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By J. A. Herbst, R. Mackelprang, J. D. Miller

Droplet size distributions were determined by stabilizing in gelatin and measurement of the distributions by means of computerized image analysis Several variables that affect droplet sizes were studied and the resulting size distributions were modeled via a population balance equation Key parameters in the model were related to the physical properties and operating conditions of the systems examined using a multiple linear approach

Jan 1, 1981

By William E. Ford, Edward Salisbury Dana

34. Axial Ratio, Axial Plane. - The crystallographic axes have been defined (Art. 22) as certain lines, usually determined by the symmetry, which are used in the description of the faces of crystals, and in the determination of their position and angular inclination. . With these objects in view, certain lengths of these axes are assumed as units to which the occurring faces are referred.

Jan 1, 1922

By L. J. Trilli

RECENT studies have indicated that there may be considerable difference in the cleanliness of the central pipe cavity of open top ingots. It has been speculated that some of this difference arises becausk of reactions that occur during reheating in the blooming mill soaking pits. In this respect, it has been recognized that an oxide coating or flow of slaggy material in the cavity could inhibit welding during hot rolling and thus reduce the amount of sound product which may be obtained. This note is intended to summarize some observations on the structure of a split ingot top which may be useful in describing a mechanism whereby the shrinkage cavity could be oxidized during reheating. A 221/2x241/2 in., big-end-down, test ingot was secured from a routine production heat. The chemical analysis representing this ingot is as follows: 0.08 pct C, 1.04 pct Mn, 0.062 pct P, 0.315 pct S, and 0.02 pct Si. The deoxidation practice used on the heat from which the experimental ingot was

Jan 1, 1957

By J. Chipman, J. B. Gero, T. B. Winkler

D. C. Hilty—This paper is a useful and timely addition to our store of quantitative data relating to manganese distribution between slag and metal in steel-making processes. For some time, many of us have suspected that the values of Kmn derived by Körber and Oelsen might be too low, and this paper now appears to confirm that suspicion. Application of the new constant to openhearth data should prove interesting. The manganese deoxidation curves of fig. 4 agree fairly well with direct measurements of isothermal solubilities of oxygen in liquid iron containing manganese that have recently been made at the Union Carbide and Carbon Research Labs., Inc. According to the solubility data just mentioned, however, the curves for 1550" and 1600°C lie much closer to each other than the authors have indicated. In other words, the solubility measurements suggest that the logarithm of the oxygen solubility is not a linear function of the reciprocal of the absolute temperature, as the authors have assumed for their calculations. It seems possible that this discrepancy may be due in part to the authors' use of the distribution coefficient, L,,, obtained from Taylor and Chipman's curve for the solubility of oxygen in iron as shown in fig. 2. Taylor and Chipman gave the equation of this curve as —6320 Log % 0 - ---T + 2.734 Their data, however, can be fitted equally well by a curve having the equation: 13.23 x104 1.18x10" Log % O = 36.350 -13.23x10 Moreover, this curve also fits the authors' oxygen distribution data somewhat better than does the linear curve of Taylor and Chipman. Its use in the derivation of Lo for calculation of the deoxidation curves by the authors' eq 15 causes the effect of temperature on the calculated curves of fig. 4 to agree quite well with the temperature effect that has been observed. An experimental determination of the temperature coefficient of the manganese reaction appears to be desirable. J. E. Stukel—Because the authors have done such an excellent job, my comments will be confined to additional information. Very shortly Professor T. L. Joseph and I will publish the results of a study of the reaction between molten manganese and silica. It was found that electrolytic manganese, melted in silica crucibles, reacted with the silica to form a silico-manganese alloy containing 18 pct silicon (1500°C) and a silica saturated MnO-SiO, slag. Therefore, the presence of silicon and its oxides in the Fe-Mn-0 system will affect the equilibrium. Carbon is also an important factor. The higher the carbon content of the iron the less ability manganese has to reduce silica. As a matter of interest, recently a study of the manganese equilibrium in carbon saturated iron and blast furnace slags was completed at Carnegie Institute of Technology. As a comparison to the present study, 2 to 5 hr were required to reach equilibrium and temperature was a much more important factor than slag composition in determining the (Mn)/(MnO) ratio. These results will also be published in the near future. It can be seen, therefore, that the results of this paper cannot be applied directly to openhearth studies without a degree of flexibility. J. Chipman (authors' reply)—With regard to the effect of silicon on the manganese constant, this effect is, of course, known. Much more data is needed in order to establish it accurately, and I hope we can look forward to having some accurate data on that. Mr. Hilty suggests that a three-term expression for the Taylor-Chipman solubility of oxygen in liquid iron might fit a little better than the two-term equation. I think that it would be too great a compliment to the precision of the data to employ a three-term equation, and I would still recommend the simple, two-term expression.

Jan 1, 1951

By Gustaf Ising

For relative gravity measurements, the author in 1918 described an instrument1 of which the essential part consists of a highly astatized, standing pendulum turning about a horizontal elastic axis. This type of instrument has been used recently in practical field work, therefore a description of its principle and technical development is given herein. Part I, describing the principle, is a direct translation of the original Swedish paper.l Part II gives an account of the practical elaboration of the instrument, together with some description of measurements performed. I. Principle of Using Astatized Pendulums1 In order to reduce the trouble and waste of time attending the use of ordinary pendulums in measuring the force of gravity, various instruments of another type have been suggested for relative determinations, as a rule arranged in such a way that the attraction of the earth on a certain body or mass of fluid is balanced by elastic force from a spring or a shut-in mass of gas. A change in the force of gravity is then made manifest as a change in the position of equilibrium of the body, in that the deformation of the spring (or compression of the gas) is changed. With such instruments there are, however, great difficulties in keeping the elastic force constant or its variation known with sufficient exactitude, because of elastic after-working, influence of temperature, and so on. The author suggests for relative determinations of gravity another type of instrument, in which two directional forces (control coefficients) rather than two forces are compared with one another. By "directional force" or "control coefficient" is meant that quantity in a rotatable system that, multiplied by the angle of deflection, indicates the moment counteracting the deflection. In this way it will be possible greatly to reduce the influence of the elastic after-working, which in instruments with spring devices acts so disturbingly. The instrument is made with a rotatable system arranged in such a way that the total control coefficient acting upon it, is composed

Jan 1, 1940

By Gustaf Ising

For relative gravity measurements, the author in 1918 described an instrument1 of which the essential part consists of a highly astatized, standing pendulum turning about a horizontal elastic axis. This type of instrument has been used recently in practical field work, therefore a description of its principle and technical development is given herein. Part I, describing the principle, is a direct translation of the original Swedish paper.l Part II gives an account of the practical elaboration of the instrument, together with some description of measurements performed. I. Principle of Using Astatized Pendulums1 In order to reduce the trouble and waste of time attending the use of ordinary pendulums in measuring the force of gravity, various instruments of another type have been suggested for relative determinations, as a rule arranged in such a way that the attraction of the earth on a certain body or mass of fluid is balanced by elastic force from a spring or a shut-in mass of gas. A change in the force of gravity is then made manifest as a change in the position of equilibrium of the body, in that the deformation of the spring (or compression of the gas) is changed. With such instruments there are, however, great difficulties in keeping the elastic force constant or its variation known with sufficient exactitude, because of elastic after-working, influence of temperature, and so on. The author suggests for relative determinations of gravity another type of instrument, in which two directional forces (control coefficients) rather than two forces are compared with one another. By "directional force" or "control coefficient" is meant that quantity in a rotatable system that, multiplied by the angle of deflection, indicates the moment counteracting the deflection. In this way it will be possible greatly to reduce the influence of the elastic after-working, which in instruments with spring devices acts so disturbingly. The instrument is made with a rotatable system arranged in such a way that the total control coefficient acting upon it, is composed

Jan 1, 1940

Mineral Industries education as an entity, again and again has sought recognition, always to be turned aside or ignored.1 The incident mentioned in Lost Chapter was only the first of a series of disappointments. The Land-Grant Act is camouflaged here and there in the name of progressive education. A new interpretation may be in the best interests of society in so far as the goose is not killed that lays the golden egg. The primary objective of the act is still "agriculture and mechanic arts" but mechanic arts is not alone mechanical arts or engineering. Mechanic arts was used in a broad sense as a symbol of creative effort in locating and winning the fundamentals of existence - plants, animals, and minerals - from Mother Earth, their primary processing and treatment into foodstuffs and raw materials of industry, their manufacture into useful articles of commerce, and, last, their distribution and utilization. Significantly the great agricultural industry is now at least half mechanic arts. The Association of Land-Grant Colleges and Universities has neglected the mineral kingdom shamefully. The Association embraces Divisions of Agriculture, Arts and Sciences, Engineering, Home Economics, and Veterinary Medicine but has no division relating to the mineral industries.

Jan 1, 1950

By D. D. Pollock, D. I. Finch

Relationships between the compositions of specially prepared manganin-type alloys and some of their electrical properties have been established. Empirical equations, based upon the electronic configurations of the alloying elements, are given which describe the resistivity, temperature coefficient, peak temperature, and thermoelectric power of Cu-Mn-Ni-Fe alloys at approximately room temperature. Calculations based upon these relationships are correlated with experimental data. SECURING suitable alloys for the construction of stable resistors, for use in precise electrical measuring devices, has been important ever since the art of electrical measurement was begun. German silver, Pt-Ag, and Platinoid are examples of some of the alloys in early use. These early alloys were un- satisfactory for one or more of the following reasons: l—insufficient stability of electrical properties with time, 2—poor reproducibility of electrical characteristics from melt to melt, 3—excessive temperature coefficients of resistance, and 4—excessive cost. Weston introduced the Cu-Mn-Fe and Cu-Mn-Ni-Fe alloys of the type now generally known as manganins. His patents are believed to represent the first disclosure of any alloy with a negative temperature coefficient of resistance. After this work, the German company, Isabellenhutte, in conjunction with Feussner and St. Lindeck, produced alloys of

Jan 1, 1957

By V. I. Purcell, R. E. Snow, S. C. Sun

FATTY acids have been used as standard collectors for floating the matrix1-8 but not the leached zone phosphate ores. After a series of investigations, Davenport9 and Tarbutton10 concluded that the Florida leached zone ores tested were unamenable to fatty acid flotation and must be upgraded by floating the siliceous gangue with Armac-T as collector. The cationic flotation was estimated11 to cost almost ten times as much as a fatty acid process that would ideally give the same flotation grades and recoveries of phosphates. Mineralization: Extensive deposits of leached zone aluminiferous phosphate occur in The ore sample used for this work was obtained from the Virginia-Carolina Chemical Corp., Clear Springs area, Florida. The ore is a potential source of uranium and aluminum as well as phosphorus. Since the uranium and the recoverable aluminum are contained in some of the phosphate minerals, an upgrading of phosphate results in a corresponding enrichment of both uranium and aluminum. The phosphate minerals in the ore are primarily pseudo- wavellite, millisite, and carbonate fluorapatite, with a minor amount of wavellite. Quartz and clay are the common gangue minerals, quartz being pre- dominant. Others present in small amounts are ilmenite, hydrated iron oxides, staurolite, monazite, and zircon.

Jan 1, 1957

By S. P. Burke

THE presence of gas in coal mines necessitates the use of costly ventila-tion arrangements and the use of expensive mining methods. On the other hand, the gas itself in many instances is of considerable intrinsic value, and its separate recovery might conceivably be a profitable under-taking. This paper is concerned with a study of the movement of gas in coal strata with the object of determining the laws governing gas move-ment and thus making at least a preliminary step toward the solution of the more practical problem of its control or recovery. Naturally, so important a subject has received much attention from scientists and engineers. In most cases, however, the studies have been concerned with the origin1?, the amount2, the composition3 and the method of retention4 of the gas in coal; or, from the more practical side, have been concerned with the phenomena and conditions associated with outbursts5 or explosions and their control. Numerous studies also have been made of the permeability of coal to gas6, of gas pressures measured in coal seams, etc. The authors are unaware, however, of any attempt to present a consistent or complete picture of the laws governing the movement of gas through the coal.

Jan 1, 1935

An educated man is distinguished neither by his clothes nor by his knowledge; he is remarkable not for the things he says, but for the way he says them. You cannot even stand with him under an archway in the rain without finding him out. He may talk only of the weather. His words and his phrases alike my be simple. What distinguishes him is the arrangement of his words; however desultory his talk, it will be methodical; he has habituated himself to foreseeing the part to be played by each word he uses and the place to be taken by each sentence he utters. On the other hand, the uneducated man, though shrewd and well-informed, will relate facts and events as they recur to him, generally in disorder; in his effort to recollect and in his attempt to rectify forgetfulness he will pause irregularly, filling the intervals with meaningless phrases, such as "and then", "and so", or "said he to me", "said I to him"; and even the continuous parts of his story will be told confusedly-because he has not learned the proper use of words.* My own experience as an editor of technical manuscripts has taught me that the misuse of 'that' and 'which' is a fruitful cause of obscurity and confusion. The relative pronouns have distinct functions, and no writer can express himself clearly until he has learned to discriminate between these functions. For example : The engineers who refused to submit were discharged. The engineers, who refused to submit, were discharged. The engineers that refused to submit were discharged.

Jan 1, 1931

By Carl F. Floe, John Chipman

A YEAR ago the authors published a paper on the solubility of sulphur dioxide in molten copper.1 The data in that paper agreed closely with that obtained by previous investigators, which, however, did not agree with the theoretical predictions for solubility of triatomic gases of this type. It has generally been assumed that the reaction of solution is: [ ] Neglecting the slight change in copper concentration and considering that the concentrations of sulphur and oxygen are both proportional to the volume of gas absorbed, it follows that this volume should be proportional to the cube root of the pressure. It was found that a plot of the volume absorbed versus the cube root of the pressure gave a straight line, but that this line did not pass through the origin. Calculations of the equilibrium constant for the reaction showed that it was fairly constant for pressures above l00 mm. but that actually it approached zero at lower pressures. The conclusion was made therefore that the solubility as determined could not be expressed by Eq. I or by any other simple equation, particularly at low pressures. Other investigators have concluded that the solubility of sulphur dioxide in copper as well as in several copper alloys is more nearly proportional to the square root of the pressure.* Such a conclusion, however, does not satisfactorily account for the results, even when rather complicated mechanisms for the solution are assumed. The highly improbable speculations of Lepp2 and of Schneider and Esch3 based upon hypothetical dissociation pressures of Cu20 and (a; 2S dissolved in copper illustrate the type of reasoning that has been used to account for the discrepancy. In the discussion of the previous paper, the authors pointed out that the discrepancy might possibly be due to impurities in the copper, although the purest form available was used. Further consideration of the problem suggested that the impurities likely to have the greatest effect on the solubility of SO2 in copper were oxygen, sulphur and carbon. Accordingly, further experiments have been made to show the effects of these elements. EXPERIMENTAL METHODS The apparatus used for measuring solubility was essentially the same as that shown in the previous paper. However, since it was desired to make measurements at low pressures, a McLeod gauge was added and 6-mm. glass tubing was substituted for the capillary. This increased the "hot volume" considerably but was necessary in order to obtain good vacuums in a reasonable length of time. A mercury diffusion pump was also installed.

Jan 1, 1942

James Douglas will long be remembered as a distinguished mining engineer and metallurgist, as a scientist with broad vision, as an executive whose understanding of practical problems coupled with his knowledge of men and affairs enabled him to weld together great mining interests and to their service build railroads and bring power. More, he will be recalled as a philosopher and a writer of charm, a historian of no mean order, and one whose catholic tastes and enthusiasms led him into the fields of education, medicine and theology. His aim was to possess that which he considered made life worth while—friends and effectiveness in whatever task he undertook. In this aim he was successful to a degree exceeded by few, as thousands of his friends and associates could and can testify. Dr. Douglas was born on November 4,1837, at Quebec, Canada. His father, James Douglas, Sr., was a prominent physician and the leading surgeon of Canada. He also established the first retreat for the insane in the Dominion, the Quebec Lunatic Asylum, to which he devoted his services for years; The son followed in the footsteps of his father and studied medicine, not only in Canada but also in Scotland and in Germany. He entered the University of Edinburgh in 1855 but after two years at that institution returned to Canada and was graduated from Queens university at Kingston, Ontario, .with the degree of Bachelor of Arts. Later he went to Lava1 University, Quebec, and subsequently became professor of chemistry at Morrin College, Quebec. After he was graduated—at the time he considered his formal education completed—he traveled with his father in Europe and in the Orient; visiting Egypt, being especially interested in that country, several times. They gathered together an important archaeological collection which later was donated by them to the Metropolitan Museum of Art in the city of New York. James Douglas was also greatly interested in photography, a hobby in which he made use of his chemical knowledge. The writer can remember distinctly the descriptions he gave more than forty years ago in Arizona of the difficulties he encountered with the old wet photographic plates, especially while traveling in Egypt, when he prepared and used his own chemicals. In his humorous manner he told how he once discovered, in the nick of time, the nurse preparing to give the baby in the party a rub-down, using for that purpose the white, limpid

Jan 1, 1934

By J. W. Marx, R. H. Langenheim

The authors are to he complimented for a timely presentation of useful information concerning application of heat to oil reservoirs to increase the rate and ultimate recovery of oil. The solution for heated area resulting from constant rate injection of a hot fluid, presented by the authors, and the fluid-flow analogy, solved by Carter,' are particularly interesting in that 110 restriction is placed on the direction of development of the heated area. It is not necessary that the heated area grow radially. Thus the solution could provide useful information for heat injection in any type of well pattern with any specified swept area data. However, it was assumed in the development that the heated region would remain at a constant clevatcd temperature. Vertical heat losses were computed on this basis. Physically. the assumption that the heated region will remain at a constant temperature (independent of distance from the injection point) appears to require that the heat injection medium be stcam, or other condensible gases near the boding point at injection pressure. If a hot liquid below the boiling point or a hot gas considerably above saturation temperature were used, the temperature of the heated region would decrease with distance from the injection well. Lauwerier2 has discussed this problem for a linear system. (See also Jenkins and Aronolsky; and McNiel and Nelson.') Thus, the authors' solution applies more nearly to steam injection than to hot water injection or hot, non-condensable gas injection. Assuming steam as the injection medium, it appears that steam would pass through the constant-temperature heated region and condense upon contact with the unheated sand. The tempcrature in the condensing region would probably be governed by the release of heat of vaporization required for phase equilibrium at the existing pressure distribution. Temperatures in the region would also be influenced by vaporization of hydrocarbon. Thus, it is doubtful that an expression for radial temperature distribution based on heat conduction only (such as the authors' Eq. 9) would have practical significance. As pointed out by Marx and Langenheim, the growth of the heated area under constant rate steam drive will eventually be limited economically by vertical heat loss. The possibility of continuing to move heat economically by cold water injection or by increasing the steam injection rate appears to be a very important consideration. (Movement of liquids away from the injection well should increase injectivity.) Fortunately, the solution presented by the authors may be generalized to thc case of a variable heat injection rate. The general solution is:

By Earl G. Alexander, W. R. Runyan

The nature of silicon growths which form on amorphous silicon dioxide during epitaxial deposition in unmasked areas was investigated. Octahedral silicon crystals nucleate and grow on the oxide at pinholes, adjacent to foreign matter, and probably at oxygen-deficient sites. Generation of these oxygen-deficient sites on the oxide surface is shown to be thermody-namically favorable under usual deposition conditions. The conditions which affect saturation (SiCl4 concentration, flow rate, temperature) determine the number of sites on which silicon will nucleate and grow. Reaction product HCl is likely responsible for suppressing silicon growths on the oxide near exposed areas of silicon. Evidence is given that the oxide-silicon interaction occurs in hydrogen only and involves water as a transport agent. SINCE shortly after the first epitaxial layers of silicon and germanium were grown over large areas, efforts have been underway to grow selectively on specified portions of the surface. Such selective growth should be particularly applicable to monolithic integrated circuits. Usually the approach has been to use a mask to delineate the desired area,'-7 but the VLS methoda can also be used, and direct programmed atomic beam deposition has been suggested.9 The masking techniques may embody two forms: in one the mask is a material which can later be removed along with any growths attached to it; in the other, attempts are made to prevent growth on the mask, so that over a whole slice the only deposition that will occur is in the unmasked regions. The present work is concerned with only the latter case, and specifically with silicon. Joyce and Baldrey1 suggested the use of thermally grown silicon dioxide as a mask material and showed that for the proper choice of operating conditions (total flow, SiCl4 concentration, total deposition time, and temperature) spurious growth could be minimized and that if the deposition times were not too large, and the criteria not too stringent, a usable process occurred. Later Schnable et al 4 observed that after extended deposition times, during which most of the silicon dioxide became covered with growth, there was always a band of oxide around the exposed silicon area that was quite free of spurious silicon crystallites. This latter observation is particularly intriguing since it extends the hope that with proper precautions masking can be much more versatile than it presently is. EXPERIMENTAL PROCEDURE Silicon substrates oriented to within 1.5 deg of a (111) plane were prepared by etch polishing lapped

Jan 1, 1967

By J. H. Henderson, H. J. Ledbetter, N. B. Gove, J. D. Griffith

INTRODUCTION The results of a series of laboratory flood tests using liquid Iso-butane to displace refined oils from test cores are pre- ented and interpreted on an empirical bask. The study Revealed the similarity of the miscible liquid displacement to that of the immiscible-fluid displacement mechanism. The efficiency of the iso-butane flood decreased markedly as the oil Viscosity increased. but the effect of injection rate on the effectiveness of the primary production stage was negligible over the range investigated. The presence of free-gas saturation prior. to iso-butane breakthrough increased the volume of iso- Butane required to recover a given percentage of the oil present. Injection of liquid i.e.-butane prior to water follicle sculpted in a marked improvement in oil recovery by the water flood. INTRODUCTION The production of petroleum is often enhanced by the injec-1i1)n of extraneous fluids into the reservoir. It has been suggeited that the injection or cycling of a wet gas or a liquefied petroleum gas would he instrumental in increasing the effic.ienr!; of oil recover! to a greater degree than the commonly injected fluids. dry pas and water. Although instances of injection of light liquid hydrocarbons are not unknown, a systrmatic recording of the result is unavailable and superfi- cially, the practice gives the impression of being economicaIIy unsound. For specific types of displacement, analytic: solutions have been advanced which aid in understanding the physical mechanisms involved. However, there is no generalized theoretical concept which permits adequate prediction of displacement efficiency from a knowledge of readily measured reservoir rock and fluid properties. The Buckley-Leverett approach to oil displacement by an immiscible phase is an admirable analysiy of that type of mechanism. Dykstra and Parsons have introduced a useful concept in their evaluation of the role of the mobility ratio and its merit in rationalizing displacement phenomena. Everett, Gooch and Calhoun3 in their report on the effect of viscosity ratio and miscibility have pointed out the lack of a comprehensive theoretical approach and the desirability of investigating the mechanism of miscible-miscible displacement. In lieu of a well-developed theory concerning miscible-111isci1)le displacement and in view of the scarcity of engineering information on actual LPG injection trials, the limited program of laboratory experimentation described below was undertaken. OBJECTIVES The original broad objectives of the program, to study the oil recovery process using LPG as the displacing phase, defined the direction and scope of the laboratory tests. The specific objective. of the experimentation were to evaluate

Jan 1, 1953

By F. N. Kissell, R. A. Jankowski

The US Bureau of Mines, in close cooperation with the mining industry, has recently established several improved face ventilation techniques for longwall mining systems. These include coursing the primary face intake from tailgate to headgate, installing curtains at the intake end of the longwall face, and identifying improved techniques for measuring face ventilation parameters on longwall sections. Applying these techniques throughout the mining industry should reduce the health hazard associated with excessive exposure to respirable dust.

Jan 1, 1984

By R. C. Troxell, S. C. Sun

EFFORTS to obtain definite quantitative data when employing the currently used bubble pick-up method', as a pre-flotation investigation tool led to the adoption of the magnifying mirror method and the microscopic method modifications. Quantitative data obtained by these two methods were converted into bubble pick-up indices and bubble pick-up coefficients. Rough correlations were established between the calculated indices and the results of contact angle testing as well as actual flotation under the same conditions.

Jan 7, 1953

By T. B. Counselman

Continuous countercurrent decantation calculations have always been a headache to the cyanide man (and the chemical engineer) because of the simultaneous equations involved. These are tedious to solve, and there is considerable opportunity for mathematical error. This paper presents a shortcut method for making these calculations, which has been in successful use for twenty years by the author, and which is particularly valuable in selecting the best of several possible flowsheets, or evaluating the effect of various assumptions.

Jan 2, 1950

By S. E. Clay, G. P. Martins, P. R. Taylor

A grain model for gas-solid reactions in cylindrical pellets is formulated and solved numerically. The numerical solution demonstrates when it is safe to assume either a thin disk or a long cylinder geometry, and when the more general two dimensional geometry must be used. The numerical solution also demonstrates the solution of a two-dimensional moving boundary problem (i.e., when the reaction is diffusion controlled and the ratio of the height to diameter dictates two dimensions).

Jan 1, 1986