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By M. M. Ali, C. Musingwini, R. A. Phillips

The long-term open-pit mine production scheduling problem (OPMPSP) is typically solved in two broad stages. The first stage entails optimisation of the ultimate pit limit, which delimits the economical final pit and maximises the undiscounted value of the deposit. The second stage entails optimisation of the long-term open-pit mine production scheduling, which maximises the discounted value of the deposit. The long-term OPMPSP is a type of an NP-hard problem, such as the travelling salesman problem (TSP), because it is intrinsically complex to solve since it incorporates both hard and soft constraints. Therefore, metaheuristic approaches have been used to find global optimum solutions to the long-term OPMPSP. This paper presents a metaheuristic approach utilising a dual interchange algorithm (DIA) to solve the long-term OPMPSP. The DIA is a parent algorithm which supervises two subservient algorithms, namely, simulated annealing (SA) and Tabu search (TS). The DIA selects the better solution generated in each run from the two algorithms, and simultaneously penalises the algorithm that produced an inferior solution, while rewarding the one that produced a better solution. The DIA was tested on the three-dimensional Arizona Copper Deposit (KD) dataset, which is publicly available for research purposes from the online MineLib repository. It was set to generate net present values (NPVs) in five-minute intervals and the solution improved until it plateaued at about 30 minutes of runtime. The NPVs compared favourably to those obtained on the same KD dataset but obtained over 24-hour runtimes, when solved using simulated annealing and Tabu search algorithms, and the NPV generated from the work of Muñoz (2012) using the modified TopoSort as found on the MineLib repository at the time of this study. Therefore, this paper demonstrates that DIAs can be developed and used to solve the three-dimensional long-term OPMPSP.

Sep 1, 2021

By Z. Botha

Sheryl Sandberg, the COO of Facebook, published a book titled 'Lean In', where she invites women to sit at the table. In summary, to 'sit at the table' entails not only embracing opportunity, but to actively seek out opportunities in leadership positions. Sandberg points out that, unfortunately, apart from institutional obstacles, a woman's own perceived limitations have a negative impact on her leadership ambitions. To create an environment where diversity will flourish, where everyone has a seat at the table and everyone has a voice, safety must be established first. To establish safety within a diverse team, there is power in relationships and connections. This study will first and foremost review theories in literature as to why diversity and inclusion are so important and the value that diversity brings to our industry. The study also recommends a few focus areas for improving diversity and inclusion in the industry.

Sep 1, 2022

By M. Pelser, J. Hagemann

"The Anglo American Platinum Rustenburg Base Metals Refiners (RBMR) is the largest producer of nickel cathode in South Africa. It treats slow-cooled matte to recover a platinum group metal-containing magnetic fraction for downstream processing and a nonmagnetic fraction to produce nickel and copper cathode, cobalt sulphate, and sodium sulphate. The refinery was recently expanded from 21 to 33 kt/a nickel. During the ramp-up phase, the sulphur removal section was found to be a constraint to throughput. Nickel hydroxide is precipitated from nickel spent electrolyte by addition of sodium hydroxide, and filtration of the nickel precipitate allows sulphur to be removed from the circuit as aqueous sodium sulphate in the filtrate. The filtration fluxes were found to be limiting the circuit’s capacity. An obvious solution would have been to install additional filtration capacity at significant capital cost and long lead times. RBMR chose rather to first initiate a collaborative project with Anglo American Technical Solutions to better understand the impact of process conditions on filtration characteristics and to evaluate whether the filtration flux could be increased by improving the quality of the nickel precipitate.A deeper understanding of nickel hydroxide precipitation chemistry and kinetics in the context of high nickel and sodium hydroxide concentrations was developed, guided by precipitation theory and confirmed by laboratory tests and plant observations. It was found that the filtration rate could be significantly improved by modifying the precipitate characteristics. A new precipitation process was engineered and retrofitted into the existing plant. Since start-up of the new precipitation system, the available capacity of the sulphur removal section has more than doubled, and control of the process has improved with the better agitation. This paper describes the theory and results used to design the new agitation system, as well as the impact of the change on capital investment, operability, and process efficiency."

Jan 1, 2016

By T. Wallmach

Heavy mineral deposits are a source of the economic important element titanium, which is contained in minerals such as ilmenite, pseudorutile and rutile. In the Ranobe heavy minerals deposit these minerals occur not only as primary unaltered minerals but also display various degrees of alteration resulting in an increased titanium content of the deposit. The presence of the various FeTi-oxide minerals? textural relations is a reflection of geological processes and may ultimately be used to unravel the geological history of the deposit. These minerals have different chemical and physical properties that influence metallurgical tests and the quality of products produced from various mineral beneficiation streams. It is therefore of metallurgical importance that the minerals are well characterized, defined and monitored during preliminary beneficiation tests.

Jan 1, 2007

By P. R. Jochens, D. D. Howat, C. W. Wolhuter

Prof D. D. Howat (Member): I am a man with a grievance this afternoon. The other authors of the paper entered into a conspiracy to ensure that I presented the paper, and finally even had the audacity to say that this was the least that could be expected from me as I had done so little of the other work. Looking at the distinguished audience here that includes some figures who have forgotten more about cement that some of us have ever learned, I realise how right the other authors were. I feel very much cast in the role of the fool who was pushed in where angels fear to tread. I hesitate to say, Sir, that I would not use the word angels in any definition of the authors of this paper. In presenting the paper, may I try to put forward a few simple points about cement that are probably as little known to the majority of the audience as they were to me-at least until very recently. Concrete, made by mixing Portland cement, sand, crushed rock and water is the most widely used constructional material in the world. Generally the production of actual Portland Cement is about two-thirds that of steel and by the time this is formed into concrete the total is about five times the world tonnage of steel produced. A very rough approximation suggests that world consumption of concrete is about one ton per person per annum. Apart from fresh water mankind consumes no other material in such quantities. What mankind originally required was a mortar to be used for joining blocks of stone or bricks. Such mortar was originally made from limestone burned in a kiln and then mixed with water. Production of mortar of this type began over 5,000 years ago. Some samples from pyramids in Egypt have shown this mortar to be in good condition although over 4,500 years old. The progress of technological development was very slow until 1824 when an English bricklayer, Joseph Aspidin, patented a new mortar material which he made by burning limestone and clay together in his kitchen stove. Aspdin called his product 'Portland Cement' because, in colour, it resembled a well-known building stone quarried in the Isle of Portland on the south coast of England. Aspid's same types of raw materials are used today in modern cement production and the product is still referred to as Portland cement. A lime-containing material such as chalk or shell and a clay-type material such as shale, slate or clay itself, are mixed in the required proportions, ground and fed to a kiln where the minimum burning temperature is about 1,500°C. The raw materials react to form hard chunks of a material known as 'clinker'. About 5 per cent of gypsum is added to the clinker and the mixture is then ground to a very fine powder, the function of the gypsum being to control the rate of setting of the concrete. The key oxides in cement production are lime, silica, alumina and iron oxide. The possible reactions occurring during and the products of the burning process are shown in Table I.

Jan 12, 1968

By J. van Deventer

Ion exchange is widely used in hydrometallurgical operations for the recovery and purification of valuable metals. Special chelating resins have a high affinity for specific elements, allowing the selective removal of low levels of impurities from a background of highly concentrated valuable metal. Solvent-impregnated resins have been added to this group more recently, expanding the treatment options. These products find application in the purification of cobalt-rich liquor prior to the production of high-purity metal or metal salts. Impurities in copper-cobalt circuits that are typically targeted by ion exchange include iron, uranium, copper, zinc, nickel, bismuth, and antimony. Ion-exchange resins can also be used to treat tailings, thereby recovering additional valuable metal and improving the Environmental and Social Governance credentials of the operation. This article takes a look at various applications of ion exchange in copper-cobalt operations

Jan 1, 2023

By C. Musingwini, T. C. Sebutsoe

"The primary objective of any mining business unit is to make profit by extracting, processing, and selling minerals from a particular mineral deposit. It is important to optimize extraction of the mineral resource given time, capital, and space constraints. Mineral extraction is often associated with uncertainty due to variable technical and human factors. Technical factors such as grade distribution, ground conditions, and equipment reliability influence the performance of a mining production system (MPS). The performance of the MPS is also impacted by human factors such as employee skills, health, and attendance. Uncertainty associated with technical and human factors often leads to actual output differing from what was planned. Therefore an in-depth analysis of the significant causes of deviations from the planned outcomes is relevant.In this paper we investigate the empirical relationships between inputs and outputs in a MPS in order to assist management in directing efforts at key production drivers. Once a MPS has been empirically characterized, more effort and resources can be focused on the key decision-making variables (DMVs) in order to meet the planned outcomes.For a typical underground, tabular, narrow-reef conventional drill-andblast platinum mine the face advance, face length mined, number of teams, team efficiencies, and team size have a statistically significant relationship with the centares (m²) produced, which is a key performance indicator (KPI). The production function derived in this study can be used to align the physical, technical, and human factors together to predict the optimal output level. The most significant production lever of the MPS is the face advance. IntroductionA mining production system (MPS) is a result of an iterative process of design, planning, and optimization of mining input variables and decision-making variables (DMVs). The MPS exists within the mineral extraction link of the complete mining value chain. It represents the stage where mining companies have the opportunity, through production, to start generating returns on the investments undertaken. Returns on shareholders’ investments can be realized at this stage."

Jan 1, 2017

By J. P. T. Kapusta

"The copper smelting industry faces increasingly stringent environmental regulations and must continue driving innovation to reduce energy usage in its production chain. With the coming of age of ‘bottom-blowing’ copper technologies in China over the last decade, much renewed attention has been brought to tuyere bath smelting for clean copper-making. Since bottom-blowing sonic injection was originally developed and implemented in steelmaking in the 1970s and lead smelting in the 1990s, the time seems appropriate to revisit the sonic injection concepts and dispel some myths. After reviewing the trends in the copper industry and providing a brief historical context of research and development related to sonic injection, the author presents a comparison of the main submerged tuyere bath smelting technologies, namely the Noranda Reactor (NR), the Teniente Converter (TC), and the Chinese Bottom-Blowing Smelting Furnace (SKS/BBS) for copper, and the Queneau-Schuhmann-Lurgi Reactor (QSL) for lead. The core of this article, however, consists of a brief description of the fundamental theories for sonic injection (jetting regime) and ‘sonic flow mapping’ to illustrate the relationship between the minimum tuyere back-pressure required to achieve sonic flow and the tuyere internal diameter. The article also offers a comparison between side- and bottom-blowing in jetting regime before concluding with the author’s vision of the new trend in submerged tuyere bath smelting. IntroductionIn the last decade, developments in the copper smelting industry have been driven by two key concepts: (1) increasing cleaner smelting capacity to meet the world’s growing copper demand, as illustrated in Figure 1, and (2) increasing process off-gas capture to comply with more stringent environmental regulations and abide by a corporate social responsibility ethic of a clean work environment (for smelter employees) and clean air (for families living around smelters). The copper smelting industry has also faced some major challenges in the last few decades, including (1) low treatment and refining charges (TC/RC or processing fees) impacting custom smelters, (2) lower concentrate quality and grades with higher levels of impurities (e.g. As and Hg), (3) higher processing and transportation costs due to the rising cost of energy, (4) restrictions in process intensification (tonnage oxygen usage) due to furnace integrity limits, and (5) scarcity of educated and trained personnel due to fewer metallurgists graduating from pyrometallurgical school programmes combined with a high retirement rate of skilled workers."

Nov 1, 2018

By G. R. Lane

A large opencast gold mining company in Africa had just completed a feasibility study for the expansion of the operations. A traditional net present value project valuation methodology had been used that showed a very positive net present value (NPV). The project team had conducted a qualitative risk assessment that identified and logged all potential project risks and had subjectively incorporated ?risk? via the discount rate and other assumption contingencies around gold price expectations, operating parameters etc. Management requested that an independent quantitative risk modelling approach be adopted to obtain a better understanding of the impact of risk on the overall project valuation and confidence level in the final valuation. A detailed project valuation model was configured in the Carbon Modelling Technology and the stochastic modelling module used to perform the Monte Carlo simulations. Analysis of historical operational performance data relating to actual achievements versus planned and budget determined the historical variability of the underlying parameters as well as variance to budget. This was done at a detailed level for mining rates and costs, processing rate, costs and recovery, and other operational efficiency measures. External market assumptions relating to gold price, exchange rate, and inflation were modelled and the dependency between them modelled using a copula. This case study will demonstrate the approach taken to building and populating the quantitative risk model and the overall results of the Monte Carlo simulation. The final valuation demonstrated that the project had a 44 percent probability of an NPV less than zero i.e. 44 percent chance of project finance loss.

Jan 1, 2012

By M. Splaine, W. Davison, D. C. Atkinson

"Since 1964, Nchanga Consolidated Copper Mines, Ltd., has been developing and using production scheduling methods for optimizing its metallurgical operations in the short term (up to one year). Medium-range metallurgical planning has been possible only by successive evaluations of a single-year L.P. A modified version of Rosenbrock's direct search technique has now been developed to allow multi-year optimization of both mining and metallurgy for medium-range planning. The modifications to the search technique have produced a marked increase in computational efffciency for this heavily constrained problem in many variables and the system is a usable planning tool, given a reasonably powerful computer. The plans are 'optimum' in the sense that they yield the maximum present value of the operations.Each of the resulting five-yearly mine plans forms the basis for further detailed metallurgical evaluation by the established production scheduling system."

Jan 1, 2014

By A. D. Pienaar

The aqueous chemistry of niobium and tantalum compounds with hydrofluoric acid is a mature field of investigation. In contrast, very little attention has been paid to reactions of these compounds in non-aqueous environments. Similarly, reactions of elemental fluorine with niobium and tantalum compounds have received little more than cursory attention in literature. This work discusses the isothermal reaction kinetics derived from thermogravimetry, using a less redundant approach than is currently the standard. We also discuss the most likely kinetic models for this reaction.

Jan 1, 2012

By A. W. S. Schumann

A. W. S. Schumann The first reason for todays success lies in the very high standard which was set by our speakers today and I think that we owe a very special vote of thanks to Messrs P. M. C. Wilson, A. A. Oakes, M. J. Deats, A. C. Coetzee, R. E. Burnton and J. G. Ferguson. This, however, cannot account for the fact that there was a tremendous response even before anyone knew who was going to speak or what they were going to speak about. I suggest that the response was enthusiastic because people expected a more informal gathering with a greater opportunity for feedback, for an exchange of views and for questions and answers. The very popularity of the idea has defeated its own ends to some extent. This gathering is too big and many people must have been left with questions unanswered. Possibly one should arrange for a number of meeting places-there are many in the Chamber and the surrounding mining houses-and then have separate meetings in the morning with joint reporting session in the afternoon. Alternatively, we could have a symposium after the colloquium, using Webster's definition of a symposium as as 'a drinking party of merry fellows'. We all know that it is easy to discuss mining into the small hours. Obviously one cannot summarise todays papers and discussions, but I would like to make a few comments on technical matters and after that I would like to speak for a moment about what I see as major trends or tendencies which arise from discussions of this nature. Regarding the question of placing a village on a mine or away from a mine, I would mention that there is quite a body of literature on the adverse sociological effects on the development of children in any community in which the great majority of parents share the same occupation. It is something to be avoided if possible. We discussed the concept of the moving stockpile. If this is proved to be successful, it could save a considerable amount of money for Escom who could then avoid the construction of very large and costly staithes which are now regarded as parts of power stations. One does not see why both a moving stockpile and a staith should be necessary. By the way, speaking of Escom, I see that Mr Bob Scott is here; he has stopped pulling my leg by saying that we sell stone to Escom; perhaps someone has told him that if this is true, it must be the cheapest stone you can buy in this country. My next point relates to the use of epidiascope. It struck me again that this instrument which provides a visual image to accompany your message, can be very effective if it is properly used. These papers speak of a great deal of hard work which went into their preparation, and one really feels that the authors should have taken a few minutes more to prepare their drawings specifically for the epidiascope. There should be nothing on the drawing which is not necessary to illustrate a particular point and there should be nothing on the drawing which is not visible from the back of the hall. In particular, if one cannot read the figures along the sides, then one

"Chairman: Professor J. de V. Lambrechts Rapporteur: Mr R. HempPapers:Computer-aided design of a flash smelting installation by M. A. T. Cocquerel, R. T. Gates and L. A. GreenPlanning of mine ventilation by E. U. Reuther and A. DohrnenCoal mine ventilation systems in the U.S.A. by R. Stefanko and R. V. Ramani Heat flow models in ventilation planning by A. J. Dickson and A. M. StarfieldThe first of the papers in the session dealt with a flash smelter installation, while the remaining three papers dealt with mine ventilation. In view of this the Chairman proposed that the paper on flash smelting be introduced and discussed first, after which the remaining three papers would be introduced followed by discussion on all three.Dr Cocquerel, who presented the first paper, used some typical flow diagrams to illustrate the process and stated that the program considered only steady-state operations and did not take into account the dynamical changes. Subjective judgment was necessary for the preparation of input data and some degree of user appreciation was required.Dr Cocquerel said that no work bad yet been done on an optimization program, but that this was being considered. Such a program would require a large amount of physical and chemical data which were not yet available. The variables to be optimized were the energy and material consumptions and the loss of metal in the slag.The Chairman opened the discussion by asking Dr Cocquerel if he could give a brief description of the flash smelting process for the benefit of those not familiar with the subject. Dr Coquerel replied that in the Outokumpu process con¬centrates were injected directly into the furnace, and that the process was somewhat analogous to flash distillation or evaporation processes, but was perhaps closer to the burning of pulverized fuel.The process had two distinct advantages. First, the heat evolved in the smelting reaction was used to heat the furnace, which resulted in less fuel consumption, and, second, the effluent gases were cheaper to treat than the effluent gases from other pyrometallurgical processes. This was important particularly in view of the current interest in pollution problems.Mr P. Marais thought that the authors had done themselves an injustice by not including details of the very complex processes taking place in the furnace."

Jan 1, 2014

By H. Vuthaluru, O. Kavianipour, C. -Z. Li, N. Suresh, D. Sahu

The paper describes a CFD model developed for 76mm Dense Medium Cyclone (DMC) using Ansys Fluent Software to gain insights into the separation of particles according to their size and density. The simulation used in this paper includes the Reynolds stress model to predict the turbulence inside the cyclone body. An Eulerian-Eulerian multiphase CFD model was applied to model a number of physical aspects such as the separation of coal particles and the segregation of magnetite media solids. The experimental results generated in a laboratory scale DMC were used for the validation purpose and the predicted results of CFD have been validated for the flow splits of water, magnetite and coal. The simulation results obtained from the Eulerian-Eulerian multiphase model revealed that the prediction agreed with experimental results, and matched reasonably well in terms of the mass flow rates of water, magnetite and coal. Furthermore, the partition curve of the DMC when mapped for the separation of - 3.0+0.50 mm size also showed to be in agreement with the experimentally obtained curves. The gravity of cut (SG50), also predicted by the Eulerian-Eulerain multiphase CFD model, was found to be in the vicinity of observed experimental value of SG50. Keywords: Dense medium cyclone, Eulerian-Eulerain multi-phase modelling, computational fluid dynamics, cut gravity (SG50)

Jan 1, 2020

By E. G. Galvez, E. Mate, L. A. Cisternas, C. Cruz, L. Cortés, M. E. Osorio

Designing and optimising flotation circuits are critical for mineral beneficiation plants, and therefore several procedures have been published. The modelling of flotation circuits with the objective of design and optimisation is problematic in that different approaches are used for the modelling of flotation units and regrinding units. Usually, regrinding models are based on particles characterised by their size, while flotation models are based on species characterised by fast and slow classes. The relationship between particle size and liberation can be used, but it requires laboratory studies that are time-consuming and costs are high. It is therefore not surprising that few studies on flotation circuit design and optimisation include flotation units and regrinding units simultaneously. In this paper, it is proposed to model the regrinding stage using species characterised by fine-slow, medium-fast, and coarse-slow flotation species. In deep, the regrinding model is derived from a population balance approach based on flotation species. The fraction of these species at the feed and product of the grinding unit is determined by flotation tests. Then, based on these data, the model parameters can be determined by a regression estimation procedure. A local ore was used to validate the model. Keywords: Modelling, flotation circuit, grinding circuit, process design

Jan 1, 2020

By S. R. Grobler, S. Bilessuris, M. E. Greenway

Ensuring that all the head ropes on a multi-rope friction winder share the load equally has been a design and maintenance issue since they were first developed. Accurate machining of the rope grooves is necessary to meet this objective. Early analysis of the change of rope loads for ropes in grooves that are mismatched in depth treated the grooves as rigid. Groove depth tolerances deduced from this analysis are stringent and difficult to achieve in practice. Test work indicated that the inherent flexibility of groove lining materials alleviates the load-sharing problem. However, no analysis of rope load changes due to mismatched flexible grooves has been published. In this paper we develop new equations for the rope load variation for flexible grooves. By allowing flexibility to reduce to zero, the corresponding rigid rope groove equations are obtained. New criteria for tolerable groove depth variations are developed from this analysis, which depend on the flexibility of the groove lining material.

Sep 1, 2021

Anglo Platinum is in the process of introducing a number of initiatives, varied in nature, on a scale that will impact operations in the short, medium and long-term. The advent of mechanization on a large scale and the introduction of new mining technologies therefore require a different approach to the design of ventilation and cooling systems. In addition, the ventilation and cooling systems serving increasingly deeper operations will place a greater demand on electrical power consumption?this in an environment where power generation is limited and increasingly expensive. Ventilation and cooling systems currently being employed in platinum operations have evolved from practices first pioneered in deep level gold mines. However, in modern and future platinum mines, deepening of operations will take place in a somewhat different environment?particularly in terms of geothermal settings and electrical power utilization. This provides the opportunities to devise novel and energy-efficient ventilation and cooling strategies tailor-made to meet these challenges. The need for refrigeration and cooling at depth will increase the capital and operational cost component of ventilation systems and the medium-term limitations on the availability of electrical power will require the provision of innovative systems. The increased use of diesel-powered vehicles underground, the viability of energy recovery and feasibility of thermal storage opportunities are discussed in this paper. Some of the challenges and the solutions currently proposed in terms of ventilation system design, refrigeration plant selection and chilled water distribution methodologies are described. Changes in strategies core to this discipline that affect the health and safety of employees and the profitability of modern operations are deemed necessary to support the transformations that are envisaged.

Jan 1, 2008

By A. Rahmanab, A. Torpy, M. I. Pownceby, M. N. Zaman, N. Haque, W. Bruckard, J. Tardio

Ilmenite is the principal feedstock for producing TiO2 - used in paints, plastics and paper and the production of Ti metal. The Brahmaputra River system in Bangladesh represents a potentially new source of ilmenite concentrated within sandbar placer deposits and the current study presents results from beneficiation and characterisation studies conducted on Brahmaputra River Fe-Ti oxides. Characteristic of fluvial systems, the ilmenite concentration is low (~0.5 wt-%), however extensive annual dredging for waterway navigation provides the economic impetus to recover saleable ilmenite. The Fe-Ti oxide suite derived from magnetic fractionation comprises homogeneous ilmenite, titanohematite and hematite, although many grains exhibit complex exsolution within the (Ilm-Hem)ss series. The textural variability results in difficulty generating a pure ilmenite feedstock with the bulk concentrate typically being low in TiO2, containing 31.0 wt-%TiO2 and 62.7 wt-% (Fe2O3). Results are interpreted via an analysis of the implications for producing a commercial-grade ilmenite concentrate suitable for TiO2 production. Keywords: Ilmenite, titanohematite, Fe-Ti oxide exsolution, TiO2 production

Jan 1, 2020

By A. Hoogervorst

Cyanide is a key contributor to the efficiency of gold extraction. Public perceptions of cyanide are emotive and following some publicised incidents involving cyanide, the gold mining industry established a voluntary cyanide code for the manufacture, transport and use of cyanide in the production of gold. The paper describes the development of the code, its content and the third party auditing of the code as a means of establishing and maintaining benchmark standards. In adhering to the cyanide code requirements, there are overlaps with other management systems such as ISO 14001 and OHASA 18001 and the paper seeks to show how existing mine systems can readily accommodate cyanide code compliance and that operating and auditing the cyanide code can assist in the integration of mine heath, safety and environmental management systems.

Jan 1, 2009

By A. Mainza, I. Govender, Elbasher M. E. Ahmed

A mechanistic description of axial segregation in rotating drum flows remains an open question. Consequently, optimal mixing of grinding balls and rocks for efficient breakage, maximum production of fines, and slurry transport is seldom achieved. Experimental and numerical studies of granular mixtures in rotating drums identify alternating axial bands that eventually coarsen in the long-term limit. Most models of axial segregation are limited to binary mixtures and cannot always predict the logarithmic coarsening effects observed experimentally. A key missing factor is a robust description of the axial free surface profile that is valid across a wide range of flow regimes. We present a practical model of the axial free surface profile by linking it to readily-derived geometric features of the cross-sectional S-shaped free surface profile. A parametric study shows good agreement with experimental measurements reported in the literature and heuristically valid trends.

May 1, 2021