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By Peter Ay, Vincenz Mehlstaubl, Satyanarayana Narra

"This study gives an insight into non-destructive testing methods suitable for porosity analysis at a small scale, firstly summarizing relevant state-of-the-art technology and pending patents, secondly realizing and documenting the start-up of the VHX-2000 digital microscope by the manufacturer Keyence and thirdly demonstrating practical application with the extended analysis of three fundamentally different agglomerates for which important application-related parameters are determined. Various analysis procedures on the different agglomerates the efficient and timesaving operational features of the digital microscope are outlined. Firstly an analysis of flax fibre agglomerates determines the process conditions best suited to produce agglomerates with specific properties by determining sphericity, cubicity, density and porosity as well as fibre damage and alteration. Secondly the production of iron-hydroxide - hemp-fibre agglomerates serves to obtain information on porosity values as a result of a cauterization of the fibres. The third analysis provides estimations of porosity values for acid-treated lignin agglomerates. Measurements were carried out with compact dry agglomerates and with soaked, expanded agglomerates, determining factors for total water carrying capacity. Practicability of non-destructive testing by digital microscopy is demonstrated and explained concerning the applications in this study by obtaining definite structural findings about the analysed agglomerates. Information on its operation is made available by a practice-oriented manual for the VHX-2000 digital microscope. Further suggestions for more in-detail analysis results with different instruments are given."

Jan 1, 2014

By Georgios N. Anastassakis

"The depletion of the high-grade chromite ores and the inefficiency of the gravimetric methods to process fine particles has shifted the industrial and research interest from gravimetric separation methods to flotation. However, the response of the minerals to various flotation schemes is not clearly established. In this work, the physicochemical behavior of chromite and olivine is studied using a fatty acid collector.The minerals used were from Vourino area (Kozani, Northern Greece). Macroscopically pure samples of chromite and olivine were selected by hand picking, crushed and subsequently ground to the required size. Each sample was then submitted to panning and magnetic separation for the removal of impurities. The mineralogical and chemical analysis of the purified samples revealed the following: a) The purity of chromite was 95% while the minor constituent was mainly magnetite. Its chemical analysis was: 12.50% Fetot, 2.50% SiO2, 54,54% Cr2O3, 9.31% Al2O3, 0.47% CaO, 13.10% MgO. These analyses show that Cr3+ has been substituted by Fe3+ and Al3+ as well as Fe2+ has been significantly substituted by Mg2+. b) Forsterite is the main constituent of olivine either as pure forsterite (Mg2SiO4), which is the prevailing form, or as iron-forsterite [(Mg,Fe)2SiO4] in less amount. Chrysotile, which belongs to the serpentine group, was also detected at a percentage of 10-15%. The chemical analysis of olivine was: 36.90% SiO2, 0,53% Cr2O3, 7,43% Fe2O3, 0.72% Al2O3, 0.91% CaO, 47.00% MgO, 6.02% LOI.The electrokinetic behavior of the minerals was studied in a electrophoretic mass transport equipment Model 1202 of Micromeritics. The reagents used were sodium hexa-metaphosphate of commercial grade and sodium oleate."

Jan 1, 2003

By M. Rudolph, E. Schach, M. Buchmann, T. Leißner, N. Krupko, U. Peuker, M. Kern

"In mineral processing, density and magnetic susceptibility are two very fundamental properties. For the beneficiation of valuables to produce a saleable concentrate a detailed understanding of these properties is essential. Especially when it comes to the processing of cassiterite, which is the main mineral for tin production, they become highly prominent. Due to the chemically inert character of cassiterite towards most industrial applied leaching agents, density and magnetic separation processes are mainly applied for its beneficiation. To guarantee an optimized utilization of the different operations not only cassiterite but also the different gangue minerals have to be considered. In this study, a skarn ore is characterized by density and magnetic susceptibility. Therefore, the material was first split into different density classes by heavy liquid separation. The obtained classes were further separated by their susceptibility to finally obtain a density-susceptibility matrix. For this purpose, an isodynamic separator was used. A more detailed characterization of the materials is done via gas pycnometer, magnetic susceptibility balance and vibrating sample magnetometer to estimate the characteristics of density and susceptibility for the various classes. Further, the determination of the chemical assay and the mineral intergrowth by mineral liberation analysis is used to generate a three dimensional database for detailed characterization of the present ore. The objective of this study is to estimate potential material streams for a modular processing plant via characterization of the material for the entire deposit by the two afore mentioned characteristic properties. The established multidimensional data matrix, enables predictions for the separation properties of the material and contributes to the characterization of the deposit within the context of geometallurgy."

Jan 1, 2018

By J. Y. ` Xie

The paper introduced bio-oxidation plants of refractory gold concentrate in China and CCGRI cultures. CCGRI cultures are composed of bacteria and archaea. Archaea Ferroplasma acidiphilum is very thick and fills whole mineral surface. It plays a more dominant role in CCGRI process. Several years of practice has proved that the operating temperature in bioreactors between 38~52?,gold recovery is almost same. Pulp concentration has been increased each year and reached to 25%.

Jan 1, 2014

By C. G. Anderson

Industrially proven and applied nitrogen species catalyzed (i.e. NSC) acidic pressure leaching has been utilized to treat a number of ores, concentrates and recycled materials. This versatile technology offers many advantages including low temperatures, low pressures, low capital costs, low operating costs, faster throughput and less costly materials of construction than conventional pressure oxidation. This paper will elucidate the application of this technology for production of gold, silver, palladium, rhodium, zinc, germanium, gallium, cobalt, copper, molybdenum, rhenium and nickel. Recycled material applications will also be elucidated. In addition, details of an application for treatment of a refractory chalcopyrite copper and gold bearing ore will be presented and pertinent economics discussed.

Jan 1, 2014

By C. B. Yu

The ore comprise of chalcopyrite, valleriite, pyrite, pyrrhotite and so on. Serpentine and talc are chief gangues. Copper phase of the tail shows that mostly valleriite be losed in tail, 70% mineral of copper is valleriite in tail, so severity reduce the recovery of copper. The traditional flow-sheet in acidity mine pulp to flotat the valleriite, but copper-sulphur separated in high alkaline pulp, so the valleriite be losed else. Explores the characters of the valleriite?s surface, studys on the flow-sheet of flotat valleriite and copper-sulphur separated. In neutral mine pulp, adopt of cationic collector to flotat valleriite, adopt of hyposulphite to depressant pyrite in low alkaline pulp can advance the recovery of copper to 5.06%.

Jan 1, 2014

By G. Atesok

Turkey?s EU affairs started in 1963 with the Ankara Treaty. On 31 December 1995 the Customs Union between Turkey and the European Union came into effect. In order to achieve the Copenhagen Criteria?s, Turkey, made laws parallel to the Framework Programs. On 3 October 2005, with the confirmation of ?Membership Framework Document? the negotiation process of Turkey has started. Coal had important role in the industrialization of EU and it is directly related to Energy and Environmental subjects. In this point many restriction became with the some protocols, in last decades. European Union composed Energy and Coal Policy, therefore engaged to reduce the emissions, and made new regulations. This regulations concern Turkey?s energy future, since large; but low quality of lignite reserves of Turkey, it is expected to have some difficulties with Union?s Environmental Legislation during Turkey?s congruity process. With the last natural gas import agreements of Turkey, the production and consumption of coal sunk in last seven years. However, considering environmental sustainability, the role of coal in energy production will rise up in the coming future. In this paper, in terms of Turkey Energy Sector, production and consumption problems of coal discussed and evaluated considering EU?s coal, energy and environmental rudiments.

Jan 1, 2014

By V. Beskrovnaya

Oxidised lead-bearing ores of one of the deposits of Russian Federation represent complex raw materials. Its main components are lead (3.2%~4.7%), silver (100~300 g/t) and zinc (0.7%~0.8%). The sample is characterized as oxidized ore. Silicium (37.4%) and iron (33.2%) are its main com-ponents and present in oxide form. Lead presents as cerussite, anglesite and plumbojarosite have a subject role. Zinc occurs as smithsonite. Certain part of zinc is associated with manganese minerals (psylomelane-pyrolusite). Close association of valuables with manganese-iron mineralization is ob-served. Silver is present in native finely dispersed form. Three options of flowsheets have been developed to recover valuables from the ore. 1. Hydrometallurgical flowsheet of lead and silver recovery on the basis of chloride leaching. 2. Flotation. 3. Flotation -flotation tailings cyanidation. Hydrometallurgical flowsheet of ore processing on the basis of chloride leaching (option I) al-lows to achieve recovery into saleable products of 68.8% Pb and 70.2% Ag. High quality lead concentrate with 71.7% lead and 700 g/t silver was obtained during flotation (option 2). Flotation tailings cyanidation (option 3) provides 24% silver additional recovery (from the ore). Technical and economical evaluation demonstrated that ore flotation (option 2) and flotation with subsequent flotation tailings cyanidation (option 3) are the most cost-efficient. Option II is the most cost-efficient variant of 1 tonne of ore processing (9.05 US dollars).

Jan 1, 2014

By H. D. Yu

In this paper, an evaluation about the process mineralogy of the rich iron mine in Mongolia with comprehensive methods is given; this provides foundation of the comprehensive recovery of iron and cobalt in the ore. The research shows that the iron?s grade is 61.23%, Co?s grade is 0.02%, the ore is rich iron ore and its Co content also reaches border mining grade. For the ore, if want get high grade iron concentrate powder, must identify the mineral composition and the occurrence of iron and take into ac-count the recovery of Co content as associated resources. Chemical phase analysis results show that 86.35% of iron exists in magnetite, which means that the theoretical recovery ratio of iron is close to 86%. Minerals spectrum analysis confirmed that the magnetite containing a small amount of Mg, Al, and other impurities, the actual iron minerals to 70.28%, which means that theoretically iron grade iron concentrate only close to 70.28%. Based on the result of iron chemical phase analysis, by investigating the liberation of sulfide minerals and gangue, the recovery index of iron can be forecasted, when grinding fineness is 35%, 45%, 55% and 65% bellow 0.074mm,theoretical content of Fe in iron concentrate separately is 67.71%,67.88%,68.78%,and 69.43%,accordingly the theoretical recovery of Fe is 87.38%,87.05%,86.92% and 86.63%. From the perspective of process mineralogy, the forecast mineral beneficiation indicators show that the ore is easy sorting ore, iron processing operations have higher recovery rate, but wants sorting of high-grade iron ore concentrate still need appropriate grinding fineness. Furthermore, associated element Co, existing mostly in pyrite and pyrrhotite, is valuable in the ore. Pyrite and pyrrhotite have ideal liberation under grinding, so cobalt can be easily enriched in sulfide concentrate.

Jan 1, 2014

By Shounak Banerjee, Suresh Babu Dasari, Dharmendr Kumar, Shashank Pardhikar, Beena Rai, Venugopal Tammishetti

Recent increase in steel demand has led to the increased usage of high grade iron ores. During the mining, crushing and washing of ore, huge quantities of fines and slimes (<150 µm) are generated, majority of which is dumped into tailing ponds owing to their high alumina contents (~10-15%). Due to rapid depletion of high grade iron ore, it is the need of the hour to utilize low grade iron ore fines and slimes. These fines and slimes contain high iron values assaying 45-60% Fe. Dumping these valuable resources in ponds locks enormous land space and causes environmental hazards. The constituent minerals present in Indian iron ores comprise of hematite, goethite, gibbsite, kaolinite and quartz. Our research is focused on beneficiation of iron ore slimes through selective flocculation. We use density functional theory simulations to screen the reagents and perform experiments with screened reagents to determine the experimental conditions for effective performance of the reagents. Our studies revealed that reagents like starch, guar gum and carboxymethyl cellulose exhibit excellent selectivity in high alkaline pH range. However, the higher cost of alkali makes the process economically less attractive. In the present study, a commercial reagent from NALCO Water (Hi-Tex 83570) has been explored as selective flocculant for beneficiation of iron ore slimes in wide pH range (4-11). However, the reagent was also found to be the most selective in alkaline pH range only. Full factorial design of experiments was done considering three parameters namely, pulp density, flocculant dosage and settling time. Regression models were developed for yield, iron grade, alumina grade and iron recovery in the concentrate, using experimental results. Further parameter optimization was done to achieve high Fe recoveries and low alumina recoveries. The optimal parameters thus arrived at were validated with the experiments. We were able to achieve a concentrate grade of 60.0% Fe and 5.1% Al2O3 with 68.8% Fe recovery from a feed containing of 53.1% Fe and 8.9% Al2O3. The performance of flocculant Hi-Tex 83570 was also compared with reagents starch and guar gum.

Jan 1, 2018

By K. Anand Rao

Extensive exploration for uranium metal by the Department of Atomic Energy in India led to the discovery of new carbonate host rock uranium deposits Tummalapalle and Gogi, which are located in southern part of India. Alkaline leaching at elevated temperature and pressure using autoclave is the central unit process in the overall strategy for extraction of uranium from these new ore deposits. Autoclave leaching is affected largely by intrinsic nature of the ore and by the process variables and the equipment settings. In order to understand such a complex process and also to be able to predict the process performance under randomly (ore grade, uranium mineralogy, gangue mineralogy etc.) and consciously perturbed variable settings (partial pressure of oxygen, temperature, concentration of leachant, stirring speed and time), sound and robust mathematical models are needed with analytical and/or numerical solutions. Present investigation attempts to develop a variant of kinetic correlation based on shrinking core model for batch alkaline leaching process in an autoclave. The leaching model incorporates particle size distribution according to popular Gates-Gaudin-Schuhmann equation, rather than assuming a single mean size for all the particles participating in the leaching reaction. Kinetic experimental data generated in autoclave leaching of Gogi uranium ore was used to evaluate various constants of the model developed. The ore sample used for experiments assayed 0.21 per cent U3O8. The partial pressure of oxygen was varied between 0.15 to 6.5 atm, temperature from 343 K to 438 K, concentration of sodium carbonate leachant in the range 0.47 g mol/L to 0.94 g mol/L, and the stirring speed from 573 rev/min to 900 rev/min in the experiments. The goodness of fit of the leaching process model is found to be >0.95. Morphological studies carried out on leached and unleached particulate matter using Scanning Electron Microscopy with Energy Dispersive X ray analyses are included in the present study.

Sep 1, 2012

By Lilong Huang, Xiang Zhao, Yuan Long, Li Shi, Guowang Zhang, Xiao Xiao

"Vertical stirred mill is expected to upgrade the iron concentrate by further size reduced efficiently. The tests for fine grinding of iron ores from Baiyun Boyu iron ore, Baogang Goup, iron contained tails of Cu-Pb-Zn ore in Wanguo ore, Xiaoshalong iron ore in Qinghai were conducted in this paper. The data of appropriate grinding fineness, optimum grinding conditions and energy consumptions obtained from the test were used to select appropriate scale of the vertical stirred mill, optimize the parameters of the grinding process and upgrade the iron concentrate of the iron ore dressing plant or design the new ore dressing plant. Based on the laboratory tests, industrial system of regrinding and magnetic separation with vertical stirred mill as the regrinding mill in micro-fine grained iron concentrate upgrading has worked out very well. The grade of Fe in iron concentrates has increased to about 65.0% (by 4.0~6.5 point percent) when the iron concentrates were reground to designed fineness and the specific energy is 5.0~7.1kWh/t. The system of regrinding and magnetic separation is proven to be eco-efficient in iron concentrate upgrading."

Jan 1, 2014

By A. A. Lavrinenko, N. I. Gluhova, L. M. Sarkisova, D. V. Makarov

"The purpose of the study is to study the feasibility of using sodium diisobutyl dithiophosphinate (DDP) to enhance the recovery of platinum group elements from low-grade refractory copper-nickel ores characterized by a portion of nickel and platinum group elements (PGEs) locked in the dominant sulfide mineral — pyrrhotite. These ore characteristics dictate the need to produce a bulk PGE-containing sulfide concentrate due to the difficulties of producing selective concentrates without a significant loss of the metals.New data on the adsorption and flotation performance of DDP-containing collectors were obtained by measuring the electrode potential, hydrophobicity of the sulfide mineral surfaces and by means of extraction spectrophotometry and flotation studies. The interaction between DDP, pyrrhotite, and pentlandite leads to the formation of surface hydrophobic layers consisting mainly of disulfide and a small amount of metal compounds. These adsorption layers are formed by the reaction with butyl xanthate. In the case of cooperite, DDP reacts with the surface, forming not only disulfide, as in the case of butyl xanthate, but also a complex platinum compound. DDP-containing agents demonstrate stronger collecting properties with respect to sulfide minerals of copper-nickel ores compared to butyl xanthate, which is explained by a stronger hydrophobizing effect and foaming properties.The addition of a combination of DDP-containing agents and butyl xanthate at a ratio of 3:1 for the recovery of sulfides and PGEs in the bulk flotation of copper-nickel ore made it possible to improve the recovery of Ni, Pt, and Pd by 13.6, 9.2, and 9.4%, respectively, relative to the recovery achieved in flotation with butyl xanthate.Bulk flotation with the addition of a composition of DDP-containing agents and butyl xanthate allows the most complete recovery of all valuable elements and is the only technologically feasible method for processing of lowgrade ores, when it is impossible to produce without loss high-grade selective concentrates suitable for smelting."

Jan 1, 2018

By T. Kumaresan

The Bayer process used for refining bauxite to smelting grade Alumina serves as the linchpin of the Aluminum production industry worldwide. Even though the process is well established, the presence of impurities in the liquor affects the quality of product and imposes a major economic cost on the industry. This review article focuses on the issues pertaining to Gibbsitic (Al2O3.3H2O) precipitation circuit. The comprehensive review provides a basic understanding of the aluminum trihydrate precipitation and agglomeration phenomena. Crystal growth rate and agglomeration play a significant role in determining the output PSD of agglomerates. Hence, the effect of certain thermodynamic and hydrodynamic parameters along with the impact of impurities (oxalates, aldols, acids etc.) and additives on hydrate growth has been reviewed.

Sep 1, 2012

By J. Quinteros-Riquelme

Despite increasing demand, the supply of silver is limited by its inadequate recovery from ores. Due to its scarcity in nature, its high ore variability and complex mineralogy of current silver deposits, its recovery by mineral froth flotation presents a real challenge for process engineers. A silver ore often contains significant amounts of five to ten different silver minerals and knowledge of the mineralogy is a necessary precursor to flotation testing. In this work, a complex silver ore was treated by using batch laboratory flotation, in an effort to find the best experimental conditions to obtain the optimal silver recovery. Mineralogical characterization of feed samples was conducted using a combination of tools, namely the mineral liberation analyzer (MLA) and inductively-coupled plasma mass spectrometry (LA-ICP-MS), which indicated that most of the silver occurred as ?invisible? silver in pyrite. The pyrite assay of the ore was approximately 4%. The ore was processed in two flotation stages. The first stage, sulphide flotation, aimed to collect a sulphide mineral rich concentrate; then a non-sulphide flotation stage was performed by adding sodium sulphide and collectors to the remaining material in the cell, with the objective of obtaining a pyrite rich concentrate. To reach the target recovery of silver from this ore, various variables were altered. In the first stage: mixed collectors, activators, dispersants, and mainstream inert grinding were assessed. In the second stage, the conditions for sulphidization and subsequent flotation were both investigated. From the grade recovery curves for silver and sulphur it was possible to observe a similar metallurgical behavior for both elements, supporting the results given by the mineralogical characterization. Keywords: silver oxide, froth flotation, sulphidization, mig

Sep 1, 2012

By M. M. Kulkarni1

Kaolin clay is used as a filler and pigment in various applications such as paper, paint, rubber, plastics, adhesives, ink and ceramics etc. Major mineral content in crude kaolin clay is kaolinite along with some accessory minerals as quartz, hematite, mica, feldspar, anatase, rutile, illemnite etc. The Fe and Ti bearing minerals play very important role in determining quality of the kaolin, as these minerals impart yellow, pink or darker color to the white kaolin. Conventional beneficiation methods used in kaolin processing are de-gritting, classification and sodium dithionite bleaching, which improves the optical properties of kaolin, wherein reduction of color imparting constituents such as Fe2O3 and TiO2 is marginal. In the present study, these coloring impurities are separated using selective adsorption of water-soluble polymers. Degritted kaolin samples after passing through 45 micron sieve is used in the study for selective flocculation by polyacrylamide based polymers. The effect of pH, zeta potential, and polymer dose on the reduction of impurities is studied. Analytical tools such as XRF, XRD, SEM, and TEM techniques are utilized and results are discussed in detail.

Sep 1, 2012

By W. H. Yong

The molybdenum ores currently utilized in China are mostly molybdenum sulfide ores, whereas oxidized molybdenum ores have not been exploited and utilized efficiently. The aim of this article is to study the applied mineralogy of an intergrown iron-molybdenum oxidized ore in order to provide a mineralogical basis for the ore beneficiation research. The ore occurred in a surface oxidized zone seam and contained 23.65% Fe2O3 and 1.58% Mo. Various characterization techniques were employed to study the chemical composition and mineral composition of the ore, the particle size distribution of the minerals in the ore, and the tex-ture and structure of the ore. The results showed that the iron-molybdenum ore occurred in a wet, soft and earthy state. The ore was composed mainly of several oxidized minerals and colloidal min-erals, including wulfenite, barite, limonite and quartz. Wulfenite was one of the major and stable components in the ore and occurred as a separate mineral. Wulfenite, barite and quartz were granu-lar minerals with particle size ranging from fine to coarse but mainly in the fine particle range. Li-monite showed a crust and honeycomb-like colloidal state and composed the framework in the ore for the formation of the other minerals. They filled the pores in the framework and the size of the pores affected the size of the minerals formed in the pores and hence the recovery of wulfenite in ore beneficiation. According to the applied mineralogical properties of the oxidized molybdenum ore, a promising ore beneficiation process was proposed as follow Magnetic separation should be carried out first to separate the magnetic iron minerals and then the molybdenum mineral will be recovered by gravity concentration or a combination process of gravity concentration and flotation, depending on the distribution of the molybdenum mineral in the product of the magnetic separation.

Jan 1, 2014

By J. Z. Yuan

The purity of rectorite is only about 70% after preliminary mineral processing, which has limited its application. In this research, different pulp densities, different dispersants, different dispersing methods and purification flow sheets were used to purify rectorite. The results show that it is possible to obtain high pure rectorite clay with the purity above 90% and concentrate yield between 24.9% and 29.4% by the purification process with composite dispersants, stirred mill grinding dispersion and so on. The obtained high pure rectorite can be further used in rectorite nanocomposites.

Jan 1, 2014

By G. S. Zeng

The mechanism of cobalt removal from zinc sulfate solution with zinc dust is still not understood well. The experiments in this paper were carried out by zinc dust from zinc sulfate solution with high cobalt to study the mechanism of cobalt removal. Residues after the cementation process from zinc sulfate solution were analyzed by SEM and Electron probe micro-analysis (EPMA) to investigate the morphology of products. Different temperature will cause different structure of particles. With the increasement of temperature, cobalt removal can be improved considerably. The mechanism of cobalt removal by zinc dust is thought as follows: firstly, the cobalt ion forms Co-Zn alloy with zinc particles, and the reaction reaches equilibrium quickly; secondly, the Co-Zn alloy continues reacting to Co with Co2+, which is the control procedure of the whole reaction.

Jan 1, 2014

By B. Peszko

There are many measuring methods of grain sizes. Taking into consideration the fine grained materials, one of the existing techniques is measurement of grain size by the Coulter Counter method. This method is based on the measuring the changes of electrolyte resistance during grains flow through the calibrated diaphragm. This method is relatively simple to apply, significantly precise and the results are given quickly. However, the problem is that the measurements are limited by the size of diaphragm. Because of this situation, measurement of grain sizes by this instrument is possible in range from 10% to 60% of diaphragm diameter. This limitation causes the necessity of re-placing diaphragm in case when the measured grain size distribution concerns the material of large size range. In purpose of determining the total grain size distribution, the synthesis of some partial distributions, given for individual grain fractions (with different diaphragm applied), is required. This is possible to do by calculating the mixture of distribution functions, created from the partial distribution functions. Two materials were selected to research: quartzite and glass, which were comminuted in cylindrical crusher, additionally grinded, screened and then analyzed by Coulter Counter for five grain fractions: 0~125 µm, 0~80 µm, 0~56 µm, 0~40 µm and 0~25 µm. For each fraction, five different partial distributions were given (in case of glass ? four), which are the basis for the creating the joined grain composition. To assure the adequacy of the given distribution function, it is necessary to obtain the possibly best approximation of partial distributions. In the work, both the parametric and non-parametric methods of density approximations of investigated random variable. For each grain fraction the approximations were done by three traditional distribution functions: RRB, GSA and log-norm distribution functions and three methods of non-parametric approximation: Gauss kernel, Epanechnikov kernel and orthogonal Fourier series method. As the criterion of the certain distribution function acceptance, the value of the residual deviation was assumed, given by the formulae:

Jan 1, 2014