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By A. C. Johnstone

The aim of the study is to determine if pit lakes are a sustainable coal mine closure option in South African. The water balance, chemistry, limnology, and bacterial population of three selected pit lakes were investigated. The lakes are in the three major coal basins of South Africa and are associated with different lithologies and mining methods. The major factors driving the water balance of the pit lakes are direct rainfall, runoff, inflow from old mine workings, and groundwater infiltration, with the major losses being evaporation or discharge onto surface. The study indicated that pit lakes can be designed as ’terminal sinks’ to provide a sustainable mine closure option. The pit lakes sampled have an alkaline pH, and mostly a sodium/calcium sulphate water with total dissolved solids content of less than 3000 mg/l. The phytoplankton and microbiological data indicates that the pit lakes support aquatic life. The study shows that correctly designed pit lakes can be an environmentally sustainable closure option for South Africa’s coal mines. A suggested design manual has been developed to assist mine owners and regulators in developing sustainable coal mine pit lakes as a closure option.

Oct 1, 2021

By J. Hadjigeorgiou, Y. Potvin

In a benchmarking paper published in the SAIMM Journal, Potvin and Hadjigeorgiou (2008) reviewed ground control strategies to manage large deformations in underground hard rock mines. Significant differences were identified in the design and acceptable tolerance between civil and mining engineering practice for squeezing ground conditions. Variations were also noted in ground support strategies between Australian and Canadian mines. In this paper we review progress in design methodology and ground support practice since 2008, including the availability of more effective deformation monitoring, new design tools, greater access to yielding ground support elements, and better installation practices. An improved understanding of technical and practical issues associated with squeezing conditions has led to the development of ground support guidelines for a range of squeezing ground conditions in hard rock mines.

Jul 5, 2023

By P. Minkkinen, R. C. A. Minnitt

Particle size distribution analysis is important to many industries for assessing product quality, process efficiency, and compliance with contractual standards, while ensuring sample accuracy, and representativeness. Inaccurate particle size distribution results, arising from fines due to excessive handling can lead to penalties and buyer-seller disputes. A non-representative particle size distribution means samples will not accurately represent the chemical components either. Sampling blasthole cuttings poses challenges due to large material volumes, segregation, and uneven settling of particles in dense gold bearing ores. Commonly used sampling methods, such as spear or sectoral sampling, may lack representativeness. Particle size distribution analysis is crucial for evaluating sampling methods by comparing test sample representativeness against a reference standard derived from multiple particle size distribution analyses. Acceptable samples must fall within the 97.5% confidence intervals of the standard; deviations analyses lying outside these boundaries indicate a strong likelihood of segregation errors. A case study demonstrates this approach by using a 6.005 kg benchmark sample to assess two additional samples (1 kg and 5 kg), highlighting particle size distribution’s role in detecting sampling errors and refining protocols. An example of three test samples collected from reverse circulation drill cuttings compared to a standard reference sample is presented to illustrate a quality assurance/quality control procedure for reverse circulation drilling.

Oct 2, 2025

By A. Mabentsela

A furnace freeze lining is necessary for safety and economic reasons in several smelting operations. The integrity of the freeze lining is put at risk by furnace power imbalances. Although numerous models have been derived to model the growth and depletion of the freeze lining due to power imbalances, no model exists for thermomechanical damage to the freeze lining. This study provides an initial pathway for modelling thermomechanical damage to the freeze lining in an ilmenite smelting furnace using information from the literature and experimental work. A methodology under the framework of continuum damage mechanics is proposed to model mechanical damage to the freeze lining due to phase changes, thermophysical changes, and constrained thermal expansion. Drill cores of solidified slag ingots were used to represent the freeze lining. The modified power law proved to be the best predictor of the softening response of the drill cores. Damage driving parameters were extracted from the raw data, and governing equations of the parameters with respect to temperature were derived for use in a finite element method (FEM) code.

Jan 26, 2023

By T. Z. Sehume, J. R. Bunt, J. C. van Dyk, C. A. Strydom

"SynopsisCo-pyrolysis of four biomass samples (hardwood chip, softwood chip, pinewood chip, and sugarcane bagasse) with an inertinite-rich medium rank C South African coal was investigated. Proximate and ultimate analyses of the chars prepared using a heating rate of 10°C/min up to 1100°C in a nitrogen atmosphere were used to compare the properties of the biomass and the coal chars. Similar gross calorific values (28.5–29.1 MJ/kg) for the woody biomass-coal blended chars were observed, which were slightly higher than that of the coal char sample (26.5 MJ/kg). CO2 surface areas of the chars of the woody biomass samples (328.1–329.4 m2/g) and of the blends (238.5–271.5 m2/g) were higher than that of the coal char sample (94.2 m2/g). Thermogravimetric (TG), differential thermal analyses (DTG), and calculated weighted averaged TG curves indicated that the influence of the biomass on the pyrolysis rate of the coal is small and vice versa. The CO2-producing reactions of the coal were slightly enhanced during co-pyrolysis.IntroductionCoal utilization has led to rising concerns about adverse impacts on the environment (global warming) caused by toxic gases (H2S, SOx, NOx, and CO2) and remaining waste. Biomass is considered to be CO2-neutral with regard to the greenhouse gas balance, and is regarded as a renewable source that assists in reducing CO2 emissions when compared with coal (McGowan, 1991; McKendry, 2002). Biomass currently makes up approximately 14% of the world’s energy sources (McGowan, 1991; McKendry, 2002). Coal combustion and gasification are prominent processes in South Africa (in producing synthesis gas and energy) due to the abundance of coal in the country. Some of South Africa’s coal resources are, however, high in ash content and also are inertinite-rich, which renders them more difficult to use industrially than vitrinite-rich low-ash coal (Jeffrey, 2005; Strydom et al., 2011). There is a need to investigate the properties and behaviour of the inertinite-rich high-ash South African coals and especially their interaction with local biomass sources. The gasification process occurs through various stages, of which pyrolysis of the char residue forms an integral part (Bunt and Waanders, 2009). Currently, particular interest is shown in the co-utilization of coal and biomass to produce synthesis gas via the gasification process (Usón et al., 2010; McGowan, 1991; McKendry, 2002). Investigating the co-pyrolysis of the South African coal and local biomass material is thus the first step in evaluating the use of local biomass sources for co-gasification purposes."

Jan 1, 2015

By S. Mudau, P. Mohasoa, M. J. Cole

Derelict and ownerless (D&O) mines in South Africa have been an ongoing concern for over a decade. While some progress has been made to rehabilitate D&O asbestos sites, significant work remains, exacerbated by the growth in illegal mining. There are concerns regarding the accuracy and transparency of the D&O database, lack of oversight, and constrained funding, and the government is increasingly looking to the private sector to fund rehabilitation. Given these challenges, a new, sustainable approach to D&O sites in South Africa is needed to bring clarity and transparency to the process, recognise responsibilities and promote partnerships, and explore economic opportunities at D&O sites to reduce the financial liability and respond to the growing demand for critical minerals. This article has three components: i) reviews of international D&O good practices; ii) it analyses South Africa’s current approach and the D&O sites; and iii) proposes a D&O framework that addresses data, risk ratings, responsibilities and rehabilitation, economic opportunities, and stakeholder engagement. The study finds that the majority of the 6,100 so-called abandoned D&O mines are quarries and mine site features, 16% are operational, at least 11% have owners, and only 752 (12%) require rehabilitation. The framework aims to bring clarity and fairness to decision-making and to assign risk and responsibility for rehabilitation and seeks to incorporate economic opportunities to reduce the financial burden of rehabilitation, create jobs for local communities, and respond to the growing demand for critical minerals for the clean energy transition.

Apr 25, 2025

By M. Hooman, W. M. Marx, R. C. W. Webber-Youngman

This paper discusses a study that investigated different underground bulk air heat exchanger (>100 m3/s) design criteria. A literature review found that no single document exists covering all design criteria for different heat exchangers, and therefore the need was identified to generate a guideline with decision analyser steps to arrive at a technical specification. The study investigated the factors influencing heat exchanger designs (spray chambers, towers, and indirect-contact heat exchangers) and the technical requirements for each. The decision analysers can be used to generate optimized, user-friendly fit-for-purpose designs for bulk air heat exchangers (air cooler and heat rejection). The study was tested against a constructed air cooler and heat rejection unit at a copper mine1. It was concluded that the decision analysers were used successfully. This tool (decision analysers) can be used by engineers for the efficient and cost-effectively design of heat exchangers.

Jan 1, 2015

By L. Diko-Makia, N. Rembuluwani

Artisanal and small-scale mining is crucial for the mineral sector in South Africa, but it also presents both opportunities and challenges. This paper presents an overview of the South African artisanal and small-scale mining sector, the socio-economic significance of artisanal and smallscale mining, the regulatory environment, and the environmental impacts, which are currently the status quo of this sector in South Africa. The paper identified ways in which artisanal and small-scale mining can support local development, create employment opportunities, and reduce poverty. The case of informatisation, ecological damage, and social insecurity has shown social conflicts, potential land degradation, and criminal behaviour. Finally, the paper discusses the future of artisanal and small-scale mining in South Africa, which will largely depend on improved legislation, enhanced personnel capacity, and more sustainable practices to ensure that the positive outcomes of the industry outweigh the negative ones. These propositions include recommendations to all stakeholders, including the government, industry, and civil society, such as stricter environmental regulations, improved working conditions for miners, and the promotion of responsible mining practices.

Dec 15, 2025

By P. T. Slatter

The efficient and accurate design of pipe and pumping plant for non-Newtonian mineral slurries remains a problem. The objective of this paper is to present the development of the Balanced Beam Tube Viscometer (BBTV), an instrument which could facilitate more accurate design for these slurries. The background of rotational and tube viscometry is reviewed and the fundamental principles and inherent advantages of the BBTV are given. The design, data acquisition and processing techniques are introduced and results showing the reliability of the new instrument are presented and discussed. It is concluded that the BBTV is a reliable and versatile instrument for both routine analyses and research work, and has the potential to provide data for the refinement of the design procedure for mineral slurry pipe and plant design.

Jan 1, 1998

By R. C. W. Webber

Mining up to depths of 5 000 m would be a world first and, accordingly, no previous experience in the determination of acceptable heat stress limits, criteria or indices is wholly applicable. However, some South African gold mines are already operating at depths beyond 3000 m and much of the knowledge gained in reaching and working at such depths will be helpful in making adequate provision for acceptable environmental control at the greater depths being contemplated. Accordingly, it is necessary to take cognisance of the industry?s experience in deep-level mining and of standards and regulations already established in South Africa and elsewhere in order to ensure acceptable working conditions and control standards, that compare favourably and defensibly with those in other mining industries. The local and international use of heat stress limits, criteria and indices were investigated as it was necessary to determine to what extent any other indices, limits or criteria would be applicable to South African deep mine conditions. In addition, it was necessary to establish whether there was a single heat stress index that could be used for South African ultra deep mining conditions. It was found that an appropriate combination of heat stress indices would be required in planning for and ultimately controlling thermal conditions in ultra-deep mining. The depths being contemplated and the concomitant potential heat hazard present too great a risk for reliance on a single environmental component of heat stress, such as wet-bulb temperature at present in common use locally. The study recommends that a heat stress index, preferably Air Cooling Power (ACP), be used to design an ultra deep mine?s ventilation system and that wet-bulb temperature be used to monitor and control the system once it is implemented.

Jan 1, 2003

By J. E. Nesset

The bubble size, together with the gas rate, determines the bubble surface area flux in flotation machines and, therefore, impacts directly on the kinetics of flotation. Understanding the variables that influence bubble size is one of the keys to understanding flotation recovery. Surprisingly, these relationships are poorly understood, largely due to the difficulties associated with gas dispersion measurements in industrial environments. The McGill gas dispersion sensors (bubble size, gas rate, gas holdup) are capable of reliably making such measurements. This paper reports on a study, using the McGill sensors in a 0.8 m3 Metso RCSTM pilot cell, which investigated the effect on bubble size of frother type and concentration, gas rate, temperature and rotor speed. A Denver laboratory cell was used to investigate the effect of elevation above sea level. Frother concentration proved to have the predominant influence on bubble size, and the effect of frother type can be normalised as a function of the critical coalescence concentration (CCC). Measurements made at many plant sites worldwide are compared to the model developed using the Metso RCSTM unit.

Jan 1, 2007

By M. -H. Fillion

"Geotechnical stability analysis and design in open pit mines requires access to representative geological, structural, hydrogeological, and rock mass models. The quality and quantity of collected geomechanical data used to build such models can have significant implications in the design of safe and economically viable slopes. Data collection is a continuous process throughout the life of a mine. The process starts from exploration to conceptual/preliminary feasibility, feasibility, design and construction, operations (early production/late production), and mine closure.As geomechanical data is often limited at an early stage of a project, the estimated variability of the geotechnical factors may be larger than the actual data variability. This can potentially result in a conservative selection of the pit slope angle. Collecting additional geomechanical data at later stages of the mine development may contribute to reducing the uncertainty associated with the geotechnical properties and a steeper slope angle may be used for the design. This can result in significant economic benefits.This paper investigates the impact of access to new data over time, following a series of geomechanical data collection campaigns, on design decisions. This is illustrated by comparing the resulting slope design, at three stages of the mine project, at a South African open pit mine. To demonstrate this, a series of slope stability analyses were performed using limit equilibrium tools. For each project stage, the slope angle design was modified until a similar probability of failure of 5% or less was obtained. The corresponding factor of safety distributions for each project stage were then compared to evaluate the impact of collecting additional data on the resulting slope design.The results showed that data variability can differ significantly from one project stage to another. Data variability may be greater at more advanced project stages (i.e. a wider distribution of the factor of safety). Furthermore, the results showed that, even for the cases with greater variability, steeper slope angles were possible at more advanced stages for the proposed design requirement (probability of failure < 5%). In this investigation, as more data became available, the uncertainty associated with data variability could be sufficiently reduced to allow the selection of less conservative slope angles. This can be used as a managerial tool in developing data collection strategies and allocating the necessary funds, and can lead to important economic benefits to the mining operation."

Jan 1, 2016

By J. Saunders, N. Pretorius, G. Krafft, T. P. Katakwa

Imagine being able to see all the future planned mining activities in real time. This is exactly what SIMS (Short Interval Management System) will enable you to do. SIMS is a real time short interval control system that continuously measures and sets optimised plans for mining production processes. The system utilises algorithm-based software that takes resources (people, equipment, tasks, and workplaces) and gives you the best achievable plan. The algorithm was developed using methods found in the field of metaheuristics. A metaheuristic is a strategy to designed to efficiently search for optimal or near optimal solutions to problems that are very complex computationally. (Glover, 1986) There are several algorithms that have been developed to solve optimisation problems which are difficult to solve using traditional techniques. This has given rise to a field called evolutionary computing. (Simon, 2013) Examples of these techniques are genetic algorithms, particle swarm optimisation, and ant colony optimisation. These methods are used to solve problems in a variety of fields. As an example, Kok and Lane (2012) used particle swarm optimisation for equipment allocation in a mechanised underground mining environment. The first SIMS deployment took place in a platinum underground tabular mechanised mine near Burgersfort (province?) from 2017-2019. This included the development from a paper-based system to a fully operational online system. SIMS then moved to a second site and is currently being tested in a second platinum underground tabular mechanised mine in South Africa. Full deployment of the system is currently being phased in. SIMS was built in response to issues that have been plaguing the mining industry for years. These issues are all aligned into one major problem statement: how can we optimise mining production processes without compromising safety? This product was developed by combining years of mining experience with software development. SIMS is based on a four-pillar concept. Safety, productivity, accountability, and business improvement. SIMS uses a set of pre-configured mining and production rules, and with the resources that are available, to determine valid short term production plans. A background algorithm then determines the optimum plan to be executed. The system continuously produces a set of job cards that are pushed to respective operators on their allocated tablets. The management portal component allows you to view and manage all current and planned work on any connected device available. This allows real time visibility of completed, current, and planned work to operators, service departments and management, i.e., seeing into the future.

Sep 1, 2021

By B. McFadzean, N. Patterson, M. Becker, S. Geldenhuys

The retreatment of secondary tailings for platinum group metals recovery presents a range of potential challenges, which vary depending on the nature of the original ore and the type of pretreatment it was subjected to. These challenges include the prevalence of ultrafine particles, which promote slimes coatings on valuable minerals, hinder bubble-particle attachment, increase pulp viscosity, and lead to higher gangue entrainment; surface oxidation of minerals, which alters their floatability; and the presence of deleterious gangue minerals and residual platinum group metal species that are inherently difficult to recover by flotation. This study compares the batch flotation performance across a range of tailings samples from the eastern and western limbs of the Bushveld Complex. A complementary mineralogical analysis was conducted to gain deeper insights into mineral behaviour. The paper evaluates these findings to identify the key factors contributing to poor platinum group metal performance.

Mar 6, 2026

By N. Theron, R. F. Meeser

The global emphasis on conserving energy resources has led to the adoption of hybrid power systems in vehicles. Optimally applied, hybrid systems can save up to 40% on fuel costs. To optimally manage a hybrid vehicle's energy flow it is necessary to know, in real time, all the energy requirements to complete a given route. The energy consumption depends mainly on the vehicle mass, speed profile, and route topography. Of these, the topographic profile is the one factor that is only route-dependent and not influenced by the vehicle or driving styles. The heading vs. distance profile is also an example of a route characteristic not influenced by the vehicle or driving style. In this study the topographic and heading profiles are used to identify routes, and are easily measured by means of digital barometric pressure and compass sensors. Correlations between the current route and previously travelled/stored routes are performed based on their topographic and heading profiles in a point-by-point manner. Above-ground tests were first performed using a road vehicle and six routes to evaluate the system. The system consistently and correctly identified a 20 km route within the first 4 km. It also proved to function correctly in underground tests through the implementation of an instrumented hand-held surveyor's wheel. This system finds direct practical application in optimizing the energy management of an underground hybrid locomotive used by the mining industry in South Africa, but can also be of benefit in applications where route identification is required and using GPS is not feasible.

Jan 26, 2023

By N. V. Moraka

In dealing with transformational aspects, the South African Constitution aspires for a balance between attempts to broadly represent the country’s demographics in the way in which organizations are structured and the need to maintain the necessary skills to ensure continual retention of skills for optimum and efficient organizational performance. Through the Mining Charter, government aimed to redress gender and racial imbalances in the mining industry as part of transforming the demographic representation of mining companies to include black people, women, and people living with disabilities in all job levels up to top management. While the Constitution calls for an appropriate balance between demographic representation and competency, this aspiration for mining poses challenges for companies due to the proficient and competent skills needed in this industry. To this end, the South African mining industry has been subjected to substantial criticism from government regarding the slow progress of transformation. The mining industry justifies its slow pace of transformation by pointing at inconsistent, varying legislation and definitions governing transformation, limited skills available from historically disadvantaged groups, and cautioning not to compromise the sustainability of mining companies by marginalizing white males and white youth with the objective of addressing racial imbalances. This lack of understanding between industry and the government has led to considerable confusion regarding the accountability of stakeholders in transforming the industry. This article presents the perspectives of ten mining executives on transformation in the South African mining industry. The theoretical description of transformation, as well as other mining stakeholders’ interpretations of transformation. is presented. Qualitative data from interviews with mining executives demonstrates their understanding of transformation through rich descriptions, interpretations, and implications in their contexts. The findings demonstrate that transformation in the mining industry is not simply about race and gender, but about cultural change, a change in mind-sets, embracing diversity, equalizing rights and opportunities, and attaining social justice. Moreover, transformation is clearly a gradual process. The article concludes with a recommendation for the definition of transformation to be corroborated with the views of mining executives and that of the Constitution of South Africa. This will create an overarching, simplified definition of transformation that can be accepted and interpreted by all stakeholders.

Jan 1, 2016

The key aspects of process metallurgy that distinguish platinum group metal (PGM) concentrate smelting from that of other base metal sulphide concentrates are presented. These differences include considerably higher input chrome and magnesia contents that directly raise the slag liquidus temperature and have the potential to increase accumulations of refractory spinels. Most importantly, the higher processing temperature required for PGM smelting and the resulting very high matte superheat lead to considerably more onerous smelting conditions than those typical of other smelting operations. This has presented challenges to furnace design and integrity, especially when coupled with the progressive intensification of smelting, involving doubling, and then redoubling, of furnace power inputs over the past 20 years. These power increases have been enabled by increasingly more advanced furnace cooling and structural technologies. Key technologies include strong constant-force spring-loaded bindings acting in three dimensions to minimize infiltration of superheated matte into brick joints, and robust well-cooled tapholes for reliably tapping the superheated matte. The result has been substantially improved productivity, and reduced smelting capital cost outlay per unit of production. A significant challenge, which was not anticipated, presented itself in the form of insidious corrosion of the furnace lining, and especially high-intensity copper cooling elements. Investigation of corrosion in related industries eventually identified chloride-accelerated sulphidation&apos;, and this term has been retained as it generically describes the most pertinent aspects of the accelerated low-temperature wear of copper coolers observed in PGM smelting. In addition to discussing the corrosion mechanism, this paper describes a number of solutions that were developed jointly by Anglo American Platinum and Hatch to address the copper corrosion problem. First, new monitoring technologies allowed furnaces to be operated more safely for a longer period of time. Second, a system for replacing corroded coolers from outside the furnace during a fast &apos;hot&apos; shutdown minimized the impact on furnace operating factor and hearth life. Finally, a corrosion-resistant graphite-protected cooler design significantly improved furnace campaign life, and heralds a more lasting solution to cooler corrosion in PGM furnaces.

Jan 1, 2013

By C. Chibwe, M. Tadie

Copper electrowinning models are useful for prediction of process performance, optimisation and improved understanding of complex phenomena such as current distribution without the expense of physical experiments. Recently, finite element models such as those generated in COMSOL Multiphysics have been developed for the electrowinning process. The predictive capacity of these models is improved by incorporating physical and chemical phenomena. Since physicochemical properties of electrolytes can have a significant effect on the quality of copper produced, this work was conducted to investigate the effect of such properties on electrodeposition at varying levels of copper, sulphuric acid concentration, iron concentration and temperature, in the presence of a smoothing agent. Tests were conducted at conditions similar to those for industrial copper electrowinning. Mathematical correlations were developed for the following properties; density, conductivity and diffusivity which were compared to existing models in literature and thereafter applied in a COMSOL generated model. The model predicts, current distribution at the cathode surface, deposit thickness (morphology), concentration of species within the cell and current efficiency. Current distribution output shows the impact of several phenomena on the deposit generated including the relationship between increasing copper concentration and current density and lack of influence of acid concentration on this property. Keywords: Electrowinning, physicochemical properties, electrolyte composition, modelling

Jan 1, 2020

By G. Akdogan, I. J. Geldenhuys, Q. G. Reynolds

Efficient recovery of titanium from titaniferous magnetite ores continues to be a promising, yet unrealised opportunity, despite significant occurrences of titaniferous magnetite deposits worldwide and the existence of established industrial-scale practices. Current smelting processes only recover iron and vanadium, while the titanium is discarded in the waste slag. These slags are stockpiled and present massive environmental challenges, while still containing value as residual iron, vanadium and titanium. The complex mineralisation of these deposits does not allow for a significant opportunity to separate the titanium and magnetite minerals through normal physical upgrading processes. An upgrading process is therefore required to derive economic value. An efficient process technology could create an opportunity to unlock all three valuable commodities. South Africa’s Bushveld Complex, the largest known deposit of its kind in the world, is known for the high vanadium content but is also associated with relatively high concentrations of titaniferous magnetite. The quality of the Bushveld Complex was one of the drivers for the establishment of the Highveld Steel and Vanadium Company in the mid-1960s. The unfortunate demise of Highveld Steel and Vanadium created an opportunity to review current practices and evaluate the best available process for this complex ore in the context of the three elements of interest. Past and current processes, the nature of these deposits and the potential to recover titanium are assessed to highlight the potential of fluxless smelting of titaniferous magnetite to produce a titania-rich slag as feedstock to the pigment, titanium metal industry in conjunction with recovery of iron and vanadium. Ilmenite smelting practices are proposed as a suitable alternative to achieve the goal, namely sustainable processing. Keywords: Titaniferous magnetite, titanomagnetite, smelting, titania slags, comprehensive processing

Jan 1, 2020

By Y. He

Early research showed that the separation of particles based upon their density was not possible in steadily rising air streams. The terminal velocities of moving spheres are determined by their aerodynamic characteristics. This prevents particle separation by density in standard air classifiers. Passive pulsing air classifiers have accelerated airflow in certain regions, which achieves density-dominant separation. Numerical simulation of airflow within passive pulsing air classifiers was done to estimate airflow patterns near a damper. Analysis of those results shows the significant acceleration of the airflow above the damper. Laboratory trial separations have shown that passive pulsing classifiers are superior to standard air classifiers. Keywords: pulsing airflow, passive, simulation of airflow models, acceleration effect, density

Jan 1, 2005