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By E. C. Pechin

Secretary's Note.—Mr. Bramwell was born in 1846, at Liverpool, England, his father being English, and his mother American, of Scotch ancestry. Brought to the United States in early childhood, he was educated at the public schools and graduated from the College of the City of New York. He subsequently studied his profession as a mining engineer at the Royal School of Mines, Berlin, Prussia, and was for a short time in the service of the Prussian government at Konigs hutte. Returning to this country in 1871, he was employed during succeeding years as mining engineer, chemist, and manager at various fnrnaces and iron works in Pennsylvania and the Southern States until, in 1883, he became actively engaged in the development of the Flat-Top coal-field of southwestern Virginia, in which region the town of Bramwell was named after him, in recognition of his services as a pioneer of the great industry which he did so much to promote. After prolonged and fluctuating ill-health, he died, July 13, 1894, at Paris, France. At the opening session of the Bridgeport meeting, the Secretary announced his death, and Mr. Pechin, being called upon, made the remarks reported below.— R. W. R. Widely as Joseph Herbert Bramwell was known, I. think I knew him somewhat better than any one else. After finishing his studies abroad, he came to this country, a very young man, seeking an engagement. This was about the beginning of 1871, as I remem ber, and I was then trying to make iron at Dunbar, at the foot of the Allegheny mountains, in Fayette county, Pa. His engaging manner and charming personnel, combined with his letters of intro duction, made me more than satisfied in securing his services as my chemist. At the outset we lived a sort of pioneer life, and for many months he was practically a member of my family, and the affection then formed remained unbroken to his dying day. I vividly recall the circular of April, 1871, signed by Eckley B. Coxe, R. P. Rothwell and Martin Coryell suggesting a meeting at Wilkes-Barre, Pa., for the purpose of forming the American Institute of Mining Engineers. At the last moment, prevented from going, I sent Bramwell; and I am more than glad that his name appears among those whose sagacity established an Institute that has become illustrious by reason of its numbers, the character of its members, its invaluable papers and its enormous influence for good

Jan 1, 1895

By J. A. Herbst, A. L. Mular

Digital simulation is defined and simulation techniques are re- viewed. Typical methodology involves the selection of models, the adjustment of raw data, the selection of a fitting strategy, the assessment of model accuracy in a statistical sense, and the development of an executive simulation structure. Simulations serve as aids for process engineers in evaluating the feasibility of a proposed flowsheet and the effect of a modification to an existing flowsheet. In addition, simulation plays an important role in off-line optimization and in the evaluation of alternative circuit control strategies. Several examples are presented to illustrate these uses for simulation including applications in crushing circuits, grinding circuits, and flotation circuits. It is the opinion of the authors that, within the next decade, mineral processing plants will be studied routinely via digital simulation during the preliminary design stage. Simulation will constitute an important component of the design decision making process.

Jan 1, 1978

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

A perspective is presented of the steel plant sulphur distribution and elimination problem from coal to liquid steel ready for teeming, giving distributions of sulphur over a range of coke sulphur content, and some methods of sulphur control, in the blast furnace, external desulphuriza-tion between blast furnace and open hearth, distribution between fuel, slag, and metal, and methods and limitations of control of sulphur in the open hearth furnace. AS a part of the 1951 AIME symposium on sulphur in steelmaking, it was thought that a discussion of the distribution of sulphur throughout the whole series of operations, from coal and ore to finished steel ingots, might have some value in giving a perspective on the whole problem. The following discussion is an attempt to present such an overall picture. The order is that of the actual plant operations, beginning with a very brief consideration of the coking process. Sulphur in Coal and Coke Since by far the largest source of sulphur entering the steelmaking cycle is in the coal used to make coke for the blast furnace, it would seem reasonable to eliminate some of it, either from the coal, or the coke, or during the coking process. This has appeared impracticable up to the present, at least, for two main reasons: the low activity of the organic sulphur in either coal or coke, and because of price limitations involved in treating a low cost material such as coke. A variable portion, usually M or less, of the sulphur is present in coal in the form of pyrites or similar compounds, and a large part of this sulphur may be removed in the coal washery. Most of the sulphur, however, is normally present as "organic" sulphur, intimately associated with the coal structure. Its distribution prevents any separation by mechanical means. Its low activity makes improbable ' any rapid chemical removal, although hydrogen will remove sulphur from both coal and coke. Thus, prolonged recirculation of coke oven gas in the coking process would tend to leave a smaller percentage of the total sulphur in the coke residue. Table I shows a typical distribution of sulphur from coal into products in the coking process. As the sulphur in the coal increases, the sulphur in the coke tends to increase in about the same proportion. Sulphur in the Blast Furnace The best picture of the situation in the blast furnace is provided by a sulphur balance of raw materials entering, and of products leaving, the furnace. The difficulties in accurate weighing and sampling of the variable solid materials entering this process, and the number of hours required for the raw materials to descend through the furnace under variable operating conditions, make it difficult to obtain an accurate balance. However, balances made over periods of weeks or months tend to average out some of these uncertainties. Table I1 presents three typical sulphur balances similar to a number that the writers have calculated. In most of these the slag volume calculated from the sulphur balance is, in some instances more, and in other instances less, than the value corresponding to the best input and output balances of the other slag constituents (lime, silica, alumina, etc.). Probably the greatest source of error in these calculations is the sulphur content of the slag. Despite some possible inaccuracies the balances of Table II show rather definitely the following points: 1—That 87 to 95 pct of the total sulphur input is in the coke and 95 to 97 pct of the total sulphur output is in the slag. Also, that if any sulphur leaves the furnace with the gas it is relatively small, amounting to a possible 1 pct or less. 2—At the lower sulphur coke level of 0.86 pct the total amount of sulphur charged is 15 Ib of sulphur per ton iron increasing to 26 lb per ton at the higher sulphur, intimately associated with the coal struc-rare burdens containing sulphur-rich ores will the total sulphur burden fail to be nearly proportional to the content in the coke used. 3—The 7 to 9 pct of the total sulphur input from the limestone of furnaces B and C is due to the relatively high sulphur content of the stone, 0.226 and 0.265 pct, respectively. In the case of furnace A, the sulphur content of the limestone was only 0.06 pct which resulted in only 3 pct of the total sulphur input coming from this source. It is rather interesting to compare the sulphur balances of a typical ferromanganese furnace with

Jan 1, 1952

By B. C. Craft

The outstanding event of the year 1932 in this area was the intensive drilling activity in the Zwolle field of Sabine Parish, Louisiana. More new development work was undertaken and completed than during the previous year. This was due in part to the increased price of oil and in a Includes 19 parishes: Beauregard, Bienville, Bossier, Caddo, Caldwell, Catn-houla, Claiborne, Concordia, DeSoto, Grant, Lincoln, Morehouse, Natchitoches, Ouachita, Red River, Richland, Sabine, Union, Webster. B H. K. Shearer: Oil and Gas Development in South Arkansas, North Louisiana and Mississippi in 1931. Trans. A. I. M. E. (1932) 98, 159. a Includes 14 counties: Ashley, Columbia, Dallas, Drew, Hot Springs, Lincoln, LnFayette, Little River, Miller, Ouachita, Pike, Sevier, Scott and Union. * H. K. Shearer: Oil and Gas Development in South Arkansas, North Louisiana and Mississippi in 1931. Trans. A. I. M. E. (1932) 98, 159. part to the fact that production is below local market demands. Exploration during 1932 resulted in the discovery of a new oil field and a new gas

Jan 1, 1933

THE discussion of this topic at the Open-hearth Conference at Detroit on Nov. 3, 1927, was opened by statements of the general principles involved, made by manufacturers of molds. Because of their general interest they are printed below. WILLIAM HAIL RAMAGE,* Hubbard, Ohio.-I did not come prepared to read a paper, thinking. the most I would be called upon to do would be to make a few, remarks by way of discussion. A number of our friends have pre-viously asked questions. similar to this which really does not permit a direct "yes" or "no" answer. If I may he pardoned for so doing, I believe it advisable to discuss rather what we consider the principal factors which in-fluence mold life. I think possibly a little undue weight has been given to the subject of the analysis of the metal. For the last eight years we have spent a lot of time, effort, and money investigating this very problem from a chemical, physical and microscopic standpoint. We have had the benefit of complete records and history of over 100,000 molds in service at a variety of plants and over long periods of time. Obviously, I cannot be so specific that I should divulge information that is really a trade secret. In our opinion the principal factors influencing mold life are: (1) design of the mold; (2) process of manu-facture, including workmanship; (3) care in the steel plant or what might be termed "the use or abuse" in the steel plant; (4) the analysis of the iron and the physical con-dition of it. Under normal conditions these factors are of importance in the order named, although in specific cases they might be rearranged in any order you choose. For instance, the best mold made can be destroyed in one or two pours by subjecting it to certain conditions which are occasion-ally met in steel plant practice. Obviously, in that case, nothing else matters but that one item. On the other hand, I have seen molds made of such a terrible design that it was a foregone conclusion it would have to be broken after the second pour of the ingot to get it out. In that case, nothing but design mattered. As to the effect of process of manufacture, let me cite a case. Some years ago there were two batches of molds sent to a certain company. They were the same sized molds, used under the same conditions and in the same kind of service. There were, 20 molds in each batch. The second batch sent gave ten times the life of the first batch. It is true that the design was changed a little, but not very much, and it was not. a bad mold to begin with. It is possible that the iron may have been better in the second consignment but not very much. In my opinion, it is con-servative to say that of the 900 per cent improvement in life, 5 per cent was probably due to better iron, 25 per cent to better design and the remaining 870 per cent to process of manufacture and workmanship.

Jan 2, 1928

By E. C. Reed

Ninety wells were completed in Nebraska during 1941, of which 42 were producers. One of them was the discovery well of a new field. All of the production to date is in Richardson County. At the close of the year 14 wells were drilling and three more were shut down for orders, hoping for future development of production. Most of the drilling activity was in Richardson County, which had 68 operations; 10 wells were drilled in Nemaha County, 5 drilled or drilling in Gage County, 3 each in Otoe and Furnas Counties, 2 each in Cass, Harlan, Johnson and Sarpy Counties, and 10 wells were completed or drilling over the remainder of the state. The most important development of the year was the discovery in October of the Barada field, sec. 36, T. 3 N., R. 16 E.,

Jan 1, 1942

By E. C. Reed

Ninety wells were completed in Nebraska during 1941, of which 42 were producers. One of them was the discovery well of a new field. All of the production to date is in Richardson County. At the close of the year 14 wells were drilling and three more were shut down for orders, hoping for future development of production. Most of the drilling activity was in Richardson County, which had 68 operations; 10 wells were drilled in Nemaha County, 5 drilled or drilling in Gage County, 3 each in Otoe and Furnas Counties, 2 each in Cass, Harlan, Johnson and Sarpy Counties, and 10 wells were completed or drilling over the remainder of the state. The most important development of the year was the discovery in October of the Barada field, sec. 36, T. 3 N., R. 16 E.,

Jan 1, 1942

By Howard N. Eavenson

WASTE of coal, or perhaps more properly the percentage of its recovery in mining, has keenly interested me during an experience of over a half century in coal mining. In the early part of that time and until a few ears after World War I the use of better mining plans and supervision resulted in an increased recovery. Since then, and particularly since controlled prices have been effective and mechanized mining has been extensively practiced, the tendency has all been toward lesser recovery in mining. The purpose of this paper is not to criticize the methods being used, which after all are the result of economic forces, nor to point to any particular mining method or to any locality where the best recovery is obtained. but to call attention to the tremendous waste of our national coal resources that is constantly continuing. A possible method of reducing this loss will be suggested. but without any illusions that it will be followed or that much attention will be paid to it.

Jan 1, 1946

By E. T. Cox

WHILE engaged, in the year 1856, in determining the oil-bearing properties of some bituminous coals from Western Kentucky, by subjecting them to dry distillation in an iron retort, which held about a pint, and to the neck of which were attached several bottles, for collecting the carbon oils, and washing and measuring the gas, I observed .that there was a very great gain in the weight of coke thus made, over that obtained from the same coal when charred in a platinum crucible, as is usual in the laboratory for making the proximate analysis of coal. It was thought at the time that the increase of coke, by the former mode of treating the coal, was due to the pressure which the retarded gas exerted upon the solid carbon, and the possible fixing of a portion of the carbon from the volatile hydrocarbons. For the purpose of obtaining a more definite idea of the effects of pressure on the coking of coals, a series of experiments were made this spring under my direction, by my assistant, Dr. G. M. Levete. The results are highly instructive and full of interest, since they point out some peculiarities in the nature of bituminous coals which

Jan 1, 1875

By AIME AIME

NAPOLEON'S dictum that an Army travels on its stomach has not changed in this present war, but the things an Army's stomach calls for would be more than strange to Napoleon. Today one of the main articles of diet for the mechanized army is oil and gasoline, and the chief and most efficient source of 'apply overland is the pipe line. It was a flexible 4-in. pipe line that supplied the voracious appetite of tanks, trucks, jeeps, and planes across the North African desert. and it is the 1800-mile pipe line from Calcutta to Kunming, Yunnan Province in China, the longest in the world, that supplied oil and 100-octane gas for the trucks, tanks, Jeeps, and planes fighting on the Assam-Burma-China front in recent months. Completion of this approximately 1800-mile pipe line to China over rivers that had never been bridged, over hill that towered 9000 ft., and over swamps

Jan 1, 1945

By C. S. Matthews, H. M. Girner, C. D. Williams, M. J. Edwardson, H. R. Parkison

Quantitative interpretation of electric logs requires knowledge of formation temperature. In this paper, methods are developed for computing changes in formation temperature caused by circulation of mud during drilling operations. The basis of the method is the mathematical solution of the differential equation of heat conduction. The solution of this equation is presented in a series of graphs. These graphs are used to determine formation temperature disturbance at various radii for arbitrary mud circulation histories. Example comparisons with field results show reasonable agreement. It is concluded that, in general, the temperature disturbances caused by circulating mud are small beyond 10 ft from the wellbore but are quite significant near the wellbore. INTRODUCTION Quantitative interpretation of electric logs requires knowledge of the formation temperature in order to establish the resistivity of the formation water with accuracy. To determine the formation temperature, the temperature disturbances produced by circulating drilling mud must be evaluated. -The objective of this investigation was to develop a method for the numerical determination of these temperature disturbances at any distance from the wellbore as a function of time. The first step was to solve the differential equation describing the temperature behavior in the formation during and after mud circulation. This solution was used to calculate a series of curves relating temperature disturbance to shut-in time for various circulating periods. An "exact" method for computing formation temperatures with the use of these temperature-disturbance plots is described, and the results of an application of this method to a well in Montana are presented. Procedures for approximating formation temperatures are also discussed. BASIC EQUATIONS AND ASSUMPTIONS With the assumptions that (1) cylindrical symmetry exists, with the borehole as the axis, (2) heat flow is 1. The formation can be treated as though it is radially infinite and homogeneous in extent, as regards heat flow. 2. The presence of mud cake can be disregarded. 3. The effect of heat generated by bit action is negligible. 4. After mud circulation ceases, the rate of radial heat flow at the wellbore is negligible and is assumed zero for purposes of calculating temperature change with time. These assumptions are believed to be reasonable because of the agreement achieved between calculated and measured borehole temperature in a well, to be discussed later. Further, the agreement at the wellbore indicates that temperatures in the formation were also approximately correct. since it is the formation temperatures that determine wellbore temperature after ceasing circulations SOLUTION OF THE DIFFERENTIAL EQUATION AND ITS APPLICATION The solution of Eq. 2 for the temperature behavior during and after a single period of mud circulation is

By R. Woods

INTRODUCTION Gaudin (1), considered that "the mechanism of mineral collection is the central problem of flotation theory". From their work on adsorption phenomena in flotation for more than fifty years (2), Gaudin and his co-workers have made major contributions to the elucidation of this mechanism, a fact acknowledged by their contemporaries in the field (e.g. (3)). Their investigations in a number of areas, e. g. demonstrating that submonolayer quantities of adsorbed collector species can induce flotation, or identifying the influence of substances which can activate or depress (1), have helped to provide the basis for and the development of a theory to account for the interaction between mineral and collector. It was established in the late fifties (4) that the presence of oxygen is a prerequisite for the flotation of metal sulfides with thiol collectors. This fact rendered inadequate the concept of simple ion adsorption being responsible for rendering these minerals hydrophobic. The requirement for oxygen has been explained (1) in terms of this species being a component in chemical reactions between the sulfide surface and the collector. Gaudin has been instrumental in identifying the products of chemical reactions at sulfide minerals (1, 5), a line of work which has been continued particularly by Finkelstein and co-workers in South Africa (6). In presenting a detailed mechanism of the mineral-collector- oxygen interaction, it has often been considered that the mineral surface [must] first be oxidized to a metal oxy-sulfur species which

Jan 1, 1976

By G. B. Gould

EVERY mathematical statement of a measure of anything (as distin-guished from a count) is followed by a qualification always implied if not explicity stated--that the statement is only an estimate, which may vary from absolute accuracy within certain limits, according to the accu-racy of the measuring instrument. This basic fact, so obvious to engineers, is not so well understood by many that make use of laboratory coal tests. The tolerances that must be allowed in measurements of coal quality are very important in both their commercial and engineering applica-tions-and the significance of these tolerances depends upon the use made of the laboratory coal test. Only in recent years has there been any intensive study of this ques-tion of tolerances as applied to measurements of coal quality. There is very little in our published literature on the subject, and what has been published is not, readily comprehended by those who sell coal, or buy coal and use it, unless they happen to have made a study of the subject them-selves. There is great need for more widespread understanding of what has been discovered and how it applies to the use of laboratory coal tests. What follows is an attempt to condense into a few pages the results of the major investigations so far reported, and to explain their meaning in simple nontechnical language, with the addition of some recent illustrative experiments.

Jan 1, 1937

By E. T. Cox

WHILe engaged, in the year 1856, in determining the oil-bearing properties of some bituminous coals from Western Kentucky, by subjecting them to dry distillation in an iron retort, which held about a pint;, and to the neck of which were attached several bottles, for collecting the carbon oils, and washing and measuring the gas, I observed that there was a very great gain in the weight of coke thus made, over that obtained from the same coal when charred in a platinum crucible, as is usual in the laboratory for making the proximate analysis of coal. It was thought at the time that the increase of coke, by the former mode of treating the coal, was due to the pressure which the retarded gas exerted upon the solid carbon, and the possible fixing of a portion of the carbon from the volatile hydrocarbons. For the purpose of obtaining a more definite idea of the effects of pressure on the coking of coals, a series of experiments were made this spring under my direction, by my assistant, Dr. G. M. Levete. The results are highly instructive and full of interest, since they point out some peculiarities in the nature of bituminous coals which

J. E. JOHNSON, JR., New York, N. Y. (communication to the Secretary*).-The discussion of Mr. Henrich's paper took place at a meeting from which I had the misfortune to be absent, and has just come to my attention. I do not wish to let the subject pass without contributing to it a little information which has come into my possession. In 1912, I published a brief note in the Transactions on The Effect of Alumina in Blast-Furnace Slags, giving in tabular form three typical slags used in iron blast-furnace practice, one of them the high alumina slag of, which Professor Richards speaks. That table is reproduced herewith. TABLE 1.-Composition of Slays. 1 3 3 4 5 6 7 CaO by Neutral Ratio Ratio Ratio Al2z03, SiO2, Differ- Substan- Cat Ca0 a A1203 Ratio Per Per ence, ces, CaO Cent. Cent. Per Per Al2O3 + Sioz Sit: SiO2 Cecil_ Cent. Virginia practice 6.5 36.0 54.0 3.5 1.27 1.6S 1.50 Lake-ore practice 13.5 33.0 50.0 3.5 1.08 1.92 1.51 Special experiment... 36.0 24.7 36.8 3.5 0.59 2.90 1.44 It supplies what seems to me convincing evidence that, for the purposes of the iron blast furnace, alumina can be regarded only as a neutral substance, dissolving in the slag and materially altering its viscosity, but not affecting its chemical activity. In subsequent experience with charcoal furnaces, I found the same rule to hold; that the chemical effect of alumina could be disregarded, providing the ratio of the lime and magnesia to the silica were kept constant. This is exactly the experience which Messrs. Dwight and Mathewson had in copper- and' lead-smelting furnaces. I have never seen any evidence or argument adequate to support the contention of Professor Richards-that the alumina acted as a base in the charcoal iron furnace and as an acid in the coke furnace, and that the function of alumina in iron-furnace slags is inherently different from that in copper- and lead-furnace slags-and I am compelled to differ absolutely with his conclusions in that respect.

Jan 9, 1917

By Albert Sauveur

My first paper dealing with the theories of the hardening of steel by rapid cooling was published in the Transactions of this Institute in 1896— 30 years ago-under the title "The Microstructure of Steel and the Current Theories of Hardening," hence, also, the title of the present paper. It was extensively discussed and was, I believe, one of the starting points of many subsequent papers and discussions of the same subject, dis-cusssions which at times became almost violent and which led to the formation of different schools such as the "Carbonists" and the "Allo-tropists." Some of the older members of the Institute, who took part in the debate or who were interested observers, will, I am sure, have kept a vivid remembrance of it. The controversy gradually quieted down, apparently for lack of ammunition, without any side acknowledging defeat. Recently, however, interest in it ha been revived through the claims of some that these vexed questions had at last been solved. In an effort to ascertain if possible the prevailing view of those best qualified to express an opinion a questionnaire was submitted to the following gentlemen: J. 0. Arnold Sir Robert Hadfield Edgar C. Bain W. H. Hatfield Carl Benedicks Samuel L. Hoyt N. Belaiew Kotaro Honda H. M. Boylston Zay Jeffries Harry Brearley H. Le Chatelier G. K. Burgess H. H. Lester William Campbell Francis F. Lucas H. C. H. Carpenter John A. Mathews Pierre Chevenard A. McCance P. Dejean Albert Portevin C. A. Edwards Walter Rosenhain F. Giolliti Bradley Stoughton G. Grenet F. C. Thompson Leon Guillet

Jan 1, 1926

By Allen J. Johnson

With a realization of the rapidly increasing importance of automatic stokers as a medium for domestic heating, the Anthracite Institute Laboratory has conducted extensive investigations, over a period of several years, to determine the various outstanding characteristics of stoker design and performance. The purpose of this paper is to summarize these findings as a guide to current practices. Stokees Investigated One or more models of the underfeed stokers listed below were included in the various phases of this investigation: Beckley Gilkoal Burnwell Iron Fireman Combustioneer Motor Stokor Electric Furnace Man Norristown Super-stoker Erie Heat Generator Whitney Fire Ball (Tests Incomplete.) Fuel Saver Principles of Construction.—The operating principle of the leading makes of underfeed stokers is similar (Fig. 1). A plunger1 or slowly revolving worm2, conveys the buckwheat or rice sizes of anthracite to the center of a specially constructed tuyere pot where air is admitted to the coal through suitable ports of varying design. As the fuel burns, it is slowly forced to the outside circumference of the tuyere by the pressure of the incoming coal; eventually the ash falls over the edge of the pot, either entirely by gravity or aided by mechanical sweepers. In at least one model, the entire firepot revolves. While various special details have been included in the design of a number of stokers, they do not affect the general principles as described above, and therefore have not been included in this discussion.

Jan 1, 1936

By Allen J. Johnson

With a realization of the rapidly increasing importance of automatic stokers as a medium for domestic heating, the Anthracite Institute Laboratory has conducted extensive investigations, over a period of several years, to determine the various outstanding characteristics of stoker design and performance. The purpose of this paper is to summarize these findings as a guide to current practices. Stokees Investigated One or more models of the underfeed stokers listed below were included in the various phases of this investigation: Beckley Gilkoal Burnwell Iron Fireman Combustioneer Motor Stokor Electric Furnace Man Norristown Super-stoker Erie Heat Generator Whitney Fire Ball (Tests Incomplete.) Fuel Saver Principles of Construction.—The operating principle of the leading makes of underfeed stokers is similar (Fig. 1). A plunger1 or slowly revolving worm2, conveys the buckwheat or rice sizes of anthracite to the center of a specially constructed tuyere pot where air is admitted to the coal through suitable ports of varying design. As the fuel burns, it is slowly forced to the outside circumference of the tuyere by the pressure of the incoming coal; eventually the ash falls over the edge of the pot, either entirely by gravity or aided by mechanical sweepers. In at least one model, the entire firepot revolves. While various special details have been included in the design of a number of stokers, they do not affect the general principles as described above, and therefore have not been included in this discussion.

Jan 1, 1936

By Ta M. Li

How do you convert a copper-zinc resource into a viable economic mining operation? This problem, unlike most, was complicated by the additional presence of a sub-arctic climate, highest labor costs in the world, low-grade copper-zinc values, depressed market prices, and a complex mineralization.

Jan 12, 1977

By M. R. Tek, K. H. Coats, D. L. Katz

A large number of boundary value problems encountcred in unsteady-state heat transfer, fluid flow through porous media, neutron diffusion and mass transfer involve the solution of a linear, parabolic partial differential equation commonly refcrred to as the diffusivity equation, where U is the dependent potential variable. K is the diffusivity (hydraulic, thermal, neutron, etc.) and 1 is the time variablc. Solutions to Eq. I are available in the literature for a wide variety of initial and houndary conditions. The great majority of these solutions are ohtained for geometric boundaries corresponding to linear. cylindrical or spherical flow models. A typical engineering application where the solution to Eq. 1 is required is the calculation of underground water encroachment across the boundaries of oil or natural gas reservoirs, In this particular area of application, the reservoir boundary is invariably approximated by circular geometry. However. the areal shape of many reservoirs can be better approximated by elliptic rather than circular boundaries. Thus the need for a general method of solving the diffusivity equation in elliptic coordinates arises in this problem as well as in other engineering applications involving elliptic boundaries. The solution to the diffusivity equation usually in volves the Error Function for the linear flow model, Bessel functions for the radial flow model and trigonometric or Legendre functions for the spherical flow model. It is well known that the general solution to the diffusivity equation in elliptic coordinates involves Mathieu functions. The significance of Mathieu functions in the analytical treatment of the diffusivity equation in elliptic coordinates is discussed in the literature However, these references do not providc analytical solutions useful in practical engineering problems. The objectives of this papcr are the development of the equations describing the unsteady-state liquid flow through a porous medium with an elliptic inner boun- dary, the development of a numerical method of solving these equations and, finally, a comparison of the water encroachment quantities calculated from the elliptic flow equation with those calculated from the radial flow equation. While the specific problem treated in this paper relates to unsteady-state liquid flow through a porous medium, the basic equations and computational techniques devcloped will apply equally well to problems occurring in the other areas of engincering interest mentioned previously. The solution given here is limited to a single case in which thc outer boundary encloses an area 100 times that of the inner boundary. DESCRIPTION OF THE FLOW MODEL Fig. 1 shows the flow model upon which the calcu lations presented in this paper are based. The inncr and outer houndaries of the flow model are represented by two confocal ellipses with major and minor axes, respectively, equal to 2a1, 2b1, and 2a, and 26,. The height of the elliptic cylinder flow model is denoted by 17. The following assumptions are employed in the development 01 the equations governing the unsteady-state liquid flow through the described flow model. 1. Uniform porosity and permeability throughout the flow model