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By T. A. Wright

THE SAMPLING of waste materials containing copper, lead and tin has taken on a new significance within recent years, and is of increasing importance, on account of the entry of some of the copper refineries into this field and the tendency towards consolidations among the smelters. In the past, such material was sold on the basis either of a rough or grab sample or of casual inspection of the lot. The two principal sources of this waste are the processes of fabrication and the scrapping of automobile, electrical, plumbing, railroad and other equipment. In both cases much material can be sorted, salvaged and sold by classification and grade to be melted "as is" or with the addition of new or virgin metal. The increasing care in classifying exercised in the large manufacturing plants tends to increase the proportion of this usable scrap; on the contrary, the increasing use of more complex and special non-ferrous alloys tends to make sorting and grading more difficult in turnings, borings, clippings, etc., and consequently is making sampling of more moment. In tin, lead, aluminum, and zinc furnace by-products such as ashes, drosses, skimmings, etc., the trade practice of evaluation has not varied much. For instance, tin drosses, battery lead, and aluminum skimmings are ordinarily sold on what is called "metallic button" or "metallic recovery" or "metallic yield." Copper and lead waste, however, is being sold more and more on actual copper and lead contents based on sample and chemical analyses. This paper will be devoted chiefly to the sampling of materials suitable only for refining and recovery of one or more of the metals present. The methods described are often based on the fact that but one metal, such as copper, is to be paid for. The loss of volatile constituents in melting does not appreciably affect the final copper result, as the "after-melting" assay can he readily calculated to the "before-melting" basis, but if smelter efficiency continues to increase, so that in time zinc, for instance, may be recovered on a paying basis, the method of sampling dirty brass, for example, will have to be changed also. As a matter of fact, we may expect that improved refining processes will make possible the partial recovery of lead and tin which are now lost.

Jan 1, 1928

By G. Sachs

According to Tammann,' the explanation of the effect of mechanical deformation in producing changes in the properties of metals is one of the most important problems of physical metallurgy, taking rank with the study of the constitution of alloy systems; and this problem has long attracted the attention of practitioner and scientist alike. Its solution was not advanced materially until Tammann2 set up his translation hypothesis with its crystallographic basis, while other hypotheses, such as the amorphous metal hypothesis of Beilby3 and Rosenhain,4 have been shown by more intensive studies to be less and less satisfactory. The principles of the Tammann translation hypothesis have proved to be useful even up to today and in spite of all of the recent accumulation of experimental data, while the differences of opinion which have arisen relate to the extent to which the hypothesis accounts for the observed changes Ïn properties and to what extent further assumptions are necessary. Stimulated by the proposals of Tammann, there soon developed a lively experimental activity which aimed at the elucidation of the structural or crystallographic changes produced by the mechanical deformation of metals. Parallel movements of crystal "blocks" along definite crystal slip planes had already been observed by Ewing and Rosenhain5 in 1900, in metals which had been moderately deformed but without shedding light on the crystallographic nature of the processes accompanying more severe deformation. The solution of this important problem was not advanced until a much later date, or until the methods of X-ray crystal analysis had become available. Shortly after the discovery of the diffraction of X-ray by crystals, symmetrical diffraction effects were

Jan 1, 1931

By Samuel S. Wyer

THE following description of methods for conducting gas-producer tests is probably the first attempt to give the subject an analytical, thorough and comprehensive treatment. In some cases where tests have been made, the methods have been so unsystematic and ambiguous that it has been impossible to secure comparable results. To eliminate this difficulty, the following provisional code of gas-producer tests-which has some resemblance to the code of boiler-trials of the American Society of Mechanical Engineers-has been developed. The primary object in testing a gas-producer is to determine whether the producer is working satisfactorily or, what is the same. thing, to see if the efficiency is as high as it should be with the type of producer in question, and also to find out if the composition of the resulting gas is adapted to the work it has to do. In order that the test shall be of any value, it must be thorough and comprehensive, and must be conducted with skill and care. When so conducted, the test will reveal the economy of the producer; and, by making suitable changes, the efficiency will often be increased very much. As a result of the tests made by Jenkins, 1 the efficiency was raised from 56.2 to 71.2 per cent. This shows the large saving that may frequently be made in the fuel-consumption by studying the results of a careful test.. The log of the test, given in Fig. 1, shows that, just as soon as the temperature became regular, the percentage of CO, decreased and that of CO increased.

Mar 1, 1905

By Robert D. Pehlke, Arden L. Bement

A mass transfer coefficient for the removal of hydrogen from liquid aluminum by inert flush degassing has been determined experimentally at 700°C. A mathematical model has been derived, assuming transport control, which adequately describes the experimental data. The removal of hydrogen is shown to increase with decreasing bubble size, and with increasing flow of flushing gas. The presence of hydrogen in liquid metals in amounts which exceed the solid solubility at one atmosphere hydrogen pressure is highly undesirable, causing unsound ingots or castings, or resulting in difficulties during further fabrication. The removal of hydrogen from liquid metals can be accomplished by several methods, the principal two being vacuum treating and inert flush degassing. The design of a process involving either of these techniques is based on the mass transfer coefficient for hydrogen between the liquid metal and a hydrogen-dilute gas phase. In an effort to extend our knowledge in this area, an investigation of the rate of removal of hydrogen from liquid metals by inert gas flushing was undertaken. In view of the difficulties encountered with gas porosity in the casting of aluminum and aluminum alloys, and because of certain experimental advantages, liquid aluminum was selected for the study. PREVIOUS INVESTIGATIONS The removal of gases from liquid metals, particularly steel, has received considerable attention in recent years, The first definitive work of a theoretical nature was presented by Geller1 who considered the case where equilibrium exists between purging gases and metal bath during flushing, and also deviations from this ideal case. Hulme2 has surveyed a number of experimental studies and discussed some of the practical limitations of removing dissolved gases from molten metals. More recently a number of studies have been reported on degassing of steel melts by flushing with an inert gas.3'4 Considerable work has also been reported on aqueous and organic liquid-gas systems, and an excellent summary prepared by Hovis.5 The treatment of liquid aluminum by a flushing gas to remove dissolved hydrogen has been known for several decades.6 A number of gases have been used for this purpose, including nitrogen, argon, and chlorine, or mixtures of these gases. Tikkanen and Erkko7 have studied the relative density changes of aluminum castings poured from melts degassed with chlorine, nitrogen, and mixtures of the two gases. Their

Jan 1, 1962

By L. L. Melton, W. T. Malone

Fluid mechanics research conducted with non-Newtonian fluid systems now permits prediction of the behavior of these fluid systems in both laminar and turbulent modes of flow through circular pipes. Present work concerns non-Newtonian fluid systems currently used in the hydraulic fracturing process. During fracturing treatments, an unsteady-state condition may frequently be encountered arising from. the reaction rate of a chemical additive. This condition must be evaluated in order to predict the actual behavior of a particular fluid during field application. Design and operation of the apparatus used to determine fluid-flow behavior permit obtaining data under such non-equilibrium conditions. This paper shows methods used to obtain rheology measurements, develop hydraulic relationships and evaluate chemical reactions producing unsteady-state conditions. Engineering application of this research is illustrated by employing measured rheological values and developed hydraulic retationships to produce frictional pressure loss (psi/100 ft) us flow rate (bbl/min) charts of common tubing and casing sizes for an example fracturing fluid. How these charts and chemical reaction rate information are then combined to predict actual turbulent hydraulic behavior during unsteady-state field conditions is also discussed INTRODUCTION Many fluids used in hydraulic fracturing contain chemical additives which impart non-Newtonian fluid properties that may drastically alter their hydraulic behavior. Equally drastic alteration in wellhead pressure, injection rate and hydraulic horsepower requirement may result from these fluid properties. Prior research conducted to relate non-Newtonian fluid properties with hydraulic behavior has not as yet produced a universal relationship, particularly for the turbulent flow region. which of the many Possible non-Newtonian fluid properties is responsible has not been conclusively established. A systematic description, suggested by Metzner, of the many possible non-Newtonian fluid properties exhibited by real-fluid behavior, and a current discussion of theoretical and applied aspects of non-Newtonian fluid technoloy can be found in Handbook of Fluid Dynamics. f Little or no research has previously been attempted with fluids exhibiting time - dependent properties. Addition of chemicals during a fracturing treatment is often accomplished by continuously mixing and displacing the fluid. This produces a time-dependent effect or unsteady-state condition while the fluid is progressing through surface and wellbore conductors. This condition is due to solution or chemical reaction of the additive. Considerable departure from conventional methods of obtaining and interpreting data was found necessary to consider these conditions. Therefore methods were developed to obtain hydraulic behavior information under these complex, unsteady-state conditions. Relationships presented in this paper to predict hydraulic behavior of non-Newtonian fluids in circular pipes were obtained by constructing and operating a small pipeline apparatus in the manner of a pilot-plant study. These relationships are suggested as scaleup equations and are not proposed as fundamental rheological parameters. While perhaps deficient from a fundamental research viewpoint, a pilot-plant study does permit the determination and convenient evaluation of variables pertinent to a process. A pilot-plant study can result in a valid engineering application procedure even when fundamental relationships are still undefined. An excellent series of articles by Bowen has appeared in the chemical engineering literature.2-8 These give a thorough review of proposed hydraulic relationships and their limitations for non-Newtonian fluid behavior in circular pipes. A graphical method is presented to scale up data for a fluid exhibiting an anomalous hydrau1ic behavior in the turbulent flow region. Considerable assistance has been obtained from these articles to interpret the anomalous behavior noted during this investigation. These articles also provided assurance that a pilot plant is a practical approach to evaluate the hydraulic behavior of non-Newtonian fracturing fluids. _____________

Jan 1, 1965

By W. D. Kruger

Methods for analyzing flooding or cycling projects by means of two-dimensional flow calculations are presented in the literature. The use of these methods allows the determination of optimum operating conditions for such projects and provides information on pattern efficiencies, relative flow rates, pressure distributions and floodout histories. The accuracy of such analyses depends on the basic reservoir data employed, especially the absolute permeability distribution in the reservoir; however, absolute permeability data normally are determined from well flow tests and/or core measurements and are so limited in quantity and quality that they do not adequately describe the reservoir flow properties. A calculation procedure is presented for determining the areal permeability distribution in the reservoir. Use of the procedure allows verification of the basic reservoir data through the matching of past reservoir conditions, thereby providing proper data for reliable predictions of future operations. The calculation procedure is a numerical technique based on a mathematical model of the reservoir and is designed for solution by means of an electronic digital computer. Results from field application of this procedure are presented. Permeability distributions from measured data are compared to those calculated to indicate the need for the analysis. Comparisons of field-measured and calculated pressure distributions are given to establish the validity of the calculation procedure. The conclusions of this paper are (I) use of the calculation provides data that allow a match of past reservoir conditions and thus reliable prediction of future operations; and (2) the procedure should, where applicable, be used to provide adequate reservoir data for use in two-dimensional flow calculations and related reservoir analyses. INTRODUCTION One means of providing more precise engineering control over reservoir operations during all stages of depletion is to develop some type of model to simulate the reservoir, using information developed from the behavior of the prototype to predict the performance of the actual reservoir. The use of mathematical rather than physical models for this purpose has been gaining in importance recently because of the application and availability of electronic computers. Mathematical models avoid instrumentation difficulties and allow easier handling of large-scale problems. At present, analyses employing two-dimensional models are common, and thought is being given to the use of models in three dimensions. The use of multi-dimensional models is necessary to show not only what is occurring in the reservoir, but also where in the reservoir the phenomena of interest are taking place. Methods for analyzing flooding or cycling projects by means of two-dimensional numerical models are presented in the literature. The use of these methods allows the determination of optimum operating conditions for such projects and provides information on pattern efficiencies, relative flow rates, pressure distributions and floodout histories. The applicability of numerical models and the results of using such analyses have been presented through field examples. The references indicated here discuss fully the data necessary to build the mathematical reservoir model and the way the model is used to study or predict the future performance of the reservoir. The accuracy of the analysis of such projects depends on the basic reservoir data employed in the construction of. the model. In order to be sure the model represents the reservoir, it must be used to match some known set of reservoir conditions, i.e., to match past reservoir performance. The accuracy of the model is directly dependent upon the absolute permeability distribution in the reservoir; however, absolute permeability data are normally determined from well flow tests and/or core measurements, and are so limited in quantity and quality that they do not adequately describe the reservoir flow properties. There is always the need for better data and for verification of existing data. The purpose of this paper is to show the use of the mathematical model to match past performance in flooding or cycling projects, the use of numerical techniques to adjust the basic absolute permeability data if necessary so that a match is obtained, and in this way to generate the effective areal permeability distribution of the reservoir. A review of the basic concepts of the numerical model will be given, followed by a discussion of the data-adjusting technique for matching past performance. TWO field examples will be cited to show

By P. G. Winchell, J. K. Jackson

The strtcture of body-centered Fe-Ni-C alloys (0 to 16.5 wt pct Ni) containing less than 0.015 wt pct C was investigated by measuring the carbon-diffusion peak at low frequencies with a torsion pendulurrr. By determining the Snoek damping in ferrite and in quenched alloys of identical composition, the effects of nickel content and heat treatment were separated, and the existence of a structure in which Snoek damping is greatly reduced was revealed in the quenched ternary alloys. The relative amount of this structure increases with increased nickel content and cooling rate. It is proposed that this structure may be highly imperfect or tetragonal. In Fe-Ni-C alloys containing only ferrite the addition of nickel (0 to 5.1 wt pet) decreases tke area of the carbon-diffusion peak and shifts its position to lower temperatures but broadens the peak only slightljl. The decrease in peak area with increasing nickel implies that short-range interactions with the nickel atoms prevent some of the carbon atoms from participating in the relaxation process. The shift in the peak position indicates a long-range effect of nickel on the movement of carbon atoms so that at 300°K the diffisivity of carbon in -ferrite is increased by about 35 pct by the addition of 5.1 tut pct Ni. The results are presented as distributions of relaxation strength 072 the activation energy under the assumption that the frequency factor is invariant. BODY-CENTERED iron-base solid solutions dissolve carbon interstitially in the octahedral sites. The occupation of an octahedral site produces a tetragonal distortion of the surrounding lattice. These sites fall into three sets according to the orientation of the tegragonal distortion associated with each and may be identified by their lattice coordinates as: type 1/2 0 0, type 0 1/2 0, type 0 0 1/2. At low carbon levels in cubic ferrite the carbon atoms are distributed at random1 among these sites. In high-carbon body-centered iron-base solid solutions, which are unstable with respect to several two-phase mixtures, a tetragonal structure occurs as the product of the diffusionless transformation of austenite. This transformation is commonly accompanied by appreciable microstrain and its product is called martensite. The tetragonality of martensite, a supersaturated solution of carbon in a iron, is maintained prior to decomposition even though local diffusion of carbon occurs as evidenced by carbide precipitation. The tetragonality is taken as evidence that one set of octahedral sites is preferentially occupied by carbon. (This preferential occupation is called ordering below.) Support for this view is presented by the absence of strain-induced carbon-atom diffusion in high-carbon solid solutions (martensite). Such diffusion occurs in ferrite and produces the 40°C internal-friction peak observed at 1 cps frequencies.' Thus, there appears to be qualitative evidence favoring the view that at sufficiently high carbon levels tetragonality arises spontaneously as a result of the ordering of the carbon atoms; however, no experimental evidence exists concerning an equilibrium between cubic and tetragonal body-centered solid solutions. Moreover, attempts to estimate the energy of ordering7-' predict a value too low to maintain the tetragonality observed at medium carbon levels, 0.3 to 0.6 wt pct C. In the present work an attempt using internal-friction measurements is made to establish the presence or absence of tetragonality in low-carbon body-centered solid solutions formed by quenching austenite. Fe-Ni alloys containing about 0.01 wt pct and 0 to 15 wt pct Ni were selected for the study. While the work was in progress Gilbert and Owen10 reported that the transformation which occurs in these alloys during quenching at rates up to 5000°C per sec is not martensitic but is more properly categorized as massive. Thus, the solid solution

Jan 1, 1964

By R. J. McNally

THERE are three primary plant nutrients—nitrogen, phosphorus, and potassium—expressed in any fertilizer compound as percent N, percent P 2 O 5, and percent K 2 O, in that order. This article will be concerned with the first two. The chief source of P 2 O 5 is phosphate rock. As found in the western states phosphate rock is essentially tricalcium phosphate-calcium fluoride (fluor-patite). As such, it is quite water insoluble.

Jan 1, 1957

By AIME AIME

AMERICAN participation in the World Engineering Congress in Japan will be generous. Nearly seventy papers have been prepared and for- warded for printing and the A. I. M. E. is well represented in the list. At our San Francisco meeting four of these will be briefly reviewed by the authors for the benefit of those who are unable to go on to Tokyo. They constitute a group dealing with the present economic status as seen by engineers of the world's iron, coal, petroleum and non-ferrous metals. Messrs. Leith, Smith and Tryon, Requa, and Probert have written papers that constitute a notable series. Messrs. Jeffries, Sweetser and Kinney, and other Institute members have written papers dealing more closely with technologic progress. The latest list of delegates and members is printed below, names of A. I. M. E. members being set in italics.

Jan 1, 1929

By C. B. Clark, W. H. Bruckner

Since the end of World War II interest has been increasing in the use of shaped charges in the mining industry and in other industries using explosives for blasting purposes. Shaped charges employ the principle known as the "Munroe effect," which was discovered by Charles E. Mun-roe more than 50 years ago (in 1888). Details of their design have been explained elsewhere.1,2,3 Fig I shows the essential features of design of two types of shaped charges (with conical and hemispherical cavities) and a schematic sketch of their action upon detonation. The following discussion deals with the behavior of the metal in the cavity liners when they are subjected to intense Pressures exerted when the explosive charge is detonated. Among the conclusions reached in research On shaped explosive charges, the following have been established concerning cavity liners:1'2 I. The optimum wall thickness of a conical cavity liner is dependent upon the apex angle of the 'One as well as On the base diameter of the charge. Acute apex angles require thinner walls for optimum perfmnance and more obtuse angles require thicker walls for the same base diameter' 2. Cones were more effective in forming a penetrating jet when the walls were tapered; that is, the thickness of the wall of the cone increased from the apex down. 3. The physical and mechanical properties of metals have a marked effect upon their performance as cavity-liner material. Boiling point, ductility, malleability, tensile strength, and hardness are among the properties that influence the effectiveness of a metal used as a cavity liner. Lead, for example, makes a wide, flat crater in steel plates. Aluminum makes a deeper crater than lead, and an aluminum alloy having a high tensile strength makes a deeper hole, but slightly smaller in diameter. Cast iron makes a deep, narrow hole. These findings, together with the following analysis under Metallographic Survey of slug, offer solutions to many of the problems involved in solving the mechanism of the formation of Munroe jets. It has been fairly well established that cavity liners collapse in a manner similar to that indicated in Fig 2, Conical liners are known to collapse upon themselves while hemispherical liners are believed to turn inside out in the process of jet formation. The first has been definitely proved by recovery of collapsed portions of cones, while liners from partly detonated charges show that hemispheres may turn inside out. Fig 3 shows a sketch of a recovered slug from a 6-in. shaped charge using a 45° cone made of cast iron. (This size of charge drilled holes up to 3 ft in depth in solid granodiorite.)

Jan 1, 1949

By C. B. Clark, W. H. Bruckner

Since the end of World War II interest has been increasing in the use of shaped charges in the mining industry and in other industries using explosives for blasting purposes. Shaped charges employ the principle known as the "Munroe effect," which was discovered by Charles E. Mun-roe more than 50 years ago (in 1888). Details of their design have been explained elsewhere.1,2,3 Fig I shows the essential features of design of two types of shaped charges (with conical and hemispherical cavities) and a schematic sketch of their action upon detonation. The following discussion deals with the behavior of the metal in the cavity liners when they are subjected to intense Pressures exerted when the explosive charge is detonated. Among the conclusions reached in research On shaped explosive charges, the following have been established concerning cavity liners:1'2 I. The optimum wall thickness of a conical cavity liner is dependent upon the apex angle of the 'One as well as On the base diameter of the charge. Acute apex angles require thinner walls for optimum perfmnance and more obtuse angles require thicker walls for the same base diameter' 2. Cones were more effective in forming a penetrating jet when the walls were tapered; that is, the thickness of the wall of the cone increased from the apex down. 3. The physical and mechanical properties of metals have a marked effect upon their performance as cavity-liner material. Boiling point, ductility, malleability, tensile strength, and hardness are among the properties that influence the effectiveness of a metal used as a cavity liner. Lead, for example, makes a wide, flat crater in steel plates. Aluminum makes a deeper crater than lead, and an aluminum alloy having a high tensile strength makes a deeper hole, but slightly smaller in diameter. Cast iron makes a deep, narrow hole. These findings, together with the following analysis under Metallographic Survey of slug, offer solutions to many of the problems involved in solving the mechanism of the formation of Munroe jets. It has been fairly well established that cavity liners collapse in a manner similar to that indicated in Fig 2, Conical liners are known to collapse upon themselves while hemispherical liners are believed to turn inside out in the process of jet formation. The first has been definitely proved by recovery of collapsed portions of cones, while liners from partly detonated charges show that hemispheres may turn inside out. Fig 3 shows a sketch of a recovered slug from a 6-in. shaped charge using a 45° cone made of cast iron. (This size of charge drilled holes up to 3 ft in depth in solid granodiorite.)

Jan 1, 1949

By David R. Mitchell, R. M. Fleming

MANY trends were evident in the coal industry during 1951. Some were favorable for the industry; others were not. Probably those having the most far-reaching consequences are those affecting coal's markets. The rapid dieselization of many railroads and the continued trend toward oil and natural gas for home heating has caused serious dislocations in various mining districts. Mines have been forced to close down or to work on a reduced time basis. Conversely the increased activity in the metallurgical industries has increased the demand for coking and special purpose coals, and the growing production of electric energy has increased the demand in certain areas for steam coals. The increased demand for metallurgical coal and for steam raising by the large utilities is more than equal tonnage-wise to the loss of coal to railroad dieselization and conversions to oil and gas in the domestic market. Since much of the expansion is at captive mines of large corporations, many commercial operators supplying railroad and domestic users have had their markets curtailed without any compensating demand from other sources. Feverish activity was in evidence in many areas by commercial operators attempting to find new markets. The Norfolk and Western Railway is the, only major American railroad committed to the burning of coal. It has an exceptionally good cost record. During 1951, fifteen additional modern coal-burning steam switching locomotives were authorized for construction in the company's Roanoke shops at a cost of about $1,450,000. When the new order is completed the road will own a total of sixty locomotives of the modern-switcher type, including thirty acquired by purchase last year. The Norfolk and Western today is the only American railroad building its own motive power. All of these locomotives burn coal. Production Coal production in the year 1951 will approximate 570 million tons of anthracite, bituminous coal, and lignite. Of this, approximately 7 pet is anthracite. Total production is approximately 30 million tons more than in 1950 and reflects the increase in general industrial activity. Exports are expected to be close to 50 million tons. The industrial stockpile is close to 75 million tons, the highest it has been since 1943. With the exception of the usual local stoppages the year 1951 was unmarked by labor disputes. It was a year of good labor-management relations with a negotiated wage increase unmarred by a work stoppage. This may be the beginning. of an era of sensible negotiations to find agreeable settlements of disputes without the usual protracted bickering and industry-wide strike. The United Mine Workers of America has condemned wildcat strikes, stating that in nearly every, instance proper grievance procedures have not been instituted. Government defense organizations, perhaps profiting by their experiences during World War II, are pursuing a more enlightened program with regard to the fuel requirements of the nation. Many prominent officials have stated that wherever possible coal should be considered the primary source of fuel in all defense expansion. With the increased consumption of steel by defense projects, approval of transportation facilities for liquid and gaseous fuels have been refused where these fuels would enter a coal producing area. Liquid and gaseous fuel producers also have failed to provide satisfactory evidence that they have sufficient reserves and are capable of supplying their product in adequate amounts for the periods of time required. Realization has at last become prevalent that the hydro-electric power installations in several areas are insufficient to handle increased demands.' Steam generating plants are being constructed in these areas and equipped to burn coal. Most notable of these, is the TWA area whose supplemental steam power plants are expected to consume 10 million tons of coal per year when completed. The Pacific Northwest is another area which is expected to follow the same pattern. Nearly all new steam-power generating plants are being equipped to burn coal either as a primary fuel or as an emergency measure. Metallurgical Coal The steel and other metallurgical industries are expanding rapidly to meet increased industrial demands and defense orders. Since these industries are dependent on coke, coke has had a year of high production and can be expected to break all existing records when steel-plant expansions are completed. Most notable of the expansion projects is that of the Delaware River area where the iron ore will be received by ship from Canada and South America. U. S. Steel is only one of several steel corporations that are planning on putting integrated steel producing units in this area. The Defense Solids Fuels Administration announced in December that through. November 30, 1951. 29 projects in eight states, with a total cost of

Jan 1, 1952

By Arthur West

It is the purpose of this paper to describe briefly some recent large power stations for blast furnaces, where the blast is exclusively supplied by gas engines using furnace gas. The stations are given in the chronological order in which they were designed, in order to indicate the various improvements suggested by experience. Fig. 1 shows the blowing-engine station of the Bethlehem Steel Co. at Bethlehem. Although the picture shows only five blowing engines, the house now contains 11 engines. In the diagrammatic plan the arrow shows the position and direction of camera. This station handles five modern blast furnaces of 450 to 500 ions capacity each. In this, as in all the other stations described in this paper, each furnace is blown by two single tandem gas engines. Ten engines are thus required for five furnaces, leaving one single tandem spare engine. This spare capacity has been shown by several years' experience at Bethlehem to be adequate to insure continuous operation. The engines in the Bethlehem station are right hand and left hand, arranged in pairs. The outboard bearings of each pair are close together so that there is not room for passageway between them. On the right of the picture are shown the inclined drums communicating with the cold-blast lines outside of building, so that any engine may be operated on any furnace. The illustration shows four cold-blast gate valves with pipes passing through wall, but since the photograph was taken the drums have been extended to take care of a fifth valve for a fifth furnace. At the extreme back of the picture can be seen the tubs of three twin vertical-horizontal Southwark steam blowing engines, which, although of modern design and as good as new, are not operated because of the greatly reduced blowing cost shown by the gas blowing engines. The next station built is shown in Fig. 2. This is the plant of the Minnesota Steel Co. at Duluth, Minn., and is at present constructed for five engines, to handle two furnaces, with one spare engine. The engines are of practically the same size as those at Bethlehem. Instead of being arranged in right-hand and left-hand pairs, as at Bethlehem, all the engines are of the same hand; this arrangement has some ad-

Jan 1, 1915

By J. Randel Hill

The 64th Texas Legislature passed the "Texas Surface Mining and Reclamation Act," Chapter 131, Texas Natural Resources Code (subsequently referred to as "the Act"), at a point in time when little surface mining had taken place within the state. Thus, enactment of regulatory control governing mining operations in our state has, for the most part, preceded widespread, intensive mining activity. I should point out that the statute applies only to the extraction of coal, lignite, uranium and uranium ore. A recent opinion rendered by the Attorney General's office has stated that extraction of the named elements, when it occurs incidental to the extraction of other materials, e.g., clay, is not not an activity subject to the Act. An important theme of the Act, frequently overlooked in other environmentally oriented legislation, is recognition of the Act's affect on the mining industry. The Act states that "the extraction of minerals by surface mining operations is a basic and essential activity making an important contribution to the economic well-being of the state and nation" and, in reference to reclamation being accomplished contemporaneously with mining, provides a recognition that "the extraction of minerals by responsible mining operations is an essential and beneficial economic activity." This legislative recognition is, of course balanced against the need for proper mined land reclamation, the rights of surface owners, the need to guard against unreasonable degradation to land and water resources and numerous other actions or events that the Legislature wanted protected. Another rather pervasive general theme that is established in the Act is the concept that the agency is entitled to obtain whatever information, or take whatever action, that appears reasonably necessary to effectuate the purposes of the Act. This concept, coupled with the detailed requirements placed on mining operations in the portions of the Act dealing with Permit Applications, Reclamation Plan and Reclamation Standards, easily insures that surface mining activities can and will be regulated to the extent necessary. Section 131.102(b)(2) is probably the most relevant provision in the Act which deals with the preclusion of mined lands becoming waste lands. This portion of the Act states that the surface mining operator shall: ". . . restore the land affected to the same or a substantially beneficial condition. ." In this language the Legislature had made it abundantly clear that the abandoned or orphaned lands found in some of the Eastern states, will not be tolerated in this State, and provided the Commission with a degree of latitude to determine what condition would be appropriate in each individual mining situation. I bring this area of the Act to your attention because it is the area which has received the most inquiries and concern by operators, i.e. what does "substantially beneficial" constitute? The Commission's staff has spent a considerable amount of time wrestling with the statutory language "substantially beneficial" to determine the legislative intent and provide operators with guidance of what they can expect. In this regard, the most definitive criteria in our opinion lies in the Act's legislative hsitory. The Legislature had before it basically three reclamation standards namely: House Bill 656, considered to provide the most stringent reclamation standard and identical to Senate Bill 66; House Bill 1717 which was supported by industry spokesmen and was considered to contain the least stringent reclamation standard; and Senate Bill 55 which was adopted by the Legislature. During the Bill's initial hearing before the House Environmental Affairs Committee, the House sponsor of the Senate Bill 55, which was the Bill eventually adopted, made the following statement: (I am quoting verbatim from the Committee's transcript although substituting the speaker's reference to the authors names with the Bill's numbers.) "An example of the differing approaches of the four bills can be seen in the area of reclamation standards and this is critical. House Bill 656, talking about restoring the surface of the land, House Bill 656 has this statement: 'At least fully capable of supporting the use to which it was capable of supporting prior to any mining or any higher or better use.' In other words you have to restore it to the level it was prior to the mining. Senate Bill 55: 'Restore the land to the same or substantially beneficial use.' Substantially beneficial use. (Emphasis his) House Bill 1717 says: 'Restore insofar as practical to appropriate beneficial post-mining use.' So that House Bill 1717 said to, 'restore insofar as practical to appropriate beneficial use. I think that's too weak, whereas I think the House Bill 656 that would call you to restore it to at least the use it had prior to the mining may also be too tight the other way. I think the Senate Bill 55 approach is down the middle of the road which says that you do restore it to the 'same or substantially beneficial use,' is a proper way to go there." Therefore, the conclusion that we have drawn from the legislative history of the bill is that the extent of reclamation required by "substantially beneficial" may be something other than the use it was prior to

Jan 1, 1979

By D. K. Deardorff, Earl T. Hayes

An improved technique is described for the accurate determination of melting points of metals in the temperature range 1500' to 2500°C. The improvements consist of gradient heating and refinements in cavity preparation to obtain true black-body conditions. The melting point of hafnium, as determined by this method, is set at 2222°±30°C. This is over 200°C higher than one of the values reported earlier. Discrepancies in the previously reported values for the melting point of hafnium are explained. The melting points of zirconium and titanium are found to be 1855O±15OC and 1668°±10°C, respectively. Reproducibility of results was ±3OC. PRIMARILY, this report is being made to correct a 1975o±25oC1 value for the melting point of hafnium, and to present an improved method for melting point determination. As a matter of interest, the technique was also applied to determining the melting point of zirconium and titanium. There is a wide difference in the melting point of hafnium as reported by competent investigators. DeBoer and Fast' reported a value of 2230°±500C, and later work of McPherson, as reported by Aden-stedt, placed the melting point at 1975o±25oC. Zwikker" calculated that the melting point was 2430°C. In resolving this discrepancy, improvements were made in the procedure for determining the melting points of refractory metals in the range 1500" to 2500°C. Special techniques are necessary to determine the melting point of these metals accurately. Since the metals oxidize in air above 900oC, the determinations must be carried out in an inert gas or vacuum. Also, every known refractory contaminates them to some extent. In addition, the attainment of true black-body conditions is always difficult. DeBoer and Fast actually determined the melting point of Hf-Zr as-deposited iodide process alloys and extrapolated to 100 pet Hf. McPherson employed a tungsten wire suspension similar to that described by Schramm, Gordon, and Kaufmann,1 and used hafnium metal produced by the iodide process. There was no apparent reason for the wide spread in melting points, since the quality of the metal was very good and the work was under the direction of highly competent investigators. Retracing some of this work revealed one understandable error in the case of hafnium, and led to development of an improved technique for determining melting points of reactive metals in the range 1500" to 2500°C. Material Crystal bar hafnium, zirconium, and titanium were used for this work, with analyses as shown in Table I. The hafnium was of the highest purity that has been prepared, though not as pure as the zirconium and titanium. At the instigation of the Pittsburgh Area Office of the AEC, a special lot of Hfo2 was purified, converted to the tetraflouride, and bomb-reduced with calcium, all at the Iowa State College Laboratories. This crude hafnium metal was then refined by the iodide process at the Foote Mineral Co. All specimens were cut with a Sic wheel, ground, pickled, and dried carefully before use. Procedure General Tungsten Suspension Method—In this method, a small specimen is suspended by a fine tungsten wire in a slender, induction-heated tungsten or tantalum cylinder, all under vacuum. The melting point is determined by heating the specimen slowly while observing it with an optical pyrometer focused upon a portion of the specimen that radiates as a black body. This means that there should be no reflected radiation, either from colder objects near the pyrometer or from hotter surfaces. The induction heater would be the only hotter object and, if it is slender enough, the specimen will attain the same temperature. Often a slender hole is drilled into the specimen as a source of black-body radiation. This was the general method used by McPherson in obtaining his value of 1975°C for the melting point of hafnium. In trying to repeat this work, it was found that the points of contact between the

Jan 1, 1957

By R. D. Begley, A. W. Khair

Introduction The technique being investigated here uses a physical mine model which consists of a plexiglass room-and-pillar model 11.4 x 11.4 cm (4.5 x 4.5 in.). It is placed at the bottom of a larger plexiglass box 25.4 x 30.5cm (10 x 12 in.). The box is then filled with a model material representing geologic overburden. Sand has been selected because of its deformability under gravitational forces. The mine model is capable of creating a cavity and induces movement on the surface of the model material. This surface disturbance can be measured using a photographic technique called holographic interferometry. This paper discusses results of an investigation of the possibility of generating subsidence criteria using physical mine models and laser holographic interferometry. Problem Statement Subsidence appears to be widespread in two states - Pennsylvania and Illinois - although serious subsidence is found in localized areas of numerous other states (HRB Singer Inc., 1977). The US Bureau of Mines estimates that more than 1618 km2 (400,000 acres) of urban areas are threatened by subsidence. Pennsylvania and West Virginia contain 57% of these threatened areas, with the remainder being spread out over 16 states (Johnson and Miller, 1979). Events of subsidence have been reported in urban areas, including Pennsylvania (Gray et al., 1977) and Illinois (Mahar and Marino, 1981). One location has observed three episodes of subsidence beginning in 1950 (Bruhn et al., 1981). These subsidence events all occurred over abandoned room-and-pillar mines. Some mines have been abandoned more than 50 years (Bauer and Hunt,1981) . Available research does not offer much on the problem related to time dependent subsidence, especially those associated with the failure of remnant coal pillars. The physical modeling technique developed at West Virginia University is capable of measuring both instantaneous and time delayed subsidence. Laser Holographic Interferometry Laser holographic interferometry, or holometry, uses the interference properties of laser light to measure small movements of objects. The object requires only optical access and no surface instrumentation. This technique is closely related to photoelasticity. Both techniques rely on the uniqueness of the light reflected off or through an object in any given state of stress. The entire measurement process is done remotely without any surface instrumentation and without using a camera. The interferogram is constructed and visual observation will yield fringes on all areas that yielded or moved. The displacement represented by one fringe was determined by Wilson et al. (1977) and is given by the relationship: [d ='1(1) 2 cos 8 where d = displacement of one fringe 6 = incident angle of laser light A = wavelength of light (632.8 x 10-9 meters for Helium-Neon lasers)] In summary, holometry is capable of detecting very small movements of objects under study. It is a very sensitive process that requires laser light and vibration free experiments. Experimental Program Lab Facilities The primary components of the facilities were the laser and the optical table. The light source was 50 milli-watt, Helium-Neon laser. The optical table was a granite slab weighing nearly one ton. All experiments were performed on the optical table. It consists of a large table sitting on air cells similar to car innertubes. Six air cells were placed on top of two layers of wood and Styrofoam. Figure 1 is a schematic diagram of the vibration free table. Even with these precautions, the experiments still had to be performed at night to reduce potential traffic related vibrations. Room-and-pillar mine models consisting of three different pillar sizes were constructed out of plexiglass. The models were designed to represent three different extraction ratios. Figure 2 shows the models. The extraction ratios were determined by the tributary loading concept (Peng, 1978). These models were the primary component of a seam extraction simulator. A schematic of this simulation device is shown in Fig. 3.

Jan 1, 1986

By C. F. Cockrell, H. E. Shafer, J. W. Leonard

A significant technological development is discussed for the processing of certain power plant flyashes that are a problem because they contain a high water-soluble mineral content and yield inferior flyash-based brick. This development is based on the addition of hydrochloric acid to the flyash-bottom ash-sodium silicate blends used for the production of flyash-based structures. Favorable changes are evi-dent in sodium silicate consumption, where about half the amount used for making brick from normal flyash is required, processing costs, and strength and appearance of the resulting freshly formed green and fired structural products. In some high cost in-stances (above the $30 or less per thousand brick estimated for normal flyash), hydrochloric acid additions can replace or reduce other costly additives normally required for problem flyashes. This results in savings of as much as $9 per thousand brick. These developments offer promising possibilities for utilizing problem flyashes as well as flyashes which may come into contact with water soluble sulfur as a result of the eventual application of certain experimental air pollution control processes. Much of the research datal-6 accrued through work by the Coal Research Bureau of West Virginia University for the Department of Interior's Office of Coal Research on converting waste power coal ash to valuable brick and other structural products, can be categorized as follows: (1) general ash-to-brick tech- nology; (2) specific ash-to-brick optimization; (3) pilot plant progress; (4) production cost estimates. A summary of the progress on bench-scale experiments indicates that coal ash can yield two types of brick. An acceptable brick is obtained by mixing 73.6% fly-ash, 23.5% bottom ash and 2.9% sodium silicate solution (dry basis). Depending on the flyash and bottom ash used, optimum brick can be manufactured by mixing 72-77% flyash, 20-25% bottom ash and 2.3-3.0% sodium silicate (dry basis) with Na2O : SiO2 ratio varied according to the composition of the flyash used. These blends are thoroughly mixed and mechanically formed into green brick shapes at pressures ranging from 1000 to the more usual 3000 lb per sq in. The green shapes then are dried and fired at temperatures ranging from 1975-2150°F using clay brick cycles that are shorter than standard. Other findings reported to date indicate that when making optimum brick, it is necessary to carefully control the top size of the moisture-releasing bottom ash so that it falls between 8 and 30 mesh, depending on the flyash used. Moreover, the chemical composition of bottom ash and flyash have been found to be of little significance with the exception of a few problem flyashes which are the subject of this paper. Finally, the pilot plant, which was previously reported under construction, is now in operation. This will indicate, among other things, the cost of brick production, which was earlier estimated for non-problem flyashes at $30 or less per thousand brick. The foregoing bench-scale technology is applicable to all the initially tested ashes as well as to the majority of ashes examined during the course of research. However, a few of the ashes yield green and fired products that may be structurally inferior or unacceptable, necessitating excessive and diverse reagent addition, and/or may be coated with unsightly surface deposits. These problem ashes generally originate in certain areas of Illinois, western Ken-

Jan 1, 1969

By S. W. Ross

In 1936 a description of the plant (Fig 1) and process employed by the Electrolytic Zinc Co. of Australasia Ltd. for the recovery of zinc from zinc concentrate by the electrolytic process was prepared. † During the twelve years which have elapsed since the preparation of the earlier paper, several major changes in the metallurgy of the process have been introduced. It is the purpose of the present paper to give a general description of these changes and thus to bring up to date the description of the plant and process. Summary The major changes in Risdon practice since 1936 have been: 1. Replacement of two stage roasting by a preliminary roast followed by the flotation of all the leach residue and the roasting of the flotation concentrate. 2. Screening of all calcine fed to the pachucas. 3. Continuous leaching of calcine and improved classification of pachuca discharge. 4. Close control of hydrogen ion concentration during purification for iron removal. 5. Recovery of cobalt as a good grade oxide. 6. Production of part of the zinc output in the form of "four nines" metal (99.99 pct purity). 7. Closer spacing of electrodes thus increasing the potential output of cathode zinc per cell by 50 pct. Changes which are in prospect and for which construction work is proceeding at the present time involve the starting up of: 1. Two suspension roasters. 2. A contact acid plant to produce 150 tons‡ of acid per day and replacing the existing Mills Packard chamber plant. 3. Extra power station capacity to permit greater current flow to existing cell room units. This will increase the output of cathode zinc from about 245 to 290 tons per day. Plans for the future envisage the building of an ammonium sulphate plant, the first unit of which will produce about 50,000 tons per year, and improved treatment of zinc plant residue for the recovery of zinc, lead and other metals. At the end of this paper tables of metallurgical data are presented relating to the year ending June 30, 1948. Details of Changed Practice Output In 1936 the production of cathode zinc amounted to about 200 tons per day. This has since been increased to about 245 tons per day while plant extensions are practically complete which will permit of an output of about 290 tons per day in the near future. Roasting Division ROASTING POLICY A major change has occurred in the roasting policy. Twelve years ago the method in use was to carry out a two stage roast in the first stage of which sulphide sulphur was reduced to about 6 pct. The pre-roast calcine was re-roasted in modified Leggo furnaces using coal as fuel, sulphide sulphur being reduced to about 0.8 pct. The whole procedure was described on pp. 482491 of the earlier paper. Although this roasting procedure had certain advantages it possessed some distinct disadvantages. For instance, it appeared uneconomical to heat up the entire input of pre-roast calcine to roasting temperature by the expenditure of fuel in order to oxidise a few percent of sulphide sulphur. It was argued that if the pre-roast calcine were leached and a process could be developed for the recovery of a zinc sulphide concentrate from the leach residue, this concentrate, small in weight compared with the pre-roost calcine, would probably roast autogenously, thus virtually eliminating the expenditure of fuel as well as greatly increasing the weight of sulphur oxidised per square foot of furnace hearth area. The obvious method of producing a suitable concentrate from leach residue was by flotation. It will be recalled (see p. 495 of the earlier paper) that when two-stage roasting was practised the leach residue was classified, the granular fraction was ground and floated while the slime fraction was thickened, filtered and dried ready for shipment to a lead smelter. This process worked quite successfully. However, when trials were made of leaching a calcine carrying several percent of sulphide sulphur the granular fraction still floated well, but the slime fraction carrying 8-10 pct sulphide sulphur yielded very poor results when subjected to flotation. This fact held up the application of "pre-roast" leaching for many years. However, successful flotation of the slime fraction of leach residue was finally achieved and in August 1940 the slime flotation plant began operation, while the leach-

Jan 1, 1950

By E. A. Gulbransen, K. F. Andrew

THIS paper. will present the results of our studies on the kinetics of the gas phase reactions of co-lumbium and tantalum with O2, N2 and H2. Studies on zirconium and titanium have been previously reported.9,11 Due to their high melting points and their many excellent physical and chemical properties, colurn-bium and tantalum have been finding their places as useful metals in science and industry. Chemically the metals are very inert to gas and liquid phase corrosion at room temperature. However, at moderate temperatures of 250°C and higher the metals react readily with oxygen and hydrogen. It is of interest to study the gas phase reactions of these metals for the following reasons: (1) to compare the rates of the several reactions with those of other metals and to correlate the rate data with theory and structure predictions, (2) to add basic information on the practical difficulties of the reduction, refining and working of columbium and tantalum, (3) to anticipate new uses for the metals and (4) to understand the role that the metals play as components in high temperature alloys. Literature Survey A number of papers exist in the literature on the gas phase reactions of columbium and tantalum with O2, N2 and H2; however, only fragmentary data have been reported on the kinetics of these reactions. Although the free energy of formation of the oxide Ta2O520at 25°C has been determined and some thermodynamic work has been reported on columbium and its oxides, the data are not sufficient to make equilibria calculations at elevated temperatures. Oxidation: 1. Kinetics: The rate of oxidation of columbium and tantalum has been studied by McAdam and Geil23 using the interference color method. Tantalum is reported to oxidize more slowly than columbium, zirconium and iron. The amounts of oxidation of columbium after 20 hr in air for several temperatures are reported by the Fansteel Metallurgical Corporation.36 Columbium starts to oxidize in air at about 200°C. The oxide is said to be adherent and prevents further oxidation until the temperature is raised. Columbium is reported not to become brittle on heating in air for short periods like tantalum because the oxide film prevents further reaction. The columbium oxides dissolve into the metal when heated in a vacuum at temperatures between red heat and 1200°C. Above 1200°C the oxides are reported to evaporate. Columbium, in the oxide free state, is an active getter in vacuum tubes. 2. Preparation and Structure: G. Grube, O. Kuba-schewski and K. Zwiauer15 have studied the decomposition of Cb2O5 in argon. They find that CbO2 is formed slowly at 1150° and rapidly at 1300°C. Cb2O5 can be reduced to CbO2 by moist H2. The reduction of this dioxide to the lower oxides is rapid. By varying conditions CbO2, Cb2O3, CbO and Cb2O can be formed. CbO2, CbO and Cb2O have independent lattices while Cb2O3 is a stoichiometric mixture of CbO and CbO2. In a later paper O. Kuba-schewski" has shown that CbO is cubic and that CbO2 has a rutile type of lattice. The most exhaustive work on the oxides of columbium is that of G. Brauer.8 The number and constitution of the solid phases in the binary system Cb-O were investigated by the preparative, analytical and X ray methods. He indicates that only the oxides Cb2O5, CbO2 and CbO exist with limited regions of homogeneity. The solid oxide Cb,O, exists

Jan 1, 1951

By R. V. Higgins

This paper presents a computer method to obtain the shape factors and equal cell volumes of the channels for any well spacing pattern from a potentiometric model. By using this program the authors have processed the data for the seven-spot, direct line-drive and the staggered line-drive patterns. The data for the five-spot pattern had been previously processed by a noncomputer method and are included for completeness. The shape factors and volumes for the channels are presented in tables for those who want to use them to process data using their own permeability relationships and viscosities of their reservoir oils. The authors have used the data and sets of representative permeability curves to process sample calculationr of waterflood performances. The comparison of the calculated results shows that the influence of well spacing is small. The permeabilities of the reservoir rock to oil and water had a greater influence on oil recovery for a given pore-volume throughput of water than the well spacing pattern. The more water-wet the reservoir rock, the better the possibility of permeabilities which are conducive to good recovery. The viscosity of the reservoir oil also influences the recovery more than the well spacing pattern. The reduction in the percentage recovery of oil with increase in viscosity of the reservoir oils is small when oil viscosities are in the range of 0.1 to 3. Above this range the reductions in recoveries are extensive. Sample comparisons of the time required for different patterns to recover the oil are presented. Results of an example calculation are given to show the effect of the permeability profile on recovery. INTRODUCTION The effect of well spacing pattern on the recovery of oil when flooding with either gas or water has been studied by many investigators. Muskat et al.1 presented an analysis using conductivity, sweep efficiency and unit mobility to the time of breakthrough. Dyes et al.2 used experimental techniques (X-ray shadowgraphs) and different mobility ratios. They presented quantitatively the relationship between mobility and sweep efficiency at and after breakthrough. Hauber3 presented a method to predict waterflood performance for arbitrary well spacing patterns and mobility ratios. Craig et al.,4 using techniques similar to Dyes et al. to determine sweep efficiency for a five-spot pattern, added the use of relative permeability curves at breakthrough and thereafter. Douglas et al.5 sed rela- tive permeabilities and continuously changing saturations throughout the entire five-spot flood pattern. In obtaining their solutions they used finite-difierence equations. Hig-gins and Leighton6,7 also used relative permeabilities and continuously changing saturations throughout the pattern before and after breakthrough. They employed techniques that process a flood-pattern calculation on the computer in about one minute. The methods of Douglas et al. and Higgins and Leighton both checked closely the laboratory results for a wide range of mobility ratios. This paper presents some sample performances calculated by the Higgins and Leighton method that show the effect on recovery of different permeabilities and viscosities using the seven-spot, the line-drive and the staggered line-drive, as well as the fivespot flood pattern. No previous paper has presented these data using different permeability curves and continuously changing saturations throughout the flood patterns. The paper also presents (1) the results and analyses of the flood-pattern prediction, (2) the computer techniques for determining the shape factors and volumes from the potentiometric models for the foregoing flood patterns, and (3) the shape factors and volumes of the channels of the flood pattern in the event reservoir engineers may like to process waterflood calculations using their own permeability curves and reservoir oils. DESCRIPTION OF METHOD VOLUMES AND SHAPE FACTORS The use of channels taken from a potentiometric model (see Fig. 1) to aid in calculating the performances of water floods of nonlinear patterns has been thoroughly explained in the literature.0,7 Therefore, very little theory, discussion, or proof regarding this phase will be repeated in this paper. The computer method presented in this paper to calculate the volumes and the shape factors of the channels of potentiometric models employs the trapezoidal rule for the volumes and the Pythagorean theorem (the hypotenuse equals the square root of the sum of the squares of the two sides of a right triangle) for the shape factors. In calculating the volume of a channel, the area of each cell in the channel is determined and then multiplied by the thickness to obtain the volume. In determining the areas of the cells, trapezoids are constructed whose vertical sides are spaced Ax apart, as shown in Fig. 2. The length of the sides is the difference between an ordinate cut off by the top and bottom of the cell — usually equipo-tentials. The coordinates of the points along an equipo-tential or streamline are obtained by Lagrange's8 equation of interpolation for which the constants are coordinate points at the intersections. Three intersections for con-

Jan 1, 1965