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By Pervez Patel, Martin G. Frohberg, Demetrios Papamantellos

Model experiments were carried out on the mass transfer of iodine between n-hexane and a 4 2 X 10-3 M aqueous sodium hydroxide solution. The influence of mechanical stirring by means of small gas bubbles (ca. 0.01 ml in volume) on mass transfer was examined in independent experiments. The results were described with the help of the penetration theory and in the case of bubble stirring with the help of a mathematical model based on this theory. S LAG-METAL reactions, especially those in steel-making processes, often occur under conditions where gas is evolved in the form of bubbles that break through the slag-metal interface. It is known that stirring by means of bubbles increases the mass transfer coefficient, but there has been very little quantitative work done in this field. Because of the difficulties that attach to experimental work at high temperatures this study was confined to model experiments at room temperature. EXPERIMENTAL APPARATUS The transport of iodine from an n-hexane layer into a weak sodium hydroxide solution was studied a) with stirring by means of bubbles and b) with mechanical stirring. The apparatus for the experiments with bubbles consisted of the cell, two Dreschel bottles filled with hexane in order to saturate the nitrogen flow, one flow meter, and a nitrogen gas cylinder fitted with a fine-regulating valve. Fig. 1 shows the cell consisting of a glass cylinder 33 cm in height with an overall diameter of 7.5 cm embedded in a copper flange. The flange is connected to a Plexiglas plate and to the gas inlet. The Plexiglas plate has holes of 0.2 mm diam bored into it which are spaced 1 cm from each other. For the experiments with mechanical stirring the bottom of the cell was sealed off by a Plexiglas plate. The upper layer was stirred by means of a glass stir-rer and the bottom layer with the help of a magnetic stirrer in a counter rotatory direction. Both stirrer speeds could be varied continuously. EXPERIMENTAL PROCEDURE Iodine dissolves in n-hexane producing a stable violet color thus enabling the iodine concentration in

Jan 1, 1970

By A. B. Cook, C. J. Walker, G. B. Spencer

Although water will displace oil from a petroleum reservoir to a greater extent than gas will, there are some reservoirs in which gas rather than water should be used for pressure maintenance. This is indicated by the high-percentage oil recovery in the case history reports on the Pickton field1 in East Texas and the Raleigh field2 in Mississippi. In gas cycling, total oil recovery includes vaporized products from the immobile oil in addition to oil produced by displacement. If a large part of the immobile oil is vaporized, total oil recovery may be higher than that obtainable from pressure maintenance by water injection. However, the amount of immobile oil vaporized may range from almost 0 to 100 percent, and no simple and reliable method has been presented in the literature for calculating oil vaporization. Development of such a method was the purpose of this study. Several papers concerning the calculation of oil vaporization have been published,3 6 and each is based on the concept dealing with K values — equilibrium constants for the various components in reservoir oils. (K for a component in a system of vapor and liquid in equilibrium is the mole fraction in the vapor phase divided by the mole fraction in the liquid phase.) This appears to be the most logical approach. Yet, the problem is too great for a perfect solution, even with modem laboratories to analyze oils and high-speed digital computers to perform calculations; the number of components in reservoir oils is too large. Because of their complexity no two reservoir oils are exactly alike. Also, K values change not only with variations in pressure and temperature but also with composition of the reservoir oil. In addition, during gas cycling, the lighter hydrocarbons tend to vaporize first. Thus, the reservoir oil becomes more dense and less volatile as gas cycling continues. Furthermore, the greatest amount of oil vaporization occurs near the injection well. Therefore, a simplified method is required for calculating oil vaporization because a rigorous method is not practical. The present methods of calculating vaporization generally require knowledge of the reservoir oil composition in which the mole fractions of the lighter components through hexanes are given; the remaining oil is described as C, + (heptanes and others of greater molecular weights). Assigning one K value for the C,+ system provides the simplest solution, but this incorrectly assumes that K for C, + does not change as gas cycling continues. This assumption can cause large errors because the K value may become less than a thousandth of the original after a large amount of oil vaporization. Another way to solve the problem is to determine changing K values for C, + according to how much of the C, + has been vaporized. To determine these values a sample of the reservoir oil is injected into a pressure-volume cell, and dry gas is batched in and out of the cell. Appropriate measurements and tests are made to determine K

Jan 1, 1970

WE the undersigned, being all persons of full age and citizens of the United States, and a majority residents of the State of New York, desiring to form a corporation pursuant to the provisions of the Membership Corporations Law for the purpose of incorporating, as provided in Section 6 of Article I. of said law, the existing unincorporated association known as American Institute of Mining Engineers, do hereby make, acknowledge and 6le this Certificate for that purpose, and do Certify as follows: I. That the American Institute of Mining Engineers is an unincorporated association organized and existing with the object of promoting the arts and sciences connected with the economic production of the useful minerals and metals and the welfare of those employed in these industries by means of meetings for social intercourse and the reading and discussion of professional papers, and to circulate by means of publications among its members and associates the information thus obtained. 11. That the persons duly appointed or designated to manage the affairs of said association are designated by the rules thereof Members of its Council; that the under- signed are all members of said Council as the same was constituted on the 29th day of December, 1904.

Jan 1, 1942

By R. A. Wattenbarger, H. J. Ramey

The trend in gas well testing has been to rely more on the early-time flow data of drawdown and buildup tests than on stabilized flow tests. The stabilized testing methods often are not adequate for complicated modern gas reservoirs because many of these reservoirs have extremely low permeabilities and the transient flow period of a well test can last for days or weeks. It has become common to augment the flow capacity of gas wells by hydraulically fracturing their producing formations. In deep reservoirs the induced fractures are generally vertical and tend to follow a single plane of weakness. The presence of a vertical fracture at the wellbore complicates the transient flow behavior of a low permeability gas well. The flow is further complicated when turbulence occurs near the wellbore. Russell and Truittl published transient drawdown solutions for vertically-fractured liquid wells. They developed methods of drawdown and buildup testing utilizing these solutions, which were based on numerical simulation. Clark2 applied the basic Russell-Truitt solutions to analyze fractured water injection wells by falloff tests. Field examples were given to substantiate the method of analysis. The Russell-Truitt solutions and well testing methods can be extended to gas well flow. Millheim and Cichowicz3 presented the drawdown equations for *Now with Scientific Software Corp., Denver. ideal gas flow, including the effects of formation damage and turbulence. Two actual field cases were presented, both of which exhibited extensive fractures and turbulent flow. For the pressure range of these well tests, the use of ideal gas equations was acceptable. These field cases were significant in being the first published data showing the occurrence of turbulence in fractured gas wells. We, also, observed such cases in practice. Our purpose here is to extend the theory of fractured gas well testing to the flow of real gases. To determine the effects of wellbore storage and turbulence on well test interpretation, we developed a finite-difference model to simulate well test conditions. Since gas wells usually are widely spaced and have high compressibility, the emphasis has been put on the early transient behavior before the effects of the outer boundary are noticeable at the wellbore. The Mathematical Model The geometry of our mathematical model is similar to that used by Prats.4 The well is centered in a circular uniform reservoir. A vertical fracture of infinite flow capacity penetrates the formation and passes through the wellbore. The wellbore itself is not important. Fig. 1 shows a sketch of the problem. Fig. 2 shows the idealized model. The equation for the flow of real gas in a porous medium with uniform porosity, permeability and thickness is5-7

Jan 1, 1970

Jan 1, 1968

Jan 1, 1967

By C. C. Harris

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

Jan 1, 1969

By R. J. Stokes

S. Floreen (international Nickel Co.)— One fairly simple way to differentiate between em brittle me nt due to surface microcracks or due to a dislocation barrier effect might be to load a brittle rock salt crystal in tension to some stress just below the fracture stress and then immerse the loaded crystal in water. If a surface film is retarding plastic flow then one should observe some plastic deformation of the rock salt as

Jan 1, 1962

By G. Beghi, G. Piatti

THE authors examined the problem of the gradual transformation of Al2O3 from the transition phases normally encountered in SAP (r -x-n)1-3 to the stable a phase. This phenomenon is well-known at temperatures above the melting point of aluminum,3"6 whereas in this paper the transformation was studied at temperatures lower than the melting point of the aluminum matrix.

Jan 1, 1967

By S. R. B. Cooke, H. S. Choi, I. Iwasaki

In a previous article1 the function of various fatty acids as collectors for iron ores was reported for the two alternate processes; (a) the flotation of iron-oxide minerals, and (b) the flotation of calcium-activated siliceous gangue. The selectivity of separation was shown to be markedly dependent not only on such common parameters as pH and the levels of collector and activator addition, but also on the structure and the degree of unsaturation of the acids used. The iron ores employed consisted primarily of hematite, goethite, and quartz and as the effectiveness of any one acid on the separation depends on the individual response of each constituent mineral toward the given collector, a comprehensive knowledge of the floatability of the individual minerals is desirable for the interpretation of the results obtained. The mechanism by which a collector ion becomes attached to a mineral-solution interface has been a subject of much investigation. From quantitative studies of the properties of the electrical double layer on mercury2 and on several solid surfaces3,4,5,6 , the adsorption of some flotation reagents at the mineral-solution interface is beginning to be understood. For example, it has been demonstrated from streaming potential measurements made on quartz5 and corundum6 that the interfacial electrical condition is strongly dependent on the pH. Thisoccurs presumably through the following electrolytic reaction at the interface, and in the presence of a flotation collector, controls the floatability of the solids. A concurrent investigation of the flotation behavior of goethite and of quartz carried out at the School of Mines and Metallurgy7 has shown a positive correlation with the electrokinetic behavior of these minerals. The purpose of the present investigation was to collect general information on the floatability of hematite, and to investigate the function of various 18-carbon aliphatic acids on the floatability of hematite, goethite, and of activated quartz. Electrophoretic mobility measurements, simplified flotation tests using a modified Hallimond tube, and contact angle and frothability measurements were employed. The results are of direct application in the interpretation of the batch flotation results on natural ores as

Jan 1, 1961

Jan 1, 1963

By Kenneth Cotingham

Many factors joined in reducing drilling activity in Ohio in 1942. Conservation Order M-68 was important, but other things, such as operators and members of drilling crews joining the armed services and going into war plants, had an important part. The reduction in number of completions was considerable. In 1941, the total of wells drilled was 1561 but in 1942 the total was 1008. The total initial daily open flow discovered by the 701 gas wells drilled in 1941 was 492,000 M cu. ft., whereas the open flow of the 471 gas wells completed in 1942 was 314,488 M cu. ft. The total initial daily production of the 333 oil wells drilled in 1941 was 10,300 bbl., and in 1942 the 181 oil wells had a total daily initial of 7570 bbl. The average daily initial per gas well therefore was somewhat less in 1942 when compared with the preceding year, but the initial per ail well increased about 36 per cent over 1941. General Activity The total depth of the 652 oil and gas wells drilled in 1942 was 1,597,000 ft. The most active county was Perry, where 63 oil wells were drilled, most of them in Clayton township. In Morgan County. where 43 gas wells were drilled, the greatest activity was in York and Marion townships. Medina County was third in number of wells drilled, although of the total of 33 gas wells in the county, 26 were in the new Hinckley township pool. The total initial open flow of these 26 wells was 39,202 M CU. ft., and was considerably larger than from any other single area. Muskingum was an active county, particularly Washington and Wayne townships, in each of which seven gas wells were drilled, and Blue Rock and Harrison townships, in each of which five gas wells were drilled.

Jan 1, 1943

By Kenneth Cotingham

Many factors joined in reducing drilling activity in Ohio in 1942. Conservation Order M-68 was important, but other things, such as operators and members of drilling crews joining the armed services and going into war plants, had an important part. The reduction in number of completions was considerable. In 1941, the total of wells drilled was 1561 but in 1942 the total was 1008. The total initial daily open flow discovered by the 701 gas wells drilled in 1941 was 492,000 M cu. ft., whereas the open flow of the 471 gas wells completed in 1942 was 314,488 M cu. ft. The total initial daily production of the 333 oil wells drilled in 1941 was 10,300 bbl., and in 1942 the 181 oil wells had a total daily initial of 7570 bbl. The average daily initial per gas well therefore was somewhat less in 1942 when compared with the preceding year, but the initial per ail well increased about 36 per cent over 1941. General Activity The total depth of the 652 oil and gas wells drilled in 1942 was 1,597,000 ft. The most active county was Perry, where 63 oil wells were drilled, most of them in Clayton township. In Morgan County. where 43 gas wells were drilled, the greatest activity was in York and Marion townships. Medina County was third in number of wells drilled, although of the total of 33 gas wells in the county, 26 were in the new Hinckley township pool. The total initial open flow of these 26 wells was 39,202 M CU. ft., and was considerably larger than from any other single area. Muskingum was an active county, particularly Washington and Wayne townships, in each of which seven gas wells were drilled, and Blue Rock and Harrison townships, in each of which five gas wells were drilled.

Jan 1, 1943

By W. H. Bassett, C. H. Davis

In the commercial annealing of cartridge brass there are four points regarding which definite data are essential. They have to do with the correct interpretation of grain count in its relation to annealing temperature and, incidentally, to Brinell hardness. These points are: 1. The comparison of the grain sizes of two cartridge-brass mixtures: 69 copper, 31 zinc, 0.376-in. (9.5-mm.) gage; and 68 copper, 32 zinc, 0.130-in. (3.3-mm.) gage 2. The comparison of the grain sizes of annealed metal that had previously been reduced by rolling varying amounts; for instance, 20.0, 36.6, 50.9, and 59.1 per cent. 3. The determination of standards for grain sizes on annealed brass of the composition 68 per cent. copper, 32 per cent. zinc and 69 copper, 31 zinc. 4. The comparison of grain size with Brinell hardness on identical samples of annealed metal. In their comprehensive and thorough investigation of the recrystal-lization of cold-worked alpha brass on annealing,l Mathewson and Phillips have discussed the relations between temperature of anneal, degree of deformation, and structural alteration in alpha brass. They have also shown certain comparisons between the ordinary physical properties and the grain size of annealed brass. The purpose of the present investigation is mainly concerned with the grain size of cartridge brass, its relation to Brinell hardness, and the publication of sufficient data to enable those engaged in the inspection of such material to have a correct foundation upon which to work. The first alloy was taken from regular mill stock that had been rolled from 0.580 in. to 0.376 in. (14.7 mm. to 9.5 mm.) gage, a reduction of 35.1 per cent. This bar (No. 1) had the following composition: Copper, 69.20 per cent.; zinc, 30.76 per cent.; lead, 0.02 per cent.; iron, 0.02 per cent. The second alloy was also taken from mill stock of 0.325 in. (8.25 mm.) gage and was rolled as shown in Table 1, in order to get four bars reduced two, four, six, and eight B. & 5. numbers respectively. This second alloy

Jan 1, 1919

By J. H. Swisher

An experimental study has been made of the possible mechanisms for foam stability in the system CaO-SiO2-Cr2O3, where Cr2O3is the foaming agent. The degree of lowering of surface tension by Cr2O3 was determined at 1600 "C for high -silica melts. Measurements were also made of foam stability under standardized conditions. The results of the two sets of measurements then were correlated by applying the Gibbs and Marangoni theories of film elasticity. It was demonstrated that the Marangoni elasticity effect is probably the largest single contributor to foam stability, although the high uiscosity of' silicate melts makes some contribution in controlling the rate of drainage of liquid from bubble lamellae. ThE tendency of metallurgical slags to foam has frequently been observed during steel-refining operations. In the open-hearth furnace,this tendency is considered undesirable because it slows down the rate of heat transfer from the burners to the molten-metal bath. On the other hand, it is sometimes desirable to have a "foamy" slag in the basic oxygen converter.' A more effective blanket over the steel bath is made by a "foamy" slag, and loss in yield of metal due to splashing is minimized. In addition, a flush slag can be removed more easily in this condition. Since slag-foaming problems in the field have been attacked largely on an empirical basis, it appeared desirable to obtain a better basic understanding of the mechanism of foam stability. The experimental work was directed almost entirely toward the behavior of Cr2O3 as the foaming agent in CaO-SiO,-Cr2O3 melts. In the first stage of the investigation, the degree of lowering of surface tension by Cr2O3 was determined at 1600°C. The second stage consisted of measuring foam stabilities under standardized conditions. The two sets of experimental results then were correlated by applying various theories of foam stability. For convenience in interpreting the surface-tension and foam-stability results reported here, a portion of the CaO-SiO2-Cr2O3 phase diagram2 is reproduced in Fig. 1. EXPERIMENTAL 1) Surface-Tension Measurements. Very little information is available in the literature on the lowering of surface tension of slags by foaming agents. Kozakevitch 3 mentioned that the addition of 1.5 pct Cr2O3 lowered the surface tension of FeO slightly. In another system, some data on the lowering of surface tension of foaming slags by P2O5 were reported by Cooper and Kitchener.4 The technique used for the surface-tension measurements was the maximum bubble-pressure method,' using a single tube immersed to various depths in the melt. This method has been used extensively in high-temperature systems, such as in liquid metals, slags, and glasses.6,7 It has the advantage that the contact angle between the liquid and the tube material need not be known. The equation used for calculating surface-tension values from maximum bubble-pressure measurements is the Schroedinger equatioq5 which can be written as follows:

Jan 1, 1964

By L. P. Durham

DuRing the early mine development period, 1929 and the first part of 1930, a 300-ton pilot-plant concentrator was built at Nkana mine of the Rhokana Corporation, Northern Rhodesia. This plant operated from May 26, 1930 until Oct. 26, 1931, treating 101,080 tons of Nkana ores and producing 4372 tons of concentrate. The plant was built, primarily, as a test unit to determine the grinding characteristics, floatability, etc. of the Nkana ores and other ores of the Rhodesian copper belt and to supply sufficient data on which to base the design of a 5000-ton concentrating plant at Nkana mine. The test unit operated wholly on development ores and no attempt was made to grade or select the ores from any section of the mine. The average assays for the 17 months of operation are given in Table 1. The first ground was broken for the 5000-ton concentrator during June, 1930, and the construction of the plant was completed in November, 1931. The plant started on Dec. 10 and has operated continuously since that date: The whole of the concentrating plant is compact and closely connected, yet with sufficient space for expansion of all sections up to 20,000 to 25,000 tons per day capacity. Mining operations are carried on along a length of about four miles on the easterly limb of the Nkana syncline. This length includes the North orebody, from which all production has thus far originated, and the Mindola orebody, now being developed for production. The ore-bearing measures in the North orebody are leached for some 75 ft. below the surface; the oxidized zone varies in vertical extent from 100 to 150 ft. Practically all the ore below the 200-ft. level is sufficiently

Jan 1, 1935

By H. R. Hanley

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

Jan 1, 1931

By Robert Thomas

DURING the past 20 years the advantages of reinforced concrete as a substitute for timbering in so-called permanent mine openings have been fully recognized, and its use has become almost general practice in the larger mines. The term "permanent mine openings" is used advisedly and is assumed to embrace such openings as ventilation shafts, material shafts and main hoisting shafts, with their adjacent stations, ore pockets, etc. In the concreting of such openings, the actual cost as compared with that of timbering has been of secondary consideration, the support of large openings, elimination of fire hazards, and prevention of inter-ruptions to production being the primary objectives. The ultimate economy, however, of lining such openings with concrete when they are to remain in use during the entire life of the mine, and especially when they are in bad ground, will seldom be questioned. During this 20-year period there have appeared in engineering publi-cations a number of interesting articles on the use of concrete in mines, nearly always describing some specific project of major proportions, such as the concreting of ore pockets, large stations or main hoisting shafts. Such articles are unquestionably of interest, since the projects require considerable engineering thought, not only in the method of excavating large openings, but in their engineering design as well. Concreting of main hoisting shafts, particularly when this is done with the least possible interruption to production, provides some interesting problems. The concreting of haulage drifts of the dimensions commonly found in mines, on the other hand, soon develops into a routine operation, there-fore it is with some hesitancy that the author attempts to prepare an article on this subject. The writer can recall only one article on the concreting of mine haulageways, although several articles have been written concerning the concreting of tunnels considerably larger in dimensions than those required in general mine practice.

Jan 1, 1936

By T. D. Parris

Within a 30-mile radius of Sudbury, Ontario, the Ontario division of the International Nickel Co. of Canada, Ltd., operates nine underground mines and two open pits. Prior to 1966, ore removal from the underground workings was mainly by scraping, using 125-hp electric slushers in blasthole and caving mining, and 30-hp or less air-operated slushers for fill- method mining. The development and evaluation of low-profile, diesel-powered front-end loaders, designed specifically for underground operations, was followed with keen interest during the period 1964-1965.

Jan 6, 1969

By H. S. Price, J. C. Cavendish, R. S. Varga

A numerical formulation of bigh-order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional problem can be obtained in the neighborhood of a sharp front without the need for a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in two dimensions. INTRODUCTION The problem of finding suitable numerical approximations for equations describing the transport of heat or mass by diffusion and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusionconvection equations, arise in describing many diverse physical processes. Of particular interest here is the equation describing the process by which one miscible liquid displaces another liquid in a one-dimensional porous medium. The behavior of such a system is described by the following parabolic partial differential equation:

Jan 1, 1969