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By D. F. Malan, J. A. L. Napier

This paper describes some important aspects associated with historical underground measurements in South African gold and coal mines. Deformation measurements were used to confirm the use of elastic theory to simulate the rock mass behaviour in the Witwatersrand gold mines in the 1960s. Although a prominent time-dependent component of stope closure was measured as early as the 1930s, it was ignored owing to the benefit of adopting elastic theory. Neglecting the time-dependent response of the rock for many decades resulted in important aspects such as the effect of mining rate, the effect of advance per blast, and the need for enhanced design criteria not being explored. Recent work is only now starting to address this gap in knowledge. In-situ measurements of large coal specimens in the 1960s and 1970s indicated that a linear formula may possibly be a better approximation of coal pillar strengths. This alternative formulation was never adopted, however, as the power law strength formula was already deeply entrenched in the industry at that stage. In spite of these apparent failures to continuously generate and adopt new knowledge, a key lesson learnt is that major advances in rock mechanics will not be possible without careful monitoring of the rock mass behaviour in experimental sites. Areas requiring further research, such as pillar strength formulae for the Bushveld Complex and enhanced design criteria for the gold mines, can only be developed using extensive undergroundmonitoring programmes.

May 1, 2021

By H. Von Michaelis

The world is rapidly becoming more ?green? i.e. energy conscious. Since mining and mineral processing are massive energy users, the mining industry is seeking ways to reduce its energy footprint. In mining and mineral processing operations, energy is often the most expensive cost item. Comminution, i.e. grinding, is frequently the most energy-intensive step between mine and metal. High pressure grinding rolls (HPGR) are being installed in a rapidly growing number of ?hard rock? mineral processing operations. In the energy conscious cement industry, HPGR grinding has been standard practice for decades. HPGR is known to be energy-efficient; the question is how energy efficient? How can the energy efficiency of HPGR be enhanced? This paper reviews recent HPGR applications for the processing particularly of copper, gold, platinum, PGM-Ni-Cu and iron ores. HPGR technology is discussed with respect to energy efficiency. The harder the ore the greater the energy savings are likely to be. Typical energy savings of 10?20% can be expected when installing HPGR vs. a SAG mill. Wipf(2005) showed why conventional Bond Work index tests on HPGR product is likely to under-estimate the energy savings that can be achieved by installing HPGR. Energy efficiency of HPGR is, however, ore-specific. The biggest energy savings of HPGR tested thus far is 9.5 kWh/tonne claimed for Vista Gold?s Mt. Todd gold ore from Western Australia, a savings of over one-third of conventional SABC comminution energy. Several commercial scale iron-ore pellet feed plants install HPGR for fine-grinding of concentrate to increase the surface area of pellet feed in a manner that reduces overall energy consumption. Energy-consumption in comminution is, however, only part of the energy savings benefit story. HPGR typically also reduces the amount of steel lost through wear of mill liners and media. Marsden(2008) showed that when the energy that would have been consumed to produce the steel that is saved by applying HPGR is factored in, then the overall energy savings of HPGR is considerably greater. Johnson et al.(1988, 2005) tested HPGR in a flowsheet in which energy savings of around 50% can be expected if HPGR product screen oversize is recirculated to the HPGR. Innovative flowsheets have been proposed by Rule et al.(2008) and by Morley (2008) which are expected to result in significantly increased energy savings. HPGR applied in flowsheets in conjunction with coarse ore separation devices, e.g. ore sorting or DMS to remove barren waste from HPGR feed promises significantly greater energy efficiencies. Fine-grinding and ultra-fine grinding of ore or concentrates in conventional ball mills results in energy consumptions that increase exponentially with product fineness. Wipf (2009) presented a flowsheet in which he proposed the installation of HPGR ahead of the Aerosion ?Disintegrator? for ultra-fine grinding of ores and/or concentrates. This dry grinding arrangement is expected to achieve ap80 = 7 µm using up to 100 kWh/t less energy than could be achieved by wet ball milling. Using conventional comminution, an ore might require grinding to p80 = 45µm (? 325 mesh), for example, to liberate the valuable components, e.g. magnetite, from gangue, e.g. silica. Inter-particle comminution in HPGR may break some ores along grain boundaries thereby liberating valuable minerals from gangue at a much coarser particle sizes. Early magnetic separation rejection of silica gangue liberated from magnetite by HPGR at coarser grain sizes could further reduce downstream comminution energy consumption.

Jan 1, 2009

By U. König, N. Norberg

Ore mineralogy defines the properties during processing and metal extraction. Fast and frequent monitoring of the mineral content and related process parameters brings value to the mining operation and material processing. Decreasing ore qualities and increasing prices for raw materials require a better control of processed ore and a more efficient use of energy. Traditionally quality control in mining industries has relied on time consuming wet chemistry or the analysis of the elemental composition. The mineralogy that defines the physical properties is often monitored infrequently, if at all. The use of highspeed detectors has turned X-ray diffraction (XRD) into an important tool for fast and accurate process control. XRD data and their interpretation make the difference in the identification of minerals, in describing their distribution in ore bodies and in their beneficiation during processing. The use of statistical techniques such as Partial Least-Squares Regression (PLSR) or Principal Component Analysis (PCA) on XRD raw data has been successfully trialled to determine raw materials and processed ores, in addition to well established methods for mineral identification and quantification. PLSR can be used to correlate process parameters directly from the XRD pattern without investigating the mineral content. Hidden information present in the XRD pattern but not determined with traditional methods can be used to improve the characterisation of the analysed materials. The practical use for the direct determination of process relevant parameters during exploration and ore beneficiation will be illustrated in several case studies from mining and ore processing. Keywords: Geometallurgy, mineral monitoring, X-ray diffraction, process efficiency

Jan 1, 2020

By M. B. C. Brandt

Richter magnitudes for seismic events were calibrated for use as a Local magnitude scale in the West Rand (WR), Far West Rand (FWR), and Klerksdorp–Orkney–Stilfontein–Hartebeesfontein (KOSH) gold mining areas. Richter magnitudes are currently calculated from seismograms recorded by local surface cluster networks using tabulated calibration values for Southern California, published in 1958. The Richter (1958) model is incorrect for distances of less than 30 km and should be applied only to crustal earthquakes in regions with similar attenuation properties to those of Southern California. When compared to the South African National Seismograph Network (SANSN), the cluster networks, on average, overestimate seismic event magnitudes by approximately 0.1 of a magnitude unit. A calibrated Local magnitude scale was derived by means of a multiple regression analysis between the Local magnitudes reported by the SANSN and the largest zero-to-peak trace amplitudes measured on the cluster network horizontal seismograms, after modelling the attenuation of the seismic waves as these progress from the epicentre to the station position. Magnitudes reported by individual stations for the same event show a significant scatter around the average magnitude owing to the near-surface amplifications of the seismic waves at the recordings sites. Average magnitude should be estimated using as many magnitudes of individual stations as possible to ensure an accurate estimate. Larger event magnitudes should be compared with those recorded by the SANSN.

Nov 1, 2021

By H. Askari-Nasab, S. Upadhyay, A. Moradi-Afrapoli

Material handling in surface mines accounts for around 50% of the operational cost. Optimum truck dispatching plays a critical role in the reduction of this operational cost in truck and shovel surface mines. Researchers in this field have presented several mathematical models to solve the truck dispatching problem optimally. However, a critical survey of the literature has shown that three significant drawbacks exist in the available truck dispatching models. The published models underestimate the importance of the interaction between truck fleet, shovel fleet, and the processing plants. They also disregard goals set by strategic-level plans. Moreover, none of the available models account for the uncertainty associated with the input parameters. In this paper we present a new truck dispatching model that covers all of these drawbacks, using a fuzzy linear programming method. The performance of the developed model was evaluated through implementatin in an active surface mining operation. The results show a significant improvement in production and fleet utilization.

Sep 1, 2021

By R. A. Wildenboer, R. F. Sandenbergh

Rare earth elements (REE) are present at concentrations of approximately 0.36 % in phosphogypsum stacks located in Phalaborwa, South Africa. The REE are present in the phosphogypsum in solid solution in the gypsum and as fluorine and aluminium rich precipitates. Recovery of the REE offers opportunities for valorization and reducing the environmental impact of the phosphogypsum waste. This paper presents the results of investigations into the leaching behaviour of rare earth elements and other impurities from phosphogypsum using aqueous sulfuric acid. The effects of acid concentration, temperature, residence time, and the influence of impurities were evaluated for their impact on the solubility of rare earth elements in a sulfate medium. It was found that a significant fraction of the REE can be leached from the phosphogypsum in a range of conditions much less severe than traditional hard rock rare earth processing conditions. The REE were leached at a sulfuric acid concentration of 110 g/l, temperature of 30°C – 40°C and a residence time of approximately 8 hours.

Nov 6, 2024

By W. R. Matizamhuka

"The consolidation of nano-sized composite materials presents a challenge using conventional hot pressing methods. Spark plasma sintering (SPS) technology has shown great promise in the successful sintering of nanoreinforced composite materials. This qualitative review seeks to impart knowledge gathered, and progress made over the years on the consolidation of nanocomposite materials using SPS technology. The review is aimed at introducing this technology to the South African science and engineering community. Emphasis is on improving the mechanical properties of structural ceramic nanocomposite materials, which over the years have shown great promise in a wide range of applications, including transport, energy, mining, and the environment. Although success has been achieved within the laboratory for research purposes, there are still great opportunities to commercialize the technology for the production of larger components with more complex shapes.IntroductionSince the pioneering work of Niihara and Nakahira (1991) over a decade ago there have been extensive attempts to harness the potential of nanostructural metals, ceramics, and composites in a number of applications (Niihara and Nakahira, 1991; Selaho et al., 1997; Sekino and Niihara, 1995; Khalil, 2012; Aalund, 2008). It is generally realized that reducing the grain size of materials to nanoscale offers a significant improvement in properties. Attempts to synthesize nanostructured compacts by conventional sintering methods have not succeeded owing to the uncontrollable high rates of grain growth driven by long dwelling times at the sintering temperatures and large powder surface areas (Sergueeva et al., 2009; Gua, Khora, and Chieng, 2004; Tang et al., 2009). Following the development of spark plasma sintering (SPS) technology, it is now possible to process nanocrystalline materials with controlled grain growth. SPS, also referred to as the field-assisted sintering technique (FAST) or pulsed electric current sintering (PECS), was developed with the aim of improving on the well-established hot pressing (HP) technology (Kessel and Hennicke, 2007;"

Jan 1, 2016

OFS Branch Minutes of the General Meeting Held at the President Steyn G.M. Ltd Canteen on Wednesday, 13th November, 1974, at 3.55 p.m. Mr D. A. Smith (Chairman of the O.F.S. Branch) was in the Chair. There were also present: Nim reports Report no. 1696 The preparation and certification of a reference sample of a preciousmetalore. Report no. 1705 The effects of various matrix elements on the efficiency of the fire-assay procedure using nickel sulphide as the collector. Report no. 1720 The initial development of processes for the direct leaching of iron-nickelcopper mattes containing platinumgroup metals, and for the production of ferronickel.

Jan 5, 1975

By C. R. Royal, B. E. Friedland

Energy & Densification Systems (EDS) has developed patented new technology crushing/milling equipment. The EDS Multishaft Mill is a compact vertical mill – utilising high-speed impacts to break down particles with large reduction ratios, low energy consumption and improved liberation. The milling action combines several processes to provide a multitude of output products to satisfy various specifications. Comminution often accounts for more than 60% of the total power consumption on a mine. Two of the key problems to be solved in comminution is to bring down the specific energy required to reduce particle sizes; and to improve liberation or improve the efficiency of the beneficiation and recovery of the required elements. The EDS Mill has shown extensive improvements in both these areas with energy efficiencies up to 75% better than conventional milling equipment and improving liberation beyond the standard beneficiation processes. Conventional comminution equipment requires massive, heavy parts to be moved or rotated, consuming a lot of energy in doing so and not inputting that energy into the breakage of particles. In the EDS Mill, ramping up the relatively small shafts to speed is all that is required. Once at speed, the momentum, together with the minimal amount of constant drive required, means that the energy can be transmitted directly and efficiently into breaking down the particles. The high-speed impacts allow for a shattering effect of the particles and this quickly generates a high ratio of fines. Furthermore, the single impact force breaks particles in a very different manner to conventional compressive force breakage. With compressive force breakage, the particle breaks from point to point. With high-speed impact force breakage, the force tends to migrate through the weakest areas of the particle, which tend to be the mineral boundaries. This, therefore, means that when the particles break, the various minerals are exposed and hence liberated. Keywords: Comminution, high-speed impact, liberation, energy efficiency, novel comminution

Jan 1, 2020

By K. A. du Preez, P. J. van Staden, M. van Rooyen

Mine-impacted water, including acid mine drainage (AMD), is a global problem. While precipitation of dissolved metals and neutralization of acidity from mine-impacted water is accomplished relatively easily with lime addition, removal of sulphate to permissible discharge limits is challenging. This paper presents a high-level comparison of four sulphate removal technologies, namely reverse osmosis, ettringite precipitation, barium carbonate addition, and biological sulphate reduction. Primarily operating costs, based on reagent and utility consumptions, are compared. Each process is shown to be subject to a unique set of constraints which might favour one over another for a specific combination of location and AMD composition. Access to and cost of reagents would be a key cost component to any of the processes studied. The total cost calculated for each process also depends on the type of effluents that are allowed to be discharged.

Oct 1, 2021

By J. G. M. Mukoya, B. Adebayo

"This work was carried out to determine the influence of rock properties and drilling machine parameters on the penetration rate at the SJ pit of Rössing Uranium Mine, Namibia. Rock properties (uniaxial compressive strength, tensile strength, and modulus of elasticity) of samples collected were determined in the laboratory. Drilling experiments were conducted in which feed pressure, air pressure, rotary speed, weight on the bit, and torque were varied to measure their effect on the penetration rate. The uniaxial compressive strength varied from 90–180 MPa for layered marble-quartzite and banded gneiss. Increases in feed pressure, weight on the bit, and rotary speed beyond the optimum level led to a decrease in penetration rate and caused the drill bit to ‘stall’. Results of the study revealed that penetration rate increases with an increase in the feed pressure and air pressure. After reaching a maximum value, the penetration rate begins to decrease despite increasing feed pressure. A very high torque causes the drill bit to stall, since the feed pressure is too high and the air pressure is not sufficient to remove the cuttings from blast-hole at maxium bailing velocities. The average penetration rate varied from 19 to 45 m/h for phase 2 of the pit, and from 17 to 68 m/h for phase 3. The optimum drilling machine parameters obtained in phase 2 and phase 3 were slightly lower than those currently being used at the mine. The trials of the optimum machine parameters will assist in reducing the cost of drilling, which varied from N$29.48 to N$36.31 per metre for the tricone bit. IntroductionDrilling and blasting contribute most to the operational cost in opencast mining and have the potential to influence the cost of downstream operations (Aipanda, 2011). The drillability of rocks depends not only on the rock properties, but also on the drilling tools and operational variables. In rotary drilling, rotational speed, thrust, and flushing are the operational variables, and are known as the controllable parameters. Rock properties and geological conditions are the uncontrollable parameters. Although, many attempts have been made to correlate drillability with rock properties, the rock characteristics affecting rotary drilling have not been entirely defined (Hartman, 1962; Kahraman, Bilgin, and Feridunoglu, 2003). The drill bit is forced into a rock surface from the rotary power head, hence creating stresses and cracks at the bit-rock interface. The performance of a particular bit in any formation i"

May 1, 2019

By M. Onifade, B. Genc, P. O. Akinseye, M. Fourie, K. O. Said

Coal is mined by both surface and underground methods and its extraction is normally characterized by numerous hazards that can lead to catastrophic accidents, which result in devastating effects such as injuries or fatalities, damage to mining assets, and destruction of mineral resources. These hazards exist due to the ability of coal to support combustion and its association with toxic, flammable, and explosive gases. Underground coal mining entails higher safety risks than opencast coal mining, chiefly because of issues relating to mine ventilation and mine collapse. Furthermore, coal mine collapses mostly occur due to crumbling of mining supports, especially in room and pillar mining systems. To avoid such adverse occurrences, safety management systems need to be in place. This study reviews the various technological safety systems and principles that are used for safe-rescue and self-escape of miners in underground coal fires, particularly in South Africa, using data obtained from Mines Rescue Services in Carltonville, South Africa. The outcome of the review shows that practising safety culture has been given priority across many South African underground coal mines through setting up safety management systems and encouraging workers to stay committed to safety principles.

May 2, 2022

By A. N. Brown

Professor D. G. Krige, of the Department of Mining Engineering of the University of the Witwatersrand, received the degree of Doctor of Engineering (honoris causa) from the University of Pretoria at the Autumn Graduation Ceremony in April 1981. He was honoured an outstanding engineer for his world-renowned pioneering work in the field of statistical methods of ore valuation, or geostatistics as it has become known. Metalliferous Ore Deposits A short course on the 'Evolution of Metalliferous Ore Deposits' will be held at Rhodes University, Grahamstown, from 8th to 13th February, 1982.

Jan 1, 1981

By J. L. Broadhurst, S. Rumjeet, S. T. L Harrison, T. Chimbganda

Post-mining land use for the production of industrial crops has the potential to mitigate environmental and socio-economic impacts of mine closure and stimulate post-mining economic growth. Fibre-rich plants are of particular interest due to their multi-product potential. In the South African context, where there are many abandoned and end-of-life mines near human settlements, an opportunity presents to assess the feasibility of transforming post-mining land into productive land and building resilient communities through the development of multi-product value chains from fibre-rich plants such as kenaf, hemp or bamboo. This paper provides an overview of the plant selection criteria, phytoremediation potential and the downstream options for the recovery of products and value from bast fibre plants (hemp and kenaf) and bamboo. Based on findings from a comprehensive review of the published literature, as well as interviews with relevant experts within South Africa, it appears that bast fibre plants are the best downstream option for producing ‘green’ textiles and high-end niche products, whereas bamboo is more suitable as a replacement for conventional timber. Although fibre plants have the potential to extract metals from soils, high concentrations may inhibit plant growth and affect the quality of the final fibre product. Further site-specific studies would thus be required to select the best route for both land remediation and optimal product recovery. Keywords: Bioremediation, bamboo, kenaf, hemp, multi-product value chains, sustainable postmining land use

Jan 1, 2020

By S. Govender, J. J. L. du Plessis, R. C. W. Webber-Youngman

Spontaneous combustion (SC) is a cold oxidation reaction that generates heat, causing a temperature rise of the reactant and leading, with limited heat dissipation, to self-ignition of the reactant, which occurs without an external heat source. Although not limited to coal, the most significant hazard of SC are the fires that occur in coal mining operations around the world. These fires pose a serious risk to the safety of workers in the mines as well as adverse effects on the environment, and can affect the quality of life for current and future generations. We investigated the occurrence of SC in raw coal storage bunkers with the purpose of compiling a decision analyser for engineers designing or working with coal storage bunkers. Specific experts were interviewed from different backgrounds and companies, followed by field research at a coal mine. The important factors affecting the possibility of SC occurring were the type of coal supplied to the bunker, the mining practice, and physical conditions around the bunker. The information presented here will assist in reducing the SC risk in raw coal storage bunkers. These findings, together with the systematic decision analyser developed, will assist design engineers and mine personnel to take early preventative steps in managing SC. The decision analyser was tested for many different scenarios and gave good guidance on how to minimize and prevent SC in bunkers

May 1, 2021

By R. Rumpold

The platinum group metals (PGMs) palladium, platinum, and rhodium represent the key materials for automotive exhaust gas treatment. Since there are currently no adequate alternatives, the importance of these metals for the automotive industry is steadily rising. The high value of PGMs in spent catalysts justifies their recycling. The state-of the-art technology is to melt the ceramic carrier and collect the precious fraction in a liquid metal bath. As the feed material has quite high melting points, huge amounts of energy are required for this process. Hydrometallurgical treatments of the spent catalysts offer the possibility to recycle the PGMs with less energy and time demands. Moreover, automotive catalysts contain further valuable materials to improve the exhaust gas treatment. These compounds, like cerium oxide, cannot be recovered in pyrometallurgical processes. This paper deals with a special process that uses hydrometallurgical principles to recycle the valuable metal fraction from spent automotive catalytic converters. Hydrochloric acid in combination with hydrogen peroxide is used as leaching agent. This combination selectively dissolves the PGMs and the cerium, which can be recovered in separate steps. As the leaching solution is regenerated and reused, wastewater generation is minimized. Furthermore, the process residues constitute potential byproducts.

Jan 1, 2012

By A. C. Lawrence, K. F. Mauer

Human factors in stope productivity -a field experiment by K. F. MAUER, M.A. (Visitor), and A. C. LAWRENCE, B.Econ., D.Com. (Member) An experiment in which three human variables were manipulated in eight stopes was conducted in a gold mine over a period of roughly six months. The three variables, which each had two levels, concerned the degree of movement from gangs, the quality of the Bantu production supervisors, and the extent to which gangs were composed of a single ethnic group (Malawians). Time was included as a variable with three levels in the analysis of the findings. On the average, stable gangs produced 12 per cent more than gangs with a high percentage of transfers, gangs with better production supervisors were more successful by nearly 9 per cent than those with poorer production supervisors, and homogeneous gangs were more productive than heterogeneous ones by as much as 15 per cent. In addition, it was found that a number of the variables exercised a joint influence on stope productivity and that the effects of the variables were accentuated by the len gth of time they had been in operation. V

Jan 10, 1973

By T. Grewar

"South Africa is a water-scarce country, rated as one of the 30 driest countries in the world. Worryingly, government expects water demand to outstrip supply as early as 2025. The South African government believes that substantial volumes of water can be made available through the reuse of treated mining-impacted water. Currently, however, large volumes of treated mining-impacted water are produced, often with no end use in mind other than discharge or disposal, when these treated effluents may have further uses in applications which are currently and unnecessarily using high-quality water. Treating the wastewaters to levels that meet ’fitness-for-use’ guidelines for alternative applications such as agriculture, sanitation, or industrial processes will reduce the treatment cost burden compared to potable water production. The primary conclusion of this review is that the most suitable option for re-using treated mine-impacted water is in agriculture, for irrigation of crops (food, forage, or energy crops), which currently and unnecessarily makes use of the majority of South Africa’s high-quality resources or potable water. Other findings include the realization that although guidelines and legislation for water re-use in South Africa exist and are readily available, they tend to be contradictory and confusing in many cases. It is imperative that these ambiguities and incongruities in the available guidelines and legislation for water re-use be clarified and updated swiftly IntroductionSouth Africa is a water-scarce country and is currently rated as one of the 30 driest countries in the world, with an average rainfall of 490 mm/a, approximately half of the global average. As an indicator of the degree of regional variability in South Africa’s water supply, it has been shown that 70% of all runoff is from approximately 20% of the land area. Regardless of the water scarcity faced in South Africa, our water conservation track record is poor, with an average consumption of 280 L/d per person, almost 60% more than the global average of 175 L/d per person. Around 40% of this allocation is utilized in watering lawns and gardens (Zhuwakinyu, 2017).The South African government predicts water demand to outstrip supply as early as 2025. On an international scale, the situation appears just as dire, with the United Nations High Level Panel on Water (HLPW) expecting a 40% water shortfall by 2030, which may affect up to 1.8 billion people based on current water demand trajectories (Zhuwakinyu, 2017)."

Mar 1, 2019

By K. Esmeli, A. Ozkan

The enhancement of shear flocculation of a galena suspension by ultrasound in the presence of sodium isopropyl xanthate and potassium ethyl xanthate was investigated by studying the effects of surfactant concentration on flocculation efficiency, contact angle, and zeta potential. Strong flocculation was obtained under alkaline conditions (pH > 9) and a high ultrasound power level (150 W). Although the ultrasonic treatment increased the negativity of the zeta potential of the galena particles, this did not decrease the flocculation efficiency, thereby showing that the hydrophobic interactions between the particle were stronger than the electrical double-layer repulsion resulting from xanthate adsorption. The findings indicate that the ultrasonic treatment promoted the adsorption of surfactants onto the mineral surfaces.

Jul 5, 2023

By M. Kumral, Y. A. Sari

Underground mining operations tend to have higher operating costs than surface mines. When metal prices decrease, profitability is jeopardized due to the high costs. Therefore, mining management harnesses new practices that increase operational efficiency. One way to manage this challenge is to invest in new mine planning practices. Stope layout optimization as a part of underground mine planning aims to identify a portion of the orebody in the form of production volumes (stopes) to maximize profit under roadway and stope dimension constraints. In this paper we propose a novel approach based on identifying ore-rich areas of the deposit and prioritizing their extraction through an iterative heuristic clustering approach. The proposed approach is compared with and validated by an exact method through a small mining example. The heuristics produced nearly identical results in a very short time. Finally, a case study was carried out using a larger data-set. The cluster-based iterative approach generated nearoptimal stope layouts in a computationally effective manner.

Mar 1, 2021