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By Forrest F. Craig

Both early and more recent1 laboratory measurements of gas-oil relative permeabilities were made by subjecting oil-saturated cores to external gas drives. In these runs it was generally assumed that the flowing gas/oil ratio was constant at every point throughout the core. Using this assumption, the resulting equation for the gas-oil relative permeability ratio becomes A where Q is the flow rate at the producing end of the system. This relative permeability was then specified for the average saturation of the core. The assumption and resulting equation fail to recognize that in an external drive, the oil flow rate varies from zero at the injection end to a maximum at the producing end. The assumption also fails to consider the existence of a saturation gradient during an external drive. This saturation gradient is due solely to non-steady-state flow. Some authors2,3,4 have used this erroneous method of calculating relative permeability ratios from external drives. Other authors5,6,7 have used relative permeability ratios in calculating expected field performance by external drive, using Equation (1) and making the incorrect assumption that the saturation is uniform within the driven portion of the reservoir. To find the correct relation between the relative permeability. ratio at the average saturation and the produced gas/oil ratio, the insertion in Equation (1) of a factor, F, to adjust it for oil velocity gradient has been considered. The equation then takes the form Assuming that the oil velocity is directly proportional to the distance from the injection point leads to the conclusion that, for a linear system, the average oil velocity is one-half that at the producing end. Thus, the average gas/oil ratio is twice the produced gas/oil ratio, and Equation (2) is transformed into However, here again the existence of a saturation gradient is not recognized. The produced gas/oil ratio is related to the relative permeability ratio at the saturation of the outlet end of the system. It can be shown quite readily that, if the true log k,/k, saturation relationship is linear over small ranges of saturation, then d log kjko Where aaxv and u2 are the average and outlet saturations, respectively. Examination of Equation (3) shows that the factor F cannot have a single value over the whole saturation range. Another method of dealing with external drives is that of Buckley and Leverett.' Their method, which assumes no gas expansion and outlet end effects, considers fully the saturation gradient present due solely to the displacement mechanism during an external drive, and applies Equation (1) to each point along this gradient. The calculation of production history using relative permeability ratio is straightforward. However, the calculation of relative permeability ratios from production history by this method is impractical. In a recent paper, Welge,1 using an approach similar to that of Buckley and Leverett, presents a simplified method for determining the saturation at the producing end of the system. At thi; saturation, the relative permeability ratio calculated by Equation (1) applies exactly, after breakthrough of the stabilizej front. By this method, relative permeability ratios or production history may be calculated with equal ease. Use of this method as a laboratory tool in determination of relative permeability ratios by external drive requires the knowledge of the total injected fluid volume; in addition to the average saturation and producing ratio. As a preliminary test of this method of calculation, laborntory external gas drives were run on a 3% in. diameter, 11 in. long sample of Nellie Bly consolidated sandstone. This core sample had a permeability of 824 md and a porosity of 28.1 per cent. The driving fluid was air. Three different oils were used, having viscosities of 1.4, 9.8, and 125 cp, respectively. The runs were made under identical pressure differentials,

Jan 1, 1952

By Robert G. Nisle

In conducting a pressure build-up test on an oil well, it is often necessary to shut-in the well for a short time prior to initiation of the test. The effect of such a short tern shut-in on the resulting pressure build-up curve is derived. The derivation is based on the Kelvin point-source solution and is a straight forward extension of the fundarnental theory. The magnitude of the effect is evaluatecl and illustrated by an example. It is shown that if the production subsequent to the short term shut-in is at least 10 times the duration of the shrct-in, the effect will he less than 10 per cent. INTRODUCTION It often happens that before a pressure build-up test can be made an additional valve must be mounted on the wellhead, or on arrival at the well, it might be learned that the well had been shut-in several hours the previous day because of a shortage of tankage. In such cases the question is How will such an interruption of production affect the results of a pressure buildup test This work explains how to take account of such a short term shut-in in order that the test can proceed without further delay. The solution presented here makes use of the Kelvin point-source solution used by Horne1 in his treatment of the pressure build-up in wells. Use is also made of the super-position theorem. MATHEMATICAL DERIVATION The well is assumed to have been in production at a constant rate for a relatively long time (several days or more). It is then shut-in for a short time, after which it is again placed in production at the previous constant rate. It continues to produce at this rate until shut-in for a pressure build-up test. The Kelvin point-source solution is expressed by Eq. 1. The sequence of production rates is listed in Table I. TABLE 1 — SEQUENCE OF PRODUCTION RATES Time interval Production Rate At time AT the pressure will be the resultant of the sequence of production rates shown in Table 1. An expression for the super-posed effects may be derived from this table. (Effect of + q for T + ? T) + (Effect of - q for T + ?T-t,) + (Effect of + q for T + ? T - t,) + (Effect of - q for ? T) = Resultant........(2) The desired result is obtained by substituting the effect defined by Eq. 1 for the terms in Eq. 2. The result is Eq. 3. When the logarithmic approximation is made for the exponential integrals and the terms collected the result is Eq. 4. Let T — t2 = T = flowing time following short term shut-in t2 — t1 = 6 = duration of short term shut-in Then T - t1 = r + 6 Substitution of these quantities in Eq. 4 gives p(rw ?T) = P,, -When d<<t - AT then d—&apos;-, T— approaches 1. r + AT In this case the effect of the short term shut-in may be ignored. In order to get some idea of the numerical magnitudes involved take the worst possible case, when AT = 0, that is at shut-in. If the error is made less than some maximum acceptable value at this time, it will be less than the maximum for all subsequent times. d + r Let the difference between the value of y-— and 1 be less than 0.1. Then —6+r — 1 < 0.1 S < O.l T Therefore, if the flowing time following a short term shut-in is at least 10 times the duration of the short term shut-in, the error introduced by ignoring the short term shut-in will not exceed 10 per cent of T+AT A T

Jan 1, 1957

By M. N. Pathasarathi, P. A. Beck, T. J. Koppenaal

EARLIER work1 indicated that in rolled and annealed copper the volume fraction of the cube-texture component may increase on continued isothermal annealing. Merlini found2 that in rolled copper the well-known increase in the amount of cube-oriented material with increasing annealing temperature3 is concurrent with grain growth, which takes place after recrystallization is complete. It was found that both in copper2 and in aluminum4 the texture resulting from primary recrystallization, even if this is followed by some grain growth, may in general comprise, in addition to the cube-texture component, four symmetrical texture components of the general type (123)[412], i.e., components not very far from the ideal orientations of the rolling texture. The volume fraction of the cube-texture component increases on further annealing at the expense of these four components.&apos; The present work was undertaken to study quantitatively the increase in the volume fraction of the cube-texture component in the course of normal grain growth, under varying conditions. Since the grain-gr3wth behavior of A1-Mn alloys has been already investigated in some detaiL5 an A1 + 0.8 pct Mn alloy was chosen for this work, in which the extent of normal grain growth can be effectively controlled by the inhibiting effect of a dispersed second phase. EXPERIMENTAL PROCEDURE High-purity aluminum of the following composition was used: A1 99.997 pct, Cu 0.0004 pct, Fe 0.0005 pct, and Si 0.001 pct. A high-purity A1-Mn master alloy was prepared by dehydrating manganese chloride at 500°C and reacting it with molten high-purity aluminum at 750°C in a graphite crucible.5 The Mn metal reduced from the salt by the molten aluminum goes into solution in the latter. The master alloy? containing 13 pct Mn, was solidified under conditions suitable for minimizing segregation. An ingot of the final alloy, to be used in the experiments, was per pared by melting high-purity aluminum and the necessary amount of master alloy in a graphite crucible, and by pouring the molten alloy into a graphite mold preheated to 700°C through a box containing graphite baffles designed to intercept most of the aluminum oxide films floating in the melt. The melt was then solidified by placing the mold on a water-cooled copper plate and by keeping the top of the mold hot, in order to minimize piping. The 6 1/2 in. long, 11/4-in.diam ingot consisted largely of a single crystal. Chemical analysis showed that there was little segregation from the bottom (0.83 pct Mn) to the top (0.81 pct Mn). Spectrographic analysis of the ingot showed that the impurity content remained essentially the same as that of the aluminum used. The ingot was processed so as to obtain a uniformly small grain size. without banding. The ini-tial steps of rolling in the form of a flat bar and of annealing, down through a thickness of 0.4 in., are shown in Table I. At this point the bar was cut into four sections, each approximately 3 in. long, which were processed separately as shown in Table If. The table also gives the penultimate grain sizes obtained in each section by annealing prior to the final rolling of 90 pct RA. During the final rolling the strips were reversed end to end between passes. Annealing treatments were given in a salt pot furnace, with the temperature controlled to 1°C. All annealing of section "6s" was carried out above the solvus point to avoid the formation of second-phase precipitate. The penultimate annealing was sufficiently short to give a relatively fine grain size in spite of the high temperature. Section "6L" was processed in a similar manner, except that the penultimate annealing was much longer, so as to allow considerable grain growth. Section "4" was given the two final

Jan 1, 1961

By M. R. Geer, H. F. Yancey

John Grifien (Pittsburgh)—I wish to congratulate the authors on this paper, which, I am sure, will promote a clearer conception of the various criteria which have been advanced as measures of coal-cleaning performance. Their clear descriptions of the more important criteria that have been advanced and explanations of their meaning and application have been sorely needed and, if widely promulgated and followed, should remove much of the confusion which, heretofore, has been evident. I suggest that much of this confusion has also arisen from lack of common agreement as to the exact meaning of the words and terms used in our discussions of this subject. I am sure I have been equally guilty in this respect and suggest it is time that we select a committee to study vocabulary as well as criteria, so that common understanding and agreement can be had. Thus, I suggest, that the title of this paper should have used the words "coal-cleaning performance" and not "coal-washery performance." My understanding is that "washery" refers to a complete plant and it has additional functions to that of coal-cleaning, such as control of size, moisture, etc. Also, I do not agree with the authors that the formulas given in their paper, the Fraser and Yancey formulas, are necessarily the only ones that can be called efficiency formulas. When a dictionary is consulted one realizes that the words "efficiency" and "efficient" have not precise meanings but only very general ones and must be further defined for each particular use, if they are to have any precise meaning. To my mind, the first of the Fraser and Yancey formulas given should never be used as expressing "efficiency" alone but only by the fuller phrase which truly expresses the kind of efficiency it states, which is "coal recovery efficiency." For complete clarity we then need only define "coal" for the particular circumstances under which the formula is used. This would recognize the fact, fully explained by the authors, that the limits in coal qualities imposed by the market have imposed a factor—and a variable factor—which has influenced any answer obtained by this formula. Such a procedure makes it entirely practical and logical to use the values of any of the qualities of coal—ash, sulphur, phosphorus, specific gravity—or a combination of them to define what is coal for that particular use of the formula. Further, it is clear to me that the criteria to be used must be dictated by the objective in view and the use for which the data are obtained. The authors clearly point out that the Fraser and Yancey coal-recovery efficiency formula is influenced by three factors, "one is determined by the coal, one is dictated by market or use conditions, and one is an inherent characteristic of the cleaning unit itself." The formula which uses the inverse of misplaced material as a measure of efficiency is also a coal recovery efficiency formula and is influenced in like manner by these three factors. As the authors point out, both are useless in obtaining data which enables a direct and valid comparison of the performance of a given cleaning unit on two coals where the other two factors vary or of the performance of two or more cleaning units of different types unless the values of the other two factors are identical. Much of the confusion in our discussions of and literature on coal cleaning has come from failure to understand clearly this fact. Many such comparisons have been made from which erroneous conclusions have been drawn. I am, therefore, particularly glad to see the authors&apos; emphasis on 4?-1e criteria which indicate the "sharpness of separation. I feel that these are the essential criteria

Jan 1, 1952

By Jay A. Carpenter

This paper deals with only one step in the treatment of ore at the West End mill; not because the other steps are repetitions of practice in other mills, but because in this particular step there is in use the Trent agitator: This device, although it has apparently failed in many mills, is here giving excellent service and has proved to be well adapted for making a thorough replacement of pregnant solution with a minimum amount of barren solution. The strong and weak points of this agitator, its present simple construction, and its use as a thickener for agitator or battery pulp are features of interest; but perhaps the point of greatest interest to the metallurgist in this day of continuous decantation is its use in a series of tanks for slime treatment by the replacement method. The agitating department of the West End mill consists of six redwood tanks, 24 ft. in diameter, 18 ft. high, each equipped with a Trent agitator, a centrifugal pump, and a motor. The pulp is transferred by a pump from the top of a flat-bottomed tank to a set of arms and nozzles in the bottom of the tank, at just sufficient pressure to cause the arms to revolve. The streams from the many nozzles keep the bottom of the tank clean, and these streams, coupled with the effect of the revolving mechanism and the ascending current, keep the pulp in constant motion and of constant gravity. These agitating tanks hold 90 tons of dry slime and 202 tons of solution with a 1.24 pulp, our ore having a specific gravity of 2.7. This capacity is about 10 per cent. in excess of the standard 15 by 45 ft. Pachuca tank. The tanks are connected in one series for continuous agitation, the pulp for each agitator being drawn by its pump through a branch suction from near the top of the preceding tank and delivered to the bottom of the tank by the agitator arms along with the regular pulp of that agitator. This method works automatically, since the flow through the branch suction varies directly with the difference in head of the two tanks. The chance for new pulp to pass out quickly is, therefore, much less than in many other systems of continuous agitation, since the new

Jan 1, 1916

By I. Javandel, P. A. Witherspoon

The finite element method was originally developed in the aircraft industry to handle problems of stress distribution in complex airframe configurations. This paper describes how the method can be extended to problems of transient flow in porous media. In this approach, the continuum is replaced by a system of finite elements. By employing the variational principle, one can obtain time dependent solutions for the potential at every point in the system by minimizing a potential energy functional. The theory of the method is reviewed. To demonstrate its validity, nonsteady-state results obtained by the finite clement method are compared with those of typical boundary value problems for which rigorous analytical solutions are available. To demonstrate the power of this approach, solutions for the more complex problem of transient flow in layered systems with crossflow are also presented. The generality of this approach with respect to arbitrary boundary conditions and changes in rock properties provides a new method of handling problems of fluid flow in complex systems. INTRODUCTION Problems of transient flow in porous media often can be handled by the methods of analytical mathematics as long as the geometry or properties of the flow system do not become too complex. When the analytical approach becomes intractable, it is customary to resort to numerical methods, and a great variety of problems have been handled in this manner. One such method relies on the finite difference approach wherein the system is divided into a network of elements, and a finite difference equation for the flow into and out of each element is developed. The solution of the resulting set of equations usually requires a high speed computer. When heterogeneous systems of arbitrary geometry must be considered, however, this approach is sometimes difficult to apply and may require large amounts of computer time. The finite element method is a new approach that avoids these difficulties. It was developed originally in the aircraft industry to provide a refined solution for stress distributions in extremely complex airframe configurations. 27 Clough has recently reviewed the application of the finite element method in the field of structural mechanics.6 The technique has been applied successfully in the stress analysis of many complex structures.l, 27, 28 Recognition that this procedure can be interpreted in terms of variational procedures involving minimizing a potential energy functional7 leads naturally to its extension to other boundary value problems. In the field of heat flow, there recently have been introduced several approximate methods of solution that are based on variational principles.2-4, 17 By employing the variational principle in conjunction with the finite element idealization, a powerful solution technique is now available for determining the potential distribution within complex bodies of arbitrary geometry. In the finite element approximation of solids, the continuum is replaced by a system of elements. An approximate solution for the potential field within each element is assumed, and flux equilibrium equations are developed at a discrete number of points within the network of finite elements. For the case of steady-state heat flow, the technique is completely described by Zienkiewicz and Cheung.33 Since the flow of fluids in porous media is analogous to the flow of heat, Zienkiewicz et al. have employed the finite element method in obtaining steady-state solutions to heterogeneous and anisotropic seepage problems. 34 Taylor and Brown have used this method to investigate steady-state flow problems involving a free surface.25 The work of Gurtin has been instrumental in laying the groundwork for the application of finite element methods to linear initial-value problems.12 As a result, Wilson and Nickell have recently

Jan 1, 1969

By M. King Hubbert, David G. Willis

A theoretical examination of the fracturing of rocks by means of pressure applied in boreholes leads to the conclusion that, regardless of whether the fracturing fluid be of the penetrating or non-penetrating type, the fractures produced should be approximately perpendicular to the axis of least stress. The general state of stress underground is that in which the three principal stresses are unequal. For tectonically relaxed areas characterized by normal faulting, the least stress should be horizontal; the fractures produced should be vertical with the injection pressure less than that of the overburden. In areas of active tectonic compression, the least stress should be vertical and equal to the pressure of the overburden; the fractures should be horizontal with injection pressures equal to or greater than the pressure of the overburden. Horizontal fractures cannot be produced by hydraulic pressures less than the total pressure of the overburden. These conclusions are compatible with field experience in fracturing and with the results of laboratory experimentation. INTRODUCTION The hydraulic-fracturing technique of well stimulation is one of the major developments in petroleum engineering of the last decade. The technique was introduced to the petroleum industry in a paper by J. B. Clark,&apos; of the Stanolind Oil and Gas Co. in 1948, and since then its use has progressively expanded so that by the end of 1955 more than 100,000 individual treatments had been performed. The technique itself is mechanically related to three other phenomena concerning which ah extensive literature had previously developed. These are: (1) pressure parting in water injection wells in secondary-recovery operations, (2) lost circulation during drilling, and (3) the breakdown of formations during squeeze-cementing operations, all of which appear to involve the formation of open fractures by pressure applied in a wellbore. The most popular interpretation of this mechanism has been that the pressure had parted the formation along a bedding plane and lifted the overburden, notwithstanding the fact that in the great majority of cases where pressures were known they were significantly less than those due to the total weight of the overburden as determined from its density. Prior to 1948, this prevalent opinion had already been queried by Dickey and Andresen,2 in a study of pressure parting; and by Walker,"&apos; who, in studies of squeeze cementing, pointed out that the pressures required were mostly less than those of the overburden, and inferred that the fractures should be vertical. J. B. Clark,&apos; in his paper introducing hydraulic fracturing, and later Howard and Fast," and Scott, Bearden, and Howard," also of Stanolind, postulated that the entire weight of the overburden need not be lifted in producing horizontal fractures, but that it was only necessary to- lift an "effective overburden," requiring a correspondingly lower pressure. Hubbert,7 in discussing the paper by Scott and associates, pointed out that the normal state of stress underground is one of unequal principal stresses; and in tectonically relaxed areas, characterized by normal faults, the least stress should be horizontal. Therefore, in most cases, fracturing should be possible with pressure less than that of the overburden and, moreover, such fractures should be vertical. Harrison, Kieschnick, and McGuire,8 also on the expectation that the least principal stress should be horizontal, argued strongly in favor of vertical fracturing. Scott, Bearden, and Howard6 observed that, when

Jan 1, 1958

By Charles C. Fatino, Joel S. Hirschhorn

Loose compacts of strained and annealed stainless-steel powders were sintered at various temperatures and times to determine the effect of residual strains on the sintering behavior. The atomized powders were prepared by annealing, ball milling, and rean-nealing. The residual strains and domain size were determined by a Fourier analysis of X-ray diffvaction dala. The specimens were prepared by siwzultaneously sintering cylinders of both strained and unstrained powder. The evaluation of the sintered compacts included density measurements, metallographic examinations, and analyses of X- ray diffraction data. The results of these studies reveal a definite increase in sintered density and sintering rate for the strained powders. THE pressure compacting of metal powders causes deformation of the particles, mechanical interlocking, and increased contact area. After the deformation, severe residual strains remain in these particles. The subsequent contributions of these strains to the sintering process have been considered to be negligible, since the recovery and recrystallization temperatures usually occur prior to the temperature required for the sintering process. Sintering is con-sidered to be the process by which the powders are transformed into solid compacts at temperatures below their melting points. The sintering operation for stainless steel usually takes place in the range of 2050" to 2400°F (0.84 to 0.95 T,, where T, is the absolute melting point or liquidus temperature) while the temperature required for recovery is approximately 1800°F (0.75 T,). However, it has been shown for several metal powders that densification becomes significant in the range of 0.35 to 0.45 T,.&apos; It is the purpose of this study to determine what effect residual strains have upon the sintering behavior of stainless-steel compacts. Loose compacts were selected for study in order to isolate the strain contributions and eliminate the variables inherent in sintering of pressure-formed compacts. Additional variables were minimized by using only one mesh size, one composition, one particle shape, and also by simultaneously sintering both types of powder. Very little work has been carried out previous to the present study to delineate the effect of strains on sintering behavior. Lenel et 01.2 observed that residual stresses in copper compacts caused shrinkage at comparatively low temperatures. Delisle3 loose-sintered ball-milled electrolytic iron and observed that bonding among particles began at a lower temperature as compared to the behavior of annealed powder. Increased densification of ball-milled oxide powders has also been noted.4&apos;5 EXPERIMENTAL PROCEDURES Powder Preparation. The powder used for this study was a commercial-type 316L stainless steel made by the atomization process. A typical chemical analysis is: C- 0.025 pct Mo- 2.2 pct Cr-18.6 pct Ni-12.5 pct Fe—Balance The atomizing process is essentially the rapid freezing of liquid droplets. This nonequilibrium cooling causes coring to occur and two distinct phases are present within the solid particle, the austenitic phase and the ferritic phase. Due to the presence of this ferritic phase, the particles tended to fracture as well as become cold-worked during the subsequent ball-milling operation. (It was also found that this type of cooling produced a broadened and shifted

Jan 1, 1968

By D. J. McAdam, G. W. Geil, D. H. Woodard, W. D. Jenkins

INTRODUCTION IN a series of papers, the authors and their associates have shown the influence of four important factors on the technical cohesion limit.3-16 By "technical cohesion limit" is meant the technically determinable resistance to fracture. The four important factors are the stress system, plastic deformation, temperature, and the strain rate. The influence of any one of these factors may be represented by a curve of cohesion limits, although only one point on the curve may represent actual fracture. As the term "fracture stress" has come into general use with the same significance as technical cohesion limit, the authors are here using the shorter term with the understanding that it does not always refer to actual fracture. The previously-mentioned series of papers contains ample evidence that plastic extension increases the fracture stress of any polycrystalline metal. In a recent paper of the series,15 moreover, it has been shown that the effect of plastic deformation on the fracture stress cannot be attributed to reorientation of internal flaws but is a work-strengthening effect, similar to the effect on the flow stress.[$] Plastic deformation of steels, however, may not only cause work hardening, but may result in additional hardness because of strain aging. Some evidence that strain aging affects not only the flow stress but also the fracture stress, has been presented in a recent paper.14 In that paper, a study is made of the influence of plastic deformation at room temperature on the fracture stress of ferritic steels at -188°C. The study is based on the results of two-stage tests. In the first stage each specimen was extended a predetermined amount at room temperature; in the second stage the specimen was tested to fracture in liquid air. In most of these tests the specimen remained in the testing machine during the interval of about one hour between the first and second stages, it was kept under a tensile load of about 500 lb and maintained at liquid air temperature. The effects of various amounts of plastic deformation on the fracture stress were thus determined and curves were obtained to represent the influence of plastic deformation at room temperature on the stress at brittle fracture in liquid air.14 Two specimens of ingot iron, however, were removed from the testing machine at the end of the first stage and several days elapsed before the specimens were tested to fracture in liquid air. As shown in Fig 8 of that paper," the points representing fracture of these specimens are far above the curve representing the results of two-stage tests without a rest interval between the stages. The evidence thus indicated that the fracture stresses of these two specimens were increased not only by the work-strengthening, but also by strain aging. The present paper presents results of further investigation of the influence of strain aging on the fracture stress. There are two kinds of aging of steel.

Jan 1, 1948

By Richard R. Zupp, David A. Stevenson

A theoretical treatment of austenitic Fe-C-Z solutions (where Z represents a substitutional solute) was developed by extending a solution model which was previously applied to austenitic Fe-C alloys. The model assumes that each carbon atom in the solution excludes H interstitial sites from occupancy by other carbon atoms rmd thai otherwise the assembly of carbon atoms is equivalent to a modified Langmuir lattice gas. The results of the treatment provide theoretical justification for a phenomenological expression previously developed to represent the influence of substitutional solutes on the thernlodynamics of carbon in austenitic Fe-C-Z alloys. The expression is: where Kz is a parameter which characterizes the Fe-C-Z system at a given temperature and Xz and xj are the concentrations of substitutional solute and interstitial carbon expressed as atom ratios. The parameter, Kz, was associated with the differ- A number of investigators have measured the influence of various substitutional solutes on the carbon activity in austenitic Fe-C-Z alloys, where Z represents any substitutional solute. Experimental information is available for alloys containing the following substitutional solutes: Mn,&apos; Si,&apos; Ni,2-5 CO,&apos;-&apos;, Cr,5, 9-11 CU,4 Al,= v,11, 12 and Ti.13 This information is essential for studying many phenomena in Fe-C-Z alloys, particularly diffusion and diffusion-controlled phase transformations For example the pffprts nf various ence between the interaction energies between carbon atorns and host-lattice atoms in the Fe-C-Z solution and the corresponding energies for the Fe-C solution. The carbon activity in a ternary Fe-C-Z alloy, ac, was obtained in terms of the composition of the alloy and the single parameter, Kz: where y° is the Henry&apos;s law constant for the binary Fe-C solution. Both y° and H crm be determined from the data for Fe-C solutions, without recourse to any experimental data for Fe-C-Z solutions. The theoretical treatment was extended to alloys containing several substitutional solutes. By neglecting interactions between the substitutional solutes, an expression for the carbon activity in multicomponent alloys was obtained in terms of the Kz values characterizing ternary Fe-C-Z systems. substitutional solutes on the rate of pearlite growth is explained14 on the basis of their effects on carbon diffusion, which result from their influences on the carbon activity in the solid phases. In addition, recent studies have emphasized the dependence of the pro-eutectoid ferrite transformation15 and other diffusional processes in austenite16 on the influence of substitutional solutes on the carbon activity in austenite. They have indicated the need for suitable analytical expressions to represent such influences. The present authors previously developed general phenomenological expressions to represent the equilibrium partitioning of carbon between Fe-C alloys and Fe-C-Z alloys.&apos;2 The expressions are applicable to all the austenitic Fe-C-Z alloys for which experimental data are available, and are sufficiently simple to recommend their general application. For alloys in

Jan 1, 1969

By Elmer R. Kaiser

COMBUSTION of coal and transfer of heat from flames and gases to boiler surfaces continue to be of great interest to engineers here and abroad. Numerous investigations have been in progress to improve furnace and boiler performance and economy. The importance of better understanding of the processes and opportunities for improvement is apparent when it is remembered that heat from at least 500 million tons of coal a year the world over is being transferred to boiler water at efficiencies ranging mostly between 50 and 90 pct. Even slight gains in efficiency, economy, and labor saving become very significant when multiplied by the enormous quantity of fuel consumed. Also the competitive position of the large coal, oil, and gas industries in satisfying the fuel consumers is greatly affected by the achievements made through technical progress with each fuel. This paper is part of a continuing activity of Bituminous Coal Research, Inc., to extend the knowledge of coal utilization for steam generation and to seek promising directions for future research and development in cooperation with others. Particularly in the latter regard, numerous interviews were held during the last three years to seek the experience and advice of boiler and combustion-equipment manufacturers, electric-utility executives, and fuel engineers. A wealth of published information was also reviewed, which together with the interviews pointed to the advisability of further work on ash and sulphur control. For the present purpose a number of factors important to efficient heat liberation and recovery have been grouped as follows: 1—combustion, temperatures, and rates of heat liberation; 2—radiation, convection, and furnace and boiler configuration; 3—sponge ash, slag, and hard-bonded deposits; 4— low-temperature deposits and corrosion (cooling flue gas below dew point and air-pollution control); 5—the limitations of coal cleaning and boiler size and cost as related to fuel characteristics; 6—future possibilities and conclusions. The development of combustion apparatus for power boilers is progressing at a lively pace. There has been no letup in improvements in design of pulverized-coal-fired boilers, and there is a strong trend at present toward improving dry-bottom units. Spreader stokers with overfire jets and dust collectors as standard equipment are gaining favor. Less than 10 years in commercial use, cyclone burners are going into numerous installations here&apos; and abroad.&apos; Underfeed and traveling-grate stokers have long since been developed for heavy-duty operation, yet new developments in overfire jets and humidification of air blast have improved their performance. A water-cooled vibrating-grate stoker of German origin is being introduced into the United States and Canada." The primary objectives of an ideal coal combustion device are: capacity to burn the variety and sizes of coals likely to be economically available during the life of the unit; capacity to burn the coals automatically for a wide load range and rapid load fluctuations and to burn the coals completely to CO2, H2O, and SO2, which means without smoke and cinders, or carbon in the refuse; capacity to control and discharge all the ash in final granular form without ash adhesion to walls or tubes, and without flue dust; minimum furnace volume; minimum labor and maintenance; low initial and operating cost. Regardless of the method of burning, the gaseous products of coal combustion are N2, CO2, O2, H20, and SO?. By way of illustration, the coal analyses in Table I is assumed from an installation described by E. McCarthy.&apos; When coal is burned with 20 pct excess air (theoretical air, 9.23 lb per lb of coal), the quantities of combustion gas shown in Table II are produced. In addition, the gases carry particles of fly ash, unconsumed cinders, soot particles, and small but significant amounts of vaporized oxides and sulphates of sodium, potassium, lithium, phosghorous, iron, and other metals. In recent years, germanium, one of the rare metals found in coal, has been shown to oxidize and vaporize at combustion temperatures and to be concentrated by reconden-sation at lower temperatures." Pulverized coal and cyclone flames" have peak temperatures of 3000&apos; to 3500°F. Temperatures in fuel beds of large underfeed stokers reach maxima of 3000°F, sufficient to fuse almost any ash and to volatilize some of it. These peak temperatures are above the optimum necessary for rapid combustion, but they hasten heat transfer for ignition as well as boiler heat absorption. Furnace and gas temperatures increase with combustion air preheat. Low excess air has the same effect. Fine coal pulverization and highly turbulent combustion shorten the distance for fuel burnout, increase flame temperature, and speed up heat transfer. Rates of combustion of pulverized coal exceeding 200,000 Btu per cu ft per hr have been demonstrated in atmospheric gas-turbine combusters,

Jan 1, 1955

By Arthur L. Crawford

PLAGER gold is perhaps the most difficult of the common mineral deposits to evaluate. Not only are the erratic pay streaks a source of never-ending uncertainty, but the spotty distribution of the gold in iso¬lated nuggets is a perpetual source of error and disappointment to the sampler. Experienced engineers have long realized this difficulty and have been wary in predicting assay values except in the most general terms, and then only after extensive and usually expensive sampling. During the past year I have had the opportunity of checking a number of gold placers, and from the results obtained I believe that the microscope may be employed to advantage in placer sampling.

Jan 1, 1933

John A. Church, New York City: It requires some courage to appear as a critic of a theory which is not only the fashion among American geologists but is usually presented by them in terms which imply that any other views are an exhibition of ignorance. Still, I am obliged to say that the theory of lateral secretion as it is stated in this and other writings of Mr. Emmons and other geologists has not added much to our real knowledge or clearness of view. In the earlier and less developed stages of the theory, when it was used as Sandberger used it, to show that certain veins were probably derived from the rocks in which they lie, or which are adjacent, it was valuable in pointing us to an immediate source of ore-deposition. When we are driven to assume the existence of undiscoverable rocks at an unknown but certainly a considerable distance and in an unknown direction from the vein, I do not see that we have improved upon the despised " unknown source in depth " with which our ignorance has been covered so long. The new theory may suffer from adolescence, and these points map be cleared up by further study, but I speak of it as it is. Differentiation in a magma, by which a metal is concentrated in one member of a series of outflows, may explain why certain ores have favored a given locality with their presence; but it is not a necessary precedent to ore-formation. Concentration in the source of supply cannot be a requirement, for the forces that have been able to take up four or five tons of gold from an extensive body of rock must be able to collect four or five thousand tons of lead, copper or nickel from a proportionately more extended body of rock. That is to say, concentration is no more essential for these metals than it is for gold. In fact, differentiation, as it is now explained, is not an advance upon old ideas, but a retreat from them. It was noticed long ago that violent eruptive phenomena, however long continued, died away in solfataras; and when a vein came to be looked upon as an extinct solfatara the inference was ready that veins are eruptive in the sense that the solfataric waters collected the metals from the unerupted residue of the magma and carried them to the veins. The early views carried differentiation further than the modern school.

Jan 1, 1894

By Walter Crafts, D. C. Hilty

The liquidus diagram for the iron field of the Fe-S-O system has been derived experimentally. The solubility of oxygen in molten Fe-S alloys has been measured at several temperatures and found first to decrease slightly and then increase rapidly with increasing sulphur content. IT is well-known that the types, distribution, and amount of inclusions in any given steel are closely allied with the oxygen and sulphur contents of the metal, so that the term "deoxidation" as conventionally applied in steelmaking refers not only to the control of oxygen and oxide inclusions but to collateral modification of sulphide inclusions as well. Inclusions commonly occur in steel as mixtures of oxides and sulphides, sometimes closely associated and sometimes apparently completely divorced. Their effects on the properties of the metal may vary widely depending on their types and distribution as well as on their quantity. The mechanism by which inclusions form, however, and the means for controlling it are exceedingly complex and not well understood. Nevertheless, as described by Benedicts and Lofquist&apos; and by Wentrup&apos; all oxide and sulphide inclusions, excluding mechanically entrained matter, must result from specific modifications of the basic equilibria of the Fe-S-O system. In an investigation of the mechanism of deoxidation and inclusion formation being carried out at the Union Carbide and Carbon Research Laboratories, it became evident that quantitative data for the fundamental Fe-S-O system were essential to a better understanding of the problem. Consequently, the liquidus surface of the portion of the Fe-S-O diagram of interest in steelmaking has been determined, and the solubility of oxygen in liquid iron containing sulphur has been measured. Fe-S-O Diagram The portion of the Fe-S-O diagram of specific interest in steelmaking is the Fe-FeS-FeO corner. Of the contiguous binary diagrams for this partial system, only those for Fe-FeO and Fe-FeS are known with reasonable certainty.3 Several attempts to derive the FeO-FeS diagram have been made, chief among them being the work of Giani as reported by Oberhoffer,4 and, more recently, that of Vogel and Fulling." Although both Giani and Vogel and Fulling published diagrams for the FeO-FeS system, their observations indicated that the oxide component approximated Fe3O4 rather than FeO, so that there is doubt whether the system is truly binary. Benedicts and Lofquist and also Wentrup published qualitative sketches of the ternary diagram based on the assumption that it was of the type described by Marsh for a ternary system in which two of the contiguous binary systems are simple eutectic systems and the third exhibits a solubility gap in the liquid. Apparently, however, the only previous effort to establish the diagram experimentally was that of Vogel and Fülling whose results are reproduced in Fig. 1. Although defining approximately the limits of the miscibility gap and the location of the ternary eutectic, point E in Fig. I, Vogel and Fülling&apos;s diagram must be considered only semiquantitative. It was derived from a limited number of observations mainly along two quasi-binary sections, and the directions of the tie-lines and the location of the critical point, K in Fig. 1, were estimated by computation. Experimental Procedure The experimental technique employed for the present investigation involved melting mixtures of ferrous sulphide and ferric oxide in contact with iron. Since it was considered that in this part of the system the vapor pressures of sulphur and SO, are so low that they would not affect the results significantly, the melting was carried out in a current of argon to sweep out products of any initial re-

Jan 1, 1953

By F. W. Jessen, H. Kazemi

A washing technique has been developed to form a spherical cavity in massive salt. The technique is, basically, a process of controlling the fluid motion in the cavity, the concentration distribution of brine, the rate of dissolution at the walls of the cavity, and the particular settings of the wash pipe, or pipes (inlet and outlet pipe) employed. For a particular arrangement of tubing and casing, fresh water enters from the casing annulus at a position above the bottom of the tubing, and the dissolved material leaves the cavity through the tubing near the bottom of the projected hole. The initial configuration of the cavity is cylindrical, representing the drilled hole situated at the axis of the projected cavity. As washing progresses the original hole is enlarged radially (any direction perpendicular to the axis of the hole). An inert fluid (generally a hydrocarbon, natural gas, or air) which serves as a blanket is injected through the annular space between the original hole and the casing. The controlled downward motion of this blanket maintains the upper edge of the brine region that is expanding on the surface of the projected sphere. A large spherical glass container and a radial model were used to investigate the flow patterns and the concentration distribution of brine occurring during the progress of solution of the salt. Aluminum flakes and dye were injected and flow patterns were observed and recorded on motion picture film. Based on the results of these model studies and other work carried out in this laboratory, the controlled washing technique was formulated. INTRODUCTION Underground cavities leached in salt beds and salt domes have been used extensively for storing hydrocarbons. Little attention has been focused on the shape of the cavities as long as large volumes were obtained and the cavities were structurally stable. The general process involves circulating fresh water and removing brine. Two distinctly different circulating systems can be employed, each of which yields characteristically different shapes. By far the most generally accepted method of washing storage caverns in salt domes is the direct circulation system. Water is introduced through the innermost pipe of a concentric pipe system, and brine is withdrawn through the annulus. In the reverse-circulation method, water enters through the annulus and is discharged through the smallesc tubing string. The size of the operation, and hence the size of tubing and casing used, depends on such factors as cavity volume, length of hole, availability of water, disposal of brine and power requirements. Underground caverns are from 100 to 1,000 ft or more in length, and when completed vary from 40 to 100 ft in diameter, providing storage of 100,000 bbl or more. The volumes of water required per volume of salt removed vary from 6 to 7 for the reverse circulation method to 10 to 11 for the direct circulation method. Up to 80,000 BWPD are often circulated. In the direct circulation method, because water enters near the bottom of the projected cavity, solution of salt is greatest at this point, particularly in view of the fact that when the washing is begun turbulent flow undoubtedly exists in the immediate vicinity of the discharge of the inlet water. As the diameter of the hole increases, flow rates also are increased, so perhaps the condition of fluid turbulence is maintained for some time. During the later stages of leaching, when capacity of equipment may be the limiting factor in maintaining flow rates, there may still exist some turbulence adjacent to the point of influx; however, for the most part laminar flow conditions prevail. There appears, therefore, rather good evidence to indicate that the state of fluid motion is responsible for the typically shaped cavity which results from such washing — namely a cylindrical form, generally somewhat larger at the bottom than at the top. Depending on the number of so-called stages which might be employed, the walls or surfaces of the salt diminish rather uniformly from bottom to

Jan 1, 1965

By L. S. Reid, R. L. Huntington, Chen Chun Ku

The simultaneous adsorption of water and hydrocarbon vapor from natural gas by three grades of alumina has been studied at atmospheric pressure and temperature. Results of this investigation reveal that the presence of water vapor in the gas inhibits the adsorption of hydrocarbon vapor, although the presence of the latter does not have a pronounced effect upon the adsorption of water vapor. Adsorption of water-hydrocarbon mixtures from gas is divided into three phases. In the first, both water and hydrocarbon vapor are adsorbed; in the second, adsorption of water vapor proceeds while desorption of hydrocarbon takes place; in the third phase, the adsorbyt approaches complete saturation with respect to both water and hydrocarbon vapors. Both theory and experimental data can be applied to design and operation of commercial dehydration plants, and an illustration is presented. Introduction Dehydration of natural gas becomes increasingly important as the operating pressures employed in gas processing and transmission are increased above 500 ib. per sq. in. Gas produced from high-pressure wells is saturated with water vapor when it leaves the reservoir and it continues into the processing plant or transmission system in a saturated condition because of the substantial decrease in temperature due to expansion at the wellhead. Usually this gas is accompanied by liquid water and hydrocarbons that have condensed with the change in temperature and pressure. Without dehydration, gas temperatures must be held at relatively high levels to prevent solidification of gas hydrates. According to Hammerschmidt&apos;s data8 on the relationship of pressure to the freezing point of natural-gas hydrates, the minimum safe operating temperatures range from 55&apos; to go°F. for pressures ranging from 500 to 3000 lb. per sq. in., respectively. Often it is difficult to maintain gas temperatures above the hydrate-freezing temperature because of climatic conditions, unless flow-line heaters are installed. Also, in processing gas for its gasoline content at high pressures, it is desirable to employ operating temperatures lower than the hydrate-freezing point. In distributing high-pressure gas, the large pressure drop from line pressure to town-gate pressure often prbduces sufficient cooling to freeze the regulators and disrupt consumer service. Therefore, it is highly essential to dehydrate high-pressure natural gas if smooth and economical operation is to be maintained at all times. Few published data are available on the simultaneous adsorption of water and hydrocarbon vapors. The literature contains many mathematical expressions for rate of adsorption in various systems but a rational correlation of the existing rate

Jan 1, 1944

By Vaughn Moyer

A method for incorporating well drawdown effect into reservoir calculations is presented in detail, together with examples of its use for widely divergent conditions that could be normally encountered in oil reservoir sands. Tentative conclusions as to the effect such factors as permeability, producing interval, well spacing, or well damage caused by completion or repair practice could have upon economic ultimate recovery are reached. It is carefully pointed out that the conclusions reached are subject to limitations indicated in the text, and placed upon all calculations made in the paper. Introduction Papers have been presentedl-3 in which theoretical depletion histories for reservoirs under equilibrium conditions and no appreciable well-bore drawdown pressure are determined. However, except for a limited number of cases, appreciable drawdown Pressures in a well bore exist, and Pressure and saturation gradients consequently vary throughout a reservoir. This Paper is an attempt to incorporate well-pressure drawdown into reservoir history, and to analyze certain factors, such as those involving recovery, well spacing, and productivity index, from the standpoint of permeability and drainage radius. BY incorporation of drawdown into reservoir history, it is believed that theoretical considerations will more closely approxi- mate actual reservoir behavior experienced in an oil field. For purposes of simplification, a solution gas-drive type field1 without primary gas cap, has been selected. Basic Data and Assumptions Details of equilibrium saturation conditions for no drawdown production for various pressures in a reservoir are deter-minable by methods described by several are permitting the construction of a curve of oil saturation vs Pressure for any selected combination of initial interstitial water and hydrocarbon saturation (Fig I). This curve is then a description of equilibrium saturation conditions at any given pressure in a reservoir. Developing the above, details of saturation and flow permeability may thus be determined for any increment or portion of a well drainage area or reservoir. In this manner the effect of pressure differential (drawdown) may be incorporated into reservoir and well drainage considerations. It is pointed out that this method thus makes use of no-drawdown equilibrium production data, superimposing a draw-hwn effect upon them. It also assumes that net flow conditions in any considered increment of reservoir are not materially altered by the entrance of fluids from other increments into them, but approach equilibrium for the existing pressure and saturation (i.e., each increment behaves in more or less independent fashion, dependent largely upon the equilibrium considerations mentioned above)

Jan 1, 1948

By Vaughn Moyer

A method for incorporating well drawdown effect into reservoir calculations is presented in detail, together with examples of its use for widely divergent conditions that could be normally encountered in oil reservoir sands. Tentative conclusions as to the effect such factors as permeability, producing interval, well spacing, or well damage caused by completion or repair practice could have upon economic ultimate recovery are reached. It is carefully pointed out that the conclusions reached are subject to limitations indicated in the text, and placed upon all calculations made in the paper. Introduction Papers have been presentedl-3 in which theoretical depletion histories for reservoirs under equilibrium conditions and no appreciable well-bore drawdown pressure are determined. However, except for a limited number of cases, appreciable drawdown Pressures in a well bore exist, and Pressure and saturation gradients consequently vary throughout a reservoir. This Paper is an attempt to incorporate well-pressure drawdown into reservoir history, and to analyze certain factors, such as those involving recovery, well spacing, and productivity index, from the standpoint of permeability and drainage radius. BY incorporation of drawdown into reservoir history, it is believed that theoretical considerations will more closely approxi- mate actual reservoir behavior experienced in an oil field. For purposes of simplification, a solution gas-drive type field1 without primary gas cap, has been selected. Basic Data and Assumptions Details of equilibrium saturation conditions for no drawdown production for various pressures in a reservoir are deter-minable by methods described by several are permitting the construction of a curve of oil saturation vs Pressure for any selected combination of initial interstitial water and hydrocarbon saturation (Fig I). This curve is then a description of equilibrium saturation conditions at any given pressure in a reservoir. Developing the above, details of saturation and flow permeability may thus be determined for any increment or portion of a well drainage area or reservoir. In this manner the effect of pressure differential (drawdown) may be incorporated into reservoir and well drainage considerations. It is pointed out that this method thus makes use of no-drawdown equilibrium production data, superimposing a draw-hwn effect upon them. It also assumes that net flow conditions in any considered increment of reservoir are not materially altered by the entrance of fluids from other increments into them, but approach equilibrium for the existing pressure and saturation (i.e., each increment behaves in more or less independent fashion, dependent largely upon the equilibrium considerations mentioned above)

Jan 1, 1948

By R. L. Huntington, L. S. Reid, Chen Chun Ku

The simultaneous adsorption of water and hydrocarbon vapor from natural gas by three grades of alumina has been studied at atmospheric pressure and temperature. Results of this investigation reveal that the presence of water vapor in the gas inhibits the adsorption of hydrocarbon vapor, although the presence of the latter does not have a pronounced effect upon the adsorption of water vapor. Adsorption of water-hydrocarbon mixtures from gas is divided into three phases. In the first, both water and hydrocarbon vapor are adsorbed; in the second, adsorption of water vapor proceeds while desorption of hydrocarbon takes place; in the third phase, the adsorbyt approaches complete saturation with respect to both water and hydrocarbon vapors. Both theory and experimental data can be applied to design and operation of commercial dehydration plants, and an illustration is presented. Introduction Dehydration of natural gas becomes increasingly important as the operating pressures employed in gas processing and transmission are increased above 500 ib. per sq. in. Gas produced from high-pressure wells is saturated with water vapor when it leaves the reservoir and it continues into the processing plant or transmission system in a saturated condition because of the substantial decrease in temperature due to expansion at the wellhead. Usually this gas is accompanied by liquid water and hydrocarbons that have condensed with the change in temperature and pressure. Without dehydration, gas temperatures must be held at relatively high levels to prevent solidification of gas hydrates. According to Hammerschmidt&apos;s data8 on the relationship of pressure to the freezing point of natural-gas hydrates, the minimum safe operating temperatures range from 55&apos; to go°F. for pressures ranging from 500 to 3000 lb. per sq. in., respectively. Often it is difficult to maintain gas temperatures above the hydrate-freezing temperature because of climatic conditions, unless flow-line heaters are installed. Also, in processing gas for its gasoline content at high pressures, it is desirable to employ operating temperatures lower than the hydrate-freezing point. In distributing high-pressure gas, the large pressure drop from line pressure to town-gate pressure often prbduces sufficient cooling to freeze the regulators and disrupt consumer service. Therefore, it is highly essential to dehydrate high-pressure natural gas if smooth and economical operation is to be maintained at all times. Few published data are available on the simultaneous adsorption of water and hydrocarbon vapors. The literature contains many mathematical expressions for rate of adsorption in various systems but a rational correlation of the existing rate

Jan 1, 1944

By E. J. Efsic, C. M. Wayman

The fcc to bcc y to a transformation in an Fe-24.5 at. pct Pt alloy was studied and found to be martensitic. Large martensite plates are formed in this alloy which are ideal for crystallographic studies; the Ms temperature is about -5°C. Ordering of the parent phase prior to transformation to martensite causes a pronounced lowering of the Ms temperature. For the first time in the case of alloys of iron, measurements of the habit plune, orientation relationship, and magnitude and direction of the shape stvain have been obtained on preselected martensite plates. It is shown that by measuring the initial and displaced (as a result of the shape strain) positions of three or more non-parallel fiducial scratches on one reference surfice the maguitude and direction of the shape strain and the state of distovtion at the parent-mavtensite interface can be determined with a reasonable degree of accuvacy. The agreement between the phenomenolog-cal crystallographic theory of martensitic transformations and the experimental results habit plune, orientation relationship, and nlagnitude and direction of the shape strain) is essentially perfect, and is thought to be the best reported to date for iron alloys. THE comparatively few investigations on binary Fe-Pt alloys have been concerned with magnetic properties, ordered phases, and the equilibrium diagram itself. Various investigations have shown that the equilibrium diagram is of the open y field type; in addition there is a peritectic transformation, a minimum melting point, and three ordered phases: Fe3Pt, FePt, and FePt3. The results of a number of investigators1-6 have shown that the temperature dependence of the y to a (and reverse) transformation in iron-rich alloys is reasonably well-established. The Fe3Pt phase was first noted by Kussman and Rittberg,7 and its existence was later confirmed by Franklin et al,8 Steijn and Donahoe,9 and Berkowitz et al.10 The FePt phase was reported by Isaac and amman&apos; and later confirmed by a host of others.3,6,7,12,13 Kussman and Rittberg7 noted the existence of the superlattice FePt3. Prior to 1956, the 1 to a transformation in iron-rich Fe-Pt alloys was termed "irreversible". In 1957 Klokholm and Donahoe" termed this transformation martensitic, presumably because of its occurrence at temperatures well below those where diffusion normally takes place. This observation served as a point of departure for the present investigation, and in the very early stages of the work to be discussed it was readily verified that the y to a transformation in iron-rich Fe-Pt alloys is indeed martensitic. Quite large martensite plates exhibiting the well-known surface tilt were observed. These plates are ideal in the sense that the boundaries between the parent and product phases were usually very straight. It thus appeared that Fe-Pt martensites would be ideal for crystallographic studies, and this was the purpose of the present investigation. Details of the crystallography of martensitic transformations (particularly for iron alloys) and the phe-nomenological crystallographic theories15-l7 have been discussed many times and thus will require very little elaboration here. The end result of these theories, given in matrix form, is

Jan 1, 1968