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By H. H. Rachford, W. P. Schultz

The use of computing machines to solve physical problems has made it imperative to represent physical data in a form computing machines can use. Although curve-fitting is an old and well-practiced art, there still seems to be some interest in questions regarding what methods can be used and why. Even though there are many good references on this subject, it may be of some value to discuss a particular method which has been found to be useful, and to make available the associated tables for its use in the hope that there can be saved a few of the tedious hours necessary to reduce many of the suggested methods to practice. Although a very brief discussion of the mathematical background of the tables is included immediately below, those interested only in their use can skip directly to the section on Use of the Tables. Direct Fit of a Set of Points A frequently encountered problem is: Given data which can be represented by a smooth curve +(x), to find an easy-to-evaluate function F(x), with only a few coefficients, which is a good approximation of +(x) on the interval 0 5 x 5 x,,. Consider F(x) as a linear combination of m + 1 arbitrary, single-valued, continuous, linearly independent functions, f,(x), Ff.r) = a,f,,(x) + a,f,(x) + . . . -+ aifi(x) + If the value of Q(x) is known or can be read from the curve at the points xj: +(x,), +(x,), . . . +(xj) . . . +(x.), and if n = m, then all the coefficients of F(x) can be determined, and F(x,) — +(x,) = 0 for j =. 0, 1, . . . n. This follows from the solution of the set of linear simultaneous equations. +(x<.) = aofu(x,) + aJI(x0) + . . . + a,,f,,(x,) =a, = f,,(x,) + a,f,(x,) + . . . + a,f,,(x,) +(x,,) = a,,f,(x,,) + a,f,(x,) + ... + a,,f.,(x,,) ...........(2) for the coefficients ai. Although the set a, from the solutions of the set of Equations 2 yields an fix) which gives an exact value of $(x,) for the n + 1 points, there is no restraint on ~(xj in the interval xi < x < xj+,. Indeed, if the points +(x,) are not sufficiently smooth, very serious differences may arise between F(x) in the interval and the smooth curve +(x). Least-Squares Fit It has been suggested&apos; that if n is made to exceed m, and the set a, is found to minimize the function the function F(x) is much less sensitive to small errors in +(xi), and usually provides a much better fit to the smooth function +(x). This is known as the best fit in the sense of the least-squares error. To minimize $, set £ = 0,i = 0,1,... iTi,.....(3) The m + 1 equations from Equation 3 are k—n k=n a.^p /,,(a) f,(xt) + a, X A: = 0 k=O k = n k = n k=0 k=0 +(x.) f,(x,),............(4) one equation being obtained for each i = 0, 1, . . . m. Inasmuch as the arbitrary functions f(x) may be evaluated for x,, k = 0, 1, . . . n, and the +(x,) are known values of the curve to be fitted, the simultaneous Equations 4 may be solved for the set a,. Particular Functions Given the functions fi(x) and the set +(xj), the solutions for require evaluating a number of sums of products as well as the solution of an m + 1 order set

Jan 1, 1956

By Jan Visman, S. J. Aresco

INTRODUCTION The accurate sampling of coal, as with most minerals, is a difficult task. Coal is a heterogeneous material made up of different types of coal and varying amounts of mineral matter. The product as mined may contain all the layers of coal and impurities found in the seam as well as portions of the stratas above and below the coal seam. The preparation plant is the device to size, crush, and/or remove impurities so the coal may be shipped as a salable product. Sampling may be needed at many stages in the preparation process. Although this chapter is primarily concerned with sampling in connection with preparation processes, the principles apply to all coal sampling. The main reasons for sampling in connection with coal preparation are: 1) To determine the washability characteristics of the raw coal by means of float and sink tests. These may be used for the design of a preparation plant, prediction of results to be obtained, or monitoring raw coal delivered to the plant. 2) To check the performance of various operating units in the plant, either by float. and sink tests or chemical analyses. 3) To determine the analytical characteristics of the products produced, such as moisture, ash, Btu, sulfur, etc. HISTORY The earliest accepted methods of sampling coal in the United States were given in US Bureau of Mines (USBM) publications by J. A. Holmes,l G. S. Pope,2 and A. C. Fieldner.3 The American Society for Testing and Materials (ASTM) more or less adopted the methods proposed by USBM in their Standard D-21-1916 and it remained a standard for many years. Briefly, this standard called for a 500 lb (227 kg) sample for coals less than 94 in. (19 mm) in size and a 1000 lb (454 kg) sample for all coals over 94 in. (19 mm) in size. No mention was made of increments, but there was an instruction that the sample should spread over the coal

Jan 1, 1979

By V. F. Parry, J. B. Goodman

The low-rank coals containing 10 to 50 pet natural bed moisture represent over half of the tonnage reserve of the available solid fuels of the United States, but only about 2 pet of United States coal production is derived from these fuels. Increased utilization of these large reserves will result if the bed moisture can be removed at the mine in high-speed low-cost processes. This paper presents a theoretical and experimental study of the factors affecting high-speed drying of coal dusts. The time required to heat particles of coal is proportional to the square of their diameter, and it is calculated that the probable time in minutes is equal to 6D2. In other words, a 11 a in. coal particle should be heated to drying temperature in 1.5 to 2.0 sec. Likewise, 1/8 in. particles should be heated in about 6 sec. The net heat required to dry subbituminous coal and lignite varies from 240 to 375 Btu per pound, depending upon the moisture removed, and this heat can be obtained from 42 to 61 cu ft of hot gases at 600°F.

Jan 1, 1949

By Ernest A. Hersam

IN every commercial establishment,&apos; it is customary and necessary to take inventory, periodically, and to account for profits and detect losses, to achieve productiveness and enhance efficiency. So, too, in the ordered life of every individual, there is the need of periodic introspection. The exercise of the Sabbath for rational thought, the utilization of pause, or rest, from the technique of daily occupation, orient the efforts. These are the constructive opportunities by which we grow always better than we were. Thus, also, there&apos; is need of a periodic analysis of the organizations to which we belong and toward which we con- tribute. So we find&apos; reason for the fitness of our loyalty. There seems to be no half-ground for the engineer. Whatever presumes to bear his name demands either the crushing pressure of his heel or the supporting pressure of his hand.

Jan 1, 1929

"The richest hill on earth" had its start as a placer gold camp in 1864. The gold was quickly worked out but then silver was discovered in the copper ores in Butte Hill. After ten years as a silver camp, rich copper ore was found in 1881 and the future of Butte was assured. In the early 1960&apos;s, the Anaconda Company began the shift from underground mining at Butte to open pit mining of the enormous, low-grade copper deposit surrounding the old high-grade veins which had been worked for over eighty years. Their first primary crusher was a 66 in. by 84 in. (168 m by 2. 13 m) jaw crusher. The pit was expanded at such a rate, however, that its location became endangered. Additionally, the new Clyde Weed concentrator had been built at Butte; and in 1969 construction of a new gyratory installation was started. The primary crusher at the Clyde Weed concentrator is one of the most unusual primary crushers ever put into operation. At times, the ore from the Berkeley pit contains large amounts of timber, pipe, rails, and so forth left in the ore body from the old underground mining operations; and the crusher installation had to be designed so this tramp material could be readily removed ahead of the crusher. Ore is delivered by 100- and 150-ton trucks to two side-by-side dump pockets, each with a capacity of 400 tons. Originally, a hydraulic grapple was installed on the dividing wall between the dump pockets; but it was found that oversize boulders large enough to block the discharge opening at the bottom of each hopper were rapidly covered with other material, and the grapple was unable to reach and remove the large boulders causing the blockage. In addition, the grapple was located in a quite vulnerable position; and it sustained damage from being struck by rock dumped from the trucks. Oversize boulders are now blasted. Underneath each pocket is a vibrating grizzly feeder with the grizzly bars set at 4 in. (102 mm). These feeder-grizzlies are 84 in. (2. 13 m) wide and 32 .ft (9. 75 m) long, with grizzly sections 16 ft (4. 88 m) long. Approximately 75% of the ore passes through the grizzlies and grizzly oversize then drops onto two parallel 6 ft (1. 83 m) wide picking belts. The belts are 3 in. (76 mm) thick and rest on idlers supported by air- filled rubber shock absorbers. At the side of each picking belt is a hydraulic grapple which is used to remove timbers too large to be removed by hand. At times, four men are used to remove timber and tramp iron. Timbers are dropped onto a transverse belt conveyor which discharges into a truck stationed outside the primary crusher building,

Jan 1, 1978

By H. LIVINGSTONE LMAN

A GOOD DEAL of information concerning flotation has come out during the patent litigation of recent years, and the legal situation has cleared considerably, to the satisfaction of Minerals Separation, Limited. Hence, Mr. H. Livingstone Sulman, one of the originators of the successful process of that company, felt at liberty to explain, on Thursday, Nov. 20, his experience and his views in a "Contribution to the Study of Flotation," before the Institution of Mining and Metallurgy. He did not give technical details. He dealt with the physical problems, so simple, as he remarked, that he had wanted 96 pages to express his views. He presented much data, partly new and based on observations of his own, bearing upon the very complex flotation phenomena in an instructive wav and made frequent references to the valuable researches especially of Prof. Edwin Edser and Mr. Emil Hatschek.

Jan 1, 1920

ARIZONA Established July 10, 1915 ROBERT W. THOMAS. Chairman ROBERT W. HUGHES, Vice-Chairman HUBERT O. WOODS, Secretary-Treasurer American Smelting & Refining Co., Hayden, Ariz. LYLE BARKER M. G. FOWLER RICHARD BRITTAIN M: C. HENRIE B. S. BUTLER F. D. 1. HONEYMAN T. G. CHAPMAN J. E. LANNING C. H. COLE A. MENDELSOHN G. C. DAVIS BRENT N. RICHARD E. Ii. DENTZER C. F. WILLIS

Jan 1, 1943

By Francis A. Thornson

WITHOUT desiring to perpetrate an Irish bull I think we may safely say that the major developments of the year in mineral industry education have taken place outside of the field itself. I refer to the remark- able and-to us of the Education Di- vision-most gratifying growth which is taking place in Institute consciousness regarding its responsibility for the recruitment and adequate training of student personnel and for the successful bridging of the difficult gap between graduation and achievement of professional status. Outstanding in this respect is President Jackling&apos;s interest in the students and schools of the mineral industry. Recalling his own hard struggle to ac- quire a technical education, Mr. Jack- ling is evidently anxious to smooth the path for the youngsters of this generation. Witness his splendid gift to the

Jan 1, 1939

By A. W. Fahrenwald

FLOTATION practice in Idaho is now about 13 years old. The advance has been steady during these 13 years. The operators have been alert to take advantage of the newest developments and they have themselves made a constant effort to improve their work. The metallurgical improvements are due to the development of (1) better equipment, (2) better reagents, and (3) better understanding and control of the process. During the eight years the writer has been in this district, there has been continuous experimentation by several companies to improve the ore dressing by flotation methods. Gravity concentration has fought hard to hold its place, and although it is true that gravity processes have been improved during this period, nevertheless, in several of the important mills of the district the few tables that managed to hang on have finally given way. On the straight-lead ores, tables and jigs are still used; and, due to better classification, their hold on this job has been strengthened. ORES OF THE DISTRICT The ores of the district are of two general classes: (1) clean lead ores, such as those from the Bunker Hill and Hecla mines, and (2) complex lead-zinc ores. The general impression prevails that the clean lead ores, class 1, are simple and that it is easy to obtain high recoveries and good grade products. This, in a way, is a false impression; there is quite a high percentage of locked middling in these ores at 100 and 150-mesh sizes. Improved classification preparatory to table treatment of this type of ore has made it possible to segregate effectively a high-grade middling which is further comminuted and floated. The complex lead-zinc ores of the district, of which the ore from the Morning mine of the Federal Mining & Smelting Co. is a typical example, always have presented a difficult problem. It always has been comparatively easy to make a clean lead concentrate, but it never has been possible to concentrate sphalerite with a-specific gravity of 4.10 from siderite of gravity 3.88. Also, a large percentage of the galena and sphalerite of this class of ore refuses to separate out until the crushing has been carried

Jan 1, 1928

By George Otis Smith

A YEAR ago, ,at the Institute&apos;s dinner, I closed my A remarks with the words: "The scientist devotes his life to the advancement of learning; the engineer gives his to the advancement of living." May 1 begin tonight where I left off then and speak first of the distinction between science and engineering? This difference is a difference both in purpose and in method: it is the sharp contrast between theory and practice; it marks the broad gap between dream and attainment. Sometimes it happens that the investigator is also the builder and applies what he discovers, yet the distinction between science and engineering serves to point out the broad dependence of the engineer upon his brother the scientist. You remember .the old - saying that "where there is no vision the people perish" translate this into modern experience and we may say that "where there is no science, the engineer fails."

Jan 1, 1929

By Albert E. Ketler

Introduction – The introduction of computerized monitoring and control (M/c) equipment into the U.S. coal mining industry may prove to be of truly revolutionary proportions in the 801s.Many of you are aware that the British have developed and have been using such sophisticated M/C equipment for performing production and safety related tasks in their coal mines for nearly a decade. The benefits of M/C systems in the mining industry are usually separated into those which improve productivity, and hence make money for the company, and those which improve safety, and hence cost money. The economic benefits of M/C are startling for the production related functions such as for monitoring and controlling conveyor belts, de-watering pumps, electrical power distribution, and the like. It doesn&apos;t take much analysis to show that hundreds and even thousands of tons of coal don&apos;t get mined every time a belt shuts down or when power is out due to circuit breakers tripping for any number of reasons. A flood, caused by a malfunctioning pump which goes undetected for lack of feedback monitoring of pumping parameters, can destroy literally millions of dollars worth of mining equipment. The M/C system would be justified or not based on whether or not the productivity benefits balanced the cost within a timespan commensurate with the return on capital philosophy of the company. The safety related benefits are easily underestimated or ignored completely and the purpose of this paper is to under-score some of these benefits, as well as to review the application of M/C to the coal mining industry. The companies involved are Transmitton Ltd of England and Minetics Corp of Pittsburgh, Pennsylvania. Transmitton is probably the most experienced manufacturer of M/C equipment in the world having some 200 installations to its credit. Minetics is the marketing and manufacturing arm for Transmitton in the U.S.A. and is responsible for all aspects of distribution and service. Plans are underway for manufacturing of the Transmitton hardware for U.S. sales to be transplanted into U.S. subcontractors by 1981. Thus our coal mining industry will benefit from British technology, but without the delays and additional costs usually involved in international shipments.

Jan 1, 1981

By Emil-Paul Lorenz

Considered as a whole, the mineral resources of the Turkish Republic (Anatolia) are in their untapped virgin state, and the little development shown is not the result of modern systematic geologic exploration but the continuation of the ancient open-pit mining methods of the Orient. When I came to Turkey in 1924, I found myself confronted by a formidable task because of the lack of preliminary geologic mapping. Today this condition has not essentially changed; less than one per cent of the country is covered by topographic mapping adequate for modern requirements. However, I was able to bring a semblance of order in a presentation of the mineral deposits of this country, by tracing the story of the mineral deposits that showed surface evidence of their existence. At present only the easily discovered mineral deposits of coal, copper, and lead-silver ores are being exploited. In the case of coal, of 3,000,000,000 metric tons available, only 100,000,000 have been mined. The same is also true for copper, lead-silver, and other ores, as shown in Table 1.

Jan 1, 1948

By James Aston

A PROPER consideration of this subject is not confined to the technical channels of production and metallurgy. It concerns an industry, and should cover economic aspects which are of material importance in any program of development. The discussion may logically be treated under three subdivisions, as follows: (1) manufacture, (2) characteristics related to quality of material; (3) adaptability and exploitation of product. MANUFACTURE OF WROUGHT IRON The manufacture of wrought iron is the most ancient art of the ferrous metal industry. The eighty years or less of existence of processes identified with our modern "Age of Steel" is a relatively brief interval compared with the several thousand years associated with the production of wrought iron by man. From the primitive early methods of production, manufacture for many centuries followed the line of direct reduction. There were developments as experience was gained through the years; but man&apos;s needs for a strong, ductile, forgeable product were met by reducing the ore to metallic iron in a single-stage operation. Following the development several centuries ago of what has become our modern blast-furnace operation, manufacture of wrought iron was gradually transformed into a two-stage method of first producing cast iron and then refining this to the desired degree of forgeability; the forerunner in principle of today&apos;s scheme of manufacture of steel. The product retained the essential characteristics of highly refined base metal with iron-silicate slag incorporation, which were associated with the material resulting from direct reduction methods. Cort&apos;s invention of the puddling process in 1784 is a landmark in wrought-iron manufacture. It consisted essentially in adapting a coal-fired reverberatory furnace to the refining operation; thus lifting the industry out of the slough of

Jan 1, 1935

By C. A. Stetefeldt

Jan 1, 1895

By JAMES A. VEASEY

SINCE the World War, excepting for a few brief periods of relief, the American petroleum industry has been obliged to meet its important economic responsibility to this nation hampered by the maladjustment that inevitably follows a marked and unwholesome over-production of its raw material, crude oil. Despite the warnings of the Federal Oil Conservation Board and the endeavors of many of the leaders in the business to stem the flow of oil from the wells of the country, the evil ,not only persists but, at the moment, seems more malignant than ever before. Unless the local and temporary restraints on drilling and production now in effect in California, Wyoming, Texas, and Oklahoma are continued, and, indeed, unless stricter and more comprehensive measures of the same sort are imposed and adhered to by the industry in those States, over-production in the enterprise during the current year will be more pronounced than in any other period in history.

Jan 1, 1929

By ARCHIBALD DOUGLAS

AT various times during the past quarter century proposals have been made that the basic Federal mining law of 1872 be repealed or amended, and that in its place a new and simpler law be enacted to provide for the entry upon, working, and the taking up of claims on the unallotted Federal mining lands in our western states and in Alaska. Serious discussion and study have been given such proposals but little has resulted there from, except perhaps a lively interchange of opinion and the conclusion that though the Apex law of 1872 is in many ways highly unsatisfactory and frequently a fomenter of strife and trouble, it still bids fair to remain with us. Under this act a greater part of the non-ferrous mining industry of the United States has developed. With all its short-comings it is the legal bridge that has carried over the innumerable caravans of adventurers and prospectors who explored our mining frontiers. For without any amendments of substance this act has for fifty-eight years been the guide in the location of our western Federal mineral claims and has stood unchanged during the most important period of our mining history. While Federal and state acts and amendments have been ground out by thousands and tens -of thousands the mining act of 1872, with a :strange tenacity, has remained as constant as the Rock of Gibraltar, and under it states, mining districts, and local customs have developed different codes for its application but its fundamentals have been unaltered. "Use doth breed a habit in a man"; until now the application of the law of 1872 has become a second mining nature.

Jan 1, 1930

Arthur Yates, lecturer in the mining department of the Royal School of Mines, London, died at Blackpool, on Feb. 14, 1923, at the age of 47 years. My first acquaintance with him was made in 1902, when taking over the management of the Redjang Lebong gold mines in Sumatra, of which he was the responsible metallurgical officer. For that position he was well qualified by reason of his previous experience with the Rand Central Ore Reduction Co., in South Africa, and particularly by his early training with the Cassel Gold Extracting Co., owners of the MacArthur-Forrest patents, and the first commercial company formed to work the then recently discovered cyanide process; in this connection, Mr. Yates had the advantage of working under J. S. MacArthur himself. Our association in Sumatra was maintained for six years, when I left. Mr. Yates remained and was eventually promoted to the management of the neighboring Soelit mine. During those six years and in the close association inseparable from life in an isolated mining camp, the sterling qualities which made him so good a man were unmistakable: industry and interest in his work, patience with everybody, a quiet and unassuming demeanor. On the exhaustion of the mine to which he had been promoted, he returned to England in 1913. In 1914, the Mining Department of the Imperial College was fortunate in securing his services. Hardly, however, had he settled down when the war broke out and he spent some years in the Cleveland District promoting the efforts of the Government toward greater output. Returning to college in 1919, Mr. Yates met the full flood of Service students. The qualities which made him a man amongst men in the world made him a man among students in the college; a senior student if you like. But, unsuspected, a dread disease had fastened itself upon him, making itself felt for the first time during the summer course he was conducting in the South Wales coal field, in 1922. Against its relentless onset he opposed the bravest optimism, and when at last he realized that he was going under, and was in fact dying, his pronounced regret was that he was leaving the boys. In his death, at the age of forty-seven, staff and students in the Mining Department have experienced a great loss, that of a reliable friend, and their deepest sympathy goes out to the young widow and son in their irreparable loss. S. J. TRuBcott.

Jan 1, 1923

Arthur Yates, lecturer in the mining department of the Royal School of Mines, London, died at Blackpool, on Feb. 14, 1923, at the age of 47 years. My first acquaintance with him was made in 1902, when taking over the management of the Redjang Lebong gold mines in Sumatra, of which he was the responsible metallurgical officer. For that position he was well qualified by reason of his previous experience with the Rand Central Ore Reduction Co., in South Africa, and particularly by his early training with the Cassel Gold Extracting Co., owners of the MacArthur-Forrest patents, and the first commercial company formed to work the then recently discovered cyanide process; in this connection, Mr. Yates had the advantage of working under J. S. MacArthur himself. Our association in Sumatra was maintained for six years, when I left. Mr. Yates remained and was eventually promoted to the management of the neighboring Soelit mine. During those six years and in the close association inseparable from life in an isolated mining camp, the sterling qualities which made him so good a man were unmistakable: industry and interest in his work, patience with everybody, a quiet and unassuming demeanor. On the exhaustion of the mine to which he had been promoted, he returned to England in 1913. In 1914, the Mining Department of the Imperial College was fortunate in securing his services. Hardly, however, had he settled down when the war broke out and he spent some years in the Cleveland District promoting the efforts of the Government toward greater output. Returning to college in 1919, Mr. Yates met the full flood of Service students. The qualities which made him a man amongst men in the world made him a man among students in the college; a senior student if you like. But, unsuspected, a dread disease had fastened itself upon him, making itself felt for the first time during the summer course he was conducting in the South Wales coal field, in 1922. Against its relentless onset he opposed the bravest optimism, and when at last he realized that he was going under, and was in fact dying, his pronounced regret was that he was leaving the boys. In his death, at the age of forty-seven, staff and students in the Mining Department have experienced a great loss, that of a reliable friend, and their deepest sympathy goes out to the young widow and son in their irreparable loss. S. J. TRuBcott.

Jan 1, 1923

By W. Rostoker, A. S. Yamamoto, K. Anderko

ALTHOUGH the corrosion resistance of magnesium and its alloys is closely related to iron content, there has been no direct measurement of the solid solubility of iron in magnesium. Bulian and Fahrenhors;1 and Mitchel]2 agree that pure iron or a limited terminal solid solution crystallizes from the Mg-rich liquid. For this reason a magnetic-moment method was selected to estimate that portion of the total iron content which is not in solid solution. Since iron in solid solution in magnesium cannot contribute to ferromagnetism, the difference between chemical and magnetic-iron analyses should yield the solid solubility. By experimentation it was found that the melting of pure sublimed magnesium (99.995 wt pet purity) in Armco-iron crucibles at about 800°C is a convenient way to introduce small amounts of iron. Melts retained 5, 10 and 20 min at 800°C analyzed 0.003,, 0.005,, and 0.018 & 0.001 weight pet Fe, respectively, after being stirred, heated to 850°C, and cast into graphite molds. The as-cast alloys were pickled in acid (dilute HC1 + HNO3), annealed at 600°C for 3 days, scalped on a lathe to remove the pitted surface, pickled again, extruded at about 100°C to 3-mm wire, reannealed 41/2 days at 500°C, and water-quenched. The specimens were again scalped, pickled, and used both for chemical and for magnetic analysis. Most of the precautions described were intended to prevent iron pickup by contact with tools or superficial iron enrichment by volatilization of magnesium during heat-treatment. It is believed that the specimens ultimately used for test were homogeneous and characteristic of phase equilibria at 500°C. Magnetic Analyses A susceptibility apparatus of the Curie type was used for magnetic analyses. Field strengths of up to 10,400 oersteds could be generated. By this method, an analytical balance measures the force of attraction which a calibrated magnetic field exerts on a suspended specimen. The force equation is as follows f/m = M dh/dy where f/m = force per unit mass of sample M = magnetic moment per unit mass dH/dy = magnetic field gradient The dH/dy characteristic of the instrument is determined by the use of a standard palladium sample, and the calibration is made independently for all values of H. Since a large finite field is required to saturate an assembly of ferromagnets, it is necessary to measure the apparent magnetic moment for increasing steps of H until a saturation value is obtained. The percentage of iron in the sample as free ferromagnetic iron may then be computed simply C= 100 (M1/M1) where C = percent content of undissolved iron in sample M1 = saturation magnetic moment of sample per unit mass M1 = saturation magnetic moment of iron per unit mass taken as 217 emu-cm per gm There is no serious difficulty in applying this method except for the unusual magnetic behavior of very fine particles of ferromagnetic substances. It has been found and is the basis for a widely accepted theory that with sufficient subdivision, the magnetic fields required to saturate and the coercive force after saturation rise to exceedingly high values. Recent work on precipitates of Fe and Co from copper solid solutions8 showed that about 5000 oersteds were necessary to approach saturation. The magnetic moments as a function of field strength measured in the present investigation are listed in Table I. Only the 0.018 wt pet Fe alloy yielded a magnetization curve with a fairly well-defined saturation plateau at 3.76x10 -2 emu-cm/ gm. This corresponds to 0.017 & 0.001 wt pet Fe in the alloy. This indicates that the solid solubility must be of the order of 0.001 wt pet Fe. The magnetic-moment data of the other two alloys are badly scattered, indicating that the amount of ferromagnetic iron in these samples is so low that the magnetic forces acting on them cannot be measured accurately by the analytical balance used. Nevertheless, the fact that even the 0.003, wt pet Fe alloy shows ferromagnetism indicates that the solid solubility must be below that value. Acknowledgment This work was sponsored by the Pitman-Dunn Laboratory of Frankford Arsenal, Philadelphia, Pa. The support and permission to publish are gratefully acknowledged. References W. Bulian and E. Fahrenhorst: Zeic. Metallkunde, 1942, vol. 34, pp. 116-170. 2 D. W. Mitchell: AIME Transactions, 1948, vol. 175, pp. 570-578. 3 G. Bate, D. Schofield, and W. Sucksmith: Philosophical Magnsine, 1955, vol. 46, pp. 621-631.

Jan 1, 1959

By D. M. James, J. C. Martin

The results are presented of a study of the application of analytical methods to the solution of two-phase flow into single wells. Approximate analytical expressions for the pressure distribution in two-phase flow are found ;;; a number of conditions. The results obtained from the analytical solutions me found to be in good agreement with results obtained by finite difference techniques using a high speed digital computer. Mathematical solutions for four sets of boundary conditions are presented. All of these solutions are composed of a short-term transient plus a steady or quasi-steady state. The rates of decay of the short-lived transients are analyzed. It is found that the durations of the short-term transients may be characterized by a parameter defined as the time constant which can be determined from simple relations. It is shown also that if the outer radius is much greater than the radius of the well. the short term transients decay at rates which are proportional to the square of the exterior radius, and the rates of decay are only slightly dependent upon the radius ratio. Numerical solutions based on finite difference techniques are presented for a number of conditions. The numerical solutions are in good agreement with the predictions based on the theoretical analysis for small and moderate drawdowns. Examples involving large drawdowns indicate that the nonlinearities in the equations of flow do not appreciably alter the longevity of the short-term transients. In all cases the time required for the short-tern transients to disappear is predicted satisfactorily. INTRODUCTION The mechanism by which oil and gas flow into a single well is of vital interest to the petroleum industry. The fundamental equations of two-phase flow which describe this mechanism are nonlinear partial differential equations. Numerical solutions of these equations describing pressure transients have been obtained with the aid of electronic computers. Although solutions obtained in this manner take into account a large number of effects, the reduction of this information to useful generalities is difficult. One method of obtaining generalities is the use of linearized approximations of the nonlinear equations. Since it is possible to obtain explicit solutions of the linearized equation, general properties of the role of pressure in the flow mechanism may be ascertained. Results obtained from this approach are limited to some extent by the linearizing assumptions. The severity of these limitations may be evaluated by comparing solutions of the linear equation with numerical solutions of the more exact nonlinear equations of two-phase flow. In the past considerable amount of work has been devoted to studying pressure build-up using the single-phase flow theory. Unfortunate1y, most prep sure build-up tests involve multiphase flow. A small amount of work has been done studying pressure build-up where the flow is two-phase. The encouraging results of these studies suggest that useful results may be found from additional studies of not only pressure build-up, but also the rapid transients associated with placing a well on production. This paper presents the basic theory of the pressure transients associated with placing a well on production and with closing it in. The paper is concerned chiefly with two-phase compressible flow; however, the results also apply to single-phase flow. The results are based on analytical solutions of the flow equations, and they are verified by numerical solutions using finite difference techniques. Much of the previous work on compressible flow into wells has been confined to single-phase flow. Some work has been done on compressible two-phase steady state flow, and solutions of the equations of flow have been found by finite difference techniques using high-speed computers. Muskat1 presents some rather general solutions to the equations of single-phase compressible flow into wells. Much work has been done on pressure build-up in wells (see, for example, Refs. 2-6). Almost all of the work on pressure build-up concerns single-phase flow with the exception of Ref. 5 and part of Ref. 2. Some work has been done on pressure fall-off in injection wells.718 Muskat9 presents the solution of the equations for radial steady state two-phase compressible flow. Handyl0 extends Muskat&apos;s solution to obtain