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By Giovanni Barla

In engineering analyses, rock is frequently treated as a homogeneous, isotropic, and linearly elastic medium. However, rock material exhibits, in most cases, physical nonlinearity, time-dependency, and anisotropy. In some instances, anelasticity due to nonmechanical effects, such as density and temperature changes, must also be taken into account. The influence of these different anomalies in the physical behavior of a rock material is often neglected. An estimate of the discrepancies which are introduced when classical theory of elasticity is used is consequently required. The easiest theoretical path by which these discrepancies can be analyzed is to vary the constitutive equations of the rock material. The constitutive equations, representing various physical experiences with different types of media, define a class of ideal substances. Assuming different constitutive equations for the description of the rock physical behavior, one can establish a sensitivity analysis to determine how much information is contained in elastic theory for reliable design purposes. The method may be applied also using numerical techniques (e.g., finite element or finite differences) which, because of the nature of the formulation, require one to assume in each case the rock constitutive equations. Theoretical and experimental work is needed to be able to perform such analyses with confidence. Adequate and practical procedures for the determination of the material functions which enter the various constitutive equations must be established. This paper presents such an attempt with respect to the simulation of some behavior patterns of rock material. It is felt worthwhile to stimulate activity in this field and it is hoped to contribute to this objective.

Jan 1, 1970

THE object of this paper is to offer some suggestions to assist metallurgists to supervise the operations of flotation plants. We do not yet thoroughly understand what "flotation" is or "why minerals float" and all attempts that have been made to explain the phenomena have been based on theory or imagination, and have not been capable of substantiation by practical demonstration, or even by laboratory tests; but it is wellknown that particular minerals will float in a liquid under certain specified conditions, and that some other minerals will not float under those conditions, and that, if a mixture of the two classes of minerals be subjected to those conditions, there will be separation of the minerals, of which one class will float and the other will remain unfloated as a residue. The duty of the flotation expert is to so control the operations and the conditions of the liquor as to obtain the best commercial result from the ore he is treating. The chief controlling factors in a flotation plant appear to fall under the following eight headings, all of which are important:1. Crushing.2. Rate and regularity of feed.3. Density of pulp.4. Temperature.5. Addition agents.6. Condition of circuit liquor.7. Agitation.8. Aeration.

Jan 1, 1920

By Persifor Frazer

TILE ore property of Mr. Augustus Roop is situated about one and a half miles south of the New Windsor Station of the Western Maryland Railroad and is connected with the town of New Windsor by the Liberty and New Windsor Turnpike. It is about forty miles distant from Baltimore, and about seven miles west from the county town of Westminster, and a little more northeast from the celebrated Liberty Copper Mines. (See accompanying Maps 1 and 2.) The property comprises about 140 acres, lying principally

Jan 1, 1881

By B. M. Larsen, T. E. Brower

INTRODUCTION THIS discussion is based on statistical manipulation and evaluation of operating data from several commercial blast furnaces which include rather wide variations in practice. We are concerned here mainly with the effect of manganese on sulphur elimination under operating conditions in commercial blast furnaces, but this effect can be outlined more clearly by including certain other variables such as silicon in iron and slag basicity, which are all in part interrelated. Real progress in understanding all of the smelting-zone reactions in this process requires more laboratory studies of slagmetal melts similar to those in the blast furnace, with careful control of oxygen pressure. Such experimental work is very difficult, however, and in the meantime, with the present scarcity of more fundamental data, such statistical trends and correlations as can be obtained from commercial furnace operating records may help in choosing laboratory test conditions as well as adding to our knowledge of operating procedures. We do not know just why the furnace data are apparently so irregular that statistical treatment of large numbers is needed to obtain accurate trends. It is .partly caused, no doubt, by the many simultaneous variables present, but there are perhaps also such factors as: (I) the continual swings up and down so characteristic of the process; (2) difficulty of accurate sampling of co-existing phases, as well as (3) the presence of both equilibrium and rate factors as a result of the fast driving of the process, In desulphurization, for example, the flow of thin layers of slag and metal over solid coke surfaces, plus the contact between droplets of iron falling through the slag layer in the hearth, would lead one to expect a close approach to equilibrium, On the other hand, most of the critical part of sulphur removal may occur in a short time and distance below the tuyeres with perhaps an accompanying decrease in oxygen pressure and the concurrent reduction of much of the silicon and manganese, so its degree of approach to equilibrium may be quite variable. PROBABLE CHEMISTRY OF DESULPHURIZATION Since there is little evidence that sulphur atoms in the iron solution are associated to any appreciable extent in molecules such as MnS, the transfer from slag to metal and back again can be written:* [(FeS) -, [FeS],[I] hence the following reactions seem most probable in the slag phase, (FeS) + (CaO) = (CaS) + (FeO), [2] and, (FeS) + (MnO) ;:2 (MnS) + (FeO). [31]

Jan 1, 1948

By Anthony Sharkey

Introduction In Europe, longwall mining accounts for over 95% of coal produced from deep mines, whereas in the U.S.A., longwall accounts for less than 5% of deep mine coal production. There are many reasons for the difference, but in general: a. U.S. mine operators have invested heavily in continuous mining or conventional equipment which continues to produce acceptable results; b. There 1s a reluctance to change from, a familiar (and proven) mining method to an unfamiliar one, despite promises of huge production increases,; c. Multi-seam mining in Europe has made longwall the only really viable mining method, whereas In the U.S.A. multi-seam mining (particularly in "below-drainage" mines), is relatively rare; d. Although some spectacular results have been achieved by longwall in the U.S.A.(I), the general performance of longwall in this country has not been good enough to generate much enthusiasm for its wide application; e. The much higher level of investment in face equipment required for longwall when compared to alternative methods (approximately $7.5 million for a medium seam longwall installation and $1 million to $1.5 million for a continuous miner section). However, declining productivity, national energy requirements and safety legislation are creating a situation in which longwall as a viable mining method is being examined more closely and in more places than hitherto. This paper, therefore, attempts to identify some of the principal criteria needing examination in any assessment of the feasibility of longwall application, the limitations of information for those criteria in the decision making process and the need for further research and development to assist the mine operators make the decisions.

Jan 1, 1981

By M. Reuter

This paper gives an overview of some criteria for the selection of processes for the recovery of lead and zinc from various raw materials and/or intermediate products. A glossary of possible processes and process variants, is presented and discussed, as well as the way in which these are interlinked. Recent developments in metallurgical process synthesis are also discussed, and the results produced by an eco-techno-.economic synthesis procedure for process routes in zinc metallurgy are given briefly. The way in which such a synthesis can complement a life-cycle assessment (LCA) of zinc-producing technologies is also described.

Jan 1, 1997

By H. A. Steane

"This presentation will cover some crushing and grinding considerations in plant design and operation. There is nothing basically new in the subject matter presented here; however, the viewpoint and interpretation may be novel and some of it controversial.The presentation is based on experience gained in the metallurgical design of plants for Placer Development Limited from 1960 to the present. The plants are listed on SLIDE No. 1.These plants have conventional crushing, followed by rod mill and ball mill grinding. I would like to think that autogenous grinding plants are the answer for eliminating secondary crushing plants and reducing the capital and operating costs; however, the autogenous plants seem to have some problems of their own and I have had no experience with them.Fineness of Grinding RequiredFirst let us consider the fineness of grinding required.The fineness of grind will affect the recovery obtained, and the relationship must be determined. If the grinding is too coarse, some of the valuable mineral, either free or locked in middling grains, will not be floated. However, as grinding is probably the greatest single operating cost, we do not want to grind any finer than is justified economically.Finer grinding should not be carried beyond the point where the net smelter return for the increment saved becomes less than the added operating cost. Even this fineness will be beyond the economic limit because of the additional capital cost for the equipment needed to achieve it. These factors will be discussed in more detail later.The Bond work index is the commonest method of expressing a measure of the effort that has to be expended in grinding. It is the kilowatt-hours required to reduce a ton of rock from an infinite size to 80% passing 100 microns. The formula allows a calculation to be made for other size ranges."

Jan 1, 1976

By I. R. Ward

"The need for energy conservation has been of increasing concern since the early 1970's, when the cost of petroleum based energy began to escalate rapidly and security of supply became uncertain. The cost of energy from alternative sources has also increased due to development costs and environmental restrictions.These developments have had a serious impact on the mining industry, which is energy intensive. The impact is magnified as lower grade ores are treated, and more energy is consumed per unit of product. In many cases increased operating cost is not compensated by the rise in product selling price.The magnitude of energy consumption in the mining industry is indicated by recent studies conducted in the U.S.A. and Canada. Based on 1970 figures, it was reported that the production of 1 ton of copper in the U.S.A. required 111. 8 x 106 BTU of energy, of which 60. 7 x 106 BTU (54.3%) was consumed in mining and processing (l)The survey of Canadian mills, conducted by CANMET and reported to the CMP meeting of 1978 (2), showed the following milling energy consumption figures (approximate averages):"

Jan 1, 1981

By Courtenay de Kalb

REVISION of the United States mining law is needed. chiefly because of the following reasons: 1. The conceptions as to the characteristics of orebodies that were held at the time the statute of 1872 was enacted have since proved to be in many respects erroneous. 2. The operation of the existing law has been found to work unnecessary hardship upon many claimants to mineral land. 3. The existing statute fails in large degree to encourage development of mineral resources. 4. The existing law often produces conflict between placer and lode-claim rights, working unnecessary injustice to placer claimants. 5. The methods of establishing the rights of locators on United States mineral land are defective, and are not uniform throughout the States within which such lands exist. LAW OF THE APEX The objections to the so-called "law of the apex" are so well known as to require no detailed statement. It is sufficient to point out that the apex rule was based upon the notion that ore deposits consisted of well-formed veins included between definite walls, like the slice of ham in a sandwich. The fact is that veins conforming to that simple type are rare. Irregularity is the rule, and the chief metal production of the United States comes from masses which bear no resemblance to the tabular form of vein. Manifestly the law of the apex fails completely in such cases. Moreover, the tendency is constantly toward the exploitation of ores of lower grade than could formerly be worked, and such deposits exist mainly in the form of large irregular impregnated zones and masses, so that the future. development of our mineral resources would be continuously embarrassed by shadowy claims to extra-lateral rights by the locators of so-called apexes

Jan 2, 1915

Discussion of the paper of COURTENAY DEKALB, presented at the New York meeting, February, 1915, and printed in Bulletin No. 98, February, 1915, pp. 331 to 337. HORACE V. WINCHELL, Minneapolis, Minn.-Mr. Dehalb summarizes the reasons which he believes are important for the revision of the mining laws, as follows: "1. The conceptions as to the characteristics of orebodies that were held at the time the statute of 1872 was enacted have since proved to be in many respects erroneous." We all appreciate the force of that suggestion. The idea of the old prospector who made rules and regulations in the early days, which were attempted to he incorporated, so far as possible, in the Mining Act of 1872, was that all of the veins in a certain district outcropped upon the surface, or approximately parallel to each other upon the surface, and maintained that parallelism beneath the surface. Therefore, it seemed perfectly proper and natural for each man to stake his claim upon an outcropping, and have the right to follow that vein in working downward, with the general expectation that his workings would always remain just as far away from those of his neighbor as the outcroppings upon the surface. This does not happen to be the case. Dr. Raymond has very ingeniously referred to the subject in one case, in words which read as follows: "If all mining properties presented this beautiful simplicity of structure (ideal location of ideal vein), and all mining locators exhibited a corresponding simplicity of purpose, the application of the law would be easy, but the naiveté of the statute fares badly between the freaks of nature and the tricks of man."

Jan 5, 1915

Environmental considerations were key factors in the design of the Ranger Uranium Mines treatment plant. The mine is located adjacent to the Kakadu National Park and has an average rainfall of 1.6m per annum. Initially, no contaminated water or liquid effluents are to be released from the project area and thus water management is a key design and operating fact. Particulate and gas emission criteria influenced design as did occupational hygiene factors (dust, radon, housekeeping, maintenance access). Equipment selection and engineering standards were conservative and resulted in the plant attaining design performance in less than three months from the date of commissioning. A number of mechanical and operational problems were experienced and are discussed in the paper, however none of these problems have had a significant effect on production.

Jan 1, 1984

By D. J. McParland

This paper will deal with some of the considerations which, were involved in the design and construction of the uranium mills in the Elliot Lake area . These plants, eleven in number, range from 1500 to 6500 tons per day milling capacity. Most references are with respect to the seven plants in the Rio Tinto Group which have a combined milling capacity in excess of 20,000 tons per day. In dealing with such abroad subject as mill design, one has the choice, of covering a few specific problems in great detail, or else attempting to cover many subjects superficially and briefly. In preparing this presentation, an attempt has been made to compromise on these extremes, and to select some interesting illustrations which are of general interest in the field of mill design, and of value to persons concerned with mill design broadly, beyond the field of uranium mills.

Jan 1, 1959

By George D. Roberts

INTRODUCTION CIVIL engineers are primarily interested in maximum core recovery. This is even more important in foundation work than in mining investigations where sludge samples are of some value. The soft materials, such as clay, gouge, or partially decomposed seams that are most readily ground up or washed away, are the materials that usually have the greatest influence on foundation and leakage conditions. Undisturbed samples of such materials should be obtained for detailed geological examination and laboratory tests. Therefore the attainable goal of sampling in connection with foundation investigations is the recovery of 99.9 pct of undisturbed core. FACTORS CAUSING CORE LOSSES The principal factors contributing to core losses are: (I) coring badly fractured or very soft material; (2) use of improperly designed coring equipment, including the use of bits of too small a diameter; (3) lack of the proper training and attitude on the part of the drillers and pertinent technical information on the part of the supervising personnel. Little can be done about the character of the material being cored but the other two factors contributing to unsatisfactory performance can be corrected. Although the title of this paper is the improvement and standardization of core barrels, it is believed that the drillers' shortcomings and the lack of pertinent technical literature are more detrimental to good core recovery than either of the other factors. USES FOR LARGE DIAMETER CORE BARRELS It is well known that the use of larger diameter bits and core barrels generally produce better cores, particularly when sampling soft or broken material. The use of larger and larger bit diameters and the insistence on better core recovery is well demonstrated by the change in practices of the various districts of the Corps of Engineers. Today NX core is the smallest diameter commonly permitted for investigational work and the more active districts are using coring bits up to 6 in. in diameter, as well as 36 and 40-in. calyx rigs. The latter size has proved to be just as economical because it provides more room for core removal and cleanup operations. The use of large diameter cores to replace the driving and jacking methods of obtaining soil samples is also increasing. The 5 7/8-in. Denison sampler, developed by H. L. Johnson to obtain samples from soft formations, is one example. The principal uses for bits larger than NX are: sampling soils and semi-consolidated rock; sampling decomposed rock or rock with soft seams; sampling concrete; obtaining samples for laboratory tests; drilling drainage holes, and other miscellaneous work. CORE DRILLING PRACTICES OF SEVERAL ENGINEER DISTRICTS Little Rock District The writer was employed in the Little Rock District from 1938 until 1942. Prior

Jan 1, 1947

By T. G. Edgeworth

Some of the historical background of the industry is reviewed, with reference to the different types of reduction cells in use for aluminum smelting. Factors to be considered in cell des1gn involve capital, power, and operating costs. The trend is apparently toward larger cells. The industry is studying new construction materials for cell linings, as well as alternate electrolyte compositions and other design changes which will improve productivity and energy efficiency. The aluminum smelting activity in other countries is briefly reviewed. TODA Y'S industrial production of aluminum is still based on the simultaneous discoveries made in 1886 by Charles Martin Hall in Ohio, and Paul Heroult in France. Up to that date, only about 50 tons of aluminum had been produced in the world, and the price at that time was about $5 per pound. Sucl1 figures must be contrasted with the present world aluminum smelting capacity of over five mil-lion tons per year, and a price of about 25 cents per pound.

Jan 1, 1962

By C. R. Austin

THE investigation described in this paper deals with the development of high-temperature alloys of the Konel series over a considerable period of time at the Research Laboratories of the Westinghouse Electric and Manufacturing Co. The word "Konel" has been coined to designate a class of alloys which are essentially nickel-cobalt-iron with other elements added, usually in small quantities. Most of these alloys contain titanium, usually about 2.5 percent. The main consideration related to strength at high temperatures but the thermionic characteristics of the alloys was also of considerable importance. It is, however, with the mechanical properties at elevated temperatures that we are herein concerned. Much attention has also been devoted to their age-hardening characteristics. There is little need to stress the utilitarian aspect of this subject, as it has been discussed frequently. Alloys exhibiting superior properties at normal or elevated temperatures rapidly find many suitable applications. There may be some question, however, as to what constitutes a "superior alloy, " or an alloy that may have good service possibilities at high temperatures. The problem is complex and various means have been employed to predict the useful life that may be expected of a material. In recent years the "creep limit" has been used as the best means of evaluation. Such tests necessarily are prolonged and hence collection of data is very slow. In a general survey of many alloys therefore the creep test is not desirable and recourse must be had to much shorter time tests in order more rapidly to select alloys which appear to show promise. For this reason the present survey has been made using the high-temperature tensile test and a bend' test developed by Howard Scott.1 In the tensile test the conditions have been standardized for all tests, so that values obtained are at least comparable, one alloy with another, although the absolute values recorded merely indicate the mechanical properties at the given temperature under a definite rate of loading.

Jan 1, 1931

By C. R. Austin

Tan investigation described in this paper deals with the development of high-temperature alloys of the Konel series over a considerable period of time at the Research Laboratories of the Westinghouse Electric and Manufacturing Co. The word "Konel" has been coined to designate a class of alloys which are essentially nickel-cobalt-iron with other elements added, usually in small quantities. Most of these alloys contain titanium, usually about 2.5 per cent. The main consideration related to strength at high temperatures but the thermionic characteristics of the alloys was also of considerable importance. It is, however, with the mechanical properties at elevated temperatures that we are herein concerned. Much attention has also been devoted to their age-hardening characteristics. There is little need to stress the utilitarian aspect of this subject, as it has been discussed frequently. Alloys exhibiting superior properties at normal or elevated temperatures rapidly find many suitable applications. There may be some question, however, as to what constitutes a "superior alloy, " or an alloy that may have good service possibilities at high temperatures. The problem is complex and various means have been employed to predict the useful life that may be expected of a material. In recent years the "creep limit" has been used as the best means of evaluation. Such tests necessarily are prolonged and hence collection of data is very slow. In a general survey of many alloys therefore the creep test is not desirable and recourse must be had to much shorter time tests in order more rapidly to select alloys which appear to show promise. For this reason the present survey has been made using the high-temperature tensile test and a bend test developed by Howard Scott.1 In the tensile test the conditions have been standardized for all tests, so that values obtained are at least comparable, one alloy with another, although the absolute values recorded merely indicate the mechanical properties at the given temperature under a definite rate of loading.

Jan 1, 1931

By L. D. Fetterolf, G. T. Mahler, W. M. Peirce, R. K. Waring

UNTIL recently, the only commercial method of producing magnesium has been fused salt electrolysis, despite a considerable amount of experimental work on the direct reduction of magnesium oxide. In this early work, a number of investigators demonstrated that magnesium oxide would react with any one of several reducing agents to produce magnesium vapor, but that the mixture of magnesium oxide and reducing agent must be heated to a high temperature even in a vacuum. In particular, it was demonstrated that silicon and aluminum are effective reducing agents and that when either of these is used the magnesium vapor can be condensed to massive metal. In spite of this knowledge, no successful commercial reduction process was developed, at least in this country, because of the operational difficulties imposed by the requirement of high temperature and vacuum and because of the high cost of the reducing agent. The outstanding experimental work done by Pidgeon, who carefully reviewed all of the known methods of making magnesium and then selected ferrosilicon reduction, overcame some of the operational difficulties. The process developed by Pidgeon comprises briquetting a mixture of calcined dolomite and reducing agent, pre-heating the briquets to about 600°C. in an externally heated alloy retort, reducing the MgO with silicon in an evacuated horizontal alloy retort heated externally to about 1150°C., and condensing the magnesium vapor to a solid in a cold extension of the retort projecting beyond the fr6nt of the furnace. A major advantage of Pidgeon's process is that the retort is not cooled down between charges. Earlier investigators had used apparatus that necessitated cooling and reheating of the retort between charges. Early in January 1942, when the War Production Board was considering the Pidgeon process for quickly increasing the production of magnesium, The New Jersey Zinc Co. offered to have a research investigation of the process carried on, with the hope of assisting in its further development. This offer was accepted, and a two-retort pilot plant was built at Palmerton and put in operation on Jan. 29, 1942. An intensive investigation of certain phases of the process was carried out in the ensuing four months. This investigation was completed in the last part of May 1942, and the pilot plant was then shut down. Subsequently, The New Jersey Zinc Co. was requested to have further development work done on the process, and a more extensive investigation was undertaken in August I942 under contract with the War production Board and under the super- vision of the War Metallurgy Committee. This paper describes the phases of the investigation that are thought to .be

Jan 1, 1944

By A. L. Garassino

Dans cet article on parle sortout de l'approche de calcule pour les pieux charges horizontalement avec particuliere reference aux curbes p-y et è une solution semi-empirique qui prende en consideration les variations typiques du module du sol avec la deformation obtenue par les données expérimentales. 1. INTRODUCTION The analysis of the piles under lateral loads depends very strictly from the pile stiffness and geometry, from the soil type and from the strain field. Different approaches were adopted in solving the problem. Some Authors developed methods for estimating the ultimate lateral resistance of piles and pile groups by conventional static approaches, as the Brom's widely known (1964) or by the plane strain solution developed by Davis (1961) or by empirical engineering con­siderations l

Jan 1, 1994

By Martin G. Casto

Mixer-settler extractors, among the oldest forms of extraction equipment, have been used since the beginning of this century. Over the past years, a very large number of mixer-settler devices have been designed and operated. Both batch and continuous flow mixer-settlers have been used. The continuous flow mixer-settler extractor offers a great number of advantages over other extraction equipment. The main advantage of the continuous flow mixer-settler is that it is a compact device. A large number of stages can be assembled in a relatively small area, and more stages can be added fairly easily. Many improvements have been made in the design of mixer-settlers. These improvements occurred in almost every phase of mixer-settler' operation; interface control, mixer design, intrastage pumping, interstage flow of the two liquid phases, and stage construction. It is believed that Barstow1 used one of the first continuous mixer-settler extractors. This mixer-settler had concurrent flow within each stage and countercurrent flow overall. The flow for both phases took place by gravity. The Barstow device and other early mixer-settlers had no separate interface control for each stage, therefore, "tilting" of the stages was necessary. In 1934 Holley and Mott2 designed a mixer-settler in which the heavy phase was recirculated within each stage by the use of a valve controlled central line instead of by a pump. Gordon and Zeigler3 in 1939 were among the first to use an adjustable valve to control the interface levels in a mixer-settler, making tilting of the extractor unnecessary. In 1952 Coplan et al. 4 used a modification of the Holley-Mott idea to build a highly improved mixer-settler with flat common walls, and with the

Jan 1, 1971

By W. Philippoff

ALTHOUGH Gaudin1 and more recently Sutherland2 have calculated the probability of collision of a falling mineral particle with a rising bubble, there is no published information concerning the details of the mechanism of attachment of a collector-coated particle to a bubble. During the past year the writer has developed a theory for the mechanism of attachment, which has been substantiated experimentally.' Funds for the investigation and for some of the equipment used have been supplied by the Mines Experiment Station of the University of Minnesota. Motion picture studies of the phenomena involved in the collision between mineral particles and bubbles, such as those of Spedden and Hannan,3 show that the contact can be completed within 0.3 millisec. Formulas developed for rigid bodies have hitherto been used' for the calculation of the motion of bubble and particle, but it is obvious that a bubble cannot be regarded as a rigid body. On the contrary, Spedden and Hannan's pictures show a great degree of deformation during the collision. The time of attachment was calculated as the time the particle drifted past the bubble. Time of Collision The theory presented in this paper enables calculation of the time of collision; using the concept that the bubble, or more generally, a liquid-air interface, acts as an elastic body. The elasticity, defined as the restoring force on a mechanical deformation, is caused by, the surface tension and is the result of the principle of the minimum of free surface energy. It is well known that an elasticity together with a mass determines a frequency of vibration. The vibrations of jets and drops caused by the elasticity of the interface are known to comply exactly with the classical theory of capillarity.5 However, the vibrations of isolated bubbles, as distinct from foams, have not been investigated previously. The following equation, presented elsewhere,' has been deduced for these frequencies: [3fB = 9.20•'./V•Vn- (n-1) • (n+2) /8[1]] in which fB is the frequency of a harmonic of the bubble in cycles per second, V the volume of the bubble in cc, n a number determining the order of the harmonic, and n = 2 the basic vibration. The first (basic) harmonic describes a change of the spherical bubble to an ellipsoidal bubble. The higher harmonics are more complicated, for the circumference of the bubble is divided approximately into as many parts' as the order of the harmonic. As an example, Spedden and Hannan's published motion picture of, a vibrating bubble corresponds to the sixth harmonic. Eq 1 shows that only the first and third harmonics are simple multiples (1 and 3), all the others being irrational fractions of the basic frequency. This means that the shape of the vibration can change with time and is in general unsymmetric in respect to the time axis. Such conditions prevail when there is a distributed elasticity or mass, as in the case of vibrating membranes or rods. The constant 9.20 is valid for water at room temperature, but a general solution involving the physical constants of the liquid has not been found. The case of the floating particle is much easier to treat I than that of the bubble. It can be assumed that the elasticity is caused exclusively by the interface and that the mass is concentrated in the particle together with some adhering water. The following expression for the frequency of a system, of one degree of freedom can be applied: [1E/m[2] fP = 27] Here f, is the frequency of the particle vibration in cycles per second, E the elasticity in dynes per cm, and m the mass in grams. The classical theory of impact phenomena gives the time of collision during the striking of a spring (in this case the surface of the bubble) by a mass, as: [t~ = 2/f = 7r\/m/E[3]] It is now possible to develop an expression for the elasticity of a floating cylindrical particle. The force equilibrium of a cylinder floating end on at the air-liquid interface is given by the well-known equation (Poisson' 1831) [aP = 4 D2.pL•g•h +7rD•y sin a[4]] which accounts for the buoyancy and the action of the surface tension where P is the force acting on the particle in dynes (weight-buoyancy), D the diameter of the cylinder in cm, pL the density of the liquid in grams per cc, g the acceleration of gravity = 981 cm per sec2, h the depression of the cylinder below the surface of-the liquid in cm, y the surface tension in dynes per cm and a the supporting angle' or the one required to insure equilibrium, a being smaller than the contact angle ?. Although demonstrated by Poisson, it has not

Jan 1, 1952