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By E. R. Johnson

THE copper concentrator of the Braden Copper Co. is at Sewell, Chile, on the western flank of the main Cordillera of the Andes, at an air distance of approximately 50 miles southeast of Santiago, the capital of Chile. The region is mountainous and space is at a premium because of the rough terrain and the danger of snowslides. The ore, a porphyry copper, is mined by a block-caving method in the El Teniente mine. It is hauled to the crushing plant, a distance of 2.6 miles, through a tunnel because of the severe climatic conditions. The usual large, coarsely set, primary reduction machines found in many ore-treating plants are not required because of the mining method employed and the limiting 12-in. spacing of the grizzlies at the head of each ore pass in the mine. It is the purpose of this paper to describe the numerous improvements made in the crushing section of this large flotation concentrator during the past 15 years to increase the efficiency of the plant and to raise the daily output from around 20,000 to 30,000 tons. The reader should keep in mind that all changes were made within the existing building because of the limited space and the high cost of the heavy mill construction required to withstand the snow loads. The present plant, as well as the former, operates on a 24-hr, 7 days a week basis. FORMER PLANT The crushing plant commenced operations at its present site in 1921, the main equipment being No. 7 1/2 gyratories, followed by 48-in. horizontal Symons disk crushers. Fines were removed by 1-in. stationary grizzlies ahead of each unit. In 1924 a third stage of crushing was provided by the addition of 55 1/2-in. rolls in open circuit. During 1925, the Symons disks were replaced by 73 1/2-in. rolls. All the rolls were changed to closed circuit, with elevators and screens to give a final product of 9 pct plus 3-mesh. The flowsheet of a typical crushing section of this period is shown in Fig I. The ore was withdrawn from a common bin serving the five crushing sections in the plant by means of any number up to six 36-in. wide-apron feeders, which discharged onto a common conveyor belt. This conveyor was equipped with a magnetic head pulley and the crusher attendant was stationed near the discharge end to pick out pieces of mine timber that might be in the ore. The ore was discharged over a 1 3/4-in. spaced bar grizzly and the oversize of this grizzly was crushed by a No. 7 1/2 McCully gyratory crusher set at 1 3/4-in. opening on the closed side. The grizzly undersize joined the gyratory discharge and was conveyed to a set of 73 1/2 by 20-in. Garfield rolls set at I in. These rolls operated in closed circuit with a bucket elevator and two vibrating screens placed in parallel.

Jan 1, 1947

By William Sellers

tested without knowing anything of their chemical composition. I had these pieces separately placed upon 10-inch bearings under a 7-gross ton lianlrner, a piece of 2½-inch round iron laid upon them as a fuller, and the hammer allotved to fall from 20 inches above the fuller, which, according to Haswell, gave a ldow of 67.75 gross tons. The pieces were then turned orcr, the fuller placed upon the convex surface, and the harnnler allowed to fall from 13 inches above the fuller, giving a blow of 58.46 gross tons. You will see that the rails do aot show ally bigrls of ruptnre, and their color at the points of torture prove them to have been absolutely cold when the test was made. I thirk these rails ought to be reasonably safe in the track. As you see by this piece of the head of one of these rails, I had it plarred, and then some teeth cut in it by a cold clrisel, and one-half of them pounded down with a hammer. The teeth of my rack did not break off. The analyses of these rails substquently made are: I. 11. Carbon,... 0.110 0.880 Silicon,..0.050 0.058 Phosphorus ,..0.086 0.082 Manganese,.. 0.042 0 840 In conclusion, I will say that Dr. Dudley's paper as a contribu tion to our knowledge of steel rails is valuable and interesting, but I protest against his conclusions king reoeived as manufacturing or comtnercial axioms. From its being presented in the form-of a report to the leading railroad of the country from one of its trusted officers, as well as from the tone of the concluding deductions, it is liable to be so received by railroad men. If any railroad com pany desires rails made under either of Dr. Dudley's formulas, and are willing to pay a price large enough to cover the loss in making them, dl and good; but so long as the rail makers are compelled to gwrrantee the wear of their rails for a given number of years, justice reqnires that. the composition of the steel from which these rails are made should be left to them. WILLIAM SELLERS, PHILADELPHIA: The very interesting paper upon the wear of steel rails that has just been read preseak the record of a series of investigations that are extreniely valuable, and the deduction that bas been drawn from the results noted seems to be unavoidable; the tests, however, to which it is proposed steel rails shall be subjected hereafter, with a view to determine their quality, sllould

Jan 1, 1881

By RICHARD MAIZE

IN this critical period of our history, the coal industry of the nation, faced with many obstacles, performed its work safely during the first ten months of 1943. Thousands of the younger mine workers were inducted into the armed forces of our country and in many instances their jobs have been filled by others without much if any experience and training. This situation alone would seem to be sufficient to interfere with the safe and orderly operation of the industry but the Government's demand for a substantial increase in coal tonnage over that produced in normal times introduced a hazard which always accompanies speed, regardless of the skill of the worker.

Jan 1, 1944

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Jan 1, 1938

By Fred W. Stiefel

To open the Fisher Hill mine of the Republic Steel Corporation, it was necessary to sink an inclined shaft into the rock and excavate stations, drifts, and ore pockets. This inclined shaft, or slope, q ft. 8 in. high by 28 ft. 2 in. wide, has compartments for two 5-ft. gauge ore-skip tracks of 120-lb. rail, one 5-ft. gauge man-skip track of 60-lb. rail, a concrete stairway, and service pipes consisting of a 12-in. compressed-air line, an 8-in. pump-discharge line, and a 4-in. water-supply line. When we were requested to investigate the work, the progress being made (by another organization) amounted to only 2 ½ ft. advance in 24 hr. We reported that progress of at least 6 ft. per day could be made. As a result, the Defense Plant Corporation, through the Republic Steel [ ] Corporation, entered into a contract with us to sink the shaft. The contract contained a penalty and bonus clause predicated upon an advance of 150 ft. per month. The average advance was 7.8 ft. per day; that in the best six-day week was 60 ft. and that in the best month, 215.5 ft. in 25 working days. The drilling was performed by 3 ½ -in. wet drifter drills mounted on five 8-ft. columns (see accompanying illustration). Even though a full-face drilling and mucking operation is customarily desirable, the steep slope, with the accumulation of water and fine debris at the face, made a modified heading and bench operation preferable. The heading and bench were of equal height. The five columns, instead

Jan 1, 1945

By WILKES BARRE

AS PREVIOUSLY announced, the semi-centennial meeting of the Institute is to be held at Wilkes-Barre, Pa., on Sept. 12, 13, 14, and 15. An interesting program of technical sessions and excursions is now being prepared by the Local Committee. At the time of going to press the program had not been completed in sufficient detail to permit its being pub¬lished in this number. The detailed program will be published in the September MINING AND METALLURGY, which will be printed in time to be distributed before the Wilkes-Barre Meeting. It may be said in advance, however, that the Com¬mittee has progressed so far in its work that we are assured of one of the -most notable meetings the Insti¬tute has held and that members will miss a rare .treat if they do not attend. The many trips and excursions, as well as the banquet, will be -given to the members as a compliment from the members of the Local Com¬mittee. . . The paper by Howard Eavenson on the Lynch Plant of -the United States Coal and Coke Co. will be illustrated by lantern slides and moving pictures. One of the most appropriate discussions will be that opened by Prof. J. F. Kemp on the Stratigraphy of the Anthracite Region; this also will be amply illustrated.

Jan 1, 1921

By J. J. Marcus

Mineral industry decision-makers are frequently required to select between competing equipment. Various criteria and methods are currently being used, and this writer would like to suggest a method recently tested on two large power shovels in an open pit mine. An initial examination of both shovel A and shovel B indicated that the latter should give the best overall results. since: 1) It swings 13% faster; 2) it carries a dipper with 23%: greater capacity; 3) purchase price is 8.6% lower; 4) shovel weight is 12.5% greater; 5) it has considerably greater bail pull under most operating conditions; 6) the electric controls are of a more modern design. However, shovel B has half the service life. With these points in mind, shovel A was selected by The Buckingham Method, an operations research technique.

Jan 9, 1965

Prospectors were digging silver in the Ray mine area in 1873, and by 1880 high-grade copper ore was feeding a 30-ton copper furnace. In 1910, D. C. Jackling and his associates organized the Ray Consolidated Copper Company, known in mining circles as Ray Con, to exploit the Ray orebody. For 50 years, all mining was underground and the Ray mine was known the world over for innovations in mining methods. It was the first to mine important tonnage (8,000 tpd) by block caving methods. The mill at Hayden, 26 miles (41.6 km) from the mine, kept pace with the new mining methods developed at Ray and in the early 1920's flotation replaced gravity concentration by the vanner machines. First mining by open pit methods began in 1950 and in January 1955 all underground mining was discontinued. The new primary crushing plant, installed in early 1967, featured design simplicity with a low capital cost. The 54 in. gyratory crusher was installed within a total height of 6 1 ft (18.65 m) from the truck dump road bed to the floor beneath the feeder-conveyor under the surge pocket. The crusher is fed by 120-ton trucks dumping from one side only into a 200-ton hopper equipped with a horizontal splitter bar to protect the crusher spider. The crushed ore pocket, of 120 -ton live capacity underneath the crusher, discharges to a 0 to 200 fpm (61.2 mpm), variable-speed, 72 in. by 80 ft long (1.83 m by 24.4 m) sloped belt feeder, probably a first in the industry. Wood-picking stations are provided on either side of the belt so that old mine timbers and tramp iron can be removed from the crushed ore and the wood can be dropped into Dempster containers located below the belt. Counter-weighted finger gates control the bed depth on the belt at 14 in. (356 mm) and a low- level radiation unit automatically maintains a minimum of a 3 ft bed (900 mm) of material on the belt directly under the surge pocket. The conveyor is skirted for its full length and is approximately 60 in. (1.52 m) wide inside its metal skirt liners. Ten impact idlers at 12 in. (305 mm) spacing are provided under the loading pocket and remaining idlers are spaced at 18 in. (457 mm). The belt is 1 ¼ in. (31. 8 mm) thick; and, while it has occasionally been damaged by tramp material, belt life averages 11 million tons. When the belt is changed, the coiled new belt is lowered into position onto one of two belt stands mounted over the feeder, the old belt is cut, and one end is attached temporarily to the new belt. The other end of

Jan 1, 1978

By P. G. Hallof

Only recently has the term Induced Polarization found its way into geophysical literature. Schlumberger first mentioned the possible use of the induced polarization effect in one of his early papers. Later, induced polarization was the subject of several university theses, such as Seigel's at the University of Toronto and Bliel's at the Michigan School of Technology. In addition, a group at the New Mexico School of Mining and Technobgy has done work on induced polarization. There is very little to be found in print on the subject of induced polarization. Several papers have been presented at meetings, the 1958 AIME and SEG* conventions, for instance, on laboratory measurements of the effect and theoretical considerations useful in the interpretation of field results. This paper is the first attempt to present actual field results which show the usefulness of induced polarization as a prospecting technique. There are several phenomena that can cause polarization when electrical current is passed through the ground. In a paper presented at the 1958 SEG* meetings Professor T. R. Madden of M. I. T. demonstrated that only two of these gave rise to appreciable effects. The first is a result of the membrane and diffusion effects in a sandstone that contains appre- ciable amounts of clay minerals. The unbonded charges associated with the clay mineral forma material that passes positive and negative ions with different mobilities. When current is passed through such a system for some period of time, polarization results. The second phenomenon, which gives rise to very much larger polarization effects, is the one used in the new prospecting technique. These effects are a result of the blocking action or polarization of metallic or electronic conductors in a medium of ionic solution conduction. This electrochemical polarization, called "the over-voltage effect" by electro-chemists, occurs whenever electrical current is passed through a volume of rock which contains metallic minerals such as base metal sulphides. This polarization at metallic interfaces in ionic solutions depends upon the chemical energies necessary to allow the ions to continue the flow of current by giving up or receiving an electron from the metallic surfaces. The energy stored in this reaction isanalogous to that stored when a condenser is charged. If a d. c. current flowing through such a system is interrupted, the polarization can be detected as a small, decaying current that flows after the applied current is discontinued. The decaying potential measured is a result of the current flow caused by charges returning to their equilibrium positions from the positions they assumed under the applied potential field. Because of the polarization, there is an excess of change at each metallic interface. There is an alternate method of measurement of the induced polarization effect. The electrical

Jan 1, 1961

By K. C. Mills, E. T. Turkdogan

A number of investigators1 6 have noted the presence of a liquid miscibility gap in the Fe-Sn binary system. However, the first attempt to measure the

Jan 1, 1964

By W. B. Plank, A. C. Callen

WAR and normalcy do not walk hand in hand, whether it be in industry, the educational field, or in the daily lives of individuals. Schools and departments offering curricula in mineral engineering have been seriously disrupted, beginning with 1942 and reaching a low during 1944. Selective Service regulations and voluntary enlistments have effectually stopped the flow of college-trained men into the mineral industry, and the flow into the colleges is little more than a dribble. The year just closed has been a most serious one for our institutions. Until March many of our faculties had been kept reasonably intact because of the large training programs of the Army and Navy. Although the number of

Jan 1, 1945

By PALMER H. TYLER

WHEN the Mid-Continent Section of the A. I. M. E. met at the roof garden dining room of the Tulsa Club on Monday evening, May 13, most of the members present came prepared with a credulity-stretching yarn of some sort or other, gathered from actual experience. Perhaps it was not realized at the time that such anecdotes would eventually burst into print, but while the excellent dinner was appeasing the inner man the word was passed around that informality and spontaneity of expression would characterize the meeting, with the result that the shackles of convention were eventually cast aside and the meeting waxed better and better until the hour grew ,late, and the chairman, L. G. E. Bignell, originator of the idea, brought to a close what was agreed to be a profitable and entertaining evening. The ball was started rolling by Paul Ruedemann of the program committee, who offered the following:

Jan 1, 1929

By C. C. Harris

An analytical solution of the integro-differential equation of batch grinding is proposed as a first appoximation to the xYt surface. The size distribution equation (xY plane) is a three parameter equation previously suggested. Possible refinements are outlined. Comminution kinetics research is presently following several related courses. Some are equivalent, in that they represent different ways of viewing the same facet of the problem, though the entire endeavor cannot be described as being completely integrated. The purpose of this note is to propose and discuss a framework into which the various approaches might be fitted. This must clearly be founded upon an analysis of the xYt surface.' (See also Fig. 1 of Reference 26.) The investigations referred to above comprise mainly researches into the following: (1) The kinetics of the creation of the finer sizes during relatively small grinding times. This is the Fahrenwald-Arbiter-Bhrany 2-3 approach which leads to equations equivalent to a special case of the charles4 relationship. It emerges that initial production of fines follow essentially zero order* kinetics. (2) Disappearance kinetics of the coarser fractions. This appears to be a first order relationship.5-6 (3) Prediction of the xYt surface via the selection and breakage functions utilizing computer-programmed or approximate solutions of the integro-differential equation of batch grinding.5,8-17 A recent discussion by Mika et al.5 references the relevant literature up to about 1966. In addition to these main-stream researches there is some work on the minor topic of extremely long grinding time considered as a limiting case.18-19 The field is supported by investigations into the size distribution of fragments produced by breakage processes, some of this work being most valuably statistical-mechanical in nature.20-25 Better fitting equations, however, are obtained by statistical and curve-fitting methods.26 This

Jan 1, 1969

By A. S. Argon, G. East

From their study of the anisotropy of grain boundary mobility in aluminum, the authors conclude that tilt boundaries have a higher mobility than twist boundaries because the atomic misfit at the pure-twist boundary is relatively small compared to the misfit at a tilt boundary for a given amount of misorientation. If this is true, then one would expect a twist boundary to have a lower grain boundary energy than the corresponding tilt boundary. Fig. 6, taken from the work of Gjostein and Rhines,10 shows that this is indeed the case for [001]-cu tilt and twist boundaries in the high-angle region, thus lending support to the authors' conclusion.

Jan 1, 1968

By W. M. Robertson

The formation of grain boundary grooves on surfaces of poly crystalline samples of cobalt and iron immersed in liquid lead has been studied. The grooves form by volume diffusion of the solutes cobalt and iron in the liquid. The diffusion coefficients of the solutes in liquid lead are derived from the measured rate of grooving. The diffusion coefficients are described by the relation D = Do exp (-Q/RT), with, for cobalt, Do = 4.6 x 10-4 sq cm per sec and Q = 5300 ± 800 cal per mole, and for iron, DO = 4.9 x 10-3 sq cm per sec and Q = 10,500 ± 1500 cal per mole. LIQUID metal-solid metal interactions occur at solid-liquid interfaces. Interfacial energy provides a driving force to change the morphology of the interface. Mullins1,2 has derived expressions for the kinetics of interface morphology changes driven by capillarity. These expressions can be applied to an isothermal system of a solid in equilibrium with a liquid saturated with the solid. Surface profile changes can occur by volume diffusion of the solute in the liquid, by volume self-diffusion in the solid, and by interfacial diffusion at the liquid-solid interface. A groove will form at the intersection of a grain boundary with a solid-liquid interface, reducing the total interfacial free energy of the system. The solid-liquid interfacial energy ? must be greater than half the grain boundary energy of the solid ?6 for Mullins' calculations to apply. If ? is less than ?b/2, then the liquid penetrates the boundaries, separating the grains rather than forming grooves. Boundary penetration did not occur in the work described here. where CO is the equilibrium volume concentration of the solid in the liquid, Dv the volume diffusion coefficient of the solid in the liquid, ? the interfacial free energy of the solid-liquid interface, O the atomic volume of the solid crystal, k Boltzmann's constant and T the absolute temperature. Eqs. [1] and [2 ] also apply to grooving by volume self-diffusion in the solid,1 with CoODv = D Self, where DSelf is the volume self-diffusion coefficient of the solid. For a grooving mechanism of interfacial diffusion at the solid-liquid interface, the groove width is given by2 where CS is the interfacial concentration of the diffusing species, and DS is the interfacial diffusion coefficient. Eqs. [1] and [3] can be used to determine the mechanism of groove growth. A t1/3 dependence of the growth indicates volume diffusion and t1/4 indicates interfacial diffusion. In some cases, volume diffusion and interfacial diffusion both can contribute substantially to the grooving process, causing the time dependence to be intermediate between t 1/3 and t1/4.3 For these cases, the relative contributions of the two processes can be separated.4 However, in many cases, one process will be dominant, and the data can be analyzed on the basis of Eq. [1] or Eq. [3] alone. The time dependences for volume diffusion in a liquid and volume self-diffusion in a solid are the same. However, the self-diffusion contribution of the solid is usually negligible compared to volume diffusion in the liquid. After the grooving mechanism has been determined, Eq. [1] or Eq. [3 ] yields the kinetic parameter A or B. The kinetic parameter can be used to calculate values for the unknown quantities in the product CD?. Usually C is known or can be estimated. If ? is known, then D can be calculated. In a measurement of grain boundary grooving of copper in liquid lead,' the time dependence indicated volume diffusion in the liquid. The quantities Co, Dv, and ? were obtained from the literature, giving excellent agreement between the observed values of A and the values calculated from Eq. [2 ].5 In a study of the grooving of several refractory metals in liquid tin and liquid silver, A1len6 educed that grooves formed by volume diffusion in the liquid. In a study of nickel in a nickel sulfide melt, Steidel, Li, and spencer7 found volume diffusion grooving kinetics. Both Dv and ? were unknown, so they could not obtain either one separately, though they did obtain a reasonable value for the temperature dependence of the product Dv ?. Several methods have been used to obtain surface profiles. It can be done by sectioning through the interface7 or by chemically removing the liquid from the solid surface after solidification of the liquid.6 However, if the liquid dewets the solid on removing the solid from the melt, then the interface can be observed directly. This method was used previously' and was utilized also in the present study. EXPERIMENTAL PROCEDURE Lead of 99.999 pct purity was obtained from American Smelting and Refining Co. Cobalt sheet was obtained from Sherritt-Gordon Mines, Ltd., with a nominal purity of 99.9 pct, the principal impurities being nickel, iron, copper, carbon, and sulfur. The sheet was

Jan 1, 1969

By E. P. Abrahamson

The effect of transition element binary solid solution additions upon the recrystallization temperature of titanium has been investigated. All additions, except columbium and tungsten, raised the recrystallization temperature. A correlation is obtained between the logarithm of the absolute rate of change of recrystallization temperature with atomic percent solute and the number of uuter d shell electrons attributed to the free atom of the solute element in its ground state. The work of Abrahamson, et aL.1-6 on dilute binary solid solution alloys indicated a correlation between the free atom ground state electron configuration of the solute and both the brittle-ductile transition and recrystallization temperature. It is shown in these studies that the number of outer s and d electrons in the solute are of prime importance in determining the absolute rate of change of recrystallization or transition temperature with composition. Furthermore, the limit of initial linearity (in the curve of recrystallization temperature vs at. pct solute) has been shown to be a function of the number of d shell electrons in the solute for dilute iron- and vanadium-base alloys. To date no complete systematic recrystallization study of dilute binary titanium base alloys has been made. This study together with that of zirconium base alloys 6 will provide the data for the comparison of the effect of the same solutes upon two solvents with the same number of outer d shell electrons.

Jan 1, 1962

By F. G. BREYER

ZINC metallurgists continue to follow with keen interest reports of successful results from the continuous retort plants at Palmerton, Pa., and Meadowbrook. W. Va. The new process had already demonstrated its ability to reduce the lead content of the metal and to effect savings in labor and fuel. The past year has shown that the retorts themselves can he built to last long enough so that the retort cost per ton of metal is lower than in past practice with small horizontal re- torts. A number of the latest type of retorts are reported to have been in continuous operation for over 18 months without a failure and are said to be still in good condition. This seems a complete answer to the last question which remained before continuous zinc smelting could he pronounced commercially practicable.

Jan 1, 1932

By ARCHIBALD DOUGLAS

BY this time almost everyone knows, in a general way, the corporate income distribution policies of the 1936 Revenue Act, and many of the practical problems arising there under. This article is not intended to be a technical discussion of that act by title, subtitle, section, or supplement, nor an analysis of inter- dependent sections of the various acts over many years woven on the fiscal loom f our Congressional machine. Such a discussion would require the spinning of a web of words too long for patience. Yet ,a general statement of the underlying plan of the 1936 Act, so far as it affects dividend policies, is perhaps useful.

Jan 1, 1937

By H. R. Hanley

The characteristics of alloyed anodes and their influence on the products of electrolysis and power consumption have been noted previosly ously in the literature.' This paper presents data in continuation of the subject given in an earlier paper by the authors.2 As noted in the authors1 previous paper, the objectives of this work are the reduction in the cell voltage and the purity of the cathode product resulting from the use of alloyed lead anodes. The fundamental facts developed in the previous paper were: (I) that the presence of a small amount of calcium lowers the anode potential approximately 50 per cent. below that of pure lead; and (2) that thallium when present to the extent of 4 per cent. rendered the anode practically stable, thereby lowering the contamination of the cathode by lead to a very insignificant amount. The specific features determined in this recent work are as follows: 1. The lowest limit of calcium in the lead which will still give maximum lowering of the anode potential. 2. The characteristics of the thallium-calcium-lead anode. 3. The characteristics of the silver-calcium-lead anode. 4. The characteristics of the thallium-silver-calciurn-lead anode. It was determined that 0.1 per cent. calcium in the anode was sufficient to lower the anode potential approximately 50 per cent. Table 1 shows the characteristics of alloyed anodes tested. The anode composed of 0.1 per cent. calcium, 4 per cent. thallium, and 95.9 per cent. * Professor of Metsllurgy, Missouri School of Mines. t Professor of Metallurgy and Ore Dressing, Hea.d of Department, Missouri School of Mines. Assistant Professor of Metallurgy, Missouri School of Mines. 1 Among others, see C. G. Fink and Li Chi Pan: Insoluble .Anodes for the Electrol-ysis of Brine [Trans. Amer. Electrochem. Soc., (1926) 49, 83-1341 and other papers by these authors. U. CJ. Taintori, 8. G. Taylor arid H. P. Ehrlinger: Lead Alloys for Anodes in Electrolytic Production of Zinc of High Purity. Truns. A. I. M. E. (1929) 192-200. 2 H. R. Hanley, C. Y. Clayton, D. F. Walsh: Investigation of Anodes for Production of Electrolytic Zinc. Trans. A. I. M. E. (1930) 275

Jan 1, 1931

By H. I. Aaronson, W. L. Winterbottom, G. M. Pound

On the basis of a statistical thermodynamic treatment of the data of smithz2 on the activity of carbon in austenite, a Darken and smith23 deduced that the interaction energy, wy, between carbon atoms occupying nearest-neighboring interstitial sites is repulsive. Aaronson, Domian, and pound 24 (ADP) have applied two different statistical thermodynamic analyses to the same data. One analysis, a composite of the treatments of Darken and Smith,23 Speiser and Spret-nak," and Kaufman, Radcliffe, and Cohen,26 primarily takes account of the positional entropic effects of wy in dilute solutions. The other, due to Lacher 27 and Fowler and Guggenheim," is also applicable to more concentrated solid solutions. The equations which both analyses produce for a,, can be rearranged so that, when In a,, is plotted against the appropriate function of the carbon content of the austenite, the plot will have a slope of unity when the correct value of wy is employed. Since no other constant need be simultaneously evaluated in order to make this determination, ascertaining the most probable value of wy from Smith's data on the bases of these analyses was an elementary problem in least-squares statistics. ADP demonstrated that the wy's obtained from both analyses vary with temperature in a statistically significant manner. McLellan, Gerrard, Horowitz, and sprague 29 (MGHS), on the other hand, have recently made a new statistical thermodynamic study of a,, and have concluded from this study that wy is independent of temperature. The equation which they derived for a,, is: where 0 = ratio of the mole fraction of carbon atoms to that of iron atoms, z = number of sites from which occupancy is excluded by the presence of a carbon atom and the repulsive character of its interaction with nearest-neighboring carbon atoms (z = zb in Ref. 24), and x = a geometric parameter taking account of overlapping of the exclusion shells of nearby carbon atoms. z is 1 when wy = 0 and 13 when wy = Q. For a given value of z there is a definite value of X . In the absence of an equation connecting z and X, MGHS computed geometrically the value of x at several integral values of z; x increases from 0.47 to 8.67 as z is increased from 2 to 5. MGHS fitted Eq. [I] to the a,, data of Ellis, Davidson, and Bodsworth30 at 925° and 1050°C and to the data of smithz2 at 1000°C. Confining z to integral values in the range 4 to 6, and also somewhat restricting AG, the values of z, X, and AG were sought which would minimize the average root mean square percent deviation, D, of the calculated a,, values from those obtained experimentally. Finding that D was almost always smallest when z = 5, they concluded that z is independent of temperature. Since:23, 26 where R = gas constant and T = absolute temperature, and w,, was considered to be similarly invariant, this statement cannot be strictly true. The temperature range employed, however, was small and the interval between the values of z tested was sufficiently large so that the results could still be consistent with a constant The statistical treatment of the a,, data applied by MGHS is clearly quite approximate. Since more than three values of a,, are available at each temperature, z, and X, and AG have actually been over deter mined, rather than underdetermined as their treatment implies. In order to make more complete and exact use of the a,, data, we shall employ the Legendre Method of Least Squares, as described by Whittaker and Robinson. This technique permits conversion of the overdetermined system of equations produced by the substitution into Eq. [I] of the available pairs of a,, and 0 data at a given temperature into an even determined system. Solution of the latter system yields the most probable value of each of the three constants. This procedure will allow a statistically rigorous test to be made of the conclusion drawn by MGHS in respect of the temperature dependence of In order to obtain linear iiequations of conditions" from Eq. [I], the exact values of the three unknown constants are replaced by G/RT = =/RT + y, where z, i, and aG are estimated values of the constants. This allows the conditional equations to be written in the form:'l where fi -f6, i, =/RT) = the right-hand side of Eq. [I] for a given pair of ayi and values. When applied to Eq. [I], these relationships become: The i equations of condition are reduced to three "normal equations" in the manner discussed by Whittaker and Robinson. The normal equations are then solved for a, ß, and y. Both of these operations are most conveniently performed by means of the computer technique of Winterbottom and Gjostein.32 This technique was used iteratively until the value of 0 (the most sensitive of the three correction factors) became 10.01. (Until they became small, a, 8, and y

Jan 1, 1969