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By Donald P. Schlick

Respirable dust provisions of the Federal Coal Mine Health and Safety Act of 1969 are designed to protect the most important resource of the coal mining industry--its workers. The Act imposes two respirable dust standards of 3.0 milligrams per cubic meter of air after June 30, 1970; and 2.0 milligrams per cubic meter of air after December 30, 1972. Under Federal regulations, each operator is required to carry out a dust sampling program established by the Secretary of the Interior and the Secretary of Health, Education, and Welfare, This program features the sampling of "high risk" face workers to establish that the exposure of each miner is below the dust standard, Mining sect ions, continuously in compliance with the regulations, may sample at less frequent intervals. In addition to requiring the mining company to take respirable dust samples, the Bureau of Mines will perform in-depth dust inspections twice a year, as well as frequent "spot inspections." The Bureau has established a semiautomatic computerized operation capable of processing over 1 million samples per year. This facility will be linked to various Bureau offices by a telecommunications system to provide automatic data printouts within hours after samples have been weighed.

Jan 1, 1970

By J. D. Livingston, H. E. Cline

Cu-Pb alloys in the vicinity of the monotectic composition have been directionally solidified under a high temperature gradient at rates up to 2 X l0-' cm per sec. Over a wide range of compositions, high growth rates yield a composite structure consisting of continuous rods of lead in a copper matrix. This range of compositions increases with increasing growth rate, in agreement with arguments based on the relative velocities of composite growth and the growth of copper dendrites or lead drops. The breakdown of the composite structure at slow growth rates is explained in terms of the relative interphase surface energies. The observed interrod spacings of the composite structure are large compared with the predictions of the Jackson-Hunt equations of eutectic growth. ThE directional solidification of many eutectic alloys produces fine composite structures of parallel lamellae or rods. There has been considerable interest not only in the fundamentals of this two-phase solidification process,'-3 but also in the interesting physical properties produced by such regular and aniso-tropic microstructures. Composite structures can be grown only over a limited range of composition, beyond which coarse primary dendrites of one phase appear. In organic eutec-tics, this composition range of composite structures has been shown to increase with increasing growth rate.7"10 These results were explained in terms of the relative velocities of composite (coupled) growth and dendritic growth. Although similar arguments should apply to metallic eutectics,11-13 suitable experimental results are lacking. In contrast to the work on eutectics, the directional solidification of monotectic alloys has received little attention. (The monotectic reaction is similar to the eutectic reaction, except that one of the resulting phases is a liquid, which subsequently solidifies in a separate reaction at a lower temperature.) Directional solidification of some monotectic alloys'4715 yields regular rodlike microstructures, whereas in other cases macroscopic separation of solid and liquid phases occurs.16 chadwick17 rationalized these results in terms of the probable relative magnitudes of the various interphase surface energies. A recent study of chill-cast Cu-Pb alloys18 revealed a fine rodlike microstructure in alloys near the monotectic composition. It was decided to investigate the directional solidification of such alloys, and to determine the range of composition and growth conditions yielding composite structures. The Cu-Pb system is convenient for such a study, both because it is simple metallurgically, with no compound formation and negligible solid solubilities, and because its basic properties are well-documented. Recent literature on the Cu-Pb system includes studies of bulk thermo-dynamic properties,'g surface energies,20"21 densi-ties,25 and diffusion constants.a6 A similar study of the directional solidification of Cu-Pb alloys was recently undertaken, independently, by Kamio and Oya." EXPERIMENTAL Alloys were prepared by melting 99.999 pct Cu and 99.999 pct Pb in a graphite crucible, stirring well, and pouring into a cold graphite mold. Rods 0.175 in. in diam were machined from the ingots. Alloy compositions studied ranged from 25 to 55 wt pct Pb. Samples were placed in graphite crucibles 5 in. long with 4 in. OD and 0.035-in. walls. They were melted under flowing argon in a vertical, two-zone. platinum -wound furnace. A voltage stabilizer was used to minimize fluctuations in input power. The narrow specimen diameter minimized convection. Directional solidification was achieved by driving the crucible downward into a +-in. hole in a water-cooled copper toroid. The toroid was located immediately below the narrow end zone of the furnace. The end zone was separately powered to maintain high local temperature. Therefore a high temperature gradient (approximately 300 deg per cm) was maintained in the specimen throughout the run. The crucible motion was screw-driven. and a wide range of drive speeds were available. The limited rate of heat removal caused a thermal lag in the specimens at high drive rates. However. temperature-time curves from thermocouples imbedded in a representative sample indicated that the average growth rate still approximately equaled the drive rate. Although the specimens were initially homogeneous, melting and re solidification redistributed the lead. producing composition variations of several percent along the specimen length. (During melting. lead melted first and ran down the sample surface. Rapid freezing tended to reproduce the resulting composition gr~dient, but slow freezing did not because a slow-moving interface tended to reject lead. as discussed later.) To determine local composition. ;-g samples were cut from regions exhibiting various microstructures and were chemically analyzed for lead content. Micrographs were taken on as-polished or lightly etched surfaces. Three-dimensional structure of the lead network was viewed with a scanning electron microscope after removal of some of the copper matrix with nitric acid. RESULTS Several different microstructures are observed, depending on composition and drive rate. Because melting and resolidification produced composition gradients, results are best presented in t&ms of final local composition, rather than initial or average composition. The ranges of local compositions and drive

Jan 1, 1970

By Franklin H. Persse

Domestic bismuth supply was projected from production figures for the ores in which it occurs--copper, lead, zinc, and tungsten. Published and unpublished data concerning host ores were examined to determine bismuth reserves and resources, and mine production. Potential recoverable bismuth occurring with reserves and resources of copper, lead, and zinc was estimated to be 36.3 million pounds. A bismuth availability diagram was developed from quantity estimates and ratios of production cost to value for these host ores. Additional recoverable bismuth occurring with domestic tungsten reserves was estimated to be 1.5 million pounds. Domestic byproduct production of bismuth will approximate 1.3 million pounds annually by 1980. This quantity will provide about 37 percent of the 3.5-million-pound estimated requirement for that year. Known world supply of bismuth appears to be sufficient for world demand to the end of this century. The study indicates that bismuth reserves and potential resources in the rest of the world have not been adequately reported.

Jan 1, 1970

By H. Colijn

Weight may be defined as the gravitational attraction or earthpull on a body (mass). Thus, from a practical viewpoint, a scale or other form of weighing device is a mans for determining mss. Weighing devices fall into two principal categories: (1) gravimetric; and (2) non-gravimetric or nuclear devices. At present, after many centuries of use, the effect of gravitational force as a meas¬ure of mass in industry and commerce remains by far the most common method for mass (weight) determination. Non-gravimetric methods are just becoming established in the industry. The general trend is toward weighing materials on the Trove. The predominant advantage of continous weighing over batch weighing operations is its ability to handle large tonnages without interrupting the material flow. The accuracy and reliability ability of continuous weighing equipment have improved greatly over the past decade. However, static weighing equipment still outsells continuous weighing equipment, because it is considered more accurate. Batch weighing can also produce a continuous weighed delivery through automation where the process requires it. In caparison with continuous weighing devices, such as belt scales, the batch-type scales have same inherent advantages affecting their accuracy. For example, weighing 300 lbs. per minute on a belt scale may mean that only 3 lbs. will be on the scale mechanism at one time. In order to obtain a scale accuracy of 1/2%, the sensitivity must be at least 1/2% of 3 lbs. or 1/4 oz. On the other hand, a batch scale may discharge a 300 lbs. load every 60 seconds through a timing mechanism. An accuracy of 1/2% can be easily obtained with a sensitivity of 1/2% of 300 lbs. or 24-oz.

Jan 1, 1970

By A. H. Larson, R. J. McClincy

The activity of Sb,03 in PbO-Sb,03 slags containing less than 50 mol pct Sb,03 was determined by the inert-gas saturation method at 700°C. In this composition range, the activity gf SbzO3 shows a strong negative deviation from ideality. The activity of PbO in these slags was calculated by application of the Gibbs-Duhem iniegration to the Sb203 activity data. The calculated activity of PbO in slags containing more than 63 mol pct PbO was found to deviate in a positive direction from ideality uthile a negative deviation was found for slags containing less PbO. The standard Gibbs free energies of formation of Sb,03 and PbO. Sb203 have been calculated and compared with existing data in the literature. The activity of Sb203 in PbO-Si0,-Sb203 (PbO/SiO, = 2) slags containing less than 25 mol pct Sb,03 was also determined by the inert-gas saturation method at 700°C. In this composition range, the activity of Sbz03 shows a very large negative deviation from ideality. VERY little experimental work has been published in the past to determine equilibrium data in the oxide systems connected with the refining of lead. These data are of value since impurities such as antimony, arsenic, and tin must be removed from lead and recovered for further treatment. Equilibrium studies on antimony and arsenic systems are also of interest for the design of new processes for lead refining and lead dross treatment. Maier and ~incke' first determined the liquidus curves for the PbO-Sb203 system and identified the compound PbO . Sb203. They found the phase diagram for this system to be two simple eutectics located on either side of the congruent melting compound. They also determined a very limited amount of vapor pressure data for Sb406 abbve PbO-Sba3 melts at 697"~. A second phase diagram investigation on this system was reported by Hennig and Kohlmeyer' who confirmed the existence of the compound PbO . Sb203 as well as the form of the diagram. A disagreement was noticed, however, in that their liquidus temperatures over nearly the entire composition range were higher than those reported by Maier and Hincke. Barthel~ and pelze14 redetermined the liquidus curve at the PbO-rich end of the PbO-Sb@, system and agreed very closely with the results of Maier and Hincke. None of the investigators mentioned above reported any mutual solid solubility in the PbO-SbD3 system. Zunkel and Larson5 have determined the phase diagram for the PbO-rich end of the PbO-Sb203 system by slag-metal equilibrium studies in the Pb-PbO-Sb203 system and by thermal analysis studies in the PbO-Sbz03 system. A maximum solid solubility of 5.6 mol pct Sb203 in PbO was observed at the eutectic temperature of 604°C. Their results for the phase diagram agree favorably with those of Maier and Hincke. The vapor pressure of Sb2O3 in the temperature range from 470" to 800°C has been determined by Hincke, using a modification of the transportation meth~d.~ His results for temperatures below the melting point of Sba3 are the only data reported in the published literature. The predominant vapor species has been shown to be Sb,06 by Norman and staley.? Myzenkov and Klushin,8 using the boiling-point method, have determined the pressure of Sb406 above liquid SbD3 in the temperature range from 715" to 1025°C. The agreement between these two studies is not very close. A portion of the discrepancy lies in the fact that Hincke used silica crucibles, which were attacked by the liquid Sbz03 at high temperatures. This fact does not account, however. for the large difference observed at the melting point. ~aier' gives a brief summary of vapor pressure data for Sb,O, above pure liquid Sbz03 which agree quite well with the data of Myzenkov and Klushin at temperatures near the melting point. This paper describes the determination of Sb2O3 activity data in the PbO-Sb203 and Pb0-Si02-Sb203 (PbO/SiOz = 2) systems by the inert-gas saturation method. These activity data are compared with the data calculated by Zunkel and arson. EXPERIMENTAL Materials. The materials used in this investigation were analytical reagent-grade PbO (99.8 pct PbO, 0.14 pct insoluble in CHsCOOH, 0.02 pct not precipitated by HB, 0.1 pct CaO, and 0.08 pct SiOz), Sbf13 (99.6 pct Sb203, 0.004 PC~ C1-, 0.005 PC~ SO;-, 0.15 p~t AS, 0.001 pct Fe, and 0.03 pct other heavy metals such as Pb), and SiOz (chromatographic grade). Apparatus for Vapor Pressure Determinations. The apparatus used in this investigation consisted of a transportation reaction system with two separate gas trains. The argon transporting gas was first mixed with a small amount of hydrogen, metered, and dried by passage through silica gel and anhydrone drying tubes arranged in series. After this preliminary drying, the argon was passed through copper wool at 500°C to convert the residual oxygen to water vapor which was removed by three anhydrone drying tubes. A second stream of argon was metered and dried and then passed around the outside of the alumina reaction tube to flush away the volatile species to pre-

Jan 1, 1970

By W. F. McDermott

The Iron Ore Industry has perhaps seen more varied and drastic changes in its approach to crushing and grinding in the last 5 years than any of the other mineral industries, Multimillion ton per year plants have become common place. Centralized control and computers have become an established benefit. Autogenous grinding along with a variety of rod and ball grinding flowsheets are in use. This morning I will discuss a few of the more important and new concepts in the control of these crushing and grinding circuits.

Jan 1, 1970

By A. K. Croteau

"At Craigmont Mines Limited, Merritt, B.C., the mining practice of trackless sub-level caving was started in the latter part of 1966. This type of mining was new to the mine personnel and the mining industry of Canada as a whole. Sub-level caving has presented many old and a number of new problems in maintaining the equipment used in this method. This paper will deal with some of these problems as they affect this particular mining operation. It will be impossible to review all phases of the equipment, but an attempt will be made to cover the most pertinent aspects in maintaining the drilling, mucking and hauling, and auxiliary equipment."

Jan 1, 1970

By Richard J. Lewandowski

Corrosion and product contamination have been difficult problems for extractive metallurgy industries in the U.S. because of the highly active chemicals used. During the past twelve years, other industries, particularly the chemical and paper segments, began to combat this problem with some degree of success using newer materials of construction, including reinforced plastics. These materials are made by combining a corrosion resistant thermosetting resin with a glass fiber reinforcing material and are used to con¬struct hoods, stacks, blowers, tanks, towers, pipe, and equipment of the most complex configurations. For service in contact with aggressive chemicals, the equipment is most often fabricated with the bisphenol-A fumarate polyester resins because of their resistance to corrosive attack. Data from both laboratory work and case histories of field applications are presented-for many common chemicals found in extractive processes. Information is also presented concerning design standards and recommended construction procedures

Jan 1, 1970

By K. C. Hong, R. B. Jensen

The problem of determining the optimum set of steam volumes and cycle lengths for a single well undergoing multicycle steam stimulation in order to maximize the cumulative discounted net income has been formulated mathematically and programmed for a digital computer. The mathematical fonnulation of the problem and the method for its solution are discussed in this paper. The oil production performance during each stimulation cycle was simulated by either a constant percentage or a harmonic decline. Using simple analytical expressions for production performance, the cumulative discounted net income and cumulative time of operation were related to the pertinent process and cost parameters and two principal process control variables (cycle length and steam injection volume). The discrete maximum principles was used to transform the equations for cumulative time of operation and cumulative discounted net income into a set of simultaneous equations. The simultaneous equations then were solved by trial and error on a digital computer to determine the set of cycle lengths and stem injection volumes that gives maximum cumulative discounted net income over the project life. INTRODUCTION Steam stimulation is a process for improving the oil recovery rate from wells producing high-viscosity crudes. The process is applied on an individual well basis and is executed in a series of cycles, each consisting of three phases: steam injection, soaking (steam condensation), and production. Significant increases in production rate following the stimulation operation result from heating the reservoir around the wellbore. As heat is removed with produced fluids and dissipated into nonproductive formations, the production rate declines, usually to near the prestimulation value. Typical production responses are given in case histories reported by Owens and Suter,l and are depicted in Fig. 1. Duration of the production phase is equal to the time for the oil production rate to decline to some specified value and is called the cycle length. (Cycle length is defined as the producing portion of the cycle and does not include the downtime required for steam injection and soaking operations.) Termination of the production phase coincides with the start of steaming for the next cycle. The process is continued, cycle by cycle, until it becomes unprofitable. Models 2-4 have been developed to simulate behavior of a single well during one cycle of steam stimulation and have been used to investigate the effects of system and operating conditions on the production responses. However, these investigations have not shown how the models can be used to determine a most profitable set of operating conditions for a multicycle stimulation project. Perhaps these models are too complicated mathematically to be adapted readily to an optimization study. This paper presents a method for optimizing a multicycle steam stimulation project based on simple production performance models. First, the performance of a steam-stimulated well during each cycle was simulated by either a constant percentage or a harmonic decline model. Using these simple analytical expressions for production performance, the cumulative discounted net income (before tax) and cumulative time of operation for each cycle were related to the pertinent process and cost parameters and two process control variables—the cycle length and the steam injection volume. Finally, the profit function to be maximized was formed. The analogy between the above problem and problems of optimizing multistage decision processes that have been reported extensively in the chemical engineering literatures5-10 in recent years has led this investigation to the use of the discrete maximum principle.5-7 This principle was used to transform the equations for cumulative discounted net income and cumulative time of operation into a set of simultaneous equations characterizing the optimum operating conditions in terms of the two control variables. These equations

Jan 1, 1970

By Ralph E. Magnuson

The quality of our environment is of great importance to all of us. Today, there is a great public concern about the degradation of our air, land, and water and the resultant slow death of our planet Earth. The miner has always been concerned with his environment. Historically, mining has been a part of the frontier. As a result, the miner has pioneered and developed many parts of the world. In the beginning, he had to hack out a place for his mine, himself, and his family. He was given accolades and praise for his accomplishments. Now, however, he is attacked as a ravager of the landscape. It was and is no different on the Marquette Iron Range in Marquette County, located in the Upper Peninsula of Michigan. Iron ore was discovered in 1845 and mining was begun by 1848. Over nearly 1 ¼ centuries, a total of 379,197,000 tons have been shipped to market. Ports, docks, railroads, and towns all had to be built in order to get that tonnage out to iron and steel makers. In those early days the environment had to be mastered, not pampered.

Jan 1, 1970

By R. V. Lundquist

Past reports by the Bureau of Mines have described the extraction of alumina from numerous aluminum silicate materials by the lime-soda sinter process. The results from pilot-plant investigations of six materials, with alumina-to-silica mole ratios from 0.3 to 8.7, indicated a relationship between the alumina-silica ratios and the processing characteristics of the sinter. A series of laboratory tests was conducted to define this relationship. Sinter and leach tests determined that an alumina-to-silica mole ratio of 0.90 or greater resulted in the best alumina and soda recoveries while leaving the leach slurries practically free from any tendency toward gelation. The widely different chemical and mineralogic properties of the various aluminous materials used in compounding the mixes had no perceptible effect on the leaching characteristics of the sinter. The blending of low-cost, high-alumina materials with aluminum silicate materials to give an alumina-to-silica mole ratio of 0.90 or greater would improve alumina and soda extraction and minimize operating problems in the lime-soda sinter process.

Jan 1, 1970

By B. H. Weber

"Industry generally has become noise-conscious and is realizing that damage to the human ear is expected to become more costly than any other compensation. Noise control has been recognized as a necessity and, fifteen years ago, in co-operation among different countries, inter-national organizations were set up to co-ordinate measures against dangerous noise and the development of silencing devices began. Rock drills for underground operations were particularly scrutinized because of the explosive nature of the noise developing from air escaping from exhaust. This paper describes the results and the present stage of development in the reduction and control of noise on rock drills, with special application to their use in mines."

Jan 1, 1970

Samples weighing O· 2 g are digested with 2 ml of 72 per cent perchloric acid for 2 hr at 180°C, and then diluted to 10 ml. with water. After settling, an aliquot of clear solution is reduced with ascorbic-citric acid, and then a zinc dithiol solution is added to form a coloured molybdenum complex which is extracted with petroleum spirit for comparison with standards.Concentrations down to 2 p.p.m. may be analysed at rates up to 150 determinations per day.

Jan 1, 1970

By Lukaszewski GM

The indiscriminate surface disposal of coal waste in the past has created a serious fire hazard in the bush of the northern New South Wales coalfields. Spontaneous or bushfire-induced combustion in large neglected dumps of this material has increased in recent years causing a rise in air pollution. Surface methods of coal waste disposal are still being practised, and in many cases new material is being added to old dumps, many of which are burning. CSI RO has conducted research on methods of preventing and controlling fires in coal waste and recommends practical methods of overcoming these problems. Some laboratory and on-site investigations are described and the hazards of improperly constructed dumps are considered. The available techniques of fire prevention and control are discussed in relation to Australian conditions. Proposals for the safe and economical disposal of coal waste are outlined.

Jan 1, 1970

By R. L. Whiting, H. J. Ramey

The basic study from which this paper was prepared was started as the result of the growing need throughout the world for increasing quantities of energy in all forms. Quite obviously, natural forms of energy that are readily available at low development cost are those in greatest demand. The underdeveloped countries — and particularly those having little or no petroleum resources — are the countries in which the most interest is being shown in the newer energy sources. One of the least expensive energy sources is natural geothermal steam.'-* Although this form of energy has been recognized for centuries, it has been only during the past 20 years that serious efforts have been made to harness it. Natural geothermal steam energy in Italy, New Zealand, Mexico, Japan, and California is now being produced through wells to drive turbines and generate electricity. Further, active exploration for natural geothermal steam is being conducted in Hawaii, Fiji, Taiwan, Chile, Russia, Greece, and Katanga. It is surprising to find that most geothermal steam exploration is in the "steam seep" stage. That is, surface studies are made and exploratory wells are drilled in the general area of steam seeps. However, in the larger geothermal steam areas, there have been efforts to apply the most modern geological and reservoir engineering principles in order to define the reservoir parameters, particularly those relating to estimates of reserves and future productivity. This paper is concerned with the development of appropriate equations and techniques to facilitate these estimates. Production of natural steam or hot water presents problems different from those experienced in the production of oil and gas. For example, steam or hot water systems may be essentially single-component systems, while hydrocarbon systems are most frequently multicomponent fluid systems. Heat effects are much larger for water than for hydrocarbon systems; and the natural steam production may or may not be isothermal, while production of petroleum reservoirs is considered normally to be isothermal. Petroleum reservoir engineering principles may be applied to natural steam or hot water reservoirs if the inherent differences in the systems are considered. The basic considerations involved in geothermal steam reservoir engineering are: thermodynamics, physical and thermal properties of water, materials and energy balances, fluid influx, and performance matching and predicting. Thermodynamics Fig. 1 is a pressure-temperature diagram for the liquid-vapor region for pure water, showing the critical point and five other points representing possible initial conditions for a geothermal steam reservoir. To generate a physical appreciation for the mechanics of natural steam production, it is helpful to consider

Jan 1, 1970

By James K. Kindig

Solvent extraction as a method of purifying and concentrating metal values from solution has become a standard hydrometallurgical operation for recovery of both uranium and vanadium, and within the past two years it has been put into operation in a commercial plant for recovery of copper from leach solutions. In this first full scale solvent extraction plant for recovery of copper operated by Ranchers Exploration and Development Corporation near Miami, Arizona, a kerosene solution of an organic compound, LIX-64® is used to selectively extract the copper from the dilute acidic copper sulfate leach liquor obtained from heap leach operations. The copper is back extracted from the solvent into a stripping solution which is fed to an electrolysis circuit producing cathode copper for sale.

Jan 1, 1970

By Fathi Habashi

CRYSTALLIZATION from aqueous solution is a simple and old method for the recovery of metal values from leach solution. Practically all the methods for the recovery of aluminum from clay are based on the crystallization of an aluminum salt from the leach solution. Differences in solubilities may be used to advantage for separating two metals from the same leach solution. The classical method of Marignac (1866), which makes use of the difference in solubilities of K2TaF7 and K2NbOF5 to separate niobium from tantalum, has been used successfully in the manufacture of these metals (Fig. 10-1). The slurry containing the metal oxides is dissolved in 70 % hydrofluoric acid, diluted, and heated near boiling. A potassium salt is added in sufficient quantity to form the complex fluorides of the niobium and tantalum present. The solution is filtered and allowed to cool. At room temperature most of the tantalum salt crystallizes out while the niobium remains in solution.

Jan 1, 1970

By Laurance L. Oden

The Bureau of Mines is developing melting data for binary combinations of the alkaline earth fluorides and yttria to facilitate the formulation of slag composition. Liquidus and solidus temperatures and solid solubilities were determined for the BaF2-Y203 system, and the information is presented as the phase diagram. The phase diagram was constructed from data obtained by ceramography, X-ray diffraction, and thermal analysis of specimens encapsulated in small molybdenum ampoules. The data indicate a simple eutectic system with maximum terminal solid solubilities of less than 1 mole pct. The eutectic temperature is 1,343° C, and the composition of the eutectic is less than 1 mole pct Y203.

Jan 1, 1970

By Rex O. McLellan

The formation of a monovacancy in a metal is simulated in an elastic model by the displacement of the surface of a small spherical cavity in a large elastic continuum. The application of linear elasticity to this distortion results in a well- known formula for the energy and an expression for the concomitant entropy change due both to the shear strain in the continuum and also to the dilation of the solid resulting from the boundary conditions at the surface of the solid. Elastic data (the sliear modulus and its temperature coelficient) are used to calculate the entropy and energy of formation for many metals. Despite the simplicity of the assumptions involved, the agreement between the calculated entropies and energies and experimental values is remarkably good. In recent years there has been a large increase in measurements of the absolute concentration of mono-vacancies in metals as a function of temperature. Hence new data for both the energy and the noncon-figurational entropy of formation of monovacancies has become available. Recent measurements&apos; of the anomalous (non-Arrhenius) self-diffusion in many bcc metals has also focused interest on the prediction of the thermodynamic parameters of mono- and multi-vacancies in those metals for which no data are available. Damask and Dienes&apos; have discussed the various theoretical calculations of the energy of formation EL, of a monovacancy. These include simple models involving the breaking of atomic bonds on moving atoms from the interior of a crystal to the surface, models combining elastic calculations with surface-energy terms and detailed quantum mechanical calculations. The simler models give the correct order of magnitude of &, but tend to overestimate it by a factor of about two. The quantum mechanical calculations4"7 have been carried out for the noble and alkali metals with generally reasonably good agreement with the available Ef data. The calculation of entropy of formation Sfv14 lnvolves a fundamental calculation of the perturbation of the phonon spectrum caused by the creation of a vacancy. Huntington, Shirn. and wajda8 have given an approximate evaluation of sJV by considering an Einstein model for the localized vibrations in the immediate neighborhood of the defect and then using elastic theory to calculate the entropy associated with the shear stress field in the distorted crystal (as originally proposed by Zenerg). They also included a term due to the dilation of the crystal. They obtained a value of 1.47k for copper, in good agreement with the experimental value (1.50k). However, Nardelli and Tetta- manzi1° have recently shown that neglecting the coupling between atoms (Einstein Model) may lead to a serious error so the agreement may be somewhat fortuitous. In this work simple linear elastic theory is used to calculate the entropy and energy of formation of mono-vacancies. Despite the simplicity of some of the assumptions involved, the agreement with the available experimental data is remarkable. However. the reasonable degree of success in the application of linear elastic calculations to the excess entropy of a solute atom in a dilute solid solution1&apos; indicates that the application of elastic theory to vacancies. where the interaction of different atomic species is not involved, may not be inappropriate. THE ELASTIC MODEL The metal is assumed to be a spherical elastic continuum. A small spherical cavity of volume V = 4i;v:&apos;/3 is cut from the center. removed. and dissolved rever-sibly in the bulk of the material. TO a good approximation no net atomic bonds are broken and the material does not undergo a volume change although the externally measured volume of the body would increase by V. The radius of the sphere of metal is much larger than r Next a negative pressure is applied to the cavity causing its surface to be displaced inward by an amount simulating the relaxation of the lattice around a monovacancy. In this model the energy and entropy accompanying the distortion are taken as 4, and <. As a first approximation the equation of state for the solid is taken as: r = ro(i + *~D LiJ where K is the bulk modulus. P the hydrostatic pressure. Vo the volume of the material at 0°K and zero pressure. and d+/dT = 30. where 0 is the linear thermal expansion coefficient. The variation of entropy with hydrostatic pressure is given by the Maxwell equation: These equations give the entropy change resulting from increasing the hydrostatic pressure from 0 to P as: and since • we have: This is the entropy arising from the dilation resulting

Jan 1, 1970

By Fun-Den Wang

A two-dimensional plane strain analysis by the stiffness matrix method was used to analyze two basic pit slope configurations. The magnitudes and the directions of stresses are given for a V-shape and for a flat-bottom pit with the slope angle varying from 30° to 90° at 10° intervals, Stress effects due to irregular pit configuration or to in-situ forces are discussed. Equations of similitude for gravity loading are given. Expressions are also given for the stresses at the toe as a function of slope angle. Procedures are given for finding the stresses in any geometrically similar pit slope of different dimensions and material properties from the stresses and equations presented in this report. Numerical results show that the V-shape pit slopes have lower stresses than the flat-bottom pit slopes; hence, they are inferred to be more stable.

Jan 1, 1970