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By Alan Morris

The investigation of the hot-working properties of metals and alloys furnishes a problem which has been attacked in many ways. Tensile, impact and hardness tests on heated specimens have furnished interesting data, but in most cases either the tests are difficult to make or the results hard to translate into terms of practical value. Some investigators have used the drop hammer to advantage. It furnishes a rapid and simple method of comparing the plasticity of metals at elevated temperatures. The indications obtained by its use are of value only in so far as their significance is understood and they can be intelligently used to predict the behavior of a given metal under given hot-working conditions. The work reported here was undertaken not only to add something to the existing volume of such data but also to contribute to our understanding of the results. Doerinckel and Trockels1 used cylindrical specimens and determined the work required to reduce the specimen to one-half the original height. Ellis2 studied the effect of constitution on the malleability of steel at high temperature. His method and that of Kent3 were similar to the one used in the present work. The tests described by Kent were made on samples of tin, zinc, aluminum, copper, lead, 70:30 and 60:40 brasses, with and without a small quantity of lead. He concluded that the test would "indicate whether a material can be forged at a given temperature without cracking." This interpretation must be used with caution. While it is probably true within limits, one must consider the severity of the forging operation. Again, it will not apply strictly to forms of hot working other than forging. For instance, an alloy running 61.0 per cent. copper, 3.25 per cent. lead, remainder zinc, will crack badly in the test, yet, it is extruded commercially. When the possibility of hot-working a new alloy is being considered a study of its drop-hammer test should help

Jan 1, 1931

E. A. Peretti (University of Notre Dame, Notre Dame, Ind.)—Mr. Deitz and Professor Halpern are to be congratulated on a fine piece of work which adds to our knowledge of precious metal—cyanide-solution reactions. I should like to offer the following comments: The authors state, "the rate plots were always linear, indicating a zero order reaction rate." This appears to conflict with the data of Fig. 6 which clearly show the rate to vary with the NaCN concentration. It seems to me that the rate curves are linear because of the experimental setup. The volume of solution is so large that over a three-hour period the change in NaCN concentration is very small. A rough calculation for the experiments of Table I where the initial NaCN strength was 0.0096 mol per liter show a change of NaCN concentration of about 0.0009 mol per liter at the end of three hours. In short, over the period of time that the experiment was in progress, the NaCN strength was practically constant; hence the rate of dissolution remained practically constant. Eq. 10, as written, specifies that 1 mol of H202 and 2 mol of hydroxyl ion are formed when 2 mol of silver react. Is this true? The effect of H,O2 on the rate was made in solutions having an oxygen pressure of 4.8 atm. Were any tests made in solutions containing hydrogen peroxide, but no oxygen? G. M. Willis (University of Melbourne, Melbourne, Australia)—The authors have shown that, under their experimental conditions of high O2 concentrations, the rate of solution of silver in aqueous cyanide solutions is controlled by diffusion of O2 or CN- to the metal surface. With diffusion control, it is generally not possible to obtain further information about the mechanism of a process.

Jan 1, 1955

By R. J. Watson

IN an earlier paper, Ehrenburg and Watson1 published the develop-ment for a potential function by which it is possible to obtain the electric potential at points on the surface of the ground when a current I passes between two electrodes and the subsoil is composed of parallel, homo-geneous layers. In view of the continued interest in the resistivity method of electrical prospecting, the author has considered it desirable to publish the results of a number of theoretical studies making use of this potential function, and also the results of a considerable number of tests with models designed to show the soundness of the mathematical theory. It is hoped that these results will prove how erroneous the various empirical rules for depth determination may be. It has always seemed puzzling to the author that one can find, in papers published comparatively recently, statements upholding empirical rules for depth determination, and attributing most of the irregularities of resistivity curves to vertical discontinuities in resistivity in the subsoil. It would indeed be fortunate for the geophysical investigator if such a simple rule held. However, numerous investigators have proved both theoretically and by the success of their field work that such a rule does not hold, even by a" rough approximation" in many cases. On the other hand, as is shown in the development to follow; if the underground conditions are favorable it is possible for such empirical rules to give a satisfactory depth determination. Authors should be more careful about making such sweeping state-ments about the accuracy of these empirical rules because the relative simplicity of the field work places the method in the hands of any inter-ested experimenter, and he is likely to become completely disgusted with the method after he has made several obviously wrong interpretations by empirical rules.

Jan 1, 1934

By Louis W. Huber

The Liberty tunnels extend through a very steep hill in Pittsburgh (locally called Mount Washington) for a distance of slightly over a mile. The two tunnels parallel each other and are 59 ft. apart, center to center (Fig. 1) They are for vehicular traffic, and in each there is ample space for two lines of vehicles. Northbound traffic, toward downtown Pitts- burgh, is confined to the east tunnel and southbound traffic to the west tunnel. The South Hills residential district is separated from downtown Pittsburgh by Mount Washington, a steeply rising escarpment south of and adjacent to the Monongahela River. The Mount Washington district immediately south of the Monongahela River has been reached, for many

Jan 1, 1927

By L. W. Pommier, F. B. Brien

In order to determine the ability of cyclones with heavy liquids to preconcentrate ores, a study of the variables which govern the behavior of the solids in the apparatus has been made. To assess the test results, the cyclone products have been examined by sink and float analysis at various specific gravities and the results examined by the Tromp type curves. This analysis has allowed the development of the SG50 unit — specific gravity of the particles, 50% of which goes to overflow and 50% to underflow — rather than the d50 (size) unit used in classification. The results of tests using the closed circuit laboratory cyclone assembly with dry feed samples show the validity of the equation: Results indicate that 83% of the feed may be rejected containing 0.1% Sn. Recoveries in excess of 80% are obtained at a grade of 1.7% Sn from feed samples assaying 0.4% Sn. As water-wetted particles may comprise the plant feed, separate tests were performed to examine the effect of water in the feed material. Results indicate that the entrained water divides into two fractions, one of which is defined as free water and is rejected to the cyclone overflow, the other fraction forms an adhering envelope of water on the ore particles, altering their effective size and density. The presence of the envelope of water on the particles increases the ease with which the heavy liquid can be recovered but decreases the concentration efficiency. It is proposed that the detrimental effect of water in the cyclone may be minimized by using a higher feed pressure cx by the use of surfactants. A higher feed pressure, by increasing shearing forces, would reduce the thickness of the water envelope on the particles. The surfactants, by adsorbing on the particles and producing hydrophobic surfaces, would displace the water envelope and allow the heavy liquids to wet the modified surfaces. In both cases a larger proportion of the entrained water would report to the cyclone overflow as free water, thus minimizing the detrimental effect of water on the concentration efficiency. Additional testing is required on the behavior of water in the system; to determine the optimum particle size for the preconcentration step; to determine final particle size for optimum liberation and subsequent treatment of the products in present plants and to determine the efficiency with which the heavy liquids may be recovered from the products. For many years tin mining in Bolivia has been very important to the economy of the country. In recent years the tin content of the ore reserves has been steadily declining. An improvement in concentration efficiency is highly desirable in order to increase productivity from the present low-grade mine ores and to allow retreatment of the many millions of tons of tailings from past operations, some of which equal the present ores in tin content. A number of investigators have been or at present are working on various phases or approaches to the problem. Gravity concentration is used in Boliva today as it has been in the past to recover tin minerals from the mine ores. An objective of the work reported in this paper was to examine the possibility of treating present low-grade ores and sink-float tailings products, at a relatively coarse grind, in order to produce a higher grade feed material to the present gravity concentration plants. The possibility of using heavy liquids in cyclones to treat large tonnages with a minimum of outlay presented itself and is of particular interest since heavy liquids are now available with physical properties which lend themselves to simple recovery methods. Tests were carried out using a closed circuit laboratory cyclone assembly. It was desirable to determine the value of using the SG50 concept and Tromp curves for analyzing the cyclone concentration test results. The effects of entrained water on the behavior of ore particles in the cyclone were examined. THEORY AND TECHNICAL CONSIDERATIONS Three types of heavy liquids are available: aqueous solutions of very soluble salts, salts or mixtures of

Jan 1, 1964

By G. Rowan, M. W. Clegg

The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including: gravity drainage of depletion-type reservoirs; multiple well systems; well interference. INTRODUCTION A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages — the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov,1 who (a) postulated a disturbed zone in the reservoir, the radius of which increases with time, and (b) replaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following: 1. That the flow is irrotational. 2. That the formation is of constant thickness. 3. Darcy's Law is valid. 4. The formation is saturated with a single homogeneous fluid.

By C. B. Murray

"OUR experience from time immemorial has been that the furnace sampler and analyst usually find one or two, and sometimes more, points less iron than the Lake Superior shippers, but, fortunately, their contracts are `iron bound'; and, generally speaking, analyses at this end are practically useless, for we are obliged to settle on basis figured out to a half dozen decimals, and, as a rule, without recourse in the event of flagrant discrepancies between the two analysts." The above is an extract from a letter received by the writer of this paper from the President of an iron company, discussing the question of sampling. It so well expresses the opinion of many blast-furnace owners and managers that it seems a suitable text for a discussion concerning the need of uniform methods of sampling. If all railroad cars were loaded the same, if all stock piles were the same shape, and if ore was received regularly each day, the problem of sampling would be comparatively simple. The conditions, however, vary widely at different places, and at the same place from day to day, and it is probably impossible to determine upon any one method of sampling which will apply at all times and in all cases. However, it would seem as though some general principles might be suggested which would tend to overcome these wide differences in samples of the same thing. The sampling of a cargo of iron ore begins at the breast of ore in the mine. The ore is sampled from the mine cars, from the stock piles, and from the railroad cars at the mines. At the lower Lake ports the ore is practically all sampled from the boat, occasionally from railroad cars. At the furnaces, the ore is nearly all sampled from railroad cars and occasionally from stock piles.

Jan 1, 1914

By M. Prats

The effect of a sand-filled vertical fracture of limited radial extent and finite capacity (fracture capacity is the product of the permeability and width of the fracture) on the flow behavior of a cylindrical reservoir producing an incompressible fluid through a centrally located well has been investigated mathematically. The shape of the lines of equal pressure near the fracture is essentially independent of the size of the reservoir, provided that the field radius is of the order of the fracture length or larger. For reasonable values of production rates and of fluid, reservoir and fracture properties, the total pressure drop between the end of the fracture and the well is gelrerally negligible compared with the pressure drop in the reservoir. With regard to production response, the effect of vertical fractures can be represented by the production response of an equivalent or effective well radius. For a high-capacity fracture, the effective well radius is a quarter of the total fracture length, decreasing with the fracture capacity. When invasion effects are simulated by decreasing the width of the damaged zone with distance from the well, the effect of formation damage around a fracture on the production response is not so seriolls as indicated by the literature. This suggests that frac fluids with a conventional filter-loss response are better than high-spurt-loss frac fluids, provided the effective permeability of the damaged zone is the same. INTRODUCTION This paper considers the effect of the fracture capacity, as well as the formation damage which can result from fracture treatments, on the productive capacity of vertically fractured wells. Other publications, notably those of van Poollen,l,2 consider these same effects. In addition to providing more general results for vertical fractures than are available from the liter- ature, the present paper gives the equivalent well radius of fractures having different lengths and capacities and, also, includes pressure distributions in and around the fractures. The effect of a damaged zone around a fracture on the production response was not found to be so great as that reported by van Poollen. 2 This difference probably stems from the fact that we consider a damaged zone which is widest (but is still small) near the well and thins out toward the extremities of the fracture, whereas van Poollen considers a damaged zone having a uniform width for the entire fracture length. Simple, but adequate, equations which describe the effect of these variables on production response are presented (in Appendixes A and B). Thus, results can easily be extended to values of the variables not specifically considered here. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The liquid is incompressible. It is assumed that, after the well is fractured, the cylindrical outer "boundary" is at a uniform potential, provided that it is not too near the fracture. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below the producing layer (reservoir).3 Fig. 1 indicates the general three-dimensional geometry of the fractured reservoir described above. If gravity effects are neglected, the flaw behavior in the reservoir will be independent of the vertical position ill the oil sand. This means that the flow behavior in the fractured reservoir can be described by the two-dimensional flow behavior shown in the horizontal cross section of the reservoir, Fig. 2. PARAMETERS The effect of a fracture on the pressure distribution in the fracture itself and in an otherwise undamaged (no skin) reservoir indicates that the pressure distribution is a function of three parameters.

By Henry M. Howe

If a bar of wrought iron or steel is suddenly cooled from a bright red-heat, the contraction which then occurs is considerably greater than the expansion previously caused by heating the bar, so that its final length is considerably less than its initial length. Mr. Thomas Wrightson has investigated this phenomenon,* and has shown that not only does this permanent contraction occur on sudden cooling, but, if the heating and cooling be repeated, say fifty times, a further permanent contraction occurs at each successive cooling. In Mr. Wrightson's opinion, this contraction cannot be explained on mechanical grounds, and he ascribes it to some molecular change caused by sudden cooling; but I think that we need not invoke mysterious molecular changes to explain this phenomenon, which appears to me to be due to very simple and purely mechanical action. Let us consider the cooling as divided into two periods, in the first of which the exterior or skin of the bar becomes almost completely cooled, and hence tends to contract almost to its initial dimensions, while the still hot arid plastic interior contracts but little; in the second period the already cool skin remains nearly constant in temperature, and hence tends to contract but little, while the interior cools from a comparatively high temperature to the ordinary temperature, and hence undergoes great contraction. The excess of the contraction of the skin over that of the interior during the first period tends to make the bar bulge in the middle of its length. This, as well as the decrease of density which occurs, Mr. Wrightson correctly explains. The interior of the bar just sufficed to fill the skin when both were in their initial condition, before being heated at all. At the end of the first period, the volume of the still hot interior is more than the now contracted skin can hold while maintaining its initial shape; and the effect is much the same as when we powerfully blom into a soft India-rubher Lag of the shape of the skin of the bar—it bulges, and tends to approach the spherical shape, in which the ratio of surface to

Jan 1, 1886

By H. Shaw

It is now generally recognized that the gravitational method of geophysical surveying is a valuable aid in elucidating the geological structure of the subsoil and enables the practical geologist to deduce the presence of useful deposits. The particular suitability of the torsion balance to the location of large and important subterranean structures has given rise to a wide application of this method by many of the important oil companies in their search for future supplies, and so intense have been these efforts that little has been heard of the method in other connections, with the exception of occasional references to its application in the location of hidden deposits of metallic ores, salt and lignite, and a number of accounts of its use in the elucidation of geological structures. Its success in the location of salt domes is well known, and establishes it as one of the most important of the geophysical methods, especially for the indirect location of oil. In the British Isles, however, no large occurrence of oil is likely to be revealed, and here the practical application of the gravitational method has been restricted almost entirely to the search for ores. This type of survey possesses well marked characteristics, and presents a gravitational problem that has necessitated the development of a special and probably unusual plan of campaign, full details of which it is hoped will shortly be published. The problems and difficulties encountered in this work are of necessity fundamentally related to those met with in the search for oil by means of the torsion balance, but they also possess several special and distinctive features which undoubtedly differentiate them from the obstacles confronting the oil prospector, and which call for special consideration and treatment. In certain of the various oil-producing areas of the world, and particularly in the Gulf Coast region, the geology may be regarded as comparatively simple, while the oil is usually associated with a large and well defined structure such as a salt dome or anticline, which is capable of giving gravitational indications sufficiently extensive to be detected by the torsion balance over a fairly wide area of the ground surface. It is customary, therefore, during the preliminary survey of an area for oil, to

Jan 1, 1929

By C. Dean Starr

The physical basis of the equation correlating electrical conductizlity and temperature coefficient of resistance of solid solution alloys has been inzlestzgated and the nature of the constants evaluated. It is shaun that the same form of correlation is valid irrespective of the magnitude of the temperature coefficient. However, the effect of therma1 expansion and change in resistivity with temberature nust be considered when the magnitude of the temperature coefficient of resistance is near zero. In a comprehensive literature survey conducted at the Armour Research Foundation in 1950, it was found that a linear relationship exists between the electrical conductivity, K, and the temperature coefficient of resistance, a, for solid solution alloys. This was expressed by the equation where B, a constant, depends on the solvent metal but not the solute. A continuing investigation by Hansen, Johnston, and Parks showed this equation was generally appropriate for Cu, Ag, Au, Pd, Pt, and Co base binary solid solution alloys as well as for many ternary alloys of copper over a range of conductivities and temperature coefficient of resistances from 0.02 to 0.60 reciprocal pohm cm and-0.00015to0.004Q per 0 per "C respectively. Eq. [I] was not always valid for alloys that exhibited a change in state (magnetic to nonmagnetic) or for alloys that had a temperature coefficient of resistance near or below zero. For example, in nickel alloys, a simple linear curve as predicted by Eq. [l] was not obtained when the conductivity was plotted against the temperature coefficient of resistance. Hansen suggested that deviations observed were caused by transitions from a ferromagnetic to a nonmagnetic state. Subsequent and more extensive data by ronin on Ni-Mn alloys confirmed Hansen's postulate. While both states independently showed a linear relationship between K and a, the value of B was different for the two states. In some copper and gold base alloys, as well as the platinum and palladium alloys investigated by Hansen, the conductivity did not extrapolate to zero at zero a as demanded by Eq. [I]. This deviation can be readily corrected by simply adding a constant to Eq. [I.] whence Eq. [2] shows the K decreases as a decreases but that K has a positive value when a = 0 which is observed for commercial solid solution alloys as well as the experimental alloys tested by Hansen. While other methods of correlation of electrical properties have been made such as the one by Starr and Wang3 who found a linear relationship between resistivity and temperature coefficient of resistance for Ni-Cr-Al-Cu alloys, the data, in general, can also be correlated by a form of Eq. [2]. Thus, Hansen's empirical equation has considerable merit. The objective of this paper is to determine the physical basis for this equation and to examine the region near zero a in more detail. ELECTRICAL CONDUCTIVITY VS TEMPERATURE COEFFICIENT OF FU3SISTANCE The resistance of a metal or alloy is defined as

Jan 1, 1963

I-Mine Ventilation A. C. CALLEN, Chairman E. A. HGLBROOK, Vice-chairman H. I. SMITH, Secretary CLARENCE E. ABBOTT ALFRED W. HESSE FRANK H. PBOBERT GAM BRIGHT G. R. JACKSON GEORGE S. RICE THOMAS H. CLAGETT JAMES F. MCCARTHY E. J. RISTEDT . OSCAR A. GLASSER W. B. PLANK GERALD F. SHERMAN Coal Mining M. D. COOPER, Chairman HOWARD N. EAVENSON JOHN A. GARCIA THOMAS D. THOMAS CHARLES ENZIAN JAMES W. PAUL C. M. YOUNG JOHN J. RUTLEDGE

Jan 1, 1929

By J. J. Gilman

WHEN a zinc crystal is immersed in liquid N, and then plastically bent, it is observed that it bends a certain amount and then fractures. This fracture occurs at a fairly reproducible radius of curvature, and the radius varies systematically with the crystal orientation. Because this phenomenon provides an unusually well-defined set of fracture conditions, it is possible to examine various possible fracture criteria and select the best one. Zinc crystals of various orientations were bent about axes lying in their glide planes (0001) and perpendicular to their glide directions [1210]. The crystals were 0.150 in. diam and had chemically polished surfaces. They were immersed in liquid N, while they were being loaded on an Instron testing machine. A three-point bending jig was used to load them. Table I lists the beam radius at which fracture occurred for each orientation. Each value is the average of at least six measurements. The normal stresses on the cleavage planes, at the time of fracture, were calculated as follows: First, from the fracture radius, R,, the orientation angle, x,, of the (0001) plane at the outer fiber was calculated. If x, = angle between the (0001) and the rod axis prior to bending, d = rod diameter, and X, = angle between the (0001) plane and a tangent to the bent crystal, then1 sinX, _ 2R, sin xn 2R, + d Since the glide shear strain is given by' y ' cot X, — Xo the glide strain at fracture, x,, can be calculated. Next, it is necessary to know the tangential stress in the outer fiber at fracture. This depends on the plastic flow curve, so a series of tensile tests were performed at —196°C on another set of zinc crystals. A curve of the average resolved shear stress versus glide strain was prepared. From this curve, shear stresses, r,, corresponding to the glide strains in the bend tests were taken. These values are listed in Table I. Then the normal stresses in the outer fibres, u,, were calculated from the relation U, = r, tan X, This analysis assumes that the fiber stresses in a bent beam are the same as the stresses in simple

Jan 1, 1959

By Jean Howard, E. V. Walker

ManY papers have been published during the last few years on the formation of cube texture in 3 1/4 pct Si-Fe, e.g., those of Assmus, Detert, Dunn, and Walter.1,5 All are concerned with the formation of the texture by a process involving secondary re-crystallization and the relative surface energies of (100) and other planes under certain conditions of annealing. The authors have found that cube texture can also be produced by heavily cold rolling strip in which the cube-on-edge texture has first been formed. Cube texture develops on annealing, but in this case it is not accompanied by secondary recrystallization. The starting material is a sintered compact, the preparation of which has been described elsewhere. Using a conventional processing schedule employed for producing commercial silicon-iron with the cube-on-edge texture, the sintered compact is cold rolled to such a thickness that, after annealing in pure hydrogen for 24 hr at 1350'~ to produce the cube-on-edge texture, the final thickness required can be obtained by giving a further cold reduction of 85 pct. Cube texture is developed by annealing at -900~~ in pure hydrogen: it has been produced in strip up to 0.024 in. thick. The microstructure of the strip consists of a mass of small primary grains within the boundaries of the much larger secondary grains which ex- hibited cube-on-edge texture before the final cold roll. If the strip is annealed at -1000 after the 85pct reduction, secondary recrystallization begins to take place, provided a strong cube texture is present. On raising the temperature more secondaries grow, complete secondary recrystallization being obtained on annealing at -1150'. From this it is concluded that the secondary crystals are produced by the texture inhibition mechanism: they show (110) and (111) planes in the surface of the strip, the former being the more numerous. It has been observed that the more perfect is the (110) [001] texture prior to cold rolling by 85 pct, the further from the ideal is the (100) [oo].] texture. A similar treatment applied to high quality commercial grain-oriented silicon-iron strip failed to produce cube texture. The tilt and rotation of the [loo] from the rolling direction was less than 10deg, but (to within a few degrees) the planes (130), (450) and (210) lay in the surface of the strip. Acknowledgment is made to the Engineer-in-Chief of the British Post Office for permission to use the information contained in this note.

Jan 1, 1962

By J. Serra, M. Turpin, R. Alberny

A new method is proposed to obtain automatically an unbiased estimate of the interdendritic spacing A. It is shown that the structure can be built by a random distribution of a rectangular basic unit. The usual definition of the interdendritic spacing yields a value 1 which, coupled with A, gives the shape of the basic rectangular unit. A study of the relationship between dendrite arm spacing and solidification rate was conducted as a part of a research program on dendritic solidification. For any comparison between theory and experiments to be meaningful, it is necessary to measure reproducible and clearly defined spacings. Fig. 1 shows clearly that even in a fairly simple structure such measurements are difficult. if we compare photos (a) and (b), we can hardly assess which one shows the larger spacing. Moreover, the quantity to be measured can be defined in several ways. On a small scale, dendrites seem to be distributed in a very regular and ordered way. However, on a larger scale, this regularity no longer exists. The usual way to measure spacings is to pick up some rows of dendrite stalks; such measurements are obviously biased since less orderly regions are not taken into account. A more objective method is to section the picture with randomly oriented straight lines and average the measured interdendritic spacings. The purpose of this paper is to show that such measurements, which can be conducted automatically, yield objective and reproducible results which are directly related to characteristic features of the dendritic structure. I) THEORETICAL ANALYSIS 1-1) Mathematical Morphology and Function of Co-variance. Matheronl and Serra2 enlarced- the definition and geometrical meaning of the notion of covariance to encompass problems in mathematical morphology. A brief review of part of their theory can be stated as follows. Let A' be an isolated particle, and let us consider a group of two scanning points h is the vector joinlng these points, h its modulus, and a its direction. Let us also define a boolean function, k(x), whose value is 1 if x lies inside the particle and 0 if it 1s outside. The total number of events in which both points are inside A' IS the covariance Ii',(h): is the measure of the intersection of A' with the particle ALh obtained by translating A' of -h Obviously K,(O) is the volume of the particle A', or its area if A' is a two-dimensional body. The average value of K,(h), when the direction of a vector h of constant modulus is rotated, is: An important property of the covariance is illustrated on Fig. 2: if the shape of a particle shows some regularity along a direction a, K,(h) generally presents a maximum. We now consider a group A, made of many particles A' distributed in space; then k(x) is taken as the realization of a stationary random f~nction.~ The volume fraction (or fraction of area if the problem is two-dimensional) occupied by A equals the average value q of k(x): meaning "average value of"). We also replace

Jan 1, 1970

By P. Bolsaitis, L. Skolnick

A model of metallic solutions is postulated which explains the energy of formation of alloys on the basis of changes in electron density around solute and solvent atoms and changes in pairwise interactions between atoms. The model is based on the Wigner-Seitz cell model of metals, with the electronegativities of the metals as the parameter for the electron-attracting power of the constituent atoms. In contrast to models based on the description of cohesive energy in terms of pairwise interactions only, which require an empirical, adjustable correction parameter to the Berthe-lot relations in order to predict negative energies of formation, the electron cell model accounts for both negative and positive energies on the basis of parameters of the pure components. Calculated heats of mixing based on published parameters of the pure metals are found to compare favorably to experimental values, despite the absence of any adjustable parameters in the model. THE formulation of an "exact" quantum mechanical model for the energy of cohesion of metals reduces ultimately to the many-body problem and for a complete solution would require the knowledge of wave functions and energy levels of all the particles in the system. The case of alloys is further complicated by the fact that the potential fields around the constituent atoms are different and the matching of boundary conditions in terms of the resulting wave functions is still an unresolved mathematical problem. It appears that in spite of the numerous investigations that have led to an elucidation of many properties of the metallic state, the formulation of the Hume-Rothery rules in 1931 marks the lone accomplishment toward simple, qualitative prediction of the energy of formation of alloys. More recently Friedel'' and Arafa have utilized the modern concepts of the electronic structure in metals in combination with the Hume-Rothery rules to arrive at quantitative estimates of the energy of formation of alloys with low solute contents. The relative success of the Hume-Rothery rules and Friedel's and Arafa's studies encouraged broaching the subject of formulation of a model based on realistic physical parameters and of conceptual and mathematical simplicity that could be applied to alloys regardless of concentration. On the basis of the free-electron theory, the cohesive energy of pure metals can be expressed as a function of "valence" (number of free electrons per atom), lattice parameter (or density), and the ioniza-tion energy. In addition to the electronic contributions to cohesion, one has to consider the pairwise exchange interaction energy between neighboring ion cores and the contribution from van der Waals forces. The energy changes upon alloying are viewed as resulting primarily from a redistribution of electron densities around the ions (a simplification of the more basic concept of a change of the wave functions of the electrons in the presence of a perturbing potential field). In addition there is a change in the pairwise interactions between neighboring ions which is usually the only factor taken into account in commonly used adaptations of the quasichemical theory. The electronegativities of the metals forming the alloy are used as the measure of the electron-attracting power of the ions in the alloy and the difference in electronegativities as the driving force for changes in electron energy on alloying. The interplay of pairwise interactions and electronic forces results in positive or negative energies of formation of alloys. Results obtained for systems to which the free electron model is best applicable are in good agreement with experimental values. 1) COHESIVE ENERGY OF METALS IN TERMS OF THE ELECTRON CELL MODEL The simplest physical model of a metal is that of a lattice of positive charges immersed in a sea of uniform, negative charge distribution represented by the

Jan 1, 1969

By N. A. Gokcen, John Chipman

SILICON is the most commonly used deoxidizer and an important alloying element in steelmak-ing; hence a detailed study of this element in liquid iron containing oxygen is of considerable interest. The equilibrium between silicon and oxygen in liquid iron has been studied by a number of investigators but generally with inconclusive or incomplete results. The variation of the activity coefficients of silicon and oxygen with composition is entirely unknown. Published investigations deal with the reaction of dissolved oxygen with silicon in liquid iron and the results are expressed in terms of a deoxidation product. For consistency and convenience in comparison of the published information, the deoxidation product as referred to the following reaction is expressed in terms of the percentage by weight of silicon and oxygen in the melt in equilibrium with solid silica: SiO (s) = Si + 2 O; K'l = [% Si] [% 012 [I] Theoretical attempts to calculate the deoxidation constant for silicon in liquid iron from the free energies of various reactions yielded results which were invariably lower than the experimental values. Thus, the deoxidation "constants" calculated by McCance,1,2 Feild,3 Schenck, and Chipman were of the order of 10, which is below the experimental values by a factor of more than 10. Experiments of Herty and coworkers" in the laboratory and steel plant resulted in an average deoxidation constant of 0.82x10 ' at about 1600°C. The technique employed in their investigation was crude and the reported temperature was quite uncertain. The concentration of silicon was obtained by subtracting silicon in the inclusions from the total. Since at least some of the inclusions resulting from chilling must represent a fraction of the silicon in solution at high temperatures, such a subtraction is not justifiable. Results of Schenck4 for K'1 from acid open-hearth plant data yielded a value of 2.8x10-5, which was later revised as 1.24x10 at 1600°C. Similarly Schenck and Bruggemann7 obtained 1.76x10-5 at 1600OC. The discrepancies and errors involved in the acid open-hearth plant data as compared with the results of more reliable laboratory techniques were attributed by these authors to the lack of equilibrium and the impurities in liquid metal and slag, and are sufficiently discussed elsewhere." Korber and Oelsen" investigated the relation between dissolved oxygen and silicon in liquid iron covered with silica-saturated slags containing varying concentrations of MnO and FeO. The deoxidation products obtained by their method scatter considerably, and their chosen average values of 1.34x10, 3.6x10-5, and 10.6x10-5 1550°, 1600°, and 1650°C, respectively, represent the best experimental results which were available until quite recently. Darken's10 plant data from a steel bath agree approximately with their data at 1575° to 1625°C. Zapffe and Sims" investigated the reaction of H2O and H2 with liquid iron containing less than 1 pct Si and obtained deoxidation products varying by a factor of more than 20. Inadequate gas-metal contact and lack of stirring in the metal bath should require a longer period of time than the 1 to 5.5 hr which they allowed for the attainment of equilibrium. Furthermore, their oxygen analyses were incomplete and irregular and confined to a few unsatisfactory preliminary samples. Their results did indeed indicate that the activity coefficient of oxygen is decreased by the presence of silicon, although they made no such simple statement. They chose to attempt to account for their anomalous data by the unlikely hypothesis that SiO is dissolved in the melt. Hilty and Crafts" investigated the reaction of liquid iron with acid slags under an atmosphere of argon, making careful determinations of silicon and oxygen contents at several temperatures. Despite erroneous interpretation of the data at very low silicon concentrations, their data represent the most dependable information on this equilibrium that has been published. In the range 0.1 to 1.0 pct Si, their data yield the following values for the deoxidation product: 1.6x10-5, 3.0x10- ', and 5.3x10 at 1550°, 1600°, and 1650°C, respectively. The purpose of the work described herein was to study the equilibrium represented by eq 1 as well as the following reactions, all in the presence of solid silica: SiO2 (s) + 2H2 (g) = Si + 2H2O (g);

Jan 1, 1953

By E. O. Kirkendall

THE investigation reported herein was undertaken to obtain data to supplement those presented in an earlier paper,' particularly for longer times at a higher temperature. This information may further develop our knowledge of the mechanism of diffusion in substitutional solid solutions. Diffusion of solute atoms interstitially dissolved is not difficult to visualize; it can take place without much disturbance of the solvent atoms. The mechanism of diffusion of solute atoms substitutionally dissolved is not so easily understood. Many theories have been advanced. Rosenhain,2.3 Frenkel,4 Smekal,5 Langmuir,3,6 Steigman, Shockley and Nix,7 and Johnson8 all proposed theories to explain diffusion. Langmuir's proposed mechanism consisted of a simple cyclic interchange of atoms involving an interchange in lattice positions of four or more atoms. This mechanism, accepted by the author at the time of his previous publication,' and apparently quite generally accepted, required a net exchange in positions of both solute and solvent atoms in equal numbers. In other words, the rate of diffusion of the solvent and solute atoms would be the same. That this cyclic interchange in lattice positions of solute and solvent atoms in equal numbers is the only true mechanism of diffusion is denied by the evidence presented in this paper. EXPERIMENTAL PROCEDURE This modification of similar procedure1 will be outlined briefly. The method consisted of the following steps: I. A pure Muntz metal (60.6 per cent Cu, 0.1 per cent total impurities) that was almost all beta brass at 780°C. was chosen. 2. The brass was turned between centers in a lathe to a diameter of 0.5965 inches. 3. After the brass was cleaned, it was electroplated with pure copper to a diameter of about one inch. 4. The sample was heated slowly to 400°C. under reduced pressure and held at that temperature for 4 hr. to remove occluded gases. It was then cooled to room temperature. 5. The sample was heated to 780°C. ±3°C. and held at that temperature for different lengths of time; then quenched. Samples were cut off each time after quenching. Metallographic samples 34 in. thick were removed at I hr., 6 hr., 24 hr., 96 hr. and 701 hr. After 96 and 701 hr. samples of about I in. were removed for X-ray analysis. 6. The metallographic samples were polished on the cross section and photo-micrographs were taken. 7. The I-in. samples were polished and etched on both ends. With the aid of a microscope each was carefully mounted in a four-jaw chuck in a lathe, so that they could be accurately bored for an arbor. 8. Turnings were taken for each X-ray sample. After each set of turnings, the sample was heated in an oil bath for about 3a hr. at 300°C. It was then etched in 25

Jan 1, 1942

By W. J. Leas, L. A. Rapoport

As a preliminary, consicleration is given to the conventional definition of relative permeability and to the conditions governing the simultaneous flow of oil and gas through porous media. For the conditions of flow prevailing throughout most of a gas drive reservoir, the oil and gas can reasonably be supposed to be in capillary equilibrium with each other. Under these conditions, and these conditions only, the relative permeability to liquid can be expressed as a function of saturation. The relative permeability to liquid in that case is dependent upon the distribution of fluids which itself is shown to be related to the capillary pressure, and, in turn, to the saturation. As a consequence, relative permeability to liquid can be expressed in terms of the volume and surface area of a network of liquid channels bounded by the rock and the gas phase. While the volume of this network can be evaluated accurately, the surface area cannot. However, for any such volume, maximum and minimum values of the corresponding surface area can be calculated from capillary pressure data. It is then possible to establish for any saturation the limits within which the value of the relative permeability to liquid must lie. As a consequence of the theoretical development, the validity of an experimental method for measuring relative permeability to liquid which utilizes a stationary gas phase is demonstrated. In this method capillary barriers are cemented to the ends of the core sample to permit the maintenance of capillary equilibrium between the two phases. At the same time, this procedure eliminates undesirable secondary phenomena such as end effects, fissure effects, etc., the presence of which adversely affect the results of other laboratory methods. The results obtained by theoretical calculations, and experimentally, are discussed. In view of the overall precision that can presently be obtained in reservoir calculations, the agreement between the calculated and measured relative permeability to liquid data can be considered satisfactory. In conclusion, for reasons of economy and simplicity, the procedure of calculating limiting relative permeability to liquid curves from capillary pressure data is indicated for general engineering purposes. It is shown that the above procedure can easily be extended to the cases where connate water is present. Its use for reservoir studies is particularly recommended in conjunction with the method for measuring relative permeability to gas' which simultaneously yields the capillary pressure data necessary for the calculations. THEORETICAL Definition of Relative Permeabilities — Basic Equations for Heterogeneous Flow The equations by which the relative permeability concept is defined and upon which the formulation of all of the gas-oil flow problems rests at the present time are expressed as: V, = — Grad PL = — Grad PL ....(la) PL µL, k k KG VG = —K - Grad PG k- Grad Pc .... (lb) Mo where ,. and G refer to liquid and gas; V is the volumetric rate of flow per unit gross area. µ the viscosity, Grad P the potential gradient. and k the specific permeability of the porous medium.* (For horizontal flow, Grad P becomes the pressure gradient; otherwise, gravity must be included.) According to these expressions, each of the constituent phases is considered similar to a homogeneous system where the volumetric rate of flow is proportional to the pressure gradient, and for each of which the constants of proportionality, k, and kG, are termed effective permeabilities. by analogy to the specific permeability as defined by Darcy's law in its original form. In order to obtain a convenient basis of comparison, the effective permeabilities are referred to the specific permeability, k, of the considered porous medium, with the help of the relations: k1. = KI. k kQ = Kn k..........(2) K,. and Kr. are defined as the relative permeabilities to the liquid and to the gas phase respectively, and frequently expressed in per cent of specific permeability. It may be seen that Equations (1); which appear to be a direct generalization of Darcy's law, correspond to the assignment at any given time of a set of "local" permeabilities to each point of the porous medium, and represent in a differential form the two fluid flow system as a simple superposition of the individual single flow systems. The above interpretation implies that the effective or relative permeabilities are independent of pressure or rate of flow,

Jan 1, 1951

By L. A. Rapoport, W. J. Leas

As a preliminary, consicleration is given to the conventional definition of relative permeability and to the conditions governing the simultaneous flow of oil and gas through porous media. For the conditions of flow prevailing throughout most of a gas drive reservoir, the oil and gas can reasonably be supposed to be in capillary equilibrium with each other. Under these conditions, and these conditions only, the relative permeability to liquid can be expressed as a function of saturation. The relative permeability to liquid in that case is dependent upon the distribution of fluids which itself is shown to be related to the capillary pressure, and, in turn, to the saturation. As a consequence, relative permeability to liquid can be expressed in terms of the volume and surface area of a network of liquid channels bounded by the rock and the gas phase. While the volume of this network can be evaluated accurately, the surface area cannot. However, for any such volume, maximum and minimum values of the corresponding surface area can be calculated from capillary pressure data. It is then possible to establish for any saturation the limits within which the value of the relative permeability to liquid must lie. As a consequence of the theoretical development, the validity of an experimental method for measuring relative permeability to liquid which utilizes a stationary gas phase is demonstrated. In this method capillary barriers are cemented to the ends of the core sample to permit the maintenance of capillary equilibrium between the two phases. At the same time, this procedure eliminates undesirable secondary phenomena such as end effects, fissure effects, etc., the presence of which adversely affect the results of other laboratory methods. The results obtained by theoretical calculations, and experimentally, are discussed. In view of the overall precision that can presently be obtained in reservoir calculations, the agreement between the calculated and measured relative permeability to liquid data can be considered satisfactory. In conclusion, for reasons of economy and simplicity, the procedure of calculating limiting relative permeability to liquid curves from capillary pressure data is indicated for general engineering purposes. It is shown that the above procedure can easily be extended to the cases where connate water is present. Its use for reservoir studies is particularly recommended in conjunction with the method for measuring relative permeability to gas' which simultaneously yields the capillary pressure data necessary for the calculations. THEORETICAL Definition of Relative Permeabilities — Basic Equations for Heterogeneous Flow The equations by which the relative permeability concept is defined and upon which the formulation of all of the gas-oil flow problems rests at the present time are expressed as: V, = — Grad PL = — Grad PL ....(la) PL µL, k k KG VG = —K - Grad PG k- Grad Pc .... (lb) Mo where ,. and G refer to liquid and gas; V is the volumetric rate of flow per unit gross area. µ the viscosity, Grad P the potential gradient. and k the specific permeability of the porous medium.* (For horizontal flow, Grad P becomes the pressure gradient; otherwise, gravity must be included.) According to these expressions, each of the constituent phases is considered similar to a homogeneous system where the volumetric rate of flow is proportional to the pressure gradient, and for each of which the constants of proportionality, k, and kG, are termed effective permeabilities. by analogy to the specific permeability as defined by Darcy's law in its original form. In order to obtain a convenient basis of comparison, the effective permeabilities are referred to the specific permeability, k, of the considered porous medium, with the help of the relations: k1. = KI. k kQ = Kn k..........(2) K,. and Kr. are defined as the relative permeabilities to the liquid and to the gas phase respectively, and frequently expressed in per cent of specific permeability. It may be seen that Equations (1); which appear to be a direct generalization of Darcy's law, correspond to the assignment at any given time of a set of "local" permeabilities to each point of the porous medium, and represent in a differential form the two fluid flow system as a simple superposition of the individual single flow systems. The above interpretation implies that the effective or relative permeabilities are independent of pressure or rate of flow,

Jan 1, 1951