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By H. Foster Bain

Jan 1, 1910

By Arthur Baker

The orebodies are long bedding-plane lenses of chalcopyrite and sphalerite, associated with garnetite masses. Most of the orebodies are within a 50-ft thickness of Cambrian limestone; other Paleozoic limestones and dolomites are locally metamorphosed but only slightly mineralized. Pre-mineral faults are numerous, but shallow folds were the main ore-localizing structures.

Jan 12, 1953

By D. J. Brown

Froth flotation is now used to recover millions of tons of fine coal,(-1/2 mm), every year. For example, in Great Britain, 4-million tons of flotation concentrate was produced in 1960, about 3% of the mechanically cleaned out- put and 2% of the total salable output. Coal flotation is not so widely practiced in the U. S. as it is in Europe since much American coal occurs in thick seams of high grade and does not require cleaning. However, in the U. S., as elsewhere, the increasing mechanization of coal mining machinery has lead to a lowering in size and grade of product from the mine and to an increasing need for cleaning of coal to improve the grade before sale. With an increasing proportion of fines in the product, froth flotation has assumed a greater importance as a cleaning process for coal. In the last three years, over 40 flotation plants have been put into operation in the Appalachian area of the U. S.80 Fine coal recovered by means of froth flotation may be used as a powdered fuel alone or mixed with coarser grades. The removal of this fine coal from washery water makes possible the recirculation and re-use of the water and also leads to improved and more consistent operation of jigs and other separating machines which function less effectively when there is a build-up of fine solids in the washery water. Finally, froth flotation is of assistance in the production of a clarified effluent which may be discharged from the washery without polluting local rivers, streams, or wells. The 1961 cost of installing a new coal flotation plant in Great Britain would be about $19,000 per ton of dry solids input per hour, including the cost of new buildings and coal and shale filtration and water clarification equipment. The figure for the gravity concentration of the large coal, the main plant, would be about $4,800 per tph. The operating cost for the new coal flotation plant would be about $1.10 per ton. Even at this level of cost the cleaning operation is economic in Great Britain for coking coals. In the U. S., a froth flotation cost of 14$per ton of cleanfilter cake was quoted early in 1958 for one operation,l though admittedly this was a special case. Assuming a cake moisture con- tent of 30%, it is estimated that this cost is equivalent to a figure of 256 per ton of dry solids input, for the incorporation of coal flotation in an existing washery. A cost oh 416 per ton of concentrate was quoted76 in a recent analysis of flotation economics, this cost being brought up to $1.11 per ton after payment of royalty, welfare and association dues and taxes. Capital cost was $120,000 for a plant producing 133 tpd of concentrate. Before considering the treatment of coal by flotation it is appropriate to survey some of the properties of coal which are of importance in the process.

Jan 1, 1962

By Wendell S. Williams

Single crystals of TIC doped with boron at high temperat~ires develop second-phase inclusions. The inclusions are in the form of lamellae lying parallel to {l 11 } planes of the Tic matrix and can be vevealed by etching [100] cleavage faces of the boron-doped Tic crystals. measurements of the critical resolved shear stress. t, of TIC crystals containing such inclusions showed that the ropid drop in t wilh increasing temperatu.ve found for pure Tic was greatly reduced. At 1600°C, the inclusion containing crystals had a value of t five limes that of pure Tic. The hardening effect also was found to inhibit hot pressing of TiC powders containing boron impurity. Thermochemical and crys-tallographic arguments suggest that the inclusions are TiB2. The boron content of the hardened carbide is a few tenths of 1 pct. TITANIUM carbide is noted for its high room-temperature hardness. However, measurements of the critical resolved shear stress, T, of single crystals of Tic at high temperatures have shown that t decreases exponentially with increasing temperature.' This temperature dependence of t is attributed to thermally activated dislocation mobility limited by a large Peierls stress.' In turn, this large Peierls stress is believed to be a consequence of strong co-valent and ionic binding in Tic. The rapid drop in 7 with increasing temperature, characteristic of pure Tic, is lessened by the introduction of second-phase inclusions. Exposure of Tic to boron at high temperatures produces such inclusions. A report of the influence of boron on t in Tic is presented, followed by a discussion of the nature and formation of boron-containing inclusions in Tic which are believed to be responsible for the hardening effect. I) CRITICAL RESOLVED SHEAR STRESS OF BORON-DOPED Tic CRYSTALS Tic single crystal boules grown by Union Carbide Corp., Linde Division, were doped with boron by placing them in a carbon crucible packed with boron powder and heating in a vacuum furnace to 2000°C for 1 hr. Samples 1 by 1 by 3 mm were cleaved from these doped boules. High-temperature compression tests' were made on these samples to determine the critical resolved shear stress as a function of temperature . The rapid drop in critical resolved shear stress found for pure Tic crystals' did not occur: tests indicated that the boron-doped crystals at 1600°C had values of T equal to those of pure Tic crystals heated to only 1100°C, Fig. 1. No difference in the shape of the individual stress-strain curves was produced by the boron doping, but the stress scale was multiplied by a large factor. The hardening effect of boron in Tic probably extends to temperatures lower than those used in the present work (1100°C minimum). At room temperature, however, the influence of boron is difficult to determine because the stress required for gross plastic flow in pure Tic is SO high that in practice fracture cccurs first. Microhardness is the only test that can give an indication of the yield stress in Tic at room temperature. such measurements were made5 on boron-doped Tic; the matrix had a hardness value

Jan 1, 1967

By H. T. Bowman

For more than 80 years the Bessemer process has depended upon the ability, skill, and judgment of the blower, although as early as the I860's it was recognized that the process would benefit by some type of instrumental contro1.l IIowever, instruments with sufficient recording speeds did not exist, so the end-point determination was always dependent upon the blower's eye. This placed an obstacle in the way of precise experimental work, because of the absence of the necessary records and reference points as reported by Work.2 The use of the photocell provided the instantaneous record of the blow that made this experimental work possible and provided more scientific control for the process as discussed by Graham,3 A brief description of the progress of the blow will illustrate the recording function of the instrument. The silicon in the iron is being oxidized during the first minutes of the blow, as pointed out by Henning4 in a table showing the order of metalloid elimination. The product of this oxidation is a solid that remains in the converter, so there is virtually no flame during this part of the blow. This low flame luminosity is illustrated by the flame recording shown in Fig. I, where the silicon blow is plainly marked. The dull flame of the silicon blow gradually gives way to the brilliant carbon flame as the increasing quantities of carbon monoxide formed inside the converter burn to dioxide upon contact with the atmosphere above the vessel. When the carbon reaches some low value under 0.20 per cent, there is insufficient menoxide formed to maintain the dioxide flame, so its intensity drops. Throughout the flame drop carbon continues to burn, until at some value around 0.05 per cent an arrest occurs in the recorded drop. This point has been designated as the end point. The period of time from this end point until the vessel is turned upon its side and blowing has stopped has been called the afterblow. This afterblow is used as a measure of the degree of metal oxidation, and uniformity demands that it be controlled within narrow limits. Iron and blowing conditions are likely to change several times during a turn, necessitating corresponding changes in end-point control. It is the purpose of this paper to discuss these changes, the factors that make them necessary, and their bearing upon the quality of steel. Carbon-FeO Relationship In the Bessemer process oxidation relationships are precise. Fig. 2 shows a graph of the carbon-FeO relationship. The familiar "bomb-test" method, described by Mccutcheon and Rautio,5 was used for obtaining these and all other dissolved FeO values herein reported, all tests being taken from the converter before deoxidation unless otherwise noted. All tests of 0.10 per. cent carbon and above were obtained from metal blown for duplexing. McGinley and Woodworth6 reported iron oxide values for blown metal, but their tests were obtained by a . different method than were those

Jan 1, 1942

By H. T. Bowman

For more than 80 years the Bessemer process has depended upon the ability, skill, and judgment of the blower, although as early as the I860's it was recognized that the process would benefit by some type of instrumental contro1.l IIowever, instruments with sufficient recording speeds did not exist, so the end-point determination was always dependent upon the blower's eye. This placed an obstacle in the way of precise experimental work, because of the absence of the necessary records and reference points as reported by Work.2 The use of the photocell provided the instantaneous record of the blow that made this experimental work possible and provided more scientific control for the process as discussed by Graham,3 A brief description of the progress of the blow will illustrate the recording function of the instrument. The silicon in the iron is being oxidized during the first minutes of the blow, as pointed out by Henning4 in a table showing the order of metalloid elimination. The product of this oxidation is a solid that remains in the converter, so there is virtually no flame during this part of the blow. This low flame luminosity is illustrated by the flame recording shown in Fig. I, where the silicon blow is plainly marked. The dull flame of the silicon blow gradually gives way to the brilliant carbon flame as the increasing quantities of carbon monoxide formed inside the converter burn to dioxide upon contact with the atmosphere above the vessel. When the carbon reaches some low value under 0.20 per cent, there is insufficient menoxide formed to maintain the dioxide flame, so its intensity drops. Throughout the flame drop carbon continues to burn, until at some value around 0.05 per cent an arrest occurs in the recorded drop. This point has been designated as the end point. The period of time from this end point until the vessel is turned upon its side and blowing has stopped has been called the afterblow. This afterblow is used as a measure of the degree of metal oxidation, and uniformity demands that it be controlled within narrow limits. Iron and blowing conditions are likely to change several times during a turn, necessitating corresponding changes in end-point control. It is the purpose of this paper to discuss these changes, the factors that make them necessary, and their bearing upon the quality of steel. Carbon-FeO Relationship In the Bessemer process oxidation relationships are precise. Fig. 2 shows a graph of the carbon-FeO relationship. The familiar "bomb-test" method, described by Mccutcheon and Rautio,5 was used for obtaining these and all other dissolved FeO values herein reported, all tests being taken from the converter before deoxidation unless otherwise noted. All tests of 0.10 per. cent carbon and above were obtained from metal blown for duplexing. McGinley and Woodworth6 reported iron oxide values for blown metal, but their tests were obtained by a . different method than were those

Jan 1, 1942

By Roland de Wit

Conclusions reached in a paper by weertmanl are amplified in a mathematical and graphical way. It is shown that in a stressed crystal a straight dis-location may be in a position of unstable equilibrium with respect to helix development, and how a small perturbation may start the dislocation off on its helical path. Helix development occurs by diffision; bulk diffusion is necessary for development from a screw, but core diffision is sufficient for development from an edge dislocation. For a mixed dislocation helix development can take place by core diffusion if the helix also climbs; a diffusion reversal takes place when the slope of the helix equals the slope of the Burgers vector. The work done on a crystal by the external forces is graphically shown to be consistent with the direction of motion of the dislocation in helix development. Finally a detailed mechanism for tangle formation from helices is presented. In the course of helix development a glide situation may be reached where a segment of each helix loop lies in a slip plane in which it can expand by glide. Thus is it visualized how a helix can deteriorate into a tangle. A paper by weertmanl proposes dislocation-tangle formation by a helical dislocation mechanism. The present paper amplifies Weertman's arguments in a mathematical and graphical way and presents some further ramifications of the proposed model. The importance of dislocation tangles is discussed in Weertman's paper and the references cited therein. The mechanism of their formation is not yet understood. Weertman proposes that in cold-worked crystals conditions exist suitable for the conversion of straight dislocations into helices and subsequently into tangles. Helix formation occurs by a diffusion process and Weertman proposes that at low temperatures only core diffusion is important. The present paper necessarily repeats some of Weertman's work. Since the mathematical development and notation used are slightly different from his, this was thought necessary for clarity. GEOMETRY OF THE HELIX weertman2 has shown that the equilibrium form of any dislocation line is a helix if it is assumed that the energy per unit length (line tension) of the dis- location is constant. For the special case that no external forces act on the dislocation, the helix reduces to a straight line. We shall assume in this paper that the dislocation has a helical form even when it is not in equilibrium. The parametric equation of a helix is, see Fig. l: where a is the radius of the cylinder tangent to the helix, and 270 the pitch of the helix. The pitch is related to v, the number of turns per unit length (along the axis) by the relation: When a = 0 Eq. [I] gives a straight line in the z direction. The growth of a straight dislocation into a helix is now described mathematically by letting a vary from zero to its final value. The tangent to the helix is given by where is the path length along the helix. This gives the direction of t and since its magnitude is unity we must have So that dcp/ds = (a2 +p2)-"'. The curvature of the helix is given by where n = -(cos + sin) is a unit vector normal to the helix. Note that the vectors k and n always point toward the axis of the helix, Fig. 1. FORCE ON THE HELIX The external force which makes the straight dis-

Jan 1, 1963

By D. W. Fuerstenau

Recently R. J. Charles1 showed that comminution of brittle or semi-brittle materials in batch operations is described more appropriately by a variable energy relationship than by the specific relationships proposed by Rittinger,2 Kick,3 and Bond.' That is, the value of n in the empirical differential operation which applies to crushing and grinding (Eq. 1) is variable. Independently, Holmes' developed an identical generalized equaton for size reduction. The experimental data of Charles and Holmes show that the value of n is determined by the material and the comminuting machine and is constant for a given combination of these two conditions. The general objective of the present article is to extend the Charles-Holmes concepts to continuous ball milling. The specific objective is to analyze the operation of a vibratory ball mill dry-grinding chromite in a continuous open circuit and to develop an equation for the comminution of chromite in this mill. Basic Principles: In a batch mill the length of time during which the material is ground is a direct measure of the energy expended on the material, except possibly for exceedingly fine grinding. However, in a continuous operation, the fraction of the total expended energy which is useful for grinding will vary according to feed rate and amount of material in the mill. If the amount of material within a continuously operating vibratory mill is low enough so that only part of the expended energy is used for grinding, the extent of size reduction at constant power input must depend upon the length of time the material is retained within the mill, namely, upon retention time. The following analysis pertains to a continuous mill operating under these conditions. The empirical differential equation that applies to the comminution of brittle materials is:" dE = -Cdx/x" [1] where dE is an infinitesimal energy expenditure, C a constant, dx an infinitesimal size change, x the particle size, and n a constant. For continuous grinding in an open circuit at constant power input, P, Eq. 1 must be written as (dtr)P = -C dx/x" [la] where dt, is an infinitesimal time during which a particle of size x is in the mill, and C' is a constant. A ground product never consists of particles of single size but always comprises a distribution of sizes. In comminution, the expenditure of energy causes a lateral shift in the size distribution.', ' " The general objective of any engineering study of comminution is to correlate the magnitude of this size distribution shift with the energy expended, the ultimate aim being to cause the greatest shift with the least amount of expended energy. Although a number of criteria can be used to measure this shift, Schuhmann's size modulus has been found convenient.' Mathematically, the Schuhmann size distribution can be expressed as follows:" y = l00(x/k) [2] or in differential form: dy = [F- xa-' dx ] K, where y is percentage of material finer than size x, a is a constant that determines weight distribution among particles of various sizes, and k the size modulus that denotes the theoretically maximum-sized particle in any size distribution. By considering the energy required to reduce an element of weight of material, dy, from size x, to size x, the element of time during which the material must remain in the mill at at power will be (dt..).= J'-Cax!(dy) [3] Summing up all the elements of weight from zero to 100 for a product that conforms to the Schuhmann size distribution yields C C f Cdx,\( 100«x°-'dx\

Jan 1, 1960

Jan 1, 1889

By J. Chipman, H. R. Larson

The data previously reported for the quantity as a function of oxygen pressure at 1550°C have been used to compute the activities of Fe, FeO, Fe2O3, and COO in slags of the ternary system. Activities of the first three have been obtained also for two quasi-ternaries involving fixed CaO:SiO2 ratios. IN a previous paper1' the authors reported the results of an investigation into the effects of oxygen pressure on the composition of various simple slags analogous to some of those which are found in steel-making practice. The ratio of ferric iron to total iron was studied at 1550°C in iron oxide melts to which lime, magnesia, lime-plus-silica, and other oxides were added. The oxygen pressures involved those of air, carbon dioxide, and carbon dioxide plus carbon monoxide in several proportions. Although very low oxygen pressures could not be used, the slag-metal equilibrium studies of Fetters and Chip-man' permitted extending the results to slags in equilibrium with iron. For the ternary system CaO-FeO-Fe,O, the oxygen pressure-composition relationship has been determined from zero percent lime to lime saturation over an oxygen pressure range from air to that represented by equilibrium with liquid iron. Lime and silica were added to iron oxide in the ratios 0.54, 1.306, and 2.235 to form three quasi-ternary systems which were also studied over the entire region of liquid melts at 1550°C. Ternary Gibbs-Duhem Equation Wagner" has developed a method by which the activities of two components of a ternary system can be calculated if' the activity of the third component is known throughout the composition range being considered. The fundamental form of the Gibbs-Duhem equation for ternary systems is N, d In a, + N2 d In a, + N3 d In a3 — 0. Wagner has developed a usable form of this equation by introducing the term y = N3/(N3 + N1) and rearranging the equation to give (? In a1/?N2)x = -y/(1-N2)2 (? In a2/?y) x2 N2/1-N2 (? In a2/?N2) y To apply this equation to the slag system CaO-FeO-Fe,O,, the activity of one of these components must be known. However, the only activity which is known from the experimental data is the activity or pressure of oxygen in the gas phase with which the slag is in equilibrium. The activity of oxygen in the slag can be defined as the square root of the oxygen pressure. In order to use oxygen as one component, the composition of the slags must be converted to the basis Fe-O-CaO. Oxygen, exclu-sive of that contained in CaO, then becomes com-ponent 2 in Eq. 1, Fe is selected as 1 and CaO as 3. Then y = Ncao/(Ncao + NFe). Eq. 1 then becomes (? In a fe/? Nx) y = -y/(1-No)2 (? In ao/?y) xp - No/ 1- No (? In ao/ O No) y. In order to evaluate Eq. 2, the boundary conditions must be known. The obvious choice of a standard state for iron is to assign slags in equilibrium with iron an activity of one. Then In a,., or log aFe, which is substituted for convenience, is determined by integrating along a line of constant y from the slag in equilibrium with iron to the composition at which log a,, is to be determined. Mathematically this can be expressed as log aFe (Nlog y) = log a'Fe + (? log a Fe/? No) dNo where the primes indicate equilibrium with liquid iron and a'Fe = 1. When Eq. 3 is integrated along a line of constant y, the following is obtained: log a Fe (No, y) = - y No ?/ No ? y ( log ao/(1- No)2) No d No - No/1-No d log ao. Lime-Iron Oxide Slags Iron Activity: The oxygen pressure of lime-iron oxide slags is shown in Fig. 1 as a function of j, de-fined as j = Fe+++/(Fe++ + Fe +++) for various constant mol percentages of lime. The j values at 0 to 60 pct lime were then determined for oxygen pressures of 1, 10-1, etc. to 10-" atm. For purposes of calculation, the line for zero percent lime was extrapolated to

Jan 1, 1955

Jan 1, 1881

By William P. Blake

Postscript to the paper read by Prof. William P. Blake at the Lake Superior meeting, September, 1904. POSTSCRIPT.*-Since the publication of my paper upon the blackening of the surface of rocks in desert regions, I have noted that similar views of the origin of superficial discoloration in the case of certain rocks from the Salt Lake basin of Utah have been expressed by Prof. George P. Merrill. In the pamphlet, he described the brown-coated weathered boulders from Tooele county, reaching the conclusion that the discoloration was due to the solution of the manganese and iron compounds 'in the interior of the boulder, while they were in the water, and the gradual bringing of this material to the surface through capillarity and its oxidation when exposed.

Mar 1, 1905

By L. H. Sjodahl, J. B. Conway

of each particle on grain boundary movement is proportional to the area it occupies on the grain boundary and this increases with the square of the particle radius. In contrast to the impression gained by Tiwari the effectiveness of a particle, as a grain boundary obstacle, increases rapidly as the particle size increases. The counterbalancing factor in a dispersion of constant volume (e.g., those considered by Zener) is of course the rapidly diminishing number of particles, a situation which does not exist in thoria dispersed materials where the volume fraction of thoria remains constant. Although it has been assumed that the Zener approach is valid for the sake of the above argument, it is open to the following objections: 1) It assumes that the effect of many small particles along a grain boundary is cumulative and does not take into account the possibility of each small particle being overcome separately. In this case the boundary will not be able to move forward until the largest particle on the boundary has been overcome. This means that the rate controlling factor will be the rate at which a few relatively large particles are overcome with the remaining small particles playing a negligible role. 2) When the dispersion consists of very small particles the interparticle spacing is usually small compared to the grain size and most particles are located away from grain boundaries where they can play no part. The most efficient dispersion, from a purely geometrical viewpoint, is one where the particle spacing is equal to the grain size so that all the particles are on grain boundaries. Tiwari's second point that voids cannot prevent re-crystallization seems to be unfounded as several workes20-23 have shown that voids alone can retard grain boundary movement and recrystallization. To further convince Tiwari, Fig. 17 shows a void in deformed TD Nichrome holding back a recrystallization interface. Tiwari's explanation for the effect of voids on recrystallization, i.e., they reduce local strain energy and hence lower the driving force for recrystallization may be valid in itself but is irrelevant to the present case where even complete recrystallization after ONE of the more recently proposed expressions for describing stress-strain behavior is that due to Voce.1,2 This equation is usually written: s =s8 ?(s8?so)e??/?c [l] where s is the true stress, ? is true strain, cC is a constant called the characteristic strain, s8 is the asymptotic stress approached at high strain values, and so is the threshold stress at which the plastic deformation begins. While this expression has not been universally accepted as yet it does contain certain features which are consistent with experimental stress-strain behavior. In a more recent publication Hsu, Davies, and Royles3 employed the concept of generalized strain proposed by seth4 and identified a method for linearizing the stress-strain plot. In this approach the generalized strain was given by: where 7 is the generalized strain, 1 is the instantaneous length, 1, the datum length, and n is a constant called the "coefficient of strain measure". A value of n was said to exist which would yield the linearity: s —s+ S[.-(tf] [3] In recent studies of both these approaches, it has been found that while the Voce equation and the gen-

Jan 1, 1970

By Carl Wagner

This paper contains solutions of the differential equation of diffusion in binary alloys if the diffusion coefficient is an exponential function of the concentration of one of the components. THE general differential equation for one-dimensional diffusion in a binary system can be written as: where c is the concentration of one of the two components in any suitable units; x, the distance from a reference plane; t, time; and D, the diffusion coefficient. Hartley and Crank&apos; have pointed out that eq 1 with conventional length units for measuring the distance x holds only if the alloy can be regarded as a rigid framework. In general, this is not the case, especially if the lattice parameter varies appreciably with composition. If a modified coordinate system is used instead of an orthodox system based on the centimeter a:: length unit, eq 52 is obtained, which has the same general form as eq 1 and is generally valid. The necessary transformations are presented in the appendix. If&apos; D is independent of concentration, analytical solutions of eq 1 are known for numerous boundary conditions. If D depends on concentration, solutions based on known analytical functions are not available. Among others, Boltzmann2 and Matano3 have shown that the concentration c depends only on (x/t1/2) in the case of diffusion between two semi- infinite spaces with uniform initial concentrations: c = c, for x < 0, t = 0, i.e., x/tl/2 = — x [21] c = c2 for x > 0, t = 0, i.e., x/tl/2 = + a Moreover, Boltzmann and Matano have shown how D may be calculated as a function of concentration over the whole range of composition from c -c, to c = c2 if c as a function of (x/t1/2) for the initial conditions stated in eq 2 has been determined experimentally. Thus, for instance, in principle, the diffusion coefficient in silver-gold alloys may be calculated as a function of concentration for the whole range of composition from only one diffusion experiment with a couple of pure silver and pure gold welded together. If the diffusion coefficient D as a function of concentration c is known, a solution of eq 1 for boundary conditions as stated in eq 2 with arbitrarily chosen values of c, and C2 may be wanted. For instance, if the diffusion coefficient in an alloy A-B as a function of concentration has been derived from an experiment with a diffusion couple involving 0 and 30 pct B as initial. concentrations, the calculation of the concentration distribution in a diffusion couple involving other initial concentrations, e.g., 5 and 20 pct B, may be desired. This problem may be solved by numerical or graphical integration of eq l. The solution of an ordinary differential equation of second order, which results from using (x/t1/2) as an independent variable, can be computed quite rapidly if two conditions for the same value of the independent variable are given. In eq 2, however, two conditions are given for two

Jan 1, 1953

By William D. Manly, John Towers, Paul A. Beck

RECENT work on grain growth in high purity aluminum and in a solid solution type alloy of aluminum and magnesium&apos; showed that the isothermal increase of the average grain diameter D with time follows the equation [D = K(to + A)-Eq I] Here [to] is that portion of the total annealing time [t], which is available for grain growth, i.e. the annealing time less the time necessary for complete recrystallization. The parameters A and n depend only on the temperature. A has the dimension of time, and is usually of the same order of magnitude as the time for recrystallization: R. In some instances R may be substituted for A in Eq I. This is the case for example at long periods of annealing or at high temperatures, where both A and R become small in comparison with t. In such instances Eq I is simplified to [D-K-tnEq2] For high purity aluminum the exponent n increases in the ratio of about 1 to 5 in the temperature range of 350° to 600°C. 70-30 brass has been long known to exhibit grain growth of the continuous type, such as occurs in pure metals. For this reason, it was of interest to determine whether grain growth in brass also follows Eq I and 2 as it does in aluminum. Isothermal grain growth data for 70-30 brass were published in recent years by R. S. French2 and by H. L. Walker3 Fig I shows a log D vs. log t plot of French&apos;s data, together with the 500, 600 and 100°C data by Walker for 42.8 pct reduction by rolling. In such a log D vs. log t plot Eq 2 corresponds to a straight line with a slope of n. As seen in Fig I, the data of French and the 600°C data of Walker can be represented by straight lines. However, the 500 and 700°C data of Walker give curved lines and the average slope of his 500°C line is very different from the slope of the line representing French&apos;s data for the same temperature. Furthermore the curvature of the 700°C line suggests a tendency for grain growth to stop after 8 hr. If confirmed, this would constitute a definite deviation from Eq 2, not explainable on the basis of the specimen thickness effect found in high purity aluminum.1 Walker&apos;s specimens were 0.125 in., i.e. 3.18 mm thick, while the deviation occurs at a grain size of approximately 0.4 mm. A further point of divergence arose recently, when J. E. Burke suggested4 on the basis of Walker&apos;s data, that, instead of Eq 2, grain growth in 70-30 brass follows the equation [D-D. =K• t,Eq3] Here D, is the grain size as recrystallized at the temperature in question. Although it was possible to show5 that, for pure aluminum, for which isothermal grain growth has been carefully investigated, Eq 3 is incompatible with the experimental facts, the situation remained somewhat obscure with respect to 70-30 brass. In view of the discrepancies between the

Jan 1, 1948

By H. O. Jahns, R. A. Greenkorn, R. E. Haring, L. K. Shallenbarger

This paper is a study of the effects of heterogeneity on flow in an analog of porous media, the Hele-Shaw model. l-3 A set of experiments in heterogeneous Hele-Shaw models showed streamlines through and around heterogeneities of various sizes, shapes and levels. (A level, we define as the ratio between the transmissibility of the heterogeneity and that of the rest of the model.) The heterogeneities were restrictions or expansions of the flow stream analogous to variations in the transmissibility of porous media. The experimental data agreed well with numerical results and with an analytical solution, which we derived for a circular heterogeneity in an infinite field. This study considers the flow-stream distortion due to the shape, size and level of heterogeneities. Size and level are much more important than shape provided the heterogeneity is not long and narrow. Our analytical solution shows that a circular heterogeneity in a large field can be replaced by an equivalent circle of either zero or infinite permeability. The radius of the. equivalent circle is a simple function of size and level of the actual circle. INTRODUCTION With the availability of high-speed computers and numerical procedures to predict reservoir behavior, we are faced with an important question. How much information about the reservoir do we need to justify the cost of the computer in any given case? To answer this question, we have to know how various reservoir parameters affect flow behavior. Reservoir heterogeneity is one of these parameters. In this study, we used a simple, two-dimensional model of porous media, the hele-shaw model, to investigate the effect of heterogeneity on flow behavior. We restricted ourselves to linear, single-phase, steady-state flow in a rectangular field with a single heterogeneity at its center. The analogy between flow in Hele-Shaw models and in porous media is easily verified. Let us first consider a homogeneous Hele-Shaw model with constant plate separation h. (The Hele-Shaw model is constructed by placing two plates, usually glass, very close together and allowing liquid to flow between them. Streamlines are made visible by introducing colored fluid into the space between the plates at a number of points across the model.) We assume a cartesian coordinate system with its origin in the middle between the plates and the z axis directed perpendicular to the glass plates (Fig. 1). The fluid flow is always in a direction parallel to the glass plates and varies from a maximum value to zero in the very small distance from the middle (z = 0) to either plate (z = h/2). For slow motion of an incompressible fluid, neglecting inertia and body forces, we have the viscous flow equation: where p is the fluid pressure, p the viscosity, and v the velocity vector with components u, v and w in the x, y and z directions, respectively. In our rase. and the derivatives of 3 with respect to x and y are small as compared with the derivative in the z direction. Therefore, approximately, with Vp and v being two-dimensional vectors in the x-y plane. Two integrations with respect to z give

Jan 1, 1965

By R. A. Greenkorn, R. C. Smith

Hele-Shaw cells are used to model creeping flow through porous media (where Darcy&apos;s law is valid). The effects of inertia on flow about obstructions in a Hele-Shaw cell can be calculated by a perturbation method if one can determine a solution to Laplace&apos;s equation. Results of a computer solution for flow about circular, square and elliptical obstructions are presented. These results show that or a modified Reynolds number of less than I, the inertia terms are small; and for values of less than 3, the average streamline predicts the ideal flow. Therefore, the analogy might be used for studying flow in porous media up to a modified Reynolds number of at least 3. INTRODUCTION The nature of fluid flow in porous media is of interest in the fields of soil mechanics, ground water flow, petroleum production, filtration and flow in packed beds. Because it is very difficult to study the phenomenological behavior of flow in porous media, homologs and analogs are used to study flow characteristics. A Hele-Shaw model, made of two closely spaced plates - usually glass — is often used as an analogy to two-dimensional flow in porous media. Hele-Shaw2 showed experimentally that the streamline configuration for creeping flow around an obstacle located between two closely spaced parallel plates is the same as for two-dimensional, ideal flow about the same obstacle. Stokes7 verified these observations mathematically. The usual equation of motion for flow in porous media is Darcy&apos;s law. The form of the mathematical statement of Darcy&apos;s law is identical, within a multiplicative constant, to the expression for the average velocity over the plate gap in the plane of a Hele-Shaw model. These models may be used to describe flow in both homogeneous and heterogeneous porous media. In the mathematical proof of the Hele-Shaw analogy it is assumed that the convective terms in the Navier-Stokes equations are negligible and that the equations of motion degenerate to Laplace&apos;s equation, with pressure the dependent variable. Whenever a Hele-Shaw model is used as an analogy to flow in porous media, .the validity of this assumption is in question. Riegels5 showed that if convection is not neglected, the velocity distribution around a cylindrical obstruction in the flow field depends on a Reynolds number, the plate spacing, and a dimension characteristic of the obstacle. Riegels&apos; solution, a perturbation solution, uses the boundary condition that the flow rate into the obstacle averaged over the plate gap at any point on the obstacle is zero. The method requires that a solution to Poisson&apos;s equation for the perturbation pressure be found. Riegels evaluated this solution for the case of the cylindrical obstruction. His method may be simplified by eliminating the need for solving Poisson&apos;s equation for the perturbation pressure. Instead, an analytic expression for the perturbation pressure gradient is obtained (valid for arbitrary shapes) and used to eliminate pressure from the equations for the perturbation velocities. The results show, for symmetrical shapes, that if NRe < 1, the convective acceleration terms are small, and that the average velocities represent ideal flow up to at least NRe = 3, where: L is a characteristic dimension of the obstacle perpendicular to flow, b is the plate spacing, u is viscosity, v, is velocity of approach and p is density. THEORY* Incompressible fluid flow through a Hele-Shaw model and a porous medium are analogous because the equations for the average velocity components in two-dimensional flow are identical if:

Jan 1, 1970

By C. Perez-Rosales

A simplified method for evaluating specific surface of porous media is described. The theory takes as its point of departure a previous statistical method. Due to its simplicity, it is considered that the proposed method has certain advantages over methods currently used. By making some simplifications, the procedure for estimating specific surface is reduced to superposing, on an enlarged reproduction of a section of the sample to be analyzed, an evenly spaced grid and counting the number of intersections of the grid lines with the perimeter of pores. An indirect test of the theory is presented and the advantages of the method are discussed. INTRODUCTION The specific surface of a porous medium, defined as the surface area of the pores per unit bulk volume, is an important parameter with regard to flow of fluids through porous materials. In particular, it has long been recognized that specific surface is closely related to permeability. Since the internal surface of any porous material is extremely complex, the measurement of specific surface can only be accomplished by indirect means. It is considered that the best method&apos; in current use for determining specific surface is the statistical method of Chalkley, Cornfield and Parka who originally applied it for estimating volume-surface ratios of morphologic components, such as cells or nuclei. This method will be outlined later. A method which has been used extensively in determining specific surface is based on the adsorption of a vapor by the internal surface of porous media. However, this method does not yield the surface pertinent to fluid flow, since it includes the tiny molecular interstices which do not influence the permeability of the porous material.&apos; This method has been employed by Brooks and Purcell measure internal surface area of sedimentary rocks. The Kozeny equation, which relates the specific surface to the permeability of a porous medium, has found wide use in determination of specific surface. However, the KOzeny equation is not strictly correct. Large discrepancies have been found between theory and experiments.&apos; Consequently. the specific surface estimated by this method is subject to serious uncertainties. Finally, there are other methods for specific surface measurements which are based upon a variety of physical and chemical properties: the heat conduction by a gas, ionic adsorption phenomena and direct chemical reactions of the surface with other substances. This paper describes a simplified statistical method for determining specific surface of porous materials. It is considered that, due to its simplicity, it has certain advantages over the methods presently used. For its validity, it is required that the porous structure of the material under study be of a spatially random nature. This is not a serious limitation since the naturally occurring porous media have this characteristic. THEORY The method proposed in this paper takes as its point of departure the interesting result obtained by Chalkley et al The method shows that if a line of length r is dropped repeatedly at random over a section of a porous medium, and counts are made of the number of times the two end points fall within pore areas (denoted by h for "hits") and the number of times the line intersects the perimeter of pores (denoted by c for "cuts"), then in a very large number of throws where S is the internal surface of the sample (surface of pores) and V, the volume of the pores (Fig. 1). Since the specific surface S, and the porosity 4 are given, respectively, by S. = Sl V, and 4 = V,l V,, where V, is the total volume of the sample, it follows from Eq. 1

By A. O. P. Casbarian

An analytical method is developed to determine the variation of the ultimate lateral resistance of a plate in a cohesionless soil with depth. This analysis is based on a modification of Rankine&apos;s classical earth pressure theory and the theory of plasticity as applied to soils. The ultimate resistance is defined as the product of the effective stress at the midpoint of the plate, the area of the plate and a dimensionless variable termed the ultimate resistance factor. This variable has been plotted us the depth ratio: i.e., the ratio of of the depth of embedment us the height of the plate. The resistance of a plate may then be calculated using the values af the ultimate resistance factor from the chart provided or the equation may be programmed for use in an analysis of anchor systems in cohesionless soils. It is emphasized that the analysis is semitheoretical. The theory has been compared with experimental results reported in the literature and results indicate general agreement. Actual field tests are necessary to further verify this theory. INTRODUCTION With the exploration for oil offshore in waters all over the world, it is of importance to determine the behavior of the various soils in relation to their ultimate resistance to deformation. Examples of such problems are the holding capacity of anchors and the lateral resistance of piles. Very little information is available in the published literature on the design and performance of anchors, in either cohesive or cohesionless soils. The analysis developed in this report is the first step in obtaining a solution for the determination of anchor holding capacity in a cohesionless material. DISCUSSION The theory used in the analysis is based on the ultimate strength of the soil and is the maximum resistance developed by the plate against further movement. In such a state the elastic deformations are disregarded in comparison with the plastic deformations. Hence, the plate can be considered as completely rigid. The theory of plasticity determines the three unknown stresses at any point by means of two equilibrium conditions for a small earth element in combination with the failure condition. However, the exact solutions can only be carried out in a few simple cases such as, for example, when the rupture or failure lines are straight (Rankine theory) or with spiral and straight rupture lines (Prandtl theory).l Kiitter 2 derived a single equation expressing the variation of the stress in any given rupture line. To utilize this equation, it is necessary to know the stress in the rupture line at a certain point. Unfortunately, this is difficult to obtain unless the rupture line intersects the free surface at a certain angle or when the earth is cohesion-less and unloaded. A method to overcome this is to consider only the boundary conditions at both ends of a rupture line without investigating the equilibrium of the earth above the rupture line. This assumes that the rupture lines meet the surface at statically correct angles, so that boundary stresses may be determined. As Kiitter&apos;s equation furnished a relation between these stresses, the unit earth pressure may be calculated at the point where the rupture line meets the wall. Another way in which Kotter&apos;s equation may be applied is in investigating the equilibrium of a soil mass above a rupture line. This method assumes that the failure or rupture line is known and that the boundary stresses in the rupture line at the ground surface can be determined. In this case it is possible to determine the earth pressure from the equations of equilibrium.

Jan 1, 1967

By James E. Lawver, James L. Wright

This paper describes the design of a cell used to measure the electrical conductivity, or the reciprocal resistivity, of granular materials. It also establishes a quantitative relationship between the electrical conductivity and the electrical separation of selected mineral samples. It is estimated&apos; that the free world produces about 13 million tons of mineral concentrates a year by electrostatic or electrodynamic methods. At least 95% of these concentrates are produced by corona discharge type machines that separate minerals by exploiting the differences in their electrical conductivity. But, what is meant by the electrical conductivity of a mineral? How large a difference in the electrical conductivity of two mineral species is necessary to effect a separation? Just what is the role of conductivity in electrical separations with regard to capacity and selectivity? There is a considerable amount of information in the literature on both the mechanisms of conduction and the values of the electrical conductivity of pure single-crystal mineral specimens. In addition, some insight into the role of mineral conductivity can be obtained by referring to a Bureau of Mines publication&apos; in which 95 minerals have been grouped as conductors or insulators as a function of temperature. The classification was made on the basis of the percent of weight that reported as "conductors" in an electrical separator. But, as is often the case in the mineral industry, practical application leads theory — and there is little useful information in the literature giving any kind of quantitative relationship between electrical conductivity and electrical separations. The problem is a bit complex because electrical conductivity per se — which can be rather easily measured on large, single grains — is not sufficiently informative to permit quantitative predictions of the separability of the mineral system. Practical "high-tension"* separations are made at such feed rates that the band of minerals on the rotor is multilayered in thickness. Thus, the effective conductivities of minerals as seen by a high-tension machine include particle-to-particle contact resistance, which is often a controlling factor in making a separation. This is particularly true when the mineral grains are coated with slime or with a dielectric surface film. From a high-tension separation viewpoint, one is actually concerned with the time required for corona-charged grains to lose enough charge so that the centrifugal force acting on the particles exceeds the image force holding the grains to the rotor. This calculation requires a knowledge of the effective mineral conductivity s. (The conductivity of the metal rotor is sufficiently greater than that of the minerals so that it need not be considered in charge decay calculations.) The charge density at time t can be approximated by the relaxation equation.3 where t 5 charge density at time t r, - charge density when t = 0 cr e total electrical conductivity £ = permittivity The time required for the charge density to fall to l/e of its initial value is called the relaxation time and is numerically equal to €/n. AS defined here, u is the conductivity of an element of homogeneous grains making contact with each other and with a grounded rotor. This paper deals with the design of a conductivity cell used to measure a, or its reciprocal p, the electrical resistivity. CELL DESIGN Initial test work was performed with a cell recommended by ASTM.4 Since the reproducibility with the ASTM cell was not satisfactory, it was decided to

Jan 1, 1970