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By J. Herbst, I. Heber-Panu, Dan G. Cojocariu, E. Ene-Danalache, G. Poling

A general model which represents not only batch but also continuous flotation in multi-cell machines is presented. This model considers both the size distribution and the distribution of flotabilities in a size class for valuable mineral particles in the pulp. The general expression is shown to reduce to several previously reported models. In addition, a number of new models are found to arise for different specific assumptions concerning the flotability distribution and/or the size distribution of the particles undergoing flotation. Several of these new models are tested experimentally using laboratory-scale batch and pilot-scale continuous data. Model parameters are estimated from this data and a comparison of calculated recoveries and those experimentally determined are presented. Finally, the influence of the feed rate, aeration rate and valuable mineral content of the feed on the recovery is examined.

Jan 1, 1976

By A. G. Royle

The reasons for constructing a model of an orebody for which both the parent population and the sampling distribution are known are discussed and details of the model are given. A brief summary of Matheron's theory of regionalized variables is included, with particular reference to his kriging estimators and their variances. Using the model, a comparison is then made between these kriging estimators and other commonly-used estimators. These are compared to find the one with the minimum variance, and an analysis is included to investigate whether any of the estimators is biased. Also included is a comparison between theoretical and experimental variances for some of the estimators.

Jan 1, 2014

By D. N. Madge

A survey is given of production scheduling systems currently in use or contemplated for future use in Cominco operations. Emphasis in this presentation is made on the chain of decisions that are made with the help of mathematical programming models. Some attention is devoted to the structural properties of the models.

Jan 1, 1966

By R. Venkataramani

Crushed stone operations have become more difficult due to rising costs of labor and equipment, with the consequent reduction in profit margins. Competition from other operators and the encroachment of vertically integrated companies place severe restrictions on available markets. The seasonal demand for stone precludes any definite production level, and changes in productivity have to be accommodated. These influence profitability, and therefore production plans must be continuously adjusted to keep pace with rapidly changing conditions. The basic problem of stone companies seeking to maximize profits is one of balancing stone production costs, costs of inventory holding and stock-piling, marketing costs and any other charges accruing over time against the selling price of stone. Because of high peak demands, slack period production must be stored and then distributed to supplement peak period production. For a multi-plant operation, there is a need to evaluate at the same time the interaction between the plants and the various marketing sectors. In a multi-plant operation with different production capacities and production costs, stone producers are interested in the answers to the following questions: 1. How much stone should be produced at each plant in each time period? 2. What should be the optimum stone shipment from each plant to each market? 3. What are the effects on net returns and shipping plans, if production capacities are altered at the plants? These questions have many different answers that, will satisfy demands. From the point of production scheduling, the minimum production cost may be the objective. From the inventory standpoint, the best solution will be to synchronize production with sales. However, when the transportation of stone is also considered, the desired end is to achieve that delivered cost which maximizes the company f s net return. Besides, only an extremely naive analysis ,wouId attempt to

Jan 1, 1971

By C. B. Manula, R. Venkataramani, T. V. Falkie

A linear programming model has been developed to solve a production planning problem for a multi-plant stone operation serving several customers. The factors considered are production capacities, inventory restrictions, demand requirements, and available stone reserves. The objective function to be optimized is the total annual net returns from the quarry operations. For the purposes of model testing, annual stone sales from each plant were divided monthwise, productwise, and assigned to a market. The concept of assigning demands to market sectors and not to any particular plant creates the self-correcting interplay between the plants and the markets. Along with the inventory restrictions, and ending inventory requirements, this interaction enables the model to minimize the external diseconomics. The model was tested against an operating prototype and a number of alternative problems were also considered. Finally, the basic linear programming model has been enlarged by the addition of another dimension and the problem has been derived in a decomposition format.

Jan 1, 1972

The mathematical models of semivariograms can be classified according to whether or not they correspond to situations where the second order stationarity conditions are satisfied (§ 3.3.4). If second order stationarity exists, the a priori variance s2 of x (z) also exists and is finite. For large distances h such that x (z) and x (z + h) are not correlated, the semi- variogram ? (h) will reach a value equal to s2 [this is a consequence of (3.27) above]. This limiting value is called the sill C of the semivariogram, and the distance at which ? (h) reaches the sill is called the range. The range of the semi- variogram obtained from a given type of sample corresponds to the distance of influence of these samples. If only the intrinsic hypothesis is satisfied, the sample variance is an increasing function of the size of the area or volume in which the samples are located, and the semivariogram ? (h) will be a continuously increasing function of h. Remark: It will be shown in Chapter 7 that the variance s2 of x (z) in the ore body O is equal to the average value of ? (h) in O. If ? (h) is a non-decreasing function with sill C, the average value of ? (h) in O will be less than C. s2 and C will be equal only if for some values of h in O, ? (h) exceeds C. In practice, if the range of the semivariogram is small with respect to the dimensions of the ore body, the difference between C and s2 will be negligible.

Jan 1, 1978

By Salim Akhtar

The operation of a mineral processing plant with a complex flowsheet is represented by a mathematical operator. Such a representation permits a quantitative evaluation and monitoring of the separation processes in different sections of a processing plant, and gives an overall separation efficiency of the plant.

Jan 1, 1973

By Salim Akhtar

The operation of a mineral processing plant with a complex flowsheet is represented by a mathematical operator. Such a representation permits a quantitative evaluation and monitoring of the separation processes in different sections of a processing plant, and gives an overall separation efficiency of the plant.

Jan 1, 1975

By I. Koliabina

In this paper, the results of mathematical modeling of casing pipes decontamination from radio-active new mineral forms directly in boreholes using hydrochloric and hydro-silicic-fluoric acids are presented. Modeling for highly and low radioactive coatings on the walls of casing pipes of Anastasievka oil-and-gas deposit was performed using GEMS software, based on the method of Gibbs energy minimization. Calculations have shown that thorianite and goethite, that are present in low-active crusts, detected in casing pipes uplifted from comparatively low depths, are well dissolved in hydrochloric acid, but radiobarite is almost insoluble. For the same acid concentrations, almost all radiobarite of high-active coatings detected in pipes uplifted from considerable depth is dissolved. To evaluate the possibility of decontamination of pipes from radiobarite of low-active coatings, modeling of crust dissolution under the action of H2SiF6 has been performed. The results have shown that for acid concentrations of more than 13 % almost all Ra and Ba are dissolved while all iron and thorium remains in goethite-thorianite solid solution. For the same acid concentration, almost all radiobarite from high-active coatings is dissolved. Therefore, no reagent allows decontamination of pipes from coatings of both types. It seems that successive or simultaneous use of both reagents (HCl and H2SiF6) in certain ratio is required. To clarify this issue, further experimental and model investigations are necessary. The issue of decontamination of casing pipes from galena that is present in high-active crusts also needs to be clarified.

Jan 1, 2005

By J. A. Herbst, D. W. Fuerstenau

An analysis of the kinetics of dolomite grinding in a dry ball mill reveals that the following approximations apply for the parameters of the batch grinding model: (1) the size discretized selection functions are proportional to the specific power input to the mill, and (2) the breakage functions can be taken as invariant. These approximations are shown to provide the basis for accurate predictions of product size distributions obtained for several mill speed, ball load, and particle load combinations within the normal ball mill operating range. The engineering implications of these findings are discussed.

Jan 1, 1974

By J. A. Herbst

Until the past decade, the energy-size reduction relationship was the dominant form of mathematical model used in the description of tumbling mill grinding processes (1,2). Typically, in these models some single measure of product fineness (e.g. the 80% passing size, the size modules, the specific surface area) is chosen as the dependent variable and the energy input per unit mass of material being ground acts as the independent variable. In some instances these models have been useful for the correlation of experimental data, but, invariably, they have been found to be inadequate for meaningful process simulation. During the last decade considerable advances have been made in the development of phenomenological grinding models derived from population balance considerations (3-7). In these models a set of dependent variables is employed, corresponding to the set of mass fractions which constitute the product size distribution, and energy input is abandoned in favor of "time in the mill" as the independent variable. The phenomenological models have been shown to provide accurate simulations of the entire size distribution of ground products produced in batch (8-10), locked cycle(11)and steady and unsteady state, open and closed circuit continuous grinding(12,13).

Jan 1, 1973

By Shoji Taniguchi, Lifeng Zhang, Kaike Cai

"In the present paper, based on the two-phase (Eulerian-Eulerian) model, the 3D fluid flows in gas-stirred systems, i.e., air-stirred water vessel and argon-stirred liquid steel ladle, are simulated. In the Eulerian-Eulerian two-phase model, gas and liquid are considered to be two different continuous fields. The phases are assumed to share space in proportion to their volume fractions. The exchange between the phases is represented by source terms in conversation equations. Turbulence simulated by the k- e model is assumed to be the property of the liquid phase. The following effects of mathematical treatments and operational factors on the fluid flow are discussed, such as interphase drag force, turbulent model, the size of bubble, and gas injection mode.Some interesting results are derived. Except the interphase drag force, the interphase lift force should be taken into accounted in order to exactly simulate the fluid flow in gas-stirred systems. Suitable turbulent model and drag coefficient have effect on the mathematical simulation. Injecting small bubbles can realize a well mixing flow condition. The distance between the two gas-injection nozzles has effect on the fluid flow."

Jan 1, 2000

By Saul Garcia Hernandez

The research work objective is to study a new ladle shroud internal design to reduce the slag emulsification in the tundish during ladle transient periods. A 3D mathematical model of the tundish was created to solve the fundamental equations, a turbulence model, and a multiphase model. The results show that using a conventional ladle shroud, the fluid is delivered with an excessive amount of kinetic energy being dissipated inside the tundish bath, generating strong mixing flow patterns and entrapping a massive amount of slag. In the other hand, the proposed ladle shroud dissipates the kinetic energy before the fluid enters the tundish promoting less intense mixing patterns; thus, the amount of slag emulsification is reduced significantly. Consequently, if the internal design of the ladle shroud is able to dissipate the kinetic energy before the fluid enters the tundish, it shall be possible to reduce considerable the slag emulsification and the slag opening area.

Aug 17, 2017

By A S. Kurilko

For mines located in the cryolitic zone, thermal conditions are an important factor for all kinds of mining works, especially for ventilation of workings. In the present paper, some models of mine air heat transfer considering air distribution in a net of mine workings are developed. A combined solution is suggested for heat transfer and air distribution problems in mine workings and other underground structures located in the cryolitic zone. A software package is also presented for calculating temperature conditions in an underground refrigeration plant and for optimisation of ventilation conditions. This software uses a special 3D mathematical model to describe heat transfer in a freezing chamber between surrounding rock massif and stored product as well as between atmospheric air and the daylight surface of rock massif. A problem requiring strengthening of a backfill material and its thermal interaction with surrounding rocks during underground deposit mining by horizontal layers in conditions of the cryolitic zone is also discussed. The developed program takes into account cement consumption, geometrical dimensions of workings filled and a sequence of deposit mining.

Jan 1, 2005

By S. Sarac

In this study, graphical and mathematical methods used to determine operating points in single or multiple fan ventilation systems are summarized. The problem is mathematically formulized for parallel connected fans in which analytical solutions are not possible. For n number of parallel connected fans, a non-linear equation system that consists of n number of non-linear equation sis derived. The Newton-Raphson method, which is a numerical analysis method, is adapted to solve this non-linear equation system and the solution steps are given as an algorithm. A computer program based on this algorithm is written and its applicability is proved.

Jan 1, 2000

By Boris K. Krstev

INTRODUCTION The flotation process, in the real industrial complexes, is characterized by the existence of the significant number of the close reciprocal interrelationship between the input and output Darameters. Traditional complexity is appeared with regard to absence of the suitable a priori information for the physico-chemical characteristics of the flotation process. It’s led to needful of taking of consideration about the undefinition which have got the place in the real conditions and application of the statistical methods. The identification of the flotation process through statistical methods is conception of the more-stages procedure, including the collection and previous analysis, the choice of the model structure and valuation method, the estimation of the model un- known parameters and the results interpretation. The essential concept for the subject of this paper belongs to the empirical models ("black-box") and deals with optimizing in constant industrial plant process. The offered method envisages the application of process operation data in order to improve the work process or constant accomplishment of better results for the target function in relation to the former achievement. The optimum process solution isn't afixed value under the influence of various factors, but "a moving point" along the response surface and deviates from the scheme solutions. The evolutive planning method constantly "investigates and approaches" the optimum solution for the selected target function. As a matter of fact the expert operates the plant without interrupting the normal work process. The essential concept of method is to attain changes in the standard conditions, but so limited that they will not effect either production or the product. This method is based on statistical concepts and availability of following the risk of error. The possible influental factors for the flotation process (m=l, 2, 3) is following: the metal content in the feed ore; production level (t/h); grinding fineness in the flotation pulp; pH-values; the consumption of collectors and modifiers The possible target functions are: the recovery of plant capacity, metal recovery in the concentration process and concentration quality. The application of the evolutive operativity method in the chalcopyrite flotation- Bucim concentrator, combines the statistical data and the operator's experience in the interpretation and making a decision which adds up to the precise automatic performance of the process. The advantages of this method in relation to the existing procedure for the flotation process are the following: • The method is unexpensive and losses are reduced to the time necessary to collect the data and record them in the computer. This method provides a constant process control and a quality decision which is better than those brought by an ex- pert with great experience ("hit-and-miss method"). • The method also provides incidental data about the effects. • The method should not be considered as a procedure for solving an incidental problem ("crash program"). • The obtained results must be interpreted by an appropriate expert in order to be considered instructive. MATHEMATICAL STATISTICS Taking into consideration the complexity of the enrichment processes it's necessary the application of "black-box" and mathematical modelling. The presence of the indefinite elements compel us towards the both theory of the probability and mathematical statistics. The mathematical statistics by the mineral processing investigation of the valuable raw materials in the last years has had greater signifance and widely spread application. The application of the mathematical statistics methods for the technological pro- cess analysis and the construction of the mathematical models is bound up by the task putting of the automatic control system de- sign (Fisher R. A. 1941, 1949). The statistical analysis in the enrichment processes is applied formerly, especially in the sampling estimation. Contemporary, in the last years is imposed the tendency of mathematical statistics application for the obtaining of quantity characteristics and quantity valuation from the individual influental factors. Above mentioned tendency in the significant level is based by the mathematical statistics development as a mathematical discipline close connected with the technical cybernetics general joining to the investigated object as a "black-box". Another essential factor in the mathematical statistics development is appeared the created mathematical theory of experiments as a new, to some extent, autonomous statistical domain. The theory of experiments has demonstrated exclusively importance for the investigation of the complex more-factorial processes in the chemistry, mineral processing, metallurgy etc. The investigation by the enrichment processes of the valuable raw materials and their management is possible by the statistical method application. In the same time, it's possible to use the wide-spread traditional methods of the statistical analysis, such as the methods of the dispersion, correlation, regression etc. Also, it may be mentioned above all the methods of the statistical planning of the extreme experiments. The more perspective are representatived some cybernetic methods which are based by the probability as an approach in the process analysis, for example, the identify of the ways and forms, the graph application as an additional instrument in the correlation analysis, the dynamic programming etc. The mathematical statistics application in the laboratory investigation scale is connected above all by the experimented material analysis and the compact representation of the obtained results. The mathematical statistics methods are more varrying and may be applied in the investigations of the wide circle of problems

Jan 1, 1996

By D. O. Ehrenburg

DURING the earlier period of electrical prospecting, the search for orebodies was by far the most important application of this method of geophysical prospecting. In the past few years, however, increasing attention has been paid to the possibilities of this method in the investigation of subsurface structures. Not only have fault zones been located, but attempts have been made to determine the depth to certain strata and thus contouring on some bed which proves to be a good "electrical marker." Many workers have published the results of field tests showing how their particular interpretations have agreed with the actual field conditions, but very few have stated their method of interpretation. Weaver, Hummel and Stefanesco and Schlumberger1 have given a more theoretical discussion of the resistivity methods. Hummel and Stefanesco and Schlumberger have published mathematical solutions for the flow of electricity in a medium involving four layers. In this paper the authors present a solution for the flow of electricity in a medium of any number of horizontal strata. By means of this solution, curves showing the relationship between the apparent resistivity and the electrode spacing can be computed for ideal cases.

Jan 1, 1931

By E. Barnea

This paper simplifies and provides a strict mathematical proof for an equation developed by R. J. Woody for the c alculation of solute recoveries in countercurrent decantation. The equation offers a convenient method for the optimization of flowsheets involving a simple countercurrent-decantation circuit.

Jan 1, 1978

By Paul L. Russell

Numbers, arithmetic, mathematics. Where did it all come from? Who needs it? We all do, but many school children would not agree. Engineers, physicists, accountants, tax collectors, and most of the general public are well aware that modern society in large part is founded on mathematics and could not exist without it. Mathematics developed over thousands of years as part of the everyday life of man. No one area, tribe, or race can claim discovery. In fact, from the records that do exist, it is evident that many facets of mathematics were discovered independently of others who made similar discoveries. While proof may be lacking, the earliest concepts of mathematics may be some 250,000 to 300,000 years old. The first use of numbers as written records is unknown. However, a bone of a young wolf deeply cut with 55 notches-arranged in two rows, one of 25 notches, the other of 30-was found in Czechoslovakia. Dated to be about 30,000 years old, it is perhaps an early record of some kind of counting system. It is assumed that mathematics was developed as an answer to man's practical needs, but anthropological studies suggest that the art of counting may have developed in connection with the needs of primitive religious ceremonial rites. Fingers probably were first used to designate sets of from one to five objects. Using both hands, up to 10 items could be indicated. As civilization developed, so did the need for more flexible systems to record items of trade, measure land, and figure taxes. Aristotle noted that the development of the decimal system comes from the fact that most people are born with 10 fingers and 10 toes. In spite of this obvious fact many early civilizations used binary, ternary, and quinary systems while vigesimal systems (with 20 as a base) and even sexagesimal systems (with 60 as a base) were used by others. A form of writing was in use in both the Mesopotamian and Nile valleys as early as 3000 to 4500 B.C. In Mesopotamia, wedge-shaped marks were impressed upon soft clay tablets that were hardened in ovens or by the sun. Many thousands of such tablets have survived. When the tablets were finally deciphered, some were found to be records of items shipped in trade, inventories, tax accounts, and mathematics exercises for teaching. Mesopotamia is also the source of an interesting board game dating back to about 2500 B.C. Its buttonlike men and triangular dice were apparently used with squares of shell, inlaid with lapis lazuli and red sandstone. Unfortunately, rules of play were not found. In Egypt, record preservation was somewhat better. Discovery of the Rosetta Stone in 1799 eventually led to deciphering the Egyptian hieroglyphics that included a numbering system at least 5,000 years old. These records show a competence in counting and numbering. A calendar of good accuracy may have been in existence as early as 4,200 years ago. Mathematics, as practiced today, developed when the Greeks added logic and proof to arithme-

Jan 12, 1982

By H. G. Wang

The production process of a coal mine can often be interrupted by all kinds of events scheduled or unscheduled and thus often needs gas drainage for a temporary or long-term sealed area. Gas drainage has a relatively high risk, therefore attention must be paid to its design to keep the process safe. Based on the conservation laws of mass and constituents, and under the conditions of the allowable gas concentration limit in the return air route, the relationship between gas concentration and gas drainage flow rate as a function of time during the drainage process is discussed in detail to build a mathematical model of gas drainage from a sealed area. With respect to the gas drainage process, two different conditions need to be considered: one with a methane free fresh air supply, the other with a low gas concentration. This paper presents the derived mathematical models for these situations and then shows that they can be incorporated into one model with some modification. This combined model can be employed to determine the optimum gas drainage flow rate with the shortest drainage time. The application is coded using the C# programming language within the .NET framework. Finally, the application program developed by the author is used to validate the mathematical model through an example of gas drainage with varying rate.

Aug 1, 2013