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By C. R. Johnson, R. A. Greenkorn, G. W. Widner

This paper describes a short, simple well-testing procedure evolved from unsteady-state concepts to appraise wellbore damage in water-input wells. Input rates are changed at specific intervals over a 30-minute period and the wellhead pressure recorded at the end of each interval. From these data, the pressure drop due to the "skin" can be computed. This pressure drop in terms of psi/B/D of input rate is referred to as the "skin factor". Two of these variable-rate tests are required to determine change of wellbore damage and to evaluate well-treatment techniques. There are several advantages to the variable-rate procedure. The necessary data can be obtained with wellhead gauges and meters. There is little interference with normal operations. The short testing period eliminates many interpretative difficulties that come from interference from adjacent wells. The shut-in period required for standard tests before the plot of pressure vs log of time attains a straight line is often 12 hours or more. Such a time period is incronvenient, especially when a well must be tested repeatedly to evaluate change in wellbore characteristics. This variable-rate procedure has been successfully npplied to water-input wells in three fields. For some of the wells, skin factors were calculated from both the standard and the variable-rate tests. These results show good agreement. THEORY OF THE VARIABLE-RATE PROCEDURE In many waterflood input wells, the pressure in the wellbore differs from the pressure in the reservoir immediately adjacent to the well because of the restriction of flow through the skin of the well. If the true pressure of the reservoir at the well, p,, is that pressure which would exist in the well if no skin were present, and p,' is the actual pressure as measured in the well, this difference, is the skin effect. Fig. 1 shows the change in skin effect with change in flow rate. As the flow is reduced, the skin effect is reduced by an amount equal to the change in wellbore pressure minus the change in reservoir pressure. The equation of pressure change without skin using the point-source solution of the diffusivity equation is (after van Everdingen')

By Wilbur I. Duvall, James F. Devine

7.4-1. Introduction. Ever since explosives were discovered and developed for mining purposes, there has existed the problem of determining what effect the air and ground vibrations resulting from blasting have on structures of various types. This problem is especially acute for surface operations such as open pit mines, quarries, and construction projects that are in the proximity of residential districts, scientific laboratories, or other vibration-sensitive structures. Because the general public is directly involved in the blasting vibration problem, it has been of major concern to local, state, and federal governments, as well as to mining and construction companies, explosive manufacturers, insurance companies, and scientific investigators. Consequently, many investigations have been conducted, both in this country and abroad, on the effects of air and ground vibration on residential and other structures. E. H. Rockwell made some of the first studies in this country on the vibrations produced by blasting and the effect that these vibrations have on structures (18). In his publications, he stressed the need for measuring the vibrations from quarry blasting in order to establish the level of vibration as a function of charge size and distance. The concept of a damage criterion for structures was developed shortly thereafter, and a displacement of less than 0.03 in. was considered safe. During the period from 1939-1942, the Bureau of Mines conducted an extensive investigation on the problem of seismic effects of quarry blasting. This work, summarized in Bulletin 442 (20) was the first major effort for establishing damage criteria for residential structures based on experimental data. An acceleration of less than 0.1 g was considered safe, 0.1 to 1.0 g was caution, and greater than 1.0 g was damage. This work

Jan 1, 1968

By Dr. O’Neil Thomas J., Donald W. Gentry

"October. This is one of the peculiarly dangerous months to speculate in stocks. The others are July, January, September, April, November, May, March, June, December, August and February." -Mark Twain Although the preceding chapters describe the type of preliminary evaluation frequently performed in the mining industry, it has been assumed that input data are known with certainty-clearly an enormous simplification. In reality, estimates of ore grade, mining cost, metal price, etc., are subject to varying degrees of uncertainty due to the inability to predict the future with much precision. Management would probably not, for example, be equally attracted to two projects each offering a 20% DCFROI-one for a new gold venture in Nevada and the other for an undersea manganese nodule operation in the Pacific. The high risk with the latter investment tells the obvious; there is more to investment decision-making than a deterministic analysis using "best estimates" of input parameters. The science of decision theory normally distinguishes between decisions under uncertainty, where probabilities of various outcomes are unknown; and decisions under risk, where such probabilities can be estimated. The first case is rarely en- countered in practical capital investment analysis in mining, so the remainder of this chapter focuses on that class of problems where rational probability distributions for the future states of input variables can be estimated. RISK IN MINING Risk in the context of this chapter is the unforeseen deviation of individual cash flows from expected values for a capital project. For a mining venture, the source of this uncertainty could be any number of factors relating to such items as ore grade, ore reserve tonnage, operating costs, product prices, etc. With conventional *Adapted from O'Neil, T.J., 1979, "Procedures for the Preliminary Financial Evaluation of Metal Mining Ventures," Computer Methods for the 80s, A. Weiss, ed., AIME, New York, pp. 556-573.

Jan 1, 1984

In 1776 two Franciscan friars, Dominguez and Escalante, started to find a direct route from Santa Fé to Monterey, and in their misguided wanderings northward they reached Timpanogos, now known as Utah lake, where they heard of a much larger body of water to the northwest, of which they were told that it was extremely salt.* They never saw this great lake, because they passed south of it on their way to the Pacific coast. The Dead Sea of North America was discovered in 1835 by James Bridger, a trapper. He and other trappers had ascended the Missouri and had reached the Bear river. A discussion arose as to the course of this river, and a bet was made, to settle which Bridger followed the Bear river to its source, in the Uintah mountains, and from the crest of the watershed he caught sight of the lake, to which he then descended.† He tasted the briny water of the inland sea and wondered if it were an arm of the Pacific Ocean. The lake was first seen by the Mormons on July 24, 1847. A band of them, numbering 145 and led by Brigham Young, had migrated from Missouri in search of a new home in the western wilderness. It is recorded that when Brigham Young stood on the summit of the pass over the Wasatch range and saw the valley of the Jordan outspread before him, he exclaimed: "It is enough. This is the right place". He had seen it in a vision, so he claimed, and foresaw the future glory of the new Zion that was to be planted in that happy

Jan 1, 1932

By James E. Lawver

Efficiency of industrial research and development depends on the ability of engineers and scientists to collect and present data to top management for decisions. In many cases, the data presented are so deceptive that management makes costly wrong decisions. Since management personnel often are selected on the basis of higher administrative quality than technical ability, it is the responsibility of the industrial scientist or research engineer to make certain that the data presented tell a completely ac- curate story of each industrial problem. It is rare that misleading data are deliberately presented, but when this is done, it is? of course, dishonest. The rise of deceptive data usually stems from the lack of knowledge of experimental design and statistical methods of analysis. Most of the better chemical engineering schools have recognized the need for courses in industrial experimentation and many of them offer complete courses at both graduate and un- dergrarluate levels. By contrast, many of the mining and metallurgical schools have not recognized the importance of industrial experimentation courses and some of them, in fact, do not allow credit toward a mining or metallurgical engineering degree for statistical courses even when taken as an elective. The sole purpose of this paper is to make a plea for mining and metallurgical schools to present required industrial. statistical courses for their graduates. It is true that fine industrial research has been accomplished in the past without the assistance of any statistical techniques. Industrial problems have also been solved without the use of digital or analog computers, electron microscopes or other modern conveniences. The fact remains, however, that the correct use of these modern tools tends to reduce the cost of research and to eliminate costly errors. A working knowledge of experimental design and statistical data analysis must simply be viewed as a valuable tool whose proper use will greatly reduce the cost of experimentation and minimize the cost of false conclusions. These statistical methods, however, simply cannot replace fundamental knowledge or, for that matter, "good horse sense". Nevertheless. an experimenter armed with fundamental knowledge and good horse sense but completely ignorant of elementary statistical techniques, may easily present his data in such a deceptive form to cause management to make incorrect decisions.

Jan 6, 1962

By Marvin P. Barnes

Definition The term mineral inventory is a relatively new designation, born of the computer age and grown to common usage with the development of computer techniques for extending drill-hole sample values to regularly gridded data blocks representing in situ mineral values. The term has been generally accepted by geologists and engineers because it has a precise meaning which avoids many of the ambiguities of ore reserve. A mineral inventory is an estimated inventory of mineral in place and is usually achieved by dividing the deposit into regularly spaced gridded blocks to which are assigned estimated values using various extension techniques. A mineral Inventory makes no presumptions about the minability of the blocks or their cutoff grade. A mineral inventory estimate may be good or poor according to how much is known about the deposit, how extensively it has been sampled, and what techniques have been employed to assign estimated grade to the blocks Such an inventory, however, does not change with time, economic conditions, or mining technology, but only with new and better data and methods. Thus, the mineral Inventory is an excellent designation for describing mineral value distributron within a deposit's geological setting

Jan 1, 1979

By George Bain

METHODS of mining building stone of any sort are planned to pro-duce as few fractures as possible, and present a strong contrast to methods of mining metallic ores, which must be crushed eventually and have fractures produced in as great profusion as possible. Shocks transmitted to a stone invariably make it more porous and more permeable and reduce its effective life, therefore methods of mining building stone involve as FIG. 1. THE SUPREME COURT BUILDING, WASHINGTON, D. C. This building illustrates the fine detail of workmanship, unprotected sharp corners and thin edges, and tall narrow columns characteristic of marble structures. These features require exceedingly coherent stone, and preservation of these features for decades or centuries demands many special quarrying precautions to keep absorption to 0.14 per cent or less, a detail not essential to stone for coarser architectural types. few shocks as possible. This is singularly true for marble because marble structures are intended to have long life, greater delicacy of line, more gracefulness and fineness of sculpture than those made from any other structural material (Fig. 1). So that delicate lines may remain in almost perfect preservation for decades, or even centuries, it is important

Jan 1, 1935

EXECUTIVE COMMITTEES OF LOCAL SECTIONS New York LOUIS D. HUNTOON, Chairman. ARTHUR S. DWIGHT, Vice-Chairman. THOMAS T. READ, Secretary, Woolworth Bldg., New York, N. Y. E. MALTBY SHIPP, Treasurer. GEORGE F. KUNZ, W. DE L. BENEDICT. Boston ROBERT H. RICHARDS, Chairman. ALBERT SAUVEUR, Vice-Chairman. AUGUSTUS H. EUSTIS, Secretary-Treasurer, 131 State St., Boston, Mass. TIMOTHY W. SPRAGUE, HENRY A. WENTWORTH.

Jan 2, 1914

Jan 1, 1953

Jan 1, 1955

Jan 1, 1953

By Y. Okahara, I. Iwasaki

Chlorination behaviors of pure iron and manganese oxides were investigated by combining a thermogravimetric analysis (TGA) technique with batch-boat roasting followed by leaching. Ferrous and manganous oxides could be chlorinated readily, but, in the absence of a reductant, the higher oxides of both iron and manganese were dificult to chlorinate. Thermogravimetric analysis curves were drawn to illustrate the complexities of the reactions, and the possible mechanisms were discussed. Then three manganif-erous materials from the Cuyuna Range of Minnesota were treated by a process involving the selective chlorination of manganese followed by leaching. The results were interpreted in the light of the chlorination mechanisms observed on the pure iron and manganese oxides. One of the unsolved problems challenging Minnesota's mining industry is the development of an economic process to utilize the manganiferous iron ores of the Cuyuna Range. Although these ores constitute one of the major domestic reserves of manganese (estimated at 432 million tons'), their grade is too low for them to be mined solely for manganese. The nearly exclusive dependence of the U.S. steelmaking industry on foreign manganese ores has stimulated research over the years in treating the low-grade Cuyuna ores. As a result, a number of processes have been proposed.'-' Among these processes is chlorination. Chlorination is the high-temperature conversion of metallic oxides to chlorides by either chlorine or hydrogen chloride gas, or by solid chlorides of sodium, calcium, or magnesium. The metallic chlorides are then recovered by either vaporization or leaching. Although chlorination is an old art, it has become popular in recent years for the recovery of such new metals as titanium, zirconium, and niobium; for the removal of impurities like copper, lead, and zinc from pyrite cinder for ironmaking raw materials;' and for the segregation roasting of coppera and nickel' ores. The chlorination processes proposed for manganifer-ous iron ores have been relatively few (hydrogen chloride chloridization-leaching, hydrogen chloride and calcium chloride chloridization-volatilization,' and chlo-rination-selective condensation6), due presumably to the economics and to the corrosion problems involved. However, the successful application of chlorination processes to other ores will undoubtedly contribute to technological advancements in the chlorination of man-

Jan 1, 1971

By A. H. Daane, J. J. Hanak, F. H. Spedding

IN the study of the rare earths at the authors' laboratory, methods have been devised for preparing pure rare earths and many of their properties'- have been determined. Very little information is found in the literature on the properties of europium metal, which is the scarcest of the naturally occurring rare earths. Since we have recently prepared europium in appreciable quantities, we are in a position to describe the method of preparation and some of the properties of this metal. Metallic europium was first prepared by Klemm and Bommera who heated a mixture of potassium and EuC1, in vacuo, and obtained a mixture of europium metal and KC1. Trombe4 electrolyzed a molten salt bath containing NaC1, KC1 and EuC1, and collected the europium in a molten cadmium electrode as a cadmium alloy. The cadmium was subsequently removed by distillation but the properties of the residue indicate that pure europium was not obtained. McCoy attempted to prepare europium by distilling mercury from europium amalgam, but the stability of the Eu-Hg intermetallic compounds prevented a complete separation of the two elements. The preparation of very pure europium was suggested by previous work in this laboratory1,4 in which ytterbium and samarium were prepared by heating the respective sesquioxides with lanthanum metal turnings to 1450°C in vacuo and distilling ytterbium and samarium from the mixtures. Subsequently, small test experiments showed that europium also could be prepared by this method. A more detailed version of this method was used in the work described in this paper. Klemm and Bommer3 found by X-ray analysis that the metal had a body-centered cubic (bcc) structure with a lattice constant a, = 4.582 ± 0.00A (converted from kX units), with a bulk density of 5.30 g per cu cm. They found europium to be weakly ferromagnetic. Spedding, et al,7 found a,, to be 4.606 ± 0.001A. Barrett' showed that the bcc structure persists throughout the temperature range 5 to 3000K with a, at 5, 78, and 300°K being 4.551 ± 0.003, 4.551 ± 0.004 and 4.577 ± 0.001A, respectively. Trombe1 described europium as being iron gray, malleable, about as hard as neodymium and not attacked by cold water. He estimated the melting point to be between 1100 and 1200°C. Previous work in this laboratory indicated that the pure metal corrodes rapidly in air and reacts with cold

Jan 1, 1959

By R. L. Vaughn, H. R. Cooper

Effective procedures for automating X-ray assaying by use of a computer are described. These methods have advanced the reliability and accuracy of X-ray data, resulting in improved assay measurements for operations and process control. Assaying by X-ray fluorescence has been increasingly used in the mining industry, both for analyzing solid H.R. COOPER and R.L. VAUGHN, Members AIME, are Systems Manager-Process Control and Senior Systems Analyst, respectively, Scientific & Engineering Computer Center, Ken-necott Copper Corp., Salt Lake City, Utah. TP 68B236. Manuscript, Sept. 16, 1968. Discussion of this paper, submitted in duplicate prior to Dec. 15, 1969, will appear in SME Transactions, March 1970, and AIME Transactions, 1970, Vol. 247. samples for smelter, mill, and mine quality control and for continuous stream analysis of mill pulps. The important advantages of X-ray methods in laboratory assaying are reduced analysis cost, and for on-stream analysis, immediate assay information on concentrator streams. Of the numerous reports concerning X-ray analysis in mining operations, one is referenced which can be noted particularly in conjunction with this report.' Continuous X-ray analyzers have been installed at Kennecott and other mining companies for operations guidance and trend information for a number of years. However, complete operational acceptance of X-ray data has been slow, due* to the difficulty in maintaining

Jan 1, 1970

By L. D. Wooddy, W D. Moore

Water drives may be natural or artificial. For the case of a natural water-drive reservoir, the volume of water influx corresponding to reservoir pressuve-production history may be calculated by means of the material bal-ance-unsteady state equation. For the artificial water drive or pattern injection water flood, the material balance equation is sufficient to determine the reservoir pressure-production, injection history. However, for both cases to reliably predict reservoir performance, it is necessary to calculate the saturation distribution of the reservoir fluids at different stages of depletion. This paper consolidates the various equations used to determine the performance of water-drive reservoirs into one approach that is applicable to any reservoir that has a natural water drive or is to be produced by water injection. The numerical procedure has been simplified considerably, and the solution is adaptable to either hand calculations or digital computer operation. INTRODUCTION A water-drive reservoir is one in which the predominant source of energy for production of oil results from the encroachment of water into the oil zone. The water drive may be either natural or artificial. Water invasion takes place by expansion of water in the aquifer as a result of the pressure decline transmitted from the oil reservoir. Because water is practically incompressible, a natural water drive requires an extensive aquifer containing a volume of water many times the volume of oil in the reservoir. Provided the permeability of the formation is reasonably high, a moderate decline of reservoir pressure usually causes adequate water influx. If the reservoir pressure decline becomes excessive because of low permeability or a high rate of production, part of the oil production may result from the liberation of dissolved gas which may cause excessive quantities of free gas to be produced along with the oil. In extreme cases, reservoir pressure may decline so rapidly as to cause the oil to be produced predominantly by dissolved gas drive, leading to abandonment of the reservoir before the advancing water has had a chance to flush the entire reservoir. In reservoirs that are not connected to extensive aquifers, an artificial water drive may be created by injecting water into or around the lower edges of a reservoir. Sometimes water is injected within the reservoir through wells interspersed among the producing wells. When injection is carried out in such fashion, the reservoir is in effect subdivided into a number of separate producing segments, in each of which the displacement of oil is simultaneously taking place, and the shapes of which are fixed by the well patterns employed. Because of the natural tendency of oil and water to segregate according to their relative densities, water will tend to migrate downdip regardless of where injected. Accordingly, the injected fluids can be retained within the intended reservoir segments only when conditions are unfavorable to their migration under the force of gravity. Pattern injections are usually confined, therefore, to reservoirs having low structural relief, or low permeability, or when production and injection rates are such that fluid movements in the reservoir are dominated by the pressure gradients rather than by the force of gravity. In the case of natural water-drive reservoirs and artificial water-drive reservoirs in which the water is in-

Jan 1, 1958

By G. E. McElroy

Changes in existing airway and fan-installation conditions offer the most common opportunities for effecting economical operation of mine-ventilating systems, but the largest possibilities for securing this result lie in the original design of the main airways for efficient service. Formulas, and charts for their graphical solution, are presented herewith for the determination of the elements of airways design when the conditions of service are known or may be roughly approximated. Although the analyses are made with particular reference to metal-mine airways, there is no essential difference between main airways in metal and coal mines, and the methods presented are applicable to both. Little attention is paid to designing airways for maximum economy in the ventilation of metal mines. Most of the openings used as main airways are originally designed and used for transportation and other operating purposes. Even where openings are made primarily for ventilation, the possibility of their future use for other operating purposes, or even custom alone, usually results in the selection of a size and shape similar to those of existing openings. Natural conditions, such as heavy ground, often seriously limit the size of opening that can be maintained without excessive expense; and various operating factors, such as velocity limitations on traveling roads, often dictate size requirements in excess of those required for economy in ventilation only. In metal mines the main airways are, as a rule, too small rather than too large for economic service, with the general result that the quantities in circulation are often less than the mine's proper requirements over long periods, even though excessive amounts of power are applied in an effort to remedy conditions. Increasing the area of an airway increases the cost for excavation and lining practically proportionately but at the same time affects a very rapid decrease in power requirements. The balancing of these two factors of the expense to give a minimum total yearly operating cost for ventilation is mainly a matter of proper original design of the main airways of the mine, since the large volumes handled through these parts of the system involve large power costs and thus present opportunities for effecting large savings through proper design. Many of the factors

Jan 1, 1932

By Henry D. Hibbard

At the beginning of a Henry M. Howe lecture it seems fitting to refer to Howe's great contributions to steel metallurgy, and particularly to the literature thereof. Most of my predecessors in this role have been able, from their association with him, to give direct testimony as to his great services; but although connected with him for years in a business way, I am less familiar with what might be called his academic side, except as it is embodied in his utterances, written and oral. Howe was adept in his genial way in extracting information in any line which, at the time, was of interest to him. In a technical lecture one expects as a rule new knowledge—something about recent advances relating to the subject. On this occasion, however, I will ask you to go back with me to the primary department of steel lore, as the title of my lecture shows, to review the more fundamental features in that field for the purpose of seeking some gaps which we can fill and so give the subject renewed interest. In thus going back one must inevitably repeat many familiar facts, but that seems to be necessary in order to reach a clear understanding of the unrecorded matters. The full interpretation of a steel analysis is at present unattainable, though much progress toward that end has been made. The ultimate analysis, giving the percentages of the five commonly determined elements, does indeed usually give a fair idea of what the steel is, and is ordinarily enough, with knowledge of its history and physical condition, to enable it to be assigned to some purpose or use for which it is reasonably suitable. But discrepancies arise. Every maker and every user of steel knows, from at least occasional examples in his own experience, that the ultimate analysis is insufficient by itself to enable one to tell positively, from its interpretation, either the mechanical properties of the steel or its adaptability to the purpose for which it was made. Further, if, while having the specified composition, it is abnormal or unsatisfactory, why it is so. Good, bad and indifferent steels may have substantially the same ultimate analysis.

By D. W. Reed, R. L. Reed, Tracht

Laboratory experiments on the reverse combustion of tar sands in a linear adiabatic system have shown that a highly upgraded oil can be produced from an exceedingly viscous, immobile oil. The dependence on the air-injection rate of peak temperature, combustion-zone velocity, oil recovery, air-oil ratio, residual coke and oil, fuel burned and distribution of product gases is shown graphically. Eflects of initial temperature, oxygen concentration, oil saturation and heat loss are discussed. Experiments bearing on the coking properties of heavy oils are mentioned and some results exhibited. Field application of the process hinges on the existence of adequate air permeability and the rate of reaction under reservoir conditions. INTRODUCTION It has been established that oil can be recovered from underground reservoirs by means of at least two fundamentally distinct methods involving in situ combustion of a certain fraction of the oil. Characteristic of both of these known methods is the production of oil from one or more wells by means of hot gases formed when a high-temperature reaction zone is advanced through the reservoir. In both cases, the reaction zone is created by heating certain of the wells to a sufficiently high temperature prior to the introduction of air, and the zone is maintained and advanced through the reservoir by appropriate control of the air-injection rate. In the first of these methods, which is called "forward combustion",' the combustion zone advances in a direction which is generally the same as that of the air flow; whereas in the second method, "reverse combustion",' the combustion zone moves in a direction generally opposite to that of the air flow. Forward combustion, on the one hand, is an ideal combustion process in the sense that a minimum of the most undesirable fraction of the oil is consumed as fuel in the form of coke, a clean sand is left behind and generated heat is used as efficiently as possible. However, the applicability of forward combustion is limited. Since the products of combustion, vaporized oil and connate water must flow into relatively cold regions of the reservoir, there is an upper limit on the viscosity of oil which can be moved by this process in a practical and economical fashion.' On the other hand, it is characteristic of reverse combustion that the vaporized oil and water together with the products of combustion are produced through sand which is already hot and has had its mobile liquid content eliminated. This means there is no upper limit on oil viscosity; indeed, the oil may be an entirely immobile semi-solid. However, fuel for the process is an intermediate fraction of the original oil, and the most undesirable fraction remains on the sand surface as a substantial deposit of coke. Since this coked material is not burned during reverse combustion, it represents energy which is available for the production of oil but is not used for this purpose. It follows that one can expect economics to be somewhat less attractive with reverse combustion than with forward combustion. Nevertheless, it is a process which is designed for reservoirs where forward combustion is impossible and, as such, has become a subject of experimental and theoretical investigation. In this paper, only experiments made with tar sands are discussed. DESCRIPTION OF THE PROCESS We proceed, then, to consider the process of reverse combustion in greater detail. Fig. 1 illustrates a temperature profile defining a combustion zone which moves from right to left when air flows from left to right. In Zone 1, the temperature is the initial reservoir temperature, and the tar sand is as yet unaltered. This statement must be modified to the extent that physical properties of the oil may be changed by low-temperature oxidation at reservoir temperature. As air passes into Zone 2, which has been warmed by conduction, it assists in vaporization of the very light ends (if there are any), and oxidation occurs at a significant rate. In this region, there is almost no production of carbon monoxide or carbon dioxide because predominantly addition-type reactions take place with the formation of oxygenated compounds such as aldehydes and acids together with water. The hydrocarbon-enriched and slightly oxygen-depleted gas stream enters Zone 3 where

By R. T. DeHoff

In any structural transfortnation which is driven by surface tension, the geometric variable of fimdamental importance is the local value of the mean surface curvatuve. Acting through the suvface free energy, this quantity determines the magtnitude of both the pressure and the chemical potential that develops in the neighborhood of an arbitrarily curved surface. A metallographic method which would permit the quaniitatiue estinzation of this propevty is of fundarnerztal irztevest to studies of such processes. In the present paper, it is shoun that the average value of the mean surface curvature in a structuve can be estimated from two simple counting measuretnents made upon a vepresentative metallograpIzic section. No simplifyirlg geonzetric assurmptions are necessary to this deviuation. It is further shoum that the result may be applied to parts of interfaces, e.g., interparticle welds in sintering, or the edge of growing platelets in a phase transformation, without loss of validity. In virtually every metallurgical process in which an interface is important, the local value of the mean surface curvature is the key structural property. This is true because the mean curvature determines the chemical potential of material adjacent to the surface, as well as the state of stress of that material. The theoretical description of such broadly different processes as sintering,1,2 grain growth,3 particle redistrib~tion4,5 and growth of Widmanstatten platelets8 all have as a central geometric variable the "local value of the mean surface curvature". The tools of quantitative metallography currently available permit the statistically precise estimation of the total or extensive geometric properties of a structure: the volume fraction of any distinguishable part:-' the total extent of any observable interface,10,11 and the total length of some three-dimensional lineal feature:' and, if some simplifying assumptions about particle shape are allowed, the total number of particles.'2"3 The size of particles in a structure, specified by a distribution or a mean value, can only be estimated if the particles are all the same shape, and if this shape is relatively simple.14-16 The relationships involved in converting measurements made upon a metallographic section to properties of the three dimensional structure of which the section is a sample are now well-established, and their utility amply demonstrated. In the present paper, another fundamental relationship is added to the tools of quantitative metallography. This relationship is fundamental in the sense that its validity depends only upon the observation of an appropriately representative sample of the structure, and not upon the geometric nature of the structure itself. It involves a new sampling procedure, devised by Rhines, called the "area tangent count". It will first be shown that the "area tangent count" is simply related to the average value of the curvature of particle outline in the two-dimensional section upon which the count is performed. The average curvature of such a section will then be shown to be proportional to the average value of the Mean surface curVature of the structure of which the section is a sample. The final result of the development is thus a relationship which permits the evaluation of the average value of the mean surface curvature from relatively simple counting measurements made upon a representative metallographic section. The result is quite independent of the geometric or even topological nature of the interface being studied. QUANTITATNE EVALUATION OF AVERAGE CURVATURE IN TWO DIMENSIONS The Area Tangent Count. Consider a two-dimen-sional structure composed of two different kinds of distinguishable areas (phases), Fig. l(a). If the system is composed of more than two "phases", it is possible to focus attention upon one phase, and consider the remaining structure as the other phase. The reference phase is separated from the rest of the structure by a set of linear boundaries, of arbitrary shapes and sizes. These boundaries may be totally smooth and continuous, or piecewise smooth and continuous. An element of such a boundary, dA, is shown in Fig. l(b). One may define the "angle subtended" by this arbitrarily curved element of arc, dO, as the angle between the normals erected at its ends, Fig. l(c). Now consider the following experiment. Let a line be swept across this two-dimensional structure, and let the number of tangents that this line forms with elements of arc in the structure be counted. This procedure constitutes the Rhines Area Tangent Count. Suppose that this experiment were repeated a large number of times, with the direction of traverse of the sweeping lines distributed uniformly over the semicircle of orientation.' Those test lines which ap- proach from orientations which lie in the range O to O + dO form a tangent with dA; those outside this range do not, see Fig. l(c). Since the lines are presumed to be uniformly distributed in direction of traverse, the fraction of test lines which form a tangent with dA is the fraction of the circumference of a semicircle which is contained in the orientation range, dO; i.e., vdO/nr or dB/n. If the number of test lines is N, the number forming tangents with dA is N(d0/n). Since each test line sweeps the entire area of the sample, the total area traversed by all N test lines is NL2. The number of tangents formed with dA, per unit area of structure sampled, is therefore

Jan 1, 1968

By W. P. Sykes

SEVEN years ago, Dr. S. L. Hoyt1 presented a masterful discussion of the hard metal carbides and cemented tungsten carbide. His lecture summarized most of the data then available in the field; many of the details relating to the production, properties and structure of cemented tungsten carbide had never been previously published, for, in fact, this information was only then being acquired. During the intervening years, the cemented carbides have assumed a place of real importance in the commercial field, with an ever widening variety of applications due to constant improvement in the control of production technique and the unremitting search for new compositions that might better the performance of the alloys in one or another type of service. The most recent summary of production methods, properties and guides for the successful application of the cemented carbides appears in the 1936 edition of the Metals Handbook. Of necessity, much detailed information relating to structures and physical properties is not included in such a summary, and it is with such details that the present discussion is concerned.

Jan 1, 1938