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Scores of visitors who had never in their lives seen a piece of rough-and-ready mining machinery flocked around the display of Russian equipment presented as part of the Soviet Exhibition at New York's Coliseum. The Russian show, which ran from June 30 to August 10, presented an impressive view of an emerging technology. The exhibitors, however, made little attempt to appeal to the visitor who came only to be amused. In the mining show, as well as elsewhere throughout the exposition, the exhibits were devastatingly highbrow. If a coal mine's hydraulic roof support were "designed to support the prop- , free face front in longwall faces of flat seams, from 1.0 to 1.6 m thick, with medium-stable side rock," the visitors were told just that.

Jan 9, 1959

By Y. Ueda, F. Noguchi, T. Yanagase

The rate of CaO dissolution into slag melts is of great importance in the field of pyrometallurgy. In this study the mechanism of CaO dissolution has been investigated by observing the melt heated on Pt-filament directly through microscope together with a help of X-ray micro analyser to find out a distribution of each element around CaO crystal. The direct observation reveals that the CaO dissolution tends to proceed by the action of molten slag entering between CaO crystal and newly formed 2Ca0. Si02 layer through cracks. The effect of slag composition on the formation of 2CaO- SiO2 layer will be shown as well as the effect of halogenides addition on the destruction of this compound layer.

Jan 1, 1976

By Bailey, Lewis Newton

Bailey, Lewis Newton, Master Engineer, Senior Grade, 4th Regiment, U. S. Engineers, Headquarters Company, died of pneumonia at Camp Merritt, N. J., on Apr. 30, 1918. Baird, Louis, Lieut., Royal Field Artillery, British Army, died on the battlefield in 1915. Ballamy, John H., Capt., 103d Engineers, killed in action near Fismes, Aug. 9, 1918. Bowles, Martin F., 2d Lieut., Co. B, 355th Infantry, killed in action, Sept. 3, 1918. Burt, Andrew, died in active service, 1916. Cobeldick, William Morley, Royal Engineers, died from gas poisoning on Oct. 7, 1915. Dougall, Ralph, 4th University Co., Princess Patricia Regiment, killed in action early in the war. Evans, Alfred Winter, Lieut.-Col., New Zealand Rifle Brigade, D. S. 0., D. C. M., killed in action on Oct. 12, 1917.

Jan 3, 1919

W: 0. HOTCHKISS, Madison, Wis. (written discussion*).-Mr. George's paper on. "The Wisconsin Zinc District" is a very complete and clear description, which it is a pleasure to read. His discussion of the origin of the ores is the least satisfactory part, and Mr. George correctly states present knowledge when he begins his paragraph on a origin with the words "the general supposition is, etc." There is still much to be learned regarding the details of the processes by which the ores have reached their present situation. In this the Wisconsin district differs not at all from other mining districts. It is my hope that Mr. George and others, equally clear-thinking, who are in close touch with these operations will be able at an early date to give us more information on this subject.

Jan 3, 1918

By Leon Hart

THE nickel-chromium alloys of importance are those containing iron and those free from iron. The most important alloys containing iron, with regard to high tonnage, are the nickel-chromium steels. Straight nickel steel was introduced about the year 1889, but it was not until 1895, when it was found that the addition of chromium greatly increased the FIG. 1.-RESISTIVITY OF NICKEL-CHROMIUM-IRON ALLOYS. strength and resistance to shock, that nickel-chromium steels came into use. The three types of nickel-chromium steels developed at this time were: Type 1, containing 3.50 per -cent. nickel and 1.50 per cent. chromium; type 2, containing 2.00 per cent. nickel and 1.00 per cent. chromium; and type 3, containing 1.50 per cent. nickel and 0.50 per cent. chromium.

Jan 1, 1921

By C. D. Stahl, R. Sandrea

This investigation was made to evaluate the comparative effects of the viscosity and the phase behavior of the buffer fluid in the composite solvent displacement of Bradford crude from waterflooded sandstone cores. Buffer slugs of propane, naphtha and other refined hydrocarbons exbihiting increasingly favorable viscosity ratios and decreasing solubility relationships, respectively, with Bradford crude, were used in long Berea sandstone cores. The secondary slug was isopropyl alcohol. The results indicate that higher oil recoveries are obtained for increasingly favorable phase relationships even when these are accompanied by unfavorable viscosity ratios within the range studied. Moreover, when propane is used as a buffer slug with an adverse viscosity ratio of 36, it gives higher oil recoveries than at a similar size slug of an any1 alcohol having a viscosity ratio of 1.09. The investigation was extended to the study of the effect of flooding rate on oil recovery. Residual Bradford crude was displaced from a 6-ft Berea sandstone core at rates varying from 0.3 to 30 ft/ day. The results show that as flooding rates were increased above or decreased below a minimum range of 1 to 2 ft /day, displacement efficiency increased considerably. INTRODUCTION In recent years, the alcohol slug process as a means of tertiary recovery has been the subject of many investigations. Gatlin and Slobod showed the advantages of the combination solvent slugs over the previous single-slug process.1 The combination slugs require less total slug material to obtain similar oil recoveries. The basic combination process involves injecting a primary or buffer solvent which is preferentially oleophilic, followed in turn by a water-driven slug of isopropyl alcohol. This procedure ensures that the buffer slug, which is miscible only with the reservoir oil, displaces and replaces the oil and is in turn displaced, together with the water, by the isopropyl alcohol slug miscible with both fluids. Likewise, for those reservoir oils that show some degree of phase incompatibility with isopropyl alcohol, as does Bradford crude, a proper buffer solvent can artificially make the alcohol process technically feasible. From an economic standpoint, propane or LPG would be the most appropriate primary solvent to be used in the field; however, Taber, et a12 have indicated that the alcohol process becomes highly inefficient in the displacement of light hydrocarbons because of the entrapment of these hydrocarbons during the stabilized bank formation. Moreover, since materials such as propane have such low viscosities, adverse viscosity ratios with most reservoir oils would be penerated, thereby possibly further reducing the efficiency of the buffer solvent. This paper presents the results of an experimental study of the comparative effects of viscosity ratio and phase relationships on the displacement efficiency of Bradford crude by means of dual solvents. The work was extended to include the effect of flooding rate on oil recovery. Rates within the range of field and laboratory values were used. EXPERIMENTAL PROCEDURE APPARATUS The porous media used in this study consisted of two 10-ft and one 6-ft Berea sandstone cores, 3-in. in diameter. These were cast in galvanized steel pipe of 31/2-in. ID and the annulus was filled with Armstrong cement. The 10-ft cores were assembled by connecting two 5-ft units, which were joined together by 8-in. steel flanges fitted to the pipe. Intimate contact between the two sections was obtained by inserting a thin disk of the same sandstone between the core faces. The outlet and inlet ends of the cores were covered with threaded pipe caps. The injected fluid was gravity fed to two variable-rate positive-displacement pumps. The fluids were filtered before entering the core and the effluent was collected in an automatic fraction collector and

Jan 1, 1966

By J. J. Taber

The turbulent flow drag coefficients, or friction factors, have been experimentally determined for the cut-tings normally encountered in drilling operations. The gas law and average drag coefficients for characteristically flat particles (limestones and Shales) and for angular to sub-rounded particles (sandstones) were used to extend Newton's equation to a more useful form. The resulting equations are expressions for the slip velocity of a particle as a function of the particle size and shape, the bottom-hole injection Pressure, and the Bottom-hole temperature. The velocities necessary to lift actual rock cuttings as observed in the laboratory were compared with velocities predicted from the derived equations. Results indicate that there is no single correct circulating velocity and that the commonly used 3,000 ft/min linear velocity may he sufficient to lift only very mall particles. The advantage, in terms of horsepower requirements, of cycling high pressure air to obtain lower compression ratios was shown. INTRODUCTION A promising departure from standard rotary hydraulic drilling is air or gas drilling. In many instances the rate of penetration and bit life increases resulting from this method have been very substantial. Further the application of air drilling in areas of lost circulation or water sensitive pay zones has been highly successful. Purpose For the drilling contractor, the practical question remains, "How much air pressure and volume should I have?' lt is necessary to have a reasonable knowledge of the air velocities and pressures required in air drilling operations for the proper design and most advantageous use of surface equipment and for adequate removal of cuttings from the borehole. Eased on experience, most operators agree that at least 3,000 ft/min linear velocity is necessary for satisfactory hole cleaning.' It was the purpose of this investigation to evaluate the drag coefficients, or friction factors, for the cuttings normally encountered in drilling operations, i.e., sandstones, limestones and shales, and to utilize the values obtained to develop an expression for the minimum velocities and pressures (hence the volumes) necessary to lift these cuttings. THEORY Several investigators have done extensive work on the ability of drilling mud to lift bit cuttings. They conclude that the ability of a drilling mud to lift cuttings is dependent on the density difference between the rock being drilled and the drilling mud, the cutting size and shape, the mud flow constants, and the flow state (laminar, transitional or turbulent) of the mud. However, drilling muds are not true fluids but have a variation of viscosity with velocity in laminar flow and a constant apparent viscosity in turbulent flow. A survey of the literature revealed that no experimental work had been done to determine the velocities necessary to lift actual rock cuttings using a compressible fluid such as air or gas. In 1953, Nicolson6 stated that in air drilling fluid mechanics the particles could be assumed spherical in shape and that ; constant drag coefficient of 0.5 could be used due to the highly turbulent flow of the circulating air. By equating the turbulent resistance on the particle to the gravitational force on the particle, he obtained the expression, ^ = 2.67^y ,.......(1) where V, is slip velocity of spherical particle, ft/sec; D is particle diameter, in.; p, is particle density, 1b/ft8; and p, is fluid density, lb/ft3. However, as stated earlier, most operators seem to agree that adequate hole cleaning can be obtained by circulating a sufficient volume of air to give a linear velocity of 3,000 ft/rnin in the annular space. Basic Equation The velocity with which a solid particle freely falls through a fluid will increase until the accelerating force, gravity, is equal to the resisting forces, buoyancy due to the fluid displaced by the particle and friction due to the relative motion of the particle through the fluid. When the accelerating and resisting forces become equal,

By Kenneth E. Gray

The turbulent flow drag coefficients, or friction factors, have been experimentally determined for the cut-tings normally encountered in drilling operations. The gas law and average drag coefficients for characteristically flat particles (limestones and Shales) and for angular to sub-rounded particles (sandstones) were used to extend Newton's equation to a more useful form. The resulting equations are expressions for the slip velocity of a particle as a function of the particle size and shape, the bottom-hole injection Pressure, and the Bottom-hole temperature. The velocities necessary to lift actual rock cuttings as observed in the laboratory were compared with velocities predicted from the derived equations. Results indicate that there is no single correct circulating velocity and that the commonly used 3,000 ft/min linear velocity may he sufficient to lift only very mall particles. The advantage, in terms of horsepower requirements, of cycling high pressure air to obtain lower compression ratios was shown. INTRODUCTION A promising departure from standard rotary hydraulic drilling is air or gas drilling. In many instances the rate of penetration and bit life increases resulting from this method have been very substantial. Further the application of air drilling in areas of lost circulation or water sensitive pay zones has been highly successful. Purpose For the drilling contractor, the practical question remains, "How much air pressure and volume should I have?' lt is necessary to have a reasonable knowledge of the air velocities and pressures required in air drilling operations for the proper design and most advantageous use of surface equipment and for adequate removal of cuttings from the borehole. Eased on experience, most operators agree that at least 3,000 ft/min linear velocity is necessary for satisfactory hole cleaning.' It was the purpose of this investigation to evaluate the drag coefficients, or friction factors, for the cuttings normally encountered in drilling operations, i.e., sandstones, limestones and shales, and to utilize the values obtained to develop an expression for the minimum velocities and pressures (hence the volumes) necessary to lift these cuttings. THEORY Several investigators have done extensive work on the ability of drilling mud to lift bit cuttings. They conclude that the ability of a drilling mud to lift cuttings is dependent on the density difference between the rock being drilled and the drilling mud, the cutting size and shape, the mud flow constants, and the flow state (laminar, transitional or turbulent) of the mud. However, drilling muds are not true fluids but have a variation of viscosity with velocity in laminar flow and a constant apparent viscosity in turbulent flow. A survey of the literature revealed that no experimental work had been done to determine the velocities necessary to lift actual rock cuttings using a compressible fluid such as air or gas. In 1953, Nicolson6 stated that in air drilling fluid mechanics the particles could be assumed spherical in shape and that ; constant drag coefficient of 0.5 could be used due to the highly turbulent flow of the circulating air. By equating the turbulent resistance on the particle to the gravitational force on the particle, he obtained the expression, ^ = 2.67^y ,.......(1) where V, is slip velocity of spherical particle, ft/sec; D is particle diameter, in.; p, is particle density, 1b/ft8; and p, is fluid density, lb/ft3. However, as stated earlier, most operators seem to agree that adequate hole cleaning can be obtained by circulating a sufficient volume of air to give a linear velocity of 3,000 ft/rnin in the annular space. Basic Equation The velocity with which a solid particle freely falls through a fluid will increase until the accelerating force, gravity, is equal to the resisting forces, buoyancy due to the fluid displaced by the particle and friction due to the relative motion of the particle through the fluid. When the accelerating and resisting forces become equal,

By E. J. Harris

Mathematical solutions to complex mine ventilation problems are possible, but often the airway network is so complex that the mathematical solution becomes tedious and impractical. A fluid network analyzer, designed and built for analyzing mine ventilation problems has been in service at the Bureau of Mines' Pittsburgh station for approximately nine years. Using this analogue as a model, the mine ventilation network is simulated electrically by a combination of series and parallel circruits laid out to conform with actual mine airways. A tungsten filament lamp, referred to as a Fluistor, is used to simulate mine airway resistance. With the analogue ventilation model established, voltages of the proper amplitude to represent ventilating pressures are impressed across the circuit at points where mine fans and airshafts are located. Comparison of mathematical solutions of complex systems against analogue results showed a maximum variance of 3% for pressures and 2% for quantities. An electrical fluid network analyzer, designed and built especially for analyzing mine ventilation distribution problems, has been in service at the Bureau of Mines, Pittsburgh, for approximately nine years. It is a nonlinear, low voltage, fluid network analyzer of the type developed by the late Malcolm S. McIlroy, Professor of Electrical Engineering, Cornell University, who cooperated with G. E. McElroy, of the Bureau of Mines, in adapting the instrument to mine ventilation systems. Several modifications have been made since the original installation, but a considerable part of this paper is drawn from G. E. McElroy's original description of the analogue.' The analysis of water or gas distribution systems2,3 led to the development of this type of network analyzer. Several similar units are now employed by utility companies for this purpose; however, the Bureau of Mines unit is the only one designed specifically for mine airflow problems. Other airflow analogues employing a similar principle have been used at the Central Research Station of the Netherlands State Mines,4 in England,5 and in South Africa.6 Information on these devices indicated they were somewhat inflexible because commercial lamps with a sufficient range of resistance are difficult to obtain. Computers for mine ventilation analysis have been developed in Germany which instead of lamps use a variable resistance to adjust for turbulent flow laws. These units are expensive, but excellent results have been reported in their application. THE ANALYZER Theory of Application: As airflow generally follows the law of turbulent fluid flow, resistance to flow is nonlinear; consequently, the problem has been to find a nonlinear resistance element of suitable range that can be used for electrically simulated airflow. Tungsten filament lamps operated on alternating or direct current approximate the square-law resistance characteristics of mine airflow over a large range below maximum or rated voltage; that is, the voltage drop varies approximately as the square of the current. Consequently, the heart of the network analyzer is a nonlinear resistor known as a Fluistor, which is simply a custom-made low voltage, tungsten filament lamp that is available in a progressive series of relative resistance values ranging from 0.05 to 500 in nominal 5% steps. However, variations in manufacturing large groups result in differences of 1 to 3%, but series arrangements required for high-loss branches can be matched within about 1%. Utilizing a combination of Fluistors and load circuits, the mine ventilation system is duplicated electrically. Intake load circuits are connected from power intake to primary points of the circuit network; segments of unregulated flow along intakes and returns are represented by Fluistors, regulated splits and leakage paths are represented by load circuits and Fluistors of proper capacity; and mine exhausts are connected to ground from the last point of the network to complete the circuit. For the special purpose of representing booster fans or natural draft conditions, boosters or separate source circuits are provided that can be inserted between any two points of a network to increase voltage to the required value. Physical Layout: The analyzer consists of three 42

Jan 1, 1963

By H. L. Shergold, O. Mellgren

Concentration of fine quartz particles at the iso-octane/water interface has been investigated under different conditions of pH and dodecylamine concentration. The results obtained from the related studies on the effect of amine concentration and pH on the interfacial tension, adsorption density, electrokinetic potential, contact angle, and concentration of the various amine species in the system are presented. A good correlation was obtained between these different variables. It is well-known that very fine mineral particles are difficult to float in conventional flotation machines. Flotation rate studies have revealed that the rate of flotation of fine mineral particles is much smaller than that of coarser size fractions. The theories accounting for this behavior have been discussed by Arbiter and Harris1 and also Meloy.2 Theoretically, a hydrophobic fine particle might never make contact with a bubble because of the presence of an energy barrier in the vicinity of the air/water interface.' This energy barrier will have electrostatic, hydrodynamic, and Van der Waals force components. It was thought that by using an oil phase instead of air the energy barrier would be decreased so that fine particles could be concentrated at the oil/water interface more readily than at the air/water interface. The technique used involves dispersing the fine particles in water, containing the appropriate chemical reagents, and injecting a fine dispersion of iso-octane oil droplets into the pulp. After vigorous conditioning, the pulp is passed into a separating column where the oil droplets coated with a layer of mineral particles rise to the surface to form a separate layer. Air is introduced into the base of the separating column to ensure that heavy agglomerates of oil and particles report with the organic layer. This technique has been described previously4 and adopted by Lai and Fuerstenau5 who studied the alumina-dodecyl sulfonate system. The interfacial phenomena in the system composed of hematite, water, and iso-octane in the presence of sodium dodecyl sulfate have been studied and the results reported" earlier. This paper describes the results obtained from investigations into the interfacial phenomena in the quartz/water/iso-octane system in the presence of dodecylamine. The technique used in measuring the interfacial tension, electrokinetic potential, contact angle, and adsorption density were similar to those described previously? Materials Selected pieces of a high purity natural quartz, from the Isle of Man, were crushed in a laboratory jaw crusher and pulverizer. The — 52+72 mesh size fraction was retained and leached with successive washes of hot concentrated hydrochloric acid to remove iron impurities. When no iron was detected in the solution by ammonium thiocyanate, the quartz was washed thoroughly with distilled water until the conductivity of the wash water assumed that of the distilled water. Samples of 25 g of the —52+72 mesh quartz were ground for 5.25 min in an agate vibratory mill. The ground product was 100% —44 m and 57% —10 am, as determined by the Andreasen pipette. The specific surface area of the sample used for the adsorption and flotation tests was 0.94 sq m g-l. This corresponds to a mean particle diameter of 2.4 am. The —44 am quartz sample was stored under vacuum in the presence of silica gel crystals. For the contact angle determination between the three phases, quartz, iso-octane, and water, a piece of the natural quartz was ground into a block about 15 mm long, 15 mm wide, and 5 mm thick. One surface of the block was then polished by successively finer grades of silicon carbide "paper." The final polishing was conducted with alumina on a "hyprocel" paper. The polished quartz specimen was cleaned using nitric acid and ethyl alcohol followed by a wash with distilled water and then stored under distilled water. This pro-

Jan 1, 1971

By Fred H. Poettmann

An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of the compressibility factor of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. Comparison of calculated results with field measured results were made on 20 dry gas wells from a given field and 11 distillate wells from different fields. The agreement between calculated and observed results is good. In addition to their use in the calculation of sandface pressures of flowing wells, the factors can be used in the direct calculation of the static bottom hole pressure of gas wells, the capacity of gas transmission lines, and in the calculation of the theoretical isothermal horsepower necessary to compress a natural gas. Examples demonstrating the use of various equations are given. INTRODUCTION Many of the equations used to calculate the sandface pressures of flowing gas wells from well head data do not take into consideration rigorously the deviation of the natural gas from ideal gas behavior. For low pressure wells this error is not serious. For high pressure wells flowing at high rates this error call be serious. Sandface pressures are used to determine the producing capacity of gas wells as shown in the U. S. Bureau of Mines' Monograph 7.' An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of compressibility of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. In addition to their use in the calculation of sandface pressures, these "factors" can also be used to give a direct solution in the calculation of the static bottom hole pressure of gas wells, the calculation of the capacity of gas transmission lines, and the calculation of the isothermal horsepower necessary to compress a gas. CALCULATION OF SANDFACE PRESSURE OF FLOWING GAS WELLS The starting point in the derivation of any specific flow equation is an energy balance on the fluid flowing between any two points in the system. This energy balance based on one pound of flowing fluid is expressed by the well known general flow equation: jp~- vdP+ Ah + -— + wt + r. = o . . (i) P, 2 g. where V = specific volume of flowing fluid in cu ft/lb P = pressure in Ib per sq ft ah = difference in height above datum plane in, ft ------ = kinetic energy change of one lb of flowing fluid, v equals velocity in ft/sec and g. = 32.174 Ws = Work done by fluid while in flow (similar to shaft work in driving a turbine) W, = Znergy losses due to the irreversibilities of the flowing fluid This equation contains no limiting assumptions and can be made the basis of any fluid flow relationship by limitations or substitutions. The energy units are usually expressed in term* of foot-pounds of energy per pound of mass. Considering the vertical flow of gas through tubing or annu-lus, the external work W, done by the gas is zero. Let us also assume that the value of the kinetic energy function is small and can be neglected. Equation (1) is reduced to: /P: VdP + Ah+Wt = 0 .......(2) Pi The energy loszes PI can be expressed in terms of the well known Fanning equatinn: 4/Lf: rf = —— ............(3) 2g,D where L = the integrated average velocity in ft/sec of vertical flow in a uniform flow string -~h = L v = the integrated average velocity in ft/sec / = dimensionless correlating function

Jan 1, 1951

By Fred H. Poettmann

An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of the compressibility factor of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. Comparison of calculated results with field measured results were made on 20 dry gas wells from a given field and 11 distillate wells from different fields. The agreement between calculated and observed results is good. In addition to their use in the calculation of sandface pressures of flowing wells, the factors can be used in the direct calculation of the static bottom hole pressure of gas wells, the capacity of gas transmission lines, and in the calculation of the theoretical isothermal horsepower necessary to compress a natural gas. Examples demonstrating the use of various equations are given. INTRODUCTION Many of the equations used to calculate the sandface pressures of flowing gas wells from well head data do not take into consideration rigorously the deviation of the natural gas from ideal gas behavior. For low pressure wells this error is not serious. For high pressure wells flowing at high rates this error call be serious. Sandface pressures are used to determine the producing capacity of gas wells as shown in the U. S. Bureau of Mines' Monograph 7.' An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of compressibility of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. In addition to their use in the calculation of sandface pressures, these "factors" can also be used to give a direct solution in the calculation of the static bottom hole pressure of gas wells, the calculation of the capacity of gas transmission lines, and the calculation of the isothermal horsepower necessary to compress a gas. CALCULATION OF SANDFACE PRESSURE OF FLOWING GAS WELLS The starting point in the derivation of any specific flow equation is an energy balance on the fluid flowing between any two points in the system. This energy balance based on one pound of flowing fluid is expressed by the well known general flow equation: jp~- vdP+ Ah + -— + wt + r. = o . . (i) P, 2 g. where V = specific volume of flowing fluid in cu ft/lb P = pressure in Ib per sq ft ah = difference in height above datum plane in, ft ------ = kinetic energy change of one lb of flowing fluid, v equals velocity in ft/sec and g. = 32.174 Ws = Work done by fluid while in flow (similar to shaft work in driving a turbine) W, = Znergy losses due to the irreversibilities of the flowing fluid This equation contains no limiting assumptions and can be made the basis of any fluid flow relationship by limitations or substitutions. The energy units are usually expressed in term* of foot-pounds of energy per pound of mass. Considering the vertical flow of gas through tubing or annu-lus, the external work W, done by the gas is zero. Let us also assume that the value of the kinetic energy function is small and can be neglected. Equation (1) is reduced to: /P: VdP + Ah+Wt = 0 .......(2) Pi The energy loszes PI can be expressed in terms of the well known Fanning equatinn: 4/Lf: rf = —— ............(3) 2g,D where L = the integrated average velocity in ft/sec of vertical flow in a uniform flow string -~h = L v = the integrated average velocity in ft/sec / = dimensionless correlating function

Jan 1, 1951

By D. E. Baldwin

A method has been devetoped for predicting the produced concentration profile for a miscible slug in a five-spot pattern. The technique consists of dividing the five-spot into radial etements and applying an approximate radial solution of the dispersion equation to each element. To test the method, laboratory experiments were conducted in five-spot models. The models were 2-ft X 2-ft X 2-in. Berea sandstone slabs simulating one quarter of a five-spot. An aqueous system was used with tritiated water as the tracer fluid. Effiuent concentration profiles were obtained for slug sizes ranging from 0.7 to 10.2 per cent of the pore volume. Agreement between predicted and experimental profiles was excellent. A prediction was also made for a field tracer test. For this case the predictive technique was modified to account for stratification by use of a layer cake model. INTRODUCTION Dispersion in porous media is of growing interest to the petroleum industry because of the increasing importance of secondary recovery operations. One of the controlling factors in the recovery of oil by miscible displacement or by the use of waterflood additives is the degree of mixing between the fluids of interest. Consequently, one of the prerequisites of any recovery prediction is an adequate method of predicting dispersion. Also, any quantitative interpretation of waterflood tracer tests is dependent upon the ability to determine the dispersion of the tracer as it flows through the reservoir. During the past few years there have been reported in the literature the results of a large number of investigations, both theoretical and experimental, concerning dispersion in porous media. (An excellent review of these results has been presented by Perkins and Johnston." Only a few of these investigations, however, deal with non-lin-ear flow systems. Raimondi et al.9 have presented an approximate solution of the dispersion equation for radial, diverging flow. Lau et al. 7 and Bentsen 1 have presented laboratory data for radial systems and have shown that the dispersion can be adequately predicted by Raimondi's solution. Bentsen has also presented a small amount of data for a radial, converging system. Brigham and Smith' have applied Raimondi's solution, with approximating assumptions, to a five-spot pattern to predict the behavior of a field tracer test. The afore-mentioned investigations have been concerned almost entirely with radial diverging flow and have neglected the effect of converging flow. For this reason the present investigation was undertaken to develop a method of predicting dispersion in a five-spot pattern including the effects of both diverging and converging flow. The method which was developed consisted of dividing the five-spot into diverging-converging radial elements and applying an approximate solution of the disperson equation to each element. DEVELOPMENT OF PREDICTION METHOD The development of the prediction method consisted of two phases: construction of a radial model of a five-spot pattern and derivation of equations to describe the dispersion in the radial model. FIVE-SPOT MODEL A radial model which simulated the flow characteristics of a five-spot pattern was constructed by dividing the five-spot into diverging-converging radial elements (Fig. 1). Because of symmetry the model represented only an octant of a complete five-spot. Each element had an included angle of l° (making a total of 45 elements) and a radius r, such that the sum of the individual element areas was equal to the area of the five-spot octant. The normal five-spot fractional flow behavior was determined by averaging the experimental five-spot data of Caudle and Witte,' Dyes, Caudle and Erickson 5 and Fay and Prats. 4 The resulting curve is shown as a solid line in Fig. 2. To match this curve with the radial model it was

Jan 1, 1967

By P. L. Raffo, T. E. Mitchell

Single crystals of Ta-Re alloys have been deformed in tension and compression as a function of solute concentration and temperature. The temperature dependence of the yield stress is decreased by alloying. At low temperatures there is a minimum in the yield stress (at about 2 at. pct) as a function of concentration. Three-stage hardening was not observed in the alloys. Linear hardening rates, where observed, were characteristic of stage II in pure tantalum. At low temperatures and high concentrations, work softening was often prominent at high strains. Electron microscopy revealed that dislocations are mostly of screw orientation, although heavily jogged and cusped, and often occurring in dipole pairs. The alloys become more prone to cleavage with increasing rhenium content, due mostly to an increasing yield stress rather than a change in cleavage strength. Alloying apparently decreases the Peierls stress of bcc metals at low temperatures by facilitating kink nucleation at solule atoms on screw dislocations. Higher solute concentrations further increase the yield stress. Quantitative calculations and comparisons with other bcc alloy systems are made to show that the predominant hardening mechanism is due to the interaction of dislocations with misfitting solute atoms close to their slip planes. The interaction is so localized that the core dilatation of dislocations could be important. CONSIDERABLE interest has arisen in recent years concerning the mechanical properties of unalloyed bcc metal single crystals.'-7 These and similar studies on polycrystalline materials have resulted in a considerable understanding of the macroscopic yielding characteristics8, 9 and the process of work hardening.'' This latter advance is due in a large part to the increased use of transmission electron microscopy in P I RAFFO junior Membar AlMF is Metallyureist Iewis such studies and the general observance of three stages of work hardening in the shear stress-shear strain curves. The properties of substitutional alloy single crystals of bcc metals have not been determined in as much detail as in the pure metals. Insufficient systematic studies have been made on the effect of alloying on the magnitude and temperature dependence of the yield stress or the stress-strain curves. The alloy systems which have been studied to some extent have been Ta-W,11, 12 Ta-Mo," Ta-Nb,12'14 Mo-Re,15 W-Re,16 and alloys of iron with nickel,17 platinum,17 vanadium,18 and aluminum.18 A large body of data exists on Fe-Si alloys, but these have mainly been a result of the experimental convenience of using this material in place of unalloyed iron, rather than a comprehensive study of the mechanical properties of the alloy system per se. The present authors have previously reported additional data on the properties of Ta-Re, Ta-W, and Ta-Mo alloy single crystals.1g Some of the data on Ta-Re alloys will be described later in this paper. Briefly, single crystals of Ta-Re (1.2 to 10.6 pct), Ta-W (6.2 and 10.2 pct), and Ta-Mo (2.9 and 9.0 pct) were grown by the floating zone method in vacuum. Compression tests were performed in the temperature range 77" to 623°K. Slip was found to occur in Ta-Re alloys at 77' and 298°K on the (~ll)[lil] system and slip lines were coarser and straighter than in unalloyed tantalum. No three-stage hardening was observed in any of the alloy systems and the curves were nearly parabolic. At temperatures above 175°K the yield stress* increased monotonically with in- creasing composition while at 77" and 175°K the yield stress initially passed through a minimum after which it began to increase again for concentrations greater than about 2 pct Re. Similar minima were observed in Ta-Mo alloys and to a lesser extent in Ta-W alloys. The minimum was interpreted as being a result of a

Jan 1, 1969

By C. E. Taylor, L. V. Fegan

With on-stream electrical-resistance measurement as a basis, Bethlehem Steel developed a system for the completely automatic control of moisture in the feed to pelletizing. This system was later successfully adapted to the sintering process, with automatic control of moisture serving to maintain the air permeability required for effective burnthrough. Control of moisture content is an important factor in the agglomeration processes by which iron ores are converted into either pellets or sinter for use in blast furnace burdens. In both pelletizing and sintering the iron ore feed must be mixed with the proper amount of water so that capillary attraction and surface tension forces can be utilized to cause the ore particles to coalesce and form larger agglomerates. The success or failure of the entire pelletizing or sintering process depends on maintaining the proper moisture. Therefore, continuous automatic control of moisture is one of the conditions for the automation and optimum use of both pelletizing and sintering processes. In pelletizing, variation in moisture can cause ball size to change and is a common cause of cycling which results in a fluctuating feed to induration. Excessive moisture can cause lumps to form at the cutter bar and contributes to the formation of buildup on the green pellet screens. Green pellets with excess moisture deform and reduce bed permeability. In sintering, accurate control of moisture is also important because moisture must be within the limits that give the bed the required permeability for good burnthrough. In response to these problems, a moisture meter, based on the principle of electrical resistance measurement, was developed. This moisture meter has been a key factor in the development of systems for providing uniform feed of concentrates in most of Bethlehem's pelletizing and sintering plants. The automatic moisture control system for pelletizing was successfully plant-tested and resulted in the following: (1) Control of moisture in the green pellets to +0.2%. (2) Up to 50% reduction of bentonite consumption. (3) Control of unscreened green pellets so that 85% of the furnace product at Cornwall was within the size range of -5/8 +3/8 in. (4) Improved quality of product with a tumble test of 94 and a "Q" index of 90 as the result of feeding a consistently uniform feed to a shaft furnace. In sintering, where sinter-mix temperature change and variable feed could cause errors in moisture measurement, the system was designed to maintain the correct moisture under virtually all plant operating conditions. It was successfully plant-tested and is now in use in most of Bethlehem's sintering plants. The system achieves: (1) Control of moisture within the desired limits for optimum production. (2) Continuous measurement and control of moisture even with intermittent feeding of mix, so that attention by the operator is seldom required. The two automatic control systems provide smoother startup, minimize upsets and contribute to the consistent control of process variables in pelletizing and sintering. MOISTURE-CONTROL SYSTEM FOR PELLETIZING Fig. 1 shows the general arrangement of the system for automatic control of moisture in pelletizing.1 In this system a bin level controller, using a single probe which regulates filter speed, maintains a uniform low concentrate level in the bin.' The moisture meter, comprised of an analyzer and control components, continuously measures the moisture in the bed and adds water to adjust to the desired level. The meter is placed in an open-loop (feed-forward) control, i.e., water addition occurs downstream from the probes. Filter vacuum must be set so as to produce a filtercake moisture which is below the optimum for pelletizing. Bentonite is added after the

Jan 1, 1969

By G. Krauss, J. M. Thompson, H. Y. Kumagai

Boron-doped p-type and arsenic-doped n-type source materials were used to deposit thin silicon films on amorphous fused quartz substrates by cathodic sputtering in argon atmospheres. All as-sputtered films were found to have high electrical resistivities of about 104 ohm-cm, despite controlled variation of the substrate temperature from 50O to 400°C. The high resistivity of the n-type films persisted even after postdeposition heat treatments as high as 1000°C, while p-type films showed a sharp decrease in re-sistivity to values on the order of 1 ohm-cm or lower after heat treatments above 650°C. X-ray and electron diffraction, together with transmission electron microscopy of the p-type films, revealed that the improved electrical properties resulted from two distinct processes. A primary crystallization or recovery within the pains, which were approximately 200Å diam in the as-sputtered films, accompanied the initial sharp drop in resistivity at 650°C. Following this process, a secondary recrystallization or discontinuous grain growth was nucleated in films heated at 1000°C. The resulting significantly increased size of certain grains and the corresponding decrease in grain boundary area accompanied an additional decrease in resistivity after heat treatment at 1000°C. The properties of thin silicon films are of interest because of their possible application in device technology. Although evaporated silicon films on amorphous substrates1-3 and both sputtered and evaporated germanium films4-' have been studied in some detail, sputtered silicon films have not been widely investigated. It has been observed that evaporated silicon films deposited at substrate temperatures below approximately 600°C are amorphous.1,3,9,10 At substrate temperatures higher than this, the films are polycrys-talline, sometimes exhibiting a fiber texture with the fiber axis normal to the substrate surface.' Similar results have been obtained with germanium films, although in this case the substrate temperature for the amorphous to crystalline transformation is reported to be as low as 150OC.5 The mechanism of film deposition by cathodic sputtering differs markedly from that of thermal evaporation. It might be expected, therefore, that the properties of sputtered and evaporated silicon films would also differ. The purpose of our study was to determine the effects of deposition conditions and heat treatments on the electrical resistivity, structure, and composition of thin silicon films sputtered on fused quartz substrates. EXPERIMENTAL DETAILS Apparatus. A schematic diagram of the sputtering apparatus is shown in Fig. 1. The vacuum chamber was enclosed by an 18-in.-diam by 12-in.-high Pyrex bell jar mounted on a stainless-steel base plate. Access holes in the base plate were used to supply argon gas to the chamber, to allow for various electrical feedthroughs, and to provide mechanical motion to the shutter between the source material and substrate. The chamber was evacuated by a pumping system consisting of a 720 liter per sec oil diffusion pump backed by a 13 cu ft per min mechanical pump. Ultimate pressures of less than 1 x 10-6 torr could be achieved. The substrates were securely mounted on the stainless-steel substrate holder by quartz tabs held in place by stainless-steel screws. Temperatures of the substrate holder were measured by a thermocouple inserted within the holder. A quartz lamp was used to heat the holder and temperatures as high as 420°C could be maintained. Deposition Conditions and Procedure. The substrates used were polished fused quartz slides 116 by 1-12 by 3 in. and were outgassed at 200°C for 1 hr at a pressure less than 1 x 10-5 torr before film deposition. This was followed by a presputtering period of 2 hr during which the movable stainless-steel shutter was so positioned that deposition on the substrate was prevented. At a sputtering voltage of 5000 v dc and an argon pressure of 0.03 torr, the cathode cur-

Jan 1, 1967

By B. Ancker, E. R. Parker

The small-angle dislocation-boundary density of nickel and some of its alloys was investigated as a function of strength. It was found that the strength is a linear function of the density for pure nickel and for Ni-Ti alloys, but not for Ni-Co alloys. The results are interpreted in terms of dislocation theory. THE processes by which metals deform are still poorly understood, even though there has been a concentrated research effort in this field during the past fifteen years. Single crystals and individual grains in polycrystalline aggregates often develop a subgrain structure when deformed. Substructure formation occurs as a result of local plastic bending and its development is augmented by annealing at moderate temperatures.' Investigations made to date, largely qualitative in nature, have shown that the substructure is composed of a network of small-angle boundaries. The purpose of the present paper is to present some quantitative data on substructure (obtained by X-ray techniques) and to correlate the results with the strength of the metal in which the substructure existed. Nickel and some of its alloys were used for the experiments. The presence of certain kinds of foreign atoms greatly alters the formation and effect of the substructure. Polycrystalline tensile specimens were prepared from high purity nickel, two Ni-Ti alloys, and two Ni-Co alloys; the composition of these materials is given in Table I. Tensile specimens having a reduced section 1/4 in. diam by 3 in. long were machined from materials which were refined, cast, forged, and swaged in the authors' laboratories. The specimens were annealed for 1/2 hr at 1150°C, a treatment which produced grains having a diameter of approximately 0.5 mm. This rather large grain size was used to facilitate later examination of the substructure. Duplicate specimens of pure nickel and of each alloy were strained various controlled amounts in tension at room temperature and recovered for 1 hr at 800°C. (This strain induced prior to recovery will be referred to hereafter as the "prestrain.") One of each of the recovered specimens was pulled in tension at room temperature to obtain the stress-strain curves. The other bar was cross-sectioned in the gage length, carefully polished, and deeply etched to remove all cold-worked material. This specimen was then examined by the X-ray microscopic technique2 to determine the nature and magnitude of the substructure. The use of duplicate specimens permitted a direct correlation between tensile properties and substructure characteristics. X-ray microscopic studies were made with an apparatus constructed to have an average resolving power of about 4 microns. The monochromatic radiation reflected from grains which were oriented favorably was recorded on fine-grain photographic plates. These were examined at Xl00 which represents the true magnification, since there was essentially none in the X-ray system. This permitted detection of angular differences between subgrains of as little as 10 min. A very large number (over 100) of grains for each specimen was examined at this magnification and the number of substructure boundaries per grain was tabulated. An accurate relative measure was obtained under fixed experimental conditions, e.g., same grain size, resolving power, magnification, etc. The greatest source of error lay in counting the subboundaries, but this proved to be surprisingly small when repeated counts were made by different investigators. Furthermore, as the subsequent analysis indicates, whatever human error was introduced was not significant. Analysis of the Substructure Data When the number of substructure boundaries per grain was plotted against the frequency, skew (asymmetrical) distribution curves of the type shown in Fig. 1 were obtained. These are smoothed curves for the 1 atomic pct Ti in nickel alloy subjected to various prestrains and are representative of the curves obtained with all the metals. Since skew-distribution curves cannot be analyzed by elementary statistical methods, detailed investigations were undertaken in order to 1—test the consistency

Jan 1, 1955

By C. G. Dunn, F. W. Daniels, M. J. Bolton

IN recent investigations1. a data on relative grain boundary energies in silicon iron have been obtained. The present investigation is a continuation of this work along similar lines for the purpose of obtaining additional information on boundary energies. One aim has been to extend the data for a (110) series, which was formerly studied,' to small differences in orientation, since a recent theory3,4 based on a dislocation model of grain boundaries predicts a E. ? (A-ln ? )* form of energy curve for small differences in orientation. Another aim has been to provide data for a (100) series—all grains having (100) planes parallel with the surface of sheet speci-mens—and to compare the results with the predicted form of energy curve. Finally, information on the nature of approach to equilibrium conditions was to be obtained through observations on the movements of grain boundaries. Recently published papers2,3,5 have treated the mathematical problem of expressing equilibrium conditions for grain boundaries for the general case when boundary energy depends upon boundary orientation. The equilibrium equations, which relate equilibrium angles and grain boundary -free energies, contain additional terms that express the variation of energy with boundary rotation. It has been necessary in the present investigation, however, to neglect these additional terms since no data for their evaluation were available. Dropping the additional terms leads to the following approximate equations which were used: E12/sin ?3 = E18/sin?2 =E28/sin?1 or those of the form: E28 + E13 cos ?3 + E12 cos ?2 = 0 Experimental Procedure Except for sample F1, all specimens were made from two lots of silicon iron called C and L respectively. The compositions of these are listed in table I. In the preparation of 12 three-grain specimens for the present investigation, a controlled grain growth technique, which has been described previously,' was used. After preparation, the specimens were notched at the boundaries to shorten them as a means of providing more rapid approach to equilibrium in the anneals. Initial grain boundary angles were determined from micrographs taken at X500. Seven anneals, totaling about two days at 1300°C and two to four days at 1400°C, were run in an atmosphere of pure dry argon as described in a previous paper.' Boundary changes generally could be followed without metallographic surface preparation, because thermal etching occurred during the anneals. These observations indicated angle changes of as much as 35" in some instances, with the major changes occurring during the first anneal. Annealing was discontinued when the angles reached stationary values. Fig. 1 gives a schematic diagram for purposes of defining crystallographic and boundary directions for three grains with common plane parallel with the sheet. Differences in orientations (expressed by A) are the angles between [00l] directions. They can be calculated with the aid of the ?'s. The grain boundary angles can be calculated from the w'S. Results Results are given in table 11. Types of crystal-lographic planes refer to those parallel with the sheet. Orientations are given in terms of [00l] direc-tions as shown in fig. 1. Except for grain 3 of specimen H6, which was unintentionally tilted 7" out of a (100) plane, all grains generally were within 1" or 2" of the plane specified. The directions of boundaries were measured generally to within 1" on both upper and lower surfaces, and the average direc- tions were determined for use in calculating the grain boundary angles. Variations in grain boundary directions refer to deviations from average directions. Boundaries near the junction point for all specimens generally were very close to perpendicu-

Jan 1, 1951

By Donald L. Johnson

REVIEWS on the applications of optical ellipsometry to the study of polycrystalline metal surfaces and oxidation-corrosion processes have been presented by Kruger and Hayfield, and white.' In view of the ability of this method to measure differences in film thicknesses to below 10A, ellipsometry was used to study nitrogen chromium surface reactions during the early stages of growth. Chromium specimens in the form of a vapor-deposited film on glass substrate were exposed to purified nitrogen for selected time increments at 700°F. This temperature was selected because it gave the best combination of exposure time and thickness change. Ellipsometer readings were taken immediately after exposure to determine polarizer and analyzer angles using monochromatic light at 5000A.2 After each set of readings was obtained, the specimens were exposed for an additional increment of time in nitrogen and a new set of readings taken. Film thickness was computed from LUCy's3 equation for nonabsorbing thin films. In Lucy's derivation, two equations for computing thickness were obtained. Peterson and Bashara4 have discussed the validity for using one of these equations when an independent value of the refractive index is used. The film was assumed to be Cr2N (Ref. 5) and was determined to be optically isotropic. Optical isotropy was determined by rotating the films about the surface normal and noting that the azimuth and phase difference did not change. Since Cr2N is reported to be hexagonal, it is assumed that optical isotropy results from randomly oriented grains of very small size. The thickness was computed using refractive indexes varying from 1.6 to 2.0. The maximum variation in computed thickness over this range for a given exposure was 20 pct. The data reported are based on a refractive index of 1.8. Reference readings on the metal surface were taken just prior to nitrogen exposure. No attempt was made to deoxidize or denitride the metal surface after vapor deposition, although specimens were observed to gain approximately 2A per day in air at room temperature. Twelve experimental exposures were made on a single sample over a total accumulative time of 132 min. Additional experiments over longer time intervals were made to determine the effect of heat up and cool down during and after exposure. As a result of these experiments, the thickness computed for each exposure was increased by 4A. Time profiles based on the linear law are similar to the oxidation profiles for many metals.' The rate of accumulation of the nitride film is rapid at first and slows down as the film thickness increases. The parabolic law x2 =k1t +k2 [11 gives a straight line in Fig. 1; therefore, it is concluded that the kinetic behavior is parabolic. Sey-bolt and Haman5 reported similar results for CrlN on chromium at 1000°C although the order of the thickness was much greater (10-3 cm). Analysis of thickness data based on other values of refractive index shows that the curves are shifted vertically but retain the same shape. According to Mott,7 it is expected that for thin films k2 is zero in Eq. [I]. Since the plot in Fig. 1 is linear through the origin, k2 is zero and the parabolic nature of the film growth is further indicated. The parabolic rate constant, calculated from the slope in Fig. 1, is 4.4 x 10-16 sq cm per sec. The corresponding work (+) required to bring an electron from the metal into the oxide is estimated to be 1.5 ev.7 The author thanks professor N. M. Bashara and D. W. Peterson, Electrical Materials Laboratory, University of Nebraska, for making the ellipsometer available and providing technical assistance. The Electrical Materials Laboratory is supported by the office of Naval Research, NONR 3718 (01). This work was partially supported by the Engineering Experiment Station and the Mechanical Engineering Department, University of Nebraska.

Jan 1, 1967

By C. F. Baes, H. H. Kellogg

The effect of dissolved sulphur on the surface tension of liquid copper has been measured by a sessile-drop method at 1120°C. Sulphur is highly "surface active" in liquid copper; the surface tension lowering is about 650 dynes per cm for a sulphur content of 0.6 pct. The degree of adsorption of sulphur on the liquid copper surface has been calculated, and various models of the surface layer have been tested against the results. THE science and technology of surface chemistry and physics at room temperature are relatively advanced, and the engineer possesses a background of information that allows him to predict and control many phenomena—such as frothing, emulsifica-tion, wetting of solids by liquids, flow through capillaries, adsorption, mineral flotation, penetration of porous bodies by liquids—which are related to the behavior at interfaces. Surface phenomena in high temperature systems are probably of equal importance to the metallurgist and ceramist, but at present are incompletely understood. Frothing of liquid slags, penetration of refractories by liquid metals and slags, infiltration of powder metallurgy "skeletons" by liquid metals, bonding of metals to ceramic oxides in the manufacture of "cermets," wetting of one metal by another for purposes of soldering and welding, are but a few of the many phenomena of metallurgy that are controlled by surface behavior. Concerning the most elementary of surface properties—the gas-liquid surface tension—the existing data on liquid metals are inadequate. For the surface tension of "pure" metals the results of different investigators differ by 10 or 20 pct. The role of surface-active agents in liquid metals is relatively unknown, and it is not unlikely that the discrepancy in the values for the surface tension of pure metals is due to the presence of traces of surface-tension-lowering impurities in the samples. Based on analogy to room-temperature systems, the highly surface-active impurities in liquid metals should be elements or compounds of limited solubility in the liquid metal and possessing considerably weaker in-termolecular bonding forces than the metal itself. The nonmetals of the V, VI, and VII groups of the periodic table might be expected to act in this way. The experimental work in this paper represents a study of the surface tension of pure liquid copper in hydrogen at 1120°C, and the effect of dissolved sulphur (which is a strongly surface-active agent in liquid copper) on the surface tension of copper. The Gibbs equation has been applied to the results and an attempt made to correlate the surface tension behavior with two models of the interfacial layer. The sessile-drop method for surface-tension measurement at high temperatures which was used in this investigation has been described by several investigators.1,2 It involves the measurements of the dimensions of a stationary liquid drop on a horizontal surface. From these dimensions and the liquid density, the surface tension of the liquid may be calculated either by means of various approximate formulas, or more accurately by means of the Bash-forth and Adams tables.- or liquid drops which exhibit a contact angle greater than 90°, by far the simplest dimensions to measure are those indicated in Fig. 1, i.e., the diameter of the maximum horizontal cross section (2X) and the maximum height (Z) above this plane. In order to calculate the contact angle and the volume of the drop, the diameter at the base of the drop (2a) and the total height (h) also must be measured. Upon investigation of the literature, the authors failed to discover a concise statement of the accuracy of the method when used in conjunction with the Bashforth and Adams tables, hence a careful examination of the tables was made and the curve shown in Fig. 2 was constructed. The "error factor," plotted as the ordinate, is the factor by which the

Jan 1, 1954