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By W. Scott Dunbar

INTRODUCTION Methods of linking boundary integral (BI) solutions with finite element (FE) solutions have been well described theoretically in other publications (e.g., Zienkiewicz, et al., 1977). The purpose of this paper is to point out some practical aspects of implementing these linkage algorithms for rock mechanics applications and to describe an alternative linkage algorithm. A not so obvious motivation for using FE-BI linkages is illustrated in Figure 1. The cost of modeling the excavation of the crusher station by the BI method can equal or exceed the cost of an equivalent FE model. This is due to the large surface area which must be modeled by the BI method. In such cases, a more economical approach might be to model the region immediately adjacent to the crusher station by finite elements and link the solution at the periphery of the FE model with a solution for the exterior region. Further economies can be realized if some flexibility in the type of exterior solution is possible. For example, it would be desirable to use a coarse discretization of the FE-BI interface for a BI solution, which would be less demanding of computer storage and time, or to use an analytical solution for the exterior region. The alternative linkage algorithm to be discussed allows such flexibility. BOUNDARY INTEGRAL METHODS There are basically two formulations of BI methods: the direct and the indirect. FE-BI linkage algorithms depend on which formulation is used. The purpose of this section is to briefly describe each method.

Jan 1, 1982

RAYMOND M. Howe (author's reply to discussion*).-Dr. Unger states that magnesite will hardly supersede dolomite in fettling after a heat; that it is hard to believe a slag composed of oxide of iron containing about 6 per cent. of phosphorus is the active corroding, agent in a boil in a furnace; and that boiling cannot be said to be the result of the chemical action of a high-phosphorus slag on a basic bottom. These are well-established facts that no one would venture to question. The authors were misinterpreted in the statement that "in the summary, it is claimed that magnesite low in lime, or of greatest purity, ranks first in resistance to slaking," etc. On the contrary, the position has been taken throughout that while a low lime content is advantageous, the presence of several per cent. of iron oxide (an impurity), is necessary in order to confer the proper bonding and sintering properties in dead-burned grain magnesite to be used for furnace hearths. We cannot subscribe to the view that pure magnesia is no better fitted for a refractory than pure lime,' nor that the pure natural varieties of magnesite; such as Grecian or Washington, must necessarily be burned with silica and oxide of iron to produce a satisfactory material. On a small scale, pure magnesia has been satisfactorily used as a refractory in the form of crucibles. In the production of dead-burned grain magnesite from the Grecian or Washington mineral, iron oxide is added but the addition of silica is neither customary nor desirable. It is possible to manufacture brick of good quality from both the Grecian and the Washington material without adding either silica or iron oxide, providing they are burned at a sufficiently high temperature. This is regularly done on a commercial scale both in America and abroad when brick of. exceptionally high refractoriness or special purity are desired. The following analysis is of an English brick of Grecian magnesite which is extremely low in all impurities.1

Jan 6, 1919

Discussion of the paper of H! M. CHANCE, presented at the Pittsburgh meeting, October, 1914, and printed in Bulletin No. 91, July, 1914, pp. 1461 to 1466. R. V. Norris, Wilkes-Barre, Pa.-In his resume of coal-taxation methods Mr. Chance has apparently confined his investigation to bituminous coal, so a summary of the anthracite situation as to coal assessment should be of interest. The anthracite region of Pennsylvania in its 484 square miles of measures lies in the counties of Wayne, Lackawanna, Luzerne, Carbon, Schuylkill, Northumberland, and Dauphin, and in each of these different methods of assessment have been adopted. The assessments for taxation in Pennsylvania are made under Acts of the Legislatures of 1841 and 1842, under which the assessors are re¬quired to "assess rate and value every subject of taxation according to the actual value thereof, and at such rates and prices as the same would bring at a bona fide sale after due notice." Under this general law coal land has been assessed as such, but up to about 25 years ago the coal was assessed at rates so low as to be unimportant, though the rates in each township and municipality, even in the same county, showed no agreement or ascertainable basis. In the late '80s, the need for greater revenue drew attention to the possibilities of collecting much larger taxes from the coal lands and, spurred by public demand, the assessing authorities in some counties undertook a reassessment of coal. The method pursued was to demand from each owner a statement of the area and average remaining thickness of coal on his lands. On this return, made in some cases under oath, an assessment based on foot-acreage was levied, with a resulting great increase in valuation, but the valuation per foot-acre was low and no serious objection was made. This assessment with occasional increases of rate remained in force until 1907, when, in the triennial assessment of that year, a complete revision of the valuation was again attempted, engineers were employed by the counties to check the accuracy of the returns from the owners, and, based on the foot-acreage thus obtained, an enormous increase, to $65 per foot-acre, was made in the valuation. This assessment was resisted in the courts by appeals resulting in very extensive litigation, and a final determination of the validity of the 1907 and all subsequent assessments has not yet been obtained. In the first appeals tried the rate calculated on a foot-acre valuation was reduced by the Luzerne County court to about $45, while the. Lackawanna County courts reduced a larger rate to $67. The county authorities were, how-

Jan 4, 1915

By S. C. Termes, P. E. Richardson, R. L. Wilfong

This paper discusses recent Federal Bureau of Mines research on the selective flocculation of various minerals with insoluble cross-linked starch xanthate (ISX). ISX is insoluble, has a shelf life of more than a year when kept cold, and has a higher molecular weight due to cross-linking than the soluble, non-cross-linked starch xanthate used in previous flocculation studies on nonsulfide minerals. ISX readily flocculates a number of sulfide mineral fines, but no silicious gangue. Floc formation, floc size, and settling rate are dependent on pH. The pH dependence, however, differs for each mineral. Bornite has been selectively flocculated from bornite-quartz mixtures, and the bornite-containing flocs were separated from the unflocculated quartz using decantation and washing procedures.

Jan 1, 1984

By Hugh P. Tiemann

The desirability of employing the centigrade scale for the measurement of temperatures is becoming more and more recognized in this country, particularly in view of the fact that this scale is used almost exclusively in technical papers and publications—exclusively in those published in French or German. There are, however, a large number of factories and laboratories which still employ the Fahrenheit scale. It therefore frequently becomes necessary to convert one into the other and the accompanying chart (Fig. I) has been found very convenient for this purpose. The formulas for conversion follow: (1) °F. = 9o /5+ 32 (2) oC = 5 (oF - 32 ) (1) 100° C X 9 900/5 =180 .'. 180 + 32 = 212° F. 5 (2) 5 (212° F - 32) = 5 X 180 =100o C 9 9

Jan 1, 1916

By J. R. Ross, J. B. Rosenbaum

Bureau of Mines research has resulted in the development of fluid- bed ion-exchange columns for recovering uranium from slime pulps. In pilot plant tests, slime slurries containing up to 6 percent solids were processed at flow rates of 10 gallons per square foot per minute with high uranium recoveries and efficient resin loadings. Uranium was effectively eluted in a laboratory fluid-bed elution column at eluate-to-resin ratios of less than 5:1 and eluate flow rates of 3 gallons per square foot per minute. Construction costs of mills using these improved ion exchange techniques are estimated to be 25 percent less than for mills using solvent extraction of clear solutions. For mills having a capacity of 2,000 to 2,500 tons of ore per day, the saving would be in the range of $2.5 million. In addition, a saving of about $0.80 per ton of ore in total operating costs is indicated.

Jan 1, 1973

By G. Hasenfus, M. Karmis, G. Goodman

Surf ace subsidence is rapidly becoming an important environmental consideration of active as well as abandoned mining operations. The damages attributed to this phenomenon have been witnessed in both rural and urban areas and include land settlement and fracturing, structural damages to surface buildings or facilities and disruption or contamination of ground water supplies. The problem of mining subsidence has gained considerable emphasis in the Appalachian coalfield. The increased mining activity over the past few years and the particular topographic conditions, lithologic and structural environment, mining methods and hydrological patterns are all factors which have contributed to specific subsidence characteristics in this region. The objective of this paper is to discuss and review the most widely accepted methods of subsidence prediction above longwall as well as partial extraction systems in Appalachia. These include an accurate and easy to apply empirical method, which has been developed entirely from field information pertaining to the Appalachian coalfield and incorporates both geometrical and geological information of the panel in question. The second method is a development of influence functions and is founded on the zone area approach. Both methods, originally derived for longwall mining, have been expanded to include room and pillar applications and have been widely used by the mining industry in the Appalachian coalfield.

Jan 1, 1984

By Waldemar Dyrssen

BEFORE discussing the relation between air pre-heaters and waste-heat boilers in conjunction with the open-hearth furnace, it is necessary to have a clear idea about what a preheater can accomplish and what effect it has on the open-hearth furnace, especially from a thermal point of view. It is generally recognized now that the heating of the combustion air in the checker chambers of the open-hearth furnaces is not as efficient as it is-desirable to make it. It is also generally recognized that heating of the air in checker chambers of ordinary construction has its limitations. These limitations are mostly the .space occupied and the cost that would be involved if checker chambers were made larger than according to present practice. There are, however, other. reasons why checker chambers cannot give the highest possible air temperatures. Checker chambers are capable of heating air to very 'high temperatures by using waste gases of extremely high temperatures and, in fact, higher temperatures can be reached with checker chambers than with any other means available. The temperature to which we can heat, air, however, is limited very materially by -the temperature at which the air enters the checkers, be-cause of the fact that checker chambers, as at present constructed, are not efficient in heating air through a wide range of temperatures. In other words, if air in-stead of entering the checker chambers at atmospheric temperatures could enter at about 1000 to 1200 deg. F., the resultant air temperature at the top of the checker chambers would be materially increased. There are now on the market metallic air preheaters which are ex-tremely efficient for preheating air to about 1200 deg. F. Such heaters occupy very little space and can usually be installed at a relatively low cost.

Jan 1, 1928

By D. G. Brubaker

IN the history of geophysical exploration by the electromagnetic method many procedures and types of equipment have been used. Source arrangements for surveying on the ground have included long wires grounded at both ends, large loops laid on the ground, small horizontal loops, and vertical loops of moderate to small sizes. Receivers have ranged from simple direction-finding coils to combinations of coils and a bridge circuit to measure quantities related to both amplitude and phase of the received signal. Primary considerations in choosing a survey method are effectiveness, speed, and economy. In work on the ground in which locating and out- lining conducting bodies is the objective, methods using a vertical coil and a simple direction-finder have been found the most satisfactory. Mason1 and his associates applied such a method by setting up a vertical source loop, fed with low frequency alternating current by an engine-generator set. With a direction-finding receiver they surveyed lines along the ground in the plane of the loop, taking measurements at each station of the azimuth and inclination of the magnetic field direction. Deviations of this direction from the normal direction, that is, from horizontal and perpendicular to the traverse line, indicated a conducting body. Recent advances in design of light-weight generating equipment have encouraged using this and similar methods.

Jan 7, 1957

By Godfrey T. Vivian

Very often important discoveries are made in one industry that may be used to advantage in another, but, owing to the rarity that men step out of one industry into another, these discoveries remain unknown outside of the trade or profession in which they are made. The present paper is written solely with the object of bringing to the notice of mining and metallurgical engineers a centrifugal ore-concentrator which eventually may have a great application. Ore- Grading. Fig. 1 represents the action which takes place when solids of different dimensions are stirred up in a box and allowed to settle as minerals of the same size but of different specific gravity. This principle is applied practically in grading china-clay and in enriching tin-slimes. In the latter case the slimes are stirred around in a tub with water and allowed to settle while at the same time the side of the tub is struck with a hammer. The colloidal slimes are then poured off and various slices taken off horizontally. The great drawback of this process is that the particles are deposited horizontally and all the colloids are lost. Fig. 2 represents the action in which the particles are deposited side by side. Here the drawback is the great length of the box needed in order to prevent excessive loss. The alternative to lengthening the box is to increase the force of gravity. Centrifugal force may be regarded practically as the force of gravity augmented. Fig. 3 illustrates the Gee centrifugal machine for grading ores. The drum, A, fitted with a box, B, is mounted on a shaft or spindle, C, the whole being suspended from a ball-bearing of special design at D, supported between girders, E. Rota-

Jan 1, 1913

By T. M. Chance

The first bituminous coal cleaning-plant to use the sand-flotation process1 was placed in operation on Oct. 1, 1925, at the tipple of the East Broad Top Railroad & Coal Co., at Mt. Union, Pa. The general principles governing the design of plants of this type and typical operating results of anthracite sand-flotation units have been previously outlined.2 I will, therefore, confine myself to a description of this particular plant and to some of the elements of design it incorporates. The Original Mt. Union Tipple The plant was installed to transfer coal.from steel narrow-gage cars of 60,000 and 70,000 lb. capacity to standard-gage equipment. The coal was hauled about 30 miles from the East Broad Top field. This coa1, mined by the Rockhill Coal & Iron Co. and other operators, is a low-volatile slow-coking coal with ash of high-fusion point. It is normally a high-grade coal that does not contain an excessive amount of free impurities, but it requires careful handling to secure a fair domestic yield, and, naturally, the need for narrow-gage transshipment increases the difficulty of the problem. The narrow-gage cars were discharged into an 8-car track hopper equipped with two feeders passing the coal to two 42-in. apron intake conveyors. Twin units were installed throughout the original plant, two lump-and-egg shaking screens, two lump picking-tables, two egg picking-tables and two stoker conveyors. The stoker coal was loaded without cleaning. The loading tracks and booms were in duplicate and, with the layout as originally installed, either unit could be run independently, each unit shipping a different class of coal or both units making the same product.

Jan 1, 1927

By Edward F. Divers

Face ventilation is generally the most critical area in the coal mine ventilation system. Various studies have repeatedly shown that good ventilation is by far the most effective and least costly method of controlling dust and gas problems created during face operations. The Bureau of Mines has recently developed and assisted in the development of several new concepts in face ventilation systems, and in methods to accurately evaluate these systems underground. A new type of scrubber and ventilation system design built into the auger support frame of a continu- ous miner was developed. The objective was to improve face ventilation and reduce respirable dust concentrations. With this arrangement, one scrubber is mounted into each side of the boom. Laboratory and underground tests of various scrubber discharge arrangements showed good to excellent dust and methane control. Combined face ventilation systems, using both blowing and exhaust curtain, potentially offer dust and gas control and deeper cutting. Bureau tests at a mine using a combined face ventilation system and remote- control-operated continuous miners showed excellent dust and methane control. The advancement of line curtain systems beyond permanent supports without exposing workmen to roof fall hazards can be accomplished only with specially designed extendable devices. Numerous concepts for line curtain extension have been developed and tested. For the past several years, the Bureau has been testing various types of extensible face ventilation systems. These extensible systems can generally be classified as floor-, roof-, machine-, and rib-mounted systems. In the past 6 years, approximately 40 systems have been evaluated for feasibility. From these, several have been selected as practical or innovative and are presented herein. A technique developed by the Bureau to accurately and safely evaluate the effectiveness of face ventilation is also presented.

Jan 1, 1982

By C. V. Millikan

When a flowing oil well is being drilled in with cable tools, it is difficult to determine when an additional streak of pay sand is drilled. By placing a gas meter on the gas-release line from the oil and gas separator, any increase in volume of gas that comes with oil production is recorded. In addition to the better knowledge of sand conditions, these data are a valuable aid in determining the size and depth of shots, plugging off water, and estimating ultimate production. The meter also gives a record of operation while the well is drilling in; and if allowed to remain on the well after completion will give information that may lead to a more efficient recovery of the oil from the sand. A VERTICAL cross section of any oil sand will show great variation in porosity and oil content. In some sands, the entire thickness contains oil, the variation being only the degree of porosity or oil content, but more often the pay streaks are distinctly separated by shells or barren streaks. These pay strata vary in number and may be thin or thick. The total thickness of the sand may be several tens of feet while the part containing oil may be only 1 or 2 ft. thick. The amount of production is not proportional to the thickness of the sand but rather the thickness of the pay streaks. If the total thickness of the sand or the amount that can be drilled is small, or if the initial production is small, the depth at which each stratum of pay is encountered can be determined sufficiently accurately for all present economic purposes. However, when drilling through sand of considerable thickness, giving large initial production, especially where conditions necessitate that the production go through the same gas trap with that from one or more wells already producing, it is difficult to determine the depth at which the pay strata are encountered even accurately enough to guide the placing of shots, and small pay streaks are apt not to be recognized at all.

Jan 9, 1925

By R. S. Bradley

WHAT are known as insulating firebrick are lightweight firebrick with low thermal conductivity designed primarily for use in direct contact with furnace gases. These are a recent development in the refractories industry, and are really an outgrowth of the type of insulating brick suitable for use behind refractory walls developed by the insulation manufacturers. To differentiate between these two types of brick, the manufacturers have recently agreed upon the use of the term "Insulating Firebrick" to apply to the type of brick suited for use in direct contact with furnace gases, and the term "Insulating Brick" for the type suitable for use behind refractory walls. Insulation was applied to steam pipes as far back as 1900. This was a low-temperature insulation, the most common type being magnesia blocks. The next development was applying insulation to the outside of furnace walls, a service which necessitated an insulating material of higher temperature resistance. Natural diatomaceous earth, both in the powdered and brick form, was used; this was efficient and is still used extensively for backing up refractory walls where the inner surface temperature of the insulation does not exceed 1600° F.

Jan 1, 1937

By Atsushi Watanabe

Japanese Operations Research Committee for Mining Industry is one of the committees in The Union of Japanese Scientists and Engineers. Members of the committee are Dowa Mining, Furukawa Mining, Mitsubishi Metal Mining, Mitsui Mining and Smelting, Nippon Mining and Sumitomo Metal Mining. This paper is an annotated summary of study reports of the committee.

Jan 1, 1969

CONTENTS CATES, LOUIS S.-Factors Affecting Bank Slopes in Steam-shovel Operations. Discussed by Fred Hellman, Louis S. Cates, W. D. B. Motter, Jr., George S. Rice 1 CRANE, W. R.-Mine Subsidence in the Red Iron Ore Mines of the Birmingham District, Alabama. Discussed by George S. Rice, W. R. Crane, Louis S. Cates : 3 Factors Affecting Bank Slopes in Steam-shovel Operations Discussion of the paper of Louis S. CATES, presented at the New York Meeting, February, 1925, and issued, as Paper No. 1360-M, with MINING AND METAL-LURGY, August, 1924. FRED HELLMAN, New York.-My experience with shovel mining has been restricted almost entirely to Chuquicamata as far as operation is concerned. We had there an unusual condition in that the rock is cohesive because of the salts in it, so that you can probably work higher banks than is possible in most places. Recently they have reduced the height of the banks and have found that lower bank affords better working condition than the higher bank. It took two years of experimentation to determine the best height of bank for that particular mine. Each mine should be considered by itself and, to state off hand the best height of bank. for any given mine is impossible. We worked the high banks in Chile without much trouble. We adopted high banks in the beginning in order to obtain the minimum expense in railway approaches at that time. The first plan was for 50 ft.; then to get production started with the smallest possible expenditure of money, the approach to the deposit was changed and the first bank had an ultimate height of about 180 ft. We worked that bank; quite success-fully, but I do not think that it could be done anywhere else.

Jan 7, 1925

By Alexander Gow

THE object of this paper is to discuss the fundamental principles of ball milling and to present some observations which have been made in laboratory and plant investigations. The discussion will be limited to wet grinding of ores in cylindrical mills, but doubtless many of the principles involved will be equally applicable in the other fields of fine grinding in which mills of the same type are used. In the concentrators of the United States there are about 650 ball, rod, or pebble mills, operating or ready to operate, and these have about 100,000 connected horsepower. For grinding in cement plants there is about three times this connected horsepower, and other industries have an unestimated amount, all used for the general type of grinding that is to be discussed here. It is true that these figures do not represent stupendous amounts of power, but when one considers that ball milling is often the "bottle neck" in a concentrator and that it often accounts for the largest single item in the cost sheet, the importance of a compre-hensive understanding of the principles of ball milling can be appreciated. Much of the past literature devoted to the subject has been concerned with the Kick-Rittinger dispute over evaluating grinding, or with dissertations on ball paths and the closely allied impact versus attrition hypotheses, or with the exclusive consideration of certain specific varia-bles, such as mill speed or circulating load, irrespective of the other factors in operation. Such subjects may be dismissed with but little discussion in what may be called, by comparison, a new philosophy in the study of ball milling. Briefly, this "new philosophy" consists of using power as an index of the mill operation, and investigating grind-ing from the standpoint of set and induced variables, with emphasis on the latter.

Jan 1, 1934

By W. H. Coghill, A. M. Gow, M. Guggenheim, A. B. Campbell

The object of this paper is to discuss the fundamental principles of ball milling and to present some observations which have been made in laboratory and plant investigations. The discussion will be limited to wet grinding of ores in cylindrical mills, but doubtless many of the principles involved will be equally applicable in the other fields of fine grinding in which mills of the same type are used. In the concentrators of the United States there are about 650 ball, rod, or pebble mills, operating or ready to operate, and these have about 100,000 connected horsepower. For grinding in cement plants there is about three times this connected horsepower, and other industries have an unestimated amount, all used for the general type of grinding that is to be discussed here. It is true that these figures do not represent stupendous amounts of power, but when one considers that ball milling is often the ((bottle neck" in a concentrator and that it often accounts for the largest single item in the cost sheet, the importance of a comprehensive understanding of the principles of ball milling can be appreciated. Much of the past literature devoted to the subject has been concerned with the Kick-Rittinger dispute over evaluating grinding, or with dissertations on ball paths and the closely allied impact versus attrition hypotheses, or with the exclusive consideration of certain specific variables, such as mill speed or circulating load, irrespective of the other factors in operation. Such subjects may be dismissed with but little discussion in what may be called, by comparison, a new philosophy in the study of ball milling. Briefly, this '(new philosophy" consists of using power as an index of the mill operation, and investigating grinding from the standpoint of set and induced varisbles, with emphasis on the latter.

Jan 1, 1935

By Bertrand S. Summers

The technologist who has devoted much attention to the foundry-business will perhaps have noticed that the present scientific development of foundry-practice is in a condition similar to that of the steel-business some twenty years ago. In many cases chemistry is looked upon with suspicion, and the old cutand-try-methods are still in vogue, especially in the smaller foundries. The larger works have laboratories and more extensive means for applying science, and have devoted considerable time and attention to the improvement of foundry-practice. This is more especially true in the East, although in the West there are numerous firms devoting considerable attention to improving their metllods through the application of chemistry to work in the iron-foundry. It is hardly necessary to dwell extensively upon the effects of the different metalloids associated with iron, in pig- or castiron, in the way they are given in most text-books. The writer has thought proper to consider more particularly new developments in eupola-practice, particularly those pertinent to the discussion of the physics of cast-iron. In this connection some new ideas which recent practice has developed will be mentioned, and some results of practical tests along the same line will be described. Yet it has been deemed advisable also, by way of introduction, to outline the effects of the metalloids as they are described in most text-books, without going into detail. The question is sometimes asked, What is the most important element governing the quality of cast-iron 'for any particular purpose ? It would seem that right here lies the basis for the greatest difference of opinion. Some twenty years or more ago, silicon was regarded by foundrymen as one of their worst

Jan 1, 1899

By Larry Kaufman

A useful method for estimating the entropy of NaCl type compounds has been developed by combining the Debye theory of specific heat with the Lindemann formula. This method pemnits calculation of the entropy from a knowledge of the atomic weight, atomic volume, and melting point of the compound. Comparison of the calculated and observed entropies of 35 NaCl type compounds over a wide temperature range yields good agreement thus establishing the usefulness of the method. Application of the method to other crystal types is also discussed. KUBASCHEWSKI~ has suggested a useful method of estimating the enthalpy of formation of inter-metallic compounds from a knowledge of the crystal structure, lattice parameter, and heats of sublimation of the component elements. This method is quite valuable in predicting the tendency for compound formation in a given system and is of particular value in considering systems for which no experimental data are available. In order to supplement the enthalpy calculation afforded by Kubschewski's method, it would be of value to be able to estimate the entropy of compounds so that the entropy and free energy of formation could be evaluated. Although a considerable body of such data determined by calorimetric and equilibration techniques exists at present, the difficulty of making such measurements (particularly at high temperatures) and the ever increasing number of new compounds which are of interest make the desirability of such a computational method clear. This paper deals with a method for calculating the entropy of a compound from a knowledge of the atomic weight, atomic volume, and melting point. Since these data are generally easier to determine and therefore more readily available than thermo-dynamic data, they should provide an additional tool in estimating the stability and reactivity of compounds at high temperatures. I) THE ENTROPY OF PURE METALS As a first approximation, the specific heat of a pure metal can be represented by a sum of terms where C,($/tI is the vibrational specific heat calculated on the basis of the Debye theory, lo4 TC, (0/T) is an approximation to the Cp - C, correction. T is the electronic specific heat, and Cp (p) is the magnetic specific heat. Although the Debye theory does not provide a complete description of the vibrational specific heat, i.e., the Debye temperature 0 is not a constant, it does provide a good approximation when used properly. Blackman6'7 has written several reviews on this subject discussing the different values of 0 determined by various methods (specific heat, X-rays, elastic constants, electrical resistivity, and so forth), as well as the observed temperature dependence of 0 with T. In the present case the 0 in Eq. [I] naturally refers to the specific heat 0. The variation of Debye 0 (as determined from specific heat measurements) with temperature is observed to exhibit its most radical behavior at very low temperatures. At higher temperatures the 8 values are observed to become more nearly constant. Blackman6'7 has presented the variation of 0 with temperature for several metals. His tabulation is reproduced in Table I. Reference to Table I shows that above a certain temperature level, 0 remains fairly constant. The value of 0 which is most useful in computing the specific heat, and hence the entropy, is the value of 0 in the range 0/6 « T « 0. In 1910, Lindemann' suggested a relation for estimating Debye 0 values. According to this formula, where V is the atomic volume, T is the melting point in 'K, M is the atomic weight, and CL is a constant. Mott and Jones9 present this formula and state that there is at present no satisfactory theoretical explanation, but that it does have an empirical basis. By using the valuesof 0 in Table I along with known1' values of M, T, and the molar volume VM, where the Lindemann constant can be evaluated. This

Jan 1, 1962