Specific Rate Of Breakage Research Articles

Assessment of different energy-based breakage distribution functions in population balance model of an industrial scale continuously operated wet stirred media mill

This study aimed at assessing various energy-based breakage distribution functions Bij within the context of a population balance model (PBM) of copper ore milling in a wet stirred media mill (WSMM). First, drop weight impact tests were performed to determine the size distribution parameters, tn. Second, various tn models called tn-fitted and tn-regenerated approaches were used to fit experimental tn, and their goodness-of-fits were compared. They were then used to construct the matrices of the cumulative breakage distributions, Bij, of the PBM, while a power-law form of the specific breakage rate function, Si, was assumed. By fitting the PBM to the product particle size distributions (PSDs) obtained under various rotor speeds, suspension volumetric flow rate, and solids content, we estimated the Si parameters. Results suggest that except the Napier-Munn model, all tn models fitted the experimental drop weight data reasonably well, which was reflected in the PBM fitting of the product PSDs. The PBM fits with Bij constructed with the tn-regenerated approach were more accurate than those with the tn-fitted approach. The PBM was validated with additional tests that were not considered in the parameter estimation. Overall, we have established that the choice of tn function and the methodology to determine Bij can affect the PBM predictions of the WSMM process significantly.

Population Balance Modeling of Milling Processes: Are We Falsifying Breakage Kinetics and Distribution via Back-Calculation Methods?

Population balance models (PBMs) for milling processes are based on two fundamental concepts: specific breakage rate function and breakage distribution function, which vary with particle size as well as design–operation conditions. The solution of the inverse problem, i.e., the estimation of these two functions’ parameters, may cause falsified kinetics and breakage distribution mechanisms. This perspective article aims to expose and mitigate various aspects of potential falsification, thus enabling the development of a robust PBM. Through an in-depth analysis of historical approaches to the PBM inverse problem and experimental observations, as well as the author’s recent contributions to the inverse methodology within the context of back-calculation methods, six principles have been offered: (i) include the governing physical phenomena and reduce errors in model building; (ii) reduce the number of model parameters via size–operation-dependent functional forms, hybrid approaches for back-calculation, and combination with CFD–DEM and other mechanistic models; (iii) generate a dense particle size distribution data set obtained at various milling times and/or locations; (iv) ensure a grid-independent solution with a sufficient number of size classes; (v) use a global optimization-based back-calculation method for parameter estimation and provide standard errors of the estimates; and (vi) test the predictive capability of the PBM. This perspective article boosts awareness of various challenges involved in the solution of the inverse PBM problem as pertinent to milling processes and provides researchers with six principles to minimize falsified kinetics.

A Novel PBM for Nanomilling of Drugs in a Recirculating Wet Stirred Media Mill: Impacts of Batch Size, Flow Rate, and Back-Mixing.

We examined the evolution of fenofibrate (FNB, drug) particle size distribution (PSD) during the production of nanosuspensions via wet stirred media milling (WSMM) with a cell-based population balance model (PBM). Our objective was to elucidate the potential impacts of batch size, suspension volumetric flow rate, and imperfect mixing in a recirculating WSMM. Various specific breakage rate functions were fitted to experimental PSD data at baseline conditions assuming perfect mixing. Then, the best function was used to simulate the PSD evolution at various batch sizes and flow rates to validate the model. A novel function, which is a product of power-law and logistic functions, fitted the evolution the best, signifying the existence of a transition particle size commensurate with a grinding limit. Although larger batches yielded coarser and wider PSDs, the suspensions had identical PSDs when milled for the same effective milling time. The flow rate had an insignificant influence on the PSD. Furthermore, the imperfect mixing in the mill chamber was simulated by considering more than one cell and different back-mixing flow ratios. The effects were weak and restricted to the first few turnovers. These insights contribute to our understanding of recirculating WSMM, providing valuable guidance for process development.

On the Similarity of Austin Model and Kotake–Kanda Model and Implications for Tumbling Ball Mill Scale-Up

The aim of this theoretical investigation is to seek any similarities between the Austin model and the Kotake–Kanda (KK) model for the specific breakage rate function in the population balance model (PBM) used for tumbling ball milling and assess feasibility of the KK model for scale-up. For both models, the limiting behavior for small particle size-to-ball size ratio and the extremum behavior for a given ball size are described by “power-law.” Motivated by this similarity, specific breakage rate data were generated using the Austin model parameters obtained from the lab-scale ball milling of coal and fitted by the KK model successfully. Then, using the Austin’s scale-up methodology, the specific breakage rate was scaled-up numerically for various mill diameter scale-up ratios and ball sizes of 30–49 mm and coal particle sizes of 0.0106–30 mm. PBM simulations suggest that the KK model predicts identical evolution of the particle size distribution to that by the Austin model prior to scale-up. Upon scale-up, the differences are relatively small. Hence, modification of the exponents in the Austin’s scale-up methodology is not warranted for scale-up with the KK model. Overall, this study has established the similarity of both models for simulation and scale-up.

Application of artificial neural network method to predict the breakage properties of PGE bearing chromite ore

In the present investigation, systematic grinding experiments were conducted in a laboratory ball mill to determine the breakage properties of low-grade PGE bearing chromite ore. The population balance modeling technique was used to study the breakage parameters such as primary breakage distribution (Bi, j) and the specific rates of breakage (Si). The breakage and selection function values were determined for six feed sizes. The results stated that the breakage follows the first-order grinding kinetics for all the feed sizes. It was observed that the coarser feed sizes exhibit higher selection function values than the finer feed size. Further, an artificial neural network was used to predict breakage characteristics of low-grade PGE bearing chromite ore. The predicted results obtained from the neural network modeling were close to the experimental results with a correlation of determination R2 = 0.99 for both product size and selection function.

The effects on the grinding parameters of chemical, morphological and mineralogical properties of three different calcites in a Hardgrove mill

Recently, the kinetic model commonly uses the specific rate of breakage (Si) and the cumulative breakage distribution (Bi,j) functions to explain the grinding behavior of the material with respect to time in a laboratory batch mill. Until now, important researches have been published on these functions, especially in a tumbling ball mill. However, so far, there has been no study determining the effects of kinetic breakage parameters on calcites in a Hardgrove mill, which is a vertical roller mill. This study presents the kinetic breakage behavior of the three different calcites in a Hardgrove mill. Firstly, the Bond and Hardgrove grindability tests were performed. Secondly, kinetic breakage parameters (Si aT, α, γ, β, and ϕj) were determined from product size distributions for short grinding times and compared for three different calcite types. From the result of the tests, contrary to expectations, different results were obtained between the grindability data and kinetic breakage parameters. The reason for these different results was found to be due to the different chemical, morphological, and mineralogical properties of calcites.

Population balance modeling approach to determining the mill diameter scale-up factor: Consideration of size distributions of the ball and particulate contents of the mill

In the past, most of the Population Balance Model-based ball mill scale-up studies were carried out using a mono-size ball charge and a single size fraction of the particulate material as the starting feed. It was reported that the breakage distribution parameters were independent of the mill diameter, specific breakage rate parameters varied with mill diameter as Dγ, and the mill diameter exponent γ was independent of particle size. However, in the industrial grinding operations, we have significantly broad size distributions for both the grinding media and the particulate contents of the mill. Therefore, experimental data generated under these conditions were taken from the literature and analyzed. These data pertain to the dry and wet batch grinding of limestone, quartz, and ores of vanadium and pyrochlore. Our analysis of the data showed that in contrast to the findings of the earlier researchers, the breakage distribution parameters were not independent of the mill diameter, and the value of γ for the specific breakage rate parameters varied with particle size in the case of multicomponent ores. Because of these complications, a single parameter, the specific rate of production of material that is finer than a specified size, was used as the basis for scale-up. It was found that the value of the mill diameter scale-up factor varied quite significantly with the breakage properties of the material, the size distribution of the particulate contents of the mill, the size of the desired product, and the mode of grinding (dry/wet). Therefore, ball mill scale-up models presently available need to be modified in light of these new findings.

Grinding characteristics of waste printed circuit boards in hammer mill using population balance model

ABSTRACT In this study, the grinding characteristics of waste printed circuit boards (PCBs) were evaluated using a population balance model (PBM). The grinding characteristics under various conditions (hammer rotational speeds of 1600, 2000, and 2400 rpm; screen aperture sizes of 0.71, 1.0, and 1.40 mm; and maximum feed sizes of 4, 6, and 8 mm) were identified. A PBM for a continuous hammer mill was used to quantitatively demonstrate the grinding characteristics. No significant difference was observed in the particle sizes of the products for various hammer rotational speeds. Nevertheless, the specific rate of breakage increased with the hammer rotational speed. As expected, a narrow screen aperture yielded a fine product. The mean residence time was increased in this case to comminute the particles to a finer size. Furthermore, a finer feed was predicted to yield a finer product; however, a coarser feed was observed to yield such a product instead. Each hammer milling condition should be controlled with respect to the hammer milling performance and target particle size of the product.

Behaviors and physical mechanism of ceftezole sodium de-agglomeration driven by ultrasound

Ultrasound-mediated method, which can effectively disperse agglomerates or even eliminate agglomeration, has received more and more attentions in industrial crystallization. However, the ultrasound-mediated de-agglomeration mechanism has not been well understood, and no general conclusions have been drawn. In this study, the crystallization and de-agglomeration process of ceftezole sodium agglomerates under ultrasound irradiation were systematically investigated. Kapur function was selected to investigate the de-agglomeration process under different ultrasonic powers. The results revealed that ultrasound could efficiently inhibit agglomeration. Besides, the de-agglomeration of large sized agglomerate particles was found to be easier to occur in comparison with small sized particles due to its higher specific breakage rate. Finally, the de-agglomeration mechanism under ultrasonic irradiation was proposed on the basis of the calculated cumulative breakage functions.

Study of the Effect of the Mineral Feed Size Distribution on a Ball Mill Using Mathematical Modeling

In this paper, the effect of the feed size distribution was studied for a ball mill. Copper ore of 0.5% was tested using batch-grinding tests. Samples were carried out using three reconstructed feed size distributions in an experimental ball mill. The size distribution was reconstructed using the double Weibull formula, and modeled using the selection and breakage functions. Copper ore samples were grinding up to 15 min. A nanosize (-19+12.5), coarse (-19+6.3 mm), medium (-4.75+0.6 mm), and fine (+0.425-0.038 mm) particle size distributions were tested. The 0.32 m diameter test mill was run at 38% volume loading and Nc (critical speed) of 68% (50.85 rpm). Breakage rate parameters were calculated and simulated for each size distribution in an industrial mill. The results show that the change in the feed size distribution has an impact on the grinding kinetics. The curves show that the mono size fraction has the highest specific rate of breakage. It is also important to note that, as the size distribution F80 decreases the power consumption increases.

Estimation of ceramic material properties using specific rate of breakage determined from grinding tests of nepheline syenite

Estimation of ceramic material properties using specific rate of breakage determined from grinding tests of nepheline syenite

THE EFFECTS OF BALL SIZE ON THE DETERMINATION OF BREAKAGE PARAMETERS OF NEPHELINE SYENITE

In this study, the changes in the specific rate of breakage and breakage distribution function of the nepheline syenite sample were investigated by using alloy steel ball in five different sizes. Specific rate of breakage and breakage distribution function values were obtained from the particle size distributions acquired after the grinding periods. As a result of grinding tests, an increase in rate of breakage is observed due to the increase in ball diameter.

Effect of particulate environment on the grinding kinetics of mixtures of minerals in ball mills

This paper presents the results of the experiments on the effect of particle size distribution and the volume ratio of components of the binary mixtures of calcite, hematite, and quartz on the grinding kinetics in ball mills. A total of forty-six grinding tests were carried out. These included forty-two dry grinding tests and four wet grinding tests. For each test, only one short-duration batch grinding experiment was carried out. The feed charge was prepared by pre-grinding the particulate material in the test ball mill. The results of these experiments have led to several important findings. Three of these findings are: (i) the size distribution of the ground product of the mixture of two minerals was practically the same as the mass-weighted sum of the size distributions of the ground product of these minerals when they were ground alone for the same duration, (ii) fine particles broke faster in the coarser environment, and (iii) while quartz fines were found to be most effective in increasing the specific breakage rate of coarse particles of quartz and hematite, in the case of calcite, hematite fines were found to be most effective. It shows that besides the hardness and density of the components, there are some other presently unknown properties that determine the fraction of the net energy input to the mill that is absorbed by the mixture.

Clasificación de la velocidad específica de fractura (Sj) mediante análisis discriminante para la molienda de cuarzo

Specific rate of breakage (Sj) is an important parameter for grinding kinetics behavior due to it isreverserelated with the process energy consumption. Size grinding media, viscosity medium, and fine particle formation are some ofmodifiablevariableforto reduce the energy in the grinding process.Nowadays, there is no model that explains the relationship among Sj and parameters described above.Aclassification model based on linear discriminant analysisfor quartz wet grinding wasproposedto identify conditions with the high Sj.Three grinding kinetic behavior groups have been found through cluster analysis and two discriminant functions that explicate difference among groups. The first function was themost powerful differentiating dimension with 89.01% of prediction percentage,and the second onerepresented an additional significant dimension with 10.99%of prediction.

Investigation on breakage behaviour of oil shale with high grinding resistance: A comparison between microwave and conventional thermal processing

Co-processing (e.g. co-combustion) of oil shale with other fuels like coal to derive energy could mitigate environmental-related problems associated with oil shale whilst enhancing thermal efficiency. However, reducing the size of oil shale particles to that similar to coal for better handling and combustion is energy-intensive given its high grinding resistance. Therefore, it is essential to improve the grindability of oil shale. In this study, two Chinese oil shales were investigated. The sieved samples were pre-treated and crushed to study the failure mechanisms, as well as changes in uniaxial peak strength and specific breakage rate in comparison to that of untreated samples; and how chemical properties were affected after either conventional or microwave-enhanced thermal processing. Results show that the uniaxial compressive strength of oil shale was significantly reduced whilst the breakage rate increased after a short exposure to microwave irradiation. The improvement was approximately 50 % higher than that observed after conventional heating. Hence, results suggest that microwave processing could lead to a more significant and rapid improvement in oil shale milling compared to conventional pre-heating, and without adversely affecting fuel quality. The simple microwave pre-treatment can potentially widen the utilization of oil shale, particularly when homogeneity is necessary during co-processing.

Energy absorption and specific breakage rate of particles under different operating conditions in dry ball milling

A detailed analysis of the specific breakage rate and mill power data on dry ball milling of three individual minerals and their binary mixtures has been carried out. The energy split model of Kapur and Fuerstenau has been examined and shown to be incorrect. A new model has been proposed. The basic postulates are: (i) only a small fraction λ of the net energy input to the mill is absorbed by the particulate material (ii) value of λ is mainly determined by the hardness and specific gravity of the material(s) being ground, volume proportion of the components of the mixture, particle size, and mill operating conditions, and (iii) in the mixture, same size particles of different components receive the same amount of energy per unit volume. An important feature of the model is the use of volume-specific mill power. It has been shown that the effect of the operating variables such as the mill speed, particle load, and mill diameter on energy absorption is significantly different for different materials. These findings have been discussed in light of our current state of understanding of the mechanics of the ball milling operation. It has been concluded that the concept of energy absorption factor can be quite useful in the analysis of grinding systems, and mill scale-up design work.

A Study of the Effect of Medium Viscosity on Breakage Parameters for Wet Grinding

The rheological behavior of mineral slurries shows the level of interaction or aggregation among particles, being a process control variable in processes such as slurry transportation, dehydration, and wet grinding systems. With the aim to analyze the effect of medium viscosity in wet grinding, a series of monosize grinding ball mill tests were performed to determine breakage parameters, according to the generally accepted kinetic approach of grinding processes. A rheological modifier (polyacrylamide, PAM) was used to modify solutions viscosity. A model was proposed by means of dimensional analysis (Buckingham’s Pi theorem) in order to determine the behavior of the specific breakage rate (Sj) for a ball grinding process in terms of the rheology of the system. In addition to this, a linear adjustment was established for the relationship between specific breakage rates with and without PAM addition, based on the reduced viscosity, μr. Furthermore, within a certain interval of viscosity, it was proved that an increment of viscosity can increase the specific breakage rate, and consequently the grinding degree.

Optimization of Solids Concentration in Iron Ore Ball Milling through Modeling and Simulation

Important advances have been made in the last 60 years or so in the modeling of ball mills using mathematical formulas and models. One approach that has gained popularity is the population balance model, in particular, when coupled to the specific breakage rate function. The paper demonstrates the application of this methodology to optimize solids concentration in ball milling of an iron ore from Brazil. The wet grinding experiments were conducted in bench (0.25 m diameter) and pilot-scale mills (0.42 m diameter), and surveys in a full-scale industrial (5.2 m diameter) mill. It is first demonstrated that the successful application of the model required recognizing the non-normalizable nature of the breakage function of the particular ore. It is then demonstrated how the model can be used to predict results of pilot-scale grinding tests under different conditions (overflow/grate discharge) based on data from batch grinding tests. Finally, the model is used to predict the effect of changing solids concentration inside the industrial mill, with good correspondence between the pilot plant and full-scale results, which demonstrated the benefit of reducing solids concentration to values between 76 and 80% in weight for the ore of interest from the 83% that was originally used in the operation.

Understanding the drawbacks of the currently popular approach to determining the breakage rate parameters for process analysis and mill scale-up design

The currently popular approach to a direct determination of the specific breakage rate of particles of a given size in the ball milling operation involves: (i) using material crushed in jaw/roll crushers as the starting feed charge, (ii) batch grinding in four–five steps for a total period of 4–8 min, and (iii) determining the slope of the best-fit straight line of a log-linear ‘first-order disappearance kinetics’ plot. In this paper, using published data on grinding of the mixtures of minerals, and effect of the mill operating variables on grinding kinetics, it has been shown that this approach is not quite suitable for carrying out a meaningful analysis of the grinding operation as the specific breakage rate of particles varies quite significantly during the period of grinding covered in the test work. It has been emphasized that the disappearance kinetics of the crushed particles as a top size should not be of any concern to us for the purpose of design and scale-up. It is the instantaneous specific breakage rate of particles produced in the test ball mill under the operating conditions of interest that needs to be determined.

Breakage modeling of needle-shaped particles using the discrete element method

This paper models the breakage of large aspect ratio particles in an attrition cell using discrete element method (DEM) and population balance (PB) models. The particles are modeled in DEM as sphero-cylinders. The stresses within each particle are calculated along the particle length using beam theory and the particle breaks into two parts if the stress exceeds a critical value. Thus, the size distribution changes with time within the DEM model. The DEM model is validated against previously published experimental data.The simulations demonstrate that particle breakage occurs primarily in front of the attrition cell blades, with the breakage rate decreasing as the particle sizes decrease. Increasing the particle elastic modulus, decreasing the particle yield strength, and increasing the attrition cell lid stress also increase the rate of breakage. Particles break most frequently at their center and the daughter size distribution normalized by the initial particle size is fit well with a Gaussian distribution. Parametric studies in which the initial particle size distribution varies demonstrate that the particle sizes approach a distribution that is independent of the initial state after a sufficient amount of work is done on the particle bed. A correlation for the specific breakage rate is developed from the DEM simulations and used within a PB model along with the daughter size distribution fit. The PB model also clearly shows that the particle size distribution becomes independent of the initial size distribution and after a sufficiently long time, is fit well with a log-normal distribution.