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By J. S. Kirkaldy, W. V. Youdelis

THE occurrence of concentration distributions in rapidly cooled castings with gradients of opposite sign to those expected on the basis of the constitution diagram and diffusion controlled kinetics, has long interested metallurgists. The work on inverse segregation to 1937 has been adequately reviewed by Vaughan, and to 1949 by ell-walpole. The general conclusion for unidirectional solidification, that the enrichment of the initially solidifying regions with lower melting point compositions is due to the flow-back along interdendritic channels of normally segregated residual liquid, has a very sound experimental basis. Perhaps the best experimental work and the most complete qualitative description of the process has been given by Adams.3 The authors accordingly assume the reader's understanding of the content of this paper. It appears that one of the principal deterrents to the complete understanding of the process has been the lack of an analytical description. Although a number of workers including Masing,4-6 Sauerwald,7 and Adams3 have attempted quantitative predictions,

Jan 1, 1959

By E. A. Sondreal, P. H. Tufte

Flue gas desulfurization when burning western U.S. coals can be expected to be easier to achieve and less costly than when burning higher sulfur eastern and central coals. However, western coals cannot generally be burned in compliance with federal and state new source emission standards without some degree of control. Western coals and lignite represent about 20% of total U.S. recoverable reserves with 90% of that strip minable, and their production is predicted to increase from the 60 million tons mined in 1973 to 300 million tons in 1985. Most will be used for electric power generation, where low sulfur content is a paramount concern. Western coals are predominantly below 1% sulfur; low compared with typical values of 2 and 3% for eastern and central coals. Coal cleaning for removal of sulfur is not commercially practiced on Western coals, which typically contain half organic and half pyritic sulfur. A fraction of the pyrite sulfur could possibly be removed, but the

Jan 1, 1976

By Y. P. Chugh, Rajiv Singh

This paper presents the feasibility analyses of terrace pit mining a single seam for depths beyond 33 m (110 feet) in a hypothetical model mine and at an operating mine in southern Illinois. One or two shovel- truck benches near the surface and a large dragline for the last bench are considered for overburden stripping in the proposed mining scheme. Mining and reclamation operations are integrated and reclamation efforts are minimized by proper spoiling techniques. The proposed mining scheme was optimized using the OPMHS (Open Pit Material Handling Simulator) model developed by Penn State. Equipment selection and economic analyses were also conducted using the simulation model. The effect of pertinent variables such as thickness of the coal seam, annual production, initial capital requirements, size of haul trucks, selling price of coal and expected rate of return were analyzed on the feasibility of terrace pit mining. Sensitivity analysis charts for the effect of these variables are presented in the paper.

Jan 1, 1983

By Orville R. Lyons

THE removal of moisture from coal has been a coal-preparation problem ever since the first wet-washing preparation plant was placed in operation. Today, when most of the coal produced in the United States is wetted at the working face when mined, or is cleaned by wet methods, the removal of moisture is a necessity for any one or a combination of the following reasons; (1) to avoid freezing difficulties during shipment, (2) to increase the B.t.u. content, (3) to effect savings in freight costs, and (4) to so improve the coal that it may be used for specific purposes-for instance, in the production of briquettes or the production of metallurgical coke-or so that it can be treated by dry cleaning methods. In general, the amount of water retained by coal particles varies with the size-consist of the coal. Fine coal has a larger surface area per unit weight than does coarse coal, so its capacity for retaining moisture is proportionately greater. The removal of moisture from coarse coal is relatively easy, while the removal of water from minus 10-mesh coal or finer is a major problem, usually requiring an individual solution at each preparation plant. As a result, a multitude of methods and a variety of equipment have been designed to solve both general and specific moisture-removal problems. Definitions In order that misunderstanding may be eliminated insofar as possible, it is necessary to define the terms to be used. The moisture content of any given increment of wet, washed coal is made up of two parts, inherent and surface moisture. Inherent moisture has been defined in various ways and as yet no standard definition has been agreed upon. However, in general, it is considered as the moisture that is present in the coal in the bed. The important point to remember is that coal with any amount of inherent moisture, whether 2 or 20 per cent, will appear dry if there is no surface moisture. Also important is the fact that inherent moisture can be removed, in part, by air drying at atmospheric temperatures. Surface moisture is that attached to the surface of the coal particles or that retained in cracks and fissures other than capillary openings in the coal substance. Essentially, insofar as the bituminous coal and anthracite indus-

Jan 1, 1950

By T. A. Boyden

The Free World uranium market is now characterized by production nearly twice that of consumption result- ing in a rapid build-up of uranium inventories. Low- cost production from high-grade deposits in Australia and Western Canada will gradually add another 10 to 15 million pounds U308 annually into Free World production throughout the 1980's. This latter material cannot be absorbed in further inventory build-up and can only displace higher cost production. The result is a restructuring of the Free World producer industry. United States and Eastern Canadian producers have the higher cost production which gradually will be displaced from the market during this decade. As United States production declines, United States utilities must increase their reliance on foreign supplies in order to meet their needs. This paper reviews the elements that will cause the restructuring of the Free World producer industry and the expected response of the United States utility industry to that developing situation.

Jan 1, 1982

By Donald N. H. Chi, Henry E. Perlee

Accident cost data bases for all mining sectors covering the period 1975 through 1980 have been created on the PDP-11/70 and VAX-11/780 computers at the Bureau of Mines Pittsburgh Research Center. The total expected costs of the following tangible elements are computed: (a) loss in personal income; (b) compensation of wages from State, Federal, and union funds for disabling injuries; (c) benefits for injuries resulting in death or permanent disability; (d) medical treatment and hospital care; (e) investigation of a fatal accident; and (f) production loss due to accidents involving fatalities or amputation injuries. Each cost record computed contains a reference to the original record in the Health and Safety Analysis Center (HSAC) accident files to permit concurrent analysis. The intended applications include examinations of annual trends in mine accident statistics on the basis of economic loss. This provides an additional perspective on mine accidents. Various examples of the use of these data bases are presented.

Jan 1, 1983

By Charles H. Mitchell

British Columbia is enjoying a period of sustained high-level activity in all aspects and sectors of its mining industry. The total value of mineral production for 1962 was in excess of $229 million. This was the greatest total value attained for any year to date, exceeding the previous record year of 1956 by more than $40 million. It appears reasonable to expect that value of production for 1963, stimulated by increased production attained during the year, and by improved metal prices for some of the major products, will be substantially in excess of that recorded for 1962. Elsewhere in this issue appears a detailed report of prospecting and exploration activity in the Province. For the purposes of this review it is sufficient to state that the level of prospecting and exploration activity during the past several years has been greater than at any previous time and that there does not appear to be any slackening in activity or interest.

Jan 12, 1963

By John R. Dyni

Colorado nahcolite deposits, estimated at 29 Gt, formed in an asymmetric permanent-lake basin whose waters were rich in algae and sulfate-reducing bacteria. Nahcolite (NaHCO3) occurs in oil shale of the Green River Formation (Eocene) as aggregates (62%), disseminated crystals (24%), and impure beds (13%). Because much of the nahcolite is nonbedded, it will be mined with oil shale and can be concentrated to marketable grade by crushing and screening. Impurities in nahcolite are kerogen, quartz, pyrite, and carbonate minerals. A promising new market for nahcolite is in the process of removal of sulfur dioxide from industrial stack gases. When heated, nahcolite reacts with sulfur dioxide to form sodium sulfate. Differences in thermally induced microporosity may explain why nahcolite is more reactive than trona, a Possible competitor with nahcolite for SO2 removal. Limited production of nahcolite at one or possibly two localities is expected in the next several years, but large-scale production is probably at least 10 years away.

Jan 1, 1982

By Alan K. Burton

The number of trucks required to perform a certain task is a function of the productive capability of one independent truck and the total tonnage required. Hourly productive capability of one truck is the product of trips, or cycles, per operating hour and effective payload. One cycle is made up of load, dump, and travel time, the sum of which comprises theoretical cycle time. This time is then derated by a productivity factor, giving an average effective cycle time which is used to calculate the number of cycles per hour. Rated payload is the truck's designed tonnage, while effective payload is the average amount that is actually carried. Manufacturers, particularly of the larger trucks, produce various body sizes designed to carry materials of different densities. However, it is not always possible to carry the rated payload because the material might be lighter than expected, a body of the required size cannot be produced, or because of spillage on particularly steep grades.

Jan 12, 1975

By D. Murgue

The circulation of air in underground workings is subject to a gradual and continuous reduction of its pressure, from intake to outlet, caused by the friction between it and the more or less rough and uneven surfaces of the passages. For comparison with the flow

Jan 1, 1894

By W. E. D. Stokes

THE application of aviation to mining and petroleum operations, on the basis of economy and attainment, has become a demonstrated fact. According to Dominion Government records, 30 Canadian companies engaged in air transportation with northland mining camps, carried 13,000 tons of freight in 1935, and in 1936 carried 16,000 tons. In addition, many mining companies used planes to transport prospectors with canoes, outboard motors, gasoline drums and supplies to remote locations, and one company maintained a plane to take radium-bearing ore from its mine on Great Bear Lake nearly 1000 miles south to the rail- way at Waterways, Alberta. Heavy pieces of mining machinery weighing 2 tons are being carried by Ford trimotor planes over the Andes, at an altitude of not less than 15,000 feet. At Bulolo, New Guinea, 17,500 tons of disassembled gold-dredging equipment has been flown over inaccessible terrain, while supplies and building materials for a community of 1000 whites have been transported by airplane. A 3 1/2-ton shaft was carried on one notable trip, while at another time an automobile, 2 tons of rice and a large safe went over the mountains in one plane. Fig. 1 shows an electric generator that was carried by plane to Bulolo. Many large American oil companies operated fleets of planes during 1936, ranging from fast, luxurious air liners for executives to elaborate geological prospecting units, ambulance planes, freight carriers and pipe-line repair crews. According to the U. S. Geological Survey, commercial firms photographed 16,291 square miles during 1935, while the Government compiled from aerial photographs by the Geological Survey in the United States 85,544 square miles. It was found that information could be determined in a fraction of the time required by surface methods. However, sounding a word of caution, a Commonwealth of Australia government report on aerial survey operations had this to say: It is necessary to repeat ad nauseam that aerial photographic survey is in no sense a substitute for geological investigation, though it is a very valuable adjunct thereto, and greatly facilitates and accelerates geological mapping. There are many details

Jan 1, 1937

By H. H. Stoek, G. W. Harries

The term " Anthracite Coal-Field " is generally used to refer to a comparatively small territory lying in the eastern-central part of Pennsylvania. This territory includes about 3,300 sq. miles of area, located chiefly in five counties in which anthracite-mining is the dominant industry. Although an area of 3,300 sq. miles is embraced in the region, less than one-fifth of this area, or only 484 sq. miles, is underlain by workable coal-seams. Moreover, this prorluctive portion is not a continuous area, but consists of a number of detached basins. These are known as the Northern, containing 176 sq. miles; Eastern-Middle, containing 133 sq. miles; Southern, containing 181 sq. miles; and western-Middle,containing 194 sq. miles. From the standpoint of business and trade, the region is divided into three districts, known as (1) the Wyoming, which includes the Northern field and the small Bernice basin; (2) the Lehigh region, which includes all of the Eastern Middle field and that part of the Southern field east of Tamaqua; and (3) the Schuylkill region, which includes the WesL ern-Middle field and that part of the Southern field west of Tamaqua. The relative commercial importance of these districts at the present time is shown by the coal-shipments as given in the following table :

Jan 1, 1904

By Walter Harvey Weed

CONTENTS. [ ] INTRODUCTION. THIS paper deals with certain ore-deposits whose structural features or mineral contents (or both) result, directly or indirectly, from igneous intrusions and their after-effects. It is largely a discussion of contact metamorphic ore-deposits based upon the physical changes in rocks due to contact action. It involves a classification of ore-deposits only so far as is abso-

Jan 1, 1913

By L. C. Graton

I. INTRODUCTION. The Relations of Scale in Geologic Work. MANY features of human accomplishment rest upon the possibility of representing natural objects on a scale of different magnitude from the actual one. Most drawings, for example, are of very different scale from that of the object depicted, and indeed are useful mainly on this account. More important than the question of mere convenience, however, is that of meeting the physical limitations of vision.1 Thus, a mural

Jan 5, 1913

By George M. Fowler, Joseph P. Lyden

PaGE Scope of this Report.................................................... 206 Production of Tri-State District.......................................... 207 Historical Sketch....................................................... 207 Location and Topography............................................... 211 Geology............................................................... 212 The Tri-State District.............................................. 214 Stratigraphy........................................................... 217 General Geological Section in Tri-State Region....................... 217 Stratigraphic Section of Mineralized part of Boone Formation........... 218 Stratigraphic Section Bird Dog Well.................................. 219 Metamorphism........................................................ 220 Structure............................................................. 225 Regional.......................................................... 225 Due to Metamorphism.............................................. 229 Ore Deposits.......................................................... 231 Types............................................................ 232 Hard-ground Deposits.......................................... 232 Brecciated................................................. 232 Stratified............................................ 232 Sheet Ground............................................. 232 In Dolomite............................................... 233 In Shale.................................................. 233 Soft-ground Deposits.......................................... 233 In the District as a Whole....................................... 233 Ore Minerah and Analysis......................................... 233 Paragenesis....................................................... 234 Conclusions........................................................... 236 Structure.......................................................... 236 Genesis of the Ore.................................................. 237 Future of the District............................................... 239 Acknowledgments..................................................... 239 Discussion............................................................ 239 Scope of This Report The Tri-State district. as outlined in this paper. refers to the entire mineralized area in sout'hwestern Missouri. southeastern Kansas and

Jan 1, 1932

By J. M. Hassler

NOWHERE can there be found a more misleading statement than the old one that "Iron can be manufactured cheaper in the South." During the past decade ironmakers and users of iron have heard varied and sundry stories of the wonderful resources of the Birmingham district, including abundant ore and coal fields, cheap labor, a wonderful climate and similar industrial Utopian statements. Birmingham represents the major portion of the iron business in the South, acid as Birmingham fares so goes the iron business of the South. The intent of this paper is to shed some light on the actual conditions surrounding ironmaking in the South, especially in the Birmingham, or North Alabama, district. By a presentation of facts it is desired to point out that the "cheap iron" of the industrial South is a thing of the past, primarily because of increased labor cost and because labor is radically different in performance to what it was years ago. Also, because most of the raw materials needed in ironmaking that would be classed better than inferior are gone.

Jan 1, 1937

By J. S. Aronofsky

The first step in a quantitative analysis ot the mechanism of oil displacement by water in a fractured reservoir is usually conceded to be the solution of the differential equation describing the saturation distribution of two immiscible fluids flowing in a porous medium, where the capillary pressure is taken into account. In such a system the production mechanism may consist of displacement of oil both by the flow of water due to natural or artificially imposed pressure gradients and by imbibition, which implies a flow of water not due to external pressure gradients. Owing to the presence of the two oil displacement mechanisms, the mathematical model given by the differential equation intended to describe the system may not properly represent the behavior of the physical system. In fact, in the reservoir the rate of water advance may be very slow, and in the case of a fractured reservoir with a great number of large fractures, the pressure difference determining the flow of water through the matrix may be much less than 1 lb/'psi over lengths of a few feet. In such a case, imbi-bition (the exchange between oil in the matrix and water in the fractures resulting from capillary forces) may become, with time. a significant element of the production mechanism. It occurred Lo the authors, however, that without going into a physical analysis of the process of production, it might be possible by means of simple abstract reasoning to throw some light on the variation of recovery with time under conditions occurring in a highly fractured oil reservoir with rising water table. The object of this paper is to present both the reasoning and its application to a reservoir of the highly fractured type. Specifically, the analysis given here was undertaken to try to explain the increase of recovery (as defined later) with time as observed in this reservoir, without having to assume unlikely variations in the reservoir parameters with depth. This attempt has been successful as will become clear upon comparison of the computed recoveries with the actual field data. ABSTRACT MODEL Let us consider a small volume. of porous matrix saturated with oil at time, t = 0. Let the process of oil displacement by water start at time, t = t,,. At some time, t, the process will have terminated. Then a volume of oil equal to or smaller than the original oil contained in the matrix will have been produced. The first basic assumption that describes the model and guides the forthcoming reasoning is that the oil production from the small volume. dv, is a continuous monotonic function of time and that it converges to a finite limit. Such an assumption is not inconsistent with the results of laboratory waterflood tests as well as with results of imbibition tests where this is, in fact, observed. The second basic assumption is that none of the properties which determine the rate of convergence change sufficiently during the process to affect this rate or the limit. Let it be assumed that the form of the function of time relative to production from the matrix volume. dv, is given by where V,(t) is the volume of oil produced up to that time t. R is the limit toward which the recovery converges, A is a constant giving the rate of convergence, and V,(t) is the volume of oil originally in place in the volume, dv. It should be noted that recovery at time I will be understood here to he It follows that r. tends to R as t tends to infinity. CONSTRUCTION OF RESERVOIR FROM ABSTRACT MODEL Let the reservoir consist of a series of identical blocks of porous matrix stacked vertically and separated by fractures. Let each of these blocks satisfy the conditions of our abstract model. These conditions are: (1) that recovery is a continuous mono-tonic function of time converging to finite limit, and (2) none of the properties that determine the rate of convergence change sufficiently to affect the rate or the limit. Let water be rising in fractures so that oil production from any part of the block starts when the water comes in conLact with it. TOTAL RECOVERY FROM THE RESERVOIR COMPARED TO RECOVERY FROM A SINGLE MODEL ELEMENT As stipulated in Eq. 1, in the case of the abstract model,

By K. E. Beu

IN the measurement of retained austenite concentrations in steels using the integrated intensity method,1 Averbach has pointed out" that the absorption factor A(0) for a flat sample making a glancing angle 4 with the incident X-ray beam can be combined with his constant factor R to obtain our constant factor" G provided that "1—There is no preferred orientation in the sample, and 2—The geometric requirements [for the sample, film, and X-ray beam] have been met precisely."' If A(0) can be combined with R to give G according to the equation G = R . A(0) [14] then the possibility for making non-compensating errors in the austenite determination has been eliminated. This has been described previously in a technical note." Because of the brevity of the previous note,3 it was impossible to emphasize the fact that the two conditions on preferred orientation and geometry can be easily met experimentally, contrary to Aver-bach's statement that: ". . . the necessary conditions must be tested experimentally for each determination, and this is done most easily by observing whether the apparent absorption has the form of Eq. 2 [the theoretical equation for A(0)]."" he way in which these two conditions can be met experimentally will be discussed briefly to help clear up this point. In addition to these two conditions, other factors such as sample shape, homogeneity, and grain size which also affect A will be included in this discussion. A (0) depends on sample shape. The theoretical function for A was derived originally for a flat surface.' In general we have found that a flat sample surface is readily obtainable." If such a surface can * The effect of surfaces which are not flat on the measurement of retained austenite will be discussed later. be obtained, it has the following advantages: 1—it is easily reproducible from sample to sample, 2—it is the form required for metallurgical examination —this type of examination being frequently desirable for this work, and 3—it is an efficient shape for diffraction purposes. Perhaps the most important of these features is that a flat sample surface is easily reproducible; hence, this requirement on the repro-ducibility of A from sample to sample is met for all such samples. Fig. 1—Schematic diagram of the quartz crystal monochromator diffraction unit. The centerline of the main beam is at 6' to the target face. The tangent to the crystal face is at 16.8' to the moin beam. The centerline of the monochromatic beam is at 33.6 to the main beam. The sample surface can be rotated in its own plane. The angle of the sample surface and the monochromatic beam can be adjusted by rotating about the vertical axis, The film holder can also be rotated about B so that the film can be exposed over the desired angular range. For Fe K, X = 1.932A. 1011 planes of quartz have d = 3.35A. A(0) depends on homogeneity and grain size. If the sample is badly segregated or the grain size is large, the effect of micro-absorption and primary extinction1. ' must be considered. It has been shown, however, that for most plain carbon or low alloy hardened steels, neither micro-absorption nor primary extinction effects are present.' A (0) depends on camera geometry. For a given angle 0, A remains theoretically constant from a geometrical viewpoint only if the following factors are kept constant: 1—the angle of inclination .+ of the sample to the X-ray beam, and 2—the centering of the sample with respect to the film cylinder. These are mechanical problems which can be solved readily if the facilities of a good machine shop are available. The arrangement used to insure that the angle 4 remains constant and the centering of the sample is reproducible is indicated schematically in Fig. 1. The sample is clamped against a thin plate with a hole in it by means of a spring-loaded pres-

Jan 1, 1954

By D. D. Smith, D. P. Squier, E. L. Dougherty

A system of differential equations describing the temperature behavior of fluid injected at constant surface temperature in a well is derived and .solved analytically. A formula for the fluid temperature at any time and depth is given, as well us a special formula valid for very large times. These formulas are used to calculate temperatures for several typical cases. The results indicate that, initially, the temperature of the water entering the formation is considerably lower than the injection temperature. This condition lasts for only a short period— less than three days for most cases of practical interest. Following this highly transient period, during which the temperature of the fluid entering the formation builds up to about 50 to 75 per cent of the injection temperature. the system enters a quasi-steady state in which the temperature changes are very slow. After severl years, the bottom-hole temperature will still be 50" to 100°F lower than the injection temperature, hilt the heat losses may he tolerable. INTRODUCTION Predicting the behavior of a hot-water flood requires that the temperature of the water entering the injection interval be estimated. This report describes the development and solution of a system of equations which describes the temperature behavior of the injected water in the wellbore with certain simplifying assumptions. The only previous means known to the authors for describing such a process is that of Moss and White.' Their results appear to be close to those obtained by our method in the practical cases which were compared; this agreement is largely due to the fact that in our method temperature soon approaches a quasi-steady state, as was assumed in their method throughout. However, our model covers all times, is continuous (whereas the Moss-White model depends on breaking the depth into discrete intervals) and. we feel. more closely describes the physical problem. FORMULATION OF THE PROBLEM PHYSICAL SYSTEM AND ASSUMPTIONS The injection procedure consists of pumping water at a fixed surface temperature T., down an infinitely long cylindrical well or tubing of inner radius Any material exterior to the water column such as mud, casing, or cement is regarded as part of the formation. The general behavior of the system may be described qualitatively as follows. When the hot water is first introduced into the system, the temperature difference between the formation and the water is large, resulting in a high rate of heat transfer. As a result, the temperature adjacent to the wellbore rises very quickly. Because the segment of the formation adjacent to the wellbore largely controls the heat transfer rate, the heat transfer rate will become relatively constant when this portion has reached a temperature close to that of the water opposite it. The temperature of the water and formation then increase very slowly with time. The length of the initial highly transient period and the temperature of the water at its conclusion will be functions of depth, injection rate, injection-string radius, surface injection temperature and the physical properties associated with the water-formation system. The following additional assumptions were made. 1. There is no heat transfer by radiation in the system. 2. There is no heat transfer by conduction in the vertical direction in either the injection stream or the formation. 3. The linear volumetric and mass flow rate of the water is constant throughout the injection stream. 4. No horizontal temperature gradient exists in the injection stream. 5. The product of density and heat capacity is constant for both the water and the formation, and the formation thermal conductivity is constant. 6. Initially, both the water in the wellbore and the reservoir are at a temperature given by the (constant) ambient surface temperature plus the product of depth and geothermal gradient (assumed constant). At large distances for the wellbore (r m), the formation will remain at this temperature. 7. The water temperature and the formation temperature at r — r,, are equal for all depths D. DERIVATION OF EQUATIONS The differential equation satisfied by the fluid temperature T,(D, t), which is obtained by writing a heat balance on a cylindrical differential of volume dV of the injection string between the depths D and D i dD, is

By Paul G. Shewmon

The migration of slightly solzrhle spherical particles through a solid under the infllrence of a temperature gradient is analylzed for the cases of various transport mechanisms. It is shown that the variation of the velocity of the particles with radius, r, depends on the dominant mechanism of matter transport around or through the inclusion. Thus the velocity varies as r-1 jor surface-dijf1~sion controlled migratzon. is independent of r for volume diffusion in either phase, and varies as if the rate is determined by an interfacial reaction (n is the order of the interfacia1 reaction). These same results hold tor the migration of a cvlinder of length much greater than its radius. and .for other types of potential gradients, e.g., an electrical field. These equations are combined with recent electron-microscopic ohservations to show that the rate of migratzolz of small helium-silled bubbles through copper is determined by surface diffusion of the metal atoms. With these equations and the temperature gradients attainahle in an electron-microscope joil, the dominant transpar/ mechanism (or any migrating pnrtzclos can he determined. AS a result of fission, rare-gas atoms are created in the hot fuel element of a nuclear reactor. These insoluble atoms precipitate to form bubbles of the gas in the fuel. The subsequent migration and coalescence of these bubbles is thought to play a dominant role in the swelling of fuel elements which in turn can limit the fuel-element life. As a result of this practical problem and because of the relative simplicity of the system, workers in several laboratories have imbedded rare-gas atoms in metals with the aid of an accelerator and studied the formation and behavior of the bubbles that form on annealing. Recently Barnes and Mazey have studied the migration of such helium-filled bubbles in copper foils using the electron microscope and the temperature gradient that can be induced in the foil by the electron beam. They found that the smaller bubbles moved faster than the larger ones. It is shown below that, if this is true, the rate-controlling step in their migration is surface diffusion of metal atoms instead of volume diffusion, vapor transport, or an interfacial reaction. The analysis given is in no way limited to gaseous inclusions so the variation of velocity with particle size for any relatively insoluble precipitate particles could be used to obtain information about the relative importance of surface diffusion, volume diffusion, and interfacial reaction in such two-phase systems. VOID MIGRATION IN A TEMPERATURE-GRADIENT ANALYSIS Barnes' studies indicate that helium does not dissolve or diffuse in metals to a measurable extent, so that when the voids move they must do so through the movement of metal atoms from the leading to the trailing sides of the void.' Thus voids might move by surface diffusion, diffusion of vacancies in the surrounding lattice, or vapor transport. We consider first the case of flow by surface diffusion alone. We assume that there is a force, in the sense of irreversible thermodynamics, tending to move the atoms around the bubble in a given direction. We designate this direction as the x axis and the force per atom as Fa. The net rate of flow of atoms from one side of the bubble to the other, under the influence of this force, is the surface flux, Js, times the cross-sectional area available for flow, A,. A, is taken as the circumference times the thickness of the high-diffusivity surface layer 6. Thus where 51 is the atomic volume. For bubbles to advance a distance dx, a volume equal to nr : dx must flow around the void so that The minus sign enters because the bubble moves in the direction opposite to that of the force on the atoms. Equations for volume diffusion and diffusion through the vapor can be obtained in the same manner. For vapor transport we have Ag = and the ratio of the densities of metal atoms in the gas and the solid (pg/pl) enters so that

Jan 1, 1964