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By K. A. Jackson, G. A. Chadwick, A. Klugert

The diffusion field ahead of a lamellar interfnce is analyzed using an electrical analog. A self-consistent solution is obtained for the shape of the interfnce and the diffusion field by an iterative process. The solutions presented here are for a 50-50 eutectoid or eutectic, The shape of the interface is found to he independent of growth velocity and lamellar spacing, and to depend on the relative values of interfacial free energies at the phase houndaries . The mode of growth of lamellar eutectics and eutectoids has been a subject of much interest for many years.1-4 Mehl and Hagel 1 have shown photomicrographs taken by Tardif when he attempted to determine experimentally the shape of an advancing pearlite interface; the results are completely ambiguous. Brandt' and schei13 have made approximate calculations of the composition ahead of a lamellar growth front. The shape of the advancing front and the composition distribution ahead of the front are difficult to calculate because one depends on the other. It is the purpose of the present paper to describe a method by which this calculation has been done. Lamellar-eutectic growth usually occurs under conditions where the growth is fairly rapid, and the interface temperature is close to the eutectic temperature. The growth rate is usually determined by heat flow. Eutectoid growth, on the other hand, can best be studied by quenching to some temperature, and allowing growth to proceed isother-mally. In both cases the growth is believed to be controlled by diffusion* rather than by the atomic kinetics of the transformation. This being the case, a single treatment of the diffusion equation will apply to both cases, provided the region of the interface in a eutectic may be considered to be isothermal. If a part of the interface could appreciably change its thermodynamic driving force by advancing ahead of or lagging behind the mean interface, then the two cases would not be similar. Eutectics normally grow in temperature gradients of the or-der of a few degrees per centimeter. The normal eutectic spacing is the order of a few microns. Part of the interface would have to extend many lamellar spacings ahead of the mean interface before it experienced sensibly different conditions. The interface temperature is usually a few tenths of a degree below the eutectic temperature so that temperature differences of the order of one-thousandths of a degree (a displacement of one lamellar spacing) would be unimportant. Protrusions large compared to the mean spacing do occur when one phase only grows into a eutectic liquid. This is usually a dendritic type of growth, and easily distinguishable from the lamellar mode of growth. A single treatment of lamellar growth will apply equally well to both eutectic and eutectoid decomposition. At the interface, which as shown above is essentially isothermal, the difference between the equilibrium eutectic temperature Teu and the actual interface temperature Ti, can be divided into two parts: 1) the composition varies across the interface, so that the local equilibrium temperature is not Teu; and 2) the interface is curved, so that the local equilibrium temperature is depressed according to the Gibbs-Thompson relationship. This undercooling can be written as Teu-Ti =?T = mAC(x) + a/r(x) [1] where ?C(X) is the departure of the composition at a point x on the interface from the eutectic composition, see Fig. 1, r(x) is the local radius of curvature at a point x on the interface, m is the slope of the liquidus line on the phase diagram, and a is a constant given by where s is the interfacial free energy, TE is the equilibrium temperature, and L is the latent heat of fusion. The calculations in this paper will be made only for the case where the phase diagram is symmetric, that is, the eutectic occurs at 50 pct, the liquidi have the same magnitude slope m at the eutectic temperature, and C,, the amount of B rejected when unit volume of a freezes, see Fig. 2(a), is the same for both phases. As shown in Fig. 2(b), the composition ahead of the a phase will be rich in B, the composition ahead of the ß phase will be rich in A. The composition at the phase boundary is the eutectic composition. The difference between the local liquidus temperature and the actual tempera-

Jan 1, 1964

By Henry M. Howe

(Cleveland Meeting, October, 1912.) Manuscript received Aug. 20, 1912. PROFESSOR RUFF'S most illuminating paper' describing his extremely valuable investigation of the carbon-iron equilibrium diagram assigns definite temperatures to certain very important lines in the diagram, viz.: the eutectic temperature EB (Fig. 1) and the " near-eutectic " temperature E'B', but without indicating clearly enough the reasoning which warrants these assignments. These temperatures are of such primary importance that it is well to indicate the apparent gaps in the reasoning, in the hope that they may be filled in. In his diagram, Fig. 1, the point X and the lines B'D'HI represent the experimental results reached by Ruff and Goecke; 2 AB, AEB, and SE represent the liquidus, the solidus, and the solubility of the carbon in solid austenite in the absence of graphite according to Gutowsky,3 AB being also in accordance with Spiecker's data;4 ,S'E' the solubility of carbon in solid austenite in the presence of graphite according to Ruer and Iljin; 5 GS the line A3 after Goerens and Meyer;6 I and EBC the true eutectic line as inferred by Prof. Ruff from Gutowsky's data. B'D' is the carbon content found in cast-iron which had apparently saturated itself with carbon in the presence of the graphite of the enclosing graphite crucibles in which the determinations were made. From its straightness and its sharp inflection near 6.67 per cent. of carbon, the carbon content of * Professor of Metallurgy in Columbia University. 1 Metallurgie, vol. viii., pp. 456 to 464 and 497 to 508 (1911). 2 Idem, vol. viii., pp. 417 to 421 (1911). 3 Idem, vol. vi., pp. 731 to 743 (1909). 4 Idem, vol. vi., p. 524 (Wüst) (1909). 5 Idem, vol. viii., pp. 97 to 101 (1911). 6 Idem, vol. vii., pp. 307 to 312 (1910).

Nov 1, 1912

By Henry Souther

It is generally admitted that a god, practical and commercial set of specifications for the use of the many purchasers of cast-iron does not exist. A few good sets are in the hands of large buyers, but even these are very different, contradictory, and not intended to cover a wide range of product. Furthermore, most of them have been prepared to cover but one branch of the industry. It is strange that purchasers of cast-iron have not looked into this question long ago and demanded good, uniform cast-irou. Probably cast-iron has been discarded from many uses because of its failure in a few instances, even though the failure was due to ignorance of the quality of the iron, or perhaps to abnormally bad iron. Some foundrymen seem to consider all gray iron from a given foundry as uniform, especially if they know that it has been made from certain well-known brands of pig. It is likely that with standard specifications and the increase of knowledge necessary for their intelligent employ-

Jan 1, 1905

By E. B. BESSELIEVRE

THE great work of Gorgas in stamping out yellow fever in Panama during the construction of the Canal was one of the chief factors contributing to the ultimate accomplishment of the task, two previous attempts having failed largely through the demoralization of workers by epidemics of contagious diseases. Mining camps, rapidly constructed during the boom days of the rushes to new fields, have been frequent sufferers from typhoid when, as so often used to be the case, little or no attention was given to bacterial examination of the water supply or to the method used for disposing of the sewage.

Jan 1, 1930

By M. L. Lissberger

entirely suitable for certain uses and even possess advantages over present solders at high temperatures, but they are certainly not suitable for all uses. However, it is conceivable that their use in many places will effect a material conservation of tin. It will probably be wiser to use cadmium in solders where good results have been obtained than in brass where this element will be a source of possible trouble. F. F. Colcord,* New York, N. Y.-—The cadmium-lead solder has a higher conductivity than the lead-tin alloy, which ought to make it advantageous in the manufacture of electrical machinery. In the case of solder for tin-can work, one of the high officials of the American Can Co. said that he fully intended to .give it a good trial, and I do not believe they are going to reach hasty conclusions and run the risk of poisoning the public. Solder, Its Use and Abuse Milton L. LissbeRgeR,† New York, N. Y.—Solder is a mechanical mixture of tin and lead, a fact which is susceptible of very simple demonstration. A bar of solder of a grade even as low as 30 per cent. tin and 70 per cent. lead, passed through a buffing machine, will show a surface practically identical with that of a bar of second-quality or reclaimed tin. The buffings, on chemical analysis, will prove to be almost pure lead. According to the best practice, solder is made in the following manner. Virgin pig lead is first melted, and when it is thoroughly liquefied, virgin pig tin is added, together with a small amount of flux; the latter is for the purpose of bringing to the surface the so-called "liver," consisting of impurities that may have remained in either the lead or the tin as a result of incomplete refining. The combined material, when completely liquid, is thoroughly stirred for some hours, and is then cast into small pigs. Just before casting, and continuously during this operation, the molten metal yields dross, consisting largely of the oxides of lead and tin; this should be carefully skimmed off. After the pigs have cooled, they are taken to a smaller kettle, re-melted, and cast into the desired shape for use; or if wires, ribbons, etc., are to be made, the solder is cast into slugs suitable for extrusion and rolling. During this second operation, the skimming of dross should be even more carefully done than at first. Hand mixing has proved to be the only reliable method for the production of the best quality of solder, irrespective of its percentages of lead and tin. The best quality of solder is not necessarily that which contains the highest percentage of tin, but rather is that composition which performs best on the required piece of work. In order to produce

Jan 1, 1919

By Irving Hartmann, H. P. Greenwald

READERS of this journal are familiar with the danger of coal-dust explosions in mines and with recommended means for preventing them. The subject was treated in a paper by R. R. Sayers in the January 1943 issue, as well as in several previous articles. Similarly the danger of dust explosions is generally recognized in the handling of grain dust, flour, starch, sugar, and other carbonaceous products. However, until recently, few people realized that an analogous, real explosion hazard exists wherever dust suspensions of certain finely divided metal powders are formed. This lack of knowledge was in part due to the fact that until a few years ago only a small number of metal powders, such as aluminum, magnesium, zinc, iron, and tungsten, were produced, in comparatively small volume, and partly because information on the explosibility of metal powders was lacking.

Jan 1, 1945

By Lyon F. Terry

ACCEPTED current practice for A the valuation of mineral property is based upon Hoskold's theory and valuation tables first published in 1877, and popularized by Herbert Hoover's "Principles of Mining" issued in 1909. In more recent years the procedure has been developed further in the course of the extensive valuation work required for Federal taxation purposes. The 1-aluation of oil and gas properties and of mines is based upon the same principle-the estimation of the present worth of future net revenues. However, in valuing oil and gas properties the method used for discounting future revenues differs somewhat from that adopted for mines, owing principally to the nature of the re- source. production from oil and gas properties usually declines from year to year, whereas coal and ore de- posits are adapted to a more uniform rate of output.

Jan 1, 1940

By D. W. Gentry, H. E. Sherer

Determining the age at which to replace a specific piece of mining equipment, and choice of the best possible replacement, is a problem common to nearly all mining operations. This paper examines the data required for intelligent equipment replacement decisions and discusses how to predict future repair cost from historic data, determine the total cost of various replacement alternatives, and evaluate the different replacement alternatives. Various operations research and computer methods were used to develop this easily-used, practical equipment replacement method.

Jan 1, 1984

By H. M. Howe

THIS paper presents an experimental verification of some of the predictions made by one of us' concerning 'the influence of certain conditions of casting upon the size and position of the pipe', and the position of the segregate, in steel ingots. The predictions which we have been able to verify are the following A. That the pipe is shortened and the segregate raised 1. By slow casting; 2. By casting with the large end up instead of down; 3. By retarding the cooling of the top, e.g., by means of a sinking-head. B. That the pipe is shortened by slow cooling. C. That the pipe and segregate lie in the last freezing-part. Our procedure was to cast ingots of wax (commercially pure stearic acid) containing a little bright green copper oleate (usually 1.5 per cent.) under varying conditions; to saw each ingot open along a longitudinal plane passing through its axis ; and to examine the longitudinal section thus laid bare. In order to make the segregate or enriched part more distinct in color from the impoverished part, a very little red cerasine was added. An experiment which will be described later showed that the cerasine itself does not segregate markedly, if at all. Hence we may assume that the strong contrasts between the green and red shown in so many of our ingots are due, not to segregation of cerasine, but to that of the green oleate, in the wax which in itself is uniformly reddened with cerasine.

Jul 1, 1907

By Howard Scott

THE GREAT advances made in mechanical engineering during recent years through the use of alloy steels, as illustrated by the development of the airplane and automobile, may be ascribed primarily to the application of nickel as an alloying element. The ternary system iron-carbon-nickel has a wide industrial application and forms, further; the fundamental basis of the even greater class of the widely used quarternary steels such as nickel-chromium, nickel-silicon, and nickel-vanadium. As is not uncommon with such cases of virile growth in an industry, the application of the alloy steel in a largely empirical manner has outstripped its scientific development so that a fertile field for metallographic research still remains. In this field, the basic nickel steels require first consideration and are the subject of the present investigation, which is offered as a preliminary to the study of the more complex alloys of industrial importance. Of the principal tools of metallography-microscopic examination, determination of -mechanical properties, and thermal analysis-the last has developed much more slowly in its application to industrial needs so that the situation often exists that heat treatment operations are based on empirically determined temperatures; that is, temperatures determined without regard to the actual transformation characteristics of the alloy. This situation is undesirable as there is no uniform standard for comparison of results. It is due, outside the usual difficulties involved in temperature measurements, to the pioneer character- of much of the published work; to the lack of definiteness of the Ac2 transformation,1 particularly in low-carbon steels; to the failure to define operating conditions completely; and to the paucity of information correlating thermal characteristics with results of heat treatment.

Jan 2, 1920

By Reginald S., Dean

The commercial availability of electrolytic manganese has greatly changed the position of manganese as a nonferrous alloying metal. Manganese metal commercially available up to about ten years ago was reduced with either silicon or aluminum and contained substantial amounts of the oxides of these metals and other impurities. Manganese was accordingly not available as an alloy base and its addition to copper-base and nickel-base alloys was generally limited to a manganese content of a few per cent. Several special alloys like manganin contained up to 12 pct of manganese.

Jan 1, 1953

By Hansheinz Portisch, Harold Margolin

A surface expansion was found to occur sometime after etching in Ti-A1 alloys containing 9.5 to 12.5 wt pct Al. The structure formed, grew, and disappeared with tzrrze. The surface expansion was followed by microscope observations and interferometric and lattice parameter measurements. Activation energy measurements for the growth of the "expansion structure" and chemical analysis indicated that the phenomenon occurred as a result of hydrogen pzckup during etching. It is proposed that hydrogen initially enters octahedral sites of Ti3Al coherent with a Ti and later shifts to the tetrahedral sites. It is postulated that expansion occurs when hydrogen enters the tetrahedral sites. The expansion structure disappeared, it is proposed, because of diffusion of hydrogen from the surface into the body of the alloy and because of loss of coherency of Ti3Al. In examining Ti-A1 alloys, Ence and arolinl observed markings on the surface of specimens. These markings did not appear after electropolishing only, but rather appeared only after etching. The markings appeared to grow as a function of time after etching and later seemed to disappear. Although the markings had some similarity to precipitates, showing, frequently, a Widmanstatten type of arrangement, the observation that other microstructural markings continue to be seen within the new structure suggested that it was actually not a precipitate. The source of this structure was unknown and an attempt was made in the present investigation to develop some understanding of its nature. It has been labeled expansion structure. 1) EXPERIMENTAL PROCEDURE A) Alloys. Alloys in the range 6.5 to 14.5 wt. pct A1 were studied. Bars, 4 in. sq, forged from Bureau of Mines (73 Bhn) titanium consumably arc-melted 4-lb ingots,' as well as 50-g arc-melted buttons of desired aluminum contents were used. The 50-g buttons also used Bureau of Mines titanium (73 Bhn) and aluminum of 99.99 pct purity. Buttons containing up to 8.5 wt pct A1 were hot-rolled from a furnace at 900' . Those with higher aluminum contents were hand-forged on a titanium anvil, and heated with an oxygen-hydrogen torch in the region of 1200" to 1300°c. Frequent reheating kept the samples at the desired temperature range. After a reduction of about 30 pct, the samples were water-quenched. To eliminate any contamination picked up either during hot rolling or forging, at least 1 mil of the surface of the sample was taken off. B) Heat Treatment. Prior to heat treatment all alloys were vacuum-annealed to remove hydrogen. Samples were annealed at 900' until a vacuum of 10'5 mm Hg was established. After this treatment, the samples were wrapped in molybdenum sheet and heat-treated in argon-filled quartz capsules, which were broken under water or iced brine at the conclusion of the heat treatment. All heat treatments under dynamic vacuum were performed in a rapid-quench furnace. This consisted of a molybdenum-lined quartz tube attached to a vacuum system and through a stopcock to a beaker of water. At the completion of the heat treatment the vacuum stopcock was closed, the furnace shut off, and dropped below the quartz tube. Then immediately the inlet stopcock was opened and water admitted until the tube was filled. The steam formed was allowed to escape through the inlet stopcock. This method was used in heat treatments up to llOO°C. C) Metallography. Specimens for metallographic examination were ground, then electropolished using a Disa Electropol machine with a perchloric acid electr01te. Specimens were etched with R-etch.3 A standard etching time of 3 min was used, with the specimen being agitated during immersion. D) Sample Preparation for X-ray Analysis. Samples 0.2 to 0.5 mm in diam were produced from heat-treated rods which were turned to 2 mm diam on a lathe and then rotation-etched. An etchant consisting of 1 pt HF, 7 pt HN03, and 12 pt HzO was satisfactory. With the rod rotating in a vertical position at 50 to 100 rpm, a needle with uniform dimensions could be obtained. Care had to be taken to insure that the rod was at the center of rotation, otherwise cavitation developed. If a small unevenness developed, it was possible to grind it off with a fine emery paper. E) X-ray Diffraction. All X-ray work was done on a North American Phillips X-ray diffraction apparatus and a Jarrell Ash microfocus X-ray unit. The Phillips unit was used with a copper target and nickel filter. The Jarrell Ash unit was fitted with a cobalt target and iron filter. For the Phillips unit 35 kva and 20 ma were used, whereas for the microfocus unit with the 100-p fixed-focus gun 40 kv and 1.5 ma were used. It was found that alignment of cameras on the Jarre11 Ash unit was very critical. The X-ray beam contains an intense area which is not the beam center. The cameras were aligned with the intense region by monitoring the beam coming out of the camera with a Geiger counter. Adjustments were made until a maximum intensity was obtained. A Phillips diffractometer with a Brown chart recorder furnished some of the lattice parameter data. The divergence slit up to 80 deg 28 was I deg; above 80 deg, a 4-deg opening was used, while the scatter slit was1 deg and the receiving slit had a 0.003-in. opening. The general scanning rate was 1 deg per min, while peaks of special interest were rescanned at -j deg per min. For elevated-temperature X-ray diffraction a Uni-cam High Temperature Camera with a film diameter

Jan 1, 1969

By T. A. Rickard

The history of this State is that of one generation. Thirtysix years only have elapsed since the birth of that beneficent industry whose footsteps were the first to traverse the wilderness of the prairies and penetrate the solitude of the mountains. So soon is history made. The men who halted on the rolling plain where Denver now stands and gazed westward at the snow-clad ranges in eager questioning of their possibilities of golden wealth, have lived to see a noble city built where once their camp-fires burned, and have participated in the discovery of a magnificent series of productive mines amid the mountains which seemed at that time the Ultima Thule of their pilgrimage. To those who investigate the workings of our complicated mining and metallurgical industries, the story of their evolution from humble beginnings will appear an instructive romance. The record that tells it presents features common to the growth of modern mining regions, but also bears aspects peculiar to

Jan 1, 1897

By Bryon W. Cheever

The greatest objection to be brought against the present methods for estimating phosphorus in iron and steel, is the time consumed in the operation. The following method, originated and perfected by Mr. E. D. Campbell, assistant in the chemical laboratory of the University of Michigan, will, 1 think, if carefully followed out, give results that are sufficiently accurate for technical purposes, and will give them in less time than they can be obtained by any other process. The process consists in comparing the color obtained by the action of stannous chloride upon a solution of the phospho-molybdate precipitate obtained from the solution of iron or steel, with that obtained from a standard solution of molybdie acid which represents a given amount of phosphorus. The operation is as follows : Take of ordinary steel 1 grm. and treat as in any of the wellknown processes until the phosphorus is precipitated as ammoniumphospho-molybdate. When precipitated, the phospho-molybdate precipitate is filtered out, thoroughly washed with an acid solution of ammonium nitrate, and, finally, twice with water, to remove most of the ammonium nitrate. The precipitate is then dissolved in about 1 c. c. of 10 per cent. potassium hydrate, which is let fall drop by drop on the filter till the precipitate is dissolved ; the filter is now washed with about 25 or 30 c. c. of water, and the potassium hydrate solution, which now consists of 25-32 c. c. of solution, is boiled until the peculiar frothy appearance, which first appears on boiling, is entirely gone. The solution is then cooled until perfectly cold, and about 10 or 15 c. c. of strong hydrochloric acid added ; if the solution is kept cold this will form no precipitate, but a light yellow or colorless solution. The solution is now ready for the addition of stannous chloride, and the subsequent comparison of color produced with that obtained from the standard molybdie acid. The comparison is made in tubes that are calibrated like carbon-tubes, but which are larger, about 5/8-inch bore, and graduated to 100 c. c.; this makes them about 18 inches long. To make the comparison of color, first transfer the now acid solu

Jan 1, 1886

By C. C. CONCANNON

POTASH is an essential. It is necessary as an ingredient in fertilizers or as a plant food, and certainly one of the great problems, and one of increasing gravity, is the maintenance of agricultural fertility that we may have adequate food supplies. When one thinks of potash one thinks of Germany. The first and most important deposits of soluble potash salts were found in Germany, and it was through Ger- man research and the application of a highly developed knowledge of chemistry that the value of these deposits was recognized and appreciated in the field of agricultural fertilization. The Germans knew that they had something of great value in their potash resources, not only because of the advantage derived from the extensive use of it in their own intensive farming operations at home, but also because of the pressing need for it on the part of other countries. Appreciating the economic value of this in international trade, they fostered the industry to the fullest extent and devoted much attention to well organized propaganda designed to bring out the essential need for potash of German origin throughout the world. Thus every one came to believe that potash was an essential commodity of only German origin and as such it held an extraordinary position in world trade prior to 1914.

Jan 1, 1926

By A. E. Nehrenberg

IN an earlier publication' it was shown that the.. shape assumed by a volume of growing austenite is inherited from the prior structure. The matrix grains of pearlitic microstructures are equiaxed, and austenite grows in an equiaxed manner in these prior structures. Martensite and bainite, on the other hand, have an acicular grain structure, and the austenite produced in such structures grows in an acicular manner. In the discussion of this paper' it was pointed out that in cold-rolled tempered martensite, austenite grows in an equiaxed, rather than an acicular manner. The results of an experiment carried out to arrive at an understanding of this behavior are illustrated by the micrographs. Fig. 1 shows the acicular microstructure produced by heating tempered martensite to a temperature within the Ac1-Ac3 range, 1350°F. When tempered martensite is cold rolled, then heated to the same temperature, the equiaxed structure shown in Fig. 2 is produced. A sample of cold-rolled tempered martensite heated at 1350°F just long enough for the formation of a small amount of austenite is shown by Fig. 3 to have an equiaxed (recrystallized) ferrite grain structure. The small particles of. austenite which formed at 1350°F in the equiaxed ferrite matrix are likewise equiaxed. Thus, the equiaxed growth of austenite in cold-worked tempered martensite is not a contradiction of the principles described in the previous paper, for the growth pattern reflects the matrix grain structure existing at the time of austenite formation. References A. E. Nehrenberg: The Growth of Austenite as Related to Prior structure. Trans. AIME (1950) 188, pp. 162-174; Journal of Metals (January 1950). 2 J. R. Vilella and R. A. Grange: Discussion of ref. 1. Trans. AIME (1950) 188, pp. 1378-1381; Journal of Metals (November 1950).

Jan 1, 1953

By WILLIAM J. LINNEY

MAGNETITE ore from Lyon Mountain, so- called "Low Phos Chateaugay," has long been known to the iron and steel industry for its almost complete absence of impurities. These magnetites occur along the northern foothills of the Adirondacks in several parallel and folded stringers which are continuous for several miles, striking generally northeast-southwest and dipping usually over 45° to the northwest, or roughly toward the St. Lawrence River. Ore of minable thickness is found in shoots of various sizes as to length and breadth, but of undetermined and possible great depth.

Jan 1, 1943

By Stanley W. Ivosevic

The South-Central Osa Peninsula produced around 30,000 kg (100,000 troy oz) of gold since prehistoric times. Gold freed by erosion from quartz lodes in the Mesozoic Nicoya ophiolite complex was reconcentrated through three generations of Quaternary sedimentary deposits. Resultant gold placers grade upward from 1 g/m3 (0.03 troy oz/m3) and are of moderate size.

Jan 1, 1980

By E. H. Mohr

At the McGill concentrator of the Nevada Consolidated Copper Co., all flotation operations have been carried out in Forrester cells since November, 1926. In respect to cost of operation, the new cell has proved to be superior to the Callow, which it supplanted, and its incorporation in the flow sheet has resulted in a marked improvement in the metallurgy. Before citing the advantages of the new machine over the old, it should be explained that the Forrester has not only replaced the Callow, but it has made possible the elimination of a number of tables as well. When Callow cells were in use, table concentration preceded flotation. The ball mill pulp was elevated, by means of air lifts, to roughing Wilfley tables situated two floors directly above the flotation floor. On an intermediate floor were additional tables which cleaned the concentrate from those above. The finished table concentrate constituted approximately 60 per cent. of the total mill production, and contained about 40 per cent, of the recovered copper. The table tailings were returned to the classifiers, the overflow from which went to the flotation cells and the oversize to the ball mills. As soon as the Forrester cells were installed, this tabling operation was discontinued. The ball mill pulp now discharges into the classifiers direct, without the intermediate tabling. It was impracticable to ernploy the present flow sheet when Callow cells were in use, for the reason that unless the great,er part of the heavy mineral was first removed, it accumulated on the mats and adversely affected the flotation operation. By tabling ahead of flotation, it was practicable to maintain the concentrator at 10 to 20 per cent. greater capacity (depending up on the pyrite contnt of the ore) than was possible by flotation alone. This was due to the necessity for finer grinding in the latter case. Even though the table conecntrates contained only 6 to 8 per cent'. copper content, the low smelting costs made it advantageous to operate at high mill capacity, at the cost of low grade of product. Standard Section equipment With the old flow sheet, a standard section consisted of four ball mills and classifiers, one air. lift, 20 roughing tables, 10 cleaner tables, 12 primary

Jan 1, 1928

By G. E. Perry, J. F. Bruskotter, D. G. Russell, J. H. Goodrich

The depletion performance of gas wells has been investigated by mathematical simulation techniques. The gas well model which was studied consisted of a single well located in the center of a bounded, cylindrical, homogeneous reservoir. Dependency of gas compressibility and viscosity on pressure was considered in studies of well performance at both constant mass flow rate and constant flowing pressure conditions. To carry out the investigation, the nonlinear, second-order, partial differential equation which describes Darcy flow of a nonideal gas through porous media was solved numerically. Some of the previous investigations of gas well performance have been of limited general use, because assumptions were introduced to simplify either the gas properties or the basic differential equation. Other studies have been rigorous in these respects but have presented a very limited set of calculated results. The present study was attempted to present a rigorous theoretical model and sufficient numerical results to permit meaningful conclusions to be drawn. It was found that all terms must be retained in the partial differential equation to make accurate predictions. The neglect of higher-order terms, e.g., terms of the order of the "gradient squared", leads to serious material balance errors at large times and to conservative estimates of gas well performance. The higher the gas flow rate and/or the lower the permeability-thickness product of the formation, the more pronounced are these deviations. For example, in a well draining 640 acres in a 25-md-ft formation (8,120 MMcf gas in place) at a constant rate of 993 Mcf/D, the rigorous solution predicts a bottom-hole pressure decline from 4,000 to 1,000 psia in 8.7 years. If higher-order terms are neglected in the differential equation, this decline in pressure is predicted to occur in 5.3 years. With the results of the numerical solution of the differential equation as a basis, simple, easy-to-use approximations for predicting gas well performance for Darcy flow conditions have been developed. These simple approximations are based on the familiar equations for flow of a single, slightly compressible fluid. The approximate methods possess a high degree of accuracy and enable the prediction of long-term gas well performance to be made quickly and accurately without the use of a digital computer. Both transient and stable flow period approximations were developed. INTRODUCTION In recent years income from the sale of natural gas and associated products has represented an ever-incre as ing fraction of the industry's total revenue from operations. To meet the surge in demand for natural gas, the industry has depended heavily upon established reserves and has actively pursued development of new reserves. The search has progressively led to reservoirs which yesterday were too tight and/or deep to yield the desired return on invested capital. More than ever before, evaluation accuracy is now required to forecast the criteria upon which engineering recommendations and management decisions are based. Considerable effort has been expended by both research and operations personnel on the development and application of methods for analyzing and predicting the performance of gas wells. Fundamentally, the problem is the familiar one of extracting data during the drilling, testing and early production life of a well and applying these data within an accurate simulation model to predict long-term behavior. During the past 30 or more years a voluminous literature dealing specifically with gas field problems has been generated. A recent book' lists a comprehensive bibliography of published material through 1959. Over 1,200 references are cited. Since then 39 additional articles on natural gas technology have been published in Transactions volumes of the Society of Petroleum Engineers of AIME. Most existing theory for predicting gas well performance requires that one or more idealizations (e.g., steady-state flow, ideal gas of constant viscosity, small and constant compressibility and constant-viscosity fluid) be applied. Although existing theory may apply directly or be adapted by various artifices to describe specific gas well and reservoir behavior, no widely applicable method is available, and existing methods appear to be subject to appreciable error unless better limits of applicability are defined. The objectives of this paper are (1) to present numerical solutions to the partial differential equation describing gas flow under conditions of general interest in gas well performance prediction work, (2) to present solutions which possess a high order of accuracy for both transient and stabilized flow periods of a well producing at constant rate or constant pressure, and (3) to develop and present simplified relationships which can be used as high-order approximations to the exact numerical results for fast and accurate predictions of gas well performance at the operating level. Combined, these objectives are designed gen-

Jan 1, 1967