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By T A. J Hopper, S Mukkisa, M L. Johns, K T. O’Neill

Technological advances the sensor design and performance for logging while drilling (LWD) instrumentation offer considerable opportunities in iron ore exploration. Here we focus on the development of a nuclear magnetic resonance (NMR) LWD tool. NMR is an advanced geotechnical logging technology capable of providing in situ measurements of the porosity, pore size and permeability of the formation of interest. The ability to provide real-time characterisation of iron-ore systems is beneficial to both drilling operations as well as real-time formation evaluation. The development of an NMR LWD tool for mineral exploration drilling has previously not been achieved due to the significant technical and environmental challenges associated with the design and operation of such a tool. The tool outlined in this work is designed to work in reverse-circulation (RC) drilling systems. As such, the tool needs to be compatible with the RC drill-string, as well as being highly robust to survive the shocks and vibrations experienced during RC drilling operations. The magnetic physics of the probe must be capable of measuring at the appropriate depth into formation whilst being subject to the array of motions experienced during drilling operations. Furthermore, conducting accurate NMR measurements in an iron ore formation is challenging and the measurement methodology must be tailored to characterise the magnetic environment. The tool also requires high-powered electronics capable of operating the necessary pulse sequences for excitation and acquisition. The resultant tool design must therefore optimise the inherent trade-offs which exist between the physical components (ie mechanical structure, magnetic components and electronic hardware) in order to operate appropriately whilst not hindering RC drilling operations. Current experimental analysis of the prototype indicates that the current probe design exhibits the required measurement specifications and is nearing field testing. Future work will look towards commercialisation of the instrument.

Nov 8, 2021

By A Horeth, E Duggan, B Versiani, A Gooch

Iron ore screening plant productivity is strongly affected by the performance of the shuttle operation, with the control system robustness and associated strategies having a direct impact on utilisation and throughput rates. A number of aspects intrinsic to the methods and technologies traditionally applied on the motion control of shuttles are often observed to decrease the efficiency of the ore distribution process across multiple screening modules, which ultimately impacts the plant’s production. Those include the number of operational scenarios that are unmanaged by conventional discrete control logics which may intermittently lead to undesired events, such as ‘high level alarms’ and ‘low level interlocks’ in the screening bins. This outcome usually demands excessive manual intervention by Control Room Operators. A Reinforcement Learning (RL) Artificial Intelligence (AI) solution was deployed to control the motion operation of the scalping and products screening shuttles in an iron ore handling plant. The RL algorithm was trained in advance to handle multiple operational scenarios and generate optimised outputs. In nominal operation, the shuttles are moved in a modulating and highly predictive fashion over the whole range of safely reachable bins. When the need occurs, the AI control agent deviates from that modulation behaviour, limiting its travel range or moving to other groups or clusters of available bins by employing ‘bin jumping’ or ‘ore break’ strategies. The RL platform capabilities include the ability to learn in real time to account for and auto-adapt to new operational patterns and process changes whilst applying improved strategies for shuttle dwelling times, target setting and speed control. The deployed Reinforcement Learning AI solution has better managed the existing challenges associated with the shuttle operation and control, and resulted in a more efficient ore delivery process, superior operations stability and increased production rates.

Nov 8, 2021

By L A. Balzan, F Nieuwenhuys

GEOSCAN on belt analysers have been widely utilised in iron ore for nearly two decades. Since the first installations in 2003, the availability of accurate and reliable grade data has become standardised across iron ore mines in the Northern Cape in South Africa, where various streams are monitored continuously to ensure the optimal functioning of plants. Assmang Khumani has been at the forefront of technology adoption, and has implemented 19 GEOSCANs at various locations throughout the plant, monitoring run-of-mine, stockyard feed, beneficiation plant feeds, and products. Over the years, collaborative advances in the technology have allowed for numerous benefits and gains to be made. In recent times, there has been a concerted effort on the development of analysis for phosphorus in the material flows, and based on Scantech’s experience in the phosphate industry, this element has now been calibrated to achieve statistically acceptable and repeatable accuracies for real time analysis of this element on many of Khumani’s GEOSCAN analysers. This paper provides an overview of GEOSCAN use at Khumani, while focusing on the development of phosphorus analysis. A case study of analysis results is presented for a range of analysers that have been calibrated for this element, as well as a discussion on the benefits obtained as a result of this analysis.

Nov 8, 2021

By CHONG WEI, TATYANA KATSAGA, DEREK APEL

This study provides an effective determination process of the interface microstructural parameters in the numerical modelling of the mechanical interaction between two different geomaterials involved with frictional-bonded interfaces, such as the interaction between backfill columns and the surrounding rocks, or the soil-structure interaction, by using the coupled finite-difference and discrete-element (FD-DE) method. Three-dimensional numerical direct shear tests were conducted on the coupled FD-DE model of the combined rock-cemented rockfill (R-CRF) specimens with the frictional-bonded interface. With the microstructural parameters determined, the macro-mechanical properties of the interface of the coupled model were calibrated to well fit with the theoretical values of the combined specimen under the direct shear condition assumed according to the Mohr–Coulomb strength criterion. Evident elastoplastic shear behaviour was observed on the interfaces, and excellent agreement was attained between the shapes of the theoretical and simulated shear stress-displacement curves of the coupled models at different servo-controlled normal stresses. Particulate-scale force chains inside the DE model of the CRF specimen were highly consistent with the force contour on the FD model of the rock specimen, demonstrating the shear stress evolution of the coupled model at different shear displacements. Parametric studies verified the effects of the deformation properties of the interface parallel bonds on the shape of shear stress-displacement curves of the coupled models. The test results validated the feasibility and rationality of the coupled FD-DE method for modelling the combined specimens with frictional-bonded interfaces or joints. Furthermore, this study provides valuable numerical modelling experience for investigating the shear behaviour at the interface between cemented backfill columns and surrounding walls by using the coupled FD-DE method.

Nov 2, 2021

By G. Akdogan, S. M. Bradshaw, A. P. Van Wyk

In the mineral processing industry, Ni-Cu-Fe-S converter matte is leached to remove base metals from a concentrate containing platinum group metals (PGMs). We investigated the atmospheric leaching process to develop a better understanding of the leaching behaviour of the base metals (Cu, Fe, and Ni) and the PGMs, in particular Ru, Ir, and Rh with respect to key process variables, acid concentration, and Cu concentration under oxidative and non-oxidative conditions. With oxidative leaching, a higher initial acid concentration resulted in higher Cu, Ni, and Fe extractions, as well as faster leaching reaction rates. A higher initial acid concentration also resulted in faster precipitation of Cu, Ru, and Ir under oxidizing conditions. Under non-oxidative conditions, Ni and Fe extraction was much slower, and the effect of initial acid concentration on Cu precipitation was less pronounced. The initial Cu concentration had a slightly smaller effect on the leaching behaviour of Fe and Ni, as well as precipitation of Ru, Ir and Rh. Moreover, a higher initial Cu concentration suppressed both Ni and Fe leaching and had a slight inhibiting effect on the precipitation behaviour of Cu, Ru, and Ir.

Nov 1, 2021

By L. Roux

This review was initially motivated by the author’s own experience in attempting to reconcile run-of mine production tons with the Coal Resource and Reserve estimation of mineable, in-situ tonnages. The primary scientific measurements and observations that are collected during exploration at the beginning of the value chain will have a profound effect on the future of the mining operation. Due diligence must be exercised during the planning of a new mine and during the life of an existing mine. The comments provided by the technical and economic assessment group of Venmyn Deloitte confirmed the need for this critical review. They found the reporting of Coal Resources in South Africa to be inconsistent. This was particularly problematic in the Waterberg Coalfield in Limpopo Province. In this coalfield there are two types of coal deposit present. The first is comprised of the thick intercalated, cyclic coal and shale/mudstone sequences of the Volksrust Formation. This overlies multiple coal seams within the Vryheid Formation, each of which is thick enough to be extracted as an individual raw coal unit. On the other hand, the interbedded coal and shale seams of the Volksrust Formation require beneficiation to separate the coal from the shale. The review examines practices and methods, investigates alternatives, provides checks and balances, and tests these against actual production reconciliations. In conclusion, the best estimates of the mineable, in-situ tonnage will be obtained from the air-dry raw material density. These estimates should be adjusted afterwards to allow for free moisture content. The adjustments are derived from reconciliation data. The greatest contributing factor to the over-estimation of Reserve tonnages is the moisture content of the raw material. This fluctuates significantly under varying conditions in situ, as well as upon exposure to the natural environment. The air-dried density of the raw material includes inherent (structurally bound) moisture within the matrix. It provides credible tonnage estimations of raw material available while also providing an indication of voids within the volume of material being assessed. The calculated solids percentage can be used to adjust the specific gravity, which is determined via the Archimedes principle. This will supply a representative estimate of the material to be mined.

Nov 1, 2021

By Víctor Vázquez, Guillermo Tiburcio, Martin Encinas, Jesús L. Valenzuela, Erick Montaño, Alejandro Valenzuela, José Parga

Merrill-Crowe is the primary process used for the concentration and purification of silver in a cyanide solution. The Langmuir Adsorption model for silver recovery in a cyanide solution was used for the study. The maximum adsorption capacity was 6.19 mmol/g for silver. This model properly adjusts the experimental results of the adsorption equilibrium with high correlation coefficients, which also favors the formation of a single layer of molecules adsorbed for the silver in the electrocoagulation- generated species. The thermodynamic parameters ΔG, ΔH, and ΔS were estimated, and it was found that the adsorption process is exothermic and spontaneous. For the kinetic study, the Lagergren second-order equation was used to study the adsorption speed. To characterize the solids coming from the electrocoagulation, X-ray diffraction (XRD), and scanning electron microscopy (SEM) with energy-dispersive X-ray spectroscopy (EDX) were used; the results of this study suggest that silver is present in aluminite, hydroniumjarosite, and alunogen.

Oct 30, 2021

By A. Halim, J. Gustafsson, A. van Wageningen, J. Johansson, · J. Lööw, K. Kocsis

A major part of the European Union’s (EU) project Sustainable Intelligent Mining System (SIMS) is investigating the development of diesel-free/carbon–neutral underground mines in order to ensure sustainable underground mining in the future. Replacing diesel machines with electric vehicles in underground hard rock mines has been widely acknowledged by the mining industry worldwide as a critical step to improve working conditions by reducing diesel exhaust–related contaminants, to reduce mine ventilation electrical power cost by reducing mine airflow quantity, and to reduce mine greenhouse gas emissions. All of these are major requirements to achieve sustainable future underground mining practices. A field trial of Epiroc’s 2nd generation of Battery Electric Vehicles (BEVs) at Agnico Eagle Finland’s Kittilä mine was conducted during 2019–2020. Vehicles tested were MT42 mine truck, ST14 Load-Haul-Dump (LHD), and Boomer E2 jumbo drill rig. This paper outlines the improvement of the working conditions observed in the field trial, and the opinions of the mine personnel at Kittilä mine on using BEVs instead of diesel machines. Measurements of atmospheric contaminants and air temperatures taken during the field trial clearly demonstrated a significant improvement of working conditions when BEVs were operating as opposed to diesel machines. This field observation was supported by the opinion of the majority of the Kittilä mine workers. However, some remaining concerns must be addressed before BEVs can replace diesel machines.

Oct 25, 2021

By Dejan Petrović, Pavle Stojković, Jelena Ivaz, Saša Stojadinović

In spite of the fact that flyrock phenomenon represents the real threat for personnel and machinery, it still remains insufficiently investigated. There are efforts, in the mining community, to explain the causes of flyrock, but the models capable to accurately predict flyrock occurrence and define the flyrock distance still do not exist. This article is an attempt to establish such a predictive model. The model utilized the adaptive nature of artificial neural networks and combined it with the accuracy of numerical modeling of physical phenomena. Necessary data were collected by high-speed camera recordings of actual blasts at three different surface mines and processed for further use for artificial network training. The result was a neuro-numerical couple which was capable to predict flyrock occurrence, estimate the launch velocity of flyrock fragments, and calculate the maximum distance of flyrock fragments. The safe distance was then calculated by multiplying the flyrock distance with a factor of safety. The article explains the procedure for data acquisition and processing, artificial network construction, training, and validation. It also explains the principles of flyrock distance calculation and, finally, provides a definition of factor of safety and safe distance. The results showed that the prediction was satisfactorily accurate and reliable.

Oct 22, 2021

By Qingwen Shi, Gang Xu, Shuwen Wang, Brijes Mishra

Pressure relief borehole is one of the most effective methods to treat coal burst in deep-buried coal seams. However, due to the large amount of randomly distributed fractures in the coal body and the varying stress environments in underground coal mines, there is no practical method to assess relief boreholes by far, which brings problems for borehole design and construction. This study focuses on the assessment of relief boreholes in a single driving entry wherein the impact of mining panels on relief boreholes is eliminated. A practical method, focusing on field monitoring of borehole deformation with a new device, is developed to evaluate the single driving entry’s relief boreholes. This method is applied to a Chinese coal mine. The application results showed that the convergence process of the boreholes showed a three-phase characteristic. Phase one involves the fastest deformation, reaching up to 8% in the first 3 to 4 days. Phase two lasts longer and converges more, reaching up to 30% in 4 months. Phase three stays balanced until the excavating face moves. The three-phase characteristic indicates that a large diameter of the borehole ensures relief efficiency because the borehole provides adequate space for deformation throughout its life span. In addition, it is verified that long boreholes and slow footage improve relief efficiency by providing adequate time for the boreholes to work. The application proposes that boreholes in the test coal mine need to be drilled larger and longer for optimization.

Oct 19, 2021

By Ana E. Bohé, Georgina De Micco, Juan P. Gaviría, FEDERICO J. POMIRO, Gastón G. Fouga

Chlorination and carbochlorination processes have industrial applications in metallurgical activity for extraction and purification of uncommon metals such as rare earth elements (REE). This study reviews the chlorination and carbochlorination reactions of light REE (LREE) oxides with chlorine considering the thermodynamic, stoichiometric, and kinetic aspects of the reactions. Lanthanides between lanthanum to gadolinium were considered as LREE, in agreement with previous literature, and the results were detailed in this review. LREE were analyzed taking into account their oxidation states and were divided into three groups accordingly. The results presented show that, although the LREE are chemically similar, their behaviors during the reactions with Cl2 and Cl2–C, and the different LREE-O–Cl compounds formed, have particularities. It was observed that gadolinium, in both thermodynamic calculations and experimental results, presents some differences, and it can be considered intermediate between light and heavy REE; i.e., reactions between LREE2O3 and Cl2(g) begin at approximately 250 °C, except for Gd2O3 that begins at 327 °C. The present work will contribute to a better understanding of the chemistry of LREE oxides and Cl2 under different conditions and the products obtained in the LREE–O––Cl2(g) and LREE–O–C–Cl2(g) systems.

Oct 15, 2021

By Huiqin Yao, Dongmei Liu

The study investigated the potential use of low activity circulating fluidized bed combustion (CFBC) ash and phosphate tailings for the production of a filling material based on the cement. The phosphate tailings were used as aggregate and consolidated by cement and CFBC ash. The results showed that the bleeding rate and fluidity of all mixtures decreased with the increasing the amount of CFBC ash. The fluidity of filling slurry is highly related to the experimental factors such as the content of sulfur solid ash, B/A ratio, and mass concentration. The replacement of CFBC ash ranging from 20 to 40% led to a significant decrease in setting time and an obvious increase in compressive strength, but the above effect was obviously weakened when reducing the mass concentration and B/A ratio. The hydrated calcium silicate gel and ettringite generated from the hydration of cement and CFBC ash were the main sources of strength for the filling materials. When the recommended proportion was 1/6 binder/phosphorus tailings ratio, 76% solid content, and 40% CFBC ash replacement, the filling material still met demands on working and mechanical properties; at the same time, the utilization of CFBC ash and phosphorus tailings was maximized.

Oct 11, 2021

By R. C. Nkhoma, K. A. Annan, S. Ngomane

The effects of welding current, electrode force, and welding time in a resistance spot weld were studied to investigate the effectiveness of welded joints between a thin EN10130: DC04 material and a thicker 817M40 part, through analysis of the microstructural and mechanical properties. All welded specimens were subjected to tensile testing at room temperature (25°C) and sub-zero temperature (–46°C) to test the strength of the welded joints. No full button failure was observed at either room temperature or sub-zero temperature after optimization of the weldng parameters. The fusion zone was observed to consist mainly of martensitic phase, due to rapid quenching, while the HAZ was composed of clusters of martensite in a ferrite and bainite matrix. The base 817M40 metal remained fully ferritic after welding. The hardness was found to increase with increasing welding current. An increase in nugget size, indicating good fusion of the weld, was observed with an increase in the welding current.

Oct 1, 2021

By D. Sethuram, P. G. Koppad, S. Iyengar, R. Shobha

TiB2 and CeO2 particle-reinforced Al6061 hybrid composites were manufactured using stir casting and hot rolling techniques. The base alloy and composites were hot-rolled at 500ºC and a 50% reduction was achieved through 12 passes. The effect of varying TiB2 and CeO2 particle additions on the microstructure and mechanical properties of the Al6061 matrix was studied. Scanning electron microscopy showed uniform dispersion of both the reinforcements, with good interfacial bonding. Microhardness and tensile properties like yield and tensile strength were found to be higher for hybrid composite with 2.5% TiB2 and 2.5% CeO2 compared to Al6061 alloy and other hybrid composites. The increased tensile strength is attributed to good dispersion and interfacial bonding between the particles and Al6061 matrix. Fracture analysis using a scanning electron microscope revealed ductile fracture for the Al6061 alloy and mixed characteristics of ductile-brittle fracture for hybrid composites

Oct 1, 2021

By Siavash Mahvelati, Alireza Kordjazi, Joseph Thomas Coe

Deep foundations are often constructed as end bearing members with a significant amount of their capacity derived from bedrock. Sites with highly variable bedrock conditions therefore present a unique challenge when attempting to estimate the design elevation of deep foundations. Typical subsurface characterization techniques such as drilling, standard penetration testing (SPT), and cone-penetration testing (CPT) may fail to adequately establish top-of-rock elevation in geologic settings prone to karst and/or significant amounts of residuum, saprolite, and deformed bedrock. Seismic surface wave geophysical methods have been increasingly used to map bedrock topography in such settings. However, the current standard of practice for these methods introduces resolution limitations in spatially variable subsurface conditions due to averaging effects from the analytical framework used to process the acquired data. This study explores the use of a tomographic approach with full waveform inversion (FWI) to map top-of-rock elevations at a numerically simulated site representative of highly variable bedrock topography. Comparisons are made with seismic refraction (SR) and a dispersion-based multichannel analysis of surface waves (MASW) approach as more commonly used in geophysical subsurface investigations. The results highlight that FWI provides a superior estimate of top-of-rock topography than SR and MASW at a cost of a significant increase in the required computational resources and time.

Oct 1, 2021

By MATT RENAUD, Mustafa Kumral

As high-quality mineral deposits are thoroughly exploited, mines of the future will have to move towards more remote and lower quality projects. With these projects comes an increased business risk to the operator. Even now, mining companies often turn to foreign countries for new deposits to produce, introducing the concept of country risk into their portfolio. Furthermore, feasibility studies often fail to accurately quantify the risks associated with various projects, and as such, this study proposes a technique of risk quantification which can both aid mining companies to evaluate the viability of investment and assist in risk disclosure to public investors for the mining industry of the future. This paper aims to quantify the effects of country risk on the valuation of potential mining projects, using a combination of modular and risk-weighting techniques to evaluate the country risk premium (CRP) required to obtain capital for operation of risky foreign mining projects. The techniques proposed in this paper are demonstrated using fictional examples and then evaluated for accuracy against a recently developed debt-funded precious metal project in a country in the Americas.

Sep 30, 2021

By Jogeshwara Sahu, Sapan Kumar Kandi, SANDEEP KUMAR JENA, Sachida Nanda Sahu, Sarada Prasad Das

The present article reports the recovery of potassium from mica scrap using carbide lime waste and NaCl. The acid leaching of mica [4 M H2SO4 at 90 °C] can recover a maximum of 25% potassium. However, in the chlorination roasting (with CLW and NaCl) water leaching process, it was possible to recover potassium to the tune of 94% [900 °C, 50 min roasting; mica:CLW:NaCl 1:0.75:1]. The formation of water-soluble phase like sylvite (KCl) and water-insoluble phases like anorthite ( CaAl2Si2O8) and nepheline ( NaAlSiO4) confirms the complete breakdown of the mica crystal structure. The extraction mechanism proceeds with the formation of a transition state followed by the release of potassium as KCl. The characterisation studies using ICP-OES, XRD, FTIR and FESEM support the proposed extraction mechanism. The salt produced by the evaporation of roast-leached liquor confirms the presence of sylvite and halite only, which can be separated to get the fertiliser grade KCl.

Sep 29, 2021

By S. Rukhaiyar, M. Ramulu, G. Pradeep, P. K. Singh, P. B. Choudhury

The drill and blast method is the most prominent rock excavation method used in mining and civil engineering projects. The ground vibration is the most undesirable effect of the blasting and needs to be controlled. The blasts are designed so that ground vibration at nearby structures must remain under the prescribed limit. Various empirical blast vibration attenuation models have been presented to estimate the ground vibration for designing purposes. These models assume ground vibration as a function of distance and maximum charge per delay. United States Bureau of Mines (USBM) model is most commonly used for the blast design. However, the uncertainty associated with these models is seldom characterized, thus diminishing its reliability. The probabilistic Bayesian approach associated with the Monte Carlo method can easily quantify uncertainties associated with the model and its parameters. Thirty-three blast vibration observation data from a granite quarry in Kerala, India, is used to estimate the probability distribution of model parameters. Markov Chain Monte Carlo (MCMC) approach is used for sampling 12,000 samples for obtaining the posterior distribution of parameters. The uncertainty associated with model parameters is quantified from the posterior distribution, and this is further used to estimate the credible and prediction limits of the model. The point estimate of the most optimum value of the parameter is estimated using maximum-aposteriori or MAP estimate. The model parameters are also obtained using the commonly used least-square error method, which is a maximum likelihood estimation or MLE approach. It has been observed that the accuracy of the model obtained using both MAP and MLE approach is almost equal. However, the MAP estimate as per Bayesian probabilistic analysis has the added advantage that it can estimate the uncertainty of parameters as well as the model, which is not possible using the MLE approach.

Sep 27, 2021

By JESSICA W. A. AZURE, Samuel Frimpong, AZUPURI G. A. KABA, FORSYTH A. KADINGDI, PROSPER E. A. AYAWAH

This study developed machine learning (ML) models for predicting rock formation reactive forces experienced by a hydraulic shovel bucket during excavation. To do this, rock formation in the form of muckpile was modeled as granular ball elements in three-dimensional particle flow code (PFC) using linear bonding logic. A typical field-size hydraulic shovel bucket modelled in AutoCAD was used to excavate the rock formation and the three-dimensional shovel bucket resistive forces required to cut through the rock formation were measured and recorded. The experiments involved different muckpile height, bulk densities, repose angles, and average fragment size. Six ML algorithms, artificial neural network (ANN), K-nearest neighbor (KNN), linear, random forest (RF), regression tree (RT), and support vector machine (SVM), were evaluated on their ability to predict the shovel bucket resistive forces. The input variables used for the prediction of the resistive forces were the muckpile bulk density, angle of repose, height, and fragment size. The results showed that the ML techniques are useful and reliable methods of predicting rock formation resistive forces based on the rock formation properties. This research is a preliminary step towards developing reliable models for accurate prediction of excavation forces/torques which could potentially enhance hydraulic shovel process automation.

Sep 27, 2021

By Xiaobo Zhang, Panshi Xie, Xiaobo Wu, Ding Lang, Jiandong Yu, Yongping Wu, Kun Feng

In a fully mechanized top coal caving mining process, the mechanical properties of the top coal deteriorate constantly, and the stability of the support is determined by the boundary position of the top coal entering the limit equilibrium state. Top coal flaking occurs more easily along the advancing direction owing to an increase in the distance of the boundary from the wall. Moreover, the strength of the coal seam is an important factor that determines the boundary position. The equation of the top coal limit equilibrium zone boundary, used in fully mechanized top coal caving mining, was determined based on the limit equilibrium theory, continuum damage mechanics, and the Hoek–Brown rock empirical strength criterion. Using the RFPA3D and UDEC numerical analysis methods, the effect of top coal strength on the mechanical behavior during the mining process was determined; and the space–time relationship between the top coal deformation failure, damage evolution, and boundary of the limit equilibrium zone was established. The results revealed that when the coal mass entered the limit equilibrium zone, accelerated crack development, accelerated damage, and axial unloading occurred. Moreover, an increase in the coal seam strength caused the boundary of the top coal limit equilibrium zone to be closer to the wall. By comparing the numerical results with the theoretical results, the overall trend was found to be consistent, allowing the verification of the rationality and reliability of the determination equation of the boundary position of the limit equilibrium zone.

Sep 21, 2021