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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 Roswell Johnson

To know the rate of decline of oil wells is very important, yet ordinarily we are prevented from getting this rate because the oil from several wells is put into one or a few tanks as soon as the wells have been connected to the power. The object of doing this is economy of tanks, pipes, labor and accounting. FIG. 1. But we are making a great sacrifice for this small economy. The advantages of the individual records are that we would know better: 1. When the well has ceased to be profitable. 2. When a well has gotten into bad condition. 3. The real record of the yield per acre and the rate of decline of the wells. The lease total is too mixed as to age of the wells for existing records to have much significance. To obtain such individual well records and more accurate logs are the great desiderata of satisfactory management of oil properties. 4. Which wells show the slowest decline-information which is valuable in locating new wells or purchasing adjoining leases.

Jan 6, 1917

By B. E. Mix, L. M. Barker

THe present methods of unloading ore and coarse-crushing at the Magna plant of the Utah Copper Co. are the developments of the pas five years. Hand dumping and breaking have given way to the rotary car dump and large gyratory; and the bucket elevators and stationary screens have been replaced by conveyors and shaking screens. The coarse-crushing plant may be broadly divided into two units the primary crushing and the secondary crushing, or primary roll, unit The former was erected in 1923; with the exception of the excavation the construction was carried on by forces of the company. Excavation was started in the middle of March, the first concrete was poured July 16, erection of the steel building started Oct. 30, and the unit was com pleted and operating Dec. 30. The unit has a capacity of over 24,001 tons of ore per day, and cost more than a million dollars. Remodeling of the primary-roll unit extended over a period of years preceding this some of the work having been done as early as 1918. Delivery of Ore from Mine, Yard Storage, and Handling Ori Cars to and from Primary Crushing Plant The ore is transported, in steel cars of 80 tons capacity, from the min&apos; to the mill, where a reserve tonnage of loaded cars is maintained. Malle locomotives, handling trains of 45 cars, or approximately 3600 tons o ore per train, deliver the trains to a storage yard which contains thre tracks—a passing track and two storage tracks. All of this trackage i equipped with an overhead trolley system, so that electric locomotive can be used in yard switching. The storage tracks have a total length of 4500 ft. and provide storag space for 140 cars. The passing track, besides permitting switchin room for locomotives, is used for storage purposes between train arrivals thus giving intermittently a total storage capacity for the yard of approxi mately 200 cars. The amount of ore stored at any one time, howeve

Jan 1, 1926

By R. N. Stevens

GRAIN boundary sliding plays a part in the high-temperature deformation of polycrystalline solids, and many investigations1-11 have been carried out to determine the deformation contributed by this mechanism. The principal method employed has been to measure one of the components of the grain boundary displacement, see Fig. 1, at a large number of boundaries, and from the mean value so obtained to calculate the elongation contributed by sliding. A formula connecting these mean values and the elongation contributed is evidently required, and many have been derived. They are usually of the form where Egh is the tensile elongation contributed by sliding, is the mean value of the component, x, N is the mean number of grain boundaries intersecting a line of unit length, and K is a dimensionless parameter known as the averaging factor. Table I summarizes the equations used by various investigators. Examination of these shows that their derivations usually involve doubtful assumptions, and are often in serious error. The purpose of this note is to point out that rigorous solutions to the problem of calculating Egb from x have been available in the literature for some years.&apos; Since this is so little known, and also the original derivations do not make some important points clear, it is proposed to rederive the basic equations. These will be extended further, and the conditions in which the extended equations may be applied will be indicated. The mean values of the components of the sliding vector (I, d, and I:, Fig. 1) will depend on the method by which boundaries are sampled in order to make measurements.3,11 Boundaries are usually sampled by marker lines parallel to the stress axis, or lines perpendicular to the stress axis, the measurement being made at the point where the marker line intercepts the boundary. We will denote the mean values determined by these various sampling methods as follows. lT, dT and vT are mean values obtained by making measurements at pointswhere a transverse line intercepts the boundary while lL, dL, and vl, are the mean values from a longitudinal line. The number of boundaries per unit length also may depend on the direction of measurement, and so we define NT and Nl to be the number of boundaries per unit length, after creep, in the transverse and longitudinal directions, respectively.

Jan 1, 1967

By H. B. Small

The Ottawa river, the northeastern boundary of the Province of Ontario, and the dividing line between the latter and the Province of Quebec, has long been famous for the rafts of timber floated over its waters from the depths of the forest, and other poets besides Moore, have immortalized its beauties in verse. But with the ad vent of the railway which now pierces the forest solitudes, the days of rafting were numbered ; and the boat song is now heard no more on "Ottawa&apos;s tide," or at such long intervals only, that, when heard, it seems an echo of the past. The lumber trade is fast leav ing Ottawa City, the old headquarters of the business; and such square timber as is now cut is gradually finding other outlets than shipment to Great Britain, and other channels of transportation than the Ottawa river. Skirting the northern shore of this great river runs a vast chain of hills, assuming at times the altitude of mountains. Their geological character marks them as remnants of the earliest land of this conti nent. Following the course of the St. Lawrence from the Gulf of that name to the confluence of the river with the Ottawa, the course of which latter they then pursue, the range has been styled " the Lau rentians." Rolling in long undulations, with rounded, rather than rugged or pointed summits, they leave a margin of 8 or 10 miles, dipping in a gentle slope to the water&apos;s edge, rich for agricultural use, and dotted here and there with farms and settlements. Beyond this margin bold hills stretch northerly to blue mountains, rich in iron, lead, plumbago, and other minerals. The Apatite District.—One of these minerals is apatite, or phos phate of lime, the use of which as a material for the manufacture of superphosphate as a fertilizer has become the basis of the industry bidding Fair to prove ere long one of the chief elements of Canadian trade. So rapid has been the growth of this new industry that if the lumber business were to disappear from this mineral district, its void would be filled by that which has sprung up in connection with phosphate.

Jan 1, 1893

By H. W. Morse

INTRODUCTION THE experimental work on the oxidized copper ore at the New Cornelia mine at Ajo, Ariz., ended on Jan. 12, 1916. On that date final decision was made on the general nature of the process to be used in the 5,000-ton leaching plant, and on many of the details, as far as experience on a40-ton scale could decide them. With the approval of the Board of Directors and the General Manager, John C. Greenway, we have compiled what seem to be the most interesting data on the results obtained during the experimental period. The most important part of the data resulted from the operation of a 1-ton and a 40-ton plant at Ajo. Experimental work on Cornelia oxidized ore dates back to April, 1912, and has been going on nearly continuously since that time. A good many variations from the original idea have been tried, but the process finally decided upon is in principle and in all of its details a simple one. The history of the leaching work on this ore has been brought down to about a year ago in papers presented to this Institute by Stuart Croasdale1 and Dr. L. D. Ricketts.2 The present paper will, therefore, deal principally with the results obtained during the last year of the work.

Jan 9, 1916

By C. F. Weinaug, P. M. Blair

Many difference equations used to approximate reservoir flow problems treat the phase pressures implicitly but not the mobility-density coefficients. Such difference equations are neither wholly explicit nor implicit, but might be described as mixed. Mixed equations are relatively easy to apply. But the associated time truncation error is relatively large, and when used to solve problems characterized by high flow rates, these equations may be unstable for practical size time steps. This paper outlines the development of a completely implicit difference analogue for reservoir simulation, along with a Newtonian iterative method for solving the resulting nonlinear set of algebraic equations that arise at each time step. While this implicit equation requires two to three times more work than does the mixed equation, it is shown to markedly decrease the time truncation error and to yield a stable solution for much larger time steps than does the mixed equation. INTRODUCTION Calculation of multiphase, multidimensional flow in porous media is generally accomplished by numerical methods that involve approximating systems of partial differential equations by systems of partial difference equations. The early development of difference equation techniques was directed toward the solution of linear differential systems. However, the equations of multiphase fluid flow through porous media are highly nonlinear in that mobility and density often are strong functions of pressure and saturation. Thus, solution of the flow problems by difference equations involves solving sets of algebraic equations whose coefficients change from step to step of the calculations. Current practice is to evaluate most coefficients at the beginning of a time step and then to apply difference methods well suited for solving linear problems. At least two major difficulties arise from the use of this practice. First, evaluation of coefficients at the old time level and evaluation of pressures at the new time level results in larger time truncation errors near displacement fronts than if all quantities in the distance derivative were evaluated at the same time level. Second, evaluation of mobility coefficients at old time levels results in an unstable difference equation in regions of high flow rate. The first of these difficulties often results in optimistic recovery behind the displacement front, and the second requires the use of small time steps or special techniques in solving coning problems1 or problems of gas percolation.2 Evaluation of all quantities in the distance differences at the new time level results in a completely implicit difference system. Such a difference system has a lower time truncation error than equations in which mobilities are evaluated at the old level and pressures evaluated at the new level. The algebraic equations resulting from the fully implicit difference equation are, in this case, nonlinear and require some iterative method for their solution. This paper develops fully implicit difference equations for two-phase flow in porous media, describes their solution by Newtonian iteration, and gives examples of problems more easily solved by the new method than by previous methods. EVOLUTION OF DIFFERENCE EQUATIONS FOR NONLINEAR PROCESSES The completely implicit difference equation used in these calculations results from evolution beginning with explicit difference equations, proceeding through what we will call "mixed" equations, and arriving now at a completely implicit equation. Let us illustrate the evolution by considering different analogs of the nonlinear equation

Jan 1, 1970

By B. D. Cullity, K. S. Sree Harsha

When a copper or aluminum single crystal is swaged into wire, the resulting deformation texture depends on the original orientation of the crystal. The<100> and <111>orientations me essentially stable, while <110> is unstable. The greater the <100> content of the deformation texture, the stronger the wire is in torsion. the greater the<111>content, the stvonger it is in tenszotz. The preferred orientation (texture) of fcc wires, either after deformation or recrystallization, is usually a double fiber texture in which some grains have <100> parallel to the wire axis and others have <111>. The relative amounts of these two texture components, as reported by different investigators for the same metal, vary considerably. Previous work in this laboratory&apos; has shown that the starting texture of a wire, i.e., the texture which it has before deformation, can have a decided influence on the texture produced by deformation. In particular, it was found that the deformation texture of copper wire is essentially a single <100> texture, if the wire before deformation contains only a <100> component. This is true even when the deformation is carried to more than 98 pct reduction in area. This paper reports on further studies of the role played by the starting texture. Copper and aluminum single crystals of various orientations have been cold swaged into wire, and quantitative measurements of the resulting deformation textures have been made. The tensile and torsional properties of the deformed wires have also been measured, and the relation between these properties has been correlated with the texture of the wire. These measurements were made in order to demonstrate that a cold-worked wire can be made relatively strong in torsion and weak in tension, or vice versa, by proper selection of the texture before deformation. MATERIALS The copper was of the tough-pitch variety, containing, by weight, 99.962 pct Cu, 0.003 pct Fe, 0.025 pct 0, and 0.0021 pct Si. The aluminum contained more than 99.99 pct .&apos;41; the only reported impurities were 0.001 pct Fe, 0.001 pct Si, and 0.003 pct Zn, by weight. Large single crystals of these metals were grown by the Bridgman method in graphite crucibles and a helium atmospliere. Cylindrical specimens of predetermined orientation, about 1.5 in. long and 0.36 in. in diameter, were machined from the as-grown crystals and then etched to 0.25 in. to remove the effects of machining. Their orientations were checked by back-reflection Laue photographs, and they were then swaged to a diameter of 0.050 in. (96 pct reduction in area). 111 order to study the "inside texture" of the deformed wires, they were etched, after swaging, to a diameter of 0.040 in. before the texture measurements were made. TEXTURE MEASUREMENTS The fiber texture which exists in wire or rod can be represented by a curve showing the relation between the pole density I, for some selected crystal-lographic plane, and the angle $ between the pole of that plane and the wire axis (fiber axis). Such a curve will show maxima at particular values of , and these values disclose the texture components which are present. The relative amounts of these components can be determined2&apos;3 from the areas under the maxima on a curve of I sin F vs F. It is seldom necesszlry to measure I over the whole range of F from 0 to 90 deg, since the existence of maxima in the low-F relgion can be inferred from measurements confined to the high-F region. The X-ray measurements were made with a General Electric XRD-5 diffractometer and filtered copper radiation, according to one or the other of the following procedures: 1) A method developed in this laboratory,4 involving diffraction from a single piece of wire. 2) A modification of the Field and Merchant method.5 This method was originally devised for the examination of sheet specimens, but it can easily be adapted to the measurement of fiber texture. Three or four short lengths of wire are held in grooves machined in the flat face of a special lucite specimen holder. The axes of the wires are parallel to the plane defined by the incident and diffracted X-ray beams, and the holder to which the wires are attached can be rotated step-wise about the diffractometer axis for measurements at various angles 9.

Jan 1, 1962

By W. E. Keck, Paul Jasberg

THE flotative properties of the principal minerals in Michigan&apos;s potential iron ores have been investigated to develop methods of beneficiation for the ores. One of these minerals, magnetite, is of considerable importance as a source of iron. Study of its flotation was similar to that of hematite1, another iron mineral in these potential ores. The present research consisted essentially of: (1) a survey of the literature on magnetite flotation, (2) the preparation of pure magnetite, (3) the development of laboratory procedure and (4) flotation experimentation. Literature on Magnetite Flotation.-For the purpose of this discussion, the results of previous work on the flotation of magnetite can be summarized as follows. The mineral has been floated with pine oil2, and it has also been separated from synthetic mixtures with silicates by the use of the same reagent2. Magnetite and apatite in iron-ore slimes have been separated by flotation of apatite with sodium palmitate3. These data were utilized in the present investigation. Preparation of Magnetite.-The pure mineral was prepared from the magnetic-separator concentrate of an iron ore (obtained from the Witherbee- Sherman Corporation, Port Henry, N. Y.). This concentrate was slowly comminuted with laboratory rolls and pulverizer to pass 65-mesh and was then tabled several times. Tabling yielded a relatively coarse magnetite concentrate, tailing and slimes. The two latter materials were rejected insofar as the present work is concerned, and treatment of the magnetite (table concentrate) was continued by separating it into -65 + 200 and -200-mesh sizes. Separation of the sizes was made by first dry screening and then wet screening. The -200-mesh material was rejected to eliminate possible introduction of error into flotation This paper describes a part of a general investigation into the flotation of Michigan&apos;s potential iron ores. A comprehensive discussion of the status of these ores and of the reasons for research on their Rotation beneficiation is available in mimeographed form on application to the Michigan allege of Mining and Technology, Houghton, Michigan. Manuscript received at the office of the Institute Nov. 12, 1936. Printfed without author&apos;s corrections. Subject to revision.

Jan 1, 1937

By Macelwane

WHAT characterizes a geophysical engineer and sets him apart from all other engineers? This is a question that is important not only for accrediting purposes but is assuming increasing importance in the licensing of professional geophysical engineers by state boards. What is the basis of distinction of one type of professional engineer from another type is answered in part by the need which brought each type into being precisely because that need could not be satisfied by pre-existing types and which nevertheless demanded the special outlook, training, and abilities of an engineer. The need which called for the civil engineer was the adaptation to the tasks of civilian life of the capabilities developed by the military engineer for the design and construction of roads, bridges, dams, earthworks, military buildings, and other structures and of mechanisms required by military operations.

Jan 4, 1954

For the convenience of members who may be trawling from place to place, the Secretary has compiled a list of libraries or other places in the United States and foreign countries here MINING AND METALLURGY or the Transactions may he consulted. This list will be sent to members if they so request.

Jan 1, 1923

For the convenience of members who may be traveling from place to place, the Secretary has compiled a list of libraries or other places In the United States and foreign countries where MINING AND METALLURGY or the Transactions may be consulted. This list will be sent to members if they so request

Jan 1, 1923

For the convenience of members who may be traveling from place to place, the Secretary has compiled a list of libraries or other places in the United States and foreign countries where MINING AND METALLURGY or the Transactions may be consulted. This list will be sent to members if they so request.

Jan 1, 1925

For the convenience of members who may be traveling from place to place, the Secretary has compiled a list of libraries or other places in the United States and foreign countries where MINING AND METALLURGY or the Transactions may be consulted. This list will be sent to members if they so request.

Jan 1, 1925

For the convenience of members who may be traveling from place to place, the Secretary has compiled a list of libraries or other places in the United States and foreign countries where MINING AND METALLURGY or the Transactions may be consulted. This list will be sent to members if they so request.

Jan 1, 1925

By F. J. Smit

The Tintic Division of Kennecott Copper Corporation operates the Burgin Mine in the East Tintic District near Eureka, Utah, approximately 75 miles south of Salt Lake City. The Division was organized in 1963 to develop concealed replacement orebodies in the area located by Bear Creek Mining Company, Kennecott&apos;s domestic exploration subsidiary. Several accounts of the discovery of these orebodies and the general geology of the area have been given (1, 2). The ore is very complex and of varying degrees of oxidation and alteration. The principal metal values are lead, zinc, silver, and cadmium. The Burgin No. 2 shaft was sunk in 1963 to provide access for mining the orebodies. Much of the ore mined after this shaft was sunk was high grade and shipped directly to smelters, but concurrently it was recognized that concentration would be required to upgrade the lower grade ore in the deposits. Accordingly, in January 1967, authorization was received to construct a 500 tpd concentrator. From the beginning it was expected that the characteristics of the mill feed would vary widely and unpredictably as mining progressed. This paper summarizes the characteristics of the ore, the test work upon which the plant design was based, the engineering and construction, and the present operation of the facility eighteen months after startup. ORE CHARACTERISTICS The Burgin ore is a complex assemblage of minerals from replacement of limestone. The replacement and alteration occurred in at

Jan 1, 1970

By Emile Hiertz

THE first press was installed in June, 1910, and the second in March,1911. They produce 1,000 briquettes per hour, weighing 5 kg. (11.05 1b.) each, under a pressure of about 400 kg. per square centimeter (5,689.2 lb. per square inch). The chloride of magnesium is received from the potash mines at Hanover in tank cars in approximately a 35 per cent. solution. The quantity of solution added to the flue dust varies between 2 and 3 per cent., so that the briquettes contain about 1 per cent of MgCl2. To fresh dust as much as 10 per cent. of coke breeze may be added and bri¬quettes made that still have a crushing strength of from 25 to 30 kg. (377 to 426.6 lb. per square inch) after 36 hr., and 45 kg. (639.9 lb. per square inch) after six days. After having treated 140,000 tons of the material, the reports of the blast furnaces show that the briquettes improved the action of a furnace and produced less dust than did Minette ore. As much as 35 per cent. of briquettes is added to the furnace charge without bad effects on operations. The presence of chlorine did not produce any corrosion in the furnace, downtakes or other piping. The presence of a certain amount of chlorine in the dust reduces the amount of chloride of magnesium which it is necessary to add for briquetting.

Jan 12, 1913

By R. H. Thorpe

Longwall working in Britain developed to overcome problems arising from depth of seams causing roof and ventilation difficulties. This system became very widespread particularly in the thicker seams, where faces of up to one mile in length were not uncommon. In the thinner sections, in order to remain competitive, the longwall undercutting machine was developed. Following the development of the coal cutter, every other aspect of the face operation was examined and mechanized as it became a bottleneck to output and efficiency. The face system itself was modified and workings concentrated with the same objectives.

Jan 1, 1977

The greatest mining center in the state of Utah is the incorporated town of Bingham about twenty-five miles southwest of Salt Lake City. The principal industry of this vicinity, prior to the early fall of 1863, was lumbering and the first saw-mill in the state was built near the mouth of Bingham Canyon. Early in the fall of that year. George B. Ogilvie discovered gold and shortly afterward the first mining district in the state was organized. Some writers place the discovery of gold in the canyon in the late fifties. If this be so, however, the matter was kept extremely quiet because no mining was engaged in until after Ogilvie&apos;s discovery. Bingham was a placer mining camp, producing about one million dollars&apos; worth of dust and nuggets from placers up and down the canyon and in Bear Gulch until early in the seventies, or about ten years after the discovery of gold, when a large body of argentiferous lead ore was discovered and soon after heavy shipments were made from a half dozen or more properties.

Jan 1, 1925

By Luckie. Peter T., Deurbouck. A. W., Harold L. Lovell, E. R. Palowitch, James K. Kindig

PART 1: DENSE MEDIUM SEPARATION by E. R. PALOWITCH and A. W. DEURBROUK INTRODUCTION During 1974, 43.9% of the 603,406,000 tons (549,099,000 mt) of bituminous coal and lignite produced was cleaned mechanically, of which 82,280,000 tons (74,875,000 mt) was cleaned by dense medium processes.l The percentage of bituminous coal and lignite cleaned mechanically by dense medium processes increased from 7 to 31 % between 1938 and 1974. Dense medium separations include those coal preparation processes which clean raw coal by immersing it in a fluid having a density inter- mediate between clean coal and reject. As there is a general correlation between ash content and specific gravity, it is possible to achieve the required degree of removal of ash-forming impurities from a raw coal by regulating the specific gravity of the separating fluid. Dense medium processes offer the following advantages over other coal cleaning processes: 1) Ability to make sharp separations at any specific gravity within the range normally required even in the presence of high percentages of the feed in the range of 2 0.1 sp gr units. 2) Ability to maintain a separating gravity that can be controlled with ± 0.005 sp gr units.

Jan 1, 1979