, these drag models are separately reduced to machine learning and conventional expressions only related to ε. The reduced models achieve almost the same predictive performance as the originals in bubbling, turbulent, and jet fluidizations. Moreover, the reduced models provide better numerical stability for coarse grid simulations. These findings provide new insights into the drag coefficient for Geldart Group A particles under full fluidization conditions.     

Effect of particle shape on hindered settling in creeping flow

Solid particles of uniform size and shape were used to study the effect of particle shape on hindered settling in creeping flow (Re_o ? 0.2), where fluid flow patterns are independent of Reynolds number and the effect of shape is most prominent. The particles of different shape studied were spherical glass beads, cubical sodium chloride crystals and ABS plastic pellets, brick-like sugar crystals, and angular (imperfect octahedral) mineral silicate crystals. The liquids used were aqueous polyethylene glycol solutions and various blends of hydrocarbon oils. Two particle sizes on the average were investigated for each particle shape, and five settling column diameters were employed, so that the overall range of column-to-particle diameter ratio covered was 22 – 226.A Richardson—Zaki type equation of the form u = u_i?ⁿ was found to correlate the constant settling rate data for each particle size and shape over the voidage range ? = 0.65 – 0.9. However, the wall effect on hindered settling rate was found in most cases to be considerably smaller than that predicted by Richardson and Zaki. The term u_i, obtained by linearly extrapolating the settling velocity u (below ? = 0.9) to ? = 1 on a log—log plot of u versus ?, was found to be measurably lower than the corresponding free settling velocity. The index n varied from an average value of 4.8 for the smooth spheres to 5.4 for the cubes to 5.8 for both the brick-like and the angular particles. These values graphically display a definite trend with settled bed voidage, ?_b, which is shape-dependent and easily measured, and may therefore be a convenient parameter for taking account of shape variation generally.The method proposed by Beranek and Klumpar for correlating fluidization data on different shaped particles, which depends on ?_b, was found to be moderately successful in correlating the present settling data for different shapes.     

Influence of the method and degree of milling on the rate of milling and leaching of industrial alumina

Conclusions An investigation was made of the influence of the method and degree of milling on the time required to obtain a given mean particle size, and on the leaching of refired industrial alumina.It was found that much more time is required for milling of alumina in a ball mill than in a vibrational mill, the more so, the higher the temperature of the preliminary firing of the alumina. At the same time, the mean surface diameter of the powder increases, together with the content of grains of radius greater than 5, while the content of grains of radius less than 2, which is important to ensure good properties of the fired ceramic, decreases.During milling in a ball mill, the amount of ground iron is increased considerably compared with the amount during vibrational milling to the same particle size of the alumina.Leaching of alumina after milling in a ball mill is less satisfactory than after milling in a vibrational mill. As a result, the Al₂O₃ content in the washed material is decreased and the amount of Na₂O, Fe₂O₃, and other impurities is increased.Translated from Ogneupory, No. 1, pp. 43–50, January, 1969.