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By John Chipman, Theodore B. Winkler

IN order to understand more fully the complexities of the reactions occurring between the liquid steel and the slag in the basic open-hearth steelmaking furnace, investigations in this country and abroad have turned to laboratory equilibrium studies. This paper is one of a series of reports on investigations being conducted at the Massachusetts Institute of Technology. The experimental work was completed early in 1942. The slag systems found in the basic open hearth are very complex, being made up of at least eight major components and as many others whose effect might be classified as minor. The first two papers of the series, by Fetters and Chipman,l,2 as well as the paper by Taylor and Chipman,3 dealt with the simplest possible slag system that would resemble those found in the basic open hearth-the slag system consisting of CaO-MgO-SiO2-FeO. (Taylor's rotating induction furnace eliminated the necessity of contamination of the slag by MgO.) Having arrived at some reasonable conclusions regarding the physical chemistry of the reactions between molten iron and the simple slags at equilibrium, the next step was to investigate more complex slag systems. The major purpose of the present investigation was to determine what factors control the distribution of phosphorus between molten iron and the more complex basic slags at equilibrium, and to what extent these factors are influential. The slags used contained CaO, MgO, Si02, FeO, Fe203, P205, MnO, A1203 and CaF2, but not all of the slag samples contained all of these components at one time. REVIEW OF LITERATURE Although a number of interesting papers have been published on the subject of dephosphorization of an iron bath, there are as yet no results from laboratory studies that give thoroughly reliable information on the dephosphorization reaction. Even for the supposedly simple Fe-O-P system involving iron-phosphorus melts and iron oxide-iron phosphate slags, the experimental results are contradictory and no formulas have been established that adequately fit the equilibrium conditions. The Fe-O-P system has been studied by Schackmann and Brings4' von Samson-Himmelstjerna,5 Oelsen and Maetz,6 Bischof and Maurer,7 Diepschlag and Schürmann,8 and Herty,9 and the results of their investigations appear in Table I. A number of different equilibrium constants and distribution ratios

Jan 1, 1946

By Michael Tenenbaum

A REVIEW of process metallurgy of iron and steel during 1944 in many ways reflects the political and military developments of the year. Early in 1944 the tremendous wartime emergency expansion program was virtually completed and the industry directed its efforts to maintaining maximum production wherever needed. The vigorous production programs inaugurated to support the Normandy invasions did not persist as Allied armies approached Germany. General public apathy which developed with the prospect of an early ending of the European fighting was reflected in a general relaxation in armament programs. Toward the end of the year, as the prospect of a prolonged war became real, the entire program revived considerably and the need for steel became acute. Between January 1940 and October 1944 more than 12,400,000 tons had been added to the nation's steelmaking capacities, and over 12,700,000 tons added to the blast-furnace facilities. The total cost of this enormous program was more than two billion dollars. By July 1, 1944, total steelmaking capacity had exceeded 94,000,000 tons, with blast-furnace capacity rated at 68,400,000. Annual steel production, which was 88,900,000 tons in 1943, increased slightly during the past year and was estimated at 89,500,000. Production of pig iron also increased somewhat from 61,800,000 tons in 1943 to an estimated output of 62,500,000 in 1944. This review deals with a few of the numerous factors involved in maintaining the wartime iron and steel program at the level necessary to accomplish this enormous output.

Jan 1, 1945

By B. J. Harlan

THE trying times experienced by the steel industry during the past four years have emphasized the necessity of producing pig iron at the lowest possible cost. The trend in both design and operation of blast furnaces has been toward this end rather than securing maximum production from a unit or group of units. In the following discussion the trend in design of stacks and auxiliary equipment will be taken up before operation is considered. Although few furnaces have been constructed in the past few years, the trend is definitely toward larger furnaces with a hearth diameter of 24 or 25 ft. Such a furnace is very flexible, being able to produce efficiently any desired daily tonnage between 500 and 1000 or more. The detail of the design of furnace stacks, with the exception of the top, has changed little in recent years.

Jan 1, 1934

By Arthur F. TQggQrt

THE approximate status of education in ore dressing in 1871 is reflected by Rossiter W. Raymond in an article written at that time presenting the curricula and descriptions of the laboratories at the various mining schools in the United States. For the most part, the curricula in which ore dressing was taught were for mining engineering. They comprised courses in geology, mining, and pyrometallurgy about equal in point or credit weights; a short course in ore dressing, and a somewhat longer course in assaying; and courses in surveying, hydraulics, power, and strength of materials-all superimposed on a broad foundation of physics, chemistry, and mathematics. Some curricula contained a fair content of the humanities. Then, as now, the emphasis on fundamentals was greater in the older colleges and universities. The report on the Columbia School of Mines, then relatively young, emphasized that a large percentage of its students were college graduates. As technical and scientific knowledge grew, first metallurgy and then geology were set up as separate curricula. So far as curriculum designation is con-

Jan 1, 1947

By D. R. Knowlton

IF I were a minister, and this were a sermon, and such a passage appeared in the Bible, I would choose for my text: "From whence cometh the oil for our war?" And no preacher was ever more serious than I am. You petroleum engineers recognize that the producing branch of the industry is passing through an era of plenty and inflated potentials into an era of scarcity and unmet allowables. Not too long ago the oil operators of Kansas, California, and Texas were appearing before operators' meetings and regulatory authorities asserting that these States were capable of producing at rates far in excess of what the demand at that time would absorb. Today every oil-producing area in the United States except the Texas Gulf Coast and West Texas is producing close to or in excess of its maximum efficient rate.

Jan 1, 1943

By Walter J. Mead

By American tradition, if not by rational decision, publicly owned natural resources have been transferred to private industry for processing. The process of transfer requires specific determination of the price which the patentee, buyer, or lessee must pay to the government in exchange for the right to develop a resource, and selection of the patentee, buyer, or lessee from among several possible contestants. Pricing of natural resources owned by the government presents a set of problems quite different from ordinary commodity pricing. Natural resources in situ are not homogeneous commodities. Within any class of natural resources, specific leasable parcels will differ in value by reason of location and accessibility, as well as quality and quantity. Thus, there is no "going price;" every lease within a specific resource class will have a different value. To complicate further the pricing and buyer selection problems, competition may not be reliable. Natural-resource markets are characterized by oligopsony conditions. In a significant number of cases, as will be shown later, there has been only one bidder for a given lease, and in most lease sales the number of bidders is equal to or less than three. As a further complication, there is great uncertainty regarding the value of specific resources, even among the best-informed buyers, as well as within the leasing agency. In some instances, particularly with regard to oil and gas resources, there may be uncertainty about the presence of resources, as well as the quantity and quality of any resources contained in the tract offered for lease. In other cases, such as oil shale and some deep ocean minerals where new technologies are involved, the presence of the resource is clearly established. There may be a relatively high degree of knowledge about the quantity and quality of the resource, but there is great uncertainty concerning the cost of production. These problems in total make it difficult for the leasing agency to appraise accurately the resource to be leased, as well as for the potential bidder to calculate a bid price that will yield him a normal profit. The pricing goal of the federal government as the seller or lessee may be to capture the economic rent; that is, the price of a resource might equal its discounted present value. A required price substantially above this amount would either generate no bids, or, in the long run, would bankrupt the buyer. A price below this amount is a disguised subsidy to the lessee. A subsidy to mineral lessees, as to any other economic interest group, as a minimum should be justified by a clear showing of net external benefits from the economic activity. In the case of timber sales by the U.S. Forest Service, the Forest Service Handbook asserts that "the aim in national forest stumpage appraisal is to determine market value,"8 and "market value" is equated to "fair market value." Regarding minerals, there is no reference in legislation or regula-

Jan 1, 1976

By Roger L. Brockenbrough

Steels available for mine-shaft applications are discussed, and the use of steel linings for round and rectangular shafts is reviewed. For rectangular shafts, frameworks constructed of square or rectangular steel tubes offer advantages. Also, corrugated steel sheeting provides a suitable alternative to timber lagging for spanning between the main shaft supports. For round shafts, self-supporting cylindrical steel liners, with circumferential stiffeners to prevent collapse, have been used to depths up to 6000 ft. Several large-diameter round shafts have been constructed with a concrete lining and an internal framework: of high-strength low-alloy steel tubes. It is concluded that in any new shaft construction steel designs should be investigated as economical alternatives to designs using other materials.

Jan 1, 1976

By Robert E. Wilson

TO show the vital importance of our future oil supplies to our economy, I will merely point out that this country, with something like 15 per cent of the world's land area and something like 7 per cent of the world's population normally uses between 65 and 70 per cent of the motor vehicles of the world, between 65 and 70 per cent of the world's rubber, and produces and refines between 65 and 70 per cent of the world's petroleum. Despite all the arguments as to which is cause and which is effect, all will agree that all three of these important industries are mutually dependent. If we had not been able to produce 65 or 70 per cent of the world's oil in this country we would not have the degree of mechanization, or the efficiency of transportation, or the high industrialization that we enjoy. Furthermore, the fact that oil has been plentiful and cheap has had a tremendous effect on all the things in which we as engineers are interested. For that reason, if for no other, it is well to look ahead to see what the outlook is for the near and far-distant future.

Jan 1, 1945

By William R. Webster

In view of the great interest now taken in the tests of castiron and details of foundry practice, with the number of investigators at work, and recent improvements in the methods of research, it would seem that the time is ripe to attempt the solution of some of the many problems with which iron founders have to contend. Recent papers and discussions on these subjects* have opened up s large field for investigation, and have emphasized the importance of many considerations generally overlooked. With the view of promoting a comprehensive and systematic discussion, I have attempted to tabulate, in conveilient form, some of the most important points for investigation, on the general plan suggested, with such good results, by our former President, H. M. Howe, for the discussion on the Physics of Steel, at the Virginia Beach meeting, last year. This table can, no doubt, be modified or enlarged to advantage, and put in shape to serve as a guide or reminder to all who are interested in this line of work. Suggested Lines for Discussion and Investigation. I.—Correspondence between chemical composition and melting point, fluidity, shrinkage, fracture, chill, micro-structure, and other physical properties. II.—Influence of: I. Cupola mixture, use of steel and other scrap, oxidized or clean material, 2. Manner of melting, flux, etc., 3. Casting temperature, A. Fracture. 4. Manner of handling melted metal and B. Micro-structure. method of casting, C. Physical properties. 5. Size and form of casting, D. Shrinkage. 6. Kind of mould, green sand (under dif- on 1 E. Chill. ferent conditions of ramming, amount F. Residual stress. of moisture, and skin-dried), dry sand, G. Condition and quan loam, and chills, tity of carbon and 7. Rate and mode of cooling castings, other elements. 8. Manner and temperature of heating for annealing, 9. Additions of nickel or aluminum,

Jan 1, 1896

By John T., Ryan

THE subject assigned me, "Safety in Mining," is a very broad one and only the high spots can be covered in this short paper. As this is a meeting of the Coal Division, these remarks will be directed largely to coal mine safety, although the principles of safety and effective accident prevention methods are relatively the same in every branch of mining or industrial work. The coal mining industry should take pride in the part it has played in the development of the accident prevention movement, and there is ample proof for the statement that the first real organized accident prevention, work, both abroad and in this country, was con- ducted in the coal mining industry.

Jan 1, 1930

By Robert Baker

IT has been demonstrated in the laboratory by A. J. Phillips1 that a transformation or conversion from beta directly to alpha may take place in a brass of 61 to 62.5 per cent copper content. The completion of the transformation is dependent on a rapid rate of cooling from the all-beta field through the alpha + beta range to the alpha field. Phillips states that in a slowly cooled alloy containing 62 per cent copper and 38 per cent zinc the copper content of the beta is progressively decreased from 62 to 56 per cent by precipitation of alpha. Rapid cooling tends to prevent the separation of alpha with a copper content of more than 62 per cent by reducing to a minimum the time required for passing through the alpha + beta range. Upon crossing beyond the (alpha + beta.)/(alpha) boundary to the all-alpha region below, the material is transformed from beta to alpha. It is essential that there be no change in composition. Phillips obtained his results by quenching a 62:38 copper-zinc alloy from 850° C. in iced brine at -8° C. No reference to the occurrence of this transformation in a manufacturing process has been found in the literature. An investigation of the microstructures of a number of hot forgings made from leaded brass rods yielded one forging in which this striking conversion from beta to alpha had occurred. The purpose of this paper is to describe the transformation and certain conditions under which it may occur during the hot pressing of brass. The conversion was noted in a shape forged from a leaded brass rod, the composition of which was: Cu, 60.26 per cent; Zn, 37.86; Pb, 1.88. In this study the lead content was disregarded and the alloy was considered to be equivalent to an alloy of 61.40 per cent copper and 38.60 per cent zinc (98.12 parts = 100 parts). The possibility that the lead might obstruct or accelerate the precipitation of alpha was not studied. The lead appears as black globules in the photomicrographs and has a negligible solubility in the alloy.

Jan 1, 1931

By H. H. Stout

IN the early fall of 1025, the writer was conducting, in the Ledoux and Co. labora-tory, New York, experiments directed to-ward ascertaining the effect on its impurity content when cathode copper was subjected to a current of various gases at an elevated temperature below the melting point. The apparatus consisted of a vertical I-in. dia. silica tube about 12 in. long, heated on the outside. The cathodes were broken to pass 3/8-in. mesh, and a current of various differ-ent gases was passed up through a 4-in. long column of these loose cathode particles in the vertical tube furnace. Temperature was maintained to 1500° to 1600°F. and the gas treatments lasted from 3 to 6 hr. The writer observed that whenever reducing gas (H2 or CO) was used, these cathode par-ticles stuck together at points of contact with each other. The cohesion was so marked and tenacious that an examination was made of the fractured surfaces, when cathode particles were broken from the cluster. Crystal growth was always found to have taken place across the surfaces where separate particles were in contact, but only after all oxides, sulphates, etc., had been completely removed. The idea of a possible new process was then conceived and was disclosed to A. M. Smoot (V. P. Ledoux and Co.) with intent to obtain his views. The first conception as given Smoot was "to gas-clean broken cathodes, put them in an extrusion press, solidify and extrude them at around 1500°F." Smoot's reaction, after a few moments considera-tion, was, "It looks perfectly all right to me." The writer and his assistants, W. H. Osborn and H. H. Stout, Jr., after confer-ring as to ways and means, decided to go ahead with preliminary laboratory experi-ments, which, it was thought, would move the new process from the possible to the probable. This program followed a course indicated by three basic considerations: I. Since brittle cathodes were often unin-tentionally produced, it seemed probable that the laws controlling this type of deposi-tion could be ascertained. 2. We had already proved the gas-clean-ing feature. 3. There remained to prove: whether a sound, homogeneous metal billet could be produced with crystal growth established over the entire surface of each individual particle in the billet; and, if so, what would be the physical properties? Osborn arranged with Columbia Uni-versity for conducting experiments in its research laboratory to prove or disprove the third item; Columbia assigned C. A. Phillipi, Jr. to assist in the project. A press with a cylinder 2 in. high and of I-in. dia. was used. It was surrounded and heated by a resistance coil of Nichrome wire. Broken tough cathodes were used.

Jan 1, 1940

By John B. Cook

Most of today's open pits take the form of conical-shaped excavations in the relatively flat or undulating terrain surrounding them. Ore is usually hauled uphill from the pit bottom by truck, rail, or skip to crushing, beneficiation, railroad loading, and stockpiling facilities near the edge of the pit. There are, however, a few orebodies whose locations in mountainous terrain causes special problems that generate different technical solutions. LAMCO's 10.5 million metric tons per year high-grade iron mine in the Nimba Range in tropical Liberia is probably one of the most extreme of these exceptions (Fig. 1). At Nimba there is a 2,000-ft difference in elevation between the upper- most levels of the orebodies and the start of the railroad below at the foot of the mountain. Haulage to the crusher is usually downhill at grades of up to 12% dictated by the topography of the rugged terrain. Ore is transported from the crusher down the mountainside by a conveyor sys- tem 10,096 ft long, whose average grade is 18%. The belt is braked by asynchronous generators, feeding power into the local network. The original crusher must be relocated at a lower elevation outside the final pit limit to allow that part of the mountain on which it now stands to be excavated. HISTORY The LAMCO Joint Venture is owned 75% by the Liberian-American- Swedish Minerals Co. (LAMCO) and 25% by Bethlehem Steel Corp. LAMCO is comprised of a 50% interest by the Liberian Government and 50% interest by Liberian Iron Ore Ltd (Fig. 2). This joint venture was launched to develop the high-grade iron ore deposits of the Nimba Range.

Jan 1, 1969

Jan 1, 1891

By Frederick Bacorn

THE paper of Reno H.. Sales on Ore Deposits at Butte, Mont.,' is a careful and painstaking work, an important contribution to the literature of the subject. As is almost inevitable in a work of such magnitude, a few misstatements occur; and some of these I shall endeavor to point out. Mr. Sales has also fallen into error, as I think, in the application of his theory of ore deposition; and while the modification which I venture to suggest is a minor one geologically, it is of great economic importance. I refer to Mr. Sales's idea of an outward migration of mineralizing solutions from a supposed central copper zone. In order that I may make myself plain, I must first epitomize the theory as a whole, which I understand to be this: All of the Butte ore deposits (save those along the Continental fault, not herein considered) are derived from solutions of magmatic origin, which, at the beginning of their ascent from a great depth, were, acid in reaction, hot, and were carriers of copper, zinc, manganese and other metals. As they departed from their point of origin they lost heat, and, because of reactions with the wall rock, became more and more alkaline. The metal deposited from these solutions in their hot condition was principally copper; and, as they cooled, they deposited less and less copper and more and more manganese and zinc, until the copper practically disappeared. With this theory in general I have no fault to find, nor have I, it must be stated, the qualifications necessary to enable me to pass upon it critically; but it seems to me to be reasonable and probable, and is one which I am quite willing to accept.

Jan 8, 1914

By H. D. Merchant

NUMEROUS publications have examined hot hardness of metals and alloys. Some have studied creep in long-time hardness tests, few of which, however, were tested under a spherical indentor. 1-3 The results of Mulhearn and Tabor' indicate a parabolic-type creep for indium and lead, whereas Moore and Tabor2 how a logarithmic-type creep for indium. Similar semilog relationships between hardness and time have been found by Goffard and wheeler3 for magnesium, aluminum, and their alloys. In this note, the elevated-temperature hardness of some complex alloys, shown in Table I, was examined to discover whether any basic information about the deformation behavior of the alloys could be obtained. One of the alloys, H-13 alloy steel, was tested for short-time creep tests to examine the hardness-time relationship at various temperatures. A standard Brinell Hardness Tester, mounted with an appropriate furnace and 99 pct A12O3 poly-crystalline indentor, was employed for the purpose. The alloys were tested at temperatures from room to 1400°F for 30 sec under 1500-kg load. The creep studies were conducted for about 8500 sec at 7l°, 500°, 1000°, and 1200°F. For constants A and B, a relationship of the type H = Aexp(-BT) [ll was obtained for all alloys between hardness, H, and temperature, T. In the semilogarithmic plot, the data points for each alloy were arranged into two groups, each group on a straight line with a specific set of A and B values. These results are in agreement with many such observations on hot hardness, notably those of Westbrook for pure metals,4 The thermal softening coefficient, B, and transition temperature, Tt, for various alloys are shown in Table I. The change in slope around the transition temperature suggests an apparent change in the mechanism of deformation, probably from the shear (below Tt) to the diffusion type of mechanism. At the temperatures above Tt, for constants A' and B, a relationship of the type was obtained between H and T, Fig. 1. The nature of the equation, where R is gas constant, suggests the possibility of a thermally activated mechanism of deformation above Tt. The constant B' may hence be defined as the apparent activation energy of indentation. The B' values of various alloys are shown in Table I. A plot of B vs B' gives a curvilinear relation shown in Fig. 2. The indentation-creep data for H-13 alloy steel is shown in Fig. 3. At 71" and 500°F, the creep initially follows the logarithmic time law. However, the indentation size stabilizes after 7 sec at 71°F, and after 10 sec at 500°F. Further increase in loading time has no effect on indentation size. The creep of 1000° and 1200°F follows the logarithmic time law for the first 1000 sec, after which the

Jan 1, 1964

By E. H. Jr. Dix

A LACK of experimental data illustrating the effect of composition, particularly in respect to impurities, on the mechanical properties and corrosion resistance of aluminum-alloy die castings induced the authors and their associates to undertake a limited study of these relations. A number of commercial alloys .and several experimental variations of a commercial alloy were used for the investigation. Mechanical property, growth, steam, paraffin, salt spray and atmospheric exposure tests were employed. This paper deals with the more accelerated tests and the results they have produced. It is concluded that a high-silicon alloy with all impurities except iron strictly limited to small amounts will develop an excellent resistance to corrosion and retain satisfactory mechanical properties and casting characteristics. As a direct result of this investiga-tion an alloy of this character with superior corrosion resistance is now available commercially. In general, copper increases the strength and decreases the elongation and corrosion resistance. Tin reduces the mechanical properties and the resistance to corrosion. Iron reduces the elongation and, if excessively high, may adversely affect the tensile strength. Even excessively high iron contents do not affect the corrosion resistance, which is fortunate for the die caster as it is this element that is the most difficult to control. Nickel and zinc in limited amounts do not seem to affect the properties or corrosion resistance of at least the high-silicon alloy. Growth tests have confirmed the previous common belief that dimensional changes under either dry or wet conditions at 100° C. are too small to be of practical importance.

Jan 1, 1935

By George A. Miller

AMERICANS like engines, but more than anything they like powerful engines, and next to that they want them quiet, silent, smooth; perhaps a slight purr might be permitted, but they must not knock. To provide a gasoline to fit all these requirements, American petroleum technologists have experimented for years, and mounting octane numbers of the gasolines the refiners have produced both before and during the war bear witness to their success. The question on everyone's mind now is what may we expect in the way of engine fuel in the postwar period? The gasoline drouth during the war rationing, the inconvenience of doing without a car were major annoyances to all motorists, but they put up with them pretty cheerfully for the most part. However, now that the war is over, and the jalopies are all out and the better cars have been taken off the jacks and brought out on the road, the call is for "Some of that good gas-fill her up, if you please."

Jan 1, 1945

By Charles E. Heinrichs

IN the last decade one of our oldest and largest non-metallic metallic mineral industries has been the subject of persistent technical research, the results of which are another example of the benefits following the systematic application of scientific principles. The consequent improvement in the technique of phosphate recovery represents the greatest material advance in the industry since the discovery of phosphate in Florida in 1881. Moreover, these successes have stimulated interest and development in numerous other nonmetallic industries. Strange as it may seem, and despite the fact that the phosphate deposits of Giles County, Tennessee, together with those of Polk and Hillsborough Counties, Florida, have yielded more than 50,000,000 long tons in the last twenty years, the mineralogy of phosphates has been given scarcely any space in our technical literature. For many years it was the general consensus that these so-called phosphate rocks represented a physical mixture composed principally of tri-calcium phosphate, together with smaller quantities of calcium carbonate, calcium fluoride, iron phosphate, aluminum phosphate and other less important constituents. More recent investigations, however, led to the conclusion that these phosphates really represent a complex mineral in which calcium, iron, aluminum, phosphorus pentoxide, carbon dioxide, and fluorine all form a definite part of the molecule. Colony, after a careful study, utilizing chemical, microscopic, and X-ray diffraction methods, reported that American phosphates consist principally of calcium-carbono-phosphate of the staffalite type.

Jan 1, 1933

By Herman Dykstra, R. L. Parsons

A numerical method for solving partial differential equations in steady state fluid flow is described. This method, known as the "relaxation method," has two advantages over analytical methods: (1) practically any problem can be solved, and (2) a solution can be obtained quickly. A disadvautage is that the solution is not general. The method is applied to core analysis and relative permeability measurement to calculate constriction effects and to calculate the true pressure drop measured by a center tap in a Hassler type relative permeability apparatus. Further applications are suggested. INTRODUCTION Many problems in fluid flow cannot be solved analytically because of the nature of the boundary conditions. For many problems, however. an exact answer is not necessary because boundary conditions are not exactly defined or the parameters describing the porous medium are not accurately known. The relaxation method can be used to obtain an approximate answer easily and quickly for the flow of incompressible fluids in porous media. The method can also be used for other types of problems, such as determining the stress in a shaft under load. or the temperature distribution during steady state heat flow. In this discussion only calculations concerned with the flow of fluids in porous media will be considered. The method was introduced by R. V. Southwell in 1935.' THEORY The treatment given here follows that given by Enimons.2 Consider a porous medium to be replaced entirely by a net of tubes of equal length and uniform cross-sectional area as shown in part in Fig. 1. Assume that the net of tubes behaves exactly like the porous medium which it replaces; that is, the net can be made fine enough to reproduce exactly the porous medium. Assume also that Darcy's Law can be used to calculate the flow from one point to another point through these tubes. The flow from point 1 to point 0 is KA . ------ P-P) .......(11 where a is the distance between points: K is the "permeability" of a tube; A is the cross-sectional area of a tube; is the viscosity of the liquid in the porous medium; and (P1 — P0) is the pressure difference between point 1 and point 0. In like manner the flow can be calculated from points 2, 3, and 4 to point 0. The net flow into point 0 is Qo = KA/µa (P1 + P2 + P33 + P4-4P0) . . (2) MB For an incompressible fluid the net flow into point 0 will be zero or, Q. = 0. This says that at point 0 fluid is neither being accumulated nor depleted. 'Therefore. P1 + P2 + P3 + P4 - 4P0 = 0 .... (3) . If. now. with specified boundary conditions. the pressure i.; known at a finite number of points in a given region, as at the points shown in Fig. 1, Equation (3) will be satisfied at every point. If, on the other hand, the pressure is not known, the pressure can be guessed at these points. Then. unless the guess is perfect. Equation (3) will not be satisfied at all of the points. When Equatiol~ (3,) is not satisfietl. let d = P1 + I?, + P, + P, - If' .,....(4) where 6 is an apparent error and is called the residual at point 0. Equation (4) shows how much the pressure guess is in error at point 0 with respect to the surrounding points. A positive residual means that the pressure is too low, and a negative residual means that the pressure is too high. To bring the residual, 6. to zero in order to satisfy Equation (3). it is necessary to make changes in the pressure guesses. Equation (4) shows that a +1 change in Po will change the residual at point 0 by -4. A +1 change in the pressure at any of the four surrounding points will change the residual at point 0 by +l. Thus it can be seen that a change at any point will affect the residual at that point and the four surrounding points. By changing the pressure from point to point, all of the residuals can eventually be brought nearly to zero and the problem will be solved. This procedure is the essence of relaxation methods and is used to relax the residuals so that Equation (3) is satisfied at every point. The procedure can be most easily explained in detail by solving a simple problem. as Southwell says, "To explain every detail of a practical technique is to risk an appearance of complexity and difficulty which may repel the reader. A

Jan 1, 1951