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By Renato G. Bautista, KNona C. Liddell

A model of the reaction of chalcopyrite, CuFeS2, with Fe3+ and with 02 is considered. Homogeneous solution phase reactions are assumed to be always at equilibrium throughout the dissolution of the mineral. The dissociation of USOb- and several complexation reactions involving cu2+, Fez+ and Fe3+ are incorporated in the model equations. The relative importance of Fe3+, 02, and H+ as oxidants and the role of solution equilibria are assessed and the implications for leaching practice discussed.

Jan 1, 1981

By J. J. Reuther

Reported are singular results of a completed combustion research program designed to reliably determine the ignition, burnout, and pollution characteristics of various ranks of pulverized coal (bituminous and lignite), chars, and water-slurries of the raw bituminous coals. Well controlled simulated boiler environments were diagnosed to do so. The standardized, systematic, and comprehensive investigation established relationships between physicochemical fuel properties (coal rank, volatile matter, and reactivity) and thermochemical combustion behavior (ignition and burnout times, and NOX pollution potential) for 31 different fuels.

Jan 1, 1985

By Stefan S. Kapusniak, Nichol Riggott, Stephen F. Smith

Stability investigations have been carried out in a number of major coal mining tunnels in the U.K. using various instrumentatcon techniques. The paper describes these instrumentation methods and presents results from studies in several tunnels. Both long and short term stability aspects are discussed. Prediction of tunnel stability is examined in relation to tunnel support requirements. Computer simulation techniques are used along with actual underground measurements to identify tunnel design parameters for influencing choice of support and prediction of long term stability.

Jan 1, 1984

By John D. Erickson, Thomas R. Carlson, Milton T. Pana, Daniel T. O’Brian

Computer use in mine planning has evolved from theory to practical application at the Bingham mine of Kennecott Copper Corp.’s Utah Copper Division. Traditional mine planning methods have been combined with computer technology to produce a more realistic calculation of ore reserves and mining plans. The techniques used are not revolutionary but are an expansion of previous mine planning methods using the calculating power of the computer. These methods can readily be adapted for planning use by practical mining men.

Jan 5, 1966

By Hemendra N. Kalia

A method is proposed for the preliminary design of coal mine openings and pillars using existing design formulas. The suggested method incorporates geophysical and core logging techniques, as well as the structural properties of the rock. Some of the existing design equations have been simplified and are included. It is anticipated that the additional cost of rock mechanics tests and logging work will be more than offset by the increased coal recovery that does not jeopardize the safety of miners,

Jan 1, 1977

By P. Raffinot, M. Rey, V. Formanek, G. Sitia

CONCENTRATION of oxidized copper and lead ores by flotation has been practiced for 30 years, but flotation of oxidized zinc ores has remained unsolved until a few years ago. This problem is, however, of great importance in mining districts where large tonnages of oxidized zinc ores exist in underground mines or in dumps. Some sulphide ores also possess a degree of oxidation high enough to justify recovery of oxide zinc minerals. Various attempts have been made to float the zinc carbonate, smithsonite.

Jan 4, 1954

By A. L. Mular. Discussion by G. A. Parks

G. A. Parks (Associate Professor, Dept. of Mineral Engineering, Stanford Univ., Stanford, Calif.) - Professor A. L. Mular in his paper Effect of Impurities on the Flotation Behavior of Zinc Oxide (T.P. 65B206) has made a most interesting and thought provoking contribution. In view of his recognized lack of measurements of the ZPC's of the doped samples and lack of direct evidence for participation of semi-conducting electrons in collector adsorption an alternate explanation should be considered. The alternate also

Jan 1, 1967

Although grinding mill liner life has been the subject of extensive studies, evaluation of the effect of liner design a power consumption, a far more significant part of total grinding costs, has been less comprehensive. The factors that effect ball mill energy efficiency are reviewed, and attention is then focused on the mechanism of grinding. The trajectory of the outer row of balls, with different lifting face angles, is derived. Basic recommendations for mill speed and liner configuration are also presented

Jan 1, 1984

By George W. Goetz

As is well known, the general use of natural gas in Pittsburgh and vicinity for domestic and industrial purposes has shown the great advantages and benefits to be derived by the use of a gaseous fuel. In order to ascertain and bring together all the facts relating to a large and economical production, purification, distribution, and consumption of fuel-gas made out of coal, the Fuel-Gas and Electric Engineering Company, Ltd., was organized, under the presidency of George westinghouse, Jr.

Jan 1, 1890

By I. Iwasaki, K. Adam

The oxidation behaviors of simple and complex chloride systems of copper, iron, nickel, cobalt, and manganese in a stream of oxygen were investigated using a thermogravimetric analysis technique. Their reaction mechanisms are discussed from stoichiometry, thermodynamic feasibility, and X-ray diffraction analysis of reaction products. The results are used to determine an optimum operating temperature of the selective oxidation step in a chlorination process for producing nonferrous metal chlorides, elemental sulfur, and ferric oxide from copper-nickel sulfide flotation concentrates.

Jan 1, 1985

By S. J. Page

Introduction Dust from cutter machines and face drills is controlled in most mines by water. However, the large quantities of water used in metal and coal mines for dust control and suppression pose serious operational problems in water soluble ores, such as salt, trona, or gypsum. This paper reviews the strategies to suppress and collect dust on cutter machines and face drills where water usage must be restricted to very low flow rates - typically 3.78 L/min (1 gpm) or less.

Jan 1, 1985

By M. L. Jeremic

Observations of ground behavior in mining the Choka Dulkan ore body have been described. Ore recovery was limited to approximately 60% because unconsolidated hydraulic fill did not permit either the widening of stope spans nor secondary ore recovery from pillars. Stope undercutting induced high stress concentrations. However, as extraction progressed, ground conditions improved but before extraction was complete they deteriorated again. After completion of a stope, the remnant rib and crown pillars continued to be deformed by other stresses for some time.

Jan 1, 1979

By A. W. Hale

In a work published in 1838, De Morgan, the author of the article on " Probabilities," in the Encyclopedia Metropolltimn, says: " The method of least squares is not yet introduced intb the affairs of common life,.....that many things which are denlonstrable only by the higher branches of mathematics are looked upon as useless." And he mentions an instance in which "a method was suggested for estimating the relative importance of boroughs, by considering their population and ~wntributions to the revenue con)-

Jan 1, 1881

By Leland L. Mink, Roy E. Williams, Lewis M. McNay, Alfred T. Wallace

In 1968 the metal-mining industry of northern Idaho installed tailings ponds as a means of handling and treating mine wastes, metallurgical process wastes and domestic wastes. The installation of the ponds constituted a crash program to minimize pollution of the Coeur d'Alene River. The ponds provided an opportunity to investigate which combinations of wastes tailings ponds can be treated. The research data pertaining to tailings pond discharge lends insight into compliance with the recent EPA guideline for effluent quality.

Jan 7, 1973

By Fred W. Gordon

The heading of this paper was prompted by the knowledge that experience, up to this time, seemed to indicate that smaller furnaces were preferable for smelting the material of this section. Since the reasonable success thus far achieved by the first furnace of this plant tends to change the above opinion, and unsettle the generally-accepted views of furnace-men as to "weak " coke, a few words as to the design of the plant and the mode of operation adopted may prove interesting.

Jan 1, 1889

By A. Gemperle

Electron diffraction and transmission electron microscopy on foils at room temperature were used to investigate the ordering of Fe-Si alloys containing 10 to 23 at. pct Si. A certain degree of DO3 order was found in all alloys. With the exception of the lowest silicon concentration for which the antiphase domains could not be clearly resolved, the alloys have a domain structure of two-domain type with boundaries having 1/4a01<111> displacement vectors for less than 12.3 at. pct Si and with boundaries having 1/2 a0<100> displacement vectors for more than 12.3 at. pct Si. The alloys with 12.3 at. pct Si have a domain structure consisting of fine domains with 1/2a&apos;o<100> boundaries within much larger domains with 1/4a&apos;o<111> boundaries. The development of these structures can be explained by transition of the alloy from the disordered state into the B2-type order and then into the D03-type order by the mechanism proposed previously for the FeSAl alloys. The existence of the B2 structure in the lower part of the investigated concentration range reported in some articles can be explained by fine domains with 1/2a&apos;o<100> boundaries formed by several disordered planes within large domains with 1/4a&apos;o<111> boundaries. The ordered structure predicted by the theory —with practically no domain boundaries —is found in the alloys having 12.3 at. pct Si where it develops in the B2 structure region. ORDERING in Fe-Si alloys was first studied by phragmenl who found that beginning with 13 at. pct Si the DO3 (Fe3Si) superlattice reflections appear in the diffraction patterns. The equilibrium diagrams constructed later by corson2 and Haughton3 from various measurements proposed the existence of a homogeneous solid solution (a phase) in the range from 0 to 25 at. pct Si. Jette and Greiner4 and Farquhar et al.5 measured the relation between lattice parameter and composition and they considered the break in the curve at 9 to 10 at. pct Si to be caused by the ordered solution a"(Fe3Si). Glaser and Ivanick6 Determined critical ordering temperatures of the alloys containing from 10.9 to 27.9 at. pct Si from the measurement of the electric resistivity of quenched samples. In all cases the critical temperature was lower than the melting point and it was highest for 25 at. pct Si. Lihl and Ebel7 measured the lattice parameter curves at various temperatures up to 1000°C. The region between two breaks on these curves, corresponding to 10 to 12.5 at. pct Si at room temperature, was considered by them to be two-phase (a + a"). They concluded by extrapolation of the measured values that a" in the alloy having 25 at. pct Si is stable up to the melting point. Davies8 studied superlattice reflections in the X-ray diffraction patterns of an alloy containing 8.7 at. pct Si. He found the B2 structure and short-range order in the slowly cooled samples and the DO3 A. GEMPERLE is Research Scientist, Institute of PhysicS, Czechoslovak Academy of Sciences, Prague, Czechoslovakia. __Manuscript submitted January 2, 1968. IMD structure in the quenched and annealed samples. This investigation first reports the presence of the B2 structure, phase a&apos;: in Fe-Si alloys. Meinhardt and krisement9,10 also found its existence in Fe-Si alloys by neutron diffraction. No order was detected by them in the alloy containing 8 at. pct Si. The onset of B2 order was observed at a composition of 9.2 at. pct Si. They found almost perfect B2-type order with partial DO3-type order at room temperature in the 10 to 12.5 at. pct Si range and almost perfect DO3-type order in the 12.5 to 25 at. pct Si range. They established the critical temperatures Tc of both the structures through measurement at higher temperatures. The critical temperature for the B2 structure was found to be always higher than the critical temperature for the DO3 structure of the same alloy. They extrapolated the curves of the critical temperatures and concluded that the alloys with more than 17 at. pct Si have the B2 structure up to the melting point and the alloys with more than 23 at. pct Si have the DO3 structure up to the melting point. The results of Meinhardt and krisement9,10 were confirmed by Dokken&apos;s measurement" of the temperature dependence of the electrical resistivity in the 10.8 to 15 at. pct Si range. On the other hand chessin12 detected by X-ray diffraction a considerably lower degree of order in the 12.7 at. pct Si alloy. ANTIPHASE DOMAIN STRUCTURE IN ALLOYS WITH B2- AND DO$-TYPE ORDER The ordered structures B2 and DO3 can be described in terms of a subdivision of the bcc lattice into four fcc sublattices with a parameter double that of the bcc lattice. Following Marcinkowski13 we will label them I. 11, 111, IV. The B2 structure in the AB alloy is formed by placing A atoms on sublattices I and II and B atoms on sublattices III and IV. In a non-stoichiometric perfectly ordered alloy having concentration (A) > (B), A atoms occupy sublattices I and 11. and A and B atoms distributed at random occupy sublattices III and IV. The DO3 structure in an A3B alloy is formed by placing A atoms on sublattices I, 11, and 111, and B atoms on sublattice IV. In a nonstoichiometric, perfectly ordered alloy having concentration (A)/3 > (B), A atoms occupy sublattices I, 11, and 111, and A and B atoms distributed at random occupy sublattice IV. As further shown in Ref. 13, two types of domains are possible in the B2 superlattice and the antiphase domain structure has associated with it boundaries with displacement vectors 1/4 a&apos;o<ll1> only. Four types of domains are possible in the DO3 superlattice and the antiphase domain structure has associated with it boundaries with displacement vectors 1/4a&apos;o<111> and 1/2a&apos;o<l00>. Bethe14 suggested on the basis of theoretical considerations that in a structure with two sublattices at low temperatures only one domain should be present in the whole crystal at equilibrium. Similarly Bragg15 concluded that at low temperatures the domain struc-

Jan 1, 1969

By S. S. Cole, H. Sigurdson

The melting points of mixtures of titanium dioxide and other titanates have been reported to a limited extent as binary systems and some results have been reported in conjunction with silicon dioxide. The limited data indicated that a low melting zone might exist in a ternary or quarternary system of CaO-MgO-TiO2-Al2O3, since the eutectics reported in the binary systems of MgO.TiO2-TiO2, Al2O3TiO2-TiO2 and CaO.TiO2-TiO2 were of about equivalent composition of TiO2. The importance of a low melting region with a high titanium oxide value in the developing of high titanium slags is fully appreciated by metallurgists. A comprehensive study was undertaken to establish basic information on low melting titanate mixtures. Compounds in the System CaO-MgO-TiO2 -Al2O3 Preparation of the titanates of CaO, MgO and Al2O3 confirmed reported data that the following compounds could be formed under oxidizing conditions in solid state reactions: CaO.TiO2, 2MgO.TiO2, MgO.TiO2, MgO-2TiO2 and Al2O3.TiO2. Under oxidizing conditions it was not possible to react MgO with TiO2 in mol ratios higher than 1:2 without having unreacted TiO2 in the product, nor was it possible to form calcium or aluminum titanates with a higher mol ratio of CaO and A12O3 to TiO2 than 1:1. By fusing mixtures of Al2O3-TiO2 and MgO-2TiO2 together a series of solid solution products were obtained, which showed an MgO-2TiO2 X ray diffraction pattern shifted to smaller inter-planar spacings. Range of Investigation Data in the literature1 indicated that these titanates had melting points which ranged from 1645 to 1860°C. Since the purpose of the study was primarily to provide useful data for smelting titaniferous ores, the work was restricted to the zone of the system which would include the crystalline phases present in the slags. The explored limits were bounded by CaO.-TiO2-MgO.TiO2-TiO2in the base plane. The system was extended to a fourth component Al2O3-TiO2 since many titaniferous ores contain appreciable amounts of A12O3. In the quarternary system CaO-TiO2-MgOTiO2-TiO2-Al2O3TiO2, a tetrahedron was used to represent graphically the components, with one component at each point of the tetrahedron. In the base plane, mixtures were pre- pared to represent fairly uniform changes of composition over the desired range expressed on a mol percent basis. Al2O3-TiO2 was brought into the system in increments of 10 mol pct up to 40 pct. Mixtures were then made up for each of the Al2O3-TiO2 planes. The ranges employed were then represented by five planes cutting the tetrahedron at 10 mol pct Al2O3-TiO2 intervals. Choice of Equipment Although it was necessary for all smelting work to be done in a strongly reducing atmosphere, it was decided that all melting point determinations should be done under oxidizing conditions. This decision was prompted by the fact that TiO2 reduced to lower oxides in a reducing atmosphere with consequent changes in melting points. The use of a micropyrometer and a platinum strip furnace for work of this type is adequately described in the literature.2 With modifications, equipment was selected which provided a fairly rapid method of determining melting points of refractory oxides. The final assembly, which was used, consisted of a platinum strip furnace and a Leeds and Northrup disappearing filament optical pyrometer attached to a special telescope which magnified the sample about 20 diam. The furnace assembly consisted of a platinum strip 0.005 X 0.3 X 2.2 in. mounted on brass posts on a refractory base. This was enclosed in a black steel shell which had a 2-in. opening at the top for purposes of sighting the telescope. The platinum strip was in series with two 0.04 ohm nichrome resistors connected to the secondary of a 2 kw transformer which supplied 17 volts. The current to the primary was varied

Jan 1, 1950

By David A. Thomas, David N. French

The lrnrdness of WC crystals has been measured with the Knoop indenter at loads of 100 and 500 g on the (0001) and (1070) planes. The hardness as tneasitred on the basal plane is 2400 kg per sq mm and shows little anisotropy. The hardness on the prism plane, however, shows a marked orientation dependence, with a low value of 1000 kg -per sq mm when the long axis of the Knoop indenter is oriented parallel to the c axis and a high value of 2400 kg per sq mm when the indenter is perpendicular to the c axis. Slip lines (Ire observed surrounding the microhardness indentations and they show slip on (1010) planes, probably in [0001] and (1120) directions. This slip behavior can be explained by the crystal structure of TVC, which is simple hexagonal with a c/a ralio of 0.976. The hardness anisotropy call be explained by [0001]{1010} and (1130) {10l0] slii) and the resolved shear-stress analysis of Daniels and Dunn. HARDNESS anisotropy is a well-known phenomenon and has been reported for many metals, with both cubic and hexagonal structure.1-6 For hexagonal tungsten carbide, WC, a wide range of hardness values is reported in the literature. For example, Schwarzkopf and Kieffer7 give a hardness of 2400 kg per sq mm and report a value of 2500 kg per sq mm by Hinnüber. Foster and coworkerss give the average Knoop microhardness as 1307 kg per sq mm with a maximum value of 2105 kg per sq mm. Although these values and the structure of WC suggest the likelihood of hardness anisotropy, no such measurements have been made. We first detected a large apparent hardness anisotropy in WC crystals about 75 p large, in over-sintered cemented tungsten carbide. Prominent slip lines also occurred around many indentations. This report presents further observations and interpretations of hardness anisotropy and slip in WC crystals obtained from Kennametal, Inc. Both Knoop and diamond pyramid indenters were used on a Wilson microhardness tester with loads of 100 and 500 g. EXPERIMENTAL RESULTS The carbide crystals tended to be triangular plates parallel to the (0001) basal plane of the hexagonal structure. The side faces were parallel to the ( 1010) prism planes. Specimens were mounted approximately parallel to these two types of faces and metallographically polished. Laue back-reflection X-ray patterns were used to orient the specimens, which werethen ground to within ±1 deg of the (0001) and (1010) planes. The Knoop hardness values measured on the basal plane are plotted in Fig. 1. There is only a small anisotropy, with a hardness range of 2240 to 2510 kg per sq mm. The additional points at angles from 52.5 to 67.5 deg confirm the sharp minimum hardness at 60-deg intervals, consistent with the sixfold hexagonal symmetry. The average hardness of all values obtained on the basal plane is 2400 kg per sq mm. While the basal plane shows only slight anisotropy, the (1010) plane exhibits marked hardness anisotropy, from 1000 to 2400 kg per sq mm. Fig. 2 shows the hardness as a function of the angle between the long axis of the indenter and the hexagonal c axis, the [0001] direction. The minimum and maximum occur when the indenter is oriented parallel and perpendicular to the [0001] direction, respectively. The anisotropy of the prism plane is contrary to that reported for hexagonal zinc and hard- However, the basal-plane anisotropy is similar to these two metals.1&apos;2 To check the accuracy and reproducibility of the measurements, a series of fifteen impressions was made at 100-g load in the same orientation in the same area of the specimen surface. The average for all was 2040 kg per sq mm, with a range of 1950 to 2130 kg per sq mm, giving an accuracy of about ± 5 pct. Thus the slight anisotropy on the basal plane is almost within experimental error. Fig. 3 shows slip lines around the Knoop indentations on the basal plane. The slip traces are in directions of the type (1130). The presence of slip steps on the basal plane indicates that the slip direction lies out of the (0001) plane. Because WC has a c/a ratio of 0.976,&apos; the shortest slip vector is [0001], which suggests slip systems of the type [0001] (1010). Fig. 4 shows slip lines around the Knoop intentations on the (1010) plane. These slip lines are inconsistent with [0001] slip but can be

Jan 1, 1965

By C. G. Dunn, M. Sharp

IN twinned crystals of the face-centered cubic metals the lattice of one twin is a mirror image of the other in a common twin boundary. When several twins appear within large grain in a sheet specimen, the twin one boundaries form a set of lines at the surface of the specimen which coincide with (111) planes of the large grain. Furthermore, for twins of the same orientation, these lines are parallel. Generally, the presence of identically oriented regions with straight parallel boundaries coinciding with a (111) plane of the surrounding crystal is strong evidence for identifying the island regions as twins of the parent crystal. However, Fig. 1, which shows the macrostructure of a large grain of copper with island regions that satisfy these conditions. is not an illustration of (111) twins. Since the reverse side of the specimen has much the same appearance, it was thought at first that these regions, which appear dark in the macrograph, actually were twins. According to X-ray data, however, these regions are second-order twins of the large crystal. With regard to their formation, these second-order twins formed by secondary recrystallization in a cube texture matrix. Growth occurred in the direction of the arrow (see Fig. 1) as the specimen moved slowly into a gradient temperature furnace as described previously.&apos; Nucleation of the second-order twins occurred, therefore, on the ends facing opposite the arrow. If the origin of the second-order twins were due to repeated twinning, some first-order twin structure should be visible on these ends. This proved to be the case, as very small twins were readily found with the aid of a microscope, and probably could have been seen, in some instances, under ideal lighting conditions without aid of a microscope. Fig. 2 shows a cross-section view taken perpendicular to both the surface and the (111) trace of the parent crystal (visible as a straight boundary in Fig. 1) at the beginning point of growth of a second-order twin and where one first-order twin was relatively thick. In the micrograph, A is the large parent grain; B is the first-order twin of A; and C, which is a first-order twin of B; is a second-order twin of A. Between A and B and between B and C the major straight portions are traces of common (111) twin boundaries. The straight portion of boundary between A and C, however, is not a common crystallographic plane to the two lattices; it is a (111) plane of A and a (115) plane of C. Without considering the mechanism of twinning itself, the origin of the second-order twins may be accounted for in terms of repeated twinning and special growth characteristics. After each nucleation, a selective growth process can be thought of as favoring growth of the first-order twin in local spots only and favoring growth of the second-order twin to an extent comparable with that of the parent grain over relatively large areas in a way similar to that described for twinning in aluminum.&apos; It has already been pointed out that the boundary between the large grain (A) and the second-order twin (C), which is responsible for the straight boundary portions in Fig. 1, involves a (111) plane of A and a (115) plane of C. The same combination of planes is not only possible in first-order twins, but actually appears quite frequently.3 Their prevalence in first-order twins and their presence here in second-order twins, together with the necessary occurrence of a large number of common lattice sites at the boundary, is an indication that this combination produces an "energy cusp"&apos; boundary. (Energy cusp boundaries have been described by Shockley and Read.") The configuration of atoms near a {Ill), (115) boundary in first-order twins is of course different from the configuration near the same type of boundary in second-order twins. References 1 M. Sharp and C. G. Dunn: Secondary Recrystallization Texture in Copper. Journal of Metals (January 1952) Trans. AIME, p. 42. 2W. G. Burgers and W. May: Stimulated Crystals and Twinning in Recrystallized Aluminum. Recueil des travaux chimiques des Pays-Bas (1945) 64, p. 5. aD. Whitwham, M. Mouflard, and P. Lacombe: Discussion of W. C. Ellis and R. G. Treuting, "Atomic Relationships in the Cubic Twinned State." Trans. AIME (1951) 191, p. 1070; Journal of Metals (October 1951). 4 W. Shockley and W. T. Read: Dislocation Models of Crystal Grain Boundaries. Physical Review (1950) 78, p. 275.

Jan 1, 1953

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.&apos; 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&apos;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. &apos;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&apos; .,....(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