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By C. T. Holland

This discussion is concerned with those com-J- paratively infrequent bumps that eject material from the failed mass with enough energy to wreck heavy machinery and seriously injure or kill people. In such cases there is a loud report, and the pressure waves set up in the mine atmosphere wreck ventilation controls several hundred feet from the site of the failure. Vibrations are induced in the earth's crust which may be detected several miles from the point of origin, and explosive gases may be released in large quantities." Usually a dense suspension of dust is produced from the failed mass, together with dust placed in suspension by the resulting air blast. These rock bursts or bumps have also been called mountain shots, bounces, pillar bursts, and crumps. Such occurrences, when extensive, are serious catastrophes, and may force abandonment of the mine.' CAUSES OF ROCK BURSTS Rock bursts in coal mines have occurred in sev-eral places under diverse geological conditions, mining methods, and/or practices. A study of many such occurrences indicates that they depend on the following conditions: 1) cover thickness of 500 ft or more, 2) cover composed of strong members and having a strong member lying close to or on the coal bed, and 3) a floor that does not readily heave. The minimum thickness of cover under which coal mine rock bursts occur is based on observation. Under unusual circumstances involving very strong roof members and/or mining systems, and/or practices especially likely to induce bursts, a burst conceivably could occur under less roof. Under special circumstances rock bursts have occurred in marble quarries almost at the surface.' Furthermore, under very heavy cover it is likely that controlling roof members need not be as strong and as close to the coal bed to cause bumps. Just what constitutes a strong roof or roof member is open to question. Obviously the strength of a roof member is determined by its thickness, the distance between bedding planes across which little or no bond exists, the spacing and development of joints, and the inherent strength of the rock material. Strata sections showing roofs under which rock bursts in coal mines have developed are illustrated in Fig. 2. In several instances the strength of rocks composing the roof of mines developing bumps has been tested. The tests (Table I) show that these rocks develop considerable strength in compression, bending, and shear and that in some cases the strengths are very high. No quantitative data are available concerning joints in roof members over coal beds where rock bursts have occurred, but the author has examined many mines subject to bursts and has received the impression that joints in the shales immediately over the coal beds are always present and well developed. In the main sandstone or conglomerate members themselves, however, where observed, he has yet to see a fracture system indicating well developed joints. Undoubtedly joints are present, but the evidence indicates that they are few, poorly developed, and widely spaced. In a mine subject to bumps, moreover, actual measurements of roof spans up to 225 ft between supports under a cover of about 800 ft have been made before roof failure occurred. Such spans signify strong roof. In coal beds subject to rock bursts, the bottom rock is usually a sandy shale. A few tests (Table I) have been made on the floor rock of coal beds. Observations in mines indicate that they are usually weaker than roof rocks and that under load they show more plastic effects. Observation in ground subject to bursts invariably indicates a relatively strong floor highly resistant to heaving, although some heaving may occur. Other factors listed as favorable to rock bursts are: 1) mountainous surface; 2) steeply dipping beds; 3) proximity of faults, folds, and other similar geologic structures. Many bursts have occurred in mines under mountainous country, although many bursts in mines under a surface with little relief indicate that thickness of cover, not relief, is the essential factor. In steeply dipping beds, many bursts occur in mines that are essentially flat, and rock bursts have been recorded in beds dipping as much as 35o Most rock bursts in the U. S. occur in beds that dip less than 10 pct. It is also true that most U. S. coal is produced from beds dipping less than 10 pct. Geologically, faults are regarded as providing stress relief, but in the vicinity of faults, as well as sharp folds, it would seem that residual stresses could be present. The fact that rock bursts occur in areas remote from faults and sharp folds suggests that their presence is not necessary for this phenomenon to take place. There have been many bursts close to faults and folds, and it may be assumed that occasionally such structures accentuate their occurrence3 The conditions and circumstances under which they occur, coupled with the effects produced, leave In the Crowsnest Pass coalfield, British Columbia, a rock burst is estimated to have released 1.8 million cu ft of methane.

Jan 1, 1959

By Julien Deby

THE sinking of a deep shaft is always a serious undertaking, especially when the strata to be traversed are of great hardness, or when they are feebly coherent or highly saturated with water. In the, first case, that of solid rock, modern appliances, such as the rock-drill, put into motion by air-compressors, along with suitable explosives, will generally answer the purposes of the mining engineer until he has reached the main water-levels. When these have

Jan 1, 1877

By Julien Deby

The sinking of a deep shaft is always a serious undertaking, especially when the strata to be traversed are of great hardness, or when they are feebly coherent or highly saturated with water. In the. first case, that of solid rock, modern appliances, such as the rock-drill, put into motion by air-compressors, along with suitable explosives, will generally answer the purposes of the mining engineer until he has reached the main water-levels. When these have

By E. T. Turkdogan, L. S. Darken

Calcium ferrite melts were equilibrated with sulfur and oxygen-bearing gases at temperatures within the range 1290°C to 1620°C. The results show that at oxygen partial pressures below 10-4 atm the sulfide-reaction occurs and at higher oxygen pressures the sulfur dissolves in the melt principally as sulfate ion. It is also observed that when the product increases, sulfate ions are converted to pyrosulfate ions and the results permit the evaluation of the pyrosulfate/sulfate ratio. It is shown that the ratio y CaSO,/rCaO increases with decreasing concentration of calcium oxide in calcium ferrite, silicate and aluminate melts. Indications are that oxy-acids, e.g. silica, alumina and ferric oxide, increase yCaS0,. A few experiments were carried out within the sul-tide range and the results are in accord with those obtained by other investigators on aluminate and silicate melts. BECAUSE of the industrial importance and theoretical interest, many investigators have studied the reactions between molten oxides, silicates or alumi-nates, and gases containing oxygen and sulfur. The early work reviewed by Schenckl and studies made by Bardenheuer and Geller2 indicates that solution of gaseous sulfur in an oxide melt resulted in the replacement of the oxide ions by the sulfide ions. Measurements made by Grant and Chipman3 on the equilibrium partition of sulfur between molten iron and complex silicate melts also demonstrated that, for a given temperature and silicate composition, the distribution of sulfur between slag and metal increased with decreasing oxygen content of the metal. Fincham and Richardson4 studied the effects of temperature and partial pressures of sulfur and oxygen on the solubility of sulfur in simple silicate and alumino-silicate melts. They observed that at oxygen partial pressures below about 10-5 atm sulfur in the gas dissolved in the melt as sulfide ions, replacing an equal number of oxide ions of the melt. At oxygen partial pressures higher than 10-3 atm, sulfur entered the melt as sulfate ions. Similar studies were made by St. Pierre and Chipman5 using calcium ferrite and calcium silico-ferrite melts. The experimental data presented in this paper were obtained in this Laboratory about 10 years ago; this is the first formal presentation although a preliminary report was given previously.6 In each of these experiments, a pure synthetic CaO-Fe2O3 melt in a platinum crucible was equilibrated with a gas mixture (SO2-O2, SO,-air, SO2-CO, or SO2-CO2-CO) at a temperature ranging from 1296°to 1620°C. The experimental technique was essentially the same as that reported by Darken and Gurry,7 the main difference being that the atmosphere in the reaction chamber contained sulfur dioxide together with oxygen, or carbon dioxide-carbon monoxide

Jan 1, 1962

By Maxwell Gensamer

Jan 1, 1960

By K. Winterton

The effect of a near-saturation unidirectional magnetic field was to decrease slightly the endurance life in Armco iron in the annealed condition. This is expected since the energy-absorbing mechanism associated with domain wall movemenfs is inhibited bv the strong field. THE factors determining failure under dynamic loading are not yet clearly recognized. One of the properties of a metal which might be thought to influence fatigue life is its damping capacity. Foppl1 conducted an alternating torsion test at a frequency of 700 cpm continuously for a period of 3 years, without the test piece breaking or showing any sign of deterioration after 1100 Million cycles of stress; the rate of dissipation of energy was sufficient to maintain the test piece at about 100°C above the ambient temperature. cazaud2 concludes from this, and others follow him, that ('the capacity for dissipation in the form of heat by internal damping is unlimited" and that therefore "damping is not directly related with fracture by fatigue." Future work may necessitate qualification of this rather uncompromising view. It is shown later that in all

Jan 1, 1960

By GEO H. DEIKE

NOT much activity has been evident during the past year in the establishment of co-operative scholarships but the interest among the coal-mining companies is more pronounced than ever. This is apparent from the replies received to a letter sent out by the Committee on the Promotion of Student Interest in Coal Mining to over 120 coalmine operating officials. With that letter was forwarded a bulletin issued by the school of mineral industries of The Pennsylvania State College entitled ?Industrial Scholarships ? a Partnership of Industry and Education.? In that circular was outlined the need for trained men in the war and postwar world, and particularly as such demands pertain to the growth of the mineral industries.

Jan 1, 1943

By N. A. Gokeen, M. Ohtani

EFFECT of elements on the solubility of carbon in liquid iron is useful in calculating and correlating a number of thermodynamic properties as shown elsewhere in detail.' Kitchener, Bockris, and spratt2, and later Turkdogan and Hancock3 have investigated the effect of sulfur on the solubility of carbon at various temperatures and concentrations. For any chosen sulfur concentration, the solubility of carbon reported by Kitchener et al. is considerably lower than that by Turkdogan and Hancock. It was therefore felt that a reinvestigation of this problem was necessary. EXPERIMENTAL METHOD An alloy of Fe-C-S .was prepared with electrolytic iron and ferrous sulfide melted by induction in a graphite crucible, 4 in. high 1 1/2 in. ID under a CO-atmosphere. The temperature, held at 1500" ± 5°C in all experiments, was measured with a Pt-Pt + 10 pct Rh thermocouple immersed into the melt. In order to assure saturation with carbon, the melt was stirred with a graphite rod. After approximately 1 hr at 1500 °C, a sample was obtained by sucking the melt into a silica tube, 4 mm ID. The sample was then analyzed for carbon and sulfur by usual methods. RESULTS The results are presented in Table I and plotted in Fig. 1 with the individual values of Turkdogan and Hancock.3 Scattering in the data is equivalent to 3 pct or less in carbon analysis. The broken line represents the empirical equation obtained by Kitchener et al. from their data. It is evident that the present data are in good agreement with those of the former investigators but in disagreement with the latter. The cause for this discrepancy cannot be readily offered since the experimental methods in the three investigations were nearly identical. Additional evidence corroborating the author's results may be presented in the following manner. The difference between the solubilities of carbon in the ternary Fe-C-X melt with a small concentration of any element X, and in the binary Fe-C melt is %?C = %CIII-%CII [1] where %?C is the change in solubility, and %CIII and %CII are the solubilities in the ternary and the binary systems respectively. The value of %Cii

Jan 1, 1961

By W. Bonfield

A study has been made of the dislocation substructures produced in hot-pressed beryllium specimens strained to various levels in the range from 800 x 10-6 In. pev in. to fracture. A number of distinctive dislocation configurations were observed in this region which had not been noted at lower levels of strain. These included dislocation-dislocation interactions to form networks, dislocation "walls", subgrain boundaries and complex arrays, interactions between dislocations and large beryllium oxide particles, and the generation of dislocations from certain particles. The nature of these differences in substructure and their relation to the stress-strain characteristics of polycrystalline beryllium are discussed. In a previous study1 of the plasticity of commercial-purity, hot-pressed beryllium a transition was found in the deformation characteristics in the mi-crostrain region. The initial plastic deformation could be represented by a parabolic stress-strain equation, but above a critical stress there was a complete departure from this relation and a reduction in the strain-hardening rate. The dislocation configurations produced by various levels of micro-strain in this region were examined by transmission electron microscopy and a general correlation was established between the observed transition in deformation characteristics and the dislocation structure of the material. The two stages in the micro-strain region distinguished in these experiments were designated as Stage A' and Stage B'. Stage A' type deformation generally was noted up to a plastic strain of -80 x 10"6 in. per in. and Stage B' type from -80 x 10-6 to -800 x 10'6 in. per in. The discovery of two stages in the microstrain region naturally posed pertinent questions as to the existence of any further distinct stages in the subsequent plastic deformation. The purpose of this paper is to present a study of the dislocation configurations produced in similar beryllium specimens strained to various levels in the range from -800 x 10 in. per in. to fracture and to discuss the relation between substructure and the stress-strain characteristics. It is concluded that this region of strain can be considered as a distinct stage in the plastic deformation of polycrystalline beryllium. Tensile specimens of gage length 1 in. and cross section 0.18 by 0.06 in. were prepared from commercial-purity, hot-pressed QMV beryllium and then annealed at 1100°C for 2 hr. 2 followed by a careful chemical polishing procedure.3 The specimens were strained at a constant rate to various levels of strain in the range from -800 x 10-6 in. per in. to fracture (at 0.5 to 2 pct elongation), using the Tuckerman strain-gage technique1 to measure plastic and total strain. Thin foils were obtained from the strained and fractured specimens by chemical polishing3 and were examined using an RCA-EMU 3 electron microscope. Considerable care waS taken to avoid both accidental deformation during the preparation of the thin foils and excessive heating during their examination. Selected-area diffraction patterns were determined for each micrograph. Tilting experiments were also performed whenever appropriate to establish the dislocation zero-contrast position and hence determine the Burgers vector. This operation was sometimes not possible due to the rapid contamination of the foils which occurred in the electron microscope. RESULTS AND DISCUSSION To enable the distinctions between the dislocation arrays at high and low strain levels to be adequately made, the main characteristics of Stage A' and Stage B' deformation are briefly reviewed. 1) Stage A'. In the annealed starting condition there was a variable density (5 x 107 to 3 x 10' cm per cu cm) of isolated dislocations within a grain. The initial deformation in a tensile specimen was heterogeneous, with the dislocation density increasing in a few grains to 5 x 10g to 1.5 x 101° cm per cu cm. The deformation occurred exclusively on the basal plane by the movement of one or more 1/3 (1130) type dislocation systems. The dislocations were long and regular in form and nearly all the intersections exhibited a simple four-point node configuration. No interactions between glide dislocations and beryllium oxide particles were observed. 2) Stage B. In Stage B' there was a large increase in the number of grains exhibiting dislocation movement and also a change in the nature of the deformation, in which jogged dislocations and elongated loops became the characteristic feature. The splitting up of the elongated loops into smaller loops and the possibility of source action from the re-

Jan 1, 1965

By I. R. Kramer, P. D. Gorsuch, D. L. Newhouse

Although many methods have been suggested for the calculation of tensile strength and yield point from chemical composition, their usefulness has been limited to a particular cooling rate or section size. Further limitations have been imposed by failure to take into account many of the alloying elements that exert a considerable influence on the tensile strength or yield point. These systems, nevertheless, have filled an important need in many instances. The value of a system for the calculation of the tensile properties from the chemical composition and cooling rate is that it permits the prediction of the effect of alloying elements and heat-treatment on the mechanical properties, so that more efficient use of alloying elements may be made. Furthermore, it aids in the choice of steel for a given application aid tends to decrease the number of steels that otherwise would have to be tested. It is the purpose of this work to present a method of calculating tensile strength and yield point from the chemical composition and to suggest a method of correlating the tensile-strength and yield-point factors and cooling rate. Steelmakers have been interested for many years in the calculation of the tensile strength of hot-rolled steels. Quest and Washburn' summarize several of the attempts that have been made to devise formulas. They show the inadequacy of each of these formulas and present one of their own. It is apparent from their work that a simple additive equation of the type TS = A + BX + Cy + DZ + . . . , where A, B, C, D, etc. are experimentally determined constants and x, y, z, etc., are the percentages of alloying elements present in the steel, is not adequate to analyze fully the situation encountered in the calculation of tensile strength. For example, it was found that the difference between the calculated and observed values became progressively greater as the carbon and manganese contents were increased. Therefore, the effect of manganese was given as a function of the carbon content. Grossmann, in his work on the harden-ability of quenched steels, and Walter, in his method of calculation of tensile strength of normalized steels, present two cases in which multiplying factors were successfully used to express the combined effect of alloying elements. The factors in both cases were obtained from the fractional increase in hardenability or tensile strength with increase of alloy content. Yield point often is more important in design than tensile strength. For quenched and tempered steels a linear correlation exists between yield 2nd tensile strength.4 This is not true, however, for normalized

Jan 1, 1948

By P. D. Gorsuch, I. R. Kramer, D. L. Newhouse

Although many methods have been suggested for the calculation of tensile strength and yield point from chemical composition, their usefulness has been limited to a particular cooling rate or section size. Further limitations have been imposed by failure to take into account many of the alloying elements that exert a considerable influence on the tensile strength or yield point. These systems, nevertheless, have filled an important need in many instances. The value of a system for the calculation of the tensile properties from the chemical composition and cooling rate is that it permits the prediction of the effect of alloying elements and heat-treatment on the mechanical properties, so that more efficient use of alloying elements may be made. Furthermore, it aids in the choice of steel for a given application aid tends to decrease the number of steels that otherwise would have to be tested. It is the purpose of this work to present a method of calculating tensile strength and yield point from the chemical composition and to suggest a method of correlating the tensile-strength and yield-point factors and cooling rate. Steelmakers have been interested for many years in the calculation of the tensile strength of hot-rolled steels. Quest and Washburn' summarize several of the attempts that have been made to devise formulas. They show the inadequacy of each of these formulas and present one of their own. It is apparent from their work that a simple additive equation of the type TS = A + BX + Cy + DZ + . . . , where A, B, C, D, etc. are experimentally determined constants and x, y, z, etc., are the percentages of alloying elements present in the steel, is not adequate to analyze fully the situation encountered in the calculation of tensile strength. For example, it was found that the difference between the calculated and observed values became progressively greater as the carbon and manganese contents were increased. Therefore, the effect of manganese was given as a function of the carbon content. Grossmann, in his work on the harden-ability of quenched steels, and Walter, in his method of calculation of tensile strength of normalized steels, present two cases in which multiplying factors were successfully used to express the combined effect of alloying elements. The factors in both cases were obtained from the fractional increase in hardenability or tensile strength with increase of alloy content. Yield point often is more important in design than tensile strength. For quenched and tempered steels a linear correlation exists between yield 2nd tensile strength.4 This is not true, however, for normalized

Jan 1, 1948

By A. D. S. Gillies

The effectiveness of coal mine face ventilation and the ability of airflow to maintain satisfactory methane and respirable dust levels in the working face area has been studied through the use of scale modeling techniques and underground testing procedures. In particular, consideration has been given to examination of the effects of operation of a high airflow, machine mounted dust scrubber unit on face air patterns. Results obtained through a cooperative study with the Peabody Coal Co. demonstrate that new mining cycles can be considered that increase production efficiency while maintaining satisfactory ventilation conditions.

Jan 1, 1986

By George Sears

UNTIL a comparatively few years ago, interest in tantalum was limited almost wholly to its scientific investigation, but its extreme resistance to the action of even the strong mineral acids, its great avidity for the common gases, hydrogen, oxygen and nitrogen, and its capacity for transforming alternating to direct current are winning for it a place in our industrial life. Tantalum is not attacked by any single acid except hydrofluoric, nor is it affected by solutions of alkalies. Aqua regia, which readily dissolves both platinum and gold, has no action on it. This resistance to the common corrosive agents is an outstanding property of tantalum and accounts for many of the applications to which it has been put. When heated, tantalum reacts readily with oxygen and nitrogen and absorbs large volumes of hydrogen, 740 volumes being taken up at a dull red heat. This capacity for absorbing gases is being utilized to an increasing extent in the manufacture of vacuum tubes, in which tantalum is used for the metal parts, especially for grid wire. It is said to be a superior metal for this purpose, particularly in power and sending tubes, because of its low secondary emission. The use of tantalum in the manufacture of electrodes, dishes, spatulas and other laboratory apparatus appears to be fairly steady. There has been some development in its use for dental instruments, surgical tools, pen points, hypodermic needles and acidproof pumps. Recently it has been used in the manufacture of spinnerets for the rayon industry, of nozzles for the manufacture of artificial casings for sausages and of lining for plant equipment.

Jan 1, 1930

Jan 1, 1923

Jan 1, 1923

By R. D. Pehlke, J. V. Gluck

The limiting activity coefficient of zinc in a "binary " solvent has been described for several dilute ternary solutions by combining literature data from recent interaction studies in the terminal regions with thermodynamic data for the central regions. Results are discussed in terms of the Alcock and Richardson random solution model. The observed behavior agrees semiquantitatively with predictions using the model. RECENT efforts in the thermodynamics of dilute ternary liquid metallic solutions have been mainly concerned with the interactions between two dilute solutes, that is, the terminal region at an apex of the composition diagram.'-' A more general case is that of the behavior of a dilute solute over the full range of a ternary system, or, more specifically, the limiting activity of solute i as projected to the k-j binary side of the i-j-k composition triangle. An implication of this is that the interaction between i and j at the k apex should connect in some way to the interaction between i and k at the j apex. A model to characterize various types of expected behavior has been discussed by Alcock and Richardson.9,10 Such behavior can range from a simple linear relation for In yi over the 1-k binary to more complex curved relations which depend on various assumptions regarding the heat of solution in the j-k binary system and the presence or absence of clustering with solute i. Quantitative data to assess these concepts have been limited primarily to the cases of nonmetallic interstitial solutes in ferrous solutions. Alcock, Richardson, and various coworkers9-13 have discussed the behavior of oxygen, sulfur, silicon, or hydrogen in dilute solution over the range of iron + metal systems; however, the only data considered by them for substitutional nonferrous metallic solutions were the results of Elliott and Chipman concerning the activity of cadmium in the Pb + Sn, Pb + Bi, and Pb + Sb systems.9 These data were limited and did not cover the terminal regions in detail. An opportunity to extend the Alcock and Richardson considerations to the dilute solution of zinc in various "binary" solvents has been afforded by combining results of several of the recent investigations of ter-minal-region interactions with previous data from the literature for the same more concentrated solutions. RESULTS AND DISCUSSION The results of the analysis are shown in Figs. 1 to 5 where the log activity coefficient, (In yZn)xzn--o, is plotted vs the composition of the "binary" solvent. In these plots the end points are the values for In yzn in the appropriate binary solution with Bi, Pb, Sn, Cd, or In at the indicated temperature—usually 550°C (823°K). For the terminal regions the slopes were drawn using the first-order interaction parameters taken from various studies reported in the literature. The data points shown in the central regions were obtained by the short extrapolation to infinite dilution of the logarithm of the activity coefficient calculated from thermodynamic studies reported in the literature for the same systems. The sources of the data are indicated on each figure. Second-order effects were not considered. Reasonably smooth curves can be drawn to connect the central region with the terminal regions for the Zn-Pb + Bi system, Fig. 1, the Zn-Pb + Zn system, Fig. 1, and the Zn-Bi + Cd and Zn-Bi + Sn systems, Fig. 2. As shown in Fig. 2, where central region data are available from two independent investigations, the agreement is quite good. The Zn-Pb + Cd system does not appear to be as well behaved as the previous four, Fig. 3; however, this type of behavior has also been

Jan 1, 1970

By Michael A. Farrugia, Marcel Michaud

Remoteness of western Canadian coals has always been a barrier to the use of Canadian coals domestically. However, the world energy crunch, increased mining costs, stricter environmental controls, labor shortages, etc., are such that by 1980 a shortfall from traditional U.S. sources will prevail. Of necessity, Canada must then bring western coals east, requiring major new transportation initiatives. CN, CP Rail and Neptune Terminals are proceeding jointly with the design of a $30 million, fully automated, coal terminal at Thunder Bay. This element forms part of a $400 million integral transportation system to open new lines, improve trackage, acquire rolling stock and ships, for an initial capacity of 3 million tpy.

Jan 1, 1977

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

By R. L. Parsons, Herman Dykstra

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

By E. D. Levine

A study was made to delineate the dislocation mechanistns controlling prismatic and basal slip at low temperatu.ves in titanium containing approximately 100 ppm interstitial impurities. Mechanical tests were performed on single crystals prepared from electron-beam zone-refined crystal bar by a thermal cycling technique. Measurements were made of critical shear stress and activation volume as functions of temperature. The study revealed that, for prismatic slip, two overlapping thermally activated processes aye rate-controlling below 300°K. From absolute zero to 220°K, reasonable agreement is obtained between experiment and the mechanism which involves overcoming 01. the Peierl's stress by the formation of double kinks in dislocation lines lying in close-packed directions. Between 220°and 300°K prismatic slip is controlled by a difierent thermally activated process which is characterized by a stronger temperature dependence of critical shear stress. Above 300OK, prismatic slip is controlled by an athermal process. Basal slip is controlled by a thermally activated process between 195' and 400°K. An approximate analysis suggests that it is the Peierl's mechanism that is rate-controlling in this range. Below 195°K, the occurrence of mechanical twinning obscures the features of basal slip. Above '400 X, basal slip is controlled by an athermal process. TITANIUM, an hcp metal with a less-than-ideal axial ratio of 1.587, deforms primarily by {1010}( 1210) slip at low temperatures.'-3 The critical shear stress is strongly affected by the presence of interstitial impurities. For example, churchman3 reported that a decrease in the combined nitrogen plus oxygen content from 0.1 to 0.01 pct decreased the critical shear stress for prismatic slip from 9.19 to 1.4 kg per sq mm at room temperature. Basal slip, however, is less strongly affected by interstitials; Churchman's data for {0001}( 1210) slip show a smaller decrease in critical shear stress from 10.90 to 6.3 kg per sq mm at room temperature, for the same decrease in interstitial content. In the present study, an attempt has been made to delineate the dislocation mechanisms controlling prismatic and basal slip in titanium. In view of the strong effects that impurities may have on at least some of these mechanisms, experiments were performed on material of reasonably high purity, so that comparisons could be made with investigations on material of both lower and higher impurity contents. EXPERIMENTAL PROCEDURE Single crystals were prepared from 1/4-in.-diam electron-beam zone-refined crystal bar, furnished by Materials Research Corp. The analysis of the zone-refined material is given in Table I, indicating that the interstitial content is of the same order as the purer of the two materials investigated by Churchman. The Vickers diamond pyramid hardness number of this material was 85.6. This value agrees quite well with the correlation between hardness and interstitial content proposed by Ogden and Jaffee.4 On the basis of hardness, the material investigated in this study is, therefore, intermediate in purity between that studied by Churchman (Vpn = 97) and that studied by Spangler and Herman (Vpn = 62.4).5 Single crystals were grown by thermal cycling through the a-ß transformation. Two-in.-long by 1/2-in.-diam bars were heat-treated in vacuum for 4 hr at 1160°C. The temperature of the furnace was then lowered to 865°C, and the specimens were maintained at this temperature for 7 days. This procedure was repeated several times, resulting in the growth of individual crystals up to 14 in. in length. Specimens for testing in direct shear along either {l010}(1210) or {0001}(1210) were obtained by electrical-discharge machining. The specimens were in the form of cylinders 1/4 in. in diam by approximately 1/2 in. in length, with a reduced section 1/8 in. in diam and a in. in length. All specimens were electropol-ished subsequent to electrical-discharge machining. The ends of the test specimens were cemented in stainless-steel collars. Shear tests were conducted in an Instron Tensile Tester at various temperatures from 4.2° to 508°K. Measurements were made of critical shear stress and activation volume. The latter measurements were performed by measuring the change in flow stress upon cycling the strain rate between ?1 = 2.62 x 10-3 per sec and ?2 = 2.62 x 10-4 per sec. Activation volume, I)*, is defined as u* =ßkT [11

Jan 1, 1967