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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 Donald B. Gillies

PROBABLY the most critical and difficult period in an engineer's career is that between the completion of his college work and his attainment of professional recognition and accepted status in the industrial world. Upon leaving school the young graduate breaks his immediate ties with his educational institution and in a sense finds himself adrift. At this time in his life a society like our Institute has both the responsibility and the privilege of meeting a vitally important need in his professional life. Some of these young men are Student Associates, in the year after graduation, and some are Junior Members, but all need advice and assistance. In some respects this situation will be accentuated by the war. Not only will thousands of young engineers be mustered out of military service but the process we describe generally as industrial reconversion will make necessary many readjustments in personnel. About two years ago the Institute recognized the opportunity of helping the younger member to help himself by appointing a new standing committee to be known as the Committee on Professional Development of Junior Engineers. Herbert Hoover is Honorary Chairman and it is my privilege to serve as Chairman. The name of the Committee is not entirely satisfactory but it expresses the general objective. The present article is in the nature of a progress report of the work of this Committee.

Jan 1, 1945

By A. L. Engel

STRICTLY speaking, ore-dressing does not commence until after the ore is in the mill bins, but where complex ores are treated and their minerals separated to make the best commercial concentrate with the lowest economical tailing, ore-dressing problems begin and often center in the mine itself. This does not refer to the grade of the ore, but rather to the frequent condition in which metallurgical results radically different from customary are obtained because of changes in the character of the ore treated. Different varieties of ore can usually be identified as coming from well-defined depths or zones in the mine, or from known sections of the orebody. Varying degrees of alteration of the ore, inclusions of harmful gangue material, or variations in the quantities of component metallic minerals are often causes of temporarily bad metallurgical results in otherwise successfully operated plants, especially those employing differential flotation to. recover several valuable minerals contained in the ore.

Jan 1, 1931

By CHARLES SIMENSTAD

COAL preparation, or coal washing, is not a new subject to the Pacific Northwest. Most of the coals mined in this state smaller than lump, and nearly all such sizes mined on the Pacific slope of the Cascade mountains, require washing before marketing. A large proportion is being put through some washing process, but coal washing in this state, has not reached the efficiency it is practicable to obtain. In fields where the coal, as mined, contains less impurities, where working conditions make it practicable to eliminate some of these impurities under- ground, or where a large proportion of the run-of-mine coal is lump, coal preparation does not present the problem it does in our local field. Here the coal is associated with considerable impurities and the slack, or steam sizes, aggregate about 80 per cent. of the output. Therefore, approximately 80 per cent. of the coal mined must be washed to become a valuable fuel. Washing then becomes a major operation, as hand picking of the 20 per cent. of lump is a standard and comparatively simple operation. Some coal beds here are so interstratified with bone, of a more or less massive nature, that hand picking of lump is not economically efficient. In such cases the coal must be crushed fine enough for it to be freed from the bone, so that the two may be separated. Coal cleaning in this field, therefore, must receive skilled attention to purify this fuel and raise its stand- ard. Without greater efficiency in cleaning, the importation of coal from outside the state will be even greater than in the past.

Jan 1, 1926

By AIME AIME

WITH a brilliance undimmed by hard times, the annual dinner on Wednesday evening, Feb. 17, was a complete success. More than 600 members, friends and ladies gathered early and filled the anterooms of the grand ballroom of the Commodore Hotel, while the president's reception was in progress, and then at the sound of the gong and with the music of the orchestra, rising above the happy din of conversation thronged to their seats at the tables. Following the "potage Ambassadeur"-probably to eliminate a very real hazard-the usual toy balloons appeared and a gay and lively scene, punctuated by multitudinous explosions, ensued, as the hundreds, of variously colored spheres bounced yon and hither until their gaseous existence was ended on the glowing end of a cigar or cigarette. On the speakers' dais sat C. E. Beecher, Chairman Petroleum Division; J. Vipond Davies, representative of A. I. M. E. on the Alfred Noble Prize Committee; Corbin T. Eddy, Alfred Noble Prize recipient; A.' S. Tuttle, Vice-president American Society of Civil Engineers; W. C. Mendenhall, Director U. S. Geological Survey; George C. Bateman, representing the Canadian Institute of Mining and Metallurgy; Paul D. Merica, Institute of Metals lecturer and Chairman James Douglas Medal Committee; Champion Herbert Mathewson, James Douglas Medalist; Robert E. Tally; retiring President; G. Temple Bridgman, toast-master; Scott Turner, incoming President; F. W. Bradley, William Lawrence Saunders Medalist; Fred Searls, Jr., Chairman William Lawrence Saunders Gold Medal Committee; Arthur Notman, President Mining and Metallurgical Society of. America; Robert J. Grant, Director U. S. Mint; Landon F. Strobel, Chairman New York Section; Howard Scott, recipient of Robert W. Hunt Prize; George. B. Waterhouse, Chairman Robert W. Hunt Prize Committee; F. M. Becket, Chairman Iron and Steel Division; Ora E. Clark, recipient of the J. E. Johnson, Jr., Award; Richard Peters, Jr., Chairman J. E. Johnson, Jr., Award Committee; E. C. Bain, Howe Lecturer; Sam Tour, Chairman Institute of Metals Division; and H. Foster Bain, Chairman. Charles F. 'Band Foundation Committee. ' Each was heartily applauded when introduced by Mr. Bridgman, as was Mrs. Tally, who also made a bow. The toastmaster made graceful reference in introducing Mr. Strobel, to the large part the latter had taken in planning the meeting.

Jan 1, 1932

By H. LIVINGSTONE LMAN

A GOOD DEAL of information concerning flotation has come out during the patent litigation of recent years, and the legal situation has cleared considerably, to the satisfaction of Minerals Separation, Limited. Hence, Mr. H. Livingstone Sulman, one of the originators of the successful process of that company, felt at liberty to explain, on Thursday, Nov. 20, his experience and his views in a "Contribution to the Study of Flotation," before the Institution of Mining and Metallurgy. He did not give technical details. He dealt with the physical problems, so simple, as he remarked, that he had wanted 96 pages to express his views. He presented much data, partly new and based on observations of his own, bearing upon the very complex flotation phenomena in an instructive wav and made frequent references to the valuable researches especially of Prof. Edwin Edser and Mr. Emil Hatschek.

Jan 1, 1920

By ROBERT R. RICHARDS

My. first paper, read at the Cobalt Meeting of the Institute,1 July, 1907, dealt with the behavior of a small Wilfley table when concentrating galena from quartz, the table being fed with natural products, with sized products; and with classified products. Since galena, having a specific gravity of 7.5, is among the heaviest minerals ordinarily concentrated by Wilfley table, it was desirable that a similar set of tests upon a mineral of, lighter weight should be made. Chalcopyrite, of a specific gravity of 4, would have been the best mineral for this purpose, on account of its great importance as an economic mineral, but in carrying out the test, however, I was obliged to content myself with a cupriferous pyrite, the specific gravity of which was 4.68 and the contents in copper 8.80 per cent. Through the kindness of Mr. H. 0. Cummins, this material was obtained from Shasta, Cal.,in very pure, clean, homogeneous condition. It was broken by rolls to 2 mm. in size, taking care to make a minimum of slimes. The quartz used to mix with it was pure, white, New England quartz, broken in the same way to the same size. In studying the accompanying tables, it should be borne in mind that the material concentrated was of low copper-content, and in many cases the quantity fed to the machine was 1 kg: or less.. As a consequence of the smallness of the 'quantity treated it was very difficult to adjust the table and get it in condition to do its best work before the feed became exhausted, and for this reason many results were obtained which appear irregular.

Sep 1, 1908

By J. Robert Van Peli

IT was a strange but highly fruitful marriage-that union of hardy explorers, seeking the rich treasures of copper in the Lake Superior wilderness, with Boston's aristocracy of brains, capital, and enterprise. This historic wedlock, which has continued happily to the present day, was the means by which the wealth of Lake Superior's copper was won for the use and benefit of an expanding America. One branch of this family tree, the western branch, begins with prehistoric Indian miners, embraces the early French explorers of the Upper Lakes, and finally includes the British and American pioneers who reduced the vague rumors of copper to known fact and actually began the mining of copper in the wilderness. The other branch, the New England line, begins

Jan 1, 1948

By Albert J. Phillips

The effect of impurities on the flowsheet in the smelting and refining circuits for copper, lead and zinc is reviewed and the interflow of by-poduct metals from copper, lead and zinc plants is pointed out. The principal points of separation of all impurities in the circuits are indicated and the additional steps added to these circuits in order to effect impurity separation me shown. In this day and age, when difficult metallurgy is mentioned, one thinks of the extraction of more refractory or possibly the alkaline earth or rare-earth metals. It is true that the extraction of metals having a high affinity for oxygen and deleterious solubility for that element in the liquid metal, affords some of the most difficult extractive metallurgy problems. When this is coupled with an affinity for hydrogen, nitrogen, and carbon a devious approach is required that sometimes rivals the complexity of the metallurgy with which this paper deals. However, in all such cases the aim is for a simple

Jan 1, 1962

By WILLIAM BOWIE

THE development of the isostatic idea during the last century would make an interesting paper in itself. But the various steps in the development have been covered in a number of papers and books which are readily available. Therefore this short paper should not give space to. the historical phases of the subject. Isostasy is really an old idea, but it is only in the last few decades that it has taken on any great prominence or importance in the study of earth phenomena. It is really remarkable how closely Dutton' sensed the isostatic principle 40 years or more ago. His general principles of isostasy as outlined by him have stood the test of time and fit in closely with the general ideas held today. It would seem, however, that some of his views with regard to the application of isostasy to geological processes must be modified or abandoned.

Jan 1, 1930

By Lyon F. Terry

ACCEPTED current practice for A the valuation of mineral property is based upon Hoskold's theory and valuation tables first published in 1877, and popularized by Herbert Hoover's "Principles of Mining" issued in 1909. In more recent years the procedure has been developed further in the course of the extensive valuation work required for Federal taxation purposes. The 1-aluation of oil and gas properties and of mines is based upon the same principle-the estimation of the present worth of future net revenues. However, in valuing oil and gas properties the method used for discounting future revenues differs somewhat from that adopted for mines, owing principally to the nature of the re- source. production from oil and gas properties usually declines from year to year, whereas coal and ore de- posits are adapted to a more uniform rate of output.

Jan 1, 1940

By FLOYD T. TAYLOR

IN 1833, less than one hundred years ago, Michael Faraday discovered and stated the laws of electrolysis. His discovery formed the foundation of a new use of metals which has now reached a variety of application not ordinarily appreciated. Furthermore, things can be done with electro-deposited coatings which have thus far defied other methods. Some of the uses of electro-deposited coatings are interesting. The tonnage of metals already finding an outlet through such a channel is considerable. In general electro-deposits have two characteristics: ornament plus utility; utility plus ornament. Both these factors are usually found associated in electro- deposits. Ornament plus utility was probably the first use. For example, the silver-plating of tableware, having its real birth in Birmingham, England, about 1844. However, this same ornament plus utility retarded the development of electro-plating, because in the beginning, electro-plated articles were branded with the odium of imitation and shunned accordingly. The gold brick of the swindler may be an exception to this statement.

Jan 1, 1929

By HAROLD VINTON COES

MANAGEMENT has been tersely defined as getting things done through the efforts of other people; but before we proceed further, let us distinguish between administration, management, and organization. Oliver Sheldon in his book, "The Philosophy of Management," offers the following definitions: Administration is the function in industry concerned in the determination of the corporate policy, the co-ordination of finance, production, and distribution, the settlement of the compass of the organization, and the ultimate control of the executive.

Jan 1, 1943

By ELLSIVOKTH DAGGETT

THE Berlin gold quartz mine is situated in Nye county, Nevada, on the west flank of the Shoshone range, about 40 miles south and 30 miles west from the town of Austin, the county-seat of Lander county. The distance from Austin is about 60 miles by stage-road. The outcrop of the vein, at the top of the incline-shaft, is situated just at the base of the mountain proper, almost exactly at the intersection of the mountain-side with the gravelly bench that slopes for about three-quarters of a mile to the flatter sage-brush plain, or desert valley, below. The -vein itself consists almost entirely of quartz, with perhaps 2 per cent. of sulphide of iron, copper, lead, zinc and antimony, and perhaps a trace of some of the compounds of tellurium with gold and silver, although none of the latter have been as yet positively identified. The relative proportion, by weight, of silver and gold in the ores, varies in different parts of the vein, from 12 silver for 1 of gold, to 7 silver for 1 of gold. The quartz vein-filling is usually frozen fast to the walls, and is very hard and compact, seldom showing the friable, fissured, or shelly structure often to be found in quartz veins. Comparatively little evidence of relative motion of one wall of the vein upon the other is to be found-a fact indicating that during the formation of the vein, and prior to the extensive movements herein described, little disturbance had taken place. No evidence whatever of metasomatic origin has been observed. On the other hand, occasional occurrences of comby structure, in which the axes of the quartz crystals are at right angles to the plane of the vein, rather indicate deposition from solution in a pre-existing fissure. Spurs, or branches, and small parallel veins, while not en-\

Mar 1, 1907

By J. D. Price, W. M. Bertholf

One of the chief difficulties with which the operator of a coal washing plant has been forced to contend is the handling of the very fine coal. First he has the problem of separating the fine coal from washery water. This is usually accomplished by the use of settling cones or Dorr thickeners: in either event the separated fine coal contains a high amount (40 to 80 pct) of water and, somewhat loosely, dependent upon the size of the coal particles present, may be known as sludge or slurry. His second problem is the satisfactory further dewatering of such separated sludge and/or slurry.

Jan 1, 1949

By Sallie McCabe Johnson

LIFE IN RUSSIA for the foreign woman is hard. It is up to her whether her days are spent in tearful longing for ironic or whether she :hakes the real effort to ferret out the interesting or amusing side of such exasperating conditions as ooh a Russian can conjure. So to begin Discard all promises.. This is perhaps the most difficult thins- to learn but promises in Russia are nut what they are at home. There are many reasons for ibis, sonic good and some bad but whatever the reason. the fact remains that they are usually worthless- as each of us must slowly learn In, disappointment after disappointment. Also we must learn that we cannot expect to get or to do a certain thing just because someone else has done so. What we did yesterday perhaps cannot be dune today. What we purchased air hour ago, oxen ten minutes ago- we cannot purchase now. What our neighbor did this morning we cannot do this afternoon. Changes arc continuous. Every person's experience is different from that of another aid this in a great measure accounts for the conflicting stories brought horns. In a land of chaos we live only from hour to hour and this is one of the exceedingly exasperating conditions to face. When I wished to visit Leningrad, an overnight nip from Moscow, more than a week's lime w as required to obtain a ticket, vi en through the efficiency of lntourist: alone I might have spent a month at it. Each day

Jan 1, 1932

By E. Sengier

At the Director's dinner of the A.I.M.E. on. April 22, Mr. Sengier of the Union Minière du Haut Katanga was a guest. Though a member of the Institute for sev-eral years this was the first occasion on which he had had opportunity to meet with it, and in response to an invitation from President DeGolyer, after expressing his pleasure at being present, he told informally of the present development of the Congo with which he has been so closely related. From his remarks the follow-ing has been prepared.-EDITOR. AS you know, the Belgian Congo, a fairly large colony compared with the territory of Belgium, is in the center of Africa. Natural waterways, railroads now built and new railroads under construc-tion, give increasing means of transportation and are helping toward development. The mining industry is by far the most important part of our colonial activity. Considering that we are all engineers, I will confine my figures to mining enterprise. First I will show you the relative importance of the mining industry com-pared with the total exports of the Belgian Congo. During 1925 there was exported 100,000 tons of cop-per, valued at about $25,000,000; 900,000 carats of diamonds, 8000 lb. of gold, 1700 tons of cassiterite, 300 .tons of cobalt, and 20 grams of radium, of a total value of about $35,000,000. Other products to a value of about $12,000,000 and consisting mainly of palm nuts, palm oil, cotton, ivory, copal, rubber, were also exported. From this you will see that mineral products account for about 72 per cent of the total exported value. Copper alone amounts to 52 per cent of the total and 72 per cent of the mineral exports. All minerals, with the exception of gold, which is mined by the government, belong practically toy two mining companies in which American and British capi-tal is interested with Belgian. These two companies (one called Forminière, interested in the diamond busi-ness, and the other called the Union Minière, interested in copper, cobalt, tin and radium) were formed twenty-five years ago. They started producing in 1911, but the war prevented normal operations, and active de-velopment could only be carried on after the Armistice.

Jan 5, 1927

By William R. Webster

I have continued the investigation of the Pottstown Iron Com pany's basic Bessemer keel plates on the line referred to in my paper of last October (Trans., xxi., 766)) and have added a study of the hardening effects of sulphur, which were not taken into account at that time. The results are so encouraging that I now give them in full. For some months we have graded all the steel at these works by the table given in my former paper, and checked the grading by the forging-tests as described in the paper of Mr. Joseph Hartshorne, read at the same meeting (Trans., xxi., 748). The steel thus graded was bloomed for the plate orders. Then one slab was rolled into a plate for the order, and a tension-test was made of this plate. By the result of this tension-test we were guided in rolling the balance of the heat. Out of several hundred heats graded in this manner, 95 per cent. met the requirements of the orders in every respect, and were accepted; but these results will no doubt be improved upon, when the influence of sulphur is also considered.. The hardening effect of sulphur seems to be about 500 pounds per each .01 per cent. of increase. In the table accompanying the pres ent paper, the ultimate strength with .06.5 per cent. sulphur agrees with the old table where sulphur was not considered; that is, for all steels with less than .065 per cent. sulphur we now get lower re sults, and for those with more sulphur, higher results. Making this allowance for sulphur, the assumed ultimate strength of a 3/8-inch plate of pure iron is reduced from 38,000 pounds per square inch to 34,750 pounds. With this as a basis, I have constructed my new table, giving each of the elements, carbon, phosphorus and manganese the same hardening effect as before, to wit : Carbon is credited with a constant effect of 800 pounds per each, 0.01 per cent. Phosphorus shows greater effect in high-carbon than in low-carbon steels. For 0.07 and 0.08 C, it has about the same effect as carbon, that is, 800 pounds for each .01 P. This increases to 1500 pounds per .01 P in 0.15 C steel, or nearly twice the effect of carbon. This

Jan 1, 1894

By J. B. Clark, A. S. Yue

PfANN1 has shown that when a cylinder of molten binary alloy freezes directionally, the solute distribution can be described analytically by cg=k(l-g)k-1 [1] in which Cg is the solute concentration in the solid at the point where a fraction g of the original liquid has frozen and Co is the initial solute concentration. The term k is the effective distribution coefficient of the solute and is defined as the ratio Cx/CL where Cl is the solute concentration in the main body of the liquid. Both the slope and the intercept at -log (1 - g) = 0 of a logarithmic plot of Eq. [I.] provide a measure of k. Under conditions of freezing where diffusion in the solid is negligible and the solute concentration in the liquid is uniform, the effective distribution coefficient k, derived as above, is equal to the equilibrium distribution coefficient taken from the phase diagram. However, under ordinary freezing conditions, even though solid-state diffusion is negligible, the solute rejected from the freezing solid does not have sufficient time to diffuse uniformly throughout the main body of the liquid. A concentration gradient is formed in a boundary layer in front of the advancing solid-liquid interface as pointed out by Rutter and Chalmers.' Thus, under these conditions, the concentration of the freezing liquid in the boundary layer is higher than that predicted from the equilibrium phase diagram and consequently the effective distribution coefficient k, obtained from directional freezing experiments, is larger than the equilibrium distribution coefficient. The symbol k will designate the effective distribution coefficient in the discussion which follows. In this paper, the determination of the effective distribution coefficient is demonstrated for several dilute binary alloys and a polynary magnesium alloy. The effect of freezing rate on the value of the derived k is presented in detail. An example for the calculation of the solidus in a dilute binary solution using the experimentally determined effective distribution coefficient is given. MATERIALS STUDIED Binary magnesium alloys of copper, nickel, and zinc containing approximately 0.10 wt pct solute and a polynary alloy containing approximately 0.10 wt pct each of several elements were selected for study. The spectrographic analyses of these alloys are given in Table I. The analyses are believed accurate to within +5 pct of the stated value. EXPERIMENTAL PROCEDURE Extruded rods of the alloys, 3/8 in. in diam and 11 in. long were used for directional freezing studies. The alloy rods were melted horizontally in a high-purity graphite boat by a Nichrome-wound tube furnace which moved parallel to the length of the rod. The boat was supported inside a Vycor tube suitably connected to provide a protective atmosphere of dry SO2. For an experimental run, the following procedure was used. The alloy rod was cleaned in dilute HNO, and dried. The graphite boat containing the alloy rod was placed within the Vycor tube and the tube connected to the gas train' The 'ystem was purged for 20 min with dry argon and then a flow of dried SO, was maintained for the duration of the run. The movable furnace was positioned and either all or a desired portion of the alloy rod was melted. After attainment of steady-state temperature gradients the molten bar was solidified from one end to the other at a controlled rate by pulling the furnace along the length of the bar at the desired speed. At the conclusion of the run, the directionally frozen alloy rod was analyzed spectrographically at 1 1/4-in. intervals.

Jan 1, 1961

By J. S. Aronofsky, J. P. Heller

This paper presents a mathematical analysis of the fluid mixing which occurs during flow through porous media. The analysis is based on the well-known diffusion equation with mass transfer term. It is pointed out that the use of this equation is justified by a single general assumption which does not specify any particular mechanism for the mixing process. Formulas are given for tracer fluid concentrations for two different boundary and initial conditions. Calculated numerical values compare closely with some published results of experiments in which no viscosity or density difference existed between displaced and displacing fluids. INTRODUCTION A phenomenological theory of the mixing and diffusion process is presented for the flow in a porous medium of two miscible liquids of equal viscosity and density. This is based on the classical diffusion equation which has been used to explain such processes as Brownian motion and heat conduction. More recently, this same differential equation has been derived (in a manner analogous to the work in this paper) for the problem of heat transfer during fluid flow in porous media.' The purpose of this paper is to show that the available experimental evidence justifies the use of this simple diffusion equation. The recent data of Koch and Slobod' and of von Rosenberg' are used to test this diffusion model by comparing concentration curve shapes and calculating "effective diffusion coefficients". The same differential equation for miscible displacement, as well as some experimental confirmation, has been published in Japan by K. Yuhara,4 and recently in this country by Day.5 Yuhara arrives at his con-clusions through analogy with turbulent diffusion, de- scribing the microscopic velocities in terms of "eddy motion" — although unlike the ordinary eddies of hydraulics. Day proceeds directly from Scheidegger's6 concept of the dispersion of velocities. After a very clear description of Scheidegger's theory, he applies it to calculate the motion first of a drop and then of a thin layer of brine into a bed of sand. For the latter case, Day derives and solves the same differential equation as Eq. 1. In our presentation no detailed assumptions are made as to the mechanism of displacement. In fact, several microscopic processes may well be involved—the only general assumption made is that their cumulative microscopic effect may be described by the addition of a second-order term to the differentirl equation of continuity. DERIVATION OF EQUATIONS Let us consider the displacement in one spatial dimension of Liquid A from a sample of porous media by Liquid B. A and B are assumed completely miscible and of the same viscosity and density. Both fluids are also assumed to be incompressible and it is specified that the total flow is the same for any cross section. It is evident that the rate at which Liquid B would be carried across such a section would be given by the product of the fluid velocity, u, and the concentration, C, of B in the fluid mixture assuming the constancy of u and C over the cross section. Since the volume of Fluid B is conserved, this leads to the one-way wave equation - u ?C/?x = ? ?C/?l where x increases in the direction of u—the "Darcy velocity", Q/A (or velocity flux per unit area), and 9 is the fractional porosity. Consider the case where fluid B is displacing Fluid A and the concentration distribution is represented by a step funcLon (with a vertical discontinuity). The solution of Eq. 1 predicts no mixing

Jan 1, 1958