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By M. Hooman, W. M. Marx, R. C. W. Webber-Youngman

This paper discusses a study that investigated different underground bulk air heat exchanger (>100 m3/s) design criteria. A literature review found that no single document exists covering all design criteria for different heat exchangers, and therefore the need was identified to generate a guideline with decision analyser steps to arrive at a technical specification. The study investigated the factors influencing heat exchanger designs (spray chambers, towers, and indirect-contact heat exchangers) and the technical requirements for each. The decision analysers can be used to generate optimized, user-friendly fit-for-purpose designs for bulk air heat exchangers (air cooler and heat rejection). The study was tested against a constructed air cooler and heat rejection unit at a copper mine1. It was concluded that the decision analysers were used successfully. This tool (decision analysers) can be used by engineers for the efficient and cost-effectively design of heat exchangers.

Jan 1, 2015

By K. Sugawara, H. -S. Jeong, Y. Ishiguro, J. Otani, Y. Obara, A. Hirata

A series of measurements of induced stress by the Compact Conical-ended Borehole Overcoring (CCBO) technique was performed within rock mass around a tunnel excavated in a region that an underground powerhouse is constructed under high initial rock stress. In the measurements, a phenomenon of core disking was observed during overcoring. The selecting method of strain data uninfluenced by the core disking was proposed, using X-ray CT scanner. Then, the state of induced stress determined from the selected strain data was discussed, comparing with the state of initial stress. Eine Serie von Messungen von induzierter Belastung mit kompaktem konisch-beendetem Überbohrverfahren (CCBO) wurde in Felsmassen um einen Tunnel ausgeführt, der in einer Region gebohrt wurde wo ein unterirdisches Kraftwert unter hoher anfänglicher Felsbelastung gebaut wird. Bei den Messungen wurde beim Überbohrverfahren ein Phänomen des Core Disking beobachtet. Die gewählte Methode der Belastungsdaten, unbeeinflusst durch Core Disking wurde vorgeschlagen, unter Verwendung von Röntgen-CT-Scanner. Dann wurde der Zustand der induzierten Belastung aus den gewählten Belastungsdaten diskutiert, unter Vergleich mit der ursprünglichen Belastung. Une série de mesures de contrainte induite effectuée avec la technique de forage de sur-carottage compacte à embout conique a été exécutée dans la masse rocheuse autour d'un tunnel excavé dans une région de construction d'une centrale électrique souterraine sous haute contrainte de roche initiale. Au cours des mesures, un phénomène de discage de carotte a été observé pendant le sur-carottage. La méthode de sélection des données de contrainte non influencée par le discage de carotte a été proposée, en utilisant un scanneur de tomographie calculée aux rayons X. Il s'ensuit que l'état de contrainte induite provoquée et déterminé à partir des données de la contrainte sélectionnées a été traitée, en effectuant une comparaison avec l'état de contrainte initiale.

Jan 1, 2003

In a gravity separation device, particle shape, size and density all play a role. The combination of these determines where each individual particle reports to in the viscous fluid, where particle crowding (solids to water ratio) also plays a role. To understand the performance of gravity separation devices in the heavy mineral industry, these particle characteristics need to be measured. There are various challenges in analysing particle density and particle size simultaneously for the purpose of quantifying gravity separator performance, not to mention particle shape. These analytical challenges include the high cost of high-density sink-float fractionation, toxicity of high-density sink-float media, inability of sink-float media to fractionate at densities greater than 4.0 g/cm3, and the time-intensive nature of these fractionations. The use of the detailed particle-by-particle output from Qemscan® particle mineral analyses (PMA) as a fast and cost-effective alternative is evaluated. The size and density outputs from the Qemscan® were employed to characterize the performance of a heavy mineral spiral concentrator as an example. Critical analytical requirements are to be addressed before the Qemscan® output data can be utilized. Keywords Gravity separator, performance, heavy minerals, Qemscan® particle mineral analysis

Jan 1, 2011

By C. Wade, A. Swart, M. Dworzanowski

AngloGold Ashanti operate a pyrite flotation plant at their East Gold Plant, Acid Plant and Flotation Plant (EGAF) metallurgical complex within their Vaal River operations in the North West province in South Africa. The flotation plant feed is reclaimed pyrite rich tailings and the concentrate produced is used as the source of sulphur for the EGAF acid plant. The sulphuric acid produced is in turn used for the recovery of uranium at the Vaal River South uranium plant. The original flotation plant built in 1960 was based on the use of many small volume flotation cells and was becoming very expensive to operate and its age severely limited further operation. A decision was taken in 2005 to replace this plant with a new one based on fewer, larger volume flotation cells. This paper describes the benefits of using larger volume flotation cells in this new flotation plant when compared to the smaller volume flotation cells in the old plant. The benefits of the new plant include a smaller plant footprint, fewer flotation cells, fewer pumps, no distribution boxes, simpler piping, a greater flotation volume and a reduction in installed power. The saving in capital cost could not be quantified because of the age of the old plant. The savings in operating cost will arise from fewer operators, less maintenance and reduced power consumption. The new plant has not been in operation long enough to quantify the benefits in operating cost and metallurgical performance. However, recovery improvements can be expected given the greater flotation volume (increased residence time) and the installation of larger, more efficient flotation cells.

Jan 1, 2007

By D. Malovichko

Seismic events and pillar failures observed in two platinum mines in the Rustenburg area, South Africa, were investigated. We studied the sources of approximately 300 largest seismic events recorded in 2009. Moment tensors of these events were estimated from amplitudes and polarities of P- and S-waves. The inverted mechanisms contain isotropic implosive components in many instances. Analyses of uncertainty of the moment tensor solutions confirmed the significance of the implosive component in general. Several documented cases of pillar failure were considered in detail. For each of these a seismic moment tensor was calculated from an elastic stress model assuming an instant removal of a failed pillar. The geometries and magnitudes of the theoretical seismic moment tensors are in agreement with the observed (i.e. inverted from seismic data). The results indicate that either pillar failure or fracturing in the proximity of the reef (within a fraction of the dominant wavelength of seismic waves) are responsible for a significant portion of the large events recorded in these mines.

Jan 1, 2012

By A. D. Storrie

An underground wet shotcrete trial was completed last year to determine the effectiveness of wet shotcrete as an alternative support mechanism to dry shotcrete in the kimberlite environment. The wet shotcrete was manufactured at the underground batch plants, transported to site using an agiecar and transferred to the shotcrete pump. The wet shotcrete is pumped to the nozzle where compressed air at 7 bar and an accelerator, Meyco SA 160, is added and enables the shotcrete to be placed successfully onto the tunnel wall. No tunnel guard was previously placed on the sidewall and hangingwall. Geotechnical testing indicates that the Grace fibre performed extremely well with the energy absorption tests at 7 kg/m3and 8kg/m3where both achieved over the mine requirement of 750joules. The Grace fibre and wet shotcrete mix also achieved a compressive strength in excess of 55 MPa, well above the 45 MPa requirement for Premier Mine. The recommended wet shotcrete mix design is therefore based upon: [Material/ CriteriaQuantity/m3Units ?Lafarge 42.5 Duratech500kg ?Condensed silica fume40kg ?Rayton Sand1.558Ton ?Delvocrete stabilizer 103.6litres ?Glenium 27CH5.4litres ?Meyco TCC 7355litres ?Water230litres ?Meyco SA16025litres ?Fibre8kg ?Cement/Water Ratio2.35? ?Water/Cement Ratio0.43? ?Flow615 mm] If the wet shotcrete system is implemented there is a potential for Premier Mine to make major savings. In South Africa there are two suitable wet shotcrete machines available for the mining industry, the Spreymec, manufactured by Tamrock and the Fermel/Meyco wet shotcrete machine. The Spreymec is a fully imported machine. The Fermel component is manufactured in South Africa and the Meyco shotcrete components are imported and supported by MBT (Master Builder Technologists) in South Africa. The Fermel/Meyco machine is assembled in South Africa. Premier Mine has allocated some R2, 100,000 towards the implementation of the wet shotcrete system. If implementation is successful there should be an internal rate of return gain of 76% with regard to this project.

Jan 1, 2001

By I. Bhattacherya

Over the last three decades, the mining industry has been moving towards underground mechanized mining methods and the number of load haul dump (LHD) vehicles utilized has increased. The growth of mechanization and automation has benefited both workers and mining companies. However, due to the design constraints with LHD vehicles and the limitations of the operating environment, restrictions to operator visibility has contributed to a number of accidents, including fatal injuries. Past researchers have used the light filament technique to collect obscuration zones around mobile equipment and have produced 2D visibility charts (shadow diagrams) of this information. The light filament method involves manually collecting visibility data, which results in errors and is time consuming, requiring around three hours to complete. This research utilizes a MENSI GS100 laser scanner along with 3D modelling software (3Dipsos) to collect and process operator visibility profiles for underground mobile equipment. Results from this test work indicate that a laser scanner can be successfully used to rapidly collect this data and utilize this information for improved mobile equipment design from a visibility perspective. Nomenclature FERIC Forest Engineering Research Institute of Canada ISO International Organization for Standardization JACK Software for digital human modelling and ergonomics LHD Load haul dump vehicle LOS Line of sight MASHA Mines and Aggregates Safety and Health Association (based in Ontario, Canada) PVD Polar visibility diagram SAMMIE3D human modelling computer aided ergonomics design system WSIB Workplace Safety and Insurance Board

Jan 1, 2006

By T. Nishi, D. Teguri, K. Saitoh

In recent years the requirement for high carbon ferro-manganese (HCFeMn), which is the key product of our company, has increased due to bullish demand for steel products. In our Tokushima works, we have 2 closed types of electric furnaces (1F and 2F), with which we are manufacturing HCFeMn. In order to meet the increase in demand, we made various modifications to these furnaces and we tried to increase the furnace operating loads. Especially for 1F, we deepened the furnace. In this paper, we report the results of this modification. The operational situations of the above-mentioned two furnaces are different because they have different individual profiles. As for 1F prior to modification, the depth of the furnace was one meter shallower than2F, because 1F was designed for the manufacturing of SiMn. On the other hand, our Tokushima works specialized in HCFeMn 14 years ago. However, the profile of 1F was not optimized for HCFeMn. In comparison to 2F, 1F was more difficult to stabilize due to the smaller tolerance for the adjustment of the electrode length, the metal / slag level, and the coke-mixing ratio. We deepened 1F by 770mm by disposing of the furnace rolling mechanism. We consequently achieved a 10% increase in production capacity. This meant the operating load was enhanced from 24.5MW to 27MW, the furnace became more stable and the tapping speed of both metal and slag increased, all which made the tapping easier than before.

Jan 1, 2007

By R. Abela

Heavy mineral sands are usually beneficiated in a spiral concentrator circuit prior to separation of heavy minerals into streams of zircon, rutile and ilmenite. The performance of the spiral concentrators is heavily affected by the presence of slimes in the rougher feed. The paper quantifies the effect of slimes content and discusses the beneficial effect of spiral feed de-sliming. The net effect of inefficient slimes removal can cause significant recovery losses of the heavy minerals present in the feed.

Jan 1, 2003

By M. Dhainaut

This paper presents computations of the electric field inside a furnace. We first look at the effect of contact between two coke particles from an electrical point of view. We give values for the amount of contact resistance in the total resistance of the system formed by the two coke particles. We show that the effect of contact resistance decreases rapidly with increasing contact area. We then look at the complete electric field inside a full scale furnace. The aim is to evaluate the amount of current that short-cuts the metal bath for different vertical positions of the electrodes. We show that electrodes high above the metal bath are undesirable since they offer a larger surface of conductive charge materials where current can flow through. On the other hand, deeper electrodes force the current to go the whole way down to the metal bath. Ohmic heating along this path ensures a high temperature of coke and slag that are beneficial for metal recovery.

Jan 1, 2004

Rockbursts continue to be a scourge in the mining industry, being responsible for accidents and damage to mining excavations. Although the problem has been present for more than a century, and although much research has been carried out, a solution is still elusive. Determination of, firstly, the demand on the support system imposed in the rockburst and, secondly, the capacity of the support system, cannot be carried out with any confidence, and therefore rockburst support cannot be designed using a conventional design approach. In contrast with the conventional approach, observations of rockburst damage in a mine have revealed a possible alternative approach to rockburst support ? sacrificial support. It was observed, in rockburst events, that a support system consisting of concrete panels restrained by grouted cables was destroyed, and the concrete panels were ejected. However, the rock behind the panels remained apparently undamaged and in place. This behaviour reignited the concept of sacrificial support, conceived more than 20 years ago, and which is described in this paper. The remedial solution implemented on the mine, involving cables wrapped over the panels and retained by grouting into boreholes, has been subjected to rockburst loading and has confirmed the validity of the sacrificial support concept. The concept of sacrificial support may be controversial, but is deliberately presented here with the aim of generating discussions and contributions, and with the ultimate aim of improving safety and reducing rockburst damage in mines. Keywords rockburst, seismicity, seismic waves, rock support, support design.

Jul 1, 2013

By K. Saito, Y. Miyauchi, T. Nishi, Y. Kizu

At Tokushima Plant of Nippon Denko Co., Ltd. high-carbon ferromanganese is produced by the electric furnace method using lump manganese ores and sintered manganese ores manufactured from fine manganese ores. However, the distribution of manganese ores is significantly uneven on earth, and their characteristics considerably vary by ore type. It is known from experience that the production of high-carbon ferromanganese by the electric furnace method depends greatly on material ores. Therefore, to maintain stable operation in the electric furnace-smelting field, it is important to identify the characteristics of each type of material ores in advance. Thus, we evaluated the high-temperature electric characteristics of manganese ores using the electrical conductivity measuring system developed by ourselves and provided the standards for determining ores and blending ratio necessary for stable operation of the electric furnace based on the evaluation results. This time we worked for optimisation of the high-temperature electric characteristics of sintered manganese ores through modification of the characteristics because the blending ratio of sintered manganese ores in the electric furnace is high. The electrical conductivity of sintered ores having various chemical components in a high-temperature region (up to 1 550°C) was measured using our electrical conductivity measuring system and high-temperature electric characteristics evaluating method reported at INFACON 9. As the result of the measurement, we revealed that the high-temperature electric characteristics of the sintered ores depend on their chemical contents and the sintered ores can be modified to those having excellent high-temperature electric characteristics by controlling their chemical contents, so that the resources can be effectively used.

Jan 1, 2004

By B. Salt, R. Sullivan, E. Roos, M. de Vries, K. McKenna, L. R. Nelson, M. J. N. Uys, P. Jacobs, E. Munnik, P. Lewarne

A review is presented of the furnace cooling system design and operating developments since the initial furnace start up in January 2001 of the Chambishi DC-arc furnace to produce ferrocobalt alloy. The furnace operating conditions present a number of challenges to the furnace designers and operators in particular to provide a sustainable lining that can withstand the highly corrosive siliceous slag (roughly 50% SiO2). This is exacerbated by the dual requirement to operate this slag in a relatively highly superheated state (?T ? 400°C when tapped at 1550°C) to ensure that the ferroalloy can be tapped (estimated 1370°C alloy liquidus temperature) prior to atomisation. The core design challenge occurs at the alloy-slag interface, where a single refractory type cannot simultaneously meet the dual requirement of both adequate alloy and slag corrosion compatibility. Progression by the fourth furnace campaign to involve a retrofit of the bath sidewall to Hatch high-intensity, water-cooled, copper Waffle coolers and tapblocks is described. This involves operation of the entire slag bath, the slag-alloy interface and a portion of the alloy within a more chemically compatible refractory/slag (and possibly at times even an alloy) freeze-lining. Due to the high furnace power density employed (up to 500 kW/m2 hearth area) coupled with the large slag superheat needed, the resulting imposed peak bath sidewall heat fluxes are substantial, especially in the alloy (typically averaging 170 kW/m2), and on occasion have exceeded 1000 kW/m2. The performance of the Hatch Copper Waffle coolers under such aggressive process pyrometallurgical conditions is described. Aspects of the design and especially operating strategies to effect sustained production at up to 38 MW power input are also discussed.

Jan 1, 2004

Poorly designed sampling protocols generally result in high project risk by artificially increasing grade variability (ie the nugget effect). Sample collection, preparation and assay protocols that suit the ore type, together with QA/QC systems, will reduce this risk. All reporting codes (eg JORC, 2004) require the Competent Person to consider the quality and implication of sampling and assaying programs. High sampling errors beget higher project uncertainty; despite this fact, sampling does not receive the attention it deserves. The case studies presented show that many sampling issues frequently go unrecognised or unaddressed, resulting in poor project decisions during exploration through to feasibility and production. The overall conclusion is that correct sampling is critical to mining project success and that there is often a need for its optimisation.

Jan 1, 2007

Assuming that a gold ore has been effectively ground to ensure maximum economic liberation of gold, the circuits that will have the most effect on the successful operation of a gold plant will be that of the leaching and carbon-in-pulp circuit (CIP). The reagent and utilities operating costs associated with leaching, adsorption, elution and regeneration would typically make up some 15% of total operating cost, whilst the capital costs associated with these areas is about 16% of the total. Although not largest (capital and operating costs associated with comminution are generally dominant) these items represent a significant proportion of capital and operating cost. Much more importantly, these plant areas represent the primary gold recovery process and their technical and operational efficiencies will have a significant impact on overall plant efficiency. The objective during process design of these sections is thus to develop a design which provides maximum technical and economic efficiency and which is robust to potential changes in ore throughput, mineralogical characteristics and head-grade. Experience has shown that, particularly for longer life and higher-grade projects typical of the South African underground gold mining industry, small changes in recovery and efficiency have a significant value over the life of the project.

Jan 1, 1999

By S. T. Johansen, C. Colomb, B. Ravary

Combustion and thermal NOx formation have been modelled using the Computational Fluid Dynamics (CFD) commercial code Fluent 6.1. The aim was to predict and give a better understanding of the combustion in electric arc furnaces for the production of ferro-silicon and silicon metal. Another important objective was to assess the impact of a number of operating parameters on the NOx formation. The CFD tools used are de-scribed with a special focus on models for species and turbulence transport. Sensitivity studies have been carried out to study the influence of several parameters: sub-models, process gas composition and distribution at the charge surface and influence of the presence of SiO. The main conclusions are: ?Using the combination of a realistic design, boundary conditions and models, the simulated NOx level is in quantitative agreement with measurements. The presence of water vapor is particularly important because of its influence on the specific heat of the mixture and resulting temperature. ?Release of significant amounts of SiO from the charge surface results in an important increase of NOx, even if we applied a simplified combustion model for SiO gas. However, because of lack of thermal and compositional data for blowing situations (when SiO flows directly from the arc cavity through open channels to the charge surface), no quantitative validation is possible. The way SiO is distributed (uniformly or as a jet) has a limited impact on the result in our computations.

Jan 1, 2007

By M. K. Elmouttie, G. V. Poropat

Mapping of exposed rock walls provides data on geological features such as joints, bedding planes, faults etc critical to the economics and the safety of mine operations. Traditional, manual methods of mapping can be time consuming, error prone and limited in their coverage. New methods have been developed to map exposed faces remotely and safely however the benefits of these methods are only fully realised when exposed structures can be quickly and easily mapped using automated methods. This paper describes the application of an automated method of analysis of 3D images that provides users with the capacity to map exposed structures automatically. The method is based on the use of vector classification methods used in some neural network systems and provides rapid and reliable identification of exposed planes. CSIRO Exploration and Mining is applying these methods to 3D images for a range of applications from pit mapping to machine automation.

Jan 1, 2006

By M. Stephton, L. A. Cornish, R. Süss, N. B. Maledi, J. H. Potgieter, L. Chown

In today?s gas turbines, the emphasis is on saving energy and reducing the amounts of pollutants emitted. This can be attained only by designing alloys with higher melting points and the capability to retain mechanical integrity at increased temperatures. The development of nickel-based superalloys (NBSAs) has reached its limit, as the alloys are operating at critical temperatures close to their melting points. Further increase in the operating temperatures will result in dissolution of the strengthening phases and even melting. Recent trends are towards developing materials that are adequately strong but have much higher melting points and corrosion resistance than the NBSAs that are currently in use. Studies conducted on Pt alloys with similar structure to the NBSAs have presented mechanical results that are promising. This has led to further development of these alloys as a replacement for NBSAs. The operations of gas turbines are usually limited by hot corrosion, which could lead to catastrophic failures. Increased temperatures have an adverse effect on the corrosion of the alloys by making the environment more aggressive. Hence, during a material selection process, the corrosion behaviour of any selected material for a highly specialized application such as gas turbines needs to be assessed. Generally, coatings are applied to combat degradation by salt deposits, hence coated and uncoated NBSAs were selected as benchmark alloys. Corrosion mechanisms and kinetic studies of the Pt alloys were conducted. High temperature hot corrosion (HTHC) tests were conducted at 950°C for540 hours. Analysis of the corrosion products was conducted using a Raman spectrometer, scanning electron microscope with energy dispersive X-ray spectroscopy (EDS) optical microscopy, and X-ray diffraction study of the phases that formed. Both coated and uncoated NBSAs failed under the test conditions. Samples were covered with a porous non-adherent greenish layer which is characteristic of NiO. Pt alloys displayed superior corrosion properties compared with the benchmark alloys.

Jan 1, 2006

By J. L. van Niekerk

The research project reported on represents the first comprehensive attempt to measure the vibration levels of a variety of tools and equipment in the South African mining industry to determine the effect thereof on the health of workers and operators. Although human vibration has been studied extensively internationally only one reference pertaining to the South African mining industry could be found. From studies conducted in Canada and elsewhere it was evident that rock drills do have potentially hazardous levels of vibration. The scope of the research included measurements on a broad spectrum of tools, machines and vehicles used in the South African mining industry. Both hand-arm vibration, where the majority of the vibration enter the body through the hands, and whole body vibration, where the whole body is exposed to vibration through contact by the buttocks or feet, were investigated and measured. More than 700 sets of vibration data measured on 70 machines at 15 mines and workshops were obtained. Data were obtained for at least 24 different types of equipment and machines that were previously judged to have medium to high levels of vibration. All measurements were conducted in accordance with the international standards: ISO 5349, on hand-arm vibration, and ISO 2631-1, for whole body vibration. The equipment with the highest vibration levels in the hand-arm category was: rock drills, both pneumatic and hydraulic; pavement breakers and jackhammers; hand-held compactors; pneumatic wrenches and electrical hammer drills. Of these the rock drills had weighted vibration levels in excess of 20 m/s2. As far as whole body vibration was concerned the highest values were measured on earth-moving equipment. The major conclusion from this study is that the measured vibration levels are sufficiently high to create an enhanced level of risk of vibration-induced disorders in a significant proportion of the operators of hand-held equipment. The most notable examples are hand-held rock drills, pavement breakers, jackhammers and certain selected workshop tools. The results of the study indicate that a comprehensive epidemiological study should be undertaken on present and past workers to determine the prevalence of vibration-induced disorders.

Jan 1, 2000

By T. Lindstad, S. Gaal, S. Hansen, S. Prytz

The SINTEF SiO-reactivity test, developed by Tuset and Raaness in the 1970?s, is a recognised method of characterising the ?reactivity? of carbonaceous reactants, used in the ferrosilicon and silicon metal industries. Recently, the test has been updated, which has resulted in the development of an extended and improved SINTEF SiO reactivity test. This has included updating the test reactor, control system, data-logging, handling and preparation of carbonaceous materials to be tested and measurement procedure. The scientific basis has been re-assessed, resulting in a new interpretation of the test results. The repeatability of the test has improved significantly, especially through system automation. In the ?old? test, the kinetics of the reaction: SiO(g) + 2C(s) = SiC(s) + CO(g) for a specific carbonaceous material was characterised solely by the ?R10? parameter, the volume of SiO(g) which passes un-reacted through the coke-bed as the concentration of CO in the outlet gas decreases from 18 to 10%. As both the kinetics and obtainable conversion of SiO to SiC for a reductant is of importance to the ferroalloy producer, the new SINTEF SiO reactivity test ranks carbon materials both according to degree of conversion and reaction rate. From research carried out on the test in the past couple of years at SINTEF and NTNU, it has also become clear that some of the reactions taking place in the test reactor are more complex than previously assumed. As an example, it was previously assumed that the condensation reaction of SiO gas in the condensation chamber of the test reactor can be described as: 2 SiO(g) = Si(l,s) + SiO2(s). However, it has been established that the side-reaction: CO(g) + 2Si(l) = SiC(s) + SiO(g) causes approximately 10% of the total volume of CO to react to SiC and SiO2 in the condensation chamber. This means that some back-reaction i.e., the reverse SiO-generator reaction 3SiO(g) + CO(g) = 2SiO2 + SiC also takes place in the condensation chamber. In the new SINTEF test, the %CO versus time curves are therefore corrected in order to describe the effect of the reaction between SiO(g) and C(s) studied i.e., SiO(g) + 2C(s) = SiC(s) + CO(g). A correction-formula for this purpose is presented.

Jan 1, 2007