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By C. Atkinson

A method is described for the numerical solution of the diffusion equation with a composition-dependent diffusion coefficient and applied to the radial growth of a cylinder; the radial growth of a sphere, and the symmetric growth of an ellipsoid. Sample applications of the method are made to the growth of particles of proeutectoid ferrite into austenite. RECENTLY&apos; we described a method for numerical solution of the diffusion equation with a composition-dependent diffusion coefficient for the case of the growth of a planar interface. In this paper we extend this method to describe the radial growth of a cylinder, the radial growth of a sphere, and the symmetric growth of an ellipsoid. In the latter case, limiting values of the axial ratios of the ellipsoid reduces the problem to one of a cylinder, a sphere, or a plane depending on the axial ratio. A check on these limiting values is made in the results section. In all of these cases we consider growth from zero size. A natural consequence of this assumption as applied to the sphere, for example, is that the radius of the sphere is proportional to the square root of the time. This is consistent with the condition that the radius is zero initially, i.e., grows from zero size. It may be argued that it is more realistic to consider particles which grow from a nucleus of finite initial size; even in this case the analysis of this paper is likely to be applicable. This can be seen if a comparison is made of the work of Cable and Evans,2 who consider a sphere of initially finite size growing by diffusion in a matrix with a constant diffusion coefficient, with the results of Scriven3 for growth from zero size. This comparison shows that the rates of growth in each case differ trivially by the time the particle has grown to about five times its initial size." This investigation is a generalization of those of Zener,4 Ham,5 and Horvay and cahn6 to the situation often encountered experimentally, in which the diffusion coefficient varies with concentration. First let us consider each of the cases separately. I) GROWTH OF SPHERICAL PARTICLES FROM ZERO SIZE In this case the differential equation in the matrix depends only on R, the radius in spherical coordinates, and can be written: ? 1 <^\ ^13D . , dt U\dRz + R 3Rj + dR dR [ J where C is the composition, t is the time, and D is the diffusion coefficient which depends on c. The boundary conditions will be: c = c, at the moving interface in the matrix, c = c, at infinity in the matrix (and at t = 0, everywhere in the matrix), c = X, is the composition in the spherical particle. Each of the above compositions is assumed constant. In addition there is the flu condition at the moving interface which can be written: , dR0 ~/3c dt \dR/H =Ra where R,, which is a function of t, is the position of the moving interface. We make the substitution q = RI~ in [I] reducing this equation to: & - m - *ws) »i where we have written D = D,F(c) or simply D,F, and Do = D(c,). Thus F[c(q0)] = 1 where q, = ~,/a is the value of the dimensionless parameter q evaluated at the interface. Multiplying Eq. [2] by dq/dc and integrating, we find: where the lower limit of the integral has been chosen so that dc/dq — 0 as c — c,, thereby satisfying the boundary condition at infinity. We require, then, to solve Eq. [3] subject to the condition c = c, when q = q, (this follows from putting R = R, at the interface) together with the flux condition which can be rewritten in terms of q as: Eqs. [3] and [4] together with the condition c = c, at q = q0 enable us to find 77, and the concentration profile c = c(q). Numerical Method. We treat Eq. [3] in the same way as we did the corresponding equation for the planar interface problem&apos; i.e., by dividing the interval c, to c, into n equal steps so that: cr = ca -rbc [5] where r takes the values 0, 1, ... n and we call no,, q1, ... nn the values of n corresponding to the compositions c,, c,, ... c,.

Jan 1, 1970

By Hollis L. Caswell

The present status of integrated circuits utilizing. superconductive switching. elements is reviewed with special attention given to fabrication techniques, methods for interconnecting completed circuits, and refrigeration requirements. Cryoelectronics has been largely an "inte- grated-circuit" technology since its conception because the switching speed of superconductive devices is attractive only when these devices are fabricated with thin-film techniques. It is true that cryotron circuits can be constructed from wires of appropriate materials (as indeed was done by Dudley Buck 1 in his early investigations) but these circuits will switch in times characteristic of milliseconds whereas similar circuits fabricated by thin-film methods have potential switching times of nanoseconds. Furthermore, cryo-electronic devices such as the cryotron lend themselves readily to fabrication by thin-film techniques since these components may be made from polycrys-talline thin films and are relatively insensitive to the presence of impurities (as measured by semiconductor standards). Therefore, during the past decade considerable effort has been devoted to developing techniques for batch fabricating circuit arrays containing superconductive switching elements. Technology had developed to the point several years ago that fabrication of cryoelectronic arrays containing up to one hundred devices was rather straightforward. However, larger arrays containing between lo4 and 106 components which are required for commercial development of cryoelectronics still pose very severe yield problems. Thus in a sense cryoelectronics found itself in 1962 at the point semiconductor technology finds itself today; namely, individual devices and small groups of integrated devices could be fabricated with acceptable yield and the outlook for building larger integrated-circuit arrays was bright. Unfortunately, problems associated largely with yield have made fabrication of these larger arrays difficult. Unlike semiconductor technology, cryoelectronics had to solve the problems of large-scale integration before it could become economically attractive. This has proven to be a sizable burden to bear. Since several reviews exist on superconductivity,2 superconductive devices,3 and cryoelectronic technology, no attempt will be made in this paper to summarize these areas. Instead a few specific topics will be dealt with in more detail. First, a brief description is given of selected superconducting switching and storage devices with special attention to several metallurgical techniques which improve the performance of these devices. Second, techniques used to fabricate cryoelectronic devices are described with emphasis on problems affecting yield. Third, techniques for interconnecting a number of cryoelectronic planes are described. And last, refrigeration of cryoelectronic components is discussed briefly since the low operating temperature of superconductive devices is an important consideration in this technology. SUPERCONDUCTING STORAGE AND SWITCHING DEVICES The basic superconductive switching device is the thin-film cryotron. The geometry of this device is attractively simple, since it involves only the intersection of two lines that are electrically insulated from each other. The switching element (gate) and control element (control) of a crossed-film cryotron are arranged as illustrated in Fig. 1. The material for the gate is selected to permit the gate to be switched from the superconducting to the normal (resistive) state by the application of a control current. Tin, which has a critical temperature (T,) of 3.7°K, is commonly used for the gate and the cryotron is operated at a temperature just below T, (for example, 3.5°K). The control material (normally lead, with T, = 7.2°K) is chosen so that the control is never driven normal during circuit operation. To improve cryotron operation, a ground plane, also of lead, is placed under all of the circuitry to act as a diamagnetic shield and improve the current-density uniformity across the width of various thin-film elements. Normally, line widths vary from 0.005 to ^ 0.020 in. and film thicknesses from 300 to 10,000A, although new fabrication techniques make narrower lines feasible. In fabricating cryotrons it is important that the edges of the gate elements be geometrically sharp to avoid undesirable switching characteristics associated with a thinner edge region, Fig. 2. One technique which has been used extensively to form patterns consists of placing a physical mask containing the film pattern between the evaporation source and the substrate and depositing through the mask. Film strips formed in this manner possess a penumbra at the film edges due to shadowing of the evapor-ant under the mask. Several techniques have been proposed for minimizing effects due to this penumbra. One of the more promising metallurgical techniques

Jan 1, 1967

By Roy J. Helfinstine

One of the most urgent needs in the field of coal combustion is the ability to predict the performance of a coal from knowledge gained from small-scale tests. Numerous types of analyses and tests are conducted on coal, including the proximate and ultimate analyses, heating value, varieties of sulphur, ash analysis, ash fusion temperatures, free swelling index, petrographic analysis, and Gieseler plasticity. The Illinois Geological Survey has made an intensive study of the value of these tests for judging the performance characteristics of the coal as actually utilized. To date, three reports 1 2 3 have been is- sued, and another is being prepared. The present paper briefly summarizes some of the results found in the studies.

Jan 1, 1949

By S. A. Falconer

D. A. Dahlstrom—Mr. Falconer has done an admirable job of proving the adaptability of the cyclone to the beneficiation of a very difficult size range in the preparation of coal. The addition of the cyclone to other coarser size methods makes it economically feasible for the coal operator to rigorously control his marketed products at a high and uniform quality level down to the 48 and 100-mesh size. Furthermore, it should be pointed out that by utilizing the cyclone as a deslimer or classifier the —3 200-mesh fraction, usually containing large percentages of undesirable material, can also be quickly and cheaply removed from the clean coal. Advantages of such a step are several: (1) Reduction of ash content, especially where clays and slimes are serious. (2) lower moisture content of the final coal. (3) easier mechanical and thermal de-watering of the fine coal due to higher cake and bed permeabilities. The author has indicated that adjustable control means should be installed on the cyclone operating on a heavy-media slurry. This also holds true for the cyclone acting as classifier, deslimer or preliminary dewatering agent and is worthy of special emphasis. Such controls, while not necessarily automatic, must be simple, rapidly adjustable, and require a minimum of attention. Three methods available to the operator are (1) throttling valve on the feed line, (2) back pressure valve on the overflow, and (3) adjustable diameter underflow nozzles. The first two, while fulfilling the specifications, suffer the disadvantage of restricting capacity if severe throttling is necessary. The third method has little effect on capacity and easily maintains an underflow of correct moisture content. Several designs are possible, but two find the widest application. The first was developed by Robert Piros of The Truax-Traer Coal Co. and uses a simple casting containing two holders for underflow nozzles which can be rotated about an axis parallel to the cyclone centerline. These holders retain the underflow nozzle flush with the conical section walls. When a change in nozzle diameter is required, the new size is placed in the spare holder and rotated into place. A second design utilizes a soft rubber cylindrical tube which is compressed or released from three sides, causing the inner diameter of the tube to decrease or increase. This is achieved by using a machine-threaded,

Jan 1, 1951

By S. A. Falconer

D. A. Dahlstrom—Mr. Falconer has done an admirable job of proving the adaptability of the cyclone to the beneficiation of a very difficult size range in the preparation of coal. The addition of the cyclone to other coarser size methods makes it economically feasible for the coal operator to rigorously control his marketed products at a high and uniform quality level down to the 48 and 100-mesh size. Furthermore, it should be pointed out that by utilizing the cyclone as a deslimer or classifier the —3 200-mesh fraction, usually containing large percentages of undesirable material, can also be quickly and cheaply removed from the clean coal. Advantages of such a step are several: (1) Reduction of ash content, especially where clays and slimes are serious. (2) lower moisture content of the final coal. (3) easier mechanical and thermal de-watering of the fine coal due to higher cake and bed permeabilities. The author has indicated that adjustable control means should be installed on the cyclone operating on a heavy-media slurry. This also holds true for the cyclone acting as classifier, deslimer or preliminary dewatering agent and is worthy of special emphasis. Such controls, while not necessarily automatic, must be simple, rapidly adjustable, and require a minimum of attention. Three methods available to the operator are (1) throttling valve on the feed line, (2) back pressure valve on the overflow, and (3) adjustable diameter underflow nozzles. The first two, while fulfilling the specifications, suffer the disadvantage of restricting capacity if severe throttling is necessary. The third method has little effect on capacity and easily maintains an underflow of correct moisture content. Several designs are possible, but two find the widest application. The first was developed by Robert Piros of The Truax-Traer Coal Co. and uses a simple casting containing two holders for underflow nozzles which can be rotated about an axis parallel to the cyclone centerline. These holders retain the underflow nozzle flush with the conical section walls. When a change in nozzle diameter is required, the new size is placed in the spare holder and rotated into place. A second design utilizes a soft rubber cylindrical tube which is compressed or released from three sides, causing the inner diameter of the tube to decrease or increase. This is achieved by using a machine-threaded,

Jan 1, 1951

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Jan 1, 1946

By Roy G. Mead

BECAUSE of its magnitude, and the type of occurrence, the deposit of boron minerals in the Kramer district, Kern County, California, is unlike any other in the world. Discovery of this vast deposit has brought about many changes in the borax industry, so a discussion of the geology, and the occurrence, would not be complete without a brief history of the development of the borax industry. Until the discovery of boron minerals in the desert regions of California and Nevada some seventy years ago, borax was imported from Tibet and India, principally for medicinal purposes, at almost prohibitive prices. Following the discovery of commercial deposits of boron minerals in California and Nevada, and the subsequent development of new uses for borax and boric acid, the consumption has steadily increased until today borax is used in some form or other in every household.

Jan 1, 1933

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Jan 1, 1902

By Oliver Bowles

PRICES of quartz sold in the United States in 1938 ranged from $1.15 to $36,000 a ton. This startling variation was due simply to the differences between glass sand and rock - crystal, materials that are virtually the same in composition. It illustrates the complexities in qualities, uses, and values of industrial minerals. It is generally stated that mineral deposits are exhaustible resources which can never be replaced when depleted. This statement is no longer true with respect to certain industrial minerals now recovered in increasing quantities from that most inexhaustible of all sources of supply, the ocean. Salt has long been obtained from sea water, particularly on the Pacific Coast, and reference was made a year ago to the large bromine recovery plant at Wilmington, N. C. Now the Westvaco Chlorine Products Corp. is operating a large plant at Newark, Calif., having a capacity sufficient to recover daily from the ocean raw materials that will produce 60 tons of various magnesium products. Lime is used to precipitate the magnesia from salt-water bittern. The products are chiefly magnesium carbonate for chemical uses, periclase and other refractories, and oxychloride cement for flooring.

Jan 1, 1939

Jan 1, 1970

United Engineering Trustees, Inc. ARTHUR S. DWIGHT GEORGE D BARRON J V. N DORR Engineering Societies Library Board ROBERT PEELE PHILIP W. HENRY GEORGE C STONE JOSEPH E. POGUE The Engineering Foundation H C BELLINGER GALEN H CLEVENGER J V N DORR

Jan 1, 1923

Jan 1, 1944