Search Documents

By H. l. Burghoff, J. S. Porter

This paper presents the results of an investigation of the effect of cold-working and annealing treatments upon the occurrence of preferred orientation in annealed brass wire. The subject has received less consideration than has the same phenomenon in annealed brass sheet or strip, and perhaps very properly so in view of the obvious and undesired results obtained in the forming or drawing of strip having preferred orientation, which results are in contrast with the lack of any spectacular visual evidence in wire. The factors of cold-working and annealing that control preferredness in strip are naturally expected to function similarly in wire. Indeed, one might expect the condition to be commoner in annealed wire, because the severity of the cold-working between anneals in commercial Practice is usually much greater for wire than for strip. For example, a wire that is to be finished in the annealed condition on a diameter of 0.040 in. is commonly subjected to a final reduction of 90 per cent between the last two anneals, the large reduction being accomplished in the successive drafts of a multiple-die machine. Wire for rivets, for instance, on 0.125-in. diameter, although given a slight reduction of about Io per cent after the final anneal, may have been given a prior reduction of 75 to 80 per cent. Barrettl states the fiber axis of annealed 70-30 brass wire to be [111], with some indication of [IOO] as a secondary fiber axis. Croft and Sachs2 have reported briefly on some differences in brass wire with random and with preferred orientations. They showed statistical plots of the twin-band angles with respect to the wire axis or direction of drawing as observed on an axial plane. They also pointed out that the resistance of the wire to corrosion-cracking when externally stressed and exposed to a mercurous nitrate solution decreased as the degree of the preferred orientation increased. Material and PRocedure All the wire for the investigation was processed from one portion as cut for extrusion from a cast billet of the following analysis: 70.13 per cent copper; 0.01 lead; 0.01 iron; 0.02, nickel; 29.83 zinc (by difference). The billet section, 8 in, in diameter, was extruded hot to 3/4-in. diameter, after which it was processed by alternate cold-drawing and annealing treatments as indicated in Fig. I. The earliest anneals in process were at about 1100°F. and the next to the last reductions by drawing were 58 per cent. The diameter of the finally annealed wire was 0.132 in. Semifinal anneals were at 800°F., 950°F. and 1100°F. and reductions of 47.5, 74, 83.5 and 89.5 per cent were employed in drawing to 0.132 in. The final anneals were at 87s°F., 1050°F. and 1250°F. and were

Jan 1, 1944

By H. N. Treaftis, J. W. Cahn

THE previous determinations of the solvus of tin in solid lead disagree with one another by as much as 40°C or almost 10 at. pct. Even determinations that appear to be careful differ considerably in both the solubility and its temperature coefficient. Recent kinetic workl -3 on the rate of precipitation of tin from lead-tin alloys and thermodynamic work4 on the heat of solution of tin in lead have given some insight into the time necessary to reach equilibrium in this system. As this time is longer than that used in almost all previous determinations of the solvus, these determinations are questionable. Since the interpretation of recent kinetic results requires accurate values of the equilibrium solubility the present work was undertaken. DISCUSSION OF PREVIOUS RESULTS The methods used in previous studies may be placed into three classes: 1) Several alloys covering a range of compositions are allowed to reach "equilibrium" at a certain temperature.5-7 Some property, such as the X-ray lattice parameter of the lead-rich phase or the re- sistivity of the alloy, is then measured as a function of composition. The solubility limit for that temperature is taken to be that composition at which a discontinuity in the property occurs. However, the recent kinetic work has shown that tin precipitates in two stages. The first stage is rapid but leaves the lead still supersaturated. The second stage is very slow, and it seems from the equilibration times quoted by the investigators of the solvus that the end of the first stage of precipitation was usually mistaken for equilibrium. This is especially true at temperatures below 100°C where the first reaction is still quite rapid (from minutes to months), but where the second stage is so slow that very long holding times are required to reach equilibrium. Alternatively micrographic work8 on several alloys of different compositions seems to have given good results if done carefully. 2) The beginning of precipitation in a homogenized alloy is measured as it cools slowly8-10. Since reactions in solids undercool easily, such determinations will always give too low a temperature for the solvus. 3) Some property of an alloy is measured as a function of temperature while the alloy is above the solvus.5,9,11 The range of temperature can usually be extended slightly into the two phase region. Then the alloy is precipitated at a much lower temperature and slowly reheated. The solvus is taken to be the lowest temperature where the property of the reheated alloy is equal to that previously determined for the homogeneous alloy. Usually one also expects a discontinuity at the solvus in the temperature variation of the property for the reheated alloy. The difficulty in this method is the sluggishness of the dissolution process, and this method may give too high a solvus temperature. EXPERIMENTAL The phase boundary or solvus temperature was determined by a modification of method 3 for a series of chemically analyzed high-purity alloys, ranging from 5.1 to 26.4 at. pct Sn. The resistance, as a function of temperature, of a sample of homogenized alloy (which could easily be undercooled) was compared with the equilibrium resistance which that alloy sample attained after precipitation at room temperature followed by long isothermal anneals at various temperatures. Before each anneal the sample was rehomogenized and reprecipitated. The solvus temperature could be located to within 1°C as the intersection of the resistance vs temperature curves for the homogenized and equilibrium alloys. No attempt was made to locate the phase boundary more accurately since the limiting factor was the precision with which the alloys could be analyzed. This precision was ±0.1 wt pct Sn and corresponds to an error of 1 deg in temperature. This use of isothermal anneals differs from some of the previous investigations in which method 3 was used with baths of slowly rising temperatures. It was found early in the investigation that solvus temperatures tended to be high if the latter method was used even at extremely slow heating rates. For example a

Jan 1, 1961

By J. S. Aronofsky, J. P. Heller

This paper presents a mathematical analysis of the fluid mixing which occurs during flow through porous media. The analysis is based on the well-known diffusion equation with mass transfer term. It is pointed out that the use of this equation is justified by a single general assumption which does not specify any particular mechanism for the mixing process. Formulas are given for tracer fluid concentrations for two different boundary and initial conditions. Calculated numerical values compare closely with some published results of experiments in which no viscosity or density difference existed between displaced and displacing fluids. INTRODUCTION A phenomenological theory of the mixing and diffusion process is presented for the flow in a porous medium of two miscible liquids of equal viscosity and density. This is based on the classical diffusion equation which has been used to explain such processes as Brownian motion and heat conduction. More recently, this same differential equation has been derived (in a manner analogous to the work in this paper) for the problem of heat transfer during fluid flow in porous media.' The purpose of this paper is to show that the available experimental evidence justifies the use of this simple diffusion equation. The recent data of Koch and Slobod' and of von Rosenberg' are used to test this diffusion model by comparing concentration curve shapes and calculating "effective diffusion coefficients". The same differential equation for miscible displacement, as well as some experimental confirmation, has been published in Japan by K. Yuhara,4 and recently in this country by Day.5 Yuhara arrives at his con-clusions through analogy with turbulent diffusion, de- scribing the microscopic velocities in terms of "eddy motion" — although unlike the ordinary eddies of hydraulics. Day proceeds directly from Scheidegger's6 concept of the dispersion of velocities. After a very clear description of Scheidegger's theory, he applies it to calculate the motion first of a drop and then of a thin layer of brine into a bed of sand. For the latter case, Day derives and solves the same differential equation as Eq. 1. In our presentation no detailed assumptions are made as to the mechanism of displacement. In fact, several microscopic processes may well be involved—the only general assumption made is that their cumulative microscopic effect may be described by the addition of a second-order term to the differentirl equation of continuity. DERIVATION OF EQUATIONS Let us consider the displacement in one spatial dimension of Liquid A from a sample of porous media by Liquid B. A and B are assumed completely miscible and of the same viscosity and density. Both fluids are also assumed to be incompressible and it is specified that the total flow is the same for any cross section. It is evident that the rate at which Liquid B would be carried across such a section would be given by the product of the fluid velocity, u, and the concentration, C, of B in the fluid mixture assuming the constancy of u and C over the cross section. Since the volume of Fluid B is conserved, this leads to the one-way wave equation - u ?C/?x = ? ?C/?l where x increases in the direction of u—the "Darcy velocity", Q/A (or velocity flux per unit area), and 9 is the fractional porosity. Consider the case where fluid B is displacing Fluid A and the concentration distribution is represented by a step funcLon (with a vertical discontinuity). The solution of Eq. 1 predicts no mixing

Jan 1, 1958

By J. E. Hilliard

A procedure is described for determining, from measurements on a section, the moments of any degree n (i.e., the mean value of Dn, where D is a linear dimension) of the size distribution of particles (or grains) in an opaque sarnple. The analysis is applicable to particles of alnzost any shape and to any mixture of shapes. However, if absolute values of the wloments are required, it is necessary to know what shapes are present and the relative frequency of their occurrence. In addition, certain shape-dependent parameters have to be evaluated before the formulas can be used. The measurement required is the distribution of intercept lengths. This can be determined either visually or with an automatic scanning instrument. Corrections me derived for the bias due to instrumental end effects. The proposed analysis has the following advantages over existing ones: 1) Confidence limits are easily estimated for the measured properties or, conversely, the number of counts required to achieve a given accuracy can be determined. 2) The analysis is not restricted to particles of a special shape. 3) The error resulting from any known degree of departure of the actual from the assumed particle shape can be estimated. 4) The computations are simple and, apart from a change in constants, are independent of particle shape. 5) The number of particles per unit volume can be determined by a simple counting procedure. MANY procedures have been described for determining the size distribution of particles from measurements on a section (or plane of polish). They differ from one another in the property measured on the section (radius, intercept length, profile area) and in the method used for the numerical solution of the integral equation relating the property to the particle size distribution. However, all existing analyses share at least two of the following disadvantages. 1) They are limited to a particular shape of particle (usually spherical). 2) It is very difficult to determine how the statistical sampling error is propagated into the computed size distribution. The investigator thus has no means of knowing the reliability of his result and, what is of equal concern, he is unable to estimate how many measurements are required to obtain a result of the required accuracy. 3) It is also difficult to assess the error resulting from any deviation between the actual and assumed particle shape. The foregoing limitations stem from the complex relationship between the particle size distribution and the distribution of the property measured on the section. As will be shown, they can be avoided by defining the size distribution in terms of discrete parameters

Jan 1, 1969

By A. S. Appleton, M. B. Bever

The stored and expended energies of an Au-Ag alloy deformed in torsion were investigated as functions of strain, strain rate, and temperature. At room temperature, the stored energy, measured by tin solution calorimetry, increased linearly up to an intermediate strain, above which it was nearly constant; at 78 °K the relation was linear up to fracture. Approximate values of the energy stored at 4 OK indicate a rapid increase below 78 OK. The energy stored by specimens deformed successively at 4o and 78oK changed slowly with increasing second strain from the curve for energy stored at 4 OK (measured after annealing at 78oK) to that for deformation at 78°K. The ratios of the stored to the expended energy at room temperature and 78 oK were unusually large. The additional amount of energy stored at the lower temperature is attributed mainly to an increase in the number and energy of dislocations. The stored energy as a function of strain rate at 4oK (measured after annealing at 78oK) increased to a sharp maximum over a narrow range of strain rates, whereas the expended energy changed only slightly. These effects are explained by the generation of point defects resulting from the enforced nonconservative movement of jogs in fast moving screw dislocations. The effect of straining successively at two strain rates at 4oK was also investigated. Stress-strain curves for 4o, 78oK, and room temperature and the results of micro examination of specimens deformed at these temperatures are also reported. THIS investigation is a continuation of a research program primarily concerned with the effects of variables on the energy relations involved in the deformation of metals. The variables investigated in earlier work have included strain, strain rate, temperature of deformation (room temperature and 78oK), and the type of deformation process. Most of this work has been carried out with Au-Ag alloys. In the investigation reported here, the deformation process was torsion, which had not been used previously in the program. The stored and expended energies involved in the deformation process were measured as functions of strain, strain rate and the temperature of deformation in the range 4" to 373" K. The microhardness and microstructure of deformed specimens were also investigated. The relation between stored energy and strain differs for different deformation processes.1,2 In general, the stored energy increases with strain and may attain a nearly constant value at large strains. This value and the rate at which it is reached, however, depend on the process of deformation. This dependence has been attributed, at least in part, to differences in the effective strain rate and rise in temperature of the specimen during deformation. The effects of the temperature of deformation on both the stored energy and the energy expended in the working process have been the subject of only a few investigations.3-6 Values of these quantities at low temperatures are of special interest as they yield information on point defects which, for many metals, is not obtainable at higher temperatures. The effect of the strain rate on the stored and expended energies has received little attention The strain rate has usually been measured only in relative terms, and hence the interpretation of its effect on the stored energy has been mainly comparative.

Jan 1, 1963

By Ian W. Wark, Keith L. Sutherland

IT has been shown' that in the absence of heavy metal salts, the nature of the alkali used to promote differential flotation-whether caustic soda, lime or sodium carbonate-is unimportant. The hydroxyl ion is the active depressant upon which differentiation depends, and the pH value determines whether a film of collector forms on any given mineral surface. Even when sodium cyanide is used in conjunction with alkalies, it is the pH value and not the nature of the alkali or its concentration that is important, for upon the pH value alone depends the fraction of the added sodium cyanide that is present as cyanide ion, which is the true depressant. When heavy metal salts become dissolved from partly oxidized ores, or are added intentionally as activators, conditions are different, and metallurgists have found that lime and sodium carbonate do not give identical results, even when added in amounts sufficient to produce the same pH value. In the few instances in which we had attempted to confirm this view, we had failed to do so, for, with nothing to guide us, we had inadvertently chosen solutions that gave contact curves of a "standard type," which correspond to conditions of ready floatability. Now, however, we find that, in the presence of copper salts, and within a restricted range of reagent concentrations, the nature of the anions present is important, and that sodium carbonate does influence the results, apart from the usual effect due to its influence on pH value. With the recognition of this anion effect, and with the control of temperature already introduced,2 it appears that the factors of major importance in connection with this set of reagents have been taken into consideration, and that the results obtained and the conclusions drawn from contact tests will be of more immediate value to the operator. It is not difficult to suggest an explanation for the influence of the carbonate ion in the presence of copper sulphate. The works of Picker-

Jan 1, 1939

By Frederick N. Rhines

Because of the nature of the environments in which metals are handled, it is natural that gases should be suspected of intruding into metallurgical operations, there to produce effects both beneficial and detrimental. The problem of tracing and controlling such effects is often made difficult by the complexity of the conditions surrounding their occurrellce, even though the fundamental mechanisms invalved may be simple and of familiar type. Many gas-metal reactions are known to be implemented by Processes of diffusion. These form a homogeneous group of phenomena which become more readily understandable when viewed together, in that the knowledge gained from one serves to interpiet the next. It is the purpose of the present Paper to present, by means of examples drawn from practice, a brief survey of the ways in which gas-metal diffusion serves and obstructs the Purposes of the metal-fabricating industries. The selection of the cases described was made by consulting more than thirty-six practicing experts whose combined experience provided an extensive coverage of the field and who were in a position to supply examples of occurrences of present day interest. The response was generous and it is regrettable that space permits the description of only a limited number of typical examples with ' but brief mention of parallel cases. The Nature of Gas-metal Diffusion The term ugas-metal diffusion," although commonly used, is ambiguous in so far as it implies that a definite group of " substances may be identified as gases and may be expected to behave in a manner distinct from other substances in association with metals. "Gases" such as oxygen and hydrogen when diffusing through a liquid or solid metal are properly components of the liquid or solid solution; they behave as alloying elements and are not to be distinguished in their diffusion characteristics from other elements, except that each has an individual rate of diffusion. On the other hand, a number of metals in certain.temperature ranges are volatile to the extent that they may appear in the form of vapor in metallurgical operations. For the sake of greater definiteness, therefore, it will be convenient to regard as "gas-metal diffusion" any process in which the result depends upon one or more of the diffusing elements appearing, at some stage, in the gaseous state, or as a compound gas. NO element, gaseous or otherwise, can diffuse through a liquid or solid metal unless it is at least slightly soluble in that metal. The elements of low atomic number

Jan 1, 1944

By Frederick N. Rhines

Because of the nature of the environments in which metals are handled, it is natural that gases should be suspected of intruding into metallurgical operations, there to produce effects both beneficial and detrimental. The problem of tracing and controlling such effects is often made difficult by the complexity of the conditions surrounding their occurrellce, even though the fundamental mechanisms invalved may be simple and of familiar type. Many gas-metal reactions are known to be implemented by Processes of diffusion. These form a homogeneous group of phenomena which become more readily understandable when viewed together, in that the knowledge gained from one serves to interpiet the next. It is the purpose of the present Paper to present, by means of examples drawn from practice, a brief survey of the ways in which gas-metal diffusion serves and obstructs the Purposes of the metal-fabricating industries. The selection of the cases described was made by consulting more than thirty-six practicing experts whose combined experience provided an extensive coverage of the field and who were in a position to supply examples of occurrences of present day interest. The response was generous and it is regrettable that space permits the description of only a limited number of typical examples with ' but brief mention of parallel cases. The Nature of Gas-metal Diffusion The term ugas-metal diffusion," although commonly used, is ambiguous in so far as it implies that a definite group of " substances may be identified as gases and may be expected to behave in a manner distinct from other substances in association with metals. "Gases" such as oxygen and hydrogen when diffusing through a liquid or solid metal are properly components of the liquid or solid solution; they behave as alloying elements and are not to be distinguished in their diffusion characteristics from other elements, except that each has an individual rate of diffusion. On the other hand, a number of metals in certain.temperature ranges are volatile to the extent that they may appear in the form of vapor in metallurgical operations. For the sake of greater definiteness, therefore, it will be convenient to regard as "gas-metal diffusion" any process in which the result depends upon one or more of the diffusing elements appearing, at some stage, in the gaseous state, or as a compound gas. NO element, gaseous or otherwise, can diffuse through a liquid or solid metal unless it is at least slightly soluble in that metal. The elements of low atomic number

Jan 1, 1944

By John Riegle

To date. utility company underground storage of gas has generally been restricted to depleted dry gas fields. The Playa del Rey project is probably the first to successfully store gas in a partially depleted oil reservoir with the recovery of large volumes of pas at high rates as the main objective. Oil recovery has been a secondary consideration. Problems encountered unusual to those of storage in a dry gas zone were: (1) the removal and retardation of formation of emulsion. (2) the upstructure movement of fluid during withdrawal periods which formed fluid blocks, and (3) reservoir shrinkage resulting from encroachment of edgewater. The solution of these problems as outlined in the paper has resulted in increasing the withdrawal rate from the original design capacity of 4,000 Mcf per hour to the current 10.000 Mcf per hour. Additional increase is anticipated as the operations continue. Migration. reservoir performance. operational procedure and a historical record are included to make this a resume of the project from its war-time inception in 1942 to the first of 1952. INTRODUCTION The Playa del Rey underground gas storage project is probably the first project in which gas has been stored in a partially depleted oil reservoir, with the recovery of large volumes of gas at high rates as the main objective and the recovery of oil a secondary consideration. Problems unusual to those of storage in dry gas zones have made this a pioneer endeavor. The total storage capacity of approximately 1,500,000 Mcf of this reservoir is small in comparison with other underground gas storage projects. However. by study and experiment the deliverability into transmission lines. operating at pressures in excess of 150 lbs., has been nearly tripled in the past three years. to the current 10.000 Mcf per hour late, without the use of compression. It is anticipated this rate will gradually increase as additional fluid is removed from the formation. LOCATION AND HISTORY Playa del Key is located on Santa Monica Bay about two miles south of the beach commuity of Venice and 15 miles southwest of the renter of the City city Los Angeles. Fig. 1. The discovery well was completed at 6,194 ft in 1929. An upper zone at 3,905 ft was found on June 18. 1930. Townlot development was very rapid and by the end of 1930 there were 141 producing wells in the field. The Union Oil Co. completed a lower zone well south of the Townlot Field in May, 1931, which was believed to be an extension, until additional drilling of about 50 wells during 1934 and 1935 indicated this area to be a separate accumulation as shown on Fig. 2. The lower oil horizon, which is now the storage zone, had no original gas cap and is believed to be a continental detrital deposit of conglomerate that accumulated in valleys and depressions of a weathered schist high. which later subsided and was overlain by an impervious nodular shale. This type of formation has rapid variation of porosity. permeability and thickness due to lack of sorting action common to marine deposition. and the uneven surface of the original schist high. Production was limited on the upstructure sides by the pinching out of the conglomerate, and downstructure by edgewater. Unrestricted production, together with high permeabilities, led to large wastage of gas from this new area as facilities were not available for transport of gas to market. The decline was very rapid and by 1942 the portion of the new area south of Ballona Creek was nearly depleted. ACQUISITION OF PROPERTY The rapid increase in industrialization of Southern California during World war 11 forced upon the local gas com-

Jan 1, 1953

By Glenn R. Zellars, J. P. Morris

The carrier gas method was used to measure the vapor pressure of copper over liquid copper and Fe-Cu alloys. From these results, the activity of copper in the alloys was determined at 146°, 1550°, and 1600°C. Calculations of the activity of iron were made from the results for copper. The activities of copper and iron show strong positive deviations from Raoult's law. KNOWLEDGE of vapor pressures of liquid metals has both practical and theoretical value. Fuming of metal baths is directly related to vapor pressure, and the data should bc particularly important to the technology of vacuum metallurgy. Theoretical applications include determinations of heats of vaporization, heats of solution, and activity coefficients in alloys. Methods for measuring vapor pressures of metals have been reviewed by Speiser and Spretnak.' High-vacuum procedures, such as the Langmuir and Knudsen methods, have been used most widely in recent years. However, these methods are not applicable to vapor pressures greater than lo-' mm Hg and often cannot be used in the temperature range of greatest interest. The purpose of this investigation was to develop a procedure for measuring vapor pressures of pure metals and alloys that would be applicable to pressures greater than lo-' mm Hg. The transportation or carrier gas method has been used. Copper was chosen for the initial work because its vapor pressure has been established fairly well and the reliability of the technique could be ascertained. Applicability to alloys was studied by determining the vapor pressure of copper over liquid Fe-Cu alloys. Experimental Procedure The tests were carried out in a graphite-spiral resistance furnace. The charge consisted of 50 g of metal contajned in a covered sintered-alumina crucible 1/2 in. deep and 13/8 in. ID. Two sintered-alumina tubes led from outside the furnace to within the crucible, passing through close-fitting holes in the crucible cover and radiation shield. The arrangement of the crucible and the two alumina tubes is shown in Fig. 1. The small-bore tube (1/16 in. ID by 5/16 in. OD) served as the inlet for the carrier gas. The tip of this tube touched the metal surface and the flow of gas imparted a rippling motion to the metal. In this manner the gas was brought into close contact with the metal and became saturated with metal vapor. In most of the tests the flow rate of the carrier gas was 560 ml per min, measured at room temperature. Approximately half of this saturated gas was allowed to escape from the crucible through the second alumina tube (3/16 in. ID by 5/16 in. OD) and was discharged to the atmosphere through a

Jan 1, 1957

By W. C. Hagel, H. J. Beattie

Phases present at 2200° and 1500°F (1204° and 816°C) were identified in sixty wrought developmental austenitic alloys possessing wide compositional variations. The bases were iron-, cobalt-, and nickel-rich as well as a base with equal parts of each. There are only two structure types previously not found in similar alloys, viz .-the Co3 W and the Fe3Al types. A molybdenum-rich monocarbide decorates dislocations with unusual clarity. Other results provide correlations between alloy composition and phas e precipitation. Grouping intermetal -lic phases into three general classes : I) geometrically close-packed, 2) topologically close-packed, and 3) bcc derivatives, the nickel base prefers the first, the iron base the second, and the cobalt base the third—each base tending to shun the other two classes. Thus, a crystal-chemistry effect previously noted by Laves and Wallbaum can he expanded for these Group VIII elemental bases. There appear to he no quaternary or higher -order structure types requiring more than three atomic species. AFTER studying aging reactions in many diverse commercial austenitic alloys1-4 as a function of detailed time-temperature variations in heat treatment, it became apparent that matrix composition influenced structure types and morphologies to a large extent. These tendencies were related4 to a crystal-chemistry effect uncovered by Laves and Wallbaum.5,6 We have now directed our efforts to exploration of the whole ternary field, covering the standard base elements—iron, cobalt. and nickel— with 10 to 20 pct Cr and moderate amounts, up to 10 pct, of the usual alloy additions. Solutioning and aging treatments were held constant with major emphasis on X-ray identification of microconstituents. The possible existence of exceptions to the Laves-Wallbaum effect, or any new phases, was sought. Phases appearing in austenitic alloys can be classified as geometrically close-packed (gcp), octahedral or nonoctahedral interstitial compounds, topologically close-packed (tcp), semicarbides, and bcc derivatives. Crystal structures of the first four classifications can be visualized as a dense packing of semirigid spheres of different sizes, while the bcc derivatives represent a less efficient packing of spheres. The Laves-Wallbaum effect involves the gcp and tcp phases. Gcp phases possess both tetrahedral and large octahedral interstices; examples are Ni3Ti(n), NiSAl(y'), Co3W(E), and other ordered AB3 compounds where B atoms are smaller than A. Their precipitation is primarily intragranular, presumably coherent with an austenitic matrix, and a chief source of strengthening. Most octahedral interstitial compounds (e.g., CbC, Tic, TiN, Mo2C) have a gcp lattice of large metal atoms which accommodate small carbon or nitrogen atoms in the octahedral interstices. The compound Ti2SC, previously designated Y phase or T phase, can, with some reservation due to its partial co-valency, be placed in this category.7 The nonoctahedral interstitials (e.g., M7C3, Fe3C) represent close-packings of spheres in structures that provide larger triangular-prism interstices for carbon, since the octahedral interstices formed by chromium and iron atoms are too small. Tcp phases possess only tetrahedral interstices; examples are Laves (Fe2Ti, Fe2Cb, Fe2W...AB2), p(Fe7MO6, Co7W8.... A8B7), and (FeCr, Co2Cr3)

Jan 1, 1965

By D. G. Westlake

NUMEROUS hydride habit planes in zirconium, Zircaloy-2, and Zircaloy-4 have been observed by various researchers and have been compiled by Louthan and Angerman.1 While only partial agreement exists on the identification of these habit planes, there is a growing consensus that a tensile stress applied on a polycrystalline rod during slow cooling will cause the hydride platelets to lie perpendicular to the tensile axis. walters2 agrees with Louthan and Angerman who reason that "The particular plane on which precipitation will occur is the plane that will result in the maximum relief of stress by precipitation of the low-density hydride." The experiment described below was initiated to test this hypothesis on a single crystal of zirconium. A single-crystalline tensile specimen of zone-refined zirconium, designated number 13 in a previous study,3 had a square cross section with one pair of parallel faces only 4 deg from the basal plane of the hcp structure. The tensile axis was about 86 deg from the (0001) pole, 14 deg from the (1010) pole, and 16 deg from the (1150) pole. The specimen was charged to 50 ppm H by allowing reaction with the correct amount of hydrogen at 1073°K, homogenizing at this temperature for 2 hr, and cooling in the furnace. Two- surface analysis revealed that hydride platelets had precipitated parallel to the three {1010} prism planes. The specimen was heated to 673°K, in air, for 16 hr to bring the hydrogen back into solid solution, and furnace-cooled to 573°K.* It was then stressed in ten- sion at 1.2 kg per sq mm. (This produced a resolved shear stress of 0.6 kg per sq mm for glide on the prism plane with the highest Schmid factor. A higher stress would have produced significant strain at this temperature because the critical resolved shear stress for prism-plane glide in this crystal was only 1.0 kg per sq mm at ambient temperature.) The poles of the three prism planes were at angles of 14, 46, and 74 deg to the tensile axis so that an applied stress would exert normal stresses on the three planes in the approximate ratios 10:5:1, respectively. As soon as the stress was applied, the sample was furnace-cooled to room temperature. It was ground on wet Sic paper and chemically polished in 50 parts HNO3, 50 parts water, and 7 parts HF before examination. The result of the experiment was negative; i.e., the three (1010) planes were of equal importance as hydride habit planes.* It must be admitted that the ex- periment is not entirely comparable to those performed by walters,2 Louthan,5 and Louthan and Marshall6 on polycrystalline materials. They have found that some critical stress (below the macroscopic yield stress) is required for stress-orienting the hydrides. The applied stress in the present experiment was, by necessity, smaller by an order of magnitude than the critical stresses they report for the various materials. If the critical stress was related to the microstructure, one might postulate that it should be decreased by increasing the degree of perfection of the metal lattice and by increasing the grain size. But the experiment above suggests that, for the extreme case of a zone-refined single crystal of zirconium, the critical stress lies above that which one can apply without introducing gross plastic strain as another variable.

Jan 1, 1967

By J. Szekely, G. P. Martins

A formulation is given and a general analytical solution is presented for the problem of spherical phase growth in multicomponent systems. The results are valid for describing the rate of bubble growth in the absence of translational motion and, therefore, should be relevant to heterogeneous nucleation processes. Typical worked examples are given, which include bubble growth due to the simultaneous diffusion of hydrogen and nitrogen in molten steel, and the growth of CO bubbles in steel, due to the reaction: C + 0 = CO. The calculations predict that the presence of additional components would enhance the growth rate and also increase the absolute rate at which the individual components are transferred into the bubble. These findings should be relevant to degassing processes. THE problem of spherical bubble growth in the absence of translational motion is appropriate to the description of the heterogeneous nucleation and subsequent growth of gas bubbles while attached to a solid surface. Such problems are of considerable practical relevance to various vacuum degassing and steelmaking processes where bubble nucleation and subsequent growth play an important part in the overall reaction mechanism. A definitive formulation of spherical phase growth has been given in the now classical paper by Scriven1 and the analogous problem for solid phase growth has been discussed by Horvay,2 furthermore a thorough review of the relevant literature has also become available.3 All this previous work was concerned with one component or binary systems. The purpose of this paper is to show that the technique developed by Scriven may be generalized to alow the consideration of multicomponent systems. The results of this generalization would be particularly relevant, as gas bubble nucleation in the metallurgical systems previously mentioned frequently involves the simultaneous diffusion of several components. In the following a general formulation is given and typical worked examples are presented. FORMULATION Consider the growth of a spherical gas bubble due to diffusion in a quiescent liquid of infinite extent, as sketched in Fig. 1. The formulation of such problems requires the statement of the following conservation equations relating to the liquid phase: Equation of continuity Equation of motion Conservation of the diffusing species—Diffusion Equation. The following assumptions are made: a) The fluid is incompressible, quiescent, and of infinite extent b) The bubble is stationary, i.e., it undergoes no translational motion c) Spherical symmetry is observed throughout d) The mass diffusivity is constant e) Chemical equilibrium exists at the surface of the bubble, and any reactions occurring there are fast compared to the diffusion process f) Inertia and surface tension effects are neglected. Equations derived on the basis of these assumptions by previous investigators were found to provide a reasonable representation of bubble growth while attached to a surface in organic and aqueous systems4,5 For gas bubble growth in molten metal systems assumptions a) to d) chould hold reasonably well. The exclusion of chemical kinetics as a possible rate limiting step may not be universally valid, but should hold for many systems operating at high temperatures. The effect of hydrodynamic inertia may be neglected for moderate growth rates; finally, the assumption neglecting the effect of interfacial tension on the bubble growth was made for the sake of mathematical convenience; its inclusion would have required the use of quite cumbersome numerical techniques. While surface tension effects could be appreciable on the initial values of the overall growth rate, especially at low absolute pressures and in case of small bubbles, they are not thought to modify the relative effects present in multicomponent systems, the study of which is the objective of this paper. Within the framework of the above assumptions

Jan 1, 1970

By Werner Engelmaier

Constant-strain rotating-bending fatigue tests were conducted on 2024-T3 aluminum alloy constant-strain McAdams-type specimens in ultrahigh vacuum, 10-lo Torr, and in atmospheric air. In the elastic strain range the ratio of vacuum-to-air fatigue life varied from about 10 at a maximum bending stress of 37 ksi to near unity at the endurance limit. After 10 pct of the fatigue life, intergvanular microcracks appeared on the surface which did not appreciably propagate further until the last 5 pct of the fatigue life, where trans-granular cracks propagating first in slip-plane cracking and then in the stress-dominated mode led to failure. No reduction in residual tensile strength resulted from the fatigue of the specimens short of fracture. The mechanism governing the fatigue life operates between the occurrence of the intergranular microcracks and the initiation of the trans granular cracks. MORE than forty years ago, Mc Adam' and Haigh2 published the first systematic studies showing that the fatigue properties of metals were greatly affected by environments. Gough and sopwith3 found in a later study that the fatigue life of metals subject to corrosion can be substantially increased by reducing the atmospheric pressure. This and their later work4'5 was, however, done in a vacuum of only 10~3 Torr. Subsequent studies6-' at a vacuum of 10- Torr showed that water vapor is the primary agent affecting the fatigue properties of aluminum alloys. Only recently was ultrahigh vacuum used for fatigue studied.l Various theories have been put forward to explain this phenomenon. Wadsworth and Hutchings12 postulated that cracks form very early in the test and that the chemical attack by the corrosive components of the atmosphere at the root of the cracks enhances the speed of the crack propagation. They also noted that chemisorbed layers of gas on the fresh crack surfaces prevent cold welding of the crack and make the separation irreversible. Ham and Reichenbach, on the other hand, reported that their specimens showed no cracks after a run in vacuum, even though the runs were long enough to have caused the specimen to fracture in atmospheric air. They also reported that in argon at 0.14 Torr the fatigue life was about 5 times longer than at 7 X lo-' Torr. Jacisinll reported that fatigue tests in dry nitrogen showed results close to those obtained in ultrahigh vacuum. The present experiment was designed to determine the extent to which cumulative fatigue damage, i.e., the accumulation of length of fatigue cracks, reduces the residual static tensile strength of an aluminum alloy in both ultrahigh vacuum and atmospheric air and to obtain fatigue-life data for this aluminum alloy at various strains. It was also hoped that information ex- tracted from these data might support one of the previously mentioned theories. 1) EXPERIMENT Test Specimens. The material tested was commer-cial 2024-T3 aluminum alloy. In addition to aluminum, this alloy contains approximately 4.4 pct Cu, 0.5 pct Fe, 0.5 pct Si, 0.6 pct Mn, 1.5 pct Mg, plus smaller amounts of other elements.13 Some typical mechanical properties of this aluminum alloy are listed in Table I.'~ The material was used in the as-received condition, i.e., solution-treated and cold-worked. Mc Adam-type1' constant-stress specimens and notched fatigue specimens were machined from 4 -in. -diam rods, Fig. 1. The critical section of the McAdam-type samples was polished with a 45-deg spiral motion to a No. 8 microfinish. The notch in the notched specimens was machined on a lathe with a forming

Jan 1, 1969

By E. Miller, K. L. Komarek

Equatiorzs have been derived which related maximum solid solubilities in semicondcting compounds having retvograde solidus curves with 1iqllidus data. Liquidus curves in Lke Pb-Te, Pb-Se, ard PB-S systetris have been defermined by thertrzal aralyses. Fvotrz expevimental maximum solibilities ia CdTe, PbTe, uxd PbS vetvograde tetrperatures hale beep1 calculated. TIze resllts support the concept that under certaiz conditiots pronounced itjlections in the liquidis curves arc indzcatiz:e of retropade solidus curves and that the tenlperatltre of the lmaximum solid solubility will be close to the temperature of the point of inflection in the liquidus curve. The maximum solid solihility within a gvoup of homologous cottrpoods SeetlLs to increase with increasing flatness ol the liquidus curvres. The phenomenon of retrograde solubility in metals, i.e., maximum solubility in a solid terminal phase in equilibrium with a liquid phase, has been theoretically predicted by Van aar' in his calculations of hypothetical binary phase diagrams. It was not until 1926, however, that retrograde solubility was experimentally verified in the Zn-Cd system.' Since then many sys- tems with retrograde solubility have been found, so that at present approximately fifty binary diagrams with a retrograde solidus curve are known.3 Meijering4 has critically appraised the accumulated data and has presented calculations which alloy predictions as to whether or not a solidus line will be retrograde. Interest in retrograde solubility increased when it was found that the solidus curves of germanium and silicon alloyed with a great many doping elements are retrrade. The problem was treated thermody-namically by odkinson,% d all available data on solid solutions in germanium and silicon were summarized by Trumbore.7 Although it is to be expected that intermetallic compounds will also exhibit retrograde solubility, experimental evidence is much more limited. Until now retrograde solidus curves have been reported for the compounds de,' be, Ps:" ne," SnSe," and SnS.13 All these compounds are semiconductors, and the solidus curves were usually obtained by determining the carrier concentration at room temperature of samples equilibrated at elevated temperatures and then quenched. It has been hown,'' however, that it is not always possible to quench in the equilibrium compositions from above a certain temperature. Theoretical discussions have either explicitly excluded intermetallic compounds4 or not treated them with respect to retrograde slubilit.j In the following a thermodynamic treatment of retrograde solu-

Jan 1, 1967

By G. C. Wallick, W. R. Foster, J. G. Savins

A comprehensive review of the salient features of the differentiation method of rheological analysis in Poiseuille flow from its inception circa 1928 is presented. Here no initial assumptions regarding the nature of the function relating rheological parameters to observed kinematical and dynamical parameters are required in the data-analysis process. In contrast, the integration method involves interpreting flow properties in terms of a particular ideal model. It is shown that, although both methods represent modes of solution of the same integral equation, being relatively bias-free, the differentiation method offers a more discriminating procedure for rheological analysis. The application to problems involving plane Poiseuille flow is also described. INTRODUCTION In most instances, the approach to the problem of interpreting the rheological properties of various compositions as they are affected by changes in chemical or physical environment, assaying the characteristics of a particular constituent of a suspension, analyzing flow behavior in terms of interactions between components in a system, to cite but a few examples, has been in terms of what Hersey1 terms the integration method. Briefly, it consists of interpreting flow properties in terms of a particular ideal model. The usual practice of the integration method is to choose a model with a minimum number of parameters because, other things being equal, it is desirable to use the simplest model which will describe the behavior of a real material and yet be mathematically tractable for the requirements of data analysis. This expression is then substituted into an equation which relates observed kinematical and dynamical quantities, such as volume flux Q and pressure gradient ], and angular velocity and torque T, in a capillary and concentric cylinder apparatus, respectively. The rheological parameters appear on integrating, in an expression relating the pairs of observable quantities such as those just given. In many instances a particular model provides a good representation of rheological behavior over a reasonable range of compositional and environmental changes. Just as often, however, it is obvious that the interpretation of rheological changes by the integration method is not providing realistic information about changes in flow behavior. A more general method of interpreting rheological data for a given material is to make no initial assumptions regarding the nature of the function relating rheological parameters to observed kinematical and dynamical quantities, e.g., flow rate and pressure drop in capillary flow or angular velocity and torque in a rotational viscometer. This general method Hersey1 terms the differentia-ation method. Instead of integrating, one differentiates the integral equation with respect to one of the limits, i.e., one of the boundary conditions; the resulting expression contains the same observable quantities just given, their derivatives, and the rheological function evaluated at that boundary. By obtaining these derivatives from experimental data, graphically or by a computer routine, they can be substituted into the differential equation and a graphical form of the function derived. THEORY OF THE DIFFERENTIATION METHOD FOR POISEUILLE-TYPE FLOWS In this introductory paper, two flow cases which are important in viscometry are considered (one for the first time) from the differentiation method of analysis, flow in a cylindrical tube and flow between fixed parallel surfaces of infinite extent, the basic integral equations being formulated in a manner analogous to the way they originally appeared in the literature. In addition, the following ideal conditions will be assumed: (1) an absence of anomalous wall effects, (2) isotropic behavior everywhere, and (3) steady laminar flow conditions. Before proceeding it would seem appropriate to preface the analysis of these flow geometries with the comment that the basic philosophy of the differentiation method could be said to stem directly from the efforts of Herzog and weissenberg2 to indicate systematically the various ways of using

By N. A. D. Parlee, N. M. El Tayeb

Fe-V alloys with small percentages of vanadium show no deviations from Sieverts' Law up to P~, = 1 atm in the 1600º to 1750ºC region. At somewhat under 8 pct V and up to at least 20pct V, at 1604" and 1661 °C, nitrogen absorption follows Sieverts' Law at low Pn2 values, but deviates from this law beyond critical values of pn2, due to formation of a nitrzde phase of approximately VN composition. The behavior below these solubility limits can be described at all temperatures by the equations: RAO and parlee1 suggested that the nitrogen absorption curve for liquid iron alloys takes the general form shown in Fig. 1 when an alloy nitride precipitate is formed. They showed that the precipitation of 6 titanium nitride from liquid Fe-Ti alloys follows this general scheme and dealt with the general theory involved. Recently, Evans and Pehlke2 reported their research on the precipitation of aluminum nitride from liquid Fe-A1 alloys. Fountain and Chipman's recent paper3 on the precipitation of boron nitride from solid solution contains a fine example of the analogous behavior which they have been studying in the solid state. Researches on the absorption of nitrogen by liquid Fe-V alloys have been described by various authors.1,4-7 Most of this work has been confined to fairly low vanadium concentrations and rather high temperatures because of difficulties variously described as, formation of surface scums, bridging, and so forth. This behavior and the rough thermo- dynamic calculations made by chipman,8 over eleven years ago, indicated that excessively high vanadium concentrations should not be required for vanadium nitride precipitation. It seemed desirable to look for and study this precipitation in order to fill out the thermodynamically interesting nitrogen absorption data on these alloys. EXPERIMENTAL METHOD The general method used was essentially the same as employed by Rao and parlee,1 in which a Sieverts type of apparatus was used and the precipitation of the nitride detected by the "break-point" in the Sieverts' Law line, and by the visual observation of the formation of the new phase on the surface of the melt. There were, however, a few differences in apparatus and technique as described below. The oil manometer used by Rao and parlee' to measure low partial pressures of nitrogen was not employed. The present authors sought to avoid as much as possible the exposure of the liquid alloy to low total pressures of gas during the 6 to 10 hr long runs because of the heavy iron evaporation-loss errors involved. This was achieved, partly by keeping evacuation times to a minimum with fast pumping but largely by the use of two burettes, one for argon and one for nitrogen, and using them to equilibrate the alloy with mixtures of argon and nitrogen rather than nitrogen alone. Each run involved a careful sequence of operations. After melting, de-

Jan 1, 1963

By C. H. Mahoney, P. E. LeGrand, A. L. Tarr

MAGNESIUM, it is now generally realized, differs in some important aspects from most other structural metals, not excepting even its close neighbors, the aluminum-base alloys. This is particularly true with respect to the effects of elevated temperatures on the two light metals when in the molten state. Aluminum, on the one hand, should not be raised above its melting point any higher than is absolutely necessary if one is to avoid danger of gas absorption and a tendency toward grain growth in the resultant casting.1 Magnesium alloys, in sharp contrast, actually benefit by being heated to temperatures considerably above their melting point. It is a commonly accepted practice to establish the desired fine grain in the industrial magnesium alloys of the aluminum-bearing group by raising the temperature of the melt after the refining treatment to some 25e C (482°F.) above the melting range. To recall briefly the typical practice for magnesium alloys, the molten metal is refined by fluxing at about 730° C. (1350 F.), and after the melt has been covered with a suitable flux, the temperature is raised to between 870°C. (1600°F.) and 930°C. (I700°F.). It is then cooled as quickly as possible to the casting temperature. This superheating, as it is called, results in a fine-grained cast metal structure which has superior mechanical properties, enhanced amenability to solution heat-treatment, and better machinability than cast metal that has not been superheated. The minimum temperature of superheating varies to some extent with the composition of the alloys to be treated and usually increases with the percentage of magnesium in alloys containing the same constituent metals.2 It should be noted that the grain-refining effect of superheating is markedly apparent only in connection with the alloys containing aluminum as an alloying constituent-the binary alloy containing r.5 per cent manganese, for example, is not refined in grain to any noticeable extent by a superheating effort. While the phenomenon of superheating is well known in the magnesium industry, the mechanism of the process has remained a mystery. During the superheating cycle something occurs that forces a fine grain upon the solidifying metal. The process has been controlled in an empirical manner, and occasionally, especially on large-scale melts, and for no apparent cause, no grain refinement is obtained from a superheating treatment. Only a few papers of importance have been published on the subject. Of these, one of the most helpful is the paper by Achenbach, Nipper and Piwowarski,3 which gives considerable data concerning the effect of superheating on various properties such as fluidity, shrinkage and hot crack formation, as well as grain size, and a discussion without any definite conclusions on the theoretical aspects of superheating. A short article by Murphy, Wells and

Jan 1, 1945

By Fred H. Poettmann

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

Jan 1, 1951

By Fred H. Poettmann

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

Jan 1, 1951