Search Documents

By A. R. MacPherson

"This paper considers four aspects of autogenous grinding. Part 1 discusses statements on power efficiency of autogenous grinding compared to conventional crushing and grinding systems. It also demonstrates how high-power efficiency can be obtained from autogenous grinding systems. Part 2 considers a simple method of testing ores to determine the type of autogenous plant which can treat these ores on a commercial basis. A comparison of testing for simple small plants vs large plants or complex ores is included. Part 3 considers important aspects which have to be considered in proper design of an autogenous grinding system. Part 4 discusses problem ores and how they can be treated in an autogenous system. INTRODUCTIONAfter about 35 years of using autogenous grinding plants in the mining industry, it is perhaps time to review where we are with this, by now, relatively old development. Autogenous grinding in this paper means grinding run-of-mine ore by fully autogenous means or by using a ball charge to assist the grinding (semi-autogenous). Also, autogenous grinding can be eitherby dry or wet.The use of autogenous grinding in the fifties was making its slow start, and it was not until the sixties that more of the mining industry would consider its use . By the seventies, most builders of new plants would have considered autogenous grinding, whether they used it or not, in the final design.In the eighties, most large plants use autogenous mills and to reduce over-all operating costs, many older conventional plants are now looking and actually retrofitting these old plants with autogenous grinding mills.It is evident that many misconceptions concerning autogenous grinding are still current. Some come from improperly designed plants which result from poor test work. There are many considerations in the proper design of autogenous plants. Some ores have been found to give problems even in a well designed plant. These ore types are beginning to be classified and recognition in the test stage allows steps to be taken in the design so that these ores can be efficiently handled in autogenous grinding plants.Therefore, this paper will address the following aspects of autogenous grinding:1. Misconceptions regarding autogenous grinding.2. Testing of ores to ensure a well designed commercial plant.3. Important considerations in mill design.4. Difficult ores, their recognition and types of design to handle such ores."

Jan 1, 1989

By J. J. Jonas

The categories of interpass time encountered in hot rolling are characterized. These are shown to fall into three broad categories: short (in rod rolling); intermediate (in strip processing); and long (in plate schedules). The effects of interpass time on recrystallization and carbonitride precipitation are described, and particularly on the interaction between these two mechanisms. The interactions, in turn, determine the final microstructure produced as well as the mechanical properties.

Jan 1, 2005

En quoi consistent les études de suivi des effets sur l?environnement (ESEE)? Les ESEE constituent une méthode de suivi scientifique que l?on peut employer pour déterminer la santé des écosystèmes aquatiques (qui peuvent être touchés par l?activité humaine) et l?efficacité des mesures de protection de l?environnement. Les ESEE sont des évaluations scientifiques itératives du poisson, de l?habitat du poisson et des ressources halieutiques. Le programme comporte des phases séquentielles de deux à six ans de suivi, d?interprétation et de compte rendu au cours desquelles la fréquence et le type de suivi dépendent des résultats des études précédentes. Le programme des ESEE pour les mines de métaux est fondé sur un accord conclu à l?unanimité entre les intervenants. Les exigences du programme et les guides techniques connexes ont été élaborés à la suite d?une consultation multilatérale à laquelle ont participé les représentants du secteur privé, du gouvernement et des collectivités. Toutes les mines assujetties au Règlement sur les effluents des mines de métaux (REMM), aux termes de la Loi sur les pê ches, doivent procéder à une ESEE dans le cadre de l?autorisation qu?ils ont reç ue de rejeter des effluents. Le REMM a été enregistré le 6 juin 2002 et publié dans la Partie II de la Gazette du Canada le 19 juin 2002 (SOR/DORS/2002-222).

Nov 1, 2002

By E. A. Davies, Trevor T. Jones

"Aspects of materials, design and operation of suspension equipment is dealt with, hazard, fatigue and special metallurgical properties of certain steels to resist are described, as is the effect of temperature in service modifying the properties of some steels to resist stress concentration and propagation of cracks.The influence of steady impact and fatigue loading and its results are described. Control of manufacture including nondestructive testing is outlined and acceptance parameters and strength calculations are given. Quality requirements and control in manufacture are outlined. A short description of the overwind detaching hook, together with illustrations of the operation of detachment are presented.MaterialsWinding engine ratings of 8000 h .p. and large capacity shafts 2000 m in depth (4000 m depths have already been reached and it may be necessary to go to 6000 m) and up to ten conveyances are not uncommon today. Loads of 100 or even 200 tons hoisted at low speeds are possible within the near future. Some of these appliances are designed to protect lives and installations in hazardous conditions when, due to human or mechanical fault, a malfunction occurs in the system. In addition, such equipment includes suspension gear of various kinds designed to anchor and apply high mechanical tensions to numerous steel ropes installed in shafts necessary either for the safe passage of conveyances or for balancing purposes when heavy loads are in motion.No unique metallurgical problems are presented in underground mining operations, but they have two special characteristics that influence the metallurgical aspects of the de sign of suspension gear and other equipment concerned in the transport of men and materials. (1) Any uncontrolled release of energy caused by the unexpected failure of a component is more likely to injure personnel and cause costly damage to installations in the narrow confines of an underground mine shaft. (2) Some mechanical equipment in mines, tends to be overloaded in service and suffer plastic distortion."

Jan 1, 1987

By J. Shahabpour

"Abstract - Breccias with angular and rounded clasts of a late-mineralization, ""barren"" hornblende porphyry dike set within a comminuted matrix consisting of rock fragments and quartz, cemented by quartz and K-feldspar of hydrothermal origin is observed in diamond drill cores from the Sar Cheshmeh porphyry copper deposit. Studies of age relationship between vein types, phyllic-altered clasts and the breccia groundmass suggest that the brecciation is a late event. The breccia clasts record evidence of hydrothermal reaction between the clasts and fluids responsible for the brecciation.Formation of bornite, in the breccia clasts of the originally barren hornblende porphyry dike, and in the breccia matrix, along with the scarcity of bornite in the Sar Cheshmeh deposit itself suggest involvement of fluids which are later and of different composition from the fluids responsible for the known mineralization at Sar Cheshmeh. These later fluids may be related to a later plutonic event and suggest the possibility of a later, and concealed deposit."

Jan 1, 1994

By Yu Chen, Charlie C. Li

The application of rock bolts for the stabilisation of underground openings needs a good understanding of the response of the bolts during rock excavation. The pull-and-shear performance of two types of rock bolts—that is, the fully encapsulated rebar and the D-Bolt—has been studied both experimentally and numerically. The rock bolts were tested under a pull-and-shear load at five displacing angles (i.e., 0°, 20°, 40°, 60°, and 90°), and at the same time, the strains on the bolts were monitored during testing. The two types of rock bolts have similar loading capacities, while the loading angle has obvious influence on the deformation capacity of the D-Bolt. The ultimate total displacement of the D-Bolt decreases from 140 mm under pure pull loading to 70 mm under shearing. The displacement capacity of the rebar is not significantly influenced by the loading angle but remains at a quite low level from 29 to 53 mm. The D-Bolt yielded in a longer section the rebar bolt under pure shear. A trilinear material model was used to simulate the strain hardening behaviour of the bolt steels in the numerical code FLAC3D. The numerical results show that the axial load in the bolt decreases exponentially with the distance from the loading position for the rebar bolts, but it remains constant in the bolt sections between the anchor positions for the D-Bolts. When the bolt is subjected to a lateral load, shear stresses are created on the D-Bolt surface within a short distance from the loading position.

Jan 1, 2015

By G. D. Mcdowell, L. B. Cochrane, B. D. Thompson

"Abstract - Exploration for deep underground copper and nickel sulfide deposits in the Sudbury mining camp is carried out with surface drilling. More detailed drilling, with an advanced underground exploration program, may be necessary to establish an accurate interpretation of the deposit geology and upgrade the resource estimate to the measured category for reserve estimation and feasibility assessment. Resource evaluation is a lengthy and costly process that could result in a decision not to proceed with the project. For projects that proceed to production, significant reductions in the operating and capital costs could be achieved with a more accurate interpretative model of the deposit geology and mineralization. Quality control procedures to ensure the integrity of the drilling data and interpretation guidelines are essential steps in the development of a deposit model. The use of block modeling techniques and the application of geostatistics for resource estimation further improve the quality of the resource model.Significant improvements to the interpretive models can also be achieved by incorporating additional information from geophysical measurements to validate and constrain the geological interpretation. The geophysical models must be consistent with known geological environments and oreforming processes, which has required the concurrent development of comparative models from exploited deposits. These geophysical techniques can also provide key geotechnical information required for the optimum design of the mining method and mining sequence. The objective of this program is to improve the accuracy of the deposit model and the quality of the resource and reserve estimates in order to optimize capital investments and reduce the development and operating costs of underground mining projects. Several of these techniques are currently being tested at INCO Limited’s Sudbury operations. The results from those tests are presented and their applications are discussed within the framework of resource/reserve classification systems and project evaluation objectives."

Jan 1, 1998

By J. O. Yeboah, V. Boakye, P. A. Bepswa, M. Lombard, G. Nutor

The comminution circuit at Tarkwa Gold Mine was commissioned as a Single Stage SAG mill operated in closed circuit with hydrocyclones designed to process 4Mtpa. The circuit was later expanded by adding a secondary ball mill in closed circuit with hydrocyclones with a view to treat 12Mtpa. The change in the circuit configuration and assigning the low aspect ratio SAG mill for primary grinding application makes the Tarkwa experience unique in terms the optimisation strategy and operating philosophy. This paper reports the findings from a series of surveys performed on this plant. The surveys reported include data obtained when the SAG mill was operated as a single stage mill in closed circuit with hydrocyclones and also SABC mode under two different feed conditions. The differences in operating philosophy between the initial single stage SAG circuit and the current SAG/Ball mill circuit are discussed.

Jan 1, 2012

By Kadir Sariiz

Abstract - The Akçayir-Yürükakçayir Miocene-Pliocene basin in the Central Anatolia hosts ancient lacustrine sepiolite deposits. The succession in this basin consists of conglomerate (Yürükakçayir conglomerate), clayey and silty limestone, dolomitic marl, sepiolite beds and lenses, and dolomitic limestone (Kepeztepe formation). The main impurities within the sepiolite rock are dolomite, opal-CT, calcite, smectite and feldspar as sepiolite + dolomite and sepiolite + opal-CT are the main mineral assemblages. Smectite and feldspar are thought to be of detrital origin. Calcite is found as a secondary filling in root tubes and casts. Sepiolite samples studied display root-marks and root tubes, soapy and massive structure indicating direct precipitation. Sepiolite deposits probably formed at a pH of 8 to 9, and Mg+2 and SiO2 concentrations in moderately saline environments of internal facies of an alkaline lake.

Jan 1, 2000

By Eugene Sun, Qiang Xu

This presentation will review current applications of composite materials in aircraft design and discuss the challenges involved in maximizing the potential of the composites in aircraft applications.

Jan 1, 2010

By J. Robitaille, P. Cousin

"Agnico-Eagle Mines Ltd, Laronde Division, is a gold operation processing ore containing copper and zinc. Copper is recovered as a concentrate prior to gold cyanidation.Copper concentrate grade was improved with the implementation of an on-stream analyzer. Further improvements to the copper concentrate grade could be achieved throughefficient zinc depression.High cyanide cost linked to the copper content in the flotation circuit tails, prompted some modifications to the flotation circuit. Economics and environmental issues were considered. A Maxwell cell was retrofitted with an Outokumpu HG cell mechanism stabilizing it's operation. Secondly a flotation column was added to increase circuit capacity.This paper will address flotation circuit operation difficulties, successes and results achieved before and after modifications.RESUMELes Mines Agnico-Eagle division Laronde sont considéré comme étant un producteur d'or. Le minerai révèle toutefois des teneurs appréciables en cuivre et zinc. Un circuit de traitement combinant des étapes de flottation du cuivre et de cyanuration de l'or a été adopte. Le rejet de la flottation est lixivie pour son contenu en or.La teneur moyenne en cuivre du concentre final fut augmentée suite a l'implantation d'un système d'analyse en continu. Une séparation plus efficace de la sphalérite se devra d'être obtenue afin d'améliorer davantage la qualité du concentre produit.Le contenu en cuivre au rejet du circuit de flottation influence directement la consommation en cyanure. Pour cette raison, certaines modifications à la flottation sont devenues nécessaires. Des facteurs d'ordre économique et environnemental ont été considérées. Une cellule existante fut transformée en cellule HG d'Outokumpu, stabilisant son opération, et une colonne de flottation fut ajoutée au circuit afin d'accroitre sa capacité.Le présent document se veut une rétrospective des problèmes d'opération vécus, des solutions apportées ainsi que des résultats obtenus et de leurs implications suite aux améliorationseffectuees."

Jan 1, 1995

By R. D. Studer

"The Snow Lake massive sulphide deposits are hosted by Aphebian volcanic rocks of the Flin Flon - Snow Lake greenstone belt, which in this area were deformed and metamorphosed to the amphibolite facies during the Hudsonian orogeny. All the known deposits occur in felsic rocks a short distance below the contact with an overlying mafic rock unit.The Stall Lake copper deposit occurs close to the top of an explosive felsic volcanic sequence. Rocks stratigraphically underlying the deposit have been hydrothermally altered by the mineralizing fluids. The sulphides were probably deposited from submarine volcanic vent(s) on the flanks of a felsic dome. The ore has been tightly folded, flattened and elongated parallel to the northerly trending lineation of the surrounding rocks during the several periods of deformation. Strike lengths of the lenses range from 60 to 300 feet and plunge-to-strike ratios are as much as 15:1.Ore lenses 1 to 4 typically consist of massive coarse-grained pyrrhotite, chalcopyrite and pyrite. A stratigraphically higher lens differs in that it contains fine-grained pyrite and chalcopyrite, with an increase in sphalerite toward its hanging wall. The No.4 orebody has a distinct footwall alteration zone characterized by brecciation and abundant chlorite-staurolite-garnet schist, with minor kyanite, anthophyllite, cordieriteand anhydrite, as well as sulphide minerals. Chemically, the alteration zone is distinguished from the unaltered felsic rocks by Na20 depletion and Mg-Fe enrichment.IntroductionThe Stall Lake copper deposit is located aproximately 4 miles southeast of Snow Lake in northern Manitoba (Fig. 1). The Snow Lake district has been a producer of gold (Rex, Bingo, Laguana and Nor-Acme Mines) and, since the fifties, of copper and zinc from deposits of Hudson Bay Mining & Smelting Co., Ltd . The district is located near the eastern limit of the Flin Flon - Snow Lake greenstone belt, which contains Amisk metavolcanic rocks of Aphebian age."

Jan 1, 1982

By M. Celikin, M. Isac, R. I. L. Guthrie, S. Ge

Horizontal Single Belt Casting (HSBC) is a near-net-shape casting process in which molten metal is directly cast into thin strips, at high cooling rates (order of several 100°C/s), with the potential for high volume, friction-free, continuous production of metal strips. This casting process was used in the present work to produce strips of Al-Mg alloys in the AA5000 series, with additions of Sc and Zr. Such aluminum alloys show exceptional potential as a structural material for transportation/aerospace applications. To demonstrate the suitability of the HSBC process to manufacture competitive strip products of Al-Mg-Sc-Zr, the mechanical properties and microstructures of the strips produced using the HSBC process were compared with conventionally cast products. The effects of annealing on the mechanical properties of the strip-cast Al-Mg-Sc-Zr alloys were also investigated.

Jan 1, 2014

By S. V. Burr

"IntroductionOne of the more puzzling geological enigmas, in the writer's opinion, is how rock can move sideways with respect to other rock. In the writer's study of regional northwest-southeast folding in North America (Burr, 1980), he may have found the answer to horizontal offset.Horizontal OffsetSideways movement or horizontal offset does not mean that the movement was, or is, horizontal. As indicated below, the actual movement may be at a high angle from the horizontal. However, it is the horizontal VECTOR, or offset, which is measured on surface by geologists. Vertical movements (thrust, normal, reverse) can be understood, as they involve blocks of rock , generally separated on four to six sides, which have been pushed up or subsided. Vertical faulting across dipping strata can produce APPARENT horizontal offsets. However, this thesis deals only with ACTUAL offsets.The ProblemFigure 1 is a plan view of a simple hypothetical horizontal offset. Three vertical dykes are intersected by a fault lineament. Two dykes exhibit no movement; the middle dyke has been offset a distance (a). Vertical faulting would have produced no horizontal offset. How can a horizontal distance (a) disappear or appear? Rock expands or contracts in infinitesimal amounts.SolutionThe map (Burr, 1980) shows the writer's interpretation of regional northwest-trending folds in Ontario. The writer made two observations about horizontal offsets during his thirty year study of these folds:1) All major horizontal fault lineaments strike at a high angle to the northwest fold axes. Many of the northwest-trending lineaments are fault s, but their movement is essentially vertical.2) Close to many of the known horizontal offsets (and several lineaments), there are areas where plunge reversals are so frequent and close together it is impossible to draw in all the fold axes at a scale of 1 inch = 4 miles, and, in the highly faulted Espanola area southwest of Sudbury, Ontario, all the indicated folds cannot be easily separated at a scale of 1 inch = 1 mile. The writer has observed folds less than 30 feet apart next to the strong Puskitamica-Chibougamau fault in northern Quebec."

Jan 1, 1982

By R. Carpenter, H. Muntanion, P. Kleespies, E. Flood, M. Tansey

Abstract - BHP Minerals discovered the Archean lode gold ULU deposit on the western margin of the High Lake Volcanic Belt in 1989. The greenschist to amphibolite facies mafic volcanic and sedimentary rocks on the property are folded into a 5 km-long anticline. There is a close spatial association of Au-As zones to the trace of this F2 ULU anticline. The largest of these auriferous zones, the Flood zone, is localized at the core of the fold. The southeast-trending Flood zone consists of several anastomosing lenses that have been traced for 435 m on surface. Overall, it dips steeply (70° to 80°) to the southwest and has been intersected to depths below 600 m. The hosting high-iron tholeitic basalt displays a lower amphibolite mineral assemblage of ferrohornblende + plagioclase + ilmenite with accessory quartz and epidote. Alteration minerals include biotite, chlorite, hornblende, actinolite-tremolite, and potassium feldspar (microcline) with minor calcite, epidote, tourmaline, and titanite. The highest gold values occur where brecciated basaltic wall-rock clasts are replaced by acicular arsenopyrite + quartz + K-feldspar. BHP Minerals completed 54,783 m of both exploration and resource development drilling before selling the ULU property to Echo Bay Mines in 1995. Development of ULU as a satellite deposit to Echo Bay’s Lupin mine began in 1996. A -15% ramp was extended to the 155-meter level, and 16,011 m of underground drilling were completed before the decline in gold prices forced a suspension of activities at ULU in August 1997. The current owners, Wolfden Resources Inc., drilled 18,569 m in 2004 to increase confidence in the resource blocks. Following this drill program, a revised resource calculated by an independent consultant determined that the ULU deposit contains an inferred/indicated mineral resource of 1,130,000 t grading 11.34 g/t gold (373,748 oz) to the 360-m level. The portal was re-opened in 2005 and a prefeasibility study is underway.

Jan 1, 2004

By Neda Zangeneh

Production efficiency from low permeability gas shales require techniques to optimize hydraulic fracture completions. One completion strategy involves the simultaneous or near simultaneous hydraulic fracturing of multiple horizontal wells to maximize the fracture network area and stimulated reservoir volume. However, changes to the in-situ stress field caused by an earlier hydrofrac on subsequent hydraulic fractures are not accounted for in traditional hydrofrac design calculations. Presented here are the results from a set of transient, coupled hydro-mechanical simulations of a rock mass containing two wellbores using the discontinuum-based distinct-element method. The results demonstrate the influence of stress-shadows generated by a hydraulic fracture on the development of subsequent hydrofracs from an adjacent well. It is shown here that these interactions have the potential to change the size and effectiveness of the hydraulic fracture stimulation by changing the extent of the hydraulic fracture around the secondary well.

Jun 1, 2012

By A. P. E. Dirige

In order to maximize the recovery of ore in variably dipping ore zones of moderate width, cemented backfill is normally placed to serve as structural support. With the intention of saving on costs, backfill of low cement content can be used, which are often supported by a sillmat of higher strength. The stability of the low cement backfill face, exposed during adjacent mining, must be carefully studied to provide very effective, safe and economic mining operations. Improper design of these stope support structures may result in fill mass failure resulting in relaxation and failure of the stope walls, with consequent losses of production, ore dilution, and in safety problems. This paper presents a study conducted to assess fill performance during adjacent pillar mining and to provide a comprehensive understanding of backfill behaviour, the failure modes that may occur and the consequences to production, ore dilution and to safety problems, and accurately predict their stability. The design study investigated the effects of stope width and height, orebody geometry and inclination, and wall roughness on the stability of cemented backfill during adjacent pillar mining. Fill properties used were based on paste fill specimens cured for 28 days. Paste fill performances were assessed based on analytical and numerical modeling studies for the different mining conditions. Analytical modeling was carried out using limiting equilibrium analysis and numerical modeling was carried out using FLAC3D. The modeling results suggested that, for stopes that are inclined with smooth wall rock conditions, backfill failure, driven by the fill self-weight, has minimum dependency on the binder content and is reduced by resisting forces developed on the footwall-fill contact. For inclined stopes with rough wall rock conditions, wall roughness contributes significantly to the stability of the backfill during adjacent pillar mining. The analytical modeling approach was demonstrated to be useful in providing some approximate parameters for predicting the behaviour of paste fill exposed faces during adjacent mining, but cannot predict the mode or mechanisms of failure. Numerical modeling not only assess the stability behaviour of paste fill free faces, but also is able to provide a better idea of paste fill failure modes and possible failure mechanisms. The depth of failure and the potential for instability for a simulated stope filled with paste fill can be predicted, and may be useful in estimating the mass of material that could possibly fail and ore dilution levels.

May 1, 2009

By N. Saracoglu, J. P. Golightly, R. H. Wallis, J. J. Brummer

"The McClean uranium deposits are in northern Saskatchewan, 11 km west-northwest of the Rabbit Lake mine. Their average grade is 1.8% UJ0 8 and they contain 6,350 tonnes of uranium oxide in seven pods of variable grade and mineral assemblage. They are hosted by both basal Athabasca Group and the regolith developed on the underlying crystalline basement. The major mineral facies are: sulphide, arsenide, 'bleached', and hematite, with the important uranium minerals being coffinite, uraninite and pitchblende.Abundances of carbon and sulphur isotopes are consistent with their derivation from pyritic graphite in basement gneisses. Isotopes of oxygen and hydrogen suggest mineralizing fluids were aqueous and at temperatures of about 200°C. Ar39/ Ar40 laser age determination of illite indicates mineralization began about 1300 m.y, ago and that significant hydrothermal alteration ceased about 1170 m.y. ago though Pb/U data suggest the deposits continued to evolve into the Mesozoic.The deposits are interpreted as the result of oxidizing ground water stripping uranium and other metals from either the Athabasca sandstone, especially the heavy mineral layers, or from the regolith, and precipitating these when encountering a warm reduced fluid emanating from the basement."

Jan 1, 1984

By Norm Clegg

The Plant Engineering and Maintenance Association of Canada PEMAC is a national, not-for-profit industry association for plant and facility maintenance, engineering and physical asset management in all industry sectors in Canada. Our mission is to provide professional development for our members and the profession in Canada

Feb 1, 2008

By P. Chu, J. A. Finch, R. Li

"Bubble size plays a key role in flotation. In this paper, we examine the possibility of using passive acoustic sending to determine bubble size. Software was developed based on Short Time Fourier Transformation (STFT) to analyze the frequency of the acoustic signal emitted from the bubble formation event. Bubble size was determined from the Minnaert frequency. The acoustic-determined bubble size responded to reagent dosage, but was less sensitive than that given by imaging. The constraints of the current acoustic approach are discussed.INTRODUCTIONBubble size is an important factor affecting flotation performance. It determines the maximum available surface area for collecting and transporting mineral particles. Of interest for process control (Gorain, Franzidis, & Manlapig, 1995), it can be determined by several techniques, such as image analysis (Grau & Heiskanen, 2002; Hernandez-Aguilar, Gomez, & Finch, 2002); conductivity (Barigou & Greaves, 1992); drift flux analysis using gas rate and gas holdup measurements (Banisi & Finch, 1994); and optical sensors (Randall, Goodall, Fairlamb, Dold, & Connor, 1989). However, all these techniques have drawbacks.Acoustic analysis of the oscillatory behaviour of bubbles to extract bubble size is another possibility (Leighton & Walton, 1987; Pandit, Varley, Thorpe, & Davidson, 1992; Al-Masry, Ali, & Aqeel, 2005; Manasseh & Ooi, 2009; Leighton & White, 2012; Spencer et al., 2012). The fundamental basis is the work of Minnaert (1933) who first correlated the bubble size with its natural frequency, Equation 1:"

Jan 1, 2016